Daily Papers

We construct the quantized electromagnetic field and the corresponding photon propagators in a rigidly rotating cylindrical reference frame. Starting from covariant Maxwell theory, we formulate the gauge-fixed field equations and solve them in cylindrical coordinates. The vector nature of the electromagnetic field requires distinguishing orbital and intrinsic angular momentum, with the total angular momentum as the natural quantum number. We construct the two independent physical photon modes in terms of Bessel functions by solving the coupled differential equations, and provide an alternative construction based on a covariant basis of vector eigenfunctions generated by a covariantly constant tetrad. After normalizing the modes with the Maxwell inner product, we quantize the electromagnetic field and define the corresponding photon states and greater and lesser two-point functions. The physical time-ordered propagator is obtained directly from the quantized modes, with its transverse dependence encoded in polarization kernels that retain the cylindrical spatial structure and its spectral poles exhibiting the rotational energy shift. The full gauge-fixed photon propagator is constructed in two complementary ways: from the scalar Green function combined with the tensor transformation of the inertial Feynman-gauge propagator, and from the covariant vector eigensystem through its spectral decomposition and the Fock--Schwinger proper-time representation. We verify that both constructions satisfy the Green-function equation for the complete gauge-fixed Maxwell operator, providing the propagator required for covariant perturbative calculations with internal photon lines.

Green's functions are a useful technique for interpreting atmospheric state responses to changes in the spatial pattern of sea surface temperature (SST). Here we train version 2 of the Ai2 Climate Emulator (ACE2) on reference historical SST simulations of the US Department of Energy's EAMv3 global atmosphere model. We compare how well the SST Green's functions generated by ACE2 match those of EAMv3, following the protocol of the Green's Function Model Intercomparison Project (GFMIP). The spatial patterns of top-of-atmosphere (TOA) radiative response from the individual GFMIP SST patch simulations are similar for ACE and the EAMv3 reference. The derived sensitivity of global net TOA radiation sensitivity to SST patch location is qualitatively similar in ACE as in EAMv3, but there are statistically significant discrepancies for some SST patches, especially over the subtropical northeast Pacific. These discrepancies may reflect insufficient diversity in the SST patterns sampled over the course of the EAMv3 AMIP simulation used for training ACE. Both ACE and EAMv3 Green's functions reconstruct the historical record of the global annual-mean TOA radiative flux from a reference EAMv3 AMIP simulation reasonably well. Notably, under our configuration and compute resources, ACE achieves these results approximately 100 times faster in wall-clock time compared to EAMv3, highlighting its potential as a powerful and efficient tool for tackling other computationally intensive problems in climate science.

We study a space-time Brownian motion with drift B(t)=(t_0+t,y_0+W(t)+t) killed at the moving boundary of the cone {(t,x):0<x<t}. This article determines the parabolic Martin boundary and all harmonic functions associated with this process. To that end, the asymptotics of Green's functions are determined along all directions. We also find the exit probabilities at the edges, the probability of remaining in the cone forever and the laws of the exit point and exit time. From this, we derive an explicit formula for the transition kernel of the process. These results arise from two different methods initially introduced to study random walks. An analytical approach, developed in the 1970s by Malyshev and based on the steepest descent method on a Riemann surface, is used to determine the asymptotics of the Green's functions. A recursive compensation approach, inspired by the method developed in the 1990s by Adan, Wessels and Zijm, is used to determine the harmonic functions.

There is extensive current interest about electronic topology in correlated settings. In strongly correlated systems, contours of Green's function zeros may develop in frequency-momentum space, and their role in correlated topology has increasingly been recognized. However, whether and how the zeros contribute to electronic properties is a matter of uncertainty. Here we address the issue in an exactly solvable model for Mott insulator. We show that the Green's function zeros contribute to several physically measurable correlation functions, in a way that does not run into inconsistencies. In particular, the physical properties remain robust to chemical potential variations up to the Mott gap as it should be based on general considerations. Our work sets the stage for further understandings on the rich interplay among topology, symmetry and strong correlations.

