k max2 and ∆k t = k max1 − k max2 is the width of the annular aperture.\"\n },\n {\n \"title\": \"Integration\",\n \"content\": \"Once the mesh, coordinates and input field set, then the argument of eq. (10) is written and the Fast Fourier Transform is calculated.\"\n },\n {\n \"title\": \"EXAMPLES\",\n \"content\": \"In the following subsections we show specific examples with different polarization states and input fields. The incident fields are described in the spatial or the angular coordinates. The magnitude of the errors and the dependency on the domain size are discussed in [2].\"\n },\n {\n \"title\": \"Linearly polarized Gaussian\",\n \"content\": \"We consider a Gaussian beam defined as:\\nwhere E 0 (set to 1) is the field amplitude before the lens, w 0 is the waist of the Gaussian, R = D/2 is the aperture radius and f 0 the filling factor for a Gaussian beam. The filling factor is defined as f 0 = ω 0 /R, with ω 0 the Gaussian waist. The same field in angular coordinates:\\nwhere the new waist is w 0 = w 0 k 1 /f . Figure 4 shows the results for a Gaussian beam initially polarized in the direction n x , focused by a lens with N A = 1.3, n = 1.518, f = 2 mm, f 0 = 1.0 and R = 1.71 mm. In the case of a collimated Gaussian beam, it is a good approximation to consider the phase as constant, so it is set to φ(x ∞ , y ∞ ) = φ(k x , k y ).\\nThen, a linear polarization state is selected with the c x and c y coefficients. In this case we select only the x component defining c x = 1 and c y = 0 in the L × L domain.\\nThe top row in Fig. 4 has the 2 dimensional amplitude, phase, aperture, x polarization and y polarization masks. The 6 frames in the bottom depict the result at the focal plane with the 3 intensities and phases.\"\n },\n {\n \"title\": \"Circular polarization\",\n \"content\": \"Same Gaussian with an optical vortex phase with φ 0x = φ ∞ m m = −1 through a quarter wave plate, where φ ∞ is the azimuthal angle in the x ∞ , y ∞ plane. The polarization state, which is defined as c x = 1, c y = i (left circular polarization with σ = 1) in the L × L domain. The input planes in the domain x ∞ , y ∞ (same plane as k x , k, y) are at the top of Fig. 5, while the resulting field at the focus with the 3 intensities and phases is in the lower part. We observe that the in the z component there is coupling between the orbital m and the polarization σ resulting in a null charge (m = 0). Figure 6 shows the polarization masks for the transverse polarization components with c x = cos φ = k x / k 2 x + k 2 y and c y = sin φ = k y / k 2 x + k 2 y . We introduced auxiliary polar coordinates φ and ρ in the x ∞ , y ∞ domain. The angle φ can also be defined as the inverse tangent of k y /k x . The calculation considers an annular mask and the same Gaussian input. Figure 7 shows the polarization masks for the transverse polarization components cx = cos φ = −k y / k 2 x + k 2 y c y = sin φ = k x / k 2 x + k 2 y . This case is interesting because there is no axial component as expected for this case, the ratio between the amplitude of the E z and the E x has an order of 10 −19 .\"\n },\n {\n \"title\": \"Radial Polarization\",\n \"content\": \"\"\n },\n {\n \"title\": \"Azimuthal Polarization\",\n \"content\": \"\"\n },\n {\n \"title\": \"Arbitrary Polarization\",\n \"content\": \"We also consider a Gaussian beam with flower (spider web) polarization state as described in [7] defined by the parameter s = 8 (−8): The polarization is c x = cos ((s/2)φ)) and c y = sin ((s/2)φ)) with s = 8, (flower). This is shown in Fig. 8. Inverting the sign in s yields a similar pattern (spiderweb).\"\n },\n {\n \"title\": \"Arbitrary beam\",\n \"content\": \"All the previous examples had considered E 0x = E 0y and φ 0x = φ 0y . Here we consider a beam that has different amplitudes, phases, and polarization profiles in the transverse components. In Fig. 9 we show a beam with parameters in the angular coordinates:\\nWhere E 0x (E 0y ) is such as the maximum value of E incx (E incy ) is one.\\nFocusing of a beam with different amplitudes, phases, and polarization profiles in the transverse components. The first group of images shows the amplitude, phase, polarization maps and the circular aperture. The second group shows the focal fields. The parameters are L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\"\n },\n {\n \"title\": \"PLANAR INTERFACE (PROPAGATING AND EVANESCENT)\",\n \"content\": \"Many applications consider these beams propagating through a planar interface like optical micromanipulation where the beam propagates through a glass-liquid interface. Due to the refractive index mismatch across the planar boundary there is the possibility of total internal reflection and having evanescent waves. In this section we reproduce the results contained in Novotny and Hetch book [6]. They consider that the field focuses at z = 0 and the boundary (xy plane) is at a height of z 0 . The refractive index at z < z 0 is n 1 and k z1 = k 2 1 − (k 2 x + k 2 y ) (k 1 = nk 0 ), above the boundary for z > z 0 , the refractive index is n 2 and k z2 = k 2 2 − (k 2 x + k 2 y ) (k 2 = n 2 k 0 ).\"\n },\n {\n \"title\": \"Fresnel Coefficients\",\n \"content\": \"The reflection and transmission Fresnel coefficients for a planar boundary (s and p components) are:\\nNormally, for a dielectric we can set µ 1 = µ 2 = 1.\\nThe field between the lens and the end of the boundary (same material) is a superposition of the focused and reflected fields E = E f (x, y, z) + E r (x, y, z) for z < z 0 :\\nIn the case of z > z 0 then E = E t where the transmitted field:\\nThe reflected E ∞,r and transmitted E ∞,t in terms of the xy components of E inc :\\nFor the other transverse polarization component (direction n y ):\\n−r s k x k y − r p k x k y k z /k 1 r s k 2 x − r p k 2 y k z1 /k 1 −r p (k 2\\nx + k 2 y )k y /k 1 )\\nThe transmitted component:\\nFor the other transverse polarization component (direction n y ).\\n(24) Figure 9 shows the case of the linearly polarized Gaussian with w 0 = R (f 0 = 1) focused by a lens with N A = 1.4 at z = 0, the position of the interface is at z 0 = −λ , 0 , λ (columns 1, 2 and 3 respectively). The first row has cross sections of the plane x, z with y = 0 and the second row plane y, z with x = 0. The colormaps represent the fourth root of total intensity in order to show more clearly the details. The case with z 0 = 0 appears in [6]. CONCLUSION We presented two simple programs to calculate tightly focused vectorial light fields: one for a propagating field and the other considers a planar interface where evanescent fields can emerge. These programs are based on previous results described in [2,6]. The main contribution is the discussion about the correction that has to be made for a shifted Fourier transform, which is a subtle detail that is easy to miss. Also, the use of Cartesian coordinates helps when making comparisons with experiments where the light is shaped by rectangular arrays which are described in [6]. FIG. 10. Intensity E · E * of a focused linearly polarized Gaussian beam with parameters f0 = 1, NA= 1.4, close to an interface glass-air. The spanned area 5λ × 8λ is the same for all the images, the (0, 0) is the focus of the lens. In the first row is shown the plane (x, z) and in the second row is shown the plane (y, z), where z. (a) The interface is located at z0 = −λ. (b) The interface is located at the focal plane z0 = 0. (c) The interface is located at z0 = λ and. We computed the field at 200 planes along the z direction . In the three cases L=2048 and N=150.\\nCONACYT LN-299057.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 7 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"FIG. 1. Representation of the focusing model. A paraxial beam propagating in a medium with refractive index n1, incident into lens that focuses at f in a medium with a refractive index n2. The lens is represented by a spherical surface of reference centered at the focal point with radius f .","content":"FIG. 1 .\n1FIG. 1. Representation of the focusing model. A paraxial beam propagating in a medium with refractive index n1, incident into lens that focuses at f in a medium with a refractive index n2. The lens is represented by a spherical surface of reference centered at the focal point with radius f ."},{"id":"","caption":"Effect of the origin translation in the phase of the resulting complex fields and correction. In the first column it is shown the phase of the x component of a focused x-linearly polarized Gaussian beam, in the second column it is shown the phase evaluated on the horizontal axis indicated on the phase map.","content":"FIG. 3 .\n3Effect of the origin translation in the phase of the resulting complex fields and correction. In the first column it is shown the phase of the x component of a focused x-linearly polarized Gaussian beam, in the second column it is shown the phase evaluated on the horizontal axis indicated on the phase map."},{"id":"","caption":"Focusing a linearly polarized Gaussian beam. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.","content":"FIG. 4 .\n4Focusing a linearly polarized Gaussian beam. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm."},{"id":"","caption":"Focusing a Gaussian beam right circular polarization, σ = 1, and vortex phase of order m = −1. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.","content":"FIG. 5 .\n5Focusing a Gaussian beam right circular polarization, σ = 1, and vortex phase of order m = −1. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm."},{"id":"","caption":"Focusing a radial polarized annular aperture (TM 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.","content":"FIG. 6 .\n6Focusing a radial polarized annular aperture (TM 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm."},{"id":"","caption":"Focusing a azimuthal polarized annular aperture (TE 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm. FIG. 8. Focusing a Gaussian beam with flower (order 8) polarization: The parameters are L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.","content":"FIG. 7 .\n7Focusing a azimuthal polarized annular aperture (TE 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm. FIG. 8. Focusing a Gaussian beam with flower (order 8) polarization: The parameters are L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"FIG. 1. Representation of the focusing model. A paraxial beam propagating in a medium with refractive index n1, incident into lens that focuses at f in a medium with a refractive index n2. The lens is represented by a spherical surface of reference centered at the focal point with radius f .\",\n \"content\": \"FIG. 1 .\\n1FIG. 1. Representation of the focusing model. A paraxial beam propagating in a medium with refractive index n1, incident into lens that focuses at f in a medium with a refractive index n2. The lens is represented by a spherical surface of reference centered at the focal point with radius f .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Effect of the origin translation in the phase of the resulting complex fields and correction. In the first column it is shown the phase of the x component of a focused x-linearly polarized Gaussian beam, in the second column it is shown the phase evaluated on the horizontal axis indicated on the phase map.\",\n \"content\": \"FIG. 3 .\\n3Effect of the origin translation in the phase of the resulting complex fields and correction. In the first column it is shown the phase of the x component of a focused x-linearly polarized Gaussian beam, in the second column it is shown the phase evaluated on the horizontal axis indicated on the phase map.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Focusing a linearly polarized Gaussian beam. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\",\n \"content\": \"FIG. 4 .\\n4Focusing a linearly polarized Gaussian beam. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Focusing a Gaussian beam right circular polarization, σ = 1, and vortex phase of order m = −1. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\",\n \"content\": \"FIG. 5 .\\n5Focusing a Gaussian beam right circular polarization, σ = 1, and vortex phase of order m = −1. The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Focusing a radial polarized annular aperture (TM 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\",\n \"content\": \"FIG. 6 .\\n6Focusing a radial polarized annular aperture (TM 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Focusing a azimuthal polarized annular aperture (TE 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm. FIG. 8. Focusing a Gaussian beam with flower (order 8) polarization: The parameters are L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\",\n \"content\": \"FIG. 7 .\\n7Focusing a azimuthal polarized annular aperture (TE 0th). The parameters are: L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm. FIG. 8. Focusing a Gaussian beam with flower (order 8) polarization: The parameters are L = 2 12 , N = 2 7 , n2 = 1.518, NA= 1.3, f = 2 mm.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"* Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro. * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx","doi":""},{"text":"Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system. B Richards, E Wolf, Proc. R. Soc. Lond.-A. R. Soc. Lond.-A253B.Richards and E. Wolf, Electromagnetic diffraction in op- tical systems, II. Structure of the image field in an apla- natic system. Proc. R. Soc. Lond.-A. 253, 358-379 (1959).","doi":""},{"text":"Fast focus field calculations. M Leutenegger, R Rao, R A Leitgeb, T Lasser, Opt. Express. 14M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser. Fast focus field calculations. Opt. Express 14, 11277-11291 (2006).","doi":""},{"text":"Flexible, fast, and benchmarked vectorial model for focused laser beams. Q Li, M Chambonneau, M Blothe, H Gross, S Nolte, Appl. Opt. 60Q. Li, M. Chambonneau, M. Blothe, H. Gross, and S. Nolte, Flexible, fast, and benchmarked vectorial model for focused laser beams. Appl. Opt. 60, 3954-3963 (2021).","doi":""},{"text":"PyFocus -A Python package for vectorial calculations of focused optical fields under realistic conditions. Application to toroidal foci. F Caprile, L A Masullo, F D Stefani, Comput. Phys. Commun. 275108315F.Caprile, L. A. Masullo, F. D. Stefani, PyFocus -A Python package for vectorial calculations of focused opti- cal fields under realistic conditions. Application to toroidal foci. Comput. Phys. Commun. 275, 108315 (2022).","doi":""},{"text":"Focal field computation of an arbitrarily polarized beam using fast Fourier transforms. B R Boruah, M A Neil, Opt. Commun. 282Boruah, B. R. and Neil, M. A. A. Focal field computa- tion of an arbitrarily polarized beam using fast Fourier transforms. Opt. Commun. 282, 4660-4667 (2009).","doi":""},{"text":"Principles of Nano-Optics. L Novotny, Hecht , Cambridge Univ. PressL. Novotny, and Hecht. Principles of Nano-Optics. (Cam- bridge Univ. Press, 2012).","doi":""},{"text":"Tailored intensity landscapes by tight focusing of singular vector beams. E Otte, K Tekce, C Denz, Opt. Express. 25E. Otte, K. Tekce, and C. Denz. Tailored intensity land- scapes by tight focusing of singular vector beams, Opt. Express 25, 20194-20201 (2017).","doi":""},{"text":"Fieldxyt=(cx. * MTy1t. * E incx. * exp(1i * phi incx)+cy. * MTy2t. * E incy. * exp(1i * phi incy)). * Hzt. Fieldxyt=(cx. * MTy1t. * E incx. * exp(1i * phi incx)+cy. * MTy2t. * E incy. * exp(1i * phi incy)). * Hzt;","doi":""},{"text":"Ey inft=ifftshift(ifft2((fftshift(Fieldxyt)))). * exp(1i * PhaseShift). Ey inft=ifftshift(ifft2((fftshift(Fieldxyt)))). * exp(1i * PhaseShift);","doi":""},{"text":"Fieldxzt=(cx. * MTz1t. * E incx. * exp(1i * phi incx)+cy. * MTz2t. * E incy. * exp(1i * phi incy)). * Hzt. Fieldxzt=(cx. * MTz1t. * E incx. * exp(1i * phi incx)+cy. * MTz2t. * E incy. * exp(1i * phi incy)). * Hzt;","doi":""},{"text":"Fieldxzt)))). * exp(1i * PhaseShift). Ez0t=ifftshift(ifft2((fftshiftEz0t=ifftshift(ifft2((fftshift(Fieldxzt)))). * exp(1i * PhaseShift);","doi":""},{"text":"j)=Ext; %field beyond the boundary. Ex, transmitted 218 Ey(:,:,j)=EytEx(:,:,j)=Ext; %field beyond the boundary, transmitted 218 Ey(:,:,j)=Eyt;","doi":""},{"text":"Ez(:,:,j)=Ezt. Ez(:,:,j)=Ezt;","doi":""},{"text":". ( Eex=permute, Ex, 23 2 1EEx=permute(Ex(xmax2+1,:,:),[3 2 1]);","doi":""},{"text":". ( Eey=permute, Ey, 23 2 1EEy=permute(Ey(xmax2+1,:,:),[3 2 1]);","doi":""},{"text":". ( Eez=permute, Ez, 23 2 1EEz=permute(Ez(xmax2+1,:,:),[3 2 1]);","doi":""},{"text":"I=abs(EEx).ˆ2+abs(EEy).ˆ2+abs(EEz).ˆ2. I=abs(EEx).ˆ2+abs(EEy).ˆ2+abs(EEz).ˆ2;","doi":""},{"text":"Imax=max(max(I)). Imax=max(max(I));","doi":""},{"text":". / In=i, Imax, 234In).ˆ(1/4)In=I/Imax; 234 imagesc((In).ˆ(1/4));","doi":""}],"string":"[\n {\n \"text\": \"* Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro. * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system. B Richards, E Wolf, Proc. R. Soc. Lond.-A. R. Soc. Lond.-A253B.Richards and E. Wolf, Electromagnetic diffraction in op- tical systems, II. Structure of the image field in an apla- natic system. Proc. R. Soc. Lond.-A. 253, 358-379 (1959).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast focus field calculations. M Leutenegger, R Rao, R A Leitgeb, T Lasser, Opt. Express. 14M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser. Fast focus field calculations. Opt. Express 14, 11277-11291 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Flexible, fast, and benchmarked vectorial model for focused laser beams. Q Li, M Chambonneau, M Blothe, H Gross, S Nolte, Appl. Opt. 60Q. Li, M. Chambonneau, M. Blothe, H. Gross, and S. Nolte, Flexible, fast, and benchmarked vectorial model for focused laser beams. Appl. Opt. 60, 3954-3963 (2021).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"PyFocus -A Python package for vectorial calculations of focused optical fields under realistic conditions. Application to toroidal foci. F Caprile, L A Masullo, F D Stefani, Comput. Phys. Commun. 275108315F.Caprile, L. A. Masullo, F. D. Stefani, PyFocus -A Python package for vectorial calculations of focused opti- cal fields under realistic conditions. Application to toroidal foci. Comput. Phys. Commun. 275, 108315 (2022).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Focal field computation of an arbitrarily polarized beam using fast Fourier transforms. B R Boruah, M A Neil, Opt. Commun. 282Boruah, B. R. and Neil, M. A. A. Focal field computa- tion of an arbitrarily polarized beam using fast Fourier transforms. Opt. Commun. 282, 4660-4667 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Principles of Nano-Optics. L Novotny, Hecht , Cambridge Univ. PressL. Novotny, and Hecht. Principles of Nano-Optics. (Cam- bridge Univ. Press, 2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tailored intensity landscapes by tight focusing of singular vector beams. E Otte, K Tekce, C Denz, Opt. Express. 25E. Otte, K. Tekce, and C. Denz. Tailored intensity land- scapes by tight focusing of singular vector beams, Opt. Express 25, 20194-20201 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fieldxyt=(cx. * MTy1t. * E incx. * exp(1i * phi incx)+cy. * MTy2t. * E incy. * exp(1i * phi incy)). * Hzt. Fieldxyt=(cx. * MTy1t. * E incx. * exp(1i * phi incx)+cy. * MTy2t. * E incy. * exp(1i * phi incy)). * Hzt;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ey inft=ifftshift(ifft2((fftshift(Fieldxyt)))). * exp(1i * PhaseShift). Ey inft=ifftshift(ifft2((fftshift(Fieldxyt)))). * exp(1i * PhaseShift);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fieldxzt=(cx. * MTz1t. * E incx. * exp(1i * phi incx)+cy. * MTz2t. * E incy. * exp(1i * phi incy)). * Hzt. Fieldxzt=(cx. * MTz1t. * E incx. * exp(1i * phi incx)+cy. * MTz2t. * E incy. * exp(1i * phi incy)). * Hzt;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fieldxzt)))). * exp(1i * PhaseShift). Ez0t=ifftshift(ifft2((fftshiftEz0t=ifftshift(ifft2((fftshift(Fieldxzt)))). * exp(1i * PhaseShift);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"j)=Ext; %field beyond the boundary. Ex, transmitted 218 Ey(:,:,j)=EytEx(:,:,j)=Ext; %field beyond the boundary, transmitted 218 Ey(:,:,j)=Eyt;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ez(:,:,j)=Ezt. Ez(:,:,j)=Ezt;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". ( Eex=permute, Ex, 23 2 1EEx=permute(Ex(xmax2+1,:,:),[3 2 1]);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". ( Eey=permute, Ey, 23 2 1EEy=permute(Ey(xmax2+1,:,:),[3 2 1]);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". ( Eez=permute, Ez, 23 2 1EEz=permute(Ez(xmax2+1,:,:),[3 2 1]);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"I=abs(EEx).ˆ2+abs(EEy).ˆ2+abs(EEz).ˆ2. I=abs(EEx).ˆ2+abs(EEy).ˆ2+abs(EEz).ˆ2;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imax=max(max(I)). Imax=max(max(I));\",\n \"doi\": \"\"\n },\n {\n \"text\": \". / In=i, Imax, 234In).ˆ(1/4)In=I/Imax; 234 imagesc((In).ˆ(1/4));\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":388,"cells":{"text":{"kind":"string","value":"\nBuilding Adversary-Resistant Deep Neural Networks without Security through Obscurity\n2016. July 2017\n\nD C Washington \nUsa \nBuilding Adversary-Resistant Deep Neural Networks without Security through Obscurity\n\nProceedings of ACM Conference\nACM ConferenceWashington, DC, USA122016. July 201710.1145/nnnnnnn.nnnnnnn978-x-xxxx-xxxx-x/YY/MM. . . $15.00\nDeep neural networks (DNNs) have proven to be quite e ective in a vast array of machine learning tasks, with recent examples in cyber security and autonomous vehicles. Despite the superior performance of DNNs in these applications, it has been recently shown that these models are susceptible to a particular type of a ack that exploits a fundamental aw in their design. is a ack consists of generating particular synthetic examples referred to as adversarial samples. ese samples are constructed by slightly manipulating real data-points in order to \"fool\" the original DNN model, forcing it to mis-classify previously correctly classi ed samples with high con dence. Addressing this aw in the model is essential if DNNs are to be used in critical applications such as those in cyber security.Previous work has provided various learning algorithms to enhance the robustness of DNN models, and they all fall into the tactic of \"security through obscurity\". is means security can be guaranteed only if one can obscure the learning algorithms from adversaries. Once the learning technique is disclosed, DNNs protected by these defense mechanisms are still susceptible to adversarial samples. In this work, we investigate this issue shared across previous research work and propose a generic approach to escalate a DNN's resistance to adversarial samples. More speci cally, our approach integrates a data transformation module with a DNN, making it robust even if we reveal the underlying learning algorithm. To demonstrate the generality of our proposed approach and its potential for handling cyber security applications, we evaluate our method and several other existing solutions on datasets publicly available, such as a large scale malware dataset, MNIST and IMDB datasets. Our results indicate that our approach typically provides superior classi cation performance and resistance in comparison with state-of-art solutions. * e rst and second author contribute equally.Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro t or commercial advantage and that copies bear this notice and the full citation on the rst page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s).\n\nINTRODUCTION\n\nBeyond highly publicized victories in automatic game-playing as in Go [34], there have been many successful applications of deep neural networks (DNN) in image and speech recognition. Recent explorations and applications include those in medical imaging [1,38] and self-driving cars [11,14]. In the domain of cybersecurity, security companies have demonstrated that deep learning could o er a far be er way to classify all types of malware [8,30,39].\n\nDespite its potential, deep neural networks (DNN), like all other machine learning approaches, are vulnerable to what is known as adversarial samples [2,18].\n\nis means that they can be easily deceived by non-obvious and potentially dangerous manipulation [26,35]. To be more speci c, an a acker could use the same training algorithm, back-propagation of errors, and a surrogate dataset to construct an auxiliary model. Since this model could provide the a acker with a capability of exploring a DNN's blind spots, he can, with minimal e ort, cra an adversarial sample -a synthetic example generated by slightly modifying a real example in order to make the deep learning system \"believe\" the sample subtly perturbed belongs to an incorrect class with high con dence.\n\nAccording to a recent study [12], adversarial samples occur in a subspace relatively broad, which means it is impractical to build a defense that can rule out all adversarial samples. As such, the design principle followed by existing defense mechanisms is not to harden a DNN model naturally resistant to any adversarial samples. Rather, they focus on hiding that subspace, making adversaries di cult in nding impactful adversarial samples. For example, representative defensesadversarial training [12] and defensive distillation [29] both increase the complexity of original DNNs with the goal of making adversarial samples -impactful for original DNNs -no longer e ective.\n\nHowever, in this work, we show that the defenses proposed are far from ideal and even considered a dangerous practice. In particular, we demonstrate existing defense mechanisms all follow the approach of \"security through obscurity\", in which security is achieved by keeping defenses obscured from adversaries. Frankly speaking, defenses following this approach indeed mitigate the adversarial sample problem. However, when applied to security critical applications such as malware classi cation, they become particularly disconcerting.\n\nIn the past, there have been a huge amount of debates on security through obscurity, and a general consensus has been reach. at is, obscurity is a perfectly valid security tactic but it cannot be trusted for security. Once design or implementation is uncovered, users totally lost all the security gained by obscurity. To regain the security through obscurity, one has to come up with a completely new design or implementation. As such, Kerckho s' principle [19] suggests obscurity can be used as a layer of defense, but it should never be used as the only layer of defense.\n\nInspired by this, we propose a new mechanism to escalate a DNN's resistance to adversarial samples. Di erent from existing defenses, our proposed approach unnecessitates model obscurity. In other words, even though we reveal the model, it will still be more than burdensome for adversaries to cra adversarial samples.\n\nMore speci cally, we arm a standard DNN with a data transformation module, which projects original data input into a new representation before they are passed through the consecutive DNN.\n\nis can be used as a defense for the following two reasons. First, data transformation can potentially stash away the space of adversarial manipulations to a carefully designed hyperspace. is makes a ackers di cult in nding adversarial samples impactful for the armed DNN. Second, as we will theoretically prove in Section 4, a data transformation module carefully designed can exponentially increase computation complexity for an a acker to cra impactful adversarial samples. is means that, even though an a acker compromises obscurity and has the full knowledge about the armed DNN model (i.e., the training algorithm, dataset and hyper-parameters), he still cannot launch the a ack -detrimental to DNNs enhanced by other adversary-resistant techniques -nor jeopardize model resistance.\n\ne approach proposed in this work is bene cial for the following reasons. First, it escalates a DNN's resistance to adversarial samples with be er security assurance. Second, our approach ensures that a DNN maintains desirable classi cation performance while requiring only minimal modi cation to existing architectures.\n\nird, while this work is primarily motivated by the need to safeguard DNN models used in critical security applications, it should be noted that the proposed technique is general and can be adopted to other applications where deep learning is popularly applied, such as image recognition and sentiment analysis. We demonstrate this applicability using publicly-available datasets in Section 5.\n\nIn summary, this work makes the following contributions.\n\n• We propose a generic approach to facilitate the development of adversary-resistant DNNs without following the tactic of security through obscurity. • Using our approach, we develop an adversary-resistant DNN, and theoretically prove its resistance cannot be jeopardized even if the model is fully disclosed. • We evaluate the classi cation performance and robustness of our adversary-resistant DNN and compare it with that ... ...\n\n\nHidden Layers\n\nSoftmax Layer Input Layer f(x) x Figure 1: A feed-forward deep neural network with three hidden layers and a so max output layer.\n\nof existing defense mechanisms. Our result shows that our DNN exhibits similar -sometimes even be er -classi cation performance but with superior model resistance. e rest of this paper is organized as follows. Section 2 introduces the background of DNNs and adversarial sample problem. Section 3 discusses existing defense mechanisms and de nes the problem scope of our research. Section 4 presents our generic approach. In Section 5, we develop and evaluate DNNs in the context of image recognition, sentiment analysis and malware classi cation. Finally, we conclude this work in Section 6.\n\n\nBACKGROUND\n\nA typical DNN architecture consists of multiple successive layers of processing elements, or so-called \"neurons\". Each processing layer can be viewed as learning a di erent, more abstract representation of the original multidimensional input distribution. As a whole, a DNN can be viewed as a highly complex function, f (·) that is capable of nonlinearly mapping an original high-dimensional data point to a lower dimensional output. In this section, we brie y introduce the well-established DNN model, followed by the description of the adversarial learning problem.\n\n\nDeep Neural Networks\n\nAs is graphically depicted in Figure 1, a DNN contains an input layer, multiple hidden layers, and an output layer. e input layer takes in each data sample in the form of a multidimensional vector. Starting from the input, computing the pre-activations of each subsequent layer simply requires, at minimum, a matrix multiplication (where a weight/parameter vector, with length equal to the number of hidden units in the target layer, is assigned to each unit of the layer below) usually followed by summation with a bias vector. is process roughly models the process of a layer of neurons integrating the information received from the layer below (i.e., computing a pre-activation) before applying an elementwise activation function 1 . is integrate-then-re process is repeated subsequently for each layer until the last layer is reached. e last layer, or output, is generally interpreted as the model's predictions for some given input data, and is o en designed to compute a parameterized likelihood distribution using the so max function (also known as multi-class regression or minimizing cross entropy). is bo omup propagation of information is also referred to as feed-forward inference [15].\n\nDuring the learning phase of the model, the DNN's predictions are evaluated by comparing them with known target labels associated with the training samples (also known as the\"ground truth\"). Speci cally, both predictions and labels are taken as the input to L(·), a selected cost function. e DNN's parameters are then optimized with respect to this cost function using the method of steepest gradient descent, minimizing prediction errors on the training set.\n\nMore formally, given (X , Y ):\n{(x 1 , 1 ), . . . , (x n , n )}, where x i ∈\nR m is a data sample and i ∈ R k is the corresponding data label, where, if categorical, is typically represented through a 1-of-k encoding, the goal of model learning is to minimize the cost function represented by\nn i=1 L(f (w; x i ); i ), where f (w; x i ) denotes the pre-\ndiction of training sample x i and w represents the weights and bias associated with the connections between neurons.\n\n\nAdversarial Sample Problem\n\nAn adversarial sample is a synthetic data sample cra ed by introducing slight perturbations to a legitimate input sample (see Figure 2). In multi-class classi cation tasks, such adversarial samples can cause a DNN to classify themselves into a random class other than the correct one (sometimes not even a reasonable alternative). Recent research [6] demonstrates that, a ackers can uncover such data samples through various approaches (e.g., [3,12,21,28,32,35]) which can all be described as solving either optimization problem\nx = ar max x L(f (x; w); ) s.t. x − x p < ε,(1)\nor optimization problem\nx = ar min x L(f (x; w);ˆ ) s.t. x − x p < ε .(2)\nHere, optimization problem (1) indicates that an a acker searches for adversarial samplex, the prediction of which is as far as its true label, whereas optimization problem (2) indicates an a acker searches for adversarial samplex so that its prediction is as close as target labelˆ whereˆ is not equal to , the true label of that adversarial sample.\n\nIn both optimization problems above, L(·) represents the aforementioned cost function and f (·) denotes the DNN model trained with the traditional learning method discussed above . · p is pnorm -sometimes also speci ed as l p distance -indicating the dissimilarity between adversarial samplex and its corresponding legit data sample x. With di erent values of p -most popularly selected in adversarial learning research -the optimization problems above can be computed in the following manners. Adversarial samples generated by following [12] Adversarial samples generated by following [6] Adversarial samples generated by following [35] (1) With p = 2, p-norm represents the measure of Euclidean distance.\n\ne constraint optimization problems above can be speci ed as unconstrained optimization problemŝ\nx = ar max x L(f (x; w); ) − c · x − x 2 (3) andx = ar min x L(f (x; w);ˆ ) + c · x − x 2 .(4)\nHere, both (3) and (4) can be solved by following either a rstorder optimization method (e.g., stochastic gradient descent [6] and L-BFGS [35]) or a second-order method (e.g., Newton-Raphson method). (2) With p = 0, p-norm indicates the number of elements in a legit data sample that an a acker needs to manipulate in order to turn it into an adversarial sample. Di erent from the computation method above, in the se ing of p = 0 where unconstrained optimization problems (3) and (4) are not di erentiable, the computation for the optimal solution has to follow an approximation method introduced in [28] or [6].\n\n(3) With p = ∞, p-norm becomes a measure indicating the maximum change to individual features. As such, the optimal solution for (1) and (2) can be approximated by following the fast gradient sign method [12], which computes perturbation ∂L(f (x; w); )/∂x (or ∂L(f (x; w);ˆ )/∂x), multiplies it by distortion scale ε and then adds the product to legitimate data sample x. Note that the way they compute that perturbation can be through back-propagation in conjunction with gradient descent.\n\nAs is illustrated in the aforementioned optimization problems, in order to generate impactful adversarial samples, an a acker needs to know either standard DNN model f (·) or know of a way to approximate f (·). A recent study [35] has revealed that an a acker could well approximate a standard DNN model using a traditional DNN training algorithm on an auxiliary training dataset. In this paper, we use \"cross-model approach\" to refer to those adversarial sample cra ing methods that rely upon the approximation of a standard DNN model.\n\n\nEXISTING DEFENSES AND PROBLEM SCOPE\n\nTo counteract the adversarial learning problem described in the section above, recent research invent various training algorithms [12,13,27,35] to improve the robustness of a DNN model. ey indicate, by using new training algorithms they design, one can improve a DNN's resistance to the adversarial samples cra ed through the aforementioned cross-model approach. is is due to the fact that their training algorithms smooth a standard DNN's decision boundary, making adversarial samples -impactful to standard DNN models -no longer su ciently e ective. In this section, we summarize these defense mechanisms and discuss their limitations. Following our summary and discussion, we also de ne the problem scope of our research.\n\n\nExisting Defense Mechanisms\n\nRecently, research in hardening deep learning mainly focuses on two di erent tactics -data augmentation and model complexity enhancement. Here, we summarize them in turn and disucss their limitations. Data augmentation. To resolve the issue of \"blind spots\" (a more informal name given to adversarial samples), many methods that could be considered as sophisticated forms of data augmentation 2 have been proposed (e.g. [12,13,27]). In principle, these methods expand their training set by combining known samples with potential blind spots, the process of which has been called adversarial training [12]. Technically speaking, adversarial training can be formally described as adding a regularization term known as DataGrad to a DNN's training loss function [27]. e regularization penalizes the directions uncovered by adversarial perturbations (introduced in Section 2.2). erefore, adversarial training can work to improve the robustness of a standard DNN. Model complexity enhancement. DNN models are already complex, with respect to both the nonlinear function that they try to approximate as well as their layered composition of many parameters. However, the underlying architecture is straightforward when it comes to facilitating the ow of information forwards and backwards, greatly alleviating the e ort in generating adversarial samples. erefore, several ideas [13,29] have been proposed to enhance the complexity of DNN models, aiming to improve the tolerance of complex DNN models with respect to adversarial samples generated from simple DNN models. For example, [29] developed a defensive distillation mechanism, which trains a DNN from data samples that are \"distilled\" from another DNN. By using the knowledge transferred from the other DNN, the learned DNN classi ers become less sensitive to adversarial samples. Similarly, [13] proposed to stack an auto-encoder together with a standard DNN. It shows that this auto-encoding enhancement increases a DNN's resistance to adversarial samples. Limitation. While the aforementioned defenses have yielded promising results in terms of increasing model resistance, the scope of the model resistance they provide is relatively limited. Once an a acker obtains the knowledge of the new training algorithms -instead of using a traditional DNN training algorithm to substitute the algorithm which the target DNN is trained with -he can build his own model with the new training algorithm, and then use it as the cross model to facilitate the cra ing of adversarial samples. As we will show in Section 5, the adversarial samples cra ed through such new cross models sustain their o ensiveness to the corresponding DNN models. is indicates that the e ectiveness of existing defense mechanisms is highly dependent upon the obscurity of training algorithms.\n\n\nProblem Scope\n\nWith the existing defenses and their limitation in mind, here we de ne the problem scope of our research. Similar to most previous research -if not all -in hardening deep learning, we assume that an a acker cra s adversarial samples by solving the aforementioned optimization problems with derivative calculation (e.g., fast sign gradient descent or Newton-Raphson method). We believe this assumption is realistic for the following reason.\n\nDerivative calculation is the most general approach for solving an optimization problem. In the future, while one might be able to derive new forms of approaches in solving the aforementioned optimization problem, he or she has to ensure the new approaches are computationally e cient. Without the aid from derivative calculation, this can be relatively di cult. Even if one could computationally e ciently resolve the aforementioned optimization problems -for example perhaps through relaxation -without derivative calculation, he still need to prove the adversarial samples derived from such an approach are impactful. Given that relaxation reshapes an optimization problem, the \"optimal\" solution may not even close to any local optima of that original optimization problem.\n\nDi erent from prior research, we also assume that an adversary has not only the access to a DNN's structure as well as the dataset(s) used to build the DNN but more importantly the algorithm used to train the network. In other words, we assume a target DNN model is no longer obscure to an adversary and rather he has the full knowledge about a DNN model that he a empts to exploit. We believe this assumption is more practical because there is li le hope of keeping an adversary-resistant training algorithm completely secret from dedicated a ackers. In the long run, any system the security of which relies upon the obscurity of its design can be doomed [19].\n\n\nOUR APPROACH\n\nTo address the problem above, we propose a new approach to harden DNN models. Technically speaking, it follows the tactic of model complexity enhancement, which improves model resistance by increasing model complexity. Di erent from the existing techniques mentioned above, our approach however goes beyond the scope of robustness they provide. It ensures that an a acker cannot perform the aforementioned a ack to generate adversarial samples impactful to our learning model even if we reveal our training algorithm. In other words, our approach escalate a DNN's resistance to adversarial samples without the requirement of obscuring training algorithms.\n\nAs is discussed in the section above, the adversarial learning problem can be viewed as an optimization problem. To resolve that optimization problem, one needs to conduct analytical computation of gradients with respect to an input data sample and perform backward propagation accordingly. erefore, we escalate a DNN's robustness not only by increasing the model complexity but, more importantly, restricting back-propagation.\n\nMore speci cally, we integrate to a DNN a data transformation module, (·) graphically indicated in Figure 3. As is illustrated in the gure, the data transformation module projects X , an input data sample to (X ), a new representation, before passing it through a consecutive DNN. is transformation increases the complexity of a DNN model and augments its resistance to adversarial samples cra ed through the aforementioned cross-model scheme. In addition, it blocks the backward ows of gradients. With this block, even if the underlying training algorithm is disclosed, an adversary cannot cra adversarial samples. In the following, we specify the design principle of our data transformation module, followed by its design detail and some necessary discussions.\n\n\nDesign Principle\n\nTo block the backward ow of gradients, the design of data transformation must satisfy three requirements. Most notably, the data transformation must be non-di erentiable. As is discussed above, cra ing adversarial samples requires the calculation of gradients as well as the back-propagation of those gradients. By making data transformation module (·) non-di erentiable, therefore, we can make gradient calculation intractable and thus obstruct the backward ow of gradients. More formally, we can choose nondi erentiable function (x), making the derivative di cult to be calculated, i.e.,\n∂ n L(f ( (x); w); ) ∂x n = ∂ n L(f ( (x); w); ) ∂ (x) n · ∂ n (x) ∂x n .(5)\nHere, f (·) represents the DNN model in tandem with the data transformation module, and L(·) denotes the cost function described in Section 2.2. e derivative can be computed using either a rstorder optimization method (e.g., gradient descent) or a second-order method (e.g., Newton-Raphson method), in which n is equal to 1 and 2, respectively. While the non-di erentiability feature restricts the cra ing of adversarial samples, an adversary might still be able to generate adversarial samples. Since end-to-end gradient ow is blocked at the input layer of the successive DNN, back-propagation can only carry error gradients to the output of the transformation module. Given (·), an adversary could construct an adversarial sample by inverting transformation module (·) and passing the manipulated transformation output through the inversion of the transformation. More formally, the adversary can construct an adversarial sample by computing\n−1 ( (x) + ∂ n L(f ( (x); w); ) ∂ (x) n ).(6)\nIn addition to making data transformation non-di erentiable, therefore, we must further ensure that the inversion of the data transformation is computationally intractable. In other words, the data transformation (·) needs to have the property of non-invertibility.\n\nSatisfying the two requirements above ensures that our proposed approach can prohibit an a acker from cra ing adversarial samples directly from the target DNN model, and one does not need to concern about the disclosure of training algorithms. However, the data transformation proposed may signi cantly jeopardize the accuracy of a DNN model if not designed carefully. Take the following extreme case for example.\n\nHash functions like MD5 and SHA1 are one-way functions which have the properties of non-di erentiability as well as non-invertibility. By simply using it as the transformation module, we can easily prohibit an a acker from cra ing adversarial samples even if he knows of which hash function we choose and how we integrate it with DNNs. However, a hash function signi cantly changes the distribution of input data samples. Armed with it, a DNN model su ers from signi cant loss in classi cation performance. Last but not least, our design therefore must ensure data transformation can preserve the distribution of data representation. is can potentially make a DNN robust without sacri cing classi cation performance.\n\n\nDesign Detail\n\nFollowing the design principle above, we choose Locally Linear Embedding (LLE) [31], a non-parametric dimensionality reduction mechanism, serving as the data transformation module. As we will discuss in the following, this representative non-parametric method is non-di erentiable. More importantly, it can be theoretically proven that inverting LLE is an NP-hard problem. Last but not least, LLE seeks low-dimensional, neighborhood-preserving map of high-dimensional input samples, and thus is a method that best suited to preserving as much information in the input as possible. In the following, we rst describe LLE and then expound upon the fact that, as a non-parametric dimensionality reduction method, LLE is non-di erentiable. Furthermore, we theoretically prove LLE is computationally non-invertible.\n\n\nLocally Linear\n\nEmbedding. LLE is a non-parametric method designed to reduce input data dimensionality and at the same time preserve local properties of high-dimensional input in a lowerdimensional space. To some extent, this can ensure the distribution of high-dimensional data samples is as close as they are in a lower-dimensional space. Technically speaking, this is achieved by representing each high-dimensional data sample via a linear combination of its nearest neighbors. More formally, this can be expressed as\nx i = N j=1 w i j · x j .\nHere, x i and x j (x i , x j ∈ R 1×m ) denote the i t h data sample and its j t h neighbor (j = 1, 2..., N ), respectively. w i j represents the weight, indicating the contribution of x j to data sample x i . As is described in [31], those weights (a.k.a. reconstruction weights) can be represented as weight matrix W and computed by solving the following optimization problem:\nmin W i x i − j w i j · x j 2 2 s.t. j w i j = 1.(7)\nIn weight matrix W , LLE deems w i j = 0 if x j is not considered as a neighbor of x i , and the total number of neighbors assigned to x i is a carefully selected hyper-parameter. e neighboring relation between x i and x j depends on the value of the l 2 distance between x i and x j . Since the reconstruction weights encode the local properties of the high-dimensional data, they can be used to preserve the data distribution at the time of performing dimensionality reduction. More speci cally, LLE imposes the corresponding reconstruction weights to each lower-dimensional data sample via a similar linear combination, and then a empts to nd Y = { 1 , 2 , . . . , N }, the lower-dimensional representation of X = {x 1 , x 2 , . . . , x N } by solving the following optimization problem:\nmin Y i i − j w i j · j 2 2 s.t. i i = 0, 1 N i T i i = I .(8)\nwhere i , j ∈ R 1×m c , indicating i , j consist of m c of elements.\n\nIn order to solve the optimization problem above, the Rayleitz-Ritz theorem [16] is typically used. It computes the eigenvectors corresponding to the smallest nonzero eigenvalues of the inner matrix product (I − W ) T · (I − W ). For a detailed explication, we refer the readers to [16]. Here, I ∈ R N ×N is an identity matrix, and W ∈ R N ×N is the aforementioned reconstruction weight matrix.\n\nLLE is speci cally designed to retain the similarity between pairs of high dimensional samples when they are mapped to lower dimensions [31]. To illustrate this property, we provide a visualization of the data before and a er being processed by LLE in Figure 4. e visualization result demonstrates that using LLE as a data transformation module satis es the last design principle discussed in Section 4.1 (i.e., preserving the distribution of the original data). More importantly, this property also helps bound the lower dimensional mapping of adversarial samples to a vicinity which is lled by mappings of original test samples that are highly similar to these adversarial samples. As a result, there is a signi cantly lower chance that an adversarial sample acts as an outliers in the lower dimensional space. In other words, LLE makes a DNN more resistant to adversarial samples. In Section 5, we empirically validate this important property.\n\n\nNon-di erentiability of LLE.\n\nExisting dimensionality reduction methods can be categorized as either parametric or nonparametric [36]. Parametric methods utilize a xed amount of parameters to specify a direct mapping from high-dimensional data samples to their low-dimensional projections (or vice versa). is direct mapping is characterized by parameters, which are typically optimized to provide the best mapping performance. is is similar to the functionality provided by a standard DNN, which maps high-dimensional data samples to the nal decision space through di erentiable function f (·). As such, the derivative of parametric methods typically can be computed in an analytically e cient manner. In other words, parametric methods are generally di erentiable, and this nature becomes a disadvantage for blocking the backward gradient ow.\n\nOn the contrary, non-parametric methods do not su er from the issue above. For any non-parametric method, (·), there is no way to express it in a closed form. erefore, the derivative of (·) can be (a) Before LLE processing.\n\n(b) A er LLE processing. Figure 4: Visualization of the MNIST data set [20] in a 2-dimensionality space before and a er proceeded by LLE.\n\ncomputed only through a numeric but not an analytical approach. More formally, this means the calculation of ∂ (x)/∂x needs to be completed through the calculation of limit lim h→0\n(x +h)− (x ) h .\nGiven that a deep neural network takes as input each individual sample, which is discrete in the sample space, it is di cult to de ne the continuity of (·) with traditional topology and thus the di erentiability of (·) cannot be guaranteed. is indicates, as a member of non-parametric methods, LLE perfectly satis es the rst design principle discussed in Section 4.1 (i.e., not capable of performing derivative calculation).\n\n\nNon-invertibility of LLE.\n\nWe validate the non-invertibility of LLE by theoretically proving that reconstructing original highdimensional data from low-dimensional representations transformed by LLE is computationally intractable. More formally, we prove that, given a set of low-dimensional data Y = { 1 , 2 , . . . , N } (Y ∈ R N ×m c ) produced by LLE, reconstructing their original highdimensional representations X = {x 1 , x 2 , . . . , x N } (X ∈ R N ×m ) from Y is at least an NP-hard problem.\n\nRecall that LLE computes weight matrix W and utilizes it to project high-dimensional data samples to a lower-dimensional space. As a result, to restore high-dimensional data from its lowerdimensional representations, one has to recover that matrix by following the calculation similar to that shown in (7), except that x i and x j are replaced by i and j .\n\nOnce weight matrix W is restored, the recovery of original highdimensional data can be viewed as solving the following optimization problem:\nmin X i x i − j w i j · x j 2 2 .(9)\nIt is not di cult to realize that Equation (9) can be de ned in the following quadratic form:\nmin X i, j m i j · (x i · x j ),(10)\nwhere m i j = δ i j − w i j − w ji + k w ki w k j . Note that δ i j = 1 if i = j and 0 otherwise. If expressing m i j as the following matrix:\nM = m 11 m 12 · · · m 1N m 21 m 22 · · · m 2N . . . . . . . . . . . . m N 1 m N 2 · · · m N N ,(11)\nwe can easily realize that M is a symmetric matrix, which has the property of M T = M. Now, with the analysis above, the validation of non-invertibility amounts to proving that solving (10) is at least an NP-hard problem. In this work, we conduct this proof by introducing several constraints to this equation. Our basic idea is to use these constraints to relax the optimization problem in (10) to a nearby problem which can be easily proved as an NP-hard problem. More speci cally, we introduce the following constraints:\ni x i = ì 0,(12)\nand\n− 1 ≤ x i j ≤ 1, ∀i ∈ N , j ∈ m,(13)\nwhere ì 0 denotes a zero vector, and x i j represents the j th element in vector x i .\n\nWith these constraints introduced to Equation 10, we can relax the optimization problem to a quadratic problem with a nonpositive semi-de nite constraint, which itself is a class of NP-hard problems [9]. In the following, we provide more details on why the involvement of the aforementioned constraints transforms the optimization problem in (10) to this class of NP-hard problems.\n\nLet A ∈ R N A ×1 denote a column vector which is the concatenation of x i for i = 1, . . . , N and N A = N × m. en, we have A = (x 1 , x 2 , · · · , x N ) T . Let q ∈ R 1×N A denote a row vector in which every element is equal to 1. We further de ne matrices P, Q ∈ R N A ×N A as follows:\nP = P 11 P 12 · · · P 1N P 21 P 22 · · · P 2N . . . . . . . . . P N 1 P N 2 · · · P N N , Q = −I,(14)\nwhere P i j = m i j · . ∈ R m×m is a matrix of ones where every element is equal to 1. Given the constraint in (12) that we introduce, it is not di cult to discover i x T i = ì 0. Since the multiplication of a vector and its transpose derives a non-negative value, we have Σ i x i x T i ≥ 0 and the constraint in (12) can be expressed as inequation\n−Σ i x i x T i + i x T i + α ≤ 0,\nindicating there exists a positive number, α that always holds the inequity. By rewriting the inequation using the notations newly de ned above, we can therefore transform the constraint in (12) into the form of A T QA + qA + α ≤ 0.\n\nGiven the constraint in (13), we can easily derive inequation\nΣ i x i x T i − N A ≤ 0, which can be further expressed as Σ i x i x T i − N A + γ ≤ 0\nindicating there alway exists a constant, γ that holds the inequity. By rewriting both the constraint itself and this inequation using newly de ned notations, we can derive constraints A ∞ ≤ 1 as well as A T IA − N A + γ ≤ 0. As such, we can transform Equation (10) and the aforementioned constraints in (12) and (13) into following form:\nmin A T PA s.t. A T QA + qA + α ≤ 0. A T IA − N A + γ ≤ 0. A ∞ ≤ 1.(15)\nHere, Q is negative semi-de nite, and thus Equation (15) is a quadratic problem with a non-positive semi-de nite constraint. According to [9,37], Equation (15) belongs to a class of NP-hard problems, which implies the non-invertability of LLE.\n\n\nDiscussion\n\nHere, we discuss some related issues and possible a acks against our proposed technique. Approximation of LLE. While the aforementioned discussion and theoretical proof have already indicated the e ectiveness of our proposed approach, intuition suggests that an adversary might still come up with an a ack. Speci cally, he might approximate LLE using a parametric mapping and then substitute LLE accordingly. Since parametric mappings do not have the property of non-di erentiability, the adversary can take advantage of the substitute, pass gradients through and eventually cra adversarial samples. However, as we will show in Section 5, even using the state-of-the-art approximation scheme, an adversary cannot cra impactful adversarial samples. Other dimensionality reduction methods. As is described above, we choose LLE, a representative non-parametric dimensionality reduction method, to serve as the data transformation module. is is due to the fact that it provides many properties needed for hardening a DNN, such as non-di erentiability, non-invertibility and the capability of preserving data distribution.\n\nGoing beyond LLE, there are other non-parametric dimensionality reduction methods that o er the same properties, e.g., t-Distributed Stochastic Neighbor Embedding (t-SNE) [23] and Sammon Mapping [33]. However, they cannot be utilized in our problem domain for the following reason.\n\nDeep neural networks exhibit superior performance when dealing with data in a relatively high dimensionality. Other non-parametric methods are typically designed more for tasks like visualization [23] where it is required that the dimensionality of the mappings is two or three. Using them as our data transformation module, they cannot provide high-dimensional data input for the DNN in tandem with the transformation, and may signi cantly jeopardize classication performance.\n\n\nEVALUATION\n\nAs is described in Section 1, adversarial training [12] and defensive distillation [29] are the most representative techniques that have been proposed to defend against adversarial samples. In this work, we use the proposed approach to train our own adversary resistant DNN (LLE-DNN), and then compare it with those enhanced by these two approaches.\n\n\nDataset\n\nWe evaluate our adversary-resistant DNN model by performing multiple experiments on several widely used datasets, including a dataset for malware detection [5], the MNIST dataset for image recognition [20] and the IMDB dataset for sentiment analysis [22]. Malware dataset: It is a collection of window audit logs, each of which ties to either a benign or malicious so ware sample. e dimensionality of the feature-space for each sample is reduced to 10,000 based on the feature selection metric in [5]. Each feature indicates the occurrence of either a single lesystem access or a sequence of access events, thus taking on the value of 0 or 1. Figure 2 illustrates a subset of features of a so ware sample. Here, 0 indicates that the sequence of events did not occur while 1 indicates the opposite. For each so ware sample, it has been labeled with either 1 or 0, indicating malicious and benign so ware, respectively. e dataset is split into 26,078 training examples, with 14,399 benign and 11,679 malicious so ware samples, and 6,000 testing samples, with benign and malicious so ware samples evenly divided. MNIST dataset: It is a large database of handwri en digits that is commonly used for training various image processing systems. It is composed of 70,000 greyscale images (of 28×28, or 784, pixels) of handwri en digits, split into a training set of 60,000 samples and a testing set of 10,000 samples. IMDB dataset: It consists of 25,000 movie reviews, with one half labeled as \"positive\" and the other \"negative\", indicating the sentiment of these reviews. We randomly split the dataset with 70% movie reviews for training and the remaining for testing. Following the procedure introduced in [24], we encoded the words in each movie review using a dictionary carrying 5,000 words most frequently used. en, we utilized a word embedding technique [24] to convert each word into a vector with a dimensionality of 600. For each movie review, we linearly combined the vectors indicating the words appearing in that review, and then treat the embedding as the representation of that movie review. Table 2: An illustration of features in malware dataset. Individual features are represented in rows. ey are either a lesystem access or a sequence of lesystem events.\n\n\nLearning Technology\nBlack Box White Box MNIST MALWARE IMDB MNIST MALWARE IMDB l ∞ l 2 l 0 l 0 l ∞ l 2 l ∞ l 2 l 0 l 0 l ∞ l 2 Standard\n\nExperimental Design\n\nFor each application described above, we train 4 DNN models using the traditional deep learning training method, adversarial training [12], defensive distillation [29] and our own approach. We specify the hyperparameters of these DNNs in Table 3. We measure their classi cation accuracy by applying the models to the corresponding testing datasets. By comparing their classi cation performance, we evaluate the in uence that our proposed approach brings to a DNN. More speci cally, we examine if LLE-DNN exhibits similar -if not the same or be er -classi cation accuracy.\n\nSince the goal of this work is to improve the robustness of a DNN model, we also evaluate our DNN models' resistance to adversarial samples. In particular, we derive adversarial samples from the aforementioned testing datasets, test them against our DNN model and compare its model resistance with those of DNNs enhanced by the other two techniques [12,29].\n\nAs is discussed in Section 2.2, an a acker cra s adversarial samples through auxiliary models. In Table 1, black-box and white-box indicate the auxiliary models trained through di erent schemes. More speci cally, black-box represents the auxiliary model trained through the standard deep learning training scheme, indicating an a acker does not have su cient knowledge about the underlying training algorithm and he can use only a standard approach to train a cross model and cra adversarial samples. White-box represents the auxiliary model trained exactly through the learning schemes proposed as a defense. is simulates a situation where a defense mechanism is publicly disclosed and an a acker exploits that mechanism to produce a highly similar -if not the same -model to cra adversarial samples. Note that, for both black-box and white-box tactics, we use the same hyperparameters and training dataset to build auxiliary models. More speci cally, our auxiliary model training shares the same hyperparameters and training dataset with the standard DNN shown in Table 3.\n\nIn addition to the methods described in Section 2.2, the cra ing of adversarial samples must ensure a slight perturbation introduced to a data sample does not undermine its semantic. In other words, we must make sure that, while misleading a classi er to output the wrong class with high con dence, the perturbation to an image should be nearly indistinguishable to the human eyes, that to a malicious so ware sample should not jeopardize so ware functionality nor break its malevolence, and that to a movie review should not break its semantic meaning. In the following, we describe how we ne-tune adversarial samples to preserve semantic for di erent applications. Malware classi cation. Recall that our malware samples are represented by features, the value of which are binary, indicating the occurrence of an lesystem access or a sequence of access events. When generating adversarial samples, we cannot simply disable Table 3: e hyperparameters of all the investigated DNN models.\n\nlesystem access events in that this might jeopardize the functionality of the so ware sample and even break down its malevolence. With this in mind, a bit of care must be taken.\n\nIn this work, our experiment follows the approach introduced in Section 2.2. To be speci c, we cra adversarial so ware samples by solving optimization problem (1) with the se ing of zero norm (i.e., l 0 ). is indicates the manipulation to a sample is restricted to ipping binary feature values. In addition, this implies the strongest a ack scenario in that optimization problem (1) carries less constraints making the adversarial samples chosen for our evaluation more impactful. Going beyond adversarial sample craing approach discussed in Section 2.2, we also restrict that the value change of a feature can be only from 0 to 1 but not the opposite.\n\nis amounts to allowing the addition of new lesystem access events only. is manipulation strategy is reasonable for the reason that malware mutation techniques (e.g., [25]) can morph a malware sample by stitching together instructions from benign programs, making the malware perform additional lesystem accesses but not undermining its maliciousness nor its functionality. Since malware manipulation is done with the intent of fooling a malware classi er driven by a DNN, it should be noticed that we do not morph a benign so ware sample, making it malicious. Image recognition. Image data samples contain less strict semantic than the malware data samples above. To preserve image semantic -making a perturbation nearly indistinguishable -we follow the approaches introduced in [6,12,35]. More speci cally, we selected l 0 , l 2 and l ∞ distance to represent the dissimilarity between an image and its corresponding adversarial sample. Especially, we restrict the l ∞ distance in a relative small range (i.e., ϵ ≤ 0.15) when cra ing adversarial samples. Sentiment Analysis. To generate adversarial samples for movie reviews, we again followed the approach introduced in Section 2.2. To be speci c, we solved optimization problem (1) and con gured p-norm with the se ing to l 2 and l ∞ . is is due to the fact that each review is encoded in a vector in which each element is a decimal, and l 2 and l ∞ distances represent the best measure for the dissimilarity between a movie review and its corresponding adversarial sample. Table 4: e comparison of classi cation accuracy on di erent datasets.\n\nAs is mentioned above, each vocabulary has been encoded in a vector with a dimensionality of 600, and we embedded a movie review by linearly combining corresponding vectors. When generating an adversarial sample by introducing a slight perturbation to the embedding, we cast the perturbation to only one vector. is ensures that we introduce only one word change to that review with the hope that it preserves the semantic meaning of that review as much as possible. However, one word change does not guarantee the invariance of the semantic meaning. For example, it would be obvious alteration to the semantic meaning if the replacement happens to be the negative word in \"… makes it the biggest disappointment I've experienced from cinema in years …\". As such, we manually choose the word that incurs minimal semantic change to that movie review. Figure 5 illustrates a movie review sample and its corresponding adversarial sample generated through this approach.\n\n\nExperimental Setup and Results\n\nOn the datasets described above, we rst measure the accuracy of all the aforementioned defense techniques. We then measure their resistance to the adversarial samples cra ed through the aforementioned tactics.\n\n\nClassification Accuracy.\n\nTo identify the optimal dimensionality to which LLE needs to map original data samples, we implemented several LLE-DNNs with di erent se ings of dimensionality of LLE mappings. Figure 6 shows the impact of dimensionality mapping upon the classi cation accuracy obtained by LLE-DNN. Across all three datasets, it is easy to observe that, the classi cation accuracy rst increases when the dimensionality of the LLE mappings rises and then starts to decrease. In our experiment, we choose the highest classi cation accuracy to represent the performance of our LLE-DNN. Table 4 presents the classi cation accuracy results obtained from all investigated DNNs with respect to the testing datasets. Note that, while prior works (e.g, [7,10]) have already demonstrated they can train a DNN with an error rate less than 1% on the MNIST benchmark, their performance improvement does not result from a DNN but model ensemble or elastic distortions added to training data. To study the in uence of our proposed approach upon a standard DNN, therefore, we did not combine models nor augment with arti cially distorted versions of the original training samples. e classi cation accuracy shown in the table has already represented the best performance that a standard DNN can achieve.\n\nSimilar to adversarial training and defensive distillation, the LLE-DNN is quite e ective in preserving classi cation accuracy.\n\nis implies our proposed approach well preserves data sample distribution. For the malware classi cation task, it can be observed that LLE-DNN appears to be be er at feature learning, achieving the highest classi cation accuracy among DNNs that we investigated. is is presumably due to the fact that malware data samples are highly sparse carrying a large amount of redundant information, and the data transformation module in LLE-DNN eliminates those redundancy and ameliorates the learning ability of a DNN. Table 1 illustrates the DNNs that we investigated as well as their accuracy in classifying adversarial samples. It can be observed that, black-box adversarial samples can cut down the accuracy of the standard DNN to 6.86%, 6.40% and 7.50% under the a acks of l ∞ , l 2 and l 0 , respectively. In contrast, all of the defense mechanisms investigated demonstrate strong resistance to these black-box adversarial samples.\n\n\nModel Resistance.\n\nis indicates, without su cient knowledge on the underlying defense mechanisms, it is di cult for an a acker to cra impactful adversarial samples. In other words, existing defense mechanisms can signi cantly escalate a DNN's resistance to adversarial samples if one can obscure the design of the defenses.\n\nDespite the improvement in model robustness, we also observe that our LLE-DNN generally exhibits the best resistance to blackbox adversarial samples, whereas the defensive distillation approach typically yields the worst resistance. is is presumably due to the fact that, the dimensionality reduction resided in LLE-DNN transforms adversarial samples into a subspace in which they no longer act as outliers, while defensive distillation smooths only a classi cation decision boundary which does not signi cantly reduce the subspace of adversarial samples.\n\nWith regard to the white-box se ing, we discover both adversarial training and defensive distillation su er from white-box adversarial samples. eir resistance to white-box adversarial samples is signi cantly worse than those created under the black-box se ing.\n\nis observation is consistent with that reported in [6]. e reason behind this is that, both techniques stash away the adversarial sample subspace, but the disclosure of defense mechanisms uncovers the path of nding that subspace.\n\nDi erent from adversarial training and defensive distillation, our LLE-DNN is naturally resistant to white-box adversarial samples. As is discussed in Section 4, our proposed approach stashes away the adversarial sample subspace and at the same time restricts derivative calculation. Even if our defense mechanism is revealed, therefore, it is still computationally di cult to nd adversarial samples.\n\nTo perform quantitative comparison with the other two approaches, however, we approximate the data transformation in the LLE-DNN -non-parametric dimensionality reduction component -using a parametric model. To be speci c, we choose a DNN to approximate LLE in that a DNN has a large amount of parameters which is typically viewed as the best approximation for non-parametric learning models [17]. With the support from this approximation, we treated the LLE-DNN as a white box and generated adversarial samples accordingly. We show its model resistance in Table 1. It can be observed that, our LLE-DNN still demonstrates strong resistance to white-box adversarial samples even if we substituted LLE to its best approximation. is implies that, there might a theoretical lower bound between a non-parametric model and its parametric approximation, which can naturally serve as a defense against white-box adversarial samples.\n\n\nCONCLUSION\n\nA Deep Neural Network is vulnerable to adversarial samples. Existing defense mechanisms improve a DNN model's resistance to adversarial samples by using the tactic of security through obscurity. Once the design of the defense is disclosed, therefore, the robustness they provide wane. Motivated by this, this work introduces a new approach to escalate the robustness of a DNN model. In particular, it integrates to a DNN model LLE, a non-parametric dimensionality reduction method. With this approach, we show that one can develop a DNN model resistant to adversarial samples even if he or she reveals its design details (i.e., the underlying training algorithm). By demonstrating the DNNs enhanced by our proposed technique across various applications, we argue the proposed approach introduces nearly no degradation in classi cation performance. In contrast, for some applications, it even exhibits performance improvement. As part of the future work, we will further explore the performance of this approach in a wider variety of applications across di erent deep neural architectures.\n\nFigure 2 :\n2Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.\n\nFigure 3 :\n3A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·).\n\nFigure 6 :\n6e variation of classi cation accuracy vs. the dimensionality of data mappings.\nere are many types of activations to choose from, including the hyperbolic tangent, the logistic sigmoid, or the linear recti ed function, etc.[4] \nData augmentation refers to arti cially expanding the data-set. 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ACM, 2014.\n"},"corpus_id":{"kind":"number","value":17676338,"string":"17,676,338"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1612.01401\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1612.01401v2.pdf\"\n ],\n \"pdf_sha\": \"91cc241de12cd4a55cdb6a1712ab390c2a55ba88\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Building Adversary-Resistant Deep Neural Networks without Security through Obscurity"},"authors":{"kind":"list like","value":["D C Washington","Usa","D C Washington","Usa"],"string":"[\n \"D C Washington\",\n \"Usa\",\n \"D C Washington\",\n \"Usa\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Deep neural networks (DNNs) have proven to be quite e ective in a vast array of machine learning tasks, with recent examples in cyber security and autonomous vehicles. Despite the superior performance of DNNs in these applications, it has been recently shown that these models are susceptible to a particular type of a ack that exploits a fundamental aw in their design. is a ack consists of generating particular synthetic examples referred to as adversarial samples. ese samples are constructed by slightly manipulating real data-points in order to \"fool\" the original DNN model, forcing it to mis-classify previously correctly classi ed samples with high con dence. Addressing this aw in the model is essential if DNNs are to be used in critical applications such as those in cyber security.Previous work has provided various learning algorithms to enhance the robustness of DNN models, and they all fall into the tactic of \"security through obscurity\". is means security can be guaranteed only if one can obscure the learning algorithms from adversaries. Once the learning technique is disclosed, DNNs protected by these defense mechanisms are still susceptible to adversarial samples. In this work, we investigate this issue shared across previous research work and propose a generic approach to escalate a DNN's resistance to adversarial samples. More speci cally, our approach integrates a data transformation module with a DNN, making it robust even if we reveal the underlying learning algorithm. To demonstrate the generality of our proposed approach and its potential for handling cyber security applications, we evaluate our method and several other existing solutions on datasets publicly available, such as a large scale malware dataset, MNIST and IMDB datasets. Our results indicate that our approach typically provides superior classi cation performance and resistance in comparison with state-of-art solutions. * e rst and second author contribute equally.Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro t or commercial advantage and that copies bear this notice and the full citation on the rst page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s)."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":""},{"title":"INTRODUCTION","content":"Beyond highly publicized victories in automatic game-playing as in Go [34], there have been many successful applications of deep neural networks (DNN) in image and speech recognition. Recent explorations and applications include those in medical imaging [1,38] and self-driving cars [11,14]. In the domain of cybersecurity, security companies have demonstrated that deep learning could o er a far be er way to classify all types of malware [8,30,39].\nBeyond highly publicized victories in automatic game-playing as in Go [34], there have been many successful applications of deep neural networks (DNN) in image and speech recognition. Recent explorations and applications include those in medical imaging [1,38] and self-driving cars [11,14]. In the domain of cybersecurity, security companies have demonstrated that deep learning could o er a far be er way to classify all types of malware [8,30,39].\nDespite its potential, deep neural networks (DNN), like all other machine learning approaches, are vulnerable to what is known as adversarial samples [2,18].\nDespite its potential, deep neural networks (DNN), like all other machine learning approaches, are vulnerable to what is known as adversarial samples [2,18].\nis means that they can be easily deceived by non-obvious and potentially dangerous manipulation [26,35]. To be more speci c, an a acker could use the same training algorithm, back-propagation of errors, and a surrogate dataset to construct an auxiliary model. Since this model could provide the a acker with a capability of exploring a DNN's blind spots, he can, with minimal e ort, cra an adversarial sample -a synthetic example generated by slightly modifying a real example in order to make the deep learning system \"believe\" the sample subtly perturbed belongs to an incorrect class with high con dence.\nis means that they can be easily deceived by non-obvious and potentially dangerous manipulation [26,35]. To be more speci c, an a acker could use the same training algorithm, back-propagation of errors, and a surrogate dataset to construct an auxiliary model. Since this model could provide the a acker with a capability of exploring a DNN's blind spots, he can, with minimal e ort, cra an adversarial sample -a synthetic example generated by slightly modifying a real example in order to make the deep learning system \"believe\" the sample subtly perturbed belongs to an incorrect class with high con dence.\nAccording to a recent study [12], adversarial samples occur in a subspace relatively broad, which means it is impractical to build a defense that can rule out all adversarial samples. As such, the design principle followed by existing defense mechanisms is not to harden a DNN model naturally resistant to any adversarial samples. Rather, they focus on hiding that subspace, making adversaries di cult in nding impactful adversarial samples. For example, representative defensesadversarial training [12] and defensive distillation [29] both increase the complexity of original DNNs with the goal of making adversarial samples -impactful for original DNNs -no longer e ective.\nAccording to a recent study [12], adversarial samples occur in a subspace relatively broad, which means it is impractical to build a defense that can rule out all adversarial samples. As such, the design principle followed by existing defense mechanisms is not to harden a DNN model naturally resistant to any adversarial samples. Rather, they focus on hiding that subspace, making adversaries di cult in nding impactful adversarial samples. For example, representative defensesadversarial training [12] and defensive distillation [29] both increase the complexity of original DNNs with the goal of making adversarial samples -impactful for original DNNs -no longer e ective.\nHowever, in this work, we show that the defenses proposed are far from ideal and even considered a dangerous practice. In particular, we demonstrate existing defense mechanisms all follow the approach of \"security through obscurity\", in which security is achieved by keeping defenses obscured from adversaries. Frankly speaking, defenses following this approach indeed mitigate the adversarial sample problem. However, when applied to security critical applications such as malware classi cation, they become particularly disconcerting.\nHowever, in this work, we show that the defenses proposed are far from ideal and even considered a dangerous practice. In particular, we demonstrate existing defense mechanisms all follow the approach of \"security through obscurity\", in which security is achieved by keeping defenses obscured from adversaries. Frankly speaking, defenses following this approach indeed mitigate the adversarial sample problem. However, when applied to security critical applications such as malware classi cation, they become particularly disconcerting.\nIn the past, there have been a huge amount of debates on security through obscurity, and a general consensus has been reach. at is, obscurity is a perfectly valid security tactic but it cannot be trusted for security. Once design or implementation is uncovered, users totally lost all the security gained by obscurity. To regain the security through obscurity, one has to come up with a completely new design or implementation. As such, Kerckho s' principle [19] suggests obscurity can be used as a layer of defense, but it should never be used as the only layer of defense.\nIn the past, there have been a huge amount of debates on security through obscurity, and a general consensus has been reach. at is, obscurity is a perfectly valid security tactic but it cannot be trusted for security. Once design or implementation is uncovered, users totally lost all the security gained by obscurity. To regain the security through obscurity, one has to come up with a completely new design or implementation. As such, Kerckho s' principle [19] suggests obscurity can be used as a layer of defense, but it should never be used as the only layer of defense.\nInspired by this, we propose a new mechanism to escalate a DNN's resistance to adversarial samples. Di erent from existing defenses, our proposed approach unnecessitates model obscurity. In other words, even though we reveal the model, it will still be more than burdensome for adversaries to cra adversarial samples.\nInspired by this, we propose a new mechanism to escalate a DNN's resistance to adversarial samples. Di erent from existing defenses, our proposed approach unnecessitates model obscurity. In other words, even though we reveal the model, it will still be more than burdensome for adversaries to cra adversarial samples.\nMore speci cally, we arm a standard DNN with a data transformation module, which projects original data input into a new representation before they are passed through the consecutive DNN.\nMore speci cally, we arm a standard DNN with a data transformation module, which projects original data input into a new representation before they are passed through the consecutive DNN.\nis can be used as a defense for the following two reasons. First, data transformation can potentially stash away the space of adversarial manipulations to a carefully designed hyperspace. is makes a ackers di cult in nding adversarial samples impactful for the armed DNN. Second, as we will theoretically prove in Section 4, a data transformation module carefully designed can exponentially increase computation complexity for an a acker to cra impactful adversarial samples. is means that, even though an a acker compromises obscurity and has the full knowledge about the armed DNN model (i.e., the training algorithm, dataset and hyper-parameters), he still cannot launch the a ack -detrimental to DNNs enhanced by other adversary-resistant techniques -nor jeopardize model resistance.\nis can be used as a defense for the following two reasons. First, data transformation can potentially stash away the space of adversarial manipulations to a carefully designed hyperspace. is makes a ackers di cult in nding adversarial samples impactful for the armed DNN. Second, as we will theoretically prove in Section 4, a data transformation module carefully designed can exponentially increase computation complexity for an a acker to cra impactful adversarial samples. is means that, even though an a acker compromises obscurity and has the full knowledge about the armed DNN model (i.e., the training algorithm, dataset and hyper-parameters), he still cannot launch the a ack -detrimental to DNNs enhanced by other adversary-resistant techniques -nor jeopardize model resistance.\ne approach proposed in this work is bene cial for the following reasons. First, it escalates a DNN's resistance to adversarial samples with be er security assurance. Second, our approach ensures that a DNN maintains desirable classi cation performance while requiring only minimal modi cation to existing architectures.\ne approach proposed in this work is bene cial for the following reasons. First, it escalates a DNN's resistance to adversarial samples with be er security assurance. Second, our approach ensures that a DNN maintains desirable classi cation performance while requiring only minimal modi cation to existing architectures.\nird, while this work is primarily motivated by the need to safeguard DNN models used in critical security applications, it should be noted that the proposed technique is general and can be adopted to other applications where deep learning is popularly applied, such as image recognition and sentiment analysis. We demonstrate this applicability using publicly-available datasets in Section 5.\nird, while this work is primarily motivated by the need to safeguard DNN models used in critical security applications, it should be noted that the proposed technique is general and can be adopted to other applications where deep learning is popularly applied, such as image recognition and sentiment analysis. We demonstrate this applicability using publicly-available datasets in Section 5.\nIn summary, this work makes the following contributions.\nIn summary, this work makes the following contributions.\n• We propose a generic approach to facilitate the development of adversary-resistant DNNs without following the tactic of security through obscurity. • Using our approach, we develop an adversary-resistant DNN, and theoretically prove its resistance cannot be jeopardized even if the model is fully disclosed. • We evaluate the classi cation performance and robustness of our adversary-resistant DNN and compare it with that ... ...\n• We propose a generic approach to facilitate the development of adversary-resistant DNNs without following the tactic of security through obscurity. • Using our approach, we develop an adversary-resistant DNN, and theoretically prove its resistance cannot be jeopardized even if the model is fully disclosed. • We evaluate the classi cation performance and robustness of our adversary-resistant DNN and compare it with that ... ..."},{"title":"Hidden Layers","content":""},{"title":"Hidden Layers","content":"Softmax Layer Input Layer f(x) x Figure 1: A feed-forward deep neural network with three hidden layers and a so max output layer.\nSoftmax Layer Input Layer f(x) x Figure 1: A feed-forward deep neural network with three hidden layers and a so max output layer.\nof existing defense mechanisms. Our result shows that our DNN exhibits similar -sometimes even be er -classi cation performance but with superior model resistance. e rest of this paper is organized as follows. Section 2 introduces the background of DNNs and adversarial sample problem. Section 3 discusses existing defense mechanisms and de nes the problem scope of our research. Section 4 presents our generic approach. In Section 5, we develop and evaluate DNNs in the context of image recognition, sentiment analysis and malware classi cation. Finally, we conclude this work in Section 6.\nof existing defense mechanisms. Our result shows that our DNN exhibits similar -sometimes even be er -classi cation performance but with superior model resistance. e rest of this paper is organized as follows. Section 2 introduces the background of DNNs and adversarial sample problem. Section 3 discusses existing defense mechanisms and de nes the problem scope of our research. Section 4 presents our generic approach. In Section 5, we develop and evaluate DNNs in the context of image recognition, sentiment analysis and malware classi cation. Finally, we conclude this work in Section 6."},{"title":"BACKGROUND","content":""},{"title":"BACKGROUND","content":"A typical DNN architecture consists of multiple successive layers of processing elements, or so-called \"neurons\". Each processing layer can be viewed as learning a di erent, more abstract representation of the original multidimensional input distribution. As a whole, a DNN can be viewed as a highly complex function, f (·) that is capable of nonlinearly mapping an original high-dimensional data point to a lower dimensional output. In this section, we brie y introduce the well-established DNN model, followed by the description of the adversarial learning problem.\nA typical DNN architecture consists of multiple successive layers of processing elements, or so-called \"neurons\". Each processing layer can be viewed as learning a di erent, more abstract representation of the original multidimensional input distribution. As a whole, a DNN can be viewed as a highly complex function, f (·) that is capable of nonlinearly mapping an original high-dimensional data point to a lower dimensional output. In this section, we brie y introduce the well-established DNN model, followed by the description of the adversarial learning problem."},{"title":"Deep Neural Networks","content":""},{"title":"Deep Neural Networks","content":"As is graphically depicted in Figure 1, a DNN contains an input layer, multiple hidden layers, and an output layer. e input layer takes in each data sample in the form of a multidimensional vector. Starting from the input, computing the pre-activations of each subsequent layer simply requires, at minimum, a matrix multiplication (where a weight/parameter vector, with length equal to the number of hidden units in the target layer, is assigned to each unit of the layer below) usually followed by summation with a bias vector. is process roughly models the process of a layer of neurons integrating the information received from the layer below (i.e., computing a pre-activation) before applying an elementwise activation function 1 . is integrate-then-re process is repeated subsequently for each layer until the last layer is reached. e last layer, or output, is generally interpreted as the model's predictions for some given input data, and is o en designed to compute a parameterized likelihood distribution using the so max function (also known as multi-class regression or minimizing cross entropy). is bo omup propagation of information is also referred to as feed-forward inference [15].\nAs is graphically depicted in Figure 1, a DNN contains an input layer, multiple hidden layers, and an output layer. e input layer takes in each data sample in the form of a multidimensional vector. Starting from the input, computing the pre-activations of each subsequent layer simply requires, at minimum, a matrix multiplication (where a weight/parameter vector, with length equal to the number of hidden units in the target layer, is assigned to each unit of the layer below) usually followed by summation with a bias vector. is process roughly models the process of a layer of neurons integrating the information received from the layer below (i.e., computing a pre-activation) before applying an elementwise activation function 1 . is integrate-then-re process is repeated subsequently for each layer until the last layer is reached. e last layer, or output, is generally interpreted as the model's predictions for some given input data, and is o en designed to compute a parameterized likelihood distribution using the so max function (also known as multi-class regression or minimizing cross entropy). is bo omup propagation of information is also referred to as feed-forward inference [15].\nDuring the learning phase of the model, the DNN's predictions are evaluated by comparing them with known target labels associated with the training samples (also known as the\"ground truth\"). Speci cally, both predictions and labels are taken as the input to L(·), a selected cost function. e DNN's parameters are then optimized with respect to this cost function using the method of steepest gradient descent, minimizing prediction errors on the training set.\nDuring the learning phase of the model, the DNN's predictions are evaluated by comparing them with known target labels associated with the training samples (also known as the\"ground truth\"). Speci cally, both predictions and labels are taken as the input to L(·), a selected cost function. e DNN's parameters are then optimized with respect to this cost function using the method of steepest gradient descent, minimizing prediction errors on the training set.\nMore formally, given (X , Y ):\nMore formally, given (X , Y ):\nR m is a data sample and i ∈ R k is the corresponding data label, where, if categorical, is typically represented through a 1-of-k encoding, the goal of model learning is to minimize the cost function represented by\nR m is a data sample and i ∈ R k is the corresponding data label, where, if categorical, is typically represented through a 1-of-k encoding, the goal of model learning is to minimize the cost function represented by\ndiction of training sample x i and w represents the weights and bias associated with the connections between neurons.\ndiction of training sample x i and w represents the weights and bias associated with the connections between neurons."},{"title":"Adversarial Sample Problem","content":""},{"title":"Adversarial Sample Problem","content":"An adversarial sample is a synthetic data sample cra ed by introducing slight perturbations to a legitimate input sample (see Figure 2). In multi-class classi cation tasks, such adversarial samples can cause a DNN to classify themselves into a random class other than the correct one (sometimes not even a reasonable alternative). Recent research [6] demonstrates that, a ackers can uncover such data samples through various approaches (e.g., [3,12,21,28,32,35]) which can all be described as solving either optimization problem\nAn adversarial sample is a synthetic data sample cra ed by introducing slight perturbations to a legitimate input sample (see Figure 2). In multi-class classi cation tasks, such adversarial samples can cause a DNN to classify themselves into a random class other than the correct one (sometimes not even a reasonable alternative). Recent research [6] demonstrates that, a ackers can uncover such data samples through various approaches (e.g., [3,12,21,28,32,35]) which can all be described as solving either optimization problem\nor optimization problem\nor optimization problem\nHere, optimization problem (1) indicates that an a acker searches for adversarial samplex, the prediction of which is as far as its true label, whereas optimization problem (2) indicates an a acker searches for adversarial samplex so that its prediction is as close as target labelˆ whereˆ is not equal to , the true label of that adversarial sample.\nHere, optimization problem (1) indicates that an a acker searches for adversarial samplex, the prediction of which is as far as its true label, whereas optimization problem (2) indicates an a acker searches for adversarial samplex so that its prediction is as close as target labelˆ whereˆ is not equal to , the true label of that adversarial sample.\nIn both optimization problems above, L(·) represents the aforementioned cost function and f (·) denotes the DNN model trained with the traditional learning method discussed above . · p is pnorm -sometimes also speci ed as l p distance -indicating the dissimilarity between adversarial samplex and its corresponding legit data sample x. With di erent values of p -most popularly selected in adversarial learning research -the optimization problems above can be computed in the following manners. Adversarial samples generated by following [12] Adversarial samples generated by following [6] Adversarial samples generated by following [35] (1) With p = 2, p-norm represents the measure of Euclidean distance.\nIn both optimization problems above, L(·) represents the aforementioned cost function and f (·) denotes the DNN model trained with the traditional learning method discussed above . · p is pnorm -sometimes also speci ed as l p distance -indicating the dissimilarity between adversarial samplex and its corresponding legit data sample x. With di erent values of p -most popularly selected in adversarial learning research -the optimization problems above can be computed in the following manners. Adversarial samples generated by following [12] Adversarial samples generated by following [6] Adversarial samples generated by following [35] (1) With p = 2, p-norm represents the measure of Euclidean distance.\ne constraint optimization problems above can be speci ed as unconstrained optimization problemŝ\ne constraint optimization problems above can be speci ed as unconstrained optimization problemŝ\nHere, both (3) and (4) can be solved by following either a rstorder optimization method (e.g., stochastic gradient descent [6] and L-BFGS [35]) or a second-order method (e.g., Newton-Raphson method). (2) With p = 0, p-norm indicates the number of elements in a legit data sample that an a acker needs to manipulate in order to turn it into an adversarial sample. Di erent from the computation method above, in the se ing of p = 0 where unconstrained optimization problems (3) and (4) are not di erentiable, the computation for the optimal solution has to follow an approximation method introduced in [28] or [6].\nHere, both (3) and (4) can be solved by following either a rstorder optimization method (e.g., stochastic gradient descent [6] and L-BFGS [35]) or a second-order method (e.g., Newton-Raphson method). (2) With p = 0, p-norm indicates the number of elements in a legit data sample that an a acker needs to manipulate in order to turn it into an adversarial sample. Di erent from the computation method above, in the se ing of p = 0 where unconstrained optimization problems (3) and (4) are not di erentiable, the computation for the optimal solution has to follow an approximation method introduced in [28] or [6].\n(3) With p = ∞, p-norm becomes a measure indicating the maximum change to individual features. As such, the optimal solution for (1) and (2) can be approximated by following the fast gradient sign method [12], which computes perturbation ∂L(f (x; w); )/∂x (or ∂L(f (x; w);ˆ )/∂x), multiplies it by distortion scale ε and then adds the product to legitimate data sample x. Note that the way they compute that perturbation can be through back-propagation in conjunction with gradient descent.\n(3) With p = ∞, p-norm becomes a measure indicating the maximum change to individual features. As such, the optimal solution for (1) and (2) can be approximated by following the fast gradient sign method [12], which computes perturbation ∂L(f (x; w); )/∂x (or ∂L(f (x; w);ˆ )/∂x), multiplies it by distortion scale ε and then adds the product to legitimate data sample x. Note that the way they compute that perturbation can be through back-propagation in conjunction with gradient descent.\nAs is illustrated in the aforementioned optimization problems, in order to generate impactful adversarial samples, an a acker needs to know either standard DNN model f (·) or know of a way to approximate f (·). A recent study [35] has revealed that an a acker could well approximate a standard DNN model using a traditional DNN training algorithm on an auxiliary training dataset. In this paper, we use \"cross-model approach\" to refer to those adversarial sample cra ing methods that rely upon the approximation of a standard DNN model.\nAs is illustrated in the aforementioned optimization problems, in order to generate impactful adversarial samples, an a acker needs to know either standard DNN model f (·) or know of a way to approximate f (·). A recent study [35] has revealed that an a acker could well approximate a standard DNN model using a traditional DNN training algorithm on an auxiliary training dataset. In this paper, we use \"cross-model approach\" to refer to those adversarial sample cra ing methods that rely upon the approximation of a standard DNN model."},{"title":"EXISTING DEFENSES AND PROBLEM SCOPE","content":""},{"title":"EXISTING DEFENSES AND PROBLEM SCOPE","content":"To counteract the adversarial learning problem described in the section above, recent research invent various training algorithms [12,13,27,35] to improve the robustness of a DNN model. ey indicate, by using new training algorithms they design, one can improve a DNN's resistance to the adversarial samples cra ed through the aforementioned cross-model approach. is is due to the fact that their training algorithms smooth a standard DNN's decision boundary, making adversarial samples -impactful to standard DNN models -no longer su ciently e ective. In this section, we summarize these defense mechanisms and discuss their limitations. Following our summary and discussion, we also de ne the problem scope of our research.\nTo counteract the adversarial learning problem described in the section above, recent research invent various training algorithms [12,13,27,35] to improve the robustness of a DNN model. ey indicate, by using new training algorithms they design, one can improve a DNN's resistance to the adversarial samples cra ed through the aforementioned cross-model approach. is is due to the fact that their training algorithms smooth a standard DNN's decision boundary, making adversarial samples -impactful to standard DNN models -no longer su ciently e ective. In this section, we summarize these defense mechanisms and discuss their limitations. Following our summary and discussion, we also de ne the problem scope of our research."},{"title":"Existing Defense Mechanisms","content":""},{"title":"Existing Defense Mechanisms","content":"Recently, research in hardening deep learning mainly focuses on two di erent tactics -data augmentation and model complexity enhancement. Here, we summarize them in turn and disucss their limitations. Data augmentation. To resolve the issue of \"blind spots\" (a more informal name given to adversarial samples), many methods that could be considered as sophisticated forms of data augmentation 2 have been proposed (e.g. [12,13,27]). In principle, these methods expand their training set by combining known samples with potential blind spots, the process of which has been called adversarial training [12]. Technically speaking, adversarial training can be formally described as adding a regularization term known as DataGrad to a DNN's training loss function [27]. e regularization penalizes the directions uncovered by adversarial perturbations (introduced in Section 2.2). erefore, adversarial training can work to improve the robustness of a standard DNN. Model complexity enhancement. DNN models are already complex, with respect to both the nonlinear function that they try to approximate as well as their layered composition of many parameters. However, the underlying architecture is straightforward when it comes to facilitating the ow of information forwards and backwards, greatly alleviating the e ort in generating adversarial samples. erefore, several ideas [13,29] have been proposed to enhance the complexity of DNN models, aiming to improve the tolerance of complex DNN models with respect to adversarial samples generated from simple DNN models. For example, [29] developed a defensive distillation mechanism, which trains a DNN from data samples that are \"distilled\" from another DNN. By using the knowledge transferred from the other DNN, the learned DNN classi ers become less sensitive to adversarial samples. Similarly, [13] proposed to stack an auto-encoder together with a standard DNN. It shows that this auto-encoding enhancement increases a DNN's resistance to adversarial samples. Limitation. While the aforementioned defenses have yielded promising results in terms of increasing model resistance, the scope of the model resistance they provide is relatively limited. Once an a acker obtains the knowledge of the new training algorithms -instead of using a traditional DNN training algorithm to substitute the algorithm which the target DNN is trained with -he can build his own model with the new training algorithm, and then use it as the cross model to facilitate the cra ing of adversarial samples. As we will show in Section 5, the adversarial samples cra ed through such new cross models sustain their o ensiveness to the corresponding DNN models. is indicates that the e ectiveness of existing defense mechanisms is highly dependent upon the obscurity of training algorithms.\nRecently, research in hardening deep learning mainly focuses on two di erent tactics -data augmentation and model complexity enhancement. Here, we summarize them in turn and disucss their limitations. Data augmentation. To resolve the issue of \"blind spots\" (a more informal name given to adversarial samples), many methods that could be considered as sophisticated forms of data augmentation 2 have been proposed (e.g. [12,13,27]). In principle, these methods expand their training set by combining known samples with potential blind spots, the process of which has been called adversarial training [12]. Technically speaking, adversarial training can be formally described as adding a regularization term known as DataGrad to a DNN's training loss function [27]. e regularization penalizes the directions uncovered by adversarial perturbations (introduced in Section 2.2). erefore, adversarial training can work to improve the robustness of a standard DNN. Model complexity enhancement. DNN models are already complex, with respect to both the nonlinear function that they try to approximate as well as their layered composition of many parameters. However, the underlying architecture is straightforward when it comes to facilitating the ow of information forwards and backwards, greatly alleviating the e ort in generating adversarial samples. erefore, several ideas [13,29] have been proposed to enhance the complexity of DNN models, aiming to improve the tolerance of complex DNN models with respect to adversarial samples generated from simple DNN models. For example, [29] developed a defensive distillation mechanism, which trains a DNN from data samples that are \"distilled\" from another DNN. By using the knowledge transferred from the other DNN, the learned DNN classi ers become less sensitive to adversarial samples. Similarly, [13] proposed to stack an auto-encoder together with a standard DNN. It shows that this auto-encoding enhancement increases a DNN's resistance to adversarial samples. Limitation. While the aforementioned defenses have yielded promising results in terms of increasing model resistance, the scope of the model resistance they provide is relatively limited. Once an a acker obtains the knowledge of the new training algorithms -instead of using a traditional DNN training algorithm to substitute the algorithm which the target DNN is trained with -he can build his own model with the new training algorithm, and then use it as the cross model to facilitate the cra ing of adversarial samples. As we will show in Section 5, the adversarial samples cra ed through such new cross models sustain their o ensiveness to the corresponding DNN models. is indicates that the e ectiveness of existing defense mechanisms is highly dependent upon the obscurity of training algorithms."},{"title":"Problem Scope","content":""},{"title":"Problem Scope","content":"With the existing defenses and their limitation in mind, here we de ne the problem scope of our research. Similar to most previous research -if not all -in hardening deep learning, we assume that an a acker cra s adversarial samples by solving the aforementioned optimization problems with derivative calculation (e.g., fast sign gradient descent or Newton-Raphson method). We believe this assumption is realistic for the following reason.\nWith the existing defenses and their limitation in mind, here we de ne the problem scope of our research. Similar to most previous research -if not all -in hardening deep learning, we assume that an a acker cra s adversarial samples by solving the aforementioned optimization problems with derivative calculation (e.g., fast sign gradient descent or Newton-Raphson method). We believe this assumption is realistic for the following reason.\nDerivative calculation is the most general approach for solving an optimization problem. In the future, while one might be able to derive new forms of approaches in solving the aforementioned optimization problem, he or she has to ensure the new approaches are computationally e cient. Without the aid from derivative calculation, this can be relatively di cult. Even if one could computationally e ciently resolve the aforementioned optimization problems -for example perhaps through relaxation -without derivative calculation, he still need to prove the adversarial samples derived from such an approach are impactful. Given that relaxation reshapes an optimization problem, the \"optimal\" solution may not even close to any local optima of that original optimization problem.\nDerivative calculation is the most general approach for solving an optimization problem. In the future, while one might be able to derive new forms of approaches in solving the aforementioned optimization problem, he or she has to ensure the new approaches are computationally e cient. Without the aid from derivative calculation, this can be relatively di cult. Even if one could computationally e ciently resolve the aforementioned optimization problems -for example perhaps through relaxation -without derivative calculation, he still need to prove the adversarial samples derived from such an approach are impactful. Given that relaxation reshapes an optimization problem, the \"optimal\" solution may not even close to any local optima of that original optimization problem.\nDi erent from prior research, we also assume that an adversary has not only the access to a DNN's structure as well as the dataset(s) used to build the DNN but more importantly the algorithm used to train the network. In other words, we assume a target DNN model is no longer obscure to an adversary and rather he has the full knowledge about a DNN model that he a empts to exploit. We believe this assumption is more practical because there is li le hope of keeping an adversary-resistant training algorithm completely secret from dedicated a ackers. In the long run, any system the security of which relies upon the obscurity of its design can be doomed [19].\nDi erent from prior research, we also assume that an adversary has not only the access to a DNN's structure as well as the dataset(s) used to build the DNN but more importantly the algorithm used to train the network. In other words, we assume a target DNN model is no longer obscure to an adversary and rather he has the full knowledge about a DNN model that he a empts to exploit. We believe this assumption is more practical because there is li le hope of keeping an adversary-resistant training algorithm completely secret from dedicated a ackers. In the long run, any system the security of which relies upon the obscurity of its design can be doomed [19]."},{"title":"OUR APPROACH","content":""},{"title":"OUR APPROACH","content":"To address the problem above, we propose a new approach to harden DNN models. Technically speaking, it follows the tactic of model complexity enhancement, which improves model resistance by increasing model complexity. Di erent from the existing techniques mentioned above, our approach however goes beyond the scope of robustness they provide. It ensures that an a acker cannot perform the aforementioned a ack to generate adversarial samples impactful to our learning model even if we reveal our training algorithm. In other words, our approach escalate a DNN's resistance to adversarial samples without the requirement of obscuring training algorithms.\nTo address the problem above, we propose a new approach to harden DNN models. Technically speaking, it follows the tactic of model complexity enhancement, which improves model resistance by increasing model complexity. Di erent from the existing techniques mentioned above, our approach however goes beyond the scope of robustness they provide. It ensures that an a acker cannot perform the aforementioned a ack to generate adversarial samples impactful to our learning model even if we reveal our training algorithm. In other words, our approach escalate a DNN's resistance to adversarial samples without the requirement of obscuring training algorithms.\nAs is discussed in the section above, the adversarial learning problem can be viewed as an optimization problem. To resolve that optimization problem, one needs to conduct analytical computation of gradients with respect to an input data sample and perform backward propagation accordingly. erefore, we escalate a DNN's robustness not only by increasing the model complexity but, more importantly, restricting back-propagation.\nAs is discussed in the section above, the adversarial learning problem can be viewed as an optimization problem. To resolve that optimization problem, one needs to conduct analytical computation of gradients with respect to an input data sample and perform backward propagation accordingly. erefore, we escalate a DNN's robustness not only by increasing the model complexity but, more importantly, restricting back-propagation.\nMore speci cally, we integrate to a DNN a data transformation module, (·) graphically indicated in Figure 3. As is illustrated in the gure, the data transformation module projects X , an input data sample to (X ), a new representation, before passing it through a consecutive DNN. is transformation increases the complexity of a DNN model and augments its resistance to adversarial samples cra ed through the aforementioned cross-model scheme. In addition, it blocks the backward ows of gradients. With this block, even if the underlying training algorithm is disclosed, an adversary cannot cra adversarial samples. In the following, we specify the design principle of our data transformation module, followed by its design detail and some necessary discussions.\nMore speci cally, we integrate to a DNN a data transformation module, (·) graphically indicated in Figure 3. As is illustrated in the gure, the data transformation module projects X , an input data sample to (X ), a new representation, before passing it through a consecutive DNN. is transformation increases the complexity of a DNN model and augments its resistance to adversarial samples cra ed through the aforementioned cross-model scheme. In addition, it blocks the backward ows of gradients. With this block, even if the underlying training algorithm is disclosed, an adversary cannot cra adversarial samples. In the following, we specify the design principle of our data transformation module, followed by its design detail and some necessary discussions."},{"title":"Design Principle","content":""},{"title":"Design Principle","content":"To block the backward ow of gradients, the design of data transformation must satisfy three requirements. Most notably, the data transformation must be non-di erentiable. As is discussed above, cra ing adversarial samples requires the calculation of gradients as well as the back-propagation of those gradients. By making data transformation module (·) non-di erentiable, therefore, we can make gradient calculation intractable and thus obstruct the backward ow of gradients. More formally, we can choose nondi erentiable function (x), making the derivative di cult to be calculated, i.e.,\nTo block the backward ow of gradients, the design of data transformation must satisfy three requirements. Most notably, the data transformation must be non-di erentiable. As is discussed above, cra ing adversarial samples requires the calculation of gradients as well as the back-propagation of those gradients. By making data transformation module (·) non-di erentiable, therefore, we can make gradient calculation intractable and thus obstruct the backward ow of gradients. More formally, we can choose nondi erentiable function (x), making the derivative di cult to be calculated, i.e.,\nHere, f (·) represents the DNN model in tandem with the data transformation module, and L(·) denotes the cost function described in Section 2.2. e derivative can be computed using either a rstorder optimization method (e.g., gradient descent) or a second-order method (e.g., Newton-Raphson method), in which n is equal to 1 and 2, respectively. While the non-di erentiability feature restricts the cra ing of adversarial samples, an adversary might still be able to generate adversarial samples. Since end-to-end gradient ow is blocked at the input layer of the successive DNN, back-propagation can only carry error gradients to the output of the transformation module. Given (·), an adversary could construct an adversarial sample by inverting transformation module (·) and passing the manipulated transformation output through the inversion of the transformation. More formally, the adversary can construct an adversarial sample by computing\nHere, f (·) represents the DNN model in tandem with the data transformation module, and L(·) denotes the cost function described in Section 2.2. e derivative can be computed using either a rstorder optimization method (e.g., gradient descent) or a second-order method (e.g., Newton-Raphson method), in which n is equal to 1 and 2, respectively. While the non-di erentiability feature restricts the cra ing of adversarial samples, an adversary might still be able to generate adversarial samples. Since end-to-end gradient ow is blocked at the input layer of the successive DNN, back-propagation can only carry error gradients to the output of the transformation module. Given (·), an adversary could construct an adversarial sample by inverting transformation module (·) and passing the manipulated transformation output through the inversion of the transformation. More formally, the adversary can construct an adversarial sample by computing\nIn addition to making data transformation non-di erentiable, therefore, we must further ensure that the inversion of the data transformation is computationally intractable. In other words, the data transformation (·) needs to have the property of non-invertibility.\nIn addition to making data transformation non-di erentiable, therefore, we must further ensure that the inversion of the data transformation is computationally intractable. In other words, the data transformation (·) needs to have the property of non-invertibility.\nSatisfying the two requirements above ensures that our proposed approach can prohibit an a acker from cra ing adversarial samples directly from the target DNN model, and one does not need to concern about the disclosure of training algorithms. However, the data transformation proposed may signi cantly jeopardize the accuracy of a DNN model if not designed carefully. Take the following extreme case for example.\nSatisfying the two requirements above ensures that our proposed approach can prohibit an a acker from cra ing adversarial samples directly from the target DNN model, and one does not need to concern about the disclosure of training algorithms. However, the data transformation proposed may signi cantly jeopardize the accuracy of a DNN model if not designed carefully. Take the following extreme case for example.\nHash functions like MD5 and SHA1 are one-way functions which have the properties of non-di erentiability as well as non-invertibility. By simply using it as the transformation module, we can easily prohibit an a acker from cra ing adversarial samples even if he knows of which hash function we choose and how we integrate it with DNNs. However, a hash function signi cantly changes the distribution of input data samples. Armed with it, a DNN model su ers from signi cant loss in classi cation performance. Last but not least, our design therefore must ensure data transformation can preserve the distribution of data representation. is can potentially make a DNN robust without sacri cing classi cation performance.\nHash functions like MD5 and SHA1 are one-way functions which have the properties of non-di erentiability as well as non-invertibility. By simply using it as the transformation module, we can easily prohibit an a acker from cra ing adversarial samples even if he knows of which hash function we choose and how we integrate it with DNNs. However, a hash function signi cantly changes the distribution of input data samples. Armed with it, a DNN model su ers from signi cant loss in classi cation performance. Last but not least, our design therefore must ensure data transformation can preserve the distribution of data representation. is can potentially make a DNN robust without sacri cing classi cation performance."},{"title":"Design Detail","content":""},{"title":"Design Detail","content":"Following the design principle above, we choose Locally Linear Embedding (LLE) [31], a non-parametric dimensionality reduction mechanism, serving as the data transformation module. As we will discuss in the following, this representative non-parametric method is non-di erentiable. More importantly, it can be theoretically proven that inverting LLE is an NP-hard problem. Last but not least, LLE seeks low-dimensional, neighborhood-preserving map of high-dimensional input samples, and thus is a method that best suited to preserving as much information in the input as possible. In the following, we rst describe LLE and then expound upon the fact that, as a non-parametric dimensionality reduction method, LLE is non-di erentiable. Furthermore, we theoretically prove LLE is computationally non-invertible.\nFollowing the design principle above, we choose Locally Linear Embedding (LLE) [31], a non-parametric dimensionality reduction mechanism, serving as the data transformation module. As we will discuss in the following, this representative non-parametric method is non-di erentiable. More importantly, it can be theoretically proven that inverting LLE is an NP-hard problem. Last but not least, LLE seeks low-dimensional, neighborhood-preserving map of high-dimensional input samples, and thus is a method that best suited to preserving as much information in the input as possible. In the following, we rst describe LLE and then expound upon the fact that, as a non-parametric dimensionality reduction method, LLE is non-di erentiable. Furthermore, we theoretically prove LLE is computationally non-invertible."},{"title":"Locally Linear","content":""},{"title":"Locally Linear","content":"Embedding. LLE is a non-parametric method designed to reduce input data dimensionality and at the same time preserve local properties of high-dimensional input in a lowerdimensional space. To some extent, this can ensure the distribution of high-dimensional data samples is as close as they are in a lower-dimensional space. Technically speaking, this is achieved by representing each high-dimensional data sample via a linear combination of its nearest neighbors. More formally, this can be expressed as\nEmbedding. LLE is a non-parametric method designed to reduce input data dimensionality and at the same time preserve local properties of high-dimensional input in a lowerdimensional space. To some extent, this can ensure the distribution of high-dimensional data samples is as close as they are in a lower-dimensional space. Technically speaking, this is achieved by representing each high-dimensional data sample via a linear combination of its nearest neighbors. More formally, this can be expressed as\nHere, x i and x j (x i , x j ∈ R 1×m ) denote the i t h data sample and its j t h neighbor (j = 1, 2..., N ), respectively. w i j represents the weight, indicating the contribution of x j to data sample x i . As is described in [31], those weights (a.k.a. reconstruction weights) can be represented as weight matrix W and computed by solving the following optimization problem:\nHere, x i and x j (x i , x j ∈ R 1×m ) denote the i t h data sample and its j t h neighbor (j = 1, 2..., N ), respectively. w i j represents the weight, indicating the contribution of x j to data sample x i . As is described in [31], those weights (a.k.a. reconstruction weights) can be represented as weight matrix W and computed by solving the following optimization problem:\nIn weight matrix W , LLE deems w i j = 0 if x j is not considered as a neighbor of x i , and the total number of neighbors assigned to x i is a carefully selected hyper-parameter. e neighboring relation between x i and x j depends on the value of the l 2 distance between x i and x j . Since the reconstruction weights encode the local properties of the high-dimensional data, they can be used to preserve the data distribution at the time of performing dimensionality reduction. More speci cally, LLE imposes the corresponding reconstruction weights to each lower-dimensional data sample via a similar linear combination, and then a empts to nd Y = { 1 , 2 , . . . , N }, the lower-dimensional representation of X = {x 1 , x 2 , . . . , x N } by solving the following optimization problem:\nIn weight matrix W , LLE deems w i j = 0 if x j is not considered as a neighbor of x i , and the total number of neighbors assigned to x i is a carefully selected hyper-parameter. e neighboring relation between x i and x j depends on the value of the l 2 distance between x i and x j . Since the reconstruction weights encode the local properties of the high-dimensional data, they can be used to preserve the data distribution at the time of performing dimensionality reduction. More speci cally, LLE imposes the corresponding reconstruction weights to each lower-dimensional data sample via a similar linear combination, and then a empts to nd Y = { 1 , 2 , . . . , N }, the lower-dimensional representation of X = {x 1 , x 2 , . . . , x N } by solving the following optimization problem:\nwhere i , j ∈ R 1×m c , indicating i , j consist of m c of elements.\nwhere i , j ∈ R 1×m c , indicating i , j consist of m c of elements.\nIn order to solve the optimization problem above, the Rayleitz-Ritz theorem [16] is typically used. It computes the eigenvectors corresponding to the smallest nonzero eigenvalues of the inner matrix product (I − W ) T · (I − W ). For a detailed explication, we refer the readers to [16]. Here, I ∈ R N ×N is an identity matrix, and W ∈ R N ×N is the aforementioned reconstruction weight matrix.\nIn order to solve the optimization problem above, the Rayleitz-Ritz theorem [16] is typically used. It computes the eigenvectors corresponding to the smallest nonzero eigenvalues of the inner matrix product (I − W ) T · (I − W ). For a detailed explication, we refer the readers to [16]. Here, I ∈ R N ×N is an identity matrix, and W ∈ R N ×N is the aforementioned reconstruction weight matrix.\nLLE is speci cally designed to retain the similarity between pairs of high dimensional samples when they are mapped to lower dimensions [31]. To illustrate this property, we provide a visualization of the data before and a er being processed by LLE in Figure 4. e visualization result demonstrates that using LLE as a data transformation module satis es the last design principle discussed in Section 4.1 (i.e., preserving the distribution of the original data). More importantly, this property also helps bound the lower dimensional mapping of adversarial samples to a vicinity which is lled by mappings of original test samples that are highly similar to these adversarial samples. As a result, there is a signi cantly lower chance that an adversarial sample acts as an outliers in the lower dimensional space. In other words, LLE makes a DNN more resistant to adversarial samples. In Section 5, we empirically validate this important property.\nLLE is speci cally designed to retain the similarity between pairs of high dimensional samples when they are mapped to lower dimensions [31]. To illustrate this property, we provide a visualization of the data before and a er being processed by LLE in Figure 4. e visualization result demonstrates that using LLE as a data transformation module satis es the last design principle discussed in Section 4.1 (i.e., preserving the distribution of the original data). More importantly, this property also helps bound the lower dimensional mapping of adversarial samples to a vicinity which is lled by mappings of original test samples that are highly similar to these adversarial samples. As a result, there is a signi cantly lower chance that an adversarial sample acts as an outliers in the lower dimensional space. In other words, LLE makes a DNN more resistant to adversarial samples. In Section 5, we empirically validate this important property."},{"title":"Non-di erentiability of LLE.","content":""},{"title":"Non-di erentiability of LLE.","content":"Existing dimensionality reduction methods can be categorized as either parametric or nonparametric [36]. Parametric methods utilize a xed amount of parameters to specify a direct mapping from high-dimensional data samples to their low-dimensional projections (or vice versa). is direct mapping is characterized by parameters, which are typically optimized to provide the best mapping performance. is is similar to the functionality provided by a standard DNN, which maps high-dimensional data samples to the nal decision space through di erentiable function f (·). As such, the derivative of parametric methods typically can be computed in an analytically e cient manner. In other words, parametric methods are generally di erentiable, and this nature becomes a disadvantage for blocking the backward gradient ow.\nExisting dimensionality reduction methods can be categorized as either parametric or nonparametric [36]. Parametric methods utilize a xed amount of parameters to specify a direct mapping from high-dimensional data samples to their low-dimensional projections (or vice versa). is direct mapping is characterized by parameters, which are typically optimized to provide the best mapping performance. is is similar to the functionality provided by a standard DNN, which maps high-dimensional data samples to the nal decision space through di erentiable function f (·). As such, the derivative of parametric methods typically can be computed in an analytically e cient manner. In other words, parametric methods are generally di erentiable, and this nature becomes a disadvantage for blocking the backward gradient ow.\nOn the contrary, non-parametric methods do not su er from the issue above. For any non-parametric method, (·), there is no way to express it in a closed form. erefore, the derivative of (·) can be (a) Before LLE processing.\nOn the contrary, non-parametric methods do not su er from the issue above. For any non-parametric method, (·), there is no way to express it in a closed form. erefore, the derivative of (·) can be (a) Before LLE processing.\n(b) A er LLE processing. Figure 4: Visualization of the MNIST data set [20] in a 2-dimensionality space before and a er proceeded by LLE.\n(b) A er LLE processing. Figure 4: Visualization of the MNIST data set [20] in a 2-dimensionality space before and a er proceeded by LLE.\ncomputed only through a numeric but not an analytical approach. More formally, this means the calculation of ∂ (x)/∂x needs to be completed through the calculation of limit lim h→0\ncomputed only through a numeric but not an analytical approach. More formally, this means the calculation of ∂ (x)/∂x needs to be completed through the calculation of limit lim h→0\nGiven that a deep neural network takes as input each individual sample, which is discrete in the sample space, it is di cult to de ne the continuity of (·) with traditional topology and thus the di erentiability of (·) cannot be guaranteed. is indicates, as a member of non-parametric methods, LLE perfectly satis es the rst design principle discussed in Section 4.1 (i.e., not capable of performing derivative calculation).\nGiven that a deep neural network takes as input each individual sample, which is discrete in the sample space, it is di cult to de ne the continuity of (·) with traditional topology and thus the di erentiability of (·) cannot be guaranteed. is indicates, as a member of non-parametric methods, LLE perfectly satis es the rst design principle discussed in Section 4.1 (i.e., not capable of performing derivative calculation)."},{"title":"Non-invertibility of LLE.","content":""},{"title":"Non-invertibility of LLE.","content":"We validate the non-invertibility of LLE by theoretically proving that reconstructing original highdimensional data from low-dimensional representations transformed by LLE is computationally intractable. More formally, we prove that, given a set of low-dimensional data Y = { 1 , 2 , . . . , N } (Y ∈ R N ×m c ) produced by LLE, reconstructing their original highdimensional representations X = {x 1 , x 2 , . . . , x N } (X ∈ R N ×m ) from Y is at least an NP-hard problem.\nWe validate the non-invertibility of LLE by theoretically proving that reconstructing original highdimensional data from low-dimensional representations transformed by LLE is computationally intractable. More formally, we prove that, given a set of low-dimensional data Y = { 1 , 2 , . . . , N } (Y ∈ R N ×m c ) produced by LLE, reconstructing their original highdimensional representations X = {x 1 , x 2 , . . . , x N } (X ∈ R N ×m ) from Y is at least an NP-hard problem.\nRecall that LLE computes weight matrix W and utilizes it to project high-dimensional data samples to a lower-dimensional space. As a result, to restore high-dimensional data from its lowerdimensional representations, one has to recover that matrix by following the calculation similar to that shown in (7), except that x i and x j are replaced by i and j .\nRecall that LLE computes weight matrix W and utilizes it to project high-dimensional data samples to a lower-dimensional space. As a result, to restore high-dimensional data from its lowerdimensional representations, one has to recover that matrix by following the calculation similar to that shown in (7), except that x i and x j are replaced by i and j .\nOnce weight matrix W is restored, the recovery of original highdimensional data can be viewed as solving the following optimization problem:\nOnce weight matrix W is restored, the recovery of original highdimensional data can be viewed as solving the following optimization problem:\nIt is not di cult to realize that Equation (9) can be de ned in the following quadratic form:\nIt is not di cult to realize that Equation (9) can be de ned in the following quadratic form:\nwhere m i j = δ i j − w i j − w ji + k w ki w k j . Note that δ i j = 1 if i = j and 0 otherwise. If expressing m i j as the following matrix:\nwhere m i j = δ i j − w i j − w ji + k w ki w k j . Note that δ i j = 1 if i = j and 0 otherwise. If expressing m i j as the following matrix:\nwe can easily realize that M is a symmetric matrix, which has the property of M T = M. Now, with the analysis above, the validation of non-invertibility amounts to proving that solving (10) is at least an NP-hard problem. In this work, we conduct this proof by introducing several constraints to this equation. Our basic idea is to use these constraints to relax the optimization problem in (10) to a nearby problem which can be easily proved as an NP-hard problem. More speci cally, we introduce the following constraints:\nwe can easily realize that M is a symmetric matrix, which has the property of M T = M. Now, with the analysis above, the validation of non-invertibility amounts to proving that solving (10) is at least an NP-hard problem. In this work, we conduct this proof by introducing several constraints to this equation. Our basic idea is to use these constraints to relax the optimization problem in (10) to a nearby problem which can be easily proved as an NP-hard problem. More speci cally, we introduce the following constraints:\nand\nand\nwhere ì 0 denotes a zero vector, and x i j represents the j th element in vector x i .\nwhere ì 0 denotes a zero vector, and x i j represents the j th element in vector x i .\nWith these constraints introduced to Equation 10, we can relax the optimization problem to a quadratic problem with a nonpositive semi-de nite constraint, which itself is a class of NP-hard problems [9]. In the following, we provide more details on why the involvement of the aforementioned constraints transforms the optimization problem in (10) to this class of NP-hard problems.\nWith these constraints introduced to Equation 10, we can relax the optimization problem to a quadratic problem with a nonpositive semi-de nite constraint, which itself is a class of NP-hard problems [9]. In the following, we provide more details on why the involvement of the aforementioned constraints transforms the optimization problem in (10) to this class of NP-hard problems.\nLet A ∈ R N A ×1 denote a column vector which is the concatenation of x i for i = 1, . . . , N and N A = N × m. en, we have A = (x 1 , x 2 , · · · , x N ) T . Let q ∈ R 1×N A denote a row vector in which every element is equal to 1. We further de ne matrices P, Q ∈ R N A ×N A as follows:\nLet A ∈ R N A ×1 denote a column vector which is the concatenation of x i for i = 1, . . . , N and N A = N × m. en, we have A = (x 1 , x 2 , · · · , x N ) T . Let q ∈ R 1×N A denote a row vector in which every element is equal to 1. We further de ne matrices P, Q ∈ R N A ×N A as follows:\nwhere P i j = m i j · . ∈ R m×m is a matrix of ones where every element is equal to 1. Given the constraint in (12) that we introduce, it is not di cult to discover i x T i = ì 0. Since the multiplication of a vector and its transpose derives a non-negative value, we have Σ i x i x T i ≥ 0 and the constraint in (12) can be expressed as inequation\nwhere P i j = m i j · . ∈ R m×m is a matrix of ones where every element is equal to 1. Given the constraint in (12) that we introduce, it is not di cult to discover i x T i = ì 0. Since the multiplication of a vector and its transpose derives a non-negative value, we have Σ i x i x T i ≥ 0 and the constraint in (12) can be expressed as inequation\nindicating there exists a positive number, α that always holds the inequity. By rewriting the inequation using the notations newly de ned above, we can therefore transform the constraint in (12) into the form of A T QA + qA + α ≤ 0.\nindicating there exists a positive number, α that always holds the inequity. By rewriting the inequation using the notations newly de ned above, we can therefore transform the constraint in (12) into the form of A T QA + qA + α ≤ 0.\nGiven the constraint in (13), we can easily derive inequation\nGiven the constraint in (13), we can easily derive inequation\nindicating there alway exists a constant, γ that holds the inequity. By rewriting both the constraint itself and this inequation using newly de ned notations, we can derive constraints A ∞ ≤ 1 as well as A T IA − N A + γ ≤ 0. As such, we can transform Equation (10) and the aforementioned constraints in (12) and (13) into following form:\nindicating there alway exists a constant, γ that holds the inequity. By rewriting both the constraint itself and this inequation using newly de ned notations, we can derive constraints A ∞ ≤ 1 as well as A T IA − N A + γ ≤ 0. As such, we can transform Equation (10) and the aforementioned constraints in (12) and (13) into following form:\nHere, Q is negative semi-de nite, and thus Equation (15) is a quadratic problem with a non-positive semi-de nite constraint. According to [9,37], Equation (15) belongs to a class of NP-hard problems, which implies the non-invertability of LLE.\nHere, Q is negative semi-de nite, and thus Equation (15) is a quadratic problem with a non-positive semi-de nite constraint. According to [9,37], Equation (15) belongs to a class of NP-hard problems, which implies the non-invertability of LLE."},{"title":"Discussion","content":""},{"title":"Discussion","content":"Here, we discuss some related issues and possible a acks against our proposed technique. Approximation of LLE. While the aforementioned discussion and theoretical proof have already indicated the e ectiveness of our proposed approach, intuition suggests that an adversary might still come up with an a ack. Speci cally, he might approximate LLE using a parametric mapping and then substitute LLE accordingly. Since parametric mappings do not have the property of non-di erentiability, the adversary can take advantage of the substitute, pass gradients through and eventually cra adversarial samples. However, as we will show in Section 5, even using the state-of-the-art approximation scheme, an adversary cannot cra impactful adversarial samples. Other dimensionality reduction methods. As is described above, we choose LLE, a representative non-parametric dimensionality reduction method, to serve as the data transformation module. is is due to the fact that it provides many properties needed for hardening a DNN, such as non-di erentiability, non-invertibility and the capability of preserving data distribution.\nHere, we discuss some related issues and possible a acks against our proposed technique. Approximation of LLE. While the aforementioned discussion and theoretical proof have already indicated the e ectiveness of our proposed approach, intuition suggests that an adversary might still come up with an a ack. Speci cally, he might approximate LLE using a parametric mapping and then substitute LLE accordingly. Since parametric mappings do not have the property of non-di erentiability, the adversary can take advantage of the substitute, pass gradients through and eventually cra adversarial samples. However, as we will show in Section 5, even using the state-of-the-art approximation scheme, an adversary cannot cra impactful adversarial samples. Other dimensionality reduction methods. As is described above, we choose LLE, a representative non-parametric dimensionality reduction method, to serve as the data transformation module. is is due to the fact that it provides many properties needed for hardening a DNN, such as non-di erentiability, non-invertibility and the capability of preserving data distribution.\nGoing beyond LLE, there are other non-parametric dimensionality reduction methods that o er the same properties, e.g., t-Distributed Stochastic Neighbor Embedding (t-SNE) [23] and Sammon Mapping [33]. However, they cannot be utilized in our problem domain for the following reason.\nGoing beyond LLE, there are other non-parametric dimensionality reduction methods that o er the same properties, e.g., t-Distributed Stochastic Neighbor Embedding (t-SNE) [23] and Sammon Mapping [33]. However, they cannot be utilized in our problem domain for the following reason.\nDeep neural networks exhibit superior performance when dealing with data in a relatively high dimensionality. Other non-parametric methods are typically designed more for tasks like visualization [23] where it is required that the dimensionality of the mappings is two or three. Using them as our data transformation module, they cannot provide high-dimensional data input for the DNN in tandem with the transformation, and may signi cantly jeopardize classication performance.\nDeep neural networks exhibit superior performance when dealing with data in a relatively high dimensionality. Other non-parametric methods are typically designed more for tasks like visualization [23] where it is required that the dimensionality of the mappings is two or three. Using them as our data transformation module, they cannot provide high-dimensional data input for the DNN in tandem with the transformation, and may signi cantly jeopardize classication performance."},{"title":"EVALUATION","content":""},{"title":"EVALUATION","content":"As is described in Section 1, adversarial training [12] and defensive distillation [29] are the most representative techniques that have been proposed to defend against adversarial samples. In this work, we use the proposed approach to train our own adversary resistant DNN (LLE-DNN), and then compare it with those enhanced by these two approaches.\nAs is described in Section 1, adversarial training [12] and defensive distillation [29] are the most representative techniques that have been proposed to defend against adversarial samples. In this work, we use the proposed approach to train our own adversary resistant DNN (LLE-DNN), and then compare it with those enhanced by these two approaches."},{"title":"Dataset","content":""},{"title":"Dataset","content":"We evaluate our adversary-resistant DNN model by performing multiple experiments on several widely used datasets, including a dataset for malware detection [5], the MNIST dataset for image recognition [20] and the IMDB dataset for sentiment analysis [22]. Malware dataset: It is a collection of window audit logs, each of which ties to either a benign or malicious so ware sample. e dimensionality of the feature-space for each sample is reduced to 10,000 based on the feature selection metric in [5]. Each feature indicates the occurrence of either a single lesystem access or a sequence of access events, thus taking on the value of 0 or 1. Figure 2 illustrates a subset of features of a so ware sample. Here, 0 indicates that the sequence of events did not occur while 1 indicates the opposite. For each so ware sample, it has been labeled with either 1 or 0, indicating malicious and benign so ware, respectively. e dataset is split into 26,078 training examples, with 14,399 benign and 11,679 malicious so ware samples, and 6,000 testing samples, with benign and malicious so ware samples evenly divided. MNIST dataset: It is a large database of handwri en digits that is commonly used for training various image processing systems. It is composed of 70,000 greyscale images (of 28×28, or 784, pixels) of handwri en digits, split into a training set of 60,000 samples and a testing set of 10,000 samples. IMDB dataset: It consists of 25,000 movie reviews, with one half labeled as \"positive\" and the other \"negative\", indicating the sentiment of these reviews. We randomly split the dataset with 70% movie reviews for training and the remaining for testing. Following the procedure introduced in [24], we encoded the words in each movie review using a dictionary carrying 5,000 words most frequently used. en, we utilized a word embedding technique [24] to convert each word into a vector with a dimensionality of 600. For each movie review, we linearly combined the vectors indicating the words appearing in that review, and then treat the embedding as the representation of that movie review. Table 2: An illustration of features in malware dataset. Individual features are represented in rows. ey are either a lesystem access or a sequence of lesystem events.\nWe evaluate our adversary-resistant DNN model by performing multiple experiments on several widely used datasets, including a dataset for malware detection [5], the MNIST dataset for image recognition [20] and the IMDB dataset for sentiment analysis [22]. Malware dataset: It is a collection of window audit logs, each of which ties to either a benign or malicious so ware sample. e dimensionality of the feature-space for each sample is reduced to 10,000 based on the feature selection metric in [5]. Each feature indicates the occurrence of either a single lesystem access or a sequence of access events, thus taking on the value of 0 or 1. Figure 2 illustrates a subset of features of a so ware sample. Here, 0 indicates that the sequence of events did not occur while 1 indicates the opposite. For each so ware sample, it has been labeled with either 1 or 0, indicating malicious and benign so ware, respectively. e dataset is split into 26,078 training examples, with 14,399 benign and 11,679 malicious so ware samples, and 6,000 testing samples, with benign and malicious so ware samples evenly divided. MNIST dataset: It is a large database of handwri en digits that is commonly used for training various image processing systems. It is composed of 70,000 greyscale images (of 28×28, or 784, pixels) of handwri en digits, split into a training set of 60,000 samples and a testing set of 10,000 samples. IMDB dataset: It consists of 25,000 movie reviews, with one half labeled as \"positive\" and the other \"negative\", indicating the sentiment of these reviews. We randomly split the dataset with 70% movie reviews for training and the remaining for testing. Following the procedure introduced in [24], we encoded the words in each movie review using a dictionary carrying 5,000 words most frequently used. en, we utilized a word embedding technique [24] to convert each word into a vector with a dimensionality of 600. For each movie review, we linearly combined the vectors indicating the words appearing in that review, and then treat the embedding as the representation of that movie review. Table 2: An illustration of features in malware dataset. Individual features are represented in rows. ey are either a lesystem access or a sequence of lesystem events."},{"title":"Learning Technology","content":""},{"title":"Learning Technology","content":""},{"title":"Experimental Design","content":""},{"title":"Experimental Design","content":"For each application described above, we train 4 DNN models using the traditional deep learning training method, adversarial training [12], defensive distillation [29] and our own approach. We specify the hyperparameters of these DNNs in Table 3. We measure their classi cation accuracy by applying the models to the corresponding testing datasets. By comparing their classi cation performance, we evaluate the in uence that our proposed approach brings to a DNN. More speci cally, we examine if LLE-DNN exhibits similar -if not the same or be er -classi cation accuracy.\nFor each application described above, we train 4 DNN models using the traditional deep learning training method, adversarial training [12], defensive distillation [29] and our own approach. We specify the hyperparameters of these DNNs in Table 3. We measure their classi cation accuracy by applying the models to the corresponding testing datasets. By comparing their classi cation performance, we evaluate the in uence that our proposed approach brings to a DNN. More speci cally, we examine if LLE-DNN exhibits similar -if not the same or be er -classi cation accuracy.\nSince the goal of this work is to improve the robustness of a DNN model, we also evaluate our DNN models' resistance to adversarial samples. In particular, we derive adversarial samples from the aforementioned testing datasets, test them against our DNN model and compare its model resistance with those of DNNs enhanced by the other two techniques [12,29].\nSince the goal of this work is to improve the robustness of a DNN model, we also evaluate our DNN models' resistance to adversarial samples. In particular, we derive adversarial samples from the aforementioned testing datasets, test them against our DNN model and compare its model resistance with those of DNNs enhanced by the other two techniques [12,29].\nAs is discussed in Section 2.2, an a acker cra s adversarial samples through auxiliary models. In Table 1, black-box and white-box indicate the auxiliary models trained through di erent schemes. More speci cally, black-box represents the auxiliary model trained through the standard deep learning training scheme, indicating an a acker does not have su cient knowledge about the underlying training algorithm and he can use only a standard approach to train a cross model and cra adversarial samples. White-box represents the auxiliary model trained exactly through the learning schemes proposed as a defense. is simulates a situation where a defense mechanism is publicly disclosed and an a acker exploits that mechanism to produce a highly similar -if not the same -model to cra adversarial samples. Note that, for both black-box and white-box tactics, we use the same hyperparameters and training dataset to build auxiliary models. More speci cally, our auxiliary model training shares the same hyperparameters and training dataset with the standard DNN shown in Table 3.\nAs is discussed in Section 2.2, an a acker cra s adversarial samples through auxiliary models. In Table 1, black-box and white-box indicate the auxiliary models trained through di erent schemes. More speci cally, black-box represents the auxiliary model trained through the standard deep learning training scheme, indicating an a acker does not have su cient knowledge about the underlying training algorithm and he can use only a standard approach to train a cross model and cra adversarial samples. White-box represents the auxiliary model trained exactly through the learning schemes proposed as a defense. is simulates a situation where a defense mechanism is publicly disclosed and an a acker exploits that mechanism to produce a highly similar -if not the same -model to cra adversarial samples. Note that, for both black-box and white-box tactics, we use the same hyperparameters and training dataset to build auxiliary models. More speci cally, our auxiliary model training shares the same hyperparameters and training dataset with the standard DNN shown in Table 3.\nIn addition to the methods described in Section 2.2, the cra ing of adversarial samples must ensure a slight perturbation introduced to a data sample does not undermine its semantic. In other words, we must make sure that, while misleading a classi er to output the wrong class with high con dence, the perturbation to an image should be nearly indistinguishable to the human eyes, that to a malicious so ware sample should not jeopardize so ware functionality nor break its malevolence, and that to a movie review should not break its semantic meaning. In the following, we describe how we ne-tune adversarial samples to preserve semantic for di erent applications. Malware classi cation. Recall that our malware samples are represented by features, the value of which are binary, indicating the occurrence of an lesystem access or a sequence of access events. When generating adversarial samples, we cannot simply disable Table 3: e hyperparameters of all the investigated DNN models.\nIn addition to the methods described in Section 2.2, the cra ing of adversarial samples must ensure a slight perturbation introduced to a data sample does not undermine its semantic. In other words, we must make sure that, while misleading a classi er to output the wrong class with high con dence, the perturbation to an image should be nearly indistinguishable to the human eyes, that to a malicious so ware sample should not jeopardize so ware functionality nor break its malevolence, and that to a movie review should not break its semantic meaning. In the following, we describe how we ne-tune adversarial samples to preserve semantic for di erent applications. Malware classi cation. Recall that our malware samples are represented by features, the value of which are binary, indicating the occurrence of an lesystem access or a sequence of access events. When generating adversarial samples, we cannot simply disable Table 3: e hyperparameters of all the investigated DNN models.\nlesystem access events in that this might jeopardize the functionality of the so ware sample and even break down its malevolence. With this in mind, a bit of care must be taken.\nlesystem access events in that this might jeopardize the functionality of the so ware sample and even break down its malevolence. With this in mind, a bit of care must be taken.\nIn this work, our experiment follows the approach introduced in Section 2.2. To be speci c, we cra adversarial so ware samples by solving optimization problem (1) with the se ing of zero norm (i.e., l 0 ). is indicates the manipulation to a sample is restricted to ipping binary feature values. In addition, this implies the strongest a ack scenario in that optimization problem (1) carries less constraints making the adversarial samples chosen for our evaluation more impactful. Going beyond adversarial sample craing approach discussed in Section 2.2, we also restrict that the value change of a feature can be only from 0 to 1 but not the opposite.\nIn this work, our experiment follows the approach introduced in Section 2.2. To be speci c, we cra adversarial so ware samples by solving optimization problem (1) with the se ing of zero norm (i.e., l 0 ). is indicates the manipulation to a sample is restricted to ipping binary feature values. In addition, this implies the strongest a ack scenario in that optimization problem (1) carries less constraints making the adversarial samples chosen for our evaluation more impactful. Going beyond adversarial sample craing approach discussed in Section 2.2, we also restrict that the value change of a feature can be only from 0 to 1 but not the opposite.\nis amounts to allowing the addition of new lesystem access events only. is manipulation strategy is reasonable for the reason that malware mutation techniques (e.g., [25]) can morph a malware sample by stitching together instructions from benign programs, making the malware perform additional lesystem accesses but not undermining its maliciousness nor its functionality. Since malware manipulation is done with the intent of fooling a malware classi er driven by a DNN, it should be noticed that we do not morph a benign so ware sample, making it malicious. Image recognition. Image data samples contain less strict semantic than the malware data samples above. To preserve image semantic -making a perturbation nearly indistinguishable -we follow the approaches introduced in [6,12,35]. More speci cally, we selected l 0 , l 2 and l ∞ distance to represent the dissimilarity between an image and its corresponding adversarial sample. Especially, we restrict the l ∞ distance in a relative small range (i.e., ϵ ≤ 0.15) when cra ing adversarial samples. Sentiment Analysis. To generate adversarial samples for movie reviews, we again followed the approach introduced in Section 2.2. To be speci c, we solved optimization problem (1) and con gured p-norm with the se ing to l 2 and l ∞ . is is due to the fact that each review is encoded in a vector in which each element is a decimal, and l 2 and l ∞ distances represent the best measure for the dissimilarity between a movie review and its corresponding adversarial sample. Table 4: e comparison of classi cation accuracy on di erent datasets.\nis amounts to allowing the addition of new lesystem access events only. is manipulation strategy is reasonable for the reason that malware mutation techniques (e.g., [25]) can morph a malware sample by stitching together instructions from benign programs, making the malware perform additional lesystem accesses but not undermining its maliciousness nor its functionality. Since malware manipulation is done with the intent of fooling a malware classi er driven by a DNN, it should be noticed that we do not morph a benign so ware sample, making it malicious. Image recognition. Image data samples contain less strict semantic than the malware data samples above. To preserve image semantic -making a perturbation nearly indistinguishable -we follow the approaches introduced in [6,12,35]. More speci cally, we selected l 0 , l 2 and l ∞ distance to represent the dissimilarity between an image and its corresponding adversarial sample. Especially, we restrict the l ∞ distance in a relative small range (i.e., ϵ ≤ 0.15) when cra ing adversarial samples. Sentiment Analysis. To generate adversarial samples for movie reviews, we again followed the approach introduced in Section 2.2. To be speci c, we solved optimization problem (1) and con gured p-norm with the se ing to l 2 and l ∞ . is is due to the fact that each review is encoded in a vector in which each element is a decimal, and l 2 and l ∞ distances represent the best measure for the dissimilarity between a movie review and its corresponding adversarial sample. Table 4: e comparison of classi cation accuracy on di erent datasets.\nAs is mentioned above, each vocabulary has been encoded in a vector with a dimensionality of 600, and we embedded a movie review by linearly combining corresponding vectors. When generating an adversarial sample by introducing a slight perturbation to the embedding, we cast the perturbation to only one vector. is ensures that we introduce only one word change to that review with the hope that it preserves the semantic meaning of that review as much as possible. However, one word change does not guarantee the invariance of the semantic meaning. For example, it would be obvious alteration to the semantic meaning if the replacement happens to be the negative word in \"… makes it the biggest disappointment I've experienced from cinema in years …\". As such, we manually choose the word that incurs minimal semantic change to that movie review. Figure 5 illustrates a movie review sample and its corresponding adversarial sample generated through this approach.\nAs is mentioned above, each vocabulary has been encoded in a vector with a dimensionality of 600, and we embedded a movie review by linearly combining corresponding vectors. When generating an adversarial sample by introducing a slight perturbation to the embedding, we cast the perturbation to only one vector. is ensures that we introduce only one word change to that review with the hope that it preserves the semantic meaning of that review as much as possible. However, one word change does not guarantee the invariance of the semantic meaning. For example, it would be obvious alteration to the semantic meaning if the replacement happens to be the negative word in \"… makes it the biggest disappointment I've experienced from cinema in years …\". As such, we manually choose the word that incurs minimal semantic change to that movie review. Figure 5 illustrates a movie review sample and its corresponding adversarial sample generated through this approach."},{"title":"Experimental Setup and Results","content":""},{"title":"Experimental Setup and Results","content":"On the datasets described above, we rst measure the accuracy of all the aforementioned defense techniques. We then measure their resistance to the adversarial samples cra ed through the aforementioned tactics.\nOn the datasets described above, we rst measure the accuracy of all the aforementioned defense techniques. We then measure their resistance to the adversarial samples cra ed through the aforementioned tactics."},{"title":"Classification Accuracy.","content":""},{"title":"Classification Accuracy.","content":"To identify the optimal dimensionality to which LLE needs to map original data samples, we implemented several LLE-DNNs with di erent se ings of dimensionality of LLE mappings. Figure 6 shows the impact of dimensionality mapping upon the classi cation accuracy obtained by LLE-DNN. Across all three datasets, it is easy to observe that, the classi cation accuracy rst increases when the dimensionality of the LLE mappings rises and then starts to decrease. In our experiment, we choose the highest classi cation accuracy to represent the performance of our LLE-DNN. Table 4 presents the classi cation accuracy results obtained from all investigated DNNs with respect to the testing datasets. Note that, while prior works (e.g, [7,10]) have already demonstrated they can train a DNN with an error rate less than 1% on the MNIST benchmark, their performance improvement does not result from a DNN but model ensemble or elastic distortions added to training data. To study the in uence of our proposed approach upon a standard DNN, therefore, we did not combine models nor augment with arti cially distorted versions of the original training samples. e classi cation accuracy shown in the table has already represented the best performance that a standard DNN can achieve.\nTo identify the optimal dimensionality to which LLE needs to map original data samples, we implemented several LLE-DNNs with di erent se ings of dimensionality of LLE mappings. Figure 6 shows the impact of dimensionality mapping upon the classi cation accuracy obtained by LLE-DNN. Across all three datasets, it is easy to observe that, the classi cation accuracy rst increases when the dimensionality of the LLE mappings rises and then starts to decrease. In our experiment, we choose the highest classi cation accuracy to represent the performance of our LLE-DNN. Table 4 presents the classi cation accuracy results obtained from all investigated DNNs with respect to the testing datasets. Note that, while prior works (e.g, [7,10]) have already demonstrated they can train a DNN with an error rate less than 1% on the MNIST benchmark, their performance improvement does not result from a DNN but model ensemble or elastic distortions added to training data. To study the in uence of our proposed approach upon a standard DNN, therefore, we did not combine models nor augment with arti cially distorted versions of the original training samples. e classi cation accuracy shown in the table has already represented the best performance that a standard DNN can achieve.\nSimilar to adversarial training and defensive distillation, the LLE-DNN is quite e ective in preserving classi cation accuracy.\nSimilar to adversarial training and defensive distillation, the LLE-DNN is quite e ective in preserving classi cation accuracy.\nis implies our proposed approach well preserves data sample distribution. For the malware classi cation task, it can be observed that LLE-DNN appears to be be er at feature learning, achieving the highest classi cation accuracy among DNNs that we investigated. is is presumably due to the fact that malware data samples are highly sparse carrying a large amount of redundant information, and the data transformation module in LLE-DNN eliminates those redundancy and ameliorates the learning ability of a DNN. Table 1 illustrates the DNNs that we investigated as well as their accuracy in classifying adversarial samples. It can be observed that, black-box adversarial samples can cut down the accuracy of the standard DNN to 6.86%, 6.40% and 7.50% under the a acks of l ∞ , l 2 and l 0 , respectively. In contrast, all of the defense mechanisms investigated demonstrate strong resistance to these black-box adversarial samples.\nis implies our proposed approach well preserves data sample distribution. For the malware classi cation task, it can be observed that LLE-DNN appears to be be er at feature learning, achieving the highest classi cation accuracy among DNNs that we investigated. is is presumably due to the fact that malware data samples are highly sparse carrying a large amount of redundant information, and the data transformation module in LLE-DNN eliminates those redundancy and ameliorates the learning ability of a DNN. Table 1 illustrates the DNNs that we investigated as well as their accuracy in classifying adversarial samples. It can be observed that, black-box adversarial samples can cut down the accuracy of the standard DNN to 6.86%, 6.40% and 7.50% under the a acks of l ∞ , l 2 and l 0 , respectively. In contrast, all of the defense mechanisms investigated demonstrate strong resistance to these black-box adversarial samples."},{"title":"Model Resistance.","content":""},{"title":"Model Resistance.","content":"is indicates, without su cient knowledge on the underlying defense mechanisms, it is di cult for an a acker to cra impactful adversarial samples. In other words, existing defense mechanisms can signi cantly escalate a DNN's resistance to adversarial samples if one can obscure the design of the defenses.\nis indicates, without su cient knowledge on the underlying defense mechanisms, it is di cult for an a acker to cra impactful adversarial samples. In other words, existing defense mechanisms can signi cantly escalate a DNN's resistance to adversarial samples if one can obscure the design of the defenses.\nDespite the improvement in model robustness, we also observe that our LLE-DNN generally exhibits the best resistance to blackbox adversarial samples, whereas the defensive distillation approach typically yields the worst resistance. is is presumably due to the fact that, the dimensionality reduction resided in LLE-DNN transforms adversarial samples into a subspace in which they no longer act as outliers, while defensive distillation smooths only a classi cation decision boundary which does not signi cantly reduce the subspace of adversarial samples.\nDespite the improvement in model robustness, we also observe that our LLE-DNN generally exhibits the best resistance to blackbox adversarial samples, whereas the defensive distillation approach typically yields the worst resistance. is is presumably due to the fact that, the dimensionality reduction resided in LLE-DNN transforms adversarial samples into a subspace in which they no longer act as outliers, while defensive distillation smooths only a classi cation decision boundary which does not signi cantly reduce the subspace of adversarial samples.\nWith regard to the white-box se ing, we discover both adversarial training and defensive distillation su er from white-box adversarial samples. eir resistance to white-box adversarial samples is signi cantly worse than those created under the black-box se ing.\nWith regard to the white-box se ing, we discover both adversarial training and defensive distillation su er from white-box adversarial samples. eir resistance to white-box adversarial samples is signi cantly worse than those created under the black-box se ing.\nis observation is consistent with that reported in [6]. e reason behind this is that, both techniques stash away the adversarial sample subspace, but the disclosure of defense mechanisms uncovers the path of nding that subspace.\nis observation is consistent with that reported in [6]. e reason behind this is that, both techniques stash away the adversarial sample subspace, but the disclosure of defense mechanisms uncovers the path of nding that subspace.\nDi erent from adversarial training and defensive distillation, our LLE-DNN is naturally resistant to white-box adversarial samples. As is discussed in Section 4, our proposed approach stashes away the adversarial sample subspace and at the same time restricts derivative calculation. Even if our defense mechanism is revealed, therefore, it is still computationally di cult to nd adversarial samples.\nDi erent from adversarial training and defensive distillation, our LLE-DNN is naturally resistant to white-box adversarial samples. As is discussed in Section 4, our proposed approach stashes away the adversarial sample subspace and at the same time restricts derivative calculation. Even if our defense mechanism is revealed, therefore, it is still computationally di cult to nd adversarial samples.\nTo perform quantitative comparison with the other two approaches, however, we approximate the data transformation in the LLE-DNN -non-parametric dimensionality reduction component -using a parametric model. To be speci c, we choose a DNN to approximate LLE in that a DNN has a large amount of parameters which is typically viewed as the best approximation for non-parametric learning models [17]. With the support from this approximation, we treated the LLE-DNN as a white box and generated adversarial samples accordingly. We show its model resistance in Table 1. It can be observed that, our LLE-DNN still demonstrates strong resistance to white-box adversarial samples even if we substituted LLE to its best approximation. is implies that, there might a theoretical lower bound between a non-parametric model and its parametric approximation, which can naturally serve as a defense against white-box adversarial samples.\nTo perform quantitative comparison with the other two approaches, however, we approximate the data transformation in the LLE-DNN -non-parametric dimensionality reduction component -using a parametric model. To be speci c, we choose a DNN to approximate LLE in that a DNN has a large amount of parameters which is typically viewed as the best approximation for non-parametric learning models [17]. With the support from this approximation, we treated the LLE-DNN as a white box and generated adversarial samples accordingly. We show its model resistance in Table 1. It can be observed that, our LLE-DNN still demonstrates strong resistance to white-box adversarial samples even if we substituted LLE to its best approximation. is implies that, there might a theoretical lower bound between a non-parametric model and its parametric approximation, which can naturally serve as a defense against white-box adversarial samples."},{"title":"CONCLUSION","content":""},{"title":"CONCLUSION","content":"A Deep Neural Network is vulnerable to adversarial samples. Existing defense mechanisms improve a DNN model's resistance to adversarial samples by using the tactic of security through obscurity. Once the design of the defense is disclosed, therefore, the robustness they provide wane. Motivated by this, this work introduces a new approach to escalate the robustness of a DNN model. In particular, it integrates to a DNN model LLE, a non-parametric dimensionality reduction method. With this approach, we show that one can develop a DNN model resistant to adversarial samples even if he or she reveals its design details (i.e., the underlying training algorithm). By demonstrating the DNNs enhanced by our proposed technique across various applications, we argue the proposed approach introduces nearly no degradation in classi cation performance. In contrast, for some applications, it even exhibits performance improvement. As part of the future work, we will further explore the performance of this approach in a wider variety of applications across di erent deep neural architectures.\nA Deep Neural Network is vulnerable to adversarial samples. Existing defense mechanisms improve a DNN model's resistance to adversarial samples by using the tactic of security through obscurity. Once the design of the defense is disclosed, therefore, the robustness they provide wane. Motivated by this, this work introduces a new approach to escalate the robustness of a DNN model. In particular, it integrates to a DNN model LLE, a non-parametric dimensionality reduction method. With this approach, we show that one can develop a DNN model resistant to adversarial samples even if he or she reveals its design details (i.e., the underlying training algorithm). By demonstrating the DNNs enhanced by our proposed technique across various applications, we argue the proposed approach introduces nearly no degradation in classi cation performance. In contrast, for some applications, it even exhibits performance improvement. As part of the future work, we will further explore the performance of this approach in a wider variety of applications across di erent deep neural architectures."},{"title":"Figure 2 :","content":""},{"title":"Figure 2 :","content":""},{"title":"Figure 3 :","content":""},{"title":"Figure 3 :","content":""},{"title":"Figure 6 :","content":""},{"title":"Figure 6 :","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"\"\n },\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"Beyond highly publicized victories in automatic game-playing as in Go [34], there have been many successful applications of deep neural networks (DNN) in image and speech recognition. Recent explorations and applications include those in medical imaging [1,38] and self-driving cars [11,14]. In the domain of cybersecurity, security companies have demonstrated that deep learning could o er a far be er way to classify all types of malware [8,30,39].\\nBeyond highly publicized victories in automatic game-playing as in Go [34], there have been many successful applications of deep neural networks (DNN) in image and speech recognition. Recent explorations and applications include those in medical imaging [1,38] and self-driving cars [11,14]. In the domain of cybersecurity, security companies have demonstrated that deep learning could o er a far be er way to classify all types of malware [8,30,39].\\nDespite its potential, deep neural networks (DNN), like all other machine learning approaches, are vulnerable to what is known as adversarial samples [2,18].\\nDespite its potential, deep neural networks (DNN), like all other machine learning approaches, are vulnerable to what is known as adversarial samples [2,18].\\nis means that they can be easily deceived by non-obvious and potentially dangerous manipulation [26,35]. To be more speci c, an a acker could use the same training algorithm, back-propagation of errors, and a surrogate dataset to construct an auxiliary model. Since this model could provide the a acker with a capability of exploring a DNN's blind spots, he can, with minimal e ort, cra an adversarial sample -a synthetic example generated by slightly modifying a real example in order to make the deep learning system \\\"believe\\\" the sample subtly perturbed belongs to an incorrect class with high con dence.\\nis means that they can be easily deceived by non-obvious and potentially dangerous manipulation [26,35]. To be more speci c, an a acker could use the same training algorithm, back-propagation of errors, and a surrogate dataset to construct an auxiliary model. Since this model could provide the a acker with a capability of exploring a DNN's blind spots, he can, with minimal e ort, cra an adversarial sample -a synthetic example generated by slightly modifying a real example in order to make the deep learning system \\\"believe\\\" the sample subtly perturbed belongs to an incorrect class with high con dence.\\nAccording to a recent study [12], adversarial samples occur in a subspace relatively broad, which means it is impractical to build a defense that can rule out all adversarial samples. As such, the design principle followed by existing defense mechanisms is not to harden a DNN model naturally resistant to any adversarial samples. Rather, they focus on hiding that subspace, making adversaries di cult in nding impactful adversarial samples. For example, representative defensesadversarial training [12] and defensive distillation [29] both increase the complexity of original DNNs with the goal of making adversarial samples -impactful for original DNNs -no longer e ective.\\nAccording to a recent study [12], adversarial samples occur in a subspace relatively broad, which means it is impractical to build a defense that can rule out all adversarial samples. As such, the design principle followed by existing defense mechanisms is not to harden a DNN model naturally resistant to any adversarial samples. Rather, they focus on hiding that subspace, making adversaries di cult in nding impactful adversarial samples. For example, representative defensesadversarial training [12] and defensive distillation [29] both increase the complexity of original DNNs with the goal of making adversarial samples -impactful for original DNNs -no longer e ective.\\nHowever, in this work, we show that the defenses proposed are far from ideal and even considered a dangerous practice. In particular, we demonstrate existing defense mechanisms all follow the approach of \\\"security through obscurity\\\", in which security is achieved by keeping defenses obscured from adversaries. Frankly speaking, defenses following this approach indeed mitigate the adversarial sample problem. However, when applied to security critical applications such as malware classi cation, they become particularly disconcerting.\\nHowever, in this work, we show that the defenses proposed are far from ideal and even considered a dangerous practice. In particular, we demonstrate existing defense mechanisms all follow the approach of \\\"security through obscurity\\\", in which security is achieved by keeping defenses obscured from adversaries. Frankly speaking, defenses following this approach indeed mitigate the adversarial sample problem. However, when applied to security critical applications such as malware classi cation, they become particularly disconcerting.\\nIn the past, there have been a huge amount of debates on security through obscurity, and a general consensus has been reach. at is, obscurity is a perfectly valid security tactic but it cannot be trusted for security. Once design or implementation is uncovered, users totally lost all the security gained by obscurity. To regain the security through obscurity, one has to come up with a completely new design or implementation. As such, Kerckho s' principle [19] suggests obscurity can be used as a layer of defense, but it should never be used as the only layer of defense.\\nIn the past, there have been a huge amount of debates on security through obscurity, and a general consensus has been reach. at is, obscurity is a perfectly valid security tactic but it cannot be trusted for security. Once design or implementation is uncovered, users totally lost all the security gained by obscurity. To regain the security through obscurity, one has to come up with a completely new design or implementation. As such, Kerckho s' principle [19] suggests obscurity can be used as a layer of defense, but it should never be used as the only layer of defense.\\nInspired by this, we propose a new mechanism to escalate a DNN's resistance to adversarial samples. Di erent from existing defenses, our proposed approach unnecessitates model obscurity. In other words, even though we reveal the model, it will still be more than burdensome for adversaries to cra adversarial samples.\\nInspired by this, we propose a new mechanism to escalate a DNN's resistance to adversarial samples. Di erent from existing defenses, our proposed approach unnecessitates model obscurity. In other words, even though we reveal the model, it will still be more than burdensome for adversaries to cra adversarial samples.\\nMore speci cally, we arm a standard DNN with a data transformation module, which projects original data input into a new representation before they are passed through the consecutive DNN.\\nMore speci cally, we arm a standard DNN with a data transformation module, which projects original data input into a new representation before they are passed through the consecutive DNN.\\nis can be used as a defense for the following two reasons. First, data transformation can potentially stash away the space of adversarial manipulations to a carefully designed hyperspace. is makes a ackers di cult in nding adversarial samples impactful for the armed DNN. Second, as we will theoretically prove in Section 4, a data transformation module carefully designed can exponentially increase computation complexity for an a acker to cra impactful adversarial samples. is means that, even though an a acker compromises obscurity and has the full knowledge about the armed DNN model (i.e., the training algorithm, dataset and hyper-parameters), he still cannot launch the a ack -detrimental to DNNs enhanced by other adversary-resistant techniques -nor jeopardize model resistance.\\nis can be used as a defense for the following two reasons. First, data transformation can potentially stash away the space of adversarial manipulations to a carefully designed hyperspace. is makes a ackers di cult in nding adversarial samples impactful for the armed DNN. Second, as we will theoretically prove in Section 4, a data transformation module carefully designed can exponentially increase computation complexity for an a acker to cra impactful adversarial samples. is means that, even though an a acker compromises obscurity and has the full knowledge about the armed DNN model (i.e., the training algorithm, dataset and hyper-parameters), he still cannot launch the a ack -detrimental to DNNs enhanced by other adversary-resistant techniques -nor jeopardize model resistance.\\ne approach proposed in this work is bene cial for the following reasons. First, it escalates a DNN's resistance to adversarial samples with be er security assurance. Second, our approach ensures that a DNN maintains desirable classi cation performance while requiring only minimal modi cation to existing architectures.\\ne approach proposed in this work is bene cial for the following reasons. First, it escalates a DNN's resistance to adversarial samples with be er security assurance. Second, our approach ensures that a DNN maintains desirable classi cation performance while requiring only minimal modi cation to existing architectures.\\nird, while this work is primarily motivated by the need to safeguard DNN models used in critical security applications, it should be noted that the proposed technique is general and can be adopted to other applications where deep learning is popularly applied, such as image recognition and sentiment analysis. We demonstrate this applicability using publicly-available datasets in Section 5.\\nird, while this work is primarily motivated by the need to safeguard DNN models used in critical security applications, it should be noted that the proposed technique is general and can be adopted to other applications where deep learning is popularly applied, such as image recognition and sentiment analysis. We demonstrate this applicability using publicly-available datasets in Section 5.\\nIn summary, this work makes the following contributions.\\nIn summary, this work makes the following contributions.\\n• We propose a generic approach to facilitate the development of adversary-resistant DNNs without following the tactic of security through obscurity. • Using our approach, we develop an adversary-resistant DNN, and theoretically prove its resistance cannot be jeopardized even if the model is fully disclosed. • We evaluate the classi cation performance and robustness of our adversary-resistant DNN and compare it with that ... ...\\n• We propose a generic approach to facilitate the development of adversary-resistant DNNs without following the tactic of security through obscurity. • Using our approach, we develop an adversary-resistant DNN, and theoretically prove its resistance cannot be jeopardized even if the model is fully disclosed. • We evaluate the classi cation performance and robustness of our adversary-resistant DNN and compare it with that ... ...\"\n },\n {\n \"title\": \"Hidden Layers\",\n \"content\": \"\"\n },\n {\n \"title\": \"Hidden Layers\",\n \"content\": \"Softmax Layer Input Layer f(x) x Figure 1: A feed-forward deep neural network with three hidden layers and a so max output layer.\\nSoftmax Layer Input Layer f(x) x Figure 1: A feed-forward deep neural network with three hidden layers and a so max output layer.\\nof existing defense mechanisms. Our result shows that our DNN exhibits similar -sometimes even be er -classi cation performance but with superior model resistance. e rest of this paper is organized as follows. Section 2 introduces the background of DNNs and adversarial sample problem. Section 3 discusses existing defense mechanisms and de nes the problem scope of our research. Section 4 presents our generic approach. In Section 5, we develop and evaluate DNNs in the context of image recognition, sentiment analysis and malware classi cation. Finally, we conclude this work in Section 6.\\nof existing defense mechanisms. Our result shows that our DNN exhibits similar -sometimes even be er -classi cation performance but with superior model resistance. e rest of this paper is organized as follows. Section 2 introduces the background of DNNs and adversarial sample problem. Section 3 discusses existing defense mechanisms and de nes the problem scope of our research. Section 4 presents our generic approach. In Section 5, we develop and evaluate DNNs in the context of image recognition, sentiment analysis and malware classi cation. Finally, we conclude this work in Section 6.\"\n },\n {\n \"title\": \"BACKGROUND\",\n \"content\": \"\"\n },\n {\n \"title\": \"BACKGROUND\",\n \"content\": \"A typical DNN architecture consists of multiple successive layers of processing elements, or so-called \\\"neurons\\\". Each processing layer can be viewed as learning a di erent, more abstract representation of the original multidimensional input distribution. As a whole, a DNN can be viewed as a highly complex function, f (·) that is capable of nonlinearly mapping an original high-dimensional data point to a lower dimensional output. In this section, we brie y introduce the well-established DNN model, followed by the description of the adversarial learning problem.\\nA typical DNN architecture consists of multiple successive layers of processing elements, or so-called \\\"neurons\\\". Each processing layer can be viewed as learning a di erent, more abstract representation of the original multidimensional input distribution. As a whole, a DNN can be viewed as a highly complex function, f (·) that is capable of nonlinearly mapping an original high-dimensional data point to a lower dimensional output. In this section, we brie y introduce the well-established DNN model, followed by the description of the adversarial learning problem.\"\n },\n {\n \"title\": \"Deep Neural Networks\",\n \"content\": \"\"\n },\n {\n \"title\": \"Deep Neural Networks\",\n \"content\": \"As is graphically depicted in Figure 1, a DNN contains an input layer, multiple hidden layers, and an output layer. e input layer takes in each data sample in the form of a multidimensional vector. Starting from the input, computing the pre-activations of each subsequent layer simply requires, at minimum, a matrix multiplication (where a weight/parameter vector, with length equal to the number of hidden units in the target layer, is assigned to each unit of the layer below) usually followed by summation with a bias vector. is process roughly models the process of a layer of neurons integrating the information received from the layer below (i.e., computing a pre-activation) before applying an elementwise activation function 1 . is integrate-then-re process is repeated subsequently for each layer until the last layer is reached. e last layer, or output, is generally interpreted as the model's predictions for some given input data, and is o en designed to compute a parameterized likelihood distribution using the so max function (also known as multi-class regression or minimizing cross entropy). is bo omup propagation of information is also referred to as feed-forward inference [15].\\nAs is graphically depicted in Figure 1, a DNN contains an input layer, multiple hidden layers, and an output layer. e input layer takes in each data sample in the form of a multidimensional vector. Starting from the input, computing the pre-activations of each subsequent layer simply requires, at minimum, a matrix multiplication (where a weight/parameter vector, with length equal to the number of hidden units in the target layer, is assigned to each unit of the layer below) usually followed by summation with a bias vector. is process roughly models the process of a layer of neurons integrating the information received from the layer below (i.e., computing a pre-activation) before applying an elementwise activation function 1 . is integrate-then-re process is repeated subsequently for each layer until the last layer is reached. e last layer, or output, is generally interpreted as the model's predictions for some given input data, and is o en designed to compute a parameterized likelihood distribution using the so max function (also known as multi-class regression or minimizing cross entropy). is bo omup propagation of information is also referred to as feed-forward inference [15].\\nDuring the learning phase of the model, the DNN's predictions are evaluated by comparing them with known target labels associated with the training samples (also known as the\\\"ground truth\\\"). Speci cally, both predictions and labels are taken as the input to L(·), a selected cost function. e DNN's parameters are then optimized with respect to this cost function using the method of steepest gradient descent, minimizing prediction errors on the training set.\\nDuring the learning phase of the model, the DNN's predictions are evaluated by comparing them with known target labels associated with the training samples (also known as the\\\"ground truth\\\"). Speci cally, both predictions and labels are taken as the input to L(·), a selected cost function. e DNN's parameters are then optimized with respect to this cost function using the method of steepest gradient descent, minimizing prediction errors on the training set.\\nMore formally, given (X , Y ):\\nMore formally, given (X , Y ):\\nR m is a data sample and i ∈ R k is the corresponding data label, where, if categorical, is typically represented through a 1-of-k encoding, the goal of model learning is to minimize the cost function represented by\\nR m is a data sample and i ∈ R k is the corresponding data label, where, if categorical, is typically represented through a 1-of-k encoding, the goal of model learning is to minimize the cost function represented by\\ndiction of training sample x i and w represents the weights and bias associated with the connections between neurons.\\ndiction of training sample x i and w represents the weights and bias associated with the connections between neurons.\"\n },\n {\n \"title\": \"Adversarial Sample Problem\",\n \"content\": \"\"\n },\n {\n \"title\": \"Adversarial Sample Problem\",\n \"content\": \"An adversarial sample is a synthetic data sample cra ed by introducing slight perturbations to a legitimate input sample (see Figure 2). In multi-class classi cation tasks, such adversarial samples can cause a DNN to classify themselves into a random class other than the correct one (sometimes not even a reasonable alternative). Recent research [6] demonstrates that, a ackers can uncover such data samples through various approaches (e.g., [3,12,21,28,32,35]) which can all be described as solving either optimization problem\\nAn adversarial sample is a synthetic data sample cra ed by introducing slight perturbations to a legitimate input sample (see Figure 2). In multi-class classi cation tasks, such adversarial samples can cause a DNN to classify themselves into a random class other than the correct one (sometimes not even a reasonable alternative). Recent research [6] demonstrates that, a ackers can uncover such data samples through various approaches (e.g., [3,12,21,28,32,35]) which can all be described as solving either optimization problem\\nor optimization problem\\nor optimization problem\\nHere, optimization problem (1) indicates that an a acker searches for adversarial samplex, the prediction of which is as far as its true label, whereas optimization problem (2) indicates an a acker searches for adversarial samplex so that its prediction is as close as target labelˆ whereˆ is not equal to , the true label of that adversarial sample.\\nHere, optimization problem (1) indicates that an a acker searches for adversarial samplex, the prediction of which is as far as its true label, whereas optimization problem (2) indicates an a acker searches for adversarial samplex so that its prediction is as close as target labelˆ whereˆ is not equal to , the true label of that adversarial sample.\\nIn both optimization problems above, L(·) represents the aforementioned cost function and f (·) denotes the DNN model trained with the traditional learning method discussed above . · p is pnorm -sometimes also speci ed as l p distance -indicating the dissimilarity between adversarial samplex and its corresponding legit data sample x. With di erent values of p -most popularly selected in adversarial learning research -the optimization problems above can be computed in the following manners. Adversarial samples generated by following [12] Adversarial samples generated by following [6] Adversarial samples generated by following [35] (1) With p = 2, p-norm represents the measure of Euclidean distance.\\nIn both optimization problems above, L(·) represents the aforementioned cost function and f (·) denotes the DNN model trained with the traditional learning method discussed above . · p is pnorm -sometimes also speci ed as l p distance -indicating the dissimilarity between adversarial samplex and its corresponding legit data sample x. With di erent values of p -most popularly selected in adversarial learning research -the optimization problems above can be computed in the following manners. Adversarial samples generated by following [12] Adversarial samples generated by following [6] Adversarial samples generated by following [35] (1) With p = 2, p-norm represents the measure of Euclidean distance.\\ne constraint optimization problems above can be speci ed as unconstrained optimization problemŝ\\ne constraint optimization problems above can be speci ed as unconstrained optimization problemŝ\\nHere, both (3) and (4) can be solved by following either a rstorder optimization method (e.g., stochastic gradient descent [6] and L-BFGS [35]) or a second-order method (e.g., Newton-Raphson method). (2) With p = 0, p-norm indicates the number of elements in a legit data sample that an a acker needs to manipulate in order to turn it into an adversarial sample. Di erent from the computation method above, in the se ing of p = 0 where unconstrained optimization problems (3) and (4) are not di erentiable, the computation for the optimal solution has to follow an approximation method introduced in [28] or [6].\\nHere, both (3) and (4) can be solved by following either a rstorder optimization method (e.g., stochastic gradient descent [6] and L-BFGS [35]) or a second-order method (e.g., Newton-Raphson method). (2) With p = 0, p-norm indicates the number of elements in a legit data sample that an a acker needs to manipulate in order to turn it into an adversarial sample. Di erent from the computation method above, in the se ing of p = 0 where unconstrained optimization problems (3) and (4) are not di erentiable, the computation for the optimal solution has to follow an approximation method introduced in [28] or [6].\\n(3) With p = ∞, p-norm becomes a measure indicating the maximum change to individual features. As such, the optimal solution for (1) and (2) can be approximated by following the fast gradient sign method [12], which computes perturbation ∂L(f (x; w); )/∂x (or ∂L(f (x; w);ˆ )/∂x), multiplies it by distortion scale ε and then adds the product to legitimate data sample x. Note that the way they compute that perturbation can be through back-propagation in conjunction with gradient descent.\\n(3) With p = ∞, p-norm becomes a measure indicating the maximum change to individual features. As such, the optimal solution for (1) and (2) can be approximated by following the fast gradient sign method [12], which computes perturbation ∂L(f (x; w); )/∂x (or ∂L(f (x; w);ˆ )/∂x), multiplies it by distortion scale ε and then adds the product to legitimate data sample x. Note that the way they compute that perturbation can be through back-propagation in conjunction with gradient descent.\\nAs is illustrated in the aforementioned optimization problems, in order to generate impactful adversarial samples, an a acker needs to know either standard DNN model f (·) or know of a way to approximate f (·). A recent study [35] has revealed that an a acker could well approximate a standard DNN model using a traditional DNN training algorithm on an auxiliary training dataset. In this paper, we use \\\"cross-model approach\\\" to refer to those adversarial sample cra ing methods that rely upon the approximation of a standard DNN model.\\nAs is illustrated in the aforementioned optimization problems, in order to generate impactful adversarial samples, an a acker needs to know either standard DNN model f (·) or know of a way to approximate f (·). A recent study [35] has revealed that an a acker could well approximate a standard DNN model using a traditional DNN training algorithm on an auxiliary training dataset. In this paper, we use \\\"cross-model approach\\\" to refer to those adversarial sample cra ing methods that rely upon the approximation of a standard DNN model.\"\n },\n {\n \"title\": \"EXISTING DEFENSES AND PROBLEM SCOPE\",\n \"content\": \"\"\n },\n {\n \"title\": \"EXISTING DEFENSES AND PROBLEM SCOPE\",\n \"content\": \"To counteract the adversarial learning problem described in the section above, recent research invent various training algorithms [12,13,27,35] to improve the robustness of a DNN model. ey indicate, by using new training algorithms they design, one can improve a DNN's resistance to the adversarial samples cra ed through the aforementioned cross-model approach. is is due to the fact that their training algorithms smooth a standard DNN's decision boundary, making adversarial samples -impactful to standard DNN models -no longer su ciently e ective. In this section, we summarize these defense mechanisms and discuss their limitations. Following our summary and discussion, we also de ne the problem scope of our research.\\nTo counteract the adversarial learning problem described in the section above, recent research invent various training algorithms [12,13,27,35] to improve the robustness of a DNN model. ey indicate, by using new training algorithms they design, one can improve a DNN's resistance to the adversarial samples cra ed through the aforementioned cross-model approach. is is due to the fact that their training algorithms smooth a standard DNN's decision boundary, making adversarial samples -impactful to standard DNN models -no longer su ciently e ective. In this section, we summarize these defense mechanisms and discuss their limitations. Following our summary and discussion, we also de ne the problem scope of our research.\"\n },\n {\n \"title\": \"Existing Defense Mechanisms\",\n \"content\": \"\"\n },\n {\n \"title\": \"Existing Defense Mechanisms\",\n \"content\": \"Recently, research in hardening deep learning mainly focuses on two di erent tactics -data augmentation and model complexity enhancement. Here, we summarize them in turn and disucss their limitations. Data augmentation. To resolve the issue of \\\"blind spots\\\" (a more informal name given to adversarial samples), many methods that could be considered as sophisticated forms of data augmentation 2 have been proposed (e.g. [12,13,27]). In principle, these methods expand their training set by combining known samples with potential blind spots, the process of which has been called adversarial training [12]. Technically speaking, adversarial training can be formally described as adding a regularization term known as DataGrad to a DNN's training loss function [27]. e regularization penalizes the directions uncovered by adversarial perturbations (introduced in Section 2.2). erefore, adversarial training can work to improve the robustness of a standard DNN. Model complexity enhancement. DNN models are already complex, with respect to both the nonlinear function that they try to approximate as well as their layered composition of many parameters. However, the underlying architecture is straightforward when it comes to facilitating the ow of information forwards and backwards, greatly alleviating the e ort in generating adversarial samples. erefore, several ideas [13,29] have been proposed to enhance the complexity of DNN models, aiming to improve the tolerance of complex DNN models with respect to adversarial samples generated from simple DNN models. For example, [29] developed a defensive distillation mechanism, which trains a DNN from data samples that are \\\"distilled\\\" from another DNN. By using the knowledge transferred from the other DNN, the learned DNN classi ers become less sensitive to adversarial samples. Similarly, [13] proposed to stack an auto-encoder together with a standard DNN. It shows that this auto-encoding enhancement increases a DNN's resistance to adversarial samples. Limitation. While the aforementioned defenses have yielded promising results in terms of increasing model resistance, the scope of the model resistance they provide is relatively limited. Once an a acker obtains the knowledge of the new training algorithms -instead of using a traditional DNN training algorithm to substitute the algorithm which the target DNN is trained with -he can build his own model with the new training algorithm, and then use it as the cross model to facilitate the cra ing of adversarial samples. As we will show in Section 5, the adversarial samples cra ed through such new cross models sustain their o ensiveness to the corresponding DNN models. is indicates that the e ectiveness of existing defense mechanisms is highly dependent upon the obscurity of training algorithms.\\nRecently, research in hardening deep learning mainly focuses on two di erent tactics -data augmentation and model complexity enhancement. Here, we summarize them in turn and disucss their limitations. Data augmentation. To resolve the issue of \\\"blind spots\\\" (a more informal name given to adversarial samples), many methods that could be considered as sophisticated forms of data augmentation 2 have been proposed (e.g. [12,13,27]). In principle, these methods expand their training set by combining known samples with potential blind spots, the process of which has been called adversarial training [12]. Technically speaking, adversarial training can be formally described as adding a regularization term known as DataGrad to a DNN's training loss function [27]. e regularization penalizes the directions uncovered by adversarial perturbations (introduced in Section 2.2). erefore, adversarial training can work to improve the robustness of a standard DNN. Model complexity enhancement. DNN models are already complex, with respect to both the nonlinear function that they try to approximate as well as their layered composition of many parameters. However, the underlying architecture is straightforward when it comes to facilitating the ow of information forwards and backwards, greatly alleviating the e ort in generating adversarial samples. erefore, several ideas [13,29] have been proposed to enhance the complexity of DNN models, aiming to improve the tolerance of complex DNN models with respect to adversarial samples generated from simple DNN models. For example, [29] developed a defensive distillation mechanism, which trains a DNN from data samples that are \\\"distilled\\\" from another DNN. By using the knowledge transferred from the other DNN, the learned DNN classi ers become less sensitive to adversarial samples. Similarly, [13] proposed to stack an auto-encoder together with a standard DNN. It shows that this auto-encoding enhancement increases a DNN's resistance to adversarial samples. Limitation. While the aforementioned defenses have yielded promising results in terms of increasing model resistance, the scope of the model resistance they provide is relatively limited. Once an a acker obtains the knowledge of the new training algorithms -instead of using a traditional DNN training algorithm to substitute the algorithm which the target DNN is trained with -he can build his own model with the new training algorithm, and then use it as the cross model to facilitate the cra ing of adversarial samples. As we will show in Section 5, the adversarial samples cra ed through such new cross models sustain their o ensiveness to the corresponding DNN models. is indicates that the e ectiveness of existing defense mechanisms is highly dependent upon the obscurity of training algorithms.\"\n },\n {\n \"title\": \"Problem Scope\",\n \"content\": \"\"\n },\n {\n \"title\": \"Problem Scope\",\n \"content\": \"With the existing defenses and their limitation in mind, here we de ne the problem scope of our research. Similar to most previous research -if not all -in hardening deep learning, we assume that an a acker cra s adversarial samples by solving the aforementioned optimization problems with derivative calculation (e.g., fast sign gradient descent or Newton-Raphson method). We believe this assumption is realistic for the following reason.\\nWith the existing defenses and their limitation in mind, here we de ne the problem scope of our research. Similar to most previous research -if not all -in hardening deep learning, we assume that an a acker cra s adversarial samples by solving the aforementioned optimization problems with derivative calculation (e.g., fast sign gradient descent or Newton-Raphson method). We believe this assumption is realistic for the following reason.\\nDerivative calculation is the most general approach for solving an optimization problem. In the future, while one might be able to derive new forms of approaches in solving the aforementioned optimization problem, he or she has to ensure the new approaches are computationally e cient. Without the aid from derivative calculation, this can be relatively di cult. Even if one could computationally e ciently resolve the aforementioned optimization problems -for example perhaps through relaxation -without derivative calculation, he still need to prove the adversarial samples derived from such an approach are impactful. Given that relaxation reshapes an optimization problem, the \\\"optimal\\\" solution may not even close to any local optima of that original optimization problem.\\nDerivative calculation is the most general approach for solving an optimization problem. In the future, while one might be able to derive new forms of approaches in solving the aforementioned optimization problem, he or she has to ensure the new approaches are computationally e cient. Without the aid from derivative calculation, this can be relatively di cult. Even if one could computationally e ciently resolve the aforementioned optimization problems -for example perhaps through relaxation -without derivative calculation, he still need to prove the adversarial samples derived from such an approach are impactful. Given that relaxation reshapes an optimization problem, the \\\"optimal\\\" solution may not even close to any local optima of that original optimization problem.