In this paper, both the pointwise space-time structures of the solution for a three-dimensional model describing gas dynamics in thermal non-equilibrium and the corresponding background stabilities within the L^p framework are investigated under the small initial perturbation assumption and the irrotationality condition. Our main tools are the modified Green's function according to the irrotationality condition, a priori energy method and a refined decay estimates of products composed of the entropy function and high-order derivatives.

Fourier Neural Operators (FNO) learn solution operators of partial differential equations by parameterizing global convolutions in the complex Fourier domain. For real-valued PDE solutions, the complex FFT carries representational redundancy through conjugate symmetry. We introduce the Hartley Neural Operator (HNO), the exact real-valued mirror of FNO: it replaces the FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode, with no complex arithmetic. Because the real Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO but one real weight where FNO carries a complex pair, so the two operators are iso-parametric at equal width and differ only in spectral basis. Our central thesis is that the best basis is a property of the operator. Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that the real Hartley multiplier diagonalizes exactly, and HNO is favored there. Time-dependent operators carry phase, from oscillation in the wave equation to transport in advection, Burgers, and Navier-Stokes, which a real diagonal multiplier cannot represent, so FNO is favored there, and increasingly so with the operator's phase content, leaving the phaseless heat equation as the borderline case. Training both operators identically and benchmarking across PDE classes, initial-condition families, and boundary conditions, we find an elliptic-versus-time-dependent split that is monotone in operator phase content and matches the Green's-function theory we develop. Rather than a universal winner, our findings give a predictive rule: match the spectral basis to the symmetry of the solution operator.

In this paper, we prove the existence of stationary pulse solutions for a bulk-surface reaction-diffusion system modeling cell polarization in two-dimensional domains. Using a singular perturbation method, we construct pulse solutions whose centers lie near non-degenerate critical points of a geometry-dependent potential expressed in terms of the Neumann Green's function, as formally predicted in watanabe2026. The main analytical difficulty arises from the half-Laplacian induced by the logarithmic singularity of the Green's function, resulting in algebraically rather than exponentially decaying inner solutions. Our analysis may also serve as a useful guide for applying singular perturbation methods to other bulk-surface type systems.

Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural operators have emerged as a promising technique to solve elliptic PDEs more efficiently by directly mapping the input to solutions. However, existing networks typically cannot handle complex geometries and inhomogeneous boundary values present in the real world. Here we introduce Boundary-Embedded Neural Operators (BENO), a novel neural operator architecture that embeds the complex geometries and inhomogeneous boundary values into the solving of elliptic PDEs. Inspired by classical Green's function, BENO consists of two branches of Graph Neural Networks (GNNs) for interior source term and boundary values, respectively. Furthermore, a Transformer encoder maps the global boundary geometry into a latent vector which influences each message passing layer of the GNNs. We test our model extensively in elliptic PDEs with various boundary conditions. We show that all existing baseline methods fail to learn the solution operator. In contrast, our model, endowed with boundary-embedded architecture, outperforms state-of-the-art neural operators and strong baselines by an average of 60.96\%. Our source code can be found https://github.com/AI4Science-WestlakeU/beno.git.

The active Brownian particle (ABP) model is one of the most widely used particle-based models in studies of scalar active fluids. A collection of interacting ABPs is known to exhibit intriguing phenomena that are absent in equilibrium counterparts. A prominent example is spontaneous phase separation, known as motility-induced phase separation (MIPS), which occurs even in the absence of explicit attractive interactions. In parallel, the large-scale behavior of scalar active fluids is often modeled using continuum descriptions, namely, scalar field theories. Such scalar active field theories are usually formulated in a top-down manner, and establishing their connections to particle-based models remains challenging, with these connections not yet fully understood. To advance the microscopic derivation of scalar active field theories, we revisit the connection between ABPs interacting via a two-body potential in two dimensions and scalar field theories that include all possible terms up to fourth order in spatial gradients. Our approach is based on a pressure expansion and the renormalization group (RG) method in the context of singular perturbation theory. The RG method provides a systematic way to eliminate fast variables and identify the dynamics on the slow invariant manifold. Within this framework, scalar field theories can be obtained as RG flow equations. We also apply this method to a chiral variant of the ABP model, the chiral ABP (cABP) model, in which a constant torque biases particle rotation to the left or right. We show that the scalar field theory corresponding to the cABP model contains two ``odd'' terms in addition to those present in the scalar field theory for ABPs. The method developed here may also be useful for deriving hydrodynamic descriptions of other types of active matter systems.