\\nDi erent from prior research, we also assume that an adversary has not only the access to a DNN's structure as well as the dataset(s) used to build the DNN but more importantly the algorithm used to train the network. In other words, we assume a target DNN model is no longer obscure to an adversary and rather he has the full knowledge about a DNN model that he a empts to exploit. We believe this assumption is more practical because there is li le hope of keeping an adversary-resistant training algorithm completely secret from dedicated a ackers. In the long run, any system the security of which relies upon the obscurity of its design can be doomed [19].\\nDi erent from prior research, we also assume that an adversary has not only the access to a DNN's structure as well as the dataset(s) used to build the DNN but more importantly the algorithm used to train the network. In other words, we assume a target DNN model is no longer obscure to an adversary and rather he has the full knowledge about a DNN model that he a empts to exploit. We believe this assumption is more practical because there is li le hope of keeping an adversary-resistant training algorithm completely secret from dedicated a ackers. In the long run, any system the security of which relies upon the obscurity of its design can be doomed [19].\"\n },\n {\n \"title\": \"OUR APPROACH\",\n \"content\": \"\"\n },\n {\n \"title\": \"OUR APPROACH\",\n \"content\": \"To address the problem above, we propose a new approach to harden DNN models. Technically speaking, it follows the tactic of model complexity enhancement, which improves model resistance by increasing model complexity. Di erent from the existing techniques mentioned above, our approach however goes beyond the scope of robustness they provide. It ensures that an a acker cannot perform the aforementioned a ack to generate adversarial samples impactful to our learning model even if we reveal our training algorithm. In other words, our approach escalate a DNN's resistance to adversarial samples without the requirement of obscuring training algorithms.\\nTo address the problem above, we propose a new approach to harden DNN models. Technically speaking, it follows the tactic of model complexity enhancement, which improves model resistance by increasing model complexity. Di erent from the existing techniques mentioned above, our approach however goes beyond the scope of robustness they provide. It ensures that an a acker cannot perform the aforementioned a ack to generate adversarial samples impactful to our learning model even if we reveal our training algorithm. In other words, our approach escalate a DNN's resistance to adversarial samples without the requirement of obscuring training algorithms.\\nAs is discussed in the section above, the adversarial learning problem can be viewed as an optimization problem. To resolve that optimization problem, one needs to conduct analytical computation of gradients with respect to an input data sample and perform backward propagation accordingly. erefore, we escalate a DNN's robustness not only by increasing the model complexity but, more importantly, restricting back-propagation.\\nAs is discussed in the section above, the adversarial learning problem can be viewed as an optimization problem. To resolve that optimization problem, one needs to conduct analytical computation of gradients with respect to an input data sample and perform backward propagation accordingly. erefore, we escalate a DNN's robustness not only by increasing the model complexity but, more importantly, restricting back-propagation.\\nMore speci cally, we integrate to a DNN a data transformation module, (·) graphically indicated in Figure 3. As is illustrated in the gure, the data transformation module projects X , an input data sample to (X ), a new representation, before passing it through a consecutive DNN. is transformation increases the complexity of a DNN model and augments its resistance to adversarial samples cra ed through the aforementioned cross-model scheme. In addition, it blocks the backward ows of gradients. With this block, even if the underlying training algorithm is disclosed, an adversary cannot cra adversarial samples. In the following, we specify the design principle of our data transformation module, followed by its design detail and some necessary discussions.\\nMore speci cally, we integrate to a DNN a data transformation module, (·) graphically indicated in Figure 3. As is illustrated in the gure, the data transformation module projects X , an input data sample to (X ), a new representation, before passing it through a consecutive DNN. is transformation increases the complexity of a DNN model and augments its resistance to adversarial samples cra ed through the aforementioned cross-model scheme. In addition, it blocks the backward ows of gradients. With this block, even if the underlying training algorithm is disclosed, an adversary cannot cra adversarial samples. In the following, we specify the design principle of our data transformation module, followed by its design detail and some necessary discussions.\"\n },\n {\n \"title\": \"Design Principle\",\n \"content\": \"\"\n },\n {\n \"title\": \"Design Principle\",\n \"content\": \"To block the backward ow of gradients, the design of data transformation must satisfy three requirements. Most notably, the data transformation must be non-di erentiable. As is discussed above, cra ing adversarial samples requires the calculation of gradients as well as the back-propagation of those gradients. By making data transformation module (·) non-di erentiable, therefore, we can make gradient calculation intractable and thus obstruct the backward ow of gradients. More formally, we can choose nondi erentiable function (x), making the derivative di cult to be calculated, i.e.,\\nTo block the backward ow of gradients, the design of data transformation must satisfy three requirements. Most notably, the data transformation must be non-di erentiable. As is discussed above, cra ing adversarial samples requires the calculation of gradients as well as the back-propagation of those gradients. By making data transformation module (·) non-di erentiable, therefore, we can make gradient calculation intractable and thus obstruct the backward ow of gradients. More formally, we can choose nondi erentiable function (x), making the derivative di cult to be calculated, i.e.,\\nHere, f (·) represents the DNN model in tandem with the data transformation module, and L(·) denotes the cost function described in Section 2.2. e derivative can be computed using either a rstorder optimization method (e.g., gradient descent) or a second-order method (e.g., Newton-Raphson method), in which n is equal to 1 and 2, respectively. While the non-di erentiability feature restricts the cra ing of adversarial samples, an adversary might still be able to generate adversarial samples. Since end-to-end gradient ow is blocked at the input layer of the successive DNN, back-propagation can only carry error gradients to the output of the transformation module. Given (·), an adversary could construct an adversarial sample by inverting transformation module (·) and passing the manipulated transformation output through the inversion of the transformation. More formally, the adversary can construct an adversarial sample by computing\\nHere, f (·) represents the DNN model in tandem with the data transformation module, and L(·) denotes the cost function described in Section 2.2. e derivative can be computed using either a rstorder optimization method (e.g., gradient descent) or a second-order method (e.g., Newton-Raphson method), in which n is equal to 1 and 2, respectively. While the non-di erentiability feature restricts the cra ing of adversarial samples, an adversary might still be able to generate adversarial samples. Since end-to-end gradient ow is blocked at the input layer of the successive DNN, back-propagation can only carry error gradients to the output of the transformation module. Given (·), an adversary could construct an adversarial sample by inverting transformation module (·) and passing the manipulated transformation output through the inversion of the transformation. More formally, the adversary can construct an adversarial sample by computing\\nIn addition to making data transformation non-di erentiable, therefore, we must further ensure that the inversion of the data transformation is computationally intractable. In other words, the data transformation (·) needs to have the property of non-invertibility.\\nIn addition to making data transformation non-di erentiable, therefore, we must further ensure that the inversion of the data transformation is computationally intractable. In other words, the data transformation (·) needs to have the property of non-invertibility.\\nSatisfying the two requirements above ensures that our proposed approach can prohibit an a acker from cra ing adversarial samples directly from the target DNN model, and one does not need to concern about the disclosure of training algorithms. However, the data transformation proposed may signi cantly jeopardize the accuracy of a DNN model if not designed carefully. Take the following extreme case for example.\\nSatisfying the two requirements above ensures that our proposed approach can prohibit an a acker from cra ing adversarial samples directly from the target DNN model, and one does not need to concern about the disclosure of training algorithms. However, the data transformation proposed may signi cantly jeopardize the accuracy of a DNN model if not designed carefully. Take the following extreme case for example.\\nHash functions like MD5 and SHA1 are one-way functions which have the properties of non-di erentiability as well as non-invertibility. By simply using it as the transformation module, we can easily prohibit an a acker from cra ing adversarial samples even if he knows of which hash function we choose and how we integrate it with DNNs. However, a hash function signi cantly changes the distribution of input data samples. Armed with it, a DNN model su ers from signi cant loss in classi cation performance. Last but not least, our design therefore must ensure data transformation can preserve the distribution of data representation. is can potentially make a DNN robust without sacri cing classi cation performance.\\nHash functions like MD5 and SHA1 are one-way functions which have the properties of non-di erentiability as well as non-invertibility. By simply using it as the transformation module, we can easily prohibit an a acker from cra ing adversarial samples even if he knows of which hash function we choose and how we integrate it with DNNs. However, a hash function signi cantly changes the distribution of input data samples. Armed with it, a DNN model su ers from signi cant loss in classi cation performance. Last but not least, our design therefore must ensure data transformation can preserve the distribution of data representation. is can potentially make a DNN robust without sacri cing classi cation performance.\"\n },\n {\n \"title\": \"Design Detail\",\n \"content\": \"\"\n },\n {\n \"title\": \"Design Detail\",\n \"content\": \"Following the design principle above, we choose Locally Linear Embedding (LLE) [31], a non-parametric dimensionality reduction mechanism, serving as the data transformation module. As we will discuss in the following, this representative non-parametric method is non-di erentiable. More importantly, it can be theoretically proven that inverting LLE is an NP-hard problem. Last but not least, LLE seeks low-dimensional, neighborhood-preserving map of high-dimensional input samples, and thus is a method that best suited to preserving as much information in the input as possible. In the following, we rst describe LLE and then expound upon the fact that, as a non-parametric dimensionality reduction method, LLE is non-di erentiable. Furthermore, we theoretically prove LLE is computationally non-invertible.\\nFollowing the design principle above, we choose Locally Linear Embedding (LLE) [31], a non-parametric dimensionality reduction mechanism, serving as the data transformation module. As we will discuss in the following, this representative non-parametric method is non-di erentiable. More importantly, it can be theoretically proven that inverting LLE is an NP-hard problem. Last but not least, LLE seeks low-dimensional, neighborhood-preserving map of high-dimensional input samples, and thus is a method that best suited to preserving as much information in the input as possible. In the following, we rst describe LLE and then expound upon the fact that, as a non-parametric dimensionality reduction method, LLE is non-di erentiable. Furthermore, we theoretically prove LLE is computationally non-invertible.\"\n },\n {\n \"title\": \"Locally Linear\",\n \"content\": \"\"\n },\n {\n \"title\": \"Locally Linear\",\n \"content\": \"Embedding. LLE is a non-parametric method designed to reduce input data dimensionality and at the same time preserve local properties of high-dimensional input in a lowerdimensional space. To some extent, this can ensure the distribution of high-dimensional data samples is as close as they are in a lower-dimensional space. Technically speaking, this is achieved by representing each high-dimensional data sample via a linear combination of its nearest neighbors. More formally, this can be expressed as\\nEmbedding. LLE is a non-parametric method designed to reduce input data dimensionality and at the same time preserve local properties of high-dimensional input in a lowerdimensional space. To some extent, this can ensure the distribution of high-dimensional data samples is as close as they are in a lower-dimensional space. Technically speaking, this is achieved by representing each high-dimensional data sample via a linear combination of its nearest neighbors. More formally, this can be expressed as\\nHere, x i and x j (x i , x j ∈ R 1×m ) denote the i t h data sample and its j t h neighbor (j = 1, 2..., N ), respectively. w i j represents the weight, indicating the contribution of x j to data sample x i . As is described in [31], those weights (a.k.a. reconstruction weights) can be represented as weight matrix W and computed by solving the following optimization problem:\\nHere, x i and x j (x i , x j ∈ R 1×m ) denote the i t h data sample and its j t h neighbor (j = 1, 2..., N ), respectively. w i j represents the weight, indicating the contribution of x j to data sample x i . As is described in [31], those weights (a.k.a. reconstruction weights) can be represented as weight matrix W and computed by solving the following optimization problem:\\nIn weight matrix W , LLE deems w i j = 0 if x j is not considered as a neighbor of x i , and the total number of neighbors assigned to x i is a carefully selected hyper-parameter. e neighboring relation between x i and x j depends on the value of the l 2 distance between x i and x j . Since the reconstruction weights encode the local properties of the high-dimensional data, they can be used to preserve the data distribution at the time of performing dimensionality reduction. More speci cally, LLE imposes the corresponding reconstruction weights to each lower-dimensional data sample via a similar linear combination, and then a empts to nd Y = { 1 , 2 , . . . , N }, the lower-dimensional representation of X = {x 1 , x 2 , . . . , x N } by solving the following optimization problem:\\nIn weight matrix W , LLE deems w i j = 0 if x j is not considered as a neighbor of x i , and the total number of neighbors assigned to x i is a carefully selected hyper-parameter. e neighboring relation between x i and x j depends on the value of the l 2 distance between x i and x j . Since the reconstruction weights encode the local properties of the high-dimensional data, they can be used to preserve the data distribution at the time of performing dimensionality reduction. More speci cally, LLE imposes the corresponding reconstruction weights to each lower-dimensional data sample via a similar linear combination, and then a empts to nd Y = { 1 , 2 , . . . , N }, the lower-dimensional representation of X = {x 1 , x 2 , . . . , x N } by solving the following optimization problem:\\nwhere i , j ∈ R 1×m c , indicating i , j consist of m c of elements.\\nwhere i , j ∈ R 1×m c , indicating i , j consist of m c of elements.\\nIn order to solve the optimization problem above, the Rayleitz-Ritz theorem [16] is typically used. It computes the eigenvectors corresponding to the smallest nonzero eigenvalues of the inner matrix product (I − W ) T · (I − W ). For a detailed explication, we refer the readers to [16]. Here, I ∈ R N ×N is an identity matrix, and W ∈ R N ×N is the aforementioned reconstruction weight matrix.\\nIn order to solve the optimization problem above, the Rayleitz-Ritz theorem [16] is typically used. It computes the eigenvectors corresponding to the smallest nonzero eigenvalues of the inner matrix product (I − W ) T · (I − W ). For a detailed explication, we refer the readers to [16]. Here, I ∈ R N ×N is an identity matrix, and W ∈ R N ×N is the aforementioned reconstruction weight matrix.\\nLLE is speci cally designed to retain the similarity between pairs of high dimensional samples when they are mapped to lower dimensions [31]. To illustrate this property, we provide a visualization of the data before and a er being processed by LLE in Figure 4. e visualization result demonstrates that using LLE as a data transformation module satis es the last design principle discussed in Section 4.1 (i.e., preserving the distribution of the original data). More importantly, this property also helps bound the lower dimensional mapping of adversarial samples to a vicinity which is lled by mappings of original test samples that are highly similar to these adversarial samples. As a result, there is a signi cantly lower chance that an adversarial sample acts as an outliers in the lower dimensional space. In other words, LLE makes a DNN more resistant to adversarial samples. In Section 5, we empirically validate this important property.\\nLLE is speci cally designed to retain the similarity between pairs of high dimensional samples when they are mapped to lower dimensions [31]. To illustrate this property, we provide a visualization of the data before and a er being processed by LLE in Figure 4. e visualization result demonstrates that using LLE as a data transformation module satis es the last design principle discussed in Section 4.1 (i.e., preserving the distribution of the original data). More importantly, this property also helps bound the lower dimensional mapping of adversarial samples to a vicinity which is lled by mappings of original test samples that are highly similar to these adversarial samples. As a result, there is a signi cantly lower chance that an adversarial sample acts as an outliers in the lower dimensional space. In other words, LLE makes a DNN more resistant to adversarial samples. In Section 5, we empirically validate this important property.\"\n },\n {\n \"title\": \"Non-di erentiability of LLE.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Non-di erentiability of LLE.\",\n \"content\": \"Existing dimensionality reduction methods can be categorized as either parametric or nonparametric [36]. Parametric methods utilize a xed amount of parameters to specify a direct mapping from high-dimensional data samples to their low-dimensional projections (or vice versa). is direct mapping is characterized by parameters, which are typically optimized to provide the best mapping performance. is is similar to the functionality provided by a standard DNN, which maps high-dimensional data samples to the nal decision space through di erentiable function f (·). As such, the derivative of parametric methods typically can be computed in an analytically e cient manner. In other words, parametric methods are generally di erentiable, and this nature becomes a disadvantage for blocking the backward gradient ow.\\nExisting dimensionality reduction methods can be categorized as either parametric or nonparametric [36]. Parametric methods utilize a xed amount of parameters to specify a direct mapping from high-dimensional data samples to their low-dimensional projections (or vice versa). is direct mapping is characterized by parameters, which are typically optimized to provide the best mapping performance. is is similar to the functionality provided by a standard DNN, which maps high-dimensional data samples to the nal decision space through di erentiable function f (·). As such, the derivative of parametric methods typically can be computed in an analytically e cient manner. In other words, parametric methods are generally di erentiable, and this nature becomes a disadvantage for blocking the backward gradient ow.\\nOn the contrary, non-parametric methods do not su er from the issue above. For any non-parametric method, (·), there is no way to express it in a closed form. erefore, the derivative of (·) can be (a) Before LLE processing.\\nOn the contrary, non-parametric methods do not su er from the issue above. For any non-parametric method, (·), there is no way to express it in a closed form. erefore, the derivative of (·) can be (a) Before LLE processing.\\n(b) A er LLE processing. Figure 4: Visualization of the MNIST data set [20] in a 2-dimensionality space before and a er proceeded by LLE.\\n(b) A er LLE processing. Figure 4: Visualization of the MNIST data set [20] in a 2-dimensionality space before and a er proceeded by LLE.\\ncomputed only through a numeric but not an analytical approach. More formally, this means the calculation of ∂ (x)/∂x needs to be completed through the calculation of limit lim h→0\\ncomputed only through a numeric but not an analytical approach. More formally, this means the calculation of ∂ (x)/∂x needs to be completed through the calculation of limit lim h→0\\nGiven that a deep neural network takes as input each individual sample, which is discrete in the sample space, it is di cult to de ne the continuity of (·) with traditional topology and thus the di erentiability of (·) cannot be guaranteed. is indicates, as a member of non-parametric methods, LLE perfectly satis es the rst design principle discussed in Section 4.1 (i.e., not capable of performing derivative calculation).\\nGiven that a deep neural network takes as input each individual sample, which is discrete in the sample space, it is di cult to de ne the continuity of (·) with traditional topology and thus the di erentiability of (·) cannot be guaranteed. is indicates, as a member of non-parametric methods, LLE perfectly satis es the rst design principle discussed in Section 4.1 (i.e., not capable of performing derivative calculation).\"\n },\n {\n \"title\": \"Non-invertibility of LLE.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Non-invertibility of LLE.\",\n \"content\": \"We validate the non-invertibility of LLE by theoretically proving that reconstructing original highdimensional data from low-dimensional representations transformed by LLE is computationally intractable. More formally, we prove that, given a set of low-dimensional data Y = { 1 , 2 , . . . , N } (Y ∈ R N ×m c ) produced by LLE, reconstructing their original highdimensional representations X = {x 1 , x 2 , . . . , x N } (X ∈ R N ×m ) from Y is at least an NP-hard problem.\\nWe validate the non-invertibility of LLE by theoretically proving that reconstructing original highdimensional data from low-dimensional representations transformed by LLE is computationally intractable. More formally, we prove that, given a set of low-dimensional data Y = { 1 , 2 , . . . , N } (Y ∈ R N ×m c ) produced by LLE, reconstructing their original highdimensional representations X = {x 1 , x 2 , . . . , x N } (X ∈ R N ×m ) from Y is at least an NP-hard problem.\\nRecall that LLE computes weight matrix W and utilizes it to project high-dimensional data samples to a lower-dimensional space. As a result, to restore high-dimensional data from its lowerdimensional representations, one has to recover that matrix by following the calculation similar to that shown in (7), except that x i and x j are replaced by i and j .\\nRecall that LLE computes weight matrix W and utilizes it to project high-dimensional data samples to a lower-dimensional space. As a result, to restore high-dimensional data from its lowerdimensional representations, one has to recover that matrix by following the calculation similar to that shown in (7), except that x i and x j are replaced by i and j .\\nOnce weight matrix W is restored, the recovery of original highdimensional data can be viewed as solving the following optimization problem:\\nOnce weight matrix W is restored, the recovery of original highdimensional data can be viewed as solving the following optimization problem:\\nIt is not di cult to realize that Equation (9) can be de ned in the following quadratic form:\\nIt is not di cult to realize that Equation (9) can be de ned in the following quadratic form:\\nwhere m i j = δ i j − w i j − w ji + k w ki w k j . Note that δ i j = 1 if i = j and 0 otherwise. If expressing m i j as the following matrix:\\nwhere m i j = δ i j − w i j − w ji + k w ki w k j . Note that δ i j = 1 if i = j and 0 otherwise. If expressing m i j as the following matrix:\\nwe can easily realize that M is a symmetric matrix, which has the property of M T = M. Now, with the analysis above, the validation of non-invertibility amounts to proving that solving (10) is at least an NP-hard problem. In this work, we conduct this proof by introducing several constraints to this equation. Our basic idea is to use these constraints to relax the optimization problem in (10) to a nearby problem which can be easily proved as an NP-hard problem. More speci cally, we introduce the following constraints:\\nwe can easily realize that M is a symmetric matrix, which has the property of M T = M. Now, with the analysis above, the validation of non-invertibility amounts to proving that solving (10) is at least an NP-hard problem. In this work, we conduct this proof by introducing several constraints to this equation. Our basic idea is to use these constraints to relax the optimization problem in (10) to a nearby problem which can be easily proved as an NP-hard problem. More speci cally, we introduce the following constraints:\\nand\\nand\\nwhere ì 0 denotes a zero vector, and x i j represents the j th element in vector x i .\\nwhere ì 0 denotes a zero vector, and x i j represents the j th element in vector x i .\\nWith these constraints introduced to Equation 10, we can relax the optimization problem to a quadratic problem with a nonpositive semi-de nite constraint, which itself is a class of NP-hard problems [9]. In the following, we provide more details on why the involvement of the aforementioned constraints transforms the optimization problem in (10) to this class of NP-hard problems.\\nWith these constraints introduced to Equation 10, we can relax the optimization problem to a quadratic problem with a nonpositive semi-de nite constraint, which itself is a class of NP-hard problems [9]. In the following, we provide more details on why the involvement of the aforementioned constraints transforms the optimization problem in (10) to this class of NP-hard problems.\\nLet A ∈ R N A ×1 denote a column vector which is the concatenation of x i for i = 1, . . . , N and N A = N × m. en, we have A = (x 1 , x 2 , · · · , x N ) T . Let q ∈ R 1×N A denote a row vector in which every element is equal to 1. We further de ne matrices P, Q ∈ R N A ×N A as follows:\\nLet A ∈ R N A ×1 denote a column vector which is the concatenation of x i for i = 1, . . . , N and N A = N × m. en, we have A = (x 1 , x 2 , · · · , x N ) T . Let q ∈ R 1×N A denote a row vector in which every element is equal to 1. We further de ne matrices P, Q ∈ R N A ×N A as follows:\\nwhere P i j = m i j · . ∈ R m×m is a matrix of ones where every element is equal to 1. Given the constraint in (12) that we introduce, it is not di cult to discover i x T i = ì 0. Since the multiplication of a vector and its transpose derives a non-negative value, we have Σ i x i x T i ≥ 0 and the constraint in (12) can be expressed as inequation\\nwhere P i j = m i j · . ∈ R m×m is a matrix of ones where every element is equal to 1. Given the constraint in (12) that we introduce, it is not di cult to discover i x T i = ì 0. Since the multiplication of a vector and its transpose derives a non-negative value, we have Σ i x i x T i ≥ 0 and the constraint in (12) can be expressed as inequation\\nindicating there exists a positive number, α that always holds the inequity. By rewriting the inequation using the notations newly de ned above, we can therefore transform the constraint in (12) into the form of A T QA + qA + α ≤ 0.\\nindicating there exists a positive number, α that always holds the inequity. By rewriting the inequation using the notations newly de ned above, we can therefore transform the constraint in (12) into the form of A T QA + qA + α ≤ 0.\\nGiven the constraint in (13), we can easily derive inequation\\nGiven the constraint in (13), we can easily derive inequation\\nindicating there alway exists a constant, γ that holds the inequity. By rewriting both the constraint itself and this inequation using newly de ned notations, we can derive constraints A ∞ ≤ 1 as well as A T IA − N A + γ ≤ 0. As such, we can transform Equation (10) and the aforementioned constraints in (12) and (13) into following form:\\nindicating there alway exists a constant, γ that holds the inequity. By rewriting both the constraint itself and this inequation using newly de ned notations, we can derive constraints A ∞ ≤ 1 as well as A T IA − N A + γ ≤ 0. As such, we can transform Equation (10) and the aforementioned constraints in (12) and (13) into following form:\\nHere, Q is negative semi-de nite, and thus Equation (15) is a quadratic problem with a non-positive semi-de nite constraint. According to [9,37], Equation (15) belongs to a class of NP-hard problems, which implies the non-invertability of LLE.\\nHere, Q is negative semi-de nite, and thus Equation (15) is a quadratic problem with a non-positive semi-de nite constraint. According to [9,37], Equation (15) belongs to a class of NP-hard problems, which implies the non-invertability of LLE.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"Here, we discuss some related issues and possible a acks against our proposed technique. Approximation of LLE. While the aforementioned discussion and theoretical proof have already indicated the e ectiveness of our proposed approach, intuition suggests that an adversary might still come up with an a ack. Speci cally, he might approximate LLE using a parametric mapping and then substitute LLE accordingly. Since parametric mappings do not have the property of non-di erentiability, the adversary can take advantage of the substitute, pass gradients through and eventually cra adversarial samples. However, as we will show in Section 5, even using the state-of-the-art approximation scheme, an adversary cannot cra impactful adversarial samples. Other dimensionality reduction methods. As is described above, we choose LLE, a representative non-parametric dimensionality reduction method, to serve as the data transformation module. is is due to the fact that it provides many properties needed for hardening a DNN, such as non-di erentiability, non-invertibility and the capability of preserving data distribution.\\nHere, we discuss some related issues and possible a acks against our proposed technique. Approximation of LLE. While the aforementioned discussion and theoretical proof have already indicated the e ectiveness of our proposed approach, intuition suggests that an adversary might still come up with an a ack. Speci cally, he might approximate LLE using a parametric mapping and then substitute LLE accordingly. Since parametric mappings do not have the property of non-di erentiability, the adversary can take advantage of the substitute, pass gradients through and eventually cra adversarial samples. However, as we will show in Section 5, even using the state-of-the-art approximation scheme, an adversary cannot cra impactful adversarial samples. Other dimensionality reduction methods. As is described above, we choose LLE, a representative non-parametric dimensionality reduction method, to serve as the data transformation module. is is due to the fact that it provides many properties needed for hardening a DNN, such as non-di erentiability, non-invertibility and the capability of preserving data distribution.\\nGoing beyond LLE, there are other non-parametric dimensionality reduction methods that o er the same properties, e.g., t-Distributed Stochastic Neighbor Embedding (t-SNE) [23] and Sammon Mapping [33]. However, they cannot be utilized in our problem domain for the following reason.\\nGoing beyond LLE, there are other non-parametric dimensionality reduction methods that o er the same properties, e.g., t-Distributed Stochastic Neighbor Embedding (t-SNE) [23] and Sammon Mapping [33]. However, they cannot be utilized in our problem domain for the following reason.\\nDeep neural networks exhibit superior performance when dealing with data in a relatively high dimensionality. Other non-parametric methods are typically designed more for tasks like visualization [23] where it is required that the dimensionality of the mappings is two or three. Using them as our data transformation module, they cannot provide high-dimensional data input for the DNN in tandem with the transformation, and may signi cantly jeopardize classication performance.\\nDeep neural networks exhibit superior performance when dealing with data in a relatively high dimensionality. Other non-parametric methods are typically designed more for tasks like visualization [23] where it is required that the dimensionality of the mappings is two or three. Using them as our data transformation module, they cannot provide high-dimensional data input for the DNN in tandem with the transformation, and may signi cantly jeopardize classication performance.\"\n },\n {\n \"title\": \"EVALUATION\",\n \"content\": \"\"\n },\n {\n \"title\": \"EVALUATION\",\n \"content\": \"As is described in Section 1, adversarial training [12] and defensive distillation [29] are the most representative techniques that have been proposed to defend against adversarial samples. In this work, we use the proposed approach to train our own adversary resistant DNN (LLE-DNN), and then compare it with those enhanced by these two approaches.\\nAs is described in Section 1, adversarial training [12] and defensive distillation [29] are the most representative techniques that have been proposed to defend against adversarial samples. In this work, we use the proposed approach to train our own adversary resistant DNN (LLE-DNN), and then compare it with those enhanced by these two approaches.\"\n },\n {\n \"title\": \"Dataset\",\n \"content\": \"\"\n },\n {\n \"title\": \"Dataset\",\n \"content\": \"We evaluate our adversary-resistant DNN model by performing multiple experiments on several widely used datasets, including a dataset for malware detection [5], the MNIST dataset for image recognition [20] and the IMDB dataset for sentiment analysis [22]. Malware dataset: It is a collection of window audit logs, each of which ties to either a benign or malicious so ware sample. e dimensionality of the feature-space for each sample is reduced to 10,000 based on the feature selection metric in [5]. Each feature indicates the occurrence of either a single lesystem access or a sequence of access events, thus taking on the value of 0 or 1. Figure 2 illustrates a subset of features of a so ware sample. Here, 0 indicates that the sequence of events did not occur while 1 indicates the opposite. For each so ware sample, it has been labeled with either 1 or 0, indicating malicious and benign so ware, respectively. e dataset is split into 26,078 training examples, with 14,399 benign and 11,679 malicious so ware samples, and 6,000 testing samples, with benign and malicious so ware samples evenly divided. MNIST dataset: It is a large database of handwri en digits that is commonly used for training various image processing systems. It is composed of 70,000 greyscale images (of 28×28, or 784, pixels) of handwri en digits, split into a training set of 60,000 samples and a testing set of 10,000 samples. IMDB dataset: It consists of 25,000 movie reviews, with one half labeled as \\\"positive\\\" and the other \\\"negative\\\", indicating the sentiment of these reviews. We randomly split the dataset with 70% movie reviews for training and the remaining for testing. Following the procedure introduced in [24], we encoded the words in each movie review using a dictionary carrying 5,000 words most frequently used. en, we utilized a word embedding technique [24] to convert each word into a vector with a dimensionality of 600. For each movie review, we linearly combined the vectors indicating the words appearing in that review, and then treat the embedding as the representation of that movie review. Table 2: An illustration of features in malware dataset. Individual features are represented in rows. ey are either a lesystem access or a sequence of lesystem events.\\nWe evaluate our adversary-resistant DNN model by performing multiple experiments on several widely used datasets, including a dataset for malware detection [5], the MNIST dataset for image recognition [20] and the IMDB dataset for sentiment analysis [22]. Malware dataset: It is a collection of window audit logs, each of which ties to either a benign or malicious so ware sample. e dimensionality of the feature-space for each sample is reduced to 10,000 based on the feature selection metric in [5]. Each feature indicates the occurrence of either a single lesystem access or a sequence of access events, thus taking on the value of 0 or 1. Figure 2 illustrates a subset of features of a so ware sample. Here, 0 indicates that the sequence of events did not occur while 1 indicates the opposite. For each so ware sample, it has been labeled with either 1 or 0, indicating malicious and benign so ware, respectively. e dataset is split into 26,078 training examples, with 14,399 benign and 11,679 malicious so ware samples, and 6,000 testing samples, with benign and malicious so ware samples evenly divided. MNIST dataset: It is a large database of handwri en digits that is commonly used for training various image processing systems. It is composed of 70,000 greyscale images (of 28×28, or 784, pixels) of handwri en digits, split into a training set of 60,000 samples and a testing set of 10,000 samples. IMDB dataset: It consists of 25,000 movie reviews, with one half labeled as \\\"positive\\\" and the other \\\"negative\\\", indicating the sentiment of these reviews. We randomly split the dataset with 70% movie reviews for training and the remaining for testing. Following the procedure introduced in [24], we encoded the words in each movie review using a dictionary carrying 5,000 words most frequently used. en, we utilized a word embedding technique [24] to convert each word into a vector with a dimensionality of 600. For each movie review, we linearly combined the vectors indicating the words appearing in that review, and then treat the embedding as the representation of that movie review. Table 2: An illustration of features in malware dataset. Individual features are represented in rows. ey are either a lesystem access or a sequence of lesystem events.\"\n },\n {\n \"title\": \"Learning Technology\",\n \"content\": \"\"\n },\n {\n \"title\": \"Learning Technology\",\n \"content\": \"\"\n },\n {\n \"title\": \"Experimental Design\",\n \"content\": \"\"\n },\n {\n \"title\": \"Experimental Design\",\n \"content\": \"For each application described above, we train 4 DNN models using the traditional deep learning training method, adversarial training [12], defensive distillation [29] and our own approach. We specify the hyperparameters of these DNNs in Table 3. We measure their classi cation accuracy by applying the models to the corresponding testing datasets. By comparing their classi cation performance, we evaluate the in uence that our proposed approach brings to a DNN. More speci cally, we examine if LLE-DNN exhibits similar -if not the same or be er -classi cation accuracy.\\nFor each application described above, we train 4 DNN models using the traditional deep learning training method, adversarial training [12], defensive distillation [29] and our own approach. We specify the hyperparameters of these DNNs in Table 3. We measure their classi cation accuracy by applying the models to the corresponding testing datasets. By comparing their classi cation performance, we evaluate the in uence that our proposed approach brings to a DNN. More speci cally, we examine if LLE-DNN exhibits similar -if not the same or be er -classi cation accuracy.\\nSince the goal of this work is to improve the robustness of a DNN model, we also evaluate our DNN models' resistance to adversarial samples. In particular, we derive adversarial samples from the aforementioned testing datasets, test them against our DNN model and compare its model resistance with those of DNNs enhanced by the other two techniques [12,29].\\nSince the goal of this work is to improve the robustness of a DNN model, we also evaluate our DNN models' resistance to adversarial samples. In particular, we derive adversarial samples from the aforementioned testing datasets, test them against our DNN model and compare its model resistance with those of DNNs enhanced by the other two techniques [12,29].\\nAs is discussed in Section 2.2, an a acker cra s adversarial samples through auxiliary models. In Table 1, black-box and white-box indicate the auxiliary models trained through di erent schemes. More speci cally, black-box represents the auxiliary model trained through the standard deep learning training scheme, indicating an a acker does not have su cient knowledge about the underlying training algorithm and he can use only a standard approach to train a cross model and cra adversarial samples. White-box represents the auxiliary model trained exactly through the learning schemes proposed as a defense. is simulates a situation where a defense mechanism is publicly disclosed and an a acker exploits that mechanism to produce a highly similar -if not the same -model to cra adversarial samples. Note that, for both black-box and white-box tactics, we use the same hyperparameters and training dataset to build auxiliary models. More speci cally, our auxiliary model training shares the same hyperparameters and training dataset with the standard DNN shown in Table 3.\\nAs is discussed in Section 2.2, an a acker cra s adversarial samples through auxiliary models. In Table 1, black-box and white-box indicate the auxiliary models trained through di erent schemes. More speci cally, black-box represents the auxiliary model trained through the standard deep learning training scheme, indicating an a acker does not have su cient knowledge about the underlying training algorithm and he can use only a standard approach to train a cross model and cra adversarial samples. White-box represents the auxiliary model trained exactly through the learning schemes proposed as a defense. is simulates a situation where a defense mechanism is publicly disclosed and an a acker exploits that mechanism to produce a highly similar -if not the same -model to cra adversarial samples. Note that, for both black-box and white-box tactics, we use the same hyperparameters and training dataset to build auxiliary models. More speci cally, our auxiliary model training shares the same hyperparameters and training dataset with the standard DNN shown in Table 3.\\nIn addition to the methods described in Section 2.2, the cra ing of adversarial samples must ensure a slight perturbation introduced to a data sample does not undermine its semantic. In other words, we must make sure that, while misleading a classi er to output the wrong class with high con dence, the perturbation to an image should be nearly indistinguishable to the human eyes, that to a malicious so ware sample should not jeopardize so ware functionality nor break its malevolence, and that to a movie review should not break its semantic meaning. In the following, we describe how we ne-tune adversarial samples to preserve semantic for di erent applications. Malware classi cation. Recall that our malware samples are represented by features, the value of which are binary, indicating the occurrence of an lesystem access or a sequence of access events. When generating adversarial samples, we cannot simply disable Table 3: e hyperparameters of all the investigated DNN models.\\nIn addition to the methods described in Section 2.2, the cra ing of adversarial samples must ensure a slight perturbation introduced to a data sample does not undermine its semantic. In other words, we must make sure that, while misleading a classi er to output the wrong class with high con dence, the perturbation to an image should be nearly indistinguishable to the human eyes, that to a malicious so ware sample should not jeopardize so ware functionality nor break its malevolence, and that to a movie review should not break its semantic meaning. In the following, we describe how we ne-tune adversarial samples to preserve semantic for di erent applications. Malware classi cation. Recall that our malware samples are represented by features, the value of which are binary, indicating the occurrence of an lesystem access or a sequence of access events. When generating adversarial samples, we cannot simply disable Table 3: e hyperparameters of all the investigated DNN models.\\nlesystem access events in that this might jeopardize the functionality of the so ware sample and even break down its malevolence. With this in mind, a bit of care must be taken.\\nlesystem access events in that this might jeopardize the functionality of the so ware sample and even break down its malevolence. With this in mind, a bit of care must be taken.\\nIn this work, our experiment follows the approach introduced in Section 2.2. To be speci c, we cra adversarial so ware samples by solving optimization problem (1) with the se ing of zero norm (i.e., l 0 ). is indicates the manipulation to a sample is restricted to ipping binary feature values. In addition, this implies the strongest a ack scenario in that optimization problem (1) carries less constraints making the adversarial samples chosen for our evaluation more impactful. Going beyond adversarial sample craing approach discussed in Section 2.2, we also restrict that the value change of a feature can be only from 0 to 1 but not the opposite.\\nIn this work, our experiment follows the approach introduced in Section 2.2. To be speci c, we cra adversarial so ware samples by solving optimization problem (1) with the se ing of zero norm (i.e., l 0 ). is indicates the manipulation to a sample is restricted to ipping binary feature values. In addition, this implies the strongest a ack scenario in that optimization problem (1) carries less constraints making the adversarial samples chosen for our evaluation more impactful. Going beyond adversarial sample craing approach discussed in Section 2.2, we also restrict that the value change of a feature can be only from 0 to 1 but not the opposite.\\nis amounts to allowing the addition of new lesystem access events only. is manipulation strategy is reasonable for the reason that malware mutation techniques (e.g., [25]) can morph a malware sample by stitching together instructions from benign programs, making the malware perform additional lesystem accesses but not undermining its maliciousness nor its functionality. Since malware manipulation is done with the intent of fooling a malware classi er driven by a DNN, it should be noticed that we do not morph a benign so ware sample, making it malicious. Image recognition. Image data samples contain less strict semantic than the malware data samples above. To preserve image semantic -making a perturbation nearly indistinguishable -we follow the approaches introduced in [6,12,35]. More speci cally, we selected l 0 , l 2 and l ∞ distance to represent the dissimilarity between an image and its corresponding adversarial sample. Especially, we restrict the l ∞ distance in a relative small range (i.e., ϵ ≤ 0.15) when cra ing adversarial samples. Sentiment Analysis. To generate adversarial samples for movie reviews, we again followed the approach introduced in Section 2.2. To be speci c, we solved optimization problem (1) and con gured p-norm with the se ing to l 2 and l ∞ . is is due to the fact that each review is encoded in a vector in which each element is a decimal, and l 2 and l ∞ distances represent the best measure for the dissimilarity between a movie review and its corresponding adversarial sample. Table 4: e comparison of classi cation accuracy on di erent datasets.\\nis amounts to allowing the addition of new lesystem access events only. is manipulation strategy is reasonable for the reason that malware mutation techniques (e.g., [25]) can morph a malware sample by stitching together instructions from benign programs, making the malware perform additional lesystem accesses but not undermining its maliciousness nor its functionality. Since malware manipulation is done with the intent of fooling a malware classi er driven by a DNN, it should be noticed that we do not morph a benign so ware sample, making it malicious. Image recognition. Image data samples contain less strict semantic than the malware data samples above. To preserve image semantic -making a perturbation nearly indistinguishable -we follow the approaches introduced in [6,12,35]. More speci cally, we selected l 0 , l 2 and l ∞ distance to represent the dissimilarity between an image and its corresponding adversarial sample. Especially, we restrict the l ∞ distance in a relative small range (i.e., ϵ ≤ 0.15) when cra ing adversarial samples. Sentiment Analysis. To generate adversarial samples for movie reviews, we again followed the approach introduced in Section 2.2. To be speci c, we solved optimization problem (1) and con gured p-norm with the se ing to l 2 and l ∞ . is is due to the fact that each review is encoded in a vector in which each element is a decimal, and l 2 and l ∞ distances represent the best measure for the dissimilarity between a movie review and its corresponding adversarial sample. Table 4: e comparison of classi cation accuracy on di erent datasets.\\nAs is mentioned above, each vocabulary has been encoded in a vector with a dimensionality of 600, and we embedded a movie review by linearly combining corresponding vectors. When generating an adversarial sample by introducing a slight perturbation to the embedding, we cast the perturbation to only one vector. is ensures that we introduce only one word change to that review with the hope that it preserves the semantic meaning of that review as much as possible. However, one word change does not guarantee the invariance of the semantic meaning. For example, it would be obvious alteration to the semantic meaning if the replacement happens to be the negative word in \\\"… makes it the biggest disappointment I've experienced from cinema in years …\\\". As such, we manually choose the word that incurs minimal semantic change to that movie review. Figure 5 illustrates a movie review sample and its corresponding adversarial sample generated through this approach.\\nAs is mentioned above, each vocabulary has been encoded in a vector with a dimensionality of 600, and we embedded a movie review by linearly combining corresponding vectors. When generating an adversarial sample by introducing a slight perturbation to the embedding, we cast the perturbation to only one vector. is ensures that we introduce only one word change to that review with the hope that it preserves the semantic meaning of that review as much as possible. However, one word change does not guarantee the invariance of the semantic meaning. For example, it would be obvious alteration to the semantic meaning if the replacement happens to be the negative word in \\\"… makes it the biggest disappointment I've experienced from cinema in years …\\\". As such, we manually choose the word that incurs minimal semantic change to that movie review. Figure 5 illustrates a movie review sample and its corresponding adversarial sample generated through this approach.\"\n },\n {\n \"title\": \"Experimental Setup and Results\",\n \"content\": \"\"\n },\n {\n \"title\": \"Experimental Setup and Results\",\n \"content\": \"On the datasets described above, we rst measure the accuracy of all the aforementioned defense techniques. We then measure their resistance to the adversarial samples cra ed through the aforementioned tactics.\\nOn the datasets described above, we rst measure the accuracy of all the aforementioned defense techniques. We then measure their resistance to the adversarial samples cra ed through the aforementioned tactics.\"\n },\n {\n \"title\": \"Classification Accuracy.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Classification Accuracy.\",\n \"content\": \"To identify the optimal dimensionality to which LLE needs to map original data samples, we implemented several LLE-DNNs with di erent se ings of dimensionality of LLE mappings. Figure 6 shows the impact of dimensionality mapping upon the classi cation accuracy obtained by LLE-DNN. Across all three datasets, it is easy to observe that, the classi cation accuracy rst increases when the dimensionality of the LLE mappings rises and then starts to decrease. In our experiment, we choose the highest classi cation accuracy to represent the performance of our LLE-DNN. Table 4 presents the classi cation accuracy results obtained from all investigated DNNs with respect to the testing datasets. Note that, while prior works (e.g, [7,10]) have already demonstrated they can train a DNN with an error rate less than 1% on the MNIST benchmark, their performance improvement does not result from a DNN but model ensemble or elastic distortions added to training data. To study the in uence of our proposed approach upon a standard DNN, therefore, we did not combine models nor augment with arti cially distorted versions of the original training samples. e classi cation accuracy shown in the table has already represented the best performance that a standard DNN can achieve.\\nTo identify the optimal dimensionality to which LLE needs to map original data samples, we implemented several LLE-DNNs with di erent se ings of dimensionality of LLE mappings. Figure 6 shows the impact of dimensionality mapping upon the classi cation accuracy obtained by LLE-DNN. Across all three datasets, it is easy to observe that, the classi cation accuracy rst increases when the dimensionality of the LLE mappings rises and then starts to decrease. In our experiment, we choose the highest classi cation accuracy to represent the performance of our LLE-DNN. Table 4 presents the classi cation accuracy results obtained from all investigated DNNs with respect to the testing datasets. Note that, while prior works (e.g, [7,10]) have already demonstrated they can train a DNN with an error rate less than 1% on the MNIST benchmark, their performance improvement does not result from a DNN but model ensemble or elastic distortions added to training data. To study the in uence of our proposed approach upon a standard DNN, therefore, we did not combine models nor augment with arti cially distorted versions of the original training samples. e classi cation accuracy shown in the table has already represented the best performance that a standard DNN can achieve.\\nSimilar to adversarial training and defensive distillation, the LLE-DNN is quite e ective in preserving classi cation accuracy.\\nSimilar to adversarial training and defensive distillation, the LLE-DNN is quite e ective in preserving classi cation accuracy.\\nis implies our proposed approach well preserves data sample distribution. For the malware classi cation task, it can be observed that LLE-DNN appears to be be er at feature learning, achieving the highest classi cation accuracy among DNNs that we investigated. is is presumably due to the fact that malware data samples are highly sparse carrying a large amount of redundant information, and the data transformation module in LLE-DNN eliminates those redundancy and ameliorates the learning ability of a DNN. Table 1 illustrates the DNNs that we investigated as well as their accuracy in classifying adversarial samples. It can be observed that, black-box adversarial samples can cut down the accuracy of the standard DNN to 6.86%, 6.40% and 7.50% under the a acks of l ∞ , l 2 and l 0 , respectively. In contrast, all of the defense mechanisms investigated demonstrate strong resistance to these black-box adversarial samples.\\nis implies our proposed approach well preserves data sample distribution. For the malware classi cation task, it can be observed that LLE-DNN appears to be be er at feature learning, achieving the highest classi cation accuracy among DNNs that we investigated. is is presumably due to the fact that malware data samples are highly sparse carrying a large amount of redundant information, and the data transformation module in LLE-DNN eliminates those redundancy and ameliorates the learning ability of a DNN. Table 1 illustrates the DNNs that we investigated as well as their accuracy in classifying adversarial samples. It can be observed that, black-box adversarial samples can cut down the accuracy of the standard DNN to 6.86%, 6.40% and 7.50% under the a acks of l ∞ , l 2 and l 0 , respectively. In contrast, all of the defense mechanisms investigated demonstrate strong resistance to these black-box adversarial samples.\"\n },\n {\n \"title\": \"Model Resistance.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Model Resistance.\",\n \"content\": \"is indicates, without su cient knowledge on the underlying defense mechanisms, it is di cult for an a acker to cra impactful adversarial samples. In other words, existing defense mechanisms can signi cantly escalate a DNN's resistance to adversarial samples if one can obscure the design of the defenses.\\nis indicates, without su cient knowledge on the underlying defense mechanisms, it is di cult for an a acker to cra impactful adversarial samples. In other words, existing defense mechanisms can signi cantly escalate a DNN's resistance to adversarial samples if one can obscure the design of the defenses.\\nDespite the improvement in model robustness, we also observe that our LLE-DNN generally exhibits the best resistance to blackbox adversarial samples, whereas the defensive distillation approach typically yields the worst resistance. is is presumably due to the fact that, the dimensionality reduction resided in LLE-DNN transforms adversarial samples into a subspace in which they no longer act as outliers, while defensive distillation smooths only a classi cation decision boundary which does not signi cantly reduce the subspace of adversarial samples.\\nDespite the improvement in model robustness, we also observe that our LLE-DNN generally exhibits the best resistance to blackbox adversarial samples, whereas the defensive distillation approach typically yields the worst resistance. is is presumably due to the fact that, the dimensionality reduction resided in LLE-DNN transforms adversarial samples into a subspace in which they no longer act as outliers, while defensive distillation smooths only a classi cation decision boundary which does not signi cantly reduce the subspace of adversarial samples.\\nWith regard to the white-box se ing, we discover both adversarial training and defensive distillation su er from white-box adversarial samples. eir resistance to white-box adversarial samples is signi cantly worse than those created under the black-box se ing.\\nWith regard to the white-box se ing, we discover both adversarial training and defensive distillation su er from white-box adversarial samples. eir resistance to white-box adversarial samples is signi cantly worse than those created under the black-box se ing.\\nis observation is consistent with that reported in [6]. e reason behind this is that, both techniques stash away the adversarial sample subspace, but the disclosure of defense mechanisms uncovers the path of nding that subspace.\\nis observation is consistent with that reported in [6]. e reason behind this is that, both techniques stash away the adversarial sample subspace, but the disclosure of defense mechanisms uncovers the path of nding that subspace.\\nDi erent from adversarial training and defensive distillation, our LLE-DNN is naturally resistant to white-box adversarial samples. As is discussed in Section 4, our proposed approach stashes away the adversarial sample subspace and at the same time restricts derivative calculation. Even if our defense mechanism is revealed, therefore, it is still computationally di cult to nd adversarial samples.\\nDi erent from adversarial training and defensive distillation, our LLE-DNN is naturally resistant to white-box adversarial samples. As is discussed in Section 4, our proposed approach stashes away the adversarial sample subspace and at the same time restricts derivative calculation. Even if our defense mechanism is revealed, therefore, it is still computationally di cult to nd adversarial samples.\\nTo perform quantitative comparison with the other two approaches, however, we approximate the data transformation in the LLE-DNN -non-parametric dimensionality reduction component -using a parametric model. To be speci c, we choose a DNN to approximate LLE in that a DNN has a large amount of parameters which is typically viewed as the best approximation for non-parametric learning models [17]. With the support from this approximation, we treated the LLE-DNN as a white box and generated adversarial samples accordingly. We show its model resistance in Table 1. It can be observed that, our LLE-DNN still demonstrates strong resistance to white-box adversarial samples even if we substituted LLE to its best approximation. is implies that, there might a theoretical lower bound between a non-parametric model and its parametric approximation, which can naturally serve as a defense against white-box adversarial samples.\\nTo perform quantitative comparison with the other two approaches, however, we approximate the data transformation in the LLE-DNN -non-parametric dimensionality reduction component -using a parametric model. To be speci c, we choose a DNN to approximate LLE in that a DNN has a large amount of parameters which is typically viewed as the best approximation for non-parametric learning models [17]. With the support from this approximation, we treated the LLE-DNN as a white box and generated adversarial samples accordingly. We show its model resistance in Table 1. It can be observed that, our LLE-DNN still demonstrates strong resistance to white-box adversarial samples even if we substituted LLE to its best approximation. is implies that, there might a theoretical lower bound between a non-parametric model and its parametric approximation, which can naturally serve as a defense against white-box adversarial samples.\"\n },\n {\n \"title\": \"CONCLUSION\",\n \"content\": \"\"\n },\n {\n \"title\": \"CONCLUSION\",\n \"content\": \"A Deep Neural Network is vulnerable to adversarial samples. Existing defense mechanisms improve a DNN model's resistance to adversarial samples by using the tactic of security through obscurity. Once the design of the defense is disclosed, therefore, the robustness they provide wane. Motivated by this, this work introduces a new approach to escalate the robustness of a DNN model. In particular, it integrates to a DNN model LLE, a non-parametric dimensionality reduction method. With this approach, we show that one can develop a DNN model resistant to adversarial samples even if he or she reveals its design details (i.e., the underlying training algorithm). By demonstrating the DNNs enhanced by our proposed technique across various applications, we argue the proposed approach introduces nearly no degradation in classi cation performance. In contrast, for some applications, it even exhibits performance improvement. As part of the future work, we will further explore the performance of this approach in a wider variety of applications across di erent deep neural architectures.\\nA Deep Neural Network is vulnerable to adversarial samples. Existing defense mechanisms improve a DNN model's resistance to adversarial samples by using the tactic of security through obscurity. Once the design of the defense is disclosed, therefore, the robustness they provide wane. Motivated by this, this work introduces a new approach to escalate the robustness of a DNN model. In particular, it integrates to a DNN model LLE, a non-parametric dimensionality reduction method. With this approach, we show that one can develop a DNN model resistant to adversarial samples even if he or she reveals its design details (i.e., the underlying training algorithm). By demonstrating the DNNs enhanced by our proposed technique across various applications, we argue the proposed approach introduces nearly no degradation in classi cation performance. In contrast, for some applications, it even exhibits performance improvement. As part of the future work, we will further explore the performance of this approach in a wider variety of applications across di erent deep neural architectures.\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.","content":"Figure 2 :\n2Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes."},{"id":"","caption":"Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.","content":"Figure 2 :\n2Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes."},{"id":"","caption":"A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·).","content":"Figure 3 :\n3A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·)."},{"id":"","caption":"A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·).","content":"Figure 3 :\n3A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·)."},{"id":"","caption":"e variation of classi cation accuracy vs. the dimensionality of data mappings.","content":"Figure 6 :\n6e variation of classi cation accuracy vs. the dimensionality of data mappings."},{"id":"","caption":"e variation of classi cation accuracy vs. the dimensionality of data mappings.","content":"Figure 6 :\n6e variation of classi cation accuracy vs. the dimensionality of data mappings."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.\",\n \"content\": \"Figure 2 :\\n2Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.\",\n \"content\": \"Figure 2 :\\n2Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·).\",\n \"content\": \"Figure 3 :\\n3A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·).\",\n \"content\": \"Figure 3 :\\n3A deep neural network with a data transformation module, projecting X , an input data sample, to new representation (X ) prior to passing it through the DNN, f (·).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"e variation of classi cation accuracy vs. the dimensionality of data mappings.\",\n \"content\": \"Figure 6 :\\n6e variation of classi cation accuracy vs. the dimensionality of data mappings.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"e variation of classi cation accuracy vs. the dimensionality of data mappings.\",\n \"content\": \"Figure 6 :\\n6e variation of classi cation accuracy vs. the dimensionality of data mappings.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Deep learning with non-medical training used for chest pathology identi cation. 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In 2015 IEEE Conference on Computer Vision and Pa ern Recognition (CVPR), pages 427-436. IEEE, 2015.","doi":""},{"text":"Unifying adversarial training algorithms with exible deep data gradient regularization. A G Ororbia, I I , C L Giles, D Kifer, arXiv:1601.07213A. G. Ororbia II, C. L. Giles, and D. Kifer. Unifying adversarial training algorithms with exible deep data gradient regularization. arXiv:1601.07213 [cs], 2016.","doi":"arXiv:1601.07213"},{"text":"e limitations of deep learning in adversarial se ings. N Papernot, P Mcdaniel, S Jha, M Fredrikson, Z B Celik, A Swami, 2016 IEEE European Symposium on Security and Privacy (EuroS&P). IEEEN. Papernot, P. McDaniel, S. Jha, M. Fredrikson, Z. B. Celik, and A. Swami. e limitations of deep learning in adversarial se ings. In 2016 IEEE European Symposium on Security and Privacy (EuroS&P), pages 372-387. IEEE, 2016.","doi":""},{"text":"Distillation as a defense to adversarial perturbations against deep neural networks. N Papernot, P Mcdaniel, X Wu, S Jha, A Swami, IEEE Symposium on Security and Privacy (S&P). IEEEN. Papernot, P. McDaniel, X. Wu, S. Jha, and A. Swami. Distillation as a defense to adversarial perturbations against deep neural networks. In IEEE Symposium on Security and Privacy (S&P). IEEE, 2016.","doi":""},{"text":"Malware classi cation with recurrent networks. R Pascanu, J W Stokes, H Sanossian, M Marinescu, A Omas, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEER. Pascanu, J. W. Stokes, H. Sanossian, M. Marinescu, and A. omas. Malware classi cation with recurrent networks. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1916-1920. IEEE, 2015.","doi":""},{"text":"Nonlinear dimensionality reduction by locally linear embedding. S T Roweis, L K Saul, Science. 2905500S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323-2326, 2000.","doi":""},{"text":"S Sabour, Y Cao, F Faghri, D J Fleet, arXiv:1511.05122Adversarial manipulation of deep representations. arXiv preprintS. Sabour, Y. Cao, F. Faghri, and D. J. Fleet. Adversarial manipulation of deep representations. arXiv preprint arXiv:1511.05122, 2015.","doi":"arXiv:1511.05122"},{"text":"A nonlinear mapping for data structure analysis. J W Sammon, IEEE Transactions on computers. 185J. W. Sammon. A nonlinear mapping for data structure analysis. IEEE Transactions on computers, 18(5):401-409, 1969.","doi":""},{"text":"Mastering the game of go with deep neural networks and tree search. D Silver, A Huang, C J Maddison, A Guez, L Sifre, G Van Den Driessche, J Schri Wieser, I Antonoglou, V Panneershelvam, M Lanctot, Nature. 5297587D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schri wieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529(7587):484- 489, 2016.","doi":""},{"text":"Intriguing properties of neural networks. C Szegedy, W Zaremba, I Sutskever, J Bruna, D Erhan, I Goodfellow, R Fergus, International Conference on Learning Representations. C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus. Intriguing properties of neural networks. In International Conference on Learning Representations, 2014.","doi":""},{"text":"L Van Der Maaten, E O Postma, H J Van Den, Herik, Dimensionality reduction: A comparative review. L. van der Maaten, E. O. Postma, and H. J. van den Herik. Dimensionality reduction: A comparative review, 2008.","doi":""},{"text":"Nonlinear Optimization: Complexity Issues. S A Vavasis, Oxford University Press, IncNew York, NY, USAS. A. Vavasis. Nonlinear Optimization: Complexity Issues. Oxford University Press, Inc., New York, NY, USA, 1991.","doi":""},{"text":"Deep learning of feature representation with multiple instance learning for medical image analysis. Y Xu, T Mo, Q Feng, P Zhong, M Lai, I Eric, C Chang, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEEY. Xu, T. Mo, Q. Feng, P. Zhong, M. Lai, I. Eric, and C. Chang. Deep learning of feature representation with multiple instance learning for medical image analysis. In 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1626-1630. IEEE, 2014.","doi":""},{"text":"Droid-sec: Deep learning in android malware detection. Z Yuan, Y Lu, Z Wang, Y Xue, ACM SIGCOMM Computer Communication Review. ACM44Z. Yuan, Y. Lu, Z. Wang, and Y. Xue. Droid-sec: Deep learning in android malware detection. In ACM SIGCOMM Computer Communication Review, volume 44, pages 371-372. ACM, 2014.","doi":""},{"text":"Deep learning with non-medical training used for chest pathology identi cation. Y Bar, I Diamant, L Wolf, H Greenspan, SPIE Medical Imaging, pages 94140V-94140V. International Society for Optics and Photonics. Y. Bar, I. Diamant, L. Wolf, and H. Greenspan. Deep learning with non-medical training used for chest pathology identi cation. In SPIE Medical Imaging, pages 94140V-94140V. International Society for Optics and Photonics, 2015.","doi":""},{"text":"Can machine learning be secure?. M Barreno, B Nelson, R Sears, A D Joseph, J D Tygar, Proceedings of the 2006 ACM Symposium on Information, computer and communications security. the 2006 ACM Symposium on Information, computer and communications securityACMM. Barreno, B. Nelson, R. Sears, A. D. Joseph, and J. D. Tygar. Can machine learning be secure? In Proceedings of the 2006 ACM Symposium on Information, computer and communications security, pages 16-25. ACM, 2006.","doi":""},{"text":"Measuring neural net robustness with constraints. O Bastani, Y Ioannou, L Lampropoulos, D Vytiniotis, A Nori, A Criminisi, Advances In Neural Information Processing Systems. O. Bastani, Y. Ioannou, L. Lampropoulos, D. Vytiniotis, A. Nori, and A. Crimin- isi. Measuring neural net robustness with constraints. In Advances In Neural Information Processing Systems, pages 2613-2621, 2016.","doi":""},{"text":"Deep learning. Book in preparation for. I G Y Bengio, A Courville, MIT PressI. G. Y. Bengio and A. Courville. Deep learning. Book in preparation for MIT Press, 2016.","doi":""},{"text":"Malicious behavior detection using windows audit logs. K Berlin, D Slater, J Saxe, Proceedings of the 8th ACM Workshop on Arti cial Intelligence and Security. the 8th ACM Workshop on Arti cial Intelligence and SecurityACMK. Berlin, D. Slater, and J. Saxe. Malicious behavior detection using windows audit logs. 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Scene parsing with multiscale feature learning, purity trees, and optimal covers. arXiv preprint arXiv:1202.2160, 2012.","doi":"arXiv:1202.2160"},{"text":"Explaining and Harnessing Adversarial Examples. I Goodfellow, J Shlens, C Szegedy, CoRRI. Goodfellow, J. Shlens, and C. Szegedy. Explaining and Harnessing Adversarial Examples. CoRR, 2014.","doi":""},{"text":"S Gu, L Rigazio, arXiv:1412.5068Towards deep neural network architectures robust to adversarial examples. S. Gu and L. Rigazio. Towards deep neural network architectures robust to adversarial examples. arXiv:1412.5068 [cs], 2014.","doi":"arXiv:1412.5068"},{"text":"Learning long-range vision for autonomous o -road driving. R Hadsell, P Sermanet, J Ben, A Erkan, M Sco Er, K Kavukcuoglu, U Muller, Y Lecun, Journal of Field Robotics. 262R. Hadsell, P. Sermanet, J. Ben, A. Erkan, M. Sco er, K. Kavukcuoglu, U. Muller, and Y. LeCun. Learning long-range vision for autonomous o -road driving. Journal of Field Robotics, 26(2):120-144, 2009.","doi":""},{"text":"Learning multiple layers of representation. G E Hinton, Trends in cognitive sciences. 1110G. E. Hinton. Learning multiple layers of representation. Trends in cognitive sciences, 11(10):428-434, 2007.","doi":""},{"text":"Matrix analysis. corrected reprint of the 1985 original. R A Horn, C R Johnson, R. A. Horn and C. R. Johnson. Matrix analysis. corrected reprint of the 1985 original, 1990.","doi":""},{"text":"Approximation capabilities of multilayer feedforward networks. K Hornik, Neural networks. 42K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural networks, 4(2):251-257, 1991.","doi":""},{"text":"Adversarial machine learning. L Huang, A D Joseph, B Nelson, B I Rubinstein, J Tygar, Proceedings of the 4th ACM workshop on Security and arti cial intelligence. the 4th ACM workshop on Security and arti cial intelligenceACML. Huang, A. D. Joseph, B. Nelson, B. I. Rubinstein, and J. Tygar. Adversarial machine learning. In Proceedings of the 4th ACM workshop on Security and arti cial intelligence, pages 43-58. ACM, 2011.","doi":""},{"text":"Kerckho s. La cryptographie militaire. A , Journal des Sciences Militaires. A. Kerckho s. La cryptographie militaire. Journal des Sciences Militaires, pages 161-191, 1883.","doi":""},{"text":"e mnist database of handwri en digits. Y Lecun, C Cortes, C J Burges, Y. LeCun, C. Cortes, and C. J. Burges. e mnist database of handwri en digits, 1998.","doi":""},{"text":"Delving into transferable adversarial examples and black-box a acks. Y Liu, X Chen, C Liu, D Song, arXiv:1611.02770arXiv preprintY. Liu, X. Chen, C. Liu, and D. Song. Delving into transferable adversarial examples and black-box a acks. arXiv preprint arXiv:1611.02770, 2016.","doi":"arXiv:1611.02770"},{"text":"Learning word vectors for sentiment analysis. A L Maas, R E Daly, P T Pham, D Huang, A Y Ng, C Po, Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies. the 49th Annual Meeting of the Association for Computational Linguistics: Human Language TechnologiesPortland, Oregon, USAAssociation for Computational LinguisticsA. L. Maas, R. E. Daly, P. T. Pham, D. Huang, A. Y. Ng, and C. Po s. Learning word vectors for sentiment analysis. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies, pages 142-150, Portland, Oregon, USA, June 2011. Association for Computational Linguistics.","doi":""},{"text":"Visualizing data using t-sne. L V D Maaten, G Hinton, Journal of Machine Learning Research. 9L. v. d. Maaten and G. Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(Nov):2579-2605, 2008.","doi":""},{"text":"Distributed representations of words and phrases and their compositionality. T Mikolov, I Sutskever, K Chen, G S Corrado, J Dean, Advances in neural information processing systems. T. Mikolov, I. Sutskever, K. Chen, G. S. Corrado, and J. Dean. Distributed rep- resentations of words and phrases and their compositionality. In Advances in neural information processing systems, pages 3111-3119, 2013.","doi":""},{"text":"Frankenstein: Stitching malware from benign binaries. V Mohan, K W Hamlen, WOOT. USENIX Association. V. Mohan and K. W. Hamlen. Frankenstein: Stitching malware from benign binaries. In WOOT. USENIX Association, 2012.","doi":""},{"text":"Deep neural networks are easily fooled: High con dence predictions for unrecognizable images. A Nguyen, J Yosinski, J Clune, 2015 IEEE Conference on Computer Vision and Pa ern Recognition (CVPR). IEEEA. Nguyen, J. Yosinski, and J. Clune. Deep neural networks are easily fooled: High con dence predictions for unrecognizable images. In 2015 IEEE Conference on Computer Vision and Pa ern Recognition (CVPR), pages 427-436. IEEE, 2015.","doi":""},{"text":"Unifying adversarial training algorithms with exible deep data gradient regularization. A G Ororbia, I I , C L Giles, D Kifer, arXiv:1601.07213A. G. Ororbia II, C. L. Giles, and D. Kifer. Unifying adversarial training algorithms with exible deep data gradient regularization. arXiv:1601.07213 [cs], 2016.","doi":"arXiv:1601.07213"},{"text":"e limitations of deep learning in adversarial se ings. N Papernot, P Mcdaniel, S Jha, M Fredrikson, Z B Celik, A Swami, 2016 IEEE European Symposium on Security and Privacy (EuroS&P). IEEEN. Papernot, P. McDaniel, S. Jha, M. Fredrikson, Z. B. Celik, and A. Swami. e limitations of deep learning in adversarial se ings. In 2016 IEEE European Symposium on Security and Privacy (EuroS&P), pages 372-387. IEEE, 2016.","doi":""},{"text":"Distillation as a defense to adversarial perturbations against deep neural networks. N Papernot, P Mcdaniel, X Wu, S Jha, A Swami, IEEE Symposium on Security and Privacy (S&P). IEEEN. Papernot, P. McDaniel, X. Wu, S. Jha, and A. Swami. Distillation as a defense to adversarial perturbations against deep neural networks. In IEEE Symposium on Security and Privacy (S&P). IEEE, 2016.","doi":""},{"text":"Malware classi cation with recurrent networks. R Pascanu, J W Stokes, H Sanossian, M Marinescu, A Omas, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEER. Pascanu, J. W. Stokes, H. Sanossian, M. Marinescu, and A. omas. Malware classi cation with recurrent networks. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1916-1920. IEEE, 2015.","doi":""},{"text":"Nonlinear dimensionality reduction by locally linear embedding. S T Roweis, L K Saul, Science. 2905500S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323-2326, 2000.","doi":""},{"text":"S Sabour, Y Cao, F Faghri, D J Fleet, arXiv:1511.05122Adversarial manipulation of deep representations. arXiv preprintS. Sabour, Y. Cao, F. Faghri, and D. J. Fleet. Adversarial manipulation of deep representations. arXiv preprint arXiv:1511.05122, 2015.","doi":"arXiv:1511.05122"},{"text":"A nonlinear mapping for data structure analysis. J W Sammon, IEEE Transactions on computers. 185J. W. Sammon. A nonlinear mapping for data structure analysis. IEEE Transactions on computers, 18(5):401-409, 1969.","doi":""},{"text":"Mastering the game of go with deep neural networks and tree search. D Silver, A Huang, C J Maddison, A Guez, L Sifre, G Van Den Driessche, J Schri Wieser, I Antonoglou, V Panneershelvam, M Lanctot, Nature. 5297587D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schri wieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529(7587):484- 489, 2016.","doi":""},{"text":"Intriguing properties of neural networks. C Szegedy, W Zaremba, I Sutskever, J Bruna, D Erhan, I Goodfellow, R Fergus, International Conference on Learning Representations. C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus. Intriguing properties of neural networks. In International Conference on Learning Representations, 2014.","doi":""},{"text":"L Van Der Maaten, E O Postma, H J Van Den, Herik, Dimensionality reduction: A comparative review. L. van der Maaten, E. O. Postma, and H. J. van den Herik. Dimensionality reduction: A comparative review, 2008.","doi":""},{"text":"Nonlinear Optimization: Complexity Issues. S A Vavasis, Oxford University Press, IncNew York, NY, USAS. A. Vavasis. Nonlinear Optimization: Complexity Issues. Oxford University Press, Inc., New York, NY, USA, 1991.","doi":""},{"text":"Deep learning of feature representation with multiple instance learning for medical image analysis. Y Xu, T Mo, Q Feng, P Zhong, M Lai, I Eric, C Chang, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEEY. Xu, T. Mo, Q. Feng, P. Zhong, M. Lai, I. Eric, and C. Chang. Deep learning of feature representation with multiple instance learning for medical image analysis. In 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1626-1630. IEEE, 2014.","doi":""},{"text":"Droid-sec: Deep learning in android malware detection. Z Yuan, Y Lu, Z Wang, Y Xue, ACM SIGCOMM Computer Communication Review. ACM44Z. Yuan, Y. Lu, Z. Wang, and Y. Xue. Droid-sec: Deep learning in android malware detection. In ACM SIGCOMM Computer Communication Review, volume 44, pages 371-372. ACM, 2014.","doi":""}],"string":"[\n {\n \"text\": \"Deep learning with non-medical training used for chest pathology identi cation. Y Bar, I Diamant, L Wolf, H Greenspan, SPIE Medical Imaging, pages 94140V-94140V. International Society for Optics and Photonics. Y. Bar, I. Diamant, L. Wolf, and H. Greenspan. Deep learning with non-medical training used for chest pathology identi cation. In SPIE Medical Imaging, pages 94140V-94140V. International Society for Optics and Photonics, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Can machine learning be secure?. M Barreno, B Nelson, R Sears, A D Joseph, J D Tygar, Proceedings of the 2006 ACM Symposium on Information, computer and communications security. the 2006 ACM Symposium on Information, computer and communications securityACMM. Barreno, B. Nelson, R. Sears, A. D. Joseph, and J. D. Tygar. Can machine learning be secure? In Proceedings of the 2006 ACM Symposium on Information, computer and communications security, pages 16-25. ACM, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measuring neural net robustness with constraints. O Bastani, Y Ioannou, L Lampropoulos, D Vytiniotis, A Nori, A Criminisi, Advances In Neural Information Processing Systems. O. Bastani, Y. Ioannou, L. Lampropoulos, D. Vytiniotis, A. Nori, and A. Crimin- isi. Measuring neural net robustness with constraints. In Advances In Neural Information Processing Systems, pages 2613-2621, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep learning. Book in preparation for. I G Y Bengio, A Courville, MIT PressI. G. Y. Bengio and A. Courville. Deep learning. Book in preparation for MIT Press, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Malicious behavior detection using windows audit logs. K Berlin, D Slater, J Saxe, Proceedings of the 8th ACM Workshop on Arti cial Intelligence and Security. the 8th ACM Workshop on Arti cial Intelligence and SecurityACMK. Berlin, D. Slater, and J. Saxe. Malicious behavior detection using windows audit logs. In Proceedings of the 8th ACM Workshop on Arti cial Intelligence and Security, pages 35-44. ACM, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Towards evaluating the robustness of neural networks. N Carlini, D Wagner, IEEE Symposium on Security and Privacy (S&P). IEEEN. Carlini and D. Wagner. Towards evaluating the robustness of neural networks. In IEEE Symposium on Security and Privacy (S&P). IEEE, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep big simple neural nets excel on handwri en digit recognition. D C Ciresan, U Meier, L M Gambardella, J Schmidhuber, abs/1003.0358CoRRD. C. Ciresan, U. Meier, L. M. Gambardella, and J. Schmidhuber. Deep big simple neural nets excel on handwri en digit recognition. CoRR, abs/1003.0358, 2010.\",\n \"doi\": \"abs/1003.0358\"\n },\n {\n \"text\": \"Large-scale malware classi cation using random projections and neural networks. G E Dahl, J W Stokes, L Deng, D Yu, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEEG. E. Dahl, J. W. Stokes, L. Deng, and D. Yu. Large-scale malware classi cation using random projections and neural networks. In 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pages 3422-3426. IEEE, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Relaxations and randomized methods for nonconvex qcqps. EE392o Class Notes. A Aspremont, S Boyd, Stanford UniversityA. d'Aspremont and S. Boyd. Relaxations and randomized methods for nonconvex qcqps. EE392o Class Notes, Stanford University, 2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep convex network: A scalable architecture for speech pa ern classi cation. L Deng, D Yu, InterspeechL. Deng and D. Yu. Deep convex network: A scalable architecture for speech pa ern classi cation. In Interspeech, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Farabet, C Couprie, L Najman, Y Lecun, arXiv:1202.2160Scene parsing with multiscale feature learning, purity trees, and optimal covers. arXiv preprintC. Farabet, C. Couprie, L. Najman, and Y. LeCun. Scene parsing with multiscale feature learning, purity trees, and optimal covers. arXiv preprint arXiv:1202.2160, 2012.\",\n \"doi\": \"arXiv:1202.2160\"\n },\n {\n \"text\": \"Explaining and Harnessing Adversarial Examples. I Goodfellow, J Shlens, C Szegedy, CoRRI. Goodfellow, J. Shlens, and C. Szegedy. Explaining and Harnessing Adversarial Examples. CoRR, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Gu, L Rigazio, arXiv:1412.5068Towards deep neural network architectures robust to adversarial examples. S. Gu and L. Rigazio. Towards deep neural network architectures robust to adversarial examples. arXiv:1412.5068 [cs], 2014.\",\n \"doi\": \"arXiv:1412.5068\"\n },\n {\n \"text\": \"Learning long-range vision for autonomous o -road driving. R Hadsell, P Sermanet, J Ben, A Erkan, M Sco Er, K Kavukcuoglu, U Muller, Y Lecun, Journal of Field Robotics. 262R. Hadsell, P. Sermanet, J. Ben, A. Erkan, M. Sco er, K. Kavukcuoglu, U. Muller, and Y. LeCun. Learning long-range vision for autonomous o -road driving. Journal of Field Robotics, 26(2):120-144, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning multiple layers of representation. G E Hinton, Trends in cognitive sciences. 1110G. E. Hinton. Learning multiple layers of representation. Trends in cognitive sciences, 11(10):428-434, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Matrix analysis. corrected reprint of the 1985 original. R A Horn, C R Johnson, R. A. Horn and C. R. Johnson. Matrix analysis. corrected reprint of the 1985 original, 1990.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Approximation capabilities of multilayer feedforward networks. K Hornik, Neural networks. 42K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural networks, 4(2):251-257, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Adversarial machine learning. 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Liu, and D. Song. Delving into transferable adversarial examples and black-box a acks. arXiv preprint arXiv:1611.02770, 2016.\",\n \"doi\": \"arXiv:1611.02770\"\n },\n {\n \"text\": \"Learning word vectors for sentiment analysis. A L Maas, R E Daly, P T Pham, D Huang, A Y Ng, C Po, Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies. the 49th Annual Meeting of the Association for Computational Linguistics: Human Language TechnologiesPortland, Oregon, USAAssociation for Computational LinguisticsA. L. Maas, R. E. Daly, P. T. Pham, D. Huang, A. Y. Ng, and C. Po s. Learning word vectors for sentiment analysis. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies, pages 142-150, Portland, Oregon, USA, June 2011. Association for Computational Linguistics.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Visualizing data using t-sne. 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ACM, 2014.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":389,"cells":{"text":{"kind":"string","value":"\nBayesian inference of the fluctuating proton shape\n\n\nHeikki Mäntysaari \nDepartment of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland\n\nBjörn Schenke \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n\nChun Shen \nDepartment of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA\n\nRIKEN BNL Research Center\nBrookhaven National Laboratory\nUpton11973NYUSA\n\nWenbin Zhao \nDepartment of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA\n\nBayesian inference of the fluctuating proton shape\n\nUsing Bayesian inference, we determine probabilistic constraints on the parameters describing the fluctuating structure of protons at high energy. We employ the color glass condensate framework supplemented with a model for the spatial structure of the proton, along with experimental data from the ZEUS and H1 Collaborations on coherent and incoherent diffractive J/ψ production in e+p collisions at HERA. This data is found to constrain most model parameters well. This work sets the stage for future global analyses, including experimental data from e+p, p+p, and p+A collisions, to constrain the fluctuating structure of nucleons along with properties of the final state.\n\nI. INTRODUCTION\n\nExtracting the multi-dimensional structure of protons and nuclei is one of the main goals of future Deep Inelastic Scattering (DIS) facilities such as the Electron-Ion Collider [1,2], LHeC/FC-he [3] and EicC [4]. Exclusive processes like J/ψ production are especially powerful probes of the hadron structure at a small longitudinal momentum fraction x for two reasons. First, the exclusive nature of the process requires at lowest order at least two gluons to be exchanged with the target, rendering the cross section approximately proportional to the squared gluon distribution [5]. Additionally, only in exclusive processes is it possible to measure the total momentum transfer, which is Fourier conjugate to the impact parameter and thereby provides access to the transverse geometry.\n\nUnderstanding the proton structure, including its event-by-event fluctuations [6], is of fundamental interest. Additionally, knowledge of the spatial structure of the colliding objects in hadronic and heavy-ion collisions is required in order to construct realistic initial conditions that can be coupled to relativistic hydrodynamic simulations to describe the space-time evolution of the produced Quark-Gluon Plasma (QGP). Besides heavyion collisions, collective phenomena that can be interpreted as signatures of QGP production have been seen in small systems such as proton/deuteron/ 3 He -nucleus [7][8][9][10][11], proton-proton [12], and even photon-nucleus collisions [13], see [14][15][16][17][18][19] for reviews. In such small collision systems with a proton projectile, the detailed fluctuation spectrum of the proton geometry is particularly important to determine if QGP is indeed produced.\n\nIt is possible to constrain the proton structure from hadronic collisions by performing a statistical analysis to extract both the transport coefficients describing the matter produced in proton-lead collisions (see e.g. Refs. [20][21][22][23][24][25][26]), as well as the proton's fluctuating geometry, by comparing with the LHC data as in Ref. [27]. Another approach, which we take in this work, is to use ex-clusive DIS data from HERA, especially exclusive vector meson production [28][29][30][31][32], as a complementary input to constrain the proton shape fluctuations, as initially suggested in Ref. [33]. In the future, the Electron-Ion Collider will provide a vast amount of precise vector meson production data with proton and nuclear targets that will provide further constraints on e.g. momentum fraction x P and the nuclear mass number A dependence. Additionally, Ultra Peripheral Collisions [34,35] at RHIC [36] and at the LHC [37][38][39][40][41][42] provide access to very high energy photoproduction processes and to effects of a nuclear environment on nucleon substructure fluctuations at high energy [43,44].\n\nIn this work, we go beyond previous studies [33,45], where model parameters were constrained \"by eye\", and perform a Bayesian analysis to extract in a statistically rigorous manner the non-perturbative parameter values allowed by the HERA data, and construct initial conditions for hadronic collisions that are compatible with the experimental DIS data. This paper is organized as follows. In Section II we review the calculation of coherent and incoherent exclusive vector meson production in the dipole picture and discuss the various aspects of our model for the proton target. In Section III we explain the procedure for our Bayesian analysis. We present results in Section IV and conclude in Section V.\n\n\nII. VECTOR MESON PRODUCTION AT HIGH ENERGY\n\nIn this work we calculate vector meson production in a framework similar to the one used in Refs. [33,45] (see also Refs. [46][47][48][49][50]), and for completeness briefly review the calculation in this section.\n\nAt high energies, DIS processes can be conveniently described in the dipole picture in the rest frame of the target proton, and interaction with the target color field is described in the Color Glass Condensate (CGC) framework [51][52][53][54]. In the proton rest frame, the lifetime of a fluctuation of the incoming virtual photon into a quarkantiquark dipole is much longer than the characteristic timescale of the dipole-target interaction. Consequently, the scattering amplitude can be factorized into a convolution of photon and vector meson wave functions and the dipole-target interaction. The scattering amplitude for exclusive vector meson V production can then be written as [55,56] \nA γ * +p→V +p = 2i d 2 r ⊥ d 2 b ⊥ dz 4π e −i[b ⊥ −( 1 2 −z)r ⊥ ]·∆⊥ × [Ψ * V Ψ γ ](Q 2 , r ⊥ , z)N Ω (r ⊥ , b ⊥ , x P ).\n(1) Here r ⊥ is the transverse size of the qq dipole, b ⊥ is the impact parameter measured relative to the proton center, and Q 2 is the photon virtuality. The fraction of the large photon plus momentum carried by the quark is given by z, and ∆ ⊥ is the transverse momentum transfer. Note that at high energies we can employ the eikonal approximation and assume that the quark transverse coordinates are fixed during the propagation through the target color field.\n\nThe γ * → qq splitting is described by the virtual photon light front wave function Ψ γ , which can be computed from QED [51]. The vector meson wave function is non-perturbative, and in this work, we use the so-called Boosted Gaussian parametrization from [55], where the model parameters are constrained by experimental data on the vector meson decay width. We note that there are multiple vector meson wave functions proposed in the literature (see e.g. Refs. [57][58][59]). Different wave functions mostly affect the overall normalization of the J/ψ production cross section, and have a much smaller effect on the |t| spectra, which we are most interested here [43,55,57]. Consequently, our results will depend only weakly on the specific wave function choice (except for the parameter controlling the overall proton density).\n\nEquation (1) is a leading order result for the vector meson production in the CGC framework (note that multiple scattering effects are resummed in the dipole amplitude N Ω ). Currently, there is rapid progress in the field toward next-to-leading order (NLO) accuracy. In particular, the cross section for the production of light mesons [60] and longitudinally polarized heavy vector mesons are now available [61] at next-to-leading order, as well as the virtual photon light front wave function [62][63][64][65][66][67] and small-x evolution equations [68][69][70][71][72][73][74][75][76][77] (see also Ref. [78] where dijet production is studied at next-toleading order). However, the NLO calculations are not yet at the level where they can be consistently used in phenomenological applications (in particular the cross section for transversely polarized heavy vector meson production is still missing). As the purpose of this work is to demonstrate the potential of Bayesian analyses to systematically extract non-perturbative parameters describing the proton event-by-event fluctuating structure, we do not expect the NLO contributions to have a large effect on the results we obtain using our leading order setup.\n\nThe coherent cross section, corresponding to the process where the target proton remains in the same quantum state, can be obtained by averaging over the target color charge configurations Ω at the amplitude level [79]:\ndσ γ * +p→V +p d|t| = 1 16π A γ * +p→V +p Ω 2 (2)\nSubtracting the coherent contribution from the total diffractive vector meson production cross section we obtain the cross section for incoherent vector meson production, in which case the final state of the target is different from the initial state [6,80,81]. Experimentally, this corresponds to processes where the target proton (or nucleus) dissociates, but the rapidity gap between the produced vector meson and the target remnants remains. The incoherent cross section can then be written as a variance\ndσ γ * +p→V +p * d|t| = 1 16π A γ * +p→V +p 2 Ω − A γ * +p→V +p Ω 2 .(3)\nDependence on the small-x structure of the target proton is included in the dipole amplitude N Ω (r ⊥ , b ⊥ , x P ), which, for a given target color charge configuration Ω, can be written as\nN Ω (r ⊥ , b ⊥ , x P ) = 1− 1 N c tr V b ⊥ + r ⊥ 2 V † b ⊥ − r ⊥ 2 .\n(4) Here V (x ⊥ ) represents a Wilson line, which describes the color rotation of a quark state when it propagates through the target field (given the target color field configuration Ω) at transverse coordinate x ⊥ . We suppressed the dependence of V on\nx P ≈ M 2 V + Q 2 W 2 + Q 2 ,(5)\nwhich is the fraction of the target longitudinal momentum transferred to the meson with mass M V in the frame where the target has a large momentum. We neglect the t dependence of x P in this work. The photon-proton center-of-mass energy is denoted by W . The Wilson lines are obtained in the same way as done in the IP-Glasma calculation [82] used e.g. in Refs. [33,45,[83][84][85][86]. The color charges ρ a are first determined using the McLerran-Venugopalan [87] model, assuming that color charges are local Gaussian random variables with expectation value zero and variance\ng 2 ρ a (x − , x ⊥ )ρ b (y − , y ⊥ ) = g 4 λ A (x − )δ ab × δ (2) (x ⊥ − y ⊥ )δ(x − − y − ) . (6)\nHere the color charge density is µ 2 = dx − λ A (x − ), and it is related to the local saturation scale Q s (x ⊥ ) determined from the IPsat parametrization [88] fitted to HERA data [89,90]. In our Bayesian analysis, the ratio\nQ s (x ⊥ ) g 2 µ ,(7)\nis a free parameter, controlling the overall proton density (see also Ref. [91] for a detailed study of this ratio). The Wilson lines V (x ⊥ ) can be obtained by solving the Yang-Mills equations, and the result reads\nV (x ⊥ ) = P − exp −ig ∞ −∞ dz − ρ a (x − , x ⊥ )t a ∇ 2 − m 2 , (8) where P − represents path ordering in the x − direction.\nHere, we introduced the infrared regulator m, which is needed to avoid the emergence of unphysical Coulomb tails, and will be another free parameter in the Bayesian analysis.\n\nIn the IPsat parametrization the saturation scale\nQ 2 s (b ⊥ ) is directly proportional to the local density T p (b ⊥ ).\nWe introduce an event-by-event fluctuating density by writing the density profile following Refs. [33,45] as:\nT p (b ⊥ ) = 1 N q Nq i=1 p i T q (b ⊥ − b ⊥,i ),(9)\nwhere\nT q (b ⊥ ) = 1 2πB q e −b 2 ⊥ /(2Bq) ,(10)\nand the coefficient p i allows for different normalizations for individual hot spots, to be discussed below. Our prescription corresponds to having N q hot spots with hot spot width B q (note that the hot spot transverse root mean square (RMS) radius in 2B q ). The hot spot positions b ⊥,i are sampled from a two-dimensional Gaussian distribution whose width is denoted by B qc , and the center-of-mass is shifted to the origin in the end. As discussed in Ref. [92,93], repulsive short-range correlations between the hot spots may explain the hollowness effect [94][95][96][97] and negative correlation between the v 2 and v 3 flow harmonics observed in highest multiplicity proton-proton collisions [98,99]. In order to study if exclusive vector meson production in DIS can be used to probe such repulsive correlations, we also introduce an additional model parameter d q,Min which controls the minimum three-dimensional distance required between any two hot spots. 1 We checked that for a large number of hot spots N q = 10 in a typical nucleon of size B qc = 4.2 GeV −2 , the model parameter d q,Min remains effective more than 90% for d q,Min ≤ 0.4 fm, meaning that in 10% of the sampled configurations the distance requirement cannot be fulfilled.\n\nFinally, we include saturation scale fluctuations by allowing the local density of each hot spot to fluctuate independently, following again Refs. [33,45] (see also Ref. [101]). These fluctuations are implemented by sampling the coefficients p i in Eq. (9) from the log-normal distribution\nP (ln p i ) = 1 √ 2πσ exp − ln 2 p i 2σ 2 .(11)\nThe sampled p i are at the end normalized by the expectation value of the distribution E[p i ] = e σ 2 /2 in order to keep the average density unmodified. The magnitude of density fluctuations is controlled by the parameter σ.\n\n\nIII. BAYESIAN ANALYSIS SETUP\n\nBayesian Inference is a general and systematic method to constrain the probability distribution of model parameters θ by comparing model calculations y(θ) with experimental measurements y exp [102]. Bayes' theorem provides the posterior distribution of model parameters as\nP(θ|y exp ) ∝ P(y exp |θ)P(θ).(12)\nHere P(y exp |θ) is the likelihood for model results with parameter θ to agree with the experimental data. We choose a multivariate normal distribution for the logarithm of the likelihood with ∆y(θ) = y(θ) − y exp [103],\nln[P(y exp |θ)] = − 1 2 ∆y(θ) T Σ −1 ∆y(θ) − 1 2 ln[(2π) n det Σ].(13)\nHere n is the number of experimental data points and Σ ≡ Σ exp + Σ model is the n × n covariance matrix, which encodes experimental and model uncertainties. In the current analysis, we assume no correlation among experimental errors of the n observables {σ i }. So the covariance matrix for experimental uncertainty takes a diagonal form,\nΣ exp = diag(σ 2 1 , · · · , σ 2 n ).(14)\nOur model uncertainties Σ model are estimated using the covariance matrix from the trained Gaussian Process (GP) Emulators [104].\n\nWe employ GP emulators [103] for our model and couple them with the Monte-Carlo Markov Chain (MCMC) method to efficiently explore the model parameter space [105,106]. The HERA measurements can be represented by five Principal Components (PC) with a residual variance of less than 0.01%, meaning that 99.99% of the variation of all studied observables within the prior parameter range are captured by the five principal components. Our GP emulators are trained to fit these five Table. I. We treat the parameter N q as a continuous real number. The fractional part of N q is treated as a probability to sample either N q or N q partons inside protons. The same approach was recently used in Ref. [107]. The experimental data included in the Bayesian analysis is the H1 data on coherent and incoherent J/ψ production cross section measured at W = 75 GeV [28]. The incoherent data is included in the |t| range 0 < |t| < 2.5 GeV 2 . We note that there is incoherent data at higher |t| also (studied in a similar context in Ref. [46]), but the highest |t| points are not included in our analysis for two reasons. First, as we determine Wilson lines at fixed x P , as discussed in Sec. II, and do not include the full x P evolution, we neglect |t| dependence in x P (see Eq. (5)). Additionally, at large |t| other effects such as DGLAP evolution [108][109][110][111] may become important.\n\n\nIV. RESULTS\n\nFor two distinct scenarios, the first with N q fixed to 3, the second with N q a free parameter, the posterior distribution of model parameters is shown in Fig. 1. Particularly for the case of N q = 3, most of the model parameters are tightly constrained by the H1 data included in the Bayesian analysis. This is due to the fact that different regions of the dataset are sensitive to different model parameters.\n\nFirst, the infrared regulator m suppresses longdistance Coulomb tails, and as such, it controls the shape of the proton at large distances. This part of the proton geometry is probed by coherent diffraction at low |t| 0.2GeV 2 [45]: impact parameter and momentum transfers are Fourier conjugates, and consequently the low-|t| region is sensitive to large distances and vice versa. On the other hand, the actual proton size controlled mostly by B qc determines the overall slope of the coherent spectrum. The hot spot size B q then determines the slope of the incoherent cross section in the |t| 1 GeV 2 region: as shown in Ref. [112] the slope of the incoherent spectra at high |t| is given by the size of the smallest fluctuating constituent.\n\nThe overall normalization is determined by the Q s /(g 2 µ) parameter and the magnitude of the cross sections constrains that, however, it can not be determined very precisely from our analysis as it is strongly correlated with many other parameters, particularly for the case that N q is a free parameter. Here we note that there is some model uncertainty related to the nonperturbative vector meson wave functions, and different phenomenological parametrizations can result in cross sections that differ by ∼ 20%, see e.g. [43,55,57]. In phenomenological analyses, the so-called skewness correction [55] is sometimes included, which can also have a numerically significant (up to ∼ 40%) effect on the cross section. However, as there are also other model uncertainties related to the overall normalization and the applicability of the skewness correction in our setup is not rigorously justified, it is not included in this work. The small real part correction discussed e.g. in Ref. [55] is also neglected.\n\nOne can see that when leaving N q variable, its value can not be constrained in our analysis, except that N q ≥ 2 is required in order to get geometry fluctuations that are necessary to describe the incoherent HERA data. At first one would expect the configurations with large N q to be so smooth that event-by-event fluctuations would not be enough to result in a large enough incoherent cross section. However, we note that there is a strong positive correlation between N q and σ, which means that large N q goes along with large hot spot density fluctuations. In this situation one can not really interpret N q as the number of hot spots, but one has to consider effective hot spots, that are generated dynamically from the sum of the N q constituents that are each strongly fluctuating in magnitude. This effective hot spot number will generally be smaller than N q . For a quantitative analysis a hot spot finding prescription, similar to jet clustering algorithms, would be needed. Additional constraints for σ originate from the incoherent cross section at small |t|, which is sensitive to the fluctuations at long-distance scales, i.e., overall density fluctuations [33,45].\n\nThe minimum distance required between the hot spots in three dimensions, d q,Min , can not be constrained at all in our analysis. This parameter is also only very weakly correlated with the other parameters, emphasizing its limited effect on the results. This means that the J/ψ production data allows, but does not require, repulsive short-range correlations used e.g. in Refs. [92,93]. We note that for large N q it is not always possible to fulfill the minimum distance requirement, reducing the effective value of d q,Min that is actually employed.\n\nLet us next study correlations between the model parameters in more detail. First, we observe that there is a clear positive correlation between the infrared regulator m and the hot spot size B q . This can be understood, as increasing m suppresses color fields at large distances and as such results in smaller hot spots. Interestingly, we find no clear correlation between the overall proton size parameter B qc and the infrared regulator m. In fact, the proton size B qc shows no clear correlation with any of the other model parameters. The infrared regulator m is strongly anti-correlated with Q s /(g 2 µ). This is again easy to understand: large m results in a reduction of the normalization of the cross section, which has to be compensated by using a larger color charge density which requires smaller Q s /(g 2 µ).\n\nWe also find a strong negative correlation between the number of hot spots N q and the proton density Q s /(g 2 µ). This correlation can be understood by considering the effect of these parameters on the incoherent cross section. When the number of hot spots with the same size increases, the incoherent slope is not significantly affected but the normalization goes down as there are smaller fluctuations in the scattering amplitude due to the smoother proton profile. This would result in the incoherent cross section being underestimated, which has to be compensated by smaller Q s /(g 2 µ).\n\nThe somewhat surprising negative correlation between the hot spot size B q and the proton density Q s /(g 2 µ) can also be understood. First, we note that large hot spots (large B q ) require a larger infrared regulator m which effectively makes the hot spots smaller (notice a positive correlation between B q and m). Then, to compensate for the effect of the larger infrared regulator on the magnitude of the cross section, a smaller Q s /(g 2 µ) is needed.\n\nThe posterior distribution of parameters B qc and B q , that determine the proton size, which can be quantified e.g. by the two dimensional RMS radius, r rms = 2(B qc + B q ), are shown in the second and third diagonal panels of Fig. 1. We find that they are sharply peaked, in particular for the case of fixed N q = 3. The extracted proton radius from this Bayesian analysis is r rms = 0.591 +0. 130 −0.071 fm (0.610 +0.065 −0.068 fm) for variable N q (N q ≡ 3) with uncertainty estimates in 90% credible intervals.\n\nWe further included the possibility of fluctuating values of B q and N q , and studied the dependence of the observables on the respective variances. We found that the considered experimental data could not constrain these parameters and decided not to include them in the presented analysis.\n\nNext, we demonstrate explicitly that sampling parameter values from the posterior distribution lead to a good description of the HERA data. The main result of the Bayesian analysis is the posterior distribution shown in Fig. 1, but it is also possible to find the so-called Maximum a Posteriori (MAP) parameter set, which is the mode of the posterior distribution. Because our prior parameter distributions are uniform, the MAP parameters shown in Table I maximize the likelihood function and provide the best fit to the experimental data. We note that statistically the expectation value for any observable is not obtained using the MAP parameters. Instead, one should calculate observables using averages over parameter samples obtained from the posterior distribution.\n\nComparison to the HERA coherent and incoherent J/ψ production data measured at W = 75 GeV [28] and used in the Bayesian analysis to constrain the model parameters are shown in panel (a) of Fig. 2. The spectra calculated by averaging the results computed using different parametrizations sampled from the posterior distribution indeed provide an excellent description of the data. We also show the statistical uncertainty, obtained by calculating the one standard deviation interval shown as red (coherent) and blue (incoherent) bands.\n\nNext, we study compatibility with experimental data not included in the Bayesian analysis. We do not include full small-x evolution e.g. by means of the JIMWLK [113][114][115][116][117][118][119][120] equation in this work. Consequently, the only center-of-mass energy W (or momentum fraction x P ) dependence comes from the x P dependence of the saturation scale Q s determined from the IPsat fit. This almost only affects the overall normalization of the calculated spectra, and misses important physical effects such as the growth of the proton with decreasing x. Consequently, we do not expect that within our setup we can describe J/ψ production at different center-of-mass energies simultaneously. Nevertheless, the geometry evolution between W = 75 GeV and W = 100 GeV should be weak enough to make predictions for the higher center-of-mass energy.\n\nThe calculated J/ψ spectra at W = 100 GeV are shown in panel (b) of Fig. 2 and compared with the data from H1 and ZEUS collaborations [29,31,32,121]. Again, we show the model prediction as the average over many samples of the posterior distribution and provide one standard deviation bands. The results are very similar to the W = 75 GeV case studied above, and the HERA data is well described (except the coherent cross section at low |t|, and high |t| in the case of the ZEUS measurement), and variation between the different parametrizations sampled from the posterior distribution is small. Similarity to the W = 75 GeV case is not surprising, as we use exactly the same fluctuating geometry, with the only difference being slightly larger Q s values, extracted from the IPsat parametrization.\n\n\nV. CONCLUSIONS\n\nWe have performed a statistically rigorous Bayesian analysis to extract posterior likelihood distributions for the non-perturbative parameters describing the event-byevent fluctuating proton geometry as constrained by exclusive J/ψ production data from HERA. We presented a comparison of the |t|-dependent coherent and incoherent cross sections, obtained from an average over many parameter samples from the posterior distribution, to the experimental data.\n\nGenerally, model parametrizations sampled from the determined posterior distribution can be used to systematically take into account uncertainties in the proton geometry as constrained by the HERA DIS data when cal- FIG. 2. Coherent and incoherent J/ψ photoproduction cross sections at W = 75 GeV (a) and W = 100 GeV (b), calculated by averaging over many parameter sets sampled from the posterior distribution, which was determined using the W = 75 GeV HERA data [28]. We compare the case with variable Nq to that with fixed Nq = 3, and to experimental data at W = 75 GeV [28] and W = 100 GeV [29][30][31][32], respectively. The bands show the one standard deviation uncertainty.\n\nculating any other observable that depends on the proton geometry, such as flow observables in high-multiplicity proton-proton and proton-nucleus collisions. To enable such studies, we provide 1,000 parametrizations sampled from the determined posterior distributions in the supplemental material.\n\nBecause the model parameters are sensitive to different aspects of the coherent and incoherent vector meson production spectra, most of them are well constrained by the Bayesian analysis. The only exceptions are the minimum distance between the hot spots (repulsive short-range correlations), and the number of hot spots N q , which cannot be well constrained by the considered HERA data. However, we note that although the analysis suggests that large N q is compatible with the HERA data, for N q 5 one should no longer simply interpret N q as the number of actual hot spots. In this regime, a good description of the data requires large fluctuations of individual hot spots' densities, and the sampled 'hot spots' can also overlap significantly. Thus, the effective number of hot spots is significantly smaller than the parameter N q might imply.\n\nThe HERA data used in this work probes the proton structure at x ∼ 10 −3 . The energy (Bjorken-x) dependence can be included in terms of JIMWLK evolution as e.g. in Refs. [86,122]. In the future we plan to extend our framework by including the full JIMWLK evolution and vector meson production data at different center-ofmass energies from HERA [28,121,123] and from the ultra peripheral proton-lead collisions measured at the LHC [39,40,124,125], allowing us to extract also the Bjorken-x dependence of the fluctuating proton geometry.\n\nIn addition to constraining the energy dependence, performing global analyses including both exclusive vector meson production data from HERA and flow harmonics from the proton-proton and proton-lead collisions measured at the LHC (including a model calculation along the lines of [126]) would allow for a powerful global analysis of the fluctuating nucleon substructure and properties of the final state. More differential DIS measurements from the future EIC such as dijet [85] or lepton-meson angular correlations [83] can also provide further constraints and can in principle be included in our framework in a straightforward manner.\n\nThe numerical framework for our physics models and the Bayesian analysis package are publicly available on Github [127,128]. Our Bayesian analysis code is developed based on the open-source numerical package by the Duke group [23]. To help visualize how observables depend on the model parameters, we provide an interactive web page with the trained GP emulators [129], where one can also find the posterior samples. expressed therein lies entirely with the authors. This research was done using resources provided by the Open Science Grid (OSG) [130,131], which is supported by the National Science Foundation award #2030508.\n\nFIG. 1 .\n1Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.\n\n\nACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views\n\nTABLE I .\nISummary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.Parameter Description \nPrior range MAP (variable Nq) MAP (Nq ≡ 3) \nm [GeV] \nInfrared regulator \n[0.05, 2] \n0.506 +1.12 \n\n−0.356 \n\n0.246 +0.162 \n\n−0.103 \n\nBqc [GeV −2 ] Proton size \n[1, 10] \n4.02 +1.73 \n\n−0.728 \n\n4.45 +0.801 \n\n−0.803 \n\nBq [GeV −2 ] Hot spot size \n[0.1, 3] \n0.474 +0.434 \n\n−0.286 \n\n0.346 +0.282 \n\n−0.202 \n\nσ \nMagnitude of Qs fluctuations \n[0, 1.5] \n0.833 +0.194 \n\n−0.441 \n\n0.563 +0.143 \n\n−0.141 \n\nQs/(g 2 µ) \nRatio of color charge density and saturation scale [0.2, 1.5] \n0.598 +0.230 \n\n−0.264 \n\n0.747 +0.0704 \n\n−0.0930 \n\ndq,Min [fm] Minimum 3D distance between hot spots \n[0, 0.5] \n0.257 +0.221 \n\n−0.231 \n\n0.254 +0.222 \n\n−0.229 \n\nNq \nNumber of hot spots \n[1, 10] \n6.79 +2.93 \n\n−4.83 \n\n3 \n\nPCs with 1,000 training simulations in the model param-\neter space. 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It uses the open-source code packages IP-Glasma https://github.com/schenke/ipglasma and the program to compute exclusive J/ψ production cross-section https: //github.com/hejajama/subnucleondiffraction.\n\nThe Open Science Grid. R Pordes, 10.1088/1742-6596/78/1/012057J. Phys. Conf. Ser. 7812057R. Pordes et. al., The Open Science Grid, J. Phys. Conf. Ser. 78 (2007) 012057.\n\nThe pilot way to Grid resources using glideinWMS. I Sfiligoi, D C Bradley, B Holzman, P Mhashilkar, S Padhi, F Wurthwrin, 10.1109/CSIE.2009.950WRI World Congress. 2428I. Sfiligoi, D. C. Bradley, B. Holzman, P. Mhashilkar, S. Padhi and F. Wurthwrin, The pilot way to Grid resources using glideinWMS, WRI World Congress 2 (2009) 428.\n"},"corpus_id":{"kind":"number","value":246608220,"string":"246,608,220"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2202.01998\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1016/j.physletb.2022.137348\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2202.01998v1.pdf\"\n ],\n \"pdf_sha\": \"499b51a3bacec895f368bebb595398fffd75dc20\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Bayesian inference of the fluctuating proton shape"},"authors":{"kind":"list like","value":["Heikki Mäntysaari \nDepartment of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland","Björn Schenke \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA","Chun Shen \nDepartment of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA\n\nRIKEN BNL Research Center\nBrookhaven National Laboratory\nUpton11973NYUSA","Wenbin Zhao \nDepartment of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA","Heikki Mäntysaari \nDepartment of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland","Björn Schenke \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA","Chun Shen \nDepartment of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA\n\nRIKEN BNL Research Center\nBrookhaven National Laboratory\nUpton11973NYUSA","Wenbin Zhao \nDepartment of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA"],"string":"[\n \"Heikki Mäntysaari \\nDepartment of Physics\\nUniversity of Jyväskylä\\nUniversity of Jyväskylä\\nP.O. Box 3540014Finland\\n\\nHelsinki Institute of Physics\\nUniversity of Helsinki\\nP.O. Box 6400014Finland\",\n \"Björn Schenke \\nPhysics Department\\nBrookhaven National Laboratory\\nUpton11973NYUSA\",\n \"Chun Shen \\nDepartment of Physics and Astronomy\\nWayne State University\\n48201DetroitMichiganUSA\\n\\nRIKEN BNL Research Center\\nBrookhaven National Laboratory\\nUpton11973NYUSA\",\n \"Wenbin Zhao \\nDepartment of Physics and Astronomy\\nWayne State University\\n48201DetroitMichiganUSA\",\n \"Heikki Mäntysaari \\nDepartment of Physics\\nUniversity of Jyväskylä\\nUniversity of Jyväskylä\\nP.O. Box 3540014Finland\\n\\nHelsinki Institute of Physics\\nUniversity of Helsinki\\nP.O. Box 6400014Finland\",\n \"Björn Schenke \\nPhysics Department\\nBrookhaven National Laboratory\\nUpton11973NYUSA\",\n \"Chun Shen \\nDepartment of Physics and Astronomy\\nWayne State University\\n48201DetroitMichiganUSA\\n\\nRIKEN BNL Research Center\\nBrookhaven National Laboratory\\nUpton11973NYUSA\",\n \"Wenbin Zhao \\nDepartment of Physics and Astronomy\\nWayne State University\\n48201DetroitMichiganUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Using Bayesian inference, we determine probabilistic constraints on the parameters describing the fluctuating structure of protons at high energy. We employ the color glass condensate framework supplemented with a model for the spatial structure of the proton, along with experimental data from the ZEUS and H1 Collaborations on coherent and incoherent diffractive J/ψ production in e+p collisions at HERA. This data is found to constrain most model parameters well. This work sets the stage for future global analyses, including experimental data from e+p, p+p, and p+A collisions, to constrain the fluctuating structure of nucleons along with properties of the final state."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":""},{"title":"I. INTRODUCTION","content":"Extracting the multi-dimensional structure of protons and nuclei is one of the main goals of future Deep Inelastic Scattering (DIS) facilities such as the Electron-Ion Collider [1,2], LHeC/FC-he [3] and EicC [4]. Exclusive processes like J/ψ production are especially powerful probes of the hadron structure at a small longitudinal momentum fraction x for two reasons. First, the exclusive nature of the process requires at lowest order at least two gluons to be exchanged with the target, rendering the cross section approximately proportional to the squared gluon distribution [5]. Additionally, only in exclusive processes is it possible to measure the total momentum transfer, which is Fourier conjugate to the impact parameter and thereby provides access to the transverse geometry.\nExtracting the multi-dimensional structure of protons and nuclei is one of the main goals of future Deep Inelastic Scattering (DIS) facilities such as the Electron-Ion Collider [1,2], LHeC/FC-he [3] and EicC [4]. Exclusive processes like J/ψ production are especially powerful probes of the hadron structure at a small longitudinal momentum fraction x for two reasons. First, the exclusive nature of the process requires at lowest order at least two gluons to be exchanged with the target, rendering the cross section approximately proportional to the squared gluon distribution [5]. Additionally, only in exclusive processes is it possible to measure the total momentum transfer, which is Fourier conjugate to the impact parameter and thereby provides access to the transverse geometry.\nUnderstanding the proton structure, including its event-by-event fluctuations [6], is of fundamental interest. Additionally, knowledge of the spatial structure of the colliding objects in hadronic and heavy-ion collisions is required in order to construct realistic initial conditions that can be coupled to relativistic hydrodynamic simulations to describe the space-time evolution of the produced Quark-Gluon Plasma (QGP). Besides heavyion collisions, collective phenomena that can be interpreted as signatures of QGP production have been seen in small systems such as proton/deuteron/ 3 He -nucleus [7][8][9][10][11], proton-proton [12], and even photon-nucleus collisions [13], see [14][15][16][17][18][19] for reviews. In such small collision systems with a proton projectile, the detailed fluctuation spectrum of the proton geometry is particularly important to determine if QGP is indeed produced.\nUnderstanding the proton structure, including its event-by-event fluctuations [6], is of fundamental interest. Additionally, knowledge of the spatial structure of the colliding objects in hadronic and heavy-ion collisions is required in order to construct realistic initial conditions that can be coupled to relativistic hydrodynamic simulations to describe the space-time evolution of the produced Quark-Gluon Plasma (QGP). Besides heavyion collisions, collective phenomena that can be interpreted as signatures of QGP production have been seen in small systems such as proton/deuteron/ 3 He -nucleus [7][8][9][10][11], proton-proton [12], and even photon-nucleus collisions [13], see [14][15][16][17][18][19] for reviews. In such small collision systems with a proton projectile, the detailed fluctuation spectrum of the proton geometry is particularly important to determine if QGP is indeed produced.\nIt is possible to constrain the proton structure from hadronic collisions by performing a statistical analysis to extract both the transport coefficients describing the matter produced in proton-lead collisions (see e.g. Refs. [20][21][22][23][24][25][26]), as well as the proton's fluctuating geometry, by comparing with the LHC data as in Ref. [27]. Another approach, which we take in this work, is to use ex-clusive DIS data from HERA, especially exclusive vector meson production [28][29][30][31][32], as a complementary input to constrain the proton shape fluctuations, as initially suggested in Ref. [33]. In the future, the Electron-Ion Collider will provide a vast amount of precise vector meson production data with proton and nuclear targets that will provide further constraints on e.g. momentum fraction x P and the nuclear mass number A dependence. Additionally, Ultra Peripheral Collisions [34,35] at RHIC [36] and at the LHC [37][38][39][40][41][42] provide access to very high energy photoproduction processes and to effects of a nuclear environment on nucleon substructure fluctuations at high energy [43,44].\nIt is possible to constrain the proton structure from hadronic collisions by performing a statistical analysis to extract both the transport coefficients describing the matter produced in proton-lead collisions (see e.g. Refs. [20][21][22][23][24][25][26]), as well as the proton's fluctuating geometry, by comparing with the LHC data as in Ref. [27]. Another approach, which we take in this work, is to use ex-clusive DIS data from HERA, especially exclusive vector meson production [28][29][30][31][32], as a complementary input to constrain the proton shape fluctuations, as initially suggested in Ref. [33]. In the future, the Electron-Ion Collider will provide a vast amount of precise vector meson production data with proton and nuclear targets that will provide further constraints on e.g. momentum fraction x P and the nuclear mass number A dependence. Additionally, Ultra Peripheral Collisions [34,35] at RHIC [36] and at the LHC [37][38][39][40][41][42] provide access to very high energy photoproduction processes and to effects of a nuclear environment on nucleon substructure fluctuations at high energy [43,44].\nIn this work, we go beyond previous studies [33,45], where model parameters were constrained \"by eye\", and perform a Bayesian analysis to extract in a statistically rigorous manner the non-perturbative parameter values allowed by the HERA data, and construct initial conditions for hadronic collisions that are compatible with the experimental DIS data. This paper is organized as follows. In Section II we review the calculation of coherent and incoherent exclusive vector meson production in the dipole picture and discuss the various aspects of our model for the proton target. In Section III we explain the procedure for our Bayesian analysis. We present results in Section IV and conclude in Section V.\nIn this work, we go beyond previous studies [33,45], where model parameters were constrained \"by eye\", and perform a Bayesian analysis to extract in a statistically rigorous manner the non-perturbative parameter values allowed by the HERA data, and construct initial conditions for hadronic collisions that are compatible with the experimental DIS data. This paper is organized as follows. In Section II we review the calculation of coherent and incoherent exclusive vector meson production in the dipole picture and discuss the various aspects of our model for the proton target. In Section III we explain the procedure for our Bayesian analysis. We present results in Section IV and conclude in Section V."},{"title":"II. VECTOR MESON PRODUCTION AT HIGH ENERGY","content":""},{"title":"II. VECTOR MESON PRODUCTION AT HIGH ENERGY","content":"In this work we calculate vector meson production in a framework similar to the one used in Refs. [33,45] (see also Refs. [46][47][48][49][50]), and for completeness briefly review the calculation in this section.\nIn this work we calculate vector meson production in a framework similar to the one used in Refs. [33,45] (see also Refs. [46][47][48][49][50]), and for completeness briefly review the calculation in this section.\nAt high energies, DIS processes can be conveniently described in the dipole picture in the rest frame of the target proton, and interaction with the target color field is described in the Color Glass Condensate (CGC) framework [51][52][53][54]. In the proton rest frame, the lifetime of a fluctuation of the incoming virtual photon into a quarkantiquark dipole is much longer than the characteristic timescale of the dipole-target interaction. Consequently, the scattering amplitude can be factorized into a convolution of photon and vector meson wave functions and the dipole-target interaction. The scattering amplitude for exclusive vector meson V production can then be written as [55,56]\nAt high energies, DIS processes can be conveniently described in the dipole picture in the rest frame of the target proton, and interaction with the target color field is described in the Color Glass Condensate (CGC) framework [51][52][53][54]. In the proton rest frame, the lifetime of a fluctuation of the incoming virtual photon into a quarkantiquark dipole is much longer than the characteristic timescale of the dipole-target interaction. Consequently, the scattering amplitude can be factorized into a convolution of photon and vector meson wave functions and the dipole-target interaction. The scattering amplitude for exclusive vector meson V production can then be written as [55,56]\n(1) Here r ⊥ is the transverse size of the qq dipole, b ⊥ is the impact parameter measured relative to the proton center, and Q 2 is the photon virtuality. The fraction of the large photon plus momentum carried by the quark is given by z, and ∆ ⊥ is the transverse momentum transfer. Note that at high energies we can employ the eikonal approximation and assume that the quark transverse coordinates are fixed during the propagation through the target color field.\n(1) Here r ⊥ is the transverse size of the qq dipole, b ⊥ is the impact parameter measured relative to the proton center, and Q 2 is the photon virtuality. The fraction of the large photon plus momentum carried by the quark is given by z, and ∆ ⊥ is the transverse momentum transfer. Note that at high energies we can employ the eikonal approximation and assume that the quark transverse coordinates are fixed during the propagation through the target color field.\nThe γ * → qq splitting is described by the virtual photon light front wave function Ψ γ , which can be computed from QED [51]. The vector meson wave function is non-perturbative, and in this work, we use the so-called Boosted Gaussian parametrization from [55], where the model parameters are constrained by experimental data on the vector meson decay width. We note that there are multiple vector meson wave functions proposed in the literature (see e.g. Refs. [57][58][59]). Different wave functions mostly affect the overall normalization of the J/ψ production cross section, and have a much smaller effect on the |t| spectra, which we are most interested here [43,55,57]. Consequently, our results will depend only weakly on the specific wave function choice (except for the parameter controlling the overall proton density).\nThe γ * → qq splitting is described by the virtual photon light front wave function Ψ γ , which can be computed from QED [51]. The vector meson wave function is non-perturbative, and in this work, we use the so-called Boosted Gaussian parametrization from [55], where the model parameters are constrained by experimental data on the vector meson decay width. We note that there are multiple vector meson wave functions proposed in the literature (see e.g. Refs. [57][58][59]). Different wave functions mostly affect the overall normalization of the J/ψ production cross section, and have a much smaller effect on the |t| spectra, which we are most interested here [43,55,57]. Consequently, our results will depend only weakly on the specific wave function choice (except for the parameter controlling the overall proton density).\nEquation (1) is a leading order result for the vector meson production in the CGC framework (note that multiple scattering effects are resummed in the dipole amplitude N Ω ). Currently, there is rapid progress in the field toward next-to-leading order (NLO) accuracy. In particular, the cross section for the production of light mesons [60] and longitudinally polarized heavy vector mesons are now available [61] at next-to-leading order, as well as the virtual photon light front wave function [62][63][64][65][66][67] and small-x evolution equations [68][69][70][71][72][73][74][75][76][77] (see also Ref. [78] where dijet production is studied at next-toleading order). However, the NLO calculations are not yet at the level where they can be consistently used in phenomenological applications (in particular the cross section for transversely polarized heavy vector meson production is still missing). As the purpose of this work is to demonstrate the potential of Bayesian analyses to systematically extract non-perturbative parameters describing the proton event-by-event fluctuating structure, we do not expect the NLO contributions to have a large effect on the results we obtain using our leading order setup.\nEquation (1) is a leading order result for the vector meson production in the CGC framework (note that multiple scattering effects are resummed in the dipole amplitude N Ω ). Currently, there is rapid progress in the field toward next-to-leading order (NLO) accuracy. In particular, the cross section for the production of light mesons [60] and longitudinally polarized heavy vector mesons are now available [61] at next-to-leading order, as well as the virtual photon light front wave function [62][63][64][65][66][67] and small-x evolution equations [68][69][70][71][72][73][74][75][76][77] (see also Ref. [78] where dijet production is studied at next-toleading order). However, the NLO calculations are not yet at the level where they can be consistently used in phenomenological applications (in particular the cross section for transversely polarized heavy vector meson production is still missing). As the purpose of this work is to demonstrate the potential of Bayesian analyses to systematically extract non-perturbative parameters describing the proton event-by-event fluctuating structure, we do not expect the NLO contributions to have a large effect on the results we obtain using our leading order setup.\nThe coherent cross section, corresponding to the process where the target proton remains in the same quantum state, can be obtained by averaging over the target color charge configurations Ω at the amplitude level [79]:\nThe coherent cross section, corresponding to the process where the target proton remains in the same quantum state, can be obtained by averaging over the target color charge configurations Ω at the amplitude level [79]:\nSubtracting the coherent contribution from the total diffractive vector meson production cross section we obtain the cross section for incoherent vector meson production, in which case the final state of the target is different from the initial state [6,80,81]. Experimentally, this corresponds to processes where the target proton (or nucleus) dissociates, but the rapidity gap between the produced vector meson and the target remnants remains. The incoherent cross section can then be written as a variance\nSubtracting the coherent contribution from the total diffractive vector meson production cross section we obtain the cross section for incoherent vector meson production, in which case the final state of the target is different from the initial state [6,80,81]. Experimentally, this corresponds to processes where the target proton (or nucleus) dissociates, but the rapidity gap between the produced vector meson and the target remnants remains. The incoherent cross section can then be written as a variance\nDependence on the small-x structure of the target proton is included in the dipole amplitude N Ω (r ⊥ , b ⊥ , x P ), which, for a given target color charge configuration Ω, can be written as\nDependence on the small-x structure of the target proton is included in the dipole amplitude N Ω (r ⊥ , b ⊥ , x P ), which, for a given target color charge configuration Ω, can be written as\n(4) Here V (x ⊥ ) represents a Wilson line, which describes the color rotation of a quark state when it propagates through the target field (given the target color field configuration Ω) at transverse coordinate x ⊥ . We suppressed the dependence of V on\n(4) Here V (x ⊥ ) represents a Wilson line, which describes the color rotation of a quark state when it propagates through the target field (given the target color field configuration Ω) at transverse coordinate x ⊥ . We suppressed the dependence of V on\nwhich is the fraction of the target longitudinal momentum transferred to the meson with mass M V in the frame where the target has a large momentum. We neglect the t dependence of x P in this work. The photon-proton center-of-mass energy is denoted by W . The Wilson lines are obtained in the same way as done in the IP-Glasma calculation [82] used e.g. in Refs. [33,45,[83][84][85][86]. The color charges ρ a are first determined using the McLerran-Venugopalan [87] model, assuming that color charges are local Gaussian random variables with expectation value zero and variance\nwhich is the fraction of the target longitudinal momentum transferred to the meson with mass M V in the frame where the target has a large momentum. We neglect the t dependence of x P in this work. The photon-proton center-of-mass energy is denoted by W . The Wilson lines are obtained in the same way as done in the IP-Glasma calculation [82] used e.g. in Refs. [33,45,[83][84][85][86]. The color charges ρ a are first determined using the McLerran-Venugopalan [87] model, assuming that color charges are local Gaussian random variables with expectation value zero and variance\nHere the color charge density is µ 2 = dx − λ A (x − ), and it is related to the local saturation scale Q s (x ⊥ ) determined from the IPsat parametrization [88] fitted to HERA data [89,90]. In our Bayesian analysis, the ratio\nHere the color charge density is µ 2 = dx − λ A (x − ), and it is related to the local saturation scale Q s (x ⊥ ) determined from the IPsat parametrization [88] fitted to HERA data [89,90]. In our Bayesian analysis, the ratio\nis a free parameter, controlling the overall proton density (see also Ref. [91] for a detailed study of this ratio). The Wilson lines V (x ⊥ ) can be obtained by solving the Yang-Mills equations, and the result reads\nis a free parameter, controlling the overall proton density (see also Ref. [91] for a detailed study of this ratio). The Wilson lines V (x ⊥ ) can be obtained by solving the Yang-Mills equations, and the result reads\nHere, we introduced the infrared regulator m, which is needed to avoid the emergence of unphysical Coulomb tails, and will be another free parameter in the Bayesian analysis.\nHere, we introduced the infrared regulator m, which is needed to avoid the emergence of unphysical Coulomb tails, and will be another free parameter in the Bayesian analysis.\nIn the IPsat parametrization the saturation scale\nIn the IPsat parametrization the saturation scale\nWe introduce an event-by-event fluctuating density by writing the density profile following Refs. [33,45] as:\nWe introduce an event-by-event fluctuating density by writing the density profile following Refs. [33,45] as:\nwhere\nwhere\nand the coefficient p i allows for different normalizations for individual hot spots, to be discussed below. Our prescription corresponds to having N q hot spots with hot spot width B q (note that the hot spot transverse root mean square (RMS) radius in 2B q ). The hot spot positions b ⊥,i are sampled from a two-dimensional Gaussian distribution whose width is denoted by B qc , and the center-of-mass is shifted to the origin in the end. As discussed in Ref. [92,93], repulsive short-range correlations between the hot spots may explain the hollowness effect [94][95][96][97] and negative correlation between the v 2 and v 3 flow harmonics observed in highest multiplicity proton-proton collisions [98,99]. In order to study if exclusive vector meson production in DIS can be used to probe such repulsive correlations, we also introduce an additional model parameter d q,Min which controls the minimum three-dimensional distance required between any two hot spots. 1 We checked that for a large number of hot spots N q = 10 in a typical nucleon of size B qc = 4.2 GeV −2 , the model parameter d q,Min remains effective more than 90% for d q,Min ≤ 0.4 fm, meaning that in 10% of the sampled configurations the distance requirement cannot be fulfilled.\nand the coefficient p i allows for different normalizations for individual hot spots, to be discussed below. Our prescription corresponds to having N q hot spots with hot spot width B q (note that the hot spot transverse root mean square (RMS) radius in 2B q ). The hot spot positions b ⊥,i are sampled from a two-dimensional Gaussian distribution whose width is denoted by B qc , and the center-of-mass is shifted to the origin in the end. As discussed in Ref. [92,93], repulsive short-range correlations between the hot spots may explain the hollowness effect [94][95][96][97] and negative correlation between the v 2 and v 3 flow harmonics observed in highest multiplicity proton-proton collisions [98,99]. In order to study if exclusive vector meson production in DIS can be used to probe such repulsive correlations, we also introduce an additional model parameter d q,Min which controls the minimum three-dimensional distance required between any two hot spots. 1 We checked that for a large number of hot spots N q = 10 in a typical nucleon of size B qc = 4.2 GeV −2 , the model parameter d q,Min remains effective more than 90% for d q,Min ≤ 0.4 fm, meaning that in 10% of the sampled configurations the distance requirement cannot be fulfilled.\nFinally, we include saturation scale fluctuations by allowing the local density of each hot spot to fluctuate independently, following again Refs. [33,45] (see also Ref. [101]). These fluctuations are implemented by sampling the coefficients p i in Eq. (9) from the log-normal distribution\nFinally, we include saturation scale fluctuations by allowing the local density of each hot spot to fluctuate independently, following again Refs. [33,45] (see also Ref. [101]). These fluctuations are implemented by sampling the coefficients p i in Eq. (9) from the log-normal distribution\nThe sampled p i are at the end normalized by the expectation value of the distribution E[p i ] = e σ 2 /2 in order to keep the average density unmodified. The magnitude of density fluctuations is controlled by the parameter σ.\nThe sampled p i are at the end normalized by the expectation value of the distribution E[p i ] = e σ 2 /2 in order to keep the average density unmodified. The magnitude of density fluctuations is controlled by the parameter σ."},{"title":"III. BAYESIAN ANALYSIS SETUP","content":""},{"title":"III. BAYESIAN ANALYSIS SETUP","content":"Bayesian Inference is a general and systematic method to constrain the probability distribution of model parameters θ by comparing model calculations y(θ) with experimental measurements y exp [102]. Bayes' theorem provides the posterior distribution of model parameters as\nBayesian Inference is a general and systematic method to constrain the probability distribution of model parameters θ by comparing model calculations y(θ) with experimental measurements y exp [102]. Bayes' theorem provides the posterior distribution of model parameters as\nHere P(y exp |θ) is the likelihood for model results with parameter θ to agree with the experimental data. We choose a multivariate normal distribution for the logarithm of the likelihood with ∆y(θ) = y(θ) − y exp [103],\nHere P(y exp |θ) is the likelihood for model results with parameter θ to agree with the experimental data. We choose a multivariate normal distribution for the logarithm of the likelihood with ∆y(θ) = y(θ) − y exp [103],\nHere n is the number of experimental data points and Σ ≡ Σ exp + Σ model is the n × n covariance matrix, which encodes experimental and model uncertainties. In the current analysis, we assume no correlation among experimental errors of the n observables {σ i }. So the covariance matrix for experimental uncertainty takes a diagonal form,\nHere n is the number of experimental data points and Σ ≡ Σ exp + Σ model is the n × n covariance matrix, which encodes experimental and model uncertainties. In the current analysis, we assume no correlation among experimental errors of the n observables {σ i }. So the covariance matrix for experimental uncertainty takes a diagonal form,\nOur model uncertainties Σ model are estimated using the covariance matrix from the trained Gaussian Process (GP) Emulators [104].\nOur model uncertainties Σ model are estimated using the covariance matrix from the trained Gaussian Process (GP) Emulators [104].\nWe employ GP emulators [103] for our model and couple them with the Monte-Carlo Markov Chain (MCMC) method to efficiently explore the model parameter space [105,106]. The HERA measurements can be represented by five Principal Components (PC) with a residual variance of less than 0.01%, meaning that 99.99% of the variation of all studied observables within the prior parameter range are captured by the five principal components. Our GP emulators are trained to fit these five Table. I. We treat the parameter N q as a continuous real number. The fractional part of N q is treated as a probability to sample either N q or N q partons inside protons. The same approach was recently used in Ref. [107]. The experimental data included in the Bayesian analysis is the H1 data on coherent and incoherent J/ψ production cross section measured at W = 75 GeV [28]. The incoherent data is included in the |t| range 0 < |t| < 2.5 GeV 2 . We note that there is incoherent data at higher |t| also (studied in a similar context in Ref. [46]), but the highest |t| points are not included in our analysis for two reasons. First, as we determine Wilson lines at fixed x P , as discussed in Sec. II, and do not include the full x P evolution, we neglect |t| dependence in x P (see Eq. (5)). Additionally, at large |t| other effects such as DGLAP evolution [108][109][110][111] may become important.\nWe employ GP emulators [103] for our model and couple them with the Monte-Carlo Markov Chain (MCMC) method to efficiently explore the model parameter space [105,106]. The HERA measurements can be represented by five Principal Components (PC) with a residual variance of less than 0.01%, meaning that 99.99% of the variation of all studied observables within the prior parameter range are captured by the five principal components. Our GP emulators are trained to fit these five Table. I. We treat the parameter N q as a continuous real number. The fractional part of N q is treated as a probability to sample either N q or N q partons inside protons. The same approach was recently used in Ref. [107]. The experimental data included in the Bayesian analysis is the H1 data on coherent and incoherent J/ψ production cross section measured at W = 75 GeV [28]. The incoherent data is included in the |t| range 0 < |t| < 2.5 GeV 2 . We note that there is incoherent data at higher |t| also (studied in a similar context in Ref. [46]), but the highest |t| points are not included in our analysis for two reasons. First, as we determine Wilson lines at fixed x P , as discussed in Sec. II, and do not include the full x P evolution, we neglect |t| dependence in x P (see Eq. (5)). Additionally, at large |t| other effects such as DGLAP evolution [108][109][110][111] may become important."},{"title":"IV. RESULTS","content":""},{"title":"IV. RESULTS","content":"For two distinct scenarios, the first with N q fixed to 3, the second with N q a free parameter, the posterior distribution of model parameters is shown in Fig. 1. Particularly for the case of N q = 3, most of the model parameters are tightly constrained by the H1 data included in the Bayesian analysis. This is due to the fact that different regions of the dataset are sensitive to different model parameters.\nFor two distinct scenarios, the first with N q fixed to 3, the second with N q a free parameter, the posterior distribution of model parameters is shown in Fig. 1. Particularly for the case of N q = 3, most of the model parameters are tightly constrained by the H1 data included in the Bayesian analysis. This is due to the fact that different regions of the dataset are sensitive to different model parameters.\nFirst, the infrared regulator m suppresses longdistance Coulomb tails, and as such, it controls the shape of the proton at large distances. This part of the proton geometry is probed by coherent diffraction at low |t| 0.2GeV 2 [45]: impact parameter and momentum transfers are Fourier conjugates, and consequently the low-|t| region is sensitive to large distances and vice versa. On the other hand, the actual proton size controlled mostly by B qc determines the overall slope of the coherent spectrum. The hot spot size B q then determines the slope of the incoherent cross section in the |t| 1 GeV 2 region: as shown in Ref. [112] the slope of the incoherent spectra at high |t| is given by the size of the smallest fluctuating constituent.\nFirst, the infrared regulator m suppresses longdistance Coulomb tails, and as such, it controls the shape of the proton at large distances. This part of the proton geometry is probed by coherent diffraction at low |t| 0.2GeV 2 [45]: impact parameter and momentum transfers are Fourier conjugates, and consequently the low-|t| region is sensitive to large distances and vice versa. On the other hand, the actual proton size controlled mostly by B qc determines the overall slope of the coherent spectrum. The hot spot size B q then determines the slope of the incoherent cross section in the |t| 1 GeV 2 region: as shown in Ref. [112] the slope of the incoherent spectra at high |t| is given by the size of the smallest fluctuating constituent.\nThe overall normalization is determined by the Q s /(g 2 µ) parameter and the magnitude of the cross sections constrains that, however, it can not be determined very precisely from our analysis as it is strongly correlated with many other parameters, particularly for the case that N q is a free parameter. Here we note that there is some model uncertainty related to the nonperturbative vector meson wave functions, and different phenomenological parametrizations can result in cross sections that differ by ∼ 20%, see e.g. [43,55,57]. In phenomenological analyses, the so-called skewness correction [55] is sometimes included, which can also have a numerically significant (up to ∼ 40%) effect on the cross section. However, as there are also other model uncertainties related to the overall normalization and the applicability of the skewness correction in our setup is not rigorously justified, it is not included in this work. The small real part correction discussed e.g. in Ref. [55] is also neglected.\nThe overall normalization is determined by the Q s /(g 2 µ) parameter and the magnitude of the cross sections constrains that, however, it can not be determined very precisely from our analysis as it is strongly correlated with many other parameters, particularly for the case that N q is a free parameter. Here we note that there is some model uncertainty related to the nonperturbative vector meson wave functions, and different phenomenological parametrizations can result in cross sections that differ by ∼ 20%, see e.g. [43,55,57]. In phenomenological analyses, the so-called skewness correction [55] is sometimes included, which can also have a numerically significant (up to ∼ 40%) effect on the cross section. However, as there are also other model uncertainties related to the overall normalization and the applicability of the skewness correction in our setup is not rigorously justified, it is not included in this work. The small real part correction discussed e.g. in Ref. [55] is also neglected.\nOne can see that when leaving N q variable, its value can not be constrained in our analysis, except that N q ≥ 2 is required in order to get geometry fluctuations that are necessary to describe the incoherent HERA data. At first one would expect the configurations with large N q to be so smooth that event-by-event fluctuations would not be enough to result in a large enough incoherent cross section. However, we note that there is a strong positive correlation between N q and σ, which means that large N q goes along with large hot spot density fluctuations. In this situation one can not really interpret N q as the number of hot spots, but one has to consider effective hot spots, that are generated dynamically from the sum of the N q constituents that are each strongly fluctuating in magnitude. This effective hot spot number will generally be smaller than N q . For a quantitative analysis a hot spot finding prescription, similar to jet clustering algorithms, would be needed. Additional constraints for σ originate from the incoherent cross section at small |t|, which is sensitive to the fluctuations at long-distance scales, i.e., overall density fluctuations [33,45].\nOne can see that when leaving N q variable, its value can not be constrained in our analysis, except that N q ≥ 2 is required in order to get geometry fluctuations that are necessary to describe the incoherent HERA data. At first one would expect the configurations with large N q to be so smooth that event-by-event fluctuations would not be enough to result in a large enough incoherent cross section. However, we note that there is a strong positive correlation between N q and σ, which means that large N q goes along with large hot spot density fluctuations. In this situation one can not really interpret N q as the number of hot spots, but one has to consider effective hot spots, that are generated dynamically from the sum of the N q constituents that are each strongly fluctuating in magnitude. This effective hot spot number will generally be smaller than N q . For a quantitative analysis a hot spot finding prescription, similar to jet clustering algorithms, would be needed. Additional constraints for σ originate from the incoherent cross section at small |t|, which is sensitive to the fluctuations at long-distance scales, i.e., overall density fluctuations [33,45].\nThe minimum distance required between the hot spots in three dimensions, d q,Min , can not be constrained at all in our analysis. This parameter is also only very weakly correlated with the other parameters, emphasizing its limited effect on the results. This means that the J/ψ production data allows, but does not require, repulsive short-range correlations used e.g. in Refs. [92,93]. We note that for large N q it is not always possible to fulfill the minimum distance requirement, reducing the effective value of d q,Min that is actually employed.\nThe minimum distance required between the hot spots in three dimensions, d q,Min , can not be constrained at all in our analysis. This parameter is also only very weakly correlated with the other parameters, emphasizing its limited effect on the results. This means that the J/ψ production data allows, but does not require, repulsive short-range correlations used e.g. in Refs. [92,93]. We note that for large N q it is not always possible to fulfill the minimum distance requirement, reducing the effective value of d q,Min that is actually employed.\nLet us next study correlations between the model parameters in more detail. First, we observe that there is a clear positive correlation between the infrared regulator m and the hot spot size B q . This can be understood, as increasing m suppresses color fields at large distances and as such results in smaller hot spots. Interestingly, we find no clear correlation between the overall proton size parameter B qc and the infrared regulator m. In fact, the proton size B qc shows no clear correlation with any of the other model parameters. The infrared regulator m is strongly anti-correlated with Q s /(g 2 µ). This is again easy to understand: large m results in a reduction of the normalization of the cross section, which has to be compensated by using a larger color charge density which requires smaller Q s /(g 2 µ).\nLet us next study correlations between the model parameters in more detail. First, we observe that there is a clear positive correlation between the infrared regulator m and the hot spot size B q . This can be understood, as increasing m suppresses color fields at large distances and as such results in smaller hot spots. Interestingly, we find no clear correlation between the overall proton size parameter B qc and the infrared regulator m. In fact, the proton size B qc shows no clear correlation with any of the other model parameters. The infrared regulator m is strongly anti-correlated with Q s /(g 2 µ). This is again easy to understand: large m results in a reduction of the normalization of the cross section, which has to be compensated by using a larger color charge density which requires smaller Q s /(g 2 µ).\nWe also find a strong negative correlation between the number of hot spots N q and the proton density Q s /(g 2 µ). This correlation can be understood by considering the effect of these parameters on the incoherent cross section. When the number of hot spots with the same size increases, the incoherent slope is not significantly affected but the normalization goes down as there are smaller fluctuations in the scattering amplitude due to the smoother proton profile. This would result in the incoherent cross section being underestimated, which has to be compensated by smaller Q s /(g 2 µ).\nWe also find a strong negative correlation between the number of hot spots N q and the proton density Q s /(g 2 µ). This correlation can be understood by considering the effect of these parameters on the incoherent cross section. When the number of hot spots with the same size increases, the incoherent slope is not significantly affected but the normalization goes down as there are smaller fluctuations in the scattering amplitude due to the smoother proton profile. This would result in the incoherent cross section being underestimated, which has to be compensated by smaller Q s /(g 2 µ).\nThe somewhat surprising negative correlation between the hot spot size B q and the proton density Q s /(g 2 µ) can also be understood. First, we note that large hot spots (large B q ) require a larger infrared regulator m which effectively makes the hot spots smaller (notice a positive correlation between B q and m). Then, to compensate for the effect of the larger infrared regulator on the magnitude of the cross section, a smaller Q s /(g 2 µ) is needed.\nThe somewhat surprising negative correlation between the hot spot size B q and the proton density Q s /(g 2 µ) can also be understood. First, we note that large hot spots (large B q ) require a larger infrared regulator m which effectively makes the hot spots smaller (notice a positive correlation between B q and m). Then, to compensate for the effect of the larger infrared regulator on the magnitude of the cross section, a smaller Q s /(g 2 µ) is needed.\nThe posterior distribution of parameters B qc and B q , that determine the proton size, which can be quantified e.g. by the two dimensional RMS radius, r rms = 2(B qc + B q ), are shown in the second and third diagonal panels of Fig. 1. We find that they are sharply peaked, in particular for the case of fixed N q = 3. The extracted proton radius from this Bayesian analysis is r rms = 0.591 +0. 130 −0.071 fm (0.610 +0.065 −0.068 fm) for variable N q (N q ≡ 3) with uncertainty estimates in 90% credible intervals.\nThe posterior distribution of parameters B qc and B q , that determine the proton size, which can be quantified e.g. by the two dimensional RMS radius, r rms = 2(B qc + B q ), are shown in the second and third diagonal panels of Fig. 1. We find that they are sharply peaked, in particular for the case of fixed N q = 3. The extracted proton radius from this Bayesian analysis is r rms = 0.591 +0. 130 −0.071 fm (0.610 +0.065 −0.068 fm) for variable N q (N q ≡ 3) with uncertainty estimates in 90% credible intervals.\nWe further included the possibility of fluctuating values of B q and N q , and studied the dependence of the observables on the respective variances. We found that the considered experimental data could not constrain these parameters and decided not to include them in the presented analysis.\nWe further included the possibility of fluctuating values of B q and N q , and studied the dependence of the observables on the respective variances. We found that the considered experimental data could not constrain these parameters and decided not to include them in the presented analysis.\nNext, we demonstrate explicitly that sampling parameter values from the posterior distribution lead to a good description of the HERA data. The main result of the Bayesian analysis is the posterior distribution shown in Fig. 1, but it is also possible to find the so-called Maximum a Posteriori (MAP) parameter set, which is the mode of the posterior distribution. Because our prior parameter distributions are uniform, the MAP parameters shown in Table I maximize the likelihood function and provide the best fit to the experimental data. We note that statistically the expectation value for any observable is not obtained using the MAP parameters. Instead, one should calculate observables using averages over parameter samples obtained from the posterior distribution.\nNext, we demonstrate explicitly that sampling parameter values from the posterior distribution lead to a good description of the HERA data. The main result of the Bayesian analysis is the posterior distribution shown in Fig. 1, but it is also possible to find the so-called Maximum a Posteriori (MAP) parameter set, which is the mode of the posterior distribution. Because our prior parameter distributions are uniform, the MAP parameters shown in Table I maximize the likelihood function and provide the best fit to the experimental data. We note that statistically the expectation value for any observable is not obtained using the MAP parameters. Instead, one should calculate observables using averages over parameter samples obtained from the posterior distribution.\nComparison to the HERA coherent and incoherent J/ψ production data measured at W = 75 GeV [28] and used in the Bayesian analysis to constrain the model parameters are shown in panel (a) of Fig. 2. The spectra calculated by averaging the results computed using different parametrizations sampled from the posterior distribution indeed provide an excellent description of the data. We also show the statistical uncertainty, obtained by calculating the one standard deviation interval shown as red (coherent) and blue (incoherent) bands.\nComparison to the HERA coherent and incoherent J/ψ production data measured at W = 75 GeV [28] and used in the Bayesian analysis to constrain the model parameters are shown in panel (a) of Fig. 2. The spectra calculated by averaging the results computed using different parametrizations sampled from the posterior distribution indeed provide an excellent description of the data. We also show the statistical uncertainty, obtained by calculating the one standard deviation interval shown as red (coherent) and blue (incoherent) bands.\nNext, we study compatibility with experimental data not included in the Bayesian analysis. We do not include full small-x evolution e.g. by means of the JIMWLK [113][114][115][116][117][118][119][120] equation in this work. Consequently, the only center-of-mass energy W (or momentum fraction x P ) dependence comes from the x P dependence of the saturation scale Q s determined from the IPsat fit. This almost only affects the overall normalization of the calculated spectra, and misses important physical effects such as the growth of the proton with decreasing x. Consequently, we do not expect that within our setup we can describe J/ψ production at different center-of-mass energies simultaneously. Nevertheless, the geometry evolution between W = 75 GeV and W = 100 GeV should be weak enough to make predictions for the higher center-of-mass energy.\nNext, we study compatibility with experimental data not included in the Bayesian analysis. We do not include full small-x evolution e.g. by means of the JIMWLK [113][114][115][116][117][118][119][120] equation in this work. Consequently, the only center-of-mass energy W (or momentum fraction x P ) dependence comes from the x P dependence of the saturation scale Q s determined from the IPsat fit. This almost only affects the overall normalization of the calculated spectra, and misses important physical effects such as the growth of the proton with decreasing x. Consequently, we do not expect that within our setup we can describe J/ψ production at different center-of-mass energies simultaneously. Nevertheless, the geometry evolution between W = 75 GeV and W = 100 GeV should be weak enough to make predictions for the higher center-of-mass energy.\nThe calculated J/ψ spectra at W = 100 GeV are shown in panel (b) of Fig. 2 and compared with the data from H1 and ZEUS collaborations [29,31,32,121]. Again, we show the model prediction as the average over many samples of the posterior distribution and provide one standard deviation bands. The results are very similar to the W = 75 GeV case studied above, and the HERA data is well described (except the coherent cross section at low |t|, and high |t| in the case of the ZEUS measurement), and variation between the different parametrizations sampled from the posterior distribution is small. Similarity to the W = 75 GeV case is not surprising, as we use exactly the same fluctuating geometry, with the only difference being slightly larger Q s values, extracted from the IPsat parametrization.\nThe calculated J/ψ spectra at W = 100 GeV are shown in panel (b) of Fig. 2 and compared with the data from H1 and ZEUS collaborations [29,31,32,121]. Again, we show the model prediction as the average over many samples of the posterior distribution and provide one standard deviation bands. The results are very similar to the W = 75 GeV case studied above, and the HERA data is well described (except the coherent cross section at low |t|, and high |t| in the case of the ZEUS measurement), and variation between the different parametrizations sampled from the posterior distribution is small. Similarity to the W = 75 GeV case is not surprising, as we use exactly the same fluctuating geometry, with the only difference being slightly larger Q s values, extracted from the IPsat parametrization."},{"title":"V. CONCLUSIONS","content":""},{"title":"V. CONCLUSIONS","content":"We have performed a statistically rigorous Bayesian analysis to extract posterior likelihood distributions for the non-perturbative parameters describing the event-byevent fluctuating proton geometry as constrained by exclusive J/ψ production data from HERA. We presented a comparison of the |t|-dependent coherent and incoherent cross sections, obtained from an average over many parameter samples from the posterior distribution, to the experimental data.\nWe have performed a statistically rigorous Bayesian analysis to extract posterior likelihood distributions for the non-perturbative parameters describing the event-byevent fluctuating proton geometry as constrained by exclusive J/ψ production data from HERA. We presented a comparison of the |t|-dependent coherent and incoherent cross sections, obtained from an average over many parameter samples from the posterior distribution, to the experimental data.\nGenerally, model parametrizations sampled from the determined posterior distribution can be used to systematically take into account uncertainties in the proton geometry as constrained by the HERA DIS data when cal- FIG. 2. Coherent and incoherent J/ψ photoproduction cross sections at W = 75 GeV (a) and W = 100 GeV (b), calculated by averaging over many parameter sets sampled from the posterior distribution, which was determined using the W = 75 GeV HERA data [28]. We compare the case with variable Nq to that with fixed Nq = 3, and to experimental data at W = 75 GeV [28] and W = 100 GeV [29][30][31][32], respectively. The bands show the one standard deviation uncertainty.\nGenerally, model parametrizations sampled from the determined posterior distribution can be used to systematically take into account uncertainties in the proton geometry as constrained by the HERA DIS data when cal- FIG. 2. Coherent and incoherent J/ψ photoproduction cross sections at W = 75 GeV (a) and W = 100 GeV (b), calculated by averaging over many parameter sets sampled from the posterior distribution, which was determined using the W = 75 GeV HERA data [28]. We compare the case with variable Nq to that with fixed Nq = 3, and to experimental data at W = 75 GeV [28] and W = 100 GeV [29][30][31][32], respectively. The bands show the one standard deviation uncertainty.\nculating any other observable that depends on the proton geometry, such as flow observables in high-multiplicity proton-proton and proton-nucleus collisions. To enable such studies, we provide 1,000 parametrizations sampled from the determined posterior distributions in the supplemental material.\nculating any other observable that depends on the proton geometry, such as flow observables in high-multiplicity proton-proton and proton-nucleus collisions. To enable such studies, we provide 1,000 parametrizations sampled from the determined posterior distributions in the supplemental material.\nBecause the model parameters are sensitive to different aspects of the coherent and incoherent vector meson production spectra, most of them are well constrained by the Bayesian analysis. The only exceptions are the minimum distance between the hot spots (repulsive short-range correlations), and the number of hot spots N q , which cannot be well constrained by the considered HERA data. However, we note that although the analysis suggests that large N q is compatible with the HERA data, for N q 5 one should no longer simply interpret N q as the number of actual hot spots. In this regime, a good description of the data requires large fluctuations of individual hot spots' densities, and the sampled 'hot spots' can also overlap significantly. Thus, the effective number of hot spots is significantly smaller than the parameter N q might imply.\nBecause the model parameters are sensitive to different aspects of the coherent and incoherent vector meson production spectra, most of them are well constrained by the Bayesian analysis. The only exceptions are the minimum distance between the hot spots (repulsive short-range correlations), and the number of hot spots N q , which cannot be well constrained by the considered HERA data. However, we note that although the analysis suggests that large N q is compatible with the HERA data, for N q 5 one should no longer simply interpret N q as the number of actual hot spots. In this regime, a good description of the data requires large fluctuations of individual hot spots' densities, and the sampled 'hot spots' can also overlap significantly. Thus, the effective number of hot spots is significantly smaller than the parameter N q might imply.\nThe HERA data used in this work probes the proton structure at x ∼ 10 −3 . The energy (Bjorken-x) dependence can be included in terms of JIMWLK evolution as e.g. in Refs. [86,122]. In the future we plan to extend our framework by including the full JIMWLK evolution and vector meson production data at different center-ofmass energies from HERA [28,121,123] and from the ultra peripheral proton-lead collisions measured at the LHC [39,40,124,125], allowing us to extract also the Bjorken-x dependence of the fluctuating proton geometry.\nThe HERA data used in this work probes the proton structure at x ∼ 10 −3 . The energy (Bjorken-x) dependence can be included in terms of JIMWLK evolution as e.g. in Refs. [86,122]. In the future we plan to extend our framework by including the full JIMWLK evolution and vector meson production data at different center-ofmass energies from HERA [28,121,123] and from the ultra peripheral proton-lead collisions measured at the LHC [39,40,124,125], allowing us to extract also the Bjorken-x dependence of the fluctuating proton geometry.\nIn addition to constraining the energy dependence, performing global analyses including both exclusive vector meson production data from HERA and flow harmonics from the proton-proton and proton-lead collisions measured at the LHC (including a model calculation along the lines of [126]) would allow for a powerful global analysis of the fluctuating nucleon substructure and properties of the final state. More differential DIS measurements from the future EIC such as dijet [85] or lepton-meson angular correlations [83] can also provide further constraints and can in principle be included in our framework in a straightforward manner.\nIn addition to constraining the energy dependence, performing global analyses including both exclusive vector meson production data from HERA and flow harmonics from the proton-proton and proton-lead collisions measured at the LHC (including a model calculation along the lines of [126]) would allow for a powerful global analysis of the fluctuating nucleon substructure and properties of the final state. More differential DIS measurements from the future EIC such as dijet [85] or lepton-meson angular correlations [83] can also provide further constraints and can in principle be included in our framework in a straightforward manner.\nThe numerical framework for our physics models and the Bayesian analysis package are publicly available on Github [127,128]. Our Bayesian analysis code is developed based on the open-source numerical package by the Duke group [23]. To help visualize how observables depend on the model parameters, we provide an interactive web page with the trained GP emulators [129], where one can also find the posterior samples. expressed therein lies entirely with the authors. This research was done using resources provided by the Open Science Grid (OSG) [130,131], which is supported by the National Science Foundation award #2030508.\nThe numerical framework for our physics models and the Bayesian analysis package are publicly available on Github [127,128]. Our Bayesian analysis code is developed based on the open-source numerical package by the Duke group [23]. To help visualize how observables depend on the model parameters, we provide an interactive web page with the trained GP emulators [129], where one can also find the posterior samples. expressed therein lies entirely with the authors. This research was done using resources provided by the Open Science Grid (OSG) [130,131], which is supported by the National Science Foundation award #2030508."},{"title":"FIG. 1 .","content":""},{"title":"FIG. 1 .","content":""},{"title":"TABLE I .","content":""},{"title":"TABLE I .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"\"\n },\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Extracting the multi-dimensional structure of protons and nuclei is one of the main goals of future Deep Inelastic Scattering (DIS) facilities such as the Electron-Ion Collider [1,2], LHeC/FC-he [3] and EicC [4]. Exclusive processes like J/ψ production are especially powerful probes of the hadron structure at a small longitudinal momentum fraction x for two reasons. First, the exclusive nature of the process requires at lowest order at least two gluons to be exchanged with the target, rendering the cross section approximately proportional to the squared gluon distribution [5]. Additionally, only in exclusive processes is it possible to measure the total momentum transfer, which is Fourier conjugate to the impact parameter and thereby provides access to the transverse geometry.\\nExtracting the multi-dimensional structure of protons and nuclei is one of the main goals of future Deep Inelastic Scattering (DIS) facilities such as the Electron-Ion Collider [1,2], LHeC/FC-he [3] and EicC [4]. Exclusive processes like J/ψ production are especially powerful probes of the hadron structure at a small longitudinal momentum fraction x for two reasons. First, the exclusive nature of the process requires at lowest order at least two gluons to be exchanged with the target, rendering the cross section approximately proportional to the squared gluon distribution [5]. Additionally, only in exclusive processes is it possible to measure the total momentum transfer, which is Fourier conjugate to the impact parameter and thereby provides access to the transverse geometry.\\nUnderstanding the proton structure, including its event-by-event fluctuations [6], is of fundamental interest. Additionally, knowledge of the spatial structure of the colliding objects in hadronic and heavy-ion collisions is required in order to construct realistic initial conditions that can be coupled to relativistic hydrodynamic simulations to describe the space-time evolution of the produced Quark-Gluon Plasma (QGP). Besides heavyion collisions, collective phenomena that can be interpreted as signatures of QGP production have been seen in small systems such as proton/deuteron/ 3 He -nucleus [7][8][9][10][11], proton-proton [12], and even photon-nucleus collisions [13], see [14][15][16][17][18][19] for reviews. In such small collision systems with a proton projectile, the detailed fluctuation spectrum of the proton geometry is particularly important to determine if QGP is indeed produced.\\nUnderstanding the proton structure, including its event-by-event fluctuations [6], is of fundamental interest. Additionally, knowledge of the spatial structure of the colliding objects in hadronic and heavy-ion collisions is required in order to construct realistic initial conditions that can be coupled to relativistic hydrodynamic simulations to describe the space-time evolution of the produced Quark-Gluon Plasma (QGP). Besides heavyion collisions, collective phenomena that can be interpreted as signatures of QGP production have been seen in small systems such as proton/deuteron/ 3 He -nucleus [7][8][9][10][11], proton-proton [12], and even photon-nucleus collisions [13], see [14][15][16][17][18][19] for reviews. In such small collision systems with a proton projectile, the detailed fluctuation spectrum of the proton geometry is particularly important to determine if QGP is indeed produced.\\nIt is possible to constrain the proton structure from hadronic collisions by performing a statistical analysis to extract both the transport coefficients describing the matter produced in proton-lead collisions (see e.g. Refs. [20][21][22][23][24][25][26]), as well as the proton's fluctuating geometry, by comparing with the LHC data as in Ref. [27]. Another approach, which we take in this work, is to use ex-clusive DIS data from HERA, especially exclusive vector meson production [28][29][30][31][32], as a complementary input to constrain the proton shape fluctuations, as initially suggested in Ref. [33]. In the future, the Electron-Ion Collider will provide a vast amount of precise vector meson production data with proton and nuclear targets that will provide further constraints on e.g. momentum fraction x P and the nuclear mass number A dependence. Additionally, Ultra Peripheral Collisions [34,35] at RHIC [36] and at the LHC [37][38][39][40][41][42] provide access to very high energy photoproduction processes and to effects of a nuclear environment on nucleon substructure fluctuations at high energy [43,44].\\nIt is possible to constrain the proton structure from hadronic collisions by performing a statistical analysis to extract both the transport coefficients describing the matter produced in proton-lead collisions (see e.g. Refs. [20][21][22][23][24][25][26]), as well as the proton's fluctuating geometry, by comparing with the LHC data as in Ref. [27]. Another approach, which we take in this work, is to use ex-clusive DIS data from HERA, especially exclusive vector meson production [28][29][30][31][32], as a complementary input to constrain the proton shape fluctuations, as initially suggested in Ref. [33]. In the future, the Electron-Ion Collider will provide a vast amount of precise vector meson production data with proton and nuclear targets that will provide further constraints on e.g. momentum fraction x P and the nuclear mass number A dependence. Additionally, Ultra Peripheral Collisions [34,35] at RHIC [36] and at the LHC [37][38][39][40][41][42] provide access to very high energy photoproduction processes and to effects of a nuclear environment on nucleon substructure fluctuations at high energy [43,44].\\nIn this work, we go beyond previous studies [33,45], where model parameters were constrained \\\"by eye\\\", and perform a Bayesian analysis to extract in a statistically rigorous manner the non-perturbative parameter values allowed by the HERA data, and construct initial conditions for hadronic collisions that are compatible with the experimental DIS data. This paper is organized as follows. In Section II we review the calculation of coherent and incoherent exclusive vector meson production in the dipole picture and discuss the various aspects of our model for the proton target. In Section III we explain the procedure for our Bayesian analysis. We present results in Section IV and conclude in Section V.\\nIn this work, we go beyond previous studies [33,45], where model parameters were constrained \\\"by eye\\\", and perform a Bayesian analysis to extract in a statistically rigorous manner the non-perturbative parameter values allowed by the HERA data, and construct initial conditions for hadronic collisions that are compatible with the experimental DIS data. This paper is organized as follows. In Section II we review the calculation of coherent and incoherent exclusive vector meson production in the dipole picture and discuss the various aspects of our model for the proton target. In Section III we explain the procedure for our Bayesian analysis. We present results in Section IV and conclude in Section V.\"\n },\n {\n \"title\": \"II. VECTOR MESON PRODUCTION AT HIGH ENERGY\",\n \"content\": \"\"\n },\n {\n \"title\": \"II. VECTOR MESON PRODUCTION AT HIGH ENERGY\",\n \"content\": \"In this work we calculate vector meson production in a framework similar to the one used in Refs. [33,45] (see also Refs. [46][47][48][49][50]), and for completeness briefly review the calculation in this section.\\nIn this work we calculate vector meson production in a framework similar to the one used in Refs. [33,45] (see also Refs. [46][47][48][49][50]), and for completeness briefly review the calculation in this section.\\nAt high energies, DIS processes can be conveniently described in the dipole picture in the rest frame of the target proton, and interaction with the target color field is described in the Color Glass Condensate (CGC) framework [51][52][53][54]. In the proton rest frame, the lifetime of a fluctuation of the incoming virtual photon into a quarkantiquark dipole is much longer than the characteristic timescale of the dipole-target interaction. Consequently, the scattering amplitude can be factorized into a convolution of photon and vector meson wave functions and the dipole-target interaction. The scattering amplitude for exclusive vector meson V production can then be written as [55,56]\\nAt high energies, DIS processes can be conveniently described in the dipole picture in the rest frame of the target proton, and interaction with the target color field is described in the Color Glass Condensate (CGC) framework [51][52][53][54]. In the proton rest frame, the lifetime of a fluctuation of the incoming virtual photon into a quarkantiquark dipole is much longer than the characteristic timescale of the dipole-target interaction. Consequently, the scattering amplitude can be factorized into a convolution of photon and vector meson wave functions and the dipole-target interaction. The scattering amplitude for exclusive vector meson V production can then be written as [55,56]\\n(1) Here r ⊥ is the transverse size of the qq dipole, b ⊥ is the impact parameter measured relative to the proton center, and Q 2 is the photon virtuality. The fraction of the large photon plus momentum carried by the quark is given by z, and ∆ ⊥ is the transverse momentum transfer. Note that at high energies we can employ the eikonal approximation and assume that the quark transverse coordinates are fixed during the propagation through the target color field.\\n(1) Here r ⊥ is the transverse size of the qq dipole, b ⊥ is the impact parameter measured relative to the proton center, and Q 2 is the photon virtuality. The fraction of the large photon plus momentum carried by the quark is given by z, and ∆ ⊥ is the transverse momentum transfer. Note that at high energies we can employ the eikonal approximation and assume that the quark transverse coordinates are fixed during the propagation through the target color field.\\nThe γ * → qq splitting is described by the virtual photon light front wave function Ψ γ , which can be computed from QED [51]. The vector meson wave function is non-perturbative, and in this work, we use the so-called Boosted Gaussian parametrization from [55], where the model parameters are constrained by experimental data on the vector meson decay width. We note that there are multiple vector meson wave functions proposed in the literature (see e.g. Refs. [57][58][59]). Different wave functions mostly affect the overall normalization of the J/ψ production cross section, and have a much smaller effect on the |t| spectra, which we are most interested here [43,55,57]. Consequently, our results will depend only weakly on the specific wave function choice (except for the parameter controlling the overall proton density).\\nThe γ * → qq splitting is described by the virtual photon light front wave function Ψ γ , which can be computed from QED [51]. The vector meson wave function is non-perturbative, and in this work, we use the so-called Boosted Gaussian parametrization from [55], where the model parameters are constrained by experimental data on the vector meson decay width. We note that there are multiple vector meson wave functions proposed in the literature (see e.g. Refs. [57][58][59]). Different wave functions mostly affect the overall normalization of the J/ψ production cross section, and have a much smaller effect on the |t| spectra, which we are most interested here [43,55,57]. Consequently, our results will depend only weakly on the specific wave function choice (except for the parameter controlling the overall proton density).\\nEquation (1) is a leading order result for the vector meson production in the CGC framework (note that multiple scattering effects are resummed in the dipole amplitude N Ω ). Currently, there is rapid progress in the field toward next-to-leading order (NLO) accuracy. In particular, the cross section for the production of light mesons [60] and longitudinally polarized heavy vector mesons are now available [61] at next-to-leading order, as well as the virtual photon light front wave function [62][63][64][65][66][67] and small-x evolution equations [68][69][70][71][72][73][74][75][76][77] (see also Ref. [78] where dijet production is studied at next-toleading order). However, the NLO calculations are not yet at the level where they can be consistently used in phenomenological applications (in particular the cross section for transversely polarized heavy vector meson production is still missing). As the purpose of this work is to demonstrate the potential of Bayesian analyses to systematically extract non-perturbative parameters describing the proton event-by-event fluctuating structure, we do not expect the NLO contributions to have a large effect on the results we obtain using our leading order setup.\\nEquation (1) is a leading order result for the vector meson production in the CGC framework (note that multiple scattering effects are resummed in the dipole amplitude N Ω ). Currently, there is rapid progress in the field toward next-to-leading order (NLO) accuracy. In particular, the cross section for the production of light mesons [60] and longitudinally polarized heavy vector mesons are now available [61] at next-to-leading order, as well as the virtual photon light front wave function [62][63][64][65][66][67] and small-x evolution equations [68][69][70][71][72][73][74][75][76][77] (see also Ref. [78] where dijet production is studied at next-toleading order). However, the NLO calculations are not yet at the level where they can be consistently used in phenomenological applications (in particular the cross section for transversely polarized heavy vector meson production is still missing). As the purpose of this work is to demonstrate the potential of Bayesian analyses to systematically extract non-perturbative parameters describing the proton event-by-event fluctuating structure, we do not expect the NLO contributions to have a large effect on the results we obtain using our leading order setup.\\nThe coherent cross section, corresponding to the process where the target proton remains in the same quantum state, can be obtained by averaging over the target color charge configurations Ω at the amplitude level [79]:\\nThe coherent cross section, corresponding to the process where the target proton remains in the same quantum state, can be obtained by averaging over the target color charge configurations Ω at the amplitude level [79]:\\nSubtracting the coherent contribution from the total diffractive vector meson production cross section we obtain the cross section for incoherent vector meson production, in which case the final state of the target is different from the initial state [6,80,81]. Experimentally, this corresponds to processes where the target proton (or nucleus) dissociates, but the rapidity gap between the produced vector meson and the target remnants remains. The incoherent cross section can then be written as a variance\\nSubtracting the coherent contribution from the total diffractive vector meson production cross section we obtain the cross section for incoherent vector meson production, in which case the final state of the target is different from the initial state [6,80,81]. Experimentally, this corresponds to processes where the target proton (or nucleus) dissociates, but the rapidity gap between the produced vector meson and the target remnants remains. The incoherent cross section can then be written as a variance\\nDependence on the small-x structure of the target proton is included in the dipole amplitude N Ω (r ⊥ , b ⊥ , x P ), which, for a given target color charge configuration Ω, can be written as\\nDependence on the small-x structure of the target proton is included in the dipole amplitude N Ω (r ⊥ , b ⊥ , x P ), which, for a given target color charge configuration Ω, can be written as\\n(4) Here V (x ⊥ ) represents a Wilson line, which describes the color rotation of a quark state when it propagates through the target field (given the target color field configuration Ω) at transverse coordinate x ⊥ . We suppressed the dependence of V on\\n(4) Here V (x ⊥ ) represents a Wilson line, which describes the color rotation of a quark state when it propagates through the target field (given the target color field configuration Ω) at transverse coordinate x ⊥ . We suppressed the dependence of V on\\nwhich is the fraction of the target longitudinal momentum transferred to the meson with mass M V in the frame where the target has a large momentum. We neglect the t dependence of x P in this work. The photon-proton center-of-mass energy is denoted by W . The Wilson lines are obtained in the same way as done in the IP-Glasma calculation [82] used e.g. in Refs. [33,45,[83][84][85][86]. The color charges ρ a are first determined using the McLerran-Venugopalan [87] model, assuming that color charges are local Gaussian random variables with expectation value zero and variance\\nwhich is the fraction of the target longitudinal momentum transferred to the meson with mass M V in the frame where the target has a large momentum. We neglect the t dependence of x P in this work. The photon-proton center-of-mass energy is denoted by W . The Wilson lines are obtained in the same way as done in the IP-Glasma calculation [82] used e.g. in Refs. [33,45,[83][84][85][86]. The color charges ρ a are first determined using the McLerran-Venugopalan [87] model, assuming that color charges are local Gaussian random variables with expectation value zero and variance\\nHere the color charge density is µ 2 = dx − λ A (x − ), and it is related to the local saturation scale Q s (x ⊥ ) determined from the IPsat parametrization [88] fitted to HERA data [89,90]. In our Bayesian analysis, the ratio\\nHere the color charge density is µ 2 = dx − λ A (x − ), and it is related to the local saturation scale Q s (x ⊥ ) determined from the IPsat parametrization [88] fitted to HERA data [89,90]. In our Bayesian analysis, the ratio\\nis a free parameter, controlling the overall proton density (see also Ref. [91] for a detailed study of this ratio). The Wilson lines V (x ⊥ ) can be obtained by solving the Yang-Mills equations, and the result reads\\nis a free parameter, controlling the overall proton density (see also Ref. [91] for a detailed study of this ratio). The Wilson lines V (x ⊥ ) can be obtained by solving the Yang-Mills equations, and the result reads\\nHere, we introduced the infrared regulator m, which is needed to avoid the emergence of unphysical Coulomb tails, and will be another free parameter in the Bayesian analysis.\\nHere, we introduced the infrared regulator m, which is needed to avoid the emergence of unphysical Coulomb tails, and will be another free parameter in the Bayesian analysis.\\nIn the IPsat parametrization the saturation scale\\nIn the IPsat parametrization the saturation scale\\nWe introduce an event-by-event fluctuating density by writing the density profile following Refs. [33,45] as:\\nWe introduce an event-by-event fluctuating density by writing the density profile following Refs. [33,45] as:\\nwhere\\nwhere\\nand the coefficient p i allows for different normalizations for individual hot spots, to be discussed below. Our prescription corresponds to having N q hot spots with hot spot width B q (note that the hot spot transverse root mean square (RMS) radius in 2B q ). The hot spot positions b ⊥,i are sampled from a two-dimensional Gaussian distribution whose width is denoted by B qc , and the center-of-mass is shifted to the origin in the end. As discussed in Ref. [92,93], repulsive short-range correlations between the hot spots may explain the hollowness effect [94][95][96][97] and negative correlation between the v 2 and v 3 flow harmonics observed in highest multiplicity proton-proton collisions [98,99]. In order to study if exclusive vector meson production in DIS can be used to probe such repulsive correlations, we also introduce an additional model parameter d q,Min which controls the minimum three-dimensional distance required between any two hot spots. 1 We checked that for a large number of hot spots N q = 10 in a typical nucleon of size B qc = 4.2 GeV −2 , the model parameter d q,Min remains effective more than 90% for d q,Min ≤ 0.4 fm, meaning that in 10% of the sampled configurations the distance requirement cannot be fulfilled.\\nand the coefficient p i allows for different normalizations for individual hot spots, to be discussed below. Our prescription corresponds to having N q hot spots with hot spot width B q (note that the hot spot transverse root mean square (RMS) radius in 2B q ). The hot spot positions b ⊥,i are sampled from a two-dimensional Gaussian distribution whose width is denoted by B qc , and the center-of-mass is shifted to the origin in the end. As discussed in Ref. [92,93], repulsive short-range correlations between the hot spots may explain the hollowness effect [94][95][96][97] and negative correlation between the v 2 and v 3 flow harmonics observed in highest multiplicity proton-proton collisions [98,99]. In order to study if exclusive vector meson production in DIS can be used to probe such repulsive correlations, we also introduce an additional model parameter d q,Min which controls the minimum three-dimensional distance required between any two hot spots. 1 We checked that for a large number of hot spots N q = 10 in a typical nucleon of size B qc = 4.2 GeV −2 , the model parameter d q,Min remains effective more than 90% for d q,Min ≤ 0.4 fm, meaning that in 10% of the sampled configurations the distance requirement cannot be fulfilled.\\nFinally, we include saturation scale fluctuations by allowing the local density of each hot spot to fluctuate independently, following again Refs. [33,45] (see also Ref. [101]). These fluctuations are implemented by sampling the coefficients p i in Eq. (9) from the log-normal distribution\\nFinally, we include saturation scale fluctuations by allowing the local density of each hot spot to fluctuate independently, following again Refs. [33,45] (see also Ref. [101]). These fluctuations are implemented by sampling the coefficients p i in Eq. (9) from the log-normal distribution\\nThe sampled p i are at the end normalized by the expectation value of the distribution E[p i ] = e σ 2 /2 in order to keep the average density unmodified. The magnitude of density fluctuations is controlled by the parameter σ.\\nThe sampled p i are at the end normalized by the expectation value of the distribution E[p i ] = e σ 2 /2 in order to keep the average density unmodified. The magnitude of density fluctuations is controlled by the parameter σ.\"\n },\n {\n \"title\": \"III. BAYESIAN ANALYSIS SETUP\",\n \"content\": \"\"\n },\n {\n \"title\": \"III. BAYESIAN ANALYSIS SETUP\",\n \"content\": \"Bayesian Inference is a general and systematic method to constrain the probability distribution of model parameters θ by comparing model calculations y(θ) with experimental measurements y exp [102]. Bayes' theorem provides the posterior distribution of model parameters as\\nBayesian Inference is a general and systematic method to constrain the probability distribution of model parameters θ by comparing model calculations y(θ) with experimental measurements y exp [102]. Bayes' theorem provides the posterior distribution of model parameters as\\nHere P(y exp |θ) is the likelihood for model results with parameter θ to agree with the experimental data. We choose a multivariate normal distribution for the logarithm of the likelihood with ∆y(θ) = y(θ) − y exp [103],\\nHere P(y exp |θ) is the likelihood for model results with parameter θ to agree with the experimental data. We choose a multivariate normal distribution for the logarithm of the likelihood with ∆y(θ) = y(θ) − y exp [103],\\nHere n is the number of experimental data points and Σ ≡ Σ exp + Σ model is the n × n covariance matrix, which encodes experimental and model uncertainties. In the current analysis, we assume no correlation among experimental errors of the n observables {σ i }. So the covariance matrix for experimental uncertainty takes a diagonal form,\\nHere n is the number of experimental data points and Σ ≡ Σ exp + Σ model is the n × n covariance matrix, which encodes experimental and model uncertainties. In the current analysis, we assume no correlation among experimental errors of the n observables {σ i }. So the covariance matrix for experimental uncertainty takes a diagonal form,\\nOur model uncertainties Σ model are estimated using the covariance matrix from the trained Gaussian Process (GP) Emulators [104].\\nOur model uncertainties Σ model are estimated using the covariance matrix from the trained Gaussian Process (GP) Emulators [104].\\nWe employ GP emulators [103] for our model and couple them with the Monte-Carlo Markov Chain (MCMC) method to efficiently explore the model parameter space [105,106]. The HERA measurements can be represented by five Principal Components (PC) with a residual variance of less than 0.01%, meaning that 99.99% of the variation of all studied observables within the prior parameter range are captured by the five principal components. Our GP emulators are trained to fit these five Table. I. We treat the parameter N q as a continuous real number. The fractional part of N q is treated as a probability to sample either N q or N q partons inside protons. The same approach was recently used in Ref. [107]. The experimental data included in the Bayesian analysis is the H1 data on coherent and incoherent J/ψ production cross section measured at W = 75 GeV [28]. The incoherent data is included in the |t| range 0 < |t| < 2.5 GeV 2 . We note that there is incoherent data at higher |t| also (studied in a similar context in Ref. [46]), but the highest |t| points are not included in our analysis for two reasons. First, as we determine Wilson lines at fixed x P , as discussed in Sec. II, and do not include the full x P evolution, we neglect |t| dependence in x P (see Eq. (5)). Additionally, at large |t| other effects such as DGLAP evolution [108][109][110][111] may become important.\\nWe employ GP emulators [103] for our model and couple them with the Monte-Carlo Markov Chain (MCMC) method to efficiently explore the model parameter space [105,106]. The HERA measurements can be represented by five Principal Components (PC) with a residual variance of less than 0.01%, meaning that 99.99% of the variation of all studied observables within the prior parameter range are captured by the five principal components. Our GP emulators are trained to fit these five Table. I. We treat the parameter N q as a continuous real number. The fractional part of N q is treated as a probability to sample either N q or N q partons inside protons. The same approach was recently used in Ref. [107]. The experimental data included in the Bayesian analysis is the H1 data on coherent and incoherent J/ψ production cross section measured at W = 75 GeV [28]. The incoherent data is included in the |t| range 0 < |t| < 2.5 GeV 2 . We note that there is incoherent data at higher |t| also (studied in a similar context in Ref. [46]), but the highest |t| points are not included in our analysis for two reasons. First, as we determine Wilson lines at fixed x P , as discussed in Sec. II, and do not include the full x P evolution, we neglect |t| dependence in x P (see Eq. (5)). Additionally, at large |t| other effects such as DGLAP evolution [108][109][110][111] may become important.\"\n },\n {\n \"title\": \"IV. RESULTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"IV. RESULTS\",\n \"content\": \"For two distinct scenarios, the first with N q fixed to 3, the second with N q a free parameter, the posterior distribution of model parameters is shown in Fig. 1. Particularly for the case of N q = 3, most of the model parameters are tightly constrained by the H1 data included in the Bayesian analysis. This is due to the fact that different regions of the dataset are sensitive to different model parameters.\\nFor two distinct scenarios, the first with N q fixed to 3, the second with N q a free parameter, the posterior distribution of model parameters is shown in Fig. 1. Particularly for the case of N q = 3, most of the model parameters are tightly constrained by the H1 data included in the Bayesian analysis. This is due to the fact that different regions of the dataset are sensitive to different model parameters.\\nFirst, the infrared regulator m suppresses longdistance Coulomb tails, and as such, it controls the shape of the proton at large distances. This part of the proton geometry is probed by coherent diffraction at low |t| 0.2GeV 2 [45]: impact parameter and momentum transfers are Fourier conjugates, and consequently the low-|t| region is sensitive to large distances and vice versa. On the other hand, the actual proton size controlled mostly by B qc determines the overall slope of the coherent spectrum. The hot spot size B q then determines the slope of the incoherent cross section in the |t| 1 GeV 2 region: as shown in Ref. [112] the slope of the incoherent spectra at high |t| is given by the size of the smallest fluctuating constituent.\\nFirst, the infrared regulator m suppresses longdistance Coulomb tails, and as such, it controls the shape of the proton at large distances. This part of the proton geometry is probed by coherent diffraction at low |t| 0.2GeV 2 [45]: impact parameter and momentum transfers are Fourier conjugates, and consequently the low-|t| region is sensitive to large distances and vice versa. On the other hand, the actual proton size controlled mostly by B qc determines the overall slope of the coherent spectrum. The hot spot size B q then determines the slope of the incoherent cross section in the |t| 1 GeV 2 region: as shown in Ref. [112] the slope of the incoherent spectra at high |t| is given by the size of the smallest fluctuating constituent.\\nThe overall normalization is determined by the Q s /(g 2 µ) parameter and the magnitude of the cross sections constrains that, however, it can not be determined very precisely from our analysis as it is strongly correlated with many other parameters, particularly for the case that N q is a free parameter. Here we note that there is some model uncertainty related to the nonperturbative vector meson wave functions, and different phenomenological parametrizations can result in cross sections that differ by ∼ 20%, see e.g. [43,55,57]. In phenomenological analyses, the so-called skewness correction [55] is sometimes included, which can also have a numerically significant (up to ∼ 40%) effect on the cross section. However, as there are also other model uncertainties related to the overall normalization and the applicability of the skewness correction in our setup is not rigorously justified, it is not included in this work. The small real part correction discussed e.g. in Ref. [55] is also neglected.\\nThe overall normalization is determined by the Q s /(g 2 µ) parameter and the magnitude of the cross sections constrains that, however, it can not be determined very precisely from our analysis as it is strongly correlated with many other parameters, particularly for the case that N q is a free parameter. Here we note that there is some model uncertainty related to the nonperturbative vector meson wave functions, and different phenomenological parametrizations can result in cross sections that differ by ∼ 20%, see e.g. [43,55,57]. In phenomenological analyses, the so-called skewness correction [55] is sometimes included, which can also have a numerically significant (up to ∼ 40%) effect on the cross section. However, as there are also other model uncertainties related to the overall normalization and the applicability of the skewness correction in our setup is not rigorously justified, it is not included in this work. The small real part correction discussed e.g. in Ref. [55] is also neglected.\\nOne can see that when leaving N q variable, its value can not be constrained in our analysis, except that N q ≥ 2 is required in order to get geometry fluctuations that are necessary to describe the incoherent HERA data. At first one would expect the configurations with large N q to be so smooth that event-by-event fluctuations would not be enough to result in a large enough incoherent cross section. However, we note that there is a strong positive correlation between N q and σ, which means that large N q goes along with large hot spot density fluctuations. In this situation one can not really interpret N q as the number of hot spots, but one has to consider effective hot spots, that are generated dynamically from the sum of the N q constituents that are each strongly fluctuating in magnitude. This effective hot spot number will generally be smaller than N q . For a quantitative analysis a hot spot finding prescription, similar to jet clustering algorithms, would be needed. Additional constraints for σ originate from the incoherent cross section at small |t|, which is sensitive to the fluctuations at long-distance scales, i.e., overall density fluctuations [33,45].