If h is a ring-valued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green matrix entries g(x,y). We also have already seen that that the determinants of L or g are both the product of the h(x). In the case where h(x) is the parity of dimension, the sum of the energy values was the standard Euler characteristic and the determinant was a unit. If h(x) was the unit in the ring then L,g are integral quadratic forms which are isospectral and inverse matrices of each other. We prove here that the quadratic energy expression summing over all pairs h(x)^* h(y) of intersecting sets is a signed sum of squares of Green function entries. The quadratic energy expression is Wu characteristic in the case when h is dimension parity. For general h, the quadratic energy expression resembles an Ising Heisenberg type interaction. The conjugate of g is the inverse of L if h takes unit values in a normed ring or in the group of unitary operators in an operator algebra.

Electron-electron (e-e) and electron-phonon (e-ph) interactions are challenging to describe in correlated materials, where their joint effects govern unconventional transport, phase transitions, and superconductivity. Here we combine first-principles e-ph calculations with dynamical mean field theory (DMFT) as a step toward a unified description of e-e and e-ph interactions in correlated materials. We compute the e-ph self-energy using the DMFT electron Green's function, and combine it with the e-e self-energy from DMFT to obtain a Green's function including both interactions. This approach captures the renormalization of quasiparticle dispersion and spectral weight on equal footing. Using our method, we study the e-ph and e-e contributions to the resistivity and spectral functions in the correlated metal Sr_2RuO_4. In this material, our results show that e-e interactions dominate transport and spectral broadening in the temperature range we study (50-310~K), while e-ph interactions are relatively weak and account for only sim10\% of the experimental resistivity. We also compute effective scattering rates, and find that the e-e interactions result in scattering several times greater than the Planckian value k_BT, whereas e-ph interactions are associated with scattering rates lower than k_BT. Our work demonstrates a first-principles approach to combine electron dynamical correlations from DMFT with e-ph interactions in a consistent way, advancing quantitative studies of correlated materials.

Climate change is profoundly affecting nearly all aspects of life on earth, including human societies, economies and health. Various human activities are responsible for significant greenhouse gas emissions, including data centres and other sources of large-scale computation. Although many important scientific milestones have been achieved thanks to the development of high-performance computing, the resultant environmental impact has been underappreciated. In this paper, we present a methodological framework to estimate the carbon footprint of any computational task in a standardised and reliable way, based on the processing time, type of computing cores, memory available and the efficiency and location of the computing facility. Metrics to interpret and contextualise greenhouse gas emissions are defined, including the equivalent distance travelled by car or plane as well as the number of tree-months necessary for carbon sequestration. We develop a freely available online tool, Green Algorithms (www.green-algorithms.org), which enables a user to estimate and report the carbon footprint of their computation. The Green Algorithms tool easily integrates with computational processes as it requires minimal information and does not interfere with existing code, while also accounting for a broad range of CPUs, GPUs, cloud computing, local servers and desktop computers. Finally, by applying Green Algorithms, we quantify the greenhouse gas emissions of algorithms used for particle physics simulations, weather forecasts and natural language processing. Taken together, this study develops a simple generalisable framework and freely available tool to quantify the carbon footprint of nearly any computation. Combined with a series of recommendations to minimise unnecessary CO2 emissions, we hope to raise awareness and facilitate greener computation.