\\nOne can see that when leaving N q variable, its value can not be constrained in our analysis, except that N q ≥ 2 is required in order to get geometry fluctuations that are necessary to describe the incoherent HERA data. At first one would expect the configurations with large N q to be so smooth that event-by-event fluctuations would not be enough to result in a large enough incoherent cross section. However, we note that there is a strong positive correlation between N q and σ, which means that large N q goes along with large hot spot density fluctuations. In this situation one can not really interpret N q as the number of hot spots, but one has to consider effective hot spots, that are generated dynamically from the sum of the N q constituents that are each strongly fluctuating in magnitude. This effective hot spot number will generally be smaller than N q . For a quantitative analysis a hot spot finding prescription, similar to jet clustering algorithms, would be needed. Additional constraints for σ originate from the incoherent cross section at small |t|, which is sensitive to the fluctuations at long-distance scales, i.e., overall density fluctuations [33,45].\\nThe minimum distance required between the hot spots in three dimensions, d q,Min , can not be constrained at all in our analysis. This parameter is also only very weakly correlated with the other parameters, emphasizing its limited effect on the results. This means that the J/ψ production data allows, but does not require, repulsive short-range correlations used e.g. in Refs. [92,93]. We note that for large N q it is not always possible to fulfill the minimum distance requirement, reducing the effective value of d q,Min that is actually employed.\\nThe minimum distance required between the hot spots in three dimensions, d q,Min , can not be constrained at all in our analysis. This parameter is also only very weakly correlated with the other parameters, emphasizing its limited effect on the results. This means that the J/ψ production data allows, but does not require, repulsive short-range correlations used e.g. in Refs. [92,93]. We note that for large N q it is not always possible to fulfill the minimum distance requirement, reducing the effective value of d q,Min that is actually employed.\\nLet us next study correlations between the model parameters in more detail. First, we observe that there is a clear positive correlation between the infrared regulator m and the hot spot size B q . This can be understood, as increasing m suppresses color fields at large distances and as such results in smaller hot spots. Interestingly, we find no clear correlation between the overall proton size parameter B qc and the infrared regulator m. In fact, the proton size B qc shows no clear correlation with any of the other model parameters. The infrared regulator m is strongly anti-correlated with Q s /(g 2 µ). This is again easy to understand: large m results in a reduction of the normalization of the cross section, which has to be compensated by using a larger color charge density which requires smaller Q s /(g 2 µ).\\nLet us next study correlations between the model parameters in more detail. First, we observe that there is a clear positive correlation between the infrared regulator m and the hot spot size B q . This can be understood, as increasing m suppresses color fields at large distances and as such results in smaller hot spots. Interestingly, we find no clear correlation between the overall proton size parameter B qc and the infrared regulator m. In fact, the proton size B qc shows no clear correlation with any of the other model parameters. The infrared regulator m is strongly anti-correlated with Q s /(g 2 µ). This is again easy to understand: large m results in a reduction of the normalization of the cross section, which has to be compensated by using a larger color charge density which requires smaller Q s /(g 2 µ).\\nWe also find a strong negative correlation between the number of hot spots N q and the proton density Q s /(g 2 µ). This correlation can be understood by considering the effect of these parameters on the incoherent cross section. When the number of hot spots with the same size increases, the incoherent slope is not significantly affected but the normalization goes down as there are smaller fluctuations in the scattering amplitude due to the smoother proton profile. This would result in the incoherent cross section being underestimated, which has to be compensated by smaller Q s /(g 2 µ).\\nWe also find a strong negative correlation between the number of hot spots N q and the proton density Q s /(g 2 µ). This correlation can be understood by considering the effect of these parameters on the incoherent cross section. When the number of hot spots with the same size increases, the incoherent slope is not significantly affected but the normalization goes down as there are smaller fluctuations in the scattering amplitude due to the smoother proton profile. This would result in the incoherent cross section being underestimated, which has to be compensated by smaller Q s /(g 2 µ).\\nThe somewhat surprising negative correlation between the hot spot size B q and the proton density Q s /(g 2 µ) can also be understood. First, we note that large hot spots (large B q ) require a larger infrared regulator m which effectively makes the hot spots smaller (notice a positive correlation between B q and m). Then, to compensate for the effect of the larger infrared regulator on the magnitude of the cross section, a smaller Q s /(g 2 µ) is needed.\\nThe somewhat surprising negative correlation between the hot spot size B q and the proton density Q s /(g 2 µ) can also be understood. First, we note that large hot spots (large B q ) require a larger infrared regulator m which effectively makes the hot spots smaller (notice a positive correlation between B q and m). Then, to compensate for the effect of the larger infrared regulator on the magnitude of the cross section, a smaller Q s /(g 2 µ) is needed.\\nThe posterior distribution of parameters B qc and B q , that determine the proton size, which can be quantified e.g. by the two dimensional RMS radius, r rms = 2(B qc + B q ), are shown in the second and third diagonal panels of Fig. 1. We find that they are sharply peaked, in particular for the case of fixed N q = 3. The extracted proton radius from this Bayesian analysis is r rms = 0.591 +0. 130 −0.071 fm (0.610 +0.065 −0.068 fm) for variable N q (N q ≡ 3) with uncertainty estimates in 90% credible intervals.\\nThe posterior distribution of parameters B qc and B q , that determine the proton size, which can be quantified e.g. by the two dimensional RMS radius, r rms = 2(B qc + B q ), are shown in the second and third diagonal panels of Fig. 1. We find that they are sharply peaked, in particular for the case of fixed N q = 3. The extracted proton radius from this Bayesian analysis is r rms = 0.591 +0. 130 −0.071 fm (0.610 +0.065 −0.068 fm) for variable N q (N q ≡ 3) with uncertainty estimates in 90% credible intervals.\\nWe further included the possibility of fluctuating values of B q and N q , and studied the dependence of the observables on the respective variances. We found that the considered experimental data could not constrain these parameters and decided not to include them in the presented analysis.\\nWe further included the possibility of fluctuating values of B q and N q , and studied the dependence of the observables on the respective variances. We found that the considered experimental data could not constrain these parameters and decided not to include them in the presented analysis.\\nNext, we demonstrate explicitly that sampling parameter values from the posterior distribution lead to a good description of the HERA data. The main result of the Bayesian analysis is the posterior distribution shown in Fig. 1, but it is also possible to find the so-called Maximum a Posteriori (MAP) parameter set, which is the mode of the posterior distribution. Because our prior parameter distributions are uniform, the MAP parameters shown in Table I maximize the likelihood function and provide the best fit to the experimental data. We note that statistically the expectation value for any observable is not obtained using the MAP parameters. Instead, one should calculate observables using averages over parameter samples obtained from the posterior distribution.\\nNext, we demonstrate explicitly that sampling parameter values from the posterior distribution lead to a good description of the HERA data. The main result of the Bayesian analysis is the posterior distribution shown in Fig. 1, but it is also possible to find the so-called Maximum a Posteriori (MAP) parameter set, which is the mode of the posterior distribution. Because our prior parameter distributions are uniform, the MAP parameters shown in Table I maximize the likelihood function and provide the best fit to the experimental data. We note that statistically the expectation value for any observable is not obtained using the MAP parameters. Instead, one should calculate observables using averages over parameter samples obtained from the posterior distribution.\\nComparison to the HERA coherent and incoherent J/ψ production data measured at W = 75 GeV [28] and used in the Bayesian analysis to constrain the model parameters are shown in panel (a) of Fig. 2. The spectra calculated by averaging the results computed using different parametrizations sampled from the posterior distribution indeed provide an excellent description of the data. We also show the statistical uncertainty, obtained by calculating the one standard deviation interval shown as red (coherent) and blue (incoherent) bands.\\nComparison to the HERA coherent and incoherent J/ψ production data measured at W = 75 GeV [28] and used in the Bayesian analysis to constrain the model parameters are shown in panel (a) of Fig. 2. The spectra calculated by averaging the results computed using different parametrizations sampled from the posterior distribution indeed provide an excellent description of the data. We also show the statistical uncertainty, obtained by calculating the one standard deviation interval shown as red (coherent) and blue (incoherent) bands.\\nNext, we study compatibility with experimental data not included in the Bayesian analysis. We do not include full small-x evolution e.g. by means of the JIMWLK [113][114][115][116][117][118][119][120] equation in this work. Consequently, the only center-of-mass energy W (or momentum fraction x P ) dependence comes from the x P dependence of the saturation scale Q s determined from the IPsat fit. This almost only affects the overall normalization of the calculated spectra, and misses important physical effects such as the growth of the proton with decreasing x. Consequently, we do not expect that within our setup we can describe J/ψ production at different center-of-mass energies simultaneously. Nevertheless, the geometry evolution between W = 75 GeV and W = 100 GeV should be weak enough to make predictions for the higher center-of-mass energy.\\nNext, we study compatibility with experimental data not included in the Bayesian analysis. We do not include full small-x evolution e.g. by means of the JIMWLK [113][114][115][116][117][118][119][120] equation in this work. Consequently, the only center-of-mass energy W (or momentum fraction x P ) dependence comes from the x P dependence of the saturation scale Q s determined from the IPsat fit. This almost only affects the overall normalization of the calculated spectra, and misses important physical effects such as the growth of the proton with decreasing x. Consequently, we do not expect that within our setup we can describe J/ψ production at different center-of-mass energies simultaneously. Nevertheless, the geometry evolution between W = 75 GeV and W = 100 GeV should be weak enough to make predictions for the higher center-of-mass energy.\\nThe calculated J/ψ spectra at W = 100 GeV are shown in panel (b) of Fig. 2 and compared with the data from H1 and ZEUS collaborations [29,31,32,121]. Again, we show the model prediction as the average over many samples of the posterior distribution and provide one standard deviation bands. The results are very similar to the W = 75 GeV case studied above, and the HERA data is well described (except the coherent cross section at low |t|, and high |t| in the case of the ZEUS measurement), and variation between the different parametrizations sampled from the posterior distribution is small. Similarity to the W = 75 GeV case is not surprising, as we use exactly the same fluctuating geometry, with the only difference being slightly larger Q s values, extracted from the IPsat parametrization.\\nThe calculated J/ψ spectra at W = 100 GeV are shown in panel (b) of Fig. 2 and compared with the data from H1 and ZEUS collaborations [29,31,32,121]. Again, we show the model prediction as the average over many samples of the posterior distribution and provide one standard deviation bands. The results are very similar to the W = 75 GeV case studied above, and the HERA data is well described (except the coherent cross section at low |t|, and high |t| in the case of the ZEUS measurement), and variation between the different parametrizations sampled from the posterior distribution is small. Similarity to the W = 75 GeV case is not surprising, as we use exactly the same fluctuating geometry, with the only difference being slightly larger Q s values, extracted from the IPsat parametrization.\"\n },\n {\n \"title\": \"V. CONCLUSIONS\",\n \"content\": \"\"\n },\n {\n \"title\": \"V. CONCLUSIONS\",\n \"content\": \"We have performed a statistically rigorous Bayesian analysis to extract posterior likelihood distributions for the non-perturbative parameters describing the event-byevent fluctuating proton geometry as constrained by exclusive J/ψ production data from HERA. We presented a comparison of the |t|-dependent coherent and incoherent cross sections, obtained from an average over many parameter samples from the posterior distribution, to the experimental data.\\nWe have performed a statistically rigorous Bayesian analysis to extract posterior likelihood distributions for the non-perturbative parameters describing the event-byevent fluctuating proton geometry as constrained by exclusive J/ψ production data from HERA. We presented a comparison of the |t|-dependent coherent and incoherent cross sections, obtained from an average over many parameter samples from the posterior distribution, to the experimental data.\\nGenerally, model parametrizations sampled from the determined posterior distribution can be used to systematically take into account uncertainties in the proton geometry as constrained by the HERA DIS data when cal- FIG. 2. Coherent and incoherent J/ψ photoproduction cross sections at W = 75 GeV (a) and W = 100 GeV (b), calculated by averaging over many parameter sets sampled from the posterior distribution, which was determined using the W = 75 GeV HERA data [28]. We compare the case with variable Nq to that with fixed Nq = 3, and to experimental data at W = 75 GeV [28] and W = 100 GeV [29][30][31][32], respectively. The bands show the one standard deviation uncertainty.\\nGenerally, model parametrizations sampled from the determined posterior distribution can be used to systematically take into account uncertainties in the proton geometry as constrained by the HERA DIS data when cal- FIG. 2. Coherent and incoherent J/ψ photoproduction cross sections at W = 75 GeV (a) and W = 100 GeV (b), calculated by averaging over many parameter sets sampled from the posterior distribution, which was determined using the W = 75 GeV HERA data [28]. We compare the case with variable Nq to that with fixed Nq = 3, and to experimental data at W = 75 GeV [28] and W = 100 GeV [29][30][31][32], respectively. The bands show the one standard deviation uncertainty.\\nculating any other observable that depends on the proton geometry, such as flow observables in high-multiplicity proton-proton and proton-nucleus collisions. To enable such studies, we provide 1,000 parametrizations sampled from the determined posterior distributions in the supplemental material.\\nculating any other observable that depends on the proton geometry, such as flow observables in high-multiplicity proton-proton and proton-nucleus collisions. To enable such studies, we provide 1,000 parametrizations sampled from the determined posterior distributions in the supplemental material.\\nBecause the model parameters are sensitive to different aspects of the coherent and incoherent vector meson production spectra, most of them are well constrained by the Bayesian analysis. The only exceptions are the minimum distance between the hot spots (repulsive short-range correlations), and the number of hot spots N q , which cannot be well constrained by the considered HERA data. However, we note that although the analysis suggests that large N q is compatible with the HERA data, for N q 5 one should no longer simply interpret N q as the number of actual hot spots. In this regime, a good description of the data requires large fluctuations of individual hot spots' densities, and the sampled 'hot spots' can also overlap significantly. Thus, the effective number of hot spots is significantly smaller than the parameter N q might imply.\\nBecause the model parameters are sensitive to different aspects of the coherent and incoherent vector meson production spectra, most of them are well constrained by the Bayesian analysis. The only exceptions are the minimum distance between the hot spots (repulsive short-range correlations), and the number of hot spots N q , which cannot be well constrained by the considered HERA data. However, we note that although the analysis suggests that large N q is compatible with the HERA data, for N q 5 one should no longer simply interpret N q as the number of actual hot spots. In this regime, a good description of the data requires large fluctuations of individual hot spots' densities, and the sampled 'hot spots' can also overlap significantly. Thus, the effective number of hot spots is significantly smaller than the parameter N q might imply.\\nThe HERA data used in this work probes the proton structure at x ∼ 10 −3 . The energy (Bjorken-x) dependence can be included in terms of JIMWLK evolution as e.g. in Refs. [86,122]. In the future we plan to extend our framework by including the full JIMWLK evolution and vector meson production data at different center-ofmass energies from HERA [28,121,123] and from the ultra peripheral proton-lead collisions measured at the LHC [39,40,124,125], allowing us to extract also the Bjorken-x dependence of the fluctuating proton geometry.\\nThe HERA data used in this work probes the proton structure at x ∼ 10 −3 . The energy (Bjorken-x) dependence can be included in terms of JIMWLK evolution as e.g. in Refs. [86,122]. In the future we plan to extend our framework by including the full JIMWLK evolution and vector meson production data at different center-ofmass energies from HERA [28,121,123] and from the ultra peripheral proton-lead collisions measured at the LHC [39,40,124,125], allowing us to extract also the Bjorken-x dependence of the fluctuating proton geometry.\\nIn addition to constraining the energy dependence, performing global analyses including both exclusive vector meson production data from HERA and flow harmonics from the proton-proton and proton-lead collisions measured at the LHC (including a model calculation along the lines of [126]) would allow for a powerful global analysis of the fluctuating nucleon substructure and properties of the final state. More differential DIS measurements from the future EIC such as dijet [85] or lepton-meson angular correlations [83] can also provide further constraints and can in principle be included in our framework in a straightforward manner.\\nIn addition to constraining the energy dependence, performing global analyses including both exclusive vector meson production data from HERA and flow harmonics from the proton-proton and proton-lead collisions measured at the LHC (including a model calculation along the lines of [126]) would allow for a powerful global analysis of the fluctuating nucleon substructure and properties of the final state. More differential DIS measurements from the future EIC such as dijet [85] or lepton-meson angular correlations [83] can also provide further constraints and can in principle be included in our framework in a straightforward manner.\\nThe numerical framework for our physics models and the Bayesian analysis package are publicly available on Github [127,128]. Our Bayesian analysis code is developed based on the open-source numerical package by the Duke group [23]. To help visualize how observables depend on the model parameters, we provide an interactive web page with the trained GP emulators [129], where one can also find the posterior samples. expressed therein lies entirely with the authors. This research was done using resources provided by the Open Science Grid (OSG) [130,131], which is supported by the National Science Foundation award #2030508.\\nThe numerical framework for our physics models and the Bayesian analysis package are publicly available on Github [127,128]. Our Bayesian analysis code is developed based on the open-source numerical package by the Duke group [23]. To help visualize how observables depend on the model parameters, we provide an interactive web page with the trained GP emulators [129], where one can also find the posterior samples. expressed therein lies entirely with the authors. This research was done using resources provided by the Open Science Grid (OSG) [130,131], which is supported by the National Science Foundation award #2030508.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I .\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.","content":"FIG. 1 .\n1Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations."},{"id":"","caption":"Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.","content":"FIG. 1 .\n1Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations."},{"id":"","caption":"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views","content":"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views"},{"id":"","caption":"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views","content":"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views"},{"id":"","caption":"Summary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.","content":"TABLE I .\nISummary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.Parameter Description \nPrior range MAP (variable Nq) MAP (Nq ≡ 3) \nm [GeV] \nInfrared regulator \n[0.05, 2] \n0.506 +1.12 \n\n−0.356 \n\n0.246 +0.162 \n\n−0.103 \n\nBqc [GeV −2 ] Proton size \n[1, 10] \n4.02 +1.73 \n\n−0.728 \n\n4.45 +0.801 \n\n−0.803 \n\nBq [GeV −2 ] Hot spot size \n[0.1, 3] \n0.474 +0.434 \n\n−0.286 \n\n0.346 +0.282 \n\n−0.202 \n\nσ \nMagnitude of Qs fluctuations \n[0, 1.5] \n0.833 +0.194 \n\n−0.441 \n\n0.563 +0.143 \n\n−0.141 \n\nQs/(g 2 µ) \nRatio of color charge density and saturation scale [0.2, 1.5] \n0.598 +0.230 \n\n−0.264 \n\n0.747 +0.0704 \n\n−0.0930 \n\ndq,Min [fm] Minimum 3D distance between hot spots \n[0, 0.5] \n0.257 +0.221 \n\n−0.231 \n\n0.254 +0.222 \n\n−0.229 \n\nNq \nNumber of hot spots \n[1, 10] \n6.79 +2.93 \n\n−4.83 \n\n3 \n\nPCs with 1,000 training simulations in the model param-\neter space. In each model parameter point, we generate \n3,000 configurations to compute the coherent and inco-\nherent cross sections. The relative statistical errors are \nwithin 5%. \nAll model parameters and their prior ranges (and Max-\nimum a Posterior values that are discussed in Sec. IV) \nare summarized in"},{"id":"","caption":"Summary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.","content":"TABLE I .\nISummary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.Parameter Description \nPrior range MAP (variable Nq) MAP (Nq ≡ 3) \nm [GeV] \nInfrared regulator \n[0.05, 2] \n0.506 +1.12 \n\n−0.356 \n\n0.246 +0.162 \n\n−0.103 \n\nBqc [GeV −2 ] Proton size \n[1, 10] \n4.02 +1.73 \n\n−0.728 \n\n4.45 +0.801 \n\n−0.803 \n\nBq [GeV −2 ] Hot spot size \n[0.1, 3] \n0.474 +0.434 \n\n−0.286 \n\n0.346 +0.282 \n\n−0.202 \n\nσ \nMagnitude of Qs fluctuations \n[0, 1.5] \n0.833 +0.194 \n\n−0.441 \n\n0.563 +0.143 \n\n−0.141 \n\nQs/(g 2 µ) \nRatio of color charge density and saturation scale [0.2, 1.5] \n0.598 +0.230 \n\n−0.264 \n\n0.747 +0.0704 \n\n−0.0930 \n\ndq,Min [fm] Minimum 3D distance between hot spots \n[0, 0.5] \n0.257 +0.221 \n\n−0.231 \n\n0.254 +0.222 \n\n−0.229 \n\nNq \nNumber of hot spots \n[1, 10] \n6.79 +2.93 \n\n−4.83 \n\n3 \n\nPCs with 1,000 training simulations in the model param-\neter space. In each model parameter point, we generate \n3,000 configurations to compute the coherent and inco-\nherent cross sections. The relative statistical errors are \nwithin 5%. \nAll model parameters and their prior ranges (and Max-\nimum a Posterior values that are discussed in Sec. IV) \nare summarized in"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.\",\n \"content\": \"FIG. 1 .\\n1Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.\",\n \"content\": \"FIG. 1 .\\n1Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views\",\n \"content\": \"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views\"\n },\n {\n \"id\": \"\",\n \"caption\": \"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views\",\n \"content\": \"ACKNOWLEDGMENTS B.P.S. and C.S. are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704 and Award No. DE-SC0021969, respectively. C.S. acknowledges a DOE Office of Science Early Career Award. H.M. is supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and projects 338263 and 346567, and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project (Grant Agreement No. 824093). W.B.Z. is supported by the National Science Foundation (NSF) under grant numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE collaboration. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Summary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.\",\n \"content\": \"TABLE I .\\nISummary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.Parameter Description \\nPrior range MAP (variable Nq) MAP (Nq ≡ 3) \\nm [GeV] \\nInfrared regulator \\n[0.05, 2] \\n0.506 +1.12 \\n\\n−0.356 \\n\\n0.246 +0.162 \\n\\n−0.103 \\n\\nBqc [GeV −2 ] Proton size \\n[1, 10] \\n4.02 +1.73 \\n\\n−0.728 \\n\\n4.45 +0.801 \\n\\n−0.803 \\n\\nBq [GeV −2 ] Hot spot size \\n[0.1, 3] \\n0.474 +0.434 \\n\\n−0.286 \\n\\n0.346 +0.282 \\n\\n−0.202 \\n\\nσ \\nMagnitude of Qs fluctuations \\n[0, 1.5] \\n0.833 +0.194 \\n\\n−0.441 \\n\\n0.563 +0.143 \\n\\n−0.141 \\n\\nQs/(g 2 µ) \\nRatio of color charge density and saturation scale [0.2, 1.5] \\n0.598 +0.230 \\n\\n−0.264 \\n\\n0.747 +0.0704 \\n\\n−0.0930 \\n\\ndq,Min [fm] Minimum 3D distance between hot spots \\n[0, 0.5] \\n0.257 +0.221 \\n\\n−0.231 \\n\\n0.254 +0.222 \\n\\n−0.229 \\n\\nNq \\nNumber of hot spots \\n[1, 10] \\n6.79 +2.93 \\n\\n−4.83 \\n\\n3 \\n\\nPCs with 1,000 training simulations in the model param-\\neter space. In each model parameter point, we generate \\n3,000 configurations to compute the coherent and inco-\\nherent cross sections. The relative statistical errors are \\nwithin 5%. \\nAll model parameters and their prior ranges (and Max-\\nimum a Posterior values that are discussed in Sec. IV) \\nare summarized in\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Summary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.\",\n \"content\": \"TABLE I .\\nISummary of model parameters, their prior ranges, and constrained maximum likelihood values with uncertainty estimates in 90% credible intervals.Parameter Description \\nPrior range MAP (variable Nq) MAP (Nq ≡ 3) \\nm [GeV] \\nInfrared regulator \\n[0.05, 2] \\n0.506 +1.12 \\n\\n−0.356 \\n\\n0.246 +0.162 \\n\\n−0.103 \\n\\nBqc [GeV −2 ] Proton size \\n[1, 10] \\n4.02 +1.73 \\n\\n−0.728 \\n\\n4.45 +0.801 \\n\\n−0.803 \\n\\nBq [GeV −2 ] Hot spot size \\n[0.1, 3] \\n0.474 +0.434 \\n\\n−0.286 \\n\\n0.346 +0.282 \\n\\n−0.202 \\n\\nσ \\nMagnitude of Qs fluctuations \\n[0, 1.5] \\n0.833 +0.194 \\n\\n−0.441 \\n\\n0.563 +0.143 \\n\\n−0.141 \\n\\nQs/(g 2 µ) \\nRatio of color charge density and saturation scale [0.2, 1.5] \\n0.598 +0.230 \\n\\n−0.264 \\n\\n0.747 +0.0704 \\n\\n−0.0930 \\n\\ndq,Min [fm] Minimum 3D distance between hot spots \\n[0, 0.5] \\n0.257 +0.221 \\n\\n−0.231 \\n\\n0.254 +0.222 \\n\\n−0.229 \\n\\nNq \\nNumber of hot spots \\n[1, 10] \\n6.79 +2.93 \\n\\n−4.83 \\n\\n3 \\n\\nPCs with 1,000 training simulations in the model param-\\neter space. In each model parameter point, we generate \\n3,000 configurations to compute the coherent and inco-\\nherent cross sections. The relative statistical errors are \\nwithin 5%. \\nAll model parameters and their prior ranges (and Max-\\nimum a Posterior values that are discussed in Sec. IV) \\nare summarized in\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"R , Abdul Khalek, arXiv:2103.05419Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report. physics.ins-detR. Abdul Khalek et. al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, arXiv:2103.05419 [physics.ins-det].","doi":"arXiv:2103.05419"},{"text":"The electron-ion collider: assessing the energy dependence of key measurements. E C Aschenauer, S Fazio, J H Lee, H Mäntysaari, B S Page, B Schenke, T Ullrich, R Venugopalan, P Zurita, 10.1088/1361-6633/aaf216arXiv:1708.01527Rept. Prog. Phys. 82224301nucl-exE. C. 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Aaij et. al., Updated measurements of exclusive J/ψ and ψ(2S) production cross-sections in pp collisions at √ s = 7 TeV, J. Phys. G 41 (2014) 055002 [arXiv:1401.3288 [hep-ex]].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Energy dependence of exclusive J/ψ photoproduction off protons in ultra-peripheral p-Pb collisions at √ sNN = 5.02 TeV. S Acharya, ALICE collaboration10.1140/epjc/s10052-019-6816-2arXiv:1809.03235Eur. Phys. J. C. 795402nucl-exALICE collaboration, S. Acharya et. al., Energy dependence of exclusive J/ψ photoproduction off protons in ultra-peripheral p-Pb collisions at √ sNN = 5.02 TeV, Eur. Phys. J. C 79 (2019) no. 5 402 [arXiv:1809.03235 [nucl-ex]].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imprints of fluctuating proton shapes on flow in proton-lead collisions at the LHC. H Mäntysaari, B Schenke, C Shen, P Tribedy, 10.1016/j.physletb.2017.07.038arXiv:1705.03177Phys. Lett. B. 772681nucl-thH. Mäntysaari, B. Schenke, C. Shen and P. Tribedy, Imprints of fluctuating proton shapes on flow in proton-lead collisions at the LHC, Phys. Lett. B 772 (2017) 681 [arXiv:1705.03177 [nucl-th]].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The overarching framework for IPGlasma + subnucleonic diffraction model is available at https: //github.com/chunshen1987/IPGlasmaFramework (v1.0.0). It uses the open-source code packagesThe overarching framework for IPGlasma + subnucleonic diffraction model is available at https: //github.com/chunshen1987/IPGlasmaFramework (v1.0.0). It uses the open-source code packages IP-Glasma https://github.com/schenke/ipglasma and the program to compute exclusive J/ψ production cross-section https: //github.com/hejajama/subnucleondiffraction.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Open Science Grid. R Pordes, 10.1088/1742-6596/78/1/012057J. Phys. Conf. Ser. 7812057R. Pordes et. al., The Open Science Grid, J. Phys. Conf. Ser. 78 (2007) 012057.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The pilot way to Grid resources using glideinWMS. I Sfiligoi, D C Bradley, B Holzman, P Mhashilkar, S Padhi, F Wurthwrin, 10.1109/CSIE.2009.950WRI World Congress. 2428I. Sfiligoi, D. C. Bradley, B. Holzman, P. Mhashilkar, S. Padhi and F. Wurthwrin, The pilot way to Grid resources using glideinWMS, WRI World Congress 2 (2009) 428.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":390,"cells":{"text":{"kind":"string","value":"\nSlide FFT on a homogeneous mesh in wafer-scale computing\nJanuary 12, 2024\n\nMaurice H P M Van Putten \nPhysics and Astronomy\nSejong University\nSeoulSouth Korea\n\nINAF-OAS Bologna\nvia P. Gobetti, 101I-40129BolognaItaly\n\nLeighton Wilson \n1237 E. Arques Ave Sunnyvale94085CerebrasCA\n\nAdam W Lavely \nLawrence Berkeley National Laboratory\n1 Cyclotron Rd Berke-ley94720CA\n\nMark Hair \n1237 E. Arques Ave Sunnyvale94085CerebrasCA\n\nSlide FFT on a homogeneous mesh in wafer-scale computing\nJanuary 12, 202430A9592A848948A1840DF0E2BB0F787CarXiv:2401.05427v1[eess.SP]\nSearches for signals at low signal-to-noise ratios frequently involve the Fast Fourier Transform (FFT).For high-throughput searches, we here consider FFT on the homogeneous mesh of Processing Elements (PEs) of a wafer-scale engine (WSE).To minimize memory overhead in the inherently non-local FFT algorithm, we introduce a new synchronous slide operation (Slide) exploiting the fast interconnect between adjacent PEs.Feasibility of compute-limited performance is demonstrated in linear scaling of Slide execution times with varying array size in preliminary benchmarks on the CS-2 WSE.The proposed implementation appears opportune to accelerate and open the full discovery potential of FFT-based signal processing in multi-messenger astronomy.\n\nI. INTRODUCTION\n\nThe Fast Fourier Transform (FFT) is central to signal processing and high-performance computing [9].Highthroughput signal analysis hereby critically depends on FFT performance.The FFT algorithm optimizes efficiency in compute, notably in the Cooley-Tukey algorithm [6] with 5n log 2 n floating point operations (FLOPs) per transform of length n [23].This performance bears out well in CPU-based implementations [e.g.8,11,13,18,19,21].However, real-time performance sensitively depends on memory overhead [16] which varies with the architecture of the computing device.\n\nOptimizing real-time signal processing has been approached on dedicated hardware including fieldprogrammable gate arrays (FPGAs) [e.g.32] and graphics processor units (GPUs) [e.g.5].These approaches can be advantageous in low latency searches for signals at low signal-to-noise ratios (SNR) by FFT-based matched filtering [22].Filtering a continuous stream in real-time requires a filter throughput faster than the input data rate.This is particularly opportune in searches of unmodeled transient signals at detector-limited sensitivity from astrophysical gravitational-wave sources in the Local Universe [26,27].\n\nThe theoretical limit of FFT compute performance is inevitably below theoretical peak compute-performance due to memory overhead on any architecture.This is not surprising since the Fourier transform is non-local.While GPUs offer substantial acceleration over CPUs, performance drops significantly when FFT-array sizes exceed the limits of Local Memory.In fact, the resulting bandwidth-limited performance is significantly below theoretical compute-performance (Fig. 1).\n\nWafer-Scale Engines (WSE) offer a radically new architecture for massively parallel computing on a homogeneous mess of Processing Elements (PEs) with fast on- chip interconnect [20].This appears promising provided memory overhead is kept low relative to compute [12].The Cerebras WSE CS-2 presents a novel architecture in a single chip consisting of 850,000 PEs (Fig. 4).CS-2 utilizes the maximum silicon square that can be cut from a 300 mm diameter wafer: a square array of 84 dies over a surface area of 550 mm 2 .The die is interconnected via a custom interconnect by on-chip routers, one per PE.It presents a homogeneous 2D grid fabric of PEs, each connected to its nearest neighbors North, South, East, and West.At 850 MHz, the 850,000 PEs of CS-2 offer a theoretical performance of 240 TFLOP/s at about 3 clock cycles per FLOP.In FFT transform sizes limited to Local Memory, 3D FFT of size 512 3 (∼ 3 TFLOPs) is realized in ∼ 1 ms [15].\n\nHere, we study memory overhead in FFT ( §2) on a homogeneous mesh ( §3) by synchronous slide of data to facilitate transforms of arbitrary size ( §4).The potential of compute-limited performance is demonstrated on the CS-2 by linear scaling of memory overhead with data-size ( §5).We summarize our outlook in ( §6).\n\n\nII. MEMORY OVERHEAD IN FFT\n\nTo revisit memory overhead in FFT, we consider an array of samples\nD = [d 0 , d 1 , • • • , d n−1 ] of size n = 2 m .\nIt starts with a permutation of indices produced by sequential partitioning in segments of size n/2 p−1 over levels p = 1, 2, • • • , m.Following this pre-processing, FFT is completed over a reverse path by concatenating adjacent elements following crossings on levels p = m, m−1,\n• • • , 1 (further below).\nTo illustrate, index permutation is evaluated over m = log 2 n steps (Appendix A), e.g.,\nI = 0 1 2 3 4 5 6 7 0 2 4 6 1 3 5 7 0 4 2 6 1 5 3 7 (1)\nfor an array of n = 8 elements over m = 3 levels.These permutations are readily tabulated, and therefore are not included in evaluation of efficiency.FFT proceeds by merging adjacent segments\nY = (E O) T of size n/2 p−1 in step p = m, m−1, • • • , 1 of even (E)L = E + O ′ , R = E − O ′ ,(2)\nequivalent to a rotation,\nY ′ = √ 2 0 1 1 0 R π/4 1 0 0 U p Y,(3)\nwhere\nR π/4 = 1 √ 2 1 1 1 −1 .(4)\nHere, the trigonometric part in FFT at level p is in the diagonal matrix\nU p = dia 1, e −i 2π N , • • • , e − 2π N (N/2−1)(5)\nof size N = 2 m−p+1 , representing N/2 samples on the semicircle of S 1 in the lower half-plane with determinant\n|U p | = 1 (p = 1) and |U p | = i (1 < p ≤ m).\nFigure 2 schematically shows\nE × O → RL,(6)\naligning E and O to shared memory for compute (3) and concatenation of output R, L. The transform (6) appears one or multiple times in each level p in mutually independent operations, amenable to embarrassingly parallel computing at each transform level (Fig. 2, Appendix A).This is ideally suited for a uniform mesh of PEs provided memory overhead in alignment is negligible (Fig. 3).\n\n\nIII. PERFORMANCE IN SLIDE FFT\n\nWhen data are distributed over a linear array of PEs, Eq. ( 6) is non-local and incurs data-transfers between the Local Memory of neighboring PEs.In this event, (2) triggers memory overhead in aligning E and O and in concatenation by sliding:\n\n• Forward: aligning E and O for compute in shared memory location;\n\n• Backward: updating the original memory locations of E and O with output L and R, respectively.\n\nIt appears natural to slide E and O to a shared memory location half-way in between (shown in Fig. 3), supporting sufficient allocation for E, O, L and R following compute in the rotation Eq. ( 2).Sliding data over a mesh incurs memory overhead that scales with n/2 p−1 in sliding data.Sliding time depends on the velocity of propagation determined by bandwidth between adjacent processors.On the CS-2, this incurs a minor latency in sending data up or down a PE's ramp connecting its CE to its router which communicates with neighboring PEs.Cumulatively over each transform level\np = m, m − 1, • • • , 1, communication overhead is n/2 + n/4 + • • • 1 ≃ n,\nwhere a and b are per datum.Given aforementioned 5n log 2 n = 5nm FLOPs per FFT, compute efficiency of Slide FFT satisfies\nη = 5bmn an + 5bmn ≃ 1 − α 5 m + O α 5 m 2 .(8)\nIn Slide FFT extending over 2 k PEs -a wave of length k on the WSE -the time to align E and O segments (Fig. 3) scales linearly with the number of array elements stored at each processor, e.g., 2 m−k .For processors with one thread compute time handling Eq. ( 2) scales with the same.While total processing time scales inversely with 2 k , efficiency Eq. ( 7) is invariant under the choice of k.By (8), it follows that efficiency of Slide FFT satisfies\nα 5 m ≪ 1.(9)\nIV. THE SLIDE OPERATION On a WSE, Eq. ( 2) is realized at Eq. ( 9) by standard operations of floating point multiply/add on data local to each PE and one radically new operation when work space exceeds Local Memory.Facilitating array sizes in excess of Local Memory, we propose sliding data arrays over a homogeneous mesh of PEs.\n\nLet p index the PE positions on a rectangular grid of a WSE.We define a Slide of data A at p 0 = &A to B at p 1 = &B, B = * Slide p (&A), (10) by translation over\np = p 1 − p 0 . (11)\nThe memory overhead Eq. ( 10) is expected to satisfy linear scaling with size of A, distributed over Local Memory of the PEs in a choice of wave length covering the desired FFT transform size.This overhead represents a waiting time prior and subsequent to compute in Eq. ( 6), Ideally, this is overhead is invariant of the choice of wave length k, provided it contains sufficient storage in Local Memory aggregated over k PEs.If so, this leaves Eq. ( 7) invariant with respect to k.\n\n\nV. SLIDES ON THE CS-2\n\nOn the CS-2, PE performance per clock cycle is as follows:\n\n• Compute: fused multiply-accomulate (FMAC) in eight 16-bit or two 32-bit FMAC operations by the CE;\n\n• Local Memory: a 128-bit read and a 64-bit write to 48 kB Local Memory (SRAM);\n\n• Router bandwidth: 32-bit packets (bi-directional) between its nearest neighbors.\n\nAdditionally, there is a modest latency in communicating between PEs in data-transfer between a CE and its router, i.e., about 2-7 cycles per a 32-bit packet transfer.\n\nTaken together, the above represents a few clock cycles for a floating point operation (FLOP).On the CS-2, Eq. ( 10) is implemented in the Cerebras Software Language (CSL) by the Cerebras Software Development Kit (SDK 1.0.0),enabling developers to write custom kernels for Cerebras systems.CSL provides direct access to low-level hardware features while providing higher-level constructs such as loops and functions.The SDK provides a CS-2 simulator that allows software to be debugged and optimized without the physical wafer and provides cycle-accurate values for performance evaluation.\n\nFig. 5 shows performance of our Slide operation Eq. ( 10) in the fabric simulator and on a real CS-2 system.Realized is the desired linear scaling of slide-time with data-size in bytes and number of number of PEs involved.\n\nHere, α = 2/3 given a = 2 for a 64 bit datum in complex single precision, while typically b = 3 including communication with Local Memory.Thus, ( 9) is satisfied by an ample margin for typical FFT transform sizes.\n\n\nVI. CONCLUSIONS\n\nThe homogeneous mesh of PEs in WSEs offers a radically new architecture for low latency high-throughput signal processing.Deep searches for signals at small signal-to-noise ratios is frequently pursued by matched filtering evaluated in the Fourier domain by FFT.\n\nSeeking to exploit WSE, we propose compute-limited FFT by a new Slide operation Eq. 10 facilitating arbitrary transform sizes.Performance Eq. ( 8) obtains by exploiting fast on-chip interconnect in Eqs.(8)(9).\n\nSliding data is required when transform sizes exceed the limitations of Local Memory per compute unit -the PE of a WSE.This approach circumvents the need for calls to Global Memory otherwise common in GPUs (Fig. 1), illustrative for the combined challenge of compute and memory overhead in seeking high throughput FFT-based signal processing on architectures different from CPUs.\n\nPreliminary benchmarks demonstrate the desired linear scaling in Slide times with array size (Fig. 4), essentially invariant to wave length -the number of PEs involved in the transform size.The proposed Slide operation hereby seems to provide a suitable starting point to ameliorate memory overhead in FFT on a homogeneous mesh architecture such as CS-2.\n\nTo exploit the full compute potential of CS-2 (240 TFLOPs), preserving the presented performance (Fig. 5) to arbitrary transform sizes extended to batch mode further calls for limiting output back to the host device by post-callback functions, originally developed for implementation on GPUs [25].With such additional steps in place to control memory overhead over the external bus, WSEs such as CS-2 hold realistic promise to accelerate the performance of high-throughput signal analysis by over an order of magnitude over conventional approaches.\n\nThis outlook is of particular interest to searches for un-modeled signals by the FFT-based butterfly matched filtering algorithm [24], nominally over 32 s segments of LIGO-Virgo-KAGRA data comprising n = 2 17 samples [25].This approach recently identified the physical trigger GW170817B -the central engine -of GRB170817A [1,[26][27][28][29].\n\nSupplementary Data.The CSL code of the Cerebras SDK generating the data of Fig. 5 is available on GitHub https://github.com/leightonw-cerebras/sliding-window.\n\nFIG. 1 .\n1\nFIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-place as a function of transform size N in batch mode, shown across memory allocations M = 64 MB and M = 1024MB in Global Memory.Efficiency is limited by memory bandwidth especially when transform sizes exceed the size of Local Memory, noticeably across N = 2 12 , leaving about 8% of theoretical peak compute performance in f32.(After Fig. D.1 of [27].)\n\n\n\n\nand odd (O) indexed elements of the array (with indices permuted).Merging is a concatenation of the transforms Y ′ = (L R) by a sum and difference,\n\n\n\n\nand double this amount including the sliding of L, R and back to the original locations of E and, respectively, O.Given raw compute performance for multiply/add, compute efficiency subject to memory overhead in data transfer across neighboring PEs depends on the number of clock cycles a per transfer and the associated compute, α = #clock cycles per transfer #clock cycles per FLOP = a b ,\n\n\nFIG. 2 .\n2\nFIG. 2. (Left panel.)Following index permutation of input data, FFT takes a path in reverse over m = log 2 n levels p = m, m − 1, • • • , 1.At each p, adjacent segments E, O are concatenated following a rotation (2), amenable to embarrassingly parallel computing of n/2 p crossings.(Right panel.)Crossing diagram of rotations (6) of adjacent segments showing (2) (wiggly line), subsequent to alignment of E and O in shared memory, followed by sliding output L, R back as input to level p − 1.\n\n\nFIG. 4 .\n4\nFIG. 4. Architecture of the wafer-scale engine CS-2 comprising a homogeneous grid of 850,000 PEs (left).Each PE consists of a Compute Element (CE) supported by 48 kB of SRAM Local Memory.Local Memory is connected over a bus to a router enabling bi-directional communication to its nearest neighbors (right).\n\n\nFIG. 5 .FIG. 6 .\n56\nFIG. 5. Performance of Slide Eq. (10) in cycles per array element versus total number of elements in individual benchmarks on the CS-2 (left top) versus simulated in the Cerebras SDK (right top).Data are distributed evenly over 8, 16, and 32 PEs, the number of elements per PE varying from 1 to 500.(Lower panel.)Results for maximal data arrays over 4k PEs (k = 2, 3, • • • , 8) shows performance to be essentially invariant to the number of PEs used by regular convergence to about 1.3 cycles per data element.\n\nACKNOWLEDGMENTSThe authors gratefully acknowledge stimulating discussions with M.A. Abchouyeh.ley 2018)Appendix A: Emulation of FFT FFT is initialized by a permutation (Fig.6) over levels p = 1, 2, • • • , m partitioned by segments of size n/2 p−1 into adjacent I ′ and I ′′ with even (E) and, respectively, odd (O) indices of I:Subsequently, FFT is evaluated by crossings (2) over m levels (2-3), illustrated in Eq. 1 for m = 3 levels.The po-tential of embarrassingly parallel computing follows from the following:In (A2), the loop v = 1 : n/N (left bar) at levels p is amenable to parallel processing on a homogeneous mesh of PEs.\n. M A Abchouyet, M H P M Van Putten, Lorenzo Amati, ApJ. 9521572023\n\n. M Abernathy, ET-0106C-10ET Science Team). 2011Tech. Rep.\n\n. R Abbott, LIGO ScientificApJ. 913L72021\n\n. T Akutsu, KAGRANature Astron. 3352019\n\n. N Bragadeesh, 2015\n\n. J W Cooley, J W Tukey, Math. Comput. 192971965\n\nM Evans, CE Techn. Rep. CE-P2100003-v7. 2021\n\nM Frigo, S G Johnson, 10.1109/JPROC.2004.840301Proc. IEEE 93. IEEE 9320052216\n\n. M Garrido, K K Parhi, J Grajal, IEEE Trans. Circ. Syst. I: Regul. Pap. 5626342009\n\n. V Iglesias, J Grajal, M A Sanchez, M López-Vallejo, IEEE Trans. Instrum. Meas. 643382015\n\nM Lee, N Malaya, R D Moser, SC'13: Proc. Int. Conf. HPC, Networking, Storage and Analysis. IEEE20131\n\nP Li, W Dong, Int. Symp. Parallel Architectures, Algorithms and Programming. 2010369\n\n. T Lu, Y.-F Chen, B Hechtman, T Wang, J Anderson, IEEE Access. 9934222021\n\n. M Maggiore, JCAP. 350\n\nM Orenes-Vera, I Sharapov, R Schreiber, 10.1145/3577193.3593708Proc. 37th Int. Conf. Supercomp. (ICS) 2023. 37th Int. Conf. Supercomp. (ICS) 20232023180\n\nH Jeong, T M Low, P Grover, P , Allerton Conf. Commununication, Control, and Computing. Allerton201888756th Ann\n\n. S M Joshi, Int. J. Comput. Appl. 11612015\n\n. D Pekurovsky, SIAM J. Sc. Comp. 34C1922012\n\nS Plimpton, fftMPI, a distributed-memory parallel FFT library. 2018\n\n. K Rocki, arXiv:2010.03660v12020\n\nD Takahashi, M Sato, T Boku, Int. Workshop OpenMP Appl. and Tools. Springer-Verlag200399\n\nAn introduction to matched filters. G L Turin, IRE T Inform. Theory. 63111960\n\nComputational frameworks for the fast Fourier transform. C Van Loan, Frontiers in Appl. Math. 1992\n\n. M H P M Van Putten, C Guidorzi, F Frontera, ApJ. 7861462014\n\n. M H P Van Putten, PTEP. 2017\n\n. M H P M Van Putten, A Levinson, F Frontera, C Guidorzi, L Amati, M Della Valle, EPJ Plus. 1345472019\n\n. M H P M Van Putten, M Della Valle, A&A. 669A372023\n\n. M H P M Van Putten, 20239279Universe\n\nM H P M Van Putten, 32nd Texas Symp. 2023\n\n. Z Wang, X Liu, B He, F Yu, IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 239732015\n\nS Wong, K Beck, Applied Reconfigurable Computing (LNCS 10216), Proc. 13th Int. Symp. L Bertels, Carro, 2017\n\nR Woods, J Mcallister, Lightbody, Y Yi, FPGAbased Implementation of Signal Processing Systems. Wi-\n"},"corpus_id":{"kind":"number","value":266933113,"string":"266,933,113"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2401.05427\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/abs-2401-05427\",\n \"doi\": \"10.48550/arxiv.2401.05427\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2401.05427v1.pdf\"\n ],\n \"pdf_sha\": \"7d36009d8f2ec065935e642c077c087c53515d86\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Slide FFT on a homogeneous mesh in wafer-scale computing"},"authors":{"kind":"list like","value":["Maurice H P M Van Putten \nPhysics and Astronomy\nSejong University\nSeoulSouth Korea\n\nINAF-OAS Bologna\nvia P. Gobetti, 101I-40129BolognaItaly","Leighton Wilson \n1237 E. Arques Ave Sunnyvale94085CerebrasCA","Adam W Lavely \nLawrence Berkeley National Laboratory\n1 Cyclotron Rd Berke-ley94720CA","Mark Hair \n1237 E. Arques Ave Sunnyvale94085CerebrasCA"],"string":"[\n \"Maurice H P M Van Putten \\nPhysics and Astronomy\\nSejong University\\nSeoulSouth Korea\\n\\nINAF-OAS Bologna\\nvia P. Gobetti, 101I-40129BolognaItaly\",\n \"Leighton Wilson \\n1237 E. Arques Ave Sunnyvale94085CerebrasCA\",\n \"Adam W Lavely \\nLawrence Berkeley National Laboratory\\n1 Cyclotron Rd Berke-ley94720CA\",\n \"Mark Hair \\n1237 E. Arques Ave Sunnyvale94085CerebrasCA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Searches for signals at low signal-to-noise ratios frequently involve the Fast Fourier Transform (FFT).For high-throughput searches, we here consider FFT on the homogeneous mesh of Processing Elements (PEs) of a wafer-scale engine (WSE).To minimize memory overhead in the inherently non-local FFT algorithm, we introduce a new synchronous slide operation (Slide) exploiting the fast interconnect between adjacent PEs.Feasibility of compute-limited performance is demonstrated in linear scaling of Slide execution times with varying array size in preliminary benchmarks on the CS-2 WSE.The proposed implementation appears opportune to accelerate and open the full discovery potential of FFT-based signal processing in multi-messenger astronomy."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"The Fast Fourier Transform (FFT) is central to signal processing and high-performance computing [9].Highthroughput signal analysis hereby critically depends on FFT performance.The FFT algorithm optimizes efficiency in compute, notably in the Cooley-Tukey algorithm [6] with 5n log 2 n floating point operations (FLOPs) per transform of length n [23].This performance bears out well in CPU-based implementations [e.g.8,11,13,18,19,21].However, real-time performance sensitively depends on memory overhead [16] which varies with the architecture of the computing device.\nOptimizing real-time signal processing has been approached on dedicated hardware including fieldprogrammable gate arrays (FPGAs) [e.g.32] and graphics processor units (GPUs) [e.g.5].These approaches can be advantageous in low latency searches for signals at low signal-to-noise ratios (SNR) by FFT-based matched filtering [22].Filtering a continuous stream in real-time requires a filter throughput faster than the input data rate.This is particularly opportune in searches of unmodeled transient signals at detector-limited sensitivity from astrophysical gravitational-wave sources in the Local Universe [26,27].\nThe theoretical limit of FFT compute performance is inevitably below theoretical peak compute-performance due to memory overhead on any architecture.This is not surprising since the Fourier transform is non-local.While GPUs offer substantial acceleration over CPUs, performance drops significantly when FFT-array sizes exceed the limits of Local Memory.In fact, the resulting bandwidth-limited performance is significantly below theoretical compute-performance (Fig. 1).\nWafer-Scale Engines (WSE) offer a radically new architecture for massively parallel computing on a homogeneous mess of Processing Elements (PEs) with fast on- chip interconnect [20].This appears promising provided memory overhead is kept low relative to compute [12].The Cerebras WSE CS-2 presents a novel architecture in a single chip consisting of 850,000 PEs (Fig. 4).CS-2 utilizes the maximum silicon square that can be cut from a 300 mm diameter wafer: a square array of 84 dies over a surface area of 550 mm 2 .The die is interconnected via a custom interconnect by on-chip routers, one per PE.It presents a homogeneous 2D grid fabric of PEs, each connected to its nearest neighbors North, South, East, and West.At 850 MHz, the 850,000 PEs of CS-2 offer a theoretical performance of 240 TFLOP/s at about 3 clock cycles per FLOP.In FFT transform sizes limited to Local Memory, 3D FFT of size 512 3 (∼ 3 TFLOPs) is realized in ∼ 1 ms [15].\nHere, we study memory overhead in FFT ( §2) on a homogeneous mesh ( §3) by synchronous slide of data to facilitate transforms of arbitrary size ( §4).The potential of compute-limited performance is demonstrated on the CS-2 by linear scaling of memory overhead with data-size ( §5).We summarize our outlook in ( §6)."},{"title":"II. MEMORY OVERHEAD IN FFT","content":"To revisit memory overhead in FFT, we consider an array of samples\nIt starts with a permutation of indices produced by sequential partitioning in segments of size n/2 p−1 over levels p = 1, 2, • • • , m.Following this pre-processing, FFT is completed over a reverse path by concatenating adjacent elements following crossings on levels p = m, m−1,\nTo illustrate, index permutation is evaluated over m = log 2 n steps (Appendix A), e.g.,\nfor an array of n = 8 elements over m = 3 levels.These permutations are readily tabulated, and therefore are not included in evaluation of efficiency.FFT proceeds by merging adjacent segments\nequivalent to a rotation,\nwhere\nHere, the trigonometric part in FFT at level p is in the diagonal matrix\nof size N = 2 m−p+1 , representing N/2 samples on the semicircle of S 1 in the lower half-plane with determinant\nFigure 2 schematically shows\naligning E and O to shared memory for compute (3) and concatenation of output R, L. The transform (6) appears one or multiple times in each level p in mutually independent operations, amenable to embarrassingly parallel computing at each transform level (Fig. 2, Appendix A).This is ideally suited for a uniform mesh of PEs provided memory overhead in alignment is negligible (Fig. 3)."},{"title":"III. PERFORMANCE IN SLIDE FFT","content":"When data are distributed over a linear array of PEs, Eq. ( 6) is non-local and incurs data-transfers between the Local Memory of neighboring PEs.In this event, (2) triggers memory overhead in aligning E and O and in concatenation by sliding:\n• Forward: aligning E and O for compute in shared memory location;\n• Backward: updating the original memory locations of E and O with output L and R, respectively.\nIt appears natural to slide E and O to a shared memory location half-way in between (shown in Fig. 3), supporting sufficient allocation for E, O, L and R following compute in the rotation Eq. ( 2).Sliding data over a mesh incurs memory overhead that scales with n/2 p−1 in sliding data.Sliding time depends on the velocity of propagation determined by bandwidth between adjacent processors.On the CS-2, this incurs a minor latency in sending data up or down a PE's ramp connecting its CE to its router which communicates with neighboring PEs.Cumulatively over each transform level\nwhere a and b are per datum.Given aforementioned 5n log 2 n = 5nm FLOPs per FFT, compute efficiency of Slide FFT satisfies\nIn Slide FFT extending over 2 k PEs -a wave of length k on the WSE -the time to align E and O segments (Fig. 3) scales linearly with the number of array elements stored at each processor, e.g., 2 m−k .For processors with one thread compute time handling Eq. ( 2) scales with the same.While total processing time scales inversely with 2 k , efficiency Eq. ( 7) is invariant under the choice of k.By (8), it follows that efficiency of Slide FFT satisfies\nIV. THE SLIDE OPERATION On a WSE, Eq. ( 2) is realized at Eq. ( 9) by standard operations of floating point multiply/add on data local to each PE and one radically new operation when work space exceeds Local Memory.Facilitating array sizes in excess of Local Memory, we propose sliding data arrays over a homogeneous mesh of PEs.\nLet p index the PE positions on a rectangular grid of a WSE.We define a Slide of data A at p 0 = &A to B at p 1 = &B, B = * Slide p (&A), (10) by translation over\nThe memory overhead Eq. ( 10) is expected to satisfy linear scaling with size of A, distributed over Local Memory of the PEs in a choice of wave length covering the desired FFT transform size.This overhead represents a waiting time prior and subsequent to compute in Eq. ( 6), Ideally, this is overhead is invariant of the choice of wave length k, provided it contains sufficient storage in Local Memory aggregated over k PEs.If so, this leaves Eq. ( 7) invariant with respect to k."},{"title":"V. SLIDES ON THE CS-2","content":"On the CS-2, PE performance per clock cycle is as follows:\n• Compute: fused multiply-accomulate (FMAC) in eight 16-bit or two 32-bit FMAC operations by the CE;\n• Local Memory: a 128-bit read and a 64-bit write to 48 kB Local Memory (SRAM);\n• Router bandwidth: 32-bit packets (bi-directional) between its nearest neighbors.\nAdditionally, there is a modest latency in communicating between PEs in data-transfer between a CE and its router, i.e., about 2-7 cycles per a 32-bit packet transfer.\nTaken together, the above represents a few clock cycles for a floating point operation (FLOP).On the CS-2, Eq. ( 10) is implemented in the Cerebras Software Language (CSL) by the Cerebras Software Development Kit (SDK 1.0.0),enabling developers to write custom kernels for Cerebras systems.CSL provides direct access to low-level hardware features while providing higher-level constructs such as loops and functions.The SDK provides a CS-2 simulator that allows software to be debugged and optimized without the physical wafer and provides cycle-accurate values for performance evaluation.\nFig. 5 shows performance of our Slide operation Eq. ( 10) in the fabric simulator and on a real CS-2 system.Realized is the desired linear scaling of slide-time with data-size in bytes and number of number of PEs involved.\nHere, α = 2/3 given a = 2 for a 64 bit datum in complex single precision, while typically b = 3 including communication with Local Memory.Thus, ( 9) is satisfied by an ample margin for typical FFT transform sizes."},{"title":"VI. CONCLUSIONS","content":"The homogeneous mesh of PEs in WSEs offers a radically new architecture for low latency high-throughput signal processing.Deep searches for signals at small signal-to-noise ratios is frequently pursued by matched filtering evaluated in the Fourier domain by FFT.\nSeeking to exploit WSE, we propose compute-limited FFT by a new Slide operation Eq. 10 facilitating arbitrary transform sizes.Performance Eq. ( 8) obtains by exploiting fast on-chip interconnect in Eqs.(8)(9).\nSliding data is required when transform sizes exceed the limitations of Local Memory per compute unit -the PE of a WSE.This approach circumvents the need for calls to Global Memory otherwise common in GPUs (Fig. 1), illustrative for the combined challenge of compute and memory overhead in seeking high throughput FFT-based signal processing on architectures different from CPUs.\nPreliminary benchmarks demonstrate the desired linear scaling in Slide times with array size (Fig. 4), essentially invariant to wave length -the number of PEs involved in the transform size.The proposed Slide operation hereby seems to provide a suitable starting point to ameliorate memory overhead in FFT on a homogeneous mesh architecture such as CS-2.\nTo exploit the full compute potential of CS-2 (240 TFLOPs), preserving the presented performance (Fig. 5) to arbitrary transform sizes extended to batch mode further calls for limiting output back to the host device by post-callback functions, originally developed for implementation on GPUs [25].With such additional steps in place to control memory overhead over the external bus, WSEs such as CS-2 hold realistic promise to accelerate the performance of high-throughput signal analysis by over an order of magnitude over conventional approaches.\nThis outlook is of particular interest to searches for un-modeled signals by the FFT-based butterfly matched filtering algorithm [24], nominally over 32 s segments of LIGO-Virgo-KAGRA data comprising n = 2 17 samples [25].This approach recently identified the physical trigger GW170817B -the central engine -of GRB170817A [1,[26][27][28][29].\nSupplementary Data.The CSL code of the Cerebras SDK generating the data of Fig. 5 is available on GitHub https://github.com/leightonw-cerebras/sliding-window."},{"title":"FIG. 1 .","content":"FIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-place as a function of transform size N in batch mode, shown across memory allocations M = 64 MB and M = 1024MB in Global Memory.Efficiency is limited by memory bandwidth especially when transform sizes exceed the size of Local Memory, noticeably across N = 2 12 , leaving about 8% of theoretical peak compute performance in f32.(After Fig. D.1 of [27].)\nand odd (O) indexed elements of the array (with indices permuted).Merging is a concatenation of the transforms Y ′ = (L R) by a sum and difference,\nand double this amount including the sliding of L, R and back to the original locations of E and, respectively, O.Given raw compute performance for multiply/add, compute efficiency subject to memory overhead in data transfer across neighboring PEs depends on the number of clock cycles a per transfer and the associated compute, α = #clock cycles per transfer #clock cycles per FLOP = a b ,"},{"title":"FIG. 2 .","content":"FIG. 2. (Left panel.)Following index permutation of input data, FFT takes a path in reverse over m = log 2 n levels p = m, m − 1, • • • , 1.At each p, adjacent segments E, O are concatenated following a rotation (2), amenable to embarrassingly parallel computing of n/2 p crossings.(Right panel.)Crossing diagram of rotations (6) of adjacent segments showing (2) (wiggly line), subsequent to alignment of E and O in shared memory, followed by sliding output L, R back as input to level p − 1."},{"title":"FIG. 4 .","content":"FIG. 4. Architecture of the wafer-scale engine CS-2 comprising a homogeneous grid of 850,000 PEs (left).Each PE consists of a Compute Element (CE) supported by 48 kB of SRAM Local Memory.Local Memory is connected over a bus to a router enabling bi-directional communication to its nearest neighbors (right)."},{"title":"FIG. 5 .FIG. 6 .","content":"FIG. 5. Performance of Slide Eq. (10) in cycles per array element versus total number of elements in individual benchmarks on the CS-2 (left top) versus simulated in the Cerebras SDK (right top).Data are distributed evenly over 8, 16, and 32 PEs, the number of elements per PE varying from 1 to 500.(Lower panel.)Results for maximal data arrays over 4k PEs (k = 2, 3, • • • , 8) shows performance to be essentially invariant to the number of PEs used by regular convergence to about 1.3 cycles per data element."}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"The Fast Fourier Transform (FFT) is central to signal processing and high-performance computing [9].Highthroughput signal analysis hereby critically depends on FFT performance.The FFT algorithm optimizes efficiency in compute, notably in the Cooley-Tukey algorithm [6] with 5n log 2 n floating point operations (FLOPs) per transform of length n [23].This performance bears out well in CPU-based implementations [e.g.8,11,13,18,19,21].However, real-time performance sensitively depends on memory overhead [16] which varies with the architecture of the computing device.\\nOptimizing real-time signal processing has been approached on dedicated hardware including fieldprogrammable gate arrays (FPGAs) [e.g.32] and graphics processor units (GPUs) [e.g.5].These approaches can be advantageous in low latency searches for signals at low signal-to-noise ratios (SNR) by FFT-based matched filtering [22].Filtering a continuous stream in real-time requires a filter throughput faster than the input data rate.This is particularly opportune in searches of unmodeled transient signals at detector-limited sensitivity from astrophysical gravitational-wave sources in the Local Universe [26,27].\\nThe theoretical limit of FFT compute performance is inevitably below theoretical peak compute-performance due to memory overhead on any architecture.This is not surprising since the Fourier transform is non-local.While GPUs offer substantial acceleration over CPUs, performance drops significantly when FFT-array sizes exceed the limits of Local Memory.In fact, the resulting bandwidth-limited performance is significantly below theoretical compute-performance (Fig. 1).\\nWafer-Scale Engines (WSE) offer a radically new architecture for massively parallel computing on a homogeneous mess of Processing Elements (PEs) with fast on- chip interconnect [20].This appears promising provided memory overhead is kept low relative to compute [12].The Cerebras WSE CS-2 presents a novel architecture in a single chip consisting of 850,000 PEs (Fig. 4).CS-2 utilizes the maximum silicon square that can be cut from a 300 mm diameter wafer: a square array of 84 dies over a surface area of 550 mm 2 .The die is interconnected via a custom interconnect by on-chip routers, one per PE.It presents a homogeneous 2D grid fabric of PEs, each connected to its nearest neighbors North, South, East, and West.At 850 MHz, the 850,000 PEs of CS-2 offer a theoretical performance of 240 TFLOP/s at about 3 clock cycles per FLOP.In FFT transform sizes limited to Local Memory, 3D FFT of size 512 3 (∼ 3 TFLOPs) is realized in ∼ 1 ms [15].\\nHere, we study memory overhead in FFT ( §2) on a homogeneous mesh ( §3) by synchronous slide of data to facilitate transforms of arbitrary size ( §4).The potential of compute-limited performance is demonstrated on the CS-2 by linear scaling of memory overhead with data-size ( §5).We summarize our outlook in ( §6).\"\n },\n {\n \"title\": \"II. MEMORY OVERHEAD IN FFT\",\n \"content\": \"To revisit memory overhead in FFT, we consider an array of samples\\nIt starts with a permutation of indices produced by sequential partitioning in segments of size n/2 p−1 over levels p = 1, 2, • • • , m.Following this pre-processing, FFT is completed over a reverse path by concatenating adjacent elements following crossings on levels p = m, m−1,\\nTo illustrate, index permutation is evaluated over m = log 2 n steps (Appendix A), e.g.,\\nfor an array of n = 8 elements over m = 3 levels.These permutations are readily tabulated, and therefore are not included in evaluation of efficiency.FFT proceeds by merging adjacent segments\\nequivalent to a rotation,\\nwhere\\nHere, the trigonometric part in FFT at level p is in the diagonal matrix\\nof size N = 2 m−p+1 , representing N/2 samples on the semicircle of S 1 in the lower half-plane with determinant\\nFigure 2 schematically shows\\naligning E and O to shared memory for compute (3) and concatenation of output R, L. The transform (6) appears one or multiple times in each level p in mutually independent operations, amenable to embarrassingly parallel computing at each transform level (Fig. 2, Appendix A).This is ideally suited for a uniform mesh of PEs provided memory overhead in alignment is negligible (Fig. 3).\"\n },\n {\n \"title\": \"III. PERFORMANCE IN SLIDE FFT\",\n \"content\": \"When data are distributed over a linear array of PEs, Eq. ( 6) is non-local and incurs data-transfers between the Local Memory of neighboring PEs.In this event, (2) triggers memory overhead in aligning E and O and in concatenation by sliding:\\n• Forward: aligning E and O for compute in shared memory location;\\n• Backward: updating the original memory locations of E and O with output L and R, respectively.\\nIt appears natural to slide E and O to a shared memory location half-way in between (shown in Fig. 3), supporting sufficient allocation for E, O, L and R following compute in the rotation Eq. ( 2).Sliding data over a mesh incurs memory overhead that scales with n/2 p−1 in sliding data.Sliding time depends on the velocity of propagation determined by bandwidth between adjacent processors.On the CS-2, this incurs a minor latency in sending data up or down a PE's ramp connecting its CE to its router which communicates with neighboring PEs.Cumulatively over each transform level\\nwhere a and b are per datum.Given aforementioned 5n log 2 n = 5nm FLOPs per FFT, compute efficiency of Slide FFT satisfies\\nIn Slide FFT extending over 2 k PEs -a wave of length k on the WSE -the time to align E and O segments (Fig. 3) scales linearly with the number of array elements stored at each processor, e.g., 2 m−k .For processors with one thread compute time handling Eq. ( 2) scales with the same.While total processing time scales inversely with 2 k , efficiency Eq. ( 7) is invariant under the choice of k.By (8), it follows that efficiency of Slide FFT satisfies\\nIV. THE SLIDE OPERATION On a WSE, Eq. ( 2) is realized at Eq. ( 9) by standard operations of floating point multiply/add on data local to each PE and one radically new operation when work space exceeds Local Memory.Facilitating array sizes in excess of Local Memory, we propose sliding data arrays over a homogeneous mesh of PEs.\\nLet p index the PE positions on a rectangular grid of a WSE.We define a Slide of data A at p 0 = &A to B at p 1 = &B, B = * Slide p (&A), (10) by translation over\\nThe memory overhead Eq. ( 10) is expected to satisfy linear scaling with size of A, distributed over Local Memory of the PEs in a choice of wave length covering the desired FFT transform size.This overhead represents a waiting time prior and subsequent to compute in Eq. ( 6), Ideally, this is overhead is invariant of the choice of wave length k, provided it contains sufficient storage in Local Memory aggregated over k PEs.If so, this leaves Eq. ( 7) invariant with respect to k.\"\n },\n {\n \"title\": \"V. SLIDES ON THE CS-2\",\n \"content\": \"On the CS-2, PE performance per clock cycle is as follows:\\n• Compute: fused multiply-accomulate (FMAC) in eight 16-bit or two 32-bit FMAC operations by the CE;\\n• Local Memory: a 128-bit read and a 64-bit write to 48 kB Local Memory (SRAM);\\n• Router bandwidth: 32-bit packets (bi-directional) between its nearest neighbors.\\nAdditionally, there is a modest latency in communicating between PEs in data-transfer between a CE and its router, i.e., about 2-7 cycles per a 32-bit packet transfer.\\nTaken together, the above represents a few clock cycles for a floating point operation (FLOP).On the CS-2, Eq. ( 10) is implemented in the Cerebras Software Language (CSL) by the Cerebras Software Development Kit (SDK 1.0.0),enabling developers to write custom kernels for Cerebras systems.CSL provides direct access to low-level hardware features while providing higher-level constructs such as loops and functions.The SDK provides a CS-2 simulator that allows software to be debugged and optimized without the physical wafer and provides cycle-accurate values for performance evaluation.\\nFig. 5 shows performance of our Slide operation Eq. ( 10) in the fabric simulator and on a real CS-2 system.Realized is the desired linear scaling of slide-time with data-size in bytes and number of number of PEs involved.\\nHere, α = 2/3 given a = 2 for a 64 bit datum in complex single precision, while typically b = 3 including communication with Local Memory.Thus, ( 9) is satisfied by an ample margin for typical FFT transform sizes.\"\n },\n {\n \"title\": \"VI. CONCLUSIONS\",\n \"content\": \"The homogeneous mesh of PEs in WSEs offers a radically new architecture for low latency high-throughput signal processing.Deep searches for signals at small signal-to-noise ratios is frequently pursued by matched filtering evaluated in the Fourier domain by FFT.\\nSeeking to exploit WSE, we propose compute-limited FFT by a new Slide operation Eq. 10 facilitating arbitrary transform sizes.Performance Eq. ( 8) obtains by exploiting fast on-chip interconnect in Eqs.(8)(9).\\nSliding data is required when transform sizes exceed the limitations of Local Memory per compute unit -the PE of a WSE.This approach circumvents the need for calls to Global Memory otherwise common in GPUs (Fig. 1), illustrative for the combined challenge of compute and memory overhead in seeking high throughput FFT-based signal processing on architectures different from CPUs.\\nPreliminary benchmarks demonstrate the desired linear scaling in Slide times with array size (Fig. 4), essentially invariant to wave length -the number of PEs involved in the transform size.The proposed Slide operation hereby seems to provide a suitable starting point to ameliorate memory overhead in FFT on a homogeneous mesh architecture such as CS-2.\\nTo exploit the full compute potential of CS-2 (240 TFLOPs), preserving the presented performance (Fig. 5) to arbitrary transform sizes extended to batch mode further calls for limiting output back to the host device by post-callback functions, originally developed for implementation on GPUs [25].With such additional steps in place to control memory overhead over the external bus, WSEs such as CS-2 hold realistic promise to accelerate the performance of high-throughput signal analysis by over an order of magnitude over conventional approaches.\\nThis outlook is of particular interest to searches for un-modeled signals by the FFT-based butterfly matched filtering algorithm [24], nominally over 32 s segments of LIGO-Virgo-KAGRA data comprising n = 2 17 samples [25].This approach recently identified the physical trigger GW170817B -the central engine -of GRB170817A [1,[26][27][28][29].\\nSupplementary Data.The CSL code of the Cerebras SDK generating the data of Fig. 5 is available on GitHub https://github.com/leightonw-cerebras/sliding-window.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"FIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-place as a function of transform size N in batch mode, shown across memory allocations M = 64 MB and M = 1024MB in Global Memory.Efficiency is limited by memory bandwidth especially when transform sizes exceed the size of Local Memory, noticeably across N = 2 12 , leaving about 8% of theoretical peak compute performance in f32.(After Fig. D.1 of [27].)\\nand odd (O) indexed elements of the array (with indices permuted).Merging is a concatenation of the transforms Y ′ = (L R) by a sum and difference,\\nand double this amount including the sliding of L, R and back to the original locations of E and, respectively, O.Given raw compute performance for multiply/add, compute efficiency subject to memory overhead in data transfer across neighboring PEs depends on the number of clock cycles a per transfer and the associated compute, α = #clock cycles per transfer #clock cycles per FLOP = a b ,\"\n },\n {\n \"title\": \"FIG. 2 .\",\n \"content\": \"FIG. 2. (Left panel.)Following index permutation of input data, FFT takes a path in reverse over m = log 2 n levels p = m, m − 1, • • • , 1.At each p, adjacent segments E, O are concatenated following a rotation (2), amenable to embarrassingly parallel computing of n/2 p crossings.(Right panel.)Crossing diagram of rotations (6) of adjacent segments showing (2) (wiggly line), subsequent to alignment of E and O in shared memory, followed by sliding output L, R back as input to level p − 1.\"\n },\n {\n \"title\": \"FIG. 4 .\",\n \"content\": \"FIG. 4. Architecture of the wafer-scale engine CS-2 comprising a homogeneous grid of 850,000 PEs (left).Each PE consists of a Compute Element (CE) supported by 48 kB of SRAM Local Memory.Local Memory is connected over a bus to a router enabling bi-directional communication to its nearest neighbors (right).\"\n },\n {\n \"title\": \"FIG. 5 .FIG. 6 .\",\n \"content\": \"FIG. 5. Performance of Slide Eq. (10) in cycles per array element versus total number of elements in individual benchmarks on the CS-2 (left top) versus simulated in the Cerebras SDK (right top).Data are distributed evenly over 8, 16, and 32 PEs, the number of elements per PE varying from 1 to 500.(Lower panel.)Results for maximal data arrays over 4k PEs (k = 2, 3, • • • , 8) shows performance to be essentially invariant to the number of PEs used by regular convergence to about 1.3 cycles per data element.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"FIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-place as a function of transform size N in batch mode, shown across memory allocations M = 64 MB and M = 1024MB in Global Memory.Efficiency is limited by memory bandwidth especially when transform sizes exceed the size of Local Memory, noticeably across N = 2 12 , leaving about 8% of theoretical peak compute performance in f32.(After Fig. D.1 of [27].)","content":"FIG. 1 .\n1\nFIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-place as a function of transform size N in batch mode, shown across memory allocations M = 64 MB and M = 1024MB in Global Memory.Efficiency is limited by memory bandwidth especially when transform sizes exceed the size of Local Memory, noticeably across N = 2 12 , leaving about 8% of theoretical peak compute performance in f32.(After Fig. D.1 of [27].)"},{"id":"","caption":"and odd (O) indexed elements of the array (with indices permuted).Merging is a concatenation of the transforms Y ′ = (L R) by a sum and difference,","content":"and odd (O) indexed elements of the array (with indices permuted).Merging is a concatenation of the transforms Y ′ = (L R) by a sum and difference,"},{"id":"","caption":"and double this amount including the sliding of L, R and back to the original locations of E and, respectively, O.Given raw compute performance for multiply/add, compute efficiency subject to memory overhead in data transfer across neighboring PEs depends on the number of clock cycles a per transfer and the associated compute, α = #clock cycles per transfer #clock cycles per FLOP = a b ,","content":"and double this amount including the sliding of L, R and back to the original locations of E and, respectively, O.Given raw compute performance for multiply/add, compute efficiency subject to memory overhead in data transfer across neighboring PEs depends on the number of clock cycles a per transfer and the associated compute, α = #clock cycles per transfer #clock cycles per FLOP = a b ,"},{"id":"","caption":"FIG. 2. (Left panel.)Following index permutation of input data, FFT takes a path in reverse over m = log 2 n levels p = m, m − 1, • • • , 1.At each p, adjacent segments E, O are concatenated following a rotation (2), amenable to embarrassingly parallel computing of n/2 p crossings.(Right panel.)Crossing diagram of rotations (6) of adjacent segments showing (2) (wiggly line), subsequent to alignment of E and O in shared memory, followed by sliding output L, R back as input to level p − 1.","content":"FIG. 2 .\n2\nFIG. 2. (Left panel.)Following index permutation of input data, FFT takes a path in reverse over m = log 2 n levels p = m, m − 1, • • • , 1.At each p, adjacent segments E, O are concatenated following a rotation (2), amenable to embarrassingly parallel computing of n/2 p crossings.(Right panel.)Crossing diagram of rotations (6) of adjacent segments showing (2) (wiggly line), subsequent to alignment of E and O in shared memory, followed by sliding output L, R back as input to level p − 1."},{"id":"","caption":"FIG. 4. Architecture of the wafer-scale engine CS-2 comprising a homogeneous grid of 850,000 PEs (left).Each PE consists of a Compute Element (CE) supported by 48 kB of SRAM Local Memory.Local Memory is connected over a bus to a router enabling bi-directional communication to its nearest neighbors (right).","content":"FIG. 4 .\n4\nFIG. 4. Architecture of the wafer-scale engine CS-2 comprising a homogeneous grid of 850,000 PEs (left).Each PE consists of a Compute Element (CE) supported by 48 kB of SRAM Local Memory.Local Memory is connected over a bus to a router enabling bi-directional communication to its nearest neighbors (right)."},{"id":"","caption":"FIG. 5. Performance of Slide Eq. (10) in cycles per array element versus total number of elements in individual benchmarks on the CS-2 (left top) versus simulated in the Cerebras SDK (right top).Data are distributed evenly over 8, 16, and 32 PEs, the number of elements per PE varying from 1 to 500.(Lower panel.)Results for maximal data arrays over 4k PEs (k = 2, 3, • • • , 8) shows performance to be essentially invariant to the number of PEs used by regular convergence to about 1.3 cycles per data element.","content":"FIG. 5 .FIG. 6 .\n56\nFIG. 5. Performance of Slide Eq. (10) in cycles per array element versus total number of elements in individual benchmarks on the CS-2 (left top) versus simulated in the Cerebras SDK (right top).Data are distributed evenly over 8, 16, and 32 PEs, the number of elements per PE varying from 1 to 500.(Lower panel.)Results for maximal data arrays over 4k PEs (k = 2, 3, • • • , 8) shows performance to be essentially invariant to the number of PEs used by regular convergence to about 1.3 cycles per data element."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"FIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-place as a function of transform size N in batch mode, shown across memory allocations M = 64 MB and M = 1024MB in Global Memory.Efficiency is limited by memory bandwidth especially when transform sizes exceed the size of Local Memory, noticeably across N = 2 12 , leaving about 8% of theoretical peak compute performance in f32.(After Fig. D.1 of [27].)\",\n \"content\": \"FIG. 1 .\\n1\\nFIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-place as a function of transform size N in batch mode, shown across memory allocations M = 64 MB and M = 1024MB in Global Memory.Efficiency is limited by memory bandwidth especially when transform sizes exceed the size of Local Memory, noticeably across N = 2 12 , leaving about 8% of theoretical peak compute performance in f32.(After Fig. D.1 of [27].)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and odd (O) indexed elements of the array (with indices permuted).Merging is a concatenation of the transforms Y ′ = (L R) by a sum and difference,\",\n \"content\": \"and odd (O) indexed elements of the array (with indices permuted).Merging is a concatenation of the transforms Y ′ = (L R) by a sum and difference,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and double this amount including the sliding of L, R and back to the original locations of E and, respectively, O.Given raw compute performance for multiply/add, compute efficiency subject to memory overhead in data transfer across neighboring PEs depends on the number of clock cycles a per transfer and the associated compute, α = #clock cycles per transfer #clock cycles per FLOP = a b ,\",\n \"content\": \"and double this amount including the sliding of L, R and back to the original locations of E and, respectively, O.Given raw compute performance for multiply/add, compute efficiency subject to memory overhead in data transfer across neighboring PEs depends on the number of clock cycles a per transfer and the associated compute, α = #clock cycles per transfer #clock cycles per FLOP = a b ,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 2. (Left panel.)Following index permutation of input data, FFT takes a path in reverse over m = log 2 n levels p = m, m − 1, • • • , 1.At each p, adjacent segments E, O are concatenated following a rotation (2), amenable to embarrassingly parallel computing of n/2 p crossings.(Right panel.)Crossing diagram of rotations (6) of adjacent segments showing (2) (wiggly line), subsequent to alignment of E and O in shared memory, followed by sliding output L, R back as input to level p − 1.\",\n \"content\": \"FIG. 2 .\\n2\\nFIG. 2. (Left panel.)Following index permutation of input data, FFT takes a path in reverse over m = log 2 n levels p = m, m − 1, • • • , 1.At each p, adjacent segments E, O are concatenated following a rotation (2), amenable to embarrassingly parallel computing of n/2 p crossings.(Right panel.)Crossing diagram of rotations (6) of adjacent segments showing (2) (wiggly line), subsequent to alignment of E and O in shared memory, followed by sliding output L, R back as input to level p − 1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 4. Architecture of the wafer-scale engine CS-2 comprising a homogeneous grid of 850,000 PEs (left).Each PE consists of a Compute Element (CE) supported by 48 kB of SRAM Local Memory.Local Memory is connected over a bus to a router enabling bi-directional communication to its nearest neighbors (right).\",\n \"content\": \"FIG. 4 .\\n4\\nFIG. 4. Architecture of the wafer-scale engine CS-2 comprising a homogeneous grid of 850,000 PEs (left).Each PE consists of a Compute Element (CE) supported by 48 kB of SRAM Local Memory.Local Memory is connected over a bus to a router enabling bi-directional communication to its nearest neighbors (right).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 5. Performance of Slide Eq. (10) in cycles per array element versus total number of elements in individual benchmarks on the CS-2 (left top) versus simulated in the Cerebras SDK (right top).Data are distributed evenly over 8, 16, and 32 PEs, the number of elements per PE varying from 1 to 500.(Lower panel.)Results for maximal data arrays over 4k PEs (k = 2, 3, • • • , 8) shows performance to be essentially invariant to the number of PEs used by regular convergence to about 1.3 cycles per data element.\",\n \"content\": \"FIG. 5 .FIG. 6 .\\n56\\nFIG. 5. Performance of Slide Eq. (10) in cycles per array element versus total number of elements in individual benchmarks on the CS-2 (left top) versus simulated in the Cerebras SDK (right top).Data are distributed evenly over 8, 16, and 32 PEs, the number of elements per PE varying from 1 to 500.(Lower panel.)Results for maximal data arrays over 4k PEs (k = 2, 3, • • • , 8) shows performance to be essentially invariant to the number of PEs used by regular convergence to about 1.3 cycles per data element.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". M A Abchouyet, M H P M Van Putten, Lorenzo Amati, ApJ. 9521572023","doi":""},{"text":". M Abernathy, ET-0106C-10ET Science Team). 2011Tech. Rep.","doi":""},{"text":". R Abbott, LIGO ScientificApJ. 913L72021","doi":""},{"text":". T Akutsu, KAGRANature Astron. 3352019","doi":""},{"text":". N Bragadeesh, 2015","doi":""},{"text":". J W Cooley, J W Tukey, Math. Comput. 192971965","doi":""},{"text":"M Evans, CE Techn. Rep. CE-P2100003-v7. 2021","doi":""},{"text":"M Frigo, S G Johnson, 10.1109/JPROC.2004.840301Proc. IEEE 93. IEEE 9320052216","doi":""},{"text":". M Garrido, K K Parhi, J Grajal, IEEE Trans. Circ. Syst. I: Regul. Pap. 5626342009","doi":""},{"text":". V Iglesias, J Grajal, M A Sanchez, M López-Vallejo, IEEE Trans. Instrum. Meas. 643382015","doi":""},{"text":"M Lee, N Malaya, R D Moser, SC'13: Proc. Int. Conf. HPC, Networking, Storage and Analysis. IEEE20131","doi":""},{"text":"P Li, W Dong, Int. Symp. Parallel Architectures, Algorithms and Programming. 2010369","doi":""},{"text":". T Lu, Y.-F Chen, B Hechtman, T Wang, J Anderson, IEEE Access. 9934222021","doi":""},{"text":". M Maggiore, JCAP. 350","doi":""},{"text":"M Orenes-Vera, I Sharapov, R Schreiber, 10.1145/3577193.3593708Proc. 37th Int. Conf. Supercomp. (ICS) 2023. 37th Int. Conf. Supercomp. (ICS) 20232023180","doi":""},{"text":"H Jeong, T M Low, P Grover, P , Allerton Conf. Commununication, Control, and Computing. Allerton201888756th Ann","doi":""},{"text":". S M Joshi, Int. J. Comput. Appl. 11612015","doi":""},{"text":". D Pekurovsky, SIAM J. Sc. Comp. 34C1922012","doi":""},{"text":"S Plimpton, fftMPI, a distributed-memory parallel FFT library. 2018","doi":""},{"text":". K Rocki, arXiv:2010.03660v12020","doi":""},{"text":"D Takahashi, M Sato, T Boku, Int. Workshop OpenMP Appl. and Tools. Springer-Verlag200399","doi":""},{"text":"An introduction to matched filters. G L Turin, IRE T Inform. Theory. 63111960","doi":""},{"text":"Computational frameworks for the fast Fourier transform. C Van Loan, Frontiers in Appl. Math. 1992","doi":""},{"text":". M H P M Van Putten, C Guidorzi, F Frontera, ApJ. 7861462014","doi":""},{"text":". M H P Van Putten, PTEP. 2017","doi":""},{"text":". M H P M Van Putten, A Levinson, F Frontera, C Guidorzi, L Amati, M Della Valle, EPJ Plus. 1345472019","doi":""},{"text":". M H P M Van Putten, M Della Valle, A&A. 669A372023","doi":""},{"text":". M H P M Van Putten, 20239279Universe","doi":""},{"text":"M H P M Van Putten, 32nd Texas Symp. 2023","doi":""},{"text":". Z Wang, X Liu, B He, F Yu, IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 239732015","doi":""},{"text":"S Wong, K Beck, Applied Reconfigurable Computing (LNCS 10216), Proc. 13th Int. Symp. L Bertels, Carro, 2017","doi":""},{"text":"R Woods, J Mcallister, Lightbody, Y Yi, FPGAbased Implementation of Signal Processing Systems. Wi-","doi":""}],"string":"[\n {\n \"text\": \". M A Abchouyet, M H P M Van Putten, Lorenzo Amati, ApJ. 9521572023\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Abernathy, ET-0106C-10ET Science Team). 2011Tech. Rep.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Abbott, LIGO ScientificApJ. 913L72021\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Akutsu, KAGRANature Astron. 3352019\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Bragadeesh, 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J W Cooley, J W Tukey, Math. Comput. 192971965\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Evans, CE Techn. Rep. CE-P2100003-v7. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Frigo, S G Johnson, 10.1109/JPROC.2004.840301Proc. IEEE 93. IEEE 9320052216\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Garrido, K K Parhi, J Grajal, IEEE Trans. Circ. Syst. I: Regul. Pap. 5626342009\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Iglesias, J Grajal, M A Sanchez, M López-Vallejo, IEEE Trans. Instrum. Meas. 643382015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Lee, N Malaya, R D Moser, SC'13: Proc. Int. Conf. HPC, Networking, Storage and Analysis. IEEE20131\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Li, W Dong, Int. Symp. Parallel Architectures, Algorithms and Programming. 2010369\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Lu, Y.-F Chen, B Hechtman, T Wang, J Anderson, IEEE Access. 9934222021\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Maggiore, JCAP. 350\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Orenes-Vera, I Sharapov, R Schreiber, 10.1145/3577193.3593708Proc. 37th Int. Conf. Supercomp. (ICS) 2023. 37th Int. Conf. Supercomp. (ICS) 20232023180\",\n \"doi\": \"\"\n },\n {\n \"text\": \"H Jeong, T M Low, P Grover, P , Allerton Conf. Commununication, Control, and Computing. Allerton201888756th Ann\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S M Joshi, Int. J. Comput. Appl. 11612015\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Pekurovsky, SIAM J. Sc. Comp. 34C1922012\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Plimpton, fftMPI, a distributed-memory parallel FFT library. 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Rocki, arXiv:2010.03660v12020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Takahashi, M Sato, T Boku, Int. Workshop OpenMP Appl. and Tools. Springer-Verlag200399\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An introduction to matched filters. G L Turin, IRE T Inform. Theory. 63111960\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Computational frameworks for the fast Fourier transform. C Van Loan, Frontiers in Appl. Math. 1992\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M H P M Van Putten, C Guidorzi, F Frontera, ApJ. 7861462014\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M H P Van Putten, PTEP. 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M H P M Van Putten, A Levinson, F Frontera, C Guidorzi, L Amati, M Della Valle, EPJ Plus. 1345472019\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M H P M Van Putten, M Della Valle, A&A. 669A372023\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M H P M Van Putten, 20239279Universe\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M H P M Van Putten, 32nd Texas Symp. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z Wang, X Liu, B He, F Yu, IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 239732015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Wong, K Beck, Applied Reconfigurable Computing (LNCS 10216), Proc. 13th Int. Symp. L Bertels, Carro, 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Woods, J Mcallister, Lightbody, Y Yi, FPGAbased Implementation of Signal Processing Systems. Wi-\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":391,"cells":{"text":{"kind":"string","value":"\nComputation of the thermal conductivity using classical and quantum molecular dynamics based methods\n9 Jul 2013 (Dated: July 8, 2013)\n\nNatalia Bedoya \nJean-Louis Barrat \nDavid Rodney \n\nLaboratory for Interdisciplinary Physics\nNanosciences Foundation\n23 rue des martyrs38000GrenobleFrance\n\n\nUMR 5588\nUniversité Grenoble\n\n\n\nSIMaP, Grenoble INP, UJF, CNRS, UMR 5266\nCNRS\nSaint Martin d'Hères38402, F-38401Saint Martin d'HèresFrance, France\n\nComputation of the thermal conductivity using classical and quantum molecular dynamics based methods\n9 Jul 2013 (Dated: July 8, 2013)\nThe thermal conductivity of a model for solid argon is investigated using nonequilibrium molecular dynamics methods, as well as the traditional Boltzmann transport equation approach with input from molecular dynamics calculations, both with classical and quantum thermostats. A surprising result is that, at low temperatures, only the classical molecular dynamics technique is in agreement with the experimental data. We argue that this agreement is due to a compensation of errors, and raise the issue of an appropriate method for calculating thermal conductivities at low (below Debye) temperatures.\n\nThe thermal conductivity of a model for solid argon is investigated using nonequilibrium molecular dynamics methods, as well as the traditional Boltzmann transport equation approach with input from molecular dynamics calculations, both with classical and quantum thermostats. A surprising result is that, at low temperatures, only the classical molecular dynamics technique is in agreement with the experimental data. We argue that this agreement is due to a compensation of errors, and raise the issue of an appropriate method for calculating thermal conductivities at low (below Debye) temperatures.\n\n\nI. INTRODUCTION\n\nDifferent methods are available for calculating thermal conductivities of crystalline solids. [1] The most standard approach involves a calculation of the phonon properties of the system, which are connected to the thermal conductivity through the Boltzmann transport equation. Alternative methods are based on equilibrium molecular dynamics (MD) and non-equilibrium molecular dynamics (NEMD). [1][2][3] The methods based on MD or NEMD are restricted to the classical limit, i.e. the limit of high temperatures. In standard MD, nuclear degrees of freedom are treated classically and quantum effects such as zero-point vibrations are not accounted for. In order to incorporate quantum effects, corrections to the thermal conductivity, based on a rescaling of the heat capacity, are commonly applied. These kind of corrections, however, are not generally accepted as reliable. Turney et al. [4] discussed their validity, and showed that this approach is oversimplified and is not generally applicable, while other authors have found an improvement of the classical thermal conductivity by applying such corrections. [5,6] Quantum effects, on the other hand, are assumed negligible depending on the capability of the classical description to describe the thermal conductivity, but independently of its limitations for predicting heat capacities or phonon lifetimes, properties directly related with the thermal conductivity. A common example is the case of solid argon. In spite of the limitations of the classical theory to predict correctly the heat capacity, a reliable description of the thermal conductivity, at temperatures well below the Debye value, is obtained from classical molecular dynamics. [7,8] Quantum effects on the thermal conductivity can be obtained from anharmonic lattice dynamics, by using the Boltzmann transport equation. [9] This methodology, nonetheless, results in much more expensive computations than molecular dynamics. It requires the full calculation of the vibrational spectrum of the system, as well as the third derivatives of the energy, something unmanageable for large or aperiodic systems. Moreover, the Boltzmann transport calculation relies on approximate theoretical expressions for the phonon lifetime and for the conductivity itself, as opposed to the MD and NEMD formalisms that are in principle exact.\n\nRecently, a Langevin type thermostat with a coloured noise was proposed [10,11] and implemented [12] by different authors in order to incorporate quantum effects in molecular dynamics. The quantum thermostat allows to recover the correct average quantum energy of a system by coupling every degree of freedom to a fictitious quantum bath, in such a way that a harmonic oscillator acquires an energy given by the Bose-Einstein distribution. As such the method is expected to provide a good description of solids in the harmonic limit, and has been shown to also work well for low temperature liquids, in terms of static properties. [10] At high temperatures, the quantum thermostat reduces to a standard Langevin thermostat. This semi-classical approach offers the possibility of performing direct thermal conductivity calculations, by using molecular dynamics, independent of the temperature regime. Savin et al. [13] have applied this methodology to the study of heat transport in low-dimensional nanostructures from non-equilibrium molecular dynamics (NEMD). In the case of a NEMD simulation there are regions of the system free of thermostat, and one will have to check the validity of the quantum thermostat under such conditions. Moreover, the quantum thermostat is not an exact representation of the quantum behavior, and for anharmonic systems suffers from \"zero-point energy leakage\" (see Ref. 11 and section III below). It is unknown if this influences thermal transport properties. Here we present an overview of the advances and challenges for using such kind of thermostats to address thermal transport studies at low temperatures (below the Debye temperature T D ). In our study we use different MD based methods for calculating the thermal conductivity of solid argon, a simple system that is well described in the literature and, as pointed out before, is particularly well described by classical MD.\n\nThe paper is organized as follows. In Sec. II we present the various methods used for estimating the thermal conductivity. In Sec. III we briefly introduce the quantum thermostat, we discuss about the zero-point energy leakage problem and its reliability for working under equilibrium and non-equilibrium conditions. In the last section we present and discuss our results for the thermal conductivity of solid Argon.\n\n\nII. METHODOLOGY\n\nThe standard methods to compute thermal conductivities are based either on molecular dynamics or lattice dynamics, or a combination of both. In this work, we used non-equilibrium molecular dynamics (NEMD) [1][2][3] and Boltzmann transport equation molecular dynamics (BTE-MD). [14,15] We did not use Green-Kubo based methods, [16,17] which have smaller size effects than NEMD, because the quantum thermostat is not compatible with this approach (see discussion below). However, the cells employed here are large enough to avoid any strong size effects in NEMD.\n\nIn NEMD, the periodic simulation cell is divided into N slabs, and a temperature gradient is imposed by coupling two selected slabs to two thermostats at different temperatures, T 1 and T 2 with T 1 < T 2 . In a periodic system, the thermostated slabs are separated by a distance equal to one half of the simulation cell length. The remaining slabs are not thermostated. The system is then allowed to reach a steady state, where on average the energy creation rate of the thermostat at T 2 is equal to the energy removal rate of the thermostat at T 1 . Calculating the heat flux j i required to maintain the gradient of temperature ∇ j T from the heat power of the source and the sink, one can estimate the thermal conductivity from Fourier's law:\nj i = −κ ij ∇ j T.(1)\nIn this work we assumed materials of isotopic symmetry. The thermal conductivity is then a scalar, and the temperature gradient and heat flux are parallel. Equation (1) can alternatively be implemented by imposing the heat flux J and calculating the resulting temperature gradient. A common approach in this case, [2,3] is to rescale the velocities, v h , of the atoms in the hot region according to\nv h ′ = v G + α( v h − v G ),(2)\nwhere v G is the velocity of the center of mass of the region, and\nα = 1 + ∆ ǫ k R .(3)\nHere ∆ ǫ is the amount of heat transferred through the system, and k R is the relative kinetic energy given by\nk R = 1 2 i∈hot m i v 2 i − 1 2 i∈hot m i v 2 G .(4)\nIn this manner, a constant heat flux\nJ = ∆ ǫ 2 A ∆ t(5)\nis imposed, where A is the cross sectional area of the simulation cell perpendicular to the heat flow, and ∆t is the time step. We implemented both NEMD methods and checked that they are fully consistent with one another. In the following, we will not distinguish between them and will simply refer to them as the NEMD approach. An alternative expression for the thermal conductivity of an isotropic material reads [9,14] \nκ = q 3 (N −1) ν C ph ( q, ν) v 2 g ( q, ν) τ ( q, ν),(6)\nwhere C ph is the volumetric phonon specific heat, v g is the phonon group velocity and τ the phonon lifetime.\n\nThe sum runs over all wave vectors, q, within the Brillouin zone of the periodic structure, and over the 3 N polarization indices, where N is the number of atoms in the elementary cell under consideration, so that contributions from all normal modes of the system are considered. In Eq. (6), the specific heats and group velocities can be computed using lattice dynamics, while the phonon lifetimes can be obtained using either lattice dynamics or a combination of lattice dynamics and molecular dynamics. [9,14] In the following, we will use only the latter approach, referred to as the BTE-MD method. Group velocities are obtained by evaluating the derivative of the dispersion curves, ω(q), over a set of 6 qpoints within a radius 0.0001Å −1 around the Γ point. Phonon frequencies and specific heats are calculated at the Γ point, in supercells that contain from 256 to 4000 atoms.\n\nPhonon lifetimes are obtained from the energy autocorrelation function of each normal mode\nE q,ν (t) = Q * ( q, ν) Q( q, ν) 2 + (7) ω 2 ( q, ν)Q * ( q, ν)Q( q, ν) 2 , with Q( q, ν) = N j=1 m j N 1/2 exp[−i q · r j,0 ] × (8) e ( q, ν) · [ r j − r j,0 ]\nthe time-dependent normal mode coordinate. The eigenvectors e( q, ν) are obtained from lattice dynamics, and the relative displacement, r j − r j,0 , of atom j, is sampled using molecular dynamics. The phonon lifetimes τ are then obtained by fitting the following relation\ne −t/τ ( q,ν) = E q,ν (t) E q,ν (0) E q,ν (0) E q,ν (0) . (9) III. QUANTUM THERMOSTAT A. Overview\nThe key idea behind the quantum thermostat is to adjust to the manner in which energy is distributed among the normal modes of a harmonic system. In the classical limit, the equipartition theorem is fulfilled and all modes have the same energy, while in the quantum regime, the energy of each mode is distributed according to Bose-Einstein statistics. The quantum Langevin thermostat enforces this distribution by using a frequency-dependent noise function (coloured noise).\n\nAs in the classical approach using a Langevin thermostat, each particle is coupled to a fictitious bath by including in the equations of motion a random force and a dissipation term related by the fluctuation-dissipation theorem. [18] Accordingly, the equation of motion of a degree of freedom x, of a particle of mass m, in presence of an external force F (x), becomes\nmẍ = −γ mẋ + F (x) + 2 m γ Θ(t),(10)\nwhere Θ(t) is a coloured noise with a power spectral density (PSD) given by the Bose-Einstein distributioñ\nΘ(ω) = e −iω t Θ(t) Θ(t ′ ) d t (11) = |ω| 1 2 + 1 e | ω | kB T − 1 ,\nincluding the zero point energy. The classical regime is recovered at high temperature, where the above PSD becomes independent of the frequency and equals k B T . [19] We also note here that the use of a Langevin equation implies the absence of local energy conservation; hence a Green-Kubo approach, based on the notion that local energy fluctuations undergo a diffusive motion, is not appropriate in a system that is coupled to a local (quantum or classical) heat bath. In practice,Θ(ω) is generated by using a signal processing method based on filtering a white noise. [20] A filterH (ω) = Θ (ω) (12) with Fourier transform H(t) is introduced, and Θ(t) is obtained by convoluting H(t) with a random white noise, r(t), of power spectral densityR(ω)=1, such that\nΘ(t) = ∞ −∞ H(s) r(t − s) ds.(13)\nThus, the power spectral density of the resulting noise is\nH (ω) 2R (ω) =Θ(ω),(14)\nwhich satisfies Eq. (11). The method is simple and can be easily implemented in a discrete molecular dynamics algorithm. From a computational point of view, the quantum thermostat does not slow down the calculations, the only difference with a classical thermostat being the convolution operation in Eq. (13). In terms of memory, the quantum thermostat is more demanding because it requires to store a finite number of past values of the white noise and of the filter in order to compute Eq. (13). However, the memory requirement for the thermostat scales linearly with the system size and is easily manageable with current computers. Moreover, it avoids generating and storing the entire time-series of random numbers as done in other implementations of the quantum thermal bath. [10] Further details concerning the method are given in Ref. 12.\n\nB. Zero-point energy leakage By coupling a system to the quantum thermostat, each harmonic mode can in principle be equilibrated at the correct quantum harmonic energy given by Eq. (11). However, as the equations that are solved describe classical coordinates, the zero point energy in these equations corresponds to the finite amplitude vibration of a classical coordinate. As such this zero point energy can be exchanged between modes, in contrast with a true quantum zero point energy.\n\nSuch an exchange becomes possible when an anharmonic coupling between the modes is introduced, and leads to the phenomenon of \"zero-point energy leakage\" (ZPE), where the zero point energy is transferred from the high-energy modes to low-energy modes, so as to homogenize the energy among the modes. [21][22][23][24][25] As the thermostat cannot fully counterbalance the leakage, an equilibrium is reached where the energy per mode is neither constant, nor as inhomogeneous as in Bose-Einstein distribution. An example is shown in Fig. 1 (left panel) in the case of a perfect crystal of aluminum at 10 K modeled with a Lennard-Jones potential. A coloured noise with power spectral densityΘ A (ω) =Θ(ω) directly from Eq. (11) was used and results in an excess of energy in modes with frequency less than about 40 THz and a deficiency in energy for modes above that value.\n\nOne way to correct for the leakage is to modify the power spectral density of the filter, such that after equilibration of the leakage, the system reaches an energymode distribution which follows the Bose-Einstein distribution. An example is shown in Fig. 1 (right panel) with the adjusted power spectral density,Θ B (ω), shown as a dotted line. The resulting energy distribution (asterisks) is in much better agreement with the desired Bose-Einstein distribution (full line) than the one obtained us- ing the original filter, shown in the left panel. The leakage is however not perfectly corrected, as can be seen from the Fourier transform of the velocity auto-correlation function (VAF) shown in Fig. 1 (bottom panel). Bear in mind that in classical MD, i.e. when a thermostat fulfilling the equipartition theorem of energy is used, the Fourier transform of the VAF equals the phonon density of states (DOS) times k B T /m; in the quantum case, we obtain the phonon DOS times the phonon population function [Eq. (11)]. Figure 1 compares results obtained with the two coloured noises,Θ A (ω) andΘ B (ω), to the exact distribution. The latter is estimated by calculating the vibrational DOS, using a classical thermal bath (CTB), multiplied byΘ(ω). In the case wherẽ Θ A (ω) =Θ(ω) (dashed blue line), the ZPE leakage results in an underpopulation of the high-energy modes (ω > 45 THz) and an overpopulation of the low-energy modes. The corrected coloured noiseΘ B (ω) (green dotted line) yields a much better, although not perfect, agreement with the exact distribution, but fitting such power spectral density is technically difficult. The corrected PSD is both system-and temperature-dependent, making the direct application of this procedure rather tedious.\n\nIn spite of the zero point energy leakage, quantum Langevin thermostats have been successfully used to map out the diamond-graphite coexistence curve, [11] as well as the proton momentum distribution in hydrogenstorage materials. [26] The quantum effects accounted for the thermostat were relevant, in these cases, for a correct description of the systems. Expecting the same degree of accuracy of the method for describing thermal conductivity properties, our calculations were performed omitting any correction concerning the leakage. However, as will be shown later, this introduces a serious limitation for the method.\n\n\nC. Quantum thermostat and NEMD\n\nDespite the zero-point energy leakage described above, the quantum thermostat allows to recover the correct temperature dependence of the equilibrium average thermal energy and heat capacity. Figure 2 shows an example in the case of solid argon. Here every degree of freedom of the system is coupled to the thermostat.\n\nIn the present semi-classical approach, we should distinguish between the temperature used as input of the thermostat, which is the true (quantum) temperature of the system, denoted as T , and the temperature measured from the kinetic energy of the system, which we call the classical temperature, T C . The relation between both, in the case of solid argon, is shown in Fig. 2. In the harmonic approximation, we have:\nT C = 1 3(N − 1)k B 3(N −1) i ω i 1 2 + 1 e ωi / kB T − 1 .(15)\nAt high temperature, the quantum temperature converges toward the classical value. When T decreases to zero, T C converges to the zero-point energy of the system (expressed in Kelvins).\n\nIn NEMD, part of the system is not thermostated, and will not be directly coupled to a quantum thermal bath. It is not straightforward if the interaction between thermalized and non-thermalized parts will transfer the frequency-dependent energy. In order to explore the evolution of the system under such conditions, we performed a test simulation with the same configuration as NEMD, but coupling both thermostated slabs to quantum thermostats at the same temperature. Figure 3 shows the instantaneous kinetic energy and the Fourier transform of the VAF, once the system has reached equilibrium. Averages were performed over atoms in different regions of the cell, either thermostated or not. As can be seen, the thermostat-free regions (left panel, dashed line) have reached an average temperature in agreement with the one imposed in the thermostated regions (full red line). The larger fluctuations in the thermostated regions are due to the fact that averages are computed over smaller numbers of atoms. Moreover, the Fourier transform of the VAF shown in Fig. (3) (right panel) shows that the same frequency-dependent energy distribution is obtained in both regions, proving that a system can be equilibrated with a mode-dependent energy by applying a quantum thermostat only to a subset of the system. It should be pointed out, however, that the thermostated regions must have a size comparable or greater than the one of the \"free\" regions. If this is not the case, the thermostated regions are not sufficient to thermostat the free part, which tends to relax towards a classical energy distribution. This is in contrast with the classical case, in which the thermostating of a few degrees of freedom is, in principle, sufficient to impose the temperature in an arbitrarily large system.\n\n\nIV. THERMAL CONDUCTIVITY\n\nThe methodologies introduced in Sec. II were applied to calculate the thermal conductivity of solid argon. This system is well documented in the literature and can be modeled with a simple Lennard-Jones interatomic potential, with parameters ǫ/k B = 120 K and σ = 3.4Å. All simulations were carried out with the T R O C A D E R O package. [28] Super-cells of 256, 1280, 5120 and 10240 atoms were used, and time-steps of 1 or 5 fs. The potential cutoff was fixed to 4σ. A comparison of our results to experimental data is shown in Fig. 4. Simulations were performed using either a classical or a quantum thermostat. NEMD was implemented with the two methodologies mentioned above, imposing either a gradient of temperature or a flux of energy. The two methods were in full agreement and are shown here with the same symbols.\n\nNEMD calculations with the quantum thermostat could not be performed below 10 K because of the difficulty to impose or measure a temperature gradient in this temperature range. To measure a temperature gradient, we first compute the profile of kinetic energy across the sample, from which we deduce the classical temperature T C , which serves to map the real temperature T by inverting Eq. (15). At low temperatures however, the classical temperature converges to the zero-point energy and becomes almost temperature independent. Temperature gradients are then difficult to estimate, requiring better statistics, i.e. larger simulation cells and longer simulation times, which limited our calculations to temperatures above about 10 K.\n\nWe can see from Fig. 4 that all approaches, classical and quantum, are in good agreement with one another and with experimental data at high temperatures, typically above 40 K.\n\nAt lower temperatures, conductivities computed with a classical thermostat remain in good agreement with experimental data [29] down to about 10 K, while the computed quantum conductivities are much lower. Fitting the experimental data to T −n , for temperatures higher than 10 K, we find n = 1.23. Our classical data present, in the same temperature range, a slightly stronger dependence with n ≈ 1.32. An agreement between classical calculations and experimental data has been obtained as well by other authors, [7] but is very surprising since quantum effects on the specific heat, which enters directly in the expression of the thermal conductivity [see Eq. (6)], start at about 40 K, as seen in Fig. 2.\n\nThe underestimation of the conductivity using the quantum thermostat is not due to an inability of the thermostat to address non-equilibrium conditions, since equivalent results are obtained with the BTE-MD, which is an equilibrium based approach. In NEMD, aside from the phonon-phonon scattering present in real materials, an additional phonon-boundary scattering is present at the boundaries between hot and cold sections (if the system in not large enough). The phonon mean free path is then reduced, i.e. the phonon lifetimes, and a lower thermal conductivity is obtained. At high temperatures such effect is less important, as the mean free path is governed by the phonon-phonon scattering. The phonon population increases with temperature, increasing the phononphonon scattering, as more phonons are present to do the scattering. [30] However, the comparison with BTE-MD results suggests that boundary scattering is not the main effect that causes the reduction in thermal conductivity when using the quantum thermostat. Indeed, size effects in BTE-MD are much less important, in the sense that mean free paths are not limited by the boundaries of the system. In this case, a system large enough must be just considered in order to ensure that all modes accessible to the system are well described in the simulation. Our simulations, have been performed for different cell sizes in order to ensure convergence.\n\nThe inability of the quantum thermostat calculations to describe correctly the thermal conductivity is more probably a consequence of the zero point energy leakage mentioned in Sec. III B. The main effect of this leakage is that the energy is distributed almost homogeneously among the modes, as in the classical regime, as seen in Fig. 5. The system with the quantum thermostat, hence, behaves almost like a classical system, but at a higher temperature. To illustrate this point, we show in Fig. 6 the evolution of the average phonon lifetime as a function of temperature, obtained with the classical and quantum thermostats. Phonon lifetimes obtained with the quantum thermostat, τ Q , are much shorter than the classical lifetimes, τ C , and the former can be obtained from the latter, by replacing the classical temperature T C by its corresponding quantum (real) temperature, T , i.e., we have:\nτ Q (T ) ∼ τ C (T C ) ,(16)\nwhere T and T C are related by Eq. 15. This correction corresponds to the usual rescaling of temperatures used for instance in Refs. 5 and 6. The agreement between the corrected classical lifetimes and the quantum lifetimes shown in Fig. 6 confirms that the system described with the quantum thermostat is equivalent to a classical system at higher temperature. Using this insight, the thermal conductivity obtained with the quantum thermostat can be predicted from the classical conductivity. Indeed, if we assume that the specific heat, group velocities and phonon lifetimes are independent, as is often done (see Refs. 5, 6), we can approximate Eq. (6) as:\nκ ∝ C ph v 2 g τ .(17)\nWe have seen in Sec. III C that the quantum thermostat allows to reproduce the average specific heat, so if we assume that the group velocity is not strongly affected by quantum effects, we can write:\nκ Q (T ) = κ C (T C ) C Q v (T ) C C v (T C ) τ Q (T ) τ C (T C ) (18) = κ C (T C ) C Q v (T ) 3 (N − 1) k B ,\nusing Eq. (16). The result of this rescaling is shown in Fig. 4, as red full circles. It is seen that this procedure closely matches the conductivities obtained with the quantum thermostat. The correction considered in Eq. (18), even if it is widely used in the literature, [5,6,31,32] is known to be oversimplified and inaccurate compared to the results of a full quantum approach. [4] On the other hand, we have shown here that this correction fully explains the results of the semi-classical Langevin quantum thermostat. One surprising result remains concerning the apparent absence of quantum effects on the thermal conductivity of argon. Some authors argue that quantum effects are not relevant for argon, even at temperatures well below the Debye value, and avoid any correction. [8] This simplification is based on the accuracy of the classical theory to describe properties such as the nearest neighbour distance, the bulk modulus, and the cohesive energy of solid noble gases. The effect of neglecting the zero point motion for these properties, is less important in solid argon than in lighter systems. [30] But, from Fig. 2, we know that the heat capacity starts to decrease at about 40 K. From Eq. (17), we see that an absence of quantum effects implies that the decrease in the specific heat is compensated by an increase of the phonon lifetime. However, an exact, or near exact compensation is not expected a priory and seems to be specific to argon, since for instance in Si, classical calculations yield conductivities higher than experimental data. [33] V. CONCLUSIONS In this work, various methods based on classical or semi classical molecular dynamics were used to obtain the thermal conductivity of a very simple system, solid argon. The results of classical nonequilibrium molecular dynamics, of NEMD using a quantum heat bath, and of the Boltzmann transport equation with lifetimes obtained from molecular dynamics were considered, and compared to experimental data. Very surprisingly, the only method that leads to results in good agreement with experimental data at low temperature is classical molecular dynamics. It must however been admitted that there are good reasons to believe this agreement in the case of Argon is partially fortuitous, and results from a cancellation of errors between heat capacity and phonon mean free path. Indeed, when an empirical description of the heat capacity is introduced in the Boltzmann transport equation, the agreement with experiments worsens. Moreover, other studies in systems such as diamond sili-con have shown that the classical MD results can actually strongly overestimate the thermal conductivity at temperatures below the Debye temperature.\n\nThe quantum heat bath method, which was originally thought to be promising, as it assigns the correct zero point energy to the phonon modes, leads to a quite poor agreement with experiments, with a strong underestimation of the thermal conductivity. The basic reason for this discrepancy, which appears both in a BTE approach and in a direct nonequilibrium calculation, is a too short lifetime for the vibrational modes. In turn, the latter can be attributed to the zero point energy leakage issue. The vibrational amplitude associated with the zero point energy motion can be exchanged between modes, and can contribute to phonon scattering, which does not correspond to the physical situation in a real quantum system.\n\nA natural question that arises as a result of this work is the existence of a reliable simulation method for computing thermal conductivities in solids below the Debye temperature. Such a method should be able, if one considers the usual formula of Eq. 6, to predict correctly normal mode heat capacities and lifetimes. At present, it appears that no method based on molecular dynamics has the ability to achieve both tasks; classical MD fails on both aspects, while the use of a quantum thermostat results in a strong underestimation of the lifetimes. Ad hoc rescaling of temperatures, or the use of classical phonon lifetimes within a BTE scheme, do not offer any guarantee in terms of reliability or accuracy, although they may work reasonably well for specific systems. For simple crystal systems, a satisfactory alternative is the use of lattice dynamics techniques for computing the phonon lifetimes, based on quantum perturbation theory and using the cubic term in the expansion of the potential energy. [9,34] Such a method is, however, computationally intensive and tedious. More importantly, it does not seem to be applicable to disordered systems, or even to crystals with complex unit cells. Therefore the calculation of heat conductivity from numerical simulations in such systems at low temperature remains an open challenge.\n\n\nenergy distribution per mode obtained from MD simulations. Left top panel: values obtained using a noise with power spectral densityΘA(ω) equal to the Bose-Einstein distributionΘ(ω) [Eq.(11)]. Right top panel: values obtained using a noise with power spectral densitỹ ΘB(ω). Bottom panel: vibrational phonon spectrum times the phonon density of state. The full gray line was obtained by calculating the DOS using a classical thermal bath and multiplying byΘ(ω). The other data were obtained directly from the Fourier transform of the velocity auto-correlation function (VAF). Calculations were performed for solid aluminum at 10 K using a Lennard-Jones potential.\n\nFIG. 2 :\n2Left panel: Scaled average kinetic energy per degree of freedom, i.e. by definition the classical temperature, T C , as a function of temperature. Right panel: Specific heat as a function of temperature obtained from molecular dynamics compared to experimental data[27] and the quantum harmonic approximation. Within the harmonic approximation we have Cv = d T C d T , see Eq. 15. The phonon spectrum was obtained from lattice dynamics calculations performed at the Γ-point, using a 1280-atoms cell.\n\nFIG. 3 :\n3NEMD simulation with T1 = T2 = 20 K for a 5120 atoms super-cell. Left panel: Twice the kinetic energy per degree of freedom as a function of time. Right panel: vibrational phonon spectrum times DOS [Eq. (11)], obtained from the Fourier transform of the velocity auto-correlation function (VAF). Averages were performed over 64 atoms coupled or not to a thermostat.\n\nFIG. 4 :\n4Thermal conductivity of solid argon as a function of temperature. Experimental data reported by Christen and Pollack.[29] \n\nFIG. 5 :\n5Scaled average kinetic energy per mode obtained from MD simulations compared to the correct quantum harmonic distribution given by Eq.(11). Crystal argon at 5 K, using a supercell of 256 atoms. The length of the MD simulation was 4 ns.\n\nFIG. 6 :\n6Average phonon lifetimes as a function of temperature obtained from BTE-LD calculations.\nAcknowledgmentsThis work has been supported by the Nanoscience Foundation of Grenoble-France.\n. S Stackhouse, L Stixrude, Rev. Mineral. Geochem. 71253S. Stackhouse and L. Stixrude, Rev. Mineral. Geochem. 71, 253 (2010).\n\n. P Jund, R Jullien, Phys. Rev. B. 5913707P. Jund and R. 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B 42, 11276 (1990).\n\n. Y Lee, R Biswas, C Soukoulis, C Wang, C Chan, K Ho, Phys. Rev. B. 436573Y. Lee, R. Biswas, C. Soukoulis, C. Wang, C. Chan, and K. Ho, Phys. Rev. B 43, 6573 (1991).\n\n. P C Howell, J. Chem. Phys. 137224111P. C. Howell, J. Chem. Phys. 137, 224111 (2012).\n\n. A Debernardi, Phys. Rev. B. 5712847A. Debernardi, Phys. Rev. B 57, 12847 (1998).\n"},"corpus_id":{"kind":"number","value":117891176,"string":"117,891,176"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1307.2409\",\n \"mag\": 1506879312,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1307.2409v1.pdf\"\n ],\n \"pdf_sha\": \"88da6b65983997aaf47fe43c0af8c33503902e15\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Computation of the thermal conductivity using classical and quantum molecular dynamics based methods"},"authors":{"kind":"list like","value":["Natalia Bedoya","Jean-Louis Barrat","David Rodney","Laboratory for Interdisciplinary Physics\nNanosciences Foundation\n23 rue des martyrs38000GrenobleFrance","UMR 5588\nUniversité Grenoble","SIMaP, Grenoble INP, UJF, CNRS, UMR 5266\nCNRS\nSaint Martin d'Hères38402, F-38401Saint Martin d'HèresFrance, France"],"string":"[\n \"Natalia Bedoya\",\n \"Jean-Louis Barrat\",\n \"David Rodney\",\n \"Laboratory for Interdisciplinary Physics\\nNanosciences Foundation\\n23 rue des martyrs38000GrenobleFrance\",\n \"UMR 5588\\nUniversité Grenoble\",\n \"SIMaP, Grenoble INP, UJF, CNRS, UMR 5266\\nCNRS\\nSaint Martin d'Hères38402, F-38401Saint Martin d'HèresFrance, France\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The thermal conductivity of a model for solid argon is investigated using nonequilibrium molecular dynamics methods, as well as the traditional Boltzmann transport equation approach with input from molecular dynamics calculations, both with classical and quantum thermostats. A surprising result is that, at low temperatures, only the classical molecular dynamics technique is in agreement with the experimental data. We argue that this agreement is due to a compensation of errors, and raise the issue of an appropriate method for calculating thermal conductivities at low (below Debye) temperatures."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Different methods are available for calculating thermal conductivities of crystalline solids. [1] The most standard approach involves a calculation of the phonon properties of the system, which are connected to the thermal conductivity through the Boltzmann transport equation. Alternative methods are based on equilibrium molecular dynamics (MD) and non-equilibrium molecular dynamics (NEMD). [1][2][3] The methods based on MD or NEMD are restricted to the classical limit, i.e. the limit of high temperatures. In standard MD, nuclear degrees of freedom are treated classically and quantum effects such as zero-point vibrations are not accounted for. In order to incorporate quantum effects, corrections to the thermal conductivity, based on a rescaling of the heat capacity, are commonly applied. These kind of corrections, however, are not generally accepted as reliable. Turney et al. [4] discussed their validity, and showed that this approach is oversimplified and is not generally applicable, while other authors have found an improvement of the classical thermal conductivity by applying such corrections. [5,6] Quantum effects, on the other hand, are assumed negligible depending on the capability of the classical description to describe the thermal conductivity, but independently of its limitations for predicting heat capacities or phonon lifetimes, properties directly related with the thermal conductivity. A common example is the case of solid argon. In spite of the limitations of the classical theory to predict correctly the heat capacity, a reliable description of the thermal conductivity, at temperatures well below the Debye value, is obtained from classical molecular dynamics. [7,8] Quantum effects on the thermal conductivity can be obtained from anharmonic lattice dynamics, by using the Boltzmann transport equation. [9] This methodology, nonetheless, results in much more expensive computations than molecular dynamics. It requires the full calculation of the vibrational spectrum of the system, as well as the third derivatives of the energy, something unmanageable for large or aperiodic systems. Moreover, the Boltzmann transport calculation relies on approximate theoretical expressions for the phonon lifetime and for the conductivity itself, as opposed to the MD and NEMD formalisms that are in principle exact.\nRecently, a Langevin type thermostat with a coloured noise was proposed [10,11] and implemented [12] by different authors in order to incorporate quantum effects in molecular dynamics. The quantum thermostat allows to recover the correct average quantum energy of a system by coupling every degree of freedom to a fictitious quantum bath, in such a way that a harmonic oscillator acquires an energy given by the Bose-Einstein distribution. As such the method is expected to provide a good description of solids in the harmonic limit, and has been shown to also work well for low temperature liquids, in terms of static properties. [10] At high temperatures, the quantum thermostat reduces to a standard Langevin thermostat. This semi-classical approach offers the possibility of performing direct thermal conductivity calculations, by using molecular dynamics, independent of the temperature regime. Savin et al. [13] have applied this methodology to the study of heat transport in low-dimensional nanostructures from non-equilibrium molecular dynamics (NEMD). In the case of a NEMD simulation there are regions of the system free of thermostat, and one will have to check the validity of the quantum thermostat under such conditions. Moreover, the quantum thermostat is not an exact representation of the quantum behavior, and for anharmonic systems suffers from \"zero-point energy leakage\" (see Ref. 11 and section III below). It is unknown if this influences thermal transport properties. Here we present an overview of the advances and challenges for using such kind of thermostats to address thermal transport studies at low temperatures (below the Debye temperature T D ). In our study we use different MD based methods for calculating the thermal conductivity of solid argon, a simple system that is well described in the literature and, as pointed out before, is particularly well described by classical MD.\nThe paper is organized as follows. In Sec. II we present the various methods used for estimating the thermal conductivity. In Sec. III we briefly introduce the quantum thermostat, we discuss about the zero-point energy leakage problem and its reliability for working under equilibrium and non-equilibrium conditions. In the last section we present and discuss our results for the thermal conductivity of solid Argon."},{"title":"II. METHODOLOGY","content":"The standard methods to compute thermal conductivities are based either on molecular dynamics or lattice dynamics, or a combination of both. In this work, we used non-equilibrium molecular dynamics (NEMD) [1][2][3] and Boltzmann transport equation molecular dynamics (BTE-MD). [14,15] We did not use Green-Kubo based methods, [16,17] which have smaller size effects than NEMD, because the quantum thermostat is not compatible with this approach (see discussion below). However, the cells employed here are large enough to avoid any strong size effects in NEMD.\nIn NEMD, the periodic simulation cell is divided into N slabs, and a temperature gradient is imposed by coupling two selected slabs to two thermostats at different temperatures, T 1 and T 2 with T 1 < T 2 . In a periodic system, the thermostated slabs are separated by a distance equal to one half of the simulation cell length. The remaining slabs are not thermostated. The system is then allowed to reach a steady state, where on average the energy creation rate of the thermostat at T 2 is equal to the energy removal rate of the thermostat at T 1 . Calculating the heat flux j i required to maintain the gradient of temperature ∇ j T from the heat power of the source and the sink, one can estimate the thermal conductivity from Fourier's law:\nIn this work we assumed materials of isotopic symmetry. The thermal conductivity is then a scalar, and the temperature gradient and heat flux are parallel. Equation (1) can alternatively be implemented by imposing the heat flux J and calculating the resulting temperature gradient. A common approach in this case, [2,3] is to rescale the velocities, v h , of the atoms in the hot region according to\nwhere v G is the velocity of the center of mass of the region, and\nHere ∆ ǫ is the amount of heat transferred through the system, and k R is the relative kinetic energy given by\nIn this manner, a constant heat flux\nis imposed, where A is the cross sectional area of the simulation cell perpendicular to the heat flow, and ∆t is the time step. We implemented both NEMD methods and checked that they are fully consistent with one another. In the following, we will not distinguish between them and will simply refer to them as the NEMD approach. An alternative expression for the thermal conductivity of an isotropic material reads [9,14]\nwhere C ph is the volumetric phonon specific heat, v g is the phonon group velocity and τ the phonon lifetime.\nThe sum runs over all wave vectors, q, within the Brillouin zone of the periodic structure, and over the 3 N polarization indices, where N is the number of atoms in the elementary cell under consideration, so that contributions from all normal modes of the system are considered. In Eq. (6), the specific heats and group velocities can be computed using lattice dynamics, while the phonon lifetimes can be obtained using either lattice dynamics or a combination of lattice dynamics and molecular dynamics. [9,14] In the following, we will use only the latter approach, referred to as the BTE-MD method. Group velocities are obtained by evaluating the derivative of the dispersion curves, ω(q), over a set of 6 qpoints within a radius 0.0001Å −1 around the Γ point. Phonon frequencies and specific heats are calculated at the Γ point, in supercells that contain from 256 to 4000 atoms.\nPhonon lifetimes are obtained from the energy autocorrelation function of each normal mode\nthe time-dependent normal mode coordinate. The eigenvectors e( q, ν) are obtained from lattice dynamics, and the relative displacement, r j − r j,0 , of atom j, is sampled using molecular dynamics. The phonon lifetimes τ are then obtained by fitting the following relation\nThe key idea behind the quantum thermostat is to adjust to the manner in which energy is distributed among the normal modes of a harmonic system. In the classical limit, the equipartition theorem is fulfilled and all modes have the same energy, while in the quantum regime, the energy of each mode is distributed according to Bose-Einstein statistics. The quantum Langevin thermostat enforces this distribution by using a frequency-dependent noise function (coloured noise).\nAs in the classical approach using a Langevin thermostat, each particle is coupled to a fictitious bath by including in the equations of motion a random force and a dissipation term related by the fluctuation-dissipation theorem. [18] Accordingly, the equation of motion of a degree of freedom x, of a particle of mass m, in presence of an external force F (x), becomes\nwhere Θ(t) is a coloured noise with a power spectral density (PSD) given by the Bose-Einstein distributioñ\nincluding the zero point energy. The classical regime is recovered at high temperature, where the above PSD becomes independent of the frequency and equals k B T . [19] We also note here that the use of a Langevin equation implies the absence of local energy conservation; hence a Green-Kubo approach, based on the notion that local energy fluctuations undergo a diffusive motion, is not appropriate in a system that is coupled to a local (quantum or classical) heat bath. In practice,Θ(ω) is generated by using a signal processing method based on filtering a white noise. [20] A filterH (ω) = Θ (ω) (12) with Fourier transform H(t) is introduced, and Θ(t) is obtained by convoluting H(t) with a random white noise, r(t), of power spectral densityR(ω)=1, such that\nThus, the power spectral density of the resulting noise is\nwhich satisfies Eq. (11). The method is simple and can be easily implemented in a discrete molecular dynamics algorithm. From a computational point of view, the quantum thermostat does not slow down the calculations, the only difference with a classical thermostat being the convolution operation in Eq. (13). In terms of memory, the quantum thermostat is more demanding because it requires to store a finite number of past values of the white noise and of the filter in order to compute Eq. (13). However, the memory requirement for the thermostat scales linearly with the system size and is easily manageable with current computers. Moreover, it avoids generating and storing the entire time-series of random numbers as done in other implementations of the quantum thermal bath. [10] Further details concerning the method are given in Ref. 12.\nB. Zero-point energy leakage By coupling a system to the quantum thermostat, each harmonic mode can in principle be equilibrated at the correct quantum harmonic energy given by Eq. (11). However, as the equations that are solved describe classical coordinates, the zero point energy in these equations corresponds to the finite amplitude vibration of a classical coordinate. As such this zero point energy can be exchanged between modes, in contrast with a true quantum zero point energy.\nSuch an exchange becomes possible when an anharmonic coupling between the modes is introduced, and leads to the phenomenon of \"zero-point energy leakage\" (ZPE), where the zero point energy is transferred from the high-energy modes to low-energy modes, so as to homogenize the energy among the modes. [21][22][23][24][25] As the thermostat cannot fully counterbalance the leakage, an equilibrium is reached where the energy per mode is neither constant, nor as inhomogeneous as in Bose-Einstein distribution. An example is shown in Fig. 1 (left panel) in the case of a perfect crystal of aluminum at 10 K modeled with a Lennard-Jones potential. A coloured noise with power spectral densityΘ A (ω) =Θ(ω) directly from Eq. (11) was used and results in an excess of energy in modes with frequency less than about 40 THz and a deficiency in energy for modes above that value.\nOne way to correct for the leakage is to modify the power spectral density of the filter, such that after equilibration of the leakage, the system reaches an energymode distribution which follows the Bose-Einstein distribution. An example is shown in Fig. 1 (right panel) with the adjusted power spectral density,Θ B (ω), shown as a dotted line. The resulting energy distribution (asterisks) is in much better agreement with the desired Bose-Einstein distribution (full line) than the one obtained us- ing the original filter, shown in the left panel. The leakage is however not perfectly corrected, as can be seen from the Fourier transform of the velocity auto-correlation function (VAF) shown in Fig. 1 (bottom panel). Bear in mind that in classical MD, i.e. when a thermostat fulfilling the equipartition theorem of energy is used, the Fourier transform of the VAF equals the phonon density of states (DOS) times k B T /m; in the quantum case, we obtain the phonon DOS times the phonon population function [Eq. (11)]. Figure 1 compares results obtained with the two coloured noises,Θ A (ω) andΘ B (ω), to the exact distribution. The latter is estimated by calculating the vibrational DOS, using a classical thermal bath (CTB), multiplied byΘ(ω). In the case wherẽ Θ A (ω) =Θ(ω) (dashed blue line), the ZPE leakage results in an underpopulation of the high-energy modes (ω > 45 THz) and an overpopulation of the low-energy modes. The corrected coloured noiseΘ B (ω) (green dotted line) yields a much better, although not perfect, agreement with the exact distribution, but fitting such power spectral density is technically difficult. The corrected PSD is both system-and temperature-dependent, making the direct application of this procedure rather tedious.\nIn spite of the zero point energy leakage, quantum Langevin thermostats have been successfully used to map out the diamond-graphite coexistence curve, [11] as well as the proton momentum distribution in hydrogenstorage materials. [26] The quantum effects accounted for the thermostat were relevant, in these cases, for a correct description of the systems. Expecting the same degree of accuracy of the method for describing thermal conductivity properties, our calculations were performed omitting any correction concerning the leakage. However, as will be shown later, this introduces a serious limitation for the method."},{"title":"C. Quantum thermostat and NEMD","content":"Despite the zero-point energy leakage described above, the quantum thermostat allows to recover the correct temperature dependence of the equilibrium average thermal energy and heat capacity. Figure 2 shows an example in the case of solid argon. Here every degree of freedom of the system is coupled to the thermostat.\nIn the present semi-classical approach, we should distinguish between the temperature used as input of the thermostat, which is the true (quantum) temperature of the system, denoted as T , and the temperature measured from the kinetic energy of the system, which we call the classical temperature, T C . The relation between both, in the case of solid argon, is shown in Fig. 2. In the harmonic approximation, we have:\nAt high temperature, the quantum temperature converges toward the classical value. When T decreases to zero, T C converges to the zero-point energy of the system (expressed in Kelvins).\nIn NEMD, part of the system is not thermostated, and will not be directly coupled to a quantum thermal bath. It is not straightforward if the interaction between thermalized and non-thermalized parts will transfer the frequency-dependent energy. In order to explore the evolution of the system under such conditions, we performed a test simulation with the same configuration as NEMD, but coupling both thermostated slabs to quantum thermostats at the same temperature. Figure 3 shows the instantaneous kinetic energy and the Fourier transform of the VAF, once the system has reached equilibrium. Averages were performed over atoms in different regions of the cell, either thermostated or not. As can be seen, the thermostat-free regions (left panel, dashed line) have reached an average temperature in agreement with the one imposed in the thermostated regions (full red line). The larger fluctuations in the thermostated regions are due to the fact that averages are computed over smaller numbers of atoms. Moreover, the Fourier transform of the VAF shown in Fig. (3) (right panel) shows that the same frequency-dependent energy distribution is obtained in both regions, proving that a system can be equilibrated with a mode-dependent energy by applying a quantum thermostat only to a subset of the system. It should be pointed out, however, that the thermostated regions must have a size comparable or greater than the one of the \"free\" regions. If this is not the case, the thermostated regions are not sufficient to thermostat the free part, which tends to relax towards a classical energy distribution. This is in contrast with the classical case, in which the thermostating of a few degrees of freedom is, in principle, sufficient to impose the temperature in an arbitrarily large system."},{"title":"IV. THERMAL CONDUCTIVITY","content":"The methodologies introduced in Sec. II were applied to calculate the thermal conductivity of solid argon. This system is well documented in the literature and can be modeled with a simple Lennard-Jones interatomic potential, with parameters ǫ/k B = 120 K and σ = 3.4Å. All simulations were carried out with the T R O C A D E R O package. [28] Super-cells of 256, 1280, 5120 and 10240 atoms were used, and time-steps of 1 or 5 fs. The potential cutoff was fixed to 4σ. A comparison of our results to experimental data is shown in Fig. 4. Simulations were performed using either a classical or a quantum thermostat. NEMD was implemented with the two methodologies mentioned above, imposing either a gradient of temperature or a flux of energy. The two methods were in full agreement and are shown here with the same symbols.\nNEMD calculations with the quantum thermostat could not be performed below 10 K because of the difficulty to impose or measure a temperature gradient in this temperature range. To measure a temperature gradient, we first compute the profile of kinetic energy across the sample, from which we deduce the classical temperature T C , which serves to map the real temperature T by inverting Eq. (15). At low temperatures however, the classical temperature converges to the zero-point energy and becomes almost temperature independent. Temperature gradients are then difficult to estimate, requiring better statistics, i.e. larger simulation cells and longer simulation times, which limited our calculations to temperatures above about 10 K.\nWe can see from Fig. 4 that all approaches, classical and quantum, are in good agreement with one another and with experimental data at high temperatures, typically above 40 K.\nAt lower temperatures, conductivities computed with a classical thermostat remain in good agreement with experimental data [29] down to about 10 K, while the computed quantum conductivities are much lower. Fitting the experimental data to T −n , for temperatures higher than 10 K, we find n = 1.23. Our classical data present, in the same temperature range, a slightly stronger dependence with n ≈ 1.32. An agreement between classical calculations and experimental data has been obtained as well by other authors, [7] but is very surprising since quantum effects on the specific heat, which enters directly in the expression of the thermal conductivity [see Eq. (6)], start at about 40 K, as seen in Fig. 2.\nThe underestimation of the conductivity using the quantum thermostat is not due to an inability of the thermostat to address non-equilibrium conditions, since equivalent results are obtained with the BTE-MD, which is an equilibrium based approach. In NEMD, aside from the phonon-phonon scattering present in real materials, an additional phonon-boundary scattering is present at the boundaries between hot and cold sections (if the system in not large enough). The phonon mean free path is then reduced, i.e. the phonon lifetimes, and a lower thermal conductivity is obtained. At high temperatures such effect is less important, as the mean free path is governed by the phonon-phonon scattering. The phonon population increases with temperature, increasing the phononphonon scattering, as more phonons are present to do the scattering. [30] However, the comparison with BTE-MD results suggests that boundary scattering is not the main effect that causes the reduction in thermal conductivity when using the quantum thermostat. Indeed, size effects in BTE-MD are much less important, in the sense that mean free paths are not limited by the boundaries of the system. In this case, a system large enough must be just considered in order to ensure that all modes accessible to the system are well described in the simulation. Our simulations, have been performed for different cell sizes in order to ensure convergence.\nThe inability of the quantum thermostat calculations to describe correctly the thermal conductivity is more probably a consequence of the zero point energy leakage mentioned in Sec. III B. The main effect of this leakage is that the energy is distributed almost homogeneously among the modes, as in the classical regime, as seen in Fig. 5. The system with the quantum thermostat, hence, behaves almost like a classical system, but at a higher temperature. To illustrate this point, we show in Fig. 6 the evolution of the average phonon lifetime as a function of temperature, obtained with the classical and quantum thermostats. Phonon lifetimes obtained with the quantum thermostat, τ Q , are much shorter than the classical lifetimes, τ C , and the former can be obtained from the latter, by replacing the classical temperature T C by its corresponding quantum (real) temperature, T , i.e., we have:\nwhere T and T C are related by Eq. 15. This correction corresponds to the usual rescaling of temperatures used for instance in Refs. 5 and 6. The agreement between the corrected classical lifetimes and the quantum lifetimes shown in Fig. 6 confirms that the system described with the quantum thermostat is equivalent to a classical system at higher temperature. Using this insight, the thermal conductivity obtained with the quantum thermostat can be predicted from the classical conductivity. Indeed, if we assume that the specific heat, group velocities and phonon lifetimes are independent, as is often done (see Refs. 5, 6), we can approximate Eq. (6) as:\nWe have seen in Sec. III C that the quantum thermostat allows to reproduce the average specific heat, so if we assume that the group velocity is not strongly affected by quantum effects, we can write:\nusing Eq. (16). The result of this rescaling is shown in Fig. 4, as red full circles. It is seen that this procedure closely matches the conductivities obtained with the quantum thermostat. The correction considered in Eq. (18), even if it is widely used in the literature, [5,6,31,32] is known to be oversimplified and inaccurate compared to the results of a full quantum approach. [4] On the other hand, we have shown here that this correction fully explains the results of the semi-classical Langevin quantum thermostat. One surprising result remains concerning the apparent absence of quantum effects on the thermal conductivity of argon. Some authors argue that quantum effects are not relevant for argon, even at temperatures well below the Debye value, and avoid any correction. [8] This simplification is based on the accuracy of the classical theory to describe properties such as the nearest neighbour distance, the bulk modulus, and the cohesive energy of solid noble gases. The effect of neglecting the zero point motion for these properties, is less important in solid argon than in lighter systems. [30] But, from Fig. 2, we know that the heat capacity starts to decrease at about 40 K. From Eq. (17), we see that an absence of quantum effects implies that the decrease in the specific heat is compensated by an increase of the phonon lifetime. However, an exact, or near exact compensation is not expected a priory and seems to be specific to argon, since for instance in Si, classical calculations yield conductivities higher than experimental data. [33] V. CONCLUSIONS In this work, various methods based on classical or semi classical molecular dynamics were used to obtain the thermal conductivity of a very simple system, solid argon. The results of classical nonequilibrium molecular dynamics, of NEMD using a quantum heat bath, and of the Boltzmann transport equation with lifetimes obtained from molecular dynamics were considered, and compared to experimental data. Very surprisingly, the only method that leads to results in good agreement with experimental data at low temperature is classical molecular dynamics. It must however been admitted that there are good reasons to believe this agreement in the case of Argon is partially fortuitous, and results from a cancellation of errors between heat capacity and phonon mean free path. Indeed, when an empirical description of the heat capacity is introduced in the Boltzmann transport equation, the agreement with experiments worsens. Moreover, other studies in systems such as diamond sili-con have shown that the classical MD results can actually strongly overestimate the thermal conductivity at temperatures below the Debye temperature.\nThe quantum heat bath method, which was originally thought to be promising, as it assigns the correct zero point energy to the phonon modes, leads to a quite poor agreement with experiments, with a strong underestimation of the thermal conductivity. The basic reason for this discrepancy, which appears both in a BTE approach and in a direct nonequilibrium calculation, is a too short lifetime for the vibrational modes. In turn, the latter can be attributed to the zero point energy leakage issue. The vibrational amplitude associated with the zero point energy motion can be exchanged between modes, and can contribute to phonon scattering, which does not correspond to the physical situation in a real quantum system.\nA natural question that arises as a result of this work is the existence of a reliable simulation method for computing thermal conductivities in solids below the Debye temperature. Such a method should be able, if one considers the usual formula of Eq. 6, to predict correctly normal mode heat capacities and lifetimes. At present, it appears that no method based on molecular dynamics has the ability to achieve both tasks; classical MD fails on both aspects, while the use of a quantum thermostat results in a strong underestimation of the lifetimes. Ad hoc rescaling of temperatures, or the use of classical phonon lifetimes within a BTE scheme, do not offer any guarantee in terms of reliability or accuracy, although they may work reasonably well for specific systems. For simple crystal systems, a satisfactory alternative is the use of lattice dynamics techniques for computing the phonon lifetimes, based on quantum perturbation theory and using the cubic term in the expansion of the potential energy. [9,34] Such a method is, however, computationally intensive and tedious. More importantly, it does not seem to be applicable to disordered systems, or even to crystals with complex unit cells. Therefore the calculation of heat conductivity from numerical simulations in such systems at low temperature remains an open challenge."},{"title":"FIG. 2 :","content":""},{"title":"FIG. 3 :","content":""},{"title":"FIG. 4 :","content":""},{"title":"FIG. 5 :","content":""},{"title":"FIG. 6 :","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Different methods are available for calculating thermal conductivities of crystalline solids. [1] The most standard approach involves a calculation of the phonon properties of the system, which are connected to the thermal conductivity through the Boltzmann transport equation. Alternative methods are based on equilibrium molecular dynamics (MD) and non-equilibrium molecular dynamics (NEMD). [1][2][3] The methods based on MD or NEMD are restricted to the classical limit, i.e. the limit of high temperatures. In standard MD, nuclear degrees of freedom are treated classically and quantum effects such as zero-point vibrations are not accounted for. In order to incorporate quantum effects, corrections to the thermal conductivity, based on a rescaling of the heat capacity, are commonly applied. These kind of corrections, however, are not generally accepted as reliable. Turney et al. [4] discussed their validity, and showed that this approach is oversimplified and is not generally applicable, while other authors have found an improvement of the classical thermal conductivity by applying such corrections. [5,6] Quantum effects, on the other hand, are assumed negligible depending on the capability of the classical description to describe the thermal conductivity, but independently of its limitations for predicting heat capacities or phonon lifetimes, properties directly related with the thermal conductivity. A common example is the case of solid argon. In spite of the limitations of the classical theory to predict correctly the heat capacity, a reliable description of the thermal conductivity, at temperatures well below the Debye value, is obtained from classical molecular dynamics. [7,8] Quantum effects on the thermal conductivity can be obtained from anharmonic lattice dynamics, by using the Boltzmann transport equation. [9] This methodology, nonetheless, results in much more expensive computations than molecular dynamics. It requires the full calculation of the vibrational spectrum of the system, as well as the third derivatives of the energy, something unmanageable for large or aperiodic systems. Moreover, the Boltzmann transport calculation relies on approximate theoretical expressions for the phonon lifetime and for the conductivity itself, as opposed to the MD and NEMD formalisms that are in principle exact.\\nRecently, a Langevin type thermostat with a coloured noise was proposed [10,11] and implemented [12] by different authors in order to incorporate quantum effects in molecular dynamics. The quantum thermostat allows to recover the correct average quantum energy of a system by coupling every degree of freedom to a fictitious quantum bath, in such a way that a harmonic oscillator acquires an energy given by the Bose-Einstein distribution. As such the method is expected to provide a good description of solids in the harmonic limit, and has been shown to also work well for low temperature liquids, in terms of static properties. [10] At high temperatures, the quantum thermostat reduces to a standard Langevin thermostat. This semi-classical approach offers the possibility of performing direct thermal conductivity calculations, by using molecular dynamics, independent of the temperature regime. Savin et al. [13] have applied this methodology to the study of heat transport in low-dimensional nanostructures from non-equilibrium molecular dynamics (NEMD). In the case of a NEMD simulation there are regions of the system free of thermostat, and one will have to check the validity of the quantum thermostat under such conditions. Moreover, the quantum thermostat is not an exact representation of the quantum behavior, and for anharmonic systems suffers from \\\"zero-point energy leakage\\\" (see Ref. 11 and section III below). It is unknown if this influences thermal transport properties. Here we present an overview of the advances and challenges for using such kind of thermostats to address thermal transport studies at low temperatures (below the Debye temperature T D ). In our study we use different MD based methods for calculating the thermal conductivity of solid argon, a simple system that is well described in the literature and, as pointed out before, is particularly well described by classical MD.\\nThe paper is organized as follows. In Sec. II we present the various methods used for estimating the thermal conductivity. In Sec. III we briefly introduce the quantum thermostat, we discuss about the zero-point energy leakage problem and its reliability for working under equilibrium and non-equilibrium conditions. In the last section we present and discuss our results for the thermal conductivity of solid Argon.\"\n },\n {\n \"title\": \"II. METHODOLOGY\",\n \"content\": \"The standard methods to compute thermal conductivities are based either on molecular dynamics or lattice dynamics, or a combination of both. In this work, we used non-equilibrium molecular dynamics (NEMD) [1][2][3] and Boltzmann transport equation molecular dynamics (BTE-MD). [14,15] We did not use Green-Kubo based methods, [16,17] which have smaller size effects than NEMD, because the quantum thermostat is not compatible with this approach (see discussion below). However, the cells employed here are large enough to avoid any strong size effects in NEMD.\\nIn NEMD, the periodic simulation cell is divided into N slabs, and a temperature gradient is imposed by coupling two selected slabs to two thermostats at different temperatures, T 1 and T 2 with T 1 < T 2 . In a periodic system, the thermostated slabs are separated by a distance equal to one half of the simulation cell length. The remaining slabs are not thermostated. The system is then allowed to reach a steady state, where on average the energy creation rate of the thermostat at T 2 is equal to the energy removal rate of the thermostat at T 1 . Calculating the heat flux j i required to maintain the gradient of temperature ∇ j T from the heat power of the source and the sink, one can estimate the thermal conductivity from Fourier's law:\\nIn this work we assumed materials of isotopic symmetry. The thermal conductivity is then a scalar, and the temperature gradient and heat flux are parallel. Equation (1) can alternatively be implemented by imposing the heat flux J and calculating the resulting temperature gradient. A common approach in this case, [2,3] is to rescale the velocities, v h , of the atoms in the hot region according to\\nwhere v G is the velocity of the center of mass of the region, and\\nHere ∆ ǫ is the amount of heat transferred through the system, and k R is the relative kinetic energy given by\\nIn this manner, a constant heat flux\\nis imposed, where A is the cross sectional area of the simulation cell perpendicular to the heat flow, and ∆t is the time step. We implemented both NEMD methods and checked that they are fully consistent with one another. In the following, we will not distinguish between them and will simply refer to them as the NEMD approach. An alternative expression for the thermal conductivity of an isotropic material reads [9,14]\\nwhere C ph is the volumetric phonon specific heat, v g is the phonon group velocity and τ the phonon lifetime.\\nThe sum runs over all wave vectors, q, within the Brillouin zone of the periodic structure, and over the 3 N polarization indices, where N is the number of atoms in the elementary cell under consideration, so that contributions from all normal modes of the system are considered. In Eq. (6), the specific heats and group velocities can be computed using lattice dynamics, while the phonon lifetimes can be obtained using either lattice dynamics or a combination of lattice dynamics and molecular dynamics. [9,14] In the following, we will use only the latter approach, referred to as the BTE-MD method. Group velocities are obtained by evaluating the derivative of the dispersion curves, ω(q), over a set of 6 qpoints within a radius 0.0001Å −1 around the Γ point. Phonon frequencies and specific heats are calculated at the Γ point, in supercells that contain from 256 to 4000 atoms.\\nPhonon lifetimes are obtained from the energy autocorrelation function of each normal mode\\nthe time-dependent normal mode coordinate. The eigenvectors e( q, ν) are obtained from lattice dynamics, and the relative displacement, r j − r j,0 , of atom j, is sampled using molecular dynamics. The phonon lifetimes τ are then obtained by fitting the following relation\\nThe key idea behind the quantum thermostat is to adjust to the manner in which energy is distributed among the normal modes of a harmonic system. In the classical limit, the equipartition theorem is fulfilled and all modes have the same energy, while in the quantum regime, the energy of each mode is distributed according to Bose-Einstein statistics. The quantum Langevin thermostat enforces this distribution by using a frequency-dependent noise function (coloured noise).\\nAs in the classical approach using a Langevin thermostat, each particle is coupled to a fictitious bath by including in the equations of motion a random force and a dissipation term related by the fluctuation-dissipation theorem. [18] Accordingly, the equation of motion of a degree of freedom x, of a particle of mass m, in presence of an external force F (x), becomes\\nwhere Θ(t) is a coloured noise with a power spectral density (PSD) given by the Bose-Einstein distributioñ\\nincluding the zero point energy. The classical regime is recovered at high temperature, where the above PSD becomes independent of the frequency and equals k B T . [19] We also note here that the use of a Langevin equation implies the absence of local energy conservation; hence a Green-Kubo approach, based on the notion that local energy fluctuations undergo a diffusive motion, is not appropriate in a system that is coupled to a local (quantum or classical) heat bath. In practice,Θ(ω) is generated by using a signal processing method based on filtering a white noise. [20] A filterH (ω) = Θ (ω) (12) with Fourier transform H(t) is introduced, and Θ(t) is obtained by convoluting H(t) with a random white noise, r(t), of power spectral densityR(ω)=1, such that\\nThus, the power spectral density of the resulting noise is\\nwhich satisfies Eq. (11). The method is simple and can be easily implemented in a discrete molecular dynamics algorithm. From a computational point of view, the quantum thermostat does not slow down the calculations, the only difference with a classical thermostat being the convolution operation in Eq. (13). In terms of memory, the quantum thermostat is more demanding because it requires to store a finite number of past values of the white noise and of the filter in order to compute Eq. (13). However, the memory requirement for the thermostat scales linearly with the system size and is easily manageable with current computers. Moreover, it avoids generating and storing the entire time-series of random numbers as done in other implementations of the quantum thermal bath. [10] Further details concerning the method are given in Ref. 12.\\nB. Zero-point energy leakage By coupling a system to the quantum thermostat, each harmonic mode can in principle be equilibrated at the correct quantum harmonic energy given by Eq. (11). However, as the equations that are solved describe classical coordinates, the zero point energy in these equations corresponds to the finite amplitude vibration of a classical coordinate. As such this zero point energy can be exchanged between modes, in contrast with a true quantum zero point energy.\\nSuch an exchange becomes possible when an anharmonic coupling between the modes is introduced, and leads to the phenomenon of \\\"zero-point energy leakage\\\" (ZPE), where the zero point energy is transferred from the high-energy modes to low-energy modes, so as to homogenize the energy among the modes. [21][22][23][24][25] As the thermostat cannot fully counterbalance the leakage, an equilibrium is reached where the energy per mode is neither constant, nor as inhomogeneous as in Bose-Einstein distribution. An example is shown in Fig. 1 (left panel) in the case of a perfect crystal of aluminum at 10 K modeled with a Lennard-Jones potential. A coloured noise with power spectral densityΘ A (ω) =Θ(ω) directly from Eq. (11) was used and results in an excess of energy in modes with frequency less than about 40 THz and a deficiency in energy for modes above that value.\\nOne way to correct for the leakage is to modify the power spectral density of the filter, such that after equilibration of the leakage, the system reaches an energymode distribution which follows the Bose-Einstein distribution. An example is shown in Fig. 1 (right panel) with the adjusted power spectral density,Θ B (ω), shown as a dotted line. The resulting energy distribution (asterisks) is in much better agreement with the desired Bose-Einstein distribution (full line) than the one obtained us- ing the original filter, shown in the left panel. The leakage is however not perfectly corrected, as can be seen from the Fourier transform of the velocity auto-correlation function (VAF) shown in Fig. 1 (bottom panel). Bear in mind that in classical MD, i.e. when a thermostat fulfilling the equipartition theorem of energy is used, the Fourier transform of the VAF equals the phonon density of states (DOS) times k B T /m; in the quantum case, we obtain the phonon DOS times the phonon population function [Eq. (11)]. Figure 1 compares results obtained with the two coloured noises,Θ A (ω) andΘ B (ω), to the exact distribution. The latter is estimated by calculating the vibrational DOS, using a classical thermal bath (CTB), multiplied byΘ(ω). In the case wherẽ Θ A (ω) =Θ(ω) (dashed blue line), the ZPE leakage results in an underpopulation of the high-energy modes (ω > 45 THz) and an overpopulation of the low-energy modes. The corrected coloured noiseΘ B (ω) (green dotted line) yields a much better, although not perfect, agreement with the exact distribution, but fitting such power spectral density is technically difficult. The corrected PSD is both system-and temperature-dependent, making the direct application of this procedure rather tedious.\\nIn spite of the zero point energy leakage, quantum Langevin thermostats have been successfully used to map out the diamond-graphite coexistence curve, [11] as well as the proton momentum distribution in hydrogenstorage materials. [26] The quantum effects accounted for the thermostat were relevant, in these cases, for a correct description of the systems. Expecting the same degree of accuracy of the method for describing thermal conductivity properties, our calculations were performed omitting any correction concerning the leakage. However, as will be shown later, this introduces a serious limitation for the method.\"\n },\n {\n \"title\": \"C. Quantum thermostat and NEMD\",\n \"content\": \"Despite the zero-point energy leakage described above, the quantum thermostat allows to recover the correct temperature dependence of the equilibrium average thermal energy and heat capacity. Figure 2 shows an example in the case of solid argon. Here every degree of freedom of the system is coupled to the thermostat.\\nIn the present semi-classical approach, we should distinguish between the temperature used as input of the thermostat, which is the true (quantum) temperature of the system, denoted as T , and the temperature measured from the kinetic energy of the system, which we call the classical temperature, T C . The relation between both, in the case of solid argon, is shown in Fig. 2. In the harmonic approximation, we have:\\nAt high temperature, the quantum temperature converges toward the classical value. When T decreases to zero, T C converges to the zero-point energy of the system (expressed in Kelvins).\\nIn NEMD, part of the system is not thermostated, and will not be directly coupled to a quantum thermal bath. It is not straightforward if the interaction between thermalized and non-thermalized parts will transfer the frequency-dependent energy. In order to explore the evolution of the system under such conditions, we performed a test simulation with the same configuration as NEMD, but coupling both thermostated slabs to quantum thermostats at the same temperature. Figure 3 shows the instantaneous kinetic energy and the Fourier transform of the VAF, once the system has reached equilibrium. Averages were performed over atoms in different regions of the cell, either thermostated or not. As can be seen, the thermostat-free regions (left panel, dashed line) have reached an average temperature in agreement with the one imposed in the thermostated regions (full red line). The larger fluctuations in the thermostated regions are due to the fact that averages are computed over smaller numbers of atoms. Moreover, the Fourier transform of the VAF shown in Fig. (3) (right panel) shows that the same frequency-dependent energy distribution is obtained in both regions, proving that a system can be equilibrated with a mode-dependent energy by applying a quantum thermostat only to a subset of the system. It should be pointed out, however, that the thermostated regions must have a size comparable or greater than the one of the \\\"free\\\" regions. If this is not the case, the thermostated regions are not sufficient to thermostat the free part, which tends to relax towards a classical energy distribution. This is in contrast with the classical case, in which the thermostating of a few degrees of freedom is, in principle, sufficient to impose the temperature in an arbitrarily large system.\"\n },\n {\n \"title\": \"IV. THERMAL CONDUCTIVITY\",\n \"content\": \"The methodologies introduced in Sec. II were applied to calculate the thermal conductivity of solid argon. This system is well documented in the literature and can be modeled with a simple Lennard-Jones interatomic potential, with parameters ǫ/k B = 120 K and σ = 3.4Å. All simulations were carried out with the T R O C A D E R O package. [28] Super-cells of 256, 1280, 5120 and 10240 atoms were used, and time-steps of 1 or 5 fs. The potential cutoff was fixed to 4σ. A comparison of our results to experimental data is shown in Fig. 4. Simulations were performed using either a classical or a quantum thermostat. NEMD was implemented with the two methodologies mentioned above, imposing either a gradient of temperature or a flux of energy. The two methods were in full agreement and are shown here with the same symbols.\\nNEMD calculations with the quantum thermostat could not be performed below 10 K because of the difficulty to impose or measure a temperature gradient in this temperature range. To measure a temperature gradient, we first compute the profile of kinetic energy across the sample, from which we deduce the classical temperature T C , which serves to map the real temperature T by inverting Eq. (15). At low temperatures however, the classical temperature converges to the zero-point energy and becomes almost temperature independent. Temperature gradients are then difficult to estimate, requiring better statistics, i.e. larger simulation cells and longer simulation times, which limited our calculations to temperatures above about 10 K.\\nWe can see from Fig. 4 that all approaches, classical and quantum, are in good agreement with one another and with experimental data at high temperatures, typically above 40 K.\\nAt lower temperatures, conductivities computed with a classical thermostat remain in good agreement with experimental data [29] down to about 10 K, while the computed quantum conductivities are much lower. Fitting the experimental data to T −n , for temperatures higher than 10 K, we find n = 1.23. Our classical data present, in the same temperature range, a slightly stronger dependence with n ≈ 1.32. An agreement between classical calculations and experimental data has been obtained as well by other authors, [7] but is very surprising since quantum effects on the specific heat, which enters directly in the expression of the thermal conductivity [see Eq. (6)], start at about 40 K, as seen in Fig. 2.\\nThe underestimation of the conductivity using the quantum thermostat is not due to an inability of the thermostat to address non-equilibrium conditions, since equivalent results are obtained with the BTE-MD, which is an equilibrium based approach. In NEMD, aside from the phonon-phonon scattering present in real materials, an additional phonon-boundary scattering is present at the boundaries between hot and cold sections (if the system in not large enough). The phonon mean free path is then reduced, i.e. the phonon lifetimes, and a lower thermal conductivity is obtained. At high temperatures such effect is less important, as the mean free path is governed by the phonon-phonon scattering. The phonon population increases with temperature, increasing the phononphonon scattering, as more phonons are present to do the scattering. [30] However, the comparison with BTE-MD results suggests that boundary scattering is not the main effect that causes the reduction in thermal conductivity when using the quantum thermostat. Indeed, size effects in BTE-MD are much less important, in the sense that mean free paths are not limited by the boundaries of the system. In this case, a system large enough must be just considered in order to ensure that all modes accessible to the system are well described in the simulation. Our simulations, have been performed for different cell sizes in order to ensure convergence.\\nThe inability of the quantum thermostat calculations to describe correctly the thermal conductivity is more probably a consequence of the zero point energy leakage mentioned in Sec. III B. The main effect of this leakage is that the energy is distributed almost homogeneously among the modes, as in the classical regime, as seen in Fig. 5. The system with the quantum thermostat, hence, behaves almost like a classical system, but at a higher temperature. To illustrate this point, we show in Fig. 6 the evolution of the average phonon lifetime as a function of temperature, obtained with the classical and quantum thermostats. Phonon lifetimes obtained with the quantum thermostat, τ Q , are much shorter than the classical lifetimes, τ C , and the former can be obtained from the latter, by replacing the classical temperature T C by its corresponding quantum (real) temperature, T , i.e., we have:\\nwhere T and T C are related by Eq. 15. This correction corresponds to the usual rescaling of temperatures used for instance in Refs. 5 and 6. The agreement between the corrected classical lifetimes and the quantum lifetimes shown in Fig. 6 confirms that the system described with the quantum thermostat is equivalent to a classical system at higher temperature. Using this insight, the thermal conductivity obtained with the quantum thermostat can be predicted from the classical conductivity. Indeed, if we assume that the specific heat, group velocities and phonon lifetimes are independent, as is often done (see Refs. 5, 6), we can approximate Eq. (6) as:\\nWe have seen in Sec. III C that the quantum thermostat allows to reproduce the average specific heat, so if we assume that the group velocity is not strongly affected by quantum effects, we can write:\\nusing Eq. (16). The result of this rescaling is shown in Fig. 4, as red full circles. It is seen that this procedure closely matches the conductivities obtained with the quantum thermostat. The correction considered in Eq. (18), even if it is widely used in the literature, [5,6,31,32] is known to be oversimplified and inaccurate compared to the results of a full quantum approach. [4] On the other hand, we have shown here that this correction fully explains the results of the semi-classical Langevin quantum thermostat. One surprising result remains concerning the apparent absence of quantum effects on the thermal conductivity of argon. Some authors argue that quantum effects are not relevant for argon, even at temperatures well below the Debye value, and avoid any correction. [8] This simplification is based on the accuracy of the classical theory to describe properties such as the nearest neighbour distance, the bulk modulus, and the cohesive energy of solid noble gases. The effect of neglecting the zero point motion for these properties, is less important in solid argon than in lighter systems. [30] But, from Fig. 2, we know that the heat capacity starts to decrease at about 40 K. From Eq. (17), we see that an absence of quantum effects implies that the decrease in the specific heat is compensated by an increase of the phonon lifetime. However, an exact, or near exact compensation is not expected a priory and seems to be specific to argon, since for instance in Si, classical calculations yield conductivities higher than experimental data. [33] V. CONCLUSIONS In this work, various methods based on classical or semi classical molecular dynamics were used to obtain the thermal conductivity of a very simple system, solid argon. The results of classical nonequilibrium molecular dynamics, of NEMD using a quantum heat bath, and of the Boltzmann transport equation with lifetimes obtained from molecular dynamics were considered, and compared to experimental data. Very surprisingly, the only method that leads to results in good agreement with experimental data at low temperature is classical molecular dynamics. It must however been admitted that there are good reasons to believe this agreement in the case of Argon is partially fortuitous, and results from a cancellation of errors between heat capacity and phonon mean free path. Indeed, when an empirical description of the heat capacity is introduced in the Boltzmann transport equation, the agreement with experiments worsens. Moreover, other studies in systems such as diamond sili-con have shown that the classical MD results can actually strongly overestimate the thermal conductivity at temperatures below the Debye temperature.\\nThe quantum heat bath method, which was originally thought to be promising, as it assigns the correct zero point energy to the phonon modes, leads to a quite poor agreement with experiments, with a strong underestimation of the thermal conductivity. The basic reason for this discrepancy, which appears both in a BTE approach and in a direct nonequilibrium calculation, is a too short lifetime for the vibrational modes. In turn, the latter can be attributed to the zero point energy leakage issue. The vibrational amplitude associated with the zero point energy motion can be exchanged between modes, and can contribute to phonon scattering, which does not correspond to the physical situation in a real quantum system.\\nA natural question that arises as a result of this work is the existence of a reliable simulation method for computing thermal conductivities in solids below the Debye temperature. Such a method should be able, if one considers the usual formula of Eq. 6, to predict correctly normal mode heat capacities and lifetimes. At present, it appears that no method based on molecular dynamics has the ability to achieve both tasks; classical MD fails on both aspects, while the use of a quantum thermostat results in a strong underestimation of the lifetimes. Ad hoc rescaling of temperatures, or the use of classical phonon lifetimes within a BTE scheme, do not offer any guarantee in terms of reliability or accuracy, although they may work reasonably well for specific systems. For simple crystal systems, a satisfactory alternative is the use of lattice dynamics techniques for computing the phonon lifetimes, based on quantum perturbation theory and using the cubic term in the expansion of the potential energy. [9,34] Such a method is, however, computationally intensive and tedious. More importantly, it does not seem to be applicable to disordered systems, or even to crystals with complex unit cells. Therefore the calculation of heat conductivity from numerical simulations in such systems at low temperature remains an open challenge.\"\n },\n {\n \"title\": \"FIG. 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 6 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"energy distribution per mode obtained from MD simulations. Left top panel: values obtained using a noise with power spectral densityΘA(ω) equal to the Bose-Einstein distributionΘ(ω) [Eq.(11)]. Right top panel: values obtained using a noise with power spectral densitỹ ΘB(ω). Bottom panel: vibrational phonon spectrum times the phonon density of state. The full gray line was obtained by calculating the DOS using a classical thermal bath and multiplying byΘ(ω). The other data were obtained directly from the Fourier transform of the velocity auto-correlation function (VAF). Calculations were performed for solid aluminum at 10 K using a Lennard-Jones potential.","content":"energy distribution per mode obtained from MD simulations. Left top panel: values obtained using a noise with power spectral densityΘA(ω) equal to the Bose-Einstein distributionΘ(ω) [Eq.(11)]. Right top panel: values obtained using a noise with power spectral densitỹ ΘB(ω). Bottom panel: vibrational phonon spectrum times the phonon density of state. The full gray line was obtained by calculating the DOS using a classical thermal bath and multiplying byΘ(ω). The other data were obtained directly from the Fourier transform of the velocity auto-correlation function (VAF). Calculations were performed for solid aluminum at 10 K using a Lennard-Jones potential."},{"id":"","caption":"Left panel: Scaled average kinetic energy per degree of freedom, i.e. by definition the classical temperature, T C , as a function of temperature. Right panel: Specific heat as a function of temperature obtained from molecular dynamics compared to experimental data[27] and the quantum harmonic approximation. Within the harmonic approximation we have Cv = d T C d T , see Eq. 15. The phonon spectrum was obtained from lattice dynamics calculations performed at the Γ-point, using a 1280-atoms cell.","content":"FIG. 2 :\n2Left panel: Scaled average kinetic energy per degree of freedom, i.e. by definition the classical temperature, T C , as a function of temperature. Right panel: Specific heat as a function of temperature obtained from molecular dynamics compared to experimental data[27] and the quantum harmonic approximation. Within the harmonic approximation we have Cv = d T C d T , see Eq. 15. The phonon spectrum was obtained from lattice dynamics calculations performed at the Γ-point, using a 1280-atoms cell."},{"id":"","caption":"NEMD simulation with T1 = T2 = 20 K for a 5120 atoms super-cell. Left panel: Twice the kinetic energy per degree of freedom as a function of time. Right panel: vibrational phonon spectrum times DOS [Eq. (11)], obtained from the Fourier transform of the velocity auto-correlation function (VAF). Averages were performed over 64 atoms coupled or not to a thermostat.","content":"FIG. 3 :\n3NEMD simulation with T1 = T2 = 20 K for a 5120 atoms super-cell. Left panel: Twice the kinetic energy per degree of freedom as a function of time. Right panel: vibrational phonon spectrum times DOS [Eq. (11)], obtained from the Fourier transform of the velocity auto-correlation function (VAF). Averages were performed over 64 atoms coupled or not to a thermostat."},{"id":"","caption":"Thermal conductivity of solid argon as a function of temperature. Experimental data reported by Christen and Pollack.[29]","content":"FIG. 4 :\n4Thermal conductivity of solid argon as a function of temperature. Experimental data reported by Christen and Pollack.[29]"},{"id":"","caption":"Scaled average kinetic energy per mode obtained from MD simulations compared to the correct quantum harmonic distribution given by Eq.(11). Crystal argon at 5 K, using a supercell of 256 atoms. The length of the MD simulation was 4 ns.","content":"FIG. 5 :\n5Scaled average kinetic energy per mode obtained from MD simulations compared to the correct quantum harmonic distribution given by Eq.(11). Crystal argon at 5 K, using a supercell of 256 atoms. The length of the MD simulation was 4 ns."},{"id":"","caption":"Average phonon lifetimes as a function of temperature obtained from BTE-LD calculations.","content":"FIG. 6 :\n6Average phonon lifetimes as a function of temperature obtained from BTE-LD calculations."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"energy distribution per mode obtained from MD simulations. Left top panel: values obtained using a noise with power spectral densityΘA(ω) equal to the Bose-Einstein distributionΘ(ω) [Eq.(11)]. Right top panel: values obtained using a noise with power spectral densitỹ ΘB(ω). Bottom panel: vibrational phonon spectrum times the phonon density of state. The full gray line was obtained by calculating the DOS using a classical thermal bath and multiplying byΘ(ω). The other data were obtained directly from the Fourier transform of the velocity auto-correlation function (VAF). Calculations were performed for solid aluminum at 10 K using a Lennard-Jones potential.\",\n \"content\": \"energy distribution per mode obtained from MD simulations. Left top panel: values obtained using a noise with power spectral densityΘA(ω) equal to the Bose-Einstein distributionΘ(ω) [Eq.(11)]. Right top panel: values obtained using a noise with power spectral densitỹ ΘB(ω). Bottom panel: vibrational phonon spectrum times the phonon density of state. The full gray line was obtained by calculating the DOS using a classical thermal bath and multiplying byΘ(ω). The other data were obtained directly from the Fourier transform of the velocity auto-correlation function (VAF). Calculations were performed for solid aluminum at 10 K using a Lennard-Jones potential.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Left panel: Scaled average kinetic energy per degree of freedom, i.e. by definition the classical temperature, T C , as a function of temperature. Right panel: Specific heat as a function of temperature obtained from molecular dynamics compared to experimental data[27] and the quantum harmonic approximation. Within the harmonic approximation we have Cv = d T C d T , see Eq. 15. The phonon spectrum was obtained from lattice dynamics calculations performed at the Γ-point, using a 1280-atoms cell.\",\n \"content\": \"FIG. 2 :\\n2Left panel: Scaled average kinetic energy per degree of freedom, i.e. by definition the classical temperature, T C , as a function of temperature. Right panel: Specific heat as a function of temperature obtained from molecular dynamics compared to experimental data[27] and the quantum harmonic approximation. Within the harmonic approximation we have Cv = d T C d T , see Eq. 15. The phonon spectrum was obtained from lattice dynamics calculations performed at the Γ-point, using a 1280-atoms cell.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"NEMD simulation with T1 = T2 = 20 K for a 5120 atoms super-cell. Left panel: Twice the kinetic energy per degree of freedom as a function of time. Right panel: vibrational phonon spectrum times DOS [Eq. (11)], obtained from the Fourier transform of the velocity auto-correlation function (VAF). Averages were performed over 64 atoms coupled or not to a thermostat.\",\n \"content\": \"FIG. 3 :\\n3NEMD simulation with T1 = T2 = 20 K for a 5120 atoms super-cell. Left panel: Twice the kinetic energy per degree of freedom as a function of time. Right panel: vibrational phonon spectrum times DOS [Eq. (11)], obtained from the Fourier transform of the velocity auto-correlation function (VAF). Averages were performed over 64 atoms coupled or not to a thermostat.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Thermal conductivity of solid argon as a function of temperature. Experimental data reported by Christen and Pollack.[29]\",\n \"content\": \"FIG. 4 :\\n4Thermal conductivity of solid argon as a function of temperature. Experimental data reported by Christen and Pollack.[29]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Scaled average kinetic energy per mode obtained from MD simulations compared to the correct quantum harmonic distribution given by Eq.(11). Crystal argon at 5 K, using a supercell of 256 atoms. The length of the MD simulation was 4 ns.\",\n \"content\": \"FIG. 5 :\\n5Scaled average kinetic energy per mode obtained from MD simulations compared to the correct quantum harmonic distribution given by Eq.(11). Crystal argon at 5 K, using a supercell of 256 atoms. The length of the MD simulation was 4 ns.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Average phonon lifetimes as a function of temperature obtained from BTE-LD calculations.\",\n \"content\": \"FIG. 6 :\\n6Average phonon lifetimes as a function of temperature obtained from BTE-LD calculations.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". S Stackhouse, L Stixrude, Rev. Mineral. Geochem. 71253S. Stackhouse and L. Stixrude, Rev. Mineral. Geochem. 71, 253 (2010).","doi":""},{"text":". P Jund, R Jullien, Phys. Rev. B. 5913707P. Jund and R. Jullien, Phys. Rev. B 59, 13707 (1999).","doi":""},{"text":". F Müller-Plathe, J. Chem. Phys. 1066082F. Müller-Plathe, J. Chem. Phys. 106, 6082 (1997).","doi":""},{"text":". J Turney, A Mcgaughey, C Amon, Phys. Rev. B. 79224305J. Turney, A. McGaughey, and C. Amon, Phys. Rev. B 79, 224305 (2009).","doi":""},{"text":". J Li, L Porter, S Yip, J. Nucl. Mater. 255139J. Li, L. Porter, and S. Yip, J. Nucl. 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J Turney, A Mcgaughey, C Amon, Phys. Rev. B. 79224305J. Turney, A. McGaughey, and C. Amon, Phys. Rev. B 79, 224305 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Li, L Porter, S Yip, J. Nucl. Mater. 255139J. Li, L. Porter, and S. Yip, J. Nucl. Mater. 255, 139 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Jund, R Jullien, Phys. Rev. B. 5913707P. Jund and R. Jullien, Phys. Rev. B 59, 13707 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Kaburaki, J Li, S Yip, H Kimizuka, J. Appl. Phys. 1026H. Kaburaki, J. Li, S. Yip, and H. Kimizuka, J. Appl. Phys. 102, 6 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Mcgaughey, M Kaviany, Int. J. Heat Mass Transfer. 471783A. McGaughey and M. Kaviany, Int. J. Heat Mass Trans- fer 47, 1783 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G P Srivastava, The Physics of phonons. Taylor & Francis GroupG. P. Srivastava, The Physics of phonons (Taylor & Francis Group, 1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Dammak, Y Chalopin, M Laroche, M Hayoun, J.-J Greffet, Phys. Rev. Lett. 103190601H. Dammak, Y. Chalopin, M. Laroche, M. Hayoun, and J.-J. Greffet, Phys. Rev. Lett. 103, 190601 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Ceriotti, G Bussi, M Parrinello, Phys. Rev. Lett. 10330603M. Ceriotti, G. Bussi, and M. Parrinello, Phys. Rev. Lett. 103, 030603 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J.-L Barrat, D Rodney, J. Stat. Phys. 144679J.-L. Barrat and D. Rodney, J. Stat. Phys. 144, 679 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Savin, Y Kosevich, A Cantarero, Phys. Rev. B. 8664305A. Savin, Y. Kosevich, and A. Cantarero, Phys. Rev. B 86, 064305 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J E Turney, E S Landry, A J H Mcgaughey, C H Amon, Phys. Rev. B. 7964301J. E. Turney, E. S. Landry, A. J. H. McGaughey, and C. H. Amon, Phys. Rev. B 79, 064301 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Mcgaughey, M Kaviany, 1098-0121Phys. Rev. B. 6994303A. McGaughey and M. Kaviany, Phys. Rev. B 69, 094303 (2004), ISSN 1098-0121.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M S Green, J. Chem. Phys. 22398M. S. Green, J. Chem. Phys. 22, 398 (1954).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Kubo, J. Phys. Soc. Jpn. 12570R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"H Risken, The Fokker-Planck Equation. Springer-VerlagH. Risken, The Fokker-Planck Equation (Springer- Verlag, 1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G S Grest, K Kremer, M Carlo, Phys. Rev. A. 333628G. S. Grest, K. Kremer, and M. Carlo, Phys. Rev. A 33, 3628 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A V Oppenheim, R W Schafer, Discrete-Time Signal Processing. Englewoods Cliffs, NJPrentice Hall2nd EditionA. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (2nd Edition) (Pren- tice Hall, Englewoods Cliffs, NJ, 1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". 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B 57, 12847 (1998).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":392,"cells":{"text":{"kind":"string","value":"\nSome Mader-perfect graph classes\n12 Oct 2022 October 13, 2022\n\nHui Lei \nSchool of Mathematical Sciences and LPMC\nNankai University\n300071TianjinChina\n\nSiyan Li \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina\n\nXiaopan Lian xiaopanlian@mail.nankai.edu.cn \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina\n\nSusu Wang \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina\n\nSome Mader-perfect graph classes\n12 Oct 2022 October 13, 2022arXiv:2210.06247v1 [math.CO]DigraphDichromatic numberSubdivisionStrongly connected\nThe dichromatic number of D, denoted by − → χ (D), is the smallest integer k such that D admits an acyclic k-coloring. We use mader χ (F ) to denote the smallest integer kWe extend octi digraphs to a larger class of digraphs and prove that it is Mader-perfect, which generalizes a result of Gishboliner, Steiner and Szabó [Dichromatic number and forced subdivisions, J. Comb. Theory, Ser. B 153 (2022) 1-30]. We also show that if K is a proper subdigraph of ↔ C 4 except for the digraph obtained from ↔ C 4 by deleting an arbitrary arc, then K is Mader-perfect.\n\nIntroduction\n\nLet γ be a (di)graph parameter. Given a (di)graph H, we use mader γ (H) to denote the smallest integer such that if γ(G) ≥ mader γ (H), then G contains a subdivision of H. There are many results and open problems about the relation between some graph parameters and the containment of topological minor. For example, let γ be the chromatic number χ. Hajós' conjecture now can be restated as mader χ (K k ) = k. Hajós' conjecture holds for k ≤ 4 by Dirac in [5] and it was disproved for k ≥ 7 by Catlin in [4]. For the cases k = 5 and k = 6, Hajós conjecture is still open. Now let γ be the minimum degree δ. For k ≤ 4, mader δ (K k ) = k − 1 which was proved by Dirac in [6]. For general case, the following was proved.\n\nTheorem 1.1. [2,8] There exists an absolute constant C > 0 such that for every integer k ∈ N, mader δ (K k ) ≤ Ck 2 .\n\nSince every graph G with chromatic number χ(G) contains a subgraph which has minimum degree at least χ(G) − 1, we can deduce from Theorem 1.1 that mader χ (K k ) ≤ Ck 2 . Therefore, if χ(G) is sufficiently large, then G contains a subdivision of any given graph.\n\nMotivated by the above discussion, Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé [1] generalized this kind of research to digraphs. Given a digraph D, an acyclic k-coloring of D can be defined as a mapping c : V (D) → {1, . . . , k} such that the color class c −1 (i) induces an acyclic subdigraph of D for each i ∈ {1, . . . , k}. The dichromatic number of D, denoted by − → χ (D), is the smallest integer k such that D admits an acyclic k-coloring. The authors in [1] proved the following result. We use (u, v) to denote an arc with head v and tail u. For a graph H, its biorientation is the digraph ↔ H obtained by replacing each edge uv of H by the pair of symmetric arcs (u, v) and (v, u), and an orientation of H is a digraph obtained from H by replacing each edge uv in H by either (u, v) or (v, u). Later the bound on mader χ ( ↔ K k ) was modified by Goshboliner, Steiner and Szabó [7]. \nΩ( k 2 log k ) ≤ mader χ ( ↔ K k ) ≤ 4 k 2 −2k+1 (k − 1) + 1.\nSince there is no analogue of Theorem 1.1 in digraphs, it seems quite challenging to obtain a polynomial bound. Goshboliner, Steiner and Szabó [7] found better bounds for special digraphs classes. One of the digraph classes is called octus, denoted by F * , and they defined it as follows.\n• K 1 ∈ F * . • Let F ∈ F * with v 0 ∈ V (F ). Let P = v 1 , . . . , v k , k ≥ 1 be an orientation of a path which is disjoint from V (F ). Let F * be obtained from F by adding the path P , both arcs (v 0 , v 1 ), (v 1 , v 0 ), and (if k ≥ 2) exactly one of the arcs (v 0 , v k ), (v k , v 0 ). Then F * ∈ F * .\n• If F ∈ F * , then every subdigraph of F is also in F * .\n\nThey established the following Theorem.\n\nTheorem 1.4. [7] For every F ∈ F * , we have mader\nχ (F ) = |V (F )|. A digraph F is called Mader-perfect if for every subgraph F ′ of F , mader− → χ (F ′ ) = |V (F ′ )|. Note that for any digraph F , we have mader χ (F ) ≥ |V (F )| since χ( ↔ K |V (F )|−1 ) = |V (F )| − 1 but ↔ K |V (F )|−1\ndoes not contain a subdivision of F . Gishboliner, Steiner and Szabó [7] showed that all the digraphs in F * are Mader-perfect and they thought it would be interesting to characterize Mader-perfect digraph classes. Let C ℓ be a cycle of length ℓ. The authors in [7] proposed the following problems.\nProblem 1.5. [7] Characterize Mader-perfect digraphs. Problem 1.6. [7] Determine mader χ ( ↔ C ℓ ).\nAs an attempt to solve the two problems, we generalize Theorem 1.4 to a larger digraph family F . In order to establish the definition of F , we call a subdigraph Q of the biorientation of P k = v 1 v 2 . . . v k is good if (a) Q is connected with |V (Q)| = k and all the digons(the pair of symmetric arcs) in Q are vertex disjoint;\n(b) d + Q (v 1 ) + d − Q (v 1 ) = 1.\nDenote the set of all the good subdigraphs of ↔ P k by S k . Let F be a family of digraphs, which can be recursively defined as follows.\n• K 1 ∈ F . • Let F ∈ F with v 0 ∈ V (F ) and Q ∈ S k . Let F * be obtained from F by adding Q, both arcs (v 0 , v 1 ), (v 1 , v 0 ), and (if k ≥ 2) exactly one of the arcs (v 0 , v k ), (v k , v 0 ). Then F * ∈ F .\n• If F ∈ F , then every subdigraph of F also belongs to F .\n\nOur main result is as follows:\nTheorem 1.7. For every F ∈ F , we have mader χ (F ) = |V (F )|.\nWe also show that the if K is a proper subdigraph of ↔ C 4 except for the digraph obtained from ↔ C 4 by deleting an arbitrary arc, then mader χ (K) = |V (K)|. The paper is organized as follows. In section 2, we list some notions and useful results. In section 3, we show that each digraph contained in the generalized family F is Maderperfect. Especially, the subdigraphs of ↔ C ℓ where the digons are vertex disjoint are contained in F . In section 4, we prove that the proper subdigraphs of ↔ C 4 except for H 3 (as shown in Figure 1) are Mader-perfect.\n\n\nPreliminary\n\nIn this section, we first introduce some notions. For a digraph D and u ∈ V (D), let δ + D (u) be the number of arcs in D with tail u and δ − D (u) be the number of arcs in D with head\nu. Let δ + (D) = min{δ + D (u) : u ∈ V (D)} and δ − (D) = min{δ − D (u) : u ∈ V (D)}.\nFor a vertex subset X in D, the subdigraph of D induced by X is denoted by D[X]. We say that W = u 0 e 1 u 1 . . . u k−1 e k u k is a diwalk from u 0 to u k in D if u i ∈ V (D) for i ∈ {0, 1, . . . , k} and e j = (u j−1 , u j ) ∈ A(D) for j ∈ {1, . . . , k}. Further, if u i = u j for i, j ∈ {0, 1, . . . , k}, then we call W a dipath from u 0 to u k . If only u 0 = u k , then we call W a dicycle of length k. A digraph D is called strongly connected if there is a dipath from u to v for every pair of vertices u and v in D. A strong component of D is a maximal induced subgraph of D which is strongly connected. And D is called k-strongly connected if the subdigraph D[V (D) \\ S] is strongly connected for any vertex subset S ⊆ V (D) with |S| < k. We say that D is k-dicritical, if χ(D) = k and for any proper subdigraph\nD ′ ⊆ D, χ(D ′ ) < k. For X, Y ⊆ V (D), an (X, Y )-dipath is a dipath which starts in a vertex of X, ends in a vertex of Y , and is internally vertex-disjoint from X ∪ Y , especially, if X = {v}, then we write (v, Y )-dipath for short. For u, v ∈ V (D), dist D (u, v)\nis defined as the length of the shortest dipath between u and v contained in D. We use D[u, v] to denote a directed path from u to v and D(u, v) to denote a directed path between u and v in D. Especially, D [u, v] and D(u, v) denote the vertex u when u = v. Let ← − D be the digraph obtained from D by reversing the orientations of all arcs.\n\nThe following results were proved in [3,7,9] and will be used in this paper.\nLemma 2.1. [7] Let F be a digraph, we have mader χ (F ) = mader χ ( ← − F ). Lemma 2.2. [7]\nLet F be the disjoint union of two digraphs F 1 and F 2 , then mader χ (F ) ≤ mader χ (F 1 ) + mader χ (F 2 ).\nLemma 2.3. [7] Let D be k-dicritical. Then δ + (D), δ − (D) ≥ k − 1 and D is strongly connected.\nTheorem 2.4. [3,9] Let k ∈ N, and let D be a k-strongly connected digraph with δ + (D),\nδ − (D) ≥ 2k. Then there is x ∈ V (D) such that D ′ = D[V (D) \\ {x}] is (also) k-strongly connected.\n3 The proof of Theorem 1.7\n\nTo prove Theorem 1.7, we first prove the following result.\nTheorem 3.1. Let F be a digraph with v 0 ∈ V (F ) and Q ∈ S k . Let F * be obtained from F by adding Q, both arcs (v 0 , v 1 ), (v 1 , v 0 ), and (if k ≥ 2) exactly one of the arcs (v 0 , v k ), (v k , v 0 ). Then mader χ (F * ) ≤ mader χ (F ) + k.\nProof. By the proof of Case (II) of Theorem 16 in [7], we only consider the case (v 0 , v k ) ∈ A(F * ). For convenience, let mader χ (F ) = M . We need to show that for any given digraph\nD with χ(D) = M + k, there is a subdivision of F * in D. Let c 0 : V (D) → {1, .\n. . , M + k} be an acyclic coloring which maximizes |c −1 0 ({1, . . . , k})|. In the rest of the proof, let\nD 1 = D[c −1 0 ({1, . . . , k})] and D 2 = D[c −1 0 ({k + 1, . . . , M + k})]\n, we have that χ(D 1 ) = k and χ(D 2 ) = M since c 0 is an acyclic coloring of D. Combining that mader χ (F ) = M , there is a subdivision S of F in D 2 and denote by x 0 the vertex in S corresponding to v 0 . In [7], the authors defined a pre-order on the acyclic colorings of D 1 with respect to x 0 as follows.\nFor each acyclic k-coloring c of D 1 , define a vector v(c) ∈ Z k with v(c) i = |N + D (x 0 ) ∩ c −1 (i)| for i ∈ [k]. Now, consider the pre-order on the set of acyclic k-colorings of D 1 , where c 1 ≺ c 2 iff v(c 1 ) < lex v(c 2 ).\nHere < lex denotes the lexicographical order on Z k . Let c denote an acyclic k-coloring of D 1 that is minimal with respect to ≺. From Claim 2 in the proof of Theorem 16 in [7], we know that there are vertices\nx 1 , . . . , x k in N + (x 0 ) ∩ V (D 1 ) such that • c(x i ) = i for i ∈ {1, . . . , k}; • there is a dicycleĈ in D containing x 0 and x 1 such that V (Ĉ) \\ {x 0 } ⊆ c −1 (1); • there is a strong component of D[c −1 ({i − 1, i})] that contains both x i−1 and x i for i ∈ {2, . . . , k}. For 2 ≤ i ≤ k, let X i−1,i be the strong component of D[c −1 ({i − 1, i})] that contains x i−1 and x i . If {(v i−1 , v i ), (v i , v i−1 )} ⊆ E(Q), then we choose a directed path P i−1,i in X i,i+1 such that P i−1,i is directed from x i−1 to x i if (v i−1 , v i ) ∈ A(Q) and from x i to x i−1 if (v i , v i−1 ) ∈ A(Q). Note that {(v 1 , v 2 ), (v 2 , v 1 )} ⊆ E(Q) since d + Q (v 1 ) + d − Q (v 1 ) = 1.\nNext, we show that there exist vertices z 1 , z 2 , . . . , z k in D 1 which satisfy\n• z 1 ∈ V (Ĉ). • for every 2 ≤ i ≤ k, there exists a dipath Q i−1,i in D 1 from z i−1 to z i if (v i−1 , v i ) ∈ A(Q), and exists a dipath Q i,i−1 in D 1 from z i to z i−1 if (v i , v i−1 ) ∈ A(Q). • the dipaths Q i−1,i and Q j,j−1 , i, j ∈ {2, . . . , k}, are pairwise internally vertex-disjoint. • either V (Ĉ) ∩ V (Q 1,2 ) = z 1 or V (Ĉ) ∩ V (Q 2,1 ) = z 1 . • V (Ĉ) ∩ V (Q i−1,i ) = ∅ and V (Ĉ) ∩ V (Q j,j−1 ) = ∅ for i, j ∈ {3, . . . , k}.\nDefine z 1 ∈ V (Ĉ) to be the unique last vertex in V (Ĉ) that we meet when traversing the trace of the path P 1,2 starting from x 1 ∈ V (Ĉ). Without loss of generality, we assume\nthat (v 1 , v 2 ) ∈ A(Q). Note that P 1,2 [z 1 , x 2 ] is a dipath from z 1 to x 2 . Now, we determine z 2 and z 3 . Suppose that {(v 2 , v 3 ), (v 3 , v 2 )} ⊆ A(Q)\n. Then for j ∈ {2, 3}, we define z j to be the first vertex of P j,j+1 that we meet when traversing the trace of the dipath P j−1,j (z j−1 , x j ) starting from z j−1 , and let\nQ i−1,i = P i−1,i (z i−1 , z i ) for i ∈ {2, 3}. Suppose that {(v 2 , v 3 ), (v 3 , v 2 )} ⊆ A(Q). Note that {(v 3 , v 4 ), (v 4 , v 3 )} ⊆ A(Q) since all the digons in Q are vertex disjoint. If there is a dicycle C in X 2,3 such that V (P 1,2 [z 1 , x 2 ]) ∩ V (C) = ∅ and V (P 3,4 ) ∩ V (C) = ∅, then let z 2 ∈ V (C) ∩ V (P 1,2 [z 1 , x 2 ]) such that dist P 1,2 (z 1 , z 2 )\nis as small as possible and let z 3 ∈ V (C)∩V (P 3,4 ) such that dist P 3,4 (z 3 , x 4 ) is as small as possible. We have that c(z i ) = i for i ∈ {2, 3} and P 3,4 [z 3 , x 4 ] is a dipath between z 3 and x 4 . We define Q 1,2 = P 1,2 (z 1 , z 2 ) and Q 2,3 ∪Q 3,2 = C. Now, assume that for any dicycle C in X 2,3 , we have that V (P 1,2 [z 1 , x 2 ])∩ V (C) = ∅ or V (P 3,4 )∩ V (C) = ∅. Since X 2,3 is a strong component, any vertex in X 2,3 lies in a dicycle. Let C 1 and C 2 be two dicyles in X 2,3 which contain x 2 and x 3 respectively. We have that\nC 1 = C 2 , V (P 1,2 [z 1 , x 2 ])∩V (C 2 ) = ∅ and V (P 3,4 )∩V (C 1 ) = ∅ by the assumption that V (P 1,2 [z 1 , x 2 ])∩V (C) = ∅ or V (P 3,4 )∩V (C) = ∅ for any dicycle C in X 2,3 . Case 1 V (C 1 ) ∩ V (C 2 ) = ∅.\nIn this case, there are two vertices u and v (it may happen that u\n= v) in V (C 1 ) ∩ V (C 2 ) satisfying C 2 [x 3 , u], C 2 [v, x 3 ], and C 1 are pairwise internally vertex disjoint. Let z 2 ∈ V (C 1 )∩V (P 1,2 [z 1 , x 2 ]) such that dist P 1,2 (z 1 , z 2 ) is as small as possible. Let z ′ 3 ∈ V (C 2 [x 3 , u]∪ C 2 [v, x 3 ]) ∩ V (P 3,4 ) such that dist P 3,4 (z ′ 3 , x 4 ) is as small as possible. If (v 3 , v 4 ) ∈ E(Q), then let z 3 = v and C 2 [v, z ′ 3 ] ∪ P 3,4 [z ′ 3 , x 4 ] is a dipath from z 3 to x 4 . If (v 4 , v 3 ) ∈ E(Q), then let z 3 = u and P 3,4 [x 4 , z ′ 3 ] ∪ C 2 [z ′ 3 , u]\nis a dipath from x 4 to z 3 . We define Q 1,2 = P 1,2 [z 1 , z 2 ] and Q 2,3 ∪ Q 3,2 = C 1 .\nCase 2 V (C 1 ) ∩ V (C 2 ) = ∅.