We study two-step recurrences with positive step sizes and nonnegative momentum coefficients whose terminal residual is the Chebyshev minimax polynomial on a prescribed spectral interval [μ,L]. These recurrences have the same terminal error on a fixed quadratic, but their first-order responses to stepwise Hessian perturbations can differ. We prove that the sine-Jacobi recurrence of Part I is asymptotically optimal for this response norm within a factor of 1.000004, uniformly over 0<μ<L. Every terminal-exact recurrence satisfies A_Nge(2c_{cert}-o(1))N^{3/2}ε_N^star/(L-μ) with c_{cert}>1.14268353624, whereas the sine-Jacobi recurrence attains the constant 2c_{sin} with c_{sin}<1.14269218. The proof reduces the gain to the diagonal energy of a Jacobi Green matrix. A centering lift represents this matrix by an exponential kernel; packing its diagonal into a density on the line preserves the energy and controls the change in the Birman-Schwinger spectrum. A calibrated identity gives a lower bound using the ground parameter and one Fredholm determinant. Two further determinants and a trace observation improve the bound through a pointwise inequality on an explicit compact state region. One rational certificate verifies this inequality by interval arithmetic. A second certificate constructs a density satisfying the selected observations and both normalizations with energy below the sine value. Finally, no C^1 function of finitely many observations of the specified kinds gives a locally sharp lower bound at the sine density.

Our aim is to contribute to quantum field theory (QFT) formalisms useful for descriptions of short time phenomena, dominant especially in heavy ion collisions. We formulate out-of-equilibrium QFT within the finite-time-path formalism (FTP) and renormalization theory (RT). The potential conflict of FTP and RT is investigated in g phi^3 QFT, by using the retarded/advanced (R/A) basis of Green functions and dimensional renormalization (DR). For example, vertices immediately after (in time) divergent self-energy loops do not conserve energy, as integrals diverge. We "repair" them, while keeping d<4, to obtain energy conservation at those vertices. Already in the S-matrix theory, the renormalized, finite part of Feynman self-energy Sigma_{F}(p_0) does not vanish when |p_0|rightarrowinfty and cannot be split to retarded and advanced parts. In the Glaser--Epstein approach, the causality is repaired in the composite object G_F(p_0)Sigma_{F}(p_0). In the FTP approach, after repairing the vertices, the corresponding composite objects are G_R(p_0)Sigma_{R}(p_0) and Sigma_{A}(p_0)G_A(p_0). In the limit drightarrow 4, one obtains causal QFT. The tadpole contribution splits into diverging and finite parts. The diverging, constant component is eliminated by the renormalization condition langle 0|phi|0rangle =0 of the S-matrix theory. The finite, oscillating energy-nonconserving tadpole contributions vanish in the limit trightarrow infty .

Describing fermionic systems with attractive interactions, such as superconductors and atomic nuclei, beyond the mean-field level remains a central challenge. In this work, we investigate the performance of Bogoliubov fully self-consistent GW (Bog.~scGW) and second-order Green's function (Bog.~scGF2) methods, implemented within a unified imaginary-time and imaginary-frequency framework. Using the attractive Hubbard model as a benchmark against full configuration interaction (FCI), we find that Bog. scGF2 provides a more accurate description than Bog.~scGW. In particular, Bog.~scGW systematically overestimates the correlation energy, with errors increasing rapidly as the attractive interaction strength grows. These results highlight the importance of exchange contributions in strongly correlated fermionic systems, even when exchange effects are included only at low order in perturbation theory.

Asymptotic Safety provides a promising framework for the quantization of Scalar-Tensor theories as non-perturbatively renormalizable quantum field theories. Over the past years, growing evidence has accumulated for the existence of non-trivial renormalization group fixed points in a variety of Scalar-Tensor models. In this work, we investigate the robustness of these fixed points under significant modifications of the computational setup. In particular, we employ a gauge condition that explicitly depends on the dynamical scalar field, going beyond a mere gauge-parameter deformation of the standard linear covariant gauges commonly used in the literature. Exploring different field parameterizations, we identify fixed points belonging to the Gaussian Matter class in four spacetime dimensions that are consistent with previous studies. The real part of the critical exponents associated with the scalar-field mass and the non-minimal scalar-curvature coupling turn out to have very small absolute value. We provide a comprehensive exploration of how the anomalous dimension of the scalar field and field parameterization impact those critical exponents showing that fixed-point candidates with two or four relevant directions emerge.