\nSince X 2,3 is a strong component, there is a shortest dipath P 1 from C 1 to C 2 and a shortest dipath P 2 from C 2 to C 1 . Denote the initial and terminal vertices of P 1 by u and v, the initial and terminal vertices of P 2 by x and y, respectively. Let z ′ 3 ∈ V (P 1 ∪C 2 )∩V (P 3,4 ) such that dist P 3,4 (z ′ 3 , x 4 ) is as small as possible. Now we need to consider the following cases. Case 2.1 z ′ 3 ∈ V (C 2 ). Denote by w the unique first vertex in V (P 1 ∪ C 1 ) \\ {v} that we meet when traversing the trace of the dipath P 2 starting from x. We have that\nP 2 [x, w] ∪ P 1 [w, v] ∪ C 2 [v, x] is a dicycle when w = y and P 2 ∪ C 1 [y, u]∪ P 1 ∪ C 2 [v, x]\nis a dicycle when w = y. Note that when w = y, we may assume\nx 2 / ∈ V (C 1 [y, u]) and x 3 / ∈ V (C 2 [v, x])\n, otherwise it can be reduced to Case 1. Denote the above dicycle by\nC 3 . Let z ′ 2 ∈ V (C 1 ∪ P 1 ∪ P 2 [x, w]) ∩ V (P 1,2 [z 1 , x 2 ]) such that dist P 1,2 (z 1 , z ′ 2 ) is as small as possible. Note that z ′ 2 = z ′ 3 since c(z ′ 2 ) = 2 and c(z ′ 3 ) = 3. Suppose that w = y. If z ′ 2 ∈ V (C 1 ), then let z 2 = w. We define Q 1,2 = P 1,2 [z 1 , z ′ 2 ] ∪ C 1 [z ′ 2 , u] ∪ P 1 [u, w] and Q 2,3 ∪ Q 3,2 = C 3 . If z ′ 2 ∈ V (P 1 [u, w]), then let z 2 = w. We define Q 1,2 = P 1,2 [z 1 , z ′ 2 ] ∪ P 1 [z ′ 2 , w] and Q 2,3 ∪ Q 3,2 = C 3 . If z ′ 2 ∈ V (P 1 [w, v] ∪ P 2 [x, w]),\nthen let z 2 = z ′ 2 . We define Q 1,2 = P 1,2 [z 1 , z 2 ] and Q 2,3 ∪ Q 3,2 = C 3 . Suppose that\nw = y. If z ′ 2 ∈ V (C 1 ) \\ V (C 3 ), then let z 2 = w. We define Q 1,2 = P 1,2 [z 1 , z ′ 2 ] ∪ C 1 [z ′ 2 , w] and Q 2,3 ∪ Q 3,2 = C 3 . If z ′ 2 ∈ V (C 3 )\n, then let z 2 = z ′ 2 and so c(z 2 ) = 2. We define\nQ 1,2 = P 1,2 [z 1 , z 2 ] and Q 2,3 ∪ Q 3,2 = C 3 . Now, we define z 3 . Suppose that V (P 3,4 (x 4 , z ′ 3 ))∩V (P 2 [x, w]∪C 2 [v, x]) = ∅. If (v 3 , v 4 ) ∈ A(Q), then let z 3 = x and C 2 [x, z ′ 3 ] ∪ P 3,4 [z ′ 3 , x 4 ] is a dipath from z 3 to x 4 . If (v 4 , v 3 ) ∈ A(Q), then let z 3 = v and P 3,4 [x 4 , z ′ 3 ] ∪ C 2 [z ′ 3 , v] is a dipath from x 4 to z 3 . Suppose that V (P 3,4 (x 4 , z ′ 3 )) ∩ V (P 2 [x, w] ∪ C 2 [v, x]) = ∅, then let z 3 ∈ V (P 3,4 (x 4 , z ′ 3 )) ∩ V (P 2 [x, w] ∪ C 2 [v,\nx]) such that dist P 3,4 (z 3 , x 4 ) is as small as possible and P 3,4 (z 3 , x 4 ) is a dipath between z 3 and x 4 . By the definition of z ′ 3 , we have that z 3 = w and c(z 3 ) = 3. Case 2.2 z ′ 3 ∈ V (P 1 ) \\ V (C 1 ). If w = y, then one of the following holds.\n\n(a) There is a vertex\nu 1 ∈ V (P 1 ) ∩ V (P 2 ) with z ′ 3 ∈ V (P 1 [u, u 1 ]) such that P 2 [u 1 , y] is internally vertex disjoint with P 1 [u, u 1 ]. (b) There is a vertex u 1 ∈ V (P 1 ) ∩ V (P 2 ) with z ′ 3 ∈ V (P 1 [u 1 , v]) \\ {u 1 } such that P 2 [x, u 1 ] is internally vertex disjoint with P 1 [u 1 , v].\n(c) There are two distinct vertices u 1 , u 2 ∈ V (P 1 ) ∩ V (P 2 ) with u 1 = z ′ 3 such that P 1 (u 1 , u 2 ) is internally vertex disjoint with P 2 (u 1 , u 2 ) and z ′ 3 ∈ V (P 1 (u 1 , u 2 )).\n\nIf w = y or (a) holds, then let C 1 = C 1 and C 2 = C 1 [y, u] ∪ P 1 [u, u 1 ] ∪ P 2 [u 1 , y]. We refer z ′ 3 as x 3 and so it can be reduced to Case 1. Suppose that (b) or (c) holds, we have\nP 2 [x, u 1 ] ∪ P 1 [u 1 , v] ∪ C 2 [v,\nx] is a dicycle or P 1 (u 1 , u 2 ) ∪ P 2 (u 1 , u 2 ) is a dicycle, respectively. Call the possible cycle C 3 . Let C 1 = C 1 and C 2 = C 3 . We refer z ′ 3 as x 3 and so it can be reduced to Case 2.1.\n\nIt is easy to check that Q 1,2 , Q 2,3 , Q 2,3 , and V (P 3,4 (z 3 , x 4 )) are pairwise internally vertex-disjoint. Therefore, recursively, we can find the corresponding z 4 , . . . , z k with respect to v 4 , . . . , v k .\n\nLet S * be the subdigraph of D formed by joining S ⊆ D[Y 2 ], the pairwise distinct vertices z 1 , . . . , z k and the connecting dipaths Q i−1,i , Q i,i−1 , i ∈ {2, . . . , k}, the two antiparallel directed pathsĈ[x 0 , z 1 ],Ĉ[z 1 , x 0 ]between x 0 and z 1 as well as the arc (x 0 , z k ). It follows that S * is isomorphic to a subdivision of F * .\n\nThis completes the proof of Theorem 3.1.\n\nBy the definition of F , we know that F * ⊆ F . We call the second item the ear addition operation and the third item taking a subdigraph operation in the definition of F . Therefore, every digraph in F can be obtained from K 1 by a sequence of operations consisting of the above two operations. We call F ∈ F maximal if it is obtained from K 1 by only using the ear addition operation. By recursively using Theorem 3.1, for F ∈ F being maximal, mader χ (F ) = |V (F )|.\n\nProof of Theorem 1.7. Let F ∈ F and let U G(F ) be the graph obtained from F by replacing every arc (u, v) with the edge uv and deleting all multiple edges between every pair of vertices apart from one. If U G(F ) is disconnected, then by Lemma 2.2, it suffices to consider the components of U G(F ). Therefore, in the following, we may assume that U G(F ) is connected. We prove Theorem 1.7 by induction on the number of cycles contained in U G(F ). Denote by ℓ the number of cycles contained in U G(F ). First, assume that ℓ = 0. Then F ∈ F * and so mader χ (F ) = |V Figure 1. In this section, we show that any graph in K is mader perfect. For any K ∈ K,\nwe have K ∈ F when K / ∈ {H 1 , ← − H 1 , H 2 , ← − H 2 },\nwhere H 1 and H 2 are as shown in Figure 1. To prove that any graph in K is mader perfect, by Theorem 1.7 and Lemma 2.1, it suffices to prove that if K ∈ {H 1 , H 2 }, then mader χ (K) = 4. \n(c) B 2 i = V i \\ (B 1 i ∪ B i ) for i ∈ {1, 2} and E = V 3 \\ (A 1 ∪ A 2 ). Now, let V ′ 1 = B 2 ∪ B 1 2 ∪ A 1 ∪ E, V ′ 2 = B 1 ∪ B 1 1 ∪ A 2 and V ′ 3 = {x} ∪ B 2 2 .\nIn order to construct an acyclic 3-coloringc of D, we need to consider the vertices in B 2 1 . Denote by T 0 the set of all the vertices in B 2 1 that are contained in some dicycle in the digraph D[B 2 1 ∪ V ′ 3 ] and let T 3 = B 2 1 \\ T 0 . Further, denote by T 1 the set of all the vertices in T 0 that are not contained in any dicycle in the digraph D[T 0 ∪ E] and let T 2 = T 0 \\ T 1 . An 3-coloringc = (Ṽ 1 ,Ṽ 2 ,Ṽ 3 ) of D can be given by settingṼ i = V ′ i ∪ T i , whereṼ i is a color class ofc for i ∈ {1, 2, 3}. Ifc is acyclic, then it leads to a contradiction as χ(D) = 4. Thus, we finish the proof by showing that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. In the following, we first list some useful claims.\nClaim 1. A 1 ∩ A 2 = ∅ for K ∈ {H 1 , H 2 }. Proof. Suppose that there is a vertex v ∈ A 1 ∩ A 2 .\nBy the partition, there is a shortest dipath P 1 from B 1 to v without passing through {x} ∪ B 2 in D. Let u ∈ V (P 1 ) ∩ B 1 . If K = H 1 , then let P 2 be a shortest dipath from v to B 2 without passing through {x} ∪ B 1 in D. If K = H 2 , then let P 2 be a shortest dipath from B 2 to v without passing through {x}∪B 1 in D. Denote the other end of P 2 by w, and so w ∈ B 2 . Since B 1 and B 2 are acyclic,\nthere are dicycles C u ⊂ D[{x} ∪ B 1 ] and C w ⊂ D[{x} ∪ B 2 ] such that {x, u} ⊆ V (C u ) and {x, w} ⊆ V (C w )\n. Therefore, C u ∪ C w ∪ P 1 ∪ P 2 contains a subdivision of K, a contradiction.\nThus, A 1 ∩ A 2 = ∅ for K ∈ {H 1 , H 2 }. Claim 2. D[V ′ i ] is acyclic for i ∈ {1, 2, 3} and K ∈ {H 1 , H 2 }. Proof. By the partition, it is obvious that D[V ′ 3 ] is acyclic. Suppose that D[V ′ 2 ]\ncontains a dicycle C. Then V (C) ∩ (B 1 ∪ B 1 1 ) = ∅ and V (C) ∩ A 2 = ∅ and so there is a dipath P from\nB 1 to V (C) ∩ A 2 without passing through {x} ∪ B 2 . This implies that (V (C) ∩ A 2 ) ⊆ A 1 , a contradiction. Suppose that D[V ′ 1 ] contains a dicycle C. Similarly, D[V ′ 1 \\ E] is acyclic. Therefore, V (C) ∩ E = ∅. Since V 3 is acyclic, V (C) ∩ (B 2 ∪ B 1 2 ) = ∅. Let u ∈ B 2 ∪ B 1 2 , then there is a dipath from B 2 to V (C) ∩ E without passing through {x} ∪ B 1 . This implies that (V (C) ∩ E) ⊆ A 2 , a contradiction. Hence, D[V ′ i ] is acyclic for i ∈ {1, 2, 3} and K ∈ {H 1 , H 2 }. Claim 3. For each vertex v in T 0 , there is a dipath from v to B 2 2 contained in D[T 0 ∪B 2 2 ]. Proof. Suppose not, then there is a vertex v ∈ T 0 such that there is no dipath from v to B 2 2 contained in D[T 0 ∪ B 2 2 ]. Since D[T 0 ∪ {x}] is acyclic and v is contained in a dicycle C in D[T 0 ∪ V ′ 3 ], there is a dipath between v and B 2 2 contained in D[T 0 ∪ B 2 2 ]. By the assumption, there is a dipath P from B 2 2 to v contained in D[T 0 ∪ B 2 2 ]. Let u ∈ V (P ) ∩ V (B 2 2 )\n. Consider a shortest dipath P 1 in D ′ from B 1 to u, the existence of P 1 can be guaranteed since D ′ is strongly connected. If V (P 1 ) ∩ B 2 = ∅, then there is diwalk P 1 ∪ P (u, v) from B 1 to v without passing through {x} ∪ B 2 which implies that v ∈ B 1 1 , a contradiction. Hence, we may assume that V (P 1 ) ∩ B 2 = ∅. Since P 1 is selected to be shortest, there is a dipath P ′ 1 ⊂ P 1 from B 2 to u without passing through {x} ∪ B 1 contradicting the fact that u ∈ B 2 2 . Therefore, for each vertex v in T 0 there is a dipath from v to B 2 2 contained in D[T 0 ∪ B 2 2 ]. \nV (C i ) ∩ T i = ∅, V (C 1 ) ∩ (V ′ 1 \\ E) = ∅ and V (C 2 ) ∩ V ′ 2 = ∅. Let v i ∈ V (C i ) ∩ T i , u 1 ∈ V (C 1 ) ∩ (V ′ 1 \\ E) and u 2 ∈ V (C 2 ) ∩ V ′ 2 for i ∈ {1, 2}\n. We first consider the vertex setṼ 1 . Suppose that u 1 ∈ B 2 ∪ B 1 2 . By Claim 3, there is a dipath P 1 from v 1 to B 2 2 contained in D[T 0 ∪ B 2 2 ]. Let w 1 ∈ V (P 1 ) ∩ B 2 2 . Therefore, by the definition of the partition, there is a diwalk W 1 = C 1 [u 1 , v 1 ] ∪ P 1 without passing through {x} ∪ B 1 . The existence of the diwalk W 1 implies that w 1 ∈ B 1 2 , a contradiction. Thus, V (C 1 ) ∩ (B 2 ∪ B 1 2 ) = ∅ and so u 1 ∈ A 1 . By the definition of the partition, there is a dipath P ′ 1 from B 1 to u 1 without passing through {x} ∪ B 2 . Thus, there is a diwalk W ′ 2 = P ′ 1 ∪ C 1 [u 1 , v 1 ] from B 1 to v 1 without passing through {x} ∪ B 2 . This implies that v 1 ∈ B 1 1 contradicting the fact that v 1 ∈ B 2 1 . Therefore, D[Ṽ 1 ] is acyclic. Now, we consider the vertex setṼ 2 . Suppose that u 2 ∈ B 1 ∪ B 1 1 , then there is a dipath containing C 2 [u 2 , v 2 ] from B 1 to v 2 without passing through {x} ∪ B 2 . It implies that v 2 ∈ B 1 1 , a contradiction. Thus, V (C 2 ) ∩ (B 1 ∪ B 1 1 ) = ∅ and so u 2 ∈ A 2 . Note that v 2 is contained in some dicycle C ′ in D[T 2 ∪ E]. Let w 2 ∈ V (C ′ ) ∩ E. Then C ′ [v 2 , w 2 ] a dipath from v 2 to w 2 without passing through {x} ∪ B 1 . Thus, there is a diwalk W 2 = C ′ [u 2 , v 2 ] ∪ C ′ [v 2 , w 2 ] from u 2 to w 2 without passing through {x} ∪ B 1 . Combining that there is a dipath from B 2 to u 2 without passing through {x} ∪ B 1 , it implies that w 2 ∈ A 2 contradicting the fact that w 2 ∈ E. Therefore, D[Ṽ 2 ] is acyclic.\n\nFinally, we conclude that D admits an acyclic 3-coloring, which is a contradiction to χ(D) = 4. So the assumption mader χ (K) > 4 is not true. Hence, mader χ (K) = 4. This completes the proof of Theorem 4.1.\n\nTheorem 1 . 2 .\n12[1] Let F be a digraph with n vertices and m arcs. Then mader χ (F ) ≤ 4 m (n − 1) + 1.\n\n\nTheorem 1.3. [7] For every k ∈ N,\n\n\n(F )| by Theorem 1.4. Now, suppose that ℓ ≥ 1. We call a cycle C an outmost cycle if there is a vertex u ∈ V (C) such that the other vertices of C cannot reach any other cycle in U G(F ) without passing u. Call u the special vertex of C. Let C be an outmost cycle in U G(F ) with vertexset {v 0 , . . . , v s } and special vertex v 0 . Let Y be the component of F [V (F ) \\ {v 1 , . . . , v s }] that contains v 0 . By the induction hypothesis, mader χ (Y ) = |V (Y )|. Let Y 1 = F [V (Y ) ∪ {v 1 , . . . , v s }]. Therefore, by Theorem 3.1, mader χ (Y 1 ) = |V (Y 1 )| since Y 1 can be obtained from Y and F [{v 1 , . . . , v s }] by the ear addition operation. For each v i with i ∈ {1, . . . , s}, Let F v i be the digraph that contains v i obtained from F by deleting the arcs adjacent to v i contained in F [{v 1 , . . . , v s }]. Since C is an outmost cycle, F v i isa spanning subdigraph of a biorientation of some tree rooted at v i , say ↔ T v i . Note that ↔ T v i can be seen as a maximal digraph of F obtained from v i by a sequence of ear addition operation (in the case, P k is an isolate vertex in the definition of F ) for i ∈ {1, . . . , s}. Let T = ↔ T v 1 ∪ · · · ∪ ↔ T vs . Denote byF the digraph obtained from Y 1 and V (T ) \\ {v 1 , . . . , v s } by a corresponding sequence of ear addition operation of each ↔ T v i , where i ∈ {1, . . . , s}. Thus, mader χ (F ) = |V (F )| by recursively using Theorem 3.1. Since F is a spanning subdigraph ofF , mader χ (F ) ≤ mader χ (F ). Therefore, mader χ (F ) = |V (F )| since mader χ (F ) ≥ |V (F )|. 4 Subdigraphs of the biorientation of C 4 Let K = {K|K be a proper subdigraph of ↔ C 4 and K ∼ = H 3 }, where H 3 is as shown in\n\nTheorem 4 . 1 .Figure 1 :\n411Let K ∈ {H 1 , H 2 }, then mader χ (K) = 4. Proof. Let K ∈ {H 1 , H 2 }. Since mader χ (K) ≥ |V (K)|, mader χ (K) ≥ 4. We prove Theorem 4.1 by contradiction. Suppose that the assertion is false, i.e., mader χ (K) > 4. Then there is a digraph D with χ(D) = 4 that does not contain a subdivision of K. Among all Three subdigraphs of the biorienitation of C 4 counterexamples we choose D so that first |V (D)| is minimum, and then |A(D)| is minimum. By the choice of D, D is connected, 4-dicritical. By Lemma 2.3 and Theorem 2.4, we know that D is strongly connected and there is a vertex x in D such that D ′ = D[V (D) \\ {x}] is strongly connected. Since D is 4-dicritical, χ(D ′ ) = 3. In the following let c = (V 1 , V 2 , V 3 ) be an acyclic 3-coloring of D ′ , where V 1 , V 2 and V 3 are the color classes. In order to show that there is an acyclic 3-coloring of D, we define the following vertex partition of D ′ with respect to x.\n\nFigure 2 :\n2The vertex partition of D ′ . (a) B i ⊆ V i consists of all the vertices that are contained in the same strong component with x in D[{x} ∪ V i ] for i ∈ {1, 2}. (b) B 1 i ⊆ V i and A i ⊆ V 3 consist of all the vertices which can be reached by some vertex of B i in D without passing through any vertex in {x} ∪ B 3−i for i ∈ {1, 2}.\n\n\nBy Claim 1, we know that {Ṽ 1 ,Ṽ 2 ,Ṽ 3 } is a partition of V (D). Now, we prove that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. Obviously, by the definition of T 3 and Claim 2, D[Ṽ 3 ] is acyclic. To get a contradiction, suppose that there is a dicycle C i contained in D[Ṽ i ] for i ∈ {1, 2}. For i ∈ {1, 2}, we have that D[V ′ i ] is acyclic by Claim 2, and since D[T 1 ∪E] and D[T 2 ] are acyclic, we have that\n\nSubdivisions in digraphs of large out-degree or large dichromatic number. P Aboulker, N Cohen, F Havet, W Lochet, P S Moura, S Thomassé, Electron. J. Comb. 26P. Aboulker, N. Cohen, F. Havet, W. Lochet, P.S. Moura, S. Thomassé, Subdivisions in digraphs of large out-degree or large dichromatic number, Electron. J. Comb. 26 (2019) 3.19.\n\nProof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs. B Bollobás, A Thomason, Eur. J. Comb. 19B. Bollobás, A. Thomason, Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, Eur. J. Comb. 19 (1998) 883-887.\n\nJ Bang-Jensen, G Z Gutin, Digraphs: Theory, Algorithms and Applications. Springer Science & Business MediaJ. Bang-Jensen, G.Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer Science & Business Media, 2008.\n\nHajós' graph coloring conjecture: variations and counterexamples. P A Catlin, J. Comb. Theory, Ser. B. 26P.A. Catlin, Hajós' graph coloring conjecture: variations and counterexamples, J. Comb. Theory, Ser. B 26 (1979) 268-274.\n\nA property of 4-chromatic graphs and some remarks on critical graphs. G A Dirac, J. Lond. Math. Soc. 27G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. Lond. Math. Soc. 27 (1952) 85-92.\n\nIn abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen. G A Dirac, Math. Nachrichten. 22G.A. Dirac. In abstrakten graphen vorhandene vollständige 4-graphen und ihre un- terteilungen, Math. Nachrichten 22 (1960) 61-85.\n\nDichromatic number and forced subdivisions. L Gishboliner, R Steiner, T Szabó, J. Comb. Theory, Ser. B. 153L. Gishboliner, R. Steiner, T. Szabó, Dichromatic number and forced subdivisions, J. Comb. Theory, Ser. B 153 (2022) 1-30.\n\nTopological cliques in graphs II. J Komlós, E Szemerédi, Comb. Probab. Comput. 5J. Komlós, E. Szemerédi, Topological cliques in graphs II, Comb. Probab. Comput. 5 (1996) 79-90.\n\nEcken von kleinem Grad in kritisch n-fach zusammenhängenden Digraphen. W Mader, J. Comb. Theory, Ser. B. 53W. Mader, Ecken von kleinem Grad in kritisch n-fach zusammenhängenden Digraphen, J. Comb. Theory, Ser. B 53 (1991) 260-272.\n"},"corpus_id":{"kind":"number","value":252846460,"string":"252,846,460"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2210.06247\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/amc/LangLLLW23\",\n \"doi\": \"10.1016/j.amc.2023.127968\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2210.06247v1.pdf\"\n ],\n \"pdf_sha\": \"fd436a3d51febd5f86b2539c656c8404a4ad44c6\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Some Mader-perfect graph classes"},"authors":{"kind":"list like","value":["Hui Lei \nSchool of Mathematical Sciences and LPMC\nNankai University\n300071TianjinChina","Siyan Li \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina","Xiaopan Lian xiaopanlian@mail.nankai.edu.cn \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina","Susu Wang \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina"],"string":"[\n \"Hui Lei \\nSchool of Mathematical Sciences and LPMC\\nNankai University\\n300071TianjinChina\",\n \"Siyan Li \\nCenter for Combinatorics and LPMC\\nNankai University\\n300071TianjinChina\",\n \"Xiaopan Lian xiaopanlian@mail.nankai.edu.cn \\nCenter for Combinatorics and LPMC\\nNankai University\\n300071TianjinChina\",\n \"Susu Wang \\nCenter for Combinatorics and LPMC\\nNankai University\\n300071TianjinChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The dichromatic number of D, denoted by − → χ (D), is the smallest integer k such that D admits an acyclic k-coloring. We use mader χ (F ) to denote the smallest integer kWe extend octi digraphs to a larger class of digraphs and prove that it is Mader-perfect, which generalizes a result of Gishboliner, Steiner and Szabó [Dichromatic number and forced subdivisions, J. Comb. Theory, Ser. B 153 (2022) 1-30]. We also show that if K is a proper subdigraph of ↔ C 4 except for the digraph obtained from ↔ C 4 by deleting an arbitrary arc, then K is Mader-perfect."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Let γ be a (di)graph parameter. Given a (di)graph H, we use mader γ (H) to denote the smallest integer such that if γ(G) ≥ mader γ (H), then G contains a subdivision of H. There are many results and open problems about the relation between some graph parameters and the containment of topological minor. For example, let γ be the chromatic number χ. Hajós' conjecture now can be restated as mader χ (K k ) = k. Hajós' conjecture holds for k ≤ 4 by Dirac in [5] and it was disproved for k ≥ 7 by Catlin in [4]. For the cases k = 5 and k = 6, Hajós conjecture is still open. Now let γ be the minimum degree δ. For k ≤ 4, mader δ (K k ) = k − 1 which was proved by Dirac in [6]. For general case, the following was proved.\nTheorem 1.1. [2,8] There exists an absolute constant C > 0 such that for every integer k ∈ N, mader δ (K k ) ≤ Ck 2 .\nSince every graph G with chromatic number χ(G) contains a subgraph which has minimum degree at least χ(G) − 1, we can deduce from Theorem 1.1 that mader χ (K k ) ≤ Ck 2 . Therefore, if χ(G) is sufficiently large, then G contains a subdivision of any given graph.\nMotivated by the above discussion, Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé [1] generalized this kind of research to digraphs. Given a digraph D, an acyclic k-coloring of D can be defined as a mapping c : V (D) → {1, . . . , k} such that the color class c −1 (i) induces an acyclic subdigraph of D for each i ∈ {1, . . . , k}. The dichromatic number of D, denoted by − → χ (D), is the smallest integer k such that D admits an acyclic k-coloring. The authors in [1] proved the following result. We use (u, v) to denote an arc with head v and tail u. For a graph H, its biorientation is the digraph ↔ H obtained by replacing each edge uv of H by the pair of symmetric arcs (u, v) and (v, u), and an orientation of H is a digraph obtained from H by replacing each edge uv in H by either (u, v) or (v, u). Later the bound on mader χ ( ↔ K k ) was modified by Goshboliner, Steiner and Szabó [7].\nSince there is no analogue of Theorem 1.1 in digraphs, it seems quite challenging to obtain a polynomial bound. Goshboliner, Steiner and Szabó [7] found better bounds for special digraphs classes. One of the digraph classes is called octus, denoted by F * , and they defined it as follows.\n• If F ∈ F * , then every subdigraph of F is also in F * .\nThey established the following Theorem.\nTheorem 1.4. [7] For every F ∈ F * , we have mader\ndoes not contain a subdivision of F . Gishboliner, Steiner and Szabó [7] showed that all the digraphs in F * are Mader-perfect and they thought it would be interesting to characterize Mader-perfect digraph classes. Let C ℓ be a cycle of length ℓ. The authors in [7] proposed the following problems.\nAs an attempt to solve the two problems, we generalize Theorem 1.4 to a larger digraph family F . In order to establish the definition of F , we call a subdigraph Q of the biorientation of P k = v 1 v 2 . . . v k is good if (a) Q is connected with |V (Q)| = k and all the digons(the pair of symmetric arcs) in Q are vertex disjoint;\nDenote the set of all the good subdigraphs of ↔ P k by S k . Let F be a family of digraphs, which can be recursively defined as follows.\n• If F ∈ F , then every subdigraph of F also belongs to F .\nOur main result is as follows:\nWe also show that the if K is a proper subdigraph of ↔ C 4 except for the digraph obtained from ↔ C 4 by deleting an arbitrary arc, then mader χ (K) = |V (K)|. The paper is organized as follows. In section 2, we list some notions and useful results. In section 3, we show that each digraph contained in the generalized family F is Maderperfect. Especially, the subdigraphs of ↔ C ℓ where the digons are vertex disjoint are contained in F . In section 4, we prove that the proper subdigraphs of ↔ C 4 except for H 3 (as shown in Figure 1) are Mader-perfect."},{"title":"Preliminary","content":"In this section, we first introduce some notions. For a digraph D and u ∈ V (D), let δ + D (u) be the number of arcs in D with tail u and δ − D (u) be the number of arcs in D with head\nFor a vertex subset X in D, the subdigraph of D induced by X is denoted by D[X]. We say that W = u 0 e 1 u 1 . . . u k−1 e k u k is a diwalk from u 0 to u k in D if u i ∈ V (D) for i ∈ {0, 1, . . . , k} and e j = (u j−1 , u j ) ∈ A(D) for j ∈ {1, . . . , k}. Further, if u i = u j for i, j ∈ {0, 1, . . . , k}, then we call W a dipath from u 0 to u k . If only u 0 = u k , then we call W a dicycle of length k. A digraph D is called strongly connected if there is a dipath from u to v for every pair of vertices u and v in D. A strong component of D is a maximal induced subgraph of D which is strongly connected. And D is called k-strongly connected if the subdigraph D[V (D) \\ S] is strongly connected for any vertex subset S ⊆ V (D) with |S| < k. We say that D is k-dicritical, if χ(D) = k and for any proper subdigraph\nis defined as the length of the shortest dipath between u and v contained in D. We use D[u, v] to denote a directed path from u to v and D(u, v) to denote a directed path between u and v in D. Especially, D [u, v] and D(u, v) denote the vertex u when u = v. Let ← − D be the digraph obtained from D by reversing the orientations of all arcs.\nThe following results were proved in [3,7,9] and will be used in this paper.\nLet F be the disjoint union of two digraphs F 1 and F 2 , then mader χ (F ) ≤ mader χ (F 1 ) + mader χ (F 2 ).\nTheorem 2.4. [3,9] Let k ∈ N, and let D be a k-strongly connected digraph with δ + (D),\n3 The proof of Theorem 1.7\nTo prove Theorem 1.7, we first prove the following result.\nProof. By the proof of Case (II) of Theorem 16 in [7], we only consider the case (v 0 , v k ) ∈ A(F * ). For convenience, let mader χ (F ) = M . We need to show that for any given digraph\n. . , M + k} be an acyclic coloring which maximizes |c −1 0 ({1, . . . , k})|. In the rest of the proof, let\n, we have that χ(D 1 ) = k and χ(D 2 ) = M since c 0 is an acyclic coloring of D. Combining that mader χ (F ) = M , there is a subdivision S of F in D 2 and denote by x 0 the vertex in S corresponding to v 0 . In [7], the authors defined a pre-order on the acyclic colorings of D 1 with respect to x 0 as follows.\nHere < lex denotes the lexicographical order on Z k . Let c denote an acyclic k-coloring of D 1 that is minimal with respect to ≺. From Claim 2 in the proof of Theorem 16 in [7], we know that there are vertices\nNext, we show that there exist vertices z 1 , z 2 , . . . , z k in D 1 which satisfy\nDefine z 1 ∈ V (Ĉ) to be the unique last vertex in V (Ĉ) that we meet when traversing the trace of the path P 1,2 starting from x 1 ∈ V (Ĉ). Without loss of generality, we assume\n. Then for j ∈ {2, 3}, we define z j to be the first vertex of P j,j+1 that we meet when traversing the trace of the dipath P j−1,j (z j−1 , x j ) starting from z j−1 , and let\nis as small as possible and let z 3 ∈ V (C)∩V (P 3,4 ) such that dist P 3,4 (z 3 , x 4 ) is as small as possible. We have that c(z i ) = i for i ∈ {2, 3} and P 3,4 [z 3 , x 4 ] is a dipath between z 3 and x 4 . We define Q 1,2 = P 1,2 (z 1 , z 2 ) and Q 2,3 ∪Q 3,2 = C. Now, assume that for any dicycle C in X 2,3 , we have that V (P 1,2 [z 1 , x 2 ])∩ V (C) = ∅ or V (P 3,4 )∩ V (C) = ∅. Since X 2,3 is a strong component, any vertex in X 2,3 lies in a dicycle. Let C 1 and C 2 be two dicyles in X 2,3 which contain x 2 and x 3 respectively. We have that\nIn this case, there are two vertices u and v (it may happen that u\nis a dipath from x 4 to z 3 . We define Q 1,2 = P 1,2 [z 1 , z 2 ] and Q 2,3 ∪ Q 3,2 = C 1 .\nSince X 2,3 is a strong component, there is a shortest dipath P 1 from C 1 to C 2 and a shortest dipath P 2 from C 2 to C 1 . Denote the initial and terminal vertices of P 1 by u and v, the initial and terminal vertices of P 2 by x and y, respectively. Let z ′ 3 ∈ V (P 1 ∪C 2 )∩V (P 3,4 ) such that dist P 3,4 (z ′ 3 , x 4 ) is as small as possible. Now we need to consider the following cases. Case 2.1 z ′ 3 ∈ V (C 2 ). Denote by w the unique first vertex in V (P 1 ∪ C 1 ) \\ {v} that we meet when traversing the trace of the dipath P 2 starting from x. We have that\nis a dicycle when w = y. Note that when w = y, we may assume\n, otherwise it can be reduced to Case 1. Denote the above dicycle by\nthen let z 2 = z ′ 2 . We define Q 1,2 = P 1,2 [z 1 , z 2 ] and Q 2,3 ∪ Q 3,2 = C 3 . Suppose that\n, then let z 2 = z ′ 2 and so c(z 2 ) = 2. We define\nx]) such that dist P 3,4 (z 3 , x 4 ) is as small as possible and P 3,4 (z 3 , x 4 ) is a dipath between z 3 and x 4 . By the definition of z ′ 3 , we have that z 3 = w and c(z 3 ) = 3. Case 2.2 z ′ 3 ∈ V (P 1 ) \\ V (C 1 ). If w = y, then one of the following holds.\n(a) There is a vertex\n(c) There are two distinct vertices u 1 , u 2 ∈ V (P 1 ) ∩ V (P 2 ) with u 1 = z ′ 3 such that P 1 (u 1 , u 2 ) is internally vertex disjoint with P 2 (u 1 , u 2 ) and z ′ 3 ∈ V (P 1 (u 1 , u 2 )).\nIf w = y or (a) holds, then let C 1 = C 1 and C 2 = C 1 [y, u] ∪ P 1 [u, u 1 ] ∪ P 2 [u 1 , y]. We refer z ′ 3 as x 3 and so it can be reduced to Case 1. Suppose that (b) or (c) holds, we have\nx] is a dicycle or P 1 (u 1 , u 2 ) ∪ P 2 (u 1 , u 2 ) is a dicycle, respectively. Call the possible cycle C 3 . Let C 1 = C 1 and C 2 = C 3 . We refer z ′ 3 as x 3 and so it can be reduced to Case 2.1.\nIt is easy to check that Q 1,2 , Q 2,3 , Q 2,3 , and V (P 3,4 (z 3 , x 4 )) are pairwise internally vertex-disjoint. Therefore, recursively, we can find the corresponding z 4 , . . . , z k with respect to v 4 , . . . , v k .\nLet S * be the subdigraph of D formed by joining S ⊆ D[Y 2 ], the pairwise distinct vertices z 1 , . . . , z k and the connecting dipaths Q i−1,i , Q i,i−1 , i ∈ {2, . . . , k}, the two antiparallel directed pathsĈ[x 0 , z 1 ],Ĉ[z 1 , x 0 ]between x 0 and z 1 as well as the arc (x 0 , z k ). It follows that S * is isomorphic to a subdivision of F * .\nThis completes the proof of Theorem 3.1.\nBy the definition of F , we know that F * ⊆ F . We call the second item the ear addition operation and the third item taking a subdigraph operation in the definition of F . Therefore, every digraph in F can be obtained from K 1 by a sequence of operations consisting of the above two operations. We call F ∈ F maximal if it is obtained from K 1 by only using the ear addition operation. By recursively using Theorem 3.1, for F ∈ F being maximal, mader χ (F ) = |V (F )|.\nProof of Theorem 1.7. Let F ∈ F and let U G(F ) be the graph obtained from F by replacing every arc (u, v) with the edge uv and deleting all multiple edges between every pair of vertices apart from one. If U G(F ) is disconnected, then by Lemma 2.2, it suffices to consider the components of U G(F ). Therefore, in the following, we may assume that U G(F ) is connected. We prove Theorem 1.7 by induction on the number of cycles contained in U G(F ). Denote by ℓ the number of cycles contained in U G(F ). First, assume that ℓ = 0. Then F ∈ F * and so mader χ (F ) = |V Figure 1. In this section, we show that any graph in K is mader perfect. For any K ∈ K,\nwhere H 1 and H 2 are as shown in Figure 1. To prove that any graph in K is mader perfect, by Theorem 1.7 and Lemma 2.1, it suffices to prove that if K ∈ {H 1 , H 2 }, then mader χ (K) = 4.\nIn order to construct an acyclic 3-coloringc of D, we need to consider the vertices in B 2 1 . Denote by T 0 the set of all the vertices in B 2 1 that are contained in some dicycle in the digraph D[B 2 1 ∪ V ′ 3 ] and let T 3 = B 2 1 \\ T 0 . Further, denote by T 1 the set of all the vertices in T 0 that are not contained in any dicycle in the digraph D[T 0 ∪ E] and let T 2 = T 0 \\ T 1 . An 3-coloringc = (Ṽ 1 ,Ṽ 2 ,Ṽ 3 ) of D can be given by settingṼ i = V ′ i ∪ T i , whereṼ i is a color class ofc for i ∈ {1, 2, 3}. Ifc is acyclic, then it leads to a contradiction as χ(D) = 4. Thus, we finish the proof by showing that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. In the following, we first list some useful claims.\nBy the partition, there is a shortest dipath P 1 from B 1 to v without passing through {x} ∪ B 2 in D. Let u ∈ V (P 1 ) ∩ B 1 . If K = H 1 , then let P 2 be a shortest dipath from v to B 2 without passing through {x} ∪ B 1 in D. If K = H 2 , then let P 2 be a shortest dipath from B 2 to v without passing through {x}∪B 1 in D. Denote the other end of P 2 by w, and so w ∈ B 2 . Since B 1 and B 2 are acyclic,\n. Therefore, C u ∪ C w ∪ P 1 ∪ P 2 contains a subdivision of K, a contradiction.\ncontains a dicycle C. Then V (C) ∩ (B 1 ∪ B 1 1 ) = ∅ and V (C) ∩ A 2 = ∅ and so there is a dipath P from\n. Consider a shortest dipath P 1 in D ′ from B 1 to u, the existence of P 1 can be guaranteed since D ′ is strongly connected. If V (P 1 ) ∩ B 2 = ∅, then there is diwalk P 1 ∪ P (u, v) from B 1 to v without passing through {x} ∪ B 2 which implies that v ∈ B 1 1 , a contradiction. Hence, we may assume that V (P 1 ) ∩ B 2 = ∅. Since P 1 is selected to be shortest, there is a dipath P ′ 1 ⊂ P 1 from B 2 to u without passing through {x} ∪ B 1 contradicting the fact that u ∈ B 2 2 . Therefore, for each vertex v in T 0 there is a dipath from v to B 2 2 contained in D[T 0 ∪ B 2 2 ].\n. We first consider the vertex setṼ 1 . Suppose that u 1 ∈ B 2 ∪ B 1 2 . By Claim 3, there is a dipath P 1 from v 1 to B 2 2 contained in D[T 0 ∪ B 2 2 ]. Let w 1 ∈ V (P 1 ) ∩ B 2 2 . Therefore, by the definition of the partition, there is a diwalk W 1 = C 1 [u 1 , v 1 ] ∪ P 1 without passing through {x} ∪ B 1 . The existence of the diwalk W 1 implies that w 1 ∈ B 1 2 , a contradiction. Thus, V (C 1 ) ∩ (B 2 ∪ B 1 2 ) = ∅ and so u 1 ∈ A 1 . By the definition of the partition, there is a dipath P ′ 1 from B 1 to u 1 without passing through {x} ∪ B 2 . Thus, there is a diwalk W ′ 2 = P ′ 1 ∪ C 1 [u 1 , v 1 ] from B 1 to v 1 without passing through {x} ∪ B 2 . This implies that v 1 ∈ B 1 1 contradicting the fact that v 1 ∈ B 2 1 . Therefore, D[Ṽ 1 ] is acyclic. Now, we consider the vertex setṼ 2 . Suppose that u 2 ∈ B 1 ∪ B 1 1 , then there is a dipath containing C 2 [u 2 , v 2 ] from B 1 to v 2 without passing through {x} ∪ B 2 . It implies that v 2 ∈ B 1 1 , a contradiction. Thus, V (C 2 ) ∩ (B 1 ∪ B 1 1 ) = ∅ and so u 2 ∈ A 2 . Note that v 2 is contained in some dicycle C ′ in D[T 2 ∪ E]. Let w 2 ∈ V (C ′ ) ∩ E. Then C ′ [v 2 , w 2 ] a dipath from v 2 to w 2 without passing through {x} ∪ B 1 . Thus, there is a diwalk W 2 = C ′ [u 2 , v 2 ] ∪ C ′ [v 2 , w 2 ] from u 2 to w 2 without passing through {x} ∪ B 1 . Combining that there is a dipath from B 2 to u 2 without passing through {x} ∪ B 1 , it implies that w 2 ∈ A 2 contradicting the fact that w 2 ∈ E. Therefore, D[Ṽ 2 ] is acyclic.\nFinally, we conclude that D admits an acyclic 3-coloring, which is a contradiction to χ(D) = 4. So the assumption mader χ (K) > 4 is not true. Hence, mader χ (K) = 4. This completes the proof of Theorem 4.1."},{"title":"Theorem 1 . 2 .","content":""},{"title":"Theorem 4 . 1 .Figure 1 :","content":""},{"title":"Figure 2 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Let γ be a (di)graph parameter. Given a (di)graph H, we use mader γ (H) to denote the smallest integer such that if γ(G) ≥ mader γ (H), then G contains a subdivision of H. There are many results and open problems about the relation between some graph parameters and the containment of topological minor. For example, let γ be the chromatic number χ. Hajós' conjecture now can be restated as mader χ (K k ) = k. Hajós' conjecture holds for k ≤ 4 by Dirac in [5] and it was disproved for k ≥ 7 by Catlin in [4]. For the cases k = 5 and k = 6, Hajós conjecture is still open. Now let γ be the minimum degree δ. For k ≤ 4, mader δ (K k ) = k − 1 which was proved by Dirac in [6]. For general case, the following was proved.\\nTheorem 1.1. [2,8] There exists an absolute constant C > 0 such that for every integer k ∈ N, mader δ (K k ) ≤ Ck 2 .\\nSince every graph G with chromatic number χ(G) contains a subgraph which has minimum degree at least χ(G) − 1, we can deduce from Theorem 1.1 that mader χ (K k ) ≤ Ck 2 . Therefore, if χ(G) is sufficiently large, then G contains a subdivision of any given graph.\\nMotivated by the above discussion, Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé [1] generalized this kind of research to digraphs. Given a digraph D, an acyclic k-coloring of D can be defined as a mapping c : V (D) → {1, . . . , k} such that the color class c −1 (i) induces an acyclic subdigraph of D for each i ∈ {1, . . . , k}. The dichromatic number of D, denoted by − → χ (D), is the smallest integer k such that D admits an acyclic k-coloring. The authors in [1] proved the following result. We use (u, v) to denote an arc with head v and tail u. For a graph H, its biorientation is the digraph ↔ H obtained by replacing each edge uv of H by the pair of symmetric arcs (u, v) and (v, u), and an orientation of H is a digraph obtained from H by replacing each edge uv in H by either (u, v) or (v, u). Later the bound on mader χ ( ↔ K k ) was modified by Goshboliner, Steiner and Szabó [7].\\nSince there is no analogue of Theorem 1.1 in digraphs, it seems quite challenging to obtain a polynomial bound. Goshboliner, Steiner and Szabó [7] found better bounds for special digraphs classes. One of the digraph classes is called octus, denoted by F * , and they defined it as follows.\\n• If F ∈ F * , then every subdigraph of F is also in F * .\\nThey established the following Theorem.\\nTheorem 1.4. [7] For every F ∈ F * , we have mader\\ndoes not contain a subdivision of F . Gishboliner, Steiner and Szabó [7] showed that all the digraphs in F * are Mader-perfect and they thought it would be interesting to characterize Mader-perfect digraph classes. Let C ℓ be a cycle of length ℓ. The authors in [7] proposed the following problems.\\nAs an attempt to solve the two problems, we generalize Theorem 1.4 to a larger digraph family F . In order to establish the definition of F , we call a subdigraph Q of the biorientation of P k = v 1 v 2 . . . v k is good if (a) Q is connected with |V (Q)| = k and all the digons(the pair of symmetric arcs) in Q are vertex disjoint;\\nDenote the set of all the good subdigraphs of ↔ P k by S k . Let F be a family of digraphs, which can be recursively defined as follows.\\n• If F ∈ F , then every subdigraph of F also belongs to F .\\nOur main result is as follows:\\nWe also show that the if K is a proper subdigraph of ↔ C 4 except for the digraph obtained from ↔ C 4 by deleting an arbitrary arc, then mader χ (K) = |V (K)|. The paper is organized as follows. In section 2, we list some notions and useful results. In section 3, we show that each digraph contained in the generalized family F is Maderperfect. Especially, the subdigraphs of ↔ C ℓ where the digons are vertex disjoint are contained in F . In section 4, we prove that the proper subdigraphs of ↔ C 4 except for H 3 (as shown in Figure 1) are Mader-perfect.\"\n },\n {\n \"title\": \"Preliminary\",\n \"content\": \"In this section, we first introduce some notions. For a digraph D and u ∈ V (D), let δ + D (u) be the number of arcs in D with tail u and δ − D (u) be the number of arcs in D with head\\nFor a vertex subset X in D, the subdigraph of D induced by X is denoted by D[X]. We say that W = u 0 e 1 u 1 . . . u k−1 e k u k is a diwalk from u 0 to u k in D if u i ∈ V (D) for i ∈ {0, 1, . . . , k} and e j = (u j−1 , u j ) ∈ A(D) for j ∈ {1, . . . , k}. Further, if u i = u j for i, j ∈ {0, 1, . . . , k}, then we call W a dipath from u 0 to u k . If only u 0 = u k , then we call W a dicycle of length k. A digraph D is called strongly connected if there is a dipath from u to v for every pair of vertices u and v in D. A strong component of D is a maximal induced subgraph of D which is strongly connected. And D is called k-strongly connected if the subdigraph D[V (D) \\\\ S] is strongly connected for any vertex subset S ⊆ V (D) with |S| < k. We say that D is k-dicritical, if χ(D) = k and for any proper subdigraph\\nis defined as the length of the shortest dipath between u and v contained in D. We use D[u, v] to denote a directed path from u to v and D(u, v) to denote a directed path between u and v in D. Especially, D [u, v] and D(u, v) denote the vertex u when u = v. Let ← − D be the digraph obtained from D by reversing the orientations of all arcs.\\nThe following results were proved in [3,7,9] and will be used in this paper.\\nLet F be the disjoint union of two digraphs F 1 and F 2 , then mader χ (F ) ≤ mader χ (F 1 ) + mader χ (F 2 ).\\nTheorem 2.4. [3,9] Let k ∈ N, and let D be a k-strongly connected digraph with δ + (D),\\n3 The proof of Theorem 1.7\\nTo prove Theorem 1.7, we first prove the following result.\\nProof. By the proof of Case (II) of Theorem 16 in [7], we only consider the case (v 0 , v k ) ∈ A(F * ). For convenience, let mader χ (F ) = M . We need to show that for any given digraph\\n. . , M + k} be an acyclic coloring which maximizes |c −1 0 ({1, . . . , k})|. In the rest of the proof, let\\n, we have that χ(D 1 ) = k and χ(D 2 ) = M since c 0 is an acyclic coloring of D. Combining that mader χ (F ) = M , there is a subdivision S of F in D 2 and denote by x 0 the vertex in S corresponding to v 0 . In [7], the authors defined a pre-order on the acyclic colorings of D 1 with respect to x 0 as follows.\\nHere < lex denotes the lexicographical order on Z k . Let c denote an acyclic k-coloring of D 1 that is minimal with respect to ≺. From Claim 2 in the proof of Theorem 16 in [7], we know that there are vertices\\nNext, we show that there exist vertices z 1 , z 2 , . . . , z k in D 1 which satisfy\\nDefine z 1 ∈ V (Ĉ) to be the unique last vertex in V (Ĉ) that we meet when traversing the trace of the path P 1,2 starting from x 1 ∈ V (Ĉ). Without loss of generality, we assume\\n. Then for j ∈ {2, 3}, we define z j to be the first vertex of P j,j+1 that we meet when traversing the trace of the dipath P j−1,j (z j−1 , x j ) starting from z j−1 , and let\\nis as small as possible and let z 3 ∈ V (C)∩V (P 3,4 ) such that dist P 3,4 (z 3 , x 4 ) is as small as possible. We have that c(z i ) = i for i ∈ {2, 3} and P 3,4 [z 3 , x 4 ] is a dipath between z 3 and x 4 . We define Q 1,2 = P 1,2 (z 1 , z 2 ) and Q 2,3 ∪Q 3,2 = C. Now, assume that for any dicycle C in X 2,3 , we have that V (P 1,2 [z 1 , x 2 ])∩ V (C) = ∅ or V (P 3,4 )∩ V (C) = ∅. Since X 2,3 is a strong component, any vertex in X 2,3 lies in a dicycle. Let C 1 and C 2 be two dicyles in X 2,3 which contain x 2 and x 3 respectively. We have that\\nIn this case, there are two vertices u and v (it may happen that u\\nis a dipath from x 4 to z 3 . We define Q 1,2 = P 1,2 [z 1 , z 2 ] and Q 2,3 ∪ Q 3,2 = C 1 .\\nSince X 2,3 is a strong component, there is a shortest dipath P 1 from C 1 to C 2 and a shortest dipath P 2 from C 2 to C 1 . Denote the initial and terminal vertices of P 1 by u and v, the initial and terminal vertices of P 2 by x and y, respectively. Let z ′ 3 ∈ V (P 1 ∪C 2 )∩V (P 3,4 ) such that dist P 3,4 (z ′ 3 , x 4 ) is as small as possible. Now we need to consider the following cases. Case 2.1 z ′ 3 ∈ V (C 2 ). Denote by w the unique first vertex in V (P 1 ∪ C 1 ) \\\\ {v} that we meet when traversing the trace of the dipath P 2 starting from x. We have that\\nis a dicycle when w = y. Note that when w = y, we may assume\\n, otherwise it can be reduced to Case 1. Denote the above dicycle by\\nthen let z 2 = z ′ 2 . We define Q 1,2 = P 1,2 [z 1 , z 2 ] and Q 2,3 ∪ Q 3,2 = C 3 . Suppose that\\n, then let z 2 = z ′ 2 and so c(z 2 ) = 2. We define\\nx]) such that dist P 3,4 (z 3 , x 4 ) is as small as possible and P 3,4 (z 3 , x 4 ) is a dipath between z 3 and x 4 . By the definition of z ′ 3 , we have that z 3 = w and c(z 3 ) = 3. Case 2.2 z ′ 3 ∈ V (P 1 ) \\\\ V (C 1 ). If w = y, then one of the following holds.\\n(a) There is a vertex\\n(c) There are two distinct vertices u 1 , u 2 ∈ V (P 1 ) ∩ V (P 2 ) with u 1 = z ′ 3 such that P 1 (u 1 , u 2 ) is internally vertex disjoint with P 2 (u 1 , u 2 ) and z ′ 3 ∈ V (P 1 (u 1 , u 2 )).\\nIf w = y or (a) holds, then let C 1 = C 1 and C 2 = C 1 [y, u] ∪ P 1 [u, u 1 ] ∪ P 2 [u 1 , y]. We refer z ′ 3 as x 3 and so it can be reduced to Case 1. Suppose that (b) or (c) holds, we have\\nx] is a dicycle or P 1 (u 1 , u 2 ) ∪ P 2 (u 1 , u 2 ) is a dicycle, respectively. Call the possible cycle C 3 . Let C 1 = C 1 and C 2 = C 3 . We refer z ′ 3 as x 3 and so it can be reduced to Case 2.1.\\nIt is easy to check that Q 1,2 , Q 2,3 , Q 2,3 , and V (P 3,4 (z 3 , x 4 )) are pairwise internally vertex-disjoint. Therefore, recursively, we can find the corresponding z 4 , . . . , z k with respect to v 4 , . . . , v k .\\nLet S * be the subdigraph of D formed by joining S ⊆ D[Y 2 ], the pairwise distinct vertices z 1 , . . . , z k and the connecting dipaths Q i−1,i , Q i,i−1 , i ∈ {2, . . . , k}, the two antiparallel directed pathsĈ[x 0 , z 1 ],Ĉ[z 1 , x 0 ]between x 0 and z 1 as well as the arc (x 0 , z k ). It follows that S * is isomorphic to a subdivision of F * .\\nThis completes the proof of Theorem 3.1.\\nBy the definition of F , we know that F * ⊆ F . We call the second item the ear addition operation and the third item taking a subdigraph operation in the definition of F . Therefore, every digraph in F can be obtained from K 1 by a sequence of operations consisting of the above two operations. We call F ∈ F maximal if it is obtained from K 1 by only using the ear addition operation. By recursively using Theorem 3.1, for F ∈ F being maximal, mader χ (F ) = |V (F )|.\\nProof of Theorem 1.7. Let F ∈ F and let U G(F ) be the graph obtained from F by replacing every arc (u, v) with the edge uv and deleting all multiple edges between every pair of vertices apart from one. If U G(F ) is disconnected, then by Lemma 2.2, it suffices to consider the components of U G(F ). Therefore, in the following, we may assume that U G(F ) is connected. We prove Theorem 1.7 by induction on the number of cycles contained in U G(F ). Denote by ℓ the number of cycles contained in U G(F ). First, assume that ℓ = 0. Then F ∈ F * and so mader χ (F ) = |V Figure 1. In this section, we show that any graph in K is mader perfect. For any K ∈ K,\\nwhere H 1 and H 2 are as shown in Figure 1. To prove that any graph in K is mader perfect, by Theorem 1.7 and Lemma 2.1, it suffices to prove that if K ∈ {H 1 , H 2 }, then mader χ (K) = 4.\\nIn order to construct an acyclic 3-coloringc of D, we need to consider the vertices in B 2 1 . Denote by T 0 the set of all the vertices in B 2 1 that are contained in some dicycle in the digraph D[B 2 1 ∪ V ′ 3 ] and let T 3 = B 2 1 \\\\ T 0 . Further, denote by T 1 the set of all the vertices in T 0 that are not contained in any dicycle in the digraph D[T 0 ∪ E] and let T 2 = T 0 \\\\ T 1 . An 3-coloringc = (Ṽ 1 ,Ṽ 2 ,Ṽ 3 ) of D can be given by settingṼ i = V ′ i ∪ T i , whereṼ i is a color class ofc for i ∈ {1, 2, 3}. Ifc is acyclic, then it leads to a contradiction as χ(D) = 4. Thus, we finish the proof by showing that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. In the following, we first list some useful claims.\\nBy the partition, there is a shortest dipath P 1 from B 1 to v without passing through {x} ∪ B 2 in D. Let u ∈ V (P 1 ) ∩ B 1 . If K = H 1 , then let P 2 be a shortest dipath from v to B 2 without passing through {x} ∪ B 1 in D. If K = H 2 , then let P 2 be a shortest dipath from B 2 to v without passing through {x}∪B 1 in D. Denote the other end of P 2 by w, and so w ∈ B 2 . Since B 1 and B 2 are acyclic,\\n. Therefore, C u ∪ C w ∪ P 1 ∪ P 2 contains a subdivision of K, a contradiction.\\ncontains a dicycle C. Then V (C) ∩ (B 1 ∪ B 1 1 ) = ∅ and V (C) ∩ A 2 = ∅ and so there is a dipath P from\\n. Consider a shortest dipath P 1 in D ′ from B 1 to u, the existence of P 1 can be guaranteed since D ′ is strongly connected. If V (P 1 ) ∩ B 2 = ∅, then there is diwalk P 1 ∪ P (u, v) from B 1 to v without passing through {x} ∪ B 2 which implies that v ∈ B 1 1 , a contradiction. Hence, we may assume that V (P 1 ) ∩ B 2 = ∅. Since P 1 is selected to be shortest, there is a dipath P ′ 1 ⊂ P 1 from B 2 to u without passing through {x} ∪ B 1 contradicting the fact that u ∈ B 2 2 . Therefore, for each vertex v in T 0 there is a dipath from v to B 2 2 contained in D[T 0 ∪ B 2 2 ].\\n. We first consider the vertex setṼ 1 . Suppose that u 1 ∈ B 2 ∪ B 1 2 . By Claim 3, there is a dipath P 1 from v 1 to B 2 2 contained in D[T 0 ∪ B 2 2 ]. Let w 1 ∈ V (P 1 ) ∩ B 2 2 . Therefore, by the definition of the partition, there is a diwalk W 1 = C 1 [u 1 , v 1 ] ∪ P 1 without passing through {x} ∪ B 1 . The existence of the diwalk W 1 implies that w 1 ∈ B 1 2 , a contradiction. Thus, V (C 1 ) ∩ (B 2 ∪ B 1 2 ) = ∅ and so u 1 ∈ A 1 . By the definition of the partition, there is a dipath P ′ 1 from B 1 to u 1 without passing through {x} ∪ B 2 . Thus, there is a diwalk W ′ 2 = P ′ 1 ∪ C 1 [u 1 , v 1 ] from B 1 to v 1 without passing through {x} ∪ B 2 . This implies that v 1 ∈ B 1 1 contradicting the fact that v 1 ∈ B 2 1 . Therefore, D[Ṽ 1 ] is acyclic. Now, we consider the vertex setṼ 2 . Suppose that u 2 ∈ B 1 ∪ B 1 1 , then there is a dipath containing C 2 [u 2 , v 2 ] from B 1 to v 2 without passing through {x} ∪ B 2 . It implies that v 2 ∈ B 1 1 , a contradiction. Thus, V (C 2 ) ∩ (B 1 ∪ B 1 1 ) = ∅ and so u 2 ∈ A 2 . Note that v 2 is contained in some dicycle C ′ in D[T 2 ∪ E]. Let w 2 ∈ V (C ′ ) ∩ E. Then C ′ [v 2 , w 2 ] a dipath from v 2 to w 2 without passing through {x} ∪ B 1 . Thus, there is a diwalk W 2 = C ′ [u 2 , v 2 ] ∪ C ′ [v 2 , w 2 ] from u 2 to w 2 without passing through {x} ∪ B 1 . Combining that there is a dipath from B 2 to u 2 without passing through {x} ∪ B 1 , it implies that w 2 ∈ A 2 contradicting the fact that w 2 ∈ E. Therefore, D[Ṽ 2 ] is acyclic.\\nFinally, we conclude that D admits an acyclic 3-coloring, which is a contradiction to χ(D) = 4. So the assumption mader χ (K) > 4 is not true. Hence, mader χ (K) = 4. This completes the proof of Theorem 4.1.\"\n },\n {\n \"title\": \"Theorem 1 . 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4 . 1 .Figure 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"[1] Let F be a digraph with n vertices and m arcs. Then mader χ (F ) ≤ 4 m (n − 1) + 1.","content":"Theorem 1 . 2 .\n12[1] Let F be a digraph with n vertices and m arcs. Then mader χ (F ) ≤ 4 m (n − 1) + 1."},{"id":"","caption":"Theorem 1.3. [7] For every k ∈ N,","content":"Theorem 1.3. [7] For every k ∈ N,"},{"id":"","caption":"(F )| by Theorem 1.4. Now, suppose that ℓ ≥ 1. We call a cycle C an outmost cycle if there is a vertex u ∈ V (C) such that the other vertices of C cannot reach any other cycle in U G(F ) without passing u. Call u the special vertex of C. Let C be an outmost cycle in U G(F ) with vertexset {v 0 , . . . , v s } and special vertex v 0 . Let Y be the component of F [V (F ) \\ {v 1 , . . . , v s }] that contains v 0 . By the induction hypothesis, mader χ (Y ) = |V (Y )|. Let Y 1 = F [V (Y ) ∪ {v 1 , . . . , v s }]. Therefore, by Theorem 3.1, mader χ (Y 1 ) = |V (Y 1 )| since Y 1 can be obtained from Y and F [{v 1 , . . . , v s }] by the ear addition operation. For each v i with i ∈ {1, . . . , s}, Let F v i be the digraph that contains v i obtained from F by deleting the arcs adjacent to v i contained in F [{v 1 , . . . , v s }]. Since C is an outmost cycle, F v i isa spanning subdigraph of a biorientation of some tree rooted at v i , say ↔ T v i . Note that ↔ T v i can be seen as a maximal digraph of F obtained from v i by a sequence of ear addition operation (in the case, P k is an isolate vertex in the definition of F ) for i ∈ {1, . . . , s}. Let T = ↔ T v 1 ∪ · · · ∪ ↔ T vs . Denote byF the digraph obtained from Y 1 and V (T ) \\ {v 1 , . . . , v s } by a corresponding sequence of ear addition operation of each ↔ T v i , where i ∈ {1, . . . , s}. Thus, mader χ (F ) = |V (F )| by recursively using Theorem 3.1. Since F is a spanning subdigraph ofF , mader χ (F ) ≤ mader χ (F ). Therefore, mader χ (F ) = |V (F )| since mader χ (F ) ≥ |V (F )|. 4 Subdigraphs of the biorientation of C 4 Let K = {K|K be a proper subdigraph of ↔ C 4 and K ∼ = H 3 }, where H 3 is as shown in","content":"(F )| by Theorem 1.4. Now, suppose that ℓ ≥ 1. We call a cycle C an outmost cycle if there is a vertex u ∈ V (C) such that the other vertices of C cannot reach any other cycle in U G(F ) without passing u. Call u the special vertex of C. Let C be an outmost cycle in U G(F ) with vertexset {v 0 , . . . , v s } and special vertex v 0 . Let Y be the component of F [V (F ) \\ {v 1 , . . . , v s }] that contains v 0 . By the induction hypothesis, mader χ (Y ) = |V (Y )|. Let Y 1 = F [V (Y ) ∪ {v 1 , . . . , v s }]. Therefore, by Theorem 3.1, mader χ (Y 1 ) = |V (Y 1 )| since Y 1 can be obtained from Y and F [{v 1 , . . . , v s }] by the ear addition operation. For each v i with i ∈ {1, . . . , s}, Let F v i be the digraph that contains v i obtained from F by deleting the arcs adjacent to v i contained in F [{v 1 , . . . , v s }]. Since C is an outmost cycle, F v i isa spanning subdigraph of a biorientation of some tree rooted at v i , say ↔ T v i . Note that ↔ T v i can be seen as a maximal digraph of F obtained from v i by a sequence of ear addition operation (in the case, P k is an isolate vertex in the definition of F ) for i ∈ {1, . . . , s}. Let T = ↔ T v 1 ∪ · · · ∪ ↔ T vs . Denote byF the digraph obtained from Y 1 and V (T ) \\ {v 1 , . . . , v s } by a corresponding sequence of ear addition operation of each ↔ T v i , where i ∈ {1, . . . , s}. Thus, mader χ (F ) = |V (F )| by recursively using Theorem 3.1. Since F is a spanning subdigraph ofF , mader χ (F ) ≤ mader χ (F ). Therefore, mader χ (F ) = |V (F )| since mader χ (F ) ≥ |V (F )|. 4 Subdigraphs of the biorientation of C 4 Let K = {K|K be a proper subdigraph of ↔ C 4 and K ∼ = H 3 }, where H 3 is as shown in"},{"id":"","caption":"Let K ∈ {H 1 , H 2 }, then mader χ (K) = 4. Proof. Let K ∈ {H 1 , H 2 }. Since mader χ (K) ≥ |V (K)|, mader χ (K) ≥ 4. We prove Theorem 4.1 by contradiction. Suppose that the assertion is false, i.e., mader χ (K) > 4. Then there is a digraph D with χ(D) = 4 that does not contain a subdivision of K. Among all Three subdigraphs of the biorienitation of C 4 counterexamples we choose D so that first |V (D)| is minimum, and then |A(D)| is minimum. By the choice of D, D is connected, 4-dicritical. By Lemma 2.3 and Theorem 2.4, we know that D is strongly connected and there is a vertex x in D such that D ′ = D[V (D) \\ {x}] is strongly connected. Since D is 4-dicritical, χ(D ′ ) = 3. In the following let c = (V 1 , V 2 , V 3 ) be an acyclic 3-coloring of D ′ , where V 1 , V 2 and V 3 are the color classes. In order to show that there is an acyclic 3-coloring of D, we define the following vertex partition of D ′ with respect to x.","content":"Theorem 4 . 1 .Figure 1 :\n411Let K ∈ {H 1 , H 2 }, then mader χ (K) = 4. Proof. Let K ∈ {H 1 , H 2 }. Since mader χ (K) ≥ |V (K)|, mader χ (K) ≥ 4. We prove Theorem 4.1 by contradiction. Suppose that the assertion is false, i.e., mader χ (K) > 4. Then there is a digraph D with χ(D) = 4 that does not contain a subdivision of K. Among all Three subdigraphs of the biorienitation of C 4 counterexamples we choose D so that first |V (D)| is minimum, and then |A(D)| is minimum. By the choice of D, D is connected, 4-dicritical. By Lemma 2.3 and Theorem 2.4, we know that D is strongly connected and there is a vertex x in D such that D ′ = D[V (D) \\ {x}] is strongly connected. Since D is 4-dicritical, χ(D ′ ) = 3. In the following let c = (V 1 , V 2 , V 3 ) be an acyclic 3-coloring of D ′ , where V 1 , V 2 and V 3 are the color classes. In order to show that there is an acyclic 3-coloring of D, we define the following vertex partition of D ′ with respect to x."},{"id":"","caption":"The vertex partition of D ′ . (a) B i ⊆ V i consists of all the vertices that are contained in the same strong component with x in D[{x} ∪ V i ] for i ∈ {1, 2}. (b) B 1 i ⊆ V i and A i ⊆ V 3 consist of all the vertices which can be reached by some vertex of B i in D without passing through any vertex in {x} ∪ B 3−i for i ∈ {1, 2}.","content":"Figure 2 :\n2The vertex partition of D ′ . (a) B i ⊆ V i consists of all the vertices that are contained in the same strong component with x in D[{x} ∪ V i ] for i ∈ {1, 2}. (b) B 1 i ⊆ V i and A i ⊆ V 3 consist of all the vertices which can be reached by some vertex of B i in D without passing through any vertex in {x} ∪ B 3−i for i ∈ {1, 2}."},{"id":"","caption":"By Claim 1, we know that {Ṽ 1 ,Ṽ 2 ,Ṽ 3 } is a partition of V (D). Now, we prove that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. Obviously, by the definition of T 3 and Claim 2, D[Ṽ 3 ] is acyclic. To get a contradiction, suppose that there is a dicycle C i contained in D[Ṽ i ] for i ∈ {1, 2}. For i ∈ {1, 2}, we have that D[V ′ i ] is acyclic by Claim 2, and since D[T 1 ∪E] and D[T 2 ] are acyclic, we have that","content":"By Claim 1, we know that {Ṽ 1 ,Ṽ 2 ,Ṽ 3 } is a partition of V (D). Now, we prove that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. Obviously, by the definition of T 3 and Claim 2, D[Ṽ 3 ] is acyclic. To get a contradiction, suppose that there is a dicycle C i contained in D[Ṽ i ] for i ∈ {1, 2}. For i ∈ {1, 2}, we have that D[V ′ i ] is acyclic by Claim 2, and since D[T 1 ∪E] and D[T 2 ] are acyclic, we have that"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"[1] Let F be a digraph with n vertices and m arcs. Then mader χ (F ) ≤ 4 m (n − 1) + 1.\",\n \"content\": \"Theorem 1 . 2 .\\n12[1] Let F be a digraph with n vertices and m arcs. Then mader χ (F ) ≤ 4 m (n − 1) + 1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 1.3. [7] For every k ∈ N,\",\n \"content\": \"Theorem 1.3. [7] For every k ∈ N,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(F )| by Theorem 1.4. Now, suppose that ℓ ≥ 1. We call a cycle C an outmost cycle if there is a vertex u ∈ V (C) such that the other vertices of C cannot reach any other cycle in U G(F ) without passing u. Call u the special vertex of C. Let C be an outmost cycle in U G(F ) with vertexset {v 0 , . . . , v s } and special vertex v 0 . Let Y be the component of F [V (F ) \\\\ {v 1 , . . . , v s }] that contains v 0 . By the induction hypothesis, mader χ (Y ) = |V (Y )|. Let Y 1 = F [V (Y ) ∪ {v 1 , . . . , v s }]. Therefore, by Theorem 3.1, mader χ (Y 1 ) = |V (Y 1 )| since Y 1 can be obtained from Y and F [{v 1 , . . . , v s }] by the ear addition operation. For each v i with i ∈ {1, . . . , s}, Let F v i be the digraph that contains v i obtained from F by deleting the arcs adjacent to v i contained in F [{v 1 , . . . , v s }]. Since C is an outmost cycle, F v i isa spanning subdigraph of a biorientation of some tree rooted at v i , say ↔ T v i . Note that ↔ T v i can be seen as a maximal digraph of F obtained from v i by a sequence of ear addition operation (in the case, P k is an isolate vertex in the definition of F ) for i ∈ {1, . . . , s}. Let T = ↔ T v 1 ∪ · · · ∪ ↔ T vs . Denote byF the digraph obtained from Y 1 and V (T ) \\\\ {v 1 , . . . , v s } by a corresponding sequence of ear addition operation of each ↔ T v i , where i ∈ {1, . . . , s}. Thus, mader χ (F ) = |V (F )| by recursively using Theorem 3.1. Since F is a spanning subdigraph ofF , mader χ (F ) ≤ mader χ (F ). Therefore, mader χ (F ) = |V (F )| since mader χ (F ) ≥ |V (F )|. 4 Subdigraphs of the biorientation of C 4 Let K = {K|K be a proper subdigraph of ↔ C 4 and K ∼ = H 3 }, where H 3 is as shown in\",\n \"content\": \"(F )| by Theorem 1.4. Now, suppose that ℓ ≥ 1. We call a cycle C an outmost cycle if there is a vertex u ∈ V (C) such that the other vertices of C cannot reach any other cycle in U G(F ) without passing u. Call u the special vertex of C. Let C be an outmost cycle in U G(F ) with vertexset {v 0 , . . . , v s } and special vertex v 0 . Let Y be the component of F [V (F ) \\\\ {v 1 , . . . , v s }] that contains v 0 . By the induction hypothesis, mader χ (Y ) = |V (Y )|. Let Y 1 = F [V (Y ) ∪ {v 1 , . . . , v s }]. Therefore, by Theorem 3.1, mader χ (Y 1 ) = |V (Y 1 )| since Y 1 can be obtained from Y and F [{v 1 , . . . , v s }] by the ear addition operation. For each v i with i ∈ {1, . . . , s}, Let F v i be the digraph that contains v i obtained from F by deleting the arcs adjacent to v i contained in F [{v 1 , . . . , v s }]. Since C is an outmost cycle, F v i isa spanning subdigraph of a biorientation of some tree rooted at v i , say ↔ T v i . Note that ↔ T v i can be seen as a maximal digraph of F obtained from v i by a sequence of ear addition operation (in the case, P k is an isolate vertex in the definition of F ) for i ∈ {1, . . . , s}. Let T = ↔ T v 1 ∪ · · · ∪ ↔ T vs . Denote byF the digraph obtained from Y 1 and V (T ) \\\\ {v 1 , . . . , v s } by a corresponding sequence of ear addition operation of each ↔ T v i , where i ∈ {1, . . . , s}. Thus, mader χ (F ) = |V (F )| by recursively using Theorem 3.1. Since F is a spanning subdigraph ofF , mader χ (F ) ≤ mader χ (F ). Therefore, mader χ (F ) = |V (F )| since mader χ (F ) ≥ |V (F )|. 4 Subdigraphs of the biorientation of C 4 Let K = {K|K be a proper subdigraph of ↔ C 4 and K ∼ = H 3 }, where H 3 is as shown in\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let K ∈ {H 1 , H 2 }, then mader χ (K) = 4. Proof. Let K ∈ {H 1 , H 2 }. Since mader χ (K) ≥ |V (K)|, mader χ (K) ≥ 4. We prove Theorem 4.1 by contradiction. Suppose that the assertion is false, i.e., mader χ (K) > 4. Then there is a digraph D with χ(D) = 4 that does not contain a subdivision of K. Among all Three subdigraphs of the biorienitation of C 4 counterexamples we choose D so that first |V (D)| is minimum, and then |A(D)| is minimum. By the choice of D, D is connected, 4-dicritical. By Lemma 2.3 and Theorem 2.4, we know that D is strongly connected and there is a vertex x in D such that D ′ = D[V (D) \\\\ {x}] is strongly connected. Since D is 4-dicritical, χ(D ′ ) = 3. In the following let c = (V 1 , V 2 , V 3 ) be an acyclic 3-coloring of D ′ , where V 1 , V 2 and V 3 are the color classes. In order to show that there is an acyclic 3-coloring of D, we define the following vertex partition of D ′ with respect to x.\",\n \"content\": \"Theorem 4 . 1 .Figure 1 :\\n411Let K ∈ {H 1 , H 2 }, then mader χ (K) = 4. Proof. Let K ∈ {H 1 , H 2 }. Since mader χ (K) ≥ |V (K)|, mader χ (K) ≥ 4. We prove Theorem 4.1 by contradiction. Suppose that the assertion is false, i.e., mader χ (K) > 4. Then there is a digraph D with χ(D) = 4 that does not contain a subdivision of K. Among all Three subdigraphs of the biorienitation of C 4 counterexamples we choose D so that first |V (D)| is minimum, and then |A(D)| is minimum. By the choice of D, D is connected, 4-dicritical. By Lemma 2.3 and Theorem 2.4, we know that D is strongly connected and there is a vertex x in D such that D ′ = D[V (D) \\\\ {x}] is strongly connected. Since D is 4-dicritical, χ(D ′ ) = 3. In the following let c = (V 1 , V 2 , V 3 ) be an acyclic 3-coloring of D ′ , where V 1 , V 2 and V 3 are the color classes. In order to show that there is an acyclic 3-coloring of D, we define the following vertex partition of D ′ with respect to x.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The vertex partition of D ′ . (a) B i ⊆ V i consists of all the vertices that are contained in the same strong component with x in D[{x} ∪ V i ] for i ∈ {1, 2}. (b) B 1 i ⊆ V i and A i ⊆ V 3 consist of all the vertices which can be reached by some vertex of B i in D without passing through any vertex in {x} ∪ B 3−i for i ∈ {1, 2}.\",\n \"content\": \"Figure 2 :\\n2The vertex partition of D ′ . (a) B i ⊆ V i consists of all the vertices that are contained in the same strong component with x in D[{x} ∪ V i ] for i ∈ {1, 2}. (b) B 1 i ⊆ V i and A i ⊆ V 3 consist of all the vertices which can be reached by some vertex of B i in D without passing through any vertex in {x} ∪ B 3−i for i ∈ {1, 2}.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"By Claim 1, we know that {Ṽ 1 ,Ṽ 2 ,Ṽ 3 } is a partition of V (D). Now, we prove that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. Obviously, by the definition of T 3 and Claim 2, D[Ṽ 3 ] is acyclic. To get a contradiction, suppose that there is a dicycle C i contained in D[Ṽ i ] for i ∈ {1, 2}. For i ∈ {1, 2}, we have that D[V ′ i ] is acyclic by Claim 2, and since D[T 1 ∪E] and D[T 2 ] are acyclic, we have that\",\n \"content\": \"By Claim 1, we know that {Ṽ 1 ,Ṽ 2 ,Ṽ 3 } is a partition of V (D). Now, we prove that D[Ṽ i ] is acyclic for i ∈ {1, 2, 3}. Obviously, by the definition of T 3 and Claim 2, D[Ṽ 3 ] is acyclic. To get a contradiction, suppose that there is a dicycle C i contained in D[Ṽ i ] for i ∈ {1, 2}. For i ∈ {1, 2}, we have that D[V ′ i ] is acyclic by Claim 2, and since D[T 1 ∪E] and D[T 2 ] are acyclic, we have that\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Subdivisions in digraphs of large out-degree or large dichromatic number. P Aboulker, N Cohen, F Havet, W Lochet, P S Moura, S Thomassé, Electron. J. Comb. 26P. Aboulker, N. Cohen, F. Havet, W. Lochet, P.S. Moura, S. Thomassé, Subdivisions in digraphs of large out-degree or large dichromatic number, Electron. J. Comb. 26 (2019) 3.19.","doi":""},{"text":"Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs. B Bollobás, A Thomason, Eur. J. Comb. 19B. Bollobás, A. Thomason, Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, Eur. J. Comb. 19 (1998) 883-887.","doi":""},{"text":"J Bang-Jensen, G Z Gutin, Digraphs: Theory, Algorithms and Applications. Springer Science & Business MediaJ. Bang-Jensen, G.Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer Science & Business Media, 2008.","doi":""},{"text":"Hajós' graph coloring conjecture: variations and counterexamples. P A Catlin, J. Comb. Theory, Ser. B. 26P.A. Catlin, Hajós' graph coloring conjecture: variations and counterexamples, J. Comb. Theory, Ser. B 26 (1979) 268-274.","doi":""},{"text":"A property of 4-chromatic graphs and some remarks on critical graphs. G A Dirac, J. Lond. Math. Soc. 27G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. Lond. Math. Soc. 27 (1952) 85-92.","doi":""},{"text":"In abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen. G A Dirac, Math. Nachrichten. 22G.A. Dirac. In abstrakten graphen vorhandene vollständige 4-graphen und ihre un- terteilungen, Math. Nachrichten 22 (1960) 61-85.","doi":""},{"text":"Dichromatic number and forced subdivisions. L Gishboliner, R Steiner, T Szabó, J. Comb. Theory, Ser. B. 153L. Gishboliner, R. Steiner, T. Szabó, Dichromatic number and forced subdivisions, J. Comb. Theory, Ser. B 153 (2022) 1-30.","doi":""},{"text":"Topological cliques in graphs II. J Komlós, E Szemerédi, Comb. Probab. Comput. 5J. Komlós, E. Szemerédi, Topological cliques in graphs II, Comb. Probab. Comput. 5 (1996) 79-90.","doi":""},{"text":"Ecken von kleinem Grad in kritisch n-fach zusammenhängenden Digraphen. W Mader, J. Comb. Theory, Ser. B. 53W. Mader, Ecken von kleinem Grad in kritisch n-fach zusammenhängenden Digraphen, J. Comb. Theory, Ser. B 53 (1991) 260-272.","doi":""}],"string":"[\n {\n \"text\": \"Subdivisions in digraphs of large out-degree or large dichromatic number. P Aboulker, N Cohen, F Havet, W Lochet, P S Moura, S Thomassé, Electron. J. Comb. 26P. Aboulker, N. Cohen, F. Havet, W. Lochet, P.S. Moura, S. Thomassé, Subdivisions in digraphs of large out-degree or large dichromatic number, Electron. J. Comb. 26 (2019) 3.19.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs. B Bollobás, A Thomason, Eur. J. Comb. 19B. Bollobás, A. Thomason, Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, Eur. J. Comb. 19 (1998) 883-887.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Bang-Jensen, G Z Gutin, Digraphs: Theory, Algorithms and Applications. Springer Science & Business MediaJ. Bang-Jensen, G.Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer Science & Business Media, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hajós' graph coloring conjecture: variations and counterexamples. P A Catlin, J. Comb. Theory, Ser. B. 26P.A. Catlin, Hajós' graph coloring conjecture: variations and counterexamples, J. Comb. Theory, Ser. B 26 (1979) 268-274.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A property of 4-chromatic graphs and some remarks on critical graphs. G A Dirac, J. Lond. Math. Soc. 27G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. Lond. Math. Soc. 27 (1952) 85-92.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"In abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen. G A Dirac, Math. Nachrichten. 22G.A. Dirac. In abstrakten graphen vorhandene vollständige 4-graphen und ihre un- terteilungen, Math. Nachrichten 22 (1960) 61-85.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dichromatic number and forced subdivisions. L Gishboliner, R Steiner, T Szabó, J. Comb. Theory, Ser. B. 153L. Gishboliner, R. Steiner, T. Szabó, Dichromatic number and forced subdivisions, J. Comb. Theory, Ser. B 153 (2022) 1-30.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Topological cliques in graphs II. J Komlós, E Szemerédi, Comb. Probab. Comput. 5J. Komlós, E. Szemerédi, Topological cliques in graphs II, Comb. Probab. Comput. 5 (1996) 79-90.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ecken von kleinem Grad in kritisch n-fach zusammenhängenden Digraphen. W Mader, J. Comb. Theory, Ser. B. 53W. Mader, Ecken von kleinem Grad in kritisch n-fach zusammenhängenden Digraphen, J. Comb. Theory, Ser. B 53 (1991) 260-272.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":393,"cells":{"text":{"kind":"string","value":"\nStructure of Business Firm Networks and Scale-Free Models\n30 Oct 2008\n\nMaksim Kitsak \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA\n\nMassimo Riccaboni \nCUSAS\nUniversity of Florence\nFlorenceItaly\n\nShlomo Havlin \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA\n\nDepartment of Physics\nMinerva Center\nBar-Ilan University\nRamat GanIsrael\n\nFabio Pammolli \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA\n\nCUSAS\nUniversity of Florence\nFlorenceItaly\n\nIMT Institute for Advanced Studies\nLuccaItaly\n\nH Eugene Stanley \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA\n\nStructure of Business Firm Networks and Scale-Free Models\n30 Oct 2008arXiv:0810.5514v1 [physics.soc-ph]PACS numbers: 8975Hc\nWe study the structure of business firm networks and scale-free models with degree distribution P (q) ∝ (q+c) −λ using the method of k-shell decomposition. We find that the Life Sciences industry network consists of three components: a \"nucleus,\" which is a small well connected subgraph, \"tendrils,\" which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus, and a \"bulk body\" which consists of the majority of nodes. At the same time we do not observe the above structure in the Information and Communication Technology sector of industry.We also conduct a systematic study of these three components in random scale-free networks. Our results suggest that the sizes of the nucleus and the tendrils decrease as λ increases and disappear for λ ≥ 3. We compare the k-shell structure of random scale-free model networks with two real world business firm networks in the Life Sciences and in the Information and Communication Technology sectors. Our results suggest that the observed behavior of the k-shell structure in the two industries is consistent with a recently proposed growth model that assumes the coexistence of both preferential and random agreements in the evolution of industrial networks.\n\nI. INTRODUCTION\n\nMany real-world complex systems are often described using the representation of graphs or networks, as sets of nodes connected by links [1,2,3,4,5,6,7]. Networks appear in various areas of science such as physics, biology, computer sciences, economics and sociology [3,4,5,6,7]. Real networks, despite their diversity, appear to have many common properties. Some real networks have been found to be \"small-world\" [3,8,9,10]: despite their large size, the typical distance between nodes is very small, of order of log(N) or less [3,11,12], where N is the number of nodes. Also, some real networks are scale-free (SF) with a power-law tail in their degree distribution\nP (q) ∝ (q + c) −λ ,(1)\nwhere q is the number of links per node, λ and c are distribution parameters [13,14,15,16].\n\nMany statistical physics and mathematical techniques have been successfully applied to study network structure including percolation [12,17,18,19,20,21], scaling [6,7,11], partitioning [22], box covering [23,24,25,26], k-core percolation [27,28] and k-shell decomposition [31] . The latter has been recently used to study the topology of the Internet at the autonomous system (AS) level [32]. It has been found that the AS Internet network consists of three distinct components:\n\n(i) The nucleus consists of highly connected nodes (hubs) and constitutes a tiny fraction of the network (≈ 0.5%).\n\n(ii) The tendrils (≈ 25%) are the subgraphs that connect to the network exclusively via the nucleus.\n\n(iii) The remaining nodes (≈ 75%) of the network constitute the bulk body. Unlike the tendrils, the nodes in the bulk remain connected even if the nucleus is removed.\n\nThe appearance of the three components led to the association of the AS Internet structure with that of a jelly-fish model [32]. For an alternative jelly-fish structure see also Ref. [33].\n\nThe nucleus plays a crucial role in data transfer throughout the Internet. If the nucleus is damaged or blocked, nodes in the tendrils cease to communicate with other nodes both in the bulk body and in other tendrils. However, the nodes in the bulk body can still communicate with each other, but less efficiently since the average path length between 2 them almost doubles [32]. The network of workplaces in Sweden [34,35] was also shown to possess a jelly-fish structure. Mean field analysis of the k-shell structure [29,30] focuses on random networks with a given degree distribution P (q).\n\nIn the present work we address the long-standing question of how long does a typical leader in an industry maintain its position. This question has attracted continuing attention in the Industrial Organization literature over the past generation [36]. Two rival views exist. The first asserts that leadership tends to persist for a 'long' time while the second one emphasizes the transience of leadership positions due to radical innovation and \"leapfrogging competition\". The Life Sciences (LS) sector is generally cited as an archetypal example of stability of industry leaders while on the contrary the Information and Communication Technology (ICT) sector is widely considered as a good example of high market turnover and instability. We study the k-shell structure of the LS and ICT industry in order to test these views and better understand the reasons behind the higher stability of the leading firms in the LS sciences and the ICT sector. The networks of LS and ICT firms are networks of nodes representing firms in the worldwide LS and ICT industries with links representing the collaborative agreements among them [26,37,38,39]. We also conduct a numerical analysis of model SF networks in order to better understand their k-shell structure. We study how the size and the connectivity of the three components of random SF model networks depend on the SF parameters. We compare the k-shell structure of SF models with that of the LS and ICT industry networks in order to get a better insight into the growth principles of the Industry networks.\n\nThe rest of the manuscript is organized as follows: in Section II we define the k-shell decomposition and apply it to both real and model SF networks. We then analyze the leadership in the industry networks from the k-shell perspective. In Section III we conduct a systematic analysis of the properties of the three components of model SF networks. In section IV we compare the results for SF models with those observed in the two industrial networks. We conclude our manuscript in Section V with a discussion and summary.\n\nII. K-SHELL, K-CORE, AND K-CRUST\n\nWe start the process of the k-shell decomposition [29,32] on a network by removing all nodes with degree q = 1. After the first iteration of pruning, there may appear new nodes 3 with degrees q = 1. We keep on pruning these nodes until only nodes with degree q ≥ 2 are left. The removed nodes along with the links connecting them form the k = 1 shell. Next, we iterate the pruning process for nodes of degree q = 2, thereby creating the k = 2 shell.\n\nWe further continue the k-shell decomposition for higher values of q until all nodes of the network are removed. As a result each node in the network is assigned a k-shell index k.\n\nThe largest shell index is called k max , which is also the total number of shells in the network, provided all shells below k max exist. The k-crust is defined as the union of all k-shells with indices smaller than k. Similarly, the k-core is defined as the union of all nodes with indices greater or equal k. (See Fig. 1 for demonstration.) As we explain later, the set of nodes in the k max shell is called nucleus provided there is a large number of nodes (tendrils) connected to the network exclusively via the k max shell.\n\nWe is consistent with the existence of a second order phase transition in the k-crust structure [32]. The type of the phase transition taking place at the k-shell decomposition is similar to that of targeted percolation in SF networks [21,40].\n\nUnlike in the ICT network, the size of the largest connected component N 1 in LS network undergoes a large jump N 1 (k = 19) − N 1 (k = 18) = 745, while the total size N 0 of the crust experiences only a small change N 0 (k = 19) − N 0 (k = 18) = 43 (See Fig. 3(a)). This can be explained as follows [32]. Approximately 10% of the he LS network are firms (which we call \"tendrils\") that prefer to sign collaborative agreements exclusively with the 43 firms that form k = k max = 19 shell (which we call nucleus). Typically, each of these firms (tendrils) signs a small number of agreements with firms in the nucleus and, therefore, has small degree.\n\nThus, the tendrils are removed in the decomposition of the first few shells. However, being connected exclusively to the nucleus in the k max = 19 shell, the tendrils do not contribute to the largest connected component of the k-crust until the last k max shell is decomposed.\n\nIt is the inclusion of the tendrils firms into N 1 (k) at the decomposition of the k max shell that results in the observed jump in N 1 (k). The appearance of the three components-the nucleus, the tendrils and the bulk body-allows one to associate the structure of the network of LS firms with that of the jelly-fish similar to the Internet at the Autonomous System level [32]. Interestingly, we do not observe the a similar jump in N 1 (k) in the ICT network (See . Thus, there is more deals between firms of the same size and age but in different technological and market areas. As a consequence, the k-shell of the ICT network is more unstable and heterogeneous which may explain why we do not detect the emergence of a nucleus such as in the LS sector.\n\nAs discussed above, one can in general calculate the size of the nucleus S n as the total number of nodes in the k max shell\n5 S n = N 0 (k max ) − N 0 (k max − 1).(2)\nThe increase in N 1 (k) at k = k max is comprised by the inclusion of tendrils S t and the nucleus S n . Thus, the size of tendrils can be calculated as\nS t = N 1 (k max ) − N 1 (k max − 1) − S n .(3)\nAs seen in Fig. 2(b), both the LS and the ICT industrial networks exhibit a SF degree distribution. Hence, in order to better understand the substructures of real networksthe nucleus, the tendrils, and the bulk components-we analyze the k-shell structure of the random SF models which were generated using the configurational approach [41]. We calculate the k-shell structure of random SF networks with c ≥ 0, and degree distribution exponent λ ∈ [2,3]. For our simulations (Fig. 4) we choose networks sizes of N = 8, 000\n\n(which is comparable to the size of the LS and ICT industry networks) and N = 10 6 in order to test the influence of finite size effects.\n\nIt is seen in Fig. 4 that both the number of shells k max and the jump in the largest connected component N 1 (k) decrease as the exponent λ increases. As the jump becomes less pronounced it becomes harder to detect it which motivates us to introduce the quantitative criterion for the emergence of the three distinct components. We define the rate of change of the largest connected component size as\nR(k) ≡ N 1 (k) − N 1 (k − 1).(4)\nWe compare the increase of the largest connected component R(k) at k = k max with that\nat k = k max − 1. The jump in N 1 (k) results in R(k max ) > R(k max − 1).(5)\nWe use Eq. (5) as a criterion for the existence of a nucleus and tendrils.\n\nBy examining R(k) plot we observe in the case of N = 8, 000 (Fig. 4c), that SF models with λ > 2.5 already do not have a nucleus and tendrils. However, for N = 10 6 we do observe nucleus and tendrils in SF models for λ ≤ 2.7. (Fig. 4d). These observations suggest that all SF networks with λ ∈ (2, 3) have a nucleus and tendrils provided N is sufficiently large. The above observations for SF models agree with the fact that we observe jelly-fish topology for LS network (λ = 2.5) and do not observe it for ICT network λ = 3.4. However, SF model networks with N = 8000 with λ = 2.5 and λ = 3.4 have k max = 5 and k max = 2 respectively, while the measured k max in LS and ICT networks (which have the same λ values) are k max = 19 and k max = 6. The observed difference in k max between industry networks and SF models with similar parameters suggests to further explore the k-shell structure of SF networks and consider SF model networks with c > 0 values in P (q) (as found in the LS and the ICT networks).\n\n\nIII. K-SHELL PROPERTIES OF SCALE-FREE NETWORKS\n\nAs shown above, SF networks may or may not have a jelly-fish structure, depending on the degree distribution P (q) and size N. In order to better understand the k-shell structure of SF networks, we calculate the sizes of the k-shells as a function of N and λ. For each pair of values N and λ we generate 10 3 − 10 4 realizations of SF models and calculate the average number of shells k max constituting the network as well as their average size S k (see Fig. 5). For small k values, S k is proportional to N. As the size of the network increases new shells start to appear. When the size increases further, the new shell growth stabilizes and becomes also proportional to N (See Fig. 5a,c). This result indicates that the size of each shell constitutes a certain finite fraction of the network and this fraction decreases with increasing k. The analytical analysis of k-core structure [29] leads to\nS k ∝ k −δ ,(6)\nwhere δ = 2 3 − λ .\n\nHence δ ≈ 2.2 for λ = 2.1 and δ ≈ 4.0 for λ = 2.5, which agrees with our simulations. One can estimate the total number of shells in a random SF network of size N as follows.\n\nSince every shell constitutes a fixed fraction of N it follows that S k ∝ Nk −δ . The last shell k max needs to possess at least one node S max ≡ S k=kmax ∼ 1, which leads to\nk max ≤ N 1/δ ,(8)\nIndeed, the total number of shells k max seems to increase as a power law with the network size N [ Fig. 6(a)]. The smaller is λ the faster is the growth of k max . As seen from Fig. 6(c), the estimated exponents 1/δ agree with those predicted by Eq. (8). Note that Eq. (8) together with\nδ = 2/(3 − λ)(9)\nis consistent with the fact that networks with λ > 3 do not have a k-shell structure.\n\nThe dependence of S max on N can be regarded as a crossover from the small N regime with N < N c (λ) where there is no nucleus to the power-law regime for N > N c (λ) where S max ∼ N τ (λ) ( Fig. 6b). We relate the observed crossover with the emergence of the nucleus and tendrils in SF networks for N > N c (λ). The critical size of SF networks, N c (λ), corresponding to the emergence of the nucleus and tendrils in SF networks seems to increase as λ increases. In the N < N c regime the size of the last shell S max increases with λ, which can be explained by the fact that SF networks with higher λ have fewer shells. On the other hand, in the power-law regime, the size of the last shell, S max , (which now becomes the nucleus S n ) is smaller for larger values of λ.\n\n\nIV. EVOLUTION AND STRUCTURE OF THE LS AND ICT INDUSTRY NET-\n\n\nWORKS\n\nWe further analyze the k-shell structure of the LS and ICT industry networks. The number of shells k max in the LS industry grows as a power-law function of its size,\nk max ∼ N θ ,(10)\nand reaches k max = 19 in 2002. Our estimates yield θ ≈ 0.6 (Fig. 7a). We find that the number of shells in ICT sector, k max , also grows and reaches k max = 6 in 2000. Note that S n for the LS network exhibits fluctuations for N ∈ [2000, 5000] and stabilizes for N > 5000 (See Fig. 7b). As seen in Fig. 7(c,d), the sizes of shells S k decrease as a function of their index k in both networks. As we notice in Section II, the observed shell sizes S k as well as the number of shells k max , measured for the LS and ICT networks, deviate from those obtained from SF models with c = 0. We expect for random SF models with c = 0, λ = 2.5 and λ = 3.4 and similar sizes as LS and ICT to find k max = 5 and k max = 2 respectively. Also, a SF network with λ = 2.5 is expected to have S k ∼ k −4. and k max ∼ N 0. 25 Size of the nucleus of the LS network, S n , as a function of N. S n exhibits fluctuations as LS network grows. Unlike in the analyzed SF models, S max becomes stable for N > 5000.\n\n(c) S k as a function of the k-shell index k for the LS network (squares). Shell sizes S k as a function of k decrease significantly slower than S k ∼ k −4 , which is expected for a random SF model with the same λ and c = 0 (triangles). However, the offset introduction of c = 4.0 in the SF degree distribution, P (q), mimics the k-shell structure of the LS network (circles). (d) S k as a function of k-shell index k for the ICT network (squares). SF model with λ = 3.4 and c = 0 does not possess a k-shell structure. However, the introduction of c = 6.0 in the SF degree distribution yields similar k-shell structure (circles), but we attribute it only to finite size effect.\n\n\nanalyze the k-shell structure of LS and ICT industrial sectors in time periods between 1990 and 2002 and between 1990 and 2000 respectively. The LS network expanded linearly since the mid-1970s while the ICT network took off in the 1990s and grew exponentially for a decade [see Fig. 2(a) and the inset]. The total number of firms in the LS is N = 6, 776 and in the ICT is N = 7, 759. These sizes refer to the largest connected component of each of the networks at the last year of observation. Both industrial networks feature SF degree distributions with λ ≈ 2.5, c ≈ 4. (LS) and λ ≈ 3.4, c ≈ 6. (ICT) [see Fig. 2(b)]. We use the Kholmogorov-Smirnov test in order to examine the goodness of fit of the degree distributions. The obtained p-values for 1000 trials are 0.24 and 0.33 for the LS and the ICT industry networks respectively.We next apply the k-shell decomposition procedure to the LS and the ICT networks. For each k-crust we calculate the total number of nodes, N 0 , comprising it, the size of the largest connected component, N 1 , in the k-crust and the size of the second largest component, N 2 , of the k-crust. As seen inFig. 3, the LS network consists of k max = 19 shells while the ICT network (which consists of comparable number of firms) has only k max = 6 shells. The size of the largest cluster in the k-crust starts growing rapidly after k = 4 and k = 2 in the LS and the ICT networks respectively. At these values of k, the size of the second largest cluster in the k-crust reaches a maximum. The above behavior of the k-crust components\n\nFig. 3 (\n3b)). In both LS and ICT industry sectors the k max shells include market leaders such as Pfizer, GSK, Novartis, J&J, Sanofi-Aventis, Bayer in the LS network and Microsoft, IBM, AT&T, Yahoo, Cisco, AOL, Time Warner and Google in the ICT network. We find that the LS network tendrils are mostly composed of the new start up firms and their university partners. On the one hand, start up firms preferentially attach to market leaders. On the other hand, market leaders compete to sign exclusive deals with new and promising start up firms. We also notice the remarkable stability of the LS nucleus: once a particular firm enters the nucleus it is very likely to remain there for many years.(See Fig. 3c) On the contrary, in the ICT sector there is more emphasis on how to integrate different technologies and markets [39]\n\n[\nFigs. 5(b) and 5(d)]. The appearance of c > 0 in the SF degree distribution significantly increases k max [see Figs. 5(e) and 5(f)]. However, the asymptotic dependence of k-shell sizes seems to remain the same as k approaches k max : S k ∼ k −δ [Figs. 5(e) and 5(f)].\n\nFIG. 2 :FIGFIG. 4 :FIG. 5 :\n245observed shell sizes [see Fig. 7(c)]. The observed differences can be explained by taking into account the offset c in SF degree distribution of both networks. The adjustment of the degree distribution of random SF models with c = 4.0 and c = 6.0 allows one to obtain similar patterns S k which are in fair agreement with the industry networks, as seen in Figs. 7(c) and 7(d).As seen above, the offset c > 0 in the SF degree distribution plays a crucial role in the formation of the k-shell structure of industry networks. A possible reason for the emergence of c > 0 in growing networks is the combination of preferential attachment with random attachment in network evolution[42]. The coexistence of preferential attachment regime with random collaborative agreements was suggested to take place in industry networks[37]. The random component is caused by the fact that sometimes firms choose exclusive relationships and novelty, and do not prefer to make deals with hub firms. Even though the coexistence of both preferential and random regimes seems to be crucial in the formation of the industry networks, it does not fully reproduce the k-shell structure of LS and ICT industries. We believe, that a better understanding of the evolution of the LS and ICT networks may be achieved by further improvements of the modeling.V. DISCUSSION AND SUMMARYWe use the k-shell decomposition to analyze the structure of the LS and ICT sectors of industry. We find that the firms in the LS industry can be naturally divided into three components: the nucleus, the tendrils and the bulk body. The nucleus of the LS industry consists mostly of the market leader firms while the tendrils are typically comprised of small start-up firms that preferentially sell their products to the nucleus firms. We show that the nucleus of the LS industry exhibits remarkable stability in time. In contrast, the ICT industry does not have a nucleus which can be explained by a high level of competition in the ICT sector. We also analyzed the dependence of the k-shell structure of SF model networks on N, λ and c. We observed the formation of the nucleus and the tendrils in SF networks only for λ < 3. The number of shells k max and the size of the nucleus S n are larger for SF networks with c > 0 compared to those with c = 0. Our results can partly explain the k-shell structure of LS and ICT industry networks. The coexistence of preferential and random attachment leads to the appearance of the offset c > 0 in the SF degree distributionP (q) ∼ (q + c) −λ [42]. Thus, the appearance of c > 0 in the degree distribution of LS and ICT networks might be explained by the interplay of random and preferential agreements among firms in the industries [37]. FIG. 1: Illustration of the k-shell decomposition method. (a) Original network. Nodes are marked by corresponding k-shell indices. Note that the k-shell index does not coincide with the node degree. (b) The 3-crust (left) and the 3-core (right) of the original network. (a) The growth of the largest connected component of LS and ICT industries. The LS industry expands almost linearly while the ICT industry exhibits exponential growth (see inset, a semi-log plot of the ICT network size as a function of time). (b) Cumulative degree distribution N(q > q 0 ) of LS and ICT networks. The inset displays cumulative degree distribution N(q > q 0 ) of LS and ICT network without offset c. . 3: (a) The k-shell structure of the LS network and (b) of the ICT network. Plots of the size of the k-crust, N 0 ; the size of the largest connected component of the k-crust, N 1 ; and the size of the second largest connected component of the k-crust, N 2 , as a function of the k-crust index k. The sizes are normalized with the total number of nodes in the network, N. N 2 is multiplied by 100. The transition of the k-crust at shell k = 18 of the LS network reveals a jelly-fish topology [32]. Since the change in F 1 ≡ N 1 /N at the k = k max is smaller in the ICT network than in the previous shell k = k max − 1, the ICT does not have a jelly-fish structure. The insets of (a,b) show the rate R of the largest connected component change as a function of shell index k. (c) The leadership persistence LP (t, t R ) as a function of years in the LS industry. We define LP (t, t R ) as a number of firms that were in the nucleus both at time t R and t. Note that most of the LS firms remain in the nucleus after they first enter it. (a,b) The k-shell structure of random SF models with (a) N = 8000 and (b) N = 10 6 nodes are shown for comparison. Note that λ = 3.4 curve overlaps with that of λ = 3.0. (c,d) The rate of the largest connected component change, R, as a function of shell index k for (c) N = 8000 and (d) N = 10 6 . Note that in order to avoid the overlap of curves in (c,d) we subsequently shift the plots with respect to each other by the additive factor 0.05 in (c) and the multiplicative factor of 10 in (d). The sizes of the nucleus and the tendrils decrease as λ increases. The nucleus and the tendrils disappear in (a,c) for N = 8000 at λ > 2.5 and in (b,d) for N = 10 6 at λ > 2.Sizes of k-shells, S k , for SF random models with c = 0 (a,b) λ = 2.1 and (c,d) λ = 2.5 as a function of (a,c) N and the k-shell index (b,d) k. Note that sizes of k-shells increases proportionally to N. (e,f) Sizes of k-shells, S k , for SF networks with c = 5.0 and (e) λ = 2.1, (f) λ = 2.5. SF models with c = 5.0 have significantly larger number of shells, k max , compared to SF models with the same λ and c = 0. S k in SF models with c = 5.0 decreases significantly slower as a function of k compared to SF models with the same λ and c = 0 in the small k region. In the large k region both types of SF models with c = 0.0 and c = 5.0 seem to have similar behavior S k ∼ k −δ , where δ = 2/(3 − λ).\n\nFIG. 6 :\n6(a) The total number of shells, k max , in SF models as a function of N. Note that k max ∝ N θ . (b) The size of the last shell S max in the SF model as a function of N. Each curve crosses over into a power law regime for N ≥ N c (λ), where N c (λ) increases with λ.(c) The calculated exponent θ as a function of λ (symbols). Our calculated values of δ agree with the mean field theory result δ ≈ 2/(3 − λ) (solid line).\n\nFIG. 7 :\n7(a) The largest shell index k max of the LS and the ICT networks as a function of N calculated for different years. The number of shells k max of the LS network increases approximately as a power-law function of the network size k max ∼ N θ , where θ ≃ 0.6. (b)\nAcknowledgmentsWe thank ONR, European project DAPHNET,the Israel Complexity Center, Merck Foundation (EPRIS project) and Israel Science Foundation for financial support. We thank L.Braunstein, S. Carmi and L. K. Gallos for valuable discussions.\n. P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6290P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6, 290 (1959). I\n\n. P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 517P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960). I\n\nB Bollobas, Random Graphs. Cambridge University PressB. Bollobas, Random Graphs (Cambridge University Press, 2001). I\n\nS Wasserman, K Faust, Social Network Analysis. Cambridge University PressS. Wasserman and K. Faust, Social Network Analysis, (Cambridge University Press, 1999). I\n\nS N Dorogovtsev, J F F Mendes, Evolution of Networks. Oxford University PressS. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks (Oxford University Press, 2003). 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Havlin, Fractals and Disordered Systems, 2nd Edition (Springer, Berlin, 1996). I\n\n. R Cohen, K Erez, D B Avraham, S Havlin, Phys. Rev. Lett. 854626R. Cohen, K. Erez, D. B. Avraham, and S. Havlin, Phys. Rev. Lett. 85, 4626 (2000). I\n\n. D S Callaway, M E J Newman, S H Strogatz, D J Watts, Phys. Rev. Lett. 855468D. S. Callaway, M. E. J. Newman, S. H. Strogatz and D. J. Watts, Phys. Rev. Lett. 85, 5468 (2000). I, II\n\n. G Paul, R Cohen, S Sreenivasan, S Havlin, H E Stanley, Phys. Rev. Lett. 99115701G. Paul, R. Cohen, S. Sreenivasan, S. Havlin and H. E. Stanley, Phys. Rev. Lett. 99, 115701 (2007);\n\n. Y Chen, E Lopez, S Havlin, H E Stanley, Phys. Rev. Lett. 9668702Y. Chen, E. Lopez, S. Havlin, and H. E. Stanley, Phys. Rev. Lett. 96, 068702 (2006).\n\n. C Song, S Havlin, H Makse, Nature. 433392C. Song, S. Havlin, and H. Makse, Nature 433, 392 (2005). I\n\n. C Song, S Havlin, H Makse, Nature Physics. 2275C. Song, S. Havlin, and H. Makse, Nature Physics 2, 275 (2006). I\n\n. K I Goh, G Salvi, B Kahng, D Kim, Phys. Rev. Lett. 9618701K. I. Goh, G. Salvi, B. Kahng, and D. Kim, Phys. Rev. Lett. 96, 018701 (2006). I\n\n. M Kitsak, S Havlin, G Paul, M Riccaboni, F Pammolli, H E Stanley, Phys. Rev. E. 7556115M. Kitsak, S. Havlin, G. Paul, M. Riccaboni, F. Pammolli, and H. E. Stanley, Phys. Rev. E. 75, 056115 (2007). I\n\nB Bollobas, Graph Theory and Combinatorics: Proceedings of the Cambridge Combinatorial Conference in honor of P. Erdös. New YorkAcademic35B. Bollobas, Graph Theory and Combinatorics: Proceedings of the Cambridge Combinatorial Conference in honor of P. Erdös, 35 (Academic, New York, 1984) I\n\n. S B Seidman, Social Networks. 5269S. B. Seidman, Social Networks, 5, 269 (1983). I\n\n. S N Dorogovtsev, A V Goltsev, J F F Mendes, Phys. Rev. Lett. 9640601S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Phys. Rev. Lett. 96, 040601 (2006).\n\n. I , I I Iii, I, II, III\n\n. A V Goltsev, S N Dorogovtsev, J F F Mendes, Phys. Rev. E. 7356101A. V. Goltsev, S. N. Dorogovtsev, and J. F. F. Mendes, Phys. Rev. E 73, 056101 (2006). I\n\n. J I Alvarez-Hamelin, L Dall'asta, A Barrat, Vespignani, Networks and Heterogeneous Media. 3371J. I. Alvarez-Hamelin, L. Dall'Asta, A. Barrat and A Vespignani, Networks and Heterogeneous Media 3, 371 (2008). I\n\nS Carmi, S Havlin, S Kirkpatrick, Y Shavitt, E Shir, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USAI, II1043S. Carmi, S. Havlin, S. Kirkpatrick, Y. Shavitt, and E. Shir, Proc. Natl. Acad. Sci. USA 104, 11150 (2007). I, II, 3\n\nL Tauro, C Palmer, G Siganos, M Faloutsos, Global Internet. L. Tauro, C. Palmer, G. Siganos, and M. Faloutsos, Global Internet (Nov. 2001). I\n\n. Scb Www, Se I, WWW.SCB.SE I\n\n. Y Chen, G Paul, R Cohen, S Havlin, S P Borgatti, F Liljeros, H E Stanley, Phys. Rev. E. 7546107Y. Chen, G. Paul, R. Cohen, S. Havlin, S. P. Borgatti, F. Liljeros and H. E. Stanley, Phys. Rev. E 75, 046107 (2007). I\n\n. J Sutton, American Economic Review. 97222J. Sutton, American Economic Review, 97, 222 (2007). I\n"},"corpus_id":{"kind":"number","value":55194194,"string":"55,194,194"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0810.5514\",\n \"mag\": 1844516428,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0810.5514v1.pdf\"\n ],\n \"pdf_sha\": \"81887a54344cbb8e6b484c038459909e9538a8cf\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Structure of Business Firm Networks and Scale-Free Models"},"authors":{"kind":"list like","value":["Maksim Kitsak \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA","Massimo Riccaboni \nCUSAS\nUniversity of Florence\nFlorenceItaly","Shlomo Havlin \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA\n\nDepartment of Physics\nMinerva Center\nBar-Ilan University\nRamat GanIsrael","Fabio Pammolli \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA\n\nCUSAS\nUniversity of Florence\nFlorenceItaly\n\nIMT Institute for Advanced Studies\nLuccaItaly","H Eugene Stanley \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA"],"string":"[\n \"Maksim Kitsak \\nCenter for Polymer Studies\\nBoston University\\n02215BostonMassachusettsUSA\",\n \"Massimo Riccaboni \\nCUSAS\\nUniversity of Florence\\nFlorenceItaly\",\n \"Shlomo Havlin \\nCenter for Polymer Studies\\nBoston University\\n02215BostonMassachusettsUSA\\n\\nDepartment of Physics\\nMinerva Center\\nBar-Ilan University\\nRamat GanIsrael\",\n \"Fabio Pammolli \\nCenter for Polymer Studies\\nBoston University\\n02215BostonMassachusettsUSA\\n\\nCUSAS\\nUniversity of Florence\\nFlorenceItaly\\n\\nIMT Institute for Advanced Studies\\nLuccaItaly\",\n \"H Eugene Stanley \\nCenter for Polymer Studies\\nBoston University\\n02215BostonMassachusettsUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study the structure of business firm networks and scale-free models with degree distribution P (q) ∝ (q+c) −λ using the method of k-shell decomposition. We find that the Life Sciences industry network consists of three components: a \"nucleus,\" which is a small well connected subgraph, \"tendrils,\" which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus, and a \"bulk body\" which consists of the majority of nodes. At the same time we do not observe the above structure in the Information and Communication Technology sector of industry.We also conduct a systematic study of these three components in random scale-free networks. Our results suggest that the sizes of the nucleus and the tendrils decrease as λ increases and disappear for λ ≥ 3. We compare the k-shell structure of random scale-free model networks with two real world business firm networks in the Life Sciences and in the Information and Communication Technology sectors. Our results suggest that the observed behavior of the k-shell structure in the two industries is consistent with a recently proposed growth model that assumes the coexistence of both preferential and random agreements in the evolution of industrial networks."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Many real-world complex systems are often described using the representation of graphs or networks, as sets of nodes connected by links [1,2,3,4,5,6,7]. Networks appear in various areas of science such as physics, biology, computer sciences, economics and sociology [3,4,5,6,7]. Real networks, despite their diversity, appear to have many common properties. Some real networks have been found to be \"small-world\" [3,8,9,10]: despite their large size, the typical distance between nodes is very small, of order of log(N) or less [3,11,12], where N is the number of nodes. Also, some real networks are scale-free (SF) with a power-law tail in their degree distribution\nwhere q is the number of links per node, λ and c are distribution parameters [13,14,15,16].\nMany statistical physics and mathematical techniques have been successfully applied to study network structure including percolation [12,17,18,19,20,21], scaling [6,7,11], partitioning [22], box covering [23,24,25,26], k-core percolation [27,28] and k-shell decomposition [31] . The latter has been recently used to study the topology of the Internet at the autonomous system (AS) level [32]. It has been found that the AS Internet network consists of three distinct components:\n(i) The nucleus consists of highly connected nodes (hubs) and constitutes a tiny fraction of the network (≈ 0.5%).\n(ii) The tendrils (≈ 25%) are the subgraphs that connect to the network exclusively via the nucleus.\n(iii) The remaining nodes (≈ 75%) of the network constitute the bulk body. Unlike the tendrils, the nodes in the bulk remain connected even if the nucleus is removed.\nThe appearance of the three components led to the association of the AS Internet structure with that of a jelly-fish model [32]. For an alternative jelly-fish structure see also Ref. [33].\nThe nucleus plays a crucial role in data transfer throughout the Internet. If the nucleus is damaged or blocked, nodes in the tendrils cease to communicate with other nodes both in the bulk body and in other tendrils. However, the nodes in the bulk body can still communicate with each other, but less efficiently since the average path length between 2 them almost doubles [32]. The network of workplaces in Sweden [34,35] was also shown to possess a jelly-fish structure. Mean field analysis of the k-shell structure [29,30] focuses on random networks with a given degree distribution P (q).\nIn the present work we address the long-standing question of how long does a typical leader in an industry maintain its position. This question has attracted continuing attention in the Industrial Organization literature over the past generation [36]. Two rival views exist. The first asserts that leadership tends to persist for a 'long' time while the second one emphasizes the transience of leadership positions due to radical innovation and \"leapfrogging competition\". The Life Sciences (LS) sector is generally cited as an archetypal example of stability of industry leaders while on the contrary the Information and Communication Technology (ICT) sector is widely considered as a good example of high market turnover and instability. We study the k-shell structure of the LS and ICT industry in order to test these views and better understand the reasons behind the higher stability of the leading firms in the LS sciences and the ICT sector. The networks of LS and ICT firms are networks of nodes representing firms in the worldwide LS and ICT industries with links representing the collaborative agreements among them [26,37,38,39]. We also conduct a numerical analysis of model SF networks in order to better understand their k-shell structure. We study how the size and the connectivity of the three components of random SF model networks depend on the SF parameters. We compare the k-shell structure of SF models with that of the LS and ICT industry networks in order to get a better insight into the growth principles of the Industry networks.\nThe rest of the manuscript is organized as follows: in Section II we define the k-shell decomposition and apply it to both real and model SF networks. We then analyze the leadership in the industry networks from the k-shell perspective. In Section III we conduct a systematic analysis of the properties of the three components of model SF networks. In section IV we compare the results for SF models with those observed in the two industrial networks. We conclude our manuscript in Section V with a discussion and summary.\nII. K-SHELL, K-CORE, AND K-CRUST\nWe start the process of the k-shell decomposition [29,32] on a network by removing all nodes with degree q = 1. After the first iteration of pruning, there may appear new nodes 3 with degrees q = 1. We keep on pruning these nodes until only nodes with degree q ≥ 2 are left. The removed nodes along with the links connecting them form the k = 1 shell. Next, we iterate the pruning process for nodes of degree q = 2, thereby creating the k = 2 shell.\nWe further continue the k-shell decomposition for higher values of q until all nodes of the network are removed. As a result each node in the network is assigned a k-shell index k.\nThe largest shell index is called k max , which is also the total number of shells in the network, provided all shells below k max exist. The k-crust is defined as the union of all k-shells with indices smaller than k. Similarly, the k-core is defined as the union of all nodes with indices greater or equal k. (See Fig. 1 for demonstration.) As we explain later, the set of nodes in the k max shell is called nucleus provided there is a large number of nodes (tendrils) connected to the network exclusively via the k max shell.\nWe is consistent with the existence of a second order phase transition in the k-crust structure [32]. The type of the phase transition taking place at the k-shell decomposition is similar to that of targeted percolation in SF networks [21,40].\nUnlike in the ICT network, the size of the largest connected component N 1 in LS network undergoes a large jump N 1 (k = 19) − N 1 (k = 18) = 745, while the total size N 0 of the crust experiences only a small change N 0 (k = 19) − N 0 (k = 18) = 43 (See Fig. 3(a)). This can be explained as follows [32]. Approximately 10% of the he LS network are firms (which we call \"tendrils\") that prefer to sign collaborative agreements exclusively with the 43 firms that form k = k max = 19 shell (which we call nucleus). Typically, each of these firms (tendrils) signs a small number of agreements with firms in the nucleus and, therefore, has small degree.\nThus, the tendrils are removed in the decomposition of the first few shells. However, being connected exclusively to the nucleus in the k max = 19 shell, the tendrils do not contribute to the largest connected component of the k-crust until the last k max shell is decomposed.\nIt is the inclusion of the tendrils firms into N 1 (k) at the decomposition of the k max shell that results in the observed jump in N 1 (k). The appearance of the three components-the nucleus, the tendrils and the bulk body-allows one to associate the structure of the network of LS firms with that of the jelly-fish similar to the Internet at the Autonomous System level [32]. Interestingly, we do not observe the a similar jump in N 1 (k) in the ICT network (See . Thus, there is more deals between firms of the same size and age but in different technological and market areas. As a consequence, the k-shell of the ICT network is more unstable and heterogeneous which may explain why we do not detect the emergence of a nucleus such as in the LS sector.\nAs discussed above, one can in general calculate the size of the nucleus S n as the total number of nodes in the k max shell\nThe increase in N 1 (k) at k = k max is comprised by the inclusion of tendrils S t and the nucleus S n . Thus, the size of tendrils can be calculated as\nAs seen in Fig. 2(b), both the LS and the ICT industrial networks exhibit a SF degree distribution. Hence, in order to better understand the substructures of real networksthe nucleus, the tendrils, and the bulk components-we analyze the k-shell structure of the random SF models which were generated using the configurational approach [41]. We calculate the k-shell structure of random SF networks with c ≥ 0, and degree distribution exponent λ ∈ [2,3]. For our simulations (Fig. 4) we choose networks sizes of N = 8, 000\n(which is comparable to the size of the LS and ICT industry networks) and N = 10 6 in order to test the influence of finite size effects.\nIt is seen in Fig. 4 that both the number of shells k max and the jump in the largest connected component N 1 (k) decrease as the exponent λ increases. As the jump becomes less pronounced it becomes harder to detect it which motivates us to introduce the quantitative criterion for the emergence of the three distinct components. We define the rate of change of the largest connected component size as\nWe compare the increase of the largest connected component R(k) at k = k max with that\nWe use Eq. (5) as a criterion for the existence of a nucleus and tendrils.\nBy examining R(k) plot we observe in the case of N = 8, 000 (Fig. 4c), that SF models with λ > 2.5 already do not have a nucleus and tendrils. However, for N = 10 6 we do observe nucleus and tendrils in SF models for λ ≤ 2.7. (Fig. 4d). These observations suggest that all SF networks with λ ∈ (2, 3) have a nucleus and tendrils provided N is sufficiently large. The above observations for SF models agree with the fact that we observe jelly-fish topology for LS network (λ = 2.5) and do not observe it for ICT network λ = 3.4. However, SF model networks with N = 8000 with λ = 2.5 and λ = 3.4 have k max = 5 and k max = 2 respectively, while the measured k max in LS and ICT networks (which have the same λ values) are k max = 19 and k max = 6. The observed difference in k max between industry networks and SF models with similar parameters suggests to further explore the k-shell structure of SF networks and consider SF model networks with c > 0 values in P (q) (as found in the LS and the ICT networks)."},{"title":"III. K-SHELL PROPERTIES OF SCALE-FREE NETWORKS","content":"As shown above, SF networks may or may not have a jelly-fish structure, depending on the degree distribution P (q) and size N. In order to better understand the k-shell structure of SF networks, we calculate the sizes of the k-shells as a function of N and λ. For each pair of values N and λ we generate 10 3 − 10 4 realizations of SF models and calculate the average number of shells k max constituting the network as well as their average size S k (see Fig. 5). For small k values, S k is proportional to N. As the size of the network increases new shells start to appear. When the size increases further, the new shell growth stabilizes and becomes also proportional to N (See Fig. 5a,c). This result indicates that the size of each shell constitutes a certain finite fraction of the network and this fraction decreases with increasing k. The analytical analysis of k-core structure [29] leads to\nwhere δ = 2 3 − λ .\nHence δ ≈ 2.2 for λ = 2.1 and δ ≈ 4.0 for λ = 2.5, which agrees with our simulations. One can estimate the total number of shells in a random SF network of size N as follows.\nSince every shell constitutes a fixed fraction of N it follows that S k ∝ Nk −δ . The last shell k max needs to possess at least one node S max ≡ S k=kmax ∼ 1, which leads to\nIndeed, the total number of shells k max seems to increase as a power law with the network size N [ Fig. 6(a)]. The smaller is λ the faster is the growth of k max . As seen from Fig. 6(c), the estimated exponents 1/δ agree with those predicted by Eq. (8). Note that Eq. (8) together with\nis consistent with the fact that networks with λ > 3 do not have a k-shell structure.\nThe dependence of S max on N can be regarded as a crossover from the small N regime with N < N c (λ) where there is no nucleus to the power-law regime for N > N c (λ) where S max ∼ N τ (λ) ( Fig. 6b). We relate the observed crossover with the emergence of the nucleus and tendrils in SF networks for N > N c (λ). The critical size of SF networks, N c (λ), corresponding to the emergence of the nucleus and tendrils in SF networks seems to increase as λ increases. In the N < N c regime the size of the last shell S max increases with λ, which can be explained by the fact that SF networks with higher λ have fewer shells. On the other hand, in the power-law regime, the size of the last shell, S max , (which now becomes the nucleus S n ) is smaller for larger values of λ."},{"title":"IV. EVOLUTION AND STRUCTURE OF THE LS AND ICT INDUSTRY NET-","content":""},{"title":"WORKS","content":"We further analyze the k-shell structure of the LS and ICT industry networks. The number of shells k max in the LS industry grows as a power-law function of its size,\nand reaches k max = 19 in 2002. Our estimates yield θ ≈ 0.6 (Fig. 7a). We find that the number of shells in ICT sector, k max , also grows and reaches k max = 6 in 2000. Note that S n for the LS network exhibits fluctuations for N ∈ [2000, 5000] and stabilizes for N > 5000 (See Fig. 7b). As seen in Fig. 7(c,d), the sizes of shells S k decrease as a function of their index k in both networks. As we notice in Section II, the observed shell sizes S k as well as the number of shells k max , measured for the LS and ICT networks, deviate from those obtained from SF models with c = 0. We expect for random SF models with c = 0, λ = 2.5 and λ = 3.4 and similar sizes as LS and ICT to find k max = 5 and k max = 2 respectively. Also, a SF network with λ = 2.5 is expected to have S k ∼ k −4. and k max ∼ N 0. 25 Size of the nucleus of the LS network, S n , as a function of N. S n exhibits fluctuations as LS network grows. Unlike in the analyzed SF models, S max becomes stable for N > 5000.\n(c) S k as a function of the k-shell index k for the LS network (squares). Shell sizes S k as a function of k decrease significantly slower than S k ∼ k −4 , which is expected for a random SF model with the same λ and c = 0 (triangles). However, the offset introduction of c = 4.0 in the SF degree distribution, P (q), mimics the k-shell structure of the LS network (circles). (d) S k as a function of k-shell index k for the ICT network (squares). SF model with λ = 3.4 and c = 0 does not possess a k-shell structure. However, the introduction of c = 6.0 in the SF degree distribution yields similar k-shell structure (circles), but we attribute it only to finite size effect."},{"title":"Fig. 3 (","content":""},{"title":"[","content":""},{"title":"FIG. 2 :FIGFIG. 4 :FIG. 5 :","content":""},{"title":"FIG. 6 :","content":""},{"title":"FIG. 7 :","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Many real-world complex systems are often described using the representation of graphs or networks, as sets of nodes connected by links [1,2,3,4,5,6,7]. Networks appear in various areas of science such as physics, biology, computer sciences, economics and sociology [3,4,5,6,7]. Real networks, despite their diversity, appear to have many common properties. Some real networks have been found to be \\\"small-world\\\" [3,8,9,10]: despite their large size, the typical distance between nodes is very small, of order of log(N) or less [3,11,12], where N is the number of nodes. Also, some real networks are scale-free (SF) with a power-law tail in their degree distribution\\nwhere q is the number of links per node, λ and c are distribution parameters [13,14,15,16].\\nMany statistical physics and mathematical techniques have been successfully applied to study network structure including percolation [12,17,18,19,20,21], scaling [6,7,11], partitioning [22], box covering [23,24,25,26], k-core percolation [27,28] and k-shell decomposition [31] . The latter has been recently used to study the topology of the Internet at the autonomous system (AS) level [32]. It has been found that the AS Internet network consists of three distinct components:\\n(i) The nucleus consists of highly connected nodes (hubs) and constitutes a tiny fraction of the network (≈ 0.5%).\\n(ii) The tendrils (≈ 25%) are the subgraphs that connect to the network exclusively via the nucleus.\\n(iii) The remaining nodes (≈ 75%) of the network constitute the bulk body. Unlike the tendrils, the nodes in the bulk remain connected even if the nucleus is removed.\\nThe appearance of the three components led to the association of the AS Internet structure with that of a jelly-fish model [32]. For an alternative jelly-fish structure see also Ref. [33].\\nThe nucleus plays a crucial role in data transfer throughout the Internet. If the nucleus is damaged or blocked, nodes in the tendrils cease to communicate with other nodes both in the bulk body and in other tendrils. However, the nodes in the bulk body can still communicate with each other, but less efficiently since the average path length between 2 them almost doubles [32]. The network of workplaces in Sweden [34,35] was also shown to possess a jelly-fish structure. Mean field analysis of the k-shell structure [29,30] focuses on random networks with a given degree distribution P (q).\\nIn the present work we address the long-standing question of how long does a typical leader in an industry maintain its position. This question has attracted continuing attention in the Industrial Organization literature over the past generation [36]. Two rival views exist. The first asserts that leadership tends to persist for a 'long' time while the second one emphasizes the transience of leadership positions due to radical innovation and \\\"leapfrogging competition\\\". The Life Sciences (LS) sector is generally cited as an archetypal example of stability of industry leaders while on the contrary the Information and Communication Technology (ICT) sector is widely considered as a good example of high market turnover and instability. We study the k-shell structure of the LS and ICT industry in order to test these views and better understand the reasons behind the higher stability of the leading firms in the LS sciences and the ICT sector. The networks of LS and ICT firms are networks of nodes representing firms in the worldwide LS and ICT industries with links representing the collaborative agreements among them [26,37,38,39]. We also conduct a numerical analysis of model SF networks in order to better understand their k-shell structure. We study how the size and the connectivity of the three components of random SF model networks depend on the SF parameters. We compare the k-shell structure of SF models with that of the LS and ICT industry networks in order to get a better insight into the growth principles of the Industry networks.\\nThe rest of the manuscript is organized as follows: in Section II we define the k-shell decomposition and apply it to both real and model SF networks. We then analyze the leadership in the industry networks from the k-shell perspective. In Section III we conduct a systematic analysis of the properties of the three components of model SF networks. In section IV we compare the results for SF models with those observed in the two industrial networks. We conclude our manuscript in Section V with a discussion and summary.\\nII. K-SHELL, K-CORE, AND K-CRUST\\nWe start the process of the k-shell decomposition [29,32] on a network by removing all nodes with degree q = 1. After the first iteration of pruning, there may appear new nodes 3 with degrees q = 1. We keep on pruning these nodes until only nodes with degree q ≥ 2 are left. The removed nodes along with the links connecting them form the k = 1 shell. Next, we iterate the pruning process for nodes of degree q = 2, thereby creating the k = 2 shell.\\nWe further continue the k-shell decomposition for higher values of q until all nodes of the network are removed. As a result each node in the network is assigned a k-shell index k.\\nThe largest shell index is called k max , which is also the total number of shells in the network, provided all shells below k max exist. The k-crust is defined as the union of all k-shells with indices smaller than k. Similarly, the k-core is defined as the union of all nodes with indices greater or equal k. (See Fig. 1 for demonstration.) As we explain later, the set of nodes in the k max shell is called nucleus provided there is a large number of nodes (tendrils) connected to the network exclusively via the k max shell.\\nWe is consistent with the existence of a second order phase transition in the k-crust structure [32]. The type of the phase transition taking place at the k-shell decomposition is similar to that of targeted percolation in SF networks [21,40].\\nUnlike in the ICT network, the size of the largest connected component N 1 in LS network undergoes a large jump N 1 (k = 19) − N 1 (k = 18) = 745, while the total size N 0 of the crust experiences only a small change N 0 (k = 19) − N 0 (k = 18) = 43 (See Fig. 3(a)). This can be explained as follows [32]. Approximately 10% of the he LS network are firms (which we call \\\"tendrils\\\") that prefer to sign collaborative agreements exclusively with the 43 firms that form k = k max = 19 shell (which we call nucleus). Typically, each of these firms (tendrils) signs a small number of agreements with firms in the nucleus and, therefore, has small degree.\\nThus, the tendrils are removed in the decomposition of the first few shells. However, being connected exclusively to the nucleus in the k max = 19 shell, the tendrils do not contribute to the largest connected component of the k-crust until the last k max shell is decomposed.\\nIt is the inclusion of the tendrils firms into N 1 (k) at the decomposition of the k max shell that results in the observed jump in N 1 (k). The appearance of the three components-the nucleus, the tendrils and the bulk body-allows one to associate the structure of the network of LS firms with that of the jelly-fish similar to the Internet at the Autonomous System level [32]. Interestingly, we do not observe the a similar jump in N 1 (k) in the ICT network (See . Thus, there is more deals between firms of the same size and age but in different technological and market areas. As a consequence, the k-shell of the ICT network is more unstable and heterogeneous which may explain why we do not detect the emergence of a nucleus such as in the LS sector.\\nAs discussed above, one can in general calculate the size of the nucleus S n as the total number of nodes in the k max shell\\nThe increase in N 1 (k) at k = k max is comprised by the inclusion of tendrils S t and the nucleus S n . Thus, the size of tendrils can be calculated as\\nAs seen in Fig. 2(b), both the LS and the ICT industrial networks exhibit a SF degree distribution. Hence, in order to better understand the substructures of real networksthe nucleus, the tendrils, and the bulk components-we analyze the k-shell structure of the random SF models which were generated using the configurational approach [41]. We calculate the k-shell structure of random SF networks with c ≥ 0, and degree distribution exponent λ ∈ [2,3]. For our simulations (Fig. 4) we choose networks sizes of N = 8, 000\\n(which is comparable to the size of the LS and ICT industry networks) and N = 10 6 in order to test the influence of finite size effects.\\nIt is seen in Fig. 4 that both the number of shells k max and the jump in the largest connected component N 1 (k) decrease as the exponent λ increases. As the jump becomes less pronounced it becomes harder to detect it which motivates us to introduce the quantitative criterion for the emergence of the three distinct components. We define the rate of change of the largest connected component size as\\nWe compare the increase of the largest connected component R(k) at k = k max with that\\nWe use Eq. (5) as a criterion for the existence of a nucleus and tendrils.\\nBy examining R(k) plot we observe in the case of N = 8, 000 (Fig. 4c), that SF models with λ > 2.5 already do not have a nucleus and tendrils. However, for N = 10 6 we do observe nucleus and tendrils in SF models for λ ≤ 2.7. (Fig. 4d). These observations suggest that all SF networks with λ ∈ (2, 3) have a nucleus and tendrils provided N is sufficiently large. The above observations for SF models agree with the fact that we observe jelly-fish topology for LS network (λ = 2.5) and do not observe it for ICT network λ = 3.4. However, SF model networks with N = 8000 with λ = 2.5 and λ = 3.4 have k max = 5 and k max = 2 respectively, while the measured k max in LS and ICT networks (which have the same λ values) are k max = 19 and k max = 6. The observed difference in k max between industry networks and SF models with similar parameters suggests to further explore the k-shell structure of SF networks and consider SF model networks with c > 0 values in P (q) (as found in the LS and the ICT networks).\"\n },\n {\n \"title\": \"III. K-SHELL PROPERTIES OF SCALE-FREE NETWORKS\",\n \"content\": \"As shown above, SF networks may or may not have a jelly-fish structure, depending on the degree distribution P (q) and size N. In order to better understand the k-shell structure of SF networks, we calculate the sizes of the k-shells as a function of N and λ. For each pair of values N and λ we generate 10 3 − 10 4 realizations of SF models and calculate the average number of shells k max constituting the network as well as their average size S k (see Fig. 5). For small k values, S k is proportional to N. As the size of the network increases new shells start to appear. When the size increases further, the new shell growth stabilizes and becomes also proportional to N (See Fig. 5a,c). This result indicates that the size of each shell constitutes a certain finite fraction of the network and this fraction decreases with increasing k. The analytical analysis of k-core structure [29] leads to\\nwhere δ = 2 3 − λ .\\nHence δ ≈ 2.2 for λ = 2.1 and δ ≈ 4.0 for λ = 2.5, which agrees with our simulations. One can estimate the total number of shells in a random SF network of size N as follows.\\nSince every shell constitutes a fixed fraction of N it follows that S k ∝ Nk −δ . The last shell k max needs to possess at least one node S max ≡ S k=kmax ∼ 1, which leads to\\nIndeed, the total number of shells k max seems to increase as a power law with the network size N [ Fig. 6(a)]. The smaller is λ the faster is the growth of k max . As seen from Fig. 6(c), the estimated exponents 1/δ agree with those predicted by Eq. (8). Note that Eq. (8) together with\\nis consistent with the fact that networks with λ > 3 do not have a k-shell structure.\\nThe dependence of S max on N can be regarded as a crossover from the small N regime with N < N c (λ) where there is no nucleus to the power-law regime for N > N c (λ) where S max ∼ N τ (λ) ( Fig. 6b). We relate the observed crossover with the emergence of the nucleus and tendrils in SF networks for N > N c (λ). The critical size of SF networks, N c (λ), corresponding to the emergence of the nucleus and tendrils in SF networks seems to increase as λ increases. In the N < N c regime the size of the last shell S max increases with λ, which can be explained by the fact that SF networks with higher λ have fewer shells. On the other hand, in the power-law regime, the size of the last shell, S max , (which now becomes the nucleus S n ) is smaller for larger values of λ.\"\n },\n {\n \"title\": \"IV. EVOLUTION AND STRUCTURE OF THE LS AND ICT INDUSTRY NET-\",\n \"content\": \"\"\n },\n {\n \"title\": \"WORKS\",\n \"content\": \"We further analyze the k-shell structure of the LS and ICT industry networks. The number of shells k max in the LS industry grows as a power-law function of its size,\\nand reaches k max = 19 in 2002. Our estimates yield θ ≈ 0.6 (Fig. 7a). We find that the number of shells in ICT sector, k max , also grows and reaches k max = 6 in 2000. Note that S n for the LS network exhibits fluctuations for N ∈ [2000, 5000] and stabilizes for N > 5000 (See Fig. 7b). As seen in Fig. 7(c,d), the sizes of shells S k decrease as a function of their index k in both networks. As we notice in Section II, the observed shell sizes S k as well as the number of shells k max , measured for the LS and ICT networks, deviate from those obtained from SF models with c = 0. We expect for random SF models with c = 0, λ = 2.5 and λ = 3.4 and similar sizes as LS and ICT to find k max = 5 and k max = 2 respectively. Also, a SF network with λ = 2.5 is expected to have S k ∼ k −4. and k max ∼ N 0. 25 Size of the nucleus of the LS network, S n , as a function of N. S n exhibits fluctuations as LS network grows. Unlike in the analyzed SF models, S max becomes stable for N > 5000.\\n(c) S k as a function of the k-shell index k for the LS network (squares). Shell sizes S k as a function of k decrease significantly slower than S k ∼ k −4 , which is expected for a random SF model with the same λ and c = 0 (triangles). However, the offset introduction of c = 4.0 in the SF degree distribution, P (q), mimics the k-shell structure of the LS network (circles). (d) S k as a function of k-shell index k for the ICT network (squares). SF model with λ = 3.4 and c = 0 does not possess a k-shell structure. However, the introduction of c = 6.0 in the SF degree distribution yields similar k-shell structure (circles), but we attribute it only to finite size effect.\"\n },\n {\n \"title\": \"Fig. 3 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"[\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 2 :FIGFIG. 4 :FIG. 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 7 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"analyze the k-shell structure of LS and ICT industrial sectors in time periods between 1990 and 2002 and between 1990 and 2000 respectively. The LS network expanded linearly since the mid-1970s while the ICT network took off in the 1990s and grew exponentially for a decade [see Fig. 2(a) and the inset]. The total number of firms in the LS is N = 6, 776 and in the ICT is N = 7, 759. These sizes refer to the largest connected component of each of the networks at the last year of observation. Both industrial networks feature SF degree distributions with λ ≈ 2.5, c ≈ 4. (LS) and λ ≈ 3.4, c ≈ 6. (ICT) [see Fig. 2(b)]. We use the Kholmogorov-Smirnov test in order to examine the goodness of fit of the degree distributions. The obtained p-values for 1000 trials are 0.24 and 0.33 for the LS and the ICT industry networks respectively.We next apply the k-shell decomposition procedure to the LS and the ICT networks. For each k-crust we calculate the total number of nodes, N 0 , comprising it, the size of the largest connected component, N 1 , in the k-crust and the size of the second largest component, N 2 , of the k-crust. As seen inFig. 3, the LS network consists of k max = 19 shells while the ICT network (which consists of comparable number of firms) has only k max = 6 shells. The size of the largest cluster in the k-crust starts growing rapidly after k = 4 and k = 2 in the LS and the ICT networks respectively. At these values of k, the size of the second largest cluster in the k-crust reaches a maximum. The above behavior of the k-crust components","content":"analyze the k-shell structure of LS and ICT industrial sectors in time periods between 1990 and 2002 and between 1990 and 2000 respectively. The LS network expanded linearly since the mid-1970s while the ICT network took off in the 1990s and grew exponentially for a decade [see Fig. 2(a) and the inset]. The total number of firms in the LS is N = 6, 776 and in the ICT is N = 7, 759. These sizes refer to the largest connected component of each of the networks at the last year of observation. Both industrial networks feature SF degree distributions with λ ≈ 2.5, c ≈ 4. (LS) and λ ≈ 3.4, c ≈ 6. (ICT) [see Fig. 2(b)]. We use the Kholmogorov-Smirnov test in order to examine the goodness of fit of the degree distributions. The obtained p-values for 1000 trials are 0.24 and 0.33 for the LS and the ICT industry networks respectively.We next apply the k-shell decomposition procedure to the LS and the ICT networks. For each k-crust we calculate the total number of nodes, N 0 , comprising it, the size of the largest connected component, N 1 , in the k-crust and the size of the second largest component, N 2 , of the k-crust. As seen inFig. 3, the LS network consists of k max = 19 shells while the ICT network (which consists of comparable number of firms) has only k max = 6 shells. The size of the largest cluster in the k-crust starts growing rapidly after k = 4 and k = 2 in the LS and the ICT networks respectively. At these values of k, the size of the second largest cluster in the k-crust reaches a maximum. The above behavior of the k-crust components"},{"id":"","caption":"b)). In both LS and ICT industry sectors the k max shells include market leaders such as Pfizer, GSK, Novartis, J&J, Sanofi-Aventis, Bayer in the LS network and Microsoft, IBM, AT&T, Yahoo, Cisco, AOL, Time Warner and Google in the ICT network. We find that the LS network tendrils are mostly composed of the new start up firms and their university partners. On the one hand, start up firms preferentially attach to market leaders. On the other hand, market leaders compete to sign exclusive deals with new and promising start up firms. We also notice the remarkable stability of the LS nucleus: once a particular firm enters the nucleus it is very likely to remain there for many years.(See Fig. 3c) On the contrary, in the ICT sector there is more emphasis on how to integrate different technologies and markets [39]","content":"Fig. 3 (\n3b)). In both LS and ICT industry sectors the k max shells include market leaders such as Pfizer, GSK, Novartis, J&J, Sanofi-Aventis, Bayer in the LS network and Microsoft, IBM, AT&T, Yahoo, Cisco, AOL, Time Warner and Google in the ICT network. We find that the LS network tendrils are mostly composed of the new start up firms and their university partners. On the one hand, start up firms preferentially attach to market leaders. On the other hand, market leaders compete to sign exclusive deals with new and promising start up firms. We also notice the remarkable stability of the LS nucleus: once a particular firm enters the nucleus it is very likely to remain there for many years.(See Fig. 3c) On the contrary, in the ICT sector there is more emphasis on how to integrate different technologies and markets [39]"},{"id":"","caption":"Figs. 5(b) and 5(d)]. The appearance of c > 0 in the SF degree distribution significantly increases k max [see Figs. 5(e) and 5(f)]. However, the asymptotic dependence of k-shell sizes seems to remain the same as k approaches k max : S k ∼ k −δ [Figs. 5(e) and 5(f)].","content":"[\nFigs. 5(b) and 5(d)]. The appearance of c > 0 in the SF degree distribution significantly increases k max [see Figs. 5(e) and 5(f)]. However, the asymptotic dependence of k-shell sizes seems to remain the same as k approaches k max : S k ∼ k −δ [Figs. 5(e) and 5(f)]."},{"id":"","caption":"observed shell sizes [see Fig. 7(c)]. The observed differences can be explained by taking into account the offset c in SF degree distribution of both networks. The adjustment of the degree distribution of random SF models with c = 4.0 and c = 6.0 allows one to obtain similar patterns S k which are in fair agreement with the industry networks, as seen in Figs. 7(c) and 7(d).As seen above, the offset c > 0 in the SF degree distribution plays a crucial role in the formation of the k-shell structure of industry networks. A possible reason for the emergence of c > 0 in growing networks is the combination of preferential attachment with random attachment in network evolution[42]. The coexistence of preferential attachment regime with random collaborative agreements was suggested to take place in industry networks[37]. The random component is caused by the fact that sometimes firms choose exclusive relationships and novelty, and do not prefer to make deals with hub firms. Even though the coexistence of both preferential and random regimes seems to be crucial in the formation of the industry networks, it does not fully reproduce the k-shell structure of LS and ICT industries. We believe, that a better understanding of the evolution of the LS and ICT networks may be achieved by further improvements of the modeling.V. DISCUSSION AND SUMMARYWe use the k-shell decomposition to analyze the structure of the LS and ICT sectors of industry. We find that the firms in the LS industry can be naturally divided into three components: the nucleus, the tendrils and the bulk body. The nucleus of the LS industry consists mostly of the market leader firms while the tendrils are typically comprised of small start-up firms that preferentially sell their products to the nucleus firms. We show that the nucleus of the LS industry exhibits remarkable stability in time. In contrast, the ICT industry does not have a nucleus which can be explained by a high level of competition in the ICT sector. We also analyzed the dependence of the k-shell structure of SF model networks on N, λ and c. We observed the formation of the nucleus and the tendrils in SF networks only for λ < 3. The number of shells k max and the size of the nucleus S n are larger for SF networks with c > 0 compared to those with c = 0. Our results can partly explain the k-shell structure of LS and ICT industry networks. The coexistence of preferential and random attachment leads to the appearance of the offset c > 0 in the SF degree distributionP (q) ∼ (q + c) −λ [42]. Thus, the appearance of c > 0 in the degree distribution of LS and ICT networks might be explained by the interplay of random and preferential agreements among firms in the industries [37]. FIG. 1: Illustration of the k-shell decomposition method. (a) Original network. Nodes are marked by corresponding k-shell indices. Note that the k-shell index does not coincide with the node degree. (b) The 3-crust (left) and the 3-core (right) of the original network. (a) The growth of the largest connected component of LS and ICT industries. The LS industry expands almost linearly while the ICT industry exhibits exponential growth (see inset, a semi-log plot of the ICT network size as a function of time). (b) Cumulative degree distribution N(q > q 0 ) of LS and ICT networks. The inset displays cumulative degree distribution N(q > q 0 ) of LS and ICT network without offset c. . 3: (a) The k-shell structure of the LS network and (b) of the ICT network. Plots of the size of the k-crust, N 0 ; the size of the largest connected component of the k-crust, N 1 ; and the size of the second largest connected component of the k-crust, N 2 , as a function of the k-crust index k. The sizes are normalized with the total number of nodes in the network, N. N 2 is multiplied by 100. The transition of the k-crust at shell k = 18 of the LS network reveals a jelly-fish topology [32]. Since the change in F 1 ≡ N 1 /N at the k = k max is smaller in the ICT network than in the previous shell k = k max − 1, the ICT does not have a jelly-fish structure. The insets of (a,b) show the rate R of the largest connected component change as a function of shell index k. (c) The leadership persistence LP (t, t R ) as a function of years in the LS industry. We define LP (t, t R ) as a number of firms that were in the nucleus both at time t R and t. Note that most of the LS firms remain in the nucleus after they first enter it. (a,b) The k-shell structure of random SF models with (a) N = 8000 and (b) N = 10 6 nodes are shown for comparison. Note that λ = 3.4 curve overlaps with that of λ = 3.0. (c,d) The rate of the largest connected component change, R, as a function of shell index k for (c) N = 8000 and (d) N = 10 6 . Note that in order to avoid the overlap of curves in (c,d) we subsequently shift the plots with respect to each other by the additive factor 0.05 in (c) and the multiplicative factor of 10 in (d). The sizes of the nucleus and the tendrils decrease as λ increases. The nucleus and the tendrils disappear in (a,c) for N = 8000 at λ > 2.5 and in (b,d) for N = 10 6 at λ > 2.Sizes of k-shells, S k , for SF random models with c = 0 (a,b) λ = 2.1 and (c,d) λ = 2.5 as a function of (a,c) N and the k-shell index (b,d) k. Note that sizes of k-shells increases proportionally to N. (e,f) Sizes of k-shells, S k , for SF networks with c = 5.0 and (e) λ = 2.1, (f) λ = 2.5. SF models with c = 5.0 have significantly larger number of shells, k max , compared to SF models with the same λ and c = 0. S k in SF models with c = 5.0 decreases significantly slower as a function of k compared to SF models with the same λ and c = 0 in the small k region. In the large k region both types of SF models with c = 0.0 and c = 5.0 seem to have similar behavior S k ∼ k −δ , where δ = 2/(3 − λ).","content":"FIG. 2 :FIGFIG. 4 :FIG. 5 :\n245observed shell sizes [see Fig. 7(c)]. The observed differences can be explained by taking into account the offset c in SF degree distribution of both networks. The adjustment of the degree distribution of random SF models with c = 4.0 and c = 6.0 allows one to obtain similar patterns S k which are in fair agreement with the industry networks, as seen in Figs. 7(c) and 7(d).As seen above, the offset c > 0 in the SF degree distribution plays a crucial role in the formation of the k-shell structure of industry networks. A possible reason for the emergence of c > 0 in growing networks is the combination of preferential attachment with random attachment in network evolution[42]. The coexistence of preferential attachment regime with random collaborative agreements was suggested to take place in industry networks[37]. The random component is caused by the fact that sometimes firms choose exclusive relationships and novelty, and do not prefer to make deals with hub firms. Even though the coexistence of both preferential and random regimes seems to be crucial in the formation of the industry networks, it does not fully reproduce the k-shell structure of LS and ICT industries. We believe, that a better understanding of the evolution of the LS and ICT networks may be achieved by further improvements of the modeling.V. DISCUSSION AND SUMMARYWe use the k-shell decomposition to analyze the structure of the LS and ICT sectors of industry. We find that the firms in the LS industry can be naturally divided into three components: the nucleus, the tendrils and the bulk body. The nucleus of the LS industry consists mostly of the market leader firms while the tendrils are typically comprised of small start-up firms that preferentially sell their products to the nucleus firms. We show that the nucleus of the LS industry exhibits remarkable stability in time. In contrast, the ICT industry does not have a nucleus which can be explained by a high level of competition in the ICT sector. We also analyzed the dependence of the k-shell structure of SF model networks on N, λ and c. We observed the formation of the nucleus and the tendrils in SF networks only for λ < 3. The number of shells k max and the size of the nucleus S n are larger for SF networks with c > 0 compared to those with c = 0. Our results can partly explain the k-shell structure of LS and ICT industry networks. The coexistence of preferential and random attachment leads to the appearance of the offset c > 0 in the SF degree distributionP (q) ∼ (q + c) −λ [42]. Thus, the appearance of c > 0 in the degree distribution of LS and ICT networks might be explained by the interplay of random and preferential agreements among firms in the industries [37]. FIG. 1: Illustration of the k-shell decomposition method. (a) Original network. Nodes are marked by corresponding k-shell indices. Note that the k-shell index does not coincide with the node degree. (b) The 3-crust (left) and the 3-core (right) of the original network. (a) The growth of the largest connected component of LS and ICT industries. The LS industry expands almost linearly while the ICT industry exhibits exponential growth (see inset, a semi-log plot of the ICT network size as a function of time). (b) Cumulative degree distribution N(q > q 0 ) of LS and ICT networks. The inset displays cumulative degree distribution N(q > q 0 ) of LS and ICT network without offset c. . 3: (a) The k-shell structure of the LS network and (b) of the ICT network. Plots of the size of the k-crust, N 0 ; the size of the largest connected component of the k-crust, N 1 ; and the size of the second largest connected component of the k-crust, N 2 , as a function of the k-crust index k. The sizes are normalized with the total number of nodes in the network, N. N 2 is multiplied by 100. The transition of the k-crust at shell k = 18 of the LS network reveals a jelly-fish topology [32]. Since the change in F 1 ≡ N 1 /N at the k = k max is smaller in the ICT network than in the previous shell k = k max − 1, the ICT does not have a jelly-fish structure. The insets of (a,b) show the rate R of the largest connected component change as a function of shell index k. (c) The leadership persistence LP (t, t R ) as a function of years in the LS industry. We define LP (t, t R ) as a number of firms that were in the nucleus both at time t R and t. Note that most of the LS firms remain in the nucleus after they first enter it. (a,b) The k-shell structure of random SF models with (a) N = 8000 and (b) N = 10 6 nodes are shown for comparison. Note that λ = 3.4 curve overlaps with that of λ = 3.0. (c,d) The rate of the largest connected component change, R, as a function of shell index k for (c) N = 8000 and (d) N = 10 6 . Note that in order to avoid the overlap of curves in (c,d) we subsequently shift the plots with respect to each other by the additive factor 0.05 in (c) and the multiplicative factor of 10 in (d). The sizes of the nucleus and the tendrils decrease as λ increases. The nucleus and the tendrils disappear in (a,c) for N = 8000 at λ > 2.5 and in (b,d) for N = 10 6 at λ > 2.Sizes of k-shells, S k , for SF random models with c = 0 (a,b) λ = 2.1 and (c,d) λ = 2.5 as a function of (a,c) N and the k-shell index (b,d) k. Note that sizes of k-shells increases proportionally to N. (e,f) Sizes of k-shells, S k , for SF networks with c = 5.0 and (e) λ = 2.1, (f) λ = 2.5. SF models with c = 5.0 have significantly larger number of shells, k max , compared to SF models with the same λ and c = 0. S k in SF models with c = 5.0 decreases significantly slower as a function of k compared to SF models with the same λ and c = 0 in the small k region. In the large k region both types of SF models with c = 0.0 and c = 5.0 seem to have similar behavior S k ∼ k −δ , where δ = 2/(3 − λ)."},{"id":"","caption":"(a) The total number of shells, k max , in SF models as a function of N. Note that k max ∝ N θ . (b) The size of the last shell S max in the SF model as a function of N. Each curve crosses over into a power law regime for N ≥ N c (λ), where N c (λ) increases with λ.(c) The calculated exponent θ as a function of λ (symbols). Our calculated values of δ agree with the mean field theory result δ ≈ 2/(3 − λ) (solid line).","content":"FIG. 6 :\n6(a) The total number of shells, k max , in SF models as a function of N. Note that k max ∝ N θ . (b) The size of the last shell S max in the SF model as a function of N. Each curve crosses over into a power law regime for N ≥ N c (λ), where N c (λ) increases with λ.(c) The calculated exponent θ as a function of λ (symbols). Our calculated values of δ agree with the mean field theory result δ ≈ 2/(3 − λ) (solid line)."},{"id":"","caption":"(a) The largest shell index k max of the LS and the ICT networks as a function of N calculated for different years. The number of shells k max of the LS network increases approximately as a power-law function of the network size k max ∼ N θ , where θ ≃ 0.6. (b)","content":"FIG. 7 :\n7(a) The largest shell index k max of the LS and the ICT networks as a function of N calculated for different years. The number of shells k max of the LS network increases approximately as a power-law function of the network size k max ∼ N θ , where θ ≃ 0.6. (b)"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"analyze the k-shell structure of LS and ICT industrial sectors in time periods between 1990 and 2002 and between 1990 and 2000 respectively. The LS network expanded linearly since the mid-1970s while the ICT network took off in the 1990s and grew exponentially for a decade [see Fig. 2(a) and the inset]. The total number of firms in the LS is N = 6, 776 and in the ICT is N = 7, 759. These sizes refer to the largest connected component of each of the networks at the last year of observation. Both industrial networks feature SF degree distributions with λ ≈ 2.5, c ≈ 4. (LS) and λ ≈ 3.4, c ≈ 6. (ICT) [see Fig. 2(b)]. We use the Kholmogorov-Smirnov test in order to examine the goodness of fit of the degree distributions. The obtained p-values for 1000 trials are 0.24 and 0.33 for the LS and the ICT industry networks respectively.We next apply the k-shell decomposition procedure to the LS and the ICT networks. For each k-crust we calculate the total number of nodes, N 0 , comprising it, the size of the largest connected component, N 1 , in the k-crust and the size of the second largest component, N 2 , of the k-crust. As seen inFig. 3, the LS network consists of k max = 19 shells while the ICT network (which consists of comparable number of firms) has only k max = 6 shells. The size of the largest cluster in the k-crust starts growing rapidly after k = 4 and k = 2 in the LS and the ICT networks respectively. At these values of k, the size of the second largest cluster in the k-crust reaches a maximum. The above behavior of the k-crust components\",\n \"content\": \"analyze the k-shell structure of LS and ICT industrial sectors in time periods between 1990 and 2002 and between 1990 and 2000 respectively. The LS network expanded linearly since the mid-1970s while the ICT network took off in the 1990s and grew exponentially for a decade [see Fig. 2(a) and the inset]. The total number of firms in the LS is N = 6, 776 and in the ICT is N = 7, 759. These sizes refer to the largest connected component of each of the networks at the last year of observation. Both industrial networks feature SF degree distributions with λ ≈ 2.5, c ≈ 4. (LS) and λ ≈ 3.4, c ≈ 6. (ICT) [see Fig. 2(b)]. We use the Kholmogorov-Smirnov test in order to examine the goodness of fit of the degree distributions. The obtained p-values for 1000 trials are 0.24 and 0.33 for the LS and the ICT industry networks respectively.We next apply the k-shell decomposition procedure to the LS and the ICT networks. For each k-crust we calculate the total number of nodes, N 0 , comprising it, the size of the largest connected component, N 1 , in the k-crust and the size of the second largest component, N 2 , of the k-crust. As seen inFig. 3, the LS network consists of k max = 19 shells while the ICT network (which consists of comparable number of firms) has only k max = 6 shells. The size of the largest cluster in the k-crust starts growing rapidly after k = 4 and k = 2 in the LS and the ICT networks respectively. At these values of k, the size of the second largest cluster in the k-crust reaches a maximum. The above behavior of the k-crust components\"\n },\n {\n \"id\": \"\",\n \"caption\": \"b)). In both LS and ICT industry sectors the k max shells include market leaders such as Pfizer, GSK, Novartis, J&J, Sanofi-Aventis, Bayer in the LS network and Microsoft, IBM, AT&T, Yahoo, Cisco, AOL, Time Warner and Google in the ICT network. We find that the LS network tendrils are mostly composed of the new start up firms and their university partners. On the one hand, start up firms preferentially attach to market leaders. On the other hand, market leaders compete to sign exclusive deals with new and promising start up firms. We also notice the remarkable stability of the LS nucleus: once a particular firm enters the nucleus it is very likely to remain there for many years.(See Fig. 3c) On the contrary, in the ICT sector there is more emphasis on how to integrate different technologies and markets [39]\",\n \"content\": \"Fig. 3 (\\n3b)). In both LS and ICT industry sectors the k max shells include market leaders such as Pfizer, GSK, Novartis, J&J, Sanofi-Aventis, Bayer in the LS network and Microsoft, IBM, AT&T, Yahoo, Cisco, AOL, Time Warner and Google in the ICT network. We find that the LS network tendrils are mostly composed of the new start up firms and their university partners. On the one hand, start up firms preferentially attach to market leaders. On the other hand, market leaders compete to sign exclusive deals with new and promising start up firms. We also notice the remarkable stability of the LS nucleus: once a particular firm enters the nucleus it is very likely to remain there for many years.(See Fig. 3c) On the contrary, in the ICT sector there is more emphasis on how to integrate different technologies and markets [39]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figs. 5(b) and 5(d)]. The appearance of c > 0 in the SF degree distribution significantly increases k max [see Figs. 5(e) and 5(f)]. However, the asymptotic dependence of k-shell sizes seems to remain the same as k approaches k max : S k ∼ k −δ [Figs. 5(e) and 5(f)].\",\n \"content\": \"[\\nFigs. 5(b) and 5(d)]. The appearance of c > 0 in the SF degree distribution significantly increases k max [see Figs. 5(e) and 5(f)]. However, the asymptotic dependence of k-shell sizes seems to remain the same as k approaches k max : S k ∼ k −δ [Figs. 5(e) and 5(f)].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"observed shell sizes [see Fig. 7(c)]. The observed differences can be explained by taking into account the offset c in SF degree distribution of both networks. The adjustment of the degree distribution of random SF models with c = 4.0 and c = 6.0 allows one to obtain similar patterns S k which are in fair agreement with the industry networks, as seen in Figs. 7(c) and 7(d).As seen above, the offset c > 0 in the SF degree distribution plays a crucial role in the formation of the k-shell structure of industry networks. A possible reason for the emergence of c > 0 in growing networks is the combination of preferential attachment with random attachment in network evolution[42]. The coexistence of preferential attachment regime with random collaborative agreements was suggested to take place in industry networks[37]. The random component is caused by the fact that sometimes firms choose exclusive relationships and novelty, and do not prefer to make deals with hub firms. Even though the coexistence of both preferential and random regimes seems to be crucial in the formation of the industry networks, it does not fully reproduce the k-shell structure of LS and ICT industries. We believe, that a better understanding of the evolution of the LS and ICT networks may be achieved by further improvements of the modeling.V. DISCUSSION AND SUMMARYWe use the k-shell decomposition to analyze the structure of the LS and ICT sectors of industry. We find that the firms in the LS industry can be naturally divided into three components: the nucleus, the tendrils and the bulk body. The nucleus of the LS industry consists mostly of the market leader firms while the tendrils are typically comprised of small start-up firms that preferentially sell their products to the nucleus firms. We show that the nucleus of the LS industry exhibits remarkable stability in time. In contrast, the ICT industry does not have a nucleus which can be explained by a high level of competition in the ICT sector. We also analyzed the dependence of the k-shell structure of SF model networks on N, λ and c. We observed the formation of the nucleus and the tendrils in SF networks only for λ < 3. The number of shells k max and the size of the nucleus S n are larger for SF networks with c > 0 compared to those with c = 0. Our results can partly explain the k-shell structure of LS and ICT industry networks. The coexistence of preferential and random attachment leads to the appearance of the offset c > 0 in the SF degree distributionP (q) ∼ (q + c) −λ [42]. Thus, the appearance of c > 0 in the degree distribution of LS and ICT networks might be explained by the interplay of random and preferential agreements among firms in the industries [37]. FIG. 1: Illustration of the k-shell decomposition method. (a) Original network. Nodes are marked by corresponding k-shell indices. Note that the k-shell index does not coincide with the node degree. (b) The 3-crust (left) and the 3-core (right) of the original network. (a) The growth of the largest connected component of LS and ICT industries. The LS industry expands almost linearly while the ICT industry exhibits exponential growth (see inset, a semi-log plot of the ICT network size as a function of time). (b) Cumulative degree distribution N(q > q 0 ) of LS and ICT networks. The inset displays cumulative degree distribution N(q > q 0 ) of LS and ICT network without offset c. . 3: (a) The k-shell structure of the LS network and (b) of the ICT network. Plots of the size of the k-crust, N 0 ; the size of the largest connected component of the k-crust, N 1 ; and the size of the second largest connected component of the k-crust, N 2 , as a function of the k-crust index k. The sizes are normalized with the total number of nodes in the network, N. N 2 is multiplied by 100. The transition of the k-crust at shell k = 18 of the LS network reveals a jelly-fish topology [32]. Since the change in F 1 ≡ N 1 /N at the k = k max is smaller in the ICT network than in the previous shell k = k max − 1, the ICT does not have a jelly-fish structure. The insets of (a,b) show the rate R of the largest connected component change as a function of shell index k. (c) The leadership persistence LP (t, t R ) as a function of years in the LS industry. We define LP (t, t R ) as a number of firms that were in the nucleus both at time t R and t. Note that most of the LS firms remain in the nucleus after they first enter it. (a,b) The k-shell structure of random SF models with (a) N = 8000 and (b) N = 10 6 nodes are shown for comparison. Note that λ = 3.4 curve overlaps with that of λ = 3.0. (c,d) The rate of the largest connected component change, R, as a function of shell index k for (c) N = 8000 and (d) N = 10 6 . Note that in order to avoid the overlap of curves in (c,d) we subsequently shift the plots with respect to each other by the additive factor 0.05 in (c) and the multiplicative factor of 10 in (d). The sizes of the nucleus and the tendrils decrease as λ increases. The nucleus and the tendrils disappear in (a,c) for N = 8000 at λ > 2.5 and in (b,d) for N = 10 6 at λ > 2.Sizes of k-shells, S k , for SF random models with c = 0 (a,b) λ = 2.1 and (c,d) λ = 2.5 as a function of (a,c) N and the k-shell index (b,d) k. Note that sizes of k-shells increases proportionally to N. (e,f) Sizes of k-shells, S k , for SF networks with c = 5.0 and (e) λ = 2.1, (f) λ = 2.5. SF models with c = 5.0 have significantly larger number of shells, k max , compared to SF models with the same λ and c = 0. S k in SF models with c = 5.0 decreases significantly slower as a function of k compared to SF models with the same λ and c = 0 in the small k region. In the large k region both types of SF models with c = 0.0 and c = 5.0 seem to have similar behavior S k ∼ k −δ , where δ = 2/(3 − λ).\",\n \"content\": \"FIG. 2 :FIGFIG. 4 :FIG. 5 :\\n245observed shell sizes [see Fig. 7(c)]. The observed differences can be explained by taking into account the offset c in SF degree distribution of both networks. The adjustment of the degree distribution of random SF models with c = 4.0 and c = 6.0 allows one to obtain similar patterns S k which are in fair agreement with the industry networks, as seen in Figs. 7(c) and 7(d).As seen above, the offset c > 0 in the SF degree distribution plays a crucial role in the formation of the k-shell structure of industry networks. A possible reason for the emergence of c > 0 in growing networks is the combination of preferential attachment with random attachment in network evolution[42]. The coexistence of preferential attachment regime with random collaborative agreements was suggested to take place in industry networks[37]. The random component is caused by the fact that sometimes firms choose exclusive relationships and novelty, and do not prefer to make deals with hub firms. Even though the coexistence of both preferential and random regimes seems to be crucial in the formation of the industry networks, it does not fully reproduce the k-shell structure of LS and ICT industries. We believe, that a better understanding of the evolution of the LS and ICT networks may be achieved by further improvements of the modeling.V. DISCUSSION AND SUMMARYWe use the k-shell decomposition to analyze the structure of the LS and ICT sectors of industry. We find that the firms in the LS industry can be naturally divided into three components: the nucleus, the tendrils and the bulk body. The nucleus of the LS industry consists mostly of the market leader firms while the tendrils are typically comprised of small start-up firms that preferentially sell their products to the nucleus firms. We show that the nucleus of the LS industry exhibits remarkable stability in time. In contrast, the ICT industry does not have a nucleus which can be explained by a high level of competition in the ICT sector. We also analyzed the dependence of the k-shell structure of SF model networks on N, λ and c. We observed the formation of the nucleus and the tendrils in SF networks only for λ < 3. The number of shells k max and the size of the nucleus S n are larger for SF networks with c > 0 compared to those with c = 0. Our results can partly explain the k-shell structure of LS and ICT industry networks. The coexistence of preferential and random attachment leads to the appearance of the offset c > 0 in the SF degree distributionP (q) ∼ (q + c) −λ [42]. Thus, the appearance of c > 0 in the degree distribution of LS and ICT networks might be explained by the interplay of random and preferential agreements among firms in the industries [37]. FIG. 1: Illustration of the k-shell decomposition method. (a) Original network. Nodes are marked by corresponding k-shell indices. Note that the k-shell index does not coincide with the node degree. (b) The 3-crust (left) and the 3-core (right) of the original network. (a) The growth of the largest connected component of LS and ICT industries. The LS industry expands almost linearly while the ICT industry exhibits exponential growth (see inset, a semi-log plot of the ICT network size as a function of time). (b) Cumulative degree distribution N(q > q 0 ) of LS and ICT networks. The inset displays cumulative degree distribution N(q > q 0 ) of LS and ICT network without offset c. . 3: (a) The k-shell structure of the LS network and (b) of the ICT network. Plots of the size of the k-crust, N 0 ; the size of the largest connected component of the k-crust, N 1 ; and the size of the second largest connected component of the k-crust, N 2 , as a function of the k-crust index k. The sizes are normalized with the total number of nodes in the network, N. N 2 is multiplied by 100. The transition of the k-crust at shell k = 18 of the LS network reveals a jelly-fish topology [32]. Since the change in F 1 ≡ N 1 /N at the k = k max is smaller in the ICT network than in the previous shell k = k max − 1, the ICT does not have a jelly-fish structure. The insets of (a,b) show the rate R of the largest connected component change as a function of shell index k. (c) The leadership persistence LP (t, t R ) as a function of years in the LS industry. We define LP (t, t R ) as a number of firms that were in the nucleus both at time t R and t. Note that most of the LS firms remain in the nucleus after they first enter it. (a,b) The k-shell structure of random SF models with (a) N = 8000 and (b) N = 10 6 nodes are shown for comparison. Note that λ = 3.4 curve overlaps with that of λ = 3.0. (c,d) The rate of the largest connected component change, R, as a function of shell index k for (c) N = 8000 and (d) N = 10 6 . Note that in order to avoid the overlap of curves in (c,d) we subsequently shift the plots with respect to each other by the additive factor 0.05 in (c) and the multiplicative factor of 10 in (d). The sizes of the nucleus and the tendrils decrease as λ increases. The nucleus and the tendrils disappear in (a,c) for N = 8000 at λ > 2.5 and in (b,d) for N = 10 6 at λ > 2.Sizes of k-shells, S k , for SF random models with c = 0 (a,b) λ = 2.1 and (c,d) λ = 2.5 as a function of (a,c) N and the k-shell index (b,d) k. Note that sizes of k-shells increases proportionally to N. (e,f) Sizes of k-shells, S k , for SF networks with c = 5.0 and (e) λ = 2.1, (f) λ = 2.5. SF models with c = 5.0 have significantly larger number of shells, k max , compared to SF models with the same λ and c = 0. S k in SF models with c = 5.0 decreases significantly slower as a function of k compared to SF models with the same λ and c = 0 in the small k region. In the large k region both types of SF models with c = 0.0 and c = 5.0 seem to have similar behavior S k ∼ k −δ , where δ = 2/(3 − λ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) The total number of shells, k max , in SF models as a function of N. Note that k max ∝ N θ . (b) The size of the last shell S max in the SF model as a function of N. Each curve crosses over into a power law regime for N ≥ N c (λ), where N c (λ) increases with λ.(c) The calculated exponent θ as a function of λ (symbols). Our calculated values of δ agree with the mean field theory result δ ≈ 2/(3 − λ) (solid line).\",\n \"content\": \"FIG. 6 :\\n6(a) The total number of shells, k max , in SF models as a function of N. Note that k max ∝ N θ . (b) The size of the last shell S max in the SF model as a function of N. Each curve crosses over into a power law regime for N ≥ N c (λ), where N c (λ) increases with λ.(c) The calculated exponent θ as a function of λ (symbols). Our calculated values of δ agree with the mean field theory result δ ≈ 2/(3 − λ) (solid line).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) The largest shell index k max of the LS and the ICT networks as a function of N calculated for different years. The number of shells k max of the LS network increases approximately as a power-law function of the network size k max ∼ N θ , where θ ≃ 0.6. (b)\",\n \"content\": \"FIG. 7 :\\n7(a) The largest shell index k max of the LS and the ICT networks as a function of N calculated for different years. The number of shells k max of the LS network increases approximately as a power-law function of the network size k max ∼ N θ , where θ ≃ 0.6. (b)\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6290P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6, 290 (1959). I","doi":""},{"text":". P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 517P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960). I","doi":""},{"text":"B Bollobas, Random Graphs. Cambridge University PressB. Bollobas, Random Graphs (Cambridge University Press, 2001). I","doi":""},{"text":"S Wasserman, K Faust, Social Network Analysis. Cambridge University PressS. Wasserman and K. Faust, Social Network Analysis, (Cambridge University Press, 1999). I","doi":""},{"text":"S N Dorogovtsev, J F F Mendes, Evolution of Networks. Oxford University PressS. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks (Oxford University Press, 2003). I","doi":""},{"text":"M Newman, A L Barabási, D J Watts, The Structure and Dynamics of Networks. 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Faloutsos, Global Internet (Nov. 2001). I","doi":""},{"text":". Scb Www, Se I, WWW.SCB.SE I","doi":""},{"text":". Y Chen, G Paul, R Cohen, S Havlin, S P Borgatti, F Liljeros, H E Stanley, Phys. Rev. E. 7546107Y. Chen, G. Paul, R. Cohen, S. Havlin, S. P. Borgatti, F. Liljeros and H. E. Stanley, Phys. Rev. E 75, 046107 (2007). I","doi":""},{"text":". J Sutton, American Economic Review. 97222J. Sutton, American Economic Review, 97, 222 (2007). I","doi":""}],"string":"[\n {\n \"text\": \". P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6290P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6, 290 (1959). I\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 517P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960). I\",\n \"doi\": \"\"\n },\n {\n \"text\": \"B Bollobas, Random Graphs. Cambridge University PressB. Bollobas, Random Graphs (Cambridge University Press, 2001). 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I\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":394,"cells":{"text":{"kind":"string","value":"\n\n21 Dec 2005\n\nJean-Philippe Préaux \nGROUPESÀ CLASSES DE CONJUGAISON INFINIES : QUELQUES EXEMPLES\n\n21 Dec 2005\nWe consider the group property of being with infinite conjugacy classes (or icc, i.e. = 1 and of which all conjugacy classes except 1 are infinite) in some particular cases, and more particulary in case of specific elementary algebraic constructions or of some 3-manifold fundamental groups. Résumé. Nousétudions la propriété pour un groupe d'êtreà classes de conjugaison infinis (ou cci, i.e. = 1 et dont toutes les classes de conjugaison autres que 1 sont infinies) dans certains cas particuliers, et plus particulièrement dans le cas de constructions algébriqueś elémentaires, ou de groupes fondamentaux de certaines 3-variétés.\n\nIntroduction\n\nLa propriété de groupe d'êtreà classes de conjugaison infinies est particulièrement simpleàénoncer (voir la définition section 2). Elle est motivée par la caractérisation de Murray et Von-Neumann des algèbres de Von-Neumann qui sont des facteurs de type II-1 (cf. [ROIV]) : Caractérisation de Murray et Von-Neumann. Soient Γ un groupe et W * λ (Γ) son algèbre de Von-Neumann. Alors W * λ (Γ) est un facteur de type II-1 si et seulement si Γ est cci.\n\nNousétudions plus particulièrement la stabilité de cette propriété par certaines constructions algébriquesélémentaires, et le cas des groupes fondamentaux de certaines 3variétés. L'étude que nous menons est somme touteélémentaire ; le lecteur intéressé pourra trouver des résultats analogues de plus grande portée dans [Co], [HP] et [Pr].\n\nCe travail aété effectué durant l'été 2004 alors que l'auteur se trouvaità l'université de Genève et aété partiellement financé par le Fonds National Suisse de la Recherche Scientifique. L'auteur tientà remercier les mathématiciens Genévois pour leur accueil, et plus particulièrement Goulnara Arjantseva et Pierre de la Harpe. L'auteur remercie encore Pierre de la Harpe de lui avoir introduit le problème, ainsi que des nombreuses discussions et des commentaires apportés.\n\n\nDéfinitions et exemples\n\nDéfinition 1. Un groupe est dità classes de conjugaisons infinies (ou cci) si il est non trivial, et si toutélément non trivial a une classe de conjugaison infinie.\n\nIl est commode de caractériser cette propriété en terme de centralisateur plutôt qu'en terme de classe de conjugaison.\n\nProposition-Définition 1. Un groupe non trivial est cci si et seulement si toutélément non trivial a un centralisateur d'indice infini. Preuve. On note G le groupe, et Z(g) le centralisateur d'unélément g ∈ G. Soient g ∈ G − {1}, et a, b ∈ G. Supposons que aga −1 = bgb −1 :\naga −1 = bgb −1 ⇐⇒ (b −1 a)g = g(b −1 a) ⇐⇒ b −1 a ∈ Z(g)\nAinsi aga −1 = bgb −1 si et seulement si a et b sont dans des right-cosets différents de G/Z(g) et donc, la classe de conjugaison de g, [[g]], est en correspondance bi-univoque avec G/Z(g). Ainsi #[[g]] = #(G/Z(g)), ce qui permet de conclure.\n\nOn peut déjàétablir quelques exemplesévidents :\n\nProposition 2.1.\n\n(1) Les groupes libres de rang ≥ 2 sont cci.\n\n(2) Les groupes Gromov-hyperboliques nonélémentaires et sans torsion sont cci.\n\n(3) Les groupes ayant un nombre fini (non nul) d'éléments de torsion (en particulier les groupes finis), ne sont pas cci. (4) Les groupes virtuellement abéliens ne sont pas cci.\n\n(5) Les groupes ayant un centre non trivial ne sont pas cci.\n\nPreuve. Dans un groupe libre le centralisateur d'unélément non trivial est cyclique infini. Puisqu'un groupe libre de rang ≥ 2 n'est pas virtuellement Z, cela prouve (1). Dans un groupe hyperbolique le centralisateur d'unélément d'ordre infini est virtuellement cyclique ( [CDP]). Ainsià l'exception des cas où le groupe est fini ou virtuellement cyclique, le centralisateur d'unélément est toujours d'ordre infini, ce qui prouve (2). Pour prouver (3) considérons unélément g de torsion (non trivial). Tous ses conjugués ont même ordre que lui, et sont donc aussi de torsion. Ainsi g n'a qu'un nombre fini de conjugués. Pour prouver (4) appelons A un sous-groupe abélien d'indice fini dans un groupe G, et considérons uń elément non trivial g ∈ A. Alors Z(g) ⊃ A, et donc Z(g) est d'indice fini dans G. Pour prouver (5) il suffit de remarquer que toutélément du centre a pour centralisateur le groupe ambient.\n\n\nStabilité par construction algébrique\n\nAfin de considérer le cas d'un produit libre, il nous est nécessaire d'établir le lemme suivant : Pour finir, si A et B ne sont pas tous deux d'ordre 2, mais ne contiennent que deś eléments d'ordre 2, alors A * B contient < x, y, z | x 2 , y 2 , (xy) 2 , z 2 >. Alors comme précédemment, a = x(yz)x et b = zx(yz)xz engendrent un groupe libre de rang 2.\nLemme 3.1. Soient A, B des groupes non triviaux. Soit A et B sont tous deux d'ordre 2, soit A * B contient le groupe libre de rang 2. Preuve. Si A et B contiennent chacun unélément d'ordre infini a ∈ A, b ∈ B, alors < a > * < b > est un sous-groupe de A * B isomorpheà F (2). Si A ou B contient unélément d'ordre > 2, A * B contient un sous-groupe < x, y | x r , y s >, avec r > 2 et s ≥ 0, s = 1. On pose a = x(xy)x −1 et b = yx(xy)x −1 y −1 .Proposition 3.1 (Produit libre). Soient A, B deux groupes non triviaux. Si A ou B est d'ordre > 2, alors A * B est cci.\nPreuve. Soit u unélément dans un des deux facteurs ; sans perte de généralité on suppose u ∈ A. Avec le théorème de commutativité dans un produit libre (cf.\n\n[MKS]), le centralisateur de u dans A * B n'est rien d'autre que le centralisateur de u dans A. Puisque A est d'indice infini dans A * B, il en va de même de Z(u). Maintenant si v est unélément qui n'est pas dans un des facteurs, son centralisateur est cyclique infini. Pour conclure il suffit d'utiliser le lemme précédent : puisque A * B contient le groupe libre de rang 2, il ne peut pasêtre virtuellement Z, et donc A * B est cci.\n\nRemarque : Dans le cas restant, celui du groupe Z 2 * Z 2 , ce dernier contient un groupe cyclique infini d'indice 2 (si l'on note A * B =< a, b | a 2 , b 2 >, considérer < ab >), et n'est donc pas cci.\nProposition 3.2 (Sous-groupe d'indice fini). Si G est cci, et si K est un sous-groupe d'indice fini de G, alors K est aussi cci. Preuve. Soit u = 1 dans K. On note respectivement Z G (u) et et Z K (u) son centralisa- teur dans G et K ; trivialement Z K (u) = Z G (u) ∩ K. Puisque G a la propriété H, Z G (u) est d'indice infini dans G. Si Z K (u)était d'indice fini dans K, il serait aussi d'indice fini dans G, ce qui est impossible puisque Z G (u) ⊃ Z K (u).\nRemarque : La réciproque est fausse ; il suffit pour voir cela de considérer le groupe F 2 × Z 2 . Elle devient vraie lorsque l'on suppose que le groupe est sans torsion. Preuve. Supposons que G ne soit pas cci, bien qu'il contienne un sous-groupe d'indice fini cci. Ainsi, il existe u = 1 dans G, tel que Z G (u) soit d'indice fini dans G, et donc Z G (u) ∩ K = K 0 est d'indice fini dans K. Nécessairement u ∈ K, car cela contredirait le fait que K est cci. Mais il est facile de voir qu'il existe n > 1 tel que u n ∈ K. L'ensemble K 0 est inclus dans Z K (u n ), qui est donc nécessairement d'indice fini dans K. Puisque K est cci, on a u n = 1, et donc G a de la torsion.\n\nProposition 3.4 (Epi-résiduellement cci). Soit G un groupe ; si pour tout g ∈ G− {1} il existe un morphisme surjectif φ sur un groupe cci, tel que φ(g) = 1 (on dira que G est epi-résiduellement H), alors G est cci.\n\nPreuve. Soit φ : G −→ K unépimorphisme sur un groupe K cci. Remarquons que puisque φ est surjectif, un conjugué de φ(g) dans K est l'image par φ d'un conjugué de g dans G. Ainsi, puisque K est cci, si φ(g) = 1 sa classe de conjugaison dans K est infinie, et donc la classe de conjugaison de g dans G est elle-même infinie.\n\nOn en déduit directement : On peut vérifie aussi la stabilité de la propriété d'être cci par extension.\n\nProposition 3.6 (Extension). Soit G l'extension de deux groupes cci K et Q :\n1 −→ K −→ G p −→ Q −→ 1\nAlors G est aussi cci.\n\nPreuve. Soit ω = 1 unélément de G. Si ω ∈ K, sa classe de conjugaison dans K est infinie, et il en est donc de même de sa classe de conjugaison dans G. Si ω ∈ K, alors p(ω) a une classe de conjugaison infinie dans Q, et donc puisque p est surjectif, la classe de conjugaison de ω dans G est aussi infinie. Soit u ∈ A * C B ; si u est dans un des facteurs, par exemple A, sa classe de conjugaison dans A * C B contient sa classe de conjugaison dans A, et est donc infinie.\n\nSupposons sans perte de généralité que C est d'indice > 2 dans A. Considérons uń elément u = 1 quelconque dans A * C B, que l'on peut supposer cycliquemet réduit. Le cas où u est dans un des facteurs ayant déjàété traité, on peut supposer que u s'écrit sous forme cycliquement réduite u = a 1 b 1 . · · · .a n b n , avec n ≥ 1. L'ensemble des right-cosets A/C contient au moins troiséléments distincts 1.C, α.C et α 2 .C. On choisit a ∈ A hors de la classe 1.C et dans une classe différente de celle de a −1 1 . On choisit b ∈ B − C. Avec ce choix, tous leséléments (ba) n .u.(ba) −n de la classe de conjugaison de u sont distincts. Ainsi toutélément non trivial de A * C B a une classe de conjugaison infinie, ce qui conclut la preuve.\n\nLe même modèle de preuve s'applique pour montrer :\n\nProposition 3.8 (Extension HNN). Si A est cci, et si C est un sous-groupe d'indice > 2 dans A, l'extension HNN A * C est cci.\n\nPreuve. Soit A * φ , avec φ : C −→ φ(C). On note t une lettre stable, telle que tct −1 = φ(t) pour tout c ∈ C. Comme précédemment on peut se ramener facilement au cas d'unélément cycliquement réduit de longueur > 1, u = a 1 t n 1 . . . . .α p t np , avec n p = 0. On a au moins 3 right-cosets dans A/C : 1.C, α.C, et α 2 .C. On choisit a ∈ A hors de 1.C et de la classe de a −1 1 . Alors tous leséléments (ta) n u(ta) −n pour n > 0 sont distincts dans la classe de conjugaison de u, ce qui permet de conclure.\n\nEt on obtient en corollaire immédiat : Proposition 3.9 (Graphes de groupes). Si Γ = π 1 (G, X) est le groupe d'un graphe de groupe dont tous les groupes de sommet sont cci, et tel que pour toute arête e d'origine s 0 et d'extrémité s 1 , G(e) est d'indice > 2 dans G(s 0 ) ou bien dans G(s 1 ), alors Γ est cci.\n\n\nGroupes de 3-variétés\n\nNous considérons d'abord le cas des groupes fondamentaux de fibrés de Seifert (cf. [Sei]). En utilisant le fait que dans un groupe de Seifert, la classe d'une fibre régulière a un centralisateur d'indice fini (cf. [JS]), onétablit immédiatement :\n\nProposition 4.1 (Fibrés de Seifert). Les groupes de fibrés de Seifert ne sont pas cci.\n\nConsidérons maintenant le cas des variétés hyperboliques de volume fini. Il montre une autre application directe de la proposition 3.4.\n\nProposition 4.2 (Hyperboliques et χ = 0). Les groupes de 3-variétés hyperboliques de volume fini nonélémentaires sont cci.\n\nPreuve. Considérons une variété M vérifiant les hypothèses : M est une 3-variété hyperboliqueà bord torique ou vide, qui n'est ni un tore solide, ni un toreépaissi. En outre, la classification des isométries hyperboliques montre que π 1 (M ) est sans torsion (cf. [BP]), et donc avec la proposition 3.3, en considérant le revêtement d'orientation de M on supposera M orientable.\n\nLe théorème de chirurgie hyperbolique de Thurston affirme qu'il existe une suite de variétés hyperboliques fermées (M n ) n , obtenues par obturation de Dehn sur M , convergeant vers M pour la topologie géométrique (cf. [BP]). Ainsi on a une suite d'épimorphismes (φ n : π 1 M −→ π 1 M n ) n . Par définition de la topologie géométrique, si u = 1 ∈ π 1 M , pour n ≫ 0, φ n (u) = 1 ; et tous les groupes π 1 M n sont hyperboliques au sens de Gromov. De plus pour n ≫ 0 ils sont nonélémentaires (ceci car leurs points fixes dans ∂H 3 convergent vers les points fixes de π 1 M et sont donc pour n ≫ 0 en nombre > 2). Ainsi avec la proposition 2.1 (2), π 1 M estépi-résiduellement cci, et la proposition 3.4 permet de conclure.\n\nNous traitons maintenant le cas des groupes de 3-variétés orientables suffisament large. Rappelons qu'un 3-variété orientable est dite suffisament large si elle contient une surface incompressibleà deux faces ; une 3-variété orientable est Haken si elle est irréductible et suffisament large. Preuve. Nous avons vu qu'un fibré de Seifert a un groupe non cci ; aussi supposons que π 1 (M ) est non cci et montrons que M est un fibré de Seifert. Rappelons que le groupe d'une 3-variété Haken est non trivial (car il contient un groupe de surface infini) ; ainsi, soit u = 1 dans π 1 (M ) ayant un centralisateur Z(u) d'indice fini, et soit p : N −→ M le revêtement fini associéà Z(u). Clairement π 1 (N ) = Z(u) a un centre non trivial. De plus nécessairement N est Haken : le corollaire 13.5 de [He] montre que N est irréductible, et de plus si S est une surface incompressibleà deux faces dans M , p −1 (S) est une surface incompressibleà deux faces dans N . Le corollaire II.5.4 de [JS] montre que N est nécessairement un fibré de Seifert et le théorème II.5.3 ( [JS]) montre qu'il en est alors de même pour M .\n\nProposition 4.4 (Le cas orientable suffisament large). Soit M une 3-variété orientable et suffisament large ayant un groupe fondamental non diédral infini ; alors soit π 1 (M ) est cci, soit la complétée de Poincaré P(M ) est un fibré de Seifert.\n\nPreuve. Considérons la décomposition de Kneser-Milnor (cf. [He]) de M : M = M 1 # · · · #M n #B 1 # · · · #B p #C 1 # · · · C q avec n, p, q ∈ N, et où les M i sont non simplement connexes et premiers, les B i sont des boules et les C i des fausses 3-sphères. Puisque M est suffisament large, π 1 (M ) est non trivial (car il contient un groupe de surface infini) et donc n > 0. La complétée de Poincaré P(M ) est alors définie commeétant : P(M ) = M 1 # · · · #M n et de plus π 1 (M ) = π 1 (P(M )) = π 1 (M 1 ) * · · · * π 1 (M n ). Ainsi avec la proposition 4.1 si P(M ) est un fibré de Seifert, alors π 1 (M ) n'est pas cci. Supposons dans la suite que π 1 (M ) n'est pas cci. Alors avec la proposition 3.1 soit π 1 (M ) est diédral infini, soit n = 1. Ainsi sous nos hypothèses P(M ) = M 1 et soit M 1 ≡ S 2 × S 1 soit M 1 est irréductible. Dans le premier cas P(M ) est un fibré de Seifert, et dans le second cas, puisque M contient une surface incompressibleà deux faces, M 1 est Haken, et la proposition 4.3 donne la conclusion.\n\nDans [HP] ces résultats ontété généralisés au cas de groupes fondamentaux de 3-variétés quelconques.\n\n\nAlors avec le théorème d'écriture sous forme réduite, un mot réduit non trivial sur a et b est non trivial dans A * B, et donc a et b engendrent un groupe libre de rang 2.\n\nProposition 3. 3 (\n3Extension finie). Si G est sans torsion, et si K est un sous-groupe d'indice fini de G cci, alors G est aussi cci.\n\nProposition 3 . 5 (\n35Produit direct). Soient A et B deux groupes non triviaux. Le produit direct A × B est cci si et seulement si A et B sont cci.Preuve. Si A ou B est non cci, trivialement A × B n'est pas cci ; aussi supposons dans la suite que A et B sont tous deux cci et montrons que A × B est cci. Soit u = ab, avec a ∈ A et b ∈ B. Si u = 1, alors sans perte de généralité on peut supposer a = 1. En considérant la projection canonique A × B −→ A, la proposition 3.4 permet de conclure.\n\nProposition 3. 7 (\n7Amalgame). Si A et B sont deux groupes cci, et si C = A ∩ B n'est pas d'indice 1 ou 2 dans chaque facteur, alors le produit amalgamé A * C B est cci.Preuve. On peut supposer que A, B = {1}, car autrement le groupe obtenu n'est autre que A ou B. De la même façon on peut supposer que C est d'indice > 1 dans A et dans B.\n\nProposition 4 . 3 (\n43Le cas Haken orientable). Soit M un 3-variété Haken orientable ; alors soit π 1 (M ) est cci soit M est un fibré de Seifert.\n\n\nCentre de recherche de l'Ecole de l'air, Ecole de l'air, F-13661 Salon de Provence air Centre de Mathématiques et d'informatique, Université de Provence, 39 rue F.Joliot-Curie, F-13453 marseille cedex 13 E-mail : preaux@cmi.univ-mrs.fr Mathematical subject classification : 20E45 (Primary), 20E06, 20E08, 20E22, 57M05 (Secondary)1 \n\n2 \n\n\n\nR Benedetti, C Petronio, Lectures on hyperbolic geometry. SpringerUniversitextR.Benedetti, and C.Petronio, Lectures on hyperbolic geometry, Universitext, Springer, 1992.\n\nInfinite conjugacy classes in groups acting on trees. Y De Cornulier, Y.de Cornulier, Infinite conjugacy classes in groups acting on trees, Prépublication 2005.\n\nM Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes, les groupes hyperboliques de Gromov. Springer-Verlag1441M.Coornaert, T.Delzant, and A.Papadopoulos, Géométrie et théorie des groupes, les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441, Springer-Verlag, 1991.\n\nP De La Harpe, J.-P Préaux, arXiv:math.GR/0509449Groupes fondamentaux des variétés de dimension 3 et algèbres d'opérateurs. 1P.de la Harpe et J.-P.Préaux, Groupes fondamentaux des variétés de dimension 3 et algèbres d'opérateurs, Prépublication arXiv:math.GR/0509449 v1 (2005).\n\nJ Hempel, Annals of mathematics studies. Priceton university press3J.Hempel, 3-manifolds, Annals of mathematics studies, Priceton university press, 1976.\n\nW Jaco, P Shalen, Seifert fibre space in 3-manifolds. 220W.Jaco and P. Shalen, Seifert fibre space in 3-manifolds, Memoirs of the Amer. Math. Society 220 (1979).\n\nOn rings of operators, IV. F J Murray, J Von Neumann, Annals of Math. 44F.J.Murray et J.von Neumann, On rings of operators, IV, Annals of Math. 44 (1943), 716-808.\n\nGroup extensions with infinite conjugacy classes. J.-P Préaux, J.-P.Préaux, Group extensions with infinite conjugacy classes, Prépublication 2005.\n\nTopology of three dimensional fibered spaces. H Seifert, Acta Mathematica. 60H.Seifert, Topology of three dimensional fibered spaces, Acta Mathematica 60 (1933), 147-288.\n"},"corpus_id":{"kind":"number","value":118090407,"string":"118,090,407"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"math/0512505\",\n \"mag\": 1655438677,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/math/0512505v1.pdf\"\n ],\n \"pdf_sha\": \"dedbaf4fc14f7ea4beddc941cad9b66815dc4b5a\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":""},"authors":{"kind":"list like","value":["Jean-Philippe Préaux \nGROUPESÀ CLASSES DE CONJUGAISON INFINIES : QUELQUES EXEMPLES"],"string":"[\n \"Jean-Philippe Préaux \\nGROUPESÀ CLASSES DE CONJUGAISON INFINIES : QUELQUES EXEMPLES\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We consider the group property of being with infinite conjugacy classes (or icc, i.e. = 1 and of which all conjugacy classes except 1 are infinite) in some particular cases, and more particulary in case of specific elementary algebraic constructions or of some 3-manifold fundamental groups. Résumé. Nousétudions la propriété pour un groupe d'êtreà classes de conjugaison infinis (ou cci, i.e. = 1 et dont toutes les classes de conjugaison autres que 1 sont infinies) dans certains cas particuliers, et plus particulièrement dans le cas de constructions algébriqueś elémentaires, ou de groupes fondamentaux de certaines 3-variétés."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"La propriété de groupe d'êtreà classes de conjugaison infinies est particulièrement simpleàénoncer (voir la définition section 2). Elle est motivée par la caractérisation de Murray et Von-Neumann des algèbres de Von-Neumann qui sont des facteurs de type II-1 (cf. [ROIV]) : Caractérisation de Murray et Von-Neumann. Soient Γ un groupe et W * λ (Γ) son algèbre de Von-Neumann. Alors W * λ (Γ) est un facteur de type II-1 si et seulement si Γ est cci.\nNousétudions plus particulièrement la stabilité de cette propriété par certaines constructions algébriquesélémentaires, et le cas des groupes fondamentaux de certaines 3variétés. L'étude que nous menons est somme touteélémentaire ; le lecteur intéressé pourra trouver des résultats analogues de plus grande portée dans [Co], [HP] et [Pr].\nCe travail aété effectué durant l'été 2004 alors que l'auteur se trouvaità l'université de Genève et aété partiellement financé par le Fonds National Suisse de la Recherche Scientifique. L'auteur tientà remercier les mathématiciens Genévois pour leur accueil, et plus particulièrement Goulnara Arjantseva et Pierre de la Harpe. L'auteur remercie encore Pierre de la Harpe de lui avoir introduit le problème, ainsi que des nombreuses discussions et des commentaires apportés."},{"title":"Définitions et exemples","content":"Définition 1. Un groupe est dità classes de conjugaisons infinies (ou cci) si il est non trivial, et si toutélément non trivial a une classe de conjugaison infinie.\nIl est commode de caractériser cette propriété en terme de centralisateur plutôt qu'en terme de classe de conjugaison.\nProposition-Définition 1. Un groupe non trivial est cci si et seulement si toutélément non trivial a un centralisateur d'indice infini. Preuve. On note G le groupe, et Z(g) le centralisateur d'unélément g ∈ G. Soient g ∈ G − {1}, et a, b ∈ G. Supposons que aga −1 = bgb −1 :\nAinsi aga −1 = bgb −1 si et seulement si a et b sont dans des right-cosets différents de G/Z(g) et donc, la classe de conjugaison de g, [[g]], est en correspondance bi-univoque avec G/Z(g). Ainsi #[[g]] = #(G/Z(g)), ce qui permet de conclure.\nOn peut déjàétablir quelques exemplesévidents :\nProposition 2.1.\n(1) Les groupes libres de rang ≥ 2 sont cci.\n(2) Les groupes Gromov-hyperboliques nonélémentaires et sans torsion sont cci.\n(3) Les groupes ayant un nombre fini (non nul) d'éléments de torsion (en particulier les groupes finis), ne sont pas cci. (4) Les groupes virtuellement abéliens ne sont pas cci.\n(5) Les groupes ayant un centre non trivial ne sont pas cci.\nPreuve. Dans un groupe libre le centralisateur d'unélément non trivial est cyclique infini. Puisqu'un groupe libre de rang ≥ 2 n'est pas virtuellement Z, cela prouve (1). Dans un groupe hyperbolique le centralisateur d'unélément d'ordre infini est virtuellement cyclique ( [CDP]). Ainsià l'exception des cas où le groupe est fini ou virtuellement cyclique, le centralisateur d'unélément est toujours d'ordre infini, ce qui prouve (2). Pour prouver (3) considérons unélément g de torsion (non trivial). Tous ses conjugués ont même ordre que lui, et sont donc aussi de torsion. Ainsi g n'a qu'un nombre fini de conjugués. Pour prouver (4) appelons A un sous-groupe abélien d'indice fini dans un groupe G, et considérons uń elément non trivial g ∈ A. Alors Z(g) ⊃ A, et donc Z(g) est d'indice fini dans G. Pour prouver (5) il suffit de remarquer que toutélément du centre a pour centralisateur le groupe ambient."},{"title":"Stabilité par construction algébrique","content":"Afin de considérer le cas d'un produit libre, il nous est nécessaire d'établir le lemme suivant : Pour finir, si A et B ne sont pas tous deux d'ordre 2, mais ne contiennent que deś eléments d'ordre 2, alors A * B contient < x, y, z | x 2 , y 2 , (xy) 2 , z 2 >. Alors comme précédemment, a = x(yz)x et b = zx(yz)xz engendrent un groupe libre de rang 2.\nPreuve. Soit u unélément dans un des deux facteurs ; sans perte de généralité on suppose u ∈ A. Avec le théorème de commutativité dans un produit libre (cf.\n[MKS]), le centralisateur de u dans A * B n'est rien d'autre que le centralisateur de u dans A. Puisque A est d'indice infini dans A * B, il en va de même de Z(u). Maintenant si v est unélément qui n'est pas dans un des facteurs, son centralisateur est cyclique infini. Pour conclure il suffit d'utiliser le lemme précédent : puisque A * B contient le groupe libre de rang 2, il ne peut pasêtre virtuellement Z, et donc A * B est cci.\nRemarque : Dans le cas restant, celui du groupe Z 2 * Z 2 , ce dernier contient un groupe cyclique infini d'indice 2 (si l'on note A * B =< a, b | a 2 , b 2 >, considérer < ab >), et n'est donc pas cci.\nRemarque : La réciproque est fausse ; il suffit pour voir cela de considérer le groupe F 2 × Z 2 . Elle devient vraie lorsque l'on suppose que le groupe est sans torsion. Preuve. Supposons que G ne soit pas cci, bien qu'il contienne un sous-groupe d'indice fini cci. Ainsi, il existe u = 1 dans G, tel que Z G (u) soit d'indice fini dans G, et donc Z G (u) ∩ K = K 0 est d'indice fini dans K. Nécessairement u ∈ K, car cela contredirait le fait que K est cci. Mais il est facile de voir qu'il existe n > 1 tel que u n ∈ K. L'ensemble K 0 est inclus dans Z K (u n ), qui est donc nécessairement d'indice fini dans K. Puisque K est cci, on a u n = 1, et donc G a de la torsion.\nProposition 3.4 (Epi-résiduellement cci). Soit G un groupe ; si pour tout g ∈ G− {1} il existe un morphisme surjectif φ sur un groupe cci, tel que φ(g) = 1 (on dira que G est epi-résiduellement H), alors G est cci.\nPreuve. Soit φ : G −→ K unépimorphisme sur un groupe K cci. Remarquons que puisque φ est surjectif, un conjugué de φ(g) dans K est l'image par φ d'un conjugué de g dans G. Ainsi, puisque K est cci, si φ(g) = 1 sa classe de conjugaison dans K est infinie, et donc la classe de conjugaison de g dans G est elle-même infinie.\nOn en déduit directement : On peut vérifie aussi la stabilité de la propriété d'être cci par extension.\nProposition 3.6 (Extension). Soit G l'extension de deux groupes cci K et Q :\nAlors G est aussi cci.\nPreuve. Soit ω = 1 unélément de G. Si ω ∈ K, sa classe de conjugaison dans K est infinie, et il en est donc de même de sa classe de conjugaison dans G. Si ω ∈ K, alors p(ω) a une classe de conjugaison infinie dans Q, et donc puisque p est surjectif, la classe de conjugaison de ω dans G est aussi infinie. Soit u ∈ A * C B ; si u est dans un des facteurs, par exemple A, sa classe de conjugaison dans A * C B contient sa classe de conjugaison dans A, et est donc infinie.\nSupposons sans perte de généralité que C est d'indice > 2 dans A. Considérons uń elément u = 1 quelconque dans A * C B, que l'on peut supposer cycliquemet réduit. Le cas où u est dans un des facteurs ayant déjàété traité, on peut supposer que u s'écrit sous forme cycliquement réduite u = a 1 b 1 . · · · .a n b n , avec n ≥ 1. L'ensemble des right-cosets A/C contient au moins troiséléments distincts 1.C, α.C et α 2 .C. On choisit a ∈ A hors de la classe 1.C et dans une classe différente de celle de a −1 1 . On choisit b ∈ B − C. Avec ce choix, tous leséléments (ba) n .u.(ba) −n de la classe de conjugaison de u sont distincts. Ainsi toutélément non trivial de A * C B a une classe de conjugaison infinie, ce qui conclut la preuve.\nLe même modèle de preuve s'applique pour montrer :\nProposition 3.8 (Extension HNN). Si A est cci, et si C est un sous-groupe d'indice > 2 dans A, l'extension HNN A * C est cci.\nPreuve. Soit A * φ , avec φ : C −→ φ(C). On note t une lettre stable, telle que tct −1 = φ(t) pour tout c ∈ C. Comme précédemment on peut se ramener facilement au cas d'unélément cycliquement réduit de longueur > 1, u = a 1 t n 1 . . . . .α p t np , avec n p = 0. On a au moins 3 right-cosets dans A/C : 1.C, α.C, et α 2 .C. On choisit a ∈ A hors de 1.C et de la classe de a −1 1 . Alors tous leséléments (ta) n u(ta) −n pour n > 0 sont distincts dans la classe de conjugaison de u, ce qui permet de conclure.\nEt on obtient en corollaire immédiat : Proposition 3.9 (Graphes de groupes). Si Γ = π 1 (G, X) est le groupe d'un graphe de groupe dont tous les groupes de sommet sont cci, et tel que pour toute arête e d'origine s 0 et d'extrémité s 1 , G(e) est d'indice > 2 dans G(s 0 ) ou bien dans G(s 1 ), alors Γ est cci."},{"title":"Groupes de 3-variétés","content":"Nous considérons d'abord le cas des groupes fondamentaux de fibrés de Seifert (cf. [Sei]). En utilisant le fait que dans un groupe de Seifert, la classe d'une fibre régulière a un centralisateur d'indice fini (cf. [JS]), onétablit immédiatement :\nProposition 4.1 (Fibrés de Seifert). Les groupes de fibrés de Seifert ne sont pas cci.\nConsidérons maintenant le cas des variétés hyperboliques de volume fini. Il montre une autre application directe de la proposition 3.4.\nProposition 4.2 (Hyperboliques et χ = 0). Les groupes de 3-variétés hyperboliques de volume fini nonélémentaires sont cci.\nPreuve. Considérons une variété M vérifiant les hypothèses : M est une 3-variété hyperboliqueà bord torique ou vide, qui n'est ni un tore solide, ni un toreépaissi. En outre, la classification des isométries hyperboliques montre que π 1 (M ) est sans torsion (cf. [BP]), et donc avec la proposition 3.3, en considérant le revêtement d'orientation de M on supposera M orientable.\nLe théorème de chirurgie hyperbolique de Thurston affirme qu'il existe une suite de variétés hyperboliques fermées (M n ) n , obtenues par obturation de Dehn sur M , convergeant vers M pour la topologie géométrique (cf. [BP]). Ainsi on a une suite d'épimorphismes (φ n : π 1 M −→ π 1 M n ) n . Par définition de la topologie géométrique, si u = 1 ∈ π 1 M , pour n ≫ 0, φ n (u) = 1 ; et tous les groupes π 1 M n sont hyperboliques au sens de Gromov. De plus pour n ≫ 0 ils sont nonélémentaires (ceci car leurs points fixes dans ∂H 3 convergent vers les points fixes de π 1 M et sont donc pour n ≫ 0 en nombre > 2). Ainsi avec la proposition 2.1 (2), π 1 M estépi-résiduellement cci, et la proposition 3.4 permet de conclure.\nNous traitons maintenant le cas des groupes de 3-variétés orientables suffisament large. Rappelons qu'un 3-variété orientable est dite suffisament large si elle contient une surface incompressibleà deux faces ; une 3-variété orientable est Haken si elle est irréductible et suffisament large. Preuve. Nous avons vu qu'un fibré de Seifert a un groupe non cci ; aussi supposons que π 1 (M ) est non cci et montrons que M est un fibré de Seifert. Rappelons que le groupe d'une 3-variété Haken est non trivial (car il contient un groupe de surface infini) ; ainsi, soit u = 1 dans π 1 (M ) ayant un centralisateur Z(u) d'indice fini, et soit p : N −→ M le revêtement fini associéà Z(u). Clairement π 1 (N ) = Z(u) a un centre non trivial. De plus nécessairement N est Haken : le corollaire 13.5 de [He] montre que N est irréductible, et de plus si S est une surface incompressibleà deux faces dans M , p −1 (S) est une surface incompressibleà deux faces dans N . Le corollaire II.5.4 de [JS] montre que N est nécessairement un fibré de Seifert et le théorème II.5.3 ( [JS]) montre qu'il en est alors de même pour M .\nProposition 4.4 (Le cas orientable suffisament large). Soit M une 3-variété orientable et suffisament large ayant un groupe fondamental non diédral infini ; alors soit π 1 (M ) est cci, soit la complétée de Poincaré P(M ) est un fibré de Seifert.\nPreuve. Considérons la décomposition de Kneser-Milnor (cf. [He]) de M : M = M 1 # · · · #M n #B 1 # · · · #B p #C 1 # · · · C q avec n, p, q ∈ N, et où les M i sont non simplement connexes et premiers, les B i sont des boules et les C i des fausses 3-sphères. Puisque M est suffisament large, π 1 (M ) est non trivial (car il contient un groupe de surface infini) et donc n > 0. La complétée de Poincaré P(M ) est alors définie commeétant : P(M ) = M 1 # · · · #M n et de plus π 1 (M ) = π 1 (P(M )) = π 1 (M 1 ) * · · · * π 1 (M n ). Ainsi avec la proposition 4.1 si P(M ) est un fibré de Seifert, alors π 1 (M ) n'est pas cci. Supposons dans la suite que π 1 (M ) n'est pas cci. Alors avec la proposition 3.1 soit π 1 (M ) est diédral infini, soit n = 1. Ainsi sous nos hypothèses P(M ) = M 1 et soit M 1 ≡ S 2 × S 1 soit M 1 est irréductible. Dans le premier cas P(M ) est un fibré de Seifert, et dans le second cas, puisque M contient une surface incompressibleà deux faces, M 1 est Haken, et la proposition 4.3 donne la conclusion.\nDans [HP] ces résultats ontété généralisés au cas de groupes fondamentaux de 3-variétés quelconques."},{"title":"Proposition 3. 3 (","content":""},{"title":"Proposition 3 . 5 (","content":""},{"title":"Proposition 3. 7 (","content":""},{"title":"Proposition 4 . 3 (","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"La propriété de groupe d'êtreà classes de conjugaison infinies est particulièrement simpleàénoncer (voir la définition section 2). Elle est motivée par la caractérisation de Murray et Von-Neumann des algèbres de Von-Neumann qui sont des facteurs de type II-1 (cf. [ROIV]) : Caractérisation de Murray et Von-Neumann. Soient Γ un groupe et W * λ (Γ) son algèbre de Von-Neumann. Alors W * λ (Γ) est un facteur de type II-1 si et seulement si Γ est cci.\\nNousétudions plus particulièrement la stabilité de cette propriété par certaines constructions algébriquesélémentaires, et le cas des groupes fondamentaux de certaines 3variétés. L'étude que nous menons est somme touteélémentaire ; le lecteur intéressé pourra trouver des résultats analogues de plus grande portée dans [Co], [HP] et [Pr].\\nCe travail aété effectué durant l'été 2004 alors que l'auteur se trouvaità l'université de Genève et aété partiellement financé par le Fonds National Suisse de la Recherche Scientifique. L'auteur tientà remercier les mathématiciens Genévois pour leur accueil, et plus particulièrement Goulnara Arjantseva et Pierre de la Harpe. L'auteur remercie encore Pierre de la Harpe de lui avoir introduit le problème, ainsi que des nombreuses discussions et des commentaires apportés.\"\n },\n {\n \"title\": \"Définitions et exemples\",\n \"content\": \"Définition 1. Un groupe est dità classes de conjugaisons infinies (ou cci) si il est non trivial, et si toutélément non trivial a une classe de conjugaison infinie.\\nIl est commode de caractériser cette propriété en terme de centralisateur plutôt qu'en terme de classe de conjugaison.\\nProposition-Définition 1. Un groupe non trivial est cci si et seulement si toutélément non trivial a un centralisateur d'indice infini. Preuve. On note G le groupe, et Z(g) le centralisateur d'unélément g ∈ G. Soient g ∈ G − {1}, et a, b ∈ G. Supposons que aga −1 = bgb −1 :\\nAinsi aga −1 = bgb −1 si et seulement si a et b sont dans des right-cosets différents de G/Z(g) et donc, la classe de conjugaison de g, [[g]], est en correspondance bi-univoque avec G/Z(g). Ainsi #[[g]] = #(G/Z(g)), ce qui permet de conclure.\\nOn peut déjàétablir quelques exemplesévidents :\\nProposition 2.1.\\n(1) Les groupes libres de rang ≥ 2 sont cci.\\n(2) Les groupes Gromov-hyperboliques nonélémentaires et sans torsion sont cci.\\n(3) Les groupes ayant un nombre fini (non nul) d'éléments de torsion (en particulier les groupes finis), ne sont pas cci. (4) Les groupes virtuellement abéliens ne sont pas cci.\\n(5) Les groupes ayant un centre non trivial ne sont pas cci.\\nPreuve. Dans un groupe libre le centralisateur d'unélément non trivial est cyclique infini. Puisqu'un groupe libre de rang ≥ 2 n'est pas virtuellement Z, cela prouve (1). Dans un groupe hyperbolique le centralisateur d'unélément d'ordre infini est virtuellement cyclique ( [CDP]). Ainsià l'exception des cas où le groupe est fini ou virtuellement cyclique, le centralisateur d'unélément est toujours d'ordre infini, ce qui prouve (2). Pour prouver (3) considérons unélément g de torsion (non trivial). Tous ses conjugués ont même ordre que lui, et sont donc aussi de torsion. Ainsi g n'a qu'un nombre fini de conjugués. Pour prouver (4) appelons A un sous-groupe abélien d'indice fini dans un groupe G, et considérons uń elément non trivial g ∈ A. Alors Z(g) ⊃ A, et donc Z(g) est d'indice fini dans G. Pour prouver (5) il suffit de remarquer que toutélément du centre a pour centralisateur le groupe ambient.\"\n },\n {\n \"title\": \"Stabilité par construction algébrique\",\n \"content\": \"Afin de considérer le cas d'un produit libre, il nous est nécessaire d'établir le lemme suivant : Pour finir, si A et B ne sont pas tous deux d'ordre 2, mais ne contiennent que deś eléments d'ordre 2, alors A * B contient < x, y, z | x 2 , y 2 , (xy) 2 , z 2 >. Alors comme précédemment, a = x(yz)x et b = zx(yz)xz engendrent un groupe libre de rang 2.\\nPreuve. Soit u unélément dans un des deux facteurs ; sans perte de généralité on suppose u ∈ A. Avec le théorème de commutativité dans un produit libre (cf.\\n[MKS]), le centralisateur de u dans A * B n'est rien d'autre que le centralisateur de u dans A. Puisque A est d'indice infini dans A * B, il en va de même de Z(u). Maintenant si v est unélément qui n'est pas dans un des facteurs, son centralisateur est cyclique infini. Pour conclure il suffit d'utiliser le lemme précédent : puisque A * B contient le groupe libre de rang 2, il ne peut pasêtre virtuellement Z, et donc A * B est cci.\\nRemarque : Dans le cas restant, celui du groupe Z 2 * Z 2 , ce dernier contient un groupe cyclique infini d'indice 2 (si l'on note A * B =< a, b | a 2 , b 2 >, considérer < ab >), et n'est donc pas cci.\\nRemarque : La réciproque est fausse ; il suffit pour voir cela de considérer le groupe F 2 × Z 2 . Elle devient vraie lorsque l'on suppose que le groupe est sans torsion. Preuve. Supposons que G ne soit pas cci, bien qu'il contienne un sous-groupe d'indice fini cci. Ainsi, il existe u = 1 dans G, tel que Z G (u) soit d'indice fini dans G, et donc Z G (u) ∩ K = K 0 est d'indice fini dans K. Nécessairement u ∈ K, car cela contredirait le fait que K est cci. Mais il est facile de voir qu'il existe n > 1 tel que u n ∈ K. L'ensemble K 0 est inclus dans Z K (u n ), qui est donc nécessairement d'indice fini dans K. Puisque K est cci, on a u n = 1, et donc G a de la torsion.\\nProposition 3.4 (Epi-résiduellement cci). Soit G un groupe ; si pour tout g ∈ G− {1} il existe un morphisme surjectif φ sur un groupe cci, tel que φ(g) = 1 (on dira que G est epi-résiduellement H), alors G est cci.\\nPreuve. Soit φ : G −→ K unépimorphisme sur un groupe K cci. Remarquons que puisque φ est surjectif, un conjugué de φ(g) dans K est l'image par φ d'un conjugué de g dans G. Ainsi, puisque K est cci, si φ(g) = 1 sa classe de conjugaison dans K est infinie, et donc la classe de conjugaison de g dans G est elle-même infinie.\\nOn en déduit directement : On peut vérifie aussi la stabilité de la propriété d'être cci par extension.\\nProposition 3.6 (Extension). Soit G l'extension de deux groupes cci K et Q :\\nAlors G est aussi cci.\\nPreuve. Soit ω = 1 unélément de G. Si ω ∈ K, sa classe de conjugaison dans K est infinie, et il en est donc de même de sa classe de conjugaison dans G. Si ω ∈ K, alors p(ω) a une classe de conjugaison infinie dans Q, et donc puisque p est surjectif, la classe de conjugaison de ω dans G est aussi infinie. Soit u ∈ A * C B ; si u est dans un des facteurs, par exemple A, sa classe de conjugaison dans A * C B contient sa classe de conjugaison dans A, et est donc infinie.\\nSupposons sans perte de généralité que C est d'indice > 2 dans A. Considérons uń elément u = 1 quelconque dans A * C B, que l'on peut supposer cycliquemet réduit. Le cas où u est dans un des facteurs ayant déjàété traité, on peut supposer que u s'écrit sous forme cycliquement réduite u = a 1 b 1 . · · · .a n b n , avec n ≥ 1. L'ensemble des right-cosets A/C contient au moins troiséléments distincts 1.C, α.C et α 2 .C. On choisit a ∈ A hors de la classe 1.C et dans une classe différente de celle de a −1 1 . On choisit b ∈ B − C. Avec ce choix, tous leséléments (ba) n .u.(ba) −n de la classe de conjugaison de u sont distincts. Ainsi toutélément non trivial de A * C B a une classe de conjugaison infinie, ce qui conclut la preuve.\\nLe même modèle de preuve s'applique pour montrer :\\nProposition 3.8 (Extension HNN). Si A est cci, et si C est un sous-groupe d'indice > 2 dans A, l'extension HNN A * C est cci.\\nPreuve. Soit A * φ , avec φ : C −→ φ(C). On note t une lettre stable, telle que tct −1 = φ(t) pour tout c ∈ C. Comme précédemment on peut se ramener facilement au cas d'unélément cycliquement réduit de longueur > 1, u = a 1 t n 1 . . . . .α p t np , avec n p = 0. On a au moins 3 right-cosets dans A/C : 1.C, α.C, et α 2 .C. On choisit a ∈ A hors de 1.C et de la classe de a −1 1 . Alors tous leséléments (ta) n u(ta) −n pour n > 0 sont distincts dans la classe de conjugaison de u, ce qui permet de conclure.\\nEt on obtient en corollaire immédiat : Proposition 3.9 (Graphes de groupes). Si Γ = π 1 (G, X) est le groupe d'un graphe de groupe dont tous les groupes de sommet sont cci, et tel que pour toute arête e d'origine s 0 et d'extrémité s 1 , G(e) est d'indice > 2 dans G(s 0 ) ou bien dans G(s 1 ), alors Γ est cci.\"\n },\n {\n \"title\": \"Groupes de 3-variétés\",\n \"content\": \"Nous considérons d'abord le cas des groupes fondamentaux de fibrés de Seifert (cf. [Sei]). En utilisant le fait que dans un groupe de Seifert, la classe d'une fibre régulière a un centralisateur d'indice fini (cf. [JS]), onétablit immédiatement :\\nProposition 4.1 (Fibrés de Seifert). Les groupes de fibrés de Seifert ne sont pas cci.\\nConsidérons maintenant le cas des variétés hyperboliques de volume fini. Il montre une autre application directe de la proposition 3.4.\\nProposition 4.2 (Hyperboliques et χ = 0). Les groupes de 3-variétés hyperboliques de volume fini nonélémentaires sont cci.\\nPreuve. Considérons une variété M vérifiant les hypothèses : M est une 3-variété hyperboliqueà bord torique ou vide, qui n'est ni un tore solide, ni un toreépaissi. En outre, la classification des isométries hyperboliques montre que π 1 (M ) est sans torsion (cf. [BP]), et donc avec la proposition 3.3, en considérant le revêtement d'orientation de M on supposera M orientable.\\nLe théorème de chirurgie hyperbolique de Thurston affirme qu'il existe une suite de variétés hyperboliques fermées (M n ) n , obtenues par obturation de Dehn sur M , convergeant vers M pour la topologie géométrique (cf. [BP]). Ainsi on a une suite d'épimorphismes (φ n : π 1 M −→ π 1 M n ) n . Par définition de la topologie géométrique, si u = 1 ∈ π 1 M , pour n ≫ 0, φ n (u) = 1 ; et tous les groupes π 1 M n sont hyperboliques au sens de Gromov. De plus pour n ≫ 0 ils sont nonélémentaires (ceci car leurs points fixes dans ∂H 3 convergent vers les points fixes de π 1 M et sont donc pour n ≫ 0 en nombre > 2). Ainsi avec la proposition 2.1 (2), π 1 M estépi-résiduellement cci, et la proposition 3.4 permet de conclure.\\nNous traitons maintenant le cas des groupes de 3-variétés orientables suffisament large. Rappelons qu'un 3-variété orientable est dite suffisament large si elle contient une surface incompressibleà deux faces ; une 3-variété orientable est Haken si elle est irréductible et suffisament large. Preuve. Nous avons vu qu'un fibré de Seifert a un groupe non cci ; aussi supposons que π 1 (M ) est non cci et montrons que M est un fibré de Seifert. Rappelons que le groupe d'une 3-variété Haken est non trivial (car il contient un groupe de surface infini) ; ainsi, soit u = 1 dans π 1 (M ) ayant un centralisateur Z(u) d'indice fini, et soit p : N −→ M le revêtement fini associéà Z(u). Clairement π 1 (N ) = Z(u) a un centre non trivial. De plus nécessairement N est Haken : le corollaire 13.5 de [He] montre que N est irréductible, et de plus si S est une surface incompressibleà deux faces dans M , p −1 (S) est une surface incompressibleà deux faces dans N . Le corollaire II.5.4 de [JS] montre que N est nécessairement un fibré de Seifert et le théorème II.5.3 ( [JS]) montre qu'il en est alors de même pour M .\\nProposition 4.4 (Le cas orientable suffisament large). Soit M une 3-variété orientable et suffisament large ayant un groupe fondamental non diédral infini ; alors soit π 1 (M ) est cci, soit la complétée de Poincaré P(M ) est un fibré de Seifert.\\nPreuve. Considérons la décomposition de Kneser-Milnor (cf. [He]) de M : M = M 1 # · · · #M n #B 1 # · · · #B p #C 1 # · · · C q avec n, p, q ∈ N, et où les M i sont non simplement connexes et premiers, les B i sont des boules et les C i des fausses 3-sphères. Puisque M est suffisament large, π 1 (M ) est non trivial (car il contient un groupe de surface infini) et donc n > 0. La complétée de Poincaré P(M ) est alors définie commeétant : P(M ) = M 1 # · · · #M n et de plus π 1 (M ) = π 1 (P(M )) = π 1 (M 1 ) * · · · * π 1 (M n ). Ainsi avec la proposition 4.1 si P(M ) est un fibré de Seifert, alors π 1 (M ) n'est pas cci. Supposons dans la suite que π 1 (M ) n'est pas cci. Alors avec la proposition 3.1 soit π 1 (M ) est diédral infini, soit n = 1. Ainsi sous nos hypothèses P(M ) = M 1 et soit M 1 ≡ S 2 × S 1 soit M 1 est irréductible. Dans le premier cas P(M ) est un fibré de Seifert, et dans le second cas, puisque M contient une surface incompressibleà deux faces, M 1 est Haken, et la proposition 4.3 donne la conclusion.\\nDans [HP] ces résultats ontété généralisés au cas de groupes fondamentaux de 3-variétés quelconques.\"\n },\n {\n \"title\": \"Proposition 3. 3 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3 . 5 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3. 7 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 4 . 3 (\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Alors avec le théorème d'écriture sous forme réduite, un mot réduit non trivial sur a et b est non trivial dans A * B, et donc a et b engendrent un groupe libre de rang 2.","content":"Alors avec le théorème d'écriture sous forme réduite, un mot réduit non trivial sur a et b est non trivial dans A * B, et donc a et b engendrent un groupe libre de rang 2."},{"id":"","caption":"Extension finie). Si G est sans torsion, et si K est un sous-groupe d'indice fini de G cci, alors G est aussi cci.","content":"Proposition 3. 3 (\n3Extension finie). Si G est sans torsion, et si K est un sous-groupe d'indice fini de G cci, alors G est aussi cci."},{"id":"","caption":"Produit direct). Soient A et B deux groupes non triviaux. Le produit direct A × B est cci si et seulement si A et B sont cci.Preuve. Si A ou B est non cci, trivialement A × B n'est pas cci ; aussi supposons dans la suite que A et B sont tous deux cci et montrons que A × B est cci. Soit u = ab, avec a ∈ A et b ∈ B. Si u = 1, alors sans perte de généralité on peut supposer a = 1. En considérant la projection canonique A × B −→ A, la proposition 3.4 permet de conclure.","content":"Proposition 3 . 5 (\n35Produit direct). Soient A et B deux groupes non triviaux. Le produit direct A × B est cci si et seulement si A et B sont cci.Preuve. Si A ou B est non cci, trivialement A × B n'est pas cci ; aussi supposons dans la suite que A et B sont tous deux cci et montrons que A × B est cci. Soit u = ab, avec a ∈ A et b ∈ B. Si u = 1, alors sans perte de généralité on peut supposer a = 1. En considérant la projection canonique A × B −→ A, la proposition 3.4 permet de conclure."},{"id":"","caption":"Amalgame). Si A et B sont deux groupes cci, et si C = A ∩ B n'est pas d'indice 1 ou 2 dans chaque facteur, alors le produit amalgamé A * C B est cci.Preuve. On peut supposer que A, B = {1}, car autrement le groupe obtenu n'est autre que A ou B. De la même façon on peut supposer que C est d'indice > 1 dans A et dans B.","content":"Proposition 3. 7 (\n7Amalgame). Si A et B sont deux groupes cci, et si C = A ∩ B n'est pas d'indice 1 ou 2 dans chaque facteur, alors le produit amalgamé A * C B est cci.Preuve. On peut supposer que A, B = {1}, car autrement le groupe obtenu n'est autre que A ou B. De la même façon on peut supposer que C est d'indice > 1 dans A et dans B."},{"id":"","caption":"Le cas Haken orientable). Soit M un 3-variété Haken orientable ; alors soit π 1 (M ) est cci soit M est un fibré de Seifert.","content":"Proposition 4 . 3 (\n43Le cas Haken orientable). Soit M un 3-variété Haken orientable ; alors soit π 1 (M ) est cci soit M est un fibré de Seifert."},{"id":"","caption":"Centre de recherche de l'Ecole de l'air, Ecole de l'air, F-13661 Salon de Provence air Centre de Mathématiques et d'informatique, Université de Provence, 39 rue F.Joliot-Curie, F-13453 marseille cedex 13 E-mail : preaux@cmi.univ-mrs.fr Mathematical subject classification : 20E45 (Primary), 20E06, 20E08, 20E22, 57M05 (Secondary)","content":"Centre de recherche de l'Ecole de l'air, Ecole de l'air, F-13661 Salon de Provence air Centre de Mathématiques et d'informatique, Université de Provence, 39 rue F.Joliot-Curie, F-13453 marseille cedex 13 E-mail : preaux@cmi.univ-mrs.fr Mathematical subject classification : 20E45 (Primary), 20E06, 20E08, 20E22, 57M05 (Secondary)1 \n\n2"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Alors avec le théorème d'écriture sous forme réduite, un mot réduit non trivial sur a et b est non trivial dans A * B, et donc a et b engendrent un groupe libre de rang 2.\",\n \"content\": \"Alors avec le théorème d'écriture sous forme réduite, un mot réduit non trivial sur a et b est non trivial dans A * B, et donc a et b engendrent un groupe libre de rang 2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Extension finie). Si G est sans torsion, et si K est un sous-groupe d'indice fini de G cci, alors G est aussi cci.\",\n \"content\": \"Proposition 3. 3 (\\n3Extension finie). Si G est sans torsion, et si K est un sous-groupe d'indice fini de G cci, alors G est aussi cci.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Produit direct). Soient A et B deux groupes non triviaux. Le produit direct A × B est cci si et seulement si A et B sont cci.Preuve. Si A ou B est non cci, trivialement A × B n'est pas cci ; aussi supposons dans la suite que A et B sont tous deux cci et montrons que A × B est cci. Soit u = ab, avec a ∈ A et b ∈ B. Si u = 1, alors sans perte de généralité on peut supposer a = 1. En considérant la projection canonique A × B −→ A, la proposition 3.4 permet de conclure.\",\n \"content\": \"Proposition 3 . 5 (\\n35Produit direct). Soient A et B deux groupes non triviaux. Le produit direct A × B est cci si et seulement si A et B sont cci.Preuve. Si A ou B est non cci, trivialement A × B n'est pas cci ; aussi supposons dans la suite que A et B sont tous deux cci et montrons que A × B est cci. Soit u = ab, avec a ∈ A et b ∈ B. Si u = 1, alors sans perte de généralité on peut supposer a = 1. En considérant la projection canonique A × B −→ A, la proposition 3.4 permet de conclure.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Amalgame). Si A et B sont deux groupes cci, et si C = A ∩ B n'est pas d'indice 1 ou 2 dans chaque facteur, alors le produit amalgamé A * C B est cci.Preuve. On peut supposer que A, B = {1}, car autrement le groupe obtenu n'est autre que A ou B. De la même façon on peut supposer que C est d'indice > 1 dans A et dans B.\",\n \"content\": \"Proposition 3. 7 (\\n7Amalgame). Si A et B sont deux groupes cci, et si C = A ∩ B n'est pas d'indice 1 ou 2 dans chaque facteur, alors le produit amalgamé A * C B est cci.Preuve. On peut supposer que A, B = {1}, car autrement le groupe obtenu n'est autre que A ou B. De la même façon on peut supposer que C est d'indice > 1 dans A et dans B.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Le cas Haken orientable). Soit M un 3-variété Haken orientable ; alors soit π 1 (M ) est cci soit M est un fibré de Seifert.\",\n \"content\": \"Proposition 4 . 3 (\\n43Le cas Haken orientable). Soit M un 3-variété Haken orientable ; alors soit π 1 (M ) est cci soit M est un fibré de Seifert.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Centre de recherche de l'Ecole de l'air, Ecole de l'air, F-13661 Salon de Provence air Centre de Mathématiques et d'informatique, Université de Provence, 39 rue F.Joliot-Curie, F-13453 marseille cedex 13 E-mail : preaux@cmi.univ-mrs.fr Mathematical subject classification : 20E45 (Primary), 20E06, 20E08, 20E22, 57M05 (Secondary)\",\n \"content\": \"Centre de recherche de l'Ecole de l'air, Ecole de l'air, F-13661 Salon de Provence air Centre de Mathématiques et d'informatique, Université de Provence, 39 rue F.Joliot-Curie, F-13453 marseille cedex 13 E-mail : preaux@cmi.univ-mrs.fr Mathematical subject classification : 20E45 (Primary), 20E06, 20E08, 20E22, 57M05 (Secondary)1 \\n\\n2\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"R Benedetti, C Petronio, Lectures on hyperbolic geometry. SpringerUniversitextR.Benedetti, and C.Petronio, Lectures on hyperbolic geometry, Universitext, Springer, 1992.","doi":""},{"text":"Infinite conjugacy classes in groups acting on trees. Y De Cornulier, Y.de Cornulier, Infinite conjugacy classes in groups acting on trees, Prépublication 2005.","doi":""},{"text":"M Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes, les groupes hyperboliques de Gromov. Springer-Verlag1441M.Coornaert, T.Delzant, and A.Papadopoulos, Géométrie et théorie des groupes, les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441, Springer-Verlag, 1991.","doi":""},{"text":"P De La Harpe, J.-P Préaux, arXiv:math.GR/0509449Groupes fondamentaux des variétés de dimension 3 et algèbres d'opérateurs. 1P.de la Harpe et J.-P.Préaux, Groupes fondamentaux des variétés de dimension 3 et algèbres d'opérateurs, Prépublication arXiv:math.GR/0509449 v1 (2005).","doi":"arXiv:math.GR/0509449"},{"text":"J Hempel, Annals of mathematics studies. Priceton university press3J.Hempel, 3-manifolds, Annals of mathematics studies, Priceton university press, 1976.","doi":""},{"text":"W Jaco, P Shalen, Seifert fibre space in 3-manifolds. 220W.Jaco and P. Shalen, Seifert fibre space in 3-manifolds, Memoirs of the Amer. Math. Society 220 (1979).","doi":""},{"text":"On rings of operators, IV. F J Murray, J Von Neumann, Annals of Math. 44F.J.Murray et J.von Neumann, On rings of operators, IV, Annals of Math. 44 (1943), 716-808.","doi":""},{"text":"Group extensions with infinite conjugacy classes. J.-P Préaux, J.-P.Préaux, Group extensions with infinite conjugacy classes, Prépublication 2005.","doi":""},{"text":"Topology of three dimensional fibered spaces. H Seifert, Acta Mathematica. 60H.Seifert, Topology of three dimensional fibered spaces, Acta Mathematica 60 (1933), 147-288.","doi":""}],"string":"[\n {\n \"text\": \"R Benedetti, C Petronio, Lectures on hyperbolic geometry. SpringerUniversitextR.Benedetti, and C.Petronio, Lectures on hyperbolic geometry, Universitext, Springer, 1992.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Infinite conjugacy classes in groups acting on trees. Y De Cornulier, Y.de Cornulier, Infinite conjugacy classes in groups acting on trees, Prépublication 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes, les groupes hyperboliques de Gromov. Springer-Verlag1441M.Coornaert, T.Delzant, and A.Papadopoulos, Géométrie et théorie des groupes, les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441, Springer-Verlag, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P De La Harpe, J.-P Préaux, arXiv:math.GR/0509449Groupes fondamentaux des variétés de dimension 3 et algèbres d'opérateurs. 1P.de la Harpe et J.-P.Préaux, Groupes fondamentaux des variétés de dimension 3 et algèbres d'opérateurs, Prépublication arXiv:math.GR/0509449 v1 (2005).\",\n \"doi\": \"arXiv:math.GR/0509449\"\n },\n {\n \"text\": \"J Hempel, Annals of mathematics studies. Priceton university press3J.Hempel, 3-manifolds, Annals of mathematics studies, Priceton university press, 1976.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W Jaco, P Shalen, Seifert fibre space in 3-manifolds. 220W.Jaco and P. Shalen, Seifert fibre space in 3-manifolds, Memoirs of the Amer. Math. Society 220 (1979).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On rings of operators, IV. F J Murray, J Von Neumann, Annals of Math. 44F.J.Murray et J.von Neumann, On rings of operators, IV, Annals of Math. 44 (1943), 716-808.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Group extensions with infinite conjugacy classes. J.-P Préaux, J.-P.Préaux, Group extensions with infinite conjugacy classes, Prépublication 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Topology of three dimensional fibered spaces. H Seifert, Acta Mathematica. 60H.Seifert, Topology of three dimensional fibered spaces, Acta Mathematica 60 (1933), 147-288.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":395,"cells":{"text":{"kind":"string","value":"\nNATURAL CONNECTIONS ON CONFORMAL RIEMANNIAN P -MANIFOLDS\n14 Sep 2011\n\nDobrinka Gribacheva \nDimitar Mekerov \nNATURAL CONNECTIONS ON CONFORMAL RIEMANNIAN P -MANIFOLDS\n14 Sep 2011Riemannian almost product manifoldRiemannian metricalmost product structurelinear connectionparallel torsion 2010 Mathematics Subject Classification: 53C1553C2553B05\nThe class W 1 of conformal Riemannian P -manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kähler manifolds in almost Hermitian geometry. In the present work 1 we study the natural connections on the manifolds (M, P, g) from the class W 1 , i.e. the linear connections preserving the almost product structure P and the Riemannian metric g. We find necessary and sufficient conditions the curvature tensor of such a connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We determine the type of the manifolds admitting a natural connection with a parallel torsion.\n\nIntroduction\n\nK. Yano initiated in [1] the study of Riemannian almost product manifolds. A Riemannian almost product manifold (M, P, g) is a differentiable manifold M with an almost product structure P and a Riemannian metric g such that P 2 x = x and g(P x, P y) = g(x, y) for any tangent vectors x and y. A. M. Naveira gave in [2] a classification of these manifolds with respect to the covariant derivative ∇P , where ∇ is the Levi-Civita connection of g. This classification is very similar to the Gray-Hervella classification in [3] of almost Hermitian manifolds. In the paper [4] M. Staikova and K. Gribachev obtained a classification of the Riemannian almost product manifolds, for which trP = 0. In this case the manifold is even-dimensional. The class W 0 from the Staikova-Gribachev classification is determined by the condition ∇P = 0. A manifold from this class is called a Riemannian P -manifold. The class W 0 is an analogue of the class of Kähler manifolds in almost Hermitian geometry.\n\nThe geometry of a Riemannian almost product manifold (M, P, g) is a geometry of both structures g and P . There are important in this geometry the linear connections in respect of which the parallel transport determine an isomorphism of the tangent spaces with the structures g and P . This is valid if and only if these structures are parallel with respect to such a connection. In the general case on a Riemannian almost product manifold there are countless number linear connections regarding which g and P are parallel. Such connections are called natural in [5].\n\nIn the present paper we consider some problems of the geometry of the natural connections on the class W 1 from the Staikova-Gribachev classification. This is the class of conformal Riemannian P -manifolds or shortly W 1 -manifolds. The class W 1 is an analogue of the class of conformal Kähler manifolds in almost Hermitian geometry.\n\nThe paper is organized as follows. In Sec. 2 we give necessary facts about Riemannian almost product manifolds, the class W 1 and the natural connections on W 1 . We recall the notion a Riemannian P -tensor on a Riemannian almost product manifold, which is an analogue of the notion of a Kähler tensor in Hermitian geometry. In Sec. 3 we obtain relations between the curvature tensors R and R ′ , the Ricci tensors ρ and ρ ′ and the scalar curvatures τ and τ ′ of the Levi-Civita connection ∇ and a natural connection ∇ ′ on a W 1 -manifold. In Sec. 4 we find some necessary and sufficient conditions the curvature tensor R ′ of ∇ ′ to be a Riemannian P -tensor. The most important result in this section is the classification Theorem 4.3. In it there are separated two special natural connections D and D, whose average connection is the canonical connection, introduced in [5]. In Sec. 5 we find conditions for a natural connection with parallel torsion. The important result here is Theorem 5.3. It characterizes a W 1 -manifold with a natural connection, which has a parallel torsion and a Riemannian P -tensor of curvature.\n\n\nPreliminaries\n\nLet (M, P, g) be a Riemannian almost product manifold, i.e. a differentiable manifold M with a tensor field P of type (1, 1) and a Riemannian metric g such that P 2 x = x, g(P x, P y) = g(x, y) for any x, y of the algebra X(M ) of the smooth vector fields on M . Further x, y, z, w will stand for arbitrary elements of X(M ) or vectors in the tangent space T p M at p ∈ M . The associated metric g of g is determined by g(x, y) = g(x, P y).\n\nIn this work we consider manifolds (M, P, g) with trP = 0. In this case M is an even-dimensional manifold. We assume that dim M = 2n.\n\nIn [2] A.M. Naveira gives a classification of Riemannian almost product manifolds with respect to the tensor F of type (0,3), defined by F (x, y, z) = g ((∇ x P ) y, z) , where ∇ is the Levi-Civita connection of g. The tensor F has the properties:\n\nF (x, y, z) = F (x, z, y) = −F (x, P y, P z), F (x, y, P z) = −F (x, P y, z).\n\nUsing the Naveira classification, in [4] M. Staikova and K. Gribachev give a classification of Riemannian almost product manifolds (M, P, g) with trP = 0. The basic classes of this classification are W 1 , W 2 and W 3 . Their intersection is the class W 0 of the Riemannian P -manifolds, determined by the condition F = 0 (or equivalently ∇P = 0) [6]. This class is an analogue of the class of Kähler manifolds in the geometry of almost Hermitian manifolds.\n\nA Riemannian almost product manifold (M, P, g) is a Riemannian product manifold if it has a local product structure, i.e. the structure P is integrable. The Riemannian product manifolds form the class W 1 ⊕ W 2 from the classification in [4]. This class is an analogue of the class of Hermitian manifolds in almost Hermitian geometry.\n\nThe class W 1 from the Staikova-Gribachev classification contains the manifolds which are locally conformal equivalent to Riemannian P -manifolds. This class plays a similar role of the role of the class of the conformal Kähler manifolds in almost Hermitian geometry. We will say that a manifold from the class W 1 is a W 1 -manifold.\n\nThe characteristic condition for the class W 1 is the following\n(2.1) F (x, y, z) = 1 2n g(x, y)θ(z) − g(x, P y)θ(P z) + g(x, z)θ(y) − g(x, P z)θ(P y) ,\nwhere the associated 1-form θ is determined by θ(x) = g ij F (e i , e j , x). Here g ij will stand for the components of the inverse matrix of g with respect to a basis\n{e i } of T p M at p ∈ M . The 1-form θ is closed, i.e. dθ = 0, if and only if (∇ x θ) y = (∇ y θ) x. Moreover, θ • P is a closed 1-form if and only if (∇ x θ) P y = (∇ y θ) P x.\nIn [4] it is proved that W 1 = W 3 ⊕ W 6 , where W 3 and W 6 are the classes from the Naveira classification determined by the following conditions:\nW 3 : F (A, B, ξ) = 1 n g(A, B)θ v (ξ), F (ξ, η, A) = 0, W 6 : F (ξ, η, A) = 1 n g(ξ, η)θ h (A), F (A, B, ξ) = 0, where A, B, ξ, η ∈ X(M ), P A = A, P B = B, P ξ = −ξ, P η = −η, θ v (x) = 1 2 (θ(x) − θ(P x)), θ h (x) = 1 2 (θ(x) + θ(P x)).\nIn the case when trP = 0, the above conditions for W 3 and W 6 can be written for any x, y, z in the following form:\n(2.2) W 3 : F (x, y, z) = 1 2n [g(x, y) + g(x, P y)] θ(z) + [g(x, z) + g(x, P z)] θ(y) , θ(P x) = −θ(x), (2.3) W 6 : F (x, y, z) = 1 2n [g(x, y) − g(x, P y)] θ(z) + [g(x, z) − g(x, P z)] θ(y) , θ(P x) = θ(x).\nIn [4], a tensor L of type (0,4) with properties\n(2.4) L(x, y, z, w) = −L(y, x, z, w) = −L(x, y, w, z), (2.5) L(x, y, z, w) + L(y, z, x, w) + L(z, x, y, w) = 0\nis called a curvature-like tensor. Such a tensor on a Riemannian almost product manifold (M, P, g) with the property (2.6) L(x, y, P z, P w) = L(x, y, z, w)\n\nis called a Riemannian P -tensor in [7]. This notion is an analogue of the notion of a Kähler tensor in Hermitian geometry. Let S be a (0,2)-tensor on a Riemannian almost product manifold. In [4] it is proved that the tensor ψ 1 (S)(x, y, z, w) = g(y, z)S(x, w)−g(x, z)S(y, w)+ S(y, z)g(x, w) − S(x, z)g(y, w) is curvature-like if and only if S(x, y) = S(y, x), and the tensor ψ 2 (S)(x, y, z, w) = ψ 1 (S)(x, y, P z, P w) is curvaturelike if and only if S(x, P y) = S(y, P x). Obviously ψ 2 (S)(x, y, P z, P w) = ψ 1 (S)(x, y, z, w). The tensors π 1 = 1 2 ψ 1 (g), π 2 = 1 2 ψ 2 (g), π 3 = ψ 1 ( g) = ψ 2 ( g) are curvature-like, and the tensors π 1 + π 2 , π 3 are Riemannian P -tensors.\n\nThe linear connections in our investigations have a torsion. Let ∇ ′ be a linear connection with a tensor Q of the transformation ∇ → ∇ ′ and a torsion T , i.e.\n∇ ′ x y = ∇ x y + Q(x, y), T (x, y) = ∇ ′ x y − ∇ ′ y x − [x, y]. The corresponding (0,3)-tensors are defined by Q(x, y, z) = g(Q(x, y), z), T (x, y, z) = g(T (x, y), z). The symmetry of ∇ implies T (x, y) = −T (y, x) = Q(x, y) − Q(y, x).\nA linear connection ∇ ′ on a Riemannian almost product manifold (M, P, g) is called a natural connection if ∇ ′ P = ∇ ′ g = 0. If ∇ ′ is a linear connection on (M, P, g), then it is a natural connection if and only if [5]: y] z and the corresponding tensor of type (0,4) is defined as follows R(x, y, z, w) = g(R(x, y)z, w). We denote the Ricci tensor and the scalar curvature for ∇ by ρ and τ , respectively, i.e. ρ(y, z) = g ij R(e i , y, z, e j ) and τ = g ij ρ(e i , e j ). Analogously there are defined the curvature tensor R ′ the Ricci tensor ρ ′ and the scalar curvature τ ′ for any connection ∇ ′ .\n(2.7) F (x, y, z) = Q(x, y, P z) − Q(x, P y, z), Q(x, y, z) = −Q(x, z, y). The curvature tensor R of ∇ is determined by R(x, y)z = ∇ x ∇ y z − ∇ y ∇ x z − ∇ [x,\nFurther ∇ ′ will stand for a natural connection on a Riemannian almost product manifold (M, P, g). Then it is valid the identity [8] R(x, y, z, w) = R ′ (x, y, z, w) − Q(T (x, y), z, w) − ∇ ′\n\nx Q (y, z, w) + (∇ y Q) (x, z, w) + g Q(x, z), Q(y, w) − g Q(y, z), Q(x, w) . (2.8)\n\nIn [9] it is established that the natural connections ∇ ′ on a W 1 -manifold (M, P, g) form a 2-parametric family, where the torsion T of ∇ ′ on (M, P, g) is determined by (2.9) T (x, y, z) = 1 2n {g(y, z)θ(P x) − g(x, z)θ(P y)} +λ {g(y, z)θ(x) − g(x, z)θ(y) + g(y, P z)θ(P x) − g(x, P z)θ(P y)} +µ {g(y, P z)θ(x) − g(x, P z)θ(y) + g(y, z)θ(P x) − g(x, z)θ(P y)} .\n\n\nCurvature properties of natural connections on W 1 -manifolds\n\nAccording to [10], for the torsion T of ∇ ′ and the tensor Q of the transformation ∇ → ∇ ′ it is valid 2Q(x, y, z) = T (x, y, z) − T (y, z, x) + T (z, x, y).\n\nThen from (2.9) for a W 1 -manifold (M, P, g) holds (3.1)\n\nQ(x, y, z) = T (z, y, x).\n\nHaving in mind (3.1), the equality (2.9) implies (3.2) Q(y, z, w) = g(y, z) λθ(w) + µ + 1 2n θ(P w) − g(y, w) λθ(z) + µ + 1 2n θ(P z) +g(y, P z) {λθ(P w) + µθ(w)} − g(y, P w) {λθ(P z) + µθ(z)} ,\n(3.3)\nQ(y, z) = g(y, P z) {λP Ω + µΩ} y − λθ(z) + µ + 1 2n θ(P z) + g(y, z) λΩ + µ + 1 2n P Ω − {λθ(P z) + µθ(z)} P y, where Ω is determined by g(Ω, x) = θ(x). According to (3.3), the connection ∇ ′ , determined by the pair of parameters (λ, µ) has the form\n(3.4) ∇ ′ x y = ∇ x y + g(x, P y) {λP Ω + µΩ} − {λθ(P y) + µθ(y)} P x +g(x, y) λΩ + µ + 1 2n P Ω − λθ(y) + µ + 1 2n θ(P y) x. We denote V (x, w) = λ (∇ ′ x θ) P w + µ (∇ ′ x θ) w, U (x, w) = λ (∇ ′ x θ) w + µ + 1 2n (∇ ′ x θ) P w. Since ∇ ′ P = ∇ ′ g = ∇ ′ g = 0, from (3.2) it follows (3.5) (∇ ′ x Q) (y, z, w) = g(y, z)U (x, w) − g(y, w)U (x, z) + g(y, P z)V (x, w) − g(y, P w)V (x, z).\nFrom (2.7) and (3.1) it follows Q(T (x, y), z, w) = −g(T (x, y), T (z, w)) which together with (3.3) and (3.5) we apply in (2.8). By appropriate calculations and using the notations (3.6) p = λΩ + µ + 1 2n P Ω, q = λP Ω + µΩ, (3.7) S ′ (y, z) = U (y, z) − 1 2n {λθ(y)θ(P z) + µθ(y)θ(z)} , (3.8) S ′′ (y, z) = V (y, P z) + 1 2n {λθ(P y)θ(P z) + µθ(P y)θ(z)} , we obtain the following Theorem 3.1. The following relation is valid\n(3.9) R = R ′ − g(p, p)π 1 − g(q, q)π 2 − g(p, q)π 3 − ψ 1 (S ′ ) − ψ 2 (S ′′ ).\nBy the traces of the tensors in (3.9) we obtain the following −tr Sg(y, P z) − 2(n − 1)S ′ (y, z) + S ′′ (y, z) + S ′′ (P y, P z),\nτ = τ ′ − 2n(2n − 1)g(p, p) + 2ng(q, q) − 2n(2n − 1)trS ′ + 2trS ′′ ,\nwhere S(y, z) = S(y, P z).\n\n\nNatural connections on W 1 -manifolds with Riemannian P -tensor of curvature\n\nBecause of ∇ ′ g = ∇ ′ P = 0, the properties (2.4) and (2.6) are valid for R ′ . Therefore R ′ is a Riemannian P -tensor if and only if R ′ satisfies (2.5). This property is valid for the curvature-like tensors R, π 1 , π 2 and π 3 . Then, according to (3.9), R ′ is a Riemannian P -tensor if and only if ψ 1 (S ′ ) and ψ 2 (S ′′ ) are curvature-like, i.e. S ′ (y, z) = S ′ (z, y) and S ′′ (y, P z) = S ′′ (z, P y). In other words, using (3.7) and (3.8), it is valid the following (3.3), there are valid the equalities \n(4.1) U (y, z) − U (z, y) + λ 2n {θ(P y)θ(z) − θ(y)θ(P z)} = 0, V (y, z) − V (z, y) + µ 2n {θ(P y)θ(z) − θ(y)θ(P z)} = 0. According to(4.2) ∇ ′ y θ z − (∇ ′ z θ) y = (∇ y θ) z − (∇ z θ) y − 1 2n {θ(P y)θ(z) − θ(y)θ(P z)} , ∇ ′ y θ P z − (∇ ′ z θ) P y = (∇ y θ) P z − (∇ z θ(4.3) λ {(∇ y θ) z − (∇ z θ) y} + µ + 1 2n {(∇ y θ) P z − (∇ z θ) P y} = 0, µ {(∇ y θ) z − (∇ z θ) y} + λ {(∇ y θ) P z − (∇ z θ) P y} = 0.\nThe conditions (4.3) form a homogeneous linear system for x 1 = (∇ y θ) z− (∇ z θ) y, x 2 = (∇ y θ) P z − (∇ z θ) P y with a determinant ∆ = λ 2 − µ 2 − µ 2n . Case I: ∆ = 0. Then (4.3) has a nonzero solution (x 1 , x 2 ) and for λ and µ are possible the following three subcases: a) λ = µ = 0; b) λ = 0, µ = − 1 2n ; c) λ = 0, λ 2 − µ 2 − µ 2n = 0. Let us consider the subcase a). If R ′ is a Riemannian P -tensor then the first equation of (4.3) implies that θ • P is closed. Since (4.3) has a nonzero solution then θ is not closed. Vice versa, if θ • P is closed, then, according to [8], R ′ is a Riemannian P -tensor.\n\nLet us consider the subcase b). If R ′ is a Riemannian P -tensor then the second equation of (4.3) implies that θ is a closed. Since (4.3) has a nonzero solution then θ • P is not closed. Vice versa, if θ is closed, then the first equation of (4.2) implies ∇ ′ y θ z − (∇ ′ z θ) y = 1 2n {θ(y)θ(P z) − θ(P y)θ(z)} and since λ = 0 the second equality of (4.3) holds. Obviously, the first equation of (4.3) is valid, too. Then, according to Proposition 4.1, R ′ is a Riemannian P -tensor.\n\nLet us consider the subcase c). Thus we have also µ = 0. If R ′ is a Riemannian P -tensor, the second equation of (4.3) yields that θ and θ • P are not closed, since the system (4.3) has a nonzero solution. In this subcase W 1 -manifold (M, P, g) is not belong to the classes W 3 and W 6 . Indeed, in the opposite case we have θ(P z) = εθ(z) (ε = ±1) and therefore the following equality is valid\n(∇ y θ) P z − (∇ z θ) P y = ε {(∇ y θ) z − (∇ z θ) y} − F (y, z, Ω) + F (z, y, Ω).\nSince (2.2) and (2.3), it follows F (y, z, Ω) = 1 2n {g(y, z) − εg(y, P z)} θ(Ω). Using the last two equalities, the system (4.3) gets the form\n(4.4) λ + ε µ + 1 2n {(∇ y θ) z − (∇ z θ) y} = 0, µ + ελ {(∇ y θ) z − (∇ z θ) y} = 0.\nBecause of λ = 0, λ 2 − µ 2 − µ 2n = 0 in the considered subcase, we have λ + ε µ + 1 2n = 0, µ + ελ = 0. Then (4.4) implies that θ is closed, which is a contradiction. Therefore (M, P, g) / ∈ W 3 and (M, P, g) / ∈ W 6 . Case II: ∆ = 0. If R ′ is a Riemannian P -tensor then the conditions (4.3) are satisfied. Since the homogeneous system has the zero solution x 1 = x 2 = 0 in this case then θ and θ • P are closed. Vice versa, if θ and θ • P are closed, then (4.2) implies the equalities ∇ ′ y θ P z − (∇ ′ z θ) P y = 0, ∇ ′ y θ z − (∇ ′ z θ) y = 1 2n {θ(y)θ(P z) − θ(P y)θ(z)} and thus conditions (4.1) hold. Then, according to Proposition 4.1, R ′ is a Riemannian P -tensor.\n\nThe obtained results we summarize in the following classification theorem for the natural connections on a W 1 -manifold. i) If ∇ ′ is the connection D determined by λ = µ = 0, then R ′ is a Riemannian P -tensor if and only if the 1-form θ is not closed and the 1-form θ • P is closed; ii) If ∇ ′ is the connection D determined by λ = 0, µ = − 1 2n , then R ′ is a Riemannian P -tensor if and only if the 1-form θ is closed and the 1-form θ • P is not closed; iii) If ∇ ′ is a connection for which λ 2 − µ 2 − µ 2n = 0, then R ′ is a Riemannian P -tensor if and only if the 1-forms θ and θ • P are closed; iv) If ∇ ′ is a connection for which λ = 0, λ 2 − µ 2 − µ 2n = 0 and R ′ is a Riemannian P -tensor then the 1-forms θ and θ • P are not closed. In this case (M, P, g) / ∈ W 3 , (M, P, g) / ∈ W 6 .\n\nLet us remark that the proposition i) of Theorem 4.3 is proved in [8], where the connection D is investigated.\n\nIn [5], it is introduced a natural connection ∇ c on an arbitrary Riemannian almost product manifold (M, P, g). This connection, called canonical, is an analogue of the Hermitian connection in Hermitian geometry. Analogues of this connection are considered on manifolds, corresponding to the class W 1 , in the geometry of almost complex manifolds with Norden metric ( [11]) and almost contact manifolds with B-metric ( [12], [13]). In [9], it is note that ∇ c on a Riemannian almost product W 1 -manifold is determined by (3.4) for λ = 0, µ = − 1 4n . Therefore, for ∇ c it is valid the condition λ 2 − µ 2 − µ 2n = 0 and then the statement iii) of Theorem 4.3 holds.\n\nHaving in mind the values of λ and µ determining the connections ∇ c , D and D, we obtain the following Calculating from (2.9) the covariant derivative ∇ ′ T and applying ∇ ′ g = ∇ ′ g = 0, we obtain the following Let ∇ ′ be a natural connection with parallel torsion and Riemannian Ptensor of curvature. Then from Proposition 5.1 we have U = V = 0. In this case Proposition 4.1 implies λW = µW = 0, where W (y, z) = θ(P y)θ(z) − θ(y)θ(P z). If W = 0 then we obtain θ(P y) = ±θ(y), i.e. (M, P, g) ∈ W 3 or (M, P, g) ∈ W 6 . Thus, we get the following Theorem 5.3. Let a W 1 -manifold (M, P, g) admits a natural connection ∇ ′ with parallel torsion and Riemannian P -tensor of curvature. Then the all possible cases are as follows:\n\ni) λ = µ = 0, W = 0, i.e. ∇ ′ = D, (M, P, g) / ∈ W 3 , (M, P, g) / ∈ W 6 ; ii) λ = µ = 0, W = 0, i.e. ∇ ′ = D, (M, P, g) ∈ W 3 or (M, P, g) ∈ W 6 ; iii) λ = 0, µ = 0, W = 0, i.e. ∇ ′ belongs to the 1-parametric family determined by (3.4) for λ = 0, (M, P, g) ∈ W 3 , (M, P, g) ∈ W 6 ; iv) λ = 0, µ = 0, W = 0, i.e. ∇ ′ belongs to the 1-parametric family determined by (3.4) for µ = 0, (M, P, g) ∈ W 3 , (M, P, g) ∈ W 6 .\n\nCorollary 3. 2 .\n2The following relations are valid ρ(y, z) = ρ ′ (y, z) − {(2n − 1)g(p, p) + trS ′ − g(q, q)} g(y, z)\n\nProposition 4 . 1 .\n41The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:\n\nProposition 4 . 2 .\n42The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:\n\nTheorem 4 . 3 .\n43Let R ′ be the curvature tensor of a natural connection ∇ ′ determined by (3.4) on a W 1 -manifold (M, P, g). Then the all possible cases are as follows:\n\nProposition 4 . 4 .\n44The canonical connection ∇ c is the average connection of D and D, i.e. ∇ c = 1 2 (D + D). 5. Natural connection on a W 1 -manifold with parallel torsion\n\n\nProposition 5.1. A natural connection ∇ ′ has a parallel torsion if and only if the 1-form θ is parallel with respect to ∇ ′ . By (4.2), (2.2) and (2.3), from Proposition 5.1 we obtain the following Corollary 5.2. If a W 1 -manifold (M, P, g) admits a natural connection with parallel torsion then the 1-form θ • P is closed. If (M, P, g) belongs to the class W 3 or the class W 6 then the 1-form θ is closed, too.\nPartially supported by project NI11-FMI-004 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria\n\n. K Yano, Pure and Applied Math. 49Pergamon Press BookYano K. Pure and Applied Math., 49, New York, Pergamon Press Book, 1965.\n\n. A M Naveira, Rend, Math, 3Naveira A. M. Rend. Math., 3, 1983, 577-592.\n\n. A Gray, L Hervella, Ann. Mat. Pura Appl. 123Gray A., L. Hervella. Ann. Mat. Pura Appl., 123, 1980, 35-58.\n\n. M Staikova, K , Gribachev. Serdica Math. P. 18Staikova M., K. Gribachev. Serdica Math. P., 18, 1992, 150-161.\n\n. V Mihova, Serdica Math. P. 15Mihova V. Serdica Math. P., 15, 1989, 351-358.\n\n. M Staikova, Plovdiv Univ. Sci. Works -Math. 323Staikova M. Plovdiv Univ. Sci. Works -Math., 32, 1987, No 3, 241-251.\n\n. D J Mekerov, Geom, 89Mekerov D. J. Geom., 89, 2008, No 1-2, 119-129.\n\n. D Gribacheva, arXiv:1103.2916Gribacheva D. arXiv:1103.2916.\n\n. D Gribacheva, Compt. rend. Acad. bulg. Sci. 646Gribacheva D. Compt. rend. Acad. bulg. Sci., 64, 2011, No 6, 799-806.\n\n. H Hayden, Proc. London. Math. Soc. 34Hayden H., Proc. London. Math. Soc., 34, 1934, 27-50.\n\n. G Ganchev, K Gribachev, V Mihova, Publ. Inst. Math. 42Ganchev G., K. Gribachev, V. Mihova. Publ. Inst. Math. (Beograd) (N.S.), 42, 1987, 107-121.\n\n. M Manev, K , Gribachev. Serdica Math. J. 20Manev M., K. Gribachev. Serdica Math. J., 20, 1994, 133-147.\n\nBulgaria Blvd, mircho@uni-plovdiv.bg4003 Plovdiv, Bulgaria e-mails: dobrinka@uni-plovdiv.bg. Bulgaria Blvd., 4003 Plovdiv, Bulgaria e-mails: dobrinka@uni-plovdiv.bg, mircho@uni-plovdiv.bg\n"},"corpus_id":{"kind":"number","value":119167696,"string":"119,167,696"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1109.2704\",\n \"mag\": 2150615818,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1109.2704v2.pdf\"\n ],\n \"pdf_sha\": \"9a31ca0422355fcf71216c0a7483156f1bdf6aa4\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"NATURAL CONNECTIONS ON CONFORMAL RIEMANNIAN P -MANIFOLDS"},"authors":{"kind":"list like","value":["Dobrinka Gribacheva","Dimitar Mekerov"],"string":"[\n \"Dobrinka Gribacheva\",\n \"Dimitar Mekerov\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The class W 1 of conformal Riemannian P -manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kähler manifolds in almost Hermitian geometry. In the present work 1 we study the natural connections on the manifolds (M, P, g) from the class W 1 , i.e. the linear connections preserving the almost product structure P and the Riemannian metric g. We find necessary and sufficient conditions the curvature tensor of such a connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We determine the type of the manifolds admitting a natural connection with a parallel torsion."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"K. Yano initiated in [1] the study of Riemannian almost product manifolds. A Riemannian almost product manifold (M, P, g) is a differentiable manifold M with an almost product structure P and a Riemannian metric g such that P 2 x = x and g(P x, P y) = g(x, y) for any tangent vectors x and y. A. M. Naveira gave in [2] a classification of these manifolds with respect to the covariant derivative ∇P , where ∇ is the Levi-Civita connection of g. This classification is very similar to the Gray-Hervella classification in [3] of almost Hermitian manifolds. In the paper [4] M. Staikova and K. Gribachev obtained a classification of the Riemannian almost product manifolds, for which trP = 0. In this case the manifold is even-dimensional. The class W 0 from the Staikova-Gribachev classification is determined by the condition ∇P = 0. A manifold from this class is called a Riemannian P -manifold. The class W 0 is an analogue of the class of Kähler manifolds in almost Hermitian geometry.\nThe geometry of a Riemannian almost product manifold (M, P, g) is a geometry of both structures g and P . There are important in this geometry the linear connections in respect of which the parallel transport determine an isomorphism of the tangent spaces with the structures g and P . This is valid if and only if these structures are parallel with respect to such a connection. In the general case on a Riemannian almost product manifold there are countless number linear connections regarding which g and P are parallel. Such connections are called natural in [5].\nIn the present paper we consider some problems of the geometry of the natural connections on the class W 1 from the Staikova-Gribachev classification. This is the class of conformal Riemannian P -manifolds or shortly W 1 -manifolds. The class W 1 is an analogue of the class of conformal Kähler manifolds in almost Hermitian geometry.\nThe paper is organized as follows. In Sec. 2 we give necessary facts about Riemannian almost product manifolds, the class W 1 and the natural connections on W 1 . We recall the notion a Riemannian P -tensor on a Riemannian almost product manifold, which is an analogue of the notion of a Kähler tensor in Hermitian geometry. In Sec. 3 we obtain relations between the curvature tensors R and R ′ , the Ricci tensors ρ and ρ ′ and the scalar curvatures τ and τ ′ of the Levi-Civita connection ∇ and a natural connection ∇ ′ on a W 1 -manifold. In Sec. 4 we find some necessary and sufficient conditions the curvature tensor R ′ of ∇ ′ to be a Riemannian P -tensor. The most important result in this section is the classification Theorem 4.3. In it there are separated two special natural connections D and D, whose average connection is the canonical connection, introduced in [5]. In Sec. 5 we find conditions for a natural connection with parallel torsion. The important result here is Theorem 5.3. It characterizes a W 1 -manifold with a natural connection, which has a parallel torsion and a Riemannian P -tensor of curvature."},{"title":"Preliminaries","content":"Let (M, P, g) be a Riemannian almost product manifold, i.e. a differentiable manifold M with a tensor field P of type (1, 1) and a Riemannian metric g such that P 2 x = x, g(P x, P y) = g(x, y) for any x, y of the algebra X(M ) of the smooth vector fields on M . Further x, y, z, w will stand for arbitrary elements of X(M ) or vectors in the tangent space T p M at p ∈ M . The associated metric g of g is determined by g(x, y) = g(x, P y).\nIn this work we consider manifolds (M, P, g) with trP = 0. In this case M is an even-dimensional manifold. We assume that dim M = 2n.\nIn [2] A.M. Naveira gives a classification of Riemannian almost product manifolds with respect to the tensor F of type (0,3), defined by F (x, y, z) = g ((∇ x P ) y, z) , where ∇ is the Levi-Civita connection of g. The tensor F has the properties:\nF (x, y, z) = F (x, z, y) = −F (x, P y, P z), F (x, y, P z) = −F (x, P y, z).\nUsing the Naveira classification, in [4] M. Staikova and K. Gribachev give a classification of Riemannian almost product manifolds (M, P, g) with trP = 0. The basic classes of this classification are W 1 , W 2 and W 3 . Their intersection is the class W 0 of the Riemannian P -manifolds, determined by the condition F = 0 (or equivalently ∇P = 0) [6]. This class is an analogue of the class of Kähler manifolds in the geometry of almost Hermitian manifolds.\nA Riemannian almost product manifold (M, P, g) is a Riemannian product manifold if it has a local product structure, i.e. the structure P is integrable. The Riemannian product manifolds form the class W 1 ⊕ W 2 from the classification in [4]. This class is an analogue of the class of Hermitian manifolds in almost Hermitian geometry.\nThe class W 1 from the Staikova-Gribachev classification contains the manifolds which are locally conformal equivalent to Riemannian P -manifolds. This class plays a similar role of the role of the class of the conformal Kähler manifolds in almost Hermitian geometry. We will say that a manifold from the class W 1 is a W 1 -manifold.\nThe characteristic condition for the class W 1 is the following\nwhere the associated 1-form θ is determined by θ(x) = g ij F (e i , e j , x). Here g ij will stand for the components of the inverse matrix of g with respect to a basis\nIn [4] it is proved that W 1 = W 3 ⊕ W 6 , where W 3 and W 6 are the classes from the Naveira classification determined by the following conditions:\nIn the case when trP = 0, the above conditions for W 3 and W 6 can be written for any x, y, z in the following form:\nIn [4], a tensor L of type (0,4) with properties\nis called a curvature-like tensor. Such a tensor on a Riemannian almost product manifold (M, P, g) with the property (2.6) L(x, y, P z, P w) = L(x, y, z, w)\nis called a Riemannian P -tensor in [7]. This notion is an analogue of the notion of a Kähler tensor in Hermitian geometry. Let S be a (0,2)-tensor on a Riemannian almost product manifold. In [4] it is proved that the tensor ψ 1 (S)(x, y, z, w) = g(y, z)S(x, w)−g(x, z)S(y, w)+ S(y, z)g(x, w) − S(x, z)g(y, w) is curvature-like if and only if S(x, y) = S(y, x), and the tensor ψ 2 (S)(x, y, z, w) = ψ 1 (S)(x, y, P z, P w) is curvaturelike if and only if S(x, P y) = S(y, P x). Obviously ψ 2 (S)(x, y, P z, P w) = ψ 1 (S)(x, y, z, w). The tensors π 1 = 1 2 ψ 1 (g), π 2 = 1 2 ψ 2 (g), π 3 = ψ 1 ( g) = ψ 2 ( g) are curvature-like, and the tensors π 1 + π 2 , π 3 are Riemannian P -tensors.\nThe linear connections in our investigations have a torsion. Let ∇ ′ be a linear connection with a tensor Q of the transformation ∇ → ∇ ′ and a torsion T , i.e.\nA linear connection ∇ ′ on a Riemannian almost product manifold (M, P, g) is called a natural connection if ∇ ′ P = ∇ ′ g = 0. If ∇ ′ is a linear connection on (M, P, g), then it is a natural connection if and only if [5]: y] z and the corresponding tensor of type (0,4) is defined as follows R(x, y, z, w) = g(R(x, y)z, w). We denote the Ricci tensor and the scalar curvature for ∇ by ρ and τ , respectively, i.e. ρ(y, z) = g ij R(e i , y, z, e j ) and τ = g ij ρ(e i , e j ). Analogously there are defined the curvature tensor R ′ the Ricci tensor ρ ′ and the scalar curvature τ ′ for any connection ∇ ′ .\nFurther ∇ ′ will stand for a natural connection on a Riemannian almost product manifold (M, P, g). Then it is valid the identity [8] R(x, y, z, w) = R ′ (x, y, z, w) − Q(T (x, y), z, w) − ∇ ′\nx Q (y, z, w) + (∇ y Q) (x, z, w) + g Q(x, z), Q(y, w) − g Q(y, z), Q(x, w) . (2.8)\nIn [9] it is established that the natural connections ∇ ′ on a W 1 -manifold (M, P, g) form a 2-parametric family, where the torsion T of ∇ ′ on (M, P, g) is determined by (2.9) T (x, y, z) = 1 2n {g(y, z)θ(P x) − g(x, z)θ(P y)} +λ {g(y, z)θ(x) − g(x, z)θ(y) + g(y, P z)θ(P x) − g(x, P z)θ(P y)} +µ {g(y, P z)θ(x) − g(x, P z)θ(y) + g(y, z)θ(P x) − g(x, z)θ(P y)} ."},{"title":"Curvature properties of natural connections on W 1 -manifolds","content":"According to [10], for the torsion T of ∇ ′ and the tensor Q of the transformation ∇ → ∇ ′ it is valid 2Q(x, y, z) = T (x, y, z) − T (y, z, x) + T (z, x, y).\nThen from (2.9) for a W 1 -manifold (M, P, g) holds (3.1)\nQ(x, y, z) = T (z, y, x).\nHaving in mind (3.1), the equality (2.9) implies (3.2) Q(y, z, w) = g(y, z) λθ(w) + µ + 1 2n θ(P w) − g(y, w) λθ(z) + µ + 1 2n θ(P z) +g(y, P z) {λθ(P w) + µθ(w)} − g(y, P w) {λθ(P z) + µθ(z)} ,\nQ(y, z) = g(y, P z) {λP Ω + µΩ} y − λθ(z) + µ + 1 2n θ(P z) + g(y, z) λΩ + µ + 1 2n P Ω − {λθ(P z) + µθ(z)} P y, where Ω is determined by g(Ω, x) = θ(x). According to (3.3), the connection ∇ ′ , determined by the pair of parameters (λ, µ) has the form\nFrom (2.7) and (3.1) it follows Q(T (x, y), z, w) = −g(T (x, y), T (z, w)) which together with (3.3) and (3.5) we apply in (2.8). By appropriate calculations and using the notations (3.6) p = λΩ + µ + 1 2n P Ω, q = λP Ω + µΩ, (3.7) S ′ (y, z) = U (y, z) − 1 2n {λθ(y)θ(P z) + µθ(y)θ(z)} , (3.8) S ′′ (y, z) = V (y, P z) + 1 2n {λθ(P y)θ(P z) + µθ(P y)θ(z)} , we obtain the following Theorem 3.1. The following relation is valid\nBy the traces of the tensors in (3.9) we obtain the following −tr Sg(y, P z) − 2(n − 1)S ′ (y, z) + S ′′ (y, z) + S ′′ (P y, P z),\nwhere S(y, z) = S(y, P z)."},{"title":"Natural connections on W 1 -manifolds with Riemannian P -tensor of curvature","content":"Because of ∇ ′ g = ∇ ′ P = 0, the properties (2.4) and (2.6) are valid for R ′ . Therefore R ′ is a Riemannian P -tensor if and only if R ′ satisfies (2.5). This property is valid for the curvature-like tensors R, π 1 , π 2 and π 3 . Then, according to (3.9), R ′ is a Riemannian P -tensor if and only if ψ 1 (S ′ ) and ψ 2 (S ′′ ) are curvature-like, i.e. S ′ (y, z) = S ′ (z, y) and S ′′ (y, P z) = S ′′ (z, P y). In other words, using (3.7) and (3.8), it is valid the following (3.3), there are valid the equalities\nThe conditions (4.3) form a homogeneous linear system for x 1 = (∇ y θ) z− (∇ z θ) y, x 2 = (∇ y θ) P z − (∇ z θ) P y with a determinant ∆ = λ 2 − µ 2 − µ 2n . Case I: ∆ = 0. Then (4.3) has a nonzero solution (x 1 , x 2 ) and for λ and µ are possible the following three subcases: a) λ = µ = 0; b) λ = 0, µ = − 1 2n ; c) λ = 0, λ 2 − µ 2 − µ 2n = 0. Let us consider the subcase a). If R ′ is a Riemannian P -tensor then the first equation of (4.3) implies that θ • P is closed. Since (4.3) has a nonzero solution then θ is not closed. Vice versa, if θ • P is closed, then, according to [8], R ′ is a Riemannian P -tensor.\nLet us consider the subcase b). If R ′ is a Riemannian P -tensor then the second equation of (4.3) implies that θ is a closed. Since (4.3) has a nonzero solution then θ • P is not closed. Vice versa, if θ is closed, then the first equation of (4.2) implies ∇ ′ y θ z − (∇ ′ z θ) y = 1 2n {θ(y)θ(P z) − θ(P y)θ(z)} and since λ = 0 the second equality of (4.3) holds. Obviously, the first equation of (4.3) is valid, too. Then, according to Proposition 4.1, R ′ is a Riemannian P -tensor.\nLet us consider the subcase c). Thus we have also µ = 0. If R ′ is a Riemannian P -tensor, the second equation of (4.3) yields that θ and θ • P are not closed, since the system (4.3) has a nonzero solution. In this subcase W 1 -manifold (M, P, g) is not belong to the classes W 3 and W 6 . Indeed, in the opposite case we have θ(P z) = εθ(z) (ε = ±1) and therefore the following equality is valid\nSince (2.2) and (2.3), it follows F (y, z, Ω) = 1 2n {g(y, z) − εg(y, P z)} θ(Ω). Using the last two equalities, the system (4.3) gets the form\nBecause of λ = 0, λ 2 − µ 2 − µ 2n = 0 in the considered subcase, we have λ + ε µ + 1 2n = 0, µ + ελ = 0. Then (4.4) implies that θ is closed, which is a contradiction. Therefore (M, P, g) / ∈ W 3 and (M, P, g) / ∈ W 6 . Case II: ∆ = 0. If R ′ is a Riemannian P -tensor then the conditions (4.3) are satisfied. Since the homogeneous system has the zero solution x 1 = x 2 = 0 in this case then θ and θ • P are closed. Vice versa, if θ and θ • P are closed, then (4.2) implies the equalities ∇ ′ y θ P z − (∇ ′ z θ) P y = 0, ∇ ′ y θ z − (∇ ′ z θ) y = 1 2n {θ(y)θ(P z) − θ(P y)θ(z)} and thus conditions (4.1) hold. Then, according to Proposition 4.1, R ′ is a Riemannian P -tensor.\nThe obtained results we summarize in the following classification theorem for the natural connections on a W 1 -manifold. i) If ∇ ′ is the connection D determined by λ = µ = 0, then R ′ is a Riemannian P -tensor if and only if the 1-form θ is not closed and the 1-form θ • P is closed; ii) If ∇ ′ is the connection D determined by λ = 0, µ = − 1 2n , then R ′ is a Riemannian P -tensor if and only if the 1-form θ is closed and the 1-form θ • P is not closed; iii) If ∇ ′ is a connection for which λ 2 − µ 2 − µ 2n = 0, then R ′ is a Riemannian P -tensor if and only if the 1-forms θ and θ • P are closed; iv) If ∇ ′ is a connection for which λ = 0, λ 2 − µ 2 − µ 2n = 0 and R ′ is a Riemannian P -tensor then the 1-forms θ and θ • P are not closed. In this case (M, P, g) / ∈ W 3 , (M, P, g) / ∈ W 6 .\nLet us remark that the proposition i) of Theorem 4.3 is proved in [8], where the connection D is investigated.\nIn [5], it is introduced a natural connection ∇ c on an arbitrary Riemannian almost product manifold (M, P, g). This connection, called canonical, is an analogue of the Hermitian connection in Hermitian geometry. Analogues of this connection are considered on manifolds, corresponding to the class W 1 , in the geometry of almost complex manifolds with Norden metric ( [11]) and almost contact manifolds with B-metric ( [12], [13]). In [9], it is note that ∇ c on a Riemannian almost product W 1 -manifold is determined by (3.4) for λ = 0, µ = − 1 4n . Therefore, for ∇ c it is valid the condition λ 2 − µ 2 − µ 2n = 0 and then the statement iii) of Theorem 4.3 holds.\nHaving in mind the values of λ and µ determining the connections ∇ c , D and D, we obtain the following Calculating from (2.9) the covariant derivative ∇ ′ T and applying ∇ ′ g = ∇ ′ g = 0, we obtain the following Let ∇ ′ be a natural connection with parallel torsion and Riemannian Ptensor of curvature. Then from Proposition 5.1 we have U = V = 0. In this case Proposition 4.1 implies λW = µW = 0, where W (y, z) = θ(P y)θ(z) − θ(y)θ(P z). If W = 0 then we obtain θ(P y) = ±θ(y), i.e. (M, P, g) ∈ W 3 or (M, P, g) ∈ W 6 . Thus, we get the following Theorem 5.3. Let a W 1 -manifold (M, P, g) admits a natural connection ∇ ′ with parallel torsion and Riemannian P -tensor of curvature. Then the all possible cases are as follows:\ni) λ = µ = 0, W = 0, i.e. ∇ ′ = D, (M, P, g) / ∈ W 3 , (M, P, g) / ∈ W 6 ; ii) λ = µ = 0, W = 0, i.e. ∇ ′ = D, (M, P, g) ∈ W 3 or (M, P, g) ∈ W 6 ; iii) λ = 0, µ = 0, W = 0, i.e. ∇ ′ belongs to the 1-parametric family determined by (3.4) for λ = 0, (M, P, g) ∈ W 3 , (M, P, g) ∈ W 6 ; iv) λ = 0, µ = 0, W = 0, i.e. ∇ ′ belongs to the 1-parametric family determined by (3.4) for µ = 0, (M, P, g) ∈ W 3 , (M, P, g) ∈ W 6 ."},{"title":"Corollary 3. 2 .","content":""},{"title":"Proposition 4 . 1 .","content":""},{"title":"Proposition 4 . 2 .","content":""},{"title":"Theorem 4 . 3 .","content":""},{"title":"Proposition 4 . 4 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"K. Yano initiated in [1] the study of Riemannian almost product manifolds. A Riemannian almost product manifold (M, P, g) is a differentiable manifold M with an almost product structure P and a Riemannian metric g such that P 2 x = x and g(P x, P y) = g(x, y) for any tangent vectors x and y. A. M. Naveira gave in [2] a classification of these manifolds with respect to the covariant derivative ∇P , where ∇ is the Levi-Civita connection of g. This classification is very similar to the Gray-Hervella classification in [3] of almost Hermitian manifolds. In the paper [4] M. Staikova and K. Gribachev obtained a classification of the Riemannian almost product manifolds, for which trP = 0. In this case the manifold is even-dimensional. The class W 0 from the Staikova-Gribachev classification is determined by the condition ∇P = 0. A manifold from this class is called a Riemannian P -manifold. The class W 0 is an analogue of the class of Kähler manifolds in almost Hermitian geometry.\\nThe geometry of a Riemannian almost product manifold (M, P, g) is a geometry of both structures g and P . There are important in this geometry the linear connections in respect of which the parallel transport determine an isomorphism of the tangent spaces with the structures g and P . This is valid if and only if these structures are parallel with respect to such a connection. In the general case on a Riemannian almost product manifold there are countless number linear connections regarding which g and P are parallel. Such connections are called natural in [5].\\nIn the present paper we consider some problems of the geometry of the natural connections on the class W 1 from the Staikova-Gribachev classification. This is the class of conformal Riemannian P -manifolds or shortly W 1 -manifolds. The class W 1 is an analogue of the class of conformal Kähler manifolds in almost Hermitian geometry.\\nThe paper is organized as follows. In Sec. 2 we give necessary facts about Riemannian almost product manifolds, the class W 1 and the natural connections on W 1 . We recall the notion a Riemannian P -tensor on a Riemannian almost product manifold, which is an analogue of the notion of a Kähler tensor in Hermitian geometry. In Sec. 3 we obtain relations between the curvature tensors R and R ′ , the Ricci tensors ρ and ρ ′ and the scalar curvatures τ and τ ′ of the Levi-Civita connection ∇ and a natural connection ∇ ′ on a W 1 -manifold. In Sec. 4 we find some necessary and sufficient conditions the curvature tensor R ′ of ∇ ′ to be a Riemannian P -tensor. The most important result in this section is the classification Theorem 4.3. In it there are separated two special natural connections D and D, whose average connection is the canonical connection, introduced in [5]. In Sec. 5 we find conditions for a natural connection with parallel torsion. The important result here is Theorem 5.3. It characterizes a W 1 -manifold with a natural connection, which has a parallel torsion and a Riemannian P -tensor of curvature.\"\n },\n {\n \"title\": \"Preliminaries\",\n \"content\": \"Let (M, P, g) be a Riemannian almost product manifold, i.e. a differentiable manifold M with a tensor field P of type (1, 1) and a Riemannian metric g such that P 2 x = x, g(P x, P y) = g(x, y) for any x, y of the algebra X(M ) of the smooth vector fields on M . Further x, y, z, w will stand for arbitrary elements of X(M ) or vectors in the tangent space T p M at p ∈ M . The associated metric g of g is determined by g(x, y) = g(x, P y).\\nIn this work we consider manifolds (M, P, g) with trP = 0. In this case M is an even-dimensional manifold. We assume that dim M = 2n.\\nIn [2] A.M. Naveira gives a classification of Riemannian almost product manifolds with respect to the tensor F of type (0,3), defined by F (x, y, z) = g ((∇ x P ) y, z) , where ∇ is the Levi-Civita connection of g. The tensor F has the properties:\\nF (x, y, z) = F (x, z, y) = −F (x, P y, P z), F (x, y, P z) = −F (x, P y, z).\\nUsing the Naveira classification, in [4] M. Staikova and K. Gribachev give a classification of Riemannian almost product manifolds (M, P, g) with trP = 0. The basic classes of this classification are W 1 , W 2 and W 3 . Their intersection is the class W 0 of the Riemannian P -manifolds, determined by the condition F = 0 (or equivalently ∇P = 0) [6]. This class is an analogue of the class of Kähler manifolds in the geometry of almost Hermitian manifolds.\\nA Riemannian almost product manifold (M, P, g) is a Riemannian product manifold if it has a local product structure, i.e. the structure P is integrable. The Riemannian product manifolds form the class W 1 ⊕ W 2 from the classification in [4]. This class is an analogue of the class of Hermitian manifolds in almost Hermitian geometry.\\nThe class W 1 from the Staikova-Gribachev classification contains the manifolds which are locally conformal equivalent to Riemannian P -manifolds. This class plays a similar role of the role of the class of the conformal Kähler manifolds in almost Hermitian geometry. We will say that a manifold from the class W 1 is a W 1 -manifold.\\nThe characteristic condition for the class W 1 is the following\\nwhere the associated 1-form θ is determined by θ(x) = g ij F (e i , e j , x). Here g ij will stand for the components of the inverse matrix of g with respect to a basis\\nIn [4] it is proved that W 1 = W 3 ⊕ W 6 , where W 3 and W 6 are the classes from the Naveira classification determined by the following conditions:\\nIn the case when trP = 0, the above conditions for W 3 and W 6 can be written for any x, y, z in the following form:\\nIn [4], a tensor L of type (0,4) with properties\\nis called a curvature-like tensor. Such a tensor on a Riemannian almost product manifold (M, P, g) with the property (2.6) L(x, y, P z, P w) = L(x, y, z, w)\\nis called a Riemannian P -tensor in [7]. This notion is an analogue of the notion of a Kähler tensor in Hermitian geometry. Let S be a (0,2)-tensor on a Riemannian almost product manifold. In [4] it is proved that the tensor ψ 1 (S)(x, y, z, w) = g(y, z)S(x, w)−g(x, z)S(y, w)+ S(y, z)g(x, w) − S(x, z)g(y, w) is curvature-like if and only if S(x, y) = S(y, x), and the tensor ψ 2 (S)(x, y, z, w) = ψ 1 (S)(x, y, P z, P w) is curvaturelike if and only if S(x, P y) = S(y, P x). Obviously ψ 2 (S)(x, y, P z, P w) = ψ 1 (S)(x, y, z, w). The tensors π 1 = 1 2 ψ 1 (g), π 2 = 1 2 ψ 2 (g), π 3 = ψ 1 ( g) = ψ 2 ( g) are curvature-like, and the tensors π 1 + π 2 , π 3 are Riemannian P -tensors.\\nThe linear connections in our investigations have a torsion. Let ∇ ′ be a linear connection with a tensor Q of the transformation ∇ → ∇ ′ and a torsion T , i.e.\\nA linear connection ∇ ′ on a Riemannian almost product manifold (M, P, g) is called a natural connection if ∇ ′ P = ∇ ′ g = 0. If ∇ ′ is a linear connection on (M, P, g), then it is a natural connection if and only if [5]: y] z and the corresponding tensor of type (0,4) is defined as follows R(x, y, z, w) = g(R(x, y)z, w). We denote the Ricci tensor and the scalar curvature for ∇ by ρ and τ , respectively, i.e. ρ(y, z) = g ij R(e i , y, z, e j ) and τ = g ij ρ(e i , e j ). Analogously there are defined the curvature tensor R ′ the Ricci tensor ρ ′ and the scalar curvature τ ′ for any connection ∇ ′ .\\nFurther ∇ ′ will stand for a natural connection on a Riemannian almost product manifold (M, P, g). Then it is valid the identity [8] R(x, y, z, w) = R ′ (x, y, z, w) − Q(T (x, y), z, w) − ∇ ′\\nx Q (y, z, w) + (∇ y Q) (x, z, w) + g Q(x, z), Q(y, w) − g Q(y, z), Q(x, w) . (2.8)\\nIn [9] it is established that the natural connections ∇ ′ on a W 1 -manifold (M, P, g) form a 2-parametric family, where the torsion T of ∇ ′ on (M, P, g) is determined by (2.9) T (x, y, z) = 1 2n {g(y, z)θ(P x) − g(x, z)θ(P y)} +λ {g(y, z)θ(x) − g(x, z)θ(y) + g(y, P z)θ(P x) − g(x, P z)θ(P y)} +µ {g(y, P z)θ(x) − g(x, P z)θ(y) + g(y, z)θ(P x) − g(x, z)θ(P y)} .\"\n },\n {\n \"title\": \"Curvature properties of natural connections on W 1 -manifolds\",\n \"content\": \"According to [10], for the torsion T of ∇ ′ and the tensor Q of the transformation ∇ → ∇ ′ it is valid 2Q(x, y, z) = T (x, y, z) − T (y, z, x) + T (z, x, y).\\nThen from (2.9) for a W 1 -manifold (M, P, g) holds (3.1)\\nQ(x, y, z) = T (z, y, x).\\nHaving in mind (3.1), the equality (2.9) implies (3.2) Q(y, z, w) = g(y, z) λθ(w) + µ + 1 2n θ(P w) − g(y, w) λθ(z) + µ + 1 2n θ(P z) +g(y, P z) {λθ(P w) + µθ(w)} − g(y, P w) {λθ(P z) + µθ(z)} ,\\nQ(y, z) = g(y, P z) {λP Ω + µΩ} y − λθ(z) + µ + 1 2n θ(P z) + g(y, z) λΩ + µ + 1 2n P Ω − {λθ(P z) + µθ(z)} P y, where Ω is determined by g(Ω, x) = θ(x). According to (3.3), the connection ∇ ′ , determined by the pair of parameters (λ, µ) has the form\\nFrom (2.7) and (3.1) it follows Q(T (x, y), z, w) = −g(T (x, y), T (z, w)) which together with (3.3) and (3.5) we apply in (2.8). By appropriate calculations and using the notations (3.6) p = λΩ + µ + 1 2n P Ω, q = λP Ω + µΩ, (3.7) S ′ (y, z) = U (y, z) − 1 2n {λθ(y)θ(P z) + µθ(y)θ(z)} , (3.8) S ′′ (y, z) = V (y, P z) + 1 2n {λθ(P y)θ(P z) + µθ(P y)θ(z)} , we obtain the following Theorem 3.1. The following relation is valid\\nBy the traces of the tensors in (3.9) we obtain the following −tr Sg(y, P z) − 2(n − 1)S ′ (y, z) + S ′′ (y, z) + S ′′ (P y, P z),\\nwhere S(y, z) = S(y, P z).\"\n },\n {\n \"title\": \"Natural connections on W 1 -manifolds with Riemannian P -tensor of curvature\",\n \"content\": \"Because of ∇ ′ g = ∇ ′ P = 0, the properties (2.4) and (2.6) are valid for R ′ . Therefore R ′ is a Riemannian P -tensor if and only if R ′ satisfies (2.5). This property is valid for the curvature-like tensors R, π 1 , π 2 and π 3 . Then, according to (3.9), R ′ is a Riemannian P -tensor if and only if ψ 1 (S ′ ) and ψ 2 (S ′′ ) are curvature-like, i.e. S ′ (y, z) = S ′ (z, y) and S ′′ (y, P z) = S ′′ (z, P y). In other words, using (3.7) and (3.8), it is valid the following (3.3), there are valid the equalities\\nThe conditions (4.3) form a homogeneous linear system for x 1 = (∇ y θ) z− (∇ z θ) y, x 2 = (∇ y θ) P z − (∇ z θ) P y with a determinant ∆ = λ 2 − µ 2 − µ 2n . Case I: ∆ = 0. Then (4.3) has a nonzero solution (x 1 , x 2 ) and for λ and µ are possible the following three subcases: a) λ = µ = 0; b) λ = 0, µ = − 1 2n ; c) λ = 0, λ 2 − µ 2 − µ 2n = 0. Let us consider the subcase a). If R ′ is a Riemannian P -tensor then the first equation of (4.3) implies that θ • P is closed. Since (4.3) has a nonzero solution then θ is not closed. Vice versa, if θ • P is closed, then, according to [8], R ′ is a Riemannian P -tensor.\\nLet us consider the subcase b). If R ′ is a Riemannian P -tensor then the second equation of (4.3) implies that θ is a closed. Since (4.3) has a nonzero solution then θ • P is not closed. Vice versa, if θ is closed, then the first equation of (4.2) implies ∇ ′ y θ z − (∇ ′ z θ) y = 1 2n {θ(y)θ(P z) − θ(P y)θ(z)} and since λ = 0 the second equality of (4.3) holds. Obviously, the first equation of (4.3) is valid, too. Then, according to Proposition 4.1, R ′ is a Riemannian P -tensor.\\nLet us consider the subcase c). Thus we have also µ = 0. If R ′ is a Riemannian P -tensor, the second equation of (4.3) yields that θ and θ • P are not closed, since the system (4.3) has a nonzero solution. In this subcase W 1 -manifold (M, P, g) is not belong to the classes W 3 and W 6 . Indeed, in the opposite case we have θ(P z) = εθ(z) (ε = ±1) and therefore the following equality is valid\\nSince (2.2) and (2.3), it follows F (y, z, Ω) = 1 2n {g(y, z) − εg(y, P z)} θ(Ω). Using the last two equalities, the system (4.3) gets the form\\nBecause of λ = 0, λ 2 − µ 2 − µ 2n = 0 in the considered subcase, we have λ + ε µ + 1 2n = 0, µ + ελ = 0. Then (4.4) implies that θ is closed, which is a contradiction. Therefore (M, P, g) / ∈ W 3 and (M, P, g) / ∈ W 6 . Case II: ∆ = 0. If R ′ is a Riemannian P -tensor then the conditions (4.3) are satisfied. Since the homogeneous system has the zero solution x 1 = x 2 = 0 in this case then θ and θ • P are closed. Vice versa, if θ and θ • P are closed, then (4.2) implies the equalities ∇ ′ y θ P z − (∇ ′ z θ) P y = 0, ∇ ′ y θ z − (∇ ′ z θ) y = 1 2n {θ(y)θ(P z) − θ(P y)θ(z)} and thus conditions (4.1) hold. Then, according to Proposition 4.1, R ′ is a Riemannian P -tensor.\\nThe obtained results we summarize in the following classification theorem for the natural connections on a W 1 -manifold. i) If ∇ ′ is the connection D determined by λ = µ = 0, then R ′ is a Riemannian P -tensor if and only if the 1-form θ is not closed and the 1-form θ • P is closed; ii) If ∇ ′ is the connection D determined by λ = 0, µ = − 1 2n , then R ′ is a Riemannian P -tensor if and only if the 1-form θ is closed and the 1-form θ • P is not closed; iii) If ∇ ′ is a connection for which λ 2 − µ 2 − µ 2n = 0, then R ′ is a Riemannian P -tensor if and only if the 1-forms θ and θ • P are closed; iv) If ∇ ′ is a connection for which λ = 0, λ 2 − µ 2 − µ 2n = 0 and R ′ is a Riemannian P -tensor then the 1-forms θ and θ • P are not closed. In this case (M, P, g) / ∈ W 3 , (M, P, g) / ∈ W 6 .\\nLet us remark that the proposition i) of Theorem 4.3 is proved in [8], where the connection D is investigated.\\nIn [5], it is introduced a natural connection ∇ c on an arbitrary Riemannian almost product manifold (M, P, g). This connection, called canonical, is an analogue of the Hermitian connection in Hermitian geometry. Analogues of this connection are considered on manifolds, corresponding to the class W 1 , in the geometry of almost complex manifolds with Norden metric ( [11]) and almost contact manifolds with B-metric ( [12], [13]). In [9], it is note that ∇ c on a Riemannian almost product W 1 -manifold is determined by (3.4) for λ = 0, µ = − 1 4n . Therefore, for ∇ c it is valid the condition λ 2 − µ 2 − µ 2n = 0 and then the statement iii) of Theorem 4.3 holds.\\nHaving in mind the values of λ and µ determining the connections ∇ c , D and D, we obtain the following Calculating from (2.9) the covariant derivative ∇ ′ T and applying ∇ ′ g = ∇ ′ g = 0, we obtain the following Let ∇ ′ be a natural connection with parallel torsion and Riemannian Ptensor of curvature. Then from Proposition 5.1 we have U = V = 0. In this case Proposition 4.1 implies λW = µW = 0, where W (y, z) = θ(P y)θ(z) − θ(y)θ(P z). If W = 0 then we obtain θ(P y) = ±θ(y), i.e. (M, P, g) ∈ W 3 or (M, P, g) ∈ W 6 . Thus, we get the following Theorem 5.3. Let a W 1 -manifold (M, P, g) admits a natural connection ∇ ′ with parallel torsion and Riemannian P -tensor of curvature. Then the all possible cases are as follows:\\ni) λ = µ = 0, W = 0, i.e. ∇ ′ = D, (M, P, g) / ∈ W 3 , (M, P, g) / ∈ W 6 ; ii) λ = µ = 0, W = 0, i.e. ∇ ′ = D, (M, P, g) ∈ W 3 or (M, P, g) ∈ W 6 ; iii) λ = 0, µ = 0, W = 0, i.e. ∇ ′ belongs to the 1-parametric family determined by (3.4) for λ = 0, (M, P, g) ∈ W 3 , (M, P, g) ∈ W 6 ; iv) λ = 0, µ = 0, W = 0, i.e. ∇ ′ belongs to the 1-parametric family determined by (3.4) for µ = 0, (M, P, g) ∈ W 3 , (M, P, g) ∈ W 6 .\"\n },\n {\n \"title\": \"Corollary 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 4 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 4 . 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4 . 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 4 . 4 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"The following relations are valid ρ(y, z) = ρ ′ (y, z) − {(2n − 1)g(p, p) + trS ′ − g(q, q)} g(y, z)","content":"Corollary 3. 2 .\n2The following relations are valid ρ(y, z) = ρ ′ (y, z) − {(2n − 1)g(p, p) + trS ′ − g(q, q)} g(y, z)"},{"id":"","caption":"The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:","content":"Proposition 4 . 1 .\n41The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:"},{"id":"","caption":"The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:","content":"Proposition 4 . 2 .\n42The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:"},{"id":"","caption":"Let R ′ be the curvature tensor of a natural connection ∇ ′ determined by (3.4) on a W 1 -manifold (M, P, g). Then the all possible cases are as follows:","content":"Theorem 4 . 3 .\n43Let R ′ be the curvature tensor of a natural connection ∇ ′ determined by (3.4) on a W 1 -manifold (M, P, g). Then the all possible cases are as follows:"},{"id":"","caption":"The canonical connection ∇ c is the average connection of D and D, i.e. ∇ c = 1 2 (D + D). 5. Natural connection on a W 1 -manifold with parallel torsion","content":"Proposition 4 . 4 .\n44The canonical connection ∇ c is the average connection of D and D, i.e. ∇ c = 1 2 (D + D). 5. Natural connection on a W 1 -manifold with parallel torsion"},{"id":"","caption":"Proposition 5.1. A natural connection ∇ ′ has a parallel torsion if and only if the 1-form θ is parallel with respect to ∇ ′ . By (4.2), (2.2) and (2.3), from Proposition 5.1 we obtain the following Corollary 5.2. If a W 1 -manifold (M, P, g) admits a natural connection with parallel torsion then the 1-form θ • P is closed. If (M, P, g) belongs to the class W 3 or the class W 6 then the 1-form θ is closed, too.","content":"Proposition 5.1. A natural connection ∇ ′ has a parallel torsion if and only if the 1-form θ is parallel with respect to ∇ ′ . By (4.2), (2.2) and (2.3), from Proposition 5.1 we obtain the following Corollary 5.2. If a W 1 -manifold (M, P, g) admits a natural connection with parallel torsion then the 1-form θ • P is closed. If (M, P, g) belongs to the class W 3 or the class W 6 then the 1-form θ is closed, too."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"The following relations are valid ρ(y, z) = ρ ′ (y, z) − {(2n − 1)g(p, p) + trS ′ − g(q, q)} g(y, z)\",\n \"content\": \"Corollary 3. 2 .\\n2The following relations are valid ρ(y, z) = ρ ′ (y, z) − {(2n − 1)g(p, p) + trS ′ − g(q, q)} g(y, z)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:\",\n \"content\": \"Proposition 4 . 1 .\\n41The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:\",\n \"content\": \"Proposition 4 . 2 .\\n42The curvature tensor of ∇ ′ , determined by(3.4), is a Riemannian P -tensor if and only if the following conditions are valid:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let R ′ be the curvature tensor of a natural connection ∇ ′ determined by (3.4) on a W 1 -manifold (M, P, g). Then the all possible cases are as follows:\",\n \"content\": \"Theorem 4 . 3 .\\n43Let R ′ be the curvature tensor of a natural connection ∇ ′ determined by (3.4) on a W 1 -manifold (M, P, g). Then the all possible cases are as follows:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The canonical connection ∇ c is the average connection of D and D, i.e. ∇ c = 1 2 (D + D). 5. Natural connection on a W 1 -manifold with parallel torsion\",\n \"content\": \"Proposition 4 . 4 .\\n44The canonical connection ∇ c is the average connection of D and D, i.e. ∇ c = 1 2 (D + D). 5. Natural connection on a W 1 -manifold with parallel torsion\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Proposition 5.1. A natural connection ∇ ′ has a parallel torsion if and only if the 1-form θ is parallel with respect to ∇ ′ . By (4.2), (2.2) and (2.3), from Proposition 5.1 we obtain the following Corollary 5.2. If a W 1 -manifold (M, P, g) admits a natural connection with parallel torsion then the 1-form θ • P is closed. If (M, P, g) belongs to the class W 3 or the class W 6 then the 1-form θ is closed, too.\",\n \"content\": \"Proposition 5.1. A natural connection ∇ ′ has a parallel torsion if and only if the 1-form θ is parallel with respect to ∇ ′ . By (4.2), (2.2) and (2.3), from Proposition 5.1 we obtain the following Corollary 5.2. If a W 1 -manifold (M, P, g) admits a natural connection with parallel torsion then the 1-form θ • P is closed. If (M, P, g) belongs to the class W 3 or the class W 6 then the 1-form θ is closed, too.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". K Yano, Pure and Applied Math. 49Pergamon Press BookYano K. Pure and Applied Math., 49, New York, Pergamon Press Book, 1965.","doi":""},{"text":". A M Naveira, Rend, Math, 3Naveira A. M. Rend. Math., 3, 1983, 577-592.","doi":""},{"text":". A Gray, L Hervella, Ann. Mat. Pura Appl. 123Gray A., L. Hervella. Ann. Mat. Pura Appl., 123, 1980, 35-58.","doi":""},{"text":". M Staikova, K , Gribachev. Serdica Math. P. 18Staikova M., K. Gribachev. Serdica Math. P., 18, 1992, 150-161.","doi":""},{"text":". V Mihova, Serdica Math. P. 15Mihova V. Serdica Math. P., 15, 1989, 351-358.","doi":""},{"text":". M Staikova, Plovdiv Univ. Sci. Works -Math. 323Staikova M. Plovdiv Univ. Sci. Works -Math., 32, 1987, No 3, 241-251.","doi":""},{"text":". D J Mekerov, Geom, 89Mekerov D. J. Geom., 89, 2008, No 1-2, 119-129.","doi":""},{"text":". D Gribacheva, arXiv:1103.2916Gribacheva D. arXiv:1103.2916.","doi":""},{"text":". D Gribacheva, Compt. rend. Acad. bulg. Sci. 646Gribacheva D. Compt. rend. Acad. bulg. Sci., 64, 2011, No 6, 799-806.","doi":""},{"text":". H Hayden, Proc. London. Math. Soc. 34Hayden H., Proc. London. Math. Soc., 34, 1934, 27-50.","doi":""},{"text":". G Ganchev, K Gribachev, V Mihova, Publ. Inst. Math. 42Ganchev G., K. Gribachev, V. Mihova. Publ. Inst. Math. (Beograd) (N.S.), 42, 1987, 107-121.","doi":""},{"text":". M Manev, K , Gribachev. Serdica Math. J. 20Manev M., K. Gribachev. Serdica Math. J., 20, 1994, 133-147.","doi":""},{"text":"Bulgaria Blvd, mircho@uni-plovdiv.bg4003 Plovdiv, Bulgaria e-mails: dobrinka@uni-plovdiv.bg. Bulgaria Blvd., 4003 Plovdiv, Bulgaria e-mails: dobrinka@uni-plovdiv.bg, mircho@uni-plovdiv.bg","doi":""}],"string":"[\n {\n \"text\": \". K Yano, Pure and Applied Math. 49Pergamon Press BookYano K. Pure and Applied Math., 49, New York, Pergamon Press Book, 1965.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A M Naveira, Rend, Math, 3Naveira A. M. Rend. Math., 3, 1983, 577-592.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Gray, L Hervella, Ann. Mat. Pura Appl. 123Gray A., L. Hervella. Ann. Mat. Pura Appl., 123, 1980, 35-58.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Staikova, K , Gribachev. Serdica Math. P. 18Staikova M., K. Gribachev. Serdica Math. P., 18, 1992, 150-161.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Mihova, Serdica Math. P. 15Mihova V. Serdica Math. P., 15, 1989, 351-358.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Staikova, Plovdiv Univ. Sci. Works -Math. 323Staikova M. Plovdiv Univ. Sci. Works -Math., 32, 1987, No 3, 241-251.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D J Mekerov, Geom, 89Mekerov D. J. Geom., 89, 2008, No 1-2, 119-129.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Gribacheva, arXiv:1103.2916Gribacheva D. arXiv:1103.2916.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Gribacheva, Compt. rend. Acad. bulg. Sci. 646Gribacheva D. Compt. rend. Acad. bulg. Sci., 64, 2011, No 6, 799-806.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Hayden, Proc. London. Math. Soc. 34Hayden H., Proc. London. Math. Soc., 34, 1934, 27-50.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Ganchev, K Gribachev, V Mihova, Publ. Inst. Math. 42Ganchev G., K. Gribachev, V. Mihova. Publ. Inst. Math. (Beograd) (N.S.), 42, 1987, 107-121.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Manev, K , Gribachev. Serdica Math. J. 20Manev M., K. Gribachev. Serdica Math. J., 20, 1994, 133-147.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bulgaria Blvd, mircho@uni-plovdiv.bg4003 Plovdiv, Bulgaria e-mails: dobrinka@uni-plovdiv.bg. Bulgaria Blvd., 4003 Plovdiv, Bulgaria e-mails: dobrinka@uni-plovdiv.bg, mircho@uni-plovdiv.bg\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":396,"cells":{"text":{"kind":"string","value":"\nInstance Segmentation with Cross-Modal Consistency\n\n\nAlex Zihao Zhu \nVincent Casser \nReza Mahjourian \nHenrik Kretzschmar \nSören Pirk \nInstance Segmentation with Cross-Modal Consistency\n\nSegmenting object instances is a key task in machine perception, with safety-critical applications in robotics and autonomous driving. We introduce a novel approach to instance segmentation that jointly leverages measurements from multiple sensor modalities, such as cameras and LiDAR. Our method learns to predict embeddings for each pixel or point that give rise to a dense segmentation of the scene. Specifically, our technique applies contrastive learning to points in the scene both across sensor modalities and the temporal domain. We demonstrate that this formulation encourages the models to learn embeddings that are invariant to viewpoint variations and consistent across sensor modalities. We further demonstrate that the embeddings are stable over time as objects move around the scene. This not only provides stable instance masks, but can also provide valuable signals to downstream tasks, such as object tracking. We evaluate our method on the Cityscapes and KITTI-360 datasets. We further conduct a number of ablation studies, demonstrating benefits when applying additional inputs for the contrastive loss.\n\nI. INTRODUCTION\n\nIdentifying and segmenting objects is a fundamental and challenging task in computer vision, and critically important for many robotics applications, such as autonomous driving. Given some sensor input such as a camera image or LiDAR scan, the goal of instance segmentation is to associate each pixel or point with a unique instance label for the object to which it corresponds. This allows for fine-grained reasoning about each object beyond sparse representations such as bounding boxes. Only recently, learning-based approaches have shown remarkable success in solving this task on a number of benchmark datasets [16], [6], [34]. Reliably identifying object instances in complex real-world data, both across different data modalities and coherent in time, remains a challenging and open problem [49], [15], [8].\n\nMany safety-critical robotics systems, such as autonomous vehicles, rely on multiple sensors that provide complementary information to perceive the environment. For instance, LiDAR sensors provide highly accurate range readings, but only in clear weather conditions and up to a limited range. On the other hand, cameras may be able to perceive objects at much longer ranges, but without any explicit range readings and only in adequate lighting conditions. As a result, there is an opportunity to improve segmentation performance for each sensor by reasoning jointly across all modalities.\n\nAnother avenue is the utilization of temporal data, which can provide motion information, varying viewpoints and context for occlusions. Temporal data comes naturally for most sensors, although sequential labeling for ground truth is more expensive. 1 Waymo LLC 2 Adobe Research (Work done while at Google Research) Fig. 1: Our approach to instance segmentation leverages information from multiple sensor modalities, such as 3D LiDAR point clouds and camera images (left). Our method learns to predict dense embeddings (depicted by the colors, right image) based on a novel cross-modal contrastive loss that uses samples from different sensor modalities and temporally consecutive frames of RGB images and LiDAR points. The resulting embeddings are stable over time and invariant to viewpoint changes.\n\nIn this paper, we aim to leverage these complementary signals and advance instance segmentation by training jointly on different data modalities and over time ( Figure 1). Specifically, we propose a novel cross-modal contrastive loss that enables us to train with sequential samples from camera images and LiDAR point clouds to learn a dense representation of object instances. Our goal is to learn embeddings that coherently represent individual objects in both modalities. By training on different modalities, our method can more meaningfully disentangle individual instances. Moreover, we show that our loss formulation is flexible enough to be applied to sequences of sensor data of both modalities, enabling learning temporally-stable embeddings for object instances. For situations where only sequential data (but no labels) are available, we propose a method for generating pseudogroundtruth using optical flow, and demonstrate that training on this data brings similar improvements in segmentation.\n\nIn summary, our contributions are as follows:\n\n• We introduce a novel cross-modal contrastive loss to obtain consistent instance embeddings for RGB images and point clouds. • We show that our loss can also be used to train on sequences of sensor data from both modalities, enabling the learning of temporally stable embeddings. • We propose a method using optical flow to generate pseudo ground-truth for video, allowing the extension of our contrastive loss to partially labeled datasets. • Our extensive experiments on the KITTI-360 [33] and Cityscapes [16] benchmarks suggest that our method significantly improves instance segmentation quality by up to 1.9 mAP when contrasting between temporal RGB data, and by up to 1.7 mAP when contrasting between temporal LiDAR and RGB data, while also improving the temporal stability of the embeddings. \n\n\nRGB\n\n\nConvNet rgb\n\n\nFig. 2:\n\nOur goal is to learn dense cross-modal embeddings via contrastive learning to segment individual object instances. Our method leverages shared information between sensors and consecutive frames by applying the contrastive loss jointly over embeddings from all input sources at training time. This global contrastive loss allows the network to share information between different data sources, and improves overall segmentation performance. Once trained, the learned embeddings can be used to obtain object IDs through mean shift clustering.\n\n\nII. RELATED WORK\n\nObject detection and segmentation have a long tradition in computer vision. While this spans a wide range of different approaches that we cannot conclusively discuss, we aim to provide an overview of learning-based methods toward instance and semantic segmentation in images and point clouds.\n\nSemantic Segmentation. The goal of semantic segmentation is to associate every pixel in an image with a discrete label that defines its semantic class. A major challenge of effectively training networks for semantic segmentation is to define precise labels, and a number of different datasets were introduced for benchmarking [16], [22], [6]. Many recent methods for semantic segmentation employ deep neural network architectures for feature aggregation based on pyramid pooling [53], attention modules [10], multi-scale activations [24], or end-to-end convolutions [36]. Furthermore, it has been recognized that dilated or atrous convolutions provide a powerful way of capturing more global features through wider receptive fields [52], [9]. Other approaches aim to solve semantic segmentation with an emphasis on learning structured representations based on conditional random fields [1], [9] or deep parsing networks [35].\n\nInstance Segmentation expands on the concept of semantic segmentation. Here the goal is to not only obtain a class label for every pixel, but to predict labels for individual object instances. Instance segmentation approaches can be separated into two different categories. In the first category, most methods rely on a two stage (or top-down) process [17], [18], [25]. A region proposal step [23] identifies object instances in a first stage and each region is then segmented into foreground and background during a second stage. The performance of top-down approaches is determined by obtaining region proposals and by the number of objects present in an image. On the contrary, bottom-up approaches aim to learn per-pixel embeddings that represent individual object instances based on recurrent instance grouping [29], watershed transformations [3], subnetwork instantiations [2], associative embeddings [39], or based on more refined loss functions such as discriminative loss [5], [26], or metric learning [20]. Bottom-up approaches are commonly implemented with more lightweight architectures, which offers more efficient training and easier integration into existing multi-task setups. However, learning meaningful embeddings from complex data to represent individual objects remains a challenging problem. Finally, a number of hybrid approaches exist that aim to learn embeddings of larger regions in images [7], [13].\n\n3D Object Detection and Segmentation: Detecting and segmenting objects in point clouds has received a considerable amount of attention. PointNets [41] directly consume raw point clouds to generate latent representations that can be used for classification or semantic segmentation tasks. By leveraging various geometric relationships and representations a number of methods have since then improved upon the performance to solve for tasks, such as object detection [54], semantic segmentation [30], [42], or instance segmentation [45], [51]. A number of approaches rely on Bird's Eye View (BEV) representations to generate bounding boxes [50], [14] that -along with pillar-based representations [31], [46] are popular in autonomous driving. More recent methods for 3D instance segmentation rely on more principled approaches. To this end 3D-MPA [19] uses an aggregation-based approach where each point votes for its object center, HAIS [11] exploits the spatial relation of points and point sets, and SSTNet [32] uses a hierarchical representation which is constructed based on learned semantic features.\n\nMultimodal Approaches. There also exists a limited number of multimodal approaches proposed in the roboticvision domain, including [48], [37]. Most of these works tackle indoor environments using closely paired RGB-D data.\n\nSimilar to these methods, our goal is to leverage multimodal sensor data. Specifically, we use a single-stage fullyconvolutional network to learn per-point and per-pixel embeddings for instance segmentation. We train these embeddings with a metric learning loss that contrasts embeddings of the same object instance with those of different objects. The key idea to our approach is that we obtain the positive and negatives samples spatially, within the same frame, across different modalities (images, points), and even across time, in the adjacent frames of a sequence. We accomplish this by proposing a novel cross-modal consistency loss that allows us to sample embedding vectors from different sources. Training a single-stage network with this loss enables us to compute dense embeddings and, consequently, more robust instance masks for each frame, while also obtaining temporally stable results for sequences.\n\n\nIII. METHOD\n\nIn this section, we introduce our model architecture, the cross-modal contrastive loss function, and implementation details. Our full pipeline is shown in Figure 2.\n\n\nA. Cross-Modal Contrastive Loss\n\nOur method consists of three main components: (1) a pair of instance segmentation networks that generate dense embeddings for RGB images and LiDAR range images. (2) a sampling based contrastive learning loss which constrains embeddings from the same instances to be similar, while forcing embeddings from different instances to be different.\n\n(3) a consistency module based on paired instance labels that introduces additional training data and correspondences between sensors and over temporally neighboring frames.\n\nContrastive Loss: The goal of the instance segmentation network is to segment object instances by learning dense embedding vectors. The network takes as input either RGB images or LiDAR range images [4], and outputs embeddings with c = 32 channels. To train the network, we use a metric learning loss for contrasting embeddings of the same instance with those of different instances. Embeddings of the same instance are pushed closer together during training, while those of different instances are pushed away in the embedding space.\n\nOur instance segmentation loss consists of the normalized temperature-scaled cross entropy (NT-XENT) loss defined in [12]. We define the per-pixel embeddings as E h×w×c , where h and w denote the spatial dimensions of the output, and c, the channels of the embedding at each pixel. For each frame, we randomly sample K 1 = 8192 embeddings, distributed evenly across all instances (visualized for RGB \n\n\nInference Embedding E 1, rgb\n\nEmbedding E 2, rgb Embedding E n, rgb ...\n\n\nEmbeddings\n\n\nFrame t1\n\nFrame t2 Fig. 4: Given a set of predicted embeddings with paired instance labels, we sample positive embedding vectors within the instance mask of an object in the current frame as well in all other frames (cross sensor and temporal). Samples from all other objects in all frames are used as negatives. We then train with our cross-modal contrastive loss to learn dense embeddings of object instances. The images at the bottom show the projected embeddings during training. frames in Figure 3). Given a set of sampled embeddings for an object instance, we generate all possible positive pairs. The samples of all other instances are used as negatives for each pair. For a set of samples U and a positive pair, e i , e j , our loss can then be defined as:\nl i,j = − log exp(sim(e i , e j )/τ ) k∈U 1 [id(i) =id(k))] exp(sim(e i , e k )/τ ) (1) with sim(x, y) = x T y x 2 y 2 ,(2)\nwhere τ denotes a temperature hyperparameter and id(i) is the mask ID of point i. The similarity function sim is used to compute the cosine similarity between two normalized embedding vectors x and y. The total loss over the instance embeddings is then defined as:\nl c = 1 U i,j∈U 1 [id(i)=id(j)] l i,j .(3)\nFurthermore, to obtain more stable results, we also apply a regularization loss over the norm of the embeddings:\nl r = 1 h × w x,y E(x, y) 2 .(4)\nOur final loss, then, is:\nl =l c + λl r ,(5)\nwhere λ = 0.01 is a weighting factor. Together, these losses allow us to learn meaningful embedding vectors as representation for object instances. Consistency Over Modalities and Time: Given a dataset with instance labels which are consistent between sensors and across time, such that an object has the same ID across all inputs, one can apply the above training loss (5) between all inputs by randomly sampling points from each set of embeddings, and using the consistent instance labels to determine positive and negative pairs. This allows the network to learn embeddings which are consistent not only within a single frame, but between all modalities and time steps seen at training.\n\nHowever, it is often the case that we do not have consistent instance labels. In these cases, we propose a method for generating pseudo-groundtruth using optical flow. Given a dataset with individually labeled frames amongst unlabeled temporal sequences, we use a pre-trained optical flow network to warp the groundtruth of the labeled frame to its temporal neighbors.\n\nWe use the predicted flow to warp the groundtruth instance and semantic labels of the current frame to the next frame and apply nearest neighbor resampling to avoid interpolating between integer label values. Following the work of Wang et al. [47], we compute occlusions according to the 'range map', defined by the number of points in the original image map to the other image. The instance and semantic labels of occluded pixels, as well as pixels that have left the image, are set to invalid, and are not sampled for the contrastive loss. The embeddings and warped labels for the next frame are then combined with those of the current frame when performing the sampling for the contrastive loss. Figure 4 illustrates the result of the warped labels as well as the sampling of embeddings from the two frames.\n\nThis pipeline has similarities to semi-supervised methods such as in the work by Chen et al. [8], which generate pseudo-groundtruth by running inference with a pre-trained model, and subsequently re-training. However, in addition to providing ground truth to temporally neighboring frames with optical flow, our method also ensures consistency between frames and thus over time. Sample outputs for our method over time can be found in Figure 5.\n\nSemantic Segmentation: We predict a semantic mask for each frame and filter out all 'stuff' classes to a single background class. During clustering, we assign semantic classes to each clustered instance with a 'majority vote' between semantic predictions within the instance. We also generate the confidence for each mask using the average semantic score. Pixels not belonging to the selected semantic class are dropped from the instance to preserve semantic boundaries. The one-hot semantic labels are concatenated to the predicted instance embeddings, so that the network only has to predict unique instance embeddings within each class. Fig. 5: Temporally-coherent embeddings: as our model is trained with a temporal consistency loss based on optical flow, we are able to obtain temporally-coherent and more stable instance embeddings. Here we show the RGB sequences of two different scenes along with the multi-dimensional per-pixel embedding vectors projected into a color space.\n\n\nInstance Mask Generation:\n\nAt inference time, we cluster the resulting per-pixel embeddings by applying a variant of the mean shift algorithm proposed by Brabandere et al. [5]. In particular, we randomly sample a point in the embedding space, and find all inlier points with cosine distance (scaled to be within [0, 1]) less than a threshold, m = 0.1, from the sampled point. We then iterate by shifting the sampled point to the mean of the set of inliers, and repeat until convergence or some maximum number of iterations is reached. We repeat this process until all pixels have been clustered. With this method, we found that erroneous masks were typically generated on the transitions between masks, generating thin artifacts along the boundaries. To compensate, we filter out masks with an area to perimeter ratio less than a threshold, r = 4. As this method computes distances across the entire image, no assumptions about the connectivity of each mask are made, and so arbitrarily distributed instances can be detected.\n\nMore complex methods are available for generating instance masks from embeddings, such as the graph cut algorithm proposed in SSAP [21] or the transformer head proposed in MaX-DeepLab [44], which would likely achieve higher performance. However, these would incur additional latency penalties, and their computational power may hide improvements to the underlying embeddings.\n\n\nB. Implementation Details\n\nIn this section, we provide details about the implementation of our network architecture.\n\nModel Architecture: We use a modified architecture based on Panoptic DeepLab [15]. Our model consists of a Xception-71 backbone, with an additional ASPP and decoder module for instance embedding prediction, with c = 32 channels, in place of the instance center and offset decoder in [15]. For our cross-modal experiments, we use two separate networks for LiDAR range image inputs and RGB image inputs.\n\nPreprocessing: For all datasets, we use full resolution images as input. During training, we augment our data by randomly flipping, scaling and cropping or padding the input images. Furthermore, we use random Gaussian blur and jitter brightness, contrast, saturation, and hue. For temporal data, we apply the same augmentation to each pair of images. For LiDAR range images, we only apply flipping, and ensure that the flipping is consistent with the camera images. 6: Cross-modal results on KITTI-360: Our models take as inputs RGB images or LiDAR range images (a), and predicts dense instance embeddings (b) which are consistent between the sensor modalities. At post-processing, we can apply a mean-shift clustering algorithm to generate instance masks for each modality (c), which we compare to the ground truth (d). Our cross-modal contrastive loss contrasts pairs between the sensor modalities, and trains the network to predict the same embedding for each object instance in each sensor, denoted by similarity in color. Each image pair shows both sensor modalities, where the LiDAR data is shown as range images (bottom).\n\nTraining: For all experiments, we use a moving average ADAM optimizer with linear warmup for 5 epochs, initial learning rate of 2.5e-4 and exponential decay with decay factor of 1.2 every 10 epochs. For the KITTI-360 dataset, we train for 10,000 steps for the ablation train/val split, and 20,000 steps for the train/test split, with an effective batch size of 128. For the Cityscapes dataset, we train for 60,000 iterations with an effective batch size of 32.\n\n\nIV. EXPERIMENTS\n\nIn this section we present and discuss a variety of experiments, including baseline comparisons, ablation studies, and qualitative results our method is able to generate. For all of our experiments, we do not perform any kind of test time augmentation.\n\nA. Datasets a) KITTI-360: The KITTI-360 dataset [33] consists of camera and LiDAR panoptic segmentation and LiDAR bounding box labels for all of the sequences of the KITTI dataset [22]. For each sequence, the instance labels are consistent both between the camera and LiDAR labels, as well as between different time steps. The dataset contains 83,000 frames and associated LiDAR scans, split into 9 training sequences and 2 test sequences with held out groundtruth. As no validation set is provided, we perform our ablations on a train/val split where sequences 00, 02, 03, 04, 05, 06, 07 are the train set and sequences 09 and 10 are the validation set.\n\nThe validation set was chosen from the two densest sequences, such that there are almost the same number of objects in the training and validation sets. For the final test set evaluation, we train on the entire training set.\n\nAs the dataset is largely transferred from primitive 3D shapes (cuboids, ellipsoids etc.), each label has a corresponding confidence value, which is used to weight the evaluation metrics. In accordance with the experiments in KITTI-360, we only use the top 70% of labels in each frame, in terms of confidence. To generate temporal frames, we randomly sample an image 2 frames before or after the current frame at each training step. To train a model on LiDAR, we use the range image representation of the point cloud generated by Bewley et al. [4]. From this 64×2048 range image, we crop a central 64×512 patch corresponding to the camera field of view, and treat this as an image into the DeepLab model. To generate per-scan groundtruth, we use nearest-neighbors to find the closest ground truth segmentation label to each point in the range image. b) Cityscapes: To test our method on a dataset without consistent instance labels and in a single-modal setting, we employ the Cityscapes dataset [16]. dataset. Cityscapes provides us with 2975, 500, and 1525 images of urban scenes for training, testing and validation, respectively. Furthermore, the dataset provides 8 'thing' and 11 'stuff' classes. In this work, we train our network on the 'fine' set of training images with ground truth instance labels. In order to generate paired groundtruth, we use the unlabeled temporal frames, and randomly sample frames from the images 2 frames before or after the labeled image. We then warp the groundtruth from the labeled frame to each temporal frame using a UFlow [27] model trained on the Waymo Open Dataset [43], consisting of 200,000 training images. To filter potential errors in the flow warping due to occlusions and other errors, we ignore warped labels based on the occlusion map generated by the flow model. For Cityscapes, we rely on a pre-trained semantic segmentation network for assigning class IDs, in particular an implementation of Panoptic Deeplab [15], with an Xception-71 backbone, trained on Cityscapes. Our implementation has a mIOU of 68.8% on the Cityscapes validation set. As embeddings along the boundary are particularly challenging, and the Cityscapes labels are typically reliable, we dedicate half of the points when sampling in the contrastive loss to embeddings that are within b = 10 pixels of the boundary of each instance mask.\n\n\nB. Results on KITTI-360\n\nIn Table I, we report an ablation study on the effects of adding additional signals for the contrastive loss, both temporally and from different modalities. Overall, we observe that instance mAP for images improves as we add each additional frame to the consistency loss. The overall trend that we observe is that the network improves significantly for rare classes such as bicycle, motorcycle, rider and truck, while regressing slightly for the more common car class. This results in a significant overall mAP improvement. Note that, because the ablation models are trained on a much smaller training set, the mAP numbers in these experiments are naturally lower than those for the test set. We show qualitative examples from our best model (Temporal RGB + Temporal LiDAR) in Figure 6, where we demonstrate that the networks are able to predict embeddings that are consistent for objects across the sensor modalities. We also show our LiDAR instance segmentation results projected to 3D in Figure 7, demonstrating that the network can generate accurate 3D instance segmentation when operating only on the range image. However, we did not observe similar improvements in 3D LiDAR mAP when training with the proposed crossmodality contrastive loss. Our observation is that, as the 2D labels are generated by projecting the 3D labels (and applying a CRF), they are typically much noisier than the 3D labels, and cannot provide useful signal for the 3D part of the model.\n\nIn Table II, we report a comparison in the mAP of our final method against the popular MaskRCNN method [25] on the held-out test set, where we outperform the ResNet-50 baseline, while approaching the performance of the ResNet-101 baseline. At the time of writing, the validation set for this dataset was not available, and better results will be achievable once the same training splits are available as for the baselines. \n\n\nC. Results on Cityscapes\n\nIn Table III, we report our instance segmentation results on Cityscapes compared to existing methods. The reported results were obtained on the Cityscapes validation set. To provide a fair comparison of the main methods themselves, we use the reported numbers for each method without test time augmentation. In addition, we report mAP for our method using groundtruth semantic segmentations in Table IV, and compare against past work which has used a similar metric. This approach allows us to evaluate the instance segmentation network independently from the performance of the semantic segmentation. Overall, the improvement from our crossmodal consistency loss allows us to strongly outperform bottom up methods such as Brabandere et al. [5] and Neven et al. [38] with groundtruth semantics. It also pushes our embedding-based method close to the previous bottom up works such as SSAP [21]. However, this work requires a complex post-processing procedure which has a significant latency, as reported in Table III. We underperform compared to Panoptic DeepLab, but note that this method, while being single shot, requires the prediction of object centers. These centers serve as top-down proposals requiring instances to have distinguishable centers and some form of NMS.\n\nWe also show visualizations of our predictions in Figure 8, where the embeddings can robustly and precisely represent a large variety of object instances.\n\nTemporal Constraints: To validate the impact of the temporal contrastive loss, we trained our model with and without contrasting over time (see Table III, Ours base vs Ours temporal). From these results, introducing the contrastive Fig. 8: Visualization of per-pixel embedding vectors on Cityscapes: our method allows us to obtain per-pixel embedding vectors from RGB input images (first row). The embeddings robustly and precisely disentangle object instances across different classes (third row). To visualize the embeddings, we project them into RGB color space (second row). In the last row, we show the ground truth instance masks. Please note that the color of the predicted and ground truth masks are not supposed to match. loss over time has a significant impact on the predicted instances masks. Adding temporal frames to the contrastive loss improves overall AP by 2%. The visualization of the predicted embeddings in Figure 5 also shows fewer artifacts.\n\nWe also compute error metrics on the stability of the instance embeddings over time for the Cityscapes dataset. Given two frames, we warp the predicted embeddings of the next frame to the current frame, and compute the cosine distance between the average embedding of each instance in the two time steps, while ignoring any pixels estimated as 'occlusion' from the optical flow. In addition, we compute as accuracy the proportion of instance pairs with cosine distance less than the clustering threshold m = 0.1. These results can be found in Table V. We observe that the majority of embeddings (75.1%) are stable over time, even without the proposed temporal contrastive loss. However, there are a number of cases, such as large motions and dense objects, where the embeddings are not stable. Overall, our proposed loss reduces the cosine distance between instances by 0.025, and improves accuracy by 8.3% to 83.4%.\n\nEmbedding Similarity: In Figure 9 we show the results of an embedding similarity experiment. We select pixels in the image and compute the distance, shown in grayscale, of the selected embeddings with all other embeddings in the image. As illustrated, the learned embeddings allow us to disentangle object instances of the same class, across different classes, and the background. Our simple clustering scheme can generate high quality masks by thresholding this similarity. \n\n\nV. LIMITATIONS AND FUTURE WORK\n\nThe main contribution of this method is the contrastive learning of instance embeddings between sensors and over time. As it stands, we believe that the learned embeddings are strong, but have chosen a relatively simple method to generate instance masks for evaluation. We use this postprocessing to highlight the improvements in the embeddings themselves, where a more complex method may obscure such improvements. However, there is room for improvement if stronger AP scores are desired, by adding a graph cut optimization such as in SSAP [21] or a transformer head on top of the embeddings as in MaX-DeepLab [44].\n\n\nVI. CONCLUSION\n\nWe have introduced a novel method for learning pixel-wise embeddings as a representation for individual object instances. Compared to other bottom-up approaches, we employ a contrastive learning scheme; embeddings of the same object instance are pushed closer together, while they are pulled away from other objects and the background. In particular, we leverage information between sensors, as well as over time, by applying our cross-modal contrastive loss on the union of all predicted embeddings for a given scene. Through quantitative experiments, we show that this contrastive loss is able to significantly improve instance segmentation performance, and coherent instance embeddings between sensors and time, which are clustered to generate high quality instance masks.\n\nFig. 3 :\n3Example of our sampling strategy where points are evenly distributed amongst instances. The bottom row shows close-ups of the used sample locations.\n\nFig. 9 :\n9Pixel similarity: for an image with objects of different classes (a) we randomly select different pixel positions in the image (yellow dot) and show the similarity of the selected embedding to all other embeddings as grayscale values. As shown, the learned embeddings allow us to meaningfully disentangle object instances, such as various different cars (b, c, d), intricate objects like bicycles (e), and small-scale objects, such as pedestrians (f).\n\nTABLE I :\nIAblation study on the effects of additional consistency on instance segmentation performance on a custom KITTI 360 train/val split. Adding additional inputs for the contrastive loss improves overall segmentation performance. Performance is measured in terms of AP (%).Fig. \n\nTABLE II :\nIIResults on the KITTI-360 held out test set.Fig. 7: Close-up analysis of our 3D LiDAR segmentation network, operating \non range images (top-left), demonstrating accurate point-cloud instance \nsegmentation. \n\nMethod \nApproach \nInput Size \nAP (%) Speed (ms) \nPanoptic FPN [28] \nTD \n512× 1024 \n33.2 \n-\nUPSNet [49] \nTD \n1024×2048 \n33.3 \n202ms \nSeamless [40] \nTD \n1024× 2048 \n33.6 \n-\nPanoptic Deeplab [15] Xception-71 \nTD 1 \n1025×2049 \n35.3 \n175ms \nSSAP [21] \nBU \n1024× 2048 \n31.5 \n260ms+ 2 \nOurs base \nBU-C \n1024× 2048 \n29.0 \n182ms \nOurs temporal \nBU-C \n1024× 2048 \n30.9 \n182ms \n\n\n\nTABLE III :\nIIICityscapes validation set results, without any test time aug-\nmentation or additional training data for all methods. TD: Top-Down. BU: \nBottom-Up. 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KITTI-360: A novel dataset and benchmarks for urban scene understanding in 2D and 3D. arXiv, 2021.\n\nMicrosoft COCO: Common objects in context. Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, C Lawrence Zitnick, ECCV. Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C. Lawrence Zitnick. Microsoft COCO: Common objects in context. In ECCV, pages 740- 755, 2014.\n\nSemantic image segmentation via deep parsing network. Z Liu, X Li, P Luo, C.-C Loy, X Tang, ICCV. USAIEEE Computer SocietyZ. Liu, X. Li, P. Luo, C.-C. Loy, and X. Tang. Semantic image segmentation via deep parsing network. In ICCV, page 1377-1385, USA, 2015. IEEE Computer Society.\n\nFully convolutional networks for semantic segmentation. J Long, E Shelhamer, T Darrell, CVPR. J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. 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Wang, A. Fathi, A. Kundu, D. A. Ross, C. Pantofaru, T. A. Funkhouser, and J. M. Solomon. Pillar-based object detection for autonomous driving. In ECCV, 2020.\n\nOcclusion aware unsupervised learning of optical flow. Y Wang, Y Yang, Z Yang, L Zhao, P Wang, W Xu, CVPR. Y. Wang, Y. Yang, Z. Yang, L. Zhao, P. Wang, and W. Xu. Occlusion aware unsupervised learning of optical flow. In CVPR, pages 4884- 4893, 2018.\n\nLearning RGB-D feature embeddings for unseen object instance segmentation. CoRL. Yu Xiang, Christopher Xie, Arsalan Mousavian, Dieter Fox, Yu Xiang, Christopher Xie, Arsalan Mousavian, and Dieter Fox. Learning RGB-D feature embeddings for unseen object instance segmentation. CoRL, 2020.\n\nUPSNet: A unified panoptic segmentation network. Y Xiong, R Liao, H Zhao, R Hu, M Bai, E Yumer, R Urtasun, CVPR. Y. Xiong, R. Liao, H. Zhao, R. Hu, M. Bai, E. Yumer, and R. Urtasun. UPSNet: A unified panoptic segmentation network. In CVPR, 2019.\n\nPIXOR: Real-time 3D object detection from point clouds. B Yang, W Luo, R Urtasun, CVPR. B. Yang, W. Luo, and R. Urtasun. PIXOR: Real-time 3D object detection from point clouds. CVPR, pages 7652-7660, 2018.\n\nGSPN: Generative shape proposal network for 3D instance segmentation in point cloud. L Yi, W Zhao, H Wang, M Sung, L J Guibas, CVPR. L. Yi, W. Zhao, H. Wang, M. Sung, and L. J. Guibas. GSPN: Generative shape proposal network for 3D instance segmentation in point cloud. CVPR, pages 3942-3951, 2019.\n\nDilated residual networks. F Yu, V Koltun, T Funkhouser, CVPR. F. Yu, V. Koltun, and T. Funkhouser. Dilated residual networks. In CVPR, pages 636-644, 2017.\n\nPyramid scene parsing network. H Zhao, J Shi, X Qi, X Wang, J Jia, CVPR. H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia. Pyramid scene parsing network. In CVPR, pages 6230-6239, 2017.\n\nVoxelNet: End-to-end learning for point cloud based 3D object detection. Y Zhou, O Tuzel, CVPR. Y. Zhou and O. Tuzel. VoxelNet: End-to-end learning for point cloud based 3D object detection. CVPR, pages 4490-4499, 2018.\n"},"corpus_id":{"kind":"number","value":252917933,"string":"252,917,933"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2210.08113\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/iros/ZhuCMKP22\",\n \"doi\": \"10.1109/iros47612.2022.9982285\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2210.08113v1.pdf\"\n ],\n \"pdf_sha\": \"3ac7c0184308b20b093f9c7748cb8f47d543d3a5\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Instance Segmentation with Cross-Modal Consistency"},"authors":{"kind":"list like","value":["Alex Zihao Zhu","Vincent Casser","Reza Mahjourian","Henrik Kretzschmar","Sören Pirk"],"string":"[\n \"Alex Zihao Zhu\",\n \"Vincent Casser\",\n \"Reza Mahjourian\",\n \"Henrik Kretzschmar\",\n \"Sören Pirk\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Segmenting object instances is a key task in machine perception, with safety-critical applications in robotics and autonomous driving. We introduce a novel approach to instance segmentation that jointly leverages measurements from multiple sensor modalities, such as cameras and LiDAR. Our method learns to predict embeddings for each pixel or point that give rise to a dense segmentation of the scene. Specifically, our technique applies contrastive learning to points in the scene both across sensor modalities and the temporal domain. We demonstrate that this formulation encourages the models to learn embeddings that are invariant to viewpoint variations and consistent across sensor modalities. We further demonstrate that the embeddings are stable over time as objects move around the scene. This not only provides stable instance masks, but can also provide valuable signals to downstream tasks, such as object tracking. We evaluate our method on the Cityscapes and KITTI-360 datasets. We further conduct a number of ablation studies, demonstrating benefits when applying additional inputs for the contrastive loss."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Identifying and segmenting objects is a fundamental and challenging task in computer vision, and critically important for many robotics applications, such as autonomous driving. Given some sensor input such as a camera image or LiDAR scan, the goal of instance segmentation is to associate each pixel or point with a unique instance label for the object to which it corresponds. This allows for fine-grained reasoning about each object beyond sparse representations such as bounding boxes. Only recently, learning-based approaches have shown remarkable success in solving this task on a number of benchmark datasets [16], [6], [34]. Reliably identifying object instances in complex real-world data, both across different data modalities and coherent in time, remains a challenging and open problem [49], [15], [8].\nMany safety-critical robotics systems, such as autonomous vehicles, rely on multiple sensors that provide complementary information to perceive the environment. For instance, LiDAR sensors provide highly accurate range readings, but only in clear weather conditions and up to a limited range. On the other hand, cameras may be able to perceive objects at much longer ranges, but without any explicit range readings and only in adequate lighting conditions. As a result, there is an opportunity to improve segmentation performance for each sensor by reasoning jointly across all modalities.\nAnother avenue is the utilization of temporal data, which can provide motion information, varying viewpoints and context for occlusions. Temporal data comes naturally for most sensors, although sequential labeling for ground truth is more expensive. 1 Waymo LLC 2 Adobe Research (Work done while at Google Research) Fig. 1: Our approach to instance segmentation leverages information from multiple sensor modalities, such as 3D LiDAR point clouds and camera images (left). Our method learns to predict dense embeddings (depicted by the colors, right image) based on a novel cross-modal contrastive loss that uses samples from different sensor modalities and temporally consecutive frames of RGB images and LiDAR points. The resulting embeddings are stable over time and invariant to viewpoint changes.\nIn this paper, we aim to leverage these complementary signals and advance instance segmentation by training jointly on different data modalities and over time ( Figure 1). Specifically, we propose a novel cross-modal contrastive loss that enables us to train with sequential samples from camera images and LiDAR point clouds to learn a dense representation of object instances. Our goal is to learn embeddings that coherently represent individual objects in both modalities. By training on different modalities, our method can more meaningfully disentangle individual instances. Moreover, we show that our loss formulation is flexible enough to be applied to sequences of sensor data of both modalities, enabling learning temporally-stable embeddings for object instances. For situations where only sequential data (but no labels) are available, we propose a method for generating pseudogroundtruth using optical flow, and demonstrate that training on this data brings similar improvements in segmentation.\nIn summary, our contributions are as follows:\n• We introduce a novel cross-modal contrastive loss to obtain consistent instance embeddings for RGB images and point clouds. • We show that our loss can also be used to train on sequences of sensor data from both modalities, enabling the learning of temporally stable embeddings. • We propose a method using optical flow to generate pseudo ground-truth for video, allowing the extension of our contrastive loss to partially labeled datasets. • Our extensive experiments on the KITTI-360 [33] and Cityscapes [16] benchmarks suggest that our method significantly improves instance segmentation quality by up to 1.9 mAP when contrasting between temporal RGB data, and by up to 1.7 mAP when contrasting between temporal LiDAR and RGB data, while also improving the temporal stability of the embeddings."},{"title":"RGB","content":""},{"title":"ConvNet rgb","content":""},{"title":"Fig. 2:","content":"Our goal is to learn dense cross-modal embeddings via contrastive learning to segment individual object instances. Our method leverages shared information between sensors and consecutive frames by applying the contrastive loss jointly over embeddings from all input sources at training time. This global contrastive loss allows the network to share information between different data sources, and improves overall segmentation performance. Once trained, the learned embeddings can be used to obtain object IDs through mean shift clustering."},{"title":"II. RELATED WORK","content":"Object detection and segmentation have a long tradition in computer vision. While this spans a wide range of different approaches that we cannot conclusively discuss, we aim to provide an overview of learning-based methods toward instance and semantic segmentation in images and point clouds.\nSemantic Segmentation. The goal of semantic segmentation is to associate every pixel in an image with a discrete label that defines its semantic class. A major challenge of effectively training networks for semantic segmentation is to define precise labels, and a number of different datasets were introduced for benchmarking [16], [22], [6]. Many recent methods for semantic segmentation employ deep neural network architectures for feature aggregation based on pyramid pooling [53], attention modules [10], multi-scale activations [24], or end-to-end convolutions [36]. Furthermore, it has been recognized that dilated or atrous convolutions provide a powerful way of capturing more global features through wider receptive fields [52], [9]. Other approaches aim to solve semantic segmentation with an emphasis on learning structured representations based on conditional random fields [1], [9] or deep parsing networks [35].\nInstance Segmentation expands on the concept of semantic segmentation. Here the goal is to not only obtain a class label for every pixel, but to predict labels for individual object instances. Instance segmentation approaches can be separated into two different categories. In the first category, most methods rely on a two stage (or top-down) process [17], [18], [25]. A region proposal step [23] identifies object instances in a first stage and each region is then segmented into foreground and background during a second stage. The performance of top-down approaches is determined by obtaining region proposals and by the number of objects present in an image. On the contrary, bottom-up approaches aim to learn per-pixel embeddings that represent individual object instances based on recurrent instance grouping [29], watershed transformations [3], subnetwork instantiations [2], associative embeddings [39], or based on more refined loss functions such as discriminative loss [5], [26], or metric learning [20]. Bottom-up approaches are commonly implemented with more lightweight architectures, which offers more efficient training and easier integration into existing multi-task setups. However, learning meaningful embeddings from complex data to represent individual objects remains a challenging problem. Finally, a number of hybrid approaches exist that aim to learn embeddings of larger regions in images [7], [13].\n3D Object Detection and Segmentation: Detecting and segmenting objects in point clouds has received a considerable amount of attention. PointNets [41] directly consume raw point clouds to generate latent representations that can be used for classification or semantic segmentation tasks. By leveraging various geometric relationships and representations a number of methods have since then improved upon the performance to solve for tasks, such as object detection [54], semantic segmentation [30], [42], or instance segmentation [45], [51]. A number of approaches rely on Bird's Eye View (BEV) representations to generate bounding boxes [50], [14] that -along with pillar-based representations [31], [46] are popular in autonomous driving. More recent methods for 3D instance segmentation rely on more principled approaches. To this end 3D-MPA [19] uses an aggregation-based approach where each point votes for its object center, HAIS [11] exploits the spatial relation of points and point sets, and SSTNet [32] uses a hierarchical representation which is constructed based on learned semantic features.\nMultimodal Approaches. There also exists a limited number of multimodal approaches proposed in the roboticvision domain, including [48], [37]. Most of these works tackle indoor environments using closely paired RGB-D data.\nSimilar to these methods, our goal is to leverage multimodal sensor data. Specifically, we use a single-stage fullyconvolutional network to learn per-point and per-pixel embeddings for instance segmentation. We train these embeddings with a metric learning loss that contrasts embeddings of the same object instance with those of different objects. The key idea to our approach is that we obtain the positive and negatives samples spatially, within the same frame, across different modalities (images, points), and even across time, in the adjacent frames of a sequence. We accomplish this by proposing a novel cross-modal consistency loss that allows us to sample embedding vectors from different sources. Training a single-stage network with this loss enables us to compute dense embeddings and, consequently, more robust instance masks for each frame, while also obtaining temporally stable results for sequences."},{"title":"III. METHOD","content":"In this section, we introduce our model architecture, the cross-modal contrastive loss function, and implementation details. Our full pipeline is shown in Figure 2."},{"title":"A. Cross-Modal Contrastive Loss","content":"Our method consists of three main components: (1) a pair of instance segmentation networks that generate dense embeddings for RGB images and LiDAR range images. (2) a sampling based contrastive learning loss which constrains embeddings from the same instances to be similar, while forcing embeddings from different instances to be different.\n(3) a consistency module based on paired instance labels that introduces additional training data and correspondences between sensors and over temporally neighboring frames.\nContrastive Loss: The goal of the instance segmentation network is to segment object instances by learning dense embedding vectors. The network takes as input either RGB images or LiDAR range images [4], and outputs embeddings with c = 32 channels. To train the network, we use a metric learning loss for contrasting embeddings of the same instance with those of different instances. Embeddings of the same instance are pushed closer together during training, while those of different instances are pushed away in the embedding space.\nOur instance segmentation loss consists of the normalized temperature-scaled cross entropy (NT-XENT) loss defined in [12]. We define the per-pixel embeddings as E h×w×c , where h and w denote the spatial dimensions of the output, and c, the channels of the embedding at each pixel. For each frame, we randomly sample K 1 = 8192 embeddings, distributed evenly across all instances (visualized for RGB"},{"title":"Inference Embedding E 1, rgb","content":"Embedding E 2, rgb Embedding E n, rgb ..."},{"title":"Embeddings","content":""},{"title":"Frame t1","content":"Frame t2 Fig. 4: Given a set of predicted embeddings with paired instance labels, we sample positive embedding vectors within the instance mask of an object in the current frame as well in all other frames (cross sensor and temporal). Samples from all other objects in all frames are used as negatives. We then train with our cross-modal contrastive loss to learn dense embeddings of object instances. The images at the bottom show the projected embeddings during training. frames in Figure 3). Given a set of sampled embeddings for an object instance, we generate all possible positive pairs. The samples of all other instances are used as negatives for each pair. For a set of samples U and a positive pair, e i , e j , our loss can then be defined as:\nwhere τ denotes a temperature hyperparameter and id(i) is the mask ID of point i. The similarity function sim is used to compute the cosine similarity between two normalized embedding vectors x and y. The total loss over the instance embeddings is then defined as:\nFurthermore, to obtain more stable results, we also apply a regularization loss over the norm of the embeddings:\nOur final loss, then, is:\nwhere λ = 0.01 is a weighting factor. Together, these losses allow us to learn meaningful embedding vectors as representation for object instances. Consistency Over Modalities and Time: Given a dataset with instance labels which are consistent between sensors and across time, such that an object has the same ID across all inputs, one can apply the above training loss (5) between all inputs by randomly sampling points from each set of embeddings, and using the consistent instance labels to determine positive and negative pairs. This allows the network to learn embeddings which are consistent not only within a single frame, but between all modalities and time steps seen at training.\nHowever, it is often the case that we do not have consistent instance labels. In these cases, we propose a method for generating pseudo-groundtruth using optical flow. Given a dataset with individually labeled frames amongst unlabeled temporal sequences, we use a pre-trained optical flow network to warp the groundtruth of the labeled frame to its temporal neighbors.\nWe use the predicted flow to warp the groundtruth instance and semantic labels of the current frame to the next frame and apply nearest neighbor resampling to avoid interpolating between integer label values. Following the work of Wang et al. [47], we compute occlusions according to the 'range map', defined by the number of points in the original image map to the other image. The instance and semantic labels of occluded pixels, as well as pixels that have left the image, are set to invalid, and are not sampled for the contrastive loss. The embeddings and warped labels for the next frame are then combined with those of the current frame when performing the sampling for the contrastive loss. Figure 4 illustrates the result of the warped labels as well as the sampling of embeddings from the two frames.\nThis pipeline has similarities to semi-supervised methods such as in the work by Chen et al. [8], which generate pseudo-groundtruth by running inference with a pre-trained model, and subsequently re-training. However, in addition to providing ground truth to temporally neighboring frames with optical flow, our method also ensures consistency between frames and thus over time. Sample outputs for our method over time can be found in Figure 5.\nSemantic Segmentation: We predict a semantic mask for each frame and filter out all 'stuff' classes to a single background class. During clustering, we assign semantic classes to each clustered instance with a 'majority vote' between semantic predictions within the instance. We also generate the confidence for each mask using the average semantic score. Pixels not belonging to the selected semantic class are dropped from the instance to preserve semantic boundaries. The one-hot semantic labels are concatenated to the predicted instance embeddings, so that the network only has to predict unique instance embeddings within each class. Fig. 5: Temporally-coherent embeddings: as our model is trained with a temporal consistency loss based on optical flow, we are able to obtain temporally-coherent and more stable instance embeddings. Here we show the RGB sequences of two different scenes along with the multi-dimensional per-pixel embedding vectors projected into a color space."},{"title":"Instance Mask Generation:","content":"At inference time, we cluster the resulting per-pixel embeddings by applying a variant of the mean shift algorithm proposed by Brabandere et al. [5]. In particular, we randomly sample a point in the embedding space, and find all inlier points with cosine distance (scaled to be within [0, 1]) less than a threshold, m = 0.1, from the sampled point. We then iterate by shifting the sampled point to the mean of the set of inliers, and repeat until convergence or some maximum number of iterations is reached. We repeat this process until all pixels have been clustered. With this method, we found that erroneous masks were typically generated on the transitions between masks, generating thin artifacts along the boundaries. To compensate, we filter out masks with an area to perimeter ratio less than a threshold, r = 4. As this method computes distances across the entire image, no assumptions about the connectivity of each mask are made, and so arbitrarily distributed instances can be detected.\nMore complex methods are available for generating instance masks from embeddings, such as the graph cut algorithm proposed in SSAP [21] or the transformer head proposed in MaX-DeepLab [44], which would likely achieve higher performance. However, these would incur additional latency penalties, and their computational power may hide improvements to the underlying embeddings."},{"title":"B. Implementation Details","content":"In this section, we provide details about the implementation of our network architecture.\nModel Architecture: We use a modified architecture based on Panoptic DeepLab [15]. Our model consists of a Xception-71 backbone, with an additional ASPP and decoder module for instance embedding prediction, with c = 32 channels, in place of the instance center and offset decoder in [15]. For our cross-modal experiments, we use two separate networks for LiDAR range image inputs and RGB image inputs.\nPreprocessing: For all datasets, we use full resolution images as input. During training, we augment our data by randomly flipping, scaling and cropping or padding the input images. Furthermore, we use random Gaussian blur and jitter brightness, contrast, saturation, and hue. For temporal data, we apply the same augmentation to each pair of images. For LiDAR range images, we only apply flipping, and ensure that the flipping is consistent with the camera images. 6: Cross-modal results on KITTI-360: Our models take as inputs RGB images or LiDAR range images (a), and predicts dense instance embeddings (b) which are consistent between the sensor modalities. At post-processing, we can apply a mean-shift clustering algorithm to generate instance masks for each modality (c), which we compare to the ground truth (d). Our cross-modal contrastive loss contrasts pairs between the sensor modalities, and trains the network to predict the same embedding for each object instance in each sensor, denoted by similarity in color. Each image pair shows both sensor modalities, where the LiDAR data is shown as range images (bottom).\nTraining: For all experiments, we use a moving average ADAM optimizer with linear warmup for 5 epochs, initial learning rate of 2.5e-4 and exponential decay with decay factor of 1.2 every 10 epochs. For the KITTI-360 dataset, we train for 10,000 steps for the ablation train/val split, and 20,000 steps for the train/test split, with an effective batch size of 128. For the Cityscapes dataset, we train for 60,000 iterations with an effective batch size of 32."},{"title":"IV. EXPERIMENTS","content":"In this section we present and discuss a variety of experiments, including baseline comparisons, ablation studies, and qualitative results our method is able to generate. For all of our experiments, we do not perform any kind of test time augmentation.\nA. Datasets a) KITTI-360: The KITTI-360 dataset [33] consists of camera and LiDAR panoptic segmentation and LiDAR bounding box labels for all of the sequences of the KITTI dataset [22]. For each sequence, the instance labels are consistent both between the camera and LiDAR labels, as well as between different time steps. The dataset contains 83,000 frames and associated LiDAR scans, split into 9 training sequences and 2 test sequences with held out groundtruth. As no validation set is provided, we perform our ablations on a train/val split where sequences 00, 02, 03, 04, 05, 06, 07 are the train set and sequences 09 and 10 are the validation set.\nThe validation set was chosen from the two densest sequences, such that there are almost the same number of objects in the training and validation sets. For the final test set evaluation, we train on the entire training set.\nAs the dataset is largely transferred from primitive 3D shapes (cuboids, ellipsoids etc.), each label has a corresponding confidence value, which is used to weight the evaluation metrics. In accordance with the experiments in KITTI-360, we only use the top 70% of labels in each frame, in terms of confidence. To generate temporal frames, we randomly sample an image 2 frames before or after the current frame at each training step. To train a model on LiDAR, we use the range image representation of the point cloud generated by Bewley et al. [4]. From this 64×2048 range image, we crop a central 64×512 patch corresponding to the camera field of view, and treat this as an image into the DeepLab model. To generate per-scan groundtruth, we use nearest-neighbors to find the closest ground truth segmentation label to each point in the range image. b) Cityscapes: To test our method on a dataset without consistent instance labels and in a single-modal setting, we employ the Cityscapes dataset [16]. dataset. Cityscapes provides us with 2975, 500, and 1525 images of urban scenes for training, testing and validation, respectively. Furthermore, the dataset provides 8 'thing' and 11 'stuff' classes. In this work, we train our network on the 'fine' set of training images with ground truth instance labels. In order to generate paired groundtruth, we use the unlabeled temporal frames, and randomly sample frames from the images 2 frames before or after the labeled image. We then warp the groundtruth from the labeled frame to each temporal frame using a UFlow [27] model trained on the Waymo Open Dataset [43], consisting of 200,000 training images. To filter potential errors in the flow warping due to occlusions and other errors, we ignore warped labels based on the occlusion map generated by the flow model. For Cityscapes, we rely on a pre-trained semantic segmentation network for assigning class IDs, in particular an implementation of Panoptic Deeplab [15], with an Xception-71 backbone, trained on Cityscapes. Our implementation has a mIOU of 68.8% on the Cityscapes validation set. As embeddings along the boundary are particularly challenging, and the Cityscapes labels are typically reliable, we dedicate half of the points when sampling in the contrastive loss to embeddings that are within b = 10 pixels of the boundary of each instance mask."},{"title":"B. Results on KITTI-360","content":"In Table I, we report an ablation study on the effects of adding additional signals for the contrastive loss, both temporally and from different modalities. Overall, we observe that instance mAP for images improves as we add each additional frame to the consistency loss. The overall trend that we observe is that the network improves significantly for rare classes such as bicycle, motorcycle, rider and truck, while regressing slightly for the more common car class. This results in a significant overall mAP improvement. Note that, because the ablation models are trained on a much smaller training set, the mAP numbers in these experiments are naturally lower than those for the test set. We show qualitative examples from our best model (Temporal RGB + Temporal LiDAR) in Figure 6, where we demonstrate that the networks are able to predict embeddings that are consistent for objects across the sensor modalities. We also show our LiDAR instance segmentation results projected to 3D in Figure 7, demonstrating that the network can generate accurate 3D instance segmentation when operating only on the range image. However, we did not observe similar improvements in 3D LiDAR mAP when training with the proposed crossmodality contrastive loss. Our observation is that, as the 2D labels are generated by projecting the 3D labels (and applying a CRF), they are typically much noisier than the 3D labels, and cannot provide useful signal for the 3D part of the model.\nIn Table II, we report a comparison in the mAP of our final method against the popular MaskRCNN method [25] on the held-out test set, where we outperform the ResNet-50 baseline, while approaching the performance of the ResNet-101 baseline. At the time of writing, the validation set for this dataset was not available, and better results will be achievable once the same training splits are available as for the baselines."},{"title":"C. Results on Cityscapes","content":"In Table III, we report our instance segmentation results on Cityscapes compared to existing methods. The reported results were obtained on the Cityscapes validation set. To provide a fair comparison of the main methods themselves, we use the reported numbers for each method without test time augmentation. In addition, we report mAP for our method using groundtruth semantic segmentations in Table IV, and compare against past work which has used a similar metric. This approach allows us to evaluate the instance segmentation network independently from the performance of the semantic segmentation. Overall, the improvement from our crossmodal consistency loss allows us to strongly outperform bottom up methods such as Brabandere et al. [5] and Neven et al. [38] with groundtruth semantics. It also pushes our embedding-based method close to the previous bottom up works such as SSAP [21]. However, this work requires a complex post-processing procedure which has a significant latency, as reported in Table III. We underperform compared to Panoptic DeepLab, but note that this method, while being single shot, requires the prediction of object centers. These centers serve as top-down proposals requiring instances to have distinguishable centers and some form of NMS.\nWe also show visualizations of our predictions in Figure 8, where the embeddings can robustly and precisely represent a large variety of object instances.\nTemporal Constraints: To validate the impact of the temporal contrastive loss, we trained our model with and without contrasting over time (see Table III, Ours base vs Ours temporal). From these results, introducing the contrastive Fig. 8: Visualization of per-pixel embedding vectors on Cityscapes: our method allows us to obtain per-pixel embedding vectors from RGB input images (first row). The embeddings robustly and precisely disentangle object instances across different classes (third row). To visualize the embeddings, we project them into RGB color space (second row). In the last row, we show the ground truth instance masks. Please note that the color of the predicted and ground truth masks are not supposed to match. loss over time has a significant impact on the predicted instances masks. Adding temporal frames to the contrastive loss improves overall AP by 2%. The visualization of the predicted embeddings in Figure 5 also shows fewer artifacts.\nWe also compute error metrics on the stability of the instance embeddings over time for the Cityscapes dataset. Given two frames, we warp the predicted embeddings of the next frame to the current frame, and compute the cosine distance between the average embedding of each instance in the two time steps, while ignoring any pixels estimated as 'occlusion' from the optical flow. In addition, we compute as accuracy the proportion of instance pairs with cosine distance less than the clustering threshold m = 0.1. These results can be found in Table V. We observe that the majority of embeddings (75.1%) are stable over time, even without the proposed temporal contrastive loss. However, there are a number of cases, such as large motions and dense objects, where the embeddings are not stable. Overall, our proposed loss reduces the cosine distance between instances by 0.025, and improves accuracy by 8.3% to 83.4%.\nEmbedding Similarity: In Figure 9 we show the results of an embedding similarity experiment. We select pixels in the image and compute the distance, shown in grayscale, of the selected embeddings with all other embeddings in the image. As illustrated, the learned embeddings allow us to disentangle object instances of the same class, across different classes, and the background. Our simple clustering scheme can generate high quality masks by thresholding this similarity."},{"title":"V. LIMITATIONS AND FUTURE WORK","content":"The main contribution of this method is the contrastive learning of instance embeddings between sensors and over time. As it stands, we believe that the learned embeddings are strong, but have chosen a relatively simple method to generate instance masks for evaluation. We use this postprocessing to highlight the improvements in the embeddings themselves, where a more complex method may obscure such improvements. However, there is room for improvement if stronger AP scores are desired, by adding a graph cut optimization such as in SSAP [21] or a transformer head on top of the embeddings as in MaX-DeepLab [44]."},{"title":"VI. CONCLUSION","content":"We have introduced a novel method for learning pixel-wise embeddings as a representation for individual object instances. Compared to other bottom-up approaches, we employ a contrastive learning scheme; embeddings of the same object instance are pushed closer together, while they are pulled away from other objects and the background. In particular, we leverage information between sensors, as well as over time, by applying our cross-modal contrastive loss on the union of all predicted embeddings for a given scene. Through quantitative experiments, we show that this contrastive loss is able to significantly improve instance segmentation performance, and coherent instance embeddings between sensors and time, which are clustered to generate high quality instance masks."},{"title":"Fig. 3 :","content":""},{"title":"Fig. 9 :","content":""},{"title":"TABLE I :","content":""},{"title":"TABLE II :","content":""},{"title":"TABLE III :","content":""},{"title":"TABLE IV :","content":""},{"title":"TABLE V :","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Identifying and segmenting objects is a fundamental and challenging task in computer vision, and critically important for many robotics applications, such as autonomous driving. Given some sensor input such as a camera image or LiDAR scan, the goal of instance segmentation is to associate each pixel or point with a unique instance label for the object to which it corresponds. This allows for fine-grained reasoning about each object beyond sparse representations such as bounding boxes. Only recently, learning-based approaches have shown remarkable success in solving this task on a number of benchmark datasets [16], [6], [34]. Reliably identifying object instances in complex real-world data, both across different data modalities and coherent in time, remains a challenging and open problem [49], [15], [8].\\nMany safety-critical robotics systems, such as autonomous vehicles, rely on multiple sensors that provide complementary information to perceive the environment. For instance, LiDAR sensors provide highly accurate range readings, but only in clear weather conditions and up to a limited range. On the other hand, cameras may be able to perceive objects at much longer ranges, but without any explicit range readings and only in adequate lighting conditions. As a result, there is an opportunity to improve segmentation performance for each sensor by reasoning jointly across all modalities.\\nAnother avenue is the utilization of temporal data, which can provide motion information, varying viewpoints and context for occlusions. Temporal data comes naturally for most sensors, although sequential labeling for ground truth is more expensive. 1 Waymo LLC 2 Adobe Research (Work done while at Google Research) Fig. 1: Our approach to instance segmentation leverages information from multiple sensor modalities, such as 3D LiDAR point clouds and camera images (left). Our method learns to predict dense embeddings (depicted by the colors, right image) based on a novel cross-modal contrastive loss that uses samples from different sensor modalities and temporally consecutive frames of RGB images and LiDAR points. The resulting embeddings are stable over time and invariant to viewpoint changes.\\nIn this paper, we aim to leverage these complementary signals and advance instance segmentation by training jointly on different data modalities and over time ( Figure 1). Specifically, we propose a novel cross-modal contrastive loss that enables us to train with sequential samples from camera images and LiDAR point clouds to learn a dense representation of object instances. Our goal is to learn embeddings that coherently represent individual objects in both modalities. By training on different modalities, our method can more meaningfully disentangle individual instances. Moreover, we show that our loss formulation is flexible enough to be applied to sequences of sensor data of both modalities, enabling learning temporally-stable embeddings for object instances. For situations where only sequential data (but no labels) are available, we propose a method for generating pseudogroundtruth using optical flow, and demonstrate that training on this data brings similar improvements in segmentation.\\nIn summary, our contributions are as follows:\\n• We introduce a novel cross-modal contrastive loss to obtain consistent instance embeddings for RGB images and point clouds. • We show that our loss can also be used to train on sequences of sensor data from both modalities, enabling the learning of temporally stable embeddings. • We propose a method using optical flow to generate pseudo ground-truth for video, allowing the extension of our contrastive loss to partially labeled datasets. • Our extensive experiments on the KITTI-360 [33] and Cityscapes [16] benchmarks suggest that our method significantly improves instance segmentation quality by up to 1.9 mAP when contrasting between temporal RGB data, and by up to 1.7 mAP when contrasting between temporal LiDAR and RGB data, while also improving the temporal stability of the embeddings.\"\n },\n {\n \"title\": \"RGB\",\n \"content\": \"\"\n },\n {\n \"title\": \"ConvNet rgb\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2:\",\n \"content\": \"Our goal is to learn dense cross-modal embeddings via contrastive learning to segment individual object instances. Our method leverages shared information between sensors and consecutive frames by applying the contrastive loss jointly over embeddings from all input sources at training time. This global contrastive loss allows the network to share information between different data sources, and improves overall segmentation performance. Once trained, the learned embeddings can be used to obtain object IDs through mean shift clustering.\"\n },\n {\n \"title\": \"II. RELATED WORK\",\n \"content\": \"Object detection and segmentation have a long tradition in computer vision. While this spans a wide range of different approaches that we cannot conclusively discuss, we aim to provide an overview of learning-based methods toward instance and semantic segmentation in images and point clouds.\\nSemantic Segmentation. The goal of semantic segmentation is to associate every pixel in an image with a discrete label that defines its semantic class. A major challenge of effectively training networks for semantic segmentation is to define precise labels, and a number of different datasets were introduced for benchmarking [16], [22], [6]. Many recent methods for semantic segmentation employ deep neural network architectures for feature aggregation based on pyramid pooling [53], attention modules [10], multi-scale activations [24], or end-to-end convolutions [36]. Furthermore, it has been recognized that dilated or atrous convolutions provide a powerful way of capturing more global features through wider receptive fields [52], [9]. Other approaches aim to solve semantic segmentation with an emphasis on learning structured representations based on conditional random fields [1], [9] or deep parsing networks [35].\\nInstance Segmentation expands on the concept of semantic segmentation. Here the goal is to not only obtain a class label for every pixel, but to predict labels for individual object instances. Instance segmentation approaches can be separated into two different categories. In the first category, most methods rely on a two stage (or top-down) process [17], [18], [25]. A region proposal step [23] identifies object instances in a first stage and each region is then segmented into foreground and background during a second stage. The performance of top-down approaches is determined by obtaining region proposals and by the number of objects present in an image. On the contrary, bottom-up approaches aim to learn per-pixel embeddings that represent individual object instances based on recurrent instance grouping [29], watershed transformations [3], subnetwork instantiations [2], associative embeddings [39], or based on more refined loss functions such as discriminative loss [5], [26], or metric learning [20]. Bottom-up approaches are commonly implemented with more lightweight architectures, which offers more efficient training and easier integration into existing multi-task setups. However, learning meaningful embeddings from complex data to represent individual objects remains a challenging problem. Finally, a number of hybrid approaches exist that aim to learn embeddings of larger regions in images [7], [13].\\n3D Object Detection and Segmentation: Detecting and segmenting objects in point clouds has received a considerable amount of attention. PointNets [41] directly consume raw point clouds to generate latent representations that can be used for classification or semantic segmentation tasks. By leveraging various geometric relationships and representations a number of methods have since then improved upon the performance to solve for tasks, such as object detection [54], semantic segmentation [30], [42], or instance segmentation [45], [51]. A number of approaches rely on Bird's Eye View (BEV) representations to generate bounding boxes [50], [14] that -along with pillar-based representations [31], [46] are popular in autonomous driving. More recent methods for 3D instance segmentation rely on more principled approaches. To this end 3D-MPA [19] uses an aggregation-based approach where each point votes for its object center, HAIS [11] exploits the spatial relation of points and point sets, and SSTNet [32] uses a hierarchical representation which is constructed based on learned semantic features.\\nMultimodal Approaches. There also exists a limited number of multimodal approaches proposed in the roboticvision domain, including [48], [37]. Most of these works tackle indoor environments using closely paired RGB-D data.\\nSimilar to these methods, our goal is to leverage multimodal sensor data. Specifically, we use a single-stage fullyconvolutional network to learn per-point and per-pixel embeddings for instance segmentation. We train these embeddings with a metric learning loss that contrasts embeddings of the same object instance with those of different objects. The key idea to our approach is that we obtain the positive and negatives samples spatially, within the same frame, across different modalities (images, points), and even across time, in the adjacent frames of a sequence. We accomplish this by proposing a novel cross-modal consistency loss that allows us to sample embedding vectors from different sources. Training a single-stage network with this loss enables us to compute dense embeddings and, consequently, more robust instance masks for each frame, while also obtaining temporally stable results for sequences.\"\n },\n {\n \"title\": \"III. METHOD\",\n \"content\": \"In this section, we introduce our model architecture, the cross-modal contrastive loss function, and implementation details. Our full pipeline is shown in Figure 2.\"\n },\n {\n \"title\": \"A. Cross-Modal Contrastive Loss\",\n \"content\": \"Our method consists of three main components: (1) a pair of instance segmentation networks that generate dense embeddings for RGB images and LiDAR range images. (2) a sampling based contrastive learning loss which constrains embeddings from the same instances to be similar, while forcing embeddings from different instances to be different.\\n(3) a consistency module based on paired instance labels that introduces additional training data and correspondences between sensors and over temporally neighboring frames.\\nContrastive Loss: The goal of the instance segmentation network is to segment object instances by learning dense embedding vectors. The network takes as input either RGB images or LiDAR range images [4], and outputs embeddings with c = 32 channels. To train the network, we use a metric learning loss for contrasting embeddings of the same instance with those of different instances. Embeddings of the same instance are pushed closer together during training, while those of different instances are pushed away in the embedding space.\\nOur instance segmentation loss consists of the normalized temperature-scaled cross entropy (NT-XENT) loss defined in [12]. We define the per-pixel embeddings as E h×w×c , where h and w denote the spatial dimensions of the output, and c, the channels of the embedding at each pixel. For each frame, we randomly sample K 1 = 8192 embeddings, distributed evenly across all instances (visualized for RGB\"\n },\n {\n \"title\": \"Inference Embedding E 1, rgb\",\n \"content\": \"Embedding E 2, rgb Embedding E n, rgb ...\"\n },\n {\n \"title\": \"Embeddings\",\n \"content\": \"\"\n },\n {\n \"title\": \"Frame t1\",\n \"content\": \"Frame t2 Fig. 4: Given a set of predicted embeddings with paired instance labels, we sample positive embedding vectors within the instance mask of an object in the current frame as well in all other frames (cross sensor and temporal). Samples from all other objects in all frames are used as negatives. We then train with our cross-modal contrastive loss to learn dense embeddings of object instances. The images at the bottom show the projected embeddings during training. frames in Figure 3). Given a set of sampled embeddings for an object instance, we generate all possible positive pairs. The samples of all other instances are used as negatives for each pair. For a set of samples U and a positive pair, e i , e j , our loss can then be defined as:\\nwhere τ denotes a temperature hyperparameter and id(i) is the mask ID of point i. The similarity function sim is used to compute the cosine similarity between two normalized embedding vectors x and y. The total loss over the instance embeddings is then defined as:\\nFurthermore, to obtain more stable results, we also apply a regularization loss over the norm of the embeddings:\\nOur final loss, then, is:\\nwhere λ = 0.01 is a weighting factor. Together, these losses allow us to learn meaningful embedding vectors as representation for object instances. Consistency Over Modalities and Time: Given a dataset with instance labels which are consistent between sensors and across time, such that an object has the same ID across all inputs, one can apply the above training loss (5) between all inputs by randomly sampling points from each set of embeddings, and using the consistent instance labels to determine positive and negative pairs. This allows the network to learn embeddings which are consistent not only within a single frame, but between all modalities and time steps seen at training.\\nHowever, it is often the case that we do not have consistent instance labels. In these cases, we propose a method for generating pseudo-groundtruth using optical flow. Given a dataset with individually labeled frames amongst unlabeled temporal sequences, we use a pre-trained optical flow network to warp the groundtruth of the labeled frame to its temporal neighbors.\\nWe use the predicted flow to warp the groundtruth instance and semantic labels of the current frame to the next frame and apply nearest neighbor resampling to avoid interpolating between integer label values. Following the work of Wang et al. [47], we compute occlusions according to the 'range map', defined by the number of points in the original image map to the other image. The instance and semantic labels of occluded pixels, as well as pixels that have left the image, are set to invalid, and are not sampled for the contrastive loss. The embeddings and warped labels for the next frame are then combined with those of the current frame when performing the sampling for the contrastive loss. Figure 4 illustrates the result of the warped labels as well as the sampling of embeddings from the two frames.\\nThis pipeline has similarities to semi-supervised methods such as in the work by Chen et al. [8], which generate pseudo-groundtruth by running inference with a pre-trained model, and subsequently re-training. However, in addition to providing ground truth to temporally neighboring frames with optical flow, our method also ensures consistency between frames and thus over time. Sample outputs for our method over time can be found in Figure 5.\\nSemantic Segmentation: We predict a semantic mask for each frame and filter out all 'stuff' classes to a single background class. During clustering, we assign semantic classes to each clustered instance with a 'majority vote' between semantic predictions within the instance. We also generate the confidence for each mask using the average semantic score. Pixels not belonging to the selected semantic class are dropped from the instance to preserve semantic boundaries. The one-hot semantic labels are concatenated to the predicted instance embeddings, so that the network only has to predict unique instance embeddings within each class. Fig. 5: Temporally-coherent embeddings: as our model is trained with a temporal consistency loss based on optical flow, we are able to obtain temporally-coherent and more stable instance embeddings. Here we show the RGB sequences of two different scenes along with the multi-dimensional per-pixel embedding vectors projected into a color space.\"\n },\n {\n \"title\": \"Instance Mask Generation:\",\n \"content\": \"At inference time, we cluster the resulting per-pixel embeddings by applying a variant of the mean shift algorithm proposed by Brabandere et al. [5]. In particular, we randomly sample a point in the embedding space, and find all inlier points with cosine distance (scaled to be within [0, 1]) less than a threshold, m = 0.1, from the sampled point. We then iterate by shifting the sampled point to the mean of the set of inliers, and repeat until convergence or some maximum number of iterations is reached. We repeat this process until all pixels have been clustered. With this method, we found that erroneous masks were typically generated on the transitions between masks, generating thin artifacts along the boundaries. To compensate, we filter out masks with an area to perimeter ratio less than a threshold, r = 4. As this method computes distances across the entire image, no assumptions about the connectivity of each mask are made, and so arbitrarily distributed instances can be detected.\\nMore complex methods are available for generating instance masks from embeddings, such as the graph cut algorithm proposed in SSAP [21] or the transformer head proposed in MaX-DeepLab [44], which would likely achieve higher performance. However, these would incur additional latency penalties, and their computational power may hide improvements to the underlying embeddings.\"\n },\n {\n \"title\": \"B. Implementation Details\",\n \"content\": \"In this section, we provide details about the implementation of our network architecture.\\nModel Architecture: We use a modified architecture based on Panoptic DeepLab [15]. Our model consists of a Xception-71 backbone, with an additional ASPP and decoder module for instance embedding prediction, with c = 32 channels, in place of the instance center and offset decoder in [15]. For our cross-modal experiments, we use two separate networks for LiDAR range image inputs and RGB image inputs.\\nPreprocessing: For all datasets, we use full resolution images as input. During training, we augment our data by randomly flipping, scaling and cropping or padding the input images. Furthermore, we use random Gaussian blur and jitter brightness, contrast, saturation, and hue. For temporal data, we apply the same augmentation to each pair of images. For LiDAR range images, we only apply flipping, and ensure that the flipping is consistent with the camera images. 6: Cross-modal results on KITTI-360: Our models take as inputs RGB images or LiDAR range images (a), and predicts dense instance embeddings (b) which are consistent between the sensor modalities. At post-processing, we can apply a mean-shift clustering algorithm to generate instance masks for each modality (c), which we compare to the ground truth (d). Our cross-modal contrastive loss contrasts pairs between the sensor modalities, and trains the network to predict the same embedding for each object instance in each sensor, denoted by similarity in color. Each image pair shows both sensor modalities, where the LiDAR data is shown as range images (bottom).\\nTraining: For all experiments, we use a moving average ADAM optimizer with linear warmup for 5 epochs, initial learning rate of 2.5e-4 and exponential decay with decay factor of 1.2 every 10 epochs. For the KITTI-360 dataset, we train for 10,000 steps for the ablation train/val split, and 20,000 steps for the train/test split, with an effective batch size of 128. For the Cityscapes dataset, we train for 60,000 iterations with an effective batch size of 32.\"\n },\n {\n \"title\": \"IV. EXPERIMENTS\",\n \"content\": \"In this section we present and discuss a variety of experiments, including baseline comparisons, ablation studies, and qualitative results our method is able to generate. For all of our experiments, we do not perform any kind of test time augmentation.\\nA. Datasets a) KITTI-360: The KITTI-360 dataset [33] consists of camera and LiDAR panoptic segmentation and LiDAR bounding box labels for all of the sequences of the KITTI dataset [22]. For each sequence, the instance labels are consistent both between the camera and LiDAR labels, as well as between different time steps. The dataset contains 83,000 frames and associated LiDAR scans, split into 9 training sequences and 2 test sequences with held out groundtruth. As no validation set is provided, we perform our ablations on a train/val split where sequences 00, 02, 03, 04, 05, 06, 07 are the train set and sequences 09 and 10 are the validation set.\\nThe validation set was chosen from the two densest sequences, such that there are almost the same number of objects in the training and validation sets. For the final test set evaluation, we train on the entire training set.\\nAs the dataset is largely transferred from primitive 3D shapes (cuboids, ellipsoids etc.), each label has a corresponding confidence value, which is used to weight the evaluation metrics. In accordance with the experiments in KITTI-360, we only use the top 70% of labels in each frame, in terms of confidence. To generate temporal frames, we randomly sample an image 2 frames before or after the current frame at each training step. To train a model on LiDAR, we use the range image representation of the point cloud generated by Bewley et al. [4]. From this 64×2048 range image, we crop a central 64×512 patch corresponding to the camera field of view, and treat this as an image into the DeepLab model. To generate per-scan groundtruth, we use nearest-neighbors to find the closest ground truth segmentation label to each point in the range image. b) Cityscapes: To test our method on a dataset without consistent instance labels and in a single-modal setting, we employ the Cityscapes dataset [16]. dataset. Cityscapes provides us with 2975, 500, and 1525 images of urban scenes for training, testing and validation, respectively. Furthermore, the dataset provides 8 'thing' and 11 'stuff' classes. In this work, we train our network on the 'fine' set of training images with ground truth instance labels. In order to generate paired groundtruth, we use the unlabeled temporal frames, and randomly sample frames from the images 2 frames before or after the labeled image. We then warp the groundtruth from the labeled frame to each temporal frame using a UFlow [27] model trained on the Waymo Open Dataset [43], consisting of 200,000 training images. To filter potential errors in the flow warping due to occlusions and other errors, we ignore warped labels based on the occlusion map generated by the flow model. For Cityscapes, we rely on a pre-trained semantic segmentation network for assigning class IDs, in particular an implementation of Panoptic Deeplab [15], with an Xception-71 backbone, trained on Cityscapes. Our implementation has a mIOU of 68.8% on the Cityscapes validation set. As embeddings along the boundary are particularly challenging, and the Cityscapes labels are typically reliable, we dedicate half of the points when sampling in the contrastive loss to embeddings that are within b = 10 pixels of the boundary of each instance mask.\"\n },\n {\n \"title\": \"B. Results on KITTI-360\",\n \"content\": \"In Table I, we report an ablation study on the effects of adding additional signals for the contrastive loss, both temporally and from different modalities. Overall, we observe that instance mAP for images improves as we add each additional frame to the consistency loss. The overall trend that we observe is that the network improves significantly for rare classes such as bicycle, motorcycle, rider and truck, while regressing slightly for the more common car class. This results in a significant overall mAP improvement. Note that, because the ablation models are trained on a much smaller training set, the mAP numbers in these experiments are naturally lower than those for the test set. We show qualitative examples from our best model (Temporal RGB + Temporal LiDAR) in Figure 6, where we demonstrate that the networks are able to predict embeddings that are consistent for objects across the sensor modalities. We also show our LiDAR instance segmentation results projected to 3D in Figure 7, demonstrating that the network can generate accurate 3D instance segmentation when operating only on the range image. However, we did not observe similar improvements in 3D LiDAR mAP when training with the proposed crossmodality contrastive loss. Our observation is that, as the 2D labels are generated by projecting the 3D labels (and applying a CRF), they are typically much noisier than the 3D labels, and cannot provide useful signal for the 3D part of the model.\\nIn Table II, we report a comparison in the mAP of our final method against the popular MaskRCNN method [25] on the held-out test set, where we outperform the ResNet-50 baseline, while approaching the performance of the ResNet-101 baseline. At the time of writing, the validation set for this dataset was not available, and better results will be achievable once the same training splits are available as for the baselines.\"\n },\n {\n \"title\": \"C. Results on Cityscapes\",\n \"content\": \"In Table III, we report our instance segmentation results on Cityscapes compared to existing methods. The reported results were obtained on the Cityscapes validation set. To provide a fair comparison of the main methods themselves, we use the reported numbers for each method without test time augmentation. In addition, we report mAP for our method using groundtruth semantic segmentations in Table IV, and compare against past work which has used a similar metric. This approach allows us to evaluate the instance segmentation network independently from the performance of the semantic segmentation. Overall, the improvement from our crossmodal consistency loss allows us to strongly outperform bottom up methods such as Brabandere et al. [5] and Neven et al. [38] with groundtruth semantics. It also pushes our embedding-based method close to the previous bottom up works such as SSAP [21]. However, this work requires a complex post-processing procedure which has a significant latency, as reported in Table III. We underperform compared to Panoptic DeepLab, but note that this method, while being single shot, requires the prediction of object centers. These centers serve as top-down proposals requiring instances to have distinguishable centers and some form of NMS.\\nWe also show visualizations of our predictions in Figure 8, where the embeddings can robustly and precisely represent a large variety of object instances.\\nTemporal Constraints: To validate the impact of the temporal contrastive loss, we trained our model with and without contrasting over time (see Table III, Ours base vs Ours temporal). From these results, introducing the contrastive Fig. 8: Visualization of per-pixel embedding vectors on Cityscapes: our method allows us to obtain per-pixel embedding vectors from RGB input images (first row). The embeddings robustly and precisely disentangle object instances across different classes (third row). To visualize the embeddings, we project them into RGB color space (second row). In the last row, we show the ground truth instance masks. Please note that the color of the predicted and ground truth masks are not supposed to match. loss over time has a significant impact on the predicted instances masks. Adding temporal frames to the contrastive loss improves overall AP by 2%. The visualization of the predicted embeddings in Figure 5 also shows fewer artifacts.\\nWe also compute error metrics on the stability of the instance embeddings over time for the Cityscapes dataset. Given two frames, we warp the predicted embeddings of the next frame to the current frame, and compute the cosine distance between the average embedding of each instance in the two time steps, while ignoring any pixels estimated as 'occlusion' from the optical flow. In addition, we compute as accuracy the proportion of instance pairs with cosine distance less than the clustering threshold m = 0.1. These results can be found in Table V. We observe that the majority of embeddings (75.1%) are stable over time, even without the proposed temporal contrastive loss. However, there are a number of cases, such as large motions and dense objects, where the embeddings are not stable. Overall, our proposed loss reduces the cosine distance between instances by 0.025, and improves accuracy by 8.3% to 83.4%.\\nEmbedding Similarity: In Figure 9 we show the results of an embedding similarity experiment. We select pixels in the image and compute the distance, shown in grayscale, of the selected embeddings with all other embeddings in the image. As illustrated, the learned embeddings allow us to disentangle object instances of the same class, across different classes, and the background. Our simple clustering scheme can generate high quality masks by thresholding this similarity.\"\n },\n {\n \"title\": \"V. LIMITATIONS AND FUTURE WORK\",\n \"content\": \"The main contribution of this method is the contrastive learning of instance embeddings between sensors and over time. As it stands, we believe that the learned embeddings are strong, but have chosen a relatively simple method to generate instance masks for evaluation. We use this postprocessing to highlight the improvements in the embeddings themselves, where a more complex method may obscure such improvements. However, there is room for improvement if stronger AP scores are desired, by adding a graph cut optimization such as in SSAP [21] or a transformer head on top of the embeddings as in MaX-DeepLab [44].\"\n },\n {\n \"title\": \"VI. CONCLUSION\",\n \"content\": \"We have introduced a novel method for learning pixel-wise embeddings as a representation for individual object instances. Compared to other bottom-up approaches, we employ a contrastive learning scheme; embeddings of the same object instance are pushed closer together, while they are pulled away from other objects and the background. In particular, we leverage information between sensors, as well as over time, by applying our cross-modal contrastive loss on the union of all predicted embeddings for a given scene. Through quantitative experiments, we show that this contrastive loss is able to significantly improve instance segmentation performance, and coherent instance embeddings between sensors and time, which are clustered to generate high quality instance masks.\"\n },\n {\n \"title\": \"Fig. 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 9 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I :\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE II :\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE III :\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE IV :\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE V :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Example of our sampling strategy where points are evenly distributed amongst instances. The bottom row shows close-ups of the used sample locations.","content":"Fig. 3 :\n3Example of our sampling strategy where points are evenly distributed amongst instances. The bottom row shows close-ups of the used sample locations."},{"id":"","caption":"Pixel similarity: for an image with objects of different classes (a) we randomly select different pixel positions in the image (yellow dot) and show the similarity of the selected embedding to all other embeddings as grayscale values. As shown, the learned embeddings allow us to meaningfully disentangle object instances, such as various different cars (b, c, d), intricate objects like bicycles (e), and small-scale objects, such as pedestrians (f).","content":"Fig. 9 :\n9Pixel similarity: for an image with objects of different classes (a) we randomly select different pixel positions in the image (yellow dot) and show the similarity of the selected embedding to all other embeddings as grayscale values. As shown, the learned embeddings allow us to meaningfully disentangle object instances, such as various different cars (b, c, d), intricate objects like bicycles (e), and small-scale objects, such as pedestrians (f)."},{"id":"","caption":"Ablation study on the effects of additional consistency on instance segmentation performance on a custom KITTI 360 train/val split. Adding additional inputs for the contrastive loss improves overall segmentation performance. Performance is measured in terms of AP (%).","content":"TABLE I :\nIAblation study on the effects of additional consistency on instance segmentation performance on a custom KITTI 360 train/val split. Adding additional inputs for the contrastive loss improves overall segmentation performance. Performance is measured in terms of AP (%).Fig."},{"id":"","caption":"Results on the KITTI-360 held out test set.","content":"TABLE II :\nIIResults on the KITTI-360 held out test set.Fig. 7: Close-up analysis of our 3D LiDAR segmentation network, operating \non range images (top-left), demonstrating accurate point-cloud instance \nsegmentation. \n\nMethod \nApproach \nInput Size \nAP (%) Speed (ms) \nPanoptic FPN [28] \nTD \n512× 1024 \n33.2 \n-\nUPSNet [49] \nTD \n1024×2048 \n33.3 \n202ms \nSeamless [40] \nTD \n1024× 2048 \n33.6 \n-\nPanoptic Deeplab [15] Xception-71 \nTD 1 \n1025×2049 \n35.3 \n175ms \nSSAP [21] \nBU \n1024× 2048 \n31.5 \n260ms+ 2 \nOurs base \nBU-C \n1024× 2048 \n29.0 \n182ms \nOurs temporal \nBU-C \n1024× 2048 \n30.9 \n182ms"},{"id":"","caption":"Cityscapes validation set results using groundtruth semantics for assigning classes to each instance. BU: Bottom-Up. BU-C: Bottom-Up via Contrastive Learning. For this setup, we significantly outperform other contrastive learning methods.","content":"TABLE III :\nIIICityscapes validation set results, without any test time aug-\nmentation or additional training data for all methods. TD: Top-Down. BU: \nBottom-Up. BU-C: Bottom-Up via Contrastive Learning. 1 Panoptic Deeplab \nis a single stage network, but relies on top-down object proposals of object \ncenters. 2 Reported time is for post-processing only and does not include the \nnetwork forward pass."},{"id":"","caption":"Temporal consistency metrics. We compute the cosine distance (in [0, 1]) between the average embedding for each instance between two temporal frames. Accuracy is the percentage of instances with cosine distance < m = 0.1 over time. Training with temporal consistency significantly increases performance.","content":"TABLE IV :\nIVCityscapes validation set results using groundtruth semantics for assigning classes to each instance. BU: Bottom-Up. BU-C: Bottom-Up via Contrastive Learning. For this setup, we significantly outperform other contrastive learning methods."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Example of our sampling strategy where points are evenly distributed amongst instances. The bottom row shows close-ups of the used sample locations.\",\n \"content\": \"Fig. 3 :\\n3Example of our sampling strategy where points are evenly distributed amongst instances. The bottom row shows close-ups of the used sample locations.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Pixel similarity: for an image with objects of different classes (a) we randomly select different pixel positions in the image (yellow dot) and show the similarity of the selected embedding to all other embeddings as grayscale values. As shown, the learned embeddings allow us to meaningfully disentangle object instances, such as various different cars (b, c, d), intricate objects like bicycles (e), and small-scale objects, such as pedestrians (f).\",\n \"content\": \"Fig. 9 :\\n9Pixel similarity: for an image with objects of different classes (a) we randomly select different pixel positions in the image (yellow dot) and show the similarity of the selected embedding to all other embeddings as grayscale values. As shown, the learned embeddings allow us to meaningfully disentangle object instances, such as various different cars (b, c, d), intricate objects like bicycles (e), and small-scale objects, such as pedestrians (f).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ablation study on the effects of additional consistency on instance segmentation performance on a custom KITTI 360 train/val split. Adding additional inputs for the contrastive loss improves overall segmentation performance. Performance is measured in terms of AP (%).\",\n \"content\": \"TABLE I :\\nIAblation study on the effects of additional consistency on instance segmentation performance on a custom KITTI 360 train/val split. Adding additional inputs for the contrastive loss improves overall segmentation performance. Performance is measured in terms of AP (%).Fig.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Results on the KITTI-360 held out test set.\",\n \"content\": \"TABLE II :\\nIIResults on the KITTI-360 held out test set.Fig. 7: Close-up analysis of our 3D LiDAR segmentation network, operating \\non range images (top-left), demonstrating accurate point-cloud instance \\nsegmentation. \\n\\nMethod \\nApproach \\nInput Size \\nAP (%) Speed (ms) \\nPanoptic FPN [28] \\nTD \\n512× 1024 \\n33.2 \\n-\\nUPSNet [49] \\nTD \\n1024×2048 \\n33.3 \\n202ms \\nSeamless [40] \\nTD \\n1024× 2048 \\n33.6 \\n-\\nPanoptic Deeplab [15] Xception-71 \\nTD 1 \\n1025×2049 \\n35.3 \\n175ms \\nSSAP [21] \\nBU \\n1024× 2048 \\n31.5 \\n260ms+ 2 \\nOurs base \\nBU-C \\n1024× 2048 \\n29.0 \\n182ms \\nOurs temporal \\nBU-C \\n1024× 2048 \\n30.9 \\n182ms\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Cityscapes validation set results using groundtruth semantics for assigning classes to each instance. BU: Bottom-Up. BU-C: Bottom-Up via Contrastive Learning. For this setup, we significantly outperform other contrastive learning methods.\",\n \"content\": \"TABLE III :\\nIIICityscapes validation set results, without any test time aug-\\nmentation or additional training data for all methods. TD: Top-Down. BU: \\nBottom-Up. BU-C: Bottom-Up via Contrastive Learning. 1 Panoptic Deeplab \\nis a single stage network, but relies on top-down object proposals of object \\ncenters. 2 Reported time is for post-processing only and does not include the \\nnetwork forward pass.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Temporal consistency metrics. We compute the cosine distance (in [0, 1]) between the average embedding for each instance between two temporal frames. Accuracy is the percentage of instances with cosine distance < m = 0.1 over time. Training with temporal consistency significantly increases performance.\",\n \"content\": \"TABLE IV :\\nIVCityscapes validation set results using groundtruth semantics for assigning classes to each instance. BU: Bottom-Up. BU-C: Bottom-Up via Contrastive Learning. For this setup, we significantly outperform other contrastive learning methods.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"A Arnab, S Jayasumana, S Zheng, P Torr, Higher order conditional random fields in deep neural networks. In ECCV. A. Arnab, S. Jayasumana, S. Zheng, and P. Torr. Higher order conditional random fields in deep neural networks. In ECCV, 2016.","doi":""},{"text":"Pixelwise instance segmentation with a dynamically instantiated network. A Arnab, P Torr, CVPRA. Arnab and P. Torr. Pixelwise instance segmentation with a dynamically instantiated network. CVPR, pages 879-888, 07 2017.","doi":""},{"text":"Deep watershed transform for instance segmentation. 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Box 13I-00044FrascatiItaly\n\nMauro Migliorati \nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\nSapienza University of Rome\nRomeItaly\n\nINFN/Roma1\nIstituto Nazionale di Fisica Nucleare\nPiazzale Aldo Moro, 200185RomeItaly\n\nLuigi Palumbo \nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\nSapienza University of Rome\nRomeItaly\n\nINFN/Roma1\nIstituto Nazionale di Fisica Nucleare\nPiazzale Aldo Moro, 200185RomeItaly\n\nBruno Spataro \nINFN\nLaboratori Nazionali di Frascati\nP.O. Box 13I-00044FrascatiItaly\n\nTheory of Diffraction by Holes of Arbitrary Sizes\n1Index Terms-Particle AccelerationLinear AcceleratorsAc- celerator applicationsAccelerator Subsystems and Technologies\nNew high-gradient accelerating RF cavities are nowadays developed in several national laboratories for highenergy physics applications. Ultra high gradients, up to the order of GV/m, can be achieved by using ultra compact accelerating structures in the sub-THz regime. Nevertheless, the experimental setup for measuring the main RF parameters for such compact structures is not trivial and can easily produce errors due to lack of accuracy. Moreover, Radio-Frequency (RF) simulations for these types of cavities can require a large amount of computational time. In particular, one of the main RF parameters that needs to be evaluated and measured for the accelerating structures is the reflection coefficient. In order to obtain a fast and accurate analytical estimation, we have developed the electromagnetic theory for the calculation of the coupling of a resonant cavity with an RF waveguide. This theory is based on the Bethe's small aperture polarization approach, also developed by Collin's. In this paper, we give an exact analytical expression of the reflection coefficient as function of the physics parameters of the cavity-waveguide system, which can be applied to any geometry, material and frequency.\n\nI. INTRODUCTION\n\nT HE concept of reflection coefficient is fundamental in high accelerating periodic structures. Electromagnetic energy may be coupled from one waveguide into another one or into a cavity resonator by a small aperture located at a suitable position in the common wall. An electromagnetic theory for the calculation of the coupling between the two devices is based on the hypothesis that the aperture is equivalent to electric or/and magnetic dipole moments [1]. These dipole moments are proportional to the normal electric and tangential magnetic field of the incident wave, respectively. The theory was originally formulated by Bethe [1], then developed by Collin [2] and modified by S. De Santis, L. Palumbo and M. Migliorati for a small hole compared to the wavelength [3], [4]. The solution procedure was based on one of Schelkunoff's field equivalence principle [2] in which the aperture is closed by a perfect magnetic wall. The incident field, which is chosen as the field in the absence of the aperture in the waveguide, will induce a magnetic current J m and a magnetic charge ρ m on the magnetic wall surface. These sources produce a scattered field that can be expressed as a field radiated by the dipole moment of the source distribution. The field radiated by this source into the waveguide and/or a cavity, together with the specified incident field, represents the total unique solution to the coupling problem. In our case there is a dielectric iris located between the waveguide and the cavity which is supported by the thin perfectly conducting screen. Babinet's principle [2], [5] allows us to relate the diffraction fields of a diffracting screen (circular wall located between the waveguide and the cavity ) to the complementary screen (aperture). The basic procedure consists in replacing the problem of diffraction conducting screen with a certain aperture as a complementary screen. The presence of the screen gives rise to transmitted and reflected fields that will be denoted as scattered field.\n\nWe introduce a theory for diffraction by holes of arbitrary sizes. We then apply the theory to the T M 01 mode of a pillbox cavity coupled with a waveguide by a small hole but comparable to the wavelength. The amplitudes of scattered fields due to polarizability of the aperture will be determined. Fig. 1. E 0 , H 0 are the primary fields in the waveguide, E sF , H sF are the scattered fields (radiation fields) due to the polarizability of the iris, and E sB , H sB are the scattered fields (radiation fields) due to the polarizability of the iris from the field entered into the cavity, reflected with a phase reversal and propagated back to the iris where they become an incoming wave from the right with a certain amplitude which can polarize the iris and, due to this polarization, generate another radiation field.\n\nBethe's diffraction theory states that the hole is equivalent to an electric and a magnetic dipole whose moments are given by\nPe = αeẑ(E0 + E sF − E sB ),(1)Pm = αmt(H0 + H sF − H sB ),(2)\nwhere α e and α m are the polarizabilites of the hole, E 0 , H 0 are the primary fields in the waveguide, E sF , H sF are the scattered fields (radiation fields) due to the polarizability of the iris, and E sB , H sB are the scattered fields (radiation fields) due to the polarizability of the iris from the field entered into the cavity, reflected with a phase reversal and propagated back to the iris where they become an incoming wave with arXiv:2207.11551v1 [physics.acc-ph] 23 Jul 2022 a certain amplitude which can polarize the iris and, due to this polarization, generate another radiation field, as shown in Figure 1. The Electro-Magnetic (EM) analytical method proposed in this paper can be applied for the estimation of the scattering parameters in any power-coupling network through small holes compared/comparable to the wavelength. In particular, through this approach we propose a viable tool for determining the main RF scattering parameters in particle accelerators made out of any material (metallic [6], [7], [8] or dielectric [9], [10], at any frequency and any temperature. For example, the reflection coefficient at the interface (such as a coupling slot) between a metallic waveguide and a resonant cavity can lose in accuracy when geometric dimensions shrink down to the order of few mm's. For example, it is not always trivial at high frequencies, especially at mm-wave range and below, to determine the scattering parameters due to the difficulty of obtaining a reliable robust measurement setup. Although these parameters can be calculated through numerical softwares, the meshing tolerances required for good accuracy can easily result in long time consuming simulations. As a specific example, we will perform the comparison between our derived analytical formulation of the reflection coefficient with the formula derived for a resonant cavity through circuit analysis [11].\n\n\nII. THE FIELDS INTO THE STRUCTURE\n\nThe structure we are going to consider is a pillbox cavity coupled by the iris to a waveguide and the cavity will be excited in a T M 01 standing-wave mode. The starting point in the process is the incident field on the iris a of Fig. (1) which polarizes the iris. The consequence of the polarization is the electric and magnetic current induction. These currents radiate scattered electric and magnetic fields into the structure (waveguide and cavity) which we have called E BF , H BF and E SF , H SF respectively. With the assumption that the only propagating mode inside the structure is the T M 01 , the fields can be written as, ξ − 01 is the T M 01 mode amplitude of scattered field propagating into the waveguide; ξ + 01 is the T M 01 mode amplitude of scattered field propagating into the cavity.\nE = E0 + E sF − E sB = ξJ0(kcr)e −jβz + ξ − 01 (t et01 −ẑ ez01)e jβnz − jξ + 01 (t et01 +ẑ ez01) e −(2α +jβz)(3)H = H0 + H sF − H sB = − ξ η01 J 0 (kcr)e −jβz +ξ − 01 (−t ht01+ẑ hz01) e jβnz −jξ + 01 (t ht01+ẑ hz01) e −(2α +jβz)(4)\nThe summation of incident field on the iris with the scattered fields are the new normal fields. They polarize again the aperture, and induce an electric and magnetic currents. Again the summation of the incident waves with the forward and backward scattered fields, polarize the iris aperture and induce currents, which will be the source of the new scattered fields. The strength of infinitesimal electric and magnetic polarization currents in cylindrical coordinates can be written as,\nPe = 1 r 0αeẑEz δ(r − r0) δ(θ − θ0) δ(z − z0) (5) Pm = − 1 r αmtHtδ(r − r0) δ(θ − θ0) δ(z − z0)(6)\nwhere, P e : infinitesimal electric polarization currents P m : infinitesimal magnetic polarization currents 0 : Electric permitivity α e : electric polarizability of the aperture α m : magnetic polarizability of the aperture The electric and magnetic polarizabilities are constants that depend on the size and shape of the aperture through the fields. Substituting Eqs. (3), 4 into the Eqs. (5), (6) we obtain:\nPe = αe[ξJ0(kcr)e −jβz − ξ − 01 J0(kcr)e jβz − jξ + 01 J0(kcr)e −(2α +jβz) ]δ(r − r0) δ(θ − θ0) δ(z − z0), (7) Pm = αm[− ξ η01 e −jβz + ξ − 01 jβ η01kc e jβz − jξ + 01 jβ η01kc e −(2α +jβz) J 0 (kcr)] δ(r − r0) δ(θ − θ0) δ(z − z0).(8)\nThe electric and magnetic moments are related to electric and magnetic current sources, J and M as follows:\nJ = jωPe (9) M = jωµ0Pm(10)\nThe fields radiated from the sources propagate into the waveguide (Ē − andH − ) and into the cavity (Ē + andH + ). They can be written as the superpositions of the modes [14],\nE+ = Σnξ + n E + n = Σnξ + n (t etn +ẑ ezn) e −jβnz ,(11)H+ = Σnξ + n H + n = Σnξ + n (t htn +ẑ hzn) e −jβnz ,(12)E− = Σnξ − n E − n = Σnξ − n (t etn −ẑ ezn) e jβnz ,(13)H− = Σnξ − n H − n = Σnξ − n (−t htn +ẑ hzn) e jβnz ,(14)\nwhere the single index n is used to represent any possible TE or TM mode. For a given current we can determine the amplitude ξ + n and ξ − n by using Lorentz reciprocity theorem, which, in our case, can be written as, (15) where S is a closed surface of the waveguide enclosing the volume V. E 1 , H 1 are the scattered fields of T M 01 mode given by Eqs. (13) and (14) propagating into the waveguide, E 2 , H 2 are the the unperturbed fields inside the waveguide,J,M are the electric and magnetic current sources related to strength of electric and magnetic moments (see Figure 2). By applying Lorentz Reciprocity Theorem to our problem we obtain,\nS (E1 × H2 − E2 × H1).nds = V (J . E2 − M . H2)dvS (E − × H + 01 − E + 01 × H − ).ẑds = V (J . E + 01 − M . H + 01 ) dv(16)\nwhere E + 01 , H + 01 are the fields of the T M 01 mode traveling in the positive z direction,\nE 2 = E + 01 = (t e t01 +ẑ e z01 ) e −jβz ,(17)H 2 = H + 01 = (t h t01 +ẑ h z01 ) e −jβz ,(18)\nwhere β is the propagation constant of the T M 01 mode. With some simplifications and considering that the tangential electric field and normal magnetic field vanishes on the waveguide walls so that surface integral of the Eq.(15) reduces to just integration over the cross section in which the aperture is located (S 1 ),\nξ − 01 S 1 (E − ×H + 01 − E + 01 ×H − ).ẑds = V (J . E + 01 −M . H + 01 ) dv.(19)\nThis equation can be written explicitly as,\nξ − 01 S 1 [(t et01 −ẑ ez01) × (t ht01 +ẑ hz01) − (t et01 +ẑ ez01) × (−tht01 +ẑ hz01)] .ẑ ds = V (J . E + 01 − M . H + 01 ) dv(20)\nfinally we obtain:\nξ − 01 = −1 P01 V (J . E + 01 − M . H + 01 ) dv = −1 P01 V [(t et01 +ẑ ez01) . J−(t ht01 +ẑ hz01) . M]e −jβnz dv(21)\nwhere P01 = 2 S 1 et01 × ht01 .ẑ ds is a normalization constant proportional to the power flow of the T M01 mode. Replacing J = jωPe and M = jωµ0Pm into the above equation we get, (22) where e z (r, φ) = (Ξ sin nφ + ξ cos nφ)J n (k c r) and h n (r, φ) = −jω kc (Ξ sin nφ + ξ cos nφ)J n (k c r) are the modal fields for T M 010 mode . Replacing Eqs. (7) and (8) into the above equation we obtain,\nξ − 01 = −1 P01 V [(t et01 +ẑ ez01) . (jωPe) − (t ht01 +ẑ hz01) . (jωµ0Pm)]e −jβnz dvξ − 01 = −1 P 01 [(t e t01 +ẑ e z01 ) e −jβnz ](ẑ jω r 0 α e [ξJ 0 (k c r)e −jβz − ξ − 01 J 0 (k c r)e jβz − jξ + 01 J 0 (k c r)e −(2α +jβz) ]) − [(t h t01 +ẑ h z01 ) e −jβnz ] . (t jω r α m [(− ξ η 01 e −jβz + ξ − 01 jβ η 01 k c e jβz − jξ + 01 jβ η 01 k c e −(2α +jβz) )J 0 (k c r)]) δ(r − r 0 ) δ(θ − θ 0 ) δ(z − z 0 ))rdrdθdz. (23)\nThe above equation can be simplified putting the aperture at the center of the wall between the waveguide and cavity so that we obtain In the next section we obtain the normalization constant proportional to the power flow of the T M 01 mode (P 01 ).\n\n\nIII. THE POWER FLOW OF THE NTH MODE\n\nAs we have mentioned earlier P 01 is the normalization constant proportional to the power flow of the T M 01 mode and it's equal to 2 S1ē t01 ×h * t01 .ẑ ds. TM modal fields as we have shown in Table 1, can be written as, \ne t01 = −j β k c J 0 (k c r) (27) h t01 = −j β η 01 k c J 0 (k c r)(28)\nSubstituting the two equations above into P 01 we have,\nP01 = 2 z e01 × h01 .ẑds = − 2β 2 η01k 2 c 2π 0 b 0 J 2 0 (kcr) r dr dθ = − 4πβ 2 η01k 2 c b 0 rJ 2 0 (kcr) dr(29)\nwhere we evaluated the integration over the cross section of the cavity. Since J 0 (k c r) = −J 1 (k c r) and replacing k c r = χ 01 we obtain,\nP01 = 2πb 2 β 2 η01k 2 c J 2 1 (χ01) = 2πb 2 η01 [1 − ( k kc ) 2 ]J 2 1 (χ01)(30)\n\nIV. CALCULATION OF ELECTRIC POLARIZABILITY\n\nAccording to Bethe's theory, the electric polarization coefficient is α = −2/3a 3 [1]. The assumption for using this coefficient is that the holes should be small compared with the wavelength. Based on this theory for the bigger holes -the size is comparable to the wavelength -we should consider the variation of the normal electric field in the Green's function which is in a factor of e ika and this correction can be of the order of (ka) 2 rather than ka. Eggimann [14] solved the problem of diffraction of arbitrary electromagnetic field by a circular perfectly conducting disk using a series representation in powers of k using the results of generalized Babinet's principle [2] by considering equivalence of the disk and the aperture problems we have\nPz = 4 3 a 3 0(1 − 3 10 (ka) 2 + j 4 9π (ka) 3 )E 0 z + 2a 5 0 15 ∂ 2 E 0 z ∂z 2 (31) = 4 3 a 3 0E 0 z − 2 5 a 3 0(ka) 2 E 0 z + 2a 5 0 15 ∂ 2 E 0 z ∂z 2 +j 16 27π a 3 0(ka) 3 E 0 z .(32)\nFor the small holes compared to the wavelength the second, third and fourth terms vanish as the radius of the holes is in the order of 6, 5 and 6 respectively. One can derive the electric polarizability of Bethe's theory by small holes. For the bigger holes all terms have contribution on electric polarizability. It should be noted that the imaginary part of the equation means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. The imaginary term is responsible for π/2 phase shift of the iris located between cavity resonators. The third term contribution where we have the second derivative of the electric field is field oscillations and converting this term in time domain we get a term with a 1 c 2 coefficient which destroys the effect of the second derivatives and the equation becomes\nPz = 4 3 a 3 0(1 − 3 10 (ka) 2 + j 4 9π (ka) 3 )E 0 z .(33)\nWhere the electric polarization coefficient can be obtained\nα = 4 3 a 3 0(1 − 3 10 (ka) 2 + j 4 9π (ka) 3 ).(34)\nWe observe a term involving (ka) 2 for the hole size comparable to the wavelength. This is exactly the correction mentioned in Bethe's theory for the diffraction by bigger holes.\n\nFor the small irises all the modes of waveguide are below cut-off and their attenuation is exponential with respect to the length of the iris. McDonald [15] has derived an electric polarizability in which the attenuation term can be considered and it is as follows,\nαe = C E α * e e −2.405h/a with C E = 0.83 and α * e = − 2 3 a 3(35)\nwhere h is the iris thickness, C E is the numerical value calculated by McDonald.\n\nGluckstern [16] has derived integral equations for the potential and field distribution within a circular hole in a plane conducting wall of finite thickness induced by uniform field. Then he obtained variational expressions for the polarizability and susceptibility of the hole, from which one can obtain the electric and magnetic dipole moments induced on the inside (far field) and outside (no far field) boundaries of the hole [16]. His results had a good agreement with McDonald's results.\n\nOne can use the Eq. (34) when the holes are comparable to the wavelength. In the next section we will calculate the amplitudes of scattered fields due to polarizability of the aperture located between waveguide and the cavity will find equations for reflection and transmission coefficients.\n\n\nV. REFLECTION COEFFICIENT\n\nFinally we can calculate the amplitude for forward and backward waves due to dipole moments,\nξ − 01 = −1 P 01 jω 0 α e ξ + −1 P 01 a 0 2π 0 h 0 jω 0 α e ξ − 01\n(−J 2 0 (k c r))e jβz δrδθδzdr dθ dz\n+ −1 P 01 a 0 2π 0 h 0 jω 0 α e [−jξ + 01 (J 2 0 (k c r))e −2α −jβz ]) δrδθδzdr dθ dz,(36)\nwhere the electric polarization coefficient is given\nαe = 4 3 a 3 0(1 − 3 10 (ka) 2 + j 4 9π (ka) 3 )(37)\nand\nP 01 = 2πb 2 η01 [1 − ( k kc ) 2 ]J 2 1 (χ 01 )\nis the normalization constant proportional to the power flow of the T M 01 mode. The imaginary part of the electric polarizability α e means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. The imaginary term is responsible for π/2 phase shift of the iris located between cavity resonators. Finally ξ − 01 and ξ + 01 are the amplitudes of the radiation fields due to iris as a dipole. They can be written as reflection and transmission coefficients.\nξ − 01 = Γ ξ,(38)ξ + 01 = (1 − Γ) ξ.(39)\nReplacing these equations into the equation above we get,\nΓ = −1 P 01 jω 0 α e (1 − j e −2α ) + −1 P 01 a 0 2π 0 h 0 jω 0 α e Γ\n(−J 2 0 (k c r))e jβz (1 + j e −2α −2jβz ) δr δθ δz dr dθ dz, (40) writing the equation above in a compact way we obtain,\nΓ = Γ0 + D V Γ f (r)g(z)h(θ)δr δθ δz dV ,(41)\nwhere\nΓ 0 = −1 P01 jω 0 α e (1 − j e −2α ), D = −1 P01 jω 0 α e , f (r) = J 2\n0 (k c r), g(z) = e jβz (1 + j e −2α −2jβz ), and h(θ): Heaviside step function. To obtain equation above we assumed that the iris is at the center of the cross section of the cavity. This assumption simplify the equation of infinitesimal electric and magnetic moments (Eq.s (5) and (6) to the equations below,\nP e = 1 r 0 α en E n δ(r) δ(θ) δ(z),(42)P m = − 1 r α mHt δ(r) δ(θ) δ(z),(43)\nthese equations allow us make a convolution of the functions at the center and simplify the equations. By considering the aperture at the center of the wall between the waveguide and the cavity and using self consistent solution method we obtain:\nΓ = −1 P01 jω 0 α e (1 − j e −2α ) 1 + j −1 P01 ω 0 α e (1 + j e −2α ) ,(44)\nwhere P 01 is a normalization constant proportional to the power flow of the T M 010 mode\nP 01 = 2πb 2 η 01 [1 − ( k k c ) 2 ]J 2 1 (χ 01 ),(45)\nand α e is the electric polarizability\nαe = 4 3 a 3 0(1 − 3 10 (ka) 2 + j 4 9π (ka) 3 ).(46)\nThe imaginary part of the electric polarizability α e means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. Finally ξ − 01 and ξ + 01 are the amplitudes of the radiation fields due to iris as a dipole. This equation is an analogous of the reflection coefficient obtained by Collin for T E 10 using the circuit theory which for a comparison we report here [2],\n1 − 4jα m β 10 /ab 1 + jX + W ,(47)\nWhere X = 2α m β 10 /ab and\nW = − 4α m k 2 0 π 2 k 2 101 abd 3 [k 2 101 − k 0 (1 + 1−j Q )](48)\nTo simplify Eq. (23) we observed that the magnetic moment has no contribution for the T M 01 mode and the second term in the equation can be totally eliminated. In the following we will prove that considering the magnetic moment for T M 01 mode leads to the zero contribution for reflection coefficient calculation. Eq. (23) for the magnetic moment part becomes Due to the properties of the Dirac delta function, the derivative of the Bessel function of the zero kind and first kind at the center of the holes is zero and leads to ξ − 01 =0, as a consequence, for the T M 01 mode, the iris aperture between the pill-box cavity coupled with a cylindrical waveguide, acts as an electric moment.\n\n\nVI. CONCLUSION\n\nWe introduced a theory for diffraction by holes of arbitrary sizes. We then applied this theory to the T M 01 mode cavity coupled with a cylindrical waveguide by a small hole comparable to the wavelength. The amplitudes of scattered fields due to polarizability of the aperture located between waveguide and the cavity have been determined and we have found equations for reflection and transmission coefficients. The Electro-Magnetic (EM) analytical method proposed in this paper can be applied for the estimation of the scattering parameters in any power-coupling network through small holes.\n\n\nPropagation constant; η 01 : Wave impedance of the T M 01 ; α: Attenuation constant per unit length; : The length of the cavity; e t01 , h t01 , : Transverse modal fields for T M 01 mode; e z01 , h z01 , : Longitudinal modal fields for T M 01 mode; ξ: Amplitude of the incident wave;\n\nFig. 2 .\n2A volume containing two sets of sources, J 1 and J 2 , which each produce fields E 1 , H 1 and E 2 , H 2 , respectively.\n\n\n[(t e t01 +ẑ e z01 ) e −jβnz ](ẑ jω r 0 α e [ξJ 0 (k c r)e −jβz − ξ − 01 J 0 (k c r)e jβz − jξ + 01 J 0 (k c r)e −(2α +jβz) ]) − [(t h t01 +ẑ h z01 ) e −jβnz ]. k c e −(2α +jβz) )J 0 (k c r)]) δr δθ δz)rdrdθdz. (24)\n\n((\ne n (r, φ) = −jβ k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (25) h n (r, φ) = −jω k c (Ξ sin nφ + ξ cos nφ)J n (k c r) = −jβ η 01 k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (26)where k c is cutoff wavenumber, n is the number of zeros in E z along the radial direction, J n (k c r) is the derivative Ξsinnφ − ξcosnφ) J n (kcr) Ξsinnφ + ξcosnφ) J n (kcr) e −jβz η η T M = βη k\n\n\nV (h t01 +ẑh z01 ) . (jωµ 0Pm )e −jβnz dv(49)substituting Eq. (8) into the above equation we obtain 01 k c e −(2α +jβz) )J 0 (k c r))δr δθ δzrdrdθdz. (50)\n\nTABLE I SUMMARY\nIOF RESULTS FOR THE WAVE PROPAGATION OF THE T Mnm MODE[13].Bessel function of the first kind and η 01 has an exactly defined \nvalue η 01 = 376.73. For T M 01 modal fields we can write, \n\n\n\nTheory of diffraction by small holes. Hans Bethe, Albrecht, Physical review. 66163Bethe, Hans Albrecht. \"Theory of diffraction by small holes.\" Physical review 66.7-8 (1944): 163.\n\nField theory of guided waves. Robert E Collin, Collin, Robert E. \"Field theory of guided waves.\" (1960).\n\nCoupling impedance of a hole in a coaxial beam pipe. S De Santis, M Migliorati, L Palumbo, M Zobov, Physical Review E. 541800De Santis, S., Migliorati, M., Palumbo, L., & Zobov, M. (1996). Coupling impedance of a hole in a coaxial beam pipe. Physical Review E, 54(1), 800.\n\nInterference effects on the coupling impedance of many holes in a coaxial beam pipe. S De Santis, A Mostacci, L Palumbo, Physical Review E. 565990De Santis, S., A. Mostacci, and L. Palumbo. \"Interference effects on the coupling impedance of many holes in a coaxial beam pipe.\" Physical Review E 56.5 (1997): 5990.\n\nClassical electrodynamics. John Jackson, David, John Wiley & SonsJackson, John David. Classical electrodynamics. John Wiley & Sons, 2007.\n\nRF breakdown tests of mm-wave metallic accelerating structures. Dal Forno, Massimo, Physical Review Accelerators and Beams. 1911301Dal Forno, Massimo, et al. \"RF breakdown tests of mm-wave metallic accelerating structures.\" Physical Review Accelerators and Beams 19.1 (2016): 011301.\n\nmm-Wave Standing-Wave Accelerating Structures for High-Gradient Tests. Emilio Nanni, 44Nanni, Emilio, et al. \"mm-Wave Standing-Wave Accelerating Structures for High-Gradient Tests.\" (2016): THPOR044.\n\nHigh Gradient Test of a mm-Wave Accelerating Cavity?. Mohamed Othman, submitted to APL manuscript 20-AR-03444Mohamed Othman et al.? High Gradient Test of a mm-Wave Accelerating Cavity?, submitted to APL manuscript 20-AR-03444\n\nDesign and Characterization of a Silicon W-Band Woodpile Photonic Crystal Waveguide. G Torrisi, IEEE Microwave and Wireless Components Letters. Torrisi, G., et al. \"Design and Characterization of a Silicon W-Band Woodpile Photonic Crystal Waveguide.\" IEEE Microwave and Wireless Components Letters (2020).\n\nTransverse forces in planar symmetric dielectric laser-driven accelerators. R England, Alexander Joel, Zhirong Ody, Huang, arXiv:1906.08316arXiv preprintEngland, R. Joel, Alexander Ody, and Zhirong Huang. \"Transverse forces in planar symmetric dielectric laser-driven accelerators.\" arXiv preprint arXiv:1906.08316 (2019).\n\nMicrowave electronics. John Slater, Clarke, Reviews of Modern Physics. 18441Slater, John Clarke. \"Microwave electronics.\" Reviews of Modern Physics 18.4 (1946): 441.\n\nPrinciples of RF Linear Accelerators. T P Wangler, John Wiley & Sons.\" IncNYWangler, T. P. \"Principles of RF Linear Accelerators, John Wiley & Sons.\" Inc., NY (1998).\n\nMicrowave engineering. David M Pozar, John Wiley & SonsPozar, David M. Microwave engineering. John Wiley & Sons, 2009.\n\nHigher-order evaluation of electromagnetic diffraction by circular disks. W H Eggimann, IRE Transactions on Microwave Theory and Techniques. 9Eggimann, W. H. \"Higher-order evaluation of electromagnetic diffrac- tion by circular disks.\" IRE Transactions on Microwave Theory and Techniques 9.5 (1961): 408-418.\n\nElectric and magnetic coupling through small apertures in shield walls of any thickness. Noel A Mcdonald, IEEE Transactions on Microwave Theory and Techniques. 20McDonald, Noel A. \"Electric and magnetic coupling through small aper- tures in shield walls of any thickness.\" IEEE Transactions on Microwave Theory and Techniques 20.10 (1972): 689-695.\n\nPenetration of fields through a circular hole in a wall of finite thickness. Robert L Gluckstern, J A Diamond, IEEE transactions on microwave theory and techniques. 39Gluckstern, Robert L., and J. A. Diamond. \"Penetration of fields through a circular hole in a wall of finite thickness.\" IEEE transactions on microwave theory and techniques 39.2 (1991): 274-279.\n"},"corpus_id":{"kind":"number","value":251040279,"string":"251,040,279"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2207.11551\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2207.11551v1.pdf\"\n ],\n \"pdf_sha\": \"86f856eea776e2723c1473aa7bef935e9125cc7b\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Theory of Diffraction by Holes of Arbitrary Sizes"},"authors":{"kind":"list like","value":["Mostafa Behtouei \nINFN\nLaboratori Nazionali di Frascati\nP.O. Box 13I-00044FrascatiItaly\n\nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\nSapienza University of Rome\nRomeItaly","Luigi Faillace \nINFN\nLaboratori Nazionali di Frascati\nP.O. Box 13I-00044FrascatiItaly","Mauro Migliorati \nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\nSapienza University of Rome\nRomeItaly\n\nINFN/Roma1\nIstituto Nazionale di Fisica Nucleare\nPiazzale Aldo Moro, 200185RomeItaly","Luigi Palumbo \nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\nSapienza University of Rome\nRomeItaly\n\nINFN/Roma1\nIstituto Nazionale di Fisica Nucleare\nPiazzale Aldo Moro, 200185RomeItaly","Bruno Spataro \nINFN\nLaboratori Nazionali di Frascati\nP.O. Box 13I-00044FrascatiItaly"],"string":"[\n \"Mostafa Behtouei \\nINFN\\nLaboratori Nazionali di Frascati\\nP.O. Box 13I-00044FrascatiItaly\\n\\nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\\nSapienza University of Rome\\nRomeItaly\",\n \"Luigi Faillace \\nINFN\\nLaboratori Nazionali di Frascati\\nP.O. Box 13I-00044FrascatiItaly\",\n \"Mauro Migliorati \\nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\\nSapienza University of Rome\\nRomeItaly\\n\\nINFN/Roma1\\nIstituto Nazionale di Fisica Nucleare\\nPiazzale Aldo Moro, 200185RomeItaly\",\n \"Luigi Palumbo \\nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\\nSapienza University of Rome\\nRomeItaly\\n\\nINFN/Roma1\\nIstituto Nazionale di Fisica Nucleare\\nPiazzale Aldo Moro, 200185RomeItaly\",\n \"Bruno Spataro \\nINFN\\nLaboratori Nazionali di Frascati\\nP.O. Box 13I-00044FrascatiItaly\"\n]","isConversational":false},"abstract":{"kind":"string","value":"New high-gradient accelerating RF cavities are nowadays developed in several national laboratories for highenergy physics applications. Ultra high gradients, up to the order of GV/m, can be achieved by using ultra compact accelerating structures in the sub-THz regime. Nevertheless, the experimental setup for measuring the main RF parameters for such compact structures is not trivial and can easily produce errors due to lack of accuracy. Moreover, Radio-Frequency (RF) simulations for these types of cavities can require a large amount of computational time. In particular, one of the main RF parameters that needs to be evaluated and measured for the accelerating structures is the reflection coefficient. In order to obtain a fast and accurate analytical estimation, we have developed the electromagnetic theory for the calculation of the coupling of a resonant cavity with an RF waveguide. This theory is based on the Bethe's small aperture polarization approach, also developed by Collin's. In this paper, we give an exact analytical expression of the reflection coefficient as function of the physics parameters of the cavity-waveguide system, which can be applied to any geometry, material and frequency."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"T HE concept of reflection coefficient is fundamental in high accelerating periodic structures. Electromagnetic energy may be coupled from one waveguide into another one or into a cavity resonator by a small aperture located at a suitable position in the common wall. An electromagnetic theory for the calculation of the coupling between the two devices is based on the hypothesis that the aperture is equivalent to electric or/and magnetic dipole moments [1]. These dipole moments are proportional to the normal electric and tangential magnetic field of the incident wave, respectively. The theory was originally formulated by Bethe [1], then developed by Collin [2] and modified by S. De Santis, L. Palumbo and M. Migliorati for a small hole compared to the wavelength [3], [4]. The solution procedure was based on one of Schelkunoff's field equivalence principle [2] in which the aperture is closed by a perfect magnetic wall. The incident field, which is chosen as the field in the absence of the aperture in the waveguide, will induce a magnetic current J m and a magnetic charge ρ m on the magnetic wall surface. These sources produce a scattered field that can be expressed as a field radiated by the dipole moment of the source distribution. The field radiated by this source into the waveguide and/or a cavity, together with the specified incident field, represents the total unique solution to the coupling problem. In our case there is a dielectric iris located between the waveguide and the cavity which is supported by the thin perfectly conducting screen. Babinet's principle [2], [5] allows us to relate the diffraction fields of a diffracting screen (circular wall located between the waveguide and the cavity ) to the complementary screen (aperture). The basic procedure consists in replacing the problem of diffraction conducting screen with a certain aperture as a complementary screen. The presence of the screen gives rise to transmitted and reflected fields that will be denoted as scattered field.