Molecular docking is critical to structure-based virtual screening, yet the throughput of such workflows is limited by the expensive optimization of scoring functions involved in most docking algorithms. We explore how machine learning can accelerate this process by learning a scoring function with a functional form that allows for more rapid optimization. Specifically, we define the scoring function to be the cross-correlation of multi-channel ligand and protein scalar fields parameterized by equivariant graph neural networks, enabling rapid optimization over rigid-body degrees of freedom with fast Fourier transforms. The runtime of our approach can be amortized at several levels of abstraction, and is particularly favorable for virtual screening settings with a common binding pocket. We benchmark our scoring functions on two simplified docking-related tasks: decoy pose scoring and rigid conformer docking. Our method attains similar but faster performance on crystal structures compared to the widely-used Vina and Gnina scoring functions, and is more robust on computationally predicted structures. Code is available at https://github.com/bjing2016/scalar-fields.

This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law A_k(a) = log(1/a) + log(k-2) + B_0 + O(a), with B_0 = log q - 2log 2, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient B(ν) and shows B_0 depends only on the conductor q, independent of the Archimedean parameter. (R2b) The degree d of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, S(a) sim (2a)^3/6. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line σ^*(a) descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue λ_1(a) is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function W(a,0;z) is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.

This thesis addresses a fundamental problem in deformation quantization: the difficulty of calculating the star-exponential, the symbol of the evolution operator, due to convergence issues. Inspired by the formalism that connects the star-exponential with the quantum propagator for bosonic systems, this work develops the analogous extension for the fermionic case. A rigorous method, based on Grassmann variables and coherent states, is constructed to obtain a closed-form expression for the fermionic star-exponential from its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, allowing for the calculation of the ground state energy directly in phase space. Finally, the method is validated by successfully applying it to the simple and driven harmonic oscillators, where it is demonstrated that a simplified ("naive") approach (with an ad-hoc "remediation") is a valid weak-coupling limit of the rigorous ("meticulous") formalism, thereby providing a new and powerful computational tool for the study of fermionic systems.

Quantum many-body methods provide a systematic route to computing electronic properties of molecules and materials, but high computational costs restrict their use in large-scale applications. Due to the complexity in many-electron wavefunctions, machine learning models capable of capturing fundamental many-body physics remain limited. Here, we present a deep learning framework targeting the many-body Green's function, which unifies predictions of electronic properties in ground and excited states, while offering physical insights into many-electron correlation effects. By learning the GW or coupled-cluster self-energy from mean-field features, our graph neural network achieves competitive performance in predicting one- and two-particle excitations and quantities derivable from one-particle density matrix. We demonstrate its high data efficiency and good transferability across chemical species, system sizes, molecular conformations, and correlation strengths in bond breaking, through multiple molecular and nanomaterial benchmarks. This work opens up opportunities for utilizing machine learning to solve many-electron problems.

We consider Proca field perturbations in a five-dimensional Schwarzschild-anti-de Sitter (Schwarzschild-AdS_{5}) black hole geometry. Using the vector spherical harmonic (VSH) method, we show that the Proca field decomposes into scalar-type and vector-type components according to their tensorial behavior on the three-sphere. Two degrees of freedom of the field are described by scalar-type components, which are coupled due to the mass term, while the remaining two degrees of freedom are described by a vector-type component, which decouples completely. Motivated by the Frolov-Krtouš-Kubizňák-Santos (FKKS) ansatz in the limit of zero spin, we use a field transformation to decouple the scalar-type components at the expense of introducing a complex separation parameter β. This parameter can be determined analytically, and its values correspond to two distinct polarizations of the scalar-type sector: "electromagnetic" and "non-electromagnetic", denoted by β_{+} and β_{-}, respectively. In the scalar-type sector, the radial differential equation for each polarization is a Fuchsian differential equation with five singularities, whereas in the vector-type sector, the radial equation has four singularities. By means of the isomonodromy method, we reformulate the boundary value problem in terms of the initial conditions of the Painlevé VI τ function and, using a series expansion of the τ function, we compute the scalar-type and vector-type quasinormal modes (QNMs) in the small horizon limit. Our results are in overall very good agreement with those obtained via the numerical integration method. This shows that the isomonodromy method is a reliable method to compute quasinormal modes in the small horizon limit with high accuracy.