\nWe introduce a theory for diffraction by holes of arbitrary sizes. We then apply the theory to the T M 01 mode of a pillbox cavity coupled with a waveguide by a small hole but comparable to the wavelength. The amplitudes of scattered fields due to polarizability of the aperture will be determined. Fig. 1. E 0 , H 0 are the primary fields in the waveguide, E sF , H sF are the scattered fields (radiation fields) due to the polarizability of the iris, and E sB , H sB are the scattered fields (radiation fields) due to the polarizability of the iris from the field entered into the cavity, reflected with a phase reversal and propagated back to the iris where they become an incoming wave from the right with a certain amplitude which can polarize the iris and, due to this polarization, generate another radiation field.\nBethe's diffraction theory states that the hole is equivalent to an electric and a magnetic dipole whose moments are given by\nwhere α e and α m are the polarizabilites of the hole, E 0 , H 0 are the primary fields in the waveguide, E sF , H sF are the scattered fields (radiation fields) due to the polarizability of the iris, and E sB , H sB are the scattered fields (radiation fields) due to the polarizability of the iris from the field entered into the cavity, reflected with a phase reversal and propagated back to the iris where they become an incoming wave with arXiv:2207.11551v1 [physics.acc-ph] 23 Jul 2022 a certain amplitude which can polarize the iris and, due to this polarization, generate another radiation field, as shown in Figure 1. The Electro-Magnetic (EM) analytical method proposed in this paper can be applied for the estimation of the scattering parameters in any power-coupling network through small holes compared/comparable to the wavelength. In particular, through this approach we propose a viable tool for determining the main RF scattering parameters in particle accelerators made out of any material (metallic [6], [7], [8] or dielectric [9], [10], at any frequency and any temperature. For example, the reflection coefficient at the interface (such as a coupling slot) between a metallic waveguide and a resonant cavity can lose in accuracy when geometric dimensions shrink down to the order of few mm's. For example, it is not always trivial at high frequencies, especially at mm-wave range and below, to determine the scattering parameters due to the difficulty of obtaining a reliable robust measurement setup. Although these parameters can be calculated through numerical softwares, the meshing tolerances required for good accuracy can easily result in long time consuming simulations. As a specific example, we will perform the comparison between our derived analytical formulation of the reflection coefficient with the formula derived for a resonant cavity through circuit analysis [11]."},{"title":"II. THE FIELDS INTO THE STRUCTURE","content":"The structure we are going to consider is a pillbox cavity coupled by the iris to a waveguide and the cavity will be excited in a T M 01 standing-wave mode. The starting point in the process is the incident field on the iris a of Fig. (1) which polarizes the iris. The consequence of the polarization is the electric and magnetic current induction. These currents radiate scattered electric and magnetic fields into the structure (waveguide and cavity) which we have called E BF , H BF and E SF , H SF respectively. With the assumption that the only propagating mode inside the structure is the T M 01 , the fields can be written as, ξ − 01 is the T M 01 mode amplitude of scattered field propagating into the waveguide; ξ + 01 is the T M 01 mode amplitude of scattered field propagating into the cavity.\nThe summation of incident field on the iris with the scattered fields are the new normal fields. They polarize again the aperture, and induce an electric and magnetic currents. Again the summation of the incident waves with the forward and backward scattered fields, polarize the iris aperture and induce currents, which will be the source of the new scattered fields. The strength of infinitesimal electric and magnetic polarization currents in cylindrical coordinates can be written as,\nwhere, P e : infinitesimal electric polarization currents P m : infinitesimal magnetic polarization currents 0 : Electric permitivity α e : electric polarizability of the aperture α m : magnetic polarizability of the aperture The electric and magnetic polarizabilities are constants that depend on the size and shape of the aperture through the fields. Substituting Eqs. (3), 4 into the Eqs. (5), (6) we obtain:\nThe electric and magnetic moments are related to electric and magnetic current sources, J and M as follows:\nThe fields radiated from the sources propagate into the waveguide (Ē − andH − ) and into the cavity (Ē + andH + ). They can be written as the superpositions of the modes [14],\nwhere the single index n is used to represent any possible TE or TM mode. For a given current we can determine the amplitude ξ + n and ξ − n by using Lorentz reciprocity theorem, which, in our case, can be written as, (15) where S is a closed surface of the waveguide enclosing the volume V. E 1 , H 1 are the scattered fields of T M 01 mode given by Eqs. (13) and (14) propagating into the waveguide, E 2 , H 2 are the the unperturbed fields inside the waveguide,J,M are the electric and magnetic current sources related to strength of electric and magnetic moments (see Figure 2). By applying Lorentz Reciprocity Theorem to our problem we obtain,\nwhere E + 01 , H + 01 are the fields of the T M 01 mode traveling in the positive z direction,\nwhere β is the propagation constant of the T M 01 mode. With some simplifications and considering that the tangential electric field and normal magnetic field vanishes on the waveguide walls so that surface integral of the Eq.(15) reduces to just integration over the cross section in which the aperture is located (S 1 ),\nThis equation can be written explicitly as,\nfinally we obtain:\nwhere P01 = 2 S 1 et01 × ht01 .ẑ ds is a normalization constant proportional to the power flow of the T M01 mode. Replacing J = jωPe and M = jωµ0Pm into the above equation we get, (22) where e z (r, φ) = (Ξ sin nφ + ξ cos nφ)J n (k c r) and h n (r, φ) = −jω kc (Ξ sin nφ + ξ cos nφ)J n (k c r) are the modal fields for T M 010 mode . Replacing Eqs. (7) and (8) into the above equation we obtain,\nThe above equation can be simplified putting the aperture at the center of the wall between the waveguide and cavity so that we obtain In the next section we obtain the normalization constant proportional to the power flow of the T M 01 mode (P 01 )."},{"title":"III. THE POWER FLOW OF THE NTH MODE","content":"As we have mentioned earlier P 01 is the normalization constant proportional to the power flow of the T M 01 mode and it's equal to 2 S1ē t01 ×h * t01 .ẑ ds. TM modal fields as we have shown in Table 1, can be written as,\nSubstituting the two equations above into P 01 we have,\nwhere we evaluated the integration over the cross section of the cavity. Since J 0 (k c r) = −J 1 (k c r) and replacing k c r = χ 01 we obtain,"},{"title":"IV. CALCULATION OF ELECTRIC POLARIZABILITY","content":"According to Bethe's theory, the electric polarization coefficient is α = −2/3a 3 [1]. The assumption for using this coefficient is that the holes should be small compared with the wavelength. Based on this theory for the bigger holes -the size is comparable to the wavelength -we should consider the variation of the normal electric field in the Green's function which is in a factor of e ika and this correction can be of the order of (ka) 2 rather than ka. Eggimann [14] solved the problem of diffraction of arbitrary electromagnetic field by a circular perfectly conducting disk using a series representation in powers of k using the results of generalized Babinet's principle [2] by considering equivalence of the disk and the aperture problems we have\nFor the small holes compared to the wavelength the second, third and fourth terms vanish as the radius of the holes is in the order of 6, 5 and 6 respectively. One can derive the electric polarizability of Bethe's theory by small holes. For the bigger holes all terms have contribution on electric polarizability. It should be noted that the imaginary part of the equation means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. The imaginary term is responsible for π/2 phase shift of the iris located between cavity resonators. The third term contribution where we have the second derivative of the electric field is field oscillations and converting this term in time domain we get a term with a 1 c 2 coefficient which destroys the effect of the second derivatives and the equation becomes\nWhere the electric polarization coefficient can be obtained\nWe observe a term involving (ka) 2 for the hole size comparable to the wavelength. This is exactly the correction mentioned in Bethe's theory for the diffraction by bigger holes.\nFor the small irises all the modes of waveguide are below cut-off and their attenuation is exponential with respect to the length of the iris. McDonald [15] has derived an electric polarizability in which the attenuation term can be considered and it is as follows,\nwhere h is the iris thickness, C E is the numerical value calculated by McDonald.\nGluckstern [16] has derived integral equations for the potential and field distribution within a circular hole in a plane conducting wall of finite thickness induced by uniform field. Then he obtained variational expressions for the polarizability and susceptibility of the hole, from which one can obtain the electric and magnetic dipole moments induced on the inside (far field) and outside (no far field) boundaries of the hole [16]. His results had a good agreement with McDonald's results.\nOne can use the Eq. (34) when the holes are comparable to the wavelength. In the next section we will calculate the amplitudes of scattered fields due to polarizability of the aperture located between waveguide and the cavity will find equations for reflection and transmission coefficients."},{"title":"V. REFLECTION COEFFICIENT","content":"Finally we can calculate the amplitude for forward and backward waves due to dipole moments,\n(−J 2 0 (k c r))e jβz δrδθδzdr dθ dz\nwhere the electric polarization coefficient is given\nand\nis the normalization constant proportional to the power flow of the T M 01 mode. The imaginary part of the electric polarizability α e means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. The imaginary term is responsible for π/2 phase shift of the iris located between cavity resonators. Finally ξ − 01 and ξ + 01 are the amplitudes of the radiation fields due to iris as a dipole. They can be written as reflection and transmission coefficients.\nReplacing these equations into the equation above we get,\n(−J 2 0 (k c r))e jβz (1 + j e −2α −2jβz ) δr δθ δz dr dθ dz, (40) writing the equation above in a compact way we obtain,\nwhere\n0 (k c r), g(z) = e jβz (1 + j e −2α −2jβz ), and h(θ): Heaviside step function. To obtain equation above we assumed that the iris is at the center of the cross section of the cavity. This assumption simplify the equation of infinitesimal electric and magnetic moments (Eq.s (5) and (6) to the equations below,\nthese equations allow us make a convolution of the functions at the center and simplify the equations. By considering the aperture at the center of the wall between the waveguide and the cavity and using self consistent solution method we obtain:\nwhere P 01 is a normalization constant proportional to the power flow of the T M 010 mode\nand α e is the electric polarizability\nThe imaginary part of the electric polarizability α e means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. Finally ξ − 01 and ξ + 01 are the amplitudes of the radiation fields due to iris as a dipole. This equation is an analogous of the reflection coefficient obtained by Collin for T E 10 using the circuit theory which for a comparison we report here [2],\nWhere X = 2α m β 10 /ab and\nTo simplify Eq. (23) we observed that the magnetic moment has no contribution for the T M 01 mode and the second term in the equation can be totally eliminated. In the following we will prove that considering the magnetic moment for T M 01 mode leads to the zero contribution for reflection coefficient calculation. Eq. (23) for the magnetic moment part becomes Due to the properties of the Dirac delta function, the derivative of the Bessel function of the zero kind and first kind at the center of the holes is zero and leads to ξ − 01 =0, as a consequence, for the T M 01 mode, the iris aperture between the pill-box cavity coupled with a cylindrical waveguide, acts as an electric moment."},{"title":"VI. CONCLUSION","content":"We introduced a theory for diffraction by holes of arbitrary sizes. We then applied this theory to the T M 01 mode cavity coupled with a cylindrical waveguide by a small hole comparable to the wavelength. The amplitudes of scattered fields due to polarizability of the aperture located between waveguide and the cavity have been determined and we have found equations for reflection and transmission coefficients. The Electro-Magnetic (EM) analytical method proposed in this paper can be applied for the estimation of the scattering parameters in any power-coupling network through small holes."},{"title":"Fig. 2 .","content":""},{"title":"((","content":""},{"title":"TABLE I SUMMARY","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"T HE concept of reflection coefficient is fundamental in high accelerating periodic structures. Electromagnetic energy may be coupled from one waveguide into another one or into a cavity resonator by a small aperture located at a suitable position in the common wall. An electromagnetic theory for the calculation of the coupling between the two devices is based on the hypothesis that the aperture is equivalent to electric or/and magnetic dipole moments [1]. These dipole moments are proportional to the normal electric and tangential magnetic field of the incident wave, respectively. The theory was originally formulated by Bethe [1], then developed by Collin [2] and modified by S. De Santis, L. Palumbo and M. Migliorati for a small hole compared to the wavelength [3], [4]. The solution procedure was based on one of Schelkunoff's field equivalence principle [2] in which the aperture is closed by a perfect magnetic wall. The incident field, which is chosen as the field in the absence of the aperture in the waveguide, will induce a magnetic current J m and a magnetic charge ρ m on the magnetic wall surface. These sources produce a scattered field that can be expressed as a field radiated by the dipole moment of the source distribution. The field radiated by this source into the waveguide and/or a cavity, together with the specified incident field, represents the total unique solution to the coupling problem. In our case there is a dielectric iris located between the waveguide and the cavity which is supported by the thin perfectly conducting screen. Babinet's principle [2], [5] allows us to relate the diffraction fields of a diffracting screen (circular wall located between the waveguide and the cavity ) to the complementary screen (aperture). The basic procedure consists in replacing the problem of diffraction conducting screen with a certain aperture as a complementary screen. The presence of the screen gives rise to transmitted and reflected fields that will be denoted as scattered field.\\nWe introduce a theory for diffraction by holes of arbitrary sizes. We then apply the theory to the T M 01 mode of a pillbox cavity coupled with a waveguide by a small hole but comparable to the wavelength. The amplitudes of scattered fields due to polarizability of the aperture will be determined. Fig. 1. E 0 , H 0 are the primary fields in the waveguide, E sF , H sF are the scattered fields (radiation fields) due to the polarizability of the iris, and E sB , H sB are the scattered fields (radiation fields) due to the polarizability of the iris from the field entered into the cavity, reflected with a phase reversal and propagated back to the iris where they become an incoming wave from the right with a certain amplitude which can polarize the iris and, due to this polarization, generate another radiation field.\\nBethe's diffraction theory states that the hole is equivalent to an electric and a magnetic dipole whose moments are given by\\nwhere α e and α m are the polarizabilites of the hole, E 0 , H 0 are the primary fields in the waveguide, E sF , H sF are the scattered fields (radiation fields) due to the polarizability of the iris, and E sB , H sB are the scattered fields (radiation fields) due to the polarizability of the iris from the field entered into the cavity, reflected with a phase reversal and propagated back to the iris where they become an incoming wave with arXiv:2207.11551v1 [physics.acc-ph] 23 Jul 2022 a certain amplitude which can polarize the iris and, due to this polarization, generate another radiation field, as shown in Figure 1. The Electro-Magnetic (EM) analytical method proposed in this paper can be applied for the estimation of the scattering parameters in any power-coupling network through small holes compared/comparable to the wavelength. In particular, through this approach we propose a viable tool for determining the main RF scattering parameters in particle accelerators made out of any material (metallic [6], [7], [8] or dielectric [9], [10], at any frequency and any temperature. For example, the reflection coefficient at the interface (such as a coupling slot) between a metallic waveguide and a resonant cavity can lose in accuracy when geometric dimensions shrink down to the order of few mm's. For example, it is not always trivial at high frequencies, especially at mm-wave range and below, to determine the scattering parameters due to the difficulty of obtaining a reliable robust measurement setup. Although these parameters can be calculated through numerical softwares, the meshing tolerances required for good accuracy can easily result in long time consuming simulations. As a specific example, we will perform the comparison between our derived analytical formulation of the reflection coefficient with the formula derived for a resonant cavity through circuit analysis [11].\"\n },\n {\n \"title\": \"II. THE FIELDS INTO THE STRUCTURE\",\n \"content\": \"The structure we are going to consider is a pillbox cavity coupled by the iris to a waveguide and the cavity will be excited in a T M 01 standing-wave mode. The starting point in the process is the incident field on the iris a of Fig. (1) which polarizes the iris. The consequence of the polarization is the electric and magnetic current induction. These currents radiate scattered electric and magnetic fields into the structure (waveguide and cavity) which we have called E BF , H BF and E SF , H SF respectively. With the assumption that the only propagating mode inside the structure is the T M 01 , the fields can be written as, ξ − 01 is the T M 01 mode amplitude of scattered field propagating into the waveguide; ξ + 01 is the T M 01 mode amplitude of scattered field propagating into the cavity.\\nThe summation of incident field on the iris with the scattered fields are the new normal fields. They polarize again the aperture, and induce an electric and magnetic currents. Again the summation of the incident waves with the forward and backward scattered fields, polarize the iris aperture and induce currents, which will be the source of the new scattered fields. The strength of infinitesimal electric and magnetic polarization currents in cylindrical coordinates can be written as,\\nwhere, P e : infinitesimal electric polarization currents P m : infinitesimal magnetic polarization currents 0 : Electric permitivity α e : electric polarizability of the aperture α m : magnetic polarizability of the aperture The electric and magnetic polarizabilities are constants that depend on the size and shape of the aperture through the fields. Substituting Eqs. (3), 4 into the Eqs. (5), (6) we obtain:\\nThe electric and magnetic moments are related to electric and magnetic current sources, J and M as follows:\\nThe fields radiated from the sources propagate into the waveguide (Ē − andH − ) and into the cavity (Ē + andH + ). They can be written as the superpositions of the modes [14],\\nwhere the single index n is used to represent any possible TE or TM mode. For a given current we can determine the amplitude ξ + n and ξ − n by using Lorentz reciprocity theorem, which, in our case, can be written as, (15) where S is a closed surface of the waveguide enclosing the volume V. E 1 , H 1 are the scattered fields of T M 01 mode given by Eqs. (13) and (14) propagating into the waveguide, E 2 , H 2 are the the unperturbed fields inside the waveguide,J,M are the electric and magnetic current sources related to strength of electric and magnetic moments (see Figure 2). By applying Lorentz Reciprocity Theorem to our problem we obtain,\\nwhere E + 01 , H + 01 are the fields of the T M 01 mode traveling in the positive z direction,\\nwhere β is the propagation constant of the T M 01 mode. With some simplifications and considering that the tangential electric field and normal magnetic field vanishes on the waveguide walls so that surface integral of the Eq.(15) reduces to just integration over the cross section in which the aperture is located (S 1 ),\\nThis equation can be written explicitly as,\\nfinally we obtain:\\nwhere P01 = 2 S 1 et01 × ht01 .ẑ ds is a normalization constant proportional to the power flow of the T M01 mode. Replacing J = jωPe and M = jωµ0Pm into the above equation we get, (22) where e z (r, φ) = (Ξ sin nφ + ξ cos nφ)J n (k c r) and h n (r, φ) = −jω kc (Ξ sin nφ + ξ cos nφ)J n (k c r) are the modal fields for T M 010 mode . Replacing Eqs. (7) and (8) into the above equation we obtain,\\nThe above equation can be simplified putting the aperture at the center of the wall between the waveguide and cavity so that we obtain In the next section we obtain the normalization constant proportional to the power flow of the T M 01 mode (P 01 ).\"\n },\n {\n \"title\": \"III. THE POWER FLOW OF THE NTH MODE\",\n \"content\": \"As we have mentioned earlier P 01 is the normalization constant proportional to the power flow of the T M 01 mode and it's equal to 2 S1ē t01 ×h * t01 .ẑ ds. TM modal fields as we have shown in Table 1, can be written as,\\nSubstituting the two equations above into P 01 we have,\\nwhere we evaluated the integration over the cross section of the cavity. Since J 0 (k c r) = −J 1 (k c r) and replacing k c r = χ 01 we obtain,\"\n },\n {\n \"title\": \"IV. CALCULATION OF ELECTRIC POLARIZABILITY\",\n \"content\": \"According to Bethe's theory, the electric polarization coefficient is α = −2/3a 3 [1]. The assumption for using this coefficient is that the holes should be small compared with the wavelength. Based on this theory for the bigger holes -the size is comparable to the wavelength -we should consider the variation of the normal electric field in the Green's function which is in a factor of e ika and this correction can be of the order of (ka) 2 rather than ka. Eggimann [14] solved the problem of diffraction of arbitrary electromagnetic field by a circular perfectly conducting disk using a series representation in powers of k using the results of generalized Babinet's principle [2] by considering equivalence of the disk and the aperture problems we have\\nFor the small holes compared to the wavelength the second, third and fourth terms vanish as the radius of the holes is in the order of 6, 5 and 6 respectively. One can derive the electric polarizability of Bethe's theory by small holes. For the bigger holes all terms have contribution on electric polarizability. It should be noted that the imaginary part of the equation means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. The imaginary term is responsible for π/2 phase shift of the iris located between cavity resonators. The third term contribution where we have the second derivative of the electric field is field oscillations and converting this term in time domain we get a term with a 1 c 2 coefficient which destroys the effect of the second derivatives and the equation becomes\\nWhere the electric polarization coefficient can be obtained\\nWe observe a term involving (ka) 2 for the hole size comparable to the wavelength. This is exactly the correction mentioned in Bethe's theory for the diffraction by bigger holes.\\nFor the small irises all the modes of waveguide are below cut-off and their attenuation is exponential with respect to the length of the iris. McDonald [15] has derived an electric polarizability in which the attenuation term can be considered and it is as follows,\\nwhere h is the iris thickness, C E is the numerical value calculated by McDonald.\\nGluckstern [16] has derived integral equations for the potential and field distribution within a circular hole in a plane conducting wall of finite thickness induced by uniform field. Then he obtained variational expressions for the polarizability and susceptibility of the hole, from which one can obtain the electric and magnetic dipole moments induced on the inside (far field) and outside (no far field) boundaries of the hole [16]. His results had a good agreement with McDonald's results.\\nOne can use the Eq. (34) when the holes are comparable to the wavelength. In the next section we will calculate the amplitudes of scattered fields due to polarizability of the aperture located between waveguide and the cavity will find equations for reflection and transmission coefficients.\"\n },\n {\n \"title\": \"V. REFLECTION COEFFICIENT\",\n \"content\": \"Finally we can calculate the amplitude for forward and backward waves due to dipole moments,\\n(−J 2 0 (k c r))e jβz δrδθδzdr dθ dz\\nwhere the electric polarization coefficient is given\\nand\\nis the normalization constant proportional to the power flow of the T M 01 mode. The imaginary part of the electric polarizability α e means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. The imaginary term is responsible for π/2 phase shift of the iris located between cavity resonators. Finally ξ − 01 and ξ + 01 are the amplitudes of the radiation fields due to iris as a dipole. They can be written as reflection and transmission coefficients.\\nReplacing these equations into the equation above we get,\\n(−J 2 0 (k c r))e jβz (1 + j e −2α −2jβz ) δr δθ δz dr dθ dz, (40) writing the equation above in a compact way we obtain,\\nwhere\\n0 (k c r), g(z) = e jβz (1 + j e −2α −2jβz ), and h(θ): Heaviside step function. To obtain equation above we assumed that the iris is at the center of the cross section of the cavity. This assumption simplify the equation of infinitesimal electric and magnetic moments (Eq.s (5) and (6) to the equations below,\\nthese equations allow us make a convolution of the functions at the center and simplify the equations. By considering the aperture at the center of the wall between the waveguide and the cavity and using self consistent solution method we obtain:\\nwhere P 01 is a normalization constant proportional to the power flow of the T M 010 mode\\nand α e is the electric polarizability\\nThe imaginary part of the electric polarizability α e means that electric moment is out of phase with the electric field and there is a phase delay delay between the application of the electric field and the induced dipole moment. Finally ξ − 01 and ξ + 01 are the amplitudes of the radiation fields due to iris as a dipole. This equation is an analogous of the reflection coefficient obtained by Collin for T E 10 using the circuit theory which for a comparison we report here [2],\\nWhere X = 2α m β 10 /ab and\\nTo simplify Eq. (23) we observed that the magnetic moment has no contribution for the T M 01 mode and the second term in the equation can be totally eliminated. In the following we will prove that considering the magnetic moment for T M 01 mode leads to the zero contribution for reflection coefficient calculation. Eq. (23) for the magnetic moment part becomes Due to the properties of the Dirac delta function, the derivative of the Bessel function of the zero kind and first kind at the center of the holes is zero and leads to ξ − 01 =0, as a consequence, for the T M 01 mode, the iris aperture between the pill-box cavity coupled with a cylindrical waveguide, acts as an electric moment.\"\n },\n {\n \"title\": \"VI. CONCLUSION\",\n \"content\": \"We introduced a theory for diffraction by holes of arbitrary sizes. We then applied this theory to the T M 01 mode cavity coupled with a cylindrical waveguide by a small hole comparable to the wavelength. The amplitudes of scattered fields due to polarizability of the aperture located between waveguide and the cavity have been determined and we have found equations for reflection and transmission coefficients. The Electro-Magnetic (EM) analytical method proposed in this paper can be applied for the estimation of the scattering parameters in any power-coupling network through small holes.\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"((\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I SUMMARY\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Propagation constant; η 01 : Wave impedance of the T M 01 ; α: Attenuation constant per unit length; : The length of the cavity; e t01 , h t01 , : Transverse modal fields for T M 01 mode; e z01 , h z01 , : Longitudinal modal fields for T M 01 mode; ξ: Amplitude of the incident wave;","content":"Propagation constant; η 01 : Wave impedance of the T M 01 ; α: Attenuation constant per unit length; : The length of the cavity; e t01 , h t01 , : Transverse modal fields for T M 01 mode; e z01 , h z01 , : Longitudinal modal fields for T M 01 mode; ξ: Amplitude of the incident wave;"},{"id":"","caption":"A volume containing two sets of sources, J 1 and J 2 , which each produce fields E 1 , H 1 and E 2 , H 2 , respectively.","content":"Fig. 2 .\n2A volume containing two sets of sources, J 1 and J 2 , which each produce fields E 1 , H 1 and E 2 , H 2 , respectively."},{"id":"","caption":"[(t e t01 +ẑ e z01 ) e −jβnz ](ẑ jω r 0 α e [ξJ 0 (k c r)e −jβz − ξ − 01 J 0 (k c r)e jβz − jξ + 01 J 0 (k c r)e −(2α +jβz) ]) − [(t h t01 +ẑ h z01 ) e −jβnz ]. k c e −(2α +jβz) )J 0 (k c r)]) δr δθ δz)rdrdθdz. (24)","content":"[(t e t01 +ẑ e z01 ) e −jβnz ](ẑ jω r 0 α e [ξJ 0 (k c r)e −jβz − ξ − 01 J 0 (k c r)e jβz − jξ + 01 J 0 (k c r)e −(2α +jβz) ]) − [(t h t01 +ẑ h z01 ) e −jβnz ]. k c e −(2α +jβz) )J 0 (k c r)]) δr δθ δz)rdrdθdz. (24)"},{"id":"","caption":"e n (r, φ) = −jβ k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (25) h n (r, φ) = −jω k c (Ξ sin nφ + ξ cos nφ)J n (k c r) = −jβ η 01 k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (26)where k c is cutoff wavenumber, n is the number of zeros in E z along the radial direction, J n (k c r) is the derivative Ξsinnφ − ξcosnφ) J n (kcr) Ξsinnφ + ξcosnφ) J n (kcr) e −jβz η η T M = βη k","content":"((\ne n (r, φ) = −jβ k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (25) h n (r, φ) = −jω k c (Ξ sin nφ + ξ cos nφ)J n (k c r) = −jβ η 01 k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (26)where k c is cutoff wavenumber, n is the number of zeros in E z along the radial direction, J n (k c r) is the derivative Ξsinnφ − ξcosnφ) J n (kcr) Ξsinnφ + ξcosnφ) J n (kcr) e −jβz η η T M = βη k"},{"id":"","caption":"V (h t01 +ẑh z01 ) . (jωµ 0Pm )e −jβnz dv(49)substituting Eq. (8) into the above equation we obtain 01 k c e −(2α +jβz) )J 0 (k c r))δr δθ δzrdrdθdz. (50)","content":"V (h t01 +ẑh z01 ) . (jωµ 0Pm )e −jβnz dv(49)substituting Eq. (8) into the above equation we obtain 01 k c e −(2α +jβz) )J 0 (k c r))δr δθ δzrdrdθdz. (50)"},{"id":"","caption":"OF RESULTS FOR THE WAVE PROPAGATION OF THE T Mnm MODE[13].","content":"TABLE I SUMMARY\nIOF RESULTS FOR THE WAVE PROPAGATION OF THE T Mnm MODE[13].Bessel function of the first kind and η 01 has an exactly defined \nvalue η 01 = 376.73. For T M 01 modal fields we can write,"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Propagation constant; η 01 : Wave impedance of the T M 01 ; α: Attenuation constant per unit length; : The length of the cavity; e t01 , h t01 , : Transverse modal fields for T M 01 mode; e z01 , h z01 , : Longitudinal modal fields for T M 01 mode; ξ: Amplitude of the incident wave;\",\n \"content\": \"Propagation constant; η 01 : Wave impedance of the T M 01 ; α: Attenuation constant per unit length; : The length of the cavity; e t01 , h t01 , : Transverse modal fields for T M 01 mode; e z01 , h z01 , : Longitudinal modal fields for T M 01 mode; ξ: Amplitude of the incident wave;\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A volume containing two sets of sources, J 1 and J 2 , which each produce fields E 1 , H 1 and E 2 , H 2 , respectively.\",\n \"content\": \"Fig. 2 .\\n2A volume containing two sets of sources, J 1 and J 2 , which each produce fields E 1 , H 1 and E 2 , H 2 , respectively.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[(t e t01 +ẑ e z01 ) e −jβnz ](ẑ jω r 0 α e [ξJ 0 (k c r)e −jβz − ξ − 01 J 0 (k c r)e jβz − jξ + 01 J 0 (k c r)e −(2α +jβz) ]) − [(t h t01 +ẑ h z01 ) e −jβnz ]. k c e −(2α +jβz) )J 0 (k c r)]) δr δθ δz)rdrdθdz. (24)\",\n \"content\": \"[(t e t01 +ẑ e z01 ) e −jβnz ](ẑ jω r 0 α e [ξJ 0 (k c r)e −jβz − ξ − 01 J 0 (k c r)e jβz − jξ + 01 J 0 (k c r)e −(2α +jβz) ]) − [(t h t01 +ẑ h z01 ) e −jβnz ]. k c e −(2α +jβz) )J 0 (k c r)]) δr δθ δz)rdrdθdz. (24)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"e n (r, φ) = −jβ k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (25) h n (r, φ) = −jω k c (Ξ sin nφ + ξ cos nφ)J n (k c r) = −jβ η 01 k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (26)where k c is cutoff wavenumber, n is the number of zeros in E z along the radial direction, J n (k c r) is the derivative Ξsinnφ − ξcosnφ) J n (kcr) Ξsinnφ + ξcosnφ) J n (kcr) e −jβz η η T M = βη k\",\n \"content\": \"((\\ne n (r, φ) = −jβ k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (25) h n (r, φ) = −jω k c (Ξ sin nφ + ξ cos nφ)J n (k c r) = −jβ η 01 k c (Ξ sin nφ + ξ cos nφ)J n (k c r) (26)where k c is cutoff wavenumber, n is the number of zeros in E z along the radial direction, J n (k c r) is the derivative Ξsinnφ − ξcosnφ) J n (kcr) Ξsinnφ + ξcosnφ) J n (kcr) e −jβz η η T M = βη k\"\n },\n {\n \"id\": \"\",\n \"caption\": \"V (h t01 +ẑh z01 ) . (jωµ 0Pm )e −jβnz dv(49)substituting Eq. (8) into the above equation we obtain 01 k c e −(2α +jβz) )J 0 (k c r))δr δθ δzrdrdθdz. (50)\",\n \"content\": \"V (h t01 +ẑh z01 ) . (jωµ 0Pm )e −jβnz dv(49)substituting Eq. (8) into the above equation we obtain 01 k c e −(2α +jβz) )J 0 (k c r))δr δθ δzrdrdθdz. (50)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"OF RESULTS FOR THE WAVE PROPAGATION OF THE T Mnm MODE[13].\",\n \"content\": \"TABLE I SUMMARY\\nIOF RESULTS FOR THE WAVE PROPAGATION OF THE T Mnm MODE[13].Bessel function of the first kind and η 01 has an exactly defined \\nvalue η 01 = 376.73. For T M 01 modal fields we can write,\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Theory of diffraction by small holes. Hans Bethe, Albrecht, Physical review. 66163Bethe, Hans Albrecht. \"Theory of diffraction by small holes.\" Physical review 66.7-8 (1944): 163.","doi":""},{"text":"Field theory of guided waves. Robert E Collin, Collin, Robert E. \"Field theory of guided waves.\" (1960).","doi":""},{"text":"Coupling impedance of a hole in a coaxial beam pipe. S De Santis, M Migliorati, L Palumbo, M Zobov, Physical Review E. 541800De Santis, S., Migliorati, M., Palumbo, L., & Zobov, M. (1996). Coupling impedance of a hole in a coaxial beam pipe. Physical Review E, 54(1), 800.","doi":""},{"text":"Interference effects on the coupling impedance of many holes in a coaxial beam pipe. S De Santis, A Mostacci, L Palumbo, Physical Review E. 565990De Santis, S., A. Mostacci, and L. Palumbo. \"Interference effects on the coupling impedance of many holes in a coaxial beam pipe.\" Physical Review E 56.5 (1997): 5990.","doi":""},{"text":"Classical electrodynamics. John Jackson, David, John Wiley & SonsJackson, John David. Classical electrodynamics. John Wiley & Sons, 2007.","doi":""},{"text":"RF breakdown tests of mm-wave metallic accelerating structures. Dal Forno, Massimo, Physical Review Accelerators and Beams. 1911301Dal Forno, Massimo, et al. \"RF breakdown tests of mm-wave metallic accelerating structures.\" Physical Review Accelerators and Beams 19.1 (2016): 011301.","doi":""},{"text":"mm-Wave Standing-Wave Accelerating Structures for High-Gradient Tests. Emilio Nanni, 44Nanni, Emilio, et al. \"mm-Wave Standing-Wave Accelerating Structures for High-Gradient Tests.\" (2016): THPOR044.","doi":""},{"text":"High Gradient Test of a mm-Wave Accelerating Cavity?. Mohamed Othman, submitted to APL manuscript 20-AR-03444Mohamed Othman et al.? High Gradient Test of a mm-Wave Accelerating Cavity?, submitted to APL manuscript 20-AR-03444","doi":""},{"text":"Design and Characterization of a Silicon W-Band Woodpile Photonic Crystal Waveguide. G Torrisi, IEEE Microwave and Wireless Components Letters. Torrisi, G., et al. \"Design and Characterization of a Silicon W-Band Woodpile Photonic Crystal Waveguide.\" IEEE Microwave and Wireless Components Letters (2020).","doi":""},{"text":"Transverse forces in planar symmetric dielectric laser-driven accelerators. R England, Alexander Joel, Zhirong Ody, Huang, arXiv:1906.08316arXiv preprintEngland, R. Joel, Alexander Ody, and Zhirong Huang. \"Transverse forces in planar symmetric dielectric laser-driven accelerators.\" arXiv preprint arXiv:1906.08316 (2019).","doi":"arXiv:1906.08316"},{"text":"Microwave electronics. John Slater, Clarke, Reviews of Modern Physics. 18441Slater, John Clarke. \"Microwave electronics.\" Reviews of Modern Physics 18.4 (1946): 441.","doi":""},{"text":"Principles of RF Linear Accelerators. T P Wangler, John Wiley & Sons.\" IncNYWangler, T. P. \"Principles of RF Linear Accelerators, John Wiley & Sons.\" Inc., NY (1998).","doi":""},{"text":"Microwave engineering. David M Pozar, John Wiley & SonsPozar, David M. Microwave engineering. John Wiley & Sons, 2009.","doi":""},{"text":"Higher-order evaluation of electromagnetic diffraction by circular disks. W H Eggimann, IRE Transactions on Microwave Theory and Techniques. 9Eggimann, W. H. \"Higher-order evaluation of electromagnetic diffrac- tion by circular disks.\" IRE Transactions on Microwave Theory and Techniques 9.5 (1961): 408-418.","doi":""},{"text":"Electric and magnetic coupling through small apertures in shield walls of any thickness. Noel A Mcdonald, IEEE Transactions on Microwave Theory and Techniques. 20McDonald, Noel A. \"Electric and magnetic coupling through small aper- tures in shield walls of any thickness.\" IEEE Transactions on Microwave Theory and Techniques 20.10 (1972): 689-695.","doi":""},{"text":"Penetration of fields through a circular hole in a wall of finite thickness. Robert L Gluckstern, J A Diamond, IEEE transactions on microwave theory and techniques. 39Gluckstern, Robert L., and J. A. Diamond. \"Penetration of fields through a circular hole in a wall of finite thickness.\" IEEE transactions on microwave theory and techniques 39.2 (1991): 274-279.","doi":""}],"string":"[\n {\n \"text\": \"Theory of diffraction by small holes. Hans Bethe, Albrecht, Physical review. 66163Bethe, Hans Albrecht. \\\"Theory of diffraction by small holes.\\\" Physical review 66.7-8 (1944): 163.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Field theory of guided waves. Robert E Collin, Collin, Robert E. \\\"Field theory of guided waves.\\\" (1960).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Coupling impedance of a hole in a coaxial beam pipe. S De Santis, M Migliorati, L Palumbo, M Zobov, Physical Review E. 541800De Santis, S., Migliorati, M., Palumbo, L., & Zobov, M. (1996). Coupling impedance of a hole in a coaxial beam pipe. Physical Review E, 54(1), 800.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interference effects on the coupling impedance of many holes in a coaxial beam pipe. S De Santis, A Mostacci, L Palumbo, Physical Review E. 565990De Santis, S., A. Mostacci, and L. Palumbo. \\\"Interference effects on the coupling impedance of many holes in a coaxial beam pipe.\\\" Physical Review E 56.5 (1997): 5990.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Classical electrodynamics. John Jackson, David, John Wiley & SonsJackson, John David. Classical electrodynamics. John Wiley & Sons, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"RF breakdown tests of mm-wave metallic accelerating structures. Dal Forno, Massimo, Physical Review Accelerators and Beams. 1911301Dal Forno, Massimo, et al. \\\"RF breakdown tests of mm-wave metallic accelerating structures.\\\" Physical Review Accelerators and Beams 19.1 (2016): 011301.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"mm-Wave Standing-Wave Accelerating Structures for High-Gradient Tests. Emilio Nanni, 44Nanni, Emilio, et al. \\\"mm-Wave Standing-Wave Accelerating Structures for High-Gradient Tests.\\\" (2016): THPOR044.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High Gradient Test of a mm-Wave Accelerating Cavity?. Mohamed Othman, submitted to APL manuscript 20-AR-03444Mohamed Othman et al.? High Gradient Test of a mm-Wave Accelerating Cavity?, submitted to APL manuscript 20-AR-03444\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Design and Characterization of a Silicon W-Band Woodpile Photonic Crystal Waveguide. G Torrisi, IEEE Microwave and Wireless Components Letters. Torrisi, G., et al. \\\"Design and Characterization of a Silicon W-Band Woodpile Photonic Crystal Waveguide.\\\" IEEE Microwave and Wireless Components Letters (2020).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transverse forces in planar symmetric dielectric laser-driven accelerators. R England, Alexander Joel, Zhirong Ody, Huang, arXiv:1906.08316arXiv preprintEngland, R. Joel, Alexander Ody, and Zhirong Huang. \\\"Transverse forces in planar symmetric dielectric laser-driven accelerators.\\\" arXiv preprint arXiv:1906.08316 (2019).\",\n \"doi\": \"arXiv:1906.08316\"\n },\n {\n \"text\": \"Microwave electronics. John Slater, Clarke, Reviews of Modern Physics. 18441Slater, John Clarke. \\\"Microwave electronics.\\\" Reviews of Modern Physics 18.4 (1946): 441.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Principles of RF Linear Accelerators. T P Wangler, John Wiley & Sons.\\\" IncNYWangler, T. P. \\\"Principles of RF Linear Accelerators, John Wiley & Sons.\\\" Inc., NY (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microwave engineering. David M Pozar, John Wiley & SonsPozar, David M. Microwave engineering. John Wiley & Sons, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Higher-order evaluation of electromagnetic diffraction by circular disks. W H Eggimann, IRE Transactions on Microwave Theory and Techniques. 9Eggimann, W. H. \\\"Higher-order evaluation of electromagnetic diffrac- tion by circular disks.\\\" IRE Transactions on Microwave Theory and Techniques 9.5 (1961): 408-418.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electric and magnetic coupling through small apertures in shield walls of any thickness. Noel A Mcdonald, IEEE Transactions on Microwave Theory and Techniques. 20McDonald, Noel A. \\\"Electric and magnetic coupling through small aper- tures in shield walls of any thickness.\\\" IEEE Transactions on Microwave Theory and Techniques 20.10 (1972): 689-695.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Penetration of fields through a circular hole in a wall of finite thickness. Robert L Gluckstern, J A Diamond, IEEE transactions on microwave theory and techniques. 39Gluckstern, Robert L., and J. A. Diamond. \\\"Penetration of fields through a circular hole in a wall of finite thickness.\\\" IEEE transactions on microwave theory and techniques 39.2 (1991): 274-279.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":398,"cells":{"text":{"kind":"string","value":"\np-hyperbolicity of homotopy groups via K-theory\n26 Nov 2021 November 29, 2021\n\nGuy Boyde \np-hyperbolicity of homotopy groups via K-theory\n26 Nov 2021 November 29, 2021arXiv:2101.04591v2 [math.AT]\nWe show that S n ∨ S m is Z/p r -hyperbolic for all primes p and all r ∈ Z + , provided n, m ≥ 2, and consequently that various spaces containing S n ∨ S m as a p-local retract are Z/p r -hyperbolic. We then give a K-theory criterion for a suspension ΣX to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian ΣGr k,n is p-hyperbolic for all odd primes p when n ≥ 3 and 0 < k < n. We obtain similar results for some related spaces.\n\nIntroduction\n\nA space X is called rationally elliptic if π * (X) ⊗ Q is finite dimensional, and rationally hyperbolic if the dimension of i≤m π i (X) ⊗ Q grows exponentially in m. It was proved in [FHT15,Chapter 33] that simply connected CW -complexes with rational homology of finite type and finite rational category are either rationally elliptic or rationally hyperbolic. In order to study the p-torsion analogue of this dichotomy, Huang and Wu [HW20] introduced the definitions of Z/p r -and p-hyperbolicity.\n\nFor p prime, by a p-torsion summand in an abelian group A, we mean a direct summand isomorphic to Z/p r for some r ≥ 1.\n\nDefinition 1.1. Let X be a space, and let p be a prime. We say that X is p-hyperbolic if the number of p-torsion summands in π * (X) grows exponentially, in the sense that\nlim inf m ln(T m ) m > 0,\nwhere T m is the number of p-torsion summands in i≤m π i (X).\n\nThe above definition counts Z/p r -summands for all values of r. It is also possible to consider only a single r, and by doing so we obtain the definition of Z/p r -hyperbolicity.\n\nDefinition 1.2. Let X be a space, let p be a prime, and fix r ∈ Z + . We say that X is Z/p r -hyperbolic if the number of Z/p r -summands in π * (X) grows exponentially, in the sense that\nlim inf m ln(t m ) m > 0,\nwhere t m is the number of Z/p r -summands in i≤m π i (X).\n\nNote that Z/p r -hyperbolicity for any r implies p-hyperbolicity. It follows immediately from a result of Henn [Hen86, Corollary of Theorem 1] that the lim infs appearing in the above definitions must be finite if X is a simply connected finite CW -complex.\n\nHuang and Wu show that for n ≥ 3, r ≥ 1 and p any prime, the Moore space P n (p r ) is Z/p rhyperbolic and Z/p r+1 -hyperbolic, and that P n (2) is also Z/8-hyperbolic [HW20, Theorem 1.6]. More generally, they give criteria in terms of a functorial loop space decomposition due to Selick and Wu [SW00; SW06] for a suspension ΣX to be Z/p r -hyperbolic.\n\nMore recently, Zhu and Pan [ZP21] use a classification of (n − 1)-connected CW -complexes of dimension at most n + 2, due to Chang [Cha50], to show that, for n ≥ 4, such a complex is Z/phyperbolic, provided that it is not contractible or a sphere after p-localization. They also prove hyperbolicity results for several so-called elementary Chang complexes.\n\nThis paper studies p-and Z/p r -hyperbolicity of certain suspensions. Our first result is as follows.\n\nTheorem 1.3. Let q 1 , q 2 ≥ 1. Then S q1+1 ∨ S q2+1 is Z/p r -hyperbolic for all primes p and all r ∈ Z + .\n\nLet p be a prime. If a space X contains a wedge of two spheres as a p-local retract, then Theorem 1.3 implies that X is Z/p r -hyperbolic for all r. Various spaces have been shown to have such a wedge as a retract -examples of this sort are given in Section 2.1. A summary is as follows:\n\n• for n, k ≥ 3, the configuration space Conf k (R n ) is Z/p r -hyperbolic for all p and r (Example 2.1);\n\n• an (n − 1)-connected 2n-dimensional manifold M , where H n (M ) is of rank at least 3, is Z/p rhyperbolic for all p and r (Example 2.2);\n\n• a generalized moment-angle complex on a simplicial complex having two minimal missing faces which are not disjoint is Z/p r -hyperbolic for all p and r (Example 2.3);\n\n• ΣCP 2 is Z/p r -hyperbolic for all p = 2 and all r, and ΣHP 2 is is Z/p r -hyperbolic for all p = 2, 3 and all r (Example 2.4).\n\nOur other result is as follows.\n\nTheorem 1.4. Let p be an odd prime, and let X be a path connected space having the p-local homotopy type of a finite CW -complex. Suppose that there exists a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX with q i ≥ 1, such that the map\nK * (ΣX) ⊗ Z/p (µ1∨µ2) * − −−−−− → K * (S q1+1 ∨ S q2+1 ) ⊗ Z/p ∼ = Z/p ⊕ Z/p\nis a surjection. Then ΣX is p-hyperbolic.\n\nThis criterion is quite different to that given by Huang and Wu [HW20, Theorem 1.5]. Their criterion is homotopical, using hypotheses on X to produce retracts of ΩΣX, whereas ours is cohomological, which makes it easier to check. On the other hand, their criterion is stronger, since it gives Z/p r -hyperbolicity, rather than just p-hyperbolicity. The examples they give, primarily various Moore spaces, differ from those we obtain, which are the suspensions of spaces related to complex projective space. More precisely, in Section 2.2, we show that the following spaces are p-hyperbolic for all p = 2:\n\n• suspended complex projective space ΣCP n for n ≥ 2 (Example 2.5), and more generally;\n\n• the suspended complex Grassmannian ΣGr k,n for n ≥ 3 and 0 < k < n (Example 2.6);\n\n• the suspended Milnor Hypersurface ΣH m,n for m ≥ 2 and n ≥ 3, (Example 2.7);\n\n• the suspended unitary group ΣU (n) for n ≥ 3 (Example 2.8).\n\nBoth Theorem 1.3 and Theorem 1.4 will be proven by constructing an exponentially growing family of classes which generate summands in the relevant homotopy groups. We think of this family as 'witnessing' the hyperbolicity. For Theorem 1.3, one can proceed directly from the Hilton-Milnor decomposition of S n ∨S m [Hil55]. For Theorem 1.4, we employ K-theoretic methods originally used by Selick [Sel83] to prove one direction of Moore's conjecture for suspensions having torsion-free homology.\n\nIf the map µ 1 ∨ µ 2 of Theorem 1.4 induces a surjection on K * ( ) ⊗ Z/p, then so does its suspension Σµ 1 ∨ Σµ 2 . The conclusion of Theorem 1.4 may therefore be strengthened in the following way.\n\nCorollary 1.5. With the hypothesis of Theorem 1.4, Σ n X is p-hyperbolic for all n ≥ 1.\n\nOne might be motivated by this observation to ask whether, in the circumstances of Theorem 1.4, the stable homotopy groups of X satisfy the growth conditions of Definition 1.1 or 1.2. In the proofs of both Theorem 1.3 and 1.4, the classes that witness the hyperbolicity are composites involving Whitehead products. The suspension of a Whitehead product is always trivial [Whi46,Theorem 3.11], so the classes we detect cannot be stable. Therefore, Corollary 1.5 does not suggest that the stable homotopy of ΣX should be p-or Z/p r -hyperbolic. On the other hand, it follows from our methods that, under the hypotheses of Theorem 1.4, the stable homotopy of ΩΣX is p-hyperbolic.\n\nBy a result of Henn [Hen83], any co-H space, and in particular any suspension, decomposes rationally as a wedge of spheres. It then follows from the Hilton-Milnor theorem [Hil55] and the computation of the rational homotopy groups of spheres [Ser51] that such a suspension is rationally hyperbolic precisely when there are at least two spheres (of dimension ≥ 2) in this decomposition.\n\nIf ΣX satisfies the hypotheses of Theorem 1.4 for any prime (including 2), then by Chern Character considerations the reduced rational homology of ΣX has dimension at least two, so ΣX is rationally a wedge of at least two spheres by the preceding discussion. This rational equivalence is a local equivalence at all but perhaps finitely many primes, so by Theorem 1.3, ΣX is Z/p r hyperbolic for all r at all but finitely many primes p. One might therefore conjecture that the conclusion of Theorem 1.4 can be strengthened to give Z/p r -hyperbolicity for all r rather than p-hyperbolicity, but we do not know whether this is possible.\n\nWe now discuss situations in which it is adequate to consider ordinary cohomology, rather than K-theory. If ΣX has torsion-free integral (co)homology, or if its cohomology is concentrated in even degrees, then the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses on the E 2 page [Hus+08]. It follows by naturality that the image of the map induced by µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX on Ktheory is identified with the image of the induced map on cohomology. We may therefore replace K-theory with cohomology in Theorem 1.4, as follows.\n\nCorollary 1.6. Let X be a path connected space having the homotopy type of a finite CW -complex, such that the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses on the E 2 page. Let p be an odd prime. Suppose that there exists a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX with q i ≥ 1, such that the map induced by µ 1 ∨ µ 2 on H * ( ) ⊗ Z/p is a surjection. Then ΣX is p-hyperbolic.\n\nOne advantage of ordinary cohomology is that it is connected to the homotopy groups integrally, via the universal coefficient theorem and Hurewicz map. We can exploit this as follows.\n\nExample 1.7. Suppose that the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses (for example, if ΣX has torsion-free homology) and that there exists q ∈ Z + so that H i (ΣX) = 0 for i ≤ q, and dim Q ( H q+1 (ΣX) ⊗ Q) ≥ 2. The Hurewicz map π q+1 (ΣX) → H q+1 (ΣX) is an isomorphism, so there exists a map µ 1 ∨ µ 2 : S q+1 ∨ S q+1 → ΣX inducing the inclusion of a Z 2 -summand in H q+1 (ΣX). By the universal coefficient theorem relating ordinary homology and cohomology, µ 1 ∨ µ 2 induces a surjection on integral cohomology, so by Corollary 1.6, ΣX is p-hyperbolic for all odd primes p.\n\nThe structure of this paper is as follows. In Section 2 we give applications of the main theorems and derive a simple lower bound for the growth of the number of Z/p r -summands in the homotopy groups of a wedge of two spheres (Corollary 2.9). The proofs of Theorems 1.3 and Theorem 1.4 may be read largely independently; Section 3 contains those preliminary results which are used in both cases. In Section 4, we give the proof of Theorem 1.3. The remainder of the paper is devoted to proving Theorem 1.4: Sections 5 and 6 give the necessary background, and Section 7 contains the proof. An overview of the proof strategy can be found at the start of Section 7.\n\nI would like to thank my PhD supervisor, Stephen Theriault, for suggesting the problems that this paper tries to address, and for many helpful conversations along the way. From a technical point of view, much is owed to papers of Huang and Wu [HW20], and of Selick [Sel83]. Neil Strickland's 'Bestiary' was extremely helpful in providing examples of Theorem 1.4. Conversations with Sam Hughes were very useful in formulating Corollary 1.6. Thanks to the anonymous reviewer for their thoughtful comments, especially for pointing out that the space in Theorem 1.4 need only be finite after p-localisation.\n\n2 Applications 2.1 Spaces having a wedge of two spheres as a retract Theorem 1.3 immediately implies that any space X which has S q1+1 ∨ S q2+1 as a retract after plocalization is Z/p r -hyperbolic for that p and all r. This implies that for all n ≥ 1, Σ n X contains S q1+n+1 ∨ S q2+n+1 as a p-local retract, and so is Z/p r -hyperbolic for all r. We first consider examples of this form.\n\nExample 2.1. It is known [Knu18, Section 3.1] that Conf k (R n ), the ordered configuration space of k points in R n , contains k−1 S n−1 as a retract. It follows that, when n, k ≥ 3, Conf k (R n ) is Z/p rhyperbolic for all p and r.\n\nExample 2.2. Let M be an (n − 1)-connected 2n-dimensional manifold. By the universal coefficient theorem, there can be no torsion in H n (M ). Suppose that the rank of H n (M ) is at least 3. By work of Beben and Theriault [BT14, Theorem 1.4], ΩM contains a wedge of two spheres as a retract after looping. Thus, M is again Z/p r -hyperbolic for all p and r.\n\nExample 2.3. Let K be a simplicial complex on the vertex set [m] = {1, . . . , m}, and let (X, A) be any sequence of pairs (D ni , S ni−1 ) with n i ≥ 2 for 1 ≤ i ≤ m. If there exist two distinct minimal missing faces of K which are not disjoint, then by work of Hao, Sun and Theriault [HST19, Theorem 4.2] the polyhedral product (X, A) K contains a wedge of two spheres as a retract after looping, and hence is Z/p r -hyperbolic for all p and all r.\n\nExample 2.4. Localized away from 2, ΣCP 2 ≃ S 3 ∨ S 5 . To see this, note that ΣCP 2 has a CWstructure consisting of one 3-cell and one 5-cell, and that π 4 (S 3 ) ∼ = Z/2 [Fre38]. This implies that the attaching map for the 5-cell is nullhomotopic after localization at an odd prime. Thus, ΣCP 2 is Z/p r -hyperbolic for all r when p = 2.\n\nSimilarly, ΣHP 2 admits a cell structure with one 5-cell and one 9-cell, and π 8 (S 5 ) ∼ = Z/24. Thus, ΣHP 2 is Z/p r -hyperbolic for all r when p = 2, 3.\n\n\nSuspensions of spaces related to CP n\n\nSuppose that one has verified the hypotheses of Theorem 1.4 for a given space X and odd prime p, using a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX. If another space Y admits a map σ : ΣX → ΣY which induces a surjection on K * ( ) ⊗ Z/p, then it is immediate that σ • (µ 1 ∨ µ 2 ) satisfies the hypotheses of Theorem 1.4, and hence that ΣY is p-hyperbolic. The slogan is that K-theory surjections allow us to generate new examples from old ones.\n\nIn this section, we will apply this idea. We have seen in Example 2.4 that, localized away from 2, ΣCP 2 ≃ S 3 ∨ S 5 , so certainly ΣCP 2 satisfies the hypotheses of Theorem 1.4 at all odd primes p. We will now consider spaces X which are known to admit maps CP 2 → X which induce surjections on integral K-theory, and hence on K * ( ) ⊗ Z/p for all odd p. It follows in each case that ΣX is p-hyperbolic, and further by Corollary 1.5, that Σ n X is p-hyperbolic for all n ≥ 1.\n\nThe inclusion of CP n into CP n+1 induces a surjection on K-theory, so it must still induce a surjection after suspending. Composing these inclusions with the local equivalence ΣCP 2 ≃ S 3 ∨ S 5 gives, for each n ≥ 2, a map S 3 ∨ S 5 → ΣCP n which still induces a surjection on K * ( ) ⊗ Z/p for all odd primes p. Applying Theorem 1.4 to this map gives the following.\n\nExample 2.5. For n ≥ 2, ΣCP n is p-hyperbolic for all p = 2. Now let Gr k,n be the Grassmannian of k-dimensional complex subspaces of C n . First note that orthogonal complement gives a homeomorphism Gr k,n ∼ = Gr n−k,n . In particular Gr n−1,n ∼ = Gr 1,n ∼ = CP n−1 , so ΣGr n−1,n is p-hyperbolic. Other Grassmannians can be treated more uniformly, as follows.\n\nLet γ k,n denote the tautological bundle over Gr k,n . Consider the inclusion ι n : C n → C n+1 (x 1 , x 2 , . . . , x n ) → (x 1 , x 2 , . . . , x n , 0). This inclusion induces a map i k,n : Gr k,n → Gr k,n+1 , defined on V ∈ Gr k,n by V → ι n (V ). It follows from this definition that i * k,n (γ k,n+1 ) = γ k,n . Letting e i denote the i-th standard basis vector in C n , we also have a map j k,n : Gr k,n → Gr k+1,n+1 , defined on V = Span(v 1 , v 2 , . . . , v k ) ∈ Gr k,n by V → Span(ι(v 1 ), ι(v 2 ), . . . , ι(v k ), e n+1 ). It follows from this definition that j * k,n (γ k+1,n+1 ) = γ k,n ⊕ C 1 , where C 1 is the 1-dimensional trivial bundle.\n\nSince K * (CP n ) is generated by the class of the tautological bundle, composing the maps i k,n and j k,n for different values of k and n will give maps CP 2 = Gr 1,3 → Gr k,n for all 1 ≤ k ≤ n − 2 and n ≥ 3 which induce surjections in integral K-theory. As in Example 2.5, this implies the following (the case k = n − 1 is Gr n−1,n , which was treated first).\n\nExample 2.6. For n ≥ 3 and 0 < k < n, the suspended complex Grassmannian ΣGr k,n is p-hyperbolic for all p = 2.\n\nFor m ≤ n, the Milnor Hypersurface H m,n is defined by\nH m,n = {([z], [w]) ∈ CP m × CP n | m i=0 z i w i = 0}.\nSuppose that m ≥ 2 and n ≥ 3. Then there is an inclusion ι : CP 2 → H m,n , defined by ι([z 0 : z 1 : z 2 ]) = ([z 0 : z 1 : z 2 : 0 : · · · : 0], [0 : · · · : 0 : 1]).\n\nWrite π 1 for the projection H m,n → CP m . Then the inclusion CP 2 → CP m factors as\nCP 2 # # • • • • • • • • • ι / / H m,n π1 CP m .\nThis implies that ι induces a surjection on integral K-theory, so we obtain the following.\n\nExample 2.7. For m ≥ 2 and n ≥ 3, the suspended Milnor Hypersurface ΣH m,n is p-hyperbolic for all p = 2.\n\nLet U (n) denote the unitary group. There is a well-known map r : ΣCP n−1 → U (n) (see for example [Whi78]) which induces a surjection on K-theory. From this we obtain Example 2.8. For n ≥ 3, the suspended unitary group ΣU (n) is p-hyperbolic for all p = 2.\n\n\nQuantitative lower bounds on growth\n\nIn Section 4, we will derive the following simple lower bound for the lim inf in the definition of Z/p rhyperbolicity, for the space S q1+1 ∨ S q2+1 .\n\nCorollary 2.9. Let p be a prime and r ∈ Z + . Let t m be the constants of Definition 1.2 for X = S q1+1 ∨ S q1+1 . Then\nlim inf m ln(t m ) m ≥ ln(2) max(q 1 , q 2 ) .\nThis implies that t m eventually exceeds ((1 − ε)2) m max(q 1 ,q 2 ) for any ε > 0. The constant 2 reflects the number of wedge summands. Note that this bound is independent of p and r.\n\nExample 2.10. Taking ε = 1 4 , we find that for all r ∈ Z + and all primes p the number of Z/p r -summands in i≤m π i (S 2 ∨ S 2 ) eventually exceeds ( 3 2 ) m . One can produce an analogous quantitative bound on the lim inf in the case of Theorem 1.4, but this bound is very weak. In particular, it depends on knowledge of the Adams operations on K * (X), and is at best ln(2) 2(p−1) .\n\n\nPreliminary results\n\nBoth Theorem 1.3 and Theorem 1.4 will be proven by means of Lemma 3.3. Our first goal is to establish this lemma. Let L be the free Lie algebra over Q on basis elements x 1 , . . . , x n . Write L k for the subset of L consisting of the basic products of the x i of weight k, in the sense of [Hil55], where the basic products of weight 1 are taken to be the x i , ordered by x 1 < x 2 < · · · < x n . The union L = ∞ k=1 L k is a vector space basis for L (see for example [Ser06,Theorem 5.3], but note that Serre uses the name Hall basis for the set of basic products).\n\nLet µ : Z + −→ {−1, 0, 1} be the Möbius inversion function, defined by\nµ(s) = 1 s = 1 0 s > 1 is not square free (−1) ℓ s > 1 is a product of ℓ distinct primes. The Witt Formula W n (k) is then defined by W n (k) = 1 k d|k µ(d)n k d .\nTheorem 3.1. [Hil55, Theorem 3.3] Let L be the free Lie algebra over Q on basis elements x 1 , . . . , x n . Then |L k | = W n (k).\n\nLemma 3.2. [BO15, Introduction] The ratio W n (k) 1 k n k tends to 1 as k tends to ∞.\n\nIn particular, this implies that for n ≥ 2, the Witt function W n (k) grows exponentially in k. It should follow that if the number of p-torsion summands in i≤k π i (Y ) exceeds W 2 (k), then Y is p-hyperbolic. The following lemma makes a slightly generalised form of this idea precise.\n\nLemma 3.3. Let Y be a space. Suppose that there exist a, b ∈ Z + such that the number of p-torsion summands (respectively, Z/p r -summands) in i≤ak+b π i (Y ) exceeds W 2 (k), for all k large enough. Then Y is p-hyperbolic (respectively, Z/p r -hyperbolic).\n\nProof. The proofs for p-and Z/p r -hyperbolicity are identical, so we give only the former. Reframing the hypothesis in terms of the sequence {T m } m of Definition 1.1, we are assuming precisely that T ak+b > W 2 (k) for sufficiently large k. We then have that\nlim inf m ln(T m ) m = lim inf k ln(T ak+b ) ak + b ≥ lim inf k ln(W 2 (k)) ak + b .\nIt then follows from Lemma 3.2 that if 1 > ε > 0, once k is large enough, we have\nW 2 (k) > (1 − ε) 1 k 2 k .\nThis implies that\nlim inf k ln(W 2 (k)) ak + b ≥ lim inf k ln((1 − ε) 1 k 2 k ) ak + b ,\nand since this holds for all ε > 0,\nlim inf m ln(T m ) m ≥ lim inf k ln(W 2 (k)) ak + b ≥ lim inf k ln( 1 k 2 k ) ak + b = lim inf k ln( 1 k ) + k ln(2) ak + b = ln(2) a ,\nwhich is greater than zero, as required.\n\n\nExistence of summands in the stable stems\n\nWe write π S j for the j-th stable stem in the homotopy groups of spheres, that is\nπ S j := lim n→∞ π n+j (S n ).\nThe proof of Theorem 1.3, depends on having, for each p and r, some j such that π S j contains a Z/p r -summand. The purpose of this subsection is to show that the existence of such a j follows from existing work of Adams and others.\n\nLemma 3.4. For any prime p and any r ∈ Z + , there exists j such that Z/p r is a direct summand in π S j . That is, for a fixed choice of such a j, Z/p r is a direct summand in π n+j (S n ) for all n ≥ j + 2. Proof. We write ν p (s) for the largest power of p dividing the integer s.\n\nCASE 1 (p odd): Set t := p r−1 (p − 1), and notice that, since (p − 1) is even, j := 4t − 1 is congruent to 7 mod 8. Theorem 1.6 of [Ada66], and the discussion immediately following it, then tells us that π S j contains a direct summand isomorphic to Z/m(2t), for a function m which Adams defines. By decomposing this subgroup into direct summands of prime power order, it suffices to show that ν p (m(2t)) = r.\n\nThe discussion after Theorem 2.5 in [Ada65] gives that since t ≡ 0 mod (p − 1),\nν p (m(2t)) = 1 + ν p (2t).\nNow, ν p (2t) is equal to (r − 1), by definition of t, so ν p (m(2t)) = r, as required. CASE 2 (p = 2, r ≥ 3): Set t := 2 r−3 , and set j := 4t − 1. From Theorem 1.5, and the discussion following Theorem 1.6 in [Ada66], π S j has a direct summand isomorphic to Z/m(2t), regardless of whether j is congruent to 3 or 7 mod 8. Again, referring to the discussion after Theorem 2.5 of [Ada65], we see that ν 2 (m(2t)) = 2 + ν 2 (2t) = 3 + ν 2 (t) = r, as required. CASE 3 (p r = 2 and p r = 4): It is known from [Fre38] \nthat π S 1 ∼ = Z/2, and from [BMT70] that π S 34 ∼ = Z/4 ⊕ (Z/2) 3 .\n\nProof of Theorem 1.3\n\nIn this section we prove Theorem 1.3, which says that the wedge of two spheres is Z/p r -hyperbolic for all p and r. We also prove Corollary 2.9, which extracts some simple quantitative information from the proof of Theorem 1.3. We first record the following simple observation.\n\nRemark 4.1. Let k 1 , . . . , k n and q 1 , . . . q n be non-negative integers. Suppose that q 1 ≤ q 2 ≤ · · · ≤ q n , and let k = n i=1 k i . Then\nkq 1 ≤ n i=1 k i q i ≤ kq n .\nProof of Theorem 1.3. Assume without loss of generality that q 1 ≤ q 2 . By Lemma 3.3 it suffices to show that there exist constants a and b such that the number of Z/p r -summands in\ni≤ak+b π i (S q1+1 ∨ S q2+1 ) exceeds W 2 (k), for k large enough.\nWe first apply the Hilton-Milnor Theorem. Since we are dealing with spheres, we need only the original form, due to Hilton in [Hil55]:\nΩ(S q1+1 ∨ S q2+1 ) ≃ ΩΣ(S q1 ∨ S q2 ) ≃ Ω B∈L S k1q1+k2q2+1 ,\nwhere, as in Section 3, L = ∞ k=1 L k is Hilton's 'basic product' basis for L, the free Lie Algebra over Q on two generators x 1 and x 2 , and k i is the number of occurrences of the generator x i in the bracket B. Recall also from Section 3 that the weight k of a bracket B is equal to k 1 + k 2 , and that the cardinality of L k is given by the Witt formula W 2 (k) by Theorem 3.1.\n\nFor fixed k ∈ Z + , consider the factor in the above product corresponding to L k ⊂ L :\nF k := Ω B∈L k S k1q1+k2q2+1 .\nThe associated subgroup of π * (S q1+1 ∨ S q2+1 ),\nB∈L k π * (S k1q1+k2q2+1 ),\nis a direct summand. We will first find a Z/p r -summand in the homotopy groups of each of the spheres appearing in F k . Since q 1 ≤ q 2 , Remark 4.1 applies, and we may lower bound the dimensions of spheres appearing in F k by k 1 q 1 + k 2 q 2 + 1 ≥ kq 1 + 1. By Lemma 3.4, there exists j ∈ Z + such that π j+ℓ (S ℓ ) has a direct summand Z/p r for ℓ ≥ j +2. Therefore, if k is large enough that kq 1 ≥ j +1, then k 1 q 1 +k 2 q 2 +1 ≥ j +2 -that is, the j-th stem is stable on all of the spheres occurring in F k . Thus, for k large enough, there is a Z/p r summand in π j+k1q1+k2q2+1 (S k1q1+k2q2+1 ) whenever k 1 + k 2 = k.\n\nWe now upper bound the dimension of the homotopy groups in which these summands appear. Since q 1 ≤ q 2 we have by Remark 4.1 that j+k 1 q 1 +k 2 q 2 +1 ≤ kq 2 +1+j, so each of the Z/p r -summands we have identified is a distinct direct summand in\ni≤kq2+1+j B∈L k π i (S k1q1+k2q2+1 ), hence in i≤kq2+1+j π i (S q1+1 ∨ S q2+1 ).\nWe have identified one such summand for each B ∈ L k , so the number of Z/p r -summands in\ni≤kq2+1+j π i (S q1+1 ∨ S q2+1 ) is at least |L k | = W 2 (k)\n. Thus, taking a = q 2 and b = 1 + j in Lemma 3.3 suffices.\n\nProof of Corollary 2.9. The last line of the proof of Lemma 3.3 shows that lim inf m ln tm m > ln 2 a . The last line of the proof of Theorem 1.3 implies that a may be taken to be q 2 , under the assumption that q 1 ≤ q 2 , which implies the result.\n\n\nK-theory and K-homology of ΩΣX\n\nThe remainder of this paper proves Theorem 1.4. Sections 5 and 6 assemble necessary background, which we will use in Section 7 to prove the result.\n\nWhen studying the homotopy groups of a suspension ΣX, as in Theorem 1.4, the following approach is natural. Since π * (ΣX) ∼ = π * −1 (ΩΣX), we may instead study ΩΣX. This is useful because ΩΣX is well understood homologically via the Bott-Samelson theorem, which decomposes its homology as the tensor algebra on H * (X). Because we will need to use Adams' e-invariant, which is defined in terms of K-theory, we wish to replace ordinary homology with K-homology.\n\nThe purpose of Section 5 is to record the version of the Bott-Samelson theorem which applies to (torsion-free) K-homology, along with a universal coefficient theorem for passing between K-theory and K-homology. All of the material here is already known (in particular much of it is in [Sel83]) so its summary here is for convenience and clarity.\n\nOur conventions on definition of K * (X) are those of [AH61]. In particular, we define K −1 (X) := K 0 (ΣX), and set K * (X) := K 0 (X) ⊕ K −1 (X). We regard K * (X) and K * (X) as being Z/2-graded.\nIt is shown in [AH61] that K * (X) is a Z/2-graded ring.\nWe will wish to work with K-theory and K-homology modulo the torsion subgroup. For a space X, write K TF * (X) and K * TF (X) for the quotients of the reduced K-homology and K-theory of X by their torsion subgroups. The same convention applies in the unreduced case.\n\n\nKünneth and universal coefficient theorems\n\nThe universal coefficient theorem for K-theory first appears in some unpublished lecture notes of Anderson [And], and is first published by Yosimura [Yos75].\n\nTheorem 5.1 (Universal coefficient theorem). For any CW -complex X and each integer n there is a short exact sequence\n0 → Ext(K n−1 (X), Z) → K n (X) → Hom(K n (X), Z) → 0.\nIn the torsion-free case, the universal coefficient theorem is as follows, where, unsurprisingly, we write Ext( K n−1 (X), Z) TF for the quotient of Ext( K n−1 (X), Z) by its torsion subgroup.\n\nCorollary 5.2.\n\n1. For any CW -complex X and each integer n there is a short exact sequence\n0 → Ext( K n−1 (X), Z) TF → K n TF (X) → Hom( K TF n (X), Z) → 0.\n2. If X is a finite CW -complex, then Ext( K n−1 (X), Z) TF = 0, and we obtain an isomorphism\nK n TF (Y ) ∼ = − → Hom( K TF n (Y ), Z).\nProof. Let T 1 denote the torsion subgroup of Ext( K n−1 (X), Z), and let T 2 be the torsion subgroup of K n (X). The Universal Coefficient Sequence of Theorem 5.1 gives an injection T 1 → K n (X), which must have image contained in T 2 , thus lift to an injection T 1 → T 2 . For any group G, Hom(G, Z) is torsion-free, so the composite T 2 → K n (X) → Hom(K n (X), Z) is trivial, and by exactness we obtain a lift T 2 → Ext( K n−1 (X), Z). The image of this map must be torsion, which is to say that it must be contained in T 1 , so the aforementioned map T 1 → T 2 is a surjection. That is, the Universal Coefficient Sequence of Theorem 5.1 has last term torsion-free and first map restricting to an isomorphism of torsion subgroups. This implies the first statement.\n\nFor the second statement, we need only note that if X is finite, then K TF n−1 (X) is finitely generated, so Ext( K n−1 (X), Z) is torsion, as required.\n\nSelick [Sel83] deduces the following from work of Atiyah [Ati62], Mislin [Mis71] and Adams [Ada69].\n\nTheorem 5.3 (Künneth theorem for K-homology). Let X and Y be of the homotopy type of finite complexes. Then there is an isomorphism of Z/2-graded Z-modules:\nK TF * (X ∧ Y ) ∼ = K TF * (X) ⊗ K TF * (Y ).\nRemark 5.4. It follows immediately from Corollary 5.2 (and knowledge of K * (S q )) that K TF * (S q ) ∼ = Z. We write ξ q for the generator of K TF * (S q ). By the Künneth Theorem (Theorem 5.3), we may choose the ξ q so that ξ n ⊗ ξ m is identified with ξ n+m under the homeomorphism S n ∧ S m ∼ = S n+m .\n\nIn the case of K-theory, the analogous result follows directly from [Ada69].\n\nTheorem 5.5 (Künneth theorem for K-theory). Let X and Y be of the homotopy type of finite complexes. Then the external product on K-theory defines an isomorphism of Z/2-graded rings:\nK * TF (X) ⊗ K * TF (Y ) ∼ = − → K * TF (X ∧ Y ).\n\nThe James construction\n\nFor a space X, let X s denote the product of s copies of X. Let ∼ be the relation on X s defined by\n(x 1 , . . . , x i−1 , * , x i+1 , x i+2 , . . . x s ) ∼ (x 1 , . . . , x i−1 , x i+1 , * , x i+2 , . . . x s ). Let J s (X) be the space X s ∼ . There is a natural inclusion J s (X) ֒− → J s+1 (X) (x 1 , . . . , x s ) → (x 1 , . . . , x s , * ).\nThe James construction JX is defined to be the colimit of the diagram consisting of the spaces J s (X) and the above inclusions. Write i s : J s (X) → JX for the map associated to the colimit. Notice that JX carries a product given by concatenation, which makes it into the free topological monoid on X, and that a topological monoid is in particular an H-space.\n\nLet X ∧i denote the smash product of i copies of X, and let η : X → ΩΣX be the unit of the adjunction Σ ⊣ Ω. Explicitly, η(x) = (t → x, t ∈ ΣX).\n\nTheorem 5.6. [Jam55] 1. There is a homotopy equivalence JX ≃ − → ΩΣX which respects the H-space structures and identifies i 1 with η.\n\n\nThere is a homotopy equivalence\n∞ i=1 ΣX ∧i ≃ − → ΣJX which restricts to a homotopy equivalence s i=1 ΣX ∧i ≃ − → ΣJ s (X) for each s ∈ Z + .\nLemma 5.7. [Sel83, Lemma 7] Let X have the homotopy type of an (r − 1)-connected CW -complex.\n1. (i s ) * : π N (J s (X)) → π N (JX) is an isomorphism for N < r(s + 1) − 1.\n2. Let x ∈ π N (J s (X)) for any N . If Σx is nontrivial then (i s ) * (x) is also nontrivial.\n\nProof. The first part follows by cellular approximation from the observation that J s (X) contains the (r(s + 1) − 1)-skeleton of JX. The second part follows from the observation that Σi s has a retraction by Theorem 5.6.\n\nFor spaces X and Y , let X * Y denote the join, which we define to be the homotopy pushout of the projections X × Y → X and X × Y → Y . The join is naturally a quotient of X × I × Y , where I denotes the unit interval. Following the treatment in [Ark11], let C 1 denote the subspace of X * Y consisting of points of the form (x, t, * ), for t ∈ I and x ∈ X, and let C 2 be the subspace consisting of points of the form ( * , t, y). The subspace\nC 1 ∪ C 2 ∼ = CX ∪ CY is contractible, so the quotient map q : X * Y → X * Y C 1 ∪ C 2 is a homotopy equivalence. The quotient X * Y C 1 ∪ C 2 is homeomorphic to ΣX ∧ Y . The suspended product Σ(X × Y ) is also a quotient of X × I × Y , and this quotient lies between X * Y and X * Y C 1 ∪ C 2 .\nThis gives a factorization of q as\nX * Y → Σ(X × Y ) → Σ(X ∧ Y )\n. Let q −1 denote any choice of homotopy inverse to q; all possible choices are homotopic. We may form a new map δ X,Y as the\ncomposite Σ(X ∧ Y ) q −1 −−→ X * Y → Σ(X × Y ). It is automatic that δ X,Y splits the quotient map π : Σ(X × Y ) → ΣX ∧ Y .\nThe homotopy class of δ X,Y is well-defined, and we will call δ X,Y the canonical splitting of π. Note that δ X,Y is natural in maps of spaces in the sense that given f :\nA → X and g : B → Y we obtain a commutative diagram ΣA ∧ B δA,B / / Σ(f ∧g) Σ(A × B) Σ(f ×g) ΣX ∧ Y δX,Y / / Σ(X × Y ).\nFor s ≥ 3, consider the quotient map ΣX s → ΣX ∧s . We define the canonical splitting of this quotient to be the composite of canonical splittings\nΣX ∧s → Σ(X × X) ∧ X ∧(s−2) → Σ((X × X) × X) ∧ X ∧(s−3) → · · · → ΣX s .\nOf course, we chose an order of multiplication here. This canonical splitting is natural as before. Write m s : (ΩΣX) s → ΩΣX for the map given by iteratively performing the standard loop multiplication on ΩΣX in any choice of order. Up to homotopy, m s is independent of this choice of order, since ΩΣX is homotopy associative.\n\nTheorem 5.6 gives the existence of a homotopy equivalence Γ :\n∞ i=1 ΣX ∧i → ΣΩΣX.\nThere are many choices of Γ, up to homotopy. The next lemma asserts that Γ can be chosen in a way which suits our purpose. Selick [Sel83] describes the composite\n∞ i=1 ΣX ∧i Γ − → ΣΩΣX ≃ − → ΣJX of Γ\nwith the homotopy equivalence of Theorem 5.6 (1). This immediately implies the following description of Γ.\n\nLemma 5.10. [Sel83] Let X be a finite CW -complex. The homotopy equivalence Γ : ∞ i=1 ΣX ∧i → ΣΩΣX may be chosen such that:\n1. S −1 (Γ * ) : T ( K TF * (X)) ∼ = − → K TF * (ΩΣX)\nis an isomorphism of algebras; 2. the restriction of Γ to ΣX ∧s is homotopic to the composite\nΣX ∧s → ΣX s Σ(η) s −−−→ Σ(ΩΣX) s Σms − −− → ΣΩΣX,\nwhere the unlabelled arrow is the canonical splitting.\n\nThe description of the map Γ in Lemma 5.10 has the following consequence. For a space Y , let ev : ΣΩY → Y be the evaluation map, which may be described explicitly by ev( γ, t ) = γ(t) for γ ∈ ΩY .\n\nLemma 5.11. Let Γ : ∞ i=1 ΣX ∧i → ΣΩΣX be the homotopy equivalence of Lemma 5.10. The composite ev • Γ is homotopic to the projection onto the first wedge summand.\n\nProof. Let ι s : ΣX ∧s → ∞ i=1 ΣX ∧i be the inclusion of the s-th wedge summand. We must show that ev • Γ • ι s ≃ 1 ΣX if s = 1, and * otherwise.\n\nThe following diagram commutes up to homotopy \nΣX Ση / / 1ΣX $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ΣΩΣXΣη ∧s O O / / Σ(X (s−1) ∧ X) Σ(η (s−1) ∧η) O O / / ΣX s . Ση s O O\nThe composite along the bottom of the diagram is Γ • ι s , so to obtain the desired factorization of Γ • ι s through v, it suffices to show that the diagram commutes up to homotopy.\n\nThe top right square commutes because m 2 • m (s−1) ≃ m s , by homotopy associativity of the Hspace ΩΣX. The remaining three squares commute by naturality of our canonical splitting. This completes the proof.\n\nLet ρ k be the projection T ( K TF * (X)) → K TF * (X) ⊗k . The next corollary is immediate from Lemma 5.11.\n\nCorollary 5.12. S −1 (ev * • Γ * ) = ρ 1 : T ( K TF * (X)) → K TF * (X).\n\n\nPrimitives and commutators\n\nIt follows from the Künneth Theorem (Theorem 5.3), and the fact that\nΣ(Y ×Y ) ≃ ΣY ∨ΣY ∨Σ(Y ∧Y ), that K TF * (Y × Y ) ∼ = K TF * (Y ) ⊗ K TF * (Y )\n. We may therefore make the following definition. A class\ny ∈ K TF * (Y ) is called primitive if ∆ * (y) = y ⊗ 1 + 1 ⊗ y, where ∆ : Y → Y × Y is the diagonal, defined by ∆(y) = (y, y). The comultiplication Y → Y ∨ Y on a co-H-space Y is a factorization of ∆ via the inclusion Y ∨ Y ֒− → Y × Y .\nFrom this point of view, the following lemma is immediate.\n\nLemma 5.13. If Y is a co-H-space, then all elements in K TF * (Y ) are primitive.\n\nIf Y is an H-group, then the multiplication m :\nY × Y → Y induces a map K TF * (Y ) ⊗ K TF * (Y ) → K TF * (Y )\n. We will denote this map by juxtaposition, so that m * (y 1 ⊗ y 2 ) = y 1 y 2 . Furthermore, the commutator Y ×Y → Y descends to a map c : Y ∧Y → Y . Expanding the definition of the commutator in terms of the K-homology Künneth Theorem (Theorem 5.3) gives the following lemma.\n\nLemma 5.14. Let Y be an H-group, and let c : Y ∧Y → Y be the commutator. If y 1 and y 2 ∈ K TF * (Y ) are primitive, then c * (y 1 ⊗ y 2 ) = y 1 y 2 − (−1) |y1||y2| y 2 y 1 .\n\n\nThe category of ψ-modules\n\nIn [Ada66], Adams defines an abelian category which we will follow Selick [Sel83] in calling ψ-modules. The e-invariant, which is our central tool, is defined by Adams in terms of ψ-modules. The purpose of this section is to record results about ψ-modules for later use.\n\nA ψ-module consists of an abelian group M , with homomorphisms\nψ ℓ : M → M\nfor each ℓ ∈ Z, satisfying the axioms of [Ada66, Section 6]. If X is a space then the group K 0 (X), together with its Adams operations, is a ψ-module. Since we defined K −1 (X) by setting K −1 (X) = K 0 (ΣX), it too has the structure of a ψ-module. Maps of spaces induce maps of ψ-modules. The Adams operation ψ ℓ on K 0 (S 2n ) is multiplication by ℓ n , so in particular Adams operations do not commute with the Bott isomorphism. For graded ψ-modules M and N we will write Hom ψ−Mod (M, N ) for the abelian group consisting of graded ψ-module homomorphisms. The unadorned notation Hom(M, N ) will mean homomorphisms of the underlying graded abelian groups. Proof. Let ϕ : M → N be a homomorphism of underlying Z-modules. If, for some k ∈ Z \\ {0}, k · ϕ is a ψ-module homomorphism, then, since N is torsion-free, ϕ is also a ψ-module homomorphism. This implies that Coker(Hom ψ−Mod (M, N ) ֒− → Hom(M, N )) is torsion-free, which implies the result. Definition 6.2 (Adams' e-invariant). Suppose that f : X → Y induces the trivial map on K * . Then the cofibre sequence of f gives a short exact sequence of ψ-modules\n0 ← K 0 (Y ) ← K 0 (C f ) ← K 0 (ΣX) ← 0.\nThe e-invariant of f is the element of Ext ψ−Mod ( K 0 (Y ), K 0 (ΣX)) represented by this exact sequence.\n\nThe e-invariant does not commute with the Bott isomorphism, but the interaction between the Bott isomorphism and the Adams operations is easy to describe, as follows. Let ψ ℓ Y be the homomorphism ψ ℓ : K 0 (Y ) → K 0 (Y ). Then, modulo the Bott isomorphism, we have ψ ℓ Σ 2 X = ℓ · ψ ℓ X . That is 'upon double suspending, the Adams operations gain a factor ℓ'. In terms of the e-invariant, all we need to know is the following.\nLemma 6.3. [Ada66, Proposition 3.4b)]\nThere is a homomorphism\nT : Ext ψ−Mod ( K 0 (Y ), K 0 (ΣX)) → Ext ψ−Mod ( K 0 (Σ 2 Y ), K 0 (Σ 3 X)),\nsuch that T (e(f )) = e(Σ 2 f ).\n\nWe will be concerned only with the e-invariants of maps whose domain is a sphere. One of the two K-groups of a sphere vanishes, in the dimension matching the parity of the sphere, but the e-invariant, as defined above, lives only in K 0 . In order to detect maps regardless of the parity of the sphere on which they are defined, we will need to keep track of the e-invariants of f and Σf , so we will use the following modified e-invariant.\n\n\nDefinition 6.4 (Double e-invariant). Let\nExt ψ−Mod ( K * (Y ), K * (ΣX)) := Ext ψ−Mod ( K 0 (Y ), K 0 (ΣX)) ⊕ Ext ψ−Mod ( K −1 (Y ), K −1 (ΣX)).\nSuppose that f : X → Y induces the trivial map on K * . Then the double e-invariant of f is e(f ) = (e(f ), e(Σf )) ∈ Ext ψ−Mod ( K * (Y ), K * (ΣX)).\n\n\nPullback of an extension along a homomorphism defines a map\nHom ψ−Mod (M, B) ⊗ Ext ψ−Mod (B, A) → Ext ψ−Mod (M, A).\nIf g : Y → Z then e(g • f ) is represented by the pullback of e(f ) and g * : K 0 (Z) → K 0 (Y ) [Ada66, Proposition 3.2 b)]. To describe e(g • f ) we need only apply this result degree-wise, as follows. For convenience, we write g * · e(f ) for the pullback of g * and e(f ). Define the map\nθ 0 (f ) : Hom ψ−Mod ( K 0 (Z), K 0 (Y )) → Ext ψ−Mod ( K 0 (Z), K 0 (ΣX)) θ 0 (f )(x) = x · e(f ). Likewise, define θ −1 (f ) : Hom ψ−Mod ( K −1 (Z), K −1 (Y )) → Ext ψ−Mod ( K −1 (Z), K −1 (ΣX)) θ −1 (f )(x) = x · e(Σf ).\nCombining these, let\nθ(f ) : Hom ψ−Mod ( K * (Z), K * (Y )) → Ext ψ−Mod ( K * (Z), K * (ΣX))\nbe the direct sum θ 0 (f ) ⊕ θ −1 (f ). These definitions, together with Adams' above result, give the following lemma.\n\nLemma 6.5. For maps f : X → Y and g : Y → Z, the following diagram commutes:\n[Y, Z] f * / / deg [X, Z] e Hom ψ−Mod ( K * (Z), K * (Y )) θ(f ) / / Ext ψ−Mod ( K * (Z), K * (ΣX)).\nFollowing [Sel83], write Z(n) for the ψ-module K 0 (S 2n ). Explicitly, Z(n) has underlying abelian group Z, and ψ ℓ acts by multiplication by ℓ n . It follows that K −1 (S 2n+1 ) := K 0 (S 2n+2 ) ∼ = Z(n + 1).\n\nLemma 6.6. [Ada66, Proposition 7.8, 7.9] If n < m then Ext ψ−Mod (Z(n), Z(m)) injects into Q Z . The e-invariant of a map f : S 2m−1 → S 2n may therefore be regarded as an element of Q Z . Furthermore, the value e(f ) in Q Z satisfies e(Σ 2 f ) = e(f ), so in particular, when f is a map between spheres, e(f ) depends only on the stable homotopy class of f .\n\nThe following theorem is the main technical component of Selick's paper [Sel83].\n\nTheorem 6.7. [Sel83, Theorem 6] Let f ′ : S 2m−1 → S 2n be such that p t−1 e(f ′ ) = 0 in Q Z , for p prime and some t ∈ Z + . Let Y have the homotopy type of a finite CW -complex and let g : S 2n → Y be such that Im(g * : K 0 (Y ) → K 0 (S 2n )) contains up s K 0 (S 2n ), for s ∈ Z + and u prime to p. If s < t, and there exists some ℓ ∈ Z + for which\nψ ℓ ⊗ Q : K 0 (Y ) ⊗ Q → K 0 (Y ) ⊗ Q\ndoes not have ℓ m as an eigenvalue, then e(g • f ′ ) = 0.\n\nThe following theorem of Gray [Gra69] will provide the map f ′ for Theorem 6.7. Specifically, this theorem provides a linearly spaced family of stems, each of which has a stable p-torsion class which is born on S 3 and detected by the e-invariant.\n\nTheorem 6.8. [Gra69, Corollary of Theorem 6.2] Let p be an odd prime and let j ∈ Z + . Then there exists a class f p,j ∈ π 2j(p−1)+2 (S 3 ) with e(f p,j ) = −1 p ∈ Q Z .\n\nThe corresponding 2-primary result is as follows. Adams [Ada66, Theorem 1.5 and Proposition 7.14] shows that, for j > 0, the (8j +3)-rd stem contains a direct summand whose 2-primary component has order 8, and that on this component the e-invariant is a surjection onto Z/4. The sphere of origin of the classes in this component was deduced by Curtis in [Cur69].\n\nTheorem 6.9. [Ada66; Cur69] Let j ∈ Z + . Then there exists a class f 2,j ∈ π 8j+6 (S 3 ) of order 4, with e(f 2,j ) = −1 2 ∈ Q Z .\n\n\nMain construction\n\nHaving assembled preliminaries in Sections 5 and 6, we can begin to work towards the proof of Theorem 1.4. Our approach is as follows. From the data of Theorem 1.4, we will construct a commutative diagram of (roughly) the following form, where B is a set and the other objects are Z-modules.\nB k / / π * (ΩΣX) I k / / Ext ψ−Mod ( K * (ΩΣX), K * (S * )).\nWe will argue that\n\n• The image of the top map consists of classes of order dividing p.\n\n• The image of the left vertical map generates a submodule isomorphic to the weight k component of the free graded Lie algebra over Z/p on two generators.\n\n• The bottom map is injective.\n\nTogether, these facts imply that there is a submodule of π * (ΩΣX) ∼ = π * +1 (ΣX), consisting of classes of order dividing p, and surjecting onto a module isomorphic to the weight k component of the free graded Lie algebra over Z/p on two generators. This submodule (which is necessarily a Z/p-vector space) must therefore have dimension at least W 2 (k) (Theorem 7.5), which will imply that ΣX is p-hyperbolic (Lemma 3.3).\n\nThe diagram will be obtained by juxtaposing three squares. Subsections 7.1, 7.2, and 7.3 each construct one of these squares. In Subsection 7.4 we put them together and prove Theorem 1.4. Roughly speaking, the top map of the diagram should be thought of as first taking a family of Samelson products and then pulling them back along some suitable map f coming from Gray's work (Theorem 6.8). The vertical maps should be thought of as passing from maps of spaces to K-theoretic invariants, and the bottom map (therefore) should be thought of as tracking the effect of the top map on these invariants.\n\nBecause of the need to work with a finite CW -complex in Selick's Theorem (Theorem 6.7) we will restrict the right hand side of the diagram to instead refer to some finite skeleton J s (X) of the James construction.\n\n\nSamelson products and their Hurewicz images in K-homology\n\nLet R be a commutative ring with unit. We take a graded Lie algebra over R to be defined as in [Nei10]. For a non-negatively graded R-module V , let L(V ) denote the free graded Lie algebra [Nei10, Section 8.5]. Write L k (V ) for the submodule of L(V ) generated by the brackets of length k in the elements of V . We will call L k (V ) the weight k component of L(V ). Note that this convention differs from Neisendorfer's -he writes L(V ) k for the weight k component.\n\nDefinition 7.1. Let Y be an H-group, and let c : Y ∧ Y → Y be the commutator of Lemma 5.14. Let α ∈ π N (Y ), and let β ∈ π M (Y ). The Samelson product of α and β, written α, β ∈ π N +M (Y ), is the composite α, β :\nS N +M ∼ = S N ∧ S M α∧β − −− → Y ∧ Y c − → Y.\nSamelson products are bilinear, graded anticommutative, and satisfy the graded Jacobi identity. They fail, however, to make π * (Y ) into a graded Lie algebra over Z in Neisendorfer's sense [Nei13,Section 7]. This is important because we want the natural map from the free Lie algebra to the corresponding tensor algebra to be an injection onto a summand (Lemma 7.4). One could define an appropriate notion of 'free graded pseudo-Lie algebra', and proceed as follows with that in place of the set B(π * (A)), which we use in what follows, but we prefer to avoid making the extra definition. which extends ν * and satisfies Φ π ν ([x, y]) = Φ π ν (x), Φ π ν (y) for all x, y ∈ B(π * (A)). For a Z/2-graded Z-module V , we define a non-negatively graded Z-module Hom( K TF * (S * ), V ), by setting\nHom( K TF * (S * ), V ) N = Hom( K TF * (S N ), V ) if N > 0, and 0 if N ≤ 0,\nwhere the homomorphisms are understood to respect the Z/2-grading on K * and V .\n\nIn the case that V = L is a Z/2-graded Lie algebra over Z, Hom( K TF * (S * ), L) inherits a nonnegatively graded Lie algebra structure as follows. Let the generators ξ N of K TF * (S N ) be as in Remark 5.4. Then the bracket [f, g] of f ∈ Hom( K TF * (S N ), L) and g ∈ Hom( K TF\n* (S M ), L) is the homomor- phism K TF * (S M ) → L carrying ξ N +M to [f (ξ N ), g(ξ M )] ∈ L.\nThe squaring operation is defined in the same way. Likewise, if V is a Z/2-graded associative algebra over Z, then Hom( K TF * (S * ), V ) inherits the structure of a non-negatively graded associative algebra.\n\nLet ν : A → ΩΣX be a map. There is a composition\nL( K TF * (A)) → T ( K TF * (A)) → K TF * (ΩΣX),\nwhere the first map is the natural map which is the identity on K TF * (A) and satisfies [x, y] → xy − (−1) |x||y| yx, and the second map is obtained by applying the universal property of the tensor algebra to ν * . Let Φ K ν : Hom( K TF * (S * ), L( K TF * (A))) → Hom( K TF * (S * ), K TF * (ΩΣX)) be the pushforward along the above composite. It is then automatic that Φ K ν is a map of non-negatively graded Lie algebras over Z, where the structures are defined as above.\n\nWe write deg : π N (Y ) → Hom( K TF * (S N ), K TF * (Y )) for the map f → f * . Let deg ′ : B(π * (A)) → Hom( K TF * (S * ), L( K TF * (A))) be the unique map which restricts to deg : π * (A) → Hom( K TF * (S * ), K TF * (A)) ⊂ Hom( K TF * (S * ), L( K TF * (A))) and carries brackets to brackets. The above maps are related as follows. Lemma 7.2. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CWcomplexes. The following diagram commutes:\nB(π * (A)) Φ π ν / / deg ′ π * (ΩΣX) deg Hom( K TF * (S * ), L( K TF * (A))) Φ K ν / / Hom( K TF * (S * ), K TF * (ΩΣX)).\nProof. By construction of B(π * (A)), it suffices to show that the restriction of the diagram to the weight 1 component B 1 (π * (A)) = π * (A) commutes, and that all maps respect the bracket operations. By definition, L 1 ( K TF * (A)) = K TF * (A). It then follows immediately from the definitions of Φ π ν and Φ K ν that restricting the left hand side of the diagram to weight 1 components gives the diagram\nπ * (A) ν * / / deg π * (ΩΣX) deg Hom( K TF * (S * ), K TF * (A)) ν * / / Hom( K TF * (S * )\n, K TF * (ΩΣX)), which commutes, since it just expresses naturality of deg.\n\nIt remains to show that all maps respect bracket operations. The maps Φ π ν and deg ′ respect the bracket operations by definition, and Φ K ν respects bracket operations by construction. We therefore only need show that deg respects brackets. Let f ∈ π N (ΩΣX), and let g ∈ π M (ΩΣX). We must show that deg( f, g ) is the commutator deg(f )deg(g) − (−1) N M deg(g)deg(f ) with respect to the algebra operation on Hom( K TF * (S * ), K TF * (ΩΣX)). Since K TF * (S N +M ) ∼ = Z, it suffices to show that the two homomorphisms agree on the generator ξ N +M (Remark 5.4). By Definition 7.1 and the Künneth Theorem (Theorem 5.3),\ndeg( f, g )(ξ N +M ) = c * • (f * ⊗ g * )(ξ N ⊗ ξ M ) = c * • (f * (ξ N ) ⊗ g * (ξ M )).\nSpheres of dimension at least 1 are co-H spaces, so by Lemma 5.13, ξ N and ξ M are primitive. By naturality of the diagonal f * (ξ N ) and g * (ξ M ) are still primitive, so by Lemma 5.14,\nc * • (f * (ξ N ) ⊗ g * (ξ M )) = f * (ξ N )g * (ξ M ) − (−1) N M g * (ξ M )f * (ξ N ),\nwhich by definition of the multiplication on Hom( K TF * (S * ), K TF * (ΩΣX)) is the result of evaluating deg(f )deg(g) − (−1) N M deg(g)deg(f ) on ξ N +M , as required.\n\nWe now lift the previous result to J s (X), thereby producing the first square of the diagram promised at the start of this section. Recall that we write i s : J s (X) → JX for the inclusion, and that by Theorem 5.6 we have a homotopy equivalence JX ≃ − → ΩΣX. We will abuse notation and also write i s for the composite J s (X) → JX ≃ − → ΩΣX.\n\nCorollary 7.3. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CWcomplexes, with X (r − 1)-connected for r ≥ 1. If N, s ∈ Z + satisfy N < r(s + 1) − 1, then (i s ) * : π N (J s X) → π N (ΩΣX) is an isomorphism and for each k ≤ s there exists a commutative diagram:\nB k N (π * (A)) Φ π ν / / deg ′ π N (J s X) deg Hom( K TF * (S N ), L k ( K TF * (A))) Φ K ν / / Hom( K TF * (S N ), K TF * (J s X)), with (i s ) * • Φ π ν = Φ π ν and Hom( K TF * (S N ), (i s ) * ) • Φ K ν = Φ K ν .\nProof. Consider the diagram of Lemma 7.2. Lemma 5.7 shows that (i s ) * is an isomorphism on π N , so let Φ π ν be the unique map such that the condition (i s ) * • Φ π ν = Φ π ν holds. By Theorem 5.6 (2) and Lemma 5.10, the map (i s ) * : K TF N (J s (X)) → K TF N (ΩΣX) is the inclusion of the tensors of length at most s. Since k ≤ s, we may therefore define Φ K ν to be the unique map such that the condition\nHom( K TF * (S N ), (i s ) * ) • Φ K ν = Φ K ν holds.\nCommutativity then follows from Lemma 7.2 by naturality of deg, since Hom( K TF * (S N ), (i s ) * ) is injective.\n\nLemma 7.4. Let V be a non-negatively-or Z/2-graded Z-module which is free and finitely generated in each dimension. Then\n\n• L(V ) and T (V ) are free Z-modules in every dimension.\n\n• The natural map L(V ) → T (V ), [x, y] → xy − (−1) |x||y| yx is an injection onto a summand.\n\nProof. The non-negatively graded case is immediate from [Nei10], Proposition 8.3.1 and p282. For the Z/2-graded case, first observe that there is a forgetful functor U from Z-graded modules to Z/2-modules which carries Z-graded (Lie) algebras to Z/2-graded (Lie) algebras, and a functor C from Z/2-graded modules to Z-modules which puts V 0 in any even dimension and V 1 in any odd dimension. Both C and U respect freeness and split injections, and there are natural isomorphisms U T (CV ) ∼ = T (V ) and U L(CV ) ∼ = L(V ). This implies the Z/2-graded result.\n\nThe graded version of Theorem 3.1 now follows immediately from Hilton's paper:\n\nTheorem 7.5. [Hil55, Theorem 3.2, 3.3] Let V be a torsion-free Zor Z/2-graded Z-module of total dimension n. Then the total dimension of L k (V ) is W n (k).\n\nLet R be a commutative ring with unit. Let M be an R-module, and as usual let T (M ) denote the tensor algebra on M . Let ι k : M ⊗k → T (M ) be the inclusion, and let ρ k : T (M ) → M ⊗k be the projection. Let τ : T (M ) → T (M ) be the composite ι 1 • ρ 1 . Given an R-algebra A, and a map ϕ : M → A, we write ϕ for the induced map T (M ) → A, that is, the unique map of algebras such that ϕ • ι 1 = ϕ. Now, let M and N be R-modules, and let ϕ : M → T (N ) be a map. In the proof of Theorem 7.7, we will wish to make a 'leading terms' style argument. This is made precise in the next Lemma, which compares ϕ with τ • ϕ. Informally, we think of τ • ϕ as the 'leading terms part' of ϕ.\n\nLemma 7.6. Let R be a commutative ring with unit. Let M and N be Zor Z/2-graded R-modules. Let ι k : M ⊗k → T (M ) be the inclusion, let ρ k : T (N ) → N ⊗k be the projection, and let τ : T (N ) → T (N ) be as above. Let ϕ : M → T (N ) be a map. Then\nρ k • ϕ • ι k = ρ k • τ • ϕ • ι k .\nProof. It suffices to check equality on basic tensors. Let v ∈ M ⊗k be a basic tensor, so that\nv = v 1 ⊗ v 2 ⊗ · · · ⊗ v k , for v i ∈ M . Then ϕ • ι k (v) = ϕ(v 1 ⊗ v 2 ⊗ · · · ⊗ v k ) = ϕ(v 1 ) ⊗ ϕ(v 2 ) ⊗ · · · ⊗ ϕ(v k ) = τ (ϕ(v 1 )) ⊗ τ (ϕ(v 2 )) ⊗ · · · ⊗ τ (ϕ(v k )) + terms of weight > k.\nApplying ρ k to both sides yields the result.\n\nTheorem 7.7. Let F = Q or Z/p for p prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes. Let ν : ΣA → ΣX be the adjoint of ν. If\nν * ⊗ F : K TF * (ΣA) ⊗ F → K TF * (ΣX) ⊗ F is an injection, then Φ K ν ⊗ F : Hom( K TF * (S * ), L( K TF * (A))) ⊗ F → Hom( K TF * (S * ), K TF * (ΩΣX)) ⊗ F\nis also an injection.\n\nRemark 7.8. In the case where ν is a suspension Σζ, we have a diagram\nΩΣA Ων / / ΩΣX A η O O ν : : ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ζ / / X, η O O\nso in particular ν * factors through the weight 1 component K TF * (X) of the tensor algebra decomposition of K TF * (ΩΣX). This dramatically simplifies the proof, removing the need for Lemma 7.6. In practice this is not a reasonable assumption -for example, the map µ : S 3 ∨ S 5 → ΣCP 2 of Example 2.5 (which plays the role of ν) does not desuspend.\n\nProof. In this proof, for a space Y , we will identify the algebras T ( K TF * (Y )) and K TF * (ΩΣY ), omitting the isomorphism S −1 Γ * of Lemma 5.10. We defined Φ K ν to be the pushforward along a certain map L( K TF * (A)) → K TF * (ΩΣX). Call this map Φ K ν ′ . It suffices to prove that Φ K ν ′ ⊗ F is an injection.\n\nThe triangle identities for the adjunction Σ ⊣ Ω give a commutative diagram\nΩΣA Ων / / ΩΣX A. η O O ν : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ Since Φ K ν\n′ is the unique map of Lie algebras extending ν, we have a commuting diagram\nT ( K TF * (A)) ∼ = K TF * (ΩΣA) (Ων) * / / K TF * (ΩΣX) L( K TF * (A)), ? O O Φ K ν ′ 5 5 ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦\nwhere we note that that by Lemma 7.4, the natural map L( K TF * (A)) → T ( K TF * (A)) is an injection onto a summand. It therefore suffices to show that (Ων) * ⊗ F is an injection.\n\nLet (ν * ) denote the extension of ν * to T ( K TF * (A)), so that (ν * ) = (Ων) * (modulo the isomorphism S −1 Γ * , as above). Since (ρ i • (ν * ) • ι k ) = 0 for i < k, it further suffices to show that (ρ k • (ν * ) • ι k ) ⊗ F is an injection for each k. By Lemma 7.6, with M = K TF * (A) and N ′ = K TF * (X), we have that\nρ k • (ν * ) • ι k = ρ k • (τ • ν * ) • ι k .\nAs previously, let ev : ΣΩY → Y denote the evaluation map. The following diagram commutes:\nΣA ν $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ Σν / / ΣΩΣX ev ΣX.\nThe hypothesis therefore implies that the composite (ev * • Σν * ) ⊗ F is an injection. Desuspending, we have that (S −1 ev * • ν * ) ⊗ F is an injection. By Lemma 5.12,\n(ρ 1 • ν * ) ⊗ F : K TF * (A) ⊗ F → K TF * (X) ⊗ F is an injection of F-vector spaces. Thus, the image (ρ 1 • ν * )( K TF * (A)) ⊗ F ⊂ K TF * (X) ⊗ F is a direct summand. Thus, the extension (τ • ν * ) ⊗ F is an injection, and (τ • ν * ) K TF * (A) ⊗k ⊂ K TF * (X) ⊗k for each k. This implies that ρ k • (τ • ν * ) • ι k is an injection for each k, as required.\nThe following corollary, which lifts the injectivity back to J s (X), is immediate from Theorem 7.7 and Lemma 7.3.\n\nCorollary 7.9. Let F = Q or Z/p for p prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r − 1)-connected for r ≥ 1. Suppose that N, s, k ∈ Z + satisfy k ≤ s, so that the map Φ K ν is as in Corollary 7.3. If\nν * ⊗ F : K TF * (ΣA) ⊗ F → K TF * (ΣX) ⊗ F is an injection, then Φ K ν ⊗ F : Hom( K TF * (S N ), L k ( K TF * (A))) ⊗ F → Hom( K TF * (S N ), K TF * (J s X)) ⊗ F\nis also an injection.\n\nWe have now established all that we will need to know about this 'first square'.\n\n\nMaps derived from the universal coefficient isomorphism\n\nIn this subsection we will build the second square of our diagram. This square is really just the Universal Coefficient theorem (Corollary 5.2) in a different form. We will write deg for both K-homological and K-theoretic degree.\n\nLemma 7.10. Let Y be a space having the homotopy type of a finite CW -complex. There exists an isomorphism U making the following diagram commute.\nπ N (Y ) deg π N (Y ) deg Hom( K TF * (S N ), K TF * (Y )) U / / Hom( K * TF (Y ), K * TF (S N )).\nProof. For β : K TF * (S N ) → K TF * (Y ), let U (β) be the unique map making the following diagram commute\nK * TF (Y ) U (β) ∼ = / / Hom( K TF * (Y ), Z) Hom(β,Z) K * TF (S N ) ∼ = / / Hom( K TF * (S N ), Z)\nwhere the isomorphisms are those of Corollary 5.2. Since K TF * (Y ) is a finitely generated free Z-module, β → Hom(β, Z) is an isomorphism, so U is also an isomorphism. Commutativity of the diagram from the statement of this lemma is by naturality of Lemma 5.2.\n\nCorollary 7.11. Let Y be a space having the homotopy type of a finite CW -complex. For a Z-module M , let τ p : M → M ⊗ Z/p be the natural map. There exists an injection U ′ making the following diagram commute.\nπ N (Y ) deg π N (Y ) deg Hom( K TF * (S N ), K TF * (Y )) τp Hom ψ−Mod ( K * TF (Y ), K * TF (S N )) τp Im(τ p • deg) H h u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ U ′ / / Im(τ p • deg) K k x x q q q q q q q q q q Hom( K TF * (S N ), K TF * (Y )) ⊗ Z/p Hom ψ−Mod ( K * TF (Y ), K * TF (S N )) ⊗ Z/p.\nProof. By Lemma 7.10, we have a commutative diagram\nπ N (Y ) deg π N (Y ) deg Hom( K TF * (S N ), K TF * (Y )) U / / Hom( K * TF (Y ), K * TF (S N )).\nwith U an isomorphism, so U ⊗ Z/p is also an isomorphism. By Lemma 6.1, the map\nHom ψ−Mod ( K * TF (Y ), K * TF (S N )) ⊗ Z/p → Hom( K * TF (Y ), K * TF (S N )) ⊗ Z/p\nis an injection. Maps of spaces induce maps of ψ-modules on K-theory, so the image of U • deg is contained in Hom ψ−Mod ( K * TF (Y ), K * TF (S N )), and hence there exists a map U ′ making the following diagram commute:\nIm(τ p • deg) _ U ′ / / Hom ψ−Mod ( K * TF (Y ), K * TF (S N )) ⊗ Z/p _ Hom( K TF * (S N ), K TF * (Y )) ⊗ Z/p U ⊗Z/p ∼ = / / Hom( K * TF (Y ), K * TF (S N )) ⊗ Z/p.\nBoth vertical maps are injections, so U ′ has the required properties.\n\n\nPulling back along classes defined on S 3\n\nLet f ∈ π j (S 3 ), and let N ≥ 3. Then, for ω ∈ π N (Y ), the composite\nS N +j−3 Σ N −3 f −−−−→ S N ω − → Y is defined. The class ω • Σ N −3 f lies in π M−1 (Y ), where M − 1 = N + j − 3. Thus motivated, we define the map f * Σ : π * (Y ) → π * (Y ) on ω ∈ π N (Y ) by setting f * Σ (ω) = (Σ N −3 f ) * ω = ω • Σ N −3 f .\nIn words, f * Σ pulls classes back along the appropriate suspension of f . Strictly speaking, f * Σ is only a partial map, because it is undefined on π N for N ≤ 2, but this will be unimportant.\n\nRecall the definition of the double e-invariant e (Definition 6.4). On π N (Y ), we have by definition that f * Σ = (Σ N −3 f ) * . By Lemma 6.5 we have a commuting square:\nπ N (Y ) f * Σ / / deg π N +j−3 (Y ) e Hom ψ−Mod ( K * TF (Y ), K * TF (S N )) θ(Σ N −3 f ) / / Ext ψ−Mod ( K * TF (Y ), K * TF (S N +j−2 )).\nMimicking the convention for f * Σ , let\nθ Σ (f ) : Hom ψ−Mod ( K * TF (Y ), K * TF (S * )) → Ext ψ−Mod ( K * TF (Y ), K * TF (S * +j−2 ))\nbe the map which is defined to be equal to θ(\nΣ N −3 f ) on the degree N component Hom ψ−Mod ( K * TF (Y ), K * TF (S N )) of Hom ψ−Mod ( K * TF (Y ), K * TF (S * )).\nLemma 7.12. Let p be a prime, and let f ∈ π j (S 3 ) with e(f ) defined. If pf = 0, then there exists a map θ p Σ (f ) making the following diagram commute for all N :\nπ N (Y ) deg f * Σ / / π N +j−3 (Y ) e Hom ψ−Mod ( K * TF (Y ), K * TF (S N )) τp θΣ(f ) / / Ext ψ−Mod ( K * TF (Y ), K * TF (S N +j−2 )) Hom ψ−Mod ( K * TF (Y ), K * TF (S N )) ⊗ Z/p. θ p Σ (f ) 3 3\nProof. Since pf = 0, we have that pe(Σ N −3 f ) = 0 for all N , which implies that θ Σ (f ) vanishes on p-divisible elements, so there exists a unique map θ p Σ (f ) making the diagram commute, as required.\n\nLemma 7.13. Let X be a finite CW -complex. Let λ X ℓ be the largest eigenvalue of the rational Adams operation\nψ ℓ ⊗ Q : K 0 (X) ⊗ Q → K 0 (X) ⊗ Q.\nThen, for i ≥ 0\n\n• the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Σ 2i J s (X)) ⊗ Q is ℓ i (λ X ℓ ) s , and\n• the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Σ 2i+1 J s (X)) ⊗ Q is ℓ i λ ΣX ℓ (λ X ℓ ) s−1 .\nProof. When i ≥ 1, Theorem 5.6 gives that ΣJ s (X) ≃ Σ s t=1 X ∧t , so Σ 2i J s (X) ≃ S 2i ∧ s t=1 X ∧t , and Σ 2i+1 J s (X) ≃ S 2i ∧ ΣX ∧ s−1 t=1 X ∧t . By the Künneth theorem (Theorem 5.5), this implies isomorphisms of rings\nK 0 TF (Σ 2i J s (X)) ∼ = s t=1 K 0 TF (S 2i ) ⊗ K 0 TF (X) ⊗t , fori ≥ 1, and K 0 TF (Σ 2i+1 J s (X)) ∼ = s−1 t=0 K 0 TF (S 2i ) ⊗ K 0 TF (ΣX) ⊗ K 0 TF (X) ⊗t fori ≥ 0.\nThe Künneth isomorphism of Theorem 5.5 is given by the external product of K-theory. Since the Adams operations are ring homomorphisms, the above isomorphisms are also isomorphisms of ψmodules. In particular, the Adams operations on the left are the tensor product of the corresponding operations on the right. These decompositions hold for K 0 TF , so they also hold for Q ⊗ K 0 , and the remaining problem is to determine the largest eigenvalue of the relevant tensor products of Adams operations. The eigenvalues of a tensor product of linear endomorphisms are precisely the products of the eigenvalues. The operation ψ ℓ acts on S 2i by multiplication by ℓ i . Together, these observations imply the result.\n\nLemma 7.14. Let p be an odd prime. Let X be an (r − 1)-connected finite CW -complex. Let N, s ∈ Z + . Consider the diagram of Lemma 7.12 for Y = J s (X) and f = f p,j ∈ π 2j(p−1)+2 (S 3 ), the map of Theorem 6.8:\nπ N (J s (X)) deg f * Σ / / π N +2j(p−1)−1 (J s (X)) e Hom ψ−Mod ( K * TF (J s (X)), K * TF (S N ))\nτp θΣ(f ) / / Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +2j(p−1) )) construction (Theorem 6.8), so e(Σ N −3 f ) = 0 by stability (Lemma 6.6). Since e(ω • Σ N −3 f ) = 0, the contrapositive of Theorem 6.7 gives that ω * has p-divisible image in K 0 (S N ). Since N is even, this implies that τ p • deg(ω) = 0, as required. CASE 2 (N odd): Write n = 2n + 1. Let f ′ = Σ N −2 f and g = Σω in Theorem 6.7, and proceed similarly to case 1. The domain of Σω • Σ N −2 f is S M−1 , where M − 1 = N + 2j(p − 1), so M is even, as is required. To check the eigenvalue hypothesis of Theorem 6.7, write M = 2m. By Lemma 7.13, the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (ΣJ s (X)) ⊗ Q is at most λ s ℓ , and ℓ m = ℓ j(p−1)+n = ℓ j(p−1)+ N −1 2 , which we assumed was greater than λ s ℓ . This means that ℓ m cannot be an eigenvalue of ψ ℓ ⊗ Q on K 0 (ΣJ s (X))⊗Q. As in the previous case, e(Σ N −2 f ) = 0. Since e(Σω •Σ N −2 f ) = 0, the contrapositive of Theorem 6.7 gives that (Σω) * has p-divisible image in K 0 (S N +1 ). Since N is odd, this implies that τ p • deg(ω) = 0, as required. This completes the case, and hence the proof.\n\n\nProof of Theorem 1.4\n\nConstruction 7.15. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r − 1)-connected for r ≥ 1. Let f ∈ π i (S 3 ) with e(f ) defined. Suppose that N, k, s ∈ Z + satisfy N < r(s + 1) − 1 and k ≤ s. The diagrams of the preceding subsections may be combined as follows.\n\nRecall the definition of deg ′ from the preamble to Lemma 7.2. Let I(A) be the submodule of Hom( K TF * (S * ), L( K TF * (A))) ⊗ Z/p generated by Im(τ p • deg ′ ). The same grading conventions as usual apply: we write I k (A) for the weight k part, we write I N (A) for the degree N part, and let\nI k N (A) = I k (A) ∩ I N (A).\nFrom Corollary 7.3, using the assumptions that N < r(s + 1) − 1 and k ≤ s (which make Φ π ν and Φ K ν well-defined) we obtain the following diagram, where the images of the vertical maps have been 'popped out' to their right.\nB k N (π * (A)) π N (J s (X)) I k N (A) Im(τ p • deg) Hom( K TF * (S N ), L k ( K TF * (A))) ⊗ Z/p Hom( K TF * (S N ), K TF * (J s (X))) ⊗ Z/p. Φ π ν τp•deg ′ τp•deg Φ K ν ⊗Z/p\nNext, from Corollary 7.11 (with Y = J s (X)) we have a diagram π N (J s (X)) π N (J s (X))\nIm(τ p • deg) Im(τ p • deg) Hom( K TF * (S N ), K TF * (J s (X))) ⊗ Z/p Hom ψ−Mod ( K * TF (J s (X)), K * TF (S N )) ⊗ Z/p. τp•deg τp•deg U ′\nLastly, we obtain the following diagram from Lemma 7.12:\nπ N (J s (X)) deg f * Σ / / π N +i−3 (J s (X)) e Hom ψ−Mod ( K * TF (J s (X)), K * TF (S N )) τp θΣ(f ) / / Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +i−2 )) Hom ψ−Mod ( K * TF (J s (X)), K * TF (S N )) ⊗ Z/p. θ p Σ (f ) 2 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡\nConcatenating these diagrams gives a diagram as follows:\nB k N (π * (A)) f * Σ • Φ π ν / / τp•deg ′ π N +i−3 (J s (X)) e I k N (A) θ p Σ (f )•U ′ •( Φ K ν ⊗Z/p) / / Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +i−2 )).\nIn this subsection, we combine the results of the previous subsections to produce results about this diagram.\n\nTheorem 7.16. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r−1)-connected for r ≥ 1. Let N, k, s ∈ Z + with N < r(s+1)−1 and k ≤ s. Let f = f p,j ∈ π 2j(p−1)+2 (S 3 ), the map of Theorem 6.8.\n\nFor ℓ ∈ Z + , let λ Y ℓ be the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Y ) ⊗ Q, and let λ ℓ = max(λ X ℓ , λ ΣX ℓ ). If\n• ν * ⊗ Z/p : K TF * (ΣA) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is\n\nan injection, and\n\n• there exists ℓ ∈ Z + such that ℓ j(p−1)+ N −1 2 > λ s ℓ ,\nthen θ p Σ (f ) • U ′ • ( Φ K ν ⊗ Z/p) : I k N (A) → Ext ψ−Mod ( K * TF (J s (X))\n, K * TF (S N +2j(p−1) )) is an injection.\n\nProof. By Corollary 7.9, since ν * ⊗ Z/p is an injection, Φ K ν ⊗ Z/p is also an injection. By Corollary 7.11 U ′ is an injection. By Lemma 7.14 the hypothesis on ℓ implies that the restriction of θ p Σ (f ) to Im(τ p • deg) is an injection. The map θ p Σ (f ) • U ′ • ( Φ K ν ⊗ Z/p) is thus a composite of injections, hence an injection, as required.\n\nIn the proof of Theorem 1.4, we will wish to restrict attention to those elements of B(π * (A)) who are brackets of classes in π * (A) in some dimensional range q min ≤ n ≤ q max . All such classes lie in dimensions kq min ≤ N ≤ kq max . Said more precisely, we have an inclusion B k ( qmax n=qmin π n (A)) ⊂ kqmax N =kqmin B k N (π * (A)). We will now study the diagram of Construction 7.15 in this dimensional range. Construction 7.17. Let p be an odd prime, ν : A → ΩΣX for finite CW -complexes A and X with X (r − 1)-connected for r ≥ 1, and f ∈ π i (S 3 ) with e(f ) defined. Let q max > q min be natural numbers. Fix k ∈ Z + , and let s = kq max +1. For N ∈ Z + with kq min ≤ N ≤ kq max , we have that N < r(s+1)−1 and k ≤ s. Combining the diagrams obtained from Construction 7.15 for this range of values of N gives the following diagram: / / kqmax N =kqmin Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +i−2 )).\n\nWe now show that by choosing a large enough c ∈ Z + , and setting f = f p,ck , the eigenvalue hypothesis of Theorem 7.16 may be satisfied across the dimensional range of Construction 7.17 for all sufficiently large k.\n\nCorollary 7.18. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X path-connected. Let q max > q min be natural numbers. Let c, k ∈ Z + . Let f = f p,ck ∈ π 2ck(p−1)+2 (S 3 ) be the map of Theorem 6.8. If ν * ⊗ Z/p : K TF * (ΣA) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection then there exists c ∈ Z + such that for large enough k ∈ Z + ,\nθ p Σ (f ) • U ′ • ( Φ K ν ⊗ Z/p) : kqmax N =kqmin I k N (A) → kqmax N =kqmin\nExt ψ−Mod ( K * TF (J s (X)), K * TF (S N +2ck(p−1) ))\n\nis an injection.\n\nProof. By Theorem 7.16, it suffices to show that for each N with kq min ≤ N ≤ kq max there exists ℓ ∈ Z + such that ℓ ck(p−1)+ N −1 2 > λ s ℓ = λ kqmax+1 ℓ . Take any ℓ ≥ 2. Since N ≥ kq min , it suffices to find c such that for large enough k we have ℓ ck(p−1)+ kq min −1 2 > λ kqmax ℓ . Taking logs on both sides, this is equivalent to (ck(p − 1) + kq min − 1 2 ) log(ℓ) > kq max log(λ ℓ ).\n\nIt is now clear that we may choose c large enough that this equation holds for large enough k, in particular, any c ≥ 1 p−1 (q max log(λ ℓ ) log(ℓ) − qmin 2 ) will do.\n\nBefore proving Theorem 1.4, we prove three lemmas. The first converts the K-theoretic hypothesis of Theorem 1.4 into the K-homological input our construction requires.\n\nLemma 7.19. Let X be a space, and let p be prime. Let µ : S q1+1 ∨ S q2+1 → ΣX be a map with q i ≥ 1, such that the map K * TF (ΣX) ⊗ Z/p µ * ⊗Z/p − −−−− → K * TF (S q1+1 ∨ S q2+1 ) ⊗ Z/p ∼ = Z/p ⊕ Z/p is a surjection. Then µ * ⊗ Z/p : K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection.\n\nProof. Naturality of the Universal Coefficient Theorem (Corollary 5.2) relative to the map µ gives a diagram of short exact sequences\n0 / / 0 / / K * TF (S q1+1 ∨ S q2+1 ) / / Hom( K TF * (S q1+1 ∨ S q2+1 ), Z) / / 0 0 / / E O O / / K * TF (ΣX) µ * O O / / Hom( K TF * (ΣX), Z) Hom(µ * ,Z) O O / / 0,\nwhere E denotes the quotient of Ext( K * −1 (ΣX), Z) by its torsion subgroup. We will argue by contrapositive. Suppose that µ * ⊗ Z/p is not injective. The Z/p-vector space K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p has dimension 2, so there exists a basis x, y where (µ * ⊗ Z/p)(y) = 0. Prior to tensoring with Z/p, this means that there exists a non-p-divisible element y of K TF * (S q1+1 ∨ S q2+1 ) such that p divides µ * ( y). This implies that any element ϕ of Hom( K TF * (S q1+1 ∨ S q2+1 ), Z) with ϕ( y) not p-divisible is not contained in the image of Hom(µ * , Z), hence, by the diagram, that µ * ⊗ Z/p is not surjective, as required.\n\nProof. For any space A, the localizing map A −→ A (p) induces an isomorphism on K * ( ) ⊗ Z/p [Mis78]. It therefore suffices to show that there exists maps µ ′ and ϕ making the following diagram commute, with the localization ϕ (p) being a homotopy equivalence.\nS q1+1 ∨ S q2+1 ϕ µ ′ / / ΣY / / ΣY (p) S q1+1 ∨ S q2+1 µ / / ΣX / / ΣX (p) . ≃ O O\nSuch maps exist by Lemma 7.21.\n\nWe are now ready to prove Theorem 1.4.\n\nProof of Theorem 1.4. Let µ = µ 1 ∨ µ 2 , with adjoint µ : S q1 ∨ S q2 → ΩΣX. Let f = f p,ck ∈ π 2ck(p−1)+2 (S 3 ). Consider the diagram of Construction 7.17, with A = S q1 ∨ S q2 , q max = max(q 1 , q 2 ), q min = min(q 1 , q 2 ), and ν = µ. We have such a diagram for each k ∈ Z + :\nkqmax N =kqmin B k N (π * (S q1 ∨ S q2 )) f * Σ • Φ π µ / / τp•deg ′ kqmax\nN =kqmin π N +2ck(p−1)−1 (J s (X)) e kqmax N =kqmin I k N (S q1 ∨ S q2 ) / / kqmax N =kqmin Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +2ck(p−1) )).\n\nBy assumption, µ * ⊗Z/p : K * (ΣX)⊗Z/p → K * (S q1+1 ∨S q2+1 )⊗Z/p is a surjection. Since K * (S q1+1 ∨ S q2+1 ) is torsion-free, Lemma 7.19 then implies that µ * ⊗ Z/p : K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection. By Lemma 7.22, we may assume without loss of generality that X has the integral homotopy type of a finite CW -complex. Thus, by Corollary 7.18, we may fix c such that for large enough k, θ p (f ) • U ′ • ( Φ K µ ⊗ Z/p) is an injection. The Hurewicz map h is a surjection π * (S q1 ∨S q2 ) → K TF * (S q1 ∨S q2 ), so the submodule generated by the image of the map h ′ : B(π * (S q1 ∨S q2 )) → L( K TF * (S q1 ∨S q2 )) of Lemma 7.20 contains the submodule generated by K TF * (S q1 ∨ S q2 ) under the bracket operation. In particular, it contains the weight k component L k ( K TF * (S q1 ∨ S q2 )) for each k. By Theorem 3.1, dim Z (L k ( K TF * (S q1 ∨ S q2 ))) = W 2 (k). Note that L k ( K TF * (S q1 ∨ S q2 )) = kqmax N =kqmin L k ( K TF * (S q1 ∨ S q2 )). It then follows from Lemma 7.20 that dim Z/p ( kqmax N =kqmin I k N (S q1 ∨ S q2 )) ≥ W 2 (k). Since θ p Σ (f ) • U ′ • ( Φ K µ ⊗ Z/p) is an injection for large enough k, it follows that the dimension of e( kqmax N =kqmin π N +2ck(p−1)−1 (J s (X))) is at least W 2 (k). By Corollary 7.3 (i s ) * is an injection, so the dimension of (i s ) * ( kqmax N =kqmin π N +2ck(p−1)−1 (J s (X))) ⊂ kqmax N =kqmin π N +2ck(p−1)−1 (ΩΣX) is also at least W 2 (k).\n\nThus, ΣX satisfies the hypotheses of Lemma 3.3 with a = 2c(p − 1) + q max = 2c(p − 1) + max(q 1 , q 2 ) and b = 0, and hence is p-hyperbolic.\n\nDefinition 5. 8 .\n8For a Z-graded (respectively Z/2-graded) module M , let T (M ) = ∞ k=1 M ⊗k denote the tensor algebra on M . The product is given by concatenation. We refer to M ⊗k as the weight k component of the tensor algebra T (M ). We define a Z-grading (respectively Z/2-grading) on T (M ) by setting |x 1 ⊗ x 2 ⊗ · · · ⊗ x k | = k i=1 |x i |.Definition 5.9. For a space Y , let σ : K TF * (Y ) ∼ = − → K TF * +1 (ΣY ) be the suspension isomorphism. Let ϕ : K TF * (ΣY ) → K TF * (ΣY ) be a homomorphism of graded groups, not necessarily induced by a map of spaces. We call the composite σ −1 • ϕ • σ the desuspension of θ, denoting it by S −1 ϕ.\n\n\n5.10, Ση = Γ • ι 1 , which implies the s = 1 statement. Now let s ≥ 2. Ganea [Gan65, Theorems 1.1 and 1.4] shows that the homotopy fibre of ev is given by Σ(ΩΣX ∧ ΩΣX) v − → ΣΩΣX ev − → ΣX, where the map v is equal to the composite Σ(ΩΣX ∧ ΩΣX) → Σ(ΩΣX × ΩΣX) Σm2 − −− → ΣΩΣX of Σm 2 with the canonical splitting. We will show that Γ • ι s factors through v, and hence composes trivially with ev. Consider the following diagram, where the unlabelled arrows are all canonical splittings: Σ(ΩΣX ∧ ΩΣX) / / v , , Σ(ΩΣX × ΩΣX) Σm2 / / ΣΩΣX Σ(ΩΣX) ∧s / / Σ((ΩΣX) (s−1) ∧ ΩΣX)\n\nLemma 6. 1 .\n1Let M and N be ψ-modules, with N torsion-free. The inclusion of Z-modules Hom ψ−Mod (M, N ) ֒− → Hom(M, N ) is an injection onto a summand.\n\n\nFor the avoidance of doubt, by the e-invariant we will always mean what Adams calls the complex e-invariant e C [Ada65; Ada66].\n\n\nFor a graded R-module V , let U (V ) denote the graded set of homogeneous elements in V . Let B(V ) be the set of formal iterated brackets of the elements of U (V ). Bracket gives a natural operation on B(V ), which we write as [x, y]. Elements of B(V ) are nonassociative words in the elements of U (V ), so we may define a grading on B(V ) which extends the grading on U (V ) via |[x, y]| = |x| + |y|. The weight of an element of B(V ) is its word length. Write B N (V ) for the subset of elements in degree N , B k (V ) for the subset of elements of weight k, and set B k N (V ) = B k (V ) ∩ B N (V ). Let ν : A → ΩΣX be a map. By definition of B(π * (A)), there exists a map Φ π ν : B(π * (A)) → π * (ΩΣX)\n\nN\n=kqmin π N +i−3 (J s (X)) e kqmax N =kqmin I k N (A) θ p Σ (f )•U ′ •( Φ K ν ⊗Z/p)\nHom ψ−Mod ( K * TF (J s (X)), K * TF (S N )) ⊗ Z/p.For ℓ ∈ Z + , let λ Y ℓ be the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Y ) ⊗ Q, and let λ ℓ = max(λ X ℓ , λ ΣX ℓ ). If there exists ℓ ∈ Z + such that ℓ j(p−1)+ N −1 2 > λ s ℓ then Ker(e • f * Σ ) ⊂ Ker(τ p • deg), and hence the restriction of θ p Σ (f ) to Im(τ p • deg) is an injection. Proof. First note that pf = 0 by Theorem 6.8, so θ p Σ (f ) is well-defined by Lemma 7.12. Let ω ∈ π N (J s (X)). Suppose that ω ∈ Ker(e • f * Σ ), that is, that the e-invariant of the compositeis trivial. By Lemma 6.3, this implies that Σ i (ω • Σ N −3 f ) has trivial e-invariant for all i. In particular, e(Σω • Σ N −2 f ) and e(ω • Σ N −3 f ) are both 0. By Lemma 7.13, the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (J s (X)) ⊗ Q is at most λ s ℓ , and the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (ΣJ s (X)) ⊗ Q is also at most λ s ℓ . We now divide into cases, based on the parity of N .CASE 1 (N even): Write N = 2n. Let f ′ = Σ N −3 f and g = ω in Theorem 6.7. The domain ofTo check the eigenvalue hypothesis of Theorem 6.7, write M = 2m. By Lemma 7.13, the largest eigenvalue of ψ ℓ ⊗Q on K 0 (J s (X)) ⊗ Q is at most λ s ℓ , and ℓ m = ℓ j(p−1)+n > ℓ j(p−1)+ N −1 2 , which we assumed was greater than λ s ℓ . This means that ℓ m cannot be an eigenvalue of ψ ℓ ⊗ Q on K 0 (J s (X)) ⊗ Q. Now, e(f ) = 0 byProof. Commutativity of the diagram. follows from the definitions. Commutativity of the diagram from the lemma statement then follows from the definition of B(π * (A)), since χ respects brackets.The third and fourth lemmas allow us to make the statement of Theorem 1.4 an entirely p-local one.Lemma 7.21. Let p be a prime, let X be a simply connected CW -complex, and let f : S q1+1 ∨ S q2+1 −→ X (p) be any map. There exists a map f : S q1+1 ∨ S q2+1 −→ X, and a map ϕ : S q1+1 ∨ S q2+1 −→ S q1+1 ∨ S q2+1 which is a homotopy equivalence after p-localization, making the diagramcommute, where the vertical arrow is the localization.Proof. Write f = f 1 ∨ f 2 , for maps f i : S qi+1 −→ X. On homotopy groups, the localizing map may be identified with the tensor map π * (X) −→ π * (X) ⊗ Z (p) ∼ = π * (X (p) ).This implies that there exist integers u 1 and u 2 , not divisible by p, such that u i f i lifts to a map f i : S qi+1 −→ X. Setting f = f 1 ∨ f 2 , and letting ϕ be the degree u i map on each wedge summand, we obtain the desired diagram.Lemma 7.22. Let p be prime, and let X and Y be connected CW -complexes. If µ : S q1+1 ∨S q2+1 −→ ΣX induces an injection on K * ( ) ⊗ Z/p, and there is a homotopy equivalence of p-localizations ΣY (p) ≃ ΣX (p) , then there exists a map µ ′ : S q1+1 ∨ S q2+1 −→ ΣY which also induces an injection on K * ( ) ⊗ Z/p.\nLet h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF * (A). As with deg ′. let h ′ : B(π * (Apreamble to the second lemma is necessary. Let h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF * (A). As with deg ′ , let h ′ : B(π * (A)) →\n\nLet M be a Z/2-graded Z-module. L( K Tf *, Let χ : Hom( K TF * (S * ), M ) → M be the map which carries ϕ ∈ Hom( K TF * (S N ), M ) to ϕ(ξ N ) ∈ M (Remark 5.4). If M = L is a Z/2-graded Lie algebra, then it follows immediately from the definition of the bracket in Hom( K TF *. → K TF * (A) ⊂ L( K TF *A)) be the unique map which restricts to h : π * (A). S * ), L) that χ is a map of Lie algebrasL( K TF * (A)) be the unique map which restricts to h : π * (A) → K TF * (A) ⊂ L( K TF * (A)) and respects brackets. Let M be a Z/2-graded Z-module. Let χ : Hom( K TF * (S * ), M ) → M be the map which carries ϕ ∈ Hom( K TF * (S N ), M ) to ϕ(ξ N ) ∈ M (Remark 5.4). If M = L is a Z/2-graded Lie algebra, then it follows immediately from the definition of the bracket in Hom( K TF * (S * ), L) that χ is a map of Lie algebras.\n\nOn the groups J(X). II. J F Adams, Topology. 3J. F. Adams. \"On the groups J(X). II\". Topology 3 (1965), pp. 137-171.\n\nOn the groups. J F Adams, J(X). IV\". Topology. 5J. F. Adams. \"On the groups J(X). IV\". Topology 5 (1966), pp. 21-71.\n\nLectures on generalised cohomology. J F Adams, Category Theory, Homology Theory and their Applications. III (Battelle Institute Conference, Seattle; BerlinSpringerJ. F. Adams. \"Lectures on generalised cohomology\". Category Theory, Homology The- ory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three). Springer, Berlin, 1969, pp. 1-138.\n\nVector bundles and homogeneous spaces. M F Atiyah, F Hirzebruch, Proc. Sympos. Pure Math. IIIAmerican Mathematical SocietyProvidence, R.I.M. F. Atiyah and F. Hirzebruch. \"Vector bundles and homogeneous spaces\". Proc. Sympos. Pure Math., Vol. III. American Mathematical Society, Providence, R.I., 1961, pp. 7-38.\n\nUniversal coefficient theorems for k-theory. mimeographed notes. D Anderson, BerkeleyD. Anderson. Universal coefficient theorems for k-theory. mimeographed notes. Berkeley.\n\nIntroduction to homotopy theory. M Arkowitz, SpringerNew YorkUniversitextM. Arkowitz. Introduction to homotopy theory. Universitext. Springer, New York, 2011.\n\nVector bundles and the Künneth formula. M F Atiyah, Topology. 1M. F. Atiyah. \"Vector bundles and the Künneth formula\". Topology 1 (1962), pp. 245-248.\n\nSome differentials in the Adams spectral sequence. II. M G Barratt, M E Mahowald, M C Tangora, Topology. 9M. G. Barratt, M. E. Mahowald, and M. C. Tangora. \"Some differentials in the Adams spectral sequence. II\". Topology 9 (1970), pp. 309-316.\n\nGrowth of subalgebras and subideals in free Lie algebras. Y Bahturin, A Olshanskii, J. Algebra. 422Y. Bahturin and A. Olshanskii. \"Growth of subalgebras and subideals in free Lie algebras\". J. Algebra 422 (2015), pp. 277-305.\n\nThe loop space homotopy type of simply-connected fourmanifolds and their generalizations. P Beben, S Theriault, Adv. Math. 262P. Beben and S. Theriault. \"The loop space homotopy type of simply-connected four- manifolds and their generalizations\". Adv. Math. 262 (2014), pp. 213-238.\n\nHomotopy invariants and continuous mappings. S.-C Chang, Proc. Roy. Soc. London Ser. A. 202S.-C. Chang. \"Homotopy invariants and continuous mappings\". Proc. Roy. Soc. London Ser. A 202 (1950), pp. 253-263.\n\nSome nonzero homotopy groups of spheres. E B Curtis, Bull. Amer. Math. Soc. 75E. B. Curtis. \"Some nonzero homotopy groups of spheres\". Bull. Amer. Math. Soc. 75 (1969), pp. 541-544.\n\nRational homotopy theory. II. Y Félix, S Halperin, J C Thomas, World Scientific Publishing Co. Pte. LtdHackensack, NJY. Félix, S. Halperin, and J. C. Thomas. Rational homotopy theory. II. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.\n\nÜber die Klassen der Sphärenabbildungen I. Große Dimensionen. H , Compositio Math. 5H. Freudenthal. \"Über die Klassen der Sphärenabbildungen I. Große Dimensionen\". Com- positio Math. 5 (1938), pp. 299-314.\n\nA generalization of the homology and homotopy suspension. T Ganea, Comment. Math. Helv. 39T. Ganea. \"A generalization of the homology and homotopy suspension\". Comment. Math. Helv. 39 (1965), pp. 295-322.\n\nOn the sphere of origin of infinite families in the homotopy groups of spheres. B Gray, Topology. 8B. Gray. \"On the sphere of origin of infinite families in the homotopy groups of spheres\". Topology 8 (1969), pp. 219-232.\n\nOn almost rational co-H-spaces. H.-W Henn, Proc. Amer. Math. Soc. 871H.-W. Henn. \"On almost rational co-H-spaces\". Proc. Amer. Math. Soc. 87.1 (1983), pp. 164-168.\n\nOn the growth of homotopy groups. H.-W Henn, Manuscripta Math. 56H.-W. Henn. \"On the growth of homotopy groups\". Manuscripta Math. 56.2 (1986), pp. 235- 245.\n\nOn the homotopy groups of the union of spheres. P J Hilton, J. London Math. Soc. 30P. J. Hilton. \"On the homotopy groups of the union of spheres\". J. London Math. Soc. 30 (1955), pp. 154-172.\n\nMoore's conjecture for polyhedral products. Y Hao, Q Sun, S Theriault, Math. Proc. Cambridge Philos. Soc. 1671Y. Hao, Q. Sun, and S. Theriault. \"Moore's conjecture for polyhedral products\". Math. Proc. Cambridge Philos. Soc. 167.1 (2019), pp. 23-33.\n\nBasic bundle theory and K-cohomology invariants. D Husemöller, Lecture Notes in Physics. With contributions by Siegfried Echterhoff, Stefan Fredenhagen and Bernhard Krötz. BerlinSpringer726D. Husemöller et al. Basic bundle theory and K-cohomology invariants. Vol. 726. Lec- ture Notes in Physics. With contributions by Siegfried Echterhoff, Stefan Fredenhagen and Bernhard Krötz. Springer, Berlin, 2008.\n\nExponential growth of homotopy groups of suspended finite complexes. R.-Z Huang, J Wu, Math. Z. 295R.-Z. Huang and J. Wu. \"Exponential growth of homotopy groups of suspended finite complexes\". Math. Z. 295.3-4 (2020), pp. 1301-1321.\n\nReduced product spaces. I M James, Ann. of Math. 2I. M. James. \"Reduced product spaces\". Ann. of Math. (2) 62 (1955), pp. 170-197.\n\nConfiguration spaces in algebraic topology. B Knudsen, arXiv:1803.11165math.ATB. Knudsen. Configuration spaces in algebraic topology. 2018. arXiv: 1803.11165 [math.AT].\n\nThe splitting of the Künneth sequence for generalized cohomology. G Mislin, Math. Z. 122G. Mislin. \"The splitting of the Künneth sequence for generalized cohomology\". Math. Z. 122 (1971), pp. 237-245.\n\nLocalization with respect to K-theory. G Mislin, J. Pure Appl. Algebra. 10G. Mislin. \"Localization with respect to K-theory\". J. Pure Appl. Algebra 10.2 (1977/78), pp. 201-213.\n\nAlgebraic methods in unstable homotopy theory. J A Neisendorfer, New Mathematical Monographs. CambridgeCambridge University Press12J. A. Neisendorfer. Algebraic methods in unstable homotopy theory. Vol. 12. New Mathe- matical Monographs. Cambridge University Press, Cambridge, 2010.\n\nSamelson products and exponents of homotopy groups. J A Neisendorfer, J. Homotopy Relat. Struct. 82J. A. Neisendorfer. \"Samelson products and exponents of homotopy groups\". J. Homotopy Relat. Struct. 8.2 (2013), pp. 239-277.\n\nOn conjectures of Moore and Serre in the case of torsion-free suspensions. P Selick, Math. Proc. Cambridge Philos. Soc. 941P. Selick. \"On conjectures of Moore and Serre in the case of torsion-free suspensions\". Math. Proc. Cambridge Philos. Soc. 94.1 (1983), pp. 53-60.\n\nLecture Notes in Mathematics. 1964 lectures given at Harvard University, Corrected fifth printing of the second. J.-P Serre, Lie algebras and Lie groups. BerlinSpringer-Verlag1500J.-P. Serre. Lie algebras and Lie groups. Vol. 1500. Lecture Notes in Mathematics. 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition. Springer-Verlag, Berlin, 2006.\n\nHomologie singulière des espaces fibrés. III. Applications homotopiques. J.-P Serre, C. R. Acad. Sci. Paris. 232J.-P. Serre. \"Homologie singulière des espaces fibrés. III. Applications homotopiques\". C. R. Acad. Sci. Paris 232 (1951), pp. 142-144.\n\nOn natural coalgebra decompositions of tensor algebras and loop suspensions. P Selick, J Wu, Mem. Amer. Math. Soc. 148109P. Selick and J. Wu. \"On natural coalgebra decompositions of tensor algebras and loop suspensions\". Mem. Amer. Math. Soc. 148.701 (2000), pp. viii+109.\n\nThe functor A min on p-local spaces. P Selick, J Wu, Math. Z. 2533P. Selick and J. Wu. \"The functor A min on p-local spaces\". Math. Z. 253.3 (2006), pp. 435- 451.\n\nOn products in homotopy groups. G W Whitehead, Ann. of Math. 2G. W. Whitehead. \"On products in homotopy groups\". Ann. of Math (2) 47 (1946), pp. 460-475.\n\nGraduate Texts in Mathematics. G W Whitehead, Springer-Verlag61New York-BerlinElements of homotopy theoryG. W. Whitehead. Elements of homotopy theory. Vol. 61. Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1978.\n\nUniversal coefficient sequences for cohomology theories of CW-spectra. Z.-I Yosimura, Osaka Math. J. 12Z.-I. Yosimura. \"Universal coefficient sequences for cohomology theories of CW-spectra\". Osaka Math. J. 12.2 (1975), pp. 305-323.\n\nThe local hyperbolicity of A 2 n -complexes. Z.-J Zhu, J.-Z Pan, Homology Homotopy Appl. 23Z.-J. Zhu and J.-Z. Pan. \"The local hyperbolicity of A 2 n -complexes\". Homology Homotopy Appl. 23.1 (2021), pp. 367-386.\n"},"corpus_id":{"kind":"number","value":231582764,"string":"231,582,764"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2101.04591\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1007/s00209-021-02917-1\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2101.04591v2.pdf\"\n ],\n \"pdf_sha\": \"49b43a1daacd20b8c35b48ccd202218d19ee138e\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"p-hyperbolicity of homotopy groups via K-theory"},"authors":{"kind":"list like","value":["Guy Boyde"],"string":"[\n \"Guy Boyde\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We show that S n ∨ S m is Z/p r -hyperbolic for all primes p and all r ∈ Z + , provided n, m ≥ 2, and consequently that various spaces containing S n ∨ S m as a p-local retract are Z/p r -hyperbolic. We then give a K-theory criterion for a suspension ΣX to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian ΣGr k,n is p-hyperbolic for all odd primes p when n ≥ 3 and 0 < k < n. We obtain similar results for some related spaces."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"A space X is called rationally elliptic if π * (X) ⊗ Q is finite dimensional, and rationally hyperbolic if the dimension of i≤m π i (X) ⊗ Q grows exponentially in m. It was proved in [FHT15,Chapter 33] that simply connected CW -complexes with rational homology of finite type and finite rational category are either rationally elliptic or rationally hyperbolic. In order to study the p-torsion analogue of this dichotomy, Huang and Wu [HW20] introduced the definitions of Z/p r -and p-hyperbolicity.\nFor p prime, by a p-torsion summand in an abelian group A, we mean a direct summand isomorphic to Z/p r for some r ≥ 1.\nDefinition 1.1. Let X be a space, and let p be a prime. We say that X is p-hyperbolic if the number of p-torsion summands in π * (X) grows exponentially, in the sense that\nwhere T m is the number of p-torsion summands in i≤m π i (X).\nThe above definition counts Z/p r -summands for all values of r. It is also possible to consider only a single r, and by doing so we obtain the definition of Z/p r -hyperbolicity.\nDefinition 1.2. Let X be a space, let p be a prime, and fix r ∈ Z + . We say that X is Z/p r -hyperbolic if the number of Z/p r -summands in π * (X) grows exponentially, in the sense that\nwhere t m is the number of Z/p r -summands in i≤m π i (X).\nNote that Z/p r -hyperbolicity for any r implies p-hyperbolicity. It follows immediately from a result of Henn [Hen86, Corollary of Theorem 1] that the lim infs appearing in the above definitions must be finite if X is a simply connected finite CW -complex.\nHuang and Wu show that for n ≥ 3, r ≥ 1 and p any prime, the Moore space P n (p r ) is Z/p rhyperbolic and Z/p r+1 -hyperbolic, and that P n (2) is also Z/8-hyperbolic [HW20, Theorem 1.6]. More generally, they give criteria in terms of a functorial loop space decomposition due to Selick and Wu [SW00; SW06] for a suspension ΣX to be Z/p r -hyperbolic.\nMore recently, Zhu and Pan [ZP21] use a classification of (n − 1)-connected CW -complexes of dimension at most n + 2, due to Chang [Cha50], to show that, for n ≥ 4, such a complex is Z/phyperbolic, provided that it is not contractible or a sphere after p-localization. They also prove hyperbolicity results for several so-called elementary Chang complexes.\nThis paper studies p-and Z/p r -hyperbolicity of certain suspensions. Our first result is as follows.\nTheorem 1.3. Let q 1 , q 2 ≥ 1. Then S q1+1 ∨ S q2+1 is Z/p r -hyperbolic for all primes p and all r ∈ Z + .\nLet p be a prime. If a space X contains a wedge of two spheres as a p-local retract, then Theorem 1.3 implies that X is Z/p r -hyperbolic for all r. Various spaces have been shown to have such a wedge as a retract -examples of this sort are given in Section 2.1. A summary is as follows:\n• for n, k ≥ 3, the configuration space Conf k (R n ) is Z/p r -hyperbolic for all p and r (Example 2.1);\n• an (n − 1)-connected 2n-dimensional manifold M , where H n (M ) is of rank at least 3, is Z/p rhyperbolic for all p and r (Example 2.2);\n• a generalized moment-angle complex on a simplicial complex having two minimal missing faces which are not disjoint is Z/p r -hyperbolic for all p and r (Example 2.3);\n• ΣCP 2 is Z/p r -hyperbolic for all p = 2 and all r, and ΣHP 2 is is Z/p r -hyperbolic for all p = 2, 3 and all r (Example 2.4).\nOur other result is as follows.\nTheorem 1.4. Let p be an odd prime, and let X be a path connected space having the p-local homotopy type of a finite CW -complex. Suppose that there exists a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX with q i ≥ 1, such that the map\nis a surjection. Then ΣX is p-hyperbolic.\nThis criterion is quite different to that given by Huang and Wu [HW20, Theorem 1.5]. Their criterion is homotopical, using hypotheses on X to produce retracts of ΩΣX, whereas ours is cohomological, which makes it easier to check. On the other hand, their criterion is stronger, since it gives Z/p r -hyperbolicity, rather than just p-hyperbolicity. The examples they give, primarily various Moore spaces, differ from those we obtain, which are the suspensions of spaces related to complex projective space. More precisely, in Section 2.2, we show that the following spaces are p-hyperbolic for all p = 2:\n• suspended complex projective space ΣCP n for n ≥ 2 (Example 2.5), and more generally;\n• the suspended complex Grassmannian ΣGr k,n for n ≥ 3 and 0 < k < n (Example 2.6);\n• the suspended Milnor Hypersurface ΣH m,n for m ≥ 2 and n ≥ 3, (Example 2.7);\n• the suspended unitary group ΣU (n) for n ≥ 3 (Example 2.8).\nBoth Theorem 1.3 and Theorem 1.4 will be proven by constructing an exponentially growing family of classes which generate summands in the relevant homotopy groups. We think of this family as 'witnessing' the hyperbolicity. For Theorem 1.3, one can proceed directly from the Hilton-Milnor decomposition of S n ∨S m [Hil55]. For Theorem 1.4, we employ K-theoretic methods originally used by Selick [Sel83] to prove one direction of Moore's conjecture for suspensions having torsion-free homology.\nIf the map µ 1 ∨ µ 2 of Theorem 1.4 induces a surjection on K * ( ) ⊗ Z/p, then so does its suspension Σµ 1 ∨ Σµ 2 . The conclusion of Theorem 1.4 may therefore be strengthened in the following way.\nCorollary 1.5. With the hypothesis of Theorem 1.4, Σ n X is p-hyperbolic for all n ≥ 1.\nOne might be motivated by this observation to ask whether, in the circumstances of Theorem 1.4, the stable homotopy groups of X satisfy the growth conditions of Definition 1.1 or 1.2. In the proofs of both Theorem 1.3 and 1.4, the classes that witness the hyperbolicity are composites involving Whitehead products. The suspension of a Whitehead product is always trivial [Whi46,Theorem 3.11], so the classes we detect cannot be stable. Therefore, Corollary 1.5 does not suggest that the stable homotopy of ΣX should be p-or Z/p r -hyperbolic. On the other hand, it follows from our methods that, under the hypotheses of Theorem 1.4, the stable homotopy of ΩΣX is p-hyperbolic.\nBy a result of Henn [Hen83], any co-H space, and in particular any suspension, decomposes rationally as a wedge of spheres. It then follows from the Hilton-Milnor theorem [Hil55] and the computation of the rational homotopy groups of spheres [Ser51] that such a suspension is rationally hyperbolic precisely when there are at least two spheres (of dimension ≥ 2) in this decomposition.\nIf ΣX satisfies the hypotheses of Theorem 1.4 for any prime (including 2), then by Chern Character considerations the reduced rational homology of ΣX has dimension at least two, so ΣX is rationally a wedge of at least two spheres by the preceding discussion. This rational equivalence is a local equivalence at all but perhaps finitely many primes, so by Theorem 1.3, ΣX is Z/p r hyperbolic for all r at all but finitely many primes p. One might therefore conjecture that the conclusion of Theorem 1.4 can be strengthened to give Z/p r -hyperbolicity for all r rather than p-hyperbolicity, but we do not know whether this is possible.\nWe now discuss situations in which it is adequate to consider ordinary cohomology, rather than K-theory. If ΣX has torsion-free integral (co)homology, or if its cohomology is concentrated in even degrees, then the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses on the E 2 page [Hus+08]. It follows by naturality that the image of the map induced by µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX on Ktheory is identified with the image of the induced map on cohomology. We may therefore replace K-theory with cohomology in Theorem 1.4, as follows.\nCorollary 1.6. Let X be a path connected space having the homotopy type of a finite CW -complex, such that the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses on the E 2 page. Let p be an odd prime. Suppose that there exists a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX with q i ≥ 1, such that the map induced by µ 1 ∨ µ 2 on H * ( ) ⊗ Z/p is a surjection. Then ΣX is p-hyperbolic.\nOne advantage of ordinary cohomology is that it is connected to the homotopy groups integrally, via the universal coefficient theorem and Hurewicz map. We can exploit this as follows.\nExample 1.7. Suppose that the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses (for example, if ΣX has torsion-free homology) and that there exists q ∈ Z + so that H i (ΣX) = 0 for i ≤ q, and dim Q ( H q+1 (ΣX) ⊗ Q) ≥ 2. The Hurewicz map π q+1 (ΣX) → H q+1 (ΣX) is an isomorphism, so there exists a map µ 1 ∨ µ 2 : S q+1 ∨ S q+1 → ΣX inducing the inclusion of a Z 2 -summand in H q+1 (ΣX). By the universal coefficient theorem relating ordinary homology and cohomology, µ 1 ∨ µ 2 induces a surjection on integral cohomology, so by Corollary 1.6, ΣX is p-hyperbolic for all odd primes p.\nThe structure of this paper is as follows. In Section 2 we give applications of the main theorems and derive a simple lower bound for the growth of the number of Z/p r -summands in the homotopy groups of a wedge of two spheres (Corollary 2.9). The proofs of Theorems 1.3 and Theorem 1.4 may be read largely independently; Section 3 contains those preliminary results which are used in both cases. In Section 4, we give the proof of Theorem 1.3. The remainder of the paper is devoted to proving Theorem 1.4: Sections 5 and 6 give the necessary background, and Section 7 contains the proof. An overview of the proof strategy can be found at the start of Section 7.\nI would like to thank my PhD supervisor, Stephen Theriault, for suggesting the problems that this paper tries to address, and for many helpful conversations along the way. From a technical point of view, much is owed to papers of Huang and Wu [HW20], and of Selick [Sel83]. Neil Strickland's 'Bestiary' was extremely helpful in providing examples of Theorem 1.4. Conversations with Sam Hughes were very useful in formulating Corollary 1.6. Thanks to the anonymous reviewer for their thoughtful comments, especially for pointing out that the space in Theorem 1.4 need only be finite after p-localisation.\n2 Applications 2.1 Spaces having a wedge of two spheres as a retract Theorem 1.3 immediately implies that any space X which has S q1+1 ∨ S q2+1 as a retract after plocalization is Z/p r -hyperbolic for that p and all r. This implies that for all n ≥ 1, Σ n X contains S q1+n+1 ∨ S q2+n+1 as a p-local retract, and so is Z/p r -hyperbolic for all r. We first consider examples of this form.\nExample 2.1. It is known [Knu18, Section 3.1] that Conf k (R n ), the ordered configuration space of k points in R n , contains k−1 S n−1 as a retract. It follows that, when n, k ≥ 3, Conf k (R n ) is Z/p rhyperbolic for all p and r.\nExample 2.2. Let M be an (n − 1)-connected 2n-dimensional manifold. By the universal coefficient theorem, there can be no torsion in H n (M ). Suppose that the rank of H n (M ) is at least 3. By work of Beben and Theriault [BT14, Theorem 1.4], ΩM contains a wedge of two spheres as a retract after looping. Thus, M is again Z/p r -hyperbolic for all p and r.\nExample 2.3. Let K be a simplicial complex on the vertex set [m] = {1, . . . , m}, and let (X, A) be any sequence of pairs (D ni , S ni−1 ) with n i ≥ 2 for 1 ≤ i ≤ m. If there exist two distinct minimal missing faces of K which are not disjoint, then by work of Hao, Sun and Theriault [HST19, Theorem 4.2] the polyhedral product (X, A) K contains a wedge of two spheres as a retract after looping, and hence is Z/p r -hyperbolic for all p and all r.\nExample 2.4. Localized away from 2, ΣCP 2 ≃ S 3 ∨ S 5 . To see this, note that ΣCP 2 has a CWstructure consisting of one 3-cell and one 5-cell, and that π 4 (S 3 ) ∼ = Z/2 [Fre38]. This implies that the attaching map for the 5-cell is nullhomotopic after localization at an odd prime. Thus, ΣCP 2 is Z/p r -hyperbolic for all r when p = 2.\nSimilarly, ΣHP 2 admits a cell structure with one 5-cell and one 9-cell, and π 8 (S 5 ) ∼ = Z/24. Thus, ΣHP 2 is Z/p r -hyperbolic for all r when p = 2, 3."},{"title":"Suspensions of spaces related to CP n","content":"Suppose that one has verified the hypotheses of Theorem 1.4 for a given space X and odd prime p, using a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX. If another space Y admits a map σ : ΣX → ΣY which induces a surjection on K * ( ) ⊗ Z/p, then it is immediate that σ • (µ 1 ∨ µ 2 ) satisfies the hypotheses of Theorem 1.4, and hence that ΣY is p-hyperbolic. The slogan is that K-theory surjections allow us to generate new examples from old ones.\nIn this section, we will apply this idea. We have seen in Example 2.4 that, localized away from 2, ΣCP 2 ≃ S 3 ∨ S 5 , so certainly ΣCP 2 satisfies the hypotheses of Theorem 1.4 at all odd primes p. We will now consider spaces X which are known to admit maps CP 2 → X which induce surjections on integral K-theory, and hence on K * ( ) ⊗ Z/p for all odd p. It follows in each case that ΣX is p-hyperbolic, and further by Corollary 1.5, that Σ n X is p-hyperbolic for all n ≥ 1.\nThe inclusion of CP n into CP n+1 induces a surjection on K-theory, so it must still induce a surjection after suspending. Composing these inclusions with the local equivalence ΣCP 2 ≃ S 3 ∨ S 5 gives, for each n ≥ 2, a map S 3 ∨ S 5 → ΣCP n which still induces a surjection on K * ( ) ⊗ Z/p for all odd primes p. Applying Theorem 1.4 to this map gives the following.\nExample 2.5. For n ≥ 2, ΣCP n is p-hyperbolic for all p = 2. Now let Gr k,n be the Grassmannian of k-dimensional complex subspaces of C n . First note that orthogonal complement gives a homeomorphism Gr k,n ∼ = Gr n−k,n . In particular Gr n−1,n ∼ = Gr 1,n ∼ = CP n−1 , so ΣGr n−1,n is p-hyperbolic. Other Grassmannians can be treated more uniformly, as follows.\nLet γ k,n denote the tautological bundle over Gr k,n . Consider the inclusion ι n : C n → C n+1 (x 1 , x 2 , . . . , x n ) → (x 1 , x 2 , . . . , x n , 0). This inclusion induces a map i k,n : Gr k,n → Gr k,n+1 , defined on V ∈ Gr k,n by V → ι n (V ). It follows from this definition that i * k,n (γ k,n+1 ) = γ k,n . Letting e i denote the i-th standard basis vector in C n , we also have a map j k,n : Gr k,n → Gr k+1,n+1 , defined on V = Span(v 1 , v 2 , . . . , v k ) ∈ Gr k,n by V → Span(ι(v 1 ), ι(v 2 ), . . . , ι(v k ), e n+1 ). It follows from this definition that j * k,n (γ k+1,n+1 ) = γ k,n ⊕ C 1 , where C 1 is the 1-dimensional trivial bundle.\nSince K * (CP n ) is generated by the class of the tautological bundle, composing the maps i k,n and j k,n for different values of k and n will give maps CP 2 = Gr 1,3 → Gr k,n for all 1 ≤ k ≤ n − 2 and n ≥ 3 which induce surjections in integral K-theory. As in Example 2.5, this implies the following (the case k = n − 1 is Gr n−1,n , which was treated first).\nExample 2.6. For n ≥ 3 and 0 < k < n, the suspended complex Grassmannian ΣGr k,n is p-hyperbolic for all p = 2.\nFor m ≤ n, the Milnor Hypersurface H m,n is defined by\nSuppose that m ≥ 2 and n ≥ 3. Then there is an inclusion ι : CP 2 → H m,n , defined by ι([z 0 : z 1 : z 2 ]) = ([z 0 : z 1 : z 2 : 0 : · · · : 0], [0 : · · · : 0 : 1]).\nWrite π 1 for the projection H m,n → CP m . Then the inclusion CP 2 → CP m factors as\nThis implies that ι induces a surjection on integral K-theory, so we obtain the following.\nExample 2.7. For m ≥ 2 and n ≥ 3, the suspended Milnor Hypersurface ΣH m,n is p-hyperbolic for all p = 2.\nLet U (n) denote the unitary group. There is a well-known map r : ΣCP n−1 → U (n) (see for example [Whi78]) which induces a surjection on K-theory. From this we obtain Example 2.8. For n ≥ 3, the suspended unitary group ΣU (n) is p-hyperbolic for all p = 2."},{"title":"Quantitative lower bounds on growth","content":"In Section 4, we will derive the following simple lower bound for the lim inf in the definition of Z/p rhyperbolicity, for the space S q1+1 ∨ S q2+1 .\nCorollary 2.9. Let p be a prime and r ∈ Z + . Let t m be the constants of Definition 1.2 for X = S q1+1 ∨ S q1+1 . Then\nThis implies that t m eventually exceeds ((1 − ε)2) m max(q 1 ,q 2 ) for any ε > 0. The constant 2 reflects the number of wedge summands. Note that this bound is independent of p and r.\nExample 2.10. Taking ε = 1 4 , we find that for all r ∈ Z + and all primes p the number of Z/p r -summands in i≤m π i (S 2 ∨ S 2 ) eventually exceeds ( 3 2 ) m . One can produce an analogous quantitative bound on the lim inf in the case of Theorem 1.4, but this bound is very weak. In particular, it depends on knowledge of the Adams operations on K * (X), and is at best ln(2) 2(p−1) ."},{"title":"Preliminary results","content":"Both Theorem 1.3 and Theorem 1.4 will be proven by means of Lemma 3.3. Our first goal is to establish this lemma. Let L be the free Lie algebra over Q on basis elements x 1 , . . . , x n . Write L k for the subset of L consisting of the basic products of the x i of weight k, in the sense of [Hil55], where the basic products of weight 1 are taken to be the x i , ordered by x 1 < x 2 < · · · < x n . The union L = ∞ k=1 L k is a vector space basis for L (see for example [Ser06,Theorem 5.3], but note that Serre uses the name Hall basis for the set of basic products).\nLet µ : Z + −→ {−1, 0, 1} be the Möbius inversion function, defined by\nTheorem 3.1. [Hil55, Theorem 3.3] Let L be the free Lie algebra over Q on basis elements x 1 , . . . , x n . Then |L k | = W n (k).\nLemma 3.2. [BO15, Introduction] The ratio W n (k) 1 k n k tends to 1 as k tends to ∞.\nIn particular, this implies that for n ≥ 2, the Witt function W n (k) grows exponentially in k. It should follow that if the number of p-torsion summands in i≤k π i (Y ) exceeds W 2 (k), then Y is p-hyperbolic. The following lemma makes a slightly generalised form of this idea precise.\nLemma 3.3. Let Y be a space. Suppose that there exist a, b ∈ Z + such that the number of p-torsion summands (respectively, Z/p r -summands) in i≤ak+b π i (Y ) exceeds W 2 (k), for all k large enough. Then Y is p-hyperbolic (respectively, Z/p r -hyperbolic).\nProof. The proofs for p-and Z/p r -hyperbolicity are identical, so we give only the former. Reframing the hypothesis in terms of the sequence {T m } m of Definition 1.1, we are assuming precisely that T ak+b > W 2 (k) for sufficiently large k. We then have that\nIt then follows from Lemma 3.2 that if 1 > ε > 0, once k is large enough, we have\nThis implies that\nand since this holds for all ε > 0,\nwhich is greater than zero, as required."},{"title":"Existence of summands in the stable stems","content":"We write π S j for the j-th stable stem in the homotopy groups of spheres, that is\nThe proof of Theorem 1.3, depends on having, for each p and r, some j such that π S j contains a Z/p r -summand. The purpose of this subsection is to show that the existence of such a j follows from existing work of Adams and others.\nLemma 3.4. For any prime p and any r ∈ Z + , there exists j such that Z/p r is a direct summand in π S j . That is, for a fixed choice of such a j, Z/p r is a direct summand in π n+j (S n ) for all n ≥ j + 2. Proof. We write ν p (s) for the largest power of p dividing the integer s.\nCASE 1 (p odd): Set t := p r−1 (p − 1), and notice that, since (p − 1) is even, j := 4t − 1 is congruent to 7 mod 8. Theorem 1.6 of [Ada66], and the discussion immediately following it, then tells us that π S j contains a direct summand isomorphic to Z/m(2t), for a function m which Adams defines. By decomposing this subgroup into direct summands of prime power order, it suffices to show that ν p (m(2t)) = r.\nThe discussion after Theorem 2.5 in [Ada65] gives that since t ≡ 0 mod (p − 1),\nNow, ν p (2t) is equal to (r − 1), by definition of t, so ν p (m(2t)) = r, as required. CASE 2 (p = 2, r ≥ 3): Set t := 2 r−3 , and set j := 4t − 1. From Theorem 1.5, and the discussion following Theorem 1.6 in [Ada66], π S j has a direct summand isomorphic to Z/m(2t), regardless of whether j is congruent to 3 or 7 mod 8. Again, referring to the discussion after Theorem 2.5 of [Ada65], we see that ν 2 (m(2t)) = 2 + ν 2 (2t) = 3 + ν 2 (t) = r, as required. CASE 3 (p r = 2 and p r = 4): It is known from [Fre38]"},{"title":"Proof of Theorem 1.3","content":"In this section we prove Theorem 1.3, which says that the wedge of two spheres is Z/p r -hyperbolic for all p and r. We also prove Corollary 2.9, which extracts some simple quantitative information from the proof of Theorem 1.3. We first record the following simple observation.\nRemark 4.1. Let k 1 , . . . , k n and q 1 , . . . q n be non-negative integers. Suppose that q 1 ≤ q 2 ≤ · · · ≤ q n , and let k = n i=1 k i . Then\nProof of Theorem 1.3. Assume without loss of generality that q 1 ≤ q 2 . By Lemma 3.3 it suffices to show that there exist constants a and b such that the number of Z/p r -summands in\nWe first apply the Hilton-Milnor Theorem. Since we are dealing with spheres, we need only the original form, due to Hilton in [Hil55]:\nwhere, as in Section 3, L = ∞ k=1 L k is Hilton's 'basic product' basis for L, the free Lie Algebra over Q on two generators x 1 and x 2 , and k i is the number of occurrences of the generator x i in the bracket B. Recall also from Section 3 that the weight k of a bracket B is equal to k 1 + k 2 , and that the cardinality of L k is given by the Witt formula W 2 (k) by Theorem 3.1.\nFor fixed k ∈ Z + , consider the factor in the above product corresponding to L k ⊂ L :\nThe associated subgroup of π * (S q1+1 ∨ S q2+1 ),\nis a direct summand. We will first find a Z/p r -summand in the homotopy groups of each of the spheres appearing in F k . Since q 1 ≤ q 2 , Remark 4.1 applies, and we may lower bound the dimensions of spheres appearing in F k by k 1 q 1 + k 2 q 2 + 1 ≥ kq 1 + 1. By Lemma 3.4, there exists j ∈ Z + such that π j+ℓ (S ℓ ) has a direct summand Z/p r for ℓ ≥ j +2. Therefore, if k is large enough that kq 1 ≥ j +1, then k 1 q 1 +k 2 q 2 +1 ≥ j +2 -that is, the j-th stem is stable on all of the spheres occurring in F k . Thus, for k large enough, there is a Z/p r summand in π j+k1q1+k2q2+1 (S k1q1+k2q2+1 ) whenever k 1 + k 2 = k.\nWe now upper bound the dimension of the homotopy groups in which these summands appear. Since q 1 ≤ q 2 we have by Remark 4.1 that j+k 1 q 1 +k 2 q 2 +1 ≤ kq 2 +1+j, so each of the Z/p r -summands we have identified is a distinct direct summand in\nWe have identified one such summand for each B ∈ L k , so the number of Z/p r -summands in\n. Thus, taking a = q 2 and b = 1 + j in Lemma 3.3 suffices.\nProof of Corollary 2.9. The last line of the proof of Lemma 3.3 shows that lim inf m ln tm m > ln 2 a . The last line of the proof of Theorem 1.3 implies that a may be taken to be q 2 , under the assumption that q 1 ≤ q 2 , which implies the result."},{"title":"K-theory and K-homology of ΩΣX","content":"The remainder of this paper proves Theorem 1.4. Sections 5 and 6 assemble necessary background, which we will use in Section 7 to prove the result.\nWhen studying the homotopy groups of a suspension ΣX, as in Theorem 1.4, the following approach is natural. Since π * (ΣX) ∼ = π * −1 (ΩΣX), we may instead study ΩΣX. This is useful because ΩΣX is well understood homologically via the Bott-Samelson theorem, which decomposes its homology as the tensor algebra on H * (X). Because we will need to use Adams' e-invariant, which is defined in terms of K-theory, we wish to replace ordinary homology with K-homology.\nThe purpose of Section 5 is to record the version of the Bott-Samelson theorem which applies to (torsion-free) K-homology, along with a universal coefficient theorem for passing between K-theory and K-homology. All of the material here is already known (in particular much of it is in [Sel83]) so its summary here is for convenience and clarity.\nOur conventions on definition of K * (X) are those of [AH61]. In particular, we define K −1 (X) := K 0 (ΣX), and set K * (X) := K 0 (X) ⊕ K −1 (X). We regard K * (X) and K * (X) as being Z/2-graded.\nWe will wish to work with K-theory and K-homology modulo the torsion subgroup. For a space X, write K TF * (X) and K * TF (X) for the quotients of the reduced K-homology and K-theory of X by their torsion subgroups. The same convention applies in the unreduced case."},{"title":"Künneth and universal coefficient theorems","content":"The universal coefficient theorem for K-theory first appears in some unpublished lecture notes of Anderson [And], and is first published by Yosimura [Yos75].\nTheorem 5.1 (Universal coefficient theorem). For any CW -complex X and each integer n there is a short exact sequence\nIn the torsion-free case, the universal coefficient theorem is as follows, where, unsurprisingly, we write Ext( K n−1 (X), Z) TF for the quotient of Ext( K n−1 (X), Z) by its torsion subgroup.\nCorollary 5.2.\n1. For any CW -complex X and each integer n there is a short exact sequence\n2. If X is a finite CW -complex, then Ext( K n−1 (X), Z) TF = 0, and we obtain an isomorphism\nProof. Let T 1 denote the torsion subgroup of Ext( K n−1 (X), Z), and let T 2 be the torsion subgroup of K n (X). The Universal Coefficient Sequence of Theorem 5.1 gives an injection T 1 → K n (X), which must have image contained in T 2 , thus lift to an injection T 1 → T 2 . For any group G, Hom(G, Z) is torsion-free, so the composite T 2 → K n (X) → Hom(K n (X), Z) is trivial, and by exactness we obtain a lift T 2 → Ext( K n−1 (X), Z). The image of this map must be torsion, which is to say that it must be contained in T 1 , so the aforementioned map T 1 → T 2 is a surjection. That is, the Universal Coefficient Sequence of Theorem 5.1 has last term torsion-free and first map restricting to an isomorphism of torsion subgroups. This implies the first statement.\nFor the second statement, we need only note that if X is finite, then K TF n−1 (X) is finitely generated, so Ext( K n−1 (X), Z) is torsion, as required.\nSelick [Sel83] deduces the following from work of Atiyah [Ati62], Mislin [Mis71] and Adams [Ada69].\nTheorem 5.3 (Künneth theorem for K-homology). Let X and Y be of the homotopy type of finite complexes. Then there is an isomorphism of Z/2-graded Z-modules:\nRemark 5.4. It follows immediately from Corollary 5.2 (and knowledge of K * (S q )) that K TF * (S q ) ∼ = Z. We write ξ q for the generator of K TF * (S q ). By the Künneth Theorem (Theorem 5.3), we may choose the ξ q so that ξ n ⊗ ξ m is identified with ξ n+m under the homeomorphism S n ∧ S m ∼ = S n+m .\nIn the case of K-theory, the analogous result follows directly from [Ada69].\nTheorem 5.5 (Künneth theorem for K-theory). Let X and Y be of the homotopy type of finite complexes. Then the external product on K-theory defines an isomorphism of Z/2-graded rings:"},{"title":"The James construction","content":"For a space X, let X s denote the product of s copies of X. Let ∼ be the relation on X s defined by\nThe James construction JX is defined to be the colimit of the diagram consisting of the spaces J s (X) and the above inclusions. Write i s : J s (X) → JX for the map associated to the colimit. Notice that JX carries a product given by concatenation, which makes it into the free topological monoid on X, and that a topological monoid is in particular an H-space.\nLet X ∧i denote the smash product of i copies of X, and let η : X → ΩΣX be the unit of the adjunction Σ ⊣ Ω. Explicitly, η(x) = (t → x, t ∈ ΣX).\nTheorem 5.6. [Jam55] 1. There is a homotopy equivalence JX ≃ − → ΩΣX which respects the H-space structures and identifies i 1 with η."},{"title":"There is a homotopy equivalence","content":"Lemma 5.7. [Sel83, Lemma 7] Let X have the homotopy type of an (r − 1)-connected CW -complex.\n2. Let x ∈ π N (J s (X)) for any N . If Σx is nontrivial then (i s ) * (x) is also nontrivial.\nProof. The first part follows by cellular approximation from the observation that J s (X) contains the (r(s + 1) − 1)-skeleton of JX. The second part follows from the observation that Σi s has a retraction by Theorem 5.6.\nFor spaces X and Y , let X * Y denote the join, which we define to be the homotopy pushout of the projections X × Y → X and X × Y → Y . The join is naturally a quotient of X × I × Y , where I denotes the unit interval. Following the treatment in [Ark11], let C 1 denote the subspace of X * Y consisting of points of the form (x, t, * ), for t ∈ I and x ∈ X, and let C 2 be the subspace consisting of points of the form ( * , t, y). The subspace\nThis gives a factorization of q as\n. Let q −1 denote any choice of homotopy inverse to q; all possible choices are homotopic. We may form a new map δ X,Y as the\nThe homotopy class of δ X,Y is well-defined, and we will call δ X,Y the canonical splitting of π. Note that δ X,Y is natural in maps of spaces in the sense that given f :\nFor s ≥ 3, consider the quotient map ΣX s → ΣX ∧s . We define the canonical splitting of this quotient to be the composite of canonical splittings\nOf course, we chose an order of multiplication here. This canonical splitting is natural as before. Write m s : (ΩΣX) s → ΩΣX for the map given by iteratively performing the standard loop multiplication on ΩΣX in any choice of order. Up to homotopy, m s is independent of this choice of order, since ΩΣX is homotopy associative.\nTheorem 5.6 gives the existence of a homotopy equivalence Γ :\nThere are many choices of Γ, up to homotopy. The next lemma asserts that Γ can be chosen in a way which suits our purpose. Selick [Sel83] describes the composite\nwith the homotopy equivalence of Theorem 5.6 (1). This immediately implies the following description of Γ.\nLemma 5.10. [Sel83] Let X be a finite CW -complex. The homotopy equivalence Γ : ∞ i=1 ΣX ∧i → ΣΩΣX may be chosen such that:\nis an isomorphism of algebras; 2. the restriction of Γ to ΣX ∧s is homotopic to the composite\nwhere the unlabelled arrow is the canonical splitting.\nThe description of the map Γ in Lemma 5.10 has the following consequence. For a space Y , let ev : ΣΩY → Y be the evaluation map, which may be described explicitly by ev( γ, t ) = γ(t) for γ ∈ ΩY .\nLemma 5.11. Let Γ : ∞ i=1 ΣX ∧i → ΣΩΣX be the homotopy equivalence of Lemma 5.10. The composite ev • Γ is homotopic to the projection onto the first wedge summand.\nProof. Let ι s : ΣX ∧s → ∞ i=1 ΣX ∧i be the inclusion of the s-th wedge summand. We must show that ev • Γ • ι s ≃ 1 ΣX if s = 1, and * otherwise.\nThe following diagram commutes up to homotopy\nThe composite along the bottom of the diagram is Γ • ι s , so to obtain the desired factorization of Γ • ι s through v, it suffices to show that the diagram commutes up to homotopy.\nThe top right square commutes because m 2 • m (s−1) ≃ m s , by homotopy associativity of the Hspace ΩΣX. The remaining three squares commute by naturality of our canonical splitting. This completes the proof.\nLet ρ k be the projection T ( K TF * (X)) → K TF * (X) ⊗k . The next corollary is immediate from Lemma 5.11.\nCorollary 5.12. S −1 (ev * • Γ * ) = ρ 1 : T ( K TF * (X)) → K TF * (X)."},{"title":"Primitives and commutators","content":"It follows from the Künneth Theorem (Theorem 5.3), and the fact that\n. We may therefore make the following definition. A class\nFrom this point of view, the following lemma is immediate.\nLemma 5.13. If Y is a co-H-space, then all elements in K TF * (Y ) are primitive.\nIf Y is an H-group, then the multiplication m :\n. We will denote this map by juxtaposition, so that m * (y 1 ⊗ y 2 ) = y 1 y 2 . Furthermore, the commutator Y ×Y → Y descends to a map c : Y ∧Y → Y . Expanding the definition of the commutator in terms of the K-homology Künneth Theorem (Theorem 5.3) gives the following lemma.\nLemma 5.14. Let Y be an H-group, and let c : Y ∧Y → Y be the commutator. If y 1 and y 2 ∈ K TF * (Y ) are primitive, then c * (y 1 ⊗ y 2 ) = y 1 y 2 − (−1) |y1||y2| y 2 y 1 ."},{"title":"The category of ψ-modules","content":"In [Ada66], Adams defines an abelian category which we will follow Selick [Sel83] in calling ψ-modules. The e-invariant, which is our central tool, is defined by Adams in terms of ψ-modules. The purpose of this section is to record results about ψ-modules for later use.\nA ψ-module consists of an abelian group M , with homomorphisms\nfor each ℓ ∈ Z, satisfying the axioms of [Ada66, Section 6]. If X is a space then the group K 0 (X), together with its Adams operations, is a ψ-module. Since we defined K −1 (X) by setting K −1 (X) = K 0 (ΣX), it too has the structure of a ψ-module. Maps of spaces induce maps of ψ-modules. The Adams operation ψ ℓ on K 0 (S 2n ) is multiplication by ℓ n , so in particular Adams operations do not commute with the Bott isomorphism. For graded ψ-modules M and N we will write Hom ψ−Mod (M, N ) for the abelian group consisting of graded ψ-module homomorphisms. The unadorned notation Hom(M, N ) will mean homomorphisms of the underlying graded abelian groups. Proof. Let ϕ : M → N be a homomorphism of underlying Z-modules. If, for some k ∈ Z \\ {0}, k · ϕ is a ψ-module homomorphism, then, since N is torsion-free, ϕ is also a ψ-module homomorphism. This implies that Coker(Hom ψ−Mod (M, N ) ֒− → Hom(M, N )) is torsion-free, which implies the result. Definition 6.2 (Adams' e-invariant). Suppose that f : X → Y induces the trivial map on K * . Then the cofibre sequence of f gives a short exact sequence of ψ-modules\nThe e-invariant of f is the element of Ext ψ−Mod ( K 0 (Y ), K 0 (ΣX)) represented by this exact sequence.\nThe e-invariant does not commute with the Bott isomorphism, but the interaction between the Bott isomorphism and the Adams operations is easy to describe, as follows. Let ψ ℓ Y be the homomorphism ψ ℓ : K 0 (Y ) → K 0 (Y ). Then, modulo the Bott isomorphism, we have ψ ℓ Σ 2 X = ℓ · ψ ℓ X . That is 'upon double suspending, the Adams operations gain a factor ℓ'. In terms of the e-invariant, all we need to know is the following.\nThere is a homomorphism\nsuch that T (e(f )) = e(Σ 2 f ).\nWe will be concerned only with the e-invariants of maps whose domain is a sphere. One of the two K-groups of a sphere vanishes, in the dimension matching the parity of the sphere, but the e-invariant, as defined above, lives only in K 0 . In order to detect maps regardless of the parity of the sphere on which they are defined, we will need to keep track of the e-invariants of f and Σf , so we will use the following modified e-invariant."},{"title":"Definition 6.4 (Double e-invariant). Let","content":"Suppose that f : X → Y induces the trivial map on K * . Then the double e-invariant of f is e(f ) = (e(f ), e(Σf )) ∈ Ext ψ−Mod ( K * (Y ), K * (ΣX))."},{"title":"Pullback of an extension along a homomorphism defines a map","content":"If g : Y → Z then e(g • f ) is represented by the pullback of e(f ) and g * : K 0 (Z) → K 0 (Y ) [Ada66, Proposition 3.2 b)]. To describe e(g • f ) we need only apply this result degree-wise, as follows. For convenience, we write g * · e(f ) for the pullback of g * and e(f ). Define the map\nCombining these, let\nbe the direct sum θ 0 (f ) ⊕ θ −1 (f ). These definitions, together with Adams' above result, give the following lemma.\nLemma 6.5. For maps f : X → Y and g : Y → Z, the following diagram commutes:\nFollowing [Sel83], write Z(n) for the ψ-module K 0 (S 2n ). Explicitly, Z(n) has underlying abelian group Z, and ψ ℓ acts by multiplication by ℓ n . It follows that K −1 (S 2n+1 ) := K 0 (S 2n+2 ) ∼ = Z(n + 1).\nLemma 6.6. [Ada66, Proposition 7.8, 7.9] If n < m then Ext ψ−Mod (Z(n), Z(m)) injects into Q Z . The e-invariant of a map f : S 2m−1 → S 2n may therefore be regarded as an element of Q Z . Furthermore, the value e(f ) in Q Z satisfies e(Σ 2 f ) = e(f ), so in particular, when f is a map between spheres, e(f ) depends only on the stable homotopy class of f .\nThe following theorem is the main technical component of Selick's paper [Sel83].\nTheorem 6.7. [Sel83, Theorem 6] Let f ′ : S 2m−1 → S 2n be such that p t−1 e(f ′ ) = 0 in Q Z , for p prime and some t ∈ Z + . Let Y have the homotopy type of a finite CW -complex and let g : S 2n → Y be such that Im(g * : K 0 (Y ) → K 0 (S 2n )) contains up s K 0 (S 2n ), for s ∈ Z + and u prime to p. If s < t, and there exists some ℓ ∈ Z + for which\ndoes not have ℓ m as an eigenvalue, then e(g • f ′ ) = 0.\nThe following theorem of Gray [Gra69] will provide the map f ′ for Theorem 6.7. Specifically, this theorem provides a linearly spaced family of stems, each of which has a stable p-torsion class which is born on S 3 and detected by the e-invariant.\nTheorem 6.8. [Gra69, Corollary of Theorem 6.2] Let p be an odd prime and let j ∈ Z + . Then there exists a class f p,j ∈ π 2j(p−1)+2 (S 3 ) with e(f p,j ) = −1 p ∈ Q Z .\nThe corresponding 2-primary result is as follows. Adams [Ada66, Theorem 1.5 and Proposition 7.14] shows that, for j > 0, the (8j +3)-rd stem contains a direct summand whose 2-primary component has order 8, and that on this component the e-invariant is a surjection onto Z/4. The sphere of origin of the classes in this component was deduced by Curtis in [Cur69].\nTheorem 6.9. [Ada66; Cur69] Let j ∈ Z + . Then there exists a class f 2,j ∈ π 8j+6 (S 3 ) of order 4, with e(f 2,j ) = −1 2 ∈ Q Z ."},{"title":"Main construction","content":"Having assembled preliminaries in Sections 5 and 6, we can begin to work towards the proof of Theorem 1.4. Our approach is as follows. From the data of Theorem 1.4, we will construct a commutative diagram of (roughly) the following form, where B is a set and the other objects are Z-modules.\nWe will argue that\n• The image of the top map consists of classes of order dividing p.\n• The image of the left vertical map generates a submodule isomorphic to the weight k component of the free graded Lie algebra over Z/p on two generators.\n• The bottom map is injective.\nTogether, these facts imply that there is a submodule of π * (ΩΣX) ∼ = π * +1 (ΣX), consisting of classes of order dividing p, and surjecting onto a module isomorphic to the weight k component of the free graded Lie algebra over Z/p on two generators. This submodule (which is necessarily a Z/p-vector space) must therefore have dimension at least W 2 (k) (Theorem 7.5), which will imply that ΣX is p-hyperbolic (Lemma 3.3).\nThe diagram will be obtained by juxtaposing three squares. Subsections 7.1, 7.2, and 7.3 each construct one of these squares. In Subsection 7.4 we put them together and prove Theorem 1.4. Roughly speaking, the top map of the diagram should be thought of as first taking a family of Samelson products and then pulling them back along some suitable map f coming from Gray's work (Theorem 6.8). The vertical maps should be thought of as passing from maps of spaces to K-theoretic invariants, and the bottom map (therefore) should be thought of as tracking the effect of the top map on these invariants.\nBecause of the need to work with a finite CW -complex in Selick's Theorem (Theorem 6.7) we will restrict the right hand side of the diagram to instead refer to some finite skeleton J s (X) of the James construction."},{"title":"Samelson products and their Hurewicz images in K-homology","content":"Let R be a commutative ring with unit. We take a graded Lie algebra over R to be defined as in [Nei10]. For a non-negatively graded R-module V , let L(V ) denote the free graded Lie algebra [Nei10, Section 8.5]. Write L k (V ) for the submodule of L(V ) generated by the brackets of length k in the elements of V . We will call L k (V ) the weight k component of L(V ). Note that this convention differs from Neisendorfer's -he writes L(V ) k for the weight k component.\nDefinition 7.1. Let Y be an H-group, and let c : Y ∧ Y → Y be the commutator of Lemma 5.14. Let α ∈ π N (Y ), and let β ∈ π M (Y ). The Samelson product of α and β, written α, β ∈ π N +M (Y ), is the composite α, β :\nSamelson products are bilinear, graded anticommutative, and satisfy the graded Jacobi identity. They fail, however, to make π * (Y ) into a graded Lie algebra over Z in Neisendorfer's sense [Nei13,Section 7]. This is important because we want the natural map from the free Lie algebra to the corresponding tensor algebra to be an injection onto a summand (Lemma 7.4). One could define an appropriate notion of 'free graded pseudo-Lie algebra', and proceed as follows with that in place of the set B(π * (A)), which we use in what follows, but we prefer to avoid making the extra definition. which extends ν * and satisfies Φ π ν ([x, y]) = Φ π ν (x), Φ π ν (y) for all x, y ∈ B(π * (A)). For a Z/2-graded Z-module V , we define a non-negatively graded Z-module Hom( K TF * (S * ), V ), by setting\nwhere the homomorphisms are understood to respect the Z/2-grading on K * and V .\nIn the case that V = L is a Z/2-graded Lie algebra over Z, Hom( K TF * (S * ), L) inherits a nonnegatively graded Lie algebra structure as follows. Let the generators ξ N of K TF * (S N ) be as in Remark 5.4. Then the bracket [f, g] of f ∈ Hom( K TF * (S N ), L) and g ∈ Hom( K TF\nThe squaring operation is defined in the same way. Likewise, if V is a Z/2-graded associative algebra over Z, then Hom( K TF * (S * ), V ) inherits the structure of a non-negatively graded associative algebra.\nLet ν : A → ΩΣX be a map. There is a composition\nwhere the first map is the natural map which is the identity on K TF * (A) and satisfies [x, y] → xy − (−1) |x||y| yx, and the second map is obtained by applying the universal property of the tensor algebra to ν * . Let Φ K ν : Hom( K TF * (S * ), L( K TF * (A))) → Hom( K TF * (S * ), K TF * (ΩΣX)) be the pushforward along the above composite. It is then automatic that Φ K ν is a map of non-negatively graded Lie algebras over Z, where the structures are defined as above.\nWe write deg : π N (Y ) → Hom( K TF * (S N ), K TF * (Y )) for the map f → f * . Let deg ′ : B(π * (A)) → Hom( K TF * (S * ), L( K TF * (A))) be the unique map which restricts to deg : π * (A) → Hom( K TF * (S * ), K TF * (A)) ⊂ Hom( K TF * (S * ), L( K TF * (A))) and carries brackets to brackets. The above maps are related as follows. Lemma 7.2. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CWcomplexes. The following diagram commutes:\nProof. By construction of B(π * (A)), it suffices to show that the restriction of the diagram to the weight 1 component B 1 (π * (A)) = π * (A) commutes, and that all maps respect the bracket operations. By definition, L 1 ( K TF * (A)) = K TF * (A). It then follows immediately from the definitions of Φ π ν and Φ K ν that restricting the left hand side of the diagram to weight 1 components gives the diagram\n, K TF * (ΩΣX)), which commutes, since it just expresses naturality of deg.\nIt remains to show that all maps respect bracket operations. The maps Φ π ν and deg ′ respect the bracket operations by definition, and Φ K ν respects bracket operations by construction. We therefore only need show that deg respects brackets. Let f ∈ π N (ΩΣX), and let g ∈ π M (ΩΣX). We must show that deg( f, g ) is the commutator deg(f )deg(g) − (−1) N M deg(g)deg(f ) with respect to the algebra operation on Hom( K TF * (S * ), K TF * (ΩΣX)). Since K TF * (S N +M ) ∼ = Z, it suffices to show that the two homomorphisms agree on the generator ξ N +M (Remark 5.4). By Definition 7.1 and the Künneth Theorem (Theorem 5.3),\nSpheres of dimension at least 1 are co-H spaces, so by Lemma 5.13, ξ N and ξ M are primitive. By naturality of the diagonal f * (ξ N ) and g * (ξ M ) are still primitive, so by Lemma 5.14,\nwhich by definition of the multiplication on Hom( K TF * (S * ), K TF * (ΩΣX)) is the result of evaluating deg(f )deg(g) − (−1) N M deg(g)deg(f ) on ξ N +M , as required.\nWe now lift the previous result to J s (X), thereby producing the first square of the diagram promised at the start of this section. Recall that we write i s : J s (X) → JX for the inclusion, and that by Theorem 5.6 we have a homotopy equivalence JX ≃ − → ΩΣX. We will abuse notation and also write i s for the composite J s (X) → JX ≃ − → ΩΣX.\nCorollary 7.3. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CWcomplexes, with X (r − 1)-connected for r ≥ 1. If N, s ∈ Z + satisfy N < r(s + 1) − 1, then (i s ) * : π N (J s X) → π N (ΩΣX) is an isomorphism and for each k ≤ s there exists a commutative diagram:\nProof. Consider the diagram of Lemma 7.2. Lemma 5.7 shows that (i s ) * is an isomorphism on π N , so let Φ π ν be the unique map such that the condition (i s ) * • Φ π ν = Φ π ν holds. By Theorem 5.6 (2) and Lemma 5.10, the map (i s ) * : K TF N (J s (X)) → K TF N (ΩΣX) is the inclusion of the tensors of length at most s. Since k ≤ s, we may therefore define Φ K ν to be the unique map such that the condition\nCommutativity then follows from Lemma 7.2 by naturality of deg, since Hom( K TF * (S N ), (i s ) * ) is injective.\nLemma 7.4. Let V be a non-negatively-or Z/2-graded Z-module which is free and finitely generated in each dimension. Then\n• L(V ) and T (V ) are free Z-modules in every dimension.\n• The natural map L(V ) → T (V ), [x, y] → xy − (−1) |x||y| yx is an injection onto a summand.\nProof. The non-negatively graded case is immediate from [Nei10], Proposition 8.3.1 and p282. For the Z/2-graded case, first observe that there is a forgetful functor U from Z-graded modules to Z/2-modules which carries Z-graded (Lie) algebras to Z/2-graded (Lie) algebras, and a functor C from Z/2-graded modules to Z-modules which puts V 0 in any even dimension and V 1 in any odd dimension. Both C and U respect freeness and split injections, and there are natural isomorphisms U T (CV ) ∼ = T (V ) and U L(CV ) ∼ = L(V ). This implies the Z/2-graded result.\nThe graded version of Theorem 3.1 now follows immediately from Hilton's paper:\nTheorem 7.5. [Hil55, Theorem 3.2, 3.3] Let V be a torsion-free Zor Z/2-graded Z-module of total dimension n. Then the total dimension of L k (V ) is W n (k).\nLet R be a commutative ring with unit. Let M be an R-module, and as usual let T (M ) denote the tensor algebra on M . Let ι k : M ⊗k → T (M ) be the inclusion, and let ρ k : T (M ) → M ⊗k be the projection. Let τ : T (M ) → T (M ) be the composite ι 1 • ρ 1 . Given an R-algebra A, and a map ϕ : M → A, we write ϕ for the induced map T (M ) → A, that is, the unique map of algebras such that ϕ • ι 1 = ϕ. Now, let M and N be R-modules, and let ϕ : M → T (N ) be a map. In the proof of Theorem 7.7, we will wish to make a 'leading terms' style argument. This is made precise in the next Lemma, which compares ϕ with τ • ϕ. Informally, we think of τ • ϕ as the 'leading terms part' of ϕ.\nLemma 7.6. Let R be a commutative ring with unit. Let M and N be Zor Z/2-graded R-modules. Let ι k : M ⊗k → T (M ) be the inclusion, let ρ k : T (N ) → N ⊗k be the projection, and let τ : T (N ) → T (N ) be as above. Let ϕ : M → T (N ) be a map. Then\nProof. It suffices to check equality on basic tensors. Let v ∈ M ⊗k be a basic tensor, so that\nApplying ρ k to both sides yields the result.\nTheorem 7.7. Let F = Q or Z/p for p prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes. Let ν : ΣA → ΣX be the adjoint of ν. If\nis also an injection.\nRemark 7.8. In the case where ν is a suspension Σζ, we have a diagram\nso in particular ν * factors through the weight 1 component K TF * (X) of the tensor algebra decomposition of K TF * (ΩΣX). This dramatically simplifies the proof, removing the need for Lemma 7.6. In practice this is not a reasonable assumption -for example, the map µ : S 3 ∨ S 5 → ΣCP 2 of Example 2.5 (which plays the role of ν) does not desuspend.\nProof. In this proof, for a space Y , we will identify the algebras T ( K TF * (Y )) and K TF * (ΩΣY ), omitting the isomorphism S −1 Γ * of Lemma 5.10. We defined Φ K ν to be the pushforward along a certain map L( K TF * (A)) → K TF * (ΩΣX). Call this map Φ K ν ′ . It suffices to prove that Φ K ν ′ ⊗ F is an injection.\nThe triangle identities for the adjunction Σ ⊣ Ω give a commutative diagram\n′ is the unique map of Lie algebras extending ν, we have a commuting diagram\nwhere we note that that by Lemma 7.4, the natural map L( K TF * (A)) → T ( K TF * (A)) is an injection onto a summand. It therefore suffices to show that (Ων) * ⊗ F is an injection.\nLet (ν * ) denote the extension of ν * to T ( K TF * (A)), so that (ν * ) = (Ων) * (modulo the isomorphism S −1 Γ * , as above). Since (ρ i • (ν * ) • ι k ) = 0 for i < k, it further suffices to show that (ρ k • (ν * ) • ι k ) ⊗ F is an injection for each k. By Lemma 7.6, with M = K TF * (A) and N ′ = K TF * (X), we have that\nAs previously, let ev : ΣΩY → Y denote the evaluation map. The following diagram commutes:\nThe hypothesis therefore implies that the composite (ev * • Σν * ) ⊗ F is an injection. Desuspending, we have that (S −1 ev * • ν * ) ⊗ F is an injection. By Lemma 5.12,\nThe following corollary, which lifts the injectivity back to J s (X), is immediate from Theorem 7.7 and Lemma 7.3.\nCorollary 7.9. Let F = Q or Z/p for p prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r − 1)-connected for r ≥ 1. Suppose that N, s, k ∈ Z + satisfy k ≤ s, so that the map Φ K ν is as in Corollary 7.3. If\nis also an injection.\nWe have now established all that we will need to know about this 'first square'."},{"title":"Maps derived from the universal coefficient isomorphism","content":"In this subsection we will build the second square of our diagram. This square is really just the Universal Coefficient theorem (Corollary 5.2) in a different form. We will write deg for both K-homological and K-theoretic degree.\nLemma 7.10. Let Y be a space having the homotopy type of a finite CW -complex. There exists an isomorphism U making the following diagram commute.\nProof. For β : K TF * (S N ) → K TF * (Y ), let U (β) be the unique map making the following diagram commute\nwhere the isomorphisms are those of Corollary 5.2. Since K TF * (Y ) is a finitely generated free Z-module, β → Hom(β, Z) is an isomorphism, so U is also an isomorphism. Commutativity of the diagram from the statement of this lemma is by naturality of Lemma 5.2.\nCorollary 7.11. Let Y be a space having the homotopy type of a finite CW -complex. For a Z-module M , let τ p : M → M ⊗ Z/p be the natural map. There exists an injection U ′ making the following diagram commute.\nProof. By Lemma 7.10, we have a commutative diagram\nwith U an isomorphism, so U ⊗ Z/p is also an isomorphism. By Lemma 6.1, the map\nis an injection. Maps of spaces induce maps of ψ-modules on K-theory, so the image of U • deg is contained in Hom ψ−Mod ( K * TF (Y ), K * TF (S N )), and hence there exists a map U ′ making the following diagram commute:\nBoth vertical maps are injections, so U ′ has the required properties."},{"title":"Pulling back along classes defined on S 3","content":"Let f ∈ π j (S 3 ), and let N ≥ 3. Then, for ω ∈ π N (Y ), the composite\nIn words, f * Σ pulls classes back along the appropriate suspension of f . Strictly speaking, f * Σ is only a partial map, because it is undefined on π N for N ≤ 2, but this will be unimportant.\nRecall the definition of the double e-invariant e (Definition 6.4). On π N (Y ), we have by definition that f * Σ = (Σ N −3 f ) * . By Lemma 6.5 we have a commuting square:\nMimicking the convention for f * Σ , let\nbe the map which is defined to be equal to θ(\nLemma 7.12. Let p be a prime, and let f ∈ π j (S 3 ) with e(f ) defined. If pf = 0, then there exists a map θ p Σ (f ) making the following diagram commute for all N :\nProof. Since pf = 0, we have that pe(Σ N −3 f ) = 0 for all N , which implies that θ Σ (f ) vanishes on p-divisible elements, so there exists a unique map θ p Σ (f ) making the diagram commute, as required.\nLemma 7.13. Let X be a finite CW -complex. Let λ X ℓ be the largest eigenvalue of the rational Adams operation\nThen, for i ≥ 0\n• the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Σ 2i J s (X)) ⊗ Q is ℓ i (λ X ℓ ) s , and\nProof. When i ≥ 1, Theorem 5.6 gives that ΣJ s (X) ≃ Σ s t=1 X ∧t , so Σ 2i J s (X) ≃ S 2i ∧ s t=1 X ∧t , and Σ 2i+1 J s (X) ≃ S 2i ∧ ΣX ∧ s−1 t=1 X ∧t . By the Künneth theorem (Theorem 5.5), this implies isomorphisms of rings\nThe Künneth isomorphism of Theorem 5.5 is given by the external product of K-theory. Since the Adams operations are ring homomorphisms, the above isomorphisms are also isomorphisms of ψmodules. In particular, the Adams operations on the left are the tensor product of the corresponding operations on the right. These decompositions hold for K 0 TF , so they also hold for Q ⊗ K 0 , and the remaining problem is to determine the largest eigenvalue of the relevant tensor products of Adams operations. The eigenvalues of a tensor product of linear endomorphisms are precisely the products of the eigenvalues. The operation ψ ℓ acts on S 2i by multiplication by ℓ i . Together, these observations imply the result.\nLemma 7.14. Let p be an odd prime. Let X be an (r − 1)-connected finite CW -complex. Let N, s ∈ Z + . Consider the diagram of Lemma 7.12 for Y = J s (X) and f = f p,j ∈ π 2j(p−1)+2 (S 3 ), the map of Theorem 6.8:\nτp θΣ(f ) / / Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +2j(p−1) )) construction (Theorem 6.8), so e(Σ N −3 f ) = 0 by stability (Lemma 6.6). Since e(ω • Σ N −3 f ) = 0, the contrapositive of Theorem 6.7 gives that ω * has p-divisible image in K 0 (S N ). Since N is even, this implies that τ p • deg(ω) = 0, as required. CASE 2 (N odd): Write n = 2n + 1. Let f ′ = Σ N −2 f and g = Σω in Theorem 6.7, and proceed similarly to case 1. The domain of Σω • Σ N −2 f is S M−1 , where M − 1 = N + 2j(p − 1), so M is even, as is required. To check the eigenvalue hypothesis of Theorem 6.7, write M = 2m. By Lemma 7.13, the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (ΣJ s (X)) ⊗ Q is at most λ s ℓ , and ℓ m = ℓ j(p−1)+n = ℓ j(p−1)+ N −1 2 , which we assumed was greater than λ s ℓ . This means that ℓ m cannot be an eigenvalue of ψ ℓ ⊗ Q on K 0 (ΣJ s (X))⊗Q. As in the previous case, e(Σ N −2 f ) = 0. Since e(Σω •Σ N −2 f ) = 0, the contrapositive of Theorem 6.7 gives that (Σω) * has p-divisible image in K 0 (S N +1 ). Since N is odd, this implies that τ p • deg(ω) = 0, as required. This completes the case, and hence the proof."},{"title":"Proof of Theorem 1.4","content":"Construction 7.15. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r − 1)-connected for r ≥ 1. Let f ∈ π i (S 3 ) with e(f ) defined. Suppose that N, k, s ∈ Z + satisfy N < r(s + 1) − 1 and k ≤ s. The diagrams of the preceding subsections may be combined as follows.\nRecall the definition of deg ′ from the preamble to Lemma 7.2. Let I(A) be the submodule of Hom( K TF * (S * ), L( K TF * (A))) ⊗ Z/p generated by Im(τ p • deg ′ ). The same grading conventions as usual apply: we write I k (A) for the weight k part, we write I N (A) for the degree N part, and let\nFrom Corollary 7.3, using the assumptions that N < r(s + 1) − 1 and k ≤ s (which make Φ π ν and Φ K ν well-defined) we obtain the following diagram, where the images of the vertical maps have been 'popped out' to their right.\nNext, from Corollary 7.11 (with Y = J s (X)) we have a diagram π N (J s (X)) π N (J s (X))\nLastly, we obtain the following diagram from Lemma 7.12:\nConcatenating these diagrams gives a diagram as follows:\nIn this subsection, we combine the results of the previous subsections to produce results about this diagram.\nTheorem 7.16. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r−1)-connected for r ≥ 1. Let N, k, s ∈ Z + with N < r(s+1)−1 and k ≤ s. Let f = f p,j ∈ π 2j(p−1)+2 (S 3 ), the map of Theorem 6.8.\nFor ℓ ∈ Z + , let λ Y ℓ be the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Y ) ⊗ Q, and let λ ℓ = max(λ X ℓ , λ ΣX ℓ ). If"},{"title":"an injection, and","content":"• there exists ℓ ∈ Z + such that ℓ j(p−1)+ N −1 2 > λ s ℓ ,\n, K * TF (S N +2j(p−1) )) is an injection.\nProof. By Corollary 7.9, since ν * ⊗ Z/p is an injection, Φ K ν ⊗ Z/p is also an injection. By Corollary 7.11 U ′ is an injection. By Lemma 7.14 the hypothesis on ℓ implies that the restriction of θ p Σ (f ) to Im(τ p • deg) is an injection. The map θ p Σ (f ) • U ′ • ( Φ K ν ⊗ Z/p) is thus a composite of injections, hence an injection, as required.\nIn the proof of Theorem 1.4, we will wish to restrict attention to those elements of B(π * (A)) who are brackets of classes in π * (A) in some dimensional range q min ≤ n ≤ q max . All such classes lie in dimensions kq min ≤ N ≤ kq max . Said more precisely, we have an inclusion B k ( qmax n=qmin π n (A)) ⊂ kqmax N =kqmin B k N (π * (A)). We will now study the diagram of Construction 7.15 in this dimensional range. Construction 7.17. Let p be an odd prime, ν : A → ΩΣX for finite CW -complexes A and X with X (r − 1)-connected for r ≥ 1, and f ∈ π i (S 3 ) with e(f ) defined. Let q max > q min be natural numbers. Fix k ∈ Z + , and let s = kq max +1. For N ∈ Z + with kq min ≤ N ≤ kq max , we have that N < r(s+1)−1 and k ≤ s. Combining the diagrams obtained from Construction 7.15 for this range of values of N gives the following diagram: / / kqmax N =kqmin Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +i−2 )).\nWe now show that by choosing a large enough c ∈ Z + , and setting f = f p,ck , the eigenvalue hypothesis of Theorem 7.16 may be satisfied across the dimensional range of Construction 7.17 for all sufficiently large k.\nCorollary 7.18. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X path-connected. Let q max > q min be natural numbers. Let c, k ∈ Z + . Let f = f p,ck ∈ π 2ck(p−1)+2 (S 3 ) be the map of Theorem 6.8. If ν * ⊗ Z/p : K TF * (ΣA) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection then there exists c ∈ Z + such that for large enough k ∈ Z + ,\nExt ψ−Mod ( K * TF (J s (X)), K * TF (S N +2ck(p−1) ))\nis an injection.\nProof. By Theorem 7.16, it suffices to show that for each N with kq min ≤ N ≤ kq max there exists ℓ ∈ Z + such that ℓ ck(p−1)+ N −1 2 > λ s ℓ = λ kqmax+1 ℓ . Take any ℓ ≥ 2. Since N ≥ kq min , it suffices to find c such that for large enough k we have ℓ ck(p−1)+ kq min −1 2 > λ kqmax ℓ . Taking logs on both sides, this is equivalent to (ck(p − 1) + kq min − 1 2 ) log(ℓ) > kq max log(λ ℓ ).\nIt is now clear that we may choose c large enough that this equation holds for large enough k, in particular, any c ≥ 1 p−1 (q max log(λ ℓ ) log(ℓ) − qmin 2 ) will do.\nBefore proving Theorem 1.4, we prove three lemmas. The first converts the K-theoretic hypothesis of Theorem 1.4 into the K-homological input our construction requires.\nLemma 7.19. Let X be a space, and let p be prime. Let µ : S q1+1 ∨ S q2+1 → ΣX be a map with q i ≥ 1, such that the map K * TF (ΣX) ⊗ Z/p µ * ⊗Z/p − −−−− → K * TF (S q1+1 ∨ S q2+1 ) ⊗ Z/p ∼ = Z/p ⊕ Z/p is a surjection. Then µ * ⊗ Z/p : K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection.\nProof. Naturality of the Universal Coefficient Theorem (Corollary 5.2) relative to the map µ gives a diagram of short exact sequences\nwhere E denotes the quotient of Ext( K * −1 (ΣX), Z) by its torsion subgroup. We will argue by contrapositive. Suppose that µ * ⊗ Z/p is not injective. The Z/p-vector space K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p has dimension 2, so there exists a basis x, y where (µ * ⊗ Z/p)(y) = 0. Prior to tensoring with Z/p, this means that there exists a non-p-divisible element y of K TF * (S q1+1 ∨ S q2+1 ) such that p divides µ * ( y). This implies that any element ϕ of Hom( K TF * (S q1+1 ∨ S q2+1 ), Z) with ϕ( y) not p-divisible is not contained in the image of Hom(µ * , Z), hence, by the diagram, that µ * ⊗ Z/p is not surjective, as required.\nProof. For any space A, the localizing map A −→ A (p) induces an isomorphism on K * ( ) ⊗ Z/p [Mis78]. It therefore suffices to show that there exists maps µ ′ and ϕ making the following diagram commute, with the localization ϕ (p) being a homotopy equivalence.\nSuch maps exist by Lemma 7.21.\nWe are now ready to prove Theorem 1.4.\nProof of Theorem 1.4. Let µ = µ 1 ∨ µ 2 , with adjoint µ : S q1 ∨ S q2 → ΩΣX. Let f = f p,ck ∈ π 2ck(p−1)+2 (S 3 ). Consider the diagram of Construction 7.17, with A = S q1 ∨ S q2 , q max = max(q 1 , q 2 ), q min = min(q 1 , q 2 ), and ν = µ. We have such a diagram for each k ∈ Z + :\nN =kqmin π N +2ck(p−1)−1 (J s (X)) e kqmax N =kqmin I k N (S q1 ∨ S q2 ) / / kqmax N =kqmin Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +2ck(p−1) )).\nBy assumption, µ * ⊗Z/p : K * (ΣX)⊗Z/p → K * (S q1+1 ∨S q2+1 )⊗Z/p is a surjection. Since K * (S q1+1 ∨ S q2+1 ) is torsion-free, Lemma 7.19 then implies that µ * ⊗ Z/p : K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection. By Lemma 7.22, we may assume without loss of generality that X has the integral homotopy type of a finite CW -complex. Thus, by Corollary 7.18, we may fix c such that for large enough k, θ p (f ) • U ′ • ( Φ K µ ⊗ Z/p) is an injection. The Hurewicz map h is a surjection π * (S q1 ∨S q2 ) → K TF * (S q1 ∨S q2 ), so the submodule generated by the image of the map h ′ : B(π * (S q1 ∨S q2 )) → L( K TF * (S q1 ∨S q2 )) of Lemma 7.20 contains the submodule generated by K TF * (S q1 ∨ S q2 ) under the bracket operation. In particular, it contains the weight k component L k ( K TF * (S q1 ∨ S q2 )) for each k. By Theorem 3.1, dim Z (L k ( K TF * (S q1 ∨ S q2 ))) = W 2 (k). Note that L k ( K TF * (S q1 ∨ S q2 )) = kqmax N =kqmin L k ( K TF * (S q1 ∨ S q2 )). It then follows from Lemma 7.20 that dim Z/p ( kqmax N =kqmin I k N (S q1 ∨ S q2 )) ≥ W 2 (k). Since θ p Σ (f ) • U ′ • ( Φ K µ ⊗ Z/p) is an injection for large enough k, it follows that the dimension of e( kqmax N =kqmin π N +2ck(p−1)−1 (J s (X))) is at least W 2 (k). By Corollary 7.3 (i s ) * is an injection, so the dimension of (i s ) * ( kqmax N =kqmin π N +2ck(p−1)−1 (J s (X))) ⊂ kqmax N =kqmin π N +2ck(p−1)−1 (ΩΣX) is also at least W 2 (k).\nThus, ΣX satisfies the hypotheses of Lemma 3.3 with a = 2c(p − 1) + q max = 2c(p − 1) + max(q 1 , q 2 ) and b = 0, and hence is p-hyperbolic."},{"title":"Definition 5. 8 .","content":""},{"title":"Lemma 6. 1 .","content":""},{"title":"N","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"A space X is called rationally elliptic if π * (X) ⊗ Q is finite dimensional, and rationally hyperbolic if the dimension of i≤m π i (X) ⊗ Q grows exponentially in m. It was proved in [FHT15,Chapter 33] that simply connected CW -complexes with rational homology of finite type and finite rational category are either rationally elliptic or rationally hyperbolic. In order to study the p-torsion analogue of this dichotomy, Huang and Wu [HW20] introduced the definitions of Z/p r -and p-hyperbolicity.\\nFor p prime, by a p-torsion summand in an abelian group A, we mean a direct summand isomorphic to Z/p r for some r ≥ 1.\\nDefinition 1.1. Let X be a space, and let p be a prime. We say that X is p-hyperbolic if the number of p-torsion summands in π * (X) grows exponentially, in the sense that\\nwhere T m is the number of p-torsion summands in i≤m π i (X).\\nThe above definition counts Z/p r -summands for all values of r. It is also possible to consider only a single r, and by doing so we obtain the definition of Z/p r -hyperbolicity.\\nDefinition 1.2. Let X be a space, let p be a prime, and fix r ∈ Z + . We say that X is Z/p r -hyperbolic if the number of Z/p r -summands in π * (X) grows exponentially, in the sense that\\nwhere t m is the number of Z/p r -summands in i≤m π i (X).\\nNote that Z/p r -hyperbolicity for any r implies p-hyperbolicity. It follows immediately from a result of Henn [Hen86, Corollary of Theorem 1] that the lim infs appearing in the above definitions must be finite if X is a simply connected finite CW -complex.\\nHuang and Wu show that for n ≥ 3, r ≥ 1 and p any prime, the Moore space P n (p r ) is Z/p rhyperbolic and Z/p r+1 -hyperbolic, and that P n (2) is also Z/8-hyperbolic [HW20, Theorem 1.6]. More generally, they give criteria in terms of a functorial loop space decomposition due to Selick and Wu [SW00; SW06] for a suspension ΣX to be Z/p r -hyperbolic.\\nMore recently, Zhu and Pan [ZP21] use a classification of (n − 1)-connected CW -complexes of dimension at most n + 2, due to Chang [Cha50], to show that, for n ≥ 4, such a complex is Z/phyperbolic, provided that it is not contractible or a sphere after p-localization. They also prove hyperbolicity results for several so-called elementary Chang complexes.\\nThis paper studies p-and Z/p r -hyperbolicity of certain suspensions. Our first result is as follows.\\nTheorem 1.3. Let q 1 , q 2 ≥ 1. Then S q1+1 ∨ S q2+1 is Z/p r -hyperbolic for all primes p and all r ∈ Z + .\\nLet p be a prime. If a space X contains a wedge of two spheres as a p-local retract, then Theorem 1.3 implies that X is Z/p r -hyperbolic for all r. Various spaces have been shown to have such a wedge as a retract -examples of this sort are given in Section 2.1. A summary is as follows:\\n• for n, k ≥ 3, the configuration space Conf k (R n ) is Z/p r -hyperbolic for all p and r (Example 2.1);\\n• an (n − 1)-connected 2n-dimensional manifold M , where H n (M ) is of rank at least 3, is Z/p rhyperbolic for all p and r (Example 2.2);\\n• a generalized moment-angle complex on a simplicial complex having two minimal missing faces which are not disjoint is Z/p r -hyperbolic for all p and r (Example 2.3);\\n• ΣCP 2 is Z/p r -hyperbolic for all p = 2 and all r, and ΣHP 2 is is Z/p r -hyperbolic for all p = 2, 3 and all r (Example 2.4).\\nOur other result is as follows.\\nTheorem 1.4. Let p be an odd prime, and let X be a path connected space having the p-local homotopy type of a finite CW -complex. Suppose that there exists a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX with q i ≥ 1, such that the map\\nis a surjection. Then ΣX is p-hyperbolic.\\nThis criterion is quite different to that given by Huang and Wu [HW20, Theorem 1.5]. Their criterion is homotopical, using hypotheses on X to produce retracts of ΩΣX, whereas ours is cohomological, which makes it easier to check. On the other hand, their criterion is stronger, since it gives Z/p r -hyperbolicity, rather than just p-hyperbolicity. The examples they give, primarily various Moore spaces, differ from those we obtain, which are the suspensions of spaces related to complex projective space. More precisely, in Section 2.2, we show that the following spaces are p-hyperbolic for all p = 2:\\n• suspended complex projective space ΣCP n for n ≥ 2 (Example 2.5), and more generally;\\n• the suspended complex Grassmannian ΣGr k,n for n ≥ 3 and 0 < k < n (Example 2.6);\\n• the suspended Milnor Hypersurface ΣH m,n for m ≥ 2 and n ≥ 3, (Example 2.7);\\n• the suspended unitary group ΣU (n) for n ≥ 3 (Example 2.8).\\nBoth Theorem 1.3 and Theorem 1.4 will be proven by constructing an exponentially growing family of classes which generate summands in the relevant homotopy groups. We think of this family as 'witnessing' the hyperbolicity. For Theorem 1.3, one can proceed directly from the Hilton-Milnor decomposition of S n ∨S m [Hil55]. For Theorem 1.4, we employ K-theoretic methods originally used by Selick [Sel83] to prove one direction of Moore's conjecture for suspensions having torsion-free homology.\\nIf the map µ 1 ∨ µ 2 of Theorem 1.4 induces a surjection on K * ( ) ⊗ Z/p, then so does its suspension Σµ 1 ∨ Σµ 2 . The conclusion of Theorem 1.4 may therefore be strengthened in the following way.\\nCorollary 1.5. With the hypothesis of Theorem 1.4, Σ n X is p-hyperbolic for all n ≥ 1.\\nOne might be motivated by this observation to ask whether, in the circumstances of Theorem 1.4, the stable homotopy groups of X satisfy the growth conditions of Definition 1.1 or 1.2. In the proofs of both Theorem 1.3 and 1.4, the classes that witness the hyperbolicity are composites involving Whitehead products. The suspension of a Whitehead product is always trivial [Whi46,Theorem 3.11], so the classes we detect cannot be stable. Therefore, Corollary 1.5 does not suggest that the stable homotopy of ΣX should be p-or Z/p r -hyperbolic. On the other hand, it follows from our methods that, under the hypotheses of Theorem 1.4, the stable homotopy of ΩΣX is p-hyperbolic.\\nBy a result of Henn [Hen83], any co-H space, and in particular any suspension, decomposes rationally as a wedge of spheres. It then follows from the Hilton-Milnor theorem [Hil55] and the computation of the rational homotopy groups of spheres [Ser51] that such a suspension is rationally hyperbolic precisely when there are at least two spheres (of dimension ≥ 2) in this decomposition.\\nIf ΣX satisfies the hypotheses of Theorem 1.4 for any prime (including 2), then by Chern Character considerations the reduced rational homology of ΣX has dimension at least two, so ΣX is rationally a wedge of at least two spheres by the preceding discussion. This rational equivalence is a local equivalence at all but perhaps finitely many primes, so by Theorem 1.3, ΣX is Z/p r hyperbolic for all r at all but finitely many primes p. One might therefore conjecture that the conclusion of Theorem 1.4 can be strengthened to give Z/p r -hyperbolicity for all r rather than p-hyperbolicity, but we do not know whether this is possible.\\nWe now discuss situations in which it is adequate to consider ordinary cohomology, rather than K-theory. If ΣX has torsion-free integral (co)homology, or if its cohomology is concentrated in even degrees, then the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses on the E 2 page [Hus+08]. It follows by naturality that the image of the map induced by µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX on Ktheory is identified with the image of the induced map on cohomology. We may therefore replace K-theory with cohomology in Theorem 1.4, as follows.\\nCorollary 1.6. Let X be a path connected space having the homotopy type of a finite CW -complex, such that the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses on the E 2 page. Let p be an odd prime. Suppose that there exists a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX with q i ≥ 1, such that the map induced by µ 1 ∨ µ 2 on H * ( ) ⊗ Z/p is a surjection. Then ΣX is p-hyperbolic.\\nOne advantage of ordinary cohomology is that it is connected to the homotopy groups integrally, via the universal coefficient theorem and Hurewicz map. We can exploit this as follows.\\nExample 1.7. Suppose that the Atiyah-Hirzebruch spectral sequence for K * (ΣX) collapses (for example, if ΣX has torsion-free homology) and that there exists q ∈ Z + so that H i (ΣX) = 0 for i ≤ q, and dim Q ( H q+1 (ΣX) ⊗ Q) ≥ 2. The Hurewicz map π q+1 (ΣX) → H q+1 (ΣX) is an isomorphism, so there exists a map µ 1 ∨ µ 2 : S q+1 ∨ S q+1 → ΣX inducing the inclusion of a Z 2 -summand in H q+1 (ΣX). By the universal coefficient theorem relating ordinary homology and cohomology, µ 1 ∨ µ 2 induces a surjection on integral cohomology, so by Corollary 1.6, ΣX is p-hyperbolic for all odd primes p.\\nThe structure of this paper is as follows. In Section 2 we give applications of the main theorems and derive a simple lower bound for the growth of the number of Z/p r -summands in the homotopy groups of a wedge of two spheres (Corollary 2.9). The proofs of Theorems 1.3 and Theorem 1.4 may be read largely independently; Section 3 contains those preliminary results which are used in both cases. In Section 4, we give the proof of Theorem 1.3. The remainder of the paper is devoted to proving Theorem 1.4: Sections 5 and 6 give the necessary background, and Section 7 contains the proof. An overview of the proof strategy can be found at the start of Section 7.\\nI would like to thank my PhD supervisor, Stephen Theriault, for suggesting the problems that this paper tries to address, and for many helpful conversations along the way. From a technical point of view, much is owed to papers of Huang and Wu [HW20], and of Selick [Sel83]. Neil Strickland's 'Bestiary' was extremely helpful in providing examples of Theorem 1.4. Conversations with Sam Hughes were very useful in formulating Corollary 1.6. Thanks to the anonymous reviewer for their thoughtful comments, especially for pointing out that the space in Theorem 1.4 need only be finite after p-localisation.\\n2 Applications 2.1 Spaces having a wedge of two spheres as a retract Theorem 1.3 immediately implies that any space X which has S q1+1 ∨ S q2+1 as a retract after plocalization is Z/p r -hyperbolic for that p and all r. This implies that for all n ≥ 1, Σ n X contains S q1+n+1 ∨ S q2+n+1 as a p-local retract, and so is Z/p r -hyperbolic for all r. We first consider examples of this form.\\nExample 2.1. It is known [Knu18, Section 3.1] that Conf k (R n ), the ordered configuration space of k points in R n , contains k−1 S n−1 as a retract. It follows that, when n, k ≥ 3, Conf k (R n ) is Z/p rhyperbolic for all p and r.\\nExample 2.2. Let M be an (n − 1)-connected 2n-dimensional manifold. By the universal coefficient theorem, there can be no torsion in H n (M ). Suppose that the rank of H n (M ) is at least 3. By work of Beben and Theriault [BT14, Theorem 1.4], ΩM contains a wedge of two spheres as a retract after looping. Thus, M is again Z/p r -hyperbolic for all p and r.\\nExample 2.3. Let K be a simplicial complex on the vertex set [m] = {1, . . . , m}, and let (X, A) be any sequence of pairs (D ni , S ni−1 ) with n i ≥ 2 for 1 ≤ i ≤ m. If there exist two distinct minimal missing faces of K which are not disjoint, then by work of Hao, Sun and Theriault [HST19, Theorem 4.2] the polyhedral product (X, A) K contains a wedge of two spheres as a retract after looping, and hence is Z/p r -hyperbolic for all p and all r.\\nExample 2.4. Localized away from 2, ΣCP 2 ≃ S 3 ∨ S 5 . To see this, note that ΣCP 2 has a CWstructure consisting of one 3-cell and one 5-cell, and that π 4 (S 3 ) ∼ = Z/2 [Fre38]. This implies that the attaching map for the 5-cell is nullhomotopic after localization at an odd prime. Thus, ΣCP 2 is Z/p r -hyperbolic for all r when p = 2.\\nSimilarly, ΣHP 2 admits a cell structure with one 5-cell and one 9-cell, and π 8 (S 5 ) ∼ = Z/24. Thus, ΣHP 2 is Z/p r -hyperbolic for all r when p = 2, 3.\"\n },\n {\n \"title\": \"Suspensions of spaces related to CP n\",\n \"content\": \"Suppose that one has verified the hypotheses of Theorem 1.4 for a given space X and odd prime p, using a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX. If another space Y admits a map σ : ΣX → ΣY which induces a surjection on K * ( ) ⊗ Z/p, then it is immediate that σ • (µ 1 ∨ µ 2 ) satisfies the hypotheses of Theorem 1.4, and hence that ΣY is p-hyperbolic. The slogan is that K-theory surjections allow us to generate new examples from old ones.\\nIn this section, we will apply this idea. We have seen in Example 2.4 that, localized away from 2, ΣCP 2 ≃ S 3 ∨ S 5 , so certainly ΣCP 2 satisfies the hypotheses of Theorem 1.4 at all odd primes p. We will now consider spaces X which are known to admit maps CP 2 → X which induce surjections on integral K-theory, and hence on K * ( ) ⊗ Z/p for all odd p. It follows in each case that ΣX is p-hyperbolic, and further by Corollary 1.5, that Σ n X is p-hyperbolic for all n ≥ 1.\\nThe inclusion of CP n into CP n+1 induces a surjection on K-theory, so it must still induce a surjection after suspending. Composing these inclusions with the local equivalence ΣCP 2 ≃ S 3 ∨ S 5 gives, for each n ≥ 2, a map S 3 ∨ S 5 → ΣCP n which still induces a surjection on K * ( ) ⊗ Z/p for all odd primes p. Applying Theorem 1.4 to this map gives the following.\\nExample 2.5. For n ≥ 2, ΣCP n is p-hyperbolic for all p = 2. Now let Gr k,n be the Grassmannian of k-dimensional complex subspaces of C n . First note that orthogonal complement gives a homeomorphism Gr k,n ∼ = Gr n−k,n . In particular Gr n−1,n ∼ = Gr 1,n ∼ = CP n−1 , so ΣGr n−1,n is p-hyperbolic. Other Grassmannians can be treated more uniformly, as follows.\\nLet γ k,n denote the tautological bundle over Gr k,n . Consider the inclusion ι n : C n → C n+1 (x 1 , x 2 , . . . , x n ) → (x 1 , x 2 , . . . , x n , 0). This inclusion induces a map i k,n : Gr k,n → Gr k,n+1 , defined on V ∈ Gr k,n by V → ι n (V ). It follows from this definition that i * k,n (γ k,n+1 ) = γ k,n . Letting e i denote the i-th standard basis vector in C n , we also have a map j k,n : Gr k,n → Gr k+1,n+1 , defined on V = Span(v 1 , v 2 , . . . , v k ) ∈ Gr k,n by V → Span(ι(v 1 ), ι(v 2 ), . . . , ι(v k ), e n+1 ). It follows from this definition that j * k,n (γ k+1,n+1 ) = γ k,n ⊕ C 1 , where C 1 is the 1-dimensional trivial bundle.\\nSince K * (CP n ) is generated by the class of the tautological bundle, composing the maps i k,n and j k,n for different values of k and n will give maps CP 2 = Gr 1,3 → Gr k,n for all 1 ≤ k ≤ n − 2 and n ≥ 3 which induce surjections in integral K-theory. As in Example 2.5, this implies the following (the case k = n − 1 is Gr n−1,n , which was treated first).\\nExample 2.6. For n ≥ 3 and 0 < k < n, the suspended complex Grassmannian ΣGr k,n is p-hyperbolic for all p = 2.\\nFor m ≤ n, the Milnor Hypersurface H m,n is defined by\\nSuppose that m ≥ 2 and n ≥ 3. Then there is an inclusion ι : CP 2 → H m,n , defined by ι([z 0 : z 1 : z 2 ]) = ([z 0 : z 1 : z 2 : 0 : · · · : 0], [0 : · · · : 0 : 1]).\\nWrite π 1 for the projection H m,n → CP m . Then the inclusion CP 2 → CP m factors as\\nThis implies that ι induces a surjection on integral K-theory, so we obtain the following.\\nExample 2.7. For m ≥ 2 and n ≥ 3, the suspended Milnor Hypersurface ΣH m,n is p-hyperbolic for all p = 2.\\nLet U (n) denote the unitary group. There is a well-known map r : ΣCP n−1 → U (n) (see for example [Whi78]) which induces a surjection on K-theory. From this we obtain Example 2.8. For n ≥ 3, the suspended unitary group ΣU (n) is p-hyperbolic for all p = 2.\"\n },\n {\n \"title\": \"Quantitative lower bounds on growth\",\n \"content\": \"In Section 4, we will derive the following simple lower bound for the lim inf in the definition of Z/p rhyperbolicity, for the space S q1+1 ∨ S q2+1 .\\nCorollary 2.9. Let p be a prime and r ∈ Z + . Let t m be the constants of Definition 1.2 for X = S q1+1 ∨ S q1+1 . Then\\nThis implies that t m eventually exceeds ((1 − ε)2) m max(q 1 ,q 2 ) for any ε > 0. The constant 2 reflects the number of wedge summands. Note that this bound is independent of p and r.\\nExample 2.10. Taking ε = 1 4 , we find that for all r ∈ Z + and all primes p the number of Z/p r -summands in i≤m π i (S 2 ∨ S 2 ) eventually exceeds ( 3 2 ) m . One can produce an analogous quantitative bound on the lim inf in the case of Theorem 1.4, but this bound is very weak. In particular, it depends on knowledge of the Adams operations on K * (X), and is at best ln(2) 2(p−1) .\"\n },\n {\n \"title\": \"Preliminary results\",\n \"content\": \"Both Theorem 1.3 and Theorem 1.4 will be proven by means of Lemma 3.3. Our first goal is to establish this lemma. Let L be the free Lie algebra over Q on basis elements x 1 , . . . , x n . Write L k for the subset of L consisting of the basic products of the x i of weight k, in the sense of [Hil55], where the basic products of weight 1 are taken to be the x i , ordered by x 1 < x 2 < · · · < x n . The union L = ∞ k=1 L k is a vector space basis for L (see for example [Ser06,Theorem 5.3], but note that Serre uses the name Hall basis for the set of basic products).\\nLet µ : Z + −→ {−1, 0, 1} be the Möbius inversion function, defined by\\nTheorem 3.1. [Hil55, Theorem 3.3] Let L be the free Lie algebra over Q on basis elements x 1 , . . . , x n . Then |L k | = W n (k).\\nLemma 3.2. [BO15, Introduction] The ratio W n (k) 1 k n k tends to 1 as k tends to ∞.\\nIn particular, this implies that for n ≥ 2, the Witt function W n (k) grows exponentially in k. It should follow that if the number of p-torsion summands in i≤k π i (Y ) exceeds W 2 (k), then Y is p-hyperbolic. The following lemma makes a slightly generalised form of this idea precise.\\nLemma 3.3. Let Y be a space. Suppose that there exist a, b ∈ Z + such that the number of p-torsion summands (respectively, Z/p r -summands) in i≤ak+b π i (Y ) exceeds W 2 (k), for all k large enough. Then Y is p-hyperbolic (respectively, Z/p r -hyperbolic).\\nProof. The proofs for p-and Z/p r -hyperbolicity are identical, so we give only the former. Reframing the hypothesis in terms of the sequence {T m } m of Definition 1.1, we are assuming precisely that T ak+b > W 2 (k) for sufficiently large k. We then have that\\nIt then follows from Lemma 3.2 that if 1 > ε > 0, once k is large enough, we have\\nThis implies that\\nand since this holds for all ε > 0,\\nwhich is greater than zero, as required.\"\n },\n {\n \"title\": \"Existence of summands in the stable stems\",\n \"content\": \"We write π S j for the j-th stable stem in the homotopy groups of spheres, that is\\nThe proof of Theorem 1.3, depends on having, for each p and r, some j such that π S j contains a Z/p r -summand. The purpose of this subsection is to show that the existence of such a j follows from existing work of Adams and others.\\nLemma 3.4. For any prime p and any r ∈ Z + , there exists j such that Z/p r is a direct summand in π S j . That is, for a fixed choice of such a j, Z/p r is a direct summand in π n+j (S n ) for all n ≥ j + 2. Proof. We write ν p (s) for the largest power of p dividing the integer s.\\nCASE 1 (p odd): Set t := p r−1 (p − 1), and notice that, since (p − 1) is even, j := 4t − 1 is congruent to 7 mod 8. Theorem 1.6 of [Ada66], and the discussion immediately following it, then tells us that π S j contains a direct summand isomorphic to Z/m(2t), for a function m which Adams defines. By decomposing this subgroup into direct summands of prime power order, it suffices to show that ν p (m(2t)) = r.\\nThe discussion after Theorem 2.5 in [Ada65] gives that since t ≡ 0 mod (p − 1),\\nNow, ν p (2t) is equal to (r − 1), by definition of t, so ν p (m(2t)) = r, as required. CASE 2 (p = 2, r ≥ 3): Set t := 2 r−3 , and set j := 4t − 1. From Theorem 1.5, and the discussion following Theorem 1.6 in [Ada66], π S j has a direct summand isomorphic to Z/m(2t), regardless of whether j is congruent to 3 or 7 mod 8. Again, referring to the discussion after Theorem 2.5 of [Ada65], we see that ν 2 (m(2t)) = 2 + ν 2 (2t) = 3 + ν 2 (t) = r, as required. CASE 3 (p r = 2 and p r = 4): It is known from [Fre38]\"\n },\n {\n \"title\": \"Proof of Theorem 1.3\",\n \"content\": \"In this section we prove Theorem 1.3, which says that the wedge of two spheres is Z/p r -hyperbolic for all p and r. We also prove Corollary 2.9, which extracts some simple quantitative information from the proof of Theorem 1.3. We first record the following simple observation.\\nRemark 4.1. Let k 1 , . . . , k n and q 1 , . . . q n be non-negative integers. Suppose that q 1 ≤ q 2 ≤ · · · ≤ q n , and let k = n i=1 k i . Then\\nProof of Theorem 1.3. Assume without loss of generality that q 1 ≤ q 2 . By Lemma 3.3 it suffices to show that there exist constants a and b such that the number of Z/p r -summands in\\nWe first apply the Hilton-Milnor Theorem. Since we are dealing with spheres, we need only the original form, due to Hilton in [Hil55]:\\nwhere, as in Section 3, L = ∞ k=1 L k is Hilton's 'basic product' basis for L, the free Lie Algebra over Q on two generators x 1 and x 2 , and k i is the number of occurrences of the generator x i in the bracket B. Recall also from Section 3 that the weight k of a bracket B is equal to k 1 + k 2 , and that the cardinality of L k is given by the Witt formula W 2 (k) by Theorem 3.1.\\nFor fixed k ∈ Z + , consider the factor in the above product corresponding to L k ⊂ L :\\nThe associated subgroup of π * (S q1+1 ∨ S q2+1 ),\\nis a direct summand. We will first find a Z/p r -summand in the homotopy groups of each of the spheres appearing in F k . Since q 1 ≤ q 2 , Remark 4.1 applies, and we may lower bound the dimensions of spheres appearing in F k by k 1 q 1 + k 2 q 2 + 1 ≥ kq 1 + 1. By Lemma 3.4, there exists j ∈ Z + such that π j+ℓ (S ℓ ) has a direct summand Z/p r for ℓ ≥ j +2. Therefore, if k is large enough that kq 1 ≥ j +1, then k 1 q 1 +k 2 q 2 +1 ≥ j +2 -that is, the j-th stem is stable on all of the spheres occurring in F k . Thus, for k large enough, there is a Z/p r summand in π j+k1q1+k2q2+1 (S k1q1+k2q2+1 ) whenever k 1 + k 2 = k.\\nWe now upper bound the dimension of the homotopy groups in which these summands appear. Since q 1 ≤ q 2 we have by Remark 4.1 that j+k 1 q 1 +k 2 q 2 +1 ≤ kq 2 +1+j, so each of the Z/p r -summands we have identified is a distinct direct summand in\\nWe have identified one such summand for each B ∈ L k , so the number of Z/p r -summands in\\n. Thus, taking a = q 2 and b = 1 + j in Lemma 3.3 suffices.\\nProof of Corollary 2.9. The last line of the proof of Lemma 3.3 shows that lim inf m ln tm m > ln 2 a . The last line of the proof of Theorem 1.3 implies that a may be taken to be q 2 , under the assumption that q 1 ≤ q 2 , which implies the result.\"\n },\n {\n \"title\": \"K-theory and K-homology of ΩΣX\",\n \"content\": \"The remainder of this paper proves Theorem 1.4. Sections 5 and 6 assemble necessary background, which we will use in Section 7 to prove the result.\\nWhen studying the homotopy groups of a suspension ΣX, as in Theorem 1.4, the following approach is natural. Since π * (ΣX) ∼ = π * −1 (ΩΣX), we may instead study ΩΣX. This is useful because ΩΣX is well understood homologically via the Bott-Samelson theorem, which decomposes its homology as the tensor algebra on H * (X). Because we will need to use Adams' e-invariant, which is defined in terms of K-theory, we wish to replace ordinary homology with K-homology.\\nThe purpose of Section 5 is to record the version of the Bott-Samelson theorem which applies to (torsion-free) K-homology, along with a universal coefficient theorem for passing between K-theory and K-homology. All of the material here is already known (in particular much of it is in [Sel83]) so its summary here is for convenience and clarity.\\nOur conventions on definition of K * (X) are those of [AH61]. In particular, we define K −1 (X) := K 0 (ΣX), and set K * (X) := K 0 (X) ⊕ K −1 (X). We regard K * (X) and K * (X) as being Z/2-graded.\\nWe will wish to work with K-theory and K-homology modulo the torsion subgroup. For a space X, write K TF * (X) and K * TF (X) for the quotients of the reduced K-homology and K-theory of X by their torsion subgroups. The same convention applies in the unreduced case.\"\n },\n {\n \"title\": \"Künneth and universal coefficient theorems\",\n \"content\": \"The universal coefficient theorem for K-theory first appears in some unpublished lecture notes of Anderson [And], and is first published by Yosimura [Yos75].\\nTheorem 5.1 (Universal coefficient theorem). For any CW -complex X and each integer n there is a short exact sequence\\nIn the torsion-free case, the universal coefficient theorem is as follows, where, unsurprisingly, we write Ext( K n−1 (X), Z) TF for the quotient of Ext( K n−1 (X), Z) by its torsion subgroup.\\nCorollary 5.2.\\n1. For any CW -complex X and each integer n there is a short exact sequence\\n2. If X is a finite CW -complex, then Ext( K n−1 (X), Z) TF = 0, and we obtain an isomorphism\\nProof. Let T 1 denote the torsion subgroup of Ext( K n−1 (X), Z), and let T 2 be the torsion subgroup of K n (X). The Universal Coefficient Sequence of Theorem 5.1 gives an injection T 1 → K n (X), which must have image contained in T 2 , thus lift to an injection T 1 → T 2 . For any group G, Hom(G, Z) is torsion-free, so the composite T 2 → K n (X) → Hom(K n (X), Z) is trivial, and by exactness we obtain a lift T 2 → Ext( K n−1 (X), Z). The image of this map must be torsion, which is to say that it must be contained in T 1 , so the aforementioned map T 1 → T 2 is a surjection. That is, the Universal Coefficient Sequence of Theorem 5.1 has last term torsion-free and first map restricting to an isomorphism of torsion subgroups. This implies the first statement.\\nFor the second statement, we need only note that if X is finite, then K TF n−1 (X) is finitely generated, so Ext( K n−1 (X), Z) is torsion, as required.\\nSelick [Sel83] deduces the following from work of Atiyah [Ati62], Mislin [Mis71] and Adams [Ada69].\\nTheorem 5.3 (Künneth theorem for K-homology). Let X and Y be of the homotopy type of finite complexes. Then there is an isomorphism of Z/2-graded Z-modules:\\nRemark 5.4. It follows immediately from Corollary 5.2 (and knowledge of K * (S q )) that K TF * (S q ) ∼ = Z. We write ξ q for the generator of K TF * (S q ). By the Künneth Theorem (Theorem 5.3), we may choose the ξ q so that ξ n ⊗ ξ m is identified with ξ n+m under the homeomorphism S n ∧ S m ∼ = S n+m .\\nIn the case of K-theory, the analogous result follows directly from [Ada69].\\nTheorem 5.5 (Künneth theorem for K-theory). Let X and Y be of the homotopy type of finite complexes. Then the external product on K-theory defines an isomorphism of Z/2-graded rings:\"\n },\n {\n \"title\": \"The James construction\",\n \"content\": \"For a space X, let X s denote the product of s copies of X. Let ∼ be the relation on X s defined by\\nThe James construction JX is defined to be the colimit of the diagram consisting of the spaces J s (X) and the above inclusions. Write i s : J s (X) → JX for the map associated to the colimit. Notice that JX carries a product given by concatenation, which makes it into the free topological monoid on X, and that a topological monoid is in particular an H-space.\\nLet X ∧i denote the smash product of i copies of X, and let η : X → ΩΣX be the unit of the adjunction Σ ⊣ Ω. Explicitly, η(x) = (t → x, t ∈ ΣX).\\nTheorem 5.6. [Jam55] 1. There is a homotopy equivalence JX ≃ − → ΩΣX which respects the H-space structures and identifies i 1 with η.\"\n },\n {\n \"title\": \"There is a homotopy equivalence\",\n \"content\": \"Lemma 5.7. [Sel83, Lemma 7] Let X have the homotopy type of an (r − 1)-connected CW -complex.\\n2. Let x ∈ π N (J s (X)) for any N . If Σx is nontrivial then (i s ) * (x) is also nontrivial.\\nProof. The first part follows by cellular approximation from the observation that J s (X) contains the (r(s + 1) − 1)-skeleton of JX. The second part follows from the observation that Σi s has a retraction by Theorem 5.6.\\nFor spaces X and Y , let X * Y denote the join, which we define to be the homotopy pushout of the projections X × Y → X and X × Y → Y . The join is naturally a quotient of X × I × Y , where I denotes the unit interval. Following the treatment in [Ark11], let C 1 denote the subspace of X * Y consisting of points of the form (x, t, * ), for t ∈ I and x ∈ X, and let C 2 be the subspace consisting of points of the form ( * , t, y). The subspace\\nThis gives a factorization of q as\\n. Let q −1 denote any choice of homotopy inverse to q; all possible choices are homotopic. We may form a new map δ X,Y as the\\nThe homotopy class of δ X,Y is well-defined, and we will call δ X,Y the canonical splitting of π. Note that δ X,Y is natural in maps of spaces in the sense that given f :\\nFor s ≥ 3, consider the quotient map ΣX s → ΣX ∧s . We define the canonical splitting of this quotient to be the composite of canonical splittings\\nOf course, we chose an order of multiplication here. This canonical splitting is natural as before. Write m s : (ΩΣX) s → ΩΣX for the map given by iteratively performing the standard loop multiplication on ΩΣX in any choice of order. Up to homotopy, m s is independent of this choice of order, since ΩΣX is homotopy associative.\\nTheorem 5.6 gives the existence of a homotopy equivalence Γ :\\nThere are many choices of Γ, up to homotopy. The next lemma asserts that Γ can be chosen in a way which suits our purpose. Selick [Sel83] describes the composite\\nwith the homotopy equivalence of Theorem 5.6 (1). This immediately implies the following description of Γ.\\nLemma 5.10. [Sel83] Let X be a finite CW -complex. The homotopy equivalence Γ : ∞ i=1 ΣX ∧i → ΣΩΣX may be chosen such that:\\nis an isomorphism of algebras; 2. the restriction of Γ to ΣX ∧s is homotopic to the composite\\nwhere the unlabelled arrow is the canonical splitting.\\nThe description of the map Γ in Lemma 5.10 has the following consequence. For a space Y , let ev : ΣΩY → Y be the evaluation map, which may be described explicitly by ev( γ, t ) = γ(t) for γ ∈ ΩY .\\nLemma 5.11. Let Γ : ∞ i=1 ΣX ∧i → ΣΩΣX be the homotopy equivalence of Lemma 5.10. The composite ev • Γ is homotopic to the projection onto the first wedge summand.\\nProof. Let ι s : ΣX ∧s → ∞ i=1 ΣX ∧i be the inclusion of the s-th wedge summand. We must show that ev • Γ • ι s ≃ 1 ΣX if s = 1, and * otherwise.\\nThe following diagram commutes up to homotopy\\nThe composite along the bottom of the diagram is Γ • ι s , so to obtain the desired factorization of Γ • ι s through v, it suffices to show that the diagram commutes up to homotopy.\\nThe top right square commutes because m 2 • m (s−1) ≃ m s , by homotopy associativity of the Hspace ΩΣX. The remaining three squares commute by naturality of our canonical splitting. This completes the proof.\\nLet ρ k be the projection T ( K TF * (X)) → K TF * (X) ⊗k . The next corollary is immediate from Lemma 5.11.\\nCorollary 5.12. S −1 (ev * • Γ * ) = ρ 1 : T ( K TF * (X)) → K TF * (X).\"\n },\n {\n \"title\": \"Primitives and commutators\",\n \"content\": \"It follows from the Künneth Theorem (Theorem 5.3), and the fact that\\n. We may therefore make the following definition. A class\\nFrom this point of view, the following lemma is immediate.\\nLemma 5.13. If Y is a co-H-space, then all elements in K TF * (Y ) are primitive.\\nIf Y is an H-group, then the multiplication m :\\n. We will denote this map by juxtaposition, so that m * (y 1 ⊗ y 2 ) = y 1 y 2 . Furthermore, the commutator Y ×Y → Y descends to a map c : Y ∧Y → Y . Expanding the definition of the commutator in terms of the K-homology Künneth Theorem (Theorem 5.3) gives the following lemma.\\nLemma 5.14. Let Y be an H-group, and let c : Y ∧Y → Y be the commutator. If y 1 and y 2 ∈ K TF * (Y ) are primitive, then c * (y 1 ⊗ y 2 ) = y 1 y 2 − (−1) |y1||y2| y 2 y 1 .\"\n },\n {\n \"title\": \"The category of ψ-modules\",\n \"content\": \"In [Ada66], Adams defines an abelian category which we will follow Selick [Sel83] in calling ψ-modules. The e-invariant, which is our central tool, is defined by Adams in terms of ψ-modules. The purpose of this section is to record results about ψ-modules for later use.\\nA ψ-module consists of an abelian group M , with homomorphisms\\nfor each ℓ ∈ Z, satisfying the axioms of [Ada66, Section 6]. If X is a space then the group K 0 (X), together with its Adams operations, is a ψ-module. Since we defined K −1 (X) by setting K −1 (X) = K 0 (ΣX), it too has the structure of a ψ-module. Maps of spaces induce maps of ψ-modules. The Adams operation ψ ℓ on K 0 (S 2n ) is multiplication by ℓ n , so in particular Adams operations do not commute with the Bott isomorphism. For graded ψ-modules M and N we will write Hom ψ−Mod (M, N ) for the abelian group consisting of graded ψ-module homomorphisms. The unadorned notation Hom(M, N ) will mean homomorphisms of the underlying graded abelian groups. Proof. Let ϕ : M → N be a homomorphism of underlying Z-modules. If, for some k ∈ Z \\\\ {0}, k · ϕ is a ψ-module homomorphism, then, since N is torsion-free, ϕ is also a ψ-module homomorphism. This implies that Coker(Hom ψ−Mod (M, N ) ֒− → Hom(M, N )) is torsion-free, which implies the result. Definition 6.2 (Adams' e-invariant). Suppose that f : X → Y induces the trivial map on K * . Then the cofibre sequence of f gives a short exact sequence of ψ-modules\\nThe e-invariant of f is the element of Ext ψ−Mod ( K 0 (Y ), K 0 (ΣX)) represented by this exact sequence.\\nThe e-invariant does not commute with the Bott isomorphism, but the interaction between the Bott isomorphism and the Adams operations is easy to describe, as follows. Let ψ ℓ Y be the homomorphism ψ ℓ : K 0 (Y ) → K 0 (Y ). Then, modulo the Bott isomorphism, we have ψ ℓ Σ 2 X = ℓ · ψ ℓ X . That is 'upon double suspending, the Adams operations gain a factor ℓ'. In terms of the e-invariant, all we need to know is the following.\\nThere is a homomorphism\\nsuch that T (e(f )) = e(Σ 2 f ).\\nWe will be concerned only with the e-invariants of maps whose domain is a sphere. One of the two K-groups of a sphere vanishes, in the dimension matching the parity of the sphere, but the e-invariant, as defined above, lives only in K 0 . In order to detect maps regardless of the parity of the sphere on which they are defined, we will need to keep track of the e-invariants of f and Σf , so we will use the following modified e-invariant.\"\n },\n {\n \"title\": \"Definition 6.4 (Double e-invariant). Let\",\n \"content\": \"Suppose that f : X → Y induces the trivial map on K * . Then the double e-invariant of f is e(f ) = (e(f ), e(Σf )) ∈ Ext ψ−Mod ( K * (Y ), K * (ΣX)).\"\n },\n {\n \"title\": \"Pullback of an extension along a homomorphism defines a map\",\n \"content\": \"If g : Y → Z then e(g • f ) is represented by the pullback of e(f ) and g * : K 0 (Z) → K 0 (Y ) [Ada66, Proposition 3.2 b)]. To describe e(g • f ) we need only apply this result degree-wise, as follows. For convenience, we write g * · e(f ) for the pullback of g * and e(f ). Define the map\\nCombining these, let\\nbe the direct sum θ 0 (f ) ⊕ θ −1 (f ). These definitions, together with Adams' above result, give the following lemma.\\nLemma 6.5. For maps f : X → Y and g : Y → Z, the following diagram commutes:\\nFollowing [Sel83], write Z(n) for the ψ-module K 0 (S 2n ). Explicitly, Z(n) has underlying abelian group Z, and ψ ℓ acts by multiplication by ℓ n . It follows that K −1 (S 2n+1 ) := K 0 (S 2n+2 ) ∼ = Z(n + 1).\\nLemma 6.6. [Ada66, Proposition 7.8, 7.9] If n < m then Ext ψ−Mod (Z(n), Z(m)) injects into Q Z . The e-invariant of a map f : S 2m−1 → S 2n may therefore be regarded as an element of Q Z . Furthermore, the value e(f ) in Q Z satisfies e(Σ 2 f ) = e(f ), so in particular, when f is a map between spheres, e(f ) depends only on the stable homotopy class of f .\\nThe following theorem is the main technical component of Selick's paper [Sel83].\\nTheorem 6.7. [Sel83, Theorem 6] Let f ′ : S 2m−1 → S 2n be such that p t−1 e(f ′ ) = 0 in Q Z , for p prime and some t ∈ Z + . Let Y have the homotopy type of a finite CW -complex and let g : S 2n → Y be such that Im(g * : K 0 (Y ) → K 0 (S 2n )) contains up s K 0 (S 2n ), for s ∈ Z + and u prime to p. If s < t, and there exists some ℓ ∈ Z + for which\\ndoes not have ℓ m as an eigenvalue, then e(g • f ′ ) = 0.\\nThe following theorem of Gray [Gra69] will provide the map f ′ for Theorem 6.7. Specifically, this theorem provides a linearly spaced family of stems, each of which has a stable p-torsion class which is born on S 3 and detected by the e-invariant.\\nTheorem 6.8. [Gra69, Corollary of Theorem 6.2] Let p be an odd prime and let j ∈ Z + . Then there exists a class f p,j ∈ π 2j(p−1)+2 (S 3 ) with e(f p,j ) = −1 p ∈ Q Z .\\nThe corresponding 2-primary result is as follows. Adams [Ada66, Theorem 1.5 and Proposition 7.14] shows that, for j > 0, the (8j +3)-rd stem contains a direct summand whose 2-primary component has order 8, and that on this component the e-invariant is a surjection onto Z/4. The sphere of origin of the classes in this component was deduced by Curtis in [Cur69].\\nTheorem 6.9. [Ada66; Cur69] Let j ∈ Z + . Then there exists a class f 2,j ∈ π 8j+6 (S 3 ) of order 4, with e(f 2,j ) = −1 2 ∈ Q Z .\"\n },\n {\n \"title\": \"Main construction\",\n \"content\": \"Having assembled preliminaries in Sections 5 and 6, we can begin to work towards the proof of Theorem 1.4. Our approach is as follows. From the data of Theorem 1.4, we will construct a commutative diagram of (roughly) the following form, where B is a set and the other objects are Z-modules.\\nWe will argue that\\n• The image of the top map consists of classes of order dividing p.\\n• The image of the left vertical map generates a submodule isomorphic to the weight k component of the free graded Lie algebra over Z/p on two generators.\\n• The bottom map is injective.\\nTogether, these facts imply that there is a submodule of π * (ΩΣX) ∼ = π * +1 (ΣX), consisting of classes of order dividing p, and surjecting onto a module isomorphic to the weight k component of the free graded Lie algebra over Z/p on two generators. This submodule (which is necessarily a Z/p-vector space) must therefore have dimension at least W 2 (k) (Theorem 7.5), which will imply that ΣX is p-hyperbolic (Lemma 3.3).\\nThe diagram will be obtained by juxtaposing three squares. Subsections 7.1, 7.2, and 7.3 each construct one of these squares. In Subsection 7.4 we put them together and prove Theorem 1.4. Roughly speaking, the top map of the diagram should be thought of as first taking a family of Samelson products and then pulling them back along some suitable map f coming from Gray's work (Theorem 6.8). The vertical maps should be thought of as passing from maps of spaces to K-theoretic invariants, and the bottom map (therefore) should be thought of as tracking the effect of the top map on these invariants.\\nBecause of the need to work with a finite CW -complex in Selick's Theorem (Theorem 6.7) we will restrict the right hand side of the diagram to instead refer to some finite skeleton J s (X) of the James construction.\"\n },\n {\n \"title\": \"Samelson products and their Hurewicz images in K-homology\",\n \"content\": \"Let R be a commutative ring with unit. We take a graded Lie algebra over R to be defined as in [Nei10]. For a non-negatively graded R-module V , let L(V ) denote the free graded Lie algebra [Nei10, Section 8.5]. Write L k (V ) for the submodule of L(V ) generated by the brackets of length k in the elements of V . We will call L k (V ) the weight k component of L(V ). Note that this convention differs from Neisendorfer's -he writes L(V ) k for the weight k component.\\nDefinition 7.1. Let Y be an H-group, and let c : Y ∧ Y → Y be the commutator of Lemma 5.14. Let α ∈ π N (Y ), and let β ∈ π M (Y ). The Samelson product of α and β, written α, β ∈ π N +M (Y ), is the composite α, β :\\nSamelson products are bilinear, graded anticommutative, and satisfy the graded Jacobi identity. They fail, however, to make π * (Y ) into a graded Lie algebra over Z in Neisendorfer's sense [Nei13,Section 7]. This is important because we want the natural map from the free Lie algebra to the corresponding tensor algebra to be an injection onto a summand (Lemma 7.4). One could define an appropriate notion of 'free graded pseudo-Lie algebra', and proceed as follows with that in place of the set B(π * (A)), which we use in what follows, but we prefer to avoid making the extra definition. which extends ν * and satisfies Φ π ν ([x, y]) = Φ π ν (x), Φ π ν (y) for all x, y ∈ B(π * (A)). For a Z/2-graded Z-module V , we define a non-negatively graded Z-module Hom( K TF * (S * ), V ), by setting\\nwhere the homomorphisms are understood to respect the Z/2-grading on K * and V .\\nIn the case that V = L is a Z/2-graded Lie algebra over Z, Hom( K TF * (S * ), L) inherits a nonnegatively graded Lie algebra structure as follows. Let the generators ξ N of K TF * (S N ) be as in Remark 5.4. Then the bracket [f, g] of f ∈ Hom( K TF * (S N ), L) and g ∈ Hom( K TF\\nThe squaring operation is defined in the same way. Likewise, if V is a Z/2-graded associative algebra over Z, then Hom( K TF * (S * ), V ) inherits the structure of a non-negatively graded associative algebra.\\nLet ν : A → ΩΣX be a map. There is a composition\\nwhere the first map is the natural map which is the identity on K TF * (A) and satisfies [x, y] → xy − (−1) |x||y| yx, and the second map is obtained by applying the universal property of the tensor algebra to ν * . Let Φ K ν : Hom( K TF * (S * ), L( K TF * (A))) → Hom( K TF * (S * ), K TF * (ΩΣX)) be the pushforward along the above composite. It is then automatic that Φ K ν is a map of non-negatively graded Lie algebras over Z, where the structures are defined as above.\\nWe write deg : π N (Y ) → Hom( K TF * (S N ), K TF * (Y )) for the map f → f * . Let deg ′ : B(π * (A)) → Hom( K TF * (S * ), L( K TF * (A))) be the unique map which restricts to deg : π * (A) → Hom( K TF * (S * ), K TF * (A)) ⊂ Hom( K TF * (S * ), L( K TF * (A))) and carries brackets to brackets. The above maps are related as follows. Lemma 7.2. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CWcomplexes. The following diagram commutes:\\nProof. By construction of B(π * (A)), it suffices to show that the restriction of the diagram to the weight 1 component B 1 (π * (A)) = π * (A) commutes, and that all maps respect the bracket operations. By definition, L 1 ( K TF * (A)) = K TF * (A). It then follows immediately from the definitions of Φ π ν and Φ K ν that restricting the left hand side of the diagram to weight 1 components gives the diagram\\n, K TF * (ΩΣX)), which commutes, since it just expresses naturality of deg.\\nIt remains to show that all maps respect bracket operations. The maps Φ π ν and deg ′ respect the bracket operations by definition, and Φ K ν respects bracket operations by construction. We therefore only need show that deg respects brackets. Let f ∈ π N (ΩΣX), and let g ∈ π M (ΩΣX). We must show that deg( f, g ) is the commutator deg(f )deg(g) − (−1) N M deg(g)deg(f ) with respect to the algebra operation on Hom( K TF * (S * ), K TF * (ΩΣX)). Since K TF * (S N +M ) ∼ = Z, it suffices to show that the two homomorphisms agree on the generator ξ N +M (Remark 5.4). By Definition 7.1 and the Künneth Theorem (Theorem 5.3),\\nSpheres of dimension at least 1 are co-H spaces, so by Lemma 5.13, ξ N and ξ M are primitive. By naturality of the diagonal f * (ξ N ) and g * (ξ M ) are still primitive, so by Lemma 5.14,\\nwhich by definition of the multiplication on Hom( K TF * (S * ), K TF * (ΩΣX)) is the result of evaluating deg(f )deg(g) − (−1) N M deg(g)deg(f ) on ξ N +M , as required.\\nWe now lift the previous result to J s (X), thereby producing the first square of the diagram promised at the start of this section. Recall that we write i s : J s (X) → JX for the inclusion, and that by Theorem 5.6 we have a homotopy equivalence JX ≃ − → ΩΣX. We will abuse notation and also write i s for the composite J s (X) → JX ≃ − → ΩΣX.\\nCorollary 7.3. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CWcomplexes, with X (r − 1)-connected for r ≥ 1. If N, s ∈ Z + satisfy N < r(s + 1) − 1, then (i s ) * : π N (J s X) → π N (ΩΣX) is an isomorphism and for each k ≤ s there exists a commutative diagram:\\nProof. Consider the diagram of Lemma 7.2. Lemma 5.7 shows that (i s ) * is an isomorphism on π N , so let Φ π ν be the unique map such that the condition (i s ) * • Φ π ν = Φ π ν holds. By Theorem 5.6 (2) and Lemma 5.10, the map (i s ) * : K TF N (J s (X)) → K TF N (ΩΣX) is the inclusion of the tensors of length at most s. Since k ≤ s, we may therefore define Φ K ν to be the unique map such that the condition\\nCommutativity then follows from Lemma 7.2 by naturality of deg, since Hom( K TF * (S N ), (i s ) * ) is injective.\\nLemma 7.4. Let V be a non-negatively-or Z/2-graded Z-module which is free and finitely generated in each dimension. Then\\n• L(V ) and T (V ) are free Z-modules in every dimension.\\n• The natural map L(V ) → T (V ), [x, y] → xy − (−1) |x||y| yx is an injection onto a summand.\\nProof. The non-negatively graded case is immediate from [Nei10], Proposition 8.3.1 and p282. For the Z/2-graded case, first observe that there is a forgetful functor U from Z-graded modules to Z/2-modules which carries Z-graded (Lie) algebras to Z/2-graded (Lie) algebras, and a functor C from Z/2-graded modules to Z-modules which puts V 0 in any even dimension and V 1 in any odd dimension. Both C and U respect freeness and split injections, and there are natural isomorphisms U T (CV ) ∼ = T (V ) and U L(CV ) ∼ = L(V ). This implies the Z/2-graded result.\\nThe graded version of Theorem 3.1 now follows immediately from Hilton's paper:\\nTheorem 7.5. [Hil55, Theorem 3.2, 3.3] Let V be a torsion-free Zor Z/2-graded Z-module of total dimension n. Then the total dimension of L k (V ) is W n (k).\\nLet R be a commutative ring with unit. Let M be an R-module, and as usual let T (M ) denote the tensor algebra on M . Let ι k : M ⊗k → T (M ) be the inclusion, and let ρ k : T (M ) → M ⊗k be the projection. Let τ : T (M ) → T (M ) be the composite ι 1 • ρ 1 . Given an R-algebra A, and a map ϕ : M → A, we write ϕ for the induced map T (M ) → A, that is, the unique map of algebras such that ϕ • ι 1 = ϕ. Now, let M and N be R-modules, and let ϕ : M → T (N ) be a map. In the proof of Theorem 7.7, we will wish to make a 'leading terms' style argument. This is made precise in the next Lemma, which compares ϕ with τ • ϕ. Informally, we think of τ • ϕ as the 'leading terms part' of ϕ.\\nLemma 7.6. Let R be a commutative ring with unit. Let M and N be Zor Z/2-graded R-modules. Let ι k : M ⊗k → T (M ) be the inclusion, let ρ k : T (N ) → N ⊗k be the projection, and let τ : T (N ) → T (N ) be as above. Let ϕ : M → T (N ) be a map. Then\\nProof. It suffices to check equality on basic tensors. Let v ∈ M ⊗k be a basic tensor, so that\\nApplying ρ k to both sides yields the result.\\nTheorem 7.7. Let F = Q or Z/p for p prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes. Let ν : ΣA → ΣX be the adjoint of ν. If\\nis also an injection.\\nRemark 7.8. In the case where ν is a suspension Σζ, we have a diagram\\nso in particular ν * factors through the weight 1 component K TF * (X) of the tensor algebra decomposition of K TF * (ΩΣX). This dramatically simplifies the proof, removing the need for Lemma 7.6. In practice this is not a reasonable assumption -for example, the map µ : S 3 ∨ S 5 → ΣCP 2 of Example 2.5 (which plays the role of ν) does not desuspend.\\nProof. In this proof, for a space Y , we will identify the algebras T ( K TF * (Y )) and K TF * (ΩΣY ), omitting the isomorphism S −1 Γ * of Lemma 5.10. We defined Φ K ν to be the pushforward along a certain map L( K TF * (A)) → K TF * (ΩΣX). Call this map Φ K ν ′ . It suffices to prove that Φ K ν ′ ⊗ F is an injection.\\nThe triangle identities for the adjunction Σ ⊣ Ω give a commutative diagram\\n′ is the unique map of Lie algebras extending ν, we have a commuting diagram\\nwhere we note that that by Lemma 7.4, the natural map L( K TF * (A)) → T ( K TF * (A)) is an injection onto a summand. It therefore suffices to show that (Ων) * ⊗ F is an injection.\\nLet (ν * ) denote the extension of ν * to T ( K TF * (A)), so that (ν * ) = (Ων) * (modulo the isomorphism S −1 Γ * , as above). Since (ρ i • (ν * ) • ι k ) = 0 for i < k, it further suffices to show that (ρ k • (ν * ) • ι k ) ⊗ F is an injection for each k. By Lemma 7.6, with M = K TF * (A) and N ′ = K TF * (X), we have that\\nAs previously, let ev : ΣΩY → Y denote the evaluation map. The following diagram commutes:\\nThe hypothesis therefore implies that the composite (ev * • Σν * ) ⊗ F is an injection. Desuspending, we have that (S −1 ev * • ν * ) ⊗ F is an injection. By Lemma 5.12,\\nThe following corollary, which lifts the injectivity back to J s (X), is immediate from Theorem 7.7 and Lemma 7.3.\\nCorollary 7.9. Let F = Q or Z/p for p prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r − 1)-connected for r ≥ 1. Suppose that N, s, k ∈ Z + satisfy k ≤ s, so that the map Φ K ν is as in Corollary 7.3. If\\nis also an injection.\\nWe have now established all that we will need to know about this 'first square'.\"\n },\n {\n \"title\": \"Maps derived from the universal coefficient isomorphism\",\n \"content\": \"In this subsection we will build the second square of our diagram. This square is really just the Universal Coefficient theorem (Corollary 5.2) in a different form. We will write deg for both K-homological and K-theoretic degree.\\nLemma 7.10. Let Y be a space having the homotopy type of a finite CW -complex. There exists an isomorphism U making the following diagram commute.\\nProof. For β : K TF * (S N ) → K TF * (Y ), let U (β) be the unique map making the following diagram commute\\nwhere the isomorphisms are those of Corollary 5.2. Since K TF * (Y ) is a finitely generated free Z-module, β → Hom(β, Z) is an isomorphism, so U is also an isomorphism. Commutativity of the diagram from the statement of this lemma is by naturality of Lemma 5.2.\\nCorollary 7.11. Let Y be a space having the homotopy type of a finite CW -complex. For a Z-module M , let τ p : M → M ⊗ Z/p be the natural map. There exists an injection U ′ making the following diagram commute.\\nProof. By Lemma 7.10, we have a commutative diagram\\nwith U an isomorphism, so U ⊗ Z/p is also an isomorphism. By Lemma 6.1, the map\\nis an injection. Maps of spaces induce maps of ψ-modules on K-theory, so the image of U • deg is contained in Hom ψ−Mod ( K * TF (Y ), K * TF (S N )), and hence there exists a map U ′ making the following diagram commute:\\nBoth vertical maps are injections, so U ′ has the required properties.\"\n },\n {\n \"title\": \"Pulling back along classes defined on S 3\",\n \"content\": \"Let f ∈ π j (S 3 ), and let N ≥ 3. Then, for ω ∈ π N (Y ), the composite\\nIn words, f * Σ pulls classes back along the appropriate suspension of f . Strictly speaking, f * Σ is only a partial map, because it is undefined on π N for N ≤ 2, but this will be unimportant.\\nRecall the definition of the double e-invariant e (Definition 6.4). On π N (Y ), we have by definition that f * Σ = (Σ N −3 f ) * . By Lemma 6.5 we have a commuting square:\\nMimicking the convention for f * Σ , let\\nbe the map which is defined to be equal to θ(\\nLemma 7.12. Let p be a prime, and let f ∈ π j (S 3 ) with e(f ) defined. If pf = 0, then there exists a map θ p Σ (f ) making the following diagram commute for all N :\\nProof. Since pf = 0, we have that pe(Σ N −3 f ) = 0 for all N , which implies that θ Σ (f ) vanishes on p-divisible elements, so there exists a unique map θ p Σ (f ) making the diagram commute, as required.\\nLemma 7.13. Let X be a finite CW -complex. Let λ X ℓ be the largest eigenvalue of the rational Adams operation\\nThen, for i ≥ 0\\n• the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Σ 2i J s (X)) ⊗ Q is ℓ i (λ X ℓ ) s , and\\nProof. When i ≥ 1, Theorem 5.6 gives that ΣJ s (X) ≃ Σ s t=1 X ∧t , so Σ 2i J s (X) ≃ S 2i ∧ s t=1 X ∧t , and Σ 2i+1 J s (X) ≃ S 2i ∧ ΣX ∧ s−1 t=1 X ∧t . By the Künneth theorem (Theorem 5.5), this implies isomorphisms of rings\\nThe Künneth isomorphism of Theorem 5.5 is given by the external product of K-theory. Since the Adams operations are ring homomorphisms, the above isomorphisms are also isomorphisms of ψmodules. In particular, the Adams operations on the left are the tensor product of the corresponding operations on the right. These decompositions hold for K 0 TF , so they also hold for Q ⊗ K 0 , and the remaining problem is to determine the largest eigenvalue of the relevant tensor products of Adams operations. The eigenvalues of a tensor product of linear endomorphisms are precisely the products of the eigenvalues. The operation ψ ℓ acts on S 2i by multiplication by ℓ i . Together, these observations imply the result.\\nLemma 7.14. Let p be an odd prime. Let X be an (r − 1)-connected finite CW -complex. Let N, s ∈ Z + . Consider the diagram of Lemma 7.12 for Y = J s (X) and f = f p,j ∈ π 2j(p−1)+2 (S 3 ), the map of Theorem 6.8:\\nτp θΣ(f ) / / Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +2j(p−1) )) construction (Theorem 6.8), so e(Σ N −3 f ) = 0 by stability (Lemma 6.6). Since e(ω • Σ N −3 f ) = 0, the contrapositive of Theorem 6.7 gives that ω * has p-divisible image in K 0 (S N ). Since N is even, this implies that τ p • deg(ω) = 0, as required. CASE 2 (N odd): Write n = 2n + 1. Let f ′ = Σ N −2 f and g = Σω in Theorem 6.7, and proceed similarly to case 1. The domain of Σω • Σ N −2 f is S M−1 , where M − 1 = N + 2j(p − 1), so M is even, as is required. To check the eigenvalue hypothesis of Theorem 6.7, write M = 2m. By Lemma 7.13, the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (ΣJ s (X)) ⊗ Q is at most λ s ℓ , and ℓ m = ℓ j(p−1)+n = ℓ j(p−1)+ N −1 2 , which we assumed was greater than λ s ℓ . This means that ℓ m cannot be an eigenvalue of ψ ℓ ⊗ Q on K 0 (ΣJ s (X))⊗Q. As in the previous case, e(Σ N −2 f ) = 0. Since e(Σω •Σ N −2 f ) = 0, the contrapositive of Theorem 6.7 gives that (Σω) * has p-divisible image in K 0 (S N +1 ). Since N is odd, this implies that τ p • deg(ω) = 0, as required. This completes the case, and hence the proof.\"\n },\n {\n \"title\": \"Proof of Theorem 1.4\",\n \"content\": \"Construction 7.15. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r − 1)-connected for r ≥ 1. Let f ∈ π i (S 3 ) with e(f ) defined. Suppose that N, k, s ∈ Z + satisfy N < r(s + 1) − 1 and k ≤ s. The diagrams of the preceding subsections may be combined as follows.\\nRecall the definition of deg ′ from the preamble to Lemma 7.2. Let I(A) be the submodule of Hom( K TF * (S * ), L( K TF * (A))) ⊗ Z/p generated by Im(τ p • deg ′ ). The same grading conventions as usual apply: we write I k (A) for the weight k part, we write I N (A) for the degree N part, and let\\nFrom Corollary 7.3, using the assumptions that N < r(s + 1) − 1 and k ≤ s (which make Φ π ν and Φ K ν well-defined) we obtain the following diagram, where the images of the vertical maps have been 'popped out' to their right.\\nNext, from Corollary 7.11 (with Y = J s (X)) we have a diagram π N (J s (X)) π N (J s (X))\\nLastly, we obtain the following diagram from Lemma 7.12:\\nConcatenating these diagrams gives a diagram as follows:\\nIn this subsection, we combine the results of the previous subsections to produce results about this diagram.\\nTheorem 7.16. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X (r−1)-connected for r ≥ 1. Let N, k, s ∈ Z + with N < r(s+1)−1 and k ≤ s. Let f = f p,j ∈ π 2j(p−1)+2 (S 3 ), the map of Theorem 6.8.\\nFor ℓ ∈ Z + , let λ Y ℓ be the largest eigenvalue of ψ ℓ ⊗ Q on K 0 (Y ) ⊗ Q, and let λ ℓ = max(λ X ℓ , λ ΣX ℓ ). If\"\n },\n {\n \"title\": \"an injection, and\",\n \"content\": \"• there exists ℓ ∈ Z + such that ℓ j(p−1)+ N −1 2 > λ s ℓ ,\\n, K * TF (S N +2j(p−1) )) is an injection.\\nProof. By Corollary 7.9, since ν * ⊗ Z/p is an injection, Φ K ν ⊗ Z/p is also an injection. By Corollary 7.11 U ′ is an injection. By Lemma 7.14 the hypothesis on ℓ implies that the restriction of θ p Σ (f ) to Im(τ p • deg) is an injection. The map θ p Σ (f ) • U ′ • ( Φ K ν ⊗ Z/p) is thus a composite of injections, hence an injection, as required.\\nIn the proof of Theorem 1.4, we will wish to restrict attention to those elements of B(π * (A)) who are brackets of classes in π * (A) in some dimensional range q min ≤ n ≤ q max . All such classes lie in dimensions kq min ≤ N ≤ kq max . Said more precisely, we have an inclusion B k ( qmax n=qmin π n (A)) ⊂ kqmax N =kqmin B k N (π * (A)). We will now study the diagram of Construction 7.15 in this dimensional range. Construction 7.17. Let p be an odd prime, ν : A → ΩΣX for finite CW -complexes A and X with X (r − 1)-connected for r ≥ 1, and f ∈ π i (S 3 ) with e(f ) defined. Let q max > q min be natural numbers. Fix k ∈ Z + , and let s = kq max +1. For N ∈ Z + with kq min ≤ N ≤ kq max , we have that N < r(s+1)−1 and k ≤ s. Combining the diagrams obtained from Construction 7.15 for this range of values of N gives the following diagram: / / kqmax N =kqmin Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +i−2 )).\\nWe now show that by choosing a large enough c ∈ Z + , and setting f = f p,ck , the eigenvalue hypothesis of Theorem 7.16 may be satisfied across the dimensional range of Construction 7.17 for all sufficiently large k.\\nCorollary 7.18. Let p be an odd prime. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CW -complexes, with X path-connected. Let q max > q min be natural numbers. Let c, k ∈ Z + . Let f = f p,ck ∈ π 2ck(p−1)+2 (S 3 ) be the map of Theorem 6.8. If ν * ⊗ Z/p : K TF * (ΣA) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection then there exists c ∈ Z + such that for large enough k ∈ Z + ,\\nExt ψ−Mod ( K * TF (J s (X)), K * TF (S N +2ck(p−1) ))\\nis an injection.\\nProof. By Theorem 7.16, it suffices to show that for each N with kq min ≤ N ≤ kq max there exists ℓ ∈ Z + such that ℓ ck(p−1)+ N −1 2 > λ s ℓ = λ kqmax+1 ℓ . Take any ℓ ≥ 2. Since N ≥ kq min , it suffices to find c such that for large enough k we have ℓ ck(p−1)+ kq min −1 2 > λ kqmax ℓ . Taking logs on both sides, this is equivalent to (ck(p − 1) + kq min − 1 2 ) log(ℓ) > kq max log(λ ℓ ).\\nIt is now clear that we may choose c large enough that this equation holds for large enough k, in particular, any c ≥ 1 p−1 (q max log(λ ℓ ) log(ℓ) − qmin 2 ) will do.\\nBefore proving Theorem 1.4, we prove three lemmas. The first converts the K-theoretic hypothesis of Theorem 1.4 into the K-homological input our construction requires.\\nLemma 7.19. Let X be a space, and let p be prime. Let µ : S q1+1 ∨ S q2+1 → ΣX be a map with q i ≥ 1, such that the map K * TF (ΣX) ⊗ Z/p µ * ⊗Z/p − −−−− → K * TF (S q1+1 ∨ S q2+1 ) ⊗ Z/p ∼ = Z/p ⊕ Z/p is a surjection. Then µ * ⊗ Z/p : K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection.\\nProof. Naturality of the Universal Coefficient Theorem (Corollary 5.2) relative to the map µ gives a diagram of short exact sequences\\nwhere E denotes the quotient of Ext( K * −1 (ΣX), Z) by its torsion subgroup. We will argue by contrapositive. Suppose that µ * ⊗ Z/p is not injective. The Z/p-vector space K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p has dimension 2, so there exists a basis x, y where (µ * ⊗ Z/p)(y) = 0. Prior to tensoring with Z/p, this means that there exists a non-p-divisible element y of K TF * (S q1+1 ∨ S q2+1 ) such that p divides µ * ( y). This implies that any element ϕ of Hom( K TF * (S q1+1 ∨ S q2+1 ), Z) with ϕ( y) not p-divisible is not contained in the image of Hom(µ * , Z), hence, by the diagram, that µ * ⊗ Z/p is not surjective, as required.\\nProof. For any space A, the localizing map A −→ A (p) induces an isomorphism on K * ( ) ⊗ Z/p [Mis78]. It therefore suffices to show that there exists maps µ ′ and ϕ making the following diagram commute, with the localization ϕ (p) being a homotopy equivalence.\\nSuch maps exist by Lemma 7.21.\\nWe are now ready to prove Theorem 1.4.\\nProof of Theorem 1.4. Let µ = µ 1 ∨ µ 2 , with adjoint µ : S q1 ∨ S q2 → ΩΣX. Let f = f p,ck ∈ π 2ck(p−1)+2 (S 3 ). Consider the diagram of Construction 7.17, with A = S q1 ∨ S q2 , q max = max(q 1 , q 2 ), q min = min(q 1 , q 2 ), and ν = µ. We have such a diagram for each k ∈ Z + :\\nN =kqmin π N +2ck(p−1)−1 (J s (X)) e kqmax N =kqmin I k N (S q1 ∨ S q2 ) / / kqmax N =kqmin Ext ψ−Mod ( K * TF (J s (X)), K * TF (S N +2ck(p−1) )).\\nBy assumption, µ * ⊗Z/p : K * (ΣX)⊗Z/p → K * (S q1+1 ∨S q2+1 )⊗Z/p is a surjection. Since K * (S q1+1 ∨ S q2+1 ) is torsion-free, Lemma 7.19 then implies that µ * ⊗ Z/p : K TF * (S q1+1 ∨ S q2+1 ) ⊗ Z/p → K TF * (ΣX) ⊗ Z/p is an injection. By Lemma 7.22, we may assume without loss of generality that X has the integral homotopy type of a finite CW -complex. Thus, by Corollary 7.18, we may fix c such that for large enough k, θ p (f ) • U ′ • ( Φ K µ ⊗ Z/p) is an injection. The Hurewicz map h is a surjection π * (S q1 ∨S q2 ) → K TF * (S q1 ∨S q2 ), so the submodule generated by the image of the map h ′ : B(π * (S q1 ∨S q2 )) → L( K TF * (S q1 ∨S q2 )) of Lemma 7.20 contains the submodule generated by K TF * (S q1 ∨ S q2 ) under the bracket operation. In particular, it contains the weight k component L k ( K TF * (S q1 ∨ S q2 )) for each k. By Theorem 3.1, dim Z (L k ( K TF * (S q1 ∨ S q2 ))) = W 2 (k). Note that L k ( K TF * (S q1 ∨ S q2 )) = kqmax N =kqmin L k ( K TF * (S q1 ∨ S q2 )). It then follows from Lemma 7.20 that dim Z/p ( kqmax N =kqmin I k N (S q1 ∨ S q2 )) ≥ W 2 (k). Since θ p Σ (f ) • U ′ • ( Φ K µ ⊗ Z/p) is an injection for large enough k, it follows that the dimension of e( kqmax N =kqmin π N +2ck(p−1)−1 (J s (X))) is at least W 2 (k). By Corollary 7.3 (i s ) * is an injection, so the dimension of (i s ) * ( kqmax N =kqmin π N +2ck(p−1)−1 (J s (X))) ⊂ kqmax N =kqmin π N +2ck(p−1)−1 (ΩΣX) is also at least W 2 (k).\\nThus, ΣX satisfies the hypotheses of Lemma 3.3 with a = 2c(p − 1) + q max = 2c(p − 1) + max(q 1 , q 2 ) and b = 0, and hence is p-hyperbolic.\"\n },\n {\n \"title\": \"Definition 5. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 6. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"N\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"For a Z-graded (respectively Z/2-graded) module M , let T (M ) = ∞ k=1 M ⊗k denote the tensor algebra on M . The product is given by concatenation. We refer to M ⊗k as the weight k component of the tensor algebra T (M ). We define a Z-grading (respectively Z/2-grading) on T (M ) by setting |x 1 ⊗ x 2 ⊗ · · · ⊗ x k | = k i=1 |x i |.Definition 5.9. For a space Y , let σ : K TF * (Y ) ∼ = − → K TF * +1 (ΣY ) be the suspension isomorphism. Let ϕ : K TF * (ΣY ) → K TF * (ΣY ) be a homomorphism of graded groups, not necessarily induced by a map of spaces. We call the composite σ −1 • ϕ • σ the desuspension of θ, denoting it by S −1 ϕ.","content":"Definition 5. 8 .\n8For a Z-graded (respectively Z/2-graded) module M , let T (M ) = ∞ k=1 M ⊗k denote the tensor algebra on M . The product is given by concatenation. We refer to M ⊗k as the weight k component of the tensor algebra T (M ). We define a Z-grading (respectively Z/2-grading) on T (M ) by setting |x 1 ⊗ x 2 ⊗ · · · ⊗ x k | = k i=1 |x i |.Definition 5.9. For a space Y , let σ : K TF * (Y ) ∼ = − → K TF * +1 (ΣY ) be the suspension isomorphism. Let ϕ : K TF * (ΣY ) → K TF * (ΣY ) be a homomorphism of graded groups, not necessarily induced by a map of spaces. We call the composite σ −1 • ϕ • σ the desuspension of θ, denoting it by S −1 ϕ."},{"id":"","caption":"5.10, Ση = Γ • ι 1 , which implies the s = 1 statement. Now let s ≥ 2. Ganea [Gan65, Theorems 1.1 and 1.4] shows that the homotopy fibre of ev is given by Σ(ΩΣX ∧ ΩΣX) v − → ΣΩΣX ev − → ΣX, where the map v is equal to the composite Σ(ΩΣX ∧ ΩΣX) → Σ(ΩΣX × ΩΣX) Σm2 − −− → ΣΩΣX of Σm 2 with the canonical splitting. We will show that Γ • ι s factors through v, and hence composes trivially with ev. Consider the following diagram, where the unlabelled arrows are all canonical splittings: Σ(ΩΣX ∧ ΩΣX) / / v , , Σ(ΩΣX × ΩΣX) Σm2 / / ΣΩΣX Σ(ΩΣX) ∧s / / Σ((ΩΣX) (s−1) ∧ ΩΣX)","content":"5.10, Ση = Γ • ι 1 , which implies the s = 1 statement. Now let s ≥ 2. Ganea [Gan65, Theorems 1.1 and 1.4] shows that the homotopy fibre of ev is given by Σ(ΩΣX ∧ ΩΣX) v − → ΣΩΣX ev − → ΣX, where the map v is equal to the composite Σ(ΩΣX ∧ ΩΣX) → Σ(ΩΣX × ΩΣX) Σm2 − −− → ΣΩΣX of Σm 2 with the canonical splitting. We will show that Γ • ι s factors through v, and hence composes trivially with ev. Consider the following diagram, where the unlabelled arrows are all canonical splittings: Σ(ΩΣX ∧ ΩΣX) / / v , , Σ(ΩΣX × ΩΣX) Σm2 / / ΣΩΣX Σ(ΩΣX) ∧s / / Σ((ΩΣX) (s−1) ∧ ΩΣX)"},{"id":"","caption":"Let M and N be ψ-modules, with N torsion-free. The inclusion of Z-modules Hom ψ−Mod (M, N ) ֒− → Hom(M, N ) is an injection onto a summand.","content":"Lemma 6. 1 .\n1Let M and N be ψ-modules, with N torsion-free. The inclusion of Z-modules Hom ψ−Mod (M, N ) ֒− → Hom(M, N ) is an injection onto a summand."},{"id":"","caption":"For the avoidance of doubt, by the e-invariant we will always mean what Adams calls the complex e-invariant e C [Ada65; Ada66].","content":"For the avoidance of doubt, by the e-invariant we will always mean what Adams calls the complex e-invariant e C [Ada65; Ada66]."},{"id":"","caption":"For a graded R-module V , let U (V ) denote the graded set of homogeneous elements in V . Let B(V ) be the set of formal iterated brackets of the elements of U (V ). Bracket gives a natural operation on B(V ), which we write as [x, y]. Elements of B(V ) are nonassociative words in the elements of U (V ), so we may define a grading on B(V ) which extends the grading on U (V ) via |[x, y]| = |x| + |y|. The weight of an element of B(V ) is its word length. Write B N (V ) for the subset of elements in degree N , B k (V ) for the subset of elements of weight k, and set B k N (V ) = B k (V ) ∩ B N (V ). Let ν : A → ΩΣX be a map. By definition of B(π * (A)), there exists a map Φ π ν : B(π * (A)) → π * (ΩΣX)","content":"For a graded R-module V , let U (V ) denote the graded set of homogeneous elements in V . Let B(V ) be the set of formal iterated brackets of the elements of U (V ). Bracket gives a natural operation on B(V ), which we write as [x, y]. Elements of B(V ) are nonassociative words in the elements of U (V ), so we may define a grading on B(V ) which extends the grading on U (V ) via |[x, y]| = |x| + |y|. The weight of an element of B(V ) is its word length. Write B N (V ) for the subset of elements in degree N , B k (V ) for the subset of elements of weight k, and set B k N (V ) = B k (V ) ∩ B N (V ). Let ν : A → ΩΣX be a map. By definition of B(π * (A)), there exists a map Φ π ν : B(π * (A)) → π * (ΩΣX)"},{"id":"","caption":"=kqmin π N +i−3 (J s (X)) e kqmax N =kqmin I k N (A) θ p Σ (f )•U ′ •( Φ K ν ⊗Z/p)","content":"N\n=kqmin π N +i−3 (J s (X)) e kqmax N =kqmin I k N (A) θ p Σ (f )•U ′ •( Φ K ν ⊗Z/p)"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"For a Z-graded (respectively Z/2-graded) module M , let T (M ) = ∞ k=1 M ⊗k denote the tensor algebra on M . The product is given by concatenation. We refer to M ⊗k as the weight k component of the tensor algebra T (M ). We define a Z-grading (respectively Z/2-grading) on T (M ) by setting |x 1 ⊗ x 2 ⊗ · · · ⊗ x k | = k i=1 |x i |.Definition 5.9. For a space Y , let σ : K TF * (Y ) ∼ = − → K TF * +1 (ΣY ) be the suspension isomorphism. Let ϕ : K TF * (ΣY ) → K TF * (ΣY ) be a homomorphism of graded groups, not necessarily induced by a map of spaces. We call the composite σ −1 • ϕ • σ the desuspension of θ, denoting it by S −1 ϕ.\",\n \"content\": \"Definition 5. 8 .\\n8For a Z-graded (respectively Z/2-graded) module M , let T (M ) = ∞ k=1 M ⊗k denote the tensor algebra on M . The product is given by concatenation. We refer to M ⊗k as the weight k component of the tensor algebra T (M ). We define a Z-grading (respectively Z/2-grading) on T (M ) by setting |x 1 ⊗ x 2 ⊗ · · · ⊗ x k | = k i=1 |x i |.Definition 5.9. For a space Y , let σ : K TF * (Y ) ∼ = − → K TF * +1 (ΣY ) be the suspension isomorphism. Let ϕ : K TF * (ΣY ) → K TF * (ΣY ) be a homomorphism of graded groups, not necessarily induced by a map of spaces. We call the composite σ −1 • ϕ • σ the desuspension of θ, denoting it by S −1 ϕ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"5.10, Ση = Γ • ι 1 , which implies the s = 1 statement. Now let s ≥ 2. Ganea [Gan65, Theorems 1.1 and 1.4] shows that the homotopy fibre of ev is given by Σ(ΩΣX ∧ ΩΣX) v − → ΣΩΣX ev − → ΣX, where the map v is equal to the composite Σ(ΩΣX ∧ ΩΣX) → Σ(ΩΣX × ΩΣX) Σm2 − −− → ΣΩΣX of Σm 2 with the canonical splitting. We will show that Γ • ι s factors through v, and hence composes trivially with ev. Consider the following diagram, where the unlabelled arrows are all canonical splittings: Σ(ΩΣX ∧ ΩΣX) / / v , , Σ(ΩΣX × ΩΣX) Σm2 / / ΣΩΣX Σ(ΩΣX) ∧s / / Σ((ΩΣX) (s−1) ∧ ΩΣX)\",\n \"content\": \"5.10, Ση = Γ • ι 1 , which implies the s = 1 statement. Now let s ≥ 2. Ganea [Gan65, Theorems 1.1 and 1.4] shows that the homotopy fibre of ev is given by Σ(ΩΣX ∧ ΩΣX) v − → ΣΩΣX ev − → ΣX, where the map v is equal to the composite Σ(ΩΣX ∧ ΩΣX) → Σ(ΩΣX × ΩΣX) Σm2 − −− → ΣΩΣX of Σm 2 with the canonical splitting. We will show that Γ • ι s factors through v, and hence composes trivially with ev. Consider the following diagram, where the unlabelled arrows are all canonical splittings: Σ(ΩΣX ∧ ΩΣX) / / v , , Σ(ΩΣX × ΩΣX) Σm2 / / ΣΩΣX Σ(ΩΣX) ∧s / / Σ((ΩΣX) (s−1) ∧ ΩΣX)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M and N be ψ-modules, with N torsion-free. The inclusion of Z-modules Hom ψ−Mod (M, N ) ֒− → Hom(M, N ) is an injection onto a summand.\",\n \"content\": \"Lemma 6. 1 .\\n1Let M and N be ψ-modules, with N torsion-free. The inclusion of Z-modules Hom ψ−Mod (M, N ) ֒− → Hom(M, N ) is an injection onto a summand.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For the avoidance of doubt, by the e-invariant we will always mean what Adams calls the complex e-invariant e C [Ada65; Ada66].\",\n \"content\": \"For the avoidance of doubt, by the e-invariant we will always mean what Adams calls the complex e-invariant e C [Ada65; Ada66].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For a graded R-module V , let U (V ) denote the graded set of homogeneous elements in V . Let B(V ) be the set of formal iterated brackets of the elements of U (V ). Bracket gives a natural operation on B(V ), which we write as [x, y]. Elements of B(V ) are nonassociative words in the elements of U (V ), so we may define a grading on B(V ) which extends the grading on U (V ) via |[x, y]| = |x| + |y|. The weight of an element of B(V ) is its word length. Write B N (V ) for the subset of elements in degree N , B k (V ) for the subset of elements of weight k, and set B k N (V ) = B k (V ) ∩ B N (V ). Let ν : A → ΩΣX be a map. By definition of B(π * (A)), there exists a map Φ π ν : B(π * (A)) → π * (ΩΣX)\",\n \"content\": \"For a graded R-module V , let U (V ) denote the graded set of homogeneous elements in V . Let B(V ) be the set of formal iterated brackets of the elements of U (V ). Bracket gives a natural operation on B(V ), which we write as [x, y]. Elements of B(V ) are nonassociative words in the elements of U (V ), so we may define a grading on B(V ) which extends the grading on U (V ) via |[x, y]| = |x| + |y|. The weight of an element of B(V ) is its word length. Write B N (V ) for the subset of elements in degree N , B k (V ) for the subset of elements of weight k, and set B k N (V ) = B k (V ) ∩ B N (V ). Let ν : A → ΩΣX be a map. By definition of B(π * (A)), there exists a map Φ π ν : B(π * (A)) → π * (ΩΣX)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"=kqmin π N +i−3 (J s (X)) e kqmax N =kqmin I k N (A) θ p Σ (f )•U ′ •( Φ K ν ⊗Z/p)\",\n \"content\": \"N\\n=kqmin π N +i−3 (J s (X)) e kqmax N =kqmin I k N (A) θ p Σ (f )•U ′ •( Φ K ν ⊗Z/p)\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Let h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF * (A). As with deg ′. let h ′ : B(π * (Apreamble to the second lemma is necessary. Let h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF * (A). As with deg ′ , let h ′ : B(π * (A)) →","doi":""},{"text":"Let M be a Z/2-graded Z-module. L( K Tf *, Let χ : Hom( K TF * (S * ), M ) → M be the map which carries ϕ ∈ Hom( K TF * (S N ), M ) to ϕ(ξ N ) ∈ M (Remark 5.4). If M = L is a Z/2-graded Lie algebra, then it follows immediately from the definition of the bracket in Hom( K TF *. → K TF * (A) ⊂ L( K TF *A)) be the unique map which restricts to h : π * (A). S * ), L) that χ is a map of Lie algebrasL( K TF * (A)) be the unique map which restricts to h : π * (A) → K TF * (A) ⊂ L( K TF * (A)) and respects brackets. Let M be a Z/2-graded Z-module. Let χ : Hom( K TF * (S * ), M ) → M be the map which carries ϕ ∈ Hom( K TF * (S N ), M ) to ϕ(ξ N ) ∈ M (Remark 5.4). If M = L is a Z/2-graded Lie algebra, then it follows immediately from the definition of the bracket in Hom( K TF * (S * ), L) that χ is a map of Lie algebras.","doi":""},{"text":"On the groups J(X). II. J F Adams, Topology. 3J. F. Adams. \"On the groups J(X). II\". Topology 3 (1965), pp. 137-171.","doi":""},{"text":"On the groups. J F Adams, J(X). IV\". Topology. 5J. F. Adams. \"On the groups J(X). IV\". Topology 5 (1966), pp. 21-71.","doi":""},{"text":"Lectures on generalised cohomology. 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Z.-J Zhu, J.-Z Pan, Homology Homotopy Appl. 23Z.-J. Zhu and J.-Z. Pan. \"The local hyperbolicity of A 2 n -complexes\". Homology Homotopy Appl. 23.1 (2021), pp. 367-386.","doi":""}],"string":"[\n {\n \"text\": \"Let h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF * (A). As with deg ′. let h ′ : B(π * (Apreamble to the second lemma is necessary. Let h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF * (A). As with deg ′ , let h ′ : B(π * (A)) →\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Let M be a Z/2-graded Z-module. L( K Tf *, Let χ : Hom( K TF * (S * ), M ) → M be the map which carries ϕ ∈ Hom( K TF * (S N ), M ) to ϕ(ξ N ) ∈ M (Remark 5.4). If M = L is a Z/2-graded Lie algebra, then it follows immediately from the definition of the bracket in Hom( K TF *. → K TF * (A) ⊂ L( K TF *A)) be the unique map which restricts to h : π * (A). S * ), L) that χ is a map of Lie algebrasL( K TF * (A)) be the unique map which restricts to h : π * (A) → K TF * (A) ⊂ L( K TF * (A)) and respects brackets. Let M be a Z/2-graded Z-module. Let χ : Hom( K TF * (S * ), M ) → M be the map which carries ϕ ∈ Hom( K TF * (S N ), M ) to ϕ(ξ N ) ∈ M (Remark 5.4). If M = L is a Z/2-graded Lie algebra, then it follows immediately from the definition of the bracket in Hom( K TF * (S * ), L) that χ is a map of Lie algebras.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the groups J(X). II. J F Adams, Topology. 3J. F. Adams. \\\"On the groups J(X). II\\\". Topology 3 (1965), pp. 137-171.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the groups. J F Adams, J(X). IV\\\". Topology. 5J. F. Adams. \\\"On the groups J(X). IV\\\". Topology 5 (1966), pp. 21-71.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lectures on generalised cohomology. J F Adams, Category Theory, Homology Theory and their Applications. 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Whitehead. \\\"On products in homotopy groups\\\". Ann. of Math (2) 47 (1946), pp. 460-475.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Graduate Texts in Mathematics. G W Whitehead, Springer-Verlag61New York-BerlinElements of homotopy theoryG. W. Whitehead. Elements of homotopy theory. Vol. 61. Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1978.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universal coefficient sequences for cohomology theories of CW-spectra. Z.-I Yosimura, Osaka Math. J. 12Z.-I. Yosimura. \\\"Universal coefficient sequences for cohomology theories of CW-spectra\\\". Osaka Math. J. 12.2 (1975), pp. 305-323.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The local hyperbolicity of A 2 n -complexes. Z.-J Zhu, J.-Z Pan, Homology Homotopy Appl. 23Z.-J. Zhu and J.-Z. Pan. \\\"The local hyperbolicity of A 2 n -complexes\\\". Homology Homotopy Appl. 23.1 (2021), pp. 367-386.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":399,"cells":{"text":{"kind":"string","value":"\nMicroVelocity: rethinking the Velocity of Money for digital currencies\nFebruary 1, 2022\n\nCarlo Campajola \nDLT Science Foundation\nLondonUnited Kingdom\n\nDepartment of Informatics\nUniversity of Zurich\nBlockchain & Distributed Ledger Technologies\nZürichSwitzerland\n\nUZH Blockchain Center\nZürichSwitzerland\n\nMarco D'errico \nEuropean Systemic Risk Board Secretariat\nEuropean Central Bank\nFrankfurtGermany\n\nClaudio J Tessone \nDepartment of Informatics\nUniversity of Zurich\nBlockchain & Distributed Ledger Technologies\nZürichSwitzerland\n\nUZH Blockchain Center\nZürichSwitzerland\n\nMicroVelocity: rethinking the Velocity of Money for digital currencies\nFebruary 1, 2022Velocity of MoneyCryptocurrencyBlockchainHeterogeneous agents JEL Classification: C81D31E41\nWe propose a novel framework to analyse the velocity of money in terms of the contribution (MicroVelocity) of each individual agent, and to uncover the distributional determinants of aggregate velocity. Leveraging on complete publicly available transactions data stored in blockchains from four cryptocurrencies, we empirically find that MicroVelocity i) is very heterogeneously distributed and ii) strongly correlates with agents' wealth. We further document the emergence of high-velocity intermediaries, thereby challenging the idea that these systems are fully decentralised. Further, our framework and results provide policy insights for the development and analysis of digital currencies.\n\nIntroduction\n\nOne of the earliest and most widespread concepts across societies is that of money. As humans we are inherently brought to organise ourselves into communities and exchange the goods and services we produce; this leads to the necessity to invent a medium of exchange that can be store of value -to be able to defer purchases to the future -and unit of account -to properly quantify the value of goods and services [1]. Any such artifact, which may take the form of anything between a large stone in the middle of the village [2], cigarettes [3], cash in the form of coins and banknotes or an encrypted sequence of bits [4], is a form of money, as long as there is more than one person willing to accept it as such.\n\nOver the years, economic theory has kept interrogating itself on how to efficiently manage this peculiar asset: who is entitled to create it, what is its cost and, most importantly, the role of money on societal and economic development. These and other questions led to the development of monetary theory, which interrogates on how the quantity of money in a system influences economic activity and macroeconomic quantities.\n\nOne of the main research areas in monetary theory is the quantity theory of money, which revolves around Fisher's equation of exchange [5] M V = P Q (1) stating the proportionality between the monetary mass M and the level of prices P through the economic output Q and V , the velocity of money. The latter is defined as the average frequency with which a given unit of money is used to purchase domestically-produced goods and services within a given time period: this makes Fisher's equation an accounting identity which holds true by construction as both sides amount to the total money flow, which we will call F . The velocity V is often seen as an indicator of money demand [6]: if money demand is low -i.e. expected yield from alternative assets is high -economic agents will convert their money into less liquid investments as soon as possible, thus leading to increased velocity, whereas in situations of high demand for money the velocity will decrease.\n\nMoney velocity is not directly measurable since it would be impossible to track every unit of currency circulating in the economy. Therefore, the equation of exchange provides a way to estimate velocity at a given point in time. The equation however holds true in aggregate, thus providing limited insights on the distributional aspects, in particular for what concerns a more granular measurement of velocity and how different economic agents (or groups of agents) contribute to the aggregate. In fact, agents are heterogeneous, have different spending habits, different levels of wealth, income and demand for money and different roles in its movement in the economy; but since their different contributions to aggregate velocity are not measurable, policy analysis and interventions cannot be targeted at a finer level.\n\nHowever, these limitations to the measurement of velocity might be not so strict anymore: over the last few years the increasing diffusion of digital payments -via electronic cards, e-banking or smartphone apps, to name some alternatives -has led to a surge in transactions data availability, meaning that it is now within reach to actually measure the velocity of money from micro-level data, at least for digital forms of money. A big role in this transition towards digital cash will be played by central banks, who are now devoting significant resources towards the development of Central Bank Digital Currencies (CBDCs) [7,8,9] to provide citizens with access to central bank money in digital form.\n\nA strong push towards digital payments has been prompted by the rise over recent years of cryptocurrencies and blockchain-based tokens. These have emerged as a family of unconventional financial assets and some of them, with Bitcoin being the first and most prominent example, claimed they could be the future of money and would take over the role of fiat currencies. While this has not happened, and outside of crypto-maximalist communities it is now commonplace not to consider Bitcoin and its many offsprings as currencies, the publicly available records of blockchains constitute a unique opportunity to explore the space of microeconomically grounded metrics for macroeconomic quantities like the velocity of money.\n\nIn particular in this paper we focus on exploiting the accounting protocol adopted by several cryptocurrencies, the Unspent Transaction Output (UTXO) protocol, to actually track specific units of currency as they move from one agent to the other. This protocol imposes that whenever funds have to be spent they are included as inputs of a transaction, and they have explicit reference to the unspent output of a previously validated transaction. This provides verification that the issuer of the new transaction has the right to spend those funds, as the output can only be accessed by means of a cryptographic private key owned by the receiver. A by-product of this mechanism is that every input to a new transaction is explicitly linked to the output of a previous one, and thus the time that those units of money stayed idle is known.\n\nAs we explain in the following, this level of detail allows to factor the total velocity V into agent-specific contributions, V i , which we call MicroVelocity and depend on agents' behavioral usage of their wealth. This description, based on the seminal work of [10], opens to several questions: how is MicroVelocity distributed cross-sectionally? How does it evolve over time? How does an agent's MicroVelocity correlate with their wealth? How does it correlate with the MicroVelocity of their partners in transactions?\n\nIn the context of cryptocurrencies, the velocity of money has been used to propose valuation models [11,12], by an argument stating that if the economy adopting cryptocurrency as its medium of exchange grows, so does the velocity due to the available supply being limited by protocol, thus an increase in velocity should also warrant an increase in pricing. Several estimators have been proposed for the velocity of cryptocurrencies, like Coin Days Destroyed [13] or Average Dormancy [14]. In particular Pernice et al. [15] propose an accurate method to estimate velocity on aggregate, which takes advantage of the UTXO protocol in similar ways to our definition of MicroVelocity.\n\nWe argue that MicroVelocity allows more detailed considerations on the state of the economy using cryptocurrencies. Specifically, statistical properties of its distribution across agents provide insight into the global level of specialization and on centralization in the flow of money: if the distribution is highly heterogeneous, agents with high MicroVelocity are likely to be intermediaries and those with low V i their clients, thus going against the common narrative that cryptocurrencies are \"decentralized economies\" [16]. Paths towards the centralization of these systems are currently attracting growing attention, with Aramonte et al. [17] explicitly talking of \"decentralization illusion\".\n\nRegardless, our framework is not necessarily limited to the context of blockchainbased tokens. The formulation of MicroVelocity general in nature and can be adapted to find application to any payment system that keeps a digital record of transactions, such as bank transfers, credit cards payments or, in the future, CBDCs. Such an instrument will then be a useful addition to economists' toolboxes when investigating effects of policy, and possibly even opening new policy channels that account for the heterogeneity of economic agents in their usage of money [18,19].\n\nMicroVelocity. We build on Wang and coauthors' theoretical framework [10], which shows that the macroeconomic velocity can be derived from a microeconomic description: here we review and expand their argument, introducing our definition of MicroVelocity.\n\nConsider the time between two transactions involving the same unit of currency, the holding time τ , and assume that τ is a random variable with probability distribution p(τ ). A condition p(0) = 0 has to be enforced since a holding time of 0 is meaningless for the description. It follows that, calling M the total number of coins in circulation, M p(τ )dτ is the number of coins with a holding time in the interval [τ, τ + dτ ).\n\nAssuming stationarity of p(τ ), the money flow F (τ ) generated by coins with holding time τ is\nF (τ ) = M 1 τ p(τ )dτ(2)\nas 1/τ is the frequency with which each coin moves. Since the total economic flow F is the integral of this quantity over all holding times, we find\nF = M V = P Q = M ∞ 0 1 τ p(τ )dτ(3)\nand it thus follows that the velocity reads\nV = +∞ 0 1 τ p(τ )dτ(4)\nIt has to be noticed that units of currency are not distinguishable and there is no reason to think that a particular coin has an inherently different holding time than any other one. Indeed the authors of [10] justify the existence of the holding times distribution by arguing that holding times depend on money ownership, i.e. the spending habits of different economic agents produce the heterogeneity in τ . While coins are indistinguishable, economic agents are not and each has their own spending, saving and speculative attitude.\n\nIn the original paper the authors only consider the aggregate effect of this heterogeneity, defining p(τ ) but neglecting to consider the individual effects of agents, as a simulation study accounting for it would have introduced too many arbitrary parameters in the description. Our goal is thus to overcome this limitation, as having access to agent-level data allows for a more fine-grained representation.\n\nLet us then define p i (τ ) as the probability distribution of holding times of agent i, with i ∈ {1, 2, . . . N }. This means that the total flow of Eq. 3 is factored in a sum of individual contributions, and then Eq. 4 transforms into\nV = N i=1 V i = N i=1 M i M ∞ 0 1 τ p i (τ )dτ (5)\nwhere M i is the amount of money held by agent i, meaning that the total velocity is a wealth-weighted average of individual contributions, the MicroVelocity V i . This equation makes it clear that, in a population with homogeneous spending preferences such that p i (τ ) ≈ p(τ ) ∀i, contributions to money velocity are fully determined by wealth. However real-world economies involve heterogeneous participants, both on the macro-scale (households, companies, business sectors) and on the micro-scale (preferences, consumption functions): our framework fully accounts for these factors, returning a detailed picture of the impact they have on the velocity.\n\nBlockchains and the UTXO protocol. To estimate MicroVelocity from real-world data one needs fine-grained transaction records, so that the holding times τ can be measured and its statistical properties estimated. The advent of blockchains and cryptocurrencies offers a unique occasion to study economic activity on a publicly available transaction ledger. A blockchain is a distributed ledger used to store information upon which a large community agrees without having a central authority that coordinates the consensus. Avoiding going into the details of how the consensus protocol works, what matters for this paper is the accounting standard that is used to store the data on the blockchain. There are several alternatives that are currently being used, but the most popular so far -and first to be implemented in Bitcoin -is the Unspent Transaction Output (UTXO).\n\nIn this protocol, schematised in Fig. 1, information is stored on the blockchain in form of transactions, transferring usage rights of some quantity of digital asset from one cryptographic address to another. These addresses consist of two parts: a public key, visible to everybody and used for verification, and a private key, which is information only available to the address owner. When an agent wants to transfer some assets to another agent, they need to point to a previously generated transaction output that has not been spent yet and, using their private key, demonstrate that they own the address that output was destined to. This shows they own the usage rights on the assets in that output, and they are able to transfer them by pointing to an address belonging to the new transaction receiver.\n\nSince all this information is stored on the blockchain, this means that the time occurring between the generation of the UTXO and its spending transaction is easily available, thus allowing to measure τ for all coins without further assumptions. These can then be used to estimate p i (τ ) depending on ownership, thus providing all the necessary information to calculate the MicroVelocity of Eq. 5.\n\nMeasuring MicroVelocity on blockchains. Events are recorded on the blockchain once every few minutes, when a new block is added. The consequence is that time is effectively discretised, and τ can be expressed in terms of number of blocks between transactions, τ ∈ N \\ {0}. Following the logic above, let us define P i (τ ) to be the probability mass function of the discrete holding times for user i's coins. Then M i P i (τ ) coins by user i have holding time τ , resulting at stationarity in a contribution to user i's flow of\nF i (τ ) = 1 τ M i P i (τ )\nIt then follows that the total velocity, given in continuous time by Eq. 5, turns into the following formula in block time\nV = N i=1 ∞ τ =1 M i M 1 τ P i (τ )(6)\nIsolating the i-th component of this sum we are then able to define user i's contribution to the macroeconomic velocity of money, i.e. the MicroVelocity, which is\nV i = ∞ τ =1 M i M 1 τ P i (τ )(7)\nThe question is then how to measure P i (τ ) to be able to compute these quantities. For the specific case of UTXO-based cryptocurrencies, a very simple estimator can be adopted: the accounting protocol itself produces the holding time τ of each unit of currency as a by-product, hence the empirical probability of these holding times conditional on the agent can be estimated. For example, in Fig. 1, the inputs of transaction 83 have the same value (4 tokens) but different holding time of two blocks (O64 [1]) and of one block (O73 [2]). This means, according to Eq. 7, that tokens in O64 [1] generate half as much velocity as the ones in O73 [2].\n\nThere are two further cautions that need to be mentioned. The first is that in principle there is no guarantee that a transaction recorded on the blockchain is actually a transfer of value from an agent to another: addresses do not have a one-to-one correspondence with agents' wallets, and in fact a wallet can contain an arbitrary number of addresses. Moreover, by protocol, inputs of transactions have to be spent in full each time: the total value of inputs has to match the total value of outputs, and the value of inputs is constrained by the UTXO they point to. The way in which this is solved is by generating an extra output, called change output, which points the excess input to an address which is actually owned by the sender. These transactions do not actually move funds between individuals, thus they should be excluded from the calculation of MicroVelocity.\n\nTo solve this issue, when we find an output that is likely to belong to the transaction sender, we do not count it on the MicroVelocity and add the weighted average holding time of the inputs of that transaction to its holding time. There are several ways to identify self-transactions through heuristic methods which take advantage of typical patterns used by cryptocurrency wallet clients. We provide further details about this in the Materials and Methods section.\n\nThe second caution that is needed is that, when calculating MicroVelocity, the measurable holding times are right-censored, or in other words one would need future information to know the holding time of current UTXOs. While there is no straightforward solution to this, it is in principle possible to devise more refined statistical methods to infer P i (τ ) so that one can reduce this issue, while also controlling for the impact of non-stationarity in the estimation, which we have neglected in this work. We choose not to tackle this problem, as it would require some arbitrary assumptions, and defer the exploration of solutions to future research.\n\n\nMaterials and Methods\n\nData processing. We directly collect data from the Litecoin, Dogecoin, Monacoin and Feathercoin blockchains by synchronizing full nodes in archive mode. The complete history has then been preprocessed using the Blocksci Python library [20], which turns raw data files into an efficiently searchable database structure.\n\nThere are many heuristic methods that have been proposed to assign addresses to unique wallets. Here we combine three heuristics, the MultiInput (MI), the Change Address (CA) and the Peeling Chains (PC):\n\n• MI takes advantage of the fact that multiple addresses appearing as inputs of the same transaction have a high likelihood of belonging to the same agent, as they had to be signed with all the private keys together. For example in Fig. 1, O64 [1] and O73 [2] are assigned to the same wallet since they appear together as inputs of transaction 83;\n\n• CA follows from the argument that, when there is more than one input and more than one output in a transaction, it is likely that the smallest output is a change address, provided it is smaller than the smallest input (otherwise such input would have not been necessary) and that it is a new address; if one such output is identified, it is considered to belong to the same agent as the input addresses. In Fig. 1 • Finally, PC identifies transaction patterns called peeling chains: sequences of transactions with a single input and two outputs, that arise when an agent needs to spend a large value UTXO (like a coinbase transaction), generating chains of transactions where the UTXO is slowly consumed. In order to identify a peeling chain, at least two subsequent transactions have to match this pattern. If that is the case, the outputs that connect the chain are assigned to the same wallet as the inputs. In Fig. 1 When using heuristics to cluster together output and input addresses, there is a significant risk of merging multiple wallets together. For this reason we adopt some logical selection rules to limit the impact of this issue. In particular we apply the following conditions:\n\n• if different heuristics disagree on the change address, no change address is selected;\n\n• if all heuristics agree on the change address or select none, that address is considered the change;\n\n• if an input address appears also in the output, it is necessarily the change and all heuristics are neglected.\n\nIn the Supplementary Information we check the robustness of the results presented in the following sections against the choice of address clustering method.\n\nEstimation of MicroVelocity. The estimation of MicroVelocity follows from the sum of Eq. 7. At a given time t, an agent owns M i (t) coins, each with some holding time τ that is estimated as the age of the UTXO they are included in when they are spent. Then the time t probability mass function of holding times of agent i, P t i (τ ), is measured as the empirical probability where w i (τ ) is the total value of UTXOs with holding time τ , ensuring that\nP t i (τ ) = w i (τ ) M i (t)τ w i (τ ) = M i (t).\nPlugging this into Eq. 5 gives the empirical value of agent i's MicroVelocity at time t\nV i (t) = τ 1 τ w i (τ ) M (t)\nwhere M (t) is the total amount of money in the system at the time. Notice that this value is known by construction, since the supply of cryptocurrencies is deterministic. We neglect additional considerations on the definition of circulating supply, as they would require arbitrary assumptions and we believe they would have a marginal impact on our results. Indeed, since the holding time τ appears at the denominator of Eq. 7, coins with a longer holding time that may be classified in less liquid monetary aggregates already contribute less to the velocity.\n\n\nResults\n\nThe distribution of MicroVelocity. We begin our analysis by checking if the total MicroVelocity we measure correctly matches the velocity calculated following Fisher's Eq. 1. In Figure 2 we compare the sum of MicroVelocities i V i , averaged weekly, with the average volume of transactions P Q divided by the total money supply M on that week. Both quantities are annualised to facilitate comparison. It appears clear that Eq. 6 holds true, with the total MicroVelocity even producing a slightly less volatile estimate of V . This is not surprising, since MicroVelocity accounts for components of V at all timescales whereas P Q/M only considers transactions happening on a specific time window (in this case a week), thus being more prone to temporary fluctuations.\n\nHowever, as discussed above, the advantage of having this microeconomic picture is that we are able to investigate the distributional aspects of the velocity of money. We then proceed to take weekly cross-sectional snapshots of the agents' V i s, which we then utilize to extract a number of statistics and test hypotheses. First, we remove values of V i (t) = 0 from our analysis. These may occur for multiple reasons: the agent may have not entered the market yet or has quit, thus they will have 0 wealth and velocity, but it may also be that they own coins which they never spend. The latter can itself be caused by a behavioral choice of holding the coins indefinitely -the so-called \"HODLers\" -, by the right-censoring of data we previously discussed or by technical problems such as lost passwords or hacks: this uncertainty leads us to exclude also agents with non-zero wealth that have zero MicroVelocity. We then proceed to analyze the cross-sectional distribution by means of the Kolmogorov-Smirnov distance D from a known distribution. This is defined as\nD = sup|P (V i ) − P (V i )| whereP (V i )\nis the empirical distribution of MicroVelocities and P (V i ) is an arbitrary probability distribution. This distance is the test statistic for the twotailed Kolmogorov-Smirnov test, which tests the null hypothesis thatP (V i ) ≡ P (V i ). Since V i ≥ 0 by definition, we choose to take as comparisons the exponential and Pareto distributions, to gain insight on whether the data is closer to a thin-or to a fat-tailed distribution. We fit the parameters of these distributions by Maximum Likelihood methods and calculate the distances D P ar and D Exp , obtaining the results summarized in Table 3. While the thin-tailed exponential null hypothesis is always rejected by the Kolmogorov-Smirnov test, it is clear that a fat-tailed Pareto distribution with tail exponent α ≈ 1.6 is a much better descriptor for the distribution generating the data. Notice that we choose a very high significance threshold of 0.2, since we want to check the robustness of the null hypothesis rather than identifying an alternative. Despite this, the test almost never rejects the Pareto null, consolidating the observation that the total velocity of money is very unevenly distributed across agents, with a small minority dominating the sum of Eq. 7.\n\nPerhaps the most striking evidence of this comes from the stacked charts of Fig. 4. In each panel's bottom chart, each colored area corresponds to the fraction of total velocity that is produced by the 5 largest contributors, compared with the total contributions by all others, tracked over time with weekly aggregation. In all the analyzed cryptocurrencies the vast majority of the total velocity is generated by less than 5 agents: most likely these -whose actual identity is unknown to us -are companies that act as intermediaries, such as exchange markets, custodians or layer 2 payment services like the Lightning Network [21]. Despite the real-world identity of these agents being unknown, we can still track them by their ID from the address clustering algorithm, which means we are able to count how frequently each agent features as a top contributor. The top charts of Fig. 4 show exactly this, counting the percentage of weeks a given agent is among the top 5 and showing their ranking with the colored bar. It appears that only one agent is always in the top 5, most of the times as top contributor, whereas others show significantly less persistence. This is most likely due to our choice of heuristics for address clustering, which possibly clusters together multiple exchanges.\n\nMicroVelocity and wealth. While we have no information about the realworld identities of agents, we can still characterize them by their activity on the blockchain and see how their features correlate with their MicroVelocity. In particular we focus on the relation arising between MicroVelocity and agent wealth M i . As mentioned when we introduced Eq. 5, in an economy where the \"consumption functions\" incorporated P i (τ ) are homogeneous, M i would be the only factor to generate diversity in MicroVelocity. The systems we analyze have high wealth concentration, which is a tendency that is common to cryptocurrencies [22,23,24,25]. This is likely due to large pseudo-banking companies that arise naturally in the environment, acting as exchange markets for fiat currencies or as vaults where crypto users securely store their tokens [26]. In Fig. 5 we show the Spearman's rank correlation coefficient ρ between V i and M i over time. We choose Spearman's coefficient to overcome the limitation of Pearson's correlations for highly heterogeneous data (as discussed in the previous section), where outliers would taint the estimation of the correlation. Fig. 5 shows that the correlation is positive and consistently above 0.6. While it is clear that M i is a strong determinant of V i , as expected from Eq. 5, there is still a significant variance around the linear trend, as we show in the inset scatterplots, and also variation over time. This residual variance is then entirely due to the heterogeneity in P i (τ ), which does not seem to be itself dependent on wealth. One interesting phenomenon seems to appear during late 2017, with ρ decreasing throughout the period of build-up of the crypto bubble that peaked in December 2017 [27].\n\nMicroVelocity and the structure of the economy. We conclude our analysis by considering the role of MicroVelocity in the structure of transaction pat- terns, introducing transaction networks. A network is an ordered pair of sets G = (V, E), called the nodes V and edges (or links) E, where the elements of E ∈ V × V represent connections between elements of V. Identifying agents as nodes and transactions as directed connections from the transaction sender to the receiver, as done for instance in [28], it is possible to obtain a topological description of the economy, highlighting with whom agents exchange tokens and how central they are in the flow of value in the economy. We construct transaction networks on weekly aggregation, which is the minimum timescale that allows to mitigate the effect of day-of-the-week seasonalities that are common in economic and financial data (e.g. weekend effects, higher volumes on Mondays/Fridays, etc.). Operationally this means that a connection from agent i to agent j is present if at least one transaction occurred between them in a given week. We then proceed to take the values of V i at the end of each week and assign them to each node as attributes, which then allows us to consider the assortativity coefficient of these networks with respect to MicroVelocity.\n\nThe assortativity coefficient, defined by [29], is the Pearson correlation coefficient of a given node-specific quantity across linked pairs in a network. It has been introduced to measure how nodes select their neighbors with respect to specific characteristics, most prominently the degree, and it is defined as\nr = (i,j)∈E (x(i) − x )(x(j) − x ) i∈V (x(i) − x ) 2 where x(i)\nis the node-specific property of interest and · indicates the arithmetic mean operator. A positive value of this coefficient indicates homophily, i.e. links mainly connect similar nodes in terms of the selected property, whereas a negative value indicates heterophily, i.e. connections mostly appear between nodes with very different values of x.\n\nIn our case we decide to use the rank in MicroVelocity as the nodes attribute x, thus computing a Spearman's correlation instead of Pearson's, again to account for the fat-tailed nature of the distribution of V i s. We report our finding in Fig. 6, where we plot this rank assortativity coefficient for MicroVelocity on weekly networks as a function of time. The resulting measure is mostly nega-tive, thus suggesting that highly-ranked nodes mostly trade with lowly-ranked ones and vice-versa.\n\n\nDiscussion\n\nIn this article we investigated the microeconomic foundations of the macroeconomic velocity of money, building upon theoretical work [10] and realizing the first empirical measurement of the velocity of money from micro-level data. We leveraged on the technological breakthrough provided by digital currencies and by cryptocurrencies in particular, which offer unprecedented detail about the movement of money in a closed economy where all coins can be followed in every transaction they take part in. We have thus been able to introduce the individual contribution to velocity by single agents, which we have called Mi-croVelocity, as a function of the agent-conditional distribution of holding times of money.\n\nWe argue that insights about the distribution and characterization of Mi-croVelocity are valuable to researchers, policymakers and to the public to gain a deeper understanding of the structure of the economy: in the particular case at hand, we provide evidence that cryptocurrency economies' claim of being \"decentralized\" is largely unjustified, as shown by the extremely skewed distributions of wealth and MicroVelocity we measure. This \"decentalization illusion\" [17] is unmasked by the highly heterogeneous MicroVelocity distribution, indicating that few entities are intermediaries to most of the transactions, since large values of MicroVelocity are easily due to high turnover in assets.\n\nThe fact that we find an extremely positive correlation between MicroVelocity and wealth, as well as negative assortativity of V i in transaction networks, is also corroborating evidence that these economies have seen the rise of pseudobanking services which centralize the supply of money and act as custodians and intermediation services, often without any scrutiny by regulators due to the lack of appropriate legislation.\n\nWe envision multiple directions that could be taken following this work, tackling some limitations of our study as well as expanding its applicability beyond the relatively narrow realm of UTXO-based cryptocurrencies.\n\nThe specific implementation we proposed here takes advantage of the fact that holding times are relatively easy to compute thanks to the age of UTXOs, but this can be easily overcome in the case where this information is not available by considering, for instance, a Last-In-First-Out (LIFO) or First-In-First-Out (FIFO) spending rule, as done for instance in [30]. We argue that a LIFO rule is possibly the most economically significant for a medium of exchange, as it would automatically exclude less liquid portions of the money supply from the calculation, but we leave this extension for future research.\n\nOur approach using empirical probabilities to estimate P i (τ ) has a limitation in the fact that holding times data is right-censored, and this becomes a bigger issue the closer one is to present times. Moreover using empirical probabilities does not fully respect the assumptions on stationarity; nonetheless the convergence between the micro and macro quantities shown in Fig. 2 leads us to think that they still are a good first approximation. We are confident that these issues could be reduced by performing some additional assumptions on the form of the holding times distribution, possibly giving it a parametric form justified by theory (e.g. the Gamma family proposed in [10]). This would mitigate the limitations and possibly allow to produce forecasts as well as predict the outcome of policy interventions.\n\nFigure 2 :\n2Comparison of total MicroVelocity and the ratio of total flow to monetary mass P Q/M , with weekly aggregation. All quantities are annualised.\n\nFigure 4 :\n4Area charts of MicroVelocity and identity of top 5 individual contributors for the four cryptocurrencies, averaged weekly. In each panel the top chart shows the frequency (vertical axis) with which the 5 most represented agents (horizontal axis) figure in the top 5 contributors, with stacked bars colored to show their ranking according to the legend. The bottom charts show the share of V contributed by the top 5 agents as a function of time. We see that one agent is always in the top 5 and that in all cases the vast majority of the total velocity is concentrated in the hands of a few agents.\n\nFigure 5 :\n5Spearman's rank correlation between the MicroVelocity Vi and the wealth Mi of the agent. Insets: scatterplots of Vi/V vs Mi/M in log-log scales at marked dates, with dashed line on the diagonal Vi/V = Mi/M .\n\nFigure 6 :\n6Rank assortativity of MicroVelocity on weekly transaction networks.\n\n\nFigure 1: Schematic representation of a UTXO blockchain. Each block contains transactions, in turn consisting of a set of inputs and outputs. Total input and output values must be matching. An output grants usage rights of some amount of tokens to an address, identified by its public key. Inputs are references to unspent outputs from previous blocks, cryptographically signed by matching the public and private keys. A coinbase transaction generates new tokens as reward to the miner of the block.# block height \n\nblock time \n\nC \n\nOX [1] \n2 \n\ntransactions \n\nspent output [position] and value, \ncryptographically signed \n\ninputs \noutputs \n\ntransaction output [position] and value, \nwith public signature of the recipient \n\nOY [3] \n4 \n\nOC [1] \nOC [2] \nOC [3] \n\n3 \n1 \n2 \n\nA \n\ncoinbase \n25 \nOA [1] \n25 \n\nB \n\nOZ [2] \n10 \nOB [1] \n10 \n\n# 8 \n\n25/03/14 03:12 \n\n83 \n\nO73 [2] \n4 \nO64 [1] \n4 \n\nO83 [1] \nO83 [2] \nO83 [3] \n\n3 \n3 \n2 \n\n81 \n\ncoinbase \n25 \nO81 [1] \n25 \n\n82 \n\nO63 [1] \n11 \nO11 [1] \n15 \n\ngeneration transaction \n\n# 7 \n\n25/03/14 02:24 \n\n73 \n\nO62 [2] \n7 \nO54 [1] \n2 \n\nO73 [1] \nO73 [2] \n\n5 \n4 \n\n71 \n\ncoinbase \n25 \nO71 [1] \n25 \n\n72 \n\nO62 [1] \n8 \nO72 [1] \n7 \n\n# 6 \n\n25/03/14 01:36 \n\n64 \n\nO23 [1] \n4 \nO64 [1] \n4 \n\n61 \n\ncoinbase \n25 \nO61 [1] \n25 \n\n63 \n\nO32 [2] \n11 \nO63 [1] \n11 \n\nO82 [1] \n10 \nO82 [2] \n1 \n\n62 \n\nO42 [1] \n15 \nO62 [1] \n8 \nO62 [2] \n7 \nO72 [2] \n1 \n\n\n\n\n, transactions 62 and 72 constitute a peeling chain, hence addresses O42[1], O62[1] and O72[1] are linked with one wallet.\n\nThe economics of money, banking and financial markets. 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Research in International Business and Finance, 54:101254, 2020.\n\nAlexandre Bovet, Carlo Campajola, Francesco Mottes, Nicolo Valerio Restocchi, Tiziano Vallarano, Claudio J Squartini, Tessone, arXiv:1907.03577The evolving liaisons between the transaction networks of bitcoin and its price dynamics. arXiv preprintAlexandre Bovet, Carlo Campajola, Francesco Mottes, Valerio Restocchi, Nicolo Vallarano, Tiziano Squartini, and Claudio J Tessone. The evolving liaisons between the transaction networks of bitcoin and its price dynamics. arXiv preprint arXiv:1907.03577, 2019.\n\nAssortative mixing in networks. E J Mark, Newman, Physical Review Letters. 8920208701Mark E.J. Newman. Assortative mixing in networks. Physical Review Letters, 89(20):208701, 2002.\n\nCarolina Mattsson, arXiv:1910.05596Networks of monetary flow at native resolution. arXiv preprintCarolina Mattsson. Networks of monetary flow at native resolution. arXiv preprint arXiv:1910.05596, 2019.\n"},"corpus_id":{"kind":"number","value":246430385,"string":"246,430,385"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2201.13416\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2201.13416v2.pdf\"\n ],\n \"pdf_sha\": \"f94fbac8bbcf684f6d03a63b57ee8ad5f8a660a7\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"MicroVelocity: rethinking the Velocity of Money for digital currencies"},"authors":{"kind":"list like","value":["Carlo Campajola \nDLT Science Foundation\nLondonUnited Kingdom\n\nDepartment of Informatics\nUniversity of Zurich\nBlockchain & Distributed Ledger Technologies\nZürichSwitzerland\n\nUZH Blockchain Center\nZürichSwitzerland","Marco D'errico \nEuropean Systemic Risk Board Secretariat\nEuropean Central Bank\nFrankfurtGermany","Claudio J Tessone \nDepartment of Informatics\nUniversity of Zurich\nBlockchain & Distributed Ledger Technologies\nZürichSwitzerland\n\nUZH Blockchain Center\nZürichSwitzerland"],"string":"[\n \"Carlo Campajola \\nDLT Science Foundation\\nLondonUnited Kingdom\\n\\nDepartment of Informatics\\nUniversity of Zurich\\nBlockchain & Distributed Ledger Technologies\\nZürichSwitzerland\\n\\nUZH Blockchain Center\\nZürichSwitzerland\",\n \"Marco D'errico \\nEuropean Systemic Risk Board Secretariat\\nEuropean Central Bank\\nFrankfurtGermany\",\n \"Claudio J Tessone \\nDepartment of Informatics\\nUniversity of Zurich\\nBlockchain & Distributed Ledger Technologies\\nZürichSwitzerland\\n\\nUZH Blockchain Center\\nZürichSwitzerland\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We propose a novel framework to analyse the velocity of money in terms of the contribution (MicroVelocity) of each individual agent, and to uncover the distributional determinants of aggregate velocity. Leveraging on complete publicly available transactions data stored in blockchains from four cryptocurrencies, we empirically find that MicroVelocity i) is very heterogeneously distributed and ii) strongly correlates with agents' wealth. We further document the emergence of high-velocity intermediaries, thereby challenging the idea that these systems are fully decentralised. Further, our framework and results provide policy insights for the development and analysis of digital currencies."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"One of the earliest and most widespread concepts across societies is that of money. As humans we are inherently brought to organise ourselves into communities and exchange the goods and services we produce; this leads to the necessity to invent a medium of exchange that can be store of value -to be able to defer purchases to the future -and unit of account -to properly quantify the value of goods and services [1]. Any such artifact, which may take the form of anything between a large stone in the middle of the village [2], cigarettes [3], cash in the form of coins and banknotes or an encrypted sequence of bits [4], is a form of money, as long as there is more than one person willing to accept it as such.\nOver the years, economic theory has kept interrogating itself on how to efficiently manage this peculiar asset: who is entitled to create it, what is its cost and, most importantly, the role of money on societal and economic development. These and other questions led to the development of monetary theory, which interrogates on how the quantity of money in a system influences economic activity and macroeconomic quantities.\nOne of the main research areas in monetary theory is the quantity theory of money, which revolves around Fisher's equation of exchange [5] M V = P Q (1) stating the proportionality between the monetary mass M and the level of prices P through the economic output Q and V , the velocity of money. The latter is defined as the average frequency with which a given unit of money is used to purchase domestically-produced goods and services within a given time period: this makes Fisher's equation an accounting identity which holds true by construction as both sides amount to the total money flow, which we will call F . The velocity V is often seen as an indicator of money demand [6]: if money demand is low -i.e. expected yield from alternative assets is high -economic agents will convert their money into less liquid investments as soon as possible, thus leading to increased velocity, whereas in situations of high demand for money the velocity will decrease.\nMoney velocity is not directly measurable since it would be impossible to track every unit of currency circulating in the economy. Therefore, the equation of exchange provides a way to estimate velocity at a given point in time. The equation however holds true in aggregate, thus providing limited insights on the distributional aspects, in particular for what concerns a more granular measurement of velocity and how different economic agents (or groups of agents) contribute to the aggregate. In fact, agents are heterogeneous, have different spending habits, different levels of wealth, income and demand for money and different roles in its movement in the economy; but since their different contributions to aggregate velocity are not measurable, policy analysis and interventions cannot be targeted at a finer level.\nHowever, these limitations to the measurement of velocity might be not so strict anymore: over the last few years the increasing diffusion of digital payments -via electronic cards, e-banking or smartphone apps, to name some alternatives -has led to a surge in transactions data availability, meaning that it is now within reach to actually measure the velocity of money from micro-level data, at least for digital forms of money. A big role in this transition towards digital cash will be played by central banks, who are now devoting significant resources towards the development of Central Bank Digital Currencies (CBDCs) [7,8,9] to provide citizens with access to central bank money in digital form.\nA strong push towards digital payments has been prompted by the rise over recent years of cryptocurrencies and blockchain-based tokens. These have emerged as a family of unconventional financial assets and some of them, with Bitcoin being the first and most prominent example, claimed they could be the future of money and would take over the role of fiat currencies. While this has not happened, and outside of crypto-maximalist communities it is now commonplace not to consider Bitcoin and its many offsprings as currencies, the publicly available records of blockchains constitute a unique opportunity to explore the space of microeconomically grounded metrics for macroeconomic quantities like the velocity of money.\nIn particular in this paper we focus on exploiting the accounting protocol adopted by several cryptocurrencies, the Unspent Transaction Output (UTXO) protocol, to actually track specific units of currency as they move from one agent to the other. This protocol imposes that whenever funds have to be spent they are included as inputs of a transaction, and they have explicit reference to the unspent output of a previously validated transaction. This provides verification that the issuer of the new transaction has the right to spend those funds, as the output can only be accessed by means of a cryptographic private key owned by the receiver. A by-product of this mechanism is that every input to a new transaction is explicitly linked to the output of a previous one, and thus the time that those units of money stayed idle is known.\nAs we explain in the following, this level of detail allows to factor the total velocity V into agent-specific contributions, V i , which we call MicroVelocity and depend on agents' behavioral usage of their wealth. This description, based on the seminal work of [10], opens to several questions: how is MicroVelocity distributed cross-sectionally? How does it evolve over time? How does an agent's MicroVelocity correlate with their wealth? How does it correlate with the MicroVelocity of their partners in transactions?\nIn the context of cryptocurrencies, the velocity of money has been used to propose valuation models [11,12], by an argument stating that if the economy adopting cryptocurrency as its medium of exchange grows, so does the velocity due to the available supply being limited by protocol, thus an increase in velocity should also warrant an increase in pricing. Several estimators have been proposed for the velocity of cryptocurrencies, like Coin Days Destroyed [13] or Average Dormancy [14]. In particular Pernice et al. [15] propose an accurate method to estimate velocity on aggregate, which takes advantage of the UTXO protocol in similar ways to our definition of MicroVelocity.\nWe argue that MicroVelocity allows more detailed considerations on the state of the economy using cryptocurrencies. Specifically, statistical properties of its distribution across agents provide insight into the global level of specialization and on centralization in the flow of money: if the distribution is highly heterogeneous, agents with high MicroVelocity are likely to be intermediaries and those with low V i their clients, thus going against the common narrative that cryptocurrencies are \"decentralized economies\" [16]. Paths towards the centralization of these systems are currently attracting growing attention, with Aramonte et al. [17] explicitly talking of \"decentralization illusion\".\nRegardless, our framework is not necessarily limited to the context of blockchainbased tokens. The formulation of MicroVelocity general in nature and can be adapted to find application to any payment system that keeps a digital record of transactions, such as bank transfers, credit cards payments or, in the future, CBDCs. Such an instrument will then be a useful addition to economists' toolboxes when investigating effects of policy, and possibly even opening new policy channels that account for the heterogeneity of economic agents in their usage of money [18,19].\nMicroVelocity. We build on Wang and coauthors' theoretical framework [10], which shows that the macroeconomic velocity can be derived from a microeconomic description: here we review and expand their argument, introducing our definition of MicroVelocity.\nConsider the time between two transactions involving the same unit of currency, the holding time τ , and assume that τ is a random variable with probability distribution p(τ ). A condition p(0) = 0 has to be enforced since a holding time of 0 is meaningless for the description. It follows that, calling M the total number of coins in circulation, M p(τ )dτ is the number of coins with a holding time in the interval [τ, τ + dτ ).\nAssuming stationarity of p(τ ), the money flow F (τ ) generated by coins with holding time τ is\nas 1/τ is the frequency with which each coin moves. Since the total economic flow F is the integral of this quantity over all holding times, we find\nand it thus follows that the velocity reads\nIt has to be noticed that units of currency are not distinguishable and there is no reason to think that a particular coin has an inherently different holding time than any other one. Indeed the authors of [10] justify the existence of the holding times distribution by arguing that holding times depend on money ownership, i.e. the spending habits of different economic agents produce the heterogeneity in τ . While coins are indistinguishable, economic agents are not and each has their own spending, saving and speculative attitude.\nIn the original paper the authors only consider the aggregate effect of this heterogeneity, defining p(τ ) but neglecting to consider the individual effects of agents, as a simulation study accounting for it would have introduced too many arbitrary parameters in the description. Our goal is thus to overcome this limitation, as having access to agent-level data allows for a more fine-grained representation.\nLet us then define p i (τ ) as the probability distribution of holding times of agent i, with i ∈ {1, 2, . . . N }. This means that the total flow of Eq. 3 is factored in a sum of individual contributions, and then Eq. 4 transforms into\nwhere M i is the amount of money held by agent i, meaning that the total velocity is a wealth-weighted average of individual contributions, the MicroVelocity V i . This equation makes it clear that, in a population with homogeneous spending preferences such that p i (τ ) ≈ p(τ ) ∀i, contributions to money velocity are fully determined by wealth. However real-world economies involve heterogeneous participants, both on the macro-scale (households, companies, business sectors) and on the micro-scale (preferences, consumption functions): our framework fully accounts for these factors, returning a detailed picture of the impact they have on the velocity.\nBlockchains and the UTXO protocol. To estimate MicroVelocity from real-world data one needs fine-grained transaction records, so that the holding times τ can be measured and its statistical properties estimated. The advent of blockchains and cryptocurrencies offers a unique occasion to study economic activity on a publicly available transaction ledger. A blockchain is a distributed ledger used to store information upon which a large community agrees without having a central authority that coordinates the consensus. Avoiding going into the details of how the consensus protocol works, what matters for this paper is the accounting standard that is used to store the data on the blockchain. There are several alternatives that are currently being used, but the most popular so far -and first to be implemented in Bitcoin -is the Unspent Transaction Output (UTXO).\nIn this protocol, schematised in Fig. 1, information is stored on the blockchain in form of transactions, transferring usage rights of some quantity of digital asset from one cryptographic address to another. These addresses consist of two parts: a public key, visible to everybody and used for verification, and a private key, which is information only available to the address owner. When an agent wants to transfer some assets to another agent, they need to point to a previously generated transaction output that has not been spent yet and, using their private key, demonstrate that they own the address that output was destined to. This shows they own the usage rights on the assets in that output, and they are able to transfer them by pointing to an address belonging to the new transaction receiver.\nSince all this information is stored on the blockchain, this means that the time occurring between the generation of the UTXO and its spending transaction is easily available, thus allowing to measure τ for all coins without further assumptions. These can then be used to estimate p i (τ ) depending on ownership, thus providing all the necessary information to calculate the MicroVelocity of Eq. 5.\nMeasuring MicroVelocity on blockchains. Events are recorded on the blockchain once every few minutes, when a new block is added. The consequence is that time is effectively discretised, and τ can be expressed in terms of number of blocks between transactions, τ ∈ N \\ {0}. Following the logic above, let us define P i (τ ) to be the probability mass function of the discrete holding times for user i's coins. Then M i P i (τ ) coins by user i have holding time τ , resulting at stationarity in a contribution to user i's flow of\nIt then follows that the total velocity, given in continuous time by Eq. 5, turns into the following formula in block time\nIsolating the i-th component of this sum we are then able to define user i's contribution to the macroeconomic velocity of money, i.e. the MicroVelocity, which is\nThe question is then how to measure P i (τ ) to be able to compute these quantities. For the specific case of UTXO-based cryptocurrencies, a very simple estimator can be adopted: the accounting protocol itself produces the holding time τ of each unit of currency as a by-product, hence the empirical probability of these holding times conditional on the agent can be estimated. For example, in Fig. 1, the inputs of transaction 83 have the same value (4 tokens) but different holding time of two blocks (O64 [1]) and of one block (O73 [2]). This means, according to Eq. 7, that tokens in O64 [1] generate half as much velocity as the ones in O73 [2].\nThere are two further cautions that need to be mentioned. The first is that in principle there is no guarantee that a transaction recorded on the blockchain is actually a transfer of value from an agent to another: addresses do not have a one-to-one correspondence with agents' wallets, and in fact a wallet can contain an arbitrary number of addresses. Moreover, by protocol, inputs of transactions have to be spent in full each time: the total value of inputs has to match the total value of outputs, and the value of inputs is constrained by the UTXO they point to. The way in which this is solved is by generating an extra output, called change output, which points the excess input to an address which is actually owned by the sender. These transactions do not actually move funds between individuals, thus they should be excluded from the calculation of MicroVelocity.\nTo solve this issue, when we find an output that is likely to belong to the transaction sender, we do not count it on the MicroVelocity and add the weighted average holding time of the inputs of that transaction to its holding time. There are several ways to identify self-transactions through heuristic methods which take advantage of typical patterns used by cryptocurrency wallet clients. We provide further details about this in the Materials and Methods section.\nThe second caution that is needed is that, when calculating MicroVelocity, the measurable holding times are right-censored, or in other words one would need future information to know the holding time of current UTXOs. While there is no straightforward solution to this, it is in principle possible to devise more refined statistical methods to infer P i (τ ) so that one can reduce this issue, while also controlling for the impact of non-stationarity in the estimation, which we have neglected in this work. We choose not to tackle this problem, as it would require some arbitrary assumptions, and defer the exploration of solutions to future research."},{"title":"Materials and Methods","content":"Data processing. We directly collect data from the Litecoin, Dogecoin, Monacoin and Feathercoin blockchains by synchronizing full nodes in archive mode. The complete history has then been preprocessed using the Blocksci Python library [20], which turns raw data files into an efficiently searchable database structure.\nThere are many heuristic methods that have been proposed to assign addresses to unique wallets. Here we combine three heuristics, the MultiInput (MI), the Change Address (CA) and the Peeling Chains (PC):\n• MI takes advantage of the fact that multiple addresses appearing as inputs of the same transaction have a high likelihood of belonging to the same agent, as they had to be signed with all the private keys together. For example in Fig. 1, O64 [1] and O73 [2] are assigned to the same wallet since they appear together as inputs of transaction 83;\n• CA follows from the argument that, when there is more than one input and more than one output in a transaction, it is likely that the smallest output is a change address, provided it is smaller than the smallest input (otherwise such input would have not been necessary) and that it is a new address; if one such output is identified, it is considered to belong to the same agent as the input addresses. In Fig. 1 • Finally, PC identifies transaction patterns called peeling chains: sequences of transactions with a single input and two outputs, that arise when an agent needs to spend a large value UTXO (like a coinbase transaction), generating chains of transactions where the UTXO is slowly consumed. In order to identify a peeling chain, at least two subsequent transactions have to match this pattern. If that is the case, the outputs that connect the chain are assigned to the same wallet as the inputs. In Fig. 1 When using heuristics to cluster together output and input addresses, there is a significant risk of merging multiple wallets together. For this reason we adopt some logical selection rules to limit the impact of this issue. In particular we apply the following conditions:\n• if different heuristics disagree on the change address, no change address is selected;\n• if all heuristics agree on the change address or select none, that address is considered the change;\n• if an input address appears also in the output, it is necessarily the change and all heuristics are neglected.\nIn the Supplementary Information we check the robustness of the results presented in the following sections against the choice of address clustering method.\nEstimation of MicroVelocity. The estimation of MicroVelocity follows from the sum of Eq. 7. At a given time t, an agent owns M i (t) coins, each with some holding time τ that is estimated as the age of the UTXO they are included in when they are spent. Then the time t probability mass function of holding times of agent i, P t i (τ ), is measured as the empirical probability where w i (τ ) is the total value of UTXOs with holding time τ , ensuring that\nPlugging this into Eq. 5 gives the empirical value of agent i's MicroVelocity at time t\nwhere M (t) is the total amount of money in the system at the time. Notice that this value is known by construction, since the supply of cryptocurrencies is deterministic. We neglect additional considerations on the definition of circulating supply, as they would require arbitrary assumptions and we believe they would have a marginal impact on our results. Indeed, since the holding time τ appears at the denominator of Eq. 7, coins with a longer holding time that may be classified in less liquid monetary aggregates already contribute less to the velocity."},{"title":"Results","content":"The distribution of MicroVelocity. We begin our analysis by checking if the total MicroVelocity we measure correctly matches the velocity calculated following Fisher's Eq. 1. In Figure 2 we compare the sum of MicroVelocities i V i , averaged weekly, with the average volume of transactions P Q divided by the total money supply M on that week. Both quantities are annualised to facilitate comparison. It appears clear that Eq. 6 holds true, with the total MicroVelocity even producing a slightly less volatile estimate of V . This is not surprising, since MicroVelocity accounts for components of V at all timescales whereas P Q/M only considers transactions happening on a specific time window (in this case a week), thus being more prone to temporary fluctuations.\nHowever, as discussed above, the advantage of having this microeconomic picture is that we are able to investigate the distributional aspects of the velocity of money. We then proceed to take weekly cross-sectional snapshots of the agents' V i s, which we then utilize to extract a number of statistics and test hypotheses. First, we remove values of V i (t) = 0 from our analysis. These may occur for multiple reasons: the agent may have not entered the market yet or has quit, thus they will have 0 wealth and velocity, but it may also be that they own coins which they never spend. The latter can itself be caused by a behavioral choice of holding the coins indefinitely -the so-called \"HODLers\" -, by the right-censoring of data we previously discussed or by technical problems such as lost passwords or hacks: this uncertainty leads us to exclude also agents with non-zero wealth that have zero MicroVelocity. We then proceed to analyze the cross-sectional distribution by means of the Kolmogorov-Smirnov distance D from a known distribution. This is defined as\nis the empirical distribution of MicroVelocities and P (V i ) is an arbitrary probability distribution. This distance is the test statistic for the twotailed Kolmogorov-Smirnov test, which tests the null hypothesis thatP (V i ) ≡ P (V i ). Since V i ≥ 0 by definition, we choose to take as comparisons the exponential and Pareto distributions, to gain insight on whether the data is closer to a thin-or to a fat-tailed distribution. We fit the parameters of these distributions by Maximum Likelihood methods and calculate the distances D P ar and D Exp , obtaining the results summarized in Table 3. While the thin-tailed exponential null hypothesis is always rejected by the Kolmogorov-Smirnov test, it is clear that a fat-tailed Pareto distribution with tail exponent α ≈ 1.6 is a much better descriptor for the distribution generating the data. Notice that we choose a very high significance threshold of 0.2, since we want to check the robustness of the null hypothesis rather than identifying an alternative. Despite this, the test almost never rejects the Pareto null, consolidating the observation that the total velocity of money is very unevenly distributed across agents, with a small minority dominating the sum of Eq. 7.\nPerhaps the most striking evidence of this comes from the stacked charts of Fig. 4. In each panel's bottom chart, each colored area corresponds to the fraction of total velocity that is produced by the 5 largest contributors, compared with the total contributions by all others, tracked over time with weekly aggregation. In all the analyzed cryptocurrencies the vast majority of the total velocity is generated by less than 5 agents: most likely these -whose actual identity is unknown to us -are companies that act as intermediaries, such as exchange markets, custodians or layer 2 payment services like the Lightning Network [21]. Despite the real-world identity of these agents being unknown, we can still track them by their ID from the address clustering algorithm, which means we are able to count how frequently each agent features as a top contributor. The top charts of Fig. 4 show exactly this, counting the percentage of weeks a given agent is among the top 5 and showing their ranking with the colored bar. It appears that only one agent is always in the top 5, most of the times as top contributor, whereas others show significantly less persistence. This is most likely due to our choice of heuristics for address clustering, which possibly clusters together multiple exchanges.\nMicroVelocity and wealth. While we have no information about the realworld identities of agents, we can still characterize them by their activity on the blockchain and see how their features correlate with their MicroVelocity. In particular we focus on the relation arising between MicroVelocity and agent wealth M i . As mentioned when we introduced Eq. 5, in an economy where the \"consumption functions\" incorporated P i (τ ) are homogeneous, M i would be the only factor to generate diversity in MicroVelocity. The systems we analyze have high wealth concentration, which is a tendency that is common to cryptocurrencies [22,23,24,25]. This is likely due to large pseudo-banking companies that arise naturally in the environment, acting as exchange markets for fiat currencies or as vaults where crypto users securely store their tokens [26]. In Fig. 5 we show the Spearman's rank correlation coefficient ρ between V i and M i over time. We choose Spearman's coefficient to overcome the limitation of Pearson's correlations for highly heterogeneous data (as discussed in the previous section), where outliers would taint the estimation of the correlation. Fig. 5 shows that the correlation is positive and consistently above 0.6. While it is clear that M i is a strong determinant of V i , as expected from Eq. 5, there is still a significant variance around the linear trend, as we show in the inset scatterplots, and also variation over time. This residual variance is then entirely due to the heterogeneity in P i (τ ), which does not seem to be itself dependent on wealth. One interesting phenomenon seems to appear during late 2017, with ρ decreasing throughout the period of build-up of the crypto bubble that peaked in December 2017 [27].\nMicroVelocity and the structure of the economy. We conclude our analysis by considering the role of MicroVelocity in the structure of transaction pat- terns, introducing transaction networks. A network is an ordered pair of sets G = (V, E), called the nodes V and edges (or links) E, where the elements of E ∈ V × V represent connections between elements of V. Identifying agents as nodes and transactions as directed connections from the transaction sender to the receiver, as done for instance in [28], it is possible to obtain a topological description of the economy, highlighting with whom agents exchange tokens and how central they are in the flow of value in the economy. We construct transaction networks on weekly aggregation, which is the minimum timescale that allows to mitigate the effect of day-of-the-week seasonalities that are common in economic and financial data (e.g. weekend effects, higher volumes on Mondays/Fridays, etc.). Operationally this means that a connection from agent i to agent j is present if at least one transaction occurred between them in a given week. We then proceed to take the values of V i at the end of each week and assign them to each node as attributes, which then allows us to consider the assortativity coefficient of these networks with respect to MicroVelocity.\nThe assortativity coefficient, defined by [29], is the Pearson correlation coefficient of a given node-specific quantity across linked pairs in a network. It has been introduced to measure how nodes select their neighbors with respect to specific characteristics, most prominently the degree, and it is defined as\nis the node-specific property of interest and · indicates the arithmetic mean operator. A positive value of this coefficient indicates homophily, i.e. links mainly connect similar nodes in terms of the selected property, whereas a negative value indicates heterophily, i.e. connections mostly appear between nodes with very different values of x.\nIn our case we decide to use the rank in MicroVelocity as the nodes attribute x, thus computing a Spearman's correlation instead of Pearson's, again to account for the fat-tailed nature of the distribution of V i s. We report our finding in Fig. 6, where we plot this rank assortativity coefficient for MicroVelocity on weekly networks as a function of time. The resulting measure is mostly nega-tive, thus suggesting that highly-ranked nodes mostly trade with lowly-ranked ones and vice-versa."},{"title":"Discussion","content":"In this article we investigated the microeconomic foundations of the macroeconomic velocity of money, building upon theoretical work [10] and realizing the first empirical measurement of the velocity of money from micro-level data. We leveraged on the technological breakthrough provided by digital currencies and by cryptocurrencies in particular, which offer unprecedented detail about the movement of money in a closed economy where all coins can be followed in every transaction they take part in. We have thus been able to introduce the individual contribution to velocity by single agents, which we have called Mi-croVelocity, as a function of the agent-conditional distribution of holding times of money.\nWe argue that insights about the distribution and characterization of Mi-croVelocity are valuable to researchers, policymakers and to the public to gain a deeper understanding of the structure of the economy: in the particular case at hand, we provide evidence that cryptocurrency economies' claim of being \"decentralized\" is largely unjustified, as shown by the extremely skewed distributions of wealth and MicroVelocity we measure. This \"decentalization illusion\" [17] is unmasked by the highly heterogeneous MicroVelocity distribution, indicating that few entities are intermediaries to most of the transactions, since large values of MicroVelocity are easily due to high turnover in assets.\nThe fact that we find an extremely positive correlation between MicroVelocity and wealth, as well as negative assortativity of V i in transaction networks, is also corroborating evidence that these economies have seen the rise of pseudobanking services which centralize the supply of money and act as custodians and intermediation services, often without any scrutiny by regulators due to the lack of appropriate legislation.\nWe envision multiple directions that could be taken following this work, tackling some limitations of our study as well as expanding its applicability beyond the relatively narrow realm of UTXO-based cryptocurrencies.\nThe specific implementation we proposed here takes advantage of the fact that holding times are relatively easy to compute thanks to the age of UTXOs, but this can be easily overcome in the case where this information is not available by considering, for instance, a Last-In-First-Out (LIFO) or First-In-First-Out (FIFO) spending rule, as done for instance in [30]. We argue that a LIFO rule is possibly the most economically significant for a medium of exchange, as it would automatically exclude less liquid portions of the money supply from the calculation, but we leave this extension for future research.\nOur approach using empirical probabilities to estimate P i (τ ) has a limitation in the fact that holding times data is right-censored, and this becomes a bigger issue the closer one is to present times. Moreover using empirical probabilities does not fully respect the assumptions on stationarity; nonetheless the convergence between the micro and macro quantities shown in Fig. 2 leads us to think that they still are a good first approximation. We are confident that these issues could be reduced by performing some additional assumptions on the form of the holding times distribution, possibly giving it a parametric form justified by theory (e.g. the Gamma family proposed in [10]). This would mitigate the limitations and possibly allow to produce forecasts as well as predict the outcome of policy interventions."},{"title":"Figure 2 :","content":""},{"title":"Figure 4 :","content":""},{"title":"Figure 5 :","content":""},{"title":"Figure 6 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"One of the earliest and most widespread concepts across societies is that of money. As humans we are inherently brought to organise ourselves into communities and exchange the goods and services we produce; this leads to the necessity to invent a medium of exchange that can be store of value -to be able to defer purchases to the future -and unit of account -to properly quantify the value of goods and services [1]. Any such artifact, which may take the form of anything between a large stone in the middle of the village [2], cigarettes [3], cash in the form of coins and banknotes or an encrypted sequence of bits [4], is a form of money, as long as there is more than one person willing to accept it as such.\\nOver the years, economic theory has kept interrogating itself on how to efficiently manage this peculiar asset: who is entitled to create it, what is its cost and, most importantly, the role of money on societal and economic development. These and other questions led to the development of monetary theory, which interrogates on how the quantity of money in a system influences economic activity and macroeconomic quantities.\\nOne of the main research areas in monetary theory is the quantity theory of money, which revolves around Fisher's equation of exchange [5] M V = P Q (1) stating the proportionality between the monetary mass M and the level of prices P through the economic output Q and V , the velocity of money. The latter is defined as the average frequency with which a given unit of money is used to purchase domestically-produced goods and services within a given time period: this makes Fisher's equation an accounting identity which holds true by construction as both sides amount to the total money flow, which we will call F . The velocity V is often seen as an indicator of money demand [6]: if money demand is low -i.e. expected yield from alternative assets is high -economic agents will convert their money into less liquid investments as soon as possible, thus leading to increased velocity, whereas in situations of high demand for money the velocity will decrease.\\nMoney velocity is not directly measurable since it would be impossible to track every unit of currency circulating in the economy. Therefore, the equation of exchange provides a way to estimate velocity at a given point in time. The equation however holds true in aggregate, thus providing limited insights on the distributional aspects, in particular for what concerns a more granular measurement of velocity and how different economic agents (or groups of agents) contribute to the aggregate. In fact, agents are heterogeneous, have different spending habits, different levels of wealth, income and demand for money and different roles in its movement in the economy; but since their different contributions to aggregate velocity are not measurable, policy analysis and interventions cannot be targeted at a finer level.\\nHowever, these limitations to the measurement of velocity might be not so strict anymore: over the last few years the increasing diffusion of digital payments -via electronic cards, e-banking or smartphone apps, to name some alternatives -has led to a surge in transactions data availability, meaning that it is now within reach to actually measure the velocity of money from micro-level data, at least for digital forms of money. A big role in this transition towards digital cash will be played by central banks, who are now devoting significant resources towards the development of Central Bank Digital Currencies (CBDCs) [7,8,9] to provide citizens with access to central bank money in digital form.\\nA strong push towards digital payments has been prompted by the rise over recent years of cryptocurrencies and blockchain-based tokens. These have emerged as a family of unconventional financial assets and some of them, with Bitcoin being the first and most prominent example, claimed they could be the future of money and would take over the role of fiat currencies. While this has not happened, and outside of crypto-maximalist communities it is now commonplace not to consider Bitcoin and its many offsprings as currencies, the publicly available records of blockchains constitute a unique opportunity to explore the space of microeconomically grounded metrics for macroeconomic quantities like the velocity of money.\\nIn particular in this paper we focus on exploiting the accounting protocol adopted by several cryptocurrencies, the Unspent Transaction Output (UTXO) protocol, to actually track specific units of currency as they move from one agent to the other. This protocol imposes that whenever funds have to be spent they are included as inputs of a transaction, and they have explicit reference to the unspent output of a previously validated transaction. This provides verification that the issuer of the new transaction has the right to spend those funds, as the output can only be accessed by means of a cryptographic private key owned by the receiver. A by-product of this mechanism is that every input to a new transaction is explicitly linked to the output of a previous one, and thus the time that those units of money stayed idle is known.\\nAs we explain in the following, this level of detail allows to factor the total velocity V into agent-specific contributions, V i , which we call MicroVelocity and depend on agents' behavioral usage of their wealth. This description, based on the seminal work of [10], opens to several questions: how is MicroVelocity distributed cross-sectionally? How does it evolve over time? How does an agent's MicroVelocity correlate with their wealth? How does it correlate with the MicroVelocity of their partners in transactions?\\nIn the context of cryptocurrencies, the velocity of money has been used to propose valuation models [11,12], by an argument stating that if the economy adopting cryptocurrency as its medium of exchange grows, so does the velocity due to the available supply being limited by protocol, thus an increase in velocity should also warrant an increase in pricing. Several estimators have been proposed for the velocity of cryptocurrencies, like Coin Days Destroyed [13] or Average Dormancy [14]. In particular Pernice et al. [15] propose an accurate method to estimate velocity on aggregate, which takes advantage of the UTXO protocol in similar ways to our definition of MicroVelocity.\\nWe argue that MicroVelocity allows more detailed considerations on the state of the economy using cryptocurrencies. Specifically, statistical properties of its distribution across agents provide insight into the global level of specialization and on centralization in the flow of money: if the distribution is highly heterogeneous, agents with high MicroVelocity are likely to be intermediaries and those with low V i their clients, thus going against the common narrative that cryptocurrencies are \\\"decentralized economies\\\" [16]. Paths towards the centralization of these systems are currently attracting growing attention, with Aramonte et al. [17] explicitly talking of \\\"decentralization illusion\\\".\\nRegardless, our framework is not necessarily limited to the context of blockchainbased tokens. The formulation of MicroVelocity general in nature and can be adapted to find application to any payment system that keeps a digital record of transactions, such as bank transfers, credit cards payments or, in the future, CBDCs. Such an instrument will then be a useful addition to economists' toolboxes when investigating effects of policy, and possibly even opening new policy channels that account for the heterogeneity of economic agents in their usage of money [18,19].\\nMicroVelocity. We build on Wang and coauthors' theoretical framework [10], which shows that the macroeconomic velocity can be derived from a microeconomic description: here we review and expand their argument, introducing our definition of MicroVelocity.\\nConsider the time between two transactions involving the same unit of currency, the holding time τ , and assume that τ is a random variable with probability distribution p(τ ). A condition p(0) = 0 has to be enforced since a holding time of 0 is meaningless for the description. It follows that, calling M the total number of coins in circulation, M p(τ )dτ is the number of coins with a holding time in the interval [τ, τ + dτ ).\\nAssuming stationarity of p(τ ), the money flow F (τ ) generated by coins with holding time τ is\\nas 1/τ is the frequency with which each coin moves. Since the total economic flow F is the integral of this quantity over all holding times, we find\\nand it thus follows that the velocity reads\\nIt has to be noticed that units of currency are not distinguishable and there is no reason to think that a particular coin has an inherently different holding time than any other one. Indeed the authors of [10] justify the existence of the holding times distribution by arguing that holding times depend on money ownership, i.e. the spending habits of different economic agents produce the heterogeneity in τ . While coins are indistinguishable, economic agents are not and each has their own spending, saving and speculative attitude.\\nIn the original paper the authors only consider the aggregate effect of this heterogeneity, defining p(τ ) but neglecting to consider the individual effects of agents, as a simulation study accounting for it would have introduced too many arbitrary parameters in the description. Our goal is thus to overcome this limitation, as having access to agent-level data allows for a more fine-grained representation.\\nLet us then define p i (τ ) as the probability distribution of holding times of agent i, with i ∈ {1, 2, . . . N }. This means that the total flow of Eq. 3 is factored in a sum of individual contributions, and then Eq. 4 transforms into\\nwhere M i is the amount of money held by agent i, meaning that the total velocity is a wealth-weighted average of individual contributions, the MicroVelocity V i . This equation makes it clear that, in a population with homogeneous spending preferences such that p i (τ ) ≈ p(τ ) ∀i, contributions to money velocity are fully determined by wealth. However real-world economies involve heterogeneous participants, both on the macro-scale (households, companies, business sectors) and on the micro-scale (preferences, consumption functions): our framework fully accounts for these factors, returning a detailed picture of the impact they have on the velocity.\\nBlockchains and the UTXO protocol. To estimate MicroVelocity from real-world data one needs fine-grained transaction records, so that the holding times τ can be measured and its statistical properties estimated. The advent of blockchains and cryptocurrencies offers a unique occasion to study economic activity on a publicly available transaction ledger. A blockchain is a distributed ledger used to store information upon which a large community agrees without having a central authority that coordinates the consensus. Avoiding going into the details of how the consensus protocol works, what matters for this paper is the accounting standard that is used to store the data on the blockchain. There are several alternatives that are currently being used, but the most popular so far -and first to be implemented in Bitcoin -is the Unspent Transaction Output (UTXO).\\nIn this protocol, schematised in Fig. 1, information is stored on the blockchain in form of transactions, transferring usage rights of some quantity of digital asset from one cryptographic address to another. These addresses consist of two parts: a public key, visible to everybody and used for verification, and a private key, which is information only available to the address owner. When an agent wants to transfer some assets to another agent, they need to point to a previously generated transaction output that has not been spent yet and, using their private key, demonstrate that they own the address that output was destined to. This shows they own the usage rights on the assets in that output, and they are able to transfer them by pointing to an address belonging to the new transaction receiver.\\nSince all this information is stored on the blockchain, this means that the time occurring between the generation of the UTXO and its spending transaction is easily available, thus allowing to measure τ for all coins without further assumptions. These can then be used to estimate p i (τ ) depending on ownership, thus providing all the necessary information to calculate the MicroVelocity of Eq. 5.\\nMeasuring MicroVelocity on blockchains. Events are recorded on the blockchain once every few minutes, when a new block is added. The consequence is that time is effectively discretised, and τ can be expressed in terms of number of blocks between transactions, τ ∈ N \\\\ {0}. Following the logic above, let us define P i (τ ) to be the probability mass function of the discrete holding times for user i's coins. Then M i P i (τ ) coins by user i have holding time τ , resulting at stationarity in a contribution to user i's flow of\\nIt then follows that the total velocity, given in continuous time by Eq. 5, turns into the following formula in block time\\nIsolating the i-th component of this sum we are then able to define user i's contribution to the macroeconomic velocity of money, i.e. the MicroVelocity, which is\\nThe question is then how to measure P i (τ ) to be able to compute these quantities. For the specific case of UTXO-based cryptocurrencies, a very simple estimator can be adopted: the accounting protocol itself produces the holding time τ of each unit of currency as a by-product, hence the empirical probability of these holding times conditional on the agent can be estimated. For example, in Fig. 1, the inputs of transaction 83 have the same value (4 tokens) but different holding time of two blocks (O64 [1]) and of one block (O73 [2]). This means, according to Eq. 7, that tokens in O64 [1] generate half as much velocity as the ones in O73 [2].\\nThere are two further cautions that need to be mentioned. The first is that in principle there is no guarantee that a transaction recorded on the blockchain is actually a transfer of value from an agent to another: addresses do not have a one-to-one correspondence with agents' wallets, and in fact a wallet can contain an arbitrary number of addresses. Moreover, by protocol, inputs of transactions have to be spent in full each time: the total value of inputs has to match the total value of outputs, and the value of inputs is constrained by the UTXO they point to. The way in which this is solved is by generating an extra output, called change output, which points the excess input to an address which is actually owned by the sender. These transactions do not actually move funds between individuals, thus they should be excluded from the calculation of MicroVelocity.\\nTo solve this issue, when we find an output that is likely to belong to the transaction sender, we do not count it on the MicroVelocity and add the weighted average holding time of the inputs of that transaction to its holding time. There are several ways to identify self-transactions through heuristic methods which take advantage of typical patterns used by cryptocurrency wallet clients. We provide further details about this in the Materials and Methods section.\\nThe second caution that is needed is that, when calculating MicroVelocity, the measurable holding times are right-censored, or in other words one would need future information to know the holding time of current UTXOs. While there is no straightforward solution to this, it is in principle possible to devise more refined statistical methods to infer P i (τ ) so that one can reduce this issue, while also controlling for the impact of non-stationarity in the estimation, which we have neglected in this work. We choose not to tackle this problem, as it would require some arbitrary assumptions, and defer the exploration of solutions to future research.\"\n },\n {\n \"title\": \"Materials and Methods\",\n \"content\": \"Data processing. We directly collect data from the Litecoin, Dogecoin, Monacoin and Feathercoin blockchains by synchronizing full nodes in archive mode. The complete history has then been preprocessed using the Blocksci Python library [20], which turns raw data files into an efficiently searchable database structure.\\nThere are many heuristic methods that have been proposed to assign addresses to unique wallets. Here we combine three heuristics, the MultiInput (MI), the Change Address (CA) and the Peeling Chains (PC):\\n• MI takes advantage of the fact that multiple addresses appearing as inputs of the same transaction have a high likelihood of belonging to the same agent, as they had to be signed with all the private keys together. For example in Fig. 1, O64 [1] and O73 [2] are assigned to the same wallet since they appear together as inputs of transaction 83;\\n• CA follows from the argument that, when there is more than one input and more than one output in a transaction, it is likely that the smallest output is a change address, provided it is smaller than the smallest input (otherwise such input would have not been necessary) and that it is a new address; if one such output is identified, it is considered to belong to the same agent as the input addresses. In Fig. 1 • Finally, PC identifies transaction patterns called peeling chains: sequences of transactions with a single input and two outputs, that arise when an agent needs to spend a large value UTXO (like a coinbase transaction), generating chains of transactions where the UTXO is slowly consumed. In order to identify a peeling chain, at least two subsequent transactions have to match this pattern. If that is the case, the outputs that connect the chain are assigned to the same wallet as the inputs. In Fig. 1 When using heuristics to cluster together output and input addresses, there is a significant risk of merging multiple wallets together. For this reason we adopt some logical selection rules to limit the impact of this issue. In particular we apply the following conditions:\\n• if different heuristics disagree on the change address, no change address is selected;\\n• if all heuristics agree on the change address or select none, that address is considered the change;\\n• if an input address appears also in the output, it is necessarily the change and all heuristics are neglected.\\nIn the Supplementary Information we check the robustness of the results presented in the following sections against the choice of address clustering method.\\nEstimation of MicroVelocity. The estimation of MicroVelocity follows from the sum of Eq. 7. At a given time t, an agent owns M i (t) coins, each with some holding time τ that is estimated as the age of the UTXO they are included in when they are spent. Then the time t probability mass function of holding times of agent i, P t i (τ ), is measured as the empirical probability where w i (τ ) is the total value of UTXOs with holding time τ , ensuring that\\nPlugging this into Eq. 5 gives the empirical value of agent i's MicroVelocity at time t\\nwhere M (t) is the total amount of money in the system at the time. Notice that this value is known by construction, since the supply of cryptocurrencies is deterministic. We neglect additional considerations on the definition of circulating supply, as they would require arbitrary assumptions and we believe they would have a marginal impact on our results. Indeed, since the holding time τ appears at the denominator of Eq. 7, coins with a longer holding time that may be classified in less liquid monetary aggregates already contribute less to the velocity.\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"The distribution of MicroVelocity. We begin our analysis by checking if the total MicroVelocity we measure correctly matches the velocity calculated following Fisher's Eq. 1. In Figure 2 we compare the sum of MicroVelocities i V i , averaged weekly, with the average volume of transactions P Q divided by the total money supply M on that week. Both quantities are annualised to facilitate comparison. It appears clear that Eq. 6 holds true, with the total MicroVelocity even producing a slightly less volatile estimate of V . This is not surprising, since MicroVelocity accounts for components of V at all timescales whereas P Q/M only considers transactions happening on a specific time window (in this case a week), thus being more prone to temporary fluctuations.\\nHowever, as discussed above, the advantage of having this microeconomic picture is that we are able to investigate the distributional aspects of the velocity of money. We then proceed to take weekly cross-sectional snapshots of the agents' V i s, which we then utilize to extract a number of statistics and test hypotheses. First, we remove values of V i (t) = 0 from our analysis. These may occur for multiple reasons: the agent may have not entered the market yet or has quit, thus they will have 0 wealth and velocity, but it may also be that they own coins which they never spend. The latter can itself be caused by a behavioral choice of holding the coins indefinitely -the so-called \\\"HODLers\\\" -, by the right-censoring of data we previously discussed or by technical problems such as lost passwords or hacks: this uncertainty leads us to exclude also agents with non-zero wealth that have zero MicroVelocity. We then proceed to analyze the cross-sectional distribution by means of the Kolmogorov-Smirnov distance D from a known distribution. This is defined as\\nis the empirical distribution of MicroVelocities and P (V i ) is an arbitrary probability distribution. This distance is the test statistic for the twotailed Kolmogorov-Smirnov test, which tests the null hypothesis thatP (V i ) ≡ P (V i ). Since V i ≥ 0 by definition, we choose to take as comparisons the exponential and Pareto distributions, to gain insight on whether the data is closer to a thin-or to a fat-tailed distribution. We fit the parameters of these distributions by Maximum Likelihood methods and calculate the distances D P ar and D Exp , obtaining the results summarized in Table 3. While the thin-tailed exponential null hypothesis is always rejected by the Kolmogorov-Smirnov test, it is clear that a fat-tailed Pareto distribution with tail exponent α ≈ 1.6 is a much better descriptor for the distribution generating the data. Notice that we choose a very high significance threshold of 0.2, since we want to check the robustness of the null hypothesis rather than identifying an alternative. Despite this, the test almost never rejects the Pareto null, consolidating the observation that the total velocity of money is very unevenly distributed across agents, with a small minority dominating the sum of Eq. 7.\\nPerhaps the most striking evidence of this comes from the stacked charts of Fig. 4. In each panel's bottom chart, each colored area corresponds to the fraction of total velocity that is produced by the 5 largest contributors, compared with the total contributions by all others, tracked over time with weekly aggregation. In all the analyzed cryptocurrencies the vast majority of the total velocity is generated by less than 5 agents: most likely these -whose actual identity is unknown to us -are companies that act as intermediaries, such as exchange markets, custodians or layer 2 payment services like the Lightning Network [21]. Despite the real-world identity of these agents being unknown, we can still track them by their ID from the address clustering algorithm, which means we are able to count how frequently each agent features as a top contributor. The top charts of Fig. 4 show exactly this, counting the percentage of weeks a given agent is among the top 5 and showing their ranking with the colored bar. It appears that only one agent is always in the top 5, most of the times as top contributor, whereas others show significantly less persistence. This is most likely due to our choice of heuristics for address clustering, which possibly clusters together multiple exchanges.\\nMicroVelocity and wealth. While we have no information about the realworld identities of agents, we can still characterize them by their activity on the blockchain and see how their features correlate with their MicroVelocity. In particular we focus on the relation arising between MicroVelocity and agent wealth M i . As mentioned when we introduced Eq. 5, in an economy where the \\\"consumption functions\\\" incorporated P i (τ ) are homogeneous, M i would be the only factor to generate diversity in MicroVelocity. The systems we analyze have high wealth concentration, which is a tendency that is common to cryptocurrencies [22,23,24,25]. This is likely due to large pseudo-banking companies that arise naturally in the environment, acting as exchange markets for fiat currencies or as vaults where crypto users securely store their tokens [26]. In Fig. 5 we show the Spearman's rank correlation coefficient ρ between V i and M i over time. We choose Spearman's coefficient to overcome the limitation of Pearson's correlations for highly heterogeneous data (as discussed in the previous section), where outliers would taint the estimation of the correlation. Fig. 5 shows that the correlation is positive and consistently above 0.6. While it is clear that M i is a strong determinant of V i , as expected from Eq. 5, there is still a significant variance around the linear trend, as we show in the inset scatterplots, and also variation over time. This residual variance is then entirely due to the heterogeneity in P i (τ ), which does not seem to be itself dependent on wealth. One interesting phenomenon seems to appear during late 2017, with ρ decreasing throughout the period of build-up of the crypto bubble that peaked in December 2017 [27].\\nMicroVelocity and the structure of the economy. We conclude our analysis by considering the role of MicroVelocity in the structure of transaction pat- terns, introducing transaction networks. A network is an ordered pair of sets G = (V, E), called the nodes V and edges (or links) E, where the elements of E ∈ V × V represent connections between elements of V. Identifying agents as nodes and transactions as directed connections from the transaction sender to the receiver, as done for instance in [28], it is possible to obtain a topological description of the economy, highlighting with whom agents exchange tokens and how central they are in the flow of value in the economy. We construct transaction networks on weekly aggregation, which is the minimum timescale that allows to mitigate the effect of day-of-the-week seasonalities that are common in economic and financial data (e.g. weekend effects, higher volumes on Mondays/Fridays, etc.). Operationally this means that a connection from agent i to agent j is present if at least one transaction occurred between them in a given week. We then proceed to take the values of V i at the end of each week and assign them to each node as attributes, which then allows us to consider the assortativity coefficient of these networks with respect to MicroVelocity.\\nThe assortativity coefficient, defined by [29], is the Pearson correlation coefficient of a given node-specific quantity across linked pairs in a network. It has been introduced to measure how nodes select their neighbors with respect to specific characteristics, most prominently the degree, and it is defined as\\nis the node-specific property of interest and · indicates the arithmetic mean operator. A positive value of this coefficient indicates homophily, i.e. links mainly connect similar nodes in terms of the selected property, whereas a negative value indicates heterophily, i.e. connections mostly appear between nodes with very different values of x.\\nIn our case we decide to use the rank in MicroVelocity as the nodes attribute x, thus computing a Spearman's correlation instead of Pearson's, again to account for the fat-tailed nature of the distribution of V i s. We report our finding in Fig. 6, where we plot this rank assortativity coefficient for MicroVelocity on weekly networks as a function of time. The resulting measure is mostly nega-tive, thus suggesting that highly-ranked nodes mostly trade with lowly-ranked ones and vice-versa.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"In this article we investigated the microeconomic foundations of the macroeconomic velocity of money, building upon theoretical work [10] and realizing the first empirical measurement of the velocity of money from micro-level data. We leveraged on the technological breakthrough provided by digital currencies and by cryptocurrencies in particular, which offer unprecedented detail about the movement of money in a closed economy where all coins can be followed in every transaction they take part in. We have thus been able to introduce the individual contribution to velocity by single agents, which we have called Mi-croVelocity, as a function of the agent-conditional distribution of holding times of money.\\nWe argue that insights about the distribution and characterization of Mi-croVelocity are valuable to researchers, policymakers and to the public to gain a deeper understanding of the structure of the economy: in the particular case at hand, we provide evidence that cryptocurrency economies' claim of being \\\"decentralized\\\" is largely unjustified, as shown by the extremely skewed distributions of wealth and MicroVelocity we measure. This \\\"decentalization illusion\\\" [17] is unmasked by the highly heterogeneous MicroVelocity distribution, indicating that few entities are intermediaries to most of the transactions, since large values of MicroVelocity are easily due to high turnover in assets.\\nThe fact that we find an extremely positive correlation between MicroVelocity and wealth, as well as negative assortativity of V i in transaction networks, is also corroborating evidence that these economies have seen the rise of pseudobanking services which centralize the supply of money and act as custodians and intermediation services, often without any scrutiny by regulators due to the lack of appropriate legislation.\\nWe envision multiple directions that could be taken following this work, tackling some limitations of our study as well as expanding its applicability beyond the relatively narrow realm of UTXO-based cryptocurrencies.\\nThe specific implementation we proposed here takes advantage of the fact that holding times are relatively easy to compute thanks to the age of UTXOs, but this can be easily overcome in the case where this information is not available by considering, for instance, a Last-In-First-Out (LIFO) or First-In-First-Out (FIFO) spending rule, as done for instance in [30]. We argue that a LIFO rule is possibly the most economically significant for a medium of exchange, as it would automatically exclude less liquid portions of the money supply from the calculation, but we leave this extension for future research.\\nOur approach using empirical probabilities to estimate P i (τ ) has a limitation in the fact that holding times data is right-censored, and this becomes a bigger issue the closer one is to present times. Moreover using empirical probabilities does not fully respect the assumptions on stationarity; nonetheless the convergence between the micro and macro quantities shown in Fig. 2 leads us to think that they still are a good first approximation. We are confident that these issues could be reduced by performing some additional assumptions on the form of the holding times distribution, possibly giving it a parametric form justified by theory (e.g. the Gamma family proposed in [10]). This would mitigate the limitations and possibly allow to produce forecasts as well as predict the outcome of policy interventions.\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Comparison of total MicroVelocity and the ratio of total flow to monetary mass P Q/M , with weekly aggregation. All quantities are annualised.","content":"Figure 2 :\n2Comparison of total MicroVelocity and the ratio of total flow to monetary mass P Q/M , with weekly aggregation. All quantities are annualised."},{"id":"","caption":"Area charts of MicroVelocity and identity of top 5 individual contributors for the four cryptocurrencies, averaged weekly. In each panel the top chart shows the frequency (vertical axis) with which the 5 most represented agents (horizontal axis) figure in the top 5 contributors, with stacked bars colored to show their ranking according to the legend. The bottom charts show the share of V contributed by the top 5 agents as a function of time. We see that one agent is always in the top 5 and that in all cases the vast majority of the total velocity is concentrated in the hands of a few agents.","content":"Figure 4 :\n4Area charts of MicroVelocity and identity of top 5 individual contributors for the four cryptocurrencies, averaged weekly. In each panel the top chart shows the frequency (vertical axis) with which the 5 most represented agents (horizontal axis) figure in the top 5 contributors, with stacked bars colored to show their ranking according to the legend. The bottom charts show the share of V contributed by the top 5 agents as a function of time. We see that one agent is always in the top 5 and that in all cases the vast majority of the total velocity is concentrated in the hands of a few agents."},{"id":"","caption":"Spearman's rank correlation between the MicroVelocity Vi and the wealth Mi of the agent. Insets: scatterplots of Vi/V vs Mi/M in log-log scales at marked dates, with dashed line on the diagonal Vi/V = Mi/M .","content":"Figure 5 :\n5Spearman's rank correlation between the MicroVelocity Vi and the wealth Mi of the agent. Insets: scatterplots of Vi/V vs Mi/M in log-log scales at marked dates, with dashed line on the diagonal Vi/V = Mi/M ."},{"id":"","caption":"Rank assortativity of MicroVelocity on weekly transaction networks.","content":"Figure 6 :\n6Rank assortativity of MicroVelocity on weekly transaction networks."},{"id":"","caption":"Figure 1: Schematic representation of a UTXO blockchain. Each block contains transactions, in turn consisting of a set of inputs and outputs. Total input and output values must be matching. An output grants usage rights of some amount of tokens to an address, identified by its public key. Inputs are references to unspent outputs from previous blocks, cryptographically signed by matching the public and private keys. A coinbase transaction generates new tokens as reward to the miner of the block.","content":"Figure 1: Schematic representation of a UTXO blockchain. Each block contains transactions, in turn consisting of a set of inputs and outputs. Total input and output values must be matching. An output grants usage rights of some amount of tokens to an address, identified by its public key. Inputs are references to unspent outputs from previous blocks, cryptographically signed by matching the public and private keys. A coinbase transaction generates new tokens as reward to the miner of the block.# block height \n\nblock time \n\nC \n\nOX [1] \n2 \n\ntransactions \n\nspent output [position] and value, \ncryptographically signed \n\ninputs \noutputs \n\ntransaction output [position] and value, \nwith public signature of the recipient \n\nOY [3] \n4 \n\nOC [1] \nOC [2] \nOC [3] \n\n3 \n1 \n2 \n\nA \n\ncoinbase \n25 \nOA [1] \n25 \n\nB \n\nOZ [2] \n10 \nOB [1] \n10 \n\n# 8 \n\n25/03/14 03:12 \n\n83 \n\nO73 [2] \n4 \nO64 [1] \n4 \n\nO83 [1] \nO83 [2] \nO83 [3] \n\n3 \n3 \n2 \n\n81 \n\ncoinbase \n25 \nO81 [1] \n25 \n\n82 \n\nO63 [1] \n11 \nO11 [1] \n15 \n\ngeneration transaction \n\n# 7 \n\n25/03/14 02:24 \n\n73 \n\nO62 [2] \n7 \nO54 [1] \n2 \n\nO73 [1] \nO73 [2] \n\n5 \n4 \n\n71 \n\ncoinbase \n25 \nO71 [1] \n25 \n\n72 \n\nO62 [1] \n8 \nO72 [1] \n7 \n\n# 6 \n\n25/03/14 01:36 \n\n64 \n\nO23 [1] \n4 \nO64 [1] \n4 \n\n61 \n\ncoinbase \n25 \nO61 [1] \n25 \n\n63 \n\nO32 [2] \n11 \nO63 [1] \n11 \n\nO82 [1] \n10 \nO82 [2] \n1 \n\n62 \n\nO42 [1] \n15 \nO62 [1] \n8 \nO62 [2] \n7 \nO72 [2] \n1"},{"id":"","caption":", transactions 62 and 72 constitute a peeling chain, hence addresses O42[1], O62[1] and O72[1] are linked with one wallet.","content":", transactions 62 and 72 constitute a peeling chain, hence addresses O42[1], O62[1] and O72[1] are linked with one wallet."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Comparison of total MicroVelocity and the ratio of total flow to monetary mass P Q/M , with weekly aggregation. All quantities are annualised.\",\n \"content\": \"Figure 2 :\\n2Comparison of total MicroVelocity and the ratio of total flow to monetary mass P Q/M , with weekly aggregation. All quantities are annualised.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Area charts of MicroVelocity and identity of top 5 individual contributors for the four cryptocurrencies, averaged weekly. In each panel the top chart shows the frequency (vertical axis) with which the 5 most represented agents (horizontal axis) figure in the top 5 contributors, with stacked bars colored to show their ranking according to the legend. The bottom charts show the share of V contributed by the top 5 agents as a function of time. We see that one agent is always in the top 5 and that in all cases the vast majority of the total velocity is concentrated in the hands of a few agents.\",\n \"content\": \"Figure 4 :\\n4Area charts of MicroVelocity and identity of top 5 individual contributors for the four cryptocurrencies, averaged weekly. In each panel the top chart shows the frequency (vertical axis) with which the 5 most represented agents (horizontal axis) figure in the top 5 contributors, with stacked bars colored to show their ranking according to the legend. The bottom charts show the share of V contributed by the top 5 agents as a function of time. We see that one agent is always in the top 5 and that in all cases the vast majority of the total velocity is concentrated in the hands of a few agents.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Spearman's rank correlation between the MicroVelocity Vi and the wealth Mi of the agent. Insets: scatterplots of Vi/V vs Mi/M in log-log scales at marked dates, with dashed line on the diagonal Vi/V = Mi/M .\",\n \"content\": \"Figure 5 :\\n5Spearman's rank correlation between the MicroVelocity Vi and the wealth Mi of the agent. Insets: scatterplots of Vi/V vs Mi/M in log-log scales at marked dates, with dashed line on the diagonal Vi/V = Mi/M .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Rank assortativity of MicroVelocity on weekly transaction networks.\",\n \"content\": \"Figure 6 :\\n6Rank assortativity of MicroVelocity on weekly transaction networks.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 1: Schematic representation of a UTXO blockchain. Each block contains transactions, in turn consisting of a set of inputs and outputs. Total input and output values must be matching. An output grants usage rights of some amount of tokens to an address, identified by its public key. Inputs are references to unspent outputs from previous blocks, cryptographically signed by matching the public and private keys. A coinbase transaction generates new tokens as reward to the miner of the block.\",\n \"content\": \"Figure 1: Schematic representation of a UTXO blockchain. Each block contains transactions, in turn consisting of a set of inputs and outputs. Total input and output values must be matching. An output grants usage rights of some amount of tokens to an address, identified by its public key. Inputs are references to unspent outputs from previous blocks, cryptographically signed by matching the public and private keys. A coinbase transaction generates new tokens as reward to the miner of the block.# block height \\n\\nblock time \\n\\nC \\n\\nOX [1] \\n2 \\n\\ntransactions \\n\\nspent output [position] and value, \\ncryptographically signed \\n\\ninputs \\noutputs \\n\\ntransaction output [position] and value, \\nwith public signature of the recipient \\n\\nOY [3] \\n4 \\n\\nOC [1] \\nOC [2] \\nOC [3] \\n\\n3 \\n1 \\n2 \\n\\nA \\n\\ncoinbase \\n25 \\nOA [1] \\n25 \\n\\nB \\n\\nOZ [2] \\n10 \\nOB [1] \\n10 \\n\\n# 8 \\n\\n25/03/14 03:12 \\n\\n83 \\n\\nO73 [2] \\n4 \\nO64 [1] \\n4 \\n\\nO83 [1] \\nO83 [2] \\nO83 [3] \\n\\n3 \\n3 \\n2 \\n\\n81 \\n\\ncoinbase \\n25 \\nO81 [1] \\n25 \\n\\n82 \\n\\nO63 [1] \\n11 \\nO11 [1] \\n15 \\n\\ngeneration transaction \\n\\n# 7 \\n\\n25/03/14 02:24 \\n\\n73 \\n\\nO62 [2] \\n7 \\nO54 [1] \\n2 \\n\\nO73 [1] \\nO73 [2] \\n\\n5 \\n4 \\n\\n71 \\n\\ncoinbase \\n25 \\nO71 [1] \\n25 \\n\\n72 \\n\\nO62 [1] \\n8 \\nO72 [1] \\n7 \\n\\n# 6 \\n\\n25/03/14 01:36 \\n\\n64 \\n\\nO23 [1] \\n4 \\nO64 [1] \\n4 \\n\\n61 \\n\\ncoinbase \\n25 \\nO61 [1] \\n25 \\n\\n63 \\n\\nO32 [2] \\n11 \\nO63 [1] \\n11 \\n\\nO82 [1] \\n10 \\nO82 [2] \\n1 \\n\\n62 \\n\\nO42 [1] \\n15 \\nO62 [1] \\n8 \\nO62 [2] \\n7 \\nO72 [2] \\n1\"\n },\n {\n \"id\": \"\",\n \"caption\": \", transactions 62 and 72 constitute a peeling chain, hence addresses O42[1], O62[1] and O72[1] are linked with one wallet.\",\n \"content\": \", transactions 62 and 72 constitute a peeling chain, hence addresses O42[1], O62[1] and O72[1] are linked with one wallet.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The economics of money, banking and financial markets. Frederic S Mishkin, Mishkin Frederic-Addison Wesley LongmanS Mishkin Frederic. The economics of money, banking and financial mar- kets. Mishkin Frederic-Addison Wesley Longman, 2004.","doi":""},{"text":"Banking on stone money: ancient antecedents to bitcoin. M Scott, Stephen Fitzpatrick, Mckeon, Economic Anthropology. 71Scott M Fitzpatrick and Stephen McKeon. Banking on stone money: an- cient antecedents to bitcoin. Economic Anthropology, 7(1):7-21, 2020.","doi":""},{"text":"Cigarette money. Kenneth Burdett, Alberto Trejos, Randall Wright, Journal of Economic Theory. 991-2Kenneth Burdett, Alberto Trejos, and Randall Wright. Cigarette money. Journal of Economic Theory, 99(1-2):117-142, 2001.","doi":""},{"text":"A peer-to-peer electronic cash system. Satoshi Nakamoto, Satoshi Nakamoto. A peer-to-peer electronic cash system. Bitcoin.-URL: https://bitcoin.org/bitcoin.pdf, 4, 2008.","doi":""},{"text":"The purchasing power of money: its' determination and relation to credit interest and crises. Irving Fisher, MacmillanNew YorkIrving Fisher. The purchasing power of money: its' determination and relation to credit interest and crises. Macmillan, New York, 1911.","doi":""},{"text":"The demand for money: some theoretical and empirical results. Milton Friedman, Journal of Political economy. 674Milton Friedman. The demand for money: some theoretical and empirical results. Journal of Political economy, 67(4):327-351, 1959.","doi":""},{"text":"Central bank digital currency: Motivations and implications. Walter Engert, Ben Siu, -Cheong Fung, Technical reportBank of Canada Staff Discussion PaperWalter Engert and Ben Siu-Cheong Fung. Central bank digital currency: Motivations and implications. Technical report, Bank of Canada Staff Dis- cussion Paper, 2017.","doi":""},{"text":"Bank of Canada, European Central Bank, Bank of Japan, Sveriges Riksbank, Swiss National Bank, Bank of England, Board of Governors of the Federal Reserve System and Bank for International Settlements. Group of central banks. Central bank digital currencies: foundational principles and core features. Technical reportGroup of central banks. Central bank digital currencies: foundational principles and core features. Technical report, Bank of Canada, Eu- ropean Central Bank, Bank of Japan, Sveriges Riksbank, Swiss Na- tional Bank, Bank of England, Board of Governors of the Federal Re- serve System and Bank for International Settlements, 2020. available at https://www.bis.org/publ/othp33.htm.","doi":""},{"text":"Report on a digital euro. European Central, Bank, ECB Report Series. European Central Bank. Report on a digital euro. ECB Report Series, 2020.","doi":""},{"text":"The circulation of money and holding time distribution. Yougui Wang, Ning Ding, Li Zhang, Physica A: Statistical Mechanics and its Applications. 3243-4Yougui Wang, Ning Ding, and Li Zhang. The circulation of money and hold- ing time distribution. Physica A: Statistical Mechanics and its Applications, 324(3-4):665-677, 2003.","doi":""},{"text":"Bitcoin pricing, adoption, and usage: Theory and evidence. Susan Athey, Ivo Parashkevov, Vishnu Sarukkai, Jing Xia, Stanford University Graduate School of Business Research PaperSusan Athey, Ivo Parashkevov, Vishnu Sarukkai, and Jing Xia. Bitcoin pricing, adoption, and usage: Theory and evidence. Stanford University Graduate School of Business Research Paper, 2016.","doi":""},{"text":"On the value of virtual currencies. Wilko Bolt, Maarten Rc Van Oordt, Journal of Money, Credit and Banking. 524Wilko Bolt and Maarten RC Van Oordt. On the value of virtual currencies. Journal of Money, Credit and Banking, 52(4):835-862, 2020.","doi":""},{"text":"The digital agenda of virtual currencies: Can bitcoin become a global currency?. Pavel Ciaian, Miroslava Rajcaniova, D'artis Kancs, Information Systems and e-Business Management. 14Pavel Ciaian, Miroslava Rajcaniova, and d'Artis Kancs. The digital agenda of virtual currencies: Can bitcoin become a global currency? Information Systems and e-Business Management, 14(4):883-919, 2016.","doi":""},{"text":"Bitcoin average dormancy: A measure of turnover and trading activity. D Reginald, Smith, Ledger, 3Reginald D Smith. Bitcoin average dormancy: A measure of turnover and trading activity. Ledger, 3, 2018.","doi":""},{"text":"Cryptocurrencies and the velocity of money. Anton Ingolf Gunnar, Georg Pernice, Hermann Gentzen, Elendner, Cryptoeconomic Systems Conference. Ingolf Gunnar Anton Pernice, Georg Gentzen, and Hermann Elendner. Cryptocurrencies and the velocity of money. In Cryptoeconomic Systems Conference, 2020.","doi":""},{"text":"Trading and arbitrage in cryptocurrency markets. Igor Makarov, Antoinette Schoar, Journal of Financial Economics. 1352Igor Makarov and Antoinette Schoar. Trading and arbitrage in cryptocur- rency markets. Journal of Financial Economics, 135(2):293-319, 2020.","doi":""},{"text":"Defi risks and the decentralisation illusion. Wenqian Sirio Aramonte, Andreas Huang, Schrimpf, BIS Quarterly Review. Sirio Aramonte, Wenqian Huang, and Andreas Schrimpf. Defi risks and the decentralisation illusion. BIS Quarterly Review, 2021.","doi":""},{"text":"Unequal scars -distributional consequences of the pandemic. Isabel Schnabel, Speech at Deutscher Juristentag 2020Isabel Schnabel. Unequal scars -distributional consequences of the pan- demic. Speech at Deutscher Juristentag 2020, 2020.","doi":""},{"text":"Understanding household wealth: linking macro and micro data to produce distributional financial accounts. H Ahnert, Kavonius, P Honkkila, Sola, ECB Statistics Paper Series. 37H Ahnert, IK Kavonius, J Honkkila, and P Sola. Understanding household wealth: linking macro and micro data to produce distributional financial accounts. ECB Statistics Paper Series, 37, 2020.","doi":""},{"text":"Blocksci: Design and applications of a blockchain analysis platform. Harry Kalodner, Malte Möser, Kevin Lee, Steven Goldfeder, Martin Plattner, Alishah Chator, Arvind Narayanan, 29th {USENIX} Security Symposium ({USENIX} Security 20). Harry Kalodner, Malte Möser, Kevin Lee, Steven Goldfeder, Martin Plat- tner, Alishah Chator, and Arvind Narayanan. Blocksci: Design and ap- plications of a blockchain analysis platform. In 29th {USENIX} Security Symposium ({USENIX} Security 20), pages 2721-2738, 2020.","doi":""},{"text":"Lightning network: a second path towards centralisation of the Bitcoin economy. Jian-Hong Lin, Kevin Primicerio, Tiziano Squartini, Christian Decker, Claudio J Tessone, New Journal of Physics. 22883022Jian-Hong Lin, Kevin Primicerio, Tiziano Squartini, Christian Decker, and Claudio J Tessone. Lightning network: a second path towards centralisation of the Bitcoin economy. New Journal of Physics, 22(8):83022, aug 2020.","doi":""},{"text":"Do the rich get richer? an empirical analysis of the bitcoin transaction network. Dániel Kondor, Márton Pósfai, István Csabai, Gábor Vattay, PLOS ONE. 9286197Dániel Kondor, Márton Pósfai, István Csabai, and Gábor Vattay. Do the rich get richer? an empirical analysis of the bitcoin transaction network. PLOS ONE, 9(2):e86197, 2014.","doi":""},{"text":"Global cryptocurrency benchmarking study. Cambridge Centre for Alternative Finance. Garrick Hileman, Michel Rauchs, 33Garrick Hileman and Michel Rauchs. Global cryptocurrency benchmarking study. Cambridge Centre for Alternative Finance, 33:33-113, 2017.","doi":""},{"text":"Bitcoin transaction networks: An overview of recent results. Nicolò Vallarano, Claudio J Tessone, Tiziano Squartini, Frontiers in Physics. 8286Nicolò Vallarano, Claudio J. Tessone, and Tiziano Squartini. Bitcoin trans- action networks: An overview of recent results. Frontiers in Physics, 8:286, 2020.","doi":""},{"text":"Heterogeneous preferential attachment in key ethereumbased cryptoassets. Francesco Maria De Collibus, Alberto Partida, Matija Piå¡korec, Claudio J Tessone, Frontiers in Physics. 92021Francesco Maria De Collibus, Alberto Partida, Matija PiÅ¡korec, and Clau- dio J. Tessone. Heterogeneous preferential attachment in key ethereum- based cryptoassets. Frontiers in Physics, 9, 2021.","doi":""},{"text":"Ashish Rajendra Sai, Jim Buckley, Brian Fitzgerald, Andrew Le Gear, Taxonomy of centralization in public blockchain systems: A systematic literature review. Information Processing & Management. 58102584Ashish Rajendra Sai, Jim Buckley, Brian Fitzgerald, and Andrew Le Gear. Taxonomy of centralization in public blockchain systems: A systematic literature review. Information Processing & Management, 58(4):102584, 2021.","doi":""},{"text":"A systematic review of the bubble dynamics of cryptocurrency prices. Nikolaos Kyriazis, Stephanos Papadamou, Shaen Corbet, Research in International Business and Finance. 54101254Nikolaos Kyriazis, Stephanos Papadamou, and Shaen Corbet. A system- atic review of the bubble dynamics of cryptocurrency prices. Research in International Business and Finance, 54:101254, 2020.","doi":""},{"text":"Alexandre Bovet, Carlo Campajola, Francesco Mottes, Nicolo Valerio Restocchi, Tiziano Vallarano, Claudio J Squartini, Tessone, arXiv:1907.03577The evolving liaisons between the transaction networks of bitcoin and its price dynamics. arXiv preprintAlexandre Bovet, Carlo Campajola, Francesco Mottes, Valerio Restocchi, Nicolo Vallarano, Tiziano Squartini, and Claudio J Tessone. The evolving liaisons between the transaction networks of bitcoin and its price dynamics. arXiv preprint arXiv:1907.03577, 2019.","doi":"arXiv:1907.03577"},{"text":"Assortative mixing in networks. E J Mark, Newman, Physical Review Letters. 8920208701Mark E.J. Newman. Assortative mixing in networks. Physical Review Letters, 89(20):208701, 2002.","doi":""},{"text":"Carolina Mattsson, arXiv:1910.05596Networks of monetary flow at native resolution. arXiv preprintCarolina Mattsson. Networks of monetary flow at native resolution. arXiv preprint arXiv:1910.05596, 2019.","doi":"arXiv:1910.05596"}],"string":"[\n {\n \"text\": \"The economics of money, banking and financial markets. Frederic S Mishkin, Mishkin Frederic-Addison Wesley LongmanS Mishkin Frederic. The economics of money, banking and financial mar- kets. Mishkin Frederic-Addison Wesley Longman, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Banking on stone money: ancient antecedents to bitcoin. M Scott, Stephen Fitzpatrick, Mckeon, Economic Anthropology. 71Scott M Fitzpatrick and Stephen McKeon. Banking on stone money: an- cient antecedents to bitcoin. Economic Anthropology, 7(1):7-21, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cigarette money. Kenneth Burdett, Alberto Trejos, Randall Wright, Journal of Economic Theory. 991-2Kenneth Burdett, Alberto Trejos, and Randall Wright. Cigarette money. Journal of Economic Theory, 99(1-2):117-142, 2001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A peer-to-peer electronic cash system. Satoshi Nakamoto, Satoshi Nakamoto. A peer-to-peer electronic cash system. Bitcoin.-URL: https://bitcoin.org/bitcoin.pdf, 4, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The purchasing power of money: its' determination and relation to credit interest and crises. Irving Fisher, MacmillanNew YorkIrving Fisher. The purchasing power of money: its' determination and relation to credit interest and crises. Macmillan, New York, 1911.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The demand for money: some theoretical and empirical results. Milton Friedman, Journal of Political economy. 674Milton Friedman. The demand for money: some theoretical and empirical results. Journal of Political economy, 67(4):327-351, 1959.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Central bank digital currency: Motivations and implications. Walter Engert, Ben Siu, -Cheong Fung, Technical reportBank of Canada Staff Discussion PaperWalter Engert and Ben Siu-Cheong Fung. Central bank digital currency: Motivations and implications. Technical report, Bank of Canada Staff Dis- cussion Paper, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bank of Canada, European Central Bank, Bank of Japan, Sveriges Riksbank, Swiss National Bank, Bank of England, Board of Governors of the Federal Reserve System and Bank for International Settlements. Group of central banks. Central bank digital currencies: foundational principles and core features. Technical reportGroup of central banks. Central bank digital currencies: foundational principles and core features. Technical report, Bank of Canada, Eu- ropean Central Bank, Bank of Japan, Sveriges Riksbank, Swiss Na- tional Bank, Bank of England, Board of Governors of the Federal Re- serve System and Bank for International Settlements, 2020. available at https://www.bis.org/publ/othp33.htm.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Report on a digital euro. European Central, Bank, ECB Report Series. European Central Bank. Report on a digital euro. ECB Report Series, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The circulation of money and holding time distribution. Yougui Wang, Ning Ding, Li Zhang, Physica A: Statistical Mechanics and its Applications. 3243-4Yougui Wang, Ning Ding, and Li Zhang. The circulation of money and hold- ing time distribution. Physica A: Statistical Mechanics and its Applications, 324(3-4):665-677, 2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bitcoin pricing, adoption, and usage: Theory and evidence. Susan Athey, Ivo Parashkevov, Vishnu Sarukkai, Jing Xia, Stanford University Graduate School of Business Research PaperSusan Athey, Ivo Parashkevov, Vishnu Sarukkai, and Jing Xia. Bitcoin pricing, adoption, and usage: Theory and evidence. Stanford University Graduate School of Business Research Paper, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the value of virtual currencies. Wilko Bolt, Maarten Rc Van Oordt, Journal of Money, Credit and Banking. 524Wilko Bolt and Maarten RC Van Oordt. On the value of virtual currencies. Journal of Money, Credit and Banking, 52(4):835-862, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The digital agenda of virtual currencies: Can bitcoin become a global currency?. Pavel Ciaian, Miroslava Rajcaniova, D'artis Kancs, Information Systems and e-Business Management. 14Pavel Ciaian, Miroslava Rajcaniova, and d'Artis Kancs. The digital agenda of virtual currencies: Can bitcoin become a global currency? Information Systems and e-Business Management, 14(4):883-919, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bitcoin average dormancy: A measure of turnover and trading activity. D Reginald, Smith, Ledger, 3Reginald D Smith. Bitcoin average dormancy: A measure of turnover and trading activity. Ledger, 3, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cryptocurrencies and the velocity of money. Anton Ingolf Gunnar, Georg Pernice, Hermann Gentzen, Elendner, Cryptoeconomic Systems Conference. Ingolf Gunnar Anton Pernice, Georg Gentzen, and Hermann Elendner. Cryptocurrencies and the velocity of money. In Cryptoeconomic Systems Conference, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Trading and arbitrage in cryptocurrency markets. Igor Makarov, Antoinette Schoar, Journal of Financial Economics. 1352Igor Makarov and Antoinette Schoar. Trading and arbitrage in cryptocur- rency markets. Journal of Financial Economics, 135(2):293-319, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Defi risks and the decentralisation illusion. Wenqian Sirio Aramonte, Andreas Huang, Schrimpf, BIS Quarterly Review. Sirio Aramonte, Wenqian Huang, and Andreas Schrimpf. Defi risks and the decentralisation illusion. BIS Quarterly Review, 2021.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unequal scars -distributional consequences of the pandemic. Isabel Schnabel, Speech at Deutscher Juristentag 2020Isabel Schnabel. Unequal scars -distributional consequences of the pan- demic. Speech at Deutscher Juristentag 2020, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Understanding household wealth: linking macro and micro data to produce distributional financial accounts. H Ahnert, Kavonius, P Honkkila, Sola, ECB Statistics Paper Series. 37H Ahnert, IK Kavonius, J Honkkila, and P Sola. Understanding household wealth: linking macro and micro data to produce distributional financial accounts. ECB Statistics Paper Series, 37, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Blocksci: Design and applications of a blockchain analysis platform. Harry Kalodner, Malte Möser, Kevin Lee, Steven Goldfeder, Martin Plattner, Alishah Chator, Arvind Narayanan, 29th {USENIX} Security Symposium ({USENIX} Security 20). Harry Kalodner, Malte Möser, Kevin Lee, Steven Goldfeder, Martin Plat- tner, Alishah Chator, and Arvind Narayanan. Blocksci: Design and ap- plications of a blockchain analysis platform. In 29th {USENIX} Security Symposium ({USENIX} Security 20), pages 2721-2738, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lightning network: a second path towards centralisation of the Bitcoin economy. Jian-Hong Lin, Kevin Primicerio, Tiziano Squartini, Christian Decker, Claudio J Tessone, New Journal of Physics. 22883022Jian-Hong Lin, Kevin Primicerio, Tiziano Squartini, Christian Decker, and Claudio J Tessone. Lightning network: a second path towards centralisation of the Bitcoin economy. New Journal of Physics, 22(8):83022, aug 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Do the rich get richer? an empirical analysis of the bitcoin transaction network. Dániel Kondor, Márton Pósfai, István Csabai, Gábor Vattay, PLOS ONE. 9286197Dániel Kondor, Márton Pósfai, István Csabai, and Gábor Vattay. Do the rich get richer? an empirical analysis of the bitcoin transaction network. PLOS ONE, 9(2):e86197, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Global cryptocurrency benchmarking study. Cambridge Centre for Alternative Finance. Garrick Hileman, Michel Rauchs, 33Garrick Hileman and Michel Rauchs. Global cryptocurrency benchmarking study. Cambridge Centre for Alternative Finance, 33:33-113, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bitcoin transaction networks: An overview of recent results. Nicolò Vallarano, Claudio J Tessone, Tiziano Squartini, Frontiers in Physics. 8286Nicolò Vallarano, Claudio J. Tessone, and Tiziano Squartini. Bitcoin trans- action networks: An overview of recent results. Frontiers in Physics, 8:286, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Heterogeneous preferential attachment in key ethereumbased cryptoassets. Francesco Maria De Collibus, Alberto Partida, Matija Piå¡korec, Claudio J Tessone, Frontiers in Physics. 92021Francesco Maria De Collibus, Alberto Partida, Matija PiÅ¡korec, and Clau- dio J. Tessone. Heterogeneous preferential attachment in key ethereum- based cryptoassets. Frontiers in Physics, 9, 2021.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ashish Rajendra Sai, Jim Buckley, Brian Fitzgerald, Andrew Le Gear, Taxonomy of centralization in public blockchain systems: A systematic literature review. Information Processing & Management. 58102584Ashish Rajendra Sai, Jim Buckley, Brian Fitzgerald, and Andrew Le Gear. Taxonomy of centralization in public blockchain systems: A systematic literature review. Information Processing & Management, 58(4):102584, 2021.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A systematic review of the bubble dynamics of cryptocurrency prices. Nikolaos Kyriazis, Stephanos Papadamou, Shaen Corbet, Research in International Business and Finance. 54101254Nikolaos Kyriazis, Stephanos Papadamou, and Shaen Corbet. A system- atic review of the bubble dynamics of cryptocurrency prices. Research in International Business and Finance, 54:101254, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Alexandre Bovet, Carlo Campajola, Francesco Mottes, Nicolo Valerio Restocchi, Tiziano Vallarano, Claudio J Squartini, Tessone, arXiv:1907.03577The evolving liaisons between the transaction networks of bitcoin and its price dynamics. arXiv preprintAlexandre Bovet, Carlo Campajola, Francesco Mottes, Valerio Restocchi, Nicolo Vallarano, Tiziano Squartini, and Claudio J Tessone. The evolving liaisons between the transaction networks of bitcoin and its price dynamics. arXiv preprint arXiv:1907.03577, 2019.\",\n \"doi\": \"arXiv:1907.03577\"\n },\n {\n \"text\": \"Assortative mixing in networks. E J Mark, Newman, Physical Review Letters. 8920208701Mark E.J. Newman. Assortative mixing in networks. Physical Review Letters, 89(20):208701, 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Carolina Mattsson, arXiv:1910.05596Networks of monetary flow at native resolution. arXiv preprintCarolina Mattsson. Networks of monetary flow at native resolution. arXiv preprint arXiv:1910.05596, 2019.\",\n \"doi\": \"arXiv:1910.05596\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":3,"numItemsPerPage":100,"numTotalItems":2579762,"offset":300,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sInJlcG9JZHMiOlsiNjc3NDI4YmI1YThjMWM5YzgyNWVmYTYwIl0sImlhdCI6MTc5NDUwNTAyNSwianRpIjoiODcwNjkxNGUtZDFmMC00ZTAxLWIwZWQtNGU0ODQwNmJkMTAzIiwic3ViIjoiL2RhdGFzZXRzL0FsZ29yaXRobWljUmVzZWFyY2hHcm91cC9zMm9yY19hcnhpdiIsImV4cCI6MTc5NDUwODYyNSwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.U5s5q_G1tR40F4GclfM4gTgooiJkM4NJsIBQ_Ds8ZRZoaZZiAZL2KmtonT4eEtc_AzY7teOnIV8xbcXVmh82BA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":2579762,"numBytesParquet":266094171612}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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|---|---|---|---|---|---|---|---|---|---|
Self-Shielded Topological Receiver Protectors
Mattis Reisner
Institut de Physique de Nice (INPHYNI)
Université Côte d'Azur
CNRS
06108NiceEUFrance
Do Hyeok Jeon
Department of Physics
Wave Transport in Complex Systems Lab
Wesleyan University
CT-06459MiddletownUSA
Carsten Schindler
Fachbereich Physik
Philipps Univ... | 204,008,674 | {
"arxiv": "1910.04684",
"mag": 3103957237,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevapplied.13.034067"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1910.04684v2.pdf"
],
"pdf_sha": "1b4a17097e815a4244a17847fd1a1a9525e0f0ac",
"open_access": {
"license": "publisher-specific, author manuscript",
"url": "https://link.aps.org/accepted/10.1103/PhysRevApplied.13.034067",
"status": "HYBRID"
}
} | Self-Shielded Topological Receiver Protectors | [
"Mattis Reisner \nInstitut de Physique de Nice (INPHYNI)\nUniversité Côte d'Azur\nCNRS\n06108NiceEUFrance",
"Do Hyeok Jeon \nDepartment of Physics\nWave Transport in Complex Systems Lab\nWesleyan University\nCT-06459MiddletownUSA",
"Carsten Schindler \nFachbereich Physik\nPhilipps Universität Marburg\n35032Marb... | Receiver protectors (RPs) shield sensitive electronics from high-power incoming signals that might damage them. Typical RP schemes range from simple fusing and PIN diodes, to superconducting circuits and plasma cells -each having a variety of drawbacks ranging from unacceptable system downtime and self-destruction to s... | [
{
"title": "I. INTRODUCTION",
"content": "The Internet of Things, where everyday devices and objects communicate with each other via electromagnetic waves, is an incipient revolution that will dramatically affect future generations. The envisaged intimate exchange of information between autonomous vehicles,... | [
{
"id": "",
"caption": "The proposed implementation of a RP based on selfinduced spontaneous C-symmetry violation of a topological defect state. (a) An application in a radar setup. (b) The RP, consisting of 17 dielectric resonators with complex eigenfrequencies νn = 6.885 GHz+i1.7 MHz structured by four di... | [
{
"text": "Gernot Hueber, Ali M Niknejad, Millimeter-Wave Circuits for 5G and Radar. CambridgeCambridge University PressGernot Hueber and Ali M. Niknejad, Millimeter-Wave Circuits for 5G and Radar (Cambridge University Press, Cambridge, 2019).",
"doi": ""
},
{
"text": "A review of solid-state radar ... |
Kondo hybridization and quantum criticality in β-YbAlB 4 by laser-ARPES
11 Jan 2018
Cédric Bareille
ISSP
University of Tokyo
277-8581KashiwaJapan
Shintaro Suzuki
ISSP
University of Tokyo
277-8581KashiwaJapan
Mitsuhiro Nakayama
ISSP
University of Tokyo
277-8581KashiwaJapan
Kenta Kuroda
ISSP
University of Tokyo
... | 119,337,131 | {
"arxiv": "1801.03696",
"mag": 2782872859,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.97.045112"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1801.03696v1.pdf"
],
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"open_access": {
"license": "publisher-specific, author manuscript",
"url": "https://link.aps.org/accepted/10.1103/PhysRevB.97.045112",
"status": "HYBRID"
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} | Kondo hybridization and quantum criticality in β-YbAlB 4 by laser-ARPES | [
"Cédric Bareille \nISSP\nUniversity of Tokyo\n277-8581KashiwaJapan",
"Shintaro Suzuki \nISSP\nUniversity of Tokyo\n277-8581KashiwaJapan",
"Mitsuhiro Nakayama \nISSP\nUniversity of Tokyo\n277-8581KashiwaJapan",
"Kenta Kuroda \nISSP\nUniversity of Tokyo\n277-8581KashiwaJapan",
"Andriy H Nevidomskyy \nDepartme... | We report an angle-resolved photoemission (ARPES) study of β-YbAlB4, which is known to harbor unconventional quantum criticality (QC) without any tuning. We directly observe a quasiparticle peak (QP), emerging from hybridization, characterized by a binding energy and an onset of coherence both at about 4 meV. This valu... | [
{
"title": "I. INTRODUCTION",
"content": "The formation of a non-magnetic Kondo-singlet ground-state commonly provides a description of heavy Fermi liquid (FL) in intermetallics rare-earth compounds. It emerges below the Kondo temperature T K , from the screening of the local moments of f orbitals by conduc... | [
{
"id": "",
"caption": "Grant No. 16H02209, No. 25707030, and No. 26105002), by Grantsin-Aid for Scientific Research on Innovative Areas \"J-Physics\" (Grant No. 15H05882 and No. 15H05883) and Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers (Grant... | [
{
"text": "The Kondo problem to Heavy Fermions. A C Hewson, C. U. PressA. C. Hewson, The Kondo problem to Heavy Fermions, edited by C. U. Press (1993).",
"doi": ""
},
{
"text": ". H V Löhneysen, A Rosch, M Vojta, P Wölfle, 10.1103/RevModPhys.79.1015Rev. Mod. Phys. 791015H. v. Löhneysen, A. Rosch, M.... |
16 Jan 2009
Laurent Decreusefond
GET
Télécom Paris
Université Paris
Université de Technologie de Compiègne
Pascal Moyal
GET
Télécom Paris
Université Paris
Université de Technologie de Compiègne
16 Jan 20093C214510B270D23ABEC680031508EDF910.1214/08-AAP518arXiv:math/0608258v3[math.PR]Received August 2006; revised... | 11,548,896 | {
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} | 16 Jan 2009
Laurent Decreusefond
GET
Télécom Paris
U | [
"té Paris\nUniversité de Technologie de Compiègne\n\n\nPascal Moyal \nGET\nTélécom Paris\nUniversité Paris\nUniversité d",
"Technologie de Compiègne\n\n16 Jan 20093C214510B270D23ABEC680031508EDF910.1214/08-AAP518arXiv:math/06082"
] | present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/∞.The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service.The convergence in law of a sequence of such processes af... | [
{
"title": "ions of some p",
"content": "formance processes in Section 5. We prove the functional central limit theorem for (µ t ) t≥0 in Section 6, and give the diffusion approximations of the performance processes in Section 6.3.\n\n\n2.\n\nPreliminaries.We denote by D b (resp.C b , C K ) the set of real-... | [
{
"id": "",
"caption": ", R), and is thus bounded on [0, T ]:\nsup s≤T | X s , φ | = sup s∈[0,T ]∩",
"content": "], R), and is thus bounded on [0, T ]:\nsup s≤T | X s , φ | = sup s∈[0,T ]∩"
},
{
"id": "",
"caption": "X s , φ | < ∞.\nTherefore, X := {X s , s ∈ [0, T ] ∩ Q} is a weakly bounded... | [
{
"text": "a diffusion process.Then, Y n tends in distribution to a continuous martingale of increasing process (γ(t)) t≥0 .\n\n10 L\n10\n. DECREUSEFOND AND P. MOYAL\n\n\n\n\nwhere μ",
"doi": ""
},
{
"text": "is the deterministic element of C([0, ∞), M + f ) defined for all t ≥ 0 and all φ ∈ D b by ... |
The anomalous dimension of the gluon-ghost mass operator in Yang-Mills theory
Jul 2003
D Dudal
H Verschelde henri.verschelde@ugent.be‡vitor@dft.if.uerj.br
M.SV E R Lemes
R Sarandy sarandy@dft.if.uerj.br
S P Sobreiro sobreiro@dft.if.uerj.br
Sorella sorella@uerj.br§marco.picariello@mi.infn.it¶jag@amtp.liv.ac.uk
M... | 119,472,582 | {
"arxiv": "hep-th/0306116",
"mag": 2053255079,
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"doi": "10.1016/j.physletb.2003.07.019"
} | {
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} | The anomalous dimension of the gluon-ghost mass operator in Yang-Mills theory | [
"D Dudal",
"H Verschelde henri.verschelde@ugent.be‡vitor@dft.if.uerj.br",
"M.SV E R Lemes",
"R Sarandy sarandy@dft.if.uerj.br",
"S P Sobreiro sobreiro@dft.if.uerj.br",
"Sorella sorella@uerj.br§marco.picariello@mi.infn.it¶jag@amtp.liv.ac.uk",
"M Picariello",
"Infn Milano",
"Italy J A Gracey",
"Depa... | The local composite gluon-ghost operator 1 2 A aµ A a µ + αc a c a is analysed in the framework of the algebraic renormalization in SU (N ) Yang-Mills theories in the Landau, Curci-Ferrari and maximal abelian gauges. We show, to all orders of perturbation theory, that this operator is multiplicatively renormalizable. F... | [
{
"title": "Introduction",
"content": "Vacuum condensates are believed to play an important role in the understanding of the nonperturbative dynamics of Yang-Mills theories. In particular, much effort has been devoted to the study of condensates of dimension two built up with gluons and ghosts. For instance... | [
{
"id": "",
"caption": "27) from which it follows that γ O (a) = − 2 γ c (a) + γ gc 2 (a) , (2.28) where γ c (a) and γ gc 2 (a) are the anomalous dimensions of the Faddeev-Popov ghost c a and of the composite operator g 2 f abc c b c c , defined as γ c (a) = µ∂ µ ln Z 1/2 c γ gc 2 (a) = µ∂ µ ln Z L γ O (a) ... | [
{
"text": ". P Boucaud, G Burgio, F Di Renzo, J P Leroy, J Micheli, C Parrinello, O Pène, C Pittori, J Rodriguez-Quintero, C Roiesnel, K Sharkey, JHEP. 00046P. Boucaud, G. Burgio, F. Di Renzo, J. P. Leroy, J. Micheli, C. Parrinello, O. Pène, C. Pittori, J. Rodriguez-Quintero, C. Roiesnel and K. Sharkey, JHEP 00... |
Nucleon axial and tensor charges with dynamical overlap quarks JLQCD collaboration: N. Yamanaka * a Nucleon axial and tensor charges with dynamical overlap quarks
July 2015
H Ohki
RIKEN BNL Research Center
Brookhaven National Laboratory
11973UptonNYUSA
S Hashimoto
High Energy Accelerator Research Organization (KEK... | 118,597,714 | {
"arxiv": "1511.04589",
"mag": 2963841905,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.22323/1.251.0121"
} | {
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"https://arxiv.org/pdf/1511.04589v1.pdf"
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"pdf_sha": "03f48537f1e62bb028b5caa12793a415d25670f7",
"open_access": {
"license": "CCBYNCND",
"url": "https://pos.sissa.it/251/121/pdf",
"status": "HYBRID"
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} | Nucleon axial and tensor charges with dynamical overlap quarks JLQCD collaboration: N. Yamanaka * a Nucleon axial and tensor charges with dynamical overlap quarks | [
"H Ohki \nRIKEN BNL Research Center\nBrookhaven National Laboratory\n11973UptonNYUSA",
"S Hashimoto \nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaJapan\n\nSchool of High Energy Accelerator Science\nThe Graduate University for Advanced Studies (Sokendai)\n305-0801TsukubaJapan",
"T Kaneko ... | We report on our calculation of the nucleon axial and tensor charges in 2+1-flavor QCD with dynamical overlap quarks. Gauge ensembles are generated at a single lattice spacing 0.12 fm and at a strange quark mass close to its physical value. We employ the all-mode-averaging technique to calculate the relevant nucleon co... | [
{
"title": "Introduction",
"content": "The nucleon charges represent the nonperturbative nature of QCD, and are also relevant to the search for new physics beyond the Standard Model. The nucleon axial charge ∆q for the quark flavor q is defined by N(p, s)|qγ µ γ 5 q|N(p, s) = 2m N s µ ∆q, (1.1) where p and ... | [
{
"id": "",
"caption": "Improvement of statistical accuracy in isovector (left panel) and isoscalar (right panel) charges.",
"content": "Figure 1 :\n1Improvement of statistical accuracy in isovector (left panel) and isoscalar (right panel) charges."
},
{
"id": "",
"caption": "Effective value... | [
{
"text": ". H Ohki, JLQCD CollaborationPhys. Rev. D. 7854502H. Ohki et al. (JLQCD Collaboration), Phys. Rev. D 78, 054502 (2008).",
"doi": ""
},
{
"text": ". K Takeda, JLQCD CollaborationPhys. Rev. D. 83114506K. Takeda et al. (JLQCD Collaboration), Phys. Rev. D 83, 114506 (2011).",
"doi": ""
... |
Bringing Yang-Mills Theory Closer to Quasiclassics
Mikhail Shifman
William I. Fine Theoretical Physics Institute
University of Minnesota
55455MinneapolisMNUSA
Bringing Yang-Mills Theory Closer to Quasiclassics
A deformation of pure Yang-Mills theory by a phantom field similar to the Faddeev-Popov ghost is conside... | 233,347,148 | {
"arxiv": "2104.10714",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevd.104.045002"
} | {
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"https://arxiv.org/pdf/2104.10714v2.pdf"
],
"pdf_sha": "31bddf42f36773fd50cdfa40ae9df94a51f371ad",
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"license": "CCBY",
"url": "http://link.aps.org/pdf/10.1103/PhysRevD.104.045002",
"status": "HYBRID"
}
} | Bringing Yang-Mills Theory Closer to Quasiclassics | [
"Mikhail Shifman \nWilliam I. Fine Theoretical Physics Institute\nUniversity of Minnesota\n55455MinneapolisMNUSA"
] | A deformation of pure Yang-Mills theory by a phantom field similar to the Faddeev-Popov ghost is considered. In this theory an Ersatz-supersymmetry is identified which results in cancellation of quantum corrections up to twoloop order. A quadruplet built from two complex fields in the adjoint -the Faddeev-Popov ghost c... | [
{
"title": "Introduction",
"content": "In the recent publication [1] I established a certain proximity between the β-function calculations in supersymmetric and non-supersymmetric versions of the Yang-Mills theory. This was achieved by changing statistics for certain fields, for instance, the \"second\" glu... | [
{
"id": "",
"caption": "Tadpole contributions to the Z factor of the ϕ field.",
"content": "Figure 3 :\n3Tadpole contributions to the Z factor of the ϕ field."
},
{
"id": "",
"caption": "cancels just a small part of the positive ϕ loop. The theory remains unitary.",
"content": "cancels j... | [
{
"text": "Perturbative proximity between supersymmetric and nonsupersymmetric theories. M Shifman, arXiv:2009.12654Phys. Rev. D. 10212125011hep-thM. Shifman, Perturbative proximity between supersymmetric and non- supersymmetric theories, Phys. Rev. D 102, no.12, 125011 (2020) [arXiv:2009.12654 [hep-th]].",
... |
UCLA UCLA Previously Published Works Title Algorithms on ensemble quantum computers. Publication Date Algorithms on ensemble quantum computers
2010
Boykin
Oscar Mor
Tal Roychowdhury
Vwani
P Oscar
Boykin Ae
Tal Mor
A E Vwani
Roychowdhury Ae
Farrokh Vatan
P O Boykin
T Mor
V Roychowdhury
F Vatan
Departmen... | 1,357,540 | {
"arxiv": "quant-ph/9907067",
"mag": 2949117517,
"acl": null,
"pubmed": "21475662v1",
"pubmedcentral": 3063506,
"dblp": "journals/nc/BoykinMRV10",
"doi": "10.1007/s11047-009-9133-0"
} | {
"pdf_urls": [],
"pdf_sha": "19125a9efc6432302a47d3cf7dc6bd63c5ef9a28",
"open_access": {
"license": "CCBYNC",
"url": "https://link.springer.com/content/pdf/10.1007/s11047-009-9133-0.pdf",
"status": "HYBRID"
}
} | UCLA UCLA Previously Published Works Title Algorithms on ensemble quantum computers. Publication Date Algorithms on ensemble quantum computers | [
"Boykin",
"Oscar Mor",
"Tal Roychowdhury",
"Vwani",
"P Oscar",
"Boykin Ae",
"Tal Mor",
"A E Vwani",
"Roychowdhury Ae",
"Farrokh Vatan",
"P O Boykin",
"T Mor",
"V Roychowdhury",
"F Vatan",
"Department of Electrical and Computer Engineering\nComputer Science Department\nElectrical Engineer... | In ensemble (or bulk) quantum computation, all computations are performed on an ensemble of computers rather than on a single computer. Measurements of qubits in an individual computer cannot be performed; instead, only expectation values (over the complete ensemble of computers) can be measured. As a result of this li... | [
{
"title": "The measurement in ensemble quantum computation",
"content": "The measurement process in quantum mechanics can be described simply as follows: To measure the state of a qubit, say jwi ¼ aj0i þ bj1i in the computation basis (j0i; j1iÞ; one measures the Hermitian operator (the observable)\nto get ... | [
{
"id": "",
"caption": ": Let B be a database of size M; then the search algorithm, with high probability, starting with the input 1 ffiffiffi M p P x2B jxi in time Oð ffiffiffiffi ffi M p Þ can determine whether there is any solutions in B or not.",
"content": ": Let B be a database of size M; then the... | [
{
"text": "Polynomial simulations of decohered quantum computers. D Aharonov, M Ben-Or, Proceedings of the 37th annual symposium on foundations of computer science, FOCS'96. the 37th annual symposium on foundations of computer science, FOCS'96Los AlamitosIEEE Computer Society PressAharonov D, Ben-Or M (1996) Po... |
DISCUSSION: CONDITIONAL GROWTH CHARTS
2006
Mary Lou Thompson
University of Washington
DISCUSSION: CONDITIONAL GROWTH CHARTS
The Annals of Statistics
3452006Main article
I will use the terms "reference centiles" or "centile charts," as the setting that I consider here is more general than that of "growth charts."L... | 14,185,683 | {
"arxiv": "math/0702642",
"mag": 2952758079,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1214/009053606000000650"
} | {
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"https://export.arxiv.org/pdf/math/0702642v1.pdf"
],
"pdf_sha": "e0d6734306ef4a2fe0e96d8cd997a1304b7de0ec",
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"license": "implied-oa",
"url": "https://doi.org/10.1214/009053606000000650",
"status": "HYBRID"
}
} | DISCUSSION: CONDITIONAL GROWTH CHARTS | [
"Mary Lou Thompson \nUniversity of Washington"
] | I will use the terms "reference centiles" or "centile charts," as the setting that I consider here is more general than that of "growth charts."Longitudinal reference centiles over some measure of time (typically age) are almost always implemented repeatedly on the same individual. In this kind of setting the notion of... | [
{
"title": "Drift.",
"content": "All of the methods of calculating conditional centiles implemented above can be expected to indicate once-off jumps in the path of an individual, but they are not able to deal satisfactorily with drift in an individual's path. WH acknowledge this when they state, for instanc... | [
{
"id": "",
"caption": "This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2006, Vol. 34, No. 5, 2119-2125. This reprint differs from the original in pagination and typographic detail.",
"content": "This is an electron... | [
{
"text": "Growth charts for both cross-sectional and longitudinal data. T J Cole, Statistics in Medicine. 13Cole, T. J. (1994). Growth charts for both cross-sectional and longitudinal data. Statistics in Medicine 13 2477-2492.",
"doi": ""
},
{
"text": "Limitations of the odds ratio in gauging the p... |
Is Multilingual BERT Fluent in Language Generation?
Samuel Rönnqvist
TurkuNLP Department of Future Technologies
University of Turku
Finland
Jenna Kanerva
TurkuNLP Department of Future Technologies
University of Turku
Finland
Tapio Salakoski
TurkuNLP Department of Future Technologies
University of Turku
Finland
... | 203,693,409 | {
"arxiv": "1910.03806",
"mag": 3088382025,
"acl": "W19-6204",
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1910-03806",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1910.03806v1.pdf"
],
"pdf_sha": "21ec087995ff5394bc78e3fe6d1bde2c194b5bd1",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Is Multilingual BERT Fluent in Language Generation? | [
"Samuel Rönnqvist \nTurkuNLP Department of Future Technologies\nUniversity of Turku\nFinland",
"Jenna Kanerva \nTurkuNLP Department of Future Technologies\nUniversity of Turku\nFinland",
"Tapio Salakoski \nTurkuNLP Department of Future Technologies\nUniversity of Turku\nFinland",
"Filip Ginter \nTurkuNLP Depa... | The multilingual BERT model is trained on 104 languages and meant to serve as a universal language model and tool for encoding sentences.We explore how well the model performs on several languages across several tasks: a diagnostic classification probing the embeddings for a particular syntactic property, a cloze task ... | [
{
"title": "Introduction",
"content": "The language representation model BERT (Bidirectional Encoder Representations from Transformers) has been shown to achieve state-of-theart performance when fine-tuned on a range of downstream tasks related to language understanding (Devlin et al., 2018), and recently a... | [
{
"id": "",
"caption": "3031 training examples.The treebank test sets yield 1002-1217 examples, with the exception of Danish with 515 examples.",
"content": "3031 training examples.The treebank test sets yield 1002-1217 examples, with the exception of Danish with 515 examples."
},
{
"id": "",
... | [
{
"text": "What does BERT look at? an analysis of BERT's attention. Kevin Clark, Urvashi Khandelwal, Omer Levy, Christopher D Manning, Proceedings of the 2019 ACL Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP. the 2019 ACL Workshop BlackboxNLP: Analyzing and Interpreting Neural Networ... |
Faster Projected GAN: Towards Faster Few-Shot Image Generation
Chuang Wang
Dept. of information
North China University of Technology
100000PekingChina
Zhengping Li
Dept. of information
North China University of Technology
100000PekingChina
Yuwen Hao
Medical Innovation Research Division
Chinese PLA General Hospi... | 268,385,074 | {
"arxiv": "2403.08778",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2403-08778",
"doi": "10.48550/arxiv.2403.08778"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2403.08778v1.pdf"
],
"pdf_sha": "6157b734fad6456ea6d4fd49f538e461f0a40651",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Faster Projected GAN: Towards Faster Few-Shot Image Generation | [
"Chuang Wang \nDept. of information\nNorth China University of Technology\n100000PekingChina",
"Zhengping Li \nDept. of information\nNorth China University of Technology\n100000PekingChina",
"Yuwen Hao \nMedical Innovation Research Division\nChinese PLA General Hospital\n100853BeijingChina",
"Lijun Wang \nDep... | In order to solve the problems of long training time, large consumption of computing resources and huge parameter amount of GAN network in image generation, this paper proposes an improved GAN network model, which is named Faster Projected GAN, based on Projected GAN.The proposed network is mainly focuses on the improv... | [
{
"title": "Introduction",
"content": "Images are of great significance to social life.A picture is worth a thousand words.Images are very important in information transmission.Especially for special fields such as aerospace, medical, and criminal investigation, high-quality images can often play an importa... | [
{
"id": "",
"caption": "Figure 2: Network structure of Generator of Faster Projected GAN.The left is the improved upblock in our model which is represented as red arrows in the right figure.The right is the structure of the Generator.",
"content": "Figure 2 :\n2\nFigure 2: Network structure of Generator... | [
{
"text": "Correction to: dl-cnn-based approach with image processing techniques for diagnosis of retinal diseases. Akash Tayal, Jivansha Gupta, Arun Solanki,",
"doi": ""
},
{
"text": ". Multimedia Systems. 2021prepublish",
"doi": ""
},
{
"text": "Generalizing from a few examples: A surv... |
Limits for learning with Language Models
21 Jun 2023
Nicholas Asher asher@irit.fr
Swarnadeep Bhar swarnadeep.bhar@irit.fr
Akshay Chaturvedi
Julie Hunter jhunter@linagora.com
Soumya Paul soumya.paul@gmail.com
CNRS
IRIT
Université de Toulouse
IRIT
Université de Toulouse
IRIT
LINAGORA
TELINDUS
Limits for... | 259,212,162 | {
"arxiv": "2306.12213",
"mag": null,
"acl": "2023.starsem-1.22",
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/starsem/AsherBCHP23",
"doi": "10.18653/v1/2023.starsem-1.22"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2306.12213v1.pdf"
],
"pdf_sha": "ec55e16e934e4a7c4527a1f54b39a680a88b392c",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Limits for learning with Language Models | [
"Nicholas Asher asher@irit.fr",
"Swarnadeep Bhar swarnadeep.bhar@irit.fr",
"Akshay Chaturvedi",
"Julie Hunter jhunter@linagora.com",
"Soumya Paul soumya.paul@gmail.com",
"CNRS\nIRIT",
"Université de Toulouse",
"IRIT\nUniversité de Toulouse",
"IRIT\nLINAGORA\nTELINDUS"
] | With the advent of large language models (LLMs), the trend in NLP has been to train LLMs on vast amounts of data to solve diverse language understanding and generation tasks. The list of LLM successes is long and varied. Nevertheless, several recent papers provide empirical evidence that LLMs fail to capture important ... | [
{
"title": "Introduction",
"content": "The success of large language models (LLMs) has led researchers in NLP to harness LLMs trained on vast amounts of data to solve a variety of language understanding and generation tasks, and some have claimed that LLMs can solve any task that can be specified via prompt... | [
{
"id": "",
"caption": "An LLM M can effectively learn a Borel set S ⊂ V ω out of a countable set of hypotheses H iff M has a uniformly consistent algorithm such that h S = inf H, as in Definition 4, and where h S : V ω → {0, 1} defines S.",
"content": "Definition 6 .\n6An LLM M can effectively learn a ... | [
{
"text": "Neural network learning: Theoretical foundations. M Anthony, P L Bartlett, P L Bartlett, cambridge university press9CambridgeAnthony, M., Bartlett, P. L., Bartlett, P. L., et al. (1999). Neural network learning: Theoretical foundations, volume 9. cambridge university press Cambridge.",
"doi": ""
... |
Enhancing Intrusion Detection in IoT Environments: An Advanced Ensemble Approach Using Kolmogorov-Arnold Networks
Amar Amouri
Abu Dhabi Polytechnic
Abu DhabiUAE
Mohamad Mahmoud
College of Computer & Information Science
King Saud University
RiyadhSaudi Arabia
Al Rahhal
College of Computer & Information Science
K... | 271,974,368 | {
"arxiv": "2408.15886",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2408-15886",
"doi": "10.1109/isncc62547.2024.10758956"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2408.15886v1.pdf"
],
"pdf_sha": "aa6eb39db3a18c7b04a52582c8f957d737e9463c",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Enhancing Intrusion Detection in IoT Environments: An Advanced Ensemble Approach Using Kolmogorov-Arnold Networks | [
"Amar Amouri \nAbu Dhabi Polytechnic\nAbu DhabiUAE",
"Mohamad Mahmoud \nCollege of Computer & Information Science\nKing Saud University\nRiyadhSaudi Arabia",
"Al Rahhal \nCollege of Computer & Information Science\nKing Saud University\nRiyadhSaudi Arabia",
"Yakoub Bazi \nCollege of Computer & Information Scie... | In recent years, the evolution of machine learning techniques has significantly impacted the field of intrusion detection, particularly within the context of the Internet of Things (IoT).As IoT networks expand, the need for robust security measures to counteract potential threats has become increasingly critical.This p... | [
{
"title": "I. INTRODUCTION",
"content": "The Internet of Things (IoT) has come a long way since Kevin Ashton first introduced the term in 1999 [1].Initially, the idea was to connect RFID devices to the internet, but today, IoT has expanded to include a wide variety of electronic devices-ranging from sensor... | [
{
"id": "",
"caption": "Fig. 2. A summary of the original N-BaIoT dataset instances per device(dataset) and per attack (class).",
"content": "Fig. 2 .\n2\nFig. 2. A summary of the original N-BaIoT dataset instances per device(dataset) and per attack (class)."
},
{
"id": "",
"caption": "Fig. ... | [
{
"text": "Multimedia internet of things: A comprehensive survey. A Nauman, Y A Qadri, M Amjad, Y B Zikria, M K Afzal, S W Kim, IEEE Access. 82020",
"doi": ""
},
{
"text": "An overview of wireless sensor networks towards internet of things. M Kocakulak, I Butun, 2017 IEEE 7th annual computing and co... |
On the power propagation time of a graph
31 Mar 2018
Chassidy Bozeman
On the power propagation time of a graph
31 Mar 2018arXiv:1608.01189v2 [math.CO]
In this paper, we give Nordhaus-Gaddum upper and lower bounds on the sum of the power propagation time of a graph and its complement, and we consider the effects of e... | 119,270,770 | {
"arxiv": "1608.01189",
"mag": 2486768534,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1608.01189v2.pdf"
],
"pdf_sha": "2d2458f13938e0603dc363da443d2d75b94c785c",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | On the power propagation time of a graph | [
"Chassidy Bozeman"
] | In this paper, we give Nordhaus-Gaddum upper and lower bounds on the sum of the power propagation time of a graph and its complement, and we consider the effects of edge subdivisions and edge contractions on the power propagation time of a graph. We also study a generalization of power propagation time, known as k−powe... | [
{
"title": "Introduction",
"content": "Phasor Measurement Units (PMUs) are machines used by energy companies to monitor the electric power grid. They are placed at selected electrical nodes (locations at which transmission lines, loads, and generators are connected) within the system. Due to the high cost o... | [
{
"id": "",
"caption": "For any t ≥ 0, there exists a graph H = (V, E) and edge e ∈ E such that ppt(H/e) ≥ ppt(H) + t. Proof. From Proposition 17, there exist graphs G and G e such that ppt(G e ) ≤ ppt(G) − t. Let H = G e and H/e = G. Then ppt(H/e) ≥ ppt(H) + t.",
"content": "Corollary 18 .\n18For any t... | [
{
"text": "n − 3, then for any minimum k−power dominating set S, |S| ≤ 3 and |N [S]| ≤ 4. It follows from Lemma 28 that ∆(G) ≤ 3. If ∆(G) ≤ 2, then G is the union of paths and cycles. Since |S| ≤ 3 for any minimum k-power dominating set S, G has at most 3 components. If G is connected. it follows from Propositi... |
Multi-View Substructure Learning for Drug-Drug Interaction Prediction
Zimeng Li
College of Information Science and Engineering
Hunan University
410086ChangshaChina
Microsoft Research Asia
10080BeijingChina
Shichao Zhu
Microsoft Research Asia
10080BeijingChina
School of Cyber Security
University of Chinese Acade... | 247,762,875 | {
"arxiv": "2203.14513",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2203-14513",
"doi": "10.48550/arxiv.2203.14513"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2203.14513v1.pdf"
],
"pdf_sha": "a0d24a815c7eadb3a661ac6b7e124a541b88cf97",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Multi-View Substructure Learning for Drug-Drug Interaction Prediction | [
"Zimeng Li \nCollege of Information Science and Engineering\nHunan University\n410086ChangshaChina\n\nMicrosoft Research Asia\n10080BeijingChina",
"Shichao Zhu \nMicrosoft Research Asia\n10080BeijingChina\n\nSchool of Cyber Security\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nInstitute of In... | Drug-drug interaction (DDI) prediction provides a drug combination strategy for systemically effective treatment. Previous studies usually model drug information constrained on a single view such as the drug itself, leading to incomplete and noisy information, which limits the accuracy of DDI prediction. In this work, ... | [
{
"title": "Introduction",
"content": "Drug combinations can provide therapeutic benefits but also increase the risk of adverse side effects, caused by the physicochemical incompatibility of the drugs [1,2,3]. The identification of drug-drug interactions (DDIs) remains a challenging task considering that th... | [
{
"id": "",
"caption": "Schematic of the MSN-DDI architecture. (A) The framework of MSN-DDI.The input drug pair is encoded by a bipartite graph followed by a series of repetitive MSN encoder blocks. For each block, substructures are extracted by the substructure module, in which substructure specific embedd... | [
{
"text": "Data-driven prediction of drug effects and interactions. N P Tatonetti, P P Ye, R Daneshjou, R B Altman, Science translational medicine. 4Tatonetti, N. P., Ye, P. P., Daneshjou, R. & Altman, R. B. Data-driven prediction of drug effects and interactions. Science translational medicine 4, 125ra31-125ra... |
A Scalable Finite Difference Method for Deep Reinforce- ment Learning
Matthew Allen mallen22@mgh.harvard.edu
Center for Systems Biology
Center for Systems Biology Massachusetts General Hospital
Center for Systems Biology Massachusetts General Hospital
Massachusetts General Hospital
John C Raisbeck jraisbeck@mgh.h... | 252,907,648 | {
"arxiv": "2210.07487",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2210-07487",
"doi": "10.48550/arxiv.2210.07487"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2210.07487v2.pdf"
],
"pdf_sha": "eae58ee87cd9455f2a53b364db9fe1e0c146a865",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A Scalable Finite Difference Method for Deep Reinforce- ment Learning | [
"Matthew Allen mallen22@mgh.harvard.edu \nCenter for Systems Biology\nCenter for Systems Biology Massachusetts General Hospital\nCenter for Systems Biology Massachusetts General Hospital\nMassachusetts General Hospital",
"John C Raisbeck jraisbeck@mgh.harvard.edu \nCenter for Systems Biology\nCenter for Systems B... | Several low-bandwidth distributable black-box optimization algorithms in the family of finite differences such as Evolution Strategies have recently been shown to perform nearly as well as tailored Reinforcement Learning methods in some Reinforcement Learning domains. One shortcoming of these black-box methods is that ... | [
{
"title": "Introduction",
"content": "Reinforcement learning (RL) is a sub-field of machine learning that is concerned with finding an optimal policy π * to direct an agent through a Markov decision process (MDP) to maximize an objective J(π). In this work, we are interested in methods relating to the poli... | [
{
"id": "",
"caption": "Methods Comparison An empirical study comparing ES, FD, and DFD in 4 continuous control tasks. Dotted lines represent the Inter-Quartile Mean and shaded regions show the point-wise 95% percentile stratified bootstrap confidence intervals over 10 random seeds.",
"content": "Figure... | [
{
"text": "Deep reinforcement learning at the edge of the statistical precipice. Rishabh Agarwal, Max Schwarzer, Pablo Samuel Castro, Aaron Courville, Marc G Bellemare, Advances in Neural Information Processing Systems. Rishabh Agarwal, Max Schwarzer, Pablo Samuel Castro, Aaron Courville, and Marc G Bellemare. ... |
ON CAT(κ) SURFACES
24 Sep 2023
Saajid Chowdhury saajid.chowdhury@stonybrook.edu
Hechen Hu
Matthew Romney mromney@stevens.edu
Adam Tsou adam.tsou@stonybrook.edu
Department of Physics and Astronomy
Stony Brook University
11794Stony BrookNYUSA
Department of Mathematics
Columbia University
2990, 10027Broadway, New... | 262,459,305 | {
"arxiv": "2309.13533",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2309.13533v1.pdf"
],
"pdf_sha": "dd32014bab2c693bcba9f6d896fd7ce501aac924",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | ON CAT(κ) SURFACES | [
"Saajid Chowdhury saajid.chowdhury@stonybrook.edu",
"Hechen Hu",
"Matthew Romney mromney@stevens.edu",
"Adam Tsou adam.tsou@stonybrook.edu",
"Department of Physics and Astronomy\nStony Brook University\n11794Stony BrookNYUSA",
"Department of Mathematics\nColumbia University\n2990, 10027Broadway, New YorkN... | We study the properties of CAT(κ) surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the CAT(κ) condition locally.The main facts about CAT(κ) surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation.W... | [
{
"title": "Introduction",
"content": "The field of Alexandrov geometry concerns two main classes of metric spaces: spaces of curvature bounded below and spaces of curvature bounded above.The latter is known at the class of CAT(κ) spaces, where κ ∈ R is some real parameter; these can be defined as spaces in... | [
{
"id": "",
"caption": "Let T = △pqr be a geodesic triangle and x an interior point.Then the geodesic [px] can be extended to intersect the opposite side [qr].",
"content": "Corollary 3 . 4 .\n34\nLet T = △pqr be a geodesic triangle and x an interior point.Then the geodesic [px] can be extended to inter... | [
{
"text": "Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces. T Adamowicz, G Veronelli, Calc. Var. Partial Differential Equations. 611302022",
"doi": ""
},
{
"text": "Intrinsic geometry of surfaces. Translated from the Russian by. A D Aleks... |
Generator versus Segmentor: Pseudo-healthy Synthesis
Yunlong Zhang
School of Informatics
Xiamen University
XiamenChina
Chenxin Li
School of Informatics
Xiamen University
XiamenChina
Xin Lin
School of Informatics
Xiamen University
XiamenChina
Liyan Sun
School of Informatics
Xiamen University
XiamenChina
Yihon... | 229,547,264 | {
"arxiv": "2009.05722",
"mag": 3085471368,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2009.05722v3.pdf"
],
"pdf_sha": "48aad31f230e9a09b5a56714dff1461353c73dd9",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Generator versus Segmentor: Pseudo-healthy Synthesis | [
"Yunlong Zhang \nSchool of Informatics\nXiamen University\nXiamenChina",
"Chenxin Li \nSchool of Informatics\nXiamen University\nXiamenChina",
"Xin Lin \nSchool of Informatics\nXiamen University\nXiamenChina",
"Liyan Sun \nSchool of Informatics\nXiamen University\nXiamenChina",
"Yihong Zhuang \nSchool of In... | This paper investigates the problem of pseudo-healthy synthesis that is defined as synthesizing a subject-specific pathology-free image from a pathological one. Recent approaches based on Generative Adversarial Network (GAN) have been developed for this task. However, these methods will inevitably fall into the trade-o... | [
{
"title": "Experiments",
"content": ""
},
{
"title": "Implementation details",
"content": "Data. Our method is mainly evaluated on BraTS2019 dataset [8,2], which contains 259 GBM (i.e., glioblastoma) and 76 LGG (i.e., lower-grade glioma) volumes. Here, we utilize the T2-weighted volumes of GBM,... | [
{
"id": "",
"caption": "(a) Framework of the pixel-level weighted cross-entropy loss. (b) The blue, yellow, green, and red boxes denote zoom-in views corresponding to the pathological image, synthetic image, segmentation prediction, and lesion annotation, respectively.",
"content": "Fig. 2 .\n2(a) Frame... | [
{
"text": "Pathology segmentation using distributional differences to images of healthy origin. S Andermatt, A Horváth, S Pezold, P Cattin, International MICCAI Brainlesion Workshop. SpringerAndermatt, S., Horváth, A., Pezold, S., Cattin, P.: Pathology segmentation using distributional differences to images of ... |
Distributed Cartesian Power Graph Segmentation for Graphon Estimation
Shitong Wei swei@ucdavis.edu
Department of Statistics UC
Department of Statistics UC
Department of Statistics UC
95618, 94720, 95618Davis Davis, Berkeley Berkeley, Davis DavisCA, CA, CA
Oscar Hernan Madrid-Padilla
Department of Statistics UC
De... | 44,072,170 | {
"arxiv": "1805.09978",
"mag": 2804401007,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1805-09978",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1805.09978v1.pdf"
],
"pdf_sha": "bba8323b6ad98d8005db25fa51988ad4868fd3c3",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Distributed Cartesian Power Graph Segmentation for Graphon Estimation | [
"Shitong Wei swei@ucdavis.edu \nDepartment of Statistics UC\nDepartment of Statistics UC\nDepartment of Statistics UC\n95618, 94720, 95618Davis Davis, Berkeley Berkeley, Davis DavisCA, CA, CA",
"Oscar Hernan Madrid-Padilla \nDepartment of Statistics UC\nDepartment of Statistics UC\nDepartment of Statistics UC\n95... | We study an extention of total variation denoising over images to over Cartesian power graphs and its applications to estimating non-parametric network models. The power graph fused lasso (PGFL) segments a matrix by exploiting a known graphical structure, G, over the rows and columns. Our main results shows that for an... | [
{
"title": "Introduction",
"content": "Total variation (TV) denoising, also known as the fused lasso, is a classical method for image denoising Chambolle and Lions (1997) that groups pixels that are adjacent to one another and have similar pixel values, a process known as segmentation. For a network, H, the... | [
{
"id": "",
"caption": "KNN-PGFL method: the 2-NN graph G 2 (left) is learned from the adjacency matrix A (middle) of the network H, then PGFL is applied to G 2 2 with the A ij dyadic labels (right).",
"content": "Figure 1 :\n1KNN-PGFL method: the 2-NN graph G 2 (left) is learned from the adjacency matr... | [
{
"text": "Stochastic blockmodel approximation of a graphon: Theory and consistent estimation. E M Airoldi, T B Costa, Chan , S H , Advances in Neural Information Processing Systems. Airoldi, E. M., Costa, T. B., and Chan, S. H. (2013). Stochastic blockmodel approximation of a graphon: Theory and consistent est... |
A Dataset for Crucial Object Recognition in Blind and Low-Vision Individuals' Navigation
23 Jul 2024
Masum Syed
Billah sbillah@psu.edu
MdTouhidul Islam
Imran Kabir
MdElena Ariel Pearce elena.pearce@drake.edu
Alimoor Reza
Syed Masum
Md Alimoor
Pennsylvania State University
USA
IMRAN KABIR
Pennsylvania State... | 271,404,417 | {
"arxiv": "2407.16777",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2407-16777",
"doi": "10.48550/arxiv.2407.16777"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2407.16777v1.pdf"
],
"pdf_sha": "afb53ab61766f748afa243a4d21365ee902c5d77",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A Dataset for Crucial Object Recognition in Blind and Low-Vision Individuals' Navigation | [
"Masum Syed",
"Billah sbillah@psu.edu",
"MdTouhidul Islam",
"Imran Kabir",
"MdElena Ariel Pearce elena.pearce@drake.edu",
"Alimoor Reza",
"Syed Masum",
"Md Alimoor",
"Pennsylvania State University\nUSA",
"IMRAN KABIR\nPennsylvania State University\nUSA",
"ELENA ARIEL PEARCE\nDrake University\nUS... | This paper introduces a dataset for improving real-time object recognition systems to aid blind and low-vision (BLV) individuals in navigation tasks.The dataset comprises 21 videos of BLV individuals navigating outdoor spaces, and a taxonomy of 90 objects crucial for BLV navigation, refined through a focus group study.... | [
{
"title": "INTRODUCTION",
"content": "With the advent of newer algorithms and large-scale, multi-modal datasets, task-specific AI models are converging into a single, task-agnostic, general-purpose model [9,20,25,25,54].These models are rapidly inching toward their large-scale deployment in various real-wo... | [
{
"id": "",
"caption": "shows the heatmap representing the classwise F1 score of all seven models.In this figure, in a cell means the corresponding model can not predict the corresponding object.The F1 score is within the [0, 1] range and is represented by the intensity of the blue color ( ) in Figure 3.The... | [
{
"text": "Recognition from Detection Model. Traditionally, detection and segmentation models predict bounding boxes and masks, respectively, for each of the categories. If an object is present in a given image, then the detection model will provide us with the pixel coordinates of bounding boxes and the catego... |
Search for neutrino signals correlated with LHAASO diffuse Galactic emission
August 23, 2024
Wenlian Li 0000-0002-4300-5130
Institute
Physics3 of , , , , Astronomy, , 6
Tian-Qi Huang 0000-0001-8510-2513
Donglian L Xu donglianxu@sjtu.edu.cn 0000-0003-1639-8829
Institute
Physics3 of , , , , Astronomy, , 6
Search ... | 271,924,468 | {
"arxiv": "2408.12123",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2408.12123v1.pdf"
],
"pdf_sha": "8adc75d738ad7ebb1205dfe618b5fc43c57117dc",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Search for neutrino signals correlated with LHAASO diffuse Galactic emission | [
"Wenlian Li 0000-0002-4300-5130\nInstitute\n\n\nPhysics3 of , , , , Astronomy, , 6",
"Tian-Qi Huang 0000-0001-8510-2513",
"Donglian L Xu donglianxu@sjtu.edu.cn 0000-0003-1639-8829\nInstitute\n\n\nPhysics3 of , , , , Astronomy, , 6"
] | The diffuse Galactic γ-ray emission originates from the interactions between cosmic rays and the interstellar medium or radiation fields within our Galaxy, where the production of neutrinos is also anticipated.Recently, the Large High Altitude Air Shower Observatory (LHAASO) reported measurements of diffuse γ-rays from... | [
{
"title": "INTRODUCTION",
"content": "The origin and acceleration mechanism of the high-energy cosmic rays remain a puzzle since the discovery of cosmic rays in 1912 (Hess 1912).During the propagation within the Galactic plane, cosmic rays interact with the dense interstellar medium through deep inelastic ... | [
{
"id": "",
"caption": "Figure 1.The muon neutrino intensity as a function of neutrino energy Eν .The best-fit flux obtained using the γ-ray flux map with > 0.5σ detection is shown as a red solid line, averaged over the LHAASO > 0.5σ region.The red shaded region indicates the 1σ uncertainty and extends over... | [
{
"text": ". M G Aartsen, 10.3847/1538-4357/aa8dfbAstrophys. J. 849672017a",
"doi": ""
},
{
"text": ". 10.1088/1748-0221/12/03/P03012JINST. 12P030122017b",
"doi": ""
},
{
"text": ". 10.3847/1538-4357/835/2/151Astrophys. J. 8351512017c",
"doi": ""
},
{
"text": ". 10.3847/1538-... |
A Detection Method of Temporally Operated Videos Using Robust Hashing
Shoko Niwa niwa-shoko@ed.tmu.ac.jp
Tokyo Metropolitan University Tokyo
Japan
Miki Tanaka tanaka-miki@ed.tmu.ac.jp
Tokyo Metropolitan University Tokyo
Japan
Hitoshi Kiya kiya@tmu.ac.jp
Tokyo Metropolitan University Tokyo
Japan
A Detection Met... | 251,467,930 | {
"arxiv": "2208.05198",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/gcce/NiwaTK22",
"doi": "10.1109/gcce56475.2022.10014233"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2208.05198v2.pdf"
],
"pdf_sha": "b8b5b8888e081213fb246066ed6d8a4b657d39c3",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A Detection Method of Temporally Operated Videos Using Robust Hashing | [
"Shoko Niwa niwa-shoko@ed.tmu.ac.jp \nTokyo Metropolitan University Tokyo\nJapan",
"Miki Tanaka tanaka-miki@ed.tmu.ac.jp \nTokyo Metropolitan University Tokyo\nJapan",
"Hitoshi Kiya kiya@tmu.ac.jp \nTokyo Metropolitan University Tokyo\nJapan"
] | SNS providers are known to carry out the recompression and resizing of uploaded videos/images, but most conventional methods for detecting tampered videos/images are not robust enough against such operations. In addition, videos are temporally operated such as the insertion of new frames and the permutation of frames, ... | [
{
"title": "I. INTRODUCTION",
"content": "Recent rapid advances in image manipulation tools and deep image synthesis techniques have made generating fake videos easy. In addition, with the spread of SNS, the existence of fake videos has become a major threat to the credibility of the international community... | [
{
"id": "",
"caption": "Example of inter-frame tampering inserting, deleting, rearranging, and replacing frames as shown inFig. 1.Frame insertion (Figure 1a) is a technique that inserts some frames from another video frames into original ones. Frame deletion(Figure 1b)is to delete some frames from the origi... | [
{
"text": "Media forensics and deepfakes: an overview. L Verdoliva, IEEE Journal of Selected Topics in Signal Processing. 145L. Verdoliva, \"Media forensics and deepfakes: an overview,\" IEEE Journal of Selected Topics in Signal Processing, vol. 14, no. 5, pp. 910-932, 2020.",
"doi": ""
},
{
"text":... |
ONE REMARK ON THE B-SEMIAMPLENESS OF THE MODULI PART
4 Nov 2013
Enrica Floris
ONE REMARK ON THE B-SEMIAMPLENESS OF THE MODULI PART
4 Nov 2013
In this short note we reduce the b-semiampleness conjecture for lc-trivial fibrations to the b-semiampleness conjecture for klt-trivial fibrations.
Introduction
An important... | 119,294,252 | {
"arxiv": "1311.0915",
"mag": 2282107431,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1311.0915v1.pdf"
],
"pdf_sha": "b53466f10e298466410ecf01a2144b6bdd2d1095",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | ONE REMARK ON THE B-SEMIAMPLENESS OF THE MODULI PART | [
"Enrica Floris"
] | In this short note we reduce the b-semiampleness conjecture for lc-trivial fibrations to the b-semiampleness conjecture for klt-trivial fibrations. | [
{
"title": "Introduction",
"content": "An important problem in birational geometry is the following: given a morphism f : X → Z and an lc pair (X, B) such that K X + B ∼ Qf 0, find a Q-divisor ∆ Z on Z such that (Z, ∆ Z ) is lc and K X + B ∼ Q f * (K Z + ∆ Z ). The problem has been solved by Ambro when (X, ... | [
{
"id": "",
"caption": "Let P ⊆ Z be a prime Weil divisor. The log canonical threshold γ P of f * (P ) with respect to the pair (X, B) is defined as follows. LetZ → Z be a resolution of Z. Let µ :X → X be the birational morphism obtained as a desingularisation of the main component of X × ZZ . Letf :X →Z. L... | [
{
"text": "The Adjunction Conjecture and its applications. F Ambro, 9903060PhD thesisThe Johns Hopkins University preprint mathF. Ambro, The Adjunction Conjecture and its applications, PhD thesis, The Johns Hopkins University preprint math.AG/9903060 (1999)",
"doi": ""
},
{
"text": "Shokurov's bound... |
Adversarial Domain Adaptation for Stance Detection
Brian Xu bwxu@mit.edu
Computer Science
Artificial Intelligence Laboratory
Massachusetts Institute of Technology Cambridge
MAUSA
Mitra Mohtarami mitram@mit.edu
Computer Science
Artificial Intelligence Laboratory
Massachusetts Institute of Technology Cambridge
MAUS... | 59,608,856 | {
"arxiv": "1902.02401",
"mag": 2913957957,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1902-02401",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1902.02401v1.pdf"
],
"pdf_sha": "865bfcf559838be4fb5f6be7d0f89c260167843e",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Adversarial Domain Adaptation for Stance Detection | [
"Brian Xu bwxu@mit.edu \nComputer Science\nArtificial Intelligence Laboratory\nMassachusetts Institute of Technology Cambridge\nMAUSA",
"Mitra Mohtarami mitram@mit.edu \nComputer Science\nArtificial Intelligence Laboratory\nMassachusetts Institute of Technology Cambridge\nMAUSA",
"James Glass glass@mit.edu \nCo... | This paper studies the problem of stance detection which aims to predict the perspective (or stance) of a given document with respect to a given claim. Stance detection is a major component of automated fact checking. As annotating stances in different domains is a tedious and costly task, automatic methods based on ma... | [
{
"title": "Introduction",
"content": "With the rise of social media and microblogs, there has been an increasing awareness of the negative influence of fake news and how it can unfairly influence public opinion on various events and policies [9,8,20]. In order to counteract these effects, various organizat... | [
{
"id": "",
"caption": "The architecture of our model with domain adaptation component for stance detection.",
"content": "Figure 1 :\n1The architecture of our model with domain adaptation component for stance detection."
},
{
"id": "",
"caption": "Accuracy: The number of correctly classifie... | [
{
"text": "Talos targets disinformation with fake news challenge victory. Sean Baird, Doug Sibley, Yuxi Pan, Fake News Challenge. Sean Baird, Doug Sibley, and Yuxi Pan. Talos targets disinformation with fake news challenge victory. In Fake News Challenge, 2017.",
"doi": ""
},
{
"text": "Integrating ... |
A Mood-based Genre Classification of Television Content *
6 Aug 2015
Humberto Jesús
Insight Centre for Data
Analytics University College Dublin
Ireland
Corona Pampín
Insight Centre for Data
Analytics University College Dublin
Ireland
Michael P O'mahony michael.omahony@insight-centre.org
Insight Centre for Data
A... | 9,336,286 | {
"arxiv": "1508.01571",
"mag": 2962918541,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/CoronaO15",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1508.01571v1.pdf"
],
"pdf_sha": "536455dea077786585ece6f641c529ca8a7895a1",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A Mood-based Genre Classification of Television Content * | [
"Humberto Jesús \nInsight Centre for Data\nAnalytics University College Dublin\nIreland",
"Corona Pampín \nInsight Centre for Data\nAnalytics University College Dublin\nIreland",
"Michael P O'mahony michael.omahony@insight-centre.org \nInsight Centre for Data\nAnalytics University College Dublin\nIreland"
] | The classification of television content helps users organise and navigate through the large list of channels and programs now available.In this paper, we address the problem of television content classification by exploiting text information extracted from program transcriptions.We present an analysis which adapts a m... | [
{
"title": "INTRODUCTION",
"content": "The problem of choice overload in television is well known.For example, in 2009 there were 2,218 television broadcast * This work was supported by Science Foundation Ireland through the CLARITY Centre for Sensor Web Technologies under grant number 07/CE/I1147 and throu... | [
{
"id": "",
"caption": "Figure 1: Valence, arousal and dominance for a year of television broadcasting per channel.",
"content": "Figure 1 :\n1\nFigure 1: Valence, arousal and dominance for a year of television broadcasting per channel."
},
{
"id": "",
"caption": "Figure 2: Valence per week ... | [
{
"text": "Affective Norms for English Words (ANEW): Instruction Manual and Affective Ratings. M Bradley, P Lang, 1999The Center for Research in Psychophysiology, University of FloridaTech. rep",
"doi": ""
},
{
"text": "Measuring the Happiness of Large-Scale Written Expression: Songs, Blogs, and Pre... |
Security Architecture for Cluster based Ad Hoc Networks
Preetida Vinayakray-Jani
DA-IICT
School of Technology and Computer Science
Tata Institute of Fundamental Research
GandhinanagerMumbaiIndia, India
Corresponding Author
DA-IICT
School of Technology and Computer Science
Tata Institute of Fundamental Research
Ga... | 8,674,554 | {
"arxiv": "1207.1701",
"mag": 1554810994,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1207-1701",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1207.1701v1.pdf"
],
"pdf_sha": "27116c8bf3329a787025f167c1a029f611d92938",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Security Architecture for Cluster based Ad Hoc Networks | [
"Preetida Vinayakray-Jani \nDA-IICT\nSchool of Technology and Computer Science\nTata Institute of Fundamental Research\nGandhinanagerMumbaiIndia, India",
"Corresponding Author \nDA-IICT\nSchool of Technology and Computer Science\nTata Institute of Fundamental Research\nGandhinanagerMumbaiIndia, India",
"Sugata ... | [
{
"title": "INTRODUCTION",
"content": "Mobile Ad hoc Networks (MANETs) allow mobile nodes to form a self-organized network without need for a permanent infrastructure. Such networks can be used in the defence applications, disaster management or in remote locations where supportive fixed network infrastruct... | [
{
"id": "",
"caption": "Creating Steg-links between CHs and updating neighbourhood",
"content": "Figure 3\n3Creating Steg-links between CHs and updating neighbourhood"
},
{
"id": "",
"caption": "Mechanism to improve convergence to form steg-links",
"content": "Figure 6\n6Mechanism to imp... | [
{
"text": "Securing ad hoc networks. Lidong Zhou, Zygmunt J Haas, IEEE Network. 136Lidong Zhou and Zygmunt J. Haas, \"Securing ad hoc networks,\" IEEE Network, Vol. 13, No. 6, pp. 24-30, 1999.",
"doi": ""
},
{
"text": "Self-securing ad hoc wireless networks. P Haiyun Luo, Jiejun Zerfos, Sonwu Kong, ... | |
The Evolution of Investor Activism in Japan
May 2, 2022
Ryo Sakai [ryo.sakai@vanderbilt.edu]
The Evolution of Investor Activism in Japan
May 2, 20221
Activist investors have gradually become a catalyst for change in Japanese companies. This study examines the impact of activist board representation on firm performan... | 249,240,436 | {
"arxiv": "2206.00640",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2206.00640v1.pdf"
],
"pdf_sha": "96d5b5a1b654fd9718a3393af9378ad7091c246c",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | The Evolution of Investor Activism in Japan | [
"Ryo Sakai [ryo.sakai@vanderbilt.edu]"
] | Activist investors have gradually become a catalyst for change in Japanese companies. This study examines the impact of activist board representation on firm performance in Japan. | [
{
"title": "Introduction",
"content": "Since Yoshiaki Murakami said this word in 2006, Japanese companies have debated whether activist funds are their enemies or allies. After the collapse of the real estate and stock market bubble in the late 1980s, foreign investors actively played important roles in the... | [
{
"id": "",
"caption": "quarter of 2011 (March 2012) and the Japanese fiscal fourth quarter of 2018 (March 2019) because the event was announced in April 2019. The post-announcement period of Kawasaki is set between the Japanese fiscal first quarter of 2019 (June 2019) and the Japanese fiscal third quarter ... | [
{
"text": "Returns to hedge fund activism: An international study. M Becht, J Franks, J Grant, H F Wagner, Review of Financial Studies. 309Becht, M., Franks, J., Grant, J., & Wagner, H. F. (2017). Returns to hedge fund activism: An international study. Review of Financial Studies, 30(9), 2933-2971.",
"doi":... |
Coverage characteristics of self-repelling random walks in mobile ad-hoc networks
23 Aug 2017 August 24, 2017
V Kulathumani *vinod.kulathumani@mail.wvu.edu
INTRODUCTION
Department of Computer Science
West Virginia University
26505MorgantownWV
M Nakagawa †manakagawa@mix.wvu.edu
INTRODUCTION
Department of Comput... | 32,180,211 | {
"arxiv": "1708.07049",
"mag": 2750505570,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1708-07049",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1708.07049v1.pdf"
],
"pdf_sha": "d8f78d8cebe6ee55ec0201ac63c760158f925476",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Coverage characteristics of self-repelling random walks in mobile ad-hoc networks | [
"V Kulathumani *vinod.kulathumani@mail.wvu.edu \nINTRODUCTION\n\n\nDepartment of Computer Science\nWest Virginia University\n26505MorgantownWV",
"M Nakagawa †manakagawa@mix.wvu.edu \nINTRODUCTION\n\n\nDepartment of Computer Science\nWest Virginia University\n26505MorgantownWV",
"A Arora \nThe Samraksh Company\n... | A self-repelling random walk of a token on a graph is one in which at each step, the token moves to a neighbor that has been visited least often (with ties broken randomly). The properties of self-repelling random walks have been analyzed for two dimensional lattices and these walks have been shown to exhibit a remarka... | [
{
"title": "Introduction",
"content": "A self-repelling random walk is one in which at each step the walk moves towards one of the neighbors that has been least visited [1]. Self-repelling random walks were introduced in the 1980s and have been studied extensively in the physics"
},
{
"title": "INTR... | [
{
"id": "",
"caption": "Distribution of number of visits at each node at different stages of exploration of a self-repelling random walk (network size 100,400 and 1000 nodes)",
"content": "Figure 1 :\n1Distribution of number of visits at each node at different stages of exploration of a self-repelling r... | [
{
"text": "On self-repelling random walks. C Byrnes, A J Guttman, Journal of Physics A: Mathemaical and General. 1717C. Byrnes and A. J. Guttman. On self-repelling random walks. Journal of Physics A: Mathemaical and General, 17(17):3335-3342, 1984.",
"doi": ""
},
{
"text": "How a random walk covers ... |
Acceleration of a polarized neutron by its weak nuclear self-interaction
M Donaire
Departamento de Física Teórica
Universidad de Valladolid
Atómica yÓptica47011ValladolidSpain
CNRS
Université Grenoble Alpes
Institut Néel
38000GrenobleFrance
Acceleration of a polarized neutron by its weak nuclear self-interaction
... | 261,557,515 | {
"arxiv": "2309.02293",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2309.02293v1.pdf"
],
"pdf_sha": "21fa1bdd66e310e44009f1e93aaf43d6b09c1e13",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Acceleration of a polarized neutron by its weak nuclear self-interaction | [
"M Donaire \nDepartamento de Física Teórica\nUniversidad de Valladolid\nAtómica yÓptica47011ValladolidSpain\n\nCNRS\nUniversité Grenoble Alpes\nInstitut Néel\n38000GrenobleFrance"
] | It is proven that the rotation of the spin of a polarized neutron is accompanied by a net nuclear force upon it. This force arises from the weak nuclear self-interaction of its constituent quarks, whose chiral nature induces the transfer of a net momentum to the fields of Z and W-bosons. This effect is linear in Fermi'... | [
{
"title": "I. INTRODUCTION",
"content": "The weak nuclear interaction of the Standard Model is chiral [1][2][3]. This implies that its Hamiltonian is not parity invariant, which manifests in the preferential coupling of left-handed particles to right-handed antiparticles. In turn, that results in a net ali... | [
{
"id": "",
"caption": "the neutral and chiral current, with g u,d v,a being the vector (v) and axial (a) couplings; and j i,µ cc = 1 2ψ d i γ µ (I − γ 5 )ψ u i is the charged and chiral current. The system of hadrons and EW bosons governed by the Hamiltonians of Eqs.(3)-(5) possesses a conserved linear mom... | [
{
"text": "J F Donoghe, E Golowich, B R Holstein, Dynamics of the Standard Model. CambridgeCambridge University PressJ.F. Donoghe, E. Golowich and B.R. Holstein, Dynam- ics of the Standard Model, Cambridge University Press, Cambridge (2022);",
"doi": ""
},
{
"text": "J C Burgess, G Moore, The Standa... |
Coherence and Sufficient Sampling Densities for Reconstruction in Compressed Sensing
2 Nov 2013
Franz J Király f.kiraly@ucl.ac.uk
Department of Statistical Science
University College London
Louis Theran theran@math.fu-berlin.de
Department of Statistical Science
University College London
Coherence and Sufficient... | 16,654,395 | {
"arxiv": "1302.2767",
"mag": 1872062320,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1302-2767",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1302.2767v2.pdf"
],
"pdf_sha": "a9d61976f4f48cc74a516455970bd55d1fa8b5fd",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Coherence and Sufficient Sampling Densities for Reconstruction in Compressed Sensing | [
"Franz J Király f.kiraly@ucl.ac.uk \nDepartment of Statistical Science\nUniversity College London",
"Louis Theran theran@math.fu-berlin.de \nDepartment of Statistical Science\nUniversity College London"
] | We give a new, very general, formulation of the compressed sensing problem in terms of coordinate projections of an analytic variety, and derive sufficient sampling rates for signal reconstruction.Our bounds are linear in the coherence of the signal space, a geometric parameter independent of the specific signal and me... | [
{
"title": "Introduction 1.Compressed Sensing, Randomness, and n log n",
"content": "Compressed sensing is the task of recovering a signal x from some low-complexity measurement(s) S(x), the samples of x.The sampling process, that is, the acquisition process of the sample S(x), is usually random and undirec... | [
{
"id": "",
"caption": "Figure 1: Schematic of coherent and incoherent spaces: (a) The projections of the coordinate vectors onto the linear space L are roughly the same size, so it has nearly minimal coherence.The flat M is a translate of the span of e 1 , giving it maximal coherence; observing the e 2 coo... | [
{
"text": "PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming. E J Candès, T Strohmer, V Voroninski, Communications on Pure and Applied Mathematics. 2012",
"doi": ""
},
{
"text": "Exact matrix completion via convex optimization. Emmanuel J Candès, Benjamin... |
PRICING COMPOUND AND EXTENDIBLE OPTIONS UNDER MIXED FRACTIONAL BROWNIAN MOTION WITH JUMPS
16 Aug 2017
Foad Shokrollahi
Department of Mathematics and Statistics
University of Vaasa
P.O. Box 700FIN-65101VaasaFINLAND
PRICING COMPOUND AND EXTENDIBLE OPTIONS UNDER MIXED FRACTIONAL BROWNIAN MOTION WITH JUMPS
16 Aug 2017
... | 102,351,277 | {
"arxiv": "1708.04829",
"mag": 2963458721,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/axioms/Shokrollahi19",
"doi": "10.3390/axioms8020039"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/1708.04829v1.pdf"
],
"pdf_sha": "da9b4063997d72200c86d6fc9556cf083aed7286",
"open_access": {
"license": "CCBY",
"url": "https://www.mdpi.com/2075-1680/8/2/39/pdf",
"status": "GOLD"
}
} | PRICING COMPOUND AND EXTENDIBLE OPTIONS UNDER MIXED FRACTIONAL BROWNIAN MOTION WITH JUMPS | [
"Foad Shokrollahi \nDepartment of Mathematics and Statistics\nUniversity of Vaasa\nP.O. Box 700FIN-65101VaasaFINLAND"
] | This study deals with the problem of pricing compound options when the underlying asset follows a mixed fractional Brownian motion with jumps. An analytic formula for compound options is derived under the risk neutral measure. Then, these results are applied to value extendible options. Moreover, some special cases of ... | [
{
"title": "Introduction",
"content": "Compound option is a standard option with mother standard option being the underlying asset. Compound options have been extensively used in corporate fiance. When the total value of a firm's assets is regarded as the risky underlying asset, the various corporate securi... | [
{
"id": "",
"caption": "If H = 1 2 , the asset price satisfies the Merton jump diffusion equation",
"content": "Corollary 4 . 1 .\n41If H = 1 2 , the asset price satisfies the Merton jump diffusion equation"
},
{
"id": "",
"caption": "formula (3.3) reduces to the diffusion case. The result i... | [
{
"text": "Retractable and extendible bonds: the canadian experience. A Ananthanarayanan, E S Schwartz, The Journal of Finance. 35A. Ananthanarayanan and E. S. Schwartz, Retractable and extendible bonds: the cana- dian experience, The Journal of Finance, 35 (1980), pp. 31-47.",
"doi": ""
},
{
"text"... |
Hydrodynamic finite-size scaling of the thermal conductivity in glasses
Alfredo Fiorentino
SISSA-Scuola Internazionale Superiore di Studi Avanzati
34136Trieste, ItalyEuropean Union
Paolo Pegolo
SISSA-Scuola Internazionale Superiore di Studi Avanzati
34136Trieste, ItalyEuropean Union
Stefano Baroni
SISSA-Scuola ... | 257,496,691 | {
"arxiv": "2303.07010",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1038/s41524-023-01116-2"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2303.07010v2.pdf"
],
"pdf_sha": "f629a3539a1e8974683e4357965020dbd9b93c5c",
"open_access": {
"license": "CCBY",
"url": "https://www.nature.com/articles/s41524-023-01116-2.pdf",
"status": "GOLD"
}
} | Hydrodynamic finite-size scaling of the thermal conductivity in glasses | [
"Alfredo Fiorentino \nSISSA-Scuola Internazionale Superiore di Studi Avanzati\n34136Trieste, ItalyEuropean Union",
"Paolo Pegolo \nSISSA-Scuola Internazionale Superiore di Studi Avanzati\n34136Trieste, ItalyEuropean Union",
"Stefano Baroni \nSISSA-Scuola Internazionale Superiore di Studi Avanzati\n34136Trieste,... | In the past few years, the theory of thermal transport in amorphous solids has been substantially extended beyond the Allen-Feldman model. The resulting formulation, based on the Green-Kubo linear response or the Wigner-transport equation, bridges this model for glasses with the traditional Boltzmann kinetic approach f... | [
{
"title": "INTRODUCTION",
"content": "The theory of thermal transport in crystalline and disordered solids has witnessed a major advancement in the past few years [1][2][3][4][5][6][7]. Until recently, the only way to simulate heat transport in glasses and complex crystals [8] was through the sheer applica... | [
{
"id": "",
"caption": "Longitudinal and transverse linewidths, Γ, computed from the DSF of an 13284-atom model of aSi at 100 K (left panel) and 500 K (right panel). The solid and dashed gray lines indicate ∼ ω 2 and ∼ ω 4 behaviors, respectively.",
"content": "FIG. 1 .\n1Longitudinal and transverse lin... | [
{
"text": "Microscopic theory and quantum simulation of atomic heat transport. A Marcolongo, P Umari, S Baroni, 10.1038/nphys3509Nat. Phys. 1280A. Marcolongo, P. Umari, and S. Baroni, Microscopic theory and quantum simulation of atomic heat transport, Nat. Phys. 12, 80 (2015).",
"doi": "10.1038/nphys3509"
... |
Holographic correlators in AdS 3
18 Mar 2019
Stefano Giusto s:stefano.giusto@pd.infn.it
School of Physics and Astronomy Queen
+− Centre for Research in String Theory
Mary University of London
Mile End RoadE1 4NSLondonUnited Kingdom
O ++ ,O −−
School of Physics and Astronomy Queen
+− Centre for Research in String T... | 119,187,338 | {
"arxiv": "1812.06479",
"mag": 2950816569,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/jhep03(2019)096"
} | {
"pdf_urls": [],
"pdf_sha": "04e9f23371558a170f993b1628bcea253c91b110",
"open_access": {
"license": "CCBY",
"url": "https://link.springer.com/content/pdf/10.1007/JHEP03(2019)096.pdf",
"status": "GOLD"
}
} | Holographic correlators in AdS 3 | [
"Stefano Giusto s:stefano.giusto@pd.infn.it \nSchool of Physics and Astronomy Queen\n+− Centre for Research in String Theory\nMary University of London\nMile End RoadE1 4NSLondonUnited Kingdom",
"O ++ ,O −− \nSchool of Physics and Astronomy Queen\n+− Centre for Research in String Theory\nMary University of London... | We derive the four-point correlators of scalar operators of dimension one in the supergravity limit of the D1D5 CFT holographically dual to string theory on AdS 3 × S 3 × M, with M either T 4 or K3. We avoid the use of Witten diagrams but deduce our result from a limit of the heavy-heavy-light-light correlators compute... | [
{
"title": "Introduction",
"content": "Recent years have seen a remarkable progress in the computation and the analysis of conformal correlators at large 't Hooft coupling and large N, where the problem becomes tractable by using holographic techniques. For the paradigmatic example of fourdimensional N = 4 ... | [
{
"id": "",
"caption": "(3.1) as O −− = [J − 0 , O +− ] and to move the current J − 0 on the other operators by the usual argument of deforming the integration contour. Using [J − 0 , O −− ] = 0 and [J − 0 , O ++ ] = −O −+ (where the minus sign is needed to have O −+ = (O +− )",
"content": "(3.1) as O −... | [
{
"text": "Holographic 4-point correlators with heavy states. A Galliani, S Giusto, R Russo, 1705.09250JHEP. 1040A. Galliani, S. Giusto, and R. Russo, \"Holographic 4-point correlators with heavy states,\" JHEP 10 (2017) 040, 1705.09250.",
"doi": "1705.09250"
},
{
"text": "Mellin amplitudes for AdS ... |
Improved quasi parton distribution through Wilson line renormalization
26 Sep 2016
Jiunn-Wei Chen
Department of Physics
Center for Theoretical Sciences
Leung Center for Cosmology and Particle Astrophysics
National Taiwan University
106TaipeiTaiwan
Center for Theoretical Physics
Massachusetts Institute of Technology... | 13,808,390 | {
"arxiv": "1609.08102",
"mag": 2527544805,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.nuclphysb.2016.12.004"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1609.08102v1.pdf"
],
"pdf_sha": "f7e4cdb11144ad9c316f61f620ae845aed12bce3",
"open_access": {
"license": "CCBY",
"url": "https://doi.org/10.1016/j.nuclphysb.2016.12.004",
"status": "GOLD"
}
} | Improved quasi parton distribution through Wilson line renormalization | [
"Jiunn-Wei Chen \nDepartment of Physics\nCenter for Theoretical Sciences\nLeung Center for Cosmology and Particle Astrophysics\nNational Taiwan University\n106TaipeiTaiwan\n\nCenter for Theoretical Physics\nMassachusetts Institute of Technology\n02139CambridgeMAUSA",
"Xiangdong Ji \nDepartment of Physics and Astr... | Recent developments showed that hadron light-cone parton distributions could be directly extracted from spacelike correlators, known as quasi parton distributions, in the large hadron momentum limit. Unlike the normal light-cone parton distribution, a quasi parton distribution contains ultraviolet (UV) power divergence... | [
{
"title": "I. INTRODUCTION",
"content": "One of the most important goals of QCD is to understand the hadron structure from its fundamental degrees of freedom -quarks and gluons. This necessarily goes to the nonperturbative regime of QCD, and it is difficult in general to obtain first principle results dire... | [
{
"id": "",
"caption": "− x. On the other hand, plugging Eq. (5) into Eq. (1), one finds that the mass One-loop diagrams for quasi quark distribution in Feynman gauge, the conjugate diagrams are not shown.",
"content": "withx = 1 FIG. 1 :\n11− x. On the other hand, plugging Eq. (5) into Eq. (1), one fin... | [
{
"text": ". J S. Alekhin, S Blumlein, Moch, arXiv:1202.2281Phys. Rev. D. 8654009hepphS. Alekhin, J. Blumlein and S. Moch, Phys. Rev. D 86, 054009 (2012) [arXiv:1202.2281 [hep- ph]].",
"doi": ""
},
{
"text": ". J Gao, M Guzzi, J Huston, H. -L Lai, Z Li, P Nadolsky, J Pumplin, D Stump, arXiv:1302.624... |
Improvement of the target sensitivity in DECIGO by optimizing its parameters for quantum noise including the effect of diffraction loss
Tomohiro Ishikawa
Department of Physics
Nagoya University
464-8602NagoyaAichiJapan
Shoki Iwaguchi
Department of Physics
Nagoya University
464-8602NagoyaAichiJapan
Yuta Michimura... | 229,348,890 | {
"arxiv": "2012.11859",
"mag": 3133159123,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3390/galaxies9010014"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2012.11859v1.pdf"
],
"pdf_sha": "083fb4cbdc047077e6793e73ee733c455086f9ff",
"open_access": {
"license": "CCBY",
"url": "https://www.mdpi.com/2075-4434/9/1/14/pdf",
"status": "GOLD"
}
} | Improvement of the target sensitivity in DECIGO by optimizing its parameters for quantum noise including the effect of diffraction loss | [
"Tomohiro Ishikawa \nDepartment of Physics\nNagoya University\n464-8602NagoyaAichiJapan",
"Shoki Iwaguchi \nDepartment of Physics\nNagoya University\n464-8602NagoyaAichiJapan",
"Yuta Michimura \nDepartment of Physics\nThe University of Tokyo\n113-0033BunkyoTokyoJapan",
"Masaki Ando \nDepartment of Physics\nTh... | DECIGO is the future Japanese gravitational wave detector in outer space. We previously set the default design parameters to provide a good target sensitivity to detect the primordial gravitational waves (GWs). However, the updated upper limit of the primordial GWs by the Planck observations motivated us for further op... | [
{
"title": "Introduction",
"content": "The existence of gravitational waves (GWs) was predicted by Einstein's theory of general relativity and verified recently by LIGO and Virgo [1] [2]. GWs are propagating spacetime waves produced by changes in the distribution of mass/energy. Examples of GW sources inclu... | [
{
"id": "",
"caption": "Schematic configuration of one cluster in DECIGO. X, Y, Z denote the three interferometers in a cluster. Also, 1, 2 represent the arms in each interferometer.",
"content": "Figure 2 :\n2Schematic configuration of one cluster in DECIGO. X, Y, Z denote the three interferometers in ... | [
{
"text": "Observation of Gravitational Waves from a Binary Black Hole Merger. B P Abbott, LIGO Scientific Collab. and Virgo Collab.Phys. Rev. Lett. 11661102LIGO Scientific Collab. and Virgo Collab. (B. P. Abbott, et al.), Observation of Gravitational Waves from a Binary Black Hole Merger Phys. Rev. Lett. 2016 ... |
ERROR DIFFUSION HALFTONING AGAINST ADVERSARIAL EXAMPLES
24 Jul 2021
Shao-Yuan Lo sylo@jhu.edu
Dept. of Electrical and Computer Engineering
Johns Hopkins University
MDUSA
Vishal M Patel vpatel36@jhu.edu
Dept. of Electrical and Computer Engineering
Johns Hopkins University
MDUSA
ERROR DIFFUSION HALFTONING AGAINST A... | 231,698,927 | {
"arxiv": "2101.09451",
"mag": 3194125766,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/icip/LoP21",
"doi": "10.1109/icip42928.2021.9506591"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2101.09451v3.pdf"
],
"pdf_sha": "3b1fa137197c334b8929a326b607359e432cb68f",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2101.09451",
"status": "GREEN"
}
} | ERROR DIFFUSION HALFTONING AGAINST ADVERSARIAL EXAMPLES | [
"Shao-Yuan Lo sylo@jhu.edu \nDept. of Electrical and Computer Engineering\nJohns Hopkins University\nMDUSA",
"Vishal M Patel vpatel36@jhu.edu \nDept. of Electrical and Computer Engineering\nJohns Hopkins University\nMDUSA"
] | Adversarial examples contain carefully crafted perturbations that can fool deep neural networks (DNNs) into making wrong predictions.Enhancing the adversarial robustness of DNNs has gained considerable interest in recent years.Although image transformation-based defenses were widely considered at an earlier time, most ... | [
{
"title": "INTRODUCTION",
"content": "The great success of computer vision methods over the recent years can be largely attributed to deep neural networks (DNNs).However, DNNs are vulnerable to adversarial attacks, in which carefully crafted imperceptible noise can mislead DNNs into misclassifying input da... | [
{
"id": "",
"caption": "Fig. 1.(a) Original image.(b) Floyd-Steinberg halftone.One can clearly see the geometric structure present in the halftone.",
"content": "Fig. 1 .\n1\nFig. 1.(a) Original image.(b) Floyd-Steinberg halftone.One can clearly see the geometric structure present in the halftone."
},... | [
{
"text": "Intriguing properties of neural networks. C Szegedy, W Zaremba, I Sutskever, J Bruna, D Erhan, I Goodfellow, R Fergus, International Conference on Learning Representations. 2014",
"doi": ""
},
{
"text": "Explaining and harnessing adversarial examples. I Goodfellow, J Shlens, C Szegedy, In... |
ON ADMISSIBLE STRATEGIES IN ROBUST UTILITY MAXIMIZATION
Keita Owari
Graduate School of Economics
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
ON ADMISSIBLE STRATEGIES IN ROBUST UTILITY MAXIMIZATION
The existence of optimal strategy in robust utility maximization is addressed when the utility f... | 122,866,305 | {
"arxiv": "1109.5512",
"mag": 2129039913,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s11579-012-0068-3"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1109.5512v2.pdf"
],
"pdf_sha": "d7d19b01a0a720d630cdc35463ec23c8acf3926b",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1109.5512",
"status": "GREEN"
}
} | ON ADMISSIBLE STRATEGIES IN ROBUST UTILITY MAXIMIZATION | [
"Keita Owari \nGraduate School of Economics\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan"
] | The existence of optimal strategy in robust utility maximization is addressed when the utility function is finite on the entire real line. A delicate problem in this case is to find a "good definition" of admissible strategies to admit an optimizer. Under certain assumptions, especially a kind of time-consistency prope... | [
{
"title": "INTRODUCTION",
"content": "This paper analyzes a qualitative aspect of the problem of robust utility maximization. Given a utility function U and a set P of probabilities which describes the model uncertainty, the basic problem of this paper is to maximize the robust utility functional exponenti... | [
{
"id": "",
"caption": "Another equivalent formulation). We have defined the classes M V and Θ V through the robust generalized entropy Q → V (Q|P). But the following equivalent formulation is sometimes useful for comparison. For each P ∈ P, we set M V (P) := {Q ∈ M loc : V (Q|P) < ∞}, (2.8) Θ V (P) := {θ ∈... | [
{
"text": "Couverture des actifs contingents et prix maximum. J.-P Ansel, C Stricker, Ann. Inst. H. Poincaré Probab. Statist. 30Ansel, J.-P., and C. Stricker (1994): Couverture des actifs contingents et prix maxi- mum. Ann. Inst. H. Poincaré Probab. Statist. 30, 303-315.",
"doi": ""
},
{
"text": "On... |
Leggett-Garg Correlation Functions from a Non-Invasive Velocity Measurement Continuous in Time
16 Nov 2016
J J Halliwell
Blackett Laboratory Imperial College
SW7 2BZLondonUK
Leggett-Garg Correlation Functions from a Non-Invasive Velocity Measurement Continuous in Time
16 Nov 2016PACS numbers: 0365Ta, 0365Ud * Elect... | 118,601,890 | {
"arxiv": "1605.09241",
"mag": 2411440090,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreva.94.052114"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1605.09241v4.pdf"
],
"pdf_sha": "9d74bd6f428a7984385d01b717ea53fdcbc9a455",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1605.09241",
"status": "GREEN"
}
} | Leggett-Garg Correlation Functions from a Non-Invasive Velocity Measurement Continuous in Time | [
"J J Halliwell \nBlackett Laboratory Imperial College\nSW7 2BZLondonUK"
] | In the Leggett-Garg approach to testing macrorealism, the two-time correlation functions, which are normally obtained by sequential measurements of a dichotomic variable Q, need to be measured in a non-invasive way in order to exclude certain types of alternative classical explanation. Here, it is shown, for a class of... | [
{
"title": "I. INTRODUCTION",
"content": "The Leggett-Garg (LG) inequalities were proposed in order to test whether certain types of realist theories could explain the observable data in the description of macroscopic systems [1,2]. In this approach, sequential measurements are made on a dichotomic variable... | [
{
"id": "",
"caption": "C 12 − C 23 − C 13 ≥ 0, (1.4) 1 − C 12 + C 23 − C 13 ≥ 0. (1.5) These inequalities are respected by certain interesting classes of macrorealistic theories, when properly measured. (For a discussion of the current status of this field see the useful review",
"content": "C 12 − C 2... | [
{
"text": ". A J Leggett, A Garg, Phys. Rev. Lett. 54857A.J.Leggett and A.Garg, Phys. Rev. Lett. 54, 857 (1985).",
"doi": ""
},
{
"text": ". A J Leggett, Foundations of Physics. 18939A. J. Leggett, Foundations of Physics, 18, 939 (1988);",
"doi": ""
},
{
"text": ". Rep. Prog. Phys. 71220... |
Theoretical derivation of Darcy's law for fluid flow in thin porous media
Francisco Javier Suárez-Grau
Theoretical derivation of Darcy's law for fluid flow in thin porous media
AMS classification numbers: 76A2076M5035B2735Q30 Keywords: HomogenizationStokes equationsDarcy's lawthin porous mediathin film fluids
In th... | 215,769,076 | {
"arxiv": "2004.07022",
"mag": 3016784964,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1002/mana.202000184"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2004.07022v1.pdf"
],
"pdf_sha": "e62ba4f257f3f49ab508c5bb4863fc079b156eaa",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2004.07022",
"status": "GREEN"
}
} | Theoretical derivation of Darcy's law for fluid flow in thin porous media | [
"Francisco Javier Suárez-Grau"
] | In this paper we study stationary incompressible Newtonian fluid flow in a thin porous media. The media under consideration is a bounded perforated 3D domain confined between two parallel plates. The description of the domain includes two small parameters: ε representing the distance between pates and aε connected to t... | [
{
"title": "Introduction",
"content": "The problem of Stokes fluid flows in a periodically perforated domains with obstacles of the same size as the period has been widely treated in the literature. As is well known, such kind of flows are generally modelled by Darcy's law, see Darcy [8]. From the mathemati... | [
{
"id": "",
"caption": "View of the domain Ω ε (left) and Q ε (right).",
"content": "Figure 2 :\n2View of the domain Ω ε (left) and Q ε (right)."
},
{
"id": "",
"caption": "Proof. Estimates (5.30) and (5.31) easily follow from Proposition 5.7 and estimates given in Lemmas 5.2 and 5.4.",
... | [
{
"text": "Homogenization of the Stokes flow in a connected porous medium. G Allaire, Asymp. Anal. 2G. Allaire, Homogenization of the Stokes flow in a connected porous medium. Asymp. Anal., 2 (1989) 203-222.",
"doi": ""
},
{
"text": "Derivation of the double porosity model of single phase flow via h... |
SPECTRAL OPTIMIZATION FOR ROBIN LAPLACIAN ON DOMAINS ADMITTING PARALLEL COORDINATES Dedicated to the memory of our friend and colleague Hagen Neidhardt
May 2022
Pavel Exner
Vladimir Lotoreichik
SPECTRAL OPTIMIZATION FOR ROBIN LAPLACIAN ON DOMAINS ADMITTING PARALLEL COORDINATES Dedicated to the memory of our friend ... | 220,968,675 | {
"arxiv": "2001.02718",
"mag": 3047205511,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1002/mana.202000013"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2001.02718v3.pdf"
],
"pdf_sha": "040be52997f9fd00764a0e204646916ce9d60d91",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2001.02718",
"status": "GREEN"
}
} | SPECTRAL OPTIMIZATION FOR ROBIN LAPLACIAN ON DOMAINS ADMITTING PARALLEL COORDINATES Dedicated to the memory of our friend and colleague Hagen Neidhardt | [
"Pavel Exner",
"Vladimir Lotoreichik"
] | In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue... | [
{
"title": "Introduction",
"content": ""
},
{
"title": "Motivation and the problem description.",
"content": "Relations between geometry and spectral properties have a rather long history. They are a trademark topic of mathematical physics at least since the celebrated Faber and Krahn proof [F23... | [
{
"id": "",
"caption": "Proof of Theorem 1.2. By assumption we have L = L • and we fix d > 0 satisfying the additional condition (1.4) in the case that κ is sign-changing. Furthermore, α ∈ R is an arbitrary fixed number. The case d = ∞ is dealt with in [KL18, Thm 1.3] and thus we may omit it here. By Propos... | [
{
"text": "Bounds and extremal domains for Robin eigenvalues with negative boundary parameter. P R S Antunes, P Freitas, D Krejčiřík, Adv. Calc. Var. 10P. R. S. Antunes, P. Freitas, and D. Krejčiřík, Bounds and extremal domains for Robin eigenvalues with negative boundary parameter, Adv. Calc. Var. 10 (2017), 3... |
FONDUE: Framework for Node Disambiguation Using Network Embeddings
Ahmad Mel ahmad.mel@ugent.be
IDLab Ghent University
GhentBelgium
Bo Kang bo.kang@ugent.be
IDLab Ghent University
GhentBelgium
Jefrey Lijffijt jefrey.lijffijt@ugent.be
IDLab Ghent University
GhentBelgium
Tijl De Bie tijl.debie@ugent.be
IDLab Gh... | 211,258,864 | {
"arxiv": "2002.10127",
"mag": 3008183086,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2002-10127",
"doi": "10.1109/dsaa49011.2020.00028"
} | {
"pdf_urls": [],
"pdf_sha": "93168df790fd8533edfca2b12612e6d684dd00df",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2002.10127",
"status": "GREEN"
}
} | FONDUE: Framework for Node Disambiguation Using Network Embeddings | [
"Ahmad Mel ahmad.mel@ugent.be \nIDLab Ghent University\nGhentBelgium",
"Bo Kang bo.kang@ugent.be \nIDLab Ghent University\nGhentBelgium",
"Jefrey Lijffijt jefrey.lijffijt@ugent.be \nIDLab Ghent University\nGhentBelgium",
"Tijl De Bie tijl.debie@ugent.be \nIDLab Ghent University\nGhentBelgium"
] | Real-world data often presents itself in the form of a network. Examples include social networks, citation networks, biological networks, and knowledge graphs. In their simplest form, networks represent real-life entities (e.g. people, papers, proteins, concepts) as nodes, and describe them in terms of their relations ... | [
{
"title": "I. INTRODUCTION",
"content": "Increasingly, data naturally takes the form of a network of interrelated entities. Examples include social networks describing social relations between people (e.g. Facebook), citation networks describing the citation relations between papers (e.g. DBLP), biological... | [
{
"id": "",
"caption": "Bar plots for the runtime performance in seconds (lower is better) for each dataset listed in",
"content": "Fig. 3 :\n3Bar plots for the runtime performance in seconds (lower is better) for each dataset listed in"
},
{
"id": "",
"caption": "The different datasets used... | [
{
"text": "The biogrid interaction database: 2008 update. B.-J Breitkreutz, C Stark, T Reguly, L Boucher, A Breitkreutz, M Livstone, R Oughtred, D H Lackner, J Bähler, V Wood, Nucleic acids research. 361supplB.-J. Breitkreutz, C. Stark, T. Reguly, L. Boucher, A. Bre- itkreutz, M. Livstone, R. Oughtred, D. H. La... |
COMPACTIFICATION OF CERTAIN CLIFFORD-KLEIN FORMS OF REDUCTIVE HOMOGENEOUS SPACES
4 Aug 2015
François Guéritaud
Olivier Guichard
ANDFanny Kassel
Anna Wienhard
COMPACTIFICATION OF CERTAIN CLIFFORD-KLEIN FORMS OF REDUCTIVE HOMOGENEOUS SPACES
4 Aug 2015
We describe smooth compactifications of certain families of redu... | 119,587,777 | {
"arxiv": "1506.03742",
"mag": 2963488639,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1307/mmj/1488510025"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/1506.03742"
],
"pdf_sha": "dc62516323da396fa537a6b998ce79b16ebf25b6",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1506.03742",
"status": "GREEN"
}
} | COMPACTIFICATION OF CERTAIN CLIFFORD-KLEIN FORMS OF REDUCTIVE HOMOGENEOUS SPACES | [
"François Guéritaud",
"Olivier Guichard",
"ANDFanny Kassel",
"Anna Wienhard"
] | We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous sp... | [
{
"title": "Introduction",
"content": "The goal of this note is two-fold. First, we describe compactifications of certain families of reductive homogeneous spaces G/H by embedding G into a larger group G ′ and realizing G/H as a G-orbit in a flag manifold of G ′ . These homogeneous spaces include:\n• group ... | [
{
"id": "",
"caption": "For 0 ≤ i ≤ n, the map π defines a fibration ofU i over F i (b) × F i (−b) with fibers isomorphic to (H i × H i )/Diag(H i ), where H i = Aut K (b V i ) is the automorphism group of the form b V i induced by b on V ⊥ b i /V i for some V i ∈ F i (b). In particular, U 0 is the unique o... | [
{
"text": "Deformations of Fuchsian AdS representations are Quasi-Fuchsian. Thierry Barbot, J. Differential Geom. 101Thierry Barbot, Deformations of Fuchsian AdS representations are Quasi-Fuchsian, J. Differential Geom. 101 (2015), 1-46.",
"doi": ""
},
{
"text": "Actions propres sur les espaces homo... |
27 Apr 2023
Kenny Ballou kennyballou@u.boisestate.edu
Boise State University
Elena Sherman elenasherman@boisestate.edu
Boise State University
27 Apr 2023A36D6B751520F265FCA41B811D1B9115arXiv:2304.14550v1[cs.LO]
Verification techniques express program states as logical formulas over program variables.For example... | 258,418,001 | {
"arxiv": "2304.14550",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/tase/BallouS23",
"doi": "10.1007/978-3-031-35257-7_13"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2304.14550v1.pdf"
],
"pdf_sha": "0c423b34b39f3eeb2e45d08bccb5cf8bbab6c0f3",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2304.14550",
"status": "GREEN"
}
} | [
"Kenny Ballou kennyballou@u.boisestate.edu \nBoise State University",
"Elena Sherman elenasherman@boisestate.edu \nBoise State University"
] | Verification techniques express program states as logical formulas over program variables.For example, symbolic execution and abstract interpretation encode program states as a set of linear integer inequalities.However, for real-world programs these formulas tend to become large, which affects scalability of analyses.... | [
{
"title": "Introduction",
"content": "Many verification techniques express a program state as a logical formula over program variables.For example, symbolic execution uses a logical formula to describe a path constraint; in abstract interpretation, relational domains such as Zones or Polyhedra use a set of... | [
{
"id": "",
"caption": "Fig. 1: Example program and two equivalent Zones",
"content": "Fig. 1 :\n1\nFig. 1: Example program and two equivalent Zones"
},
{
"id": "",
"caption": "1: function MinChangedSet(Z, dv, de, Method) end function de.A DFA framework can directly provide the sets dv and d... | [
{
"text": "An extended account of trace-relating compiler correctness and secure compilation. C Abate, R Blanco, C Ciobâcȃ, A Durier, D Garg, C Hritt ¸cu, M Patrignani, E Tanter, J Thibault, 10.1145/34608602898 1002 1464 237 167 2032 CC 1194 2706 1324 1930 473 173 NN 1191 2709 1322 1933 473 172 MN 1164 2736 130... | |
HARDY-LITTLEWOOD INEQUALITIES FOR MULTIPOLYNOMIALS
26 Jan 2018
Daniel Tomaz
HARDY-LITTLEWOOD INEQUALITIES FOR MULTIPOLYNOMIALS
26 Jan 2018
The notion of multipolynomials was recently introduced and explored by T. Velanga in[10]as an attempt to encompass the theories of polynomials and multilinear operators. In the p... | 54,513,784 | {
"arxiv": "1801.08786",
"mag": 2963454967,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.15352/aot.1808-1413"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1801.08786v1.pdf"
],
"pdf_sha": "db6f105fde63aa655aa41be86c0b5c0bc6ed1c49",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1801.08786",
"status": "GREEN"
}
} | HARDY-LITTLEWOOD INEQUALITIES FOR MULTIPOLYNOMIALS | [
"Daniel Tomaz"
] | The notion of multipolynomials was recently introduced and explored by T. Velanga in[10]as an attempt to encompass the theories of polynomials and multilinear operators. In the present paper we push this subject further, by proving Hardy-Littlewood inequalities for multipolynomials and, en passant, a variant of the Kah... | [
{
"title": "Introduction",
"content": "Throughout this paper, E 1 , . . . , E m , F will be Banach spaces over the scalar field K, which can be either R or C. We recall that a map P : E −→ F is called m-homogeneous polynomial when there is an m-linear operator T : E × · · · × E −→ F such that P (x) := T (x,... | [
{
"id": "",
"caption": "Let n, m be positive integers and p ≥ 1. Then there exists a m-homogeneous polynomial P : ℓ p −→ K of the form",
"content": "Theorem 2 . 3 .\n23Let n, m be positive integers and p ≥ 1. Then there exists a m-homogeneous polynomial P : ℓ p −→ K of the form"
},
{
"id": "",
... | [
{
"text": "Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators. N Albuquerque, F Bayart, D Pellegrino, J B Seoane-Sepulveda, Israel J. Math. 211N. Albuquerque, F. Bayart, D. Pellegrino, J. B. Seoane-Sepulveda, Optimal Hardy-Littlewood type inequalities for polynomials and multil... |
Electron-hole coherent states for the Bogoliubov-de Gennes equation
23 Sep 2015
Sven Gnutzmann sven.gnutzmann@nottingham.ac.uk
School of Mathematical Sciences
University of Nottingham
UK
Marek Kuś marek.kus@cft.edu.pl
Center for Theoretical Physics
Polish Academy of Sciences
WarsawPoland
Jordan Langham-Lopez
Sch... | 119,610,139 | {
"arxiv": "1509.06936",
"mag": 3105401064,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1751-8113/49/8/085302"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1509.06936v1.pdf"
],
"pdf_sha": "5a168ec546f7e73e18d716b078f475e4c9e09f53",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1509.06936",
"status": "GREEN"
}
} | Electron-hole coherent states for the Bogoliubov-de Gennes equation | [
"Sven Gnutzmann sven.gnutzmann@nottingham.ac.uk \nSchool of Mathematical Sciences\nUniversity of Nottingham\nUK",
"Marek Kuś marek.kus@cft.edu.pl \nCenter for Theoretical Physics\nPolish Academy of Sciences\nWarsawPoland",
"Jordan Langham-Lopez \nSchool of Mathematical Sciences\nUniversity of Nottingham\nUK"
] | We construct a new set of generalized coherent states, the electron-hole coherent states, for a (quasi-)spin particle on the infinite line.The definition is inspired by applications to the Bogoliubovde Gennes equations where the quasi-spin refers to electron-and hole-like components of electronic excitations in a super... | [
{
"title": "I. INTRODUCTION",
"content": "Coherent states in their multiple variants and generalizations have become an indispensable tool in various branches of quantum mechanics and quantum field theory.The standard coherent states are strongly connected to the quantum description of a one-dimensional har... | [
{
"id": "",
"caption": "FIG. 1. Reduced phase space functions h (red) ⊗,ρ (α, α * ) (top figures with α = (q + ip)/ √ 2) and h (red) ,ρ (α, α * ) (bottom figures with α = (q + iv)/ √ 2) for product coherent state ρ = |α0 ⊗ β0 α0 ⊗ β0| (left figures) and an electron-hole coherent state ρ = |α0 β0 α0 β0| (wit... | [
{
"text": "Coherent and incoherent states of the radiation field. R J Glauber, Phys. Rev. 131627661963",
"doi": ""
},
{
"text": "The quantum theory of optical coherence. R J Glauber, Phys. Rev. 130625291963",
"doi": ""
},
{
"text": "Equivalence of semiclassical and quantum mechanical des... |
Enhanced space charge limited current for curved electron emitters
Gaurav Singh
Bhabha Atomic Research Centre
400 085MumbaiINDIA
Homi Bhabha National Institute
400 094MumbaiINDIA
Raghwendra Kumar
Bhabha Atomic Research Centre
400 085MumbaiINDIA
Debabrata Biswas
Bhabha Atomic Research Centre
400 085MumbaiINDIA
... | 220,250,483 | {
"arxiv": "2006.16207",
"mag": 3104445457,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1063/5.0028007"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2006.16207v2.pdf"
],
"pdf_sha": "156e1dd9f7887a718dafd35a6fa9ecc6d3c9fb8c",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2006.16207",
"status": "GREEN"
}
} | Enhanced space charge limited current for curved electron emitters | [
"Gaurav Singh \nBhabha Atomic Research Centre\n400 085MumbaiINDIA\n\nHomi Bhabha National Institute\n400 094MumbaiINDIA",
"Raghwendra Kumar \nBhabha Atomic Research Centre\n400 085MumbaiINDIA",
"Debabrata Biswas \nBhabha Atomic Research Centre\n400 085MumbaiINDIA\n\nHomi Bhabha National Institute\n400 094Mumbai... | The maximum current that can be transported across a vacuum diode is limited by forces arising due to space charge. In a planar diode, the space charge limited (SCL) current density from an emitting patch is given by the Child-Langmuir (CL) law J CL ∼ V 3/2 g /D 2 where V g is the potential difference across the diode ... | [
{
"title": "I. INTRODUCTION",
"content": "The maximum current that can be transported across a vacuum diode is a quantity of immense interest in vacuum micro-electronics, space navigation and especially in high-power vacuum electronic devices such as magnetron, gyrotron, vircator or relativistic backward wa... | [
{
"id": "",
"caption": "A schematic of a planar diode with a hemi-ellipsoidal emitter along with Gaussian surfaces marked by dashed lines.",
"content": "FIG. 2 .\n2A schematic of a planar diode with a hemi-ellipsoidal emitter along with Gaussian surfaces marked by dashed lines."
},
{
"id": "",
... | [
{
"text": "High Power Microwaves. J Benford, J A Swegle, E Schamiloglu, Taylor & Francis GroupNew YorkJ. Benford, J. A. Swegle, and E. Schamiloglu, High Power Mi- crowaves, Taylor & Francis Group, New York (2007).",
"doi": ""
},
{
"text": ". C D Child, Phys. Rev. 32492C. D. Child, Phys. Rev. 32, 492... |
Improving the compromise between accuracy, interpretability and personalization of rule-based machine learning in medical problems
Francisco Valente
Jorge Henriques
Simão Paredes
Teresa Rocha
Paulo De Carvalho
João Morais
Improving the compromise between accuracy, interpretability and personalization of rule-b... | 235,436,150 | {
"arxiv": "2106.07827",
"mag": null,
"acl": null,
"pubmed": "34891710",
"pubmedcentral": null,
"dblp": "conf/embc/ValenteHPRCM21",
"doi": "10.1109/embc46164.2021.9630184"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2106.07827v2.pdf"
],
"pdf_sha": "f4a3692b0efc91acc9f172e9e54aedc381483758",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2106.07827",
"status": "GREEN"
}
} | Improving the compromise between accuracy, interpretability and personalization of rule-based machine learning in medical problems | [
"Francisco Valente",
"Jorge Henriques",
"Simão Paredes",
"Teresa Rocha",
"Paulo De Carvalho",
"João Morais"
] | One of the key challenges when developing a predictive model is the capability to describe the domain knowledge and the cause-effect relationships in a simple way. Decision rules are a useful and important methodology in this context, justifying their application in several areas, particularly in clinical practice. Sev... | [
{
"title": "I. INTRODUCTION",
"content": "Physicians usually incorporate clinical decision rules (CDRs) in several domains of their practice, such as bedside diagnostic or therapeutic choices. CDRs appeared as a tool to help the decision-making of the clinical staff, reducing its uncertainty and making it m... | [
{
"id": "",
"caption": "Example of a decision tree. The split nodes are represented by the blue square boxes, and the leaf (or terminal) nodes are represented by the grey round boxes. x1 and x2 are two attributes (features), and y1 and y2 are two arbitrary classes (labels).",
"content": "Fig. 1 .\n1Exam... | [
{
"text": "Implementation of Clinical Decision Rules in the Emergency Department. I G Stiell, C Bennett, Academic Emergency Medicine. 1411I. G. Stiell and C. Bennett, \"Implementation of Clinical Decision Rules in the Emergency Department,\" Academic Emergency Medicine, vol. 14, no. 11, pp. 955-959, nov 2007.",... |
Applying deep neural networks to the detection and space parameter estimation of compact binary coalescence with a network of gravitational wave detectors
November 6, 2018
Xilong Fan
Jin Li
Xin Li
Yuanhong Zhong
Junwei Cao
Department of Physics and Astronomy
Hubei University of Education
430205WuhanChina
Depa... | 119,230,219 | {
"arxiv": "1811.01380",
"mag": 3098570185,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s11433-018-9321-7"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1811.01380v2.pdf"
],
"pdf_sha": "071e872840549952d31df5eab19eea4efb17edde",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/1811.01380",
"status": "GREEN"
}
} | Applying deep neural networks to the detection and space parameter estimation of compact binary coalescence with a network of gravitational wave detectors | [
"Xilong Fan",
"Jin Li",
"Xin Li",
"Yuanhong Zhong",
"Junwei Cao",
"Department of Physics and Astronomy\nHubei University of Education\n430205WuhanChina",
"Department of Physics\nChongqing University\n401331ChongqingChina",
"Department of Physics\nChongqing University\n401331chongqingChina",
"College... | In this paper, we study an application of deep learning to the advanced LIGO and advanced Virgo coincident detection of gravitational waves (GWs) from compact binary star mergers.This deep learning method is an extension of the Deep Filtering method used by George and Huerta (2017) for multi-inputs of network detectors... | [
{
"title": "I. INTRODUCTION",
"content": "As the authority of gravitational wave (GW) detection, last year, the advanced Laser Interferometer Gravitational-Wave Observatory (advanced LIGO) and advanced Virgo performed the first three-detector detection of a GW signal from a binary neutron star coalescence [... | [
{
"id": "",
"caption": "FIG.1: Top panel: Blue curves are simulated time series in the advanced LIGO and advanced Virgo detectors, respectively, corresponding to the similar GW event and inputted to the DNN together.Bottom panel: Zoom plots corresponding to the top plots around injection time, which varies ... | [
{
"text": ". B P Abbott, LIGO Scientific Collaboration ; Virgo CollaborationPhys. Rev. Lett. 1191611012017",
"doi": ""
},
{
"text": ". B P Abbott, LIGO Scientific Collaboration ; Virgo CollaborationPhys. Rev. Lett. 1191411012017",
"doi": ""
},
{
"text": ". B P Abbott, LIGO Scientific Col... |
Universal low-frequency vibrational modes in silica glasses
arXiv:2003.07614v1 [cond-mat.soft] 17 Mar 2020
Silvia Bonfanti
Center for Complexity and Biosystems
Department of Physics
University of Milan
via Celoria 1620133MilanoItaly
Roberto Guerra
Center for Complexity and Biosystems
Department of Physics
Universi... | 212,736,904 | {
"arxiv": "2003.07614",
"mag": 3011271215,
"acl": null,
"pubmed": "32909803",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevlett.125.085501"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2003.07614v1.pdf"
],
"pdf_sha": "3403ade7aca47802464972e233fb50abd56f2c69",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2003.07614",
"status": "GREEN"
}
} | Universal low-frequency vibrational modes in silica glasses | [
"Silvia Bonfanti \nCenter for Complexity and Biosystems\nDepartment of Physics\nUniversity of Milan\nvia Celoria 1620133MilanoItaly",
"Roberto Guerra \nCenter for Complexity and Biosystems\nDepartment of Physics\nUniversity of Milan\nvia Celoria 1620133MilanoItaly",
"Chandana Mondal \nDept. of Chemical Physics\... | It was recently shown that different simple models of glass formers with binary interactions define a universality class in terms of the density of states of their quasi-localized low-frequency modes. Explicitly, once the hybridization with standard Debye (extended) modes is avoided, a number of such models exhibit a u... | [
{
"title": "•",
"content": ""
},
{
"title": "FIG. 1 .",
"content": ""
},
{
"title": "FIG. 2 .",
"content": ""
},
{
"title": "FIG. 3 .",
"content": ""
},
{
"title": "FIG. 4 .FIG. 5 .",
"content": ""
}
] | [
{
"id": "",
"caption": "N = 222 atoms composed by N Si = 74 silicon atoms and N O = 148 oxygen atoms. Box length L = 15Å, 1000 configurations.",
"content": "•\nN = 222 atoms composed by N Si = 74 silicon atoms and N O = 148 oxygen atoms. Box length L = 15Å, 1000 configurations."
},
{
"id": "",
... | [
{
"text": "Anharmonic potentials and vibrational localization in glasses. U Buchenau, Yu M Galperin, V L Gurevich, H R Schober, Phys. Rev. B. 43U. Buchenau, Yu. M. Galperin, V. L. Gurevich, and H. R. Schober. Anharmonic potentials and vibrational localiza- tion in glasses. Phys. Rev. B, 43:5039-5045, 1991.",
... |
Enhanced Superconductivity in Bilayer PtTe 2 by Alkali-Metal Intercalations
Danhong Wu
Department of Physics
Siyuan Laboratory, Guangzhou Key Laboratory of Vacuum Coating Technologies and New Energy Materials
Jinan University
510632GuangzhouChina
Yiping Lin
Department of Physics
Siyuan Laboratory, Guangzhou Key L... | 235,253,901 | {
"arxiv": "2105.14243",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.103.224502"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2105.14243v1.pdf"
],
"pdf_sha": "2e957350235f600ba664770e8937dd5df65347b2",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2105.14243",
"status": "GREEN"
}
} | Enhanced Superconductivity in Bilayer PtTe 2 by Alkali-Metal Intercalations | [
"Danhong Wu \nDepartment of Physics\nSiyuan Laboratory, Guangzhou Key Laboratory of Vacuum Coating Technologies and New Energy Materials\nJinan University\n510632GuangzhouChina",
"Yiping Lin \nDepartment of Physics\nSiyuan Laboratory, Guangzhou Key Laboratory of Vacuum Coating Technologies and New Energy Material... | Layered platinum tellurium (PtTe2) were recently synthesized with controllable layer numbers down to monolayer limit. Using ab initio calculation based on anisotropic Midgal-Eliashberg formalism, we show that by rubidium (Rb) intercalation, weak superconductivity in bilayer PtTe2 can be significantly boosted with super... | [
{
"title": "I. INTRODUCTION",
"content": "Transition metal dichalcogenides (TMDCs) exhibit intriguing physical properties including superconductivity, charge-density wave, Dirac semimetals, among which the two-dimensional (2D) superconductivity has received growing attention, where the reduced dimensionalit... | [
{
"id": "",
"caption": "(a) Side and top views of a slab model of rubidium intercalated bilayer PtTe2 (schematic diagram). The black rhombus indicates a two-dimensional hexagonal unit cell. (b) Formation energies as a function of Rb concentration (x) calculated in a 3×3 supercell via density-functional theo... | [
{
"text": "Xiong contribute equally to this work. D H Wu, Y P Lin, L X , * fpzheng phy@email.jnu.edu.cnD.H. Wu, Y.P. Lin and L.X. Xiong contribute equally to this work. * fpzheng phy@email.jnu.edu.cn",
"doi": ""
},
{
"text": ". X Xi, L Zhao, Z Wang, H Berger, L Forró, J Shan, K F Mak, Nat. Nanotechn... |
Broadband-laser-diode pumped PPKTP-Sagnac polarization-entangled photon source
8 Oct 2021
Neng Cai
Hubei Key Laboratory of Optical Information and Pattern Recognition
Wuhan Institute of Technology
430205WuhanChina
Wu-Hao Cai
Shun Wang
Hubei Key Laboratory of Optical Information and Pattern Recognition
Wuhan Insti... | 233,481,266 | {
"arxiv": "2105.00454",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1364/josab.437808"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2105.00454v2.pdf"
],
"pdf_sha": "8ab0dfa6a3f68b03e2abcad4bdedef588652c369",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2105.00454",
"status": "GREEN"
}
} | Broadband-laser-diode pumped PPKTP-Sagnac polarization-entangled photon source | [
"Neng Cai \nHubei Key Laboratory of Optical Information and Pattern Recognition\nWuhan Institute of Technology\n430205WuhanChina",
"Wu-Hao Cai",
"Shun Wang \nHubei Key Laboratory of Optical Information and Pattern Recognition\nWuhan Institute of Technology\n430205WuhanChina",
"Fang Li \nHubei Key Laboratory o... | We experimentally demonstrate a polarization-entangled photon source at 810 nm using a type-II phase-matched PPKTP crystal pumped by a low-cost, broadband laser diode with a central wavelength of 405 nm and a typical bandwidth of 0.53 nm. The PPKTP crystal is placed in a Sagnac-loop to realize the compact size and high... | [
{
"title": "I. INTRODUCTION",
"content": "Quantum entanglement, which represents a nonclassical correlation among several quantum subsystems, is a critical feature of quantum information science. High-quality entangled photon pairs play an important role in many quantum information technologies, such as qua... | [
{
"id": "",
"caption": "Experimental setup for the LD pumped type-II-PPKTP-Sagnac entangled photon source. SMFC=singlemode fiber coupler, SMF=single-mode fiber, HWP=halfwave plate, QWP=quarter-wave plate, SPF=short-pass filter, L=lens, DM=dichroic mirror, PBS=polarization beam splitter, DPBS=dual-wavelength... | [
{
"text": "Experimental two-photon, three-dimensional entanglement for quantum communication. Alipasha Vaziri, Gregor Weihs, Anton Zeilinger, Phys. Rev. Lett. 89240401Alipasha Vaziri, Gregor Weihs, and Anton Zeilinger, \"Experimental two-photon, three-dimensional entangle- ment for quantum communication,\" Phys... |
Circular conversion dichroism in planar chiral metamaterials
28 Apr 2006
V A Fedotov
School of Physics and Astronomy
EPSRC Nanophotonics Portfolio Centre
University of Southampton
SO17 1BJUK
P L Mladyonov
Institute of Radio Astronomy
National Academy of Sciences of Ukraine
61002KharkovUkraine
S L Prosvirnin
Inst... | 119,436,346 | {
"arxiv": "physics/0604234",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevlett.97.167401"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/physics/0604234v1.pdf"
],
"pdf_sha": "e9056bb4a1b0ab909462c3423cb7fc2b4358fd67",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/physics/0604234",
"status": "GREEN"
}
} | Circular conversion dichroism in planar chiral metamaterials | [
"V A Fedotov \nSchool of Physics and Astronomy\nEPSRC Nanophotonics Portfolio Centre\nUniversity of Southampton\nSO17 1BJUK",
"P L Mladyonov \nInstitute of Radio Astronomy\nNational Academy of Sciences of Ukraine\n61002KharkovUkraine",
"S L Prosvirnin \nInstitute of Radio Astronomy\nNational Academy of Sciences... | We report the first experiential observation and theoretical analysis of the new phenomenon of planar chiral circular conversion dichroism, which in some aspects resembles the Faraday effect in magnetized media, but does not require the presence of a magnetic field for its observation. It results from the interaction o... | [
{
"title": "Once again",
"content": "This indicates that the perceived sense of rotation of the planar chiral structure (which depends on whether the structure is observed from one side or another) controls the sing of the effect. In other words it matters if the wave vector of the incident wave is parallel... | [
{
"id": "",
"caption": "Intensities and phase retardations of these polarization components are the same Total intensities of transmitted waves are different FIG. 1: Propagation of circularly polarized waves through (a) three dimensional chiral medium (b) Faraday and (c) planar chiral media. The colored cir... | [
{
"text": ". n.i.zheludev@soton.ac.uk. n.i.zheludev@soton.ac.uk;",
"doi": ""
},
{
"text": ". L Hetch, L D Barron, Chem. Phys. Lett. 225525L. Hetch and L. D. Barron, Chem. Phys. Lett 225, 525 (1994).",
"doi": ""
},
{
"text": "L R Arnaut, L E Davis, Proceedings of the International Confere... |
ADMEOOD: Out-of-Distribution Benchmark for Drug Property Prediction
Shuoying Wei
College of Informatics
Huazhong Agricultural University
430070WuhanChina
Shenzhen Institute of Nutrition and Health
Huazhong Agricultural University
518000ShenzhenChina
Shenzhen Branch
Guangdong Laboratory for Lingnan Modern Agricult... | 263,834,912 | {
"arxiv": "2310.07253",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/bibm/WeiLGZWZ23",
"doi": "10.1109/bibm58861.2023.10385708"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2310.07253v1.pdf"
],
"pdf_sha": "5041411b6e6112822b70e405b23d6080bf323a51",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2310.07253",
"status": "GREEN"
}
} | ADMEOOD: Out-of-Distribution Benchmark for Drug Property Prediction | [
"Shuoying Wei \nCollege of Informatics\nHuazhong Agricultural University\n430070WuhanChina\n\nShenzhen Institute of Nutrition and Health\nHuazhong Agricultural University\n518000ShenzhenChina\n\nShenzhen Branch\nGuangdong Laboratory for Lingnan Modern Agriculture, Genome Analysis Laboratory\n\n\nMinistry of Agricul... | Obtaining accurate and valid information for drug molecules is a crucial and challenging task.However, chemical knowledge and information have been accumulated over the past 100 years from various regions, laboratories, and experimental purposes.Little has been explored in terms of the out-of-distribution (OOD) problem... | [
{
"title": "Introduction",
"content": "The traditional drug discovery process is extremely time-consuming and expensive.Especially, up to 50% of clinical trial failures have been attributed to deficiencies in ADMET (Absorption, Distribution, Metabolism, Excretion, Toxicity) properties (Kennedy 1997;Kola and... | [
{
"id": "",
"caption": "Figure 2: Normalization process of the drug properties prediction basis data.",
"content": "Figure 2 :\n2\nFigure 2: Normalization process of the drug properties prediction basis data."
},
{
"id": "",
"caption": "Figure 3: Flowchart of drug property data label definit... | [
{
"text": "E Alexov, E L Mehler, N A Baker, D J Schwab, . ; Huang, F Milletti, J Nielsen, D Farrell, T Carstensen, M Olsson, G F Ayne, J Warwicker, S Williams, J M Word, rogress in the prediction of pKa values in proteins. 201179",
"doi": ""
},
{
"text": "Invariant risk minimization. M Arjovsky, L B... |
Do Intermediate-Mass Black Holes Exist in Globular Clusters?
13 Aug 2013
Mou-Yuan Sun
Ya-Ling Jin
Wei-Min Gu
Tong Liu
Da-Bin Lin
Ju-Fu Lu
Do Intermediate-Mass Black Holes Exist in Globular Clusters?
13 Aug 2013Subject headings: accretion, accretion disks -black hole physics -globular clus- ters: general
The exi... | 118,628,372 | {
"arxiv": "1308.2869",
"mag": 2079890781,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0004-637x/776/2/118"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1308.2869v1.pdf"
],
"pdf_sha": "9634543ba50e5aad34eda3d62b166c2500f1df43",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1308.2869",
"status": "GREEN"
}
} | Do Intermediate-Mass Black Holes Exist in Globular Clusters? | [
"Mou-Yuan Sun",
"Ya-Ling Jin",
"Wei-Min Gu",
"Tong Liu",
"Da-Bin Lin",
"Ju-Fu Lu"
] | The existence of intermediate-mass black holes (IMBHs) in globular clusters (GCs) remains a crucial problem. Searching IMBHs in GCs reveals a discrepancy between radio observations and dynamical modelings: the upper mass limits constrained by radio observations are systematically lower than that of dynamical modelings.... | [
{
"title": "Introduction",
"content": "Over the past decades, astronomers have suspected that there might be a population of intermediate-mass black holes (IMBHs), which connect the stellar-mass black holes and the supermassive black holes. On the observational side, accretion of IMBHs could be the energy s... | [
{
"id": "",
"caption": "-Radio luminosity L R as a function of black hole mass M BH . The filled circles represent the 3σ upper limit of L R for each source, where the black hole masses are constrained via dynamical modelings. The dashed line represents the predicted L R as a function of M BH according toSt... | [
{
"text": ". F N Bash, K Gebhardt, W M Goss, P A Bout, AJ. 135182Bash, F. N., Gebhardt, K., Goss, W. M., & Vanden Bout, P. A. 2008, AJ, 135, 182",
"doi": ""
},
{
"text": ". R D Blandford, A Köumlnigl, ApJ. 23234Blandford, R. D., & Köumlnigl, A. 1979, ApJ, 232, 34",
"doi": ""
},
{
"text":... |
Bismuth-surfactant-induced growth and structure of InAs/GaAs(110) quantum dots
Published 16 September 2019
Ryan B Lewis rlewis@mcmaster.ca
Paul-Drude-Institut für Festkörperelektronik
Leibniz-Institut im Forschungsverbund Berlin e.V
Hausvogteiplatz 5-7D-10117BerlinGermany
Department of Engineering Physics
McMaster ... | 153,312,604 | {
"arxiv": "1905.05303",
"mag": 2969898265,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1361-6641/ab3c23"
} | {
"pdf_urls": [],
"pdf_sha": "bef7ac6c8ce38e27490225affa7f89626b206015",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1905.05303",
"status": "GREEN"
}
} | Bismuth-surfactant-induced growth and structure of InAs/GaAs(110) quantum dots | [
"Ryan B Lewis rlewis@mcmaster.ca \nPaul-Drude-Institut für Festkörperelektronik\nLeibniz-Institut im Forschungsverbund Berlin e.V\nHausvogteiplatz 5-7D-10117BerlinGermany\n\nDepartment of Engineering Physics\nMcMaster University\nL8S 4L7HamiltonCanada",
"Achim Trampert \nPaul-Drude-Institut für Festkörperelektron... | We explore the Bi-surfactant-directed self-assembly and structure of InAs quantum dots (QDs) grown on GaAs(110) by molecular beam epitaxy. The addition of a Bi flux during InAs deposition changes the InAs growth mode from two-dimensional (2D) Frank-van der Merwe to Stranski-Krastanov (SK), resulting in the formation of... | [
{
"title": "Introduction",
"content": "(In, Ga)As quantum dots (QDs) are of high interest for quantum optics applications requiring single photon emission [1,2] and emission of entangled photon pairs [3,4]. While the properties of QDs are intimately linked to their symmetryand thus the substrate orientation... | [
{
"id": "",
"caption": "under the presence of a Bi flux.Figures 1(a)-(g) displays atomic force microscopy (AFM) topographs for various InAs thicknesses (increasing left-right) and Bi fluxes (increasing topbottom). In the absence of a Bi flux (figure 1(a)), deposition of 2.1 ML of InAs results in a smooth 2D... | [
{
"text": "Triggered single photons from a quantum dot. C Santori, M Pelton, G Solomon, Y Dale, Y Yamamoto, 10.1103/PhysRevLett.86.1502Phys. Rev. Lett. 86Santori C, Pelton M, Solomon G, Dale Y and Yamamoto Y 2001 Triggered single photons from a quantum dot Phys. Rev. Lett. 86 1502-5",
"doi": "10.1103/PhysRe... |
Magnetization plateau in S = 3/2 antiferromagnetic Heisenberg chain with anisotropy
30 Oct 1997
Tôru Sakai
Faculty of Science
Himeji Institute of Technology
678-12Kamigori, Ako-gun, HyogoJapan
Minoru Takahashi
Institute for Solid State Physics
University of Tokyo
Minato-ku106Roppongi, TokyoJapan
Magnetization pla... | 119,376,526 | {
"arxiv": "cond-mat/9710327",
"mag": 2053676334,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.57.r3201"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/cond-mat/9710327v1.pdf"
],
"pdf_sha": "d131269b79845ec0283cfdb79260787878dd44ab",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/cond-mat/9710327",
"status": "GREEN"
}
} | Magnetization plateau in S = 3/2 antiferromagnetic Heisenberg chain with anisotropy | [
"Tôru Sakai \nFaculty of Science\nHimeji Institute of Technology\n678-12Kamigori, Ako-gun, HyogoJapan",
"Minoru Takahashi \nInstitute for Solid State Physics\nUniversity of Tokyo\nMinato-ku106Roppongi, TokyoJapan"
] | The magnetization process of the S=3/2 antiferromagnetic Heisenberg chain with the single-ion anisotropy D at T = 0 is investigated by the exact diagonalization of finite clusters and finite-size scaling analyses. It is found that a magnetization plateau appears at m = 1/2 for D > Dc = 0.93 ± 0.01. The phase transition... | [
{
"title": "FIG. 1 .FIG. 2 .",
"content": ""
},
{
"title": "FIG. 3 .",
"content": ""
},
{
"title": "FIG. 4 ., 1 Fig. 4 .",
"content": ""
},
{
"title": "Fig. 4 .FIG. 5 .",
"content": ""
}
] | [
{
"id": "",
"caption": "Scaled gap L∆ versus the single-ion anisotropy D. L-dependent fixed point DcL,L+2 is plotted versus 1/L to determine Dc in the thermodynamic limit. The estimated value is Dc = 0.93 ± 0.01.",
"content": "FIG. 1 .FIG. 2 .\n12Scaled gap L∆ versus the single-ion anisotropy D. L-depen... | [
{
"text": ". F D M Haldane, Phys. Lett. 93464F. D. M. Haldane, Phys. Lett. 93A,464 (1993);",
"doi": ""
},
{
"text": ". Phys. Rev. Lett. 501153Phys. Rev. Lett. 50,1153 (1983).",
"doi": ""
},
{
"text": ". K Katsumata, Phys. Rev. Lett. 6386K. Katsumata et al. Phys. Rev. Lett. 63, 86 (1989).... |
Modeling Cognitive Processes in Social Tagging to Improve Tag Recommendations
Dominik Kowald dkowald@know-center.at
Supervised by: Prof. Stefanie Lindstaedt (Know-Center) Know-Center
Graz University of Technology Inffeldgasse 13
GrazAustria
Modeling Cognitive Processes in Social Tagging to Improve Tag Recommendati... | 17,488,810 | {
"arxiv": "1805.11878",
"mag": 2950861510,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1805-11878",
"doi": "10.1145/2740908.2741746"
} | {
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"https://arxiv.org/pdf/1805.11878v1.pdf"
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"url": "https://arxiv.org/pdf/1805.11878",
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} | Modeling Cognitive Processes in Social Tagging to Improve Tag Recommendations | [
"Dominik Kowald dkowald@know-center.at \nSupervised by: Prof. Stefanie Lindstaedt (Know-Center) Know-Center\nGraz University of Technology Inffeldgasse 13\nGrazAustria"
] | With the emergence of Web 2.0, tag recommenders have become important tools, which aim to support users in finding descriptive tags for their bookmarked resources. Although current algorithms provide good results in terms of tag prediction accuracy, they are often designed in a data-driven way and thus, lack a thorough... | [
{
"title": "INTRODUCTION",
"content": "Social tagging systems enable users to collaboratively assign freely chosen keywords, so-called tags, to resources.\nCopyright is held by the author. Copyright 20XX ACM X-XXXXX-XX-X/XX/XX ...$15.00.\nThese tags can then be used for navigating, searching, organizing and... | [
{
"id": "",
"caption": "Schematic illustration of our basic 3Layers (3L) approach showing the connections between the semantic matrix (MS) encoding the latent topics and the lexical matrix (ML) encoding the tags.",
"content": "Figure 1 :\n1Schematic illustration of our basic 3Layers (3L) approach showin... | [
{
"text": "An integrated theory of the mind. J R Anderson, M D Byrne, S Douglass, C Lebiere, Y Qin, Psychological Review. 1114J. R. Anderson, M. D. Byrne, S. Douglass, C. Lebiere, and Y. Qin. An integrated theory of the mind. Psychological Review, 111(4):1036-1050, 2004.",
"doi": ""
},
{
"text": "Re... |
LevelMerge: Collaborative Game Level Editing by Merging Labeled Graphs
2 Mar 2016
Christian Santoni
Gabriele Salvati
Valentina Tibaldo
Fabio Pellacini
LevelMerge: Collaborative Game Level Editing by Merging Labeled Graphs
2 Mar 2016E573ECC970D6FEC120A5FBFA16DD6C48arXiv:1603.00713v1[cs.GR]Game DesignCollaborative ... | 16,493,986 | {
"arxiv": "1603.00713",
"mag": 2962873173,
"acl": null,
"pubmed": "29975191v1",
"pubmedcentral": null,
"dblp": "journals/cga/SantoniSTP18",
"doi": "10.1109/mcg.2018.042731660"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1603.00713v1.pdf"
],
"pdf_sha": "befee33a2c7a4f17eb3dd62cc825dc0bca44b733",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/1603.00713",
"status": "GREEN"
}
} | LevelMerge: Collaborative Game Level Editing by Merging Labeled Graphs | [
"Christian Santoni",
"Gabriele Salvati",
"Valentina Tibaldo",
"Fabio Pellacini"
] | Game development is commonly seen as a collaborative effort, with teams of people cooperating on the same project.Nowadays, a variety of cloud-based services have shown the benefits of performing tasks in real-time collaboration with others.In this paper, we present a system for collaborative game levels editing.We mod... | [
{
"title": "INTRODUCTION",
"content": "G AME level editing is the process of constructing a full game level starting from 3D asset libraries, e.g.3d models, textures, shaders, scripts.In level editing, designers define the look and behavior of the whole level by placing objects, assigning materials and ligh... | [
{
"id": "",
"caption": "Fig.1.Example merge computed with our algorithm.The original scene (left) is concurrently edited by two artists (middle), and automatically merged by our algorithm (right).Our merge algorithm maintains the semantic of user edits making the game playable after merging.The edits are co... | [
{
"text": ". S Chacon, Pro Git. APress. 2009",
"doi": ""
},
{
"text": "Plasticscm documentation. Plasticscm, 2015",
"doi": ""
},
{
"text": "Clara.io documentation. Clara Io, 2015",
"doi": ""
},
{
"text": "Unity3d documentation. Unity3d, 2015",
"doi": ""
},
{
"text... |
Cognitive and Physical Activities Impair Perception of Smartphone Vibrations
11 May 2023
Student Member, IEEEKyle T Yoshida
Joel X Kiernan
Student Member, IEEERachel A G Adenekan
Steven H Trinh
Alexis J Lowber
Fellow, IEEEAllison M Okamura
Member, IEEECara M Nunez
Computer Science Department
Stanford Universi... | 258,302,330 | {
"arxiv": "2305.06556",
"mag": null,
"acl": null,
"pubmed": "37220041v1",
"pubmedcentral": null,
"dblp": "journals/toh/YoshidaKATLON23",
"doi": "10.1109/toh.2023.3279201"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2305.06556v1.pdf"
],
"pdf_sha": "41594cbf62e7b24e5739040f17c0c1ad02b04e63",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2305.06556",
"status": "GREEN"
}
} | Cognitive and Physical Activities Impair Perception of Smartphone Vibrations | [
"Student Member, IEEEKyle T Yoshida",
"Joel X Kiernan",
"Student Member, IEEERachel A G Adenekan",
"Steven H Trinh",
"Alexis J Lowber",
"Fellow, IEEEAllison M Okamura",
"Member, IEEECara M Nunez",
"Computer Science Department\nStanford University\n94305StanfordCA",
"Sibley School of Mechanical and A... | Vibration feedback is common in everyday devices, from virtual reality systems to smartphones.However, cognitive and physical activities may impede our ability to sense vibrations from devices.In this study, we develop and characterize a smartphone platform to investigate how a shape-memory task (cognitive activity) an... | [
{
"title": "I. INTRODUCTION",
"content": "Real-life, out-of-lab haptic applications, including virtual reality, social interactions, wearables, and smartphones, are becoming increasingly common.However, the transition from in-lab to out-of-lab can present a variety of challenges for researchers [1].Traditio... | [
{
"id": "",
"caption": "Fig. 3. Vibration perception thresholds were measured while the participant had low (c) and high (C) cognitive activity, combined with low (p) or high (P) physical activity.This yielded four test conditions.A shape-recall task introduced cognitive activity (C), while walking on a tre... | [
{
"text": "Getting Your Hands Dirty Outside the Lab: A Practical Primer for Conducting Wearable Vibrotactile Haptics Research. J R Blum, IEEE Transactions on Haptics. 1232019",
"doi": ""
},
{
"text": "Perception of vibrotactile stimuli during motor activity in human subjects. L Post, I Zompa, C Chap... |
arXiv:physics/0109026v1 [physics.plasm-ph] 12 Sep 2001 Rotational kinetics of absorbing dust grains in neutral gas
A M Ignatov
General Physics Institute
MoscowRussia
S A Trigger
Humboldt University
BerlinGermany
S A Maiorov
General Physics Institute
MoscowRussia
W Ebeling
Humboldt University
BerlinGermany
ar... | 46,099,215 | {
"arxiv": "physics/0109026",
"mag": 1970570388,
"acl": null,
"pubmed": "12006029",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreve.65.046413"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/physics/0109026v1.pdf"
],
"pdf_sha": "fab2075a22ca51a46f4ef6683ba7f5530d911d63",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/physics/0109026",
"status": "GREEN"
}
} | arXiv:physics/0109026v1 [physics.plasm-ph] 12 Sep 2001 Rotational kinetics of absorbing dust grains in neutral gas | [
"A M Ignatov \nGeneral Physics Institute\nMoscowRussia",
"S A Trigger \nHumboldt University\nBerlinGermany",
"S A Maiorov \nGeneral Physics Institute\nMoscowRussia",
"W Ebeling \nHumboldt University\nBerlinGermany"
] | We study the rotational and translational kinetics of massive particulates (dust grains) absorbing the ambient gas. Equations for microscopic phase densities are deduced resulting in the Fokker-Planck equation for the dust component. It is shown that although there is no stationary distribution, the translational and r... | [
{
"title": "I. INTRODUCTION",
"content": "Recently there was growing interest to the investigation of composite media, called dusty plasmas, consisting of aerosol particles in a gas discharge. Besides numerous industrial applications dusty plasmas provide ample opportunities to studying in situ phase transi... | [
{
"id": "",
"caption": "where τ iα is the instant of collision of the i-th atom with the α-th grain. At the time intervals between the collisions Eqs. (7,8) describe plain Newtonian dynamics. However, at the instant of collision the microscopic phase densities change abruptly:",
"content": "where τ iα i... | [
{
"text": ". R A Quinn, J Goree, Phys. Plasmas. 73904R. A. Quinn and J. Goree, Phys. Plasmas 7, 3904 (2000).",
"doi": ""
},
{
"text": ". R A Quinn, J Goree, Phys. Rev. E. 613033R. A. Quinn and J. Goree, Phys. Rev. E 61 3033 (2000).",
"doi": ""
},
{
"text": ". A G Zagorodny, P P J M Schra... |
HOMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES
25 Jun 2014
Enxin Wu
HOMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES
25 Jun 2014arXiv:1406.6717v1 [math.KT]
Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M. Souriau and his mathematical group in the 1980's. Diffeological ... | 54,775,712 | {
"arxiv": "1406.6717",
"mag": 2963894934,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.4310/hha.2015.v17.n1.a17"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1406.6717v1.pdf"
],
"pdf_sha": "93bb5076cde6160ea5cc685231f5577ad7f5a20a",
"open_access": {
"license": null,
"url": "http://www.intlpress.com/site/pub/files/_fulltext/journals/hha/2015/0017/0001/HHA-2015-0017-0001-a017.pdf",
"status": "BRONZE"
}
} | HOMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES | [
"Enxin Wu"
] | Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M. Souriau and his mathematical group in the 1980's. Diffeological vector spaces (especially fine diffeological vector spaces) were first used by P. Iglesias-Zemmour to model some infinite dimensional spaces in [I1, I2]. K. Costello a... | [
{
"title": "Introduction",
"content": "The concept of a diffeological space and the terminology was formulated by J.M. Souriau and his mathematical group in the 1980's ( [So1,So2]) as follows:\nDefinition 1.1. A diffeological space is a set X together with a specified set D X of functions U → X (called plot... | [
{
"id": "",
"caption": "The category DVect is cocomplete.",
"content": "Theorem 3 . 3 .\n33The category DVect is cocomplete."
},
{
"id": "",
"caption": "⇒ (2) are clear. (1) ⇒ (3): By [H, The Short Five Lemma 1.17], (r, p) : V → A × B is an isomorphism in Vect.",
"content": "⇒ (2) are cl... | [
{
"text": "Convenient categories of smooth spaces. J C Baez, A E Hoffnung, Trans. Amer. Math. Soc. 36311Baez, J.C., Hoffnung, A.E.: Convenient categories of smooth spaces. Trans. Amer. Math. Soc. 363(11), 5789- 5825 (2011).",
"doi": ""
},
{
"text": "Factorization algebras in quantum field theory. K ... |
Numerical Energy Dissipation for Time-Fractional Phase-Field Equations
September 15, 2020
Chaoyu Quan quancy@sustech.edu.cn
SUSTech International Center for Mathematics
Southern University of Science and Technology
ShenzhenChina
Tao Tang ttang@uic.edu.cn
Division of Science and Technology
BNU-HKBU United Internati... | 221,655,172 | {
"arxiv": "2009.06178",
"mag": 3085674002,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2009-06178",
"doi": "10.4208/jcm.2311-m2021-0199"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2009.06178v1.pdf"
],
"pdf_sha": "51a6dc224b53ecec94e721e6716a8ce3ae5f7545",
"open_access": {
"license": null,
"url": "https://global-sci.org/intro/online/preview?online_id=2221&pdf=https://doc.global-sci.org/uploads/admin/article_pdf/20240311/dfc82bcfe0895cdb33... | Numerical Energy Dissipation for Time-Fractional Phase-Field Equations | [
"Chaoyu Quan quancy@sustech.edu.cn \nSUSTech International Center for Mathematics\nSouthern University of Science and Technology\nShenzhenChina",
"Tao Tang ttang@uic.edu.cn \nDivision of Science and Technology\nBNU-HKBU United International College\nZhuhai, GuangdongChina\n\nGuangdong Provincial Key Laboratory of... | The energy dissipation is an important and essential property of classical phase-field equations.However, it is still unknown if the phase-field models with Caputo time-fractional derivative preserve this property, which is challenging due to the existence of both nonlocality and nonlinearity.Our recent work shows that... | [
{
"title": "Introduction",
"content": "In recent years, to model memory effects and subdiffusive regimes, there has been an increasing interest in time-fractional differential equations, where the classical time derivative is replaced by a fractional one, typically a Caputo or a Riemann-Liouville derivative... | [
{
"id": "",
"caption": ".31) we obtain(3.26)and then(3.24).Therefore, f (x) is monotonically increasing.As a consequence, (3.21) is true and the property (P3) holds.In summary, M satisfies (P1)-(P3) in Lemma 3.2.Then, we claim that M is positive definite and B + B T is also positive definite.According to(3.... | [
{
"text": "A parabolic problem with a fractional time derivative. Luis A Mark Ryan Allen, Alexis Caffarelli, Vasseur, Archive for Rational Mechanics and Analysis. 2212016",
"doi": ""
},
{
"text": "On the maximum principle for a time-fractional diffusion equation. Yuri Luchko, Masahiro Yamamoto, Frac... |
GRB021125: the first GRB imaged by INTEGRAL ⋆
Jul 2003 November 2, 2018
G Malaguti
Istituto di Astrofisica Spaziale e Fisica Cosmica (IASF)
CNR
Sezione di Bologna
Via Gobetti 10140129BolognaItaly
A Bazzano
Istituto di Astrofisica Spaziale e Fisica Cosmica (IASF)
CNR
Sezione di Roma
Via Fosso del Cavaliere 10000133... | 16,914,765 | {
"arxiv": "astro-ph/0307523",
"mag": 2153804813,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1051/0004-6361:20031155"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/astro-ph/0307523v1.pdf"
],
"pdf_sha": "ab530ec2c0db5f67bbad5fd452bd168568188d9e",
"open_access": {
"license": null,
"url": "https://www.aanda.org/articles/aa/pdf/2003/43/aaINTEGRAL28.pdf",
"status": "BRONZE"
}
} | GRB021125: the first GRB imaged by INTEGRAL ⋆ | [
"G Malaguti \nIstituto di Astrofisica Spaziale e Fisica Cosmica (IASF)\nCNR\nSezione di Bologna\nVia Gobetti 10140129BolognaItaly",
"A Bazzano \nIstituto di Astrofisica Spaziale e Fisica Cosmica (IASF)\nCNR\nSezione di Roma\nVia Fosso del Cavaliere 10000133RomaItaly",
"V Beckmann \nINTEGRAL Science Data Centre\... | In the late afternoon of November 25 th , 2002 a gamma-ray burst (GRB) was detected in the partially coded field of view (about 7.3 • from the centre) of the imager IBIS on board the INTEGRAL satellite. The instruments on-board INTEGRAL allowed, for the first time, the observation of the prompt gamma-ray emission over ... | [
{
"title": "Introduction",
"content": "INTEGRAL is the ESA satellite dedicated to the astrophysics in the X-and γ-ray domain, launched on October 17 th , 2002th , (Winkler et al. 2003. It is composed of two main high-energy telescopes (IBIS, SPI) coupled with two monitors, one in the X-ray energy band (JEM-... | [
{
"id": "",
"caption": "GRB021125 lightcurves: (top) IBIS/ISGRI lightcurve in the whole energy range, heavily affected by telemetry gaps; time starts from 17:58:00 UT. (bottom) SPI lightcurve (in Crab units) obtained from the detector count rates in the energy range 0.02 − 8 MeV, which are part of the scien... | [
{
"text": ". A Bazzano, A Paizis, 1706Bazzano, A., Paizis, A., 2002, GCN 1706",
"doi": ""
},
{
"text": ". Di Cocco, G Caroli, E Celesti, E , A&A. 1714Di Cocco, G., Caroli, E., Celesti, E., et al., 2003, A&A, this issue Goldwurm, A., David, P., Foschini, L., et al., 2003, A&A, this issue Gros, A., Pr... |
Galaxy cluster mass density profile derived using the submillimetre galaxies magnification bias
Fernandez L
Cueli M M
González-Nuevo J
Bonavera L
Crespo D
Casas J M
Lapi A
Mathematisches Institute
Universität Bern
Sidlerstrasse 53012BernSwitzerland
Departamento de Fisica
Universidad de Oviedo
C. Federico G... | 243,938,390 | {
"arxiv": "2111.05422",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1051/0004-6361/202141905"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2111.05422v1.pdf"
],
"pdf_sha": "67a16a1f6dbe9601b4a1dc48fde32fe4b94b0829",
"open_access": {
"license": null,
"url": "https://www.aanda.org/articles/aa/pdf/forth/aa41905-21.pdf",
"status": "BRONZE"
}
} | Galaxy cluster mass density profile derived using the submillimetre galaxies magnification bias | [
"Fernandez L",
"Cueli M M",
"González-Nuevo J",
"Bonavera L",
"Crespo D",
"Casas J M",
"Lapi A",
"Mathematisches Institute\nUniversität Bern\nSidlerstrasse 53012BernSwitzerland",
"Departamento de Fisica\nUniversidad de Oviedo\nC. Federico Garcia Lorca 1833007OviedoSpain",
"Instituto Universitario ... | Context. The magnification bias is a gravitational lensing effect that produces an increase or decrease in the detection probability of background sources near the position of a lense. The special properties of the submillimetre galaxies (SMGs; steep source number counts, high redshift, and a very low cross-contaminati... | [
{
"title": "Introduction",
"content": "The magnification bias (Schneider et al. 1992) is a gravitational lensing effect that consists of an increase or decrease in the detection probability of background sources near the positions of lenses. It is due to a modification of the integrated source number counts... | [
{
"id": "",
"caption": "Top: Redshift distribution of the target lenses from the Wen et al. (2012) catalogue (in red) and the background sources from the H-ATLAS sample (in black). Bottom: Redshift distribution of the galaxy clusters for each of the five richness bins.",
"content": "Fig. 1 .\n1Top: Reds... | [
{
"text": ". S W Allen, A E Evrard, A B Mantz, ARA&A. 49409Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409",
"doi": ""
},
{
"text": ". M Bartelmann, P Schneider, Phys. Rep. 340291Bartelmann, M. & Schneider, P. 2001, Phys. Rep., 340, 291",
"doi": ""
},
{
"text": ". A H Ba... |
X-ray photon correlation spectroscopy under flow
March 10, 2008
Andrei Fluerasu
European Synchrotron Radiation Facility
GrenobleFrance
Abdellatif Moussaïd
European Synchrotron Radiation Facility
GrenobleFrance
Henri Gleyzolle
European Synchrotron Radiation Facility
GrenobleFrance
Péter Falus
Institut Laue-Lang... | 206,352,427 | {
"arxiv": "0803.1437",
"mag": 2963558944,
"acl": null,
"pubmed": "18552431",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1107/s0909049508006420"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0803.1437v1.pdf"
],
"pdf_sha": "4e55d79b4e3180f3fb56fb5cd6771cca15d20774",
"open_access": {
"license": null,
"url": "http://journals.iucr.org/s/issues/2008/04/00/ot5581/ot5581.pdf",
"status": "BRONZE"
}
} | X-ray photon correlation spectroscopy under flow | [
"Andrei Fluerasu \nEuropean Synchrotron Radiation Facility\nGrenobleFrance",
"Abdellatif Moussaïd \nEuropean Synchrotron Radiation Facility\nGrenobleFrance",
"Henri Gleyzolle \nEuropean Synchrotron Radiation Facility\nGrenobleFrance",
"Péter Falus \nInstitut Laue-Langevin\nGrenobleFrance",
"Anders Madsen \n... | X-ray photon correlation spectroscopy was used to probe the diffusive dynamics of colloidal particles in a shear flow.Combining X-ray techniques with microfluidics is an experimental strategy that reduces the risk of x-ray induced beam damage and also allows time-resolved studies of processes taking place in flowcells.... | [
{
"title": "I. INTRODUCTION",
"content": "Over the past decade, X-ray photon correlation spectroscopy (XPCS) has emerged as an unique experimental tool that allows the direct measurement of fluctuations in a large number of condensed matter systems [1].It provides a method complementary to Dynamic Light Sca... | [
{
"id": "",
"caption": "FIG.1:(a) Small-angle X-ray experimental setup and flow chamber for the XPCS experiments.Static scattering, Ix, Iz, for the two flow geometries considered here -(b) transverse, q ⊥ v and respectively -(c) longitudinal, q v.The data were recorded at several flow rates (v=0-200 µm/s).I... | [
{
"text": "M Sutton, Third-Generation Hard X-Ray Synchrotron Radiation Sources: Source Properties, Optics, and Experimental Techniques. D Millis, New YorkJohn Wiley & Sons, Inc2002",
"doi": ""
},
{
"text": "B Berne, R Pecora, Dynamic Light Scattering. New YorkDover2000",
"doi": ""
},
{
"... |
Finitely presented algebras and groups defined by permutation relations *
2 Oct 2008
Ferran Cedó
Eric Jespers
Jan Okniński
Finitely presented algebras and groups defined by permutation relations *
2 Oct 2008symmetric presentationfinitely presentedsemigroup algebramonoidgroup
The class of finitely presented algebra... | 16,009,764 | {
"arxiv": "0810.0352",
"mag": 2949185412,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.jpaa.2009.09.015"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0810.0352v1.pdf"
],
"pdf_sha": "d568d63791c7fa251aa1ec26ebd6e221a0174042",
"open_access": {
"license": "elsevier-specific: oa user license",
"url": "https://doi.org/10.1016/j.jpaa.2009.09.015",
"status": "BRONZE"
}
} | Finitely presented algebras and groups defined by permutation relations * | [
"Ferran Cedó",
"Eric Jespers",
"Jan Okniński"
] | The class of finitely presented algebras over a field K with a set of generators a1, . . . , an and defined by homogeneous relations of the form a1a2 · · · an = a σ(a) a σ(2) · · · a σ(n) , where σ runs through a subset H of the symmetric group Sym n of degree n, is introduced. The emphasis is on the case of a cyclic s... | [
{
"title": "Introduction",
"content": "Recently there has been extensive interest in some finitely presented algebras A defined by homogeneous semigroup relations, that is, relations of the form w = v, where w and v are words of the same length in a generating set of the algebra. A particular intriguing cla... | [
{
"id": "",
"caption": "b 1 , . . . , b m are elements of a monoid M then we denote by b 1 , . . . , b m the submonoid generated by b 1 , . . . , b m . If M is a group then gr(b 1 , . . . , b m ) denotes the subgroup of M generated by b 1 , . . . , b m .",
"content": "b 1 , . . . , b m are elements of a... | [
{
"text": "Is it a residually finite group? 2. When does every element of S n (H) have a unique canonical form. Describe the structure of G n (H). as is the case of the monoid S n (gr({(1 , 2. considered in Section 2?Describe the structure of G n (H). Is it a residually finite group? 2. When does every element ... |
Mergers of Systems Containing Gas
13 Oct 1993
F R Pearce
Astronomy Centre
Sussex University
BN1 9QHFalmer, Brighton
P A Thomas
Astronomy Centre
Sussex University
BN1 9QHFalmer, Brighton
H M P Couchman
Department of Astronomy
University of Western Ontario
N6A 3KVLondonOntarioCanada
Mergers of Systems Containing ... | 118,949,927 | {
"arxiv": "astro-ph/9310023",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1093/mnras/268.4.953"
} | {
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"https://export.arxiv.org/pdf/astro-ph/9310023v1.pdf"
],
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"license": null,
"url": "https://academic.oup.com/mnras/article-pdf/268/4/953/3281384/mnras268-0953.pdf",
"status": "BRONZE"
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} | Mergers of Systems Containing Gas | [
"F R Pearce \nAstronomy Centre\nSussex University\nBN1 9QHFalmer, Brighton",
"P A Thomas \nAstronomy Centre\nSussex University\nBN1 9QHFalmer, Brighton",
"H M P Couchman \nDepartment of Astronomy\nUniversity of Western Ontario\nN6A 3KVLondonOntarioCanada"
] | Several simple mergers between model galaxy clusters containing a mixture of gas and dark matter are examined, testing the coupling of the gas to the underlying collisionless material. The gas is shocked, irreversibly dissipating the energy fed into it by the collisionless component and forms a resolved constant-densit... | [
{
"title": "Introduction",
"content": "In this paper we examine the behaviour of the intra-cluster medium in galaxy clusters. Cluster gas is important because it is the dominant form of baryonic matter in these objects and observations of its X-ray emission provide some of the best currently available const... | [
{
"id": "",
"caption": "Energy Evolution of the gas and dark matter density profiles.",
"content": "Figure 1 :Figure 2 :\n12Energy Evolution of the gas and dark matter density profiles."
},
{
"id": "",
"caption": "Evolution of the gas temperature and the range in temperatures at the endpoint... | [
{
"text": ". E Bertschinger, ApJS. 5839Bertschinger, E., 1985, ApJS, 58, 39",
"doi": ""
},
{
"text": "Galactic Dynamics. J Binney, S Tremaine, Princeton University PressPrincetonBinney, J., Tremaine, S., 1987, Galactic Dynamics. Princeton University Press, Princeton",
"doi": ""
},
{
"tex... |
H-NMR studies of supercooled and glassy aspirin
R Nath
Fachbereich Physik and Interdisziplinäres Zentrum für Magnetische Resonanz
Universität Dortmund
44221DortmundGermany
B Geil
Fachbereich Physik and Interdisziplinäres Zentrum für Magnetische Resonanz
Universität Dortmund
44221DortmundGermany
R Böhmer
Fachber... | 97,427,438 | {
"arxiv": "cond-mat/0701201",
"mag": 2100365940,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.jnoncrysol.2007.05.180"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/cond-mat/0701201v1.pdf"
],
"pdf_sha": "def58bed1c6268620e744b0f2e3ddc170d0c3421",
"open_access": {
"license": null,
"url": null,
"status": "CLOSED"
}
} | H-NMR studies of supercooled and glassy aspirin | [
"R Nath \nFachbereich Physik and Interdisziplinäres Zentrum für Magnetische Resonanz\nUniversität Dortmund\n44221DortmundGermany",
"B Geil \nFachbereich Physik and Interdisziplinäres Zentrum für Magnetische Resonanz\nUniversität Dortmund\n44221DortmundGermany",
"R Böhmer \nFachbereich Physik and Interdisziplinä... | Acetyl salicylic acid, deuterated at the methyl group, was investigated using 2 H-NMR in its supercooled and glassy states. Just above the glass transition temperature the molecular reorientations, studied using stimulated-echo spectroscopy, demonstrated a large degree of similarity with other glass formers. Deep in th... | [
{
"title": "Introduction",
"content": "There are numerous examples for relaxations in complex systems, with the molecular dynamics in viscous and vitreous pharmaceuticals being a relatively recent issue that attracts interest in the scientific community dealing with supercooled liquids and glasses. In view ... | [
{
"id": "",
"caption": "Fig. 4",
"content": "Fig. 4\n4Fig. 4"
},
{
"id": "",
"caption": "Stimulated-echo decay times τ 2 for supercooled aspirin-d 3 are shown as a function of the evolution time t p . The experimental data obtained for two temperatures are represented by the symbols. The sol... | [
{
"text": "J Bernstein, Polymorphism in Molecular Crystals. Clarendon, Oxfordsee also PJ. Bernstein, Polymorphism in Molecular Crystals (Clarendon, Oxford, 2002); see also P.",
"doi": ""
},
{
"text": ". J A Vishweshwar, M Mcmahon, M L Oliveira, M J Peterson, Zaworotko, J. Am. Chem. Soc. 12716802Vish... |
Augmenting Language Models with Long-Term Memory
12 Jun 2023
Weizhi Wang weizhiwang@ucsb.edu
University of California
Santa Barbara ‡ Microsoft Research
Li Dong lidong1@microsoft.com
University of California
Santa Barbara ‡ Microsoft Research
Hao Cheng haocheng@microsoft.com
University of California
Santa Barb... | 259,137,816 | {
"arxiv": "2306.07174",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/nips/Wang0CLYGW23",
"doi": "10.48550/arxiv.2306.07174"
} | {
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"https://export.arxiv.org/pdf/2306.07174v1.pdf"
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"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2306.07174",
"status": "CLOSED"
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} | Augmenting Language Models with Long-Term Memory | [
"Weizhi Wang weizhiwang@ucsb.edu \nUniversity of California\nSanta Barbara ‡ Microsoft Research",
"Li Dong lidong1@microsoft.com \nUniversity of California\nSanta Barbara ‡ Microsoft Research",
"Hao Cheng haocheng@microsoft.com \nUniversity of California\nSanta Barbara ‡ Microsoft Research",
"Xiaodong Liu \nU... | Existing large language models (LLMs) can only afford fix-sized inputs due to the input length limit, preventing them from utilizing rich long-context information from past inputs.To address this, we propose a framework, Language Models Augmented with Long-Term Memory (LONGMEM), which enables LLMs to memorize long hist... | [
{
"title": "Introduction",
"content": "Large language models (LLMs) have revolutionized natural language processing with great successes in advancing the state-of-the-art on various understanding and generation tasks [DCLT19, RWC + 19, LOG + 19, YDY + 19, BMR + 20, RSR + 20].Most LLMs benefit from self-supe... | [
{
"id": "",
"caption": "In-Context Learning LLMs have the emerging capability of in-context learning (ICL) via learning knowledge nonparametrically from few-shot demonstration examples in the local context.However, conventional in-context learning is heavily restricted by input context length, rendering it ... | [
{
"text": "Dbpedia: A nucleus for a web of open data. Sören Auer, Christian Bizer, Georgi Kobilarov, Jens Lehmann, Richard Cyganiak, Zachary Ives, ABK + 07The semantic web. Springer2007",
"doi": ""
},
{
"text": "Improving language models by retrieving from trillions of tokens. Sebastian Borgeaud, Ar... |
THE DAMPING TAIL OF CMB ANISOTROPIES
Sep 1996
Wayne Hu
Institute for Advanced Study
School of Natural Sciences
Enrico Fermi Institute
University of Chicago
08540, 60637Princeton, ChicagoNJ, IL
Martin White
Institute for Advanced Study
School of Natural Sciences
Enrico Fermi Institute
University of Chicago
08540, 6... | 14,601,866 | {
"arxiv": "astro-ph/9609079",
"mag": 2951042161,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1086/303928"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/astro-ph/9609079v1.pdf"
],
"pdf_sha": "4189929afb5e893a0b1a3eaba76cb83d53991bd6",
"open_access": {
"license": null,
"url": null,
"status": "CLOSED"
}
} | THE DAMPING TAIL OF CMB ANISOTROPIES | [
"Wayne Hu \nInstitute for Advanced Study\nSchool of Natural Sciences\nEnrico Fermi Institute\nUniversity of Chicago\n08540, 60637Princeton, ChicagoNJ, IL",
"Martin White \nInstitute for Advanced Study\nSchool of Natural Sciences\nEnrico Fermi Institute\nUniversity of Chicago\n08540, 60637Princeton, ChicagoNJ, IL"... | By decomposing the damping tail of CMB anisotropies into a series of transfer functions representing individual physical effects, we provide ingredients that will aid in the reconstruction of the cosmological model from small-scale CMB anisotropy data. We accurately calibrate the model-independent effects of diffusion ... | [
{
"title": "Introduction",
"content": "Much effort is being expended to measure the angular power spectrum of the cosmic microwave background (CMB) anisotropy on increasingly smaller angular scales. For many types of models for structure formation, the spectrum can be predicted to a precision of about 1% , ... | [
{
"id": "",
"caption": "-Diffusion damping calibration. In the absence of both diffusion damping and gravitational sources, the rms temperature fluctuation at recombination (short-dashed line) exhibits simple acoustic oscillations. These are mapped onto anisotropies on the sky in a near one to one fashion (... | [
{
"text": ". N Aghanim, F X Desert, J L Puget, R Gispert, astro-ph/9605083A&A. 3111Aghanim, N., Desert, F. X., Puget, J. L. & Gispert, R. 1996, A&A, 311, 1 [astro-ph/9605083]",
"doi": "astro-ph/9605083"
},
{
"text": ". A Albrecht, D Coulson, P Ferreira, J Magueijo, astro-ph/9505030Phys. Rev. Lett. 7... |
CONSTRAINTS ON CARDASSIAN EXPANSION FROM DISTANT TYPE Ia SUPERNOVAE
Zong-Hong Zhu zong-hong.zhu@nao.ac.jp
National Astronomical Observatory
2-21-1181-8588OsawaMitaka, TokyoJapan
Masa-Katsu Fujimoto fujimoto.masa-katsu@nao.ac.jp
National Astronomical Observatory
2-21-1181-8588OsawaMitaka, TokyoJapan
CONSTRAINTS O... | 6,811,850 | {
"arxiv": "astro-ph/0303021",
"mag": 3101343048,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1086/346002"
} | {
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"url": null,
"status": "CLOSED"
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} | CONSTRAINTS ON CARDASSIAN EXPANSION FROM DISTANT TYPE Ia SUPERNOVAE | [
"Zong-Hong Zhu zong-hong.zhu@nao.ac.jp \nNational Astronomical Observatory\n2-21-1181-8588OsawaMitaka, TokyoJapan",
"Masa-Katsu Fujimoto fujimoto.masa-katsu@nao.ac.jp \nNational Astronomical Observatory\n2-21-1181-8588OsawaMitaka, TokyoJapan"
] | The distant Type Ia supernovae data compiled by Perlmutter et al. are used to analyze the Cardassian expansion scenario, which was recently proposed by Freese & Lewis as an alternative to a cosmological constant (or more generally a dark energy component) in explaining the currently accelerating universe. We show that ... | [
{
"title": "INTRODUCTION",
"content": "A major development in modern cosmology is the discovery of the acceleration of the universe through observations of distant Type Ia supernovae (Perlmutter et al. 1998(Perlmutter et al. , 1999Riess et al. 1998Riess et al. , 2001Leibundgut 2001). It is well known that a... | [
{
"id": "",
"caption": "-Hubble diagram for 42 high-redshift SNe Ia from the Supernova Cosmology Project and 18 low-redshift SNe Ia from the Calań /Tololo Supernova Survey.",
"content": "Fig. 1 .\n1-Hubble diagram for 42 high-redshift SNe Ia from the Supernova Cosmology Project and 18 low-redshift SNe I... | [
{
"text": ". J S Alcaniz, J A S Lima, ApJ. 52187Alcaniz, J. S., & Lima, J. A. S. 1999, ApJ, 521, L87",
"doi": ""
},
{
"text": ". P P Avelino, J P M De Carvalho, C J A P Martins, Phys. Rev. D. 6463505Avelino, P. P., de Carvalho, J. P. M., & Martins, C. J. A. P. 2001, Phys. Rev. D, 64, 063505",
"d... |
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
22 Feb 2024
H. H. Wills Physics Laboratory
University of Bristol
BristolUnited Kingdom
Cavendish Laboratory
University of Cambridge
CambridgeUnited Kingdom
Department of Physics
University of Warwick
CoventryUnited Kingdom
STFC Rutherford Appleton Laboratory
Di... | 264,305,801 | {
"arxiv": "2310.12278",
"mag": null,
"acl": null,
"pubmed": "38457697v1",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevlett.132.081901"
} | {
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"https://export.arxiv.org/pdf/2310.12278v2.pdf"
],
"pdf_sha": "a73ddd4d7881407f351123425d2058cf40a9c551",
"open_access": {
"license": "CCBY",
"url": "http://link.aps.org/pdf/10.1103/PhysRevLett.132.081901",
"status": "HYBRID"
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} | EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) | [
"H. H. Wills Physics Laboratory\nUniversity of Bristol\nBristolUnited Kingdom",
"Cavendish Laboratory\nUniversity of Cambridge\nCambridgeUnited Kingdom",
"Department of Physics\nUniversity of Warwick\nCoventryUnited Kingdom",
"STFC Rutherford Appleton Laboratory\nDidcotUnited Kingdom",
"School of Physics an... | The production rate of Λ 0 b baryons relative to B 0 mesons in pp collisions at a center-of-mass energy √ s = 13 TeV is measured by the LHCb experiment.The ratio of Λ 0 b to B 0 production cross-sections shows a significant dependence on both the transverse momentum and the measured charged-particle multiplicity.At low... | [
{
"title": "Figure 1 :",
"content": "Figure 1: Invariant mass spectra in the transverse momentum interval 10 < p T < 12 GeV/c for the (a) J/ψpK − and (b) J/ψπ − K + final states, integrated over multiplicity.Fit projections including signal and background components are shown."
},
{
"title": "bFigur... | [
{
"id": "",
"caption": "Figure 1: Invariant mass spectra in the transverse momentum interval 10 < p T < 12 GeV/c for the (a) J/ψpK − and (b) J/ψπ − K + final states, integrated over multiplicity.Fit projections including signal and background components are shown.",
"content": "Figure 1 :\n1\nFigure 1: ... | [
{
"text": "Confinement of quarks. K G Wilson, 10.1103/PhysRevD.10.2445Phys. Rev. 1024451974",
"doi": ""
},
{
"text": "Factorization of hard processes in QCD. J C Collins, D E Soper, G F Sterman, 10.1142/9789814503266_0001arXiv:hep-ph/0409313Adv. Ser. Direct. High Energy Phys. 511989",
"doi": ""
... |
Time-dependent density functional theory for a unified description of ultrafast dynamics: pulsed light, electrons, and atoms in crystalline solids
15 Oct 2018
Atsushi Yamada
Center for Computational Sciences
University of Tsukuba
1-1-1 Tennodai305-8577TsukubaIbarakiJapan
Kazuhiro Yabana
Center for Computational Sc... | 53,530,821 | {
"arxiv": "1810.06168",
"mag": 2948104537,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.99.245103"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1810.06168v1.pdf"
],
"pdf_sha": "028fe59df184385567ede40cb469823abd37a2e1",
"open_access": {
"license": "publisher-specific, author manuscript",
"url": "https://link.aps.org/accepted/10.1103/PhysRevB.99.245103",
"status": "HYBRID"
}
} | Time-dependent density functional theory for a unified description of ultrafast dynamics: pulsed light, electrons, and atoms in crystalline solids | [
"Atsushi Yamada \nCenter for Computational Sciences\nUniversity of Tsukuba\n1-1-1 Tennodai305-8577TsukubaIbarakiJapan",
"Kazuhiro Yabana \nCenter for Computational Sciences\nUniversity of Tsukuba\n1-1-1 Tennodai305-8577TsukubaIbarakiJapan"
] | We have developed a novel multiscale computational scheme to describe coupled dynamics of light electromagnetic field with electrons and atoms in crystalline solids, where first-principles molecular dynamics based on time-dependent density functional theory is used to describe the microscopic dynamics. The method is ap... | [
{
"title": "FIG. 1 .",
"content": ""
},
{
"title": "FIG. 2 .",
"content": ""
},
{
"title": "FIG. 4 .",
"content": ""
},
{
"title": "FIG. 5 .",
"content": ""
}
] | [
{
"id": "",
"caption": "Schematic illustration of the multiscale scheme for a light propagation through a material.",
"content": "FIG. 1 .\n1Schematic illustration of the multiscale scheme for a light propagation through a material."
},
{
"id": "",
"caption": "(a) Snapshots of the electric f... | [
{
"text": "The Principles of Nonlinear Optics. Y R Shen, Wiley-Interscience1st editionY. R. Shen. The Principles of Nonlinear Optics. Wiley- Interscience, 1st edition, 1984.",
"doi": ""
},
{
"text": ". Nicolaas Bloembergen. Nonlinear Optics. World Scientific. 4th editionNicolaas Bloembergen. Nonline... |
Electrical control of hybrid exciton transport in a van der Waals heterostructure
Fedele Tagarelli
Institute of Electrical and Microengineering
École Polytechnique Fédérale de Lausanne (EPFL)
CH-1015LausanneSwitzerland
Institute of Materials Science and Engineering
École Polytechnique Fédérale de Lausanne (EPFL)
C... | 257,255,263 | {
"arxiv": "2303.00419",
"mag": null,
"acl": null,
"pubmed": "37426431v1",
"pubmedcentral": 10322698,
"dblp": null,
"doi": "10.1038/s41566-023-01198-w"
} | {
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"https://export.arxiv.org/pdf/2303.00419v1.pdf"
],
"pdf_sha": "db053d639ead452905baa860d03cac0fcf3aa619",
"open_access": {
"license": "CCBY",
"url": "https://www.nature.com/articles/s41566-023-01198-w.pdf",
"status": "HYBRID"
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} | Electrical control of hybrid exciton transport in a van der Waals heterostructure | [
"Fedele Tagarelli \nInstitute of Electrical and Microengineering\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nInstitute of Materials Science and Engineering\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nCH-1015LausanneSwitzerland",
"Edoardo Lopriore \nInstitute of Electrical... | Interactions between out-of-plane dipoles in bosonic gases enable the long-range propagation of excitons.The lack of direct control over collective dipolar properties has hitherto limited the degrees of tunability and the microscopic understanding of exciton transport.In this work, we modulate the layer hybridization a... | [
{
"title": "INTRODUCTION",
"content": "Exciton transport has been indicated as a potential basis for realizing scaled optical interconnects and modulators in chip-scale optical processing systems 1 .Strongly-bound and long-lived propagating excitons can act as carriers of information within a semiconductor,... | [
{
"id": "",
"caption": "The potential energy of out-of-plane electron-hole pairs with fixed dipole lengths can be modulated by the linear quantum-confined Stark effect as ≈ .The degree of interlayer character of hIXs is highly tunable via an application of a vertical electric field.It can then be observ... | [
{
"text": "Excitonic interconnects. M Baldo, V Stojanović, Nat. Photonics. 32009",
"doi": ""
},
{
"text": "Excitonic devices. L V Butov, Superlattices Microstruct. 1082017",
"doi": ""
},
{
"text": "Exciton optics, dynamics and transport in atomically thin semiconductors. R Perea-Causin, ... |
Fast micron-scale 3D printing with a resonant-scanning two photon microscope
Benjamin W Pearre
Department of Biology
Boston University
02215BostonMAUSA
Christos Michas
Department of Biomedical Engineering
Boston University
02215BostonMAUSA
Jean-Marc Tsang
Department of Biomedical Engineering
Boston University
0... | 4,951,227 | {
"arxiv": "1803.07135",
"mag": 2980218440,
"acl": null,
"pubmed": "32864346",
"pubmedcentral": 7450988,
"dblp": null,
"doi": "10.1016/j.addma.2019.100887"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1803.07135v1.pdf"
],
"pdf_sha": "a92aa888abf24af3e8926b1684ef85d54f9462a0",
"open_access": {
"license": "CCBYNCND",
"url": "https://doi.org/10.1016/j.addma.2019.100887",
"status": "HYBRID"
}
} | Fast micron-scale 3D printing with a resonant-scanning two photon microscope | [
"Benjamin W Pearre \nDepartment of Biology\nBoston University\n02215BostonMAUSA",
"Christos Michas \nDepartment of Biomedical Engineering\nBoston University\n02215BostonMAUSA",
"Jean-Marc Tsang \nDepartment of Biomedical Engineering\nBoston University\n02215BostonMAUSA",
"Timothy J Gardner \nDepartment of Bio... | 3D printing allows rapid fabrication of complex objects from digital designs. One 3D-printing process, direct laser writing, polymerises a light-sensitive material by steering a focused laser beam through the shape of the object to be created. The highest-resolution direct laser writing systems use a femtosecond laser,... | [
{
"title": "Introduction",
"content": "Direct laser writing (DLW) lithography [1] is a 3D-printing technology that can be used to fabricate small-scale objects with complex geometries by programmatically exposing a light-sensitive material to a focused laser beam [2][3][4][5][6]. Using femtosecond laser pul... | [
{
"id": "",
"caption": "Overview of the resonant rDLW printer. (a) Schematic of the optical path from laser source to printed object. Solid-outline box shows components comprising a standard resonant scanning 2-photon microscope. Dash-outline box shows zoomed view of build plate stack. (b) The raster scanne... | [
{
"text": "Three-dimensional microfabrication with two-photon absorbed photopolymerization. S Maruo, O Nakamura, S Kawata, 10.1364/OL.22.000132Opt. Lett. 22PubMed: 18183126Maruo S, Nakamura O, Kawata S, Three-dimensional microfabrication with two-photon absorbed photopolymerization, Opt. Lett 22 (1997) 132-134,... |
Multi-reference perturbation theory with Cholesky decomposition for the density matrix renormalization group. Supporting Information
Leon Freitag
Laboratorium für Physikalische Chemie
ETH Zürich
Vladimir-Prelog-Weg 28093ZürichSwitzerland
Stefan Knecht
Laboratorium für Physikalische Chemie
ETH Zürich
Vladimir-Prel... | 3,921,419 | {
"arxiv": "1608.02006",
"mag": 2520021794,
"acl": null,
"pubmed": "28094988",
"pubmedcentral": 5312874,
"dblp": null,
"doi": "10.1021/acs.jctc.6b00778"
} | {
"pdf_urls": [],
"pdf_sha": "bd43712c37d269fb03ff56bcbd58af7c79298772",
"open_access": {
"license": "CCBY",
"url": "https://pubs.acs.org/doi/pdf/10.1021/acs.jctc.6b00778",
"status": "HYBRID"
}
} | Multi-reference perturbation theory with Cholesky decomposition for the density matrix renormalization group. Supporting Information | [
"Leon Freitag \nLaboratorium für Physikalische Chemie\nETH Zürich\nVladimir-Prelog-Weg 28093ZürichSwitzerland",
"Stefan Knecht \nLaboratorium für Physikalische Chemie\nETH Zürich\nVladimir-Prelog-Weg 28093ZürichSwitzerland",
"Celestino Angeli \nDipartimento di Scienze Chimiche e Farmaceutiche\nUniversità di Fer... | [
{
"title": "I. ACTIVE SPACE SELECTION",
"content": "A. model 2 complex (14,18)), including four π, π * pairs on the ligands (denoted π 1 to π 4 and π * 1 to π * 4 ), five metal 3d orbitals and five second 3d shells (denoted 3d ) to account for the double-shell effect. A hybrid procedure including manual or... | [
{
"id": "",
"caption": ". S1 shows the active space chosen for all DMRG-NEVPT2 calculations of the model 2 complex. It comprises 14 electrons in 18 orbitals (denoted",
"content": "Fig\n. S1 shows the active space chosen for all DMRG-NEVPT2 calculations of the model 2 complex. It comprises 14 electrons i... | [
{
"text": ". C J Stein, M Reiher, J. Chem. Theory Comput. 121760C. J. Stein and M. Reiher, J. Chem. Theory Comput. 12, 1760 (2016).",
"doi": ""
},
{
"text": ". 2ö, J Legeza, Sólyom, Phys. Rev. B. 681951162Ö . Legeza and J. Sólyom, Phys. Rev. B 68, 195116 (2003).",
"doi": ""
},
{
"text": ... | |
Thermally activated diffusion and lattice relaxation in (Si)GeSn materials
Nils Von Den Driesch
JARA-Institut Green IT
RWTH Aachen
Sommerfeldstraße 1452074Germany
Peter Grünberg Institute (PGI
Stephan Wirths
Peter Grünberg Institute (PGI
Rene Troitsch
Peter Grünberg Institute (PGI
Gregor Mussler
Peter Grü... | 207,794,174 | {
"arxiv": "1911.02334",
"mag": 3104569966,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevmaterials.4.033604"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1911.02334v1.pdf"
],
"pdf_sha": "f7917f32317fee2154be0291727463efd368dda0",
"open_access": {
"license": "CCBY",
"url": "http://link.aps.org/pdf/10.1103/PhysRevMaterials.4.033604",
"status": "HYBRID"
}
} | Thermally activated diffusion and lattice relaxation in (Si)GeSn materials | [
"Nils Von Den Driesch \nJARA-Institut Green IT\nRWTH Aachen\nSommerfeldstraße 1452074Germany\n\nPeter Grünberg Institute (PGI",
"Stephan Wirths \nPeter Grünberg Institute (PGI",
"Rene Troitsch \nPeter Grünberg Institute (PGI",
"Gregor Mussler \nPeter Grünberg Institute (PGI",
"Uwe Breuer \nZentralinstitut f... | Germanium-tin (GeSn) alloys have emerged as a promising material for future optoelectronics, energy harvesting, and nanoelectronics owing to their direct band gap and compatibility with existing Si-based electronics. Yet, their metastability poses significant challenges calling for in-depth investigations of their ther... | [
{
"title": "I. INTRODUCTION",
"content": "The Si-Ge-Sn semiconductor system has sparked a great deal of interest among scientists because of its unique properties within group IV materials. The demonstrated fundamental direct band gap of this truly silicon-compatible material can pave the way for numerous n... | [
{
"id": "",
"caption": "Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.",
"content": "Publi... | [
{
"text": ". S Wirths, R Geiger, N Von Den Driesch, G Mussier, T Stoica, S Mantl, Z Ikonić, M Luysberg, S Chiussi, J.-M Hartmann, 10.1038/nphoton.2014.321Nat. Photon. 988S. Wirths, R. Geiger, N. von den Driesch, G. Mussier, T. Stoica, S. Mantl, Z. Ikonić, M. Luysberg, S. Chiussi, J.-M. Hartmann et al., Nat. Pho... |
A Joint Model for Word Embedding and Word Morphology
8 Jun 2016
Kris Cao
Computer Lab
University of Cambridge United Kingdom
Marek Rei
Computer Lab
University of Cambridge United Kingdom
A Joint Model for Word Embedding and Word Morphology
8 Jun 20169D4D4721C4B3ED9AF8B8AB952E5F5CEDarXiv:1606.02601v1[cs.CL]
This... | 7,511,759 | {
"arxiv": "1606.02601",
"mag": 2952928559,
"acl": "W16-1603",
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/rep4nlp/CaoR16",
"doi": "10.18653/v1/w16-1603"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1606.02601v1.pdf"
],
"pdf_sha": "2505d4bc7cc3d80a5567d6b7dc4b8b11ad898d0b",
"open_access": {
"license": "CCBY",
"url": "https://aclanthology.org/W16-1603.pdf",
"status": "HYBRID"
}
} | A Joint Model for Word Embedding and Word Morphology | [
"Kris Cao \nComputer Lab\nUniversity of Cambridge United Kingdom",
"Marek Rei \nComputer Lab\nUniversity of Cambridge United Kingdom"
] | This paper presents a joint model for performing unsupervised morphological analysis on words, and learning a character-level composition function from morphemes to word embeddings.Our model splits individual words into segments, and weights each segment according to its ability to predict context words.Our morphologic... | [
{
"title": "Introduction",
"content": "Word embedding models associate each word in a corpus with a vector in a semantic space.These vectors can either be learnt to optimize performance in a downstream task (Bengio et al., 2003;Collobert et al., 2011) or learnt via the distributional hypothesis: words with ... | [
{
"id": "",
"caption": "Figure 2: An illustration of Char2Vec.A bidirectional LSTM reads the word (start and end of word symbols represented by ˆand $ respectively), outputting a sequence of hidden states.These are then passed through a feed-forward layer (not shown), weighted by an attention model (the squ... | [
{
"text": "Improved transitionbased parsing by modeling characters instead of words with LSTMs. Baayen, ICLR. [Ballesteros et al.2015Linguistic Data Consortium. David A Balota, Melvin J Yap, Keith A Hutchison, Michael J Cortese, Brett Kessler, Bjorn Loftis, James H Neely, Douglas L Nelson, Greg B Simpson, Rebec... |
Cuspy and fractured black hole shadows in a toy model with axisymmetry
10 Jan 2022
Wei-Liang Qian
Center for Gravitation and Cosmology
College of Physical Science and Technology
Yangzhou University
225009YangzhouChina
Escola de Engenharia de Lorena
Universidade de São Paulo
12602-810LorenaSPBrazil
Institute for th... | 231,846,862 | {
"arxiv": "2102.03820",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1140/epjc/s10052-022-10033-5"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2102.03820v4.pdf"
],
"pdf_sha": "323659007b968448c45e2e74e91f52b6c41012d1",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Cuspy and fractured black hole shadows in a toy model with axisymmetry | [
"Wei-Liang Qian \nCenter for Gravitation and Cosmology\nCollege of Physical Science and Technology\nYangzhou University\n225009YangzhouChina\n\nEscola de Engenharia de Lorena\nUniversidade de São Paulo\n12602-810LorenaSPBrazil\n\nInstitute for theoretical physics and cosmology\nZhejiang University of Technology\n31... | Cuspy shadow was first reported for hairy rotating black holes, whose metrics deviate significantly from the Kerr one. The non-smooth edge of the shadow is attributed to a transition between different branches of unstable but bounded orbits, known as the fundamental photon orbits, which end up at the light rings.In sea... | [
{
"title": "I. INTRODUCTION",
"content": "The bending of light rays owing to the spacetime curvature constitutes one of the most influential predictions of General Relativity. At its extreme form, the shadow casted by a black hole [1][2][3][4][5][6] is widely considered as an essential observable in the ele... | [
{
"id": "",
"caption": "totic behavior of the metric, the FPOs are determined by spacetime properties in the vicinity of the horizon. Although all the FPOs for our metric are spherical orbits, the presence of a rotating thin shell leads to some interesting implications. In what follows, let us elaborate on ... | [
{
"text": "The Escape of Photons from Gravitationally Intense Stars. J L Synge, 10.1093/mnras/131.3.463Mon. Not. Roy. Astron. Soc. 131J. L. Synge, The Escape of Photons from Gravitationally Intense Stars, Mon. Not. Roy. Astron. Soc. 131 (1966) 463-466.",
"doi": "10.1093/mnras/131.3.463"
},
{
"text":... |
Your Day in Your Pocket: Complex Activity Recognition from Smartphone Accelerometers
Emma Bouton-Bessac
Lakmal Meegahapola
Daniel Gatica-Perez
Idiap Reseach Institute
Switzerland
EPFL
Switzerland
Your Day in Your Pocket: Complex Activity Recognition from Smartphone Accelerometers
smartphone sensinghuman activ... | 254,404,976 | {
"arxiv": "2301.06993",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/ph/Bouton-BessacMG22",
"doi": "10.48550/arxiv.2301.06993"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2301.06993v1.pdf"
],
"pdf_sha": "58d67ef4c5029408d75da113687ab460b84bc24d",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Your Day in Your Pocket: Complex Activity Recognition from Smartphone Accelerometers | [
"Emma Bouton-Bessac",
"Lakmal Meegahapola",
"Daniel Gatica-Perez",
"Idiap Reseach Institute\nSwitzerland",
"EPFL\nSwitzerland"
] | Human Activity Recognition (HAR) enables context-aware user experiences where mobile apps can alter content and interactions depending on user activities. Hence, smartphones have become valuable for HAR as they allow large, and diversified data collection. Although previous work in HAR managed to detect simple activiti... | [
{
"title": "Introduction",
"content": "In Human Activity Recognition (HAR), various human activities such as walking, running, sitting, [...], cooking, driving are recognized. The data can be collected from wearable sensors or accelerometer or through video frames or images [9]. HAR is possible thanks to se... | [
{
"id": "",
"caption": "distribution of the dataset. (b) Age distribution of the dataset.",
"content": "distribution of the dataset. (b) Age distribution of the dataset."
},
{
"id": "",
"caption": "Gender and age distribution of the dataset.",
"content": "Fig. 1 :\n1Gender and age distri... | [
{
"text": ". Final model of diversity. Final model of diversity, https://www.internetofus.eu/wp-content/uploads/ sites/38/2021/03/D1.3-Final-model-of-diversity.pdf",
"doi": ""
},
{
"text": "Human activity recognition: A review. O C Ann, L B Theng, 10.1109/ICCSCE.2014.7072750IEEE International Confer... |
Breathing mode of a quantum droplet in a quasi-one-dimensional dipolar Bose gas
April 17, 2024
E Orignac
Univ Lyon
Ens de Lyon
Laboratoire de Physique
CNRS
F-69342LyonFrance
S De Palo
CNR-IOM-Democritos National UDS
Via Bonomea 265I-34136TriesteItaly
Dipartimento di Fisica Teorica
Università Trieste
Strada Costi... | 266,844,334 | {
"arxiv": "2401.03918",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreva.109.043316"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2401.03918v2.pdf"
],
"pdf_sha": "b5187a0ebc37fabe7557a1c94d3b33a8c92adfa2",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Breathing mode of a quantum droplet in a quasi-one-dimensional dipolar Bose gas | [
"E Orignac \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nCNRS\nF-69342LyonFrance",
"S De Palo \nCNR-IOM-Democritos National UDS\nVia Bonomea 265I-34136TriesteItaly\n\nDipartimento di Fisica Teorica\nUniversità Trieste\nStrada Costiera 11I-34014TriesteItaly",
"L Salasnich \nDipartimento di Fisica e Astron... | We investigate the breathing mode and the stability of a quantum droplet in a tightly trapped onedimensional dipolar gas of bosonic atoms.When the droplet with a flat-top density profile is formed, the breathing mode frequency scales as the inverse of the number of atoms in the cloud.This is straightforwardly derived w... | [
{
"title": "I. INTRODUCTION",
"content": "In the last years the research activity on quantum droplets in atomic gases has become very intense and fruitful.The achievement of Bose-Einstein condensation of lanthanide atoms with large magnetic moments (like 164 Dy or 166 Er) 1 , where the competition of short ... | [
{
"id": "",
"caption": "FIG. 1.In panel a) we show the evolution of the density profile as the number of the particles in the cloud is increased for the case a1D = −7000a0, for some selected cases i.e.N = 50, 100, 200 and 400.In panel b) we show, for several scattering lengths a 1d , the ratio between the d... | [
{
"text": ". F F Lauriane Chomaz, Igor Ferrier-Barbut, B L L Bruno Laburthe-Tolra, T Pfau, Rep. Prog. Phys. 86264012023",
"doi": ""
},
{
"text": ". M Schmitt, M Wenzel, F Böttcher, I Ferrier-Barbut, T Pfau, Nature. 5392592016",
"doi": ""
},
{
"text": ". H Kadau, M Schmitt, M Wenzel, C Wi... |
Real-Time Elderly Monitoring for Senior Safety by Lightweight Human Action Recognition
Han Sun
Department of Electrical and Computer Engineering
Binghamton University
13902BinghamtonNYUSA
Yu Chen ychen@binghamton.edu
Department of Electrical and Computer Engineering
Binghamton University
13902BinghamtonNYUSA
Rea... | 250,660,850 | {
"arxiv": "2207.10519",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/ismict/Sun022",
"doi": "10.1109/ismict56646.2022.9828343"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2207.10519v1.pdf"
],
"pdf_sha": "c807f4408ac53f7510cce848b625ec7d74db85a8",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Real-Time Elderly Monitoring for Senior Safety by Lightweight Human Action Recognition | [
"Han Sun \nDepartment of Electrical and Computer Engineering\nBinghamton University\n13902BinghamtonNYUSA",
"Yu Chen ychen@binghamton.edu \nDepartment of Electrical and Computer Engineering\nBinghamton University\n13902BinghamtonNYUSA"
] | With an increasing number of elders living alone, care-giving from a distance becomes a compelling need, particularly for safety. Real-time monitoring and action recognition are essential to raise an alert timely when abnormal behaviors or unusual activities occur. While wearable sensors are widely recognized as a prom... | [
{
"title": "I. INTRODUCTION",
"content": "According to the U.S. Department of Health and Human Services (HHS), the population age 65 and over has increased from 37.2 million in 2006 to 49.2 million in 2016, and is is expected to nearly double to 98 million in 2060 [19]. Of these, about 28% (13.8 million) of... | [
{
"id": "",
"caption": "An overview of the REMS scheme.",
"content": "Fig. 1 .\n1An overview of the REMS scheme."
},
{
"id": "",
"caption": "Configuration of the 20 joint in skeleton image, The labels of the joints are: 1.hip center, 2.middle-spine, 3.shoulder center, 4.Head, 5.Leftshoulder,... | [
{
"text": "Validation of accuracy of svm-based fall detection system using real-world fall and non-fall datasets. O Aziz, J Klenk, L Schwickert, L Chiari, C Becker, E J Park, G Mori, S N Robinovitch, PLoS one. 127180318O. Aziz, J. Klenk, L. Schwickert, L. Chiari, C. Becker, E. J. Park, G. Mori, and S. N. Robino... |
Recent developments in quarkonium and open flavour production calculations 1 Heavy Quarkonium Production 1.1 Introduction
8 Sep 2011
Mathias Butenschoen
II. Institut für Theoretische Physik
Universität Hamburg Luruper Chaussee 149
22761HamburgGermany
Recent developments in quarkonium and open flavour production cal... | 116,934,452 | {
"arxiv": "1109.1740",
"mag": 2963659567,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1109.1740v1.pdf"
],
"pdf_sha": "0f166bce7629dcb983c6e7cbbf34baebb51f4148",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Recent developments in quarkonium and open flavour production calculations 1 Heavy Quarkonium Production 1.1 Introduction | [
"Mathias Butenschoen \nII. Institut für Theoretische Physik\nUniversität Hamburg Luruper Chaussee 149\n22761HamburgGermany"
] | This report reviews recent theory progress in the field of heavy quarkonium and open heavy flavour production calculations.The calculation of next-to-leading order (NLO) corrections to the short distance cross sections of the intermediate CO states, especially to the 3 P [8] J states, have proven challenging. But as fo... | [
{
"title": "Heavy Quarkonium Production",
"content": ""
},
{
"title": "Introduction",
"content": "Heavy quarkonia are bound states of a heavy quark and its antiquark. There are charmonia and bottomonia. According to the factorization theorem of nonrelativistic QCD (NRQCD) [1], the cross section ... | [
{
"id": "",
"caption": "Plots u-z (continuation): Results of global fit[9] compared to H1[20,21] data. The blue bands are the color singlet model predictions, the yellow bands include the color octet contributions.Set A: Do not mind feed-downs Set B: Subtract feed-downs first O J/ψ ( 1 S 97 ± 0.44) × 10 −2 ... | [
{
"text": ". G T Bodwin, E Braaten, G P Lepage, Phys. Rev. D. 511125Erratum-ibid. D 55, 5853 (1997)G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Erratum-ibid. D 55, 5853 (1997)].",
"doi": ""
},
{
"text": ". A Petrelli, M Cacciari, M Greco, F Maltoni, M L Mangano, Nucl. Phy... |
A few points suffice: Efficient large-scale computation of brain voxel-wise functional connectomes from a sparse spatio-temporal point-process
Enzo Tagliazucchi
Department of Neurology and Brain Imaging Center
Goethe University
Frankfurt am MainGermany
Helmut Laufs
Department of Neurology and Brain Imaging Center... | 17,905,138 | {
"arxiv": "1409.6378",
"mag": 2279919164,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1409.6378v1.pdf"
],
"pdf_sha": "bd75c0e75173689d00968cee3ca308f4495f3ab6",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A few points suffice: Efficient large-scale computation of brain voxel-wise functional connectomes from a sparse spatio-temporal point-process | [
"Enzo Tagliazucchi \nDepartment of Neurology and Brain Imaging Center\nGoethe University\nFrankfurt am MainGermany",
"Helmut Laufs \nDepartment of Neurology and Brain Imaging Center\nGoethe University\nFrankfurt am MainGermany\n\nDepartment of Neurology\nUniversity Hospital Schleswig Holstein\nKielGermany",
"Da... | Large efforts are currently under way to systematically map functional connectivity between all pairs of millimeter-scale brain regions using big volumes of neuroimaging data. Functional magnetic resonance imaging (fMRI) can produce these functional connectomes, however, large amounts of data and lengthy computation ti... | [
{
"title": "I. INTRODUCTION",
"content": "The human brain comprises an interconnected network of cortical and sub-cortical regions globally linked by long-range tracts of anatomical connections. The mapping of such connections at a particular spatial scale (dubbed connectome in contemporary neuroscience; Sp... | [
{
"id": "",
"caption": "Eguiluz et al., 2005, Salvador et al., 2005, Sporns et al., 2004) by computing the statistical covariance between all pairs of Blood Oxygen Level Dependent (BOLD) signals. This functional connectome contains dynamical information on how all pairs of regions (at a certain spatial scal... | [
{
"text": "Hubs of brain functional networks are radically reorganized in comatose patients. S Achard, C Delon-Martin, P E Vrtes, F Renard, M Schenck, F Schneider, C Heinrich, S Kremer, E T Bullmore, Proc. Natl. Acad. Sci. USA. 10950Achard, S., Delon-Martin, C., Vrtes, P. E., Renard, F., Schenck, M., Schneider,... |
A law of motion for spherical shells in special relativity
200x
L Zaninetti
Dipartimento di Fisica Generale
Università degli Studi di Torino
via P. Giuria 110125TorinoItaly
A law of motion for spherical shells in special relativity
Adv. Studies Theor. Phys
xx200x4775+f5227Ny9838MzKeywords: Relativistic fluid dynam... | 14,484,981 | {
"arxiv": "1011.2912",
"mag": 2098617332,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [],
"pdf_sha": "1538a0792f5eea7fa84a29d8c5a20e98ccb0da0f",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A law of motion for spherical shells in special relativity | [
"L Zaninetti \nDipartimento di Fisica Generale\nUniversità degli Studi di Torino\nvia P. Giuria 110125TorinoItaly"
] | Self-similar solutions to the problem of a special relativistic law of motion for thin shells of matter are calculated. These solutions represent the special relativistic generalization of momentum conservation for the thin layer approximation in classical physics. The analytical and numerical results are applied to Su... | [
{
"title": "Introduction",
"content": "The study of the relativistic dynamics of thin shells of matter is a current subject of investigation when the framework of general relativity is adopted [1,2,3,4]. Here, we will explore how the framework of special relativity can produce a law of motion which can be c... | [
{
"id": "",
"caption": "Velocity as a function of radius when R 0 = 1, c = 1, v 0 /c = 0.99.",
"content": "Figure 1 :\n1Velocity as a function of radius when R 0 = 1, c = 1, v 0 /c = 0.99."
},
{
"id": "",
"caption": "Plot of the hypergeometric function 2 F 1 (−1/2, 1/6; 7/6; x) as a function... | [
{
"text": "Singular hypersurfaces and thin shells in general relativity. W Israel, Nuovo Cimento B Serie. 44W. Israel, Singular hypersurfaces and thin shells in general relativity, Nuovo Cimento B Serie 44 (1966), 1-14.",
"doi": ""
},
{
"text": "Gravitational Collapse and Causality. W Israel, Physic... |
27 Mar 2020
27 Mar 2020SURGERY OPERATIONS TO FOLD MAPS TO CONSTRUCT FOLD MAPS WHOSE SINGULAR VALUE SETS MAY HAVE CROSSINGS NAOKI KITAZAWAand phrases Singularities of differentiable maps; generic maps Differential topol- ogy Reeb spaces
Constructing Morse functions and their higher dimensional versions or fold maps is... | 212,634,788 | {
"arxiv": "2003.04147",
"mag": 3009800092,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2003.04147v4.pdf"
],
"pdf_sha": "b251a86051101fb4d0cbc6d595d592d66e7b29c7",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | [] | Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions, fold maps and more general generic maps. It is one of important and interesting bra... | [
{
"title": "2.",
"content": "Bubbling operations and fold maps such that preimages of regular values are disjoint unions of spheres."
},
{
"title": "2.1.",
"content": "Definitions and fundamental properties of Reeb spaces and bubbling operations. Throughout this paper, let m > n ≥ 1 be integers,... | [
{
"id": "",
"caption": "The set of all the crossings of the family {c j | Sj : S j → P } l j=1 of the immersions is a finite set.(4) Let {p j ′ } l ′ j ′ =1 be the set of all the crossings of the family {c j | Sj : S j → P } l j=1 of the immersions. The set of all crossings of the family {c j | ∂N (Sj ) :",... | [
{
"text": "Stable mappings and their singularities. M Golubitsky, V Guillemin, Graduate Texts in Mathematics. 14Springer-VerlagM. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics (14), Springer-Verlag (1974).",
"doi": ""
},
{
"text": "On round fold ... | |
The BCS theory of superconductivity revisited: quasiparticle imbalance in equilibrium and multiple solutions of the energy gap equation
26 Sep 2016
Dragoş-Victor Anghel
Horia Hulubei National Institute for Physics and Nuclear Engineering
P.O. Box MG-6077126Mȃgurele, IlfovRomania
The BCS theory of superconductivity ... | 119,275,861 | {
"arxiv": "1609.07931",
"mag": 2526196314,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1609.07931v3.pdf"
],
"pdf_sha": "a4c9fc528e7c320c1919d47a81f6000baae20341",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | The BCS theory of superconductivity revisited: quasiparticle imbalance in equilibrium and multiple solutions of the energy gap equation | [
"Dragoş-Victor Anghel \nHoria Hulubei National Institute for Physics and Nuclear Engineering\nP.O. Box MG-6077126Mȃgurele, IlfovRomania",
"Dragoş-Victor Anghel \nHoria Hulubei National Institute for Physics and Nuclear Engineering\nP.O. Box MG-6077126Mȃgurele, IlfovRomania"
] | We examine the equilibrium solutions of the BCS theory of superconductivity in the low temperature limit, allowing the attraction band to be asymmetric with respect to the chemical potential of the system µR. If we denote by µ the middle of the attraction band, we observe that the superconducting phase is formed only i... | [
{
"title": "INTRODUCTION",
"content": ""
},
{
"title": "INTRODUCTION",
"content": "In Ref. [1], the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [2,3] has been revisited under the assumption that the attraction band is asymmetric with respect to the chemical potential of the syste... | [
{
"id": "",
"caption": "(Color online) The r.h.s. of Eq. (8b), for −1 ≤ ab ≤ 0, continued by (9b), for ab ≤ −1, plotted vs. b, for different values of a. The straight black line is b vs b.",
"content": "FIG. 1 .\n1(Color online) The r.h.s. of Eq. (8b), for −1 ≤ ab ≤ 0, continued by (9b), for ab ≤ −1, pl... | [
{
"text": ". D V Anghel, G A Nemnes, Physica A. 46474D. V. Anghel and G. A. Nemnes. Physica A, 464:74, 2016.",
"doi": ""
},
{
"text": ". J Bardeen, L N Cooper, J R Schrieffer, Phys. Rev. 1081175J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Phys. Rev., 108:1175, 1957.",
"doi": ""
},
{
"... |
Employing Layerwised Unsupervised Learning to Lessen Data and Loss Requirements in Forward-Forward Algorithms
23 Apr 2024
Taewook Hwang taewook5295@gmail.com
Department of Computer Science and Engineering
Chungnam National University
Hyein Seo
Department of Computer Science and Engineering
Chungnam National Unive... | 269,302,558 | {
"arxiv": "2404.14664",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2404-14664",
"doi": "10.48550/arxiv.2404.14664"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2404.14664v1.pdf"
],
"pdf_sha": "bc8a56cbf8cad1cfa6c47fe5b647e5841c8a26c8",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Employing Layerwised Unsupervised Learning to Lessen Data and Loss Requirements in Forward-Forward Algorithms | [
"Taewook Hwang taewook5295@gmail.com \nDepartment of Computer Science and Engineering\nChungnam National University",
"Hyein Seo \nDepartment of Computer Science and Engineering\nChungnam National University",
"Sangkeun Jung \nDepartment of Computer Science and Engineering\nChungnam National University"
] | Recent deep learning models such as ChatGPT utilizing the back-propagation algorithm have exhibited remarkable performance.However, the disparity between the biological brain processes and the back-propagation algorithm has been noted.The Forward-Forward algorithm, which trains deep learning models solely through the f... | [
{
"title": "Introduction",
"content": "Recently, large language models, including ChatGPT, have demonstrated performance surpassing that of human capabilities.This achievement is largely attributed to the back-propagation algorithm(BP) [Kelley, 1960;Linnainmaa, 1970], a highly effective learning algorithm, ... | [
{
"id": "",
"caption": "Figure 1: Using Data and Forward-Forward algorithm and Unsupervised learning Forward-Forward model.",
"content": "Figure 1 :\n1\nFigure 1: Using Data and Forward-Forward algorithm and Unsupervised learning Forward-Forward model."
},
{
"id": "",
"caption": "Figure 2: U... | [
{
"text": "Ba, arXiv:1607.06450Lukas Biewald et al. Experiment tracking with weights and biases. Software available from wandb. com. 2016. 2016. 20202233Layer normalization. arXiv preprintBiewald and others, 2020",
"doi": ""
},
{
"text": "The recent excitement about neural networks. Francis Crick, ;... |
Simple computer program to calculate arbitrary tightly focused (propagating and evanescent) vector light fields
Isael Herrera
Instituto de Ciencias Nucleares
Universidad Nacional Autónoma de México
Apartado Postal 70-54304510Cd. MxMéxico
Pedro A Quinto-Su
Instituto de Ciencias Nucleares
Universidad Nacional Autón... | 253,510,549 | {
"arxiv": "2211.06725",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
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"https://export.arxiv.org/pdf/2211.06725v1.pdf"
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"pdf_sha": "ca5531217beefabd7e379c7884e024cbf616a192",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Simple computer program to calculate arbitrary tightly focused (propagating and evanescent) vector light fields | [
"Isael Herrera \nInstituto de Ciencias Nucleares\nUniversidad Nacional Autónoma de México\nApartado Postal 70-54304510Cd. MxMéxico",
"Pedro A Quinto-Su \nInstituto de Ciencias Nucleares\nUniversidad Nacional Autónoma de México\nApartado Postal 70-54304510Cd. MxMéxico"
] | [
{
"title": "INTRODUCTION",
"content": "In general, tightly focused light has significant polarization components in the 3 spatial directions and can be described by the vectorial model that Richards and Wolf developed in 1956 [1]. Their model has become even more relevant, as some of the most important opti... | [
{
"id": "",
"caption": "FIG. 1. Representation of the focusing model. A paraxial beam propagating in a medium with refractive index n1, incident into lens that focuses at f in a medium with a refractive index n2. The lens is represented by a spherical surface of reference centered at the focal point with ra... | [
{
"text": "* Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro. * Current address: Fresnel Institute, isael.herrera@fresnel.fr † pedro.quinto@nucleares.unam.mx * Current address: Fresnel Institute,... | |
Building Adversary-Resistant Deep Neural Networks without Security through Obscurity
2016. July 2017
D C Washington
Usa
Building Adversary-Resistant Deep Neural Networks without Security through Obscurity
Proceedings of ACM Conference
ACM ConferenceWashington, DC, USA122016. July 201710.1145/nnnnnnn.nnnnnnn978-x-x... | 17,676,338 | {
"arxiv": "1612.01401",
"mag": null,
"acl": null,
"pubmed": null,
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"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1612.01401v2.pdf"
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"open_access": {
"license": null,
"url": null,
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} | Building Adversary-Resistant Deep Neural Networks without Security through Obscurity | [
"D C Washington",
"Usa",
"D C Washington",
"Usa"
] | Deep neural networks (DNNs) have proven to be quite e ective in a vast array of machine learning tasks, with recent examples in cyber security and autonomous vehicles. Despite the superior performance of DNNs in these applications, it has been recently shown that these models are susceptible to a particular type of a a... | [
{
"title": "INTRODUCTION",
"content": ""
},
{
"title": "INTRODUCTION",
"content": "Beyond highly publicized victories in automatic game-playing as in Go [34], there have been many successful applications of deep neural networks (DNN) in image and speech recognition. Recent explorations and appli... | [
{
"id": "",
"caption": "Legitimate samples and their corresponding adversarial samples generated by following various approaches. e adversarial samples are cra ed by introducing to the legitimate samples pertubations nearly indistinguishable to the human eyes.",
"content": "Figure 2 :\n2Legitimate sampl... | [
{
"text": "Deep learning with non-medical training used for chest pathology identi cation. Y Bar, I Diamant, L Wolf, H Greenspan, SPIE Medical Imaging, pages 94140V-94140V. International Society for Optics and Photonics. Y. Bar, I. Diamant, L. Wolf, and H. Greenspan. Deep learning with non-medical training used... |
Bayesian inference of the fluctuating proton shape
Heikki Mäntysaari
Department of Physics
University of Jyväskylä
University of Jyväskylä
P.O. Box 3540014Finland
Helsinki Institute of Physics
University of Helsinki
P.O. Box 6400014Finland
Björn Schenke
Physics Department
Brookhaven National Laboratory
Upton1197... | 246,608,220 | {
"arxiv": "2202.01998",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.physletb.2022.137348"
} | {
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"https://export.arxiv.org/pdf/2202.01998v1.pdf"
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"pdf_sha": "499b51a3bacec895f368bebb595398fffd75dc20",
"open_access": {
"license": null,
"url": null,
"status": null
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} | Bayesian inference of the fluctuating proton shape | [
"Heikki Mäntysaari \nDepartment of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland",
"Björn Schenke \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA",
"Chun Shen \nDepartment... | Using Bayesian inference, we determine probabilistic constraints on the parameters describing the fluctuating structure of protons at high energy. We employ the color glass condensate framework supplemented with a model for the spatial structure of the proton, along with experimental data from the ZEUS and H1 Collabora... | [
{
"title": "I. INTRODUCTION",
"content": ""
},
{
"title": "I. INTRODUCTION",
"content": "Extracting the multi-dimensional structure of protons and nuclei is one of the main goals of future Deep Inelastic Scattering (DIS) facilities such as the Electron-Ion Collider [1,2], LHeC/FC-he [3] and EicC... | [
{
"id": "",
"caption": "Bayesian posterior distributions of the model parameters. The diagonal panels show the probability distributions for individual parameters, and off-diagonal panels illustrate their pairwise correlations.",
"content": "FIG. 1 .\n1Bayesian posterior distributions of the model param... | [
{
"text": "R , Abdul Khalek, arXiv:2103.05419Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report. physics.ins-detR. Abdul Khalek et. al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, arXiv:2103.05419 [physics.ins-det].",
... |
Slide FFT on a homogeneous mesh in wafer-scale computing
January 12, 2024
Maurice H P M Van Putten
Physics and Astronomy
Sejong University
SeoulSouth Korea
INAF-OAS Bologna
via P. Gobetti, 101I-40129BolognaItaly
Leighton Wilson
1237 E. Arques Ave Sunnyvale94085CerebrasCA
Adam W Lavely
Lawrence Berkeley National... | 266,933,113 | {
"arxiv": "2401.05427",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2401-05427",
"doi": "10.48550/arxiv.2401.05427"
} | {
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"https://export.arxiv.org/pdf/2401.05427v1.pdf"
],
"pdf_sha": "7d36009d8f2ec065935e642c077c087c53515d86",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Slide FFT on a homogeneous mesh in wafer-scale computing | [
"Maurice H P M Van Putten \nPhysics and Astronomy\nSejong University\nSeoulSouth Korea\n\nINAF-OAS Bologna\nvia P. Gobetti, 101I-40129BolognaItaly",
"Leighton Wilson \n1237 E. Arques Ave Sunnyvale94085CerebrasCA",
"Adam W Lavely \nLawrence Berkeley National Laboratory\n1 Cyclotron Rd Berke-ley94720CA",
"Mark ... | Searches for signals at low signal-to-noise ratios frequently involve the Fast Fourier Transform (FFT).For high-throughput searches, we here consider FFT on the homogeneous mesh of Processing Elements (PEs) of a wafer-scale engine (WSE).To minimize memory overhead in the inherently non-local FFT algorithm, we introduce... | [
{
"title": "I. INTRODUCTION",
"content": "The Fast Fourier Transform (FFT) is central to signal processing and high-performance computing [9].Highthroughput signal analysis hereby critically depends on FFT performance.The FFT algorithm optimizes efficiency in compute, notably in the Cooley-Tukey algorithm [... | [
{
"id": "",
"caption": "FIG. 1. FFT performance on a GPU (Radeon VII with HBM2) by efficiency (left scale) and throughput (right scale) in matched filtering, expressed by the product of transform size N = 2 m and FFT transform rate.Results are computed with clFFT [5] in complex single-precision and out-of-p... | [
{
"text": ". M A Abchouyet, M H P M Van Putten, Lorenzo Amati, ApJ. 9521572023",
"doi": ""
},
{
"text": ". M Abernathy, ET-0106C-10ET Science Team). 2011Tech. Rep.",
"doi": ""
},
{
"text": ". R Abbott, LIGO ScientificApJ. 913L72021",
"doi": ""
},
{
"text": ". T Akutsu, KAGRAN... |
Computation of the thermal conductivity using classical and quantum molecular dynamics based methods
9 Jul 2013 (Dated: July 8, 2013)
Natalia Bedoya
Jean-Louis Barrat
David Rodney
Laboratory for Interdisciplinary Physics
Nanosciences Foundation
23 rue des martyrs38000GrenobleFrance
UMR 5588
Université Grenoble
... | 117,891,176 | {
"arxiv": "1307.2409",
"mag": 1506879312,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1307.2409v1.pdf"
],
"pdf_sha": "88da6b65983997aaf47fe43c0af8c33503902e15",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Computation of the thermal conductivity using classical and quantum molecular dynamics based methods | [
"Natalia Bedoya",
"Jean-Louis Barrat",
"David Rodney",
"Laboratory for Interdisciplinary Physics\nNanosciences Foundation\n23 rue des martyrs38000GrenobleFrance",
"UMR 5588\nUniversité Grenoble",
"SIMaP, Grenoble INP, UJF, CNRS, UMR 5266\nCNRS\nSaint Martin d'Hères38402, F-38401Saint Martin d'HèresFrance,... | The thermal conductivity of a model for solid argon is investigated using nonequilibrium molecular dynamics methods, as well as the traditional Boltzmann transport equation approach with input from molecular dynamics calculations, both with classical and quantum thermostats. A surprising result is that, at low temperat... | [
{
"title": "I. INTRODUCTION",
"content": "Different methods are available for calculating thermal conductivities of crystalline solids. [1] The most standard approach involves a calculation of the phonon properties of the system, which are connected to the thermal conductivity through the Boltzmann transpor... | [
{
"id": "",
"caption": "energy distribution per mode obtained from MD simulations. Left top panel: values obtained using a noise with power spectral densityΘA(ω) equal to the Bose-Einstein distributionΘ(ω) [Eq.(11)]. Right top panel: values obtained using a noise with power spectral densitỹ ΘB(ω). Bottom pa... | [
{
"text": ". S Stackhouse, L Stixrude, Rev. Mineral. Geochem. 71253S. Stackhouse and L. Stixrude, Rev. Mineral. Geochem. 71, 253 (2010).",
"doi": ""
},
{
"text": ". P Jund, R Jullien, Phys. Rev. B. 5913707P. Jund and R. Jullien, Phys. Rev. B 59, 13707 (1999).",
"doi": ""
},
{
"text": ". ... |
Some Mader-perfect graph classes
12 Oct 2022 October 13, 2022
Hui Lei
School of Mathematical Sciences and LPMC
Nankai University
300071TianjinChina
Siyan Li
Center for Combinatorics and LPMC
Nankai University
300071TianjinChina
Xiaopan Lian xiaopanlian@mail.nankai.edu.cn
Center for Combinatorics and LPMC
Nankai ... | 252,846,460 | {
"arxiv": "2210.06247",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/amc/LangLLLW23",
"doi": "10.1016/j.amc.2023.127968"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2210.06247v1.pdf"
],
"pdf_sha": "fd436a3d51febd5f86b2539c656c8404a4ad44c6",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Some Mader-perfect graph classes | [
"Hui Lei \nSchool of Mathematical Sciences and LPMC\nNankai University\n300071TianjinChina",
"Siyan Li \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina",
"Xiaopan Lian xiaopanlian@mail.nankai.edu.cn \nCenter for Combinatorics and LPMC\nNankai University\n300071TianjinChina",
"Susu Wa... | The dichromatic number of D, denoted by − → χ (D), is the smallest integer k such that D admits an acyclic k-coloring. We use mader χ (F ) to denote the smallest integer kWe extend octi digraphs to a larger class of digraphs and prove that it is Mader-perfect, which generalizes a result of Gishboliner, Steiner and Szab... | [
{
"title": "Introduction",
"content": "Let γ be a (di)graph parameter. Given a (di)graph H, we use mader γ (H) to denote the smallest integer such that if γ(G) ≥ mader γ (H), then G contains a subdivision of H. There are many results and open problems about the relation between some graph parameters and the... | [
{
"id": "",
"caption": "[1] Let F be a digraph with n vertices and m arcs. Then mader χ (F ) ≤ 4 m (n − 1) + 1.",
"content": "Theorem 1 . 2 .\n12[1] Let F be a digraph with n vertices and m arcs. Then mader χ (F ) ≤ 4 m (n − 1) + 1."
},
{
"id": "",
"caption": "Theorem 1.3. [7] For every k ∈ ... | [
{
"text": "Subdivisions in digraphs of large out-degree or large dichromatic number. P Aboulker, N Cohen, F Havet, W Lochet, P S Moura, S Thomassé, Electron. J. Comb. 26P. Aboulker, N. Cohen, F. Havet, W. Lochet, P.S. Moura, S. Thomassé, Subdivisions in digraphs of large out-degree or large dichromatic number, ... |
Structure of Business Firm Networks and Scale-Free Models
30 Oct 2008
Maksim Kitsak
Center for Polymer Studies
Boston University
02215BostonMassachusettsUSA
Massimo Riccaboni
CUSAS
University of Florence
FlorenceItaly
Shlomo Havlin
Center for Polymer Studies
Boston University
02215BostonMassachusettsUSA
Departm... | 55,194,194 | {
"arxiv": "0810.5514",
"mag": 1844516428,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0810.5514v1.pdf"
],
"pdf_sha": "81887a54344cbb8e6b484c038459909e9538a8cf",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Structure of Business Firm Networks and Scale-Free Models | [
"Maksim Kitsak \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA",
"Massimo Riccaboni \nCUSAS\nUniversity of Florence\nFlorenceItaly",
"Shlomo Havlin \nCenter for Polymer Studies\nBoston University\n02215BostonMassachusettsUSA\n\nDepartment of Physics\nMinerva Center\nBar-Ilan Univers... | We study the structure of business firm networks and scale-free models with degree distribution P (q) ∝ (q+c) −λ using the method of k-shell decomposition. We find that the Life Sciences industry network consists of three components: a "nucleus," which is a small well connected subgraph, "tendrils," which are small sub... | [
{
"title": "I. INTRODUCTION",
"content": "Many real-world complex systems are often described using the representation of graphs or networks, as sets of nodes connected by links [1,2,3,4,5,6,7]. Networks appear in various areas of science such as physics, biology, computer sciences, economics and sociology ... | [
{
"id": "",
"caption": "analyze the k-shell structure of LS and ICT industrial sectors in time periods between 1990 and 2002 and between 1990 and 2000 respectively. The LS network expanded linearly since the mid-1970s while the ICT network took off in the 1990s and grew exponentially for a decade [see Fig. ... | [
{
"text": ". P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6290P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 6, 290 (1959). I",
"doi": ""
},
{
"text": ". P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 517P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960).... |
21 Dec 2005
Jean-Philippe Préaux
GROUPESÀ CLASSES DE CONJUGAISON INFINIES : QUELQUES EXEMPLES
21 Dec 2005
We consider the group property of being with infinite conjugacy classes (or icc, i.e. = 1 and of which all conjugacy classes except 1 are infinite) in some particular cases, and more particulary in case of spe... | 118,090,407 | {
"arxiv": "math/0512505",
"mag": 1655438677,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/math/0512505v1.pdf"
],
"pdf_sha": "dedbaf4fc14f7ea4beddc941cad9b66815dc4b5a",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | [
"Jean-Philippe Préaux \nGROUPESÀ CLASSES DE CONJUGAISON INFINIES : QUELQUES EXEMPLES"
] | We consider the group property of being with infinite conjugacy classes (or icc, i.e. = 1 and of which all conjugacy classes except 1 are infinite) in some particular cases, and more particulary in case of specific elementary algebraic constructions or of some 3-manifold fundamental groups. Résumé. Nousétudions la prop... | [
{
"title": "Introduction",
"content": "La propriété de groupe d'êtreà classes de conjugaison infinies est particulièrement simpleàénoncer (voir la définition section 2). Elle est motivée par la caractérisation de Murray et Von-Neumann des algèbres de Von-Neumann qui sont des facteurs de type II-1 (cf. [ROIV... | [
{
"id": "",
"caption": "Alors avec le théorème d'écriture sous forme réduite, un mot réduit non trivial sur a et b est non trivial dans A * B, et donc a et b engendrent un groupe libre de rang 2.",
"content": "Alors avec le théorème d'écriture sous forme réduite, un mot réduit non trivial sur a et b est... | [
{
"text": "R Benedetti, C Petronio, Lectures on hyperbolic geometry. SpringerUniversitextR.Benedetti, and C.Petronio, Lectures on hyperbolic geometry, Universitext, Springer, 1992.",
"doi": ""
},
{
"text": "Infinite conjugacy classes in groups acting on trees. Y De Cornulier, Y.de Cornulier, Infinit... | |
NATURAL CONNECTIONS ON CONFORMAL RIEMANNIAN P -MANIFOLDS
14 Sep 2011
Dobrinka Gribacheva
Dimitar Mekerov
NATURAL CONNECTIONS ON CONFORMAL RIEMANNIAN P -MANIFOLDS
14 Sep 2011Riemannian almost product manifoldRiemannian metricalmost product structurelinear connectionparallel torsion 2010 Mathematics Subject Classific... | 119,167,696 | {
"arxiv": "1109.2704",
"mag": 2150615818,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1109.2704v2.pdf"
],
"pdf_sha": "9a31ca0422355fcf71216c0a7483156f1bdf6aa4",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | NATURAL CONNECTIONS ON CONFORMAL RIEMANNIAN P -MANIFOLDS | [
"Dobrinka Gribacheva",
"Dimitar Mekerov"
] | The class W 1 of conformal Riemannian P -manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kähler manifolds in almost Hermitian geometry. In the p... | [
{
"title": "Introduction",
"content": "K. Yano initiated in [1] the study of Riemannian almost product manifolds. A Riemannian almost product manifold (M, P, g) is a differentiable manifold M with an almost product structure P and a Riemannian metric g such that P 2 x = x and g(P x, P y) = g(x, y) for any t... | [
{
"id": "",
"caption": "The following relations are valid ρ(y, z) = ρ ′ (y, z) − {(2n − 1)g(p, p) + trS ′ − g(q, q)} g(y, z)",
"content": "Corollary 3. 2 .\n2The following relations are valid ρ(y, z) = ρ ′ (y, z) − {(2n − 1)g(p, p) + trS ′ − g(q, q)} g(y, z)"
},
{
"id": "",
"caption": "The c... | [
{
"text": ". K Yano, Pure and Applied Math. 49Pergamon Press BookYano K. Pure and Applied Math., 49, New York, Pergamon Press Book, 1965.",
"doi": ""
},
{
"text": ". A M Naveira, Rend, Math, 3Naveira A. M. Rend. Math., 3, 1983, 577-592.",
"doi": ""
},
{
"text": ". A Gray, L Hervella, Ann... |
Instance Segmentation with Cross-Modal Consistency
Alex Zihao Zhu
Vincent Casser
Reza Mahjourian
Henrik Kretzschmar
Sören Pirk
Instance Segmentation with Cross-Modal Consistency
Segmenting object instances is a key task in machine perception, with safety-critical applications in robotics and autonomous driving... | 252,917,933 | {
"arxiv": "2210.08113",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/iros/ZhuCMKP22",
"doi": "10.1109/iros47612.2022.9982285"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2210.08113v1.pdf"
],
"pdf_sha": "3ac7c0184308b20b093f9c7748cb8f47d543d3a5",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Instance Segmentation with Cross-Modal Consistency | [
"Alex Zihao Zhu",
"Vincent Casser",
"Reza Mahjourian",
"Henrik Kretzschmar",
"Sören Pirk"
] | Segmenting object instances is a key task in machine perception, with safety-critical applications in robotics and autonomous driving. We introduce a novel approach to instance segmentation that jointly leverages measurements from multiple sensor modalities, such as cameras and LiDAR. Our method learns to predict embed... | [
{
"title": "I. INTRODUCTION",
"content": "Identifying and segmenting objects is a fundamental and challenging task in computer vision, and critically important for many robotics applications, such as autonomous driving. Given some sensor input such as a camera image or LiDAR scan, the goal of instance segme... | [
{
"id": "",
"caption": "Example of our sampling strategy where points are evenly distributed amongst instances. The bottom row shows close-ups of the used sample locations.",
"content": "Fig. 3 :\n3Example of our sampling strategy where points are evenly distributed amongst instances. The bottom row sho... | [
{
"text": "A Arnab, S Jayasumana, S Zheng, P Torr, Higher order conditional random fields in deep neural networks. In ECCV. A. Arnab, S. Jayasumana, S. Zheng, and P. Torr. Higher order conditional random fields in deep neural networks. In ECCV, 2016.",
"doi": ""
},
{
"text": "Pixelwise instance segm... |
Theory of Diffraction by Holes of Arbitrary Sizes
Mostafa Behtouei
INFN
Laboratori Nazionali di Frascati
P.O. Box 13I-00044FrascatiItaly
Dipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)
Sapienza University of Rome
RomeItaly
Luigi Faillace
INFN
Laboratori Nazionali di Frascati
P.O. Box 13I-0004... | 251,040,279 | {
"arxiv": "2207.11551",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2207.11551v1.pdf"
],
"pdf_sha": "86f856eea776e2723c1473aa7bef935e9125cc7b",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Theory of Diffraction by Holes of Arbitrary Sizes | [
"Mostafa Behtouei \nINFN\nLaboratori Nazionali di Frascati\nP.O. Box 13I-00044FrascatiItaly\n\nDipartimento di Scienze di Base e Applicate per l'Ingegneria (SBAI)\nSapienza University of Rome\nRomeItaly",
"Luigi Faillace \nINFN\nLaboratori Nazionali di Frascati\nP.O. Box 13I-00044FrascatiItaly",
"Mauro Migliora... | New high-gradient accelerating RF cavities are nowadays developed in several national laboratories for highenergy physics applications. Ultra high gradients, up to the order of GV/m, can be achieved by using ultra compact accelerating structures in the sub-THz regime. Nevertheless, the experimental setup for measuring ... | [
{
"title": "I. INTRODUCTION",
"content": "T HE concept of reflection coefficient is fundamental in high accelerating periodic structures. Electromagnetic energy may be coupled from one waveguide into another one or into a cavity resonator by a small aperture located at a suitable position in the common wall... | [
{
"id": "",
"caption": "Propagation constant; η 01 : Wave impedance of the T M 01 ; α: Attenuation constant per unit length; : The length of the cavity; e t01 , h t01 , : Transverse modal fields for T M 01 mode; e z01 , h z01 , : Longitudinal modal fields for T M 01 mode; ξ: Amplitude of the incident wave;"... | [
{
"text": "Theory of diffraction by small holes. Hans Bethe, Albrecht, Physical review. 66163Bethe, Hans Albrecht. \"Theory of diffraction by small holes.\" Physical review 66.7-8 (1944): 163.",
"doi": ""
},
{
"text": "Field theory of guided waves. Robert E Collin, Collin, Robert E. \"Field theory o... |
p-hyperbolicity of homotopy groups via K-theory
26 Nov 2021 November 29, 2021
Guy Boyde
p-hyperbolicity of homotopy groups via K-theory
26 Nov 2021 November 29, 2021arXiv:2101.04591v2 [math.AT]
We show that S n ∨ S m is Z/p r -hyperbolic for all primes p and all r ∈ Z + , provided n, m ≥ 2, and consequently that var... | 231,582,764 | {
"arxiv": "2101.04591",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s00209-021-02917-1"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2101.04591v2.pdf"
],
"pdf_sha": "49b43a1daacd20b8c35b48ccd202218d19ee138e",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | p-hyperbolicity of homotopy groups via K-theory | [
"Guy Boyde"
] | We show that S n ∨ S m is Z/p r -hyperbolic for all primes p and all r ∈ Z + , provided n, m ≥ 2, and consequently that various spaces containing S n ∨ S m as a p-local retract are Z/p r -hyperbolic. We then give a K-theory criterion for a suspension ΣX to be p-hyperbolic, and use it to deduce that the suspension of a ... | [
{
"title": "Introduction",
"content": "A space X is called rationally elliptic if π * (X) ⊗ Q is finite dimensional, and rationally hyperbolic if the dimension of i≤m π i (X) ⊗ Q grows exponentially in m. It was proved in [FHT15,Chapter 33] that simply connected CW -complexes with rational homology of finit... | [
{
"id": "",
"caption": "For a Z-graded (respectively Z/2-graded) module M , let T (M ) = ∞ k=1 M ⊗k denote the tensor algebra on M . The product is given by concatenation. We refer to M ⊗k as the weight k component of the tensor algebra T (M ). We define a Z-grading (respectively Z/2-grading) on T (M ) by s... | [
{
"text": "Let h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF * (A). As with deg ′. let h ′ : B(π * (Apreamble to the second lemma is necessary. Let h : π * (A) → K TF * (A) be the K-homological Hurewicz map, which sends f ∈ π N (A) to f * (ξ N ) ∈ K TF ... |
MicroVelocity: rethinking the Velocity of Money for digital currencies
February 1, 2022
Carlo Campajola
DLT Science Foundation
LondonUnited Kingdom
Department of Informatics
University of Zurich
Blockchain & Distributed Ledger Technologies
ZürichSwitzerland
UZH Blockchain Center
ZürichSwitzerland
Marco D'err... | 246,430,385 | {
"arxiv": "2201.13416",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2201.13416v2.pdf"
],
"pdf_sha": "f94fbac8bbcf684f6d03a63b57ee8ad5f8a660a7",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | MicroVelocity: rethinking the Velocity of Money for digital currencies | [
"Carlo Campajola \nDLT Science Foundation\nLondonUnited Kingdom\n\nDepartment of Informatics\nUniversity of Zurich\nBlockchain & Distributed Ledger Technologies\nZürichSwitzerland\n\nUZH Blockchain Center\nZürichSwitzerland",
"Marco D'errico \nEuropean Systemic Risk Board Secretariat\nEuropean Central Bank\n... | We propose a novel framework to analyse the velocity of money in terms of the contribution (MicroVelocity) of each individual agent, and to uncover the distributional determinants of aggregate velocity. Leveraging on complete publicly available transactions data stored in blockchains from four cryptocurrencies, we empi... | [
{
"title": "Introduction",
"content": "One of the earliest and most widespread concepts across societies is that of money. As humans we are inherently brought to organise ourselves into communities and exchange the goods and services we produce; this leads to the necessity to invent a medium of exchange tha... | [
{
"id": "",
"caption": "Comparison of total MicroVelocity and the ratio of total flow to monetary mass P Q/M , with weekly aggregation. All quantities are annualised.",
"content": "Figure 2 :\n2Comparison of total MicroVelocity and the ratio of total flow to monetary mass P Q/M , with weekly aggregation... | [
{
"text": "The economics of money, banking and financial markets. Frederic S Mishkin, Mishkin Frederic-Addison Wesley LongmanS Mishkin Frederic. The economics of money, banking and financial mar- kets. Mishkin Frederic-Addison Wesley Longman, 2004.",
"doi": ""
},
{
"text": "Banking on stone money: a... |