One way to model the strange metal phase of certain materials is via a holographic description in terms of probe D-branes in a Lifshitz spacetime, characterised by a dynamical exponent z. The background geometry is dual to a strongly-interacting quantum critical theory while the probe D-branes are dual to a finite density of charge carriers that can exhibit the characteristic properties of strange metals. We compute holographically the low-frequency and low-momentum form of the charge density and current retarded Green's functions in these systems for massless charge carriers. The results reveal a quasi-particle excitation when z<2, which in analogy with Landau Fermi liquids we call zero sound. The real part of the dispersion relation depends on momentum k linearly, while the imaginary part goes as k^2/z. When z is greater than or equal to 2 the zero sound is not a well-defined quasi-particle. We also compute the frequency-dependent conductivity in arbitrary spacetime dimensions. Using that as a measure of the charge current spectral function, we find that the zero sound appears only when the spectral function consists of a single delta function at zero frequency.

We study the inverse adjoint problem for a perturbative bosonic outgoing map. Within the polynomial bosonic canonical-commutation-relation (CCR) algebra, we prove a constructive integrability theorem: order-by-order preservation of the CCRs is sufficient to reconstruct recursively a formal self-adjoint generator whose exponential implements the outgoing map by adjoint conjugation. At each order, the Baker--Campbell--Hausdorff contribution determined by previously reconstructed generators is subtracted from the outgoing coefficient, and the remaining homogeneous CCR conditions determine the next generator explicitly. We apply the construction to graviton scattering in the null-surface formulation (NSF), specialized to the flat solution-space sector with Minkowski metric, null generators, affine parameter, and cone measure. The NSF outgoing map preserves the Ashtekar radiative CCRs through (δa_3), corresponding to order (\varepsilon^2). At this order, the scalar CCR sector uniquely selects the symmetric ordering (c=1) for the nontrivial interacting kernel. The associated linear contraction, generated by the same cubic kernel, cancels the double-contraction contribution from ([δa_2,δa_2^\dagger]). The theorem then reconstructs the self-adjoint generators (δT_1) and (δT_2), providing a formal perturbative unitary implementation. The resulting canonical organization also separates the direct (2\to2) contribution into irreducible-loop, one-particle-reducible, and factorized sectors.

We develop a perturbative framework for systems coupled quadratically to bosonic environments with quadratic Hamiltonians and linear Lindblad operators. The method applies to both single-mode and multimode environments whose initial states may be non-Gaussian and nonstationary. By combining linear evolution in a doubled operator space with a characteristic-function representation, we express ordered multitime environmental correlation functions in terms of derivatives of the initial characteristic function. These correlation functions enter a recursive construction of the time-convolutionless generator at arbitrary finite order. We illustrate the method with a qubit coupled to a damped mode via a two-photon interaction. Initial Fock and thermal states with identical first and second field moments produce different second-order generators through their fourth moments.

The magnetization in a superconductor induced due to the inverse proximity effect is investigated in hybrid bilayers containing a superconductor and a ferromagnetic insulator or a strongly spin-polarized ferromagnetic metal. The study is performed within a quasiclassical Green function framework, wherein Usadel equations are solved with boundary conditions appropriate for strongly spin-polarized ferromagnetic materials. A comparison with recent experimental data is presented. The singlet to triplet conversion of the superconducting correlations as a result of the proximity effect with a ferromagnet is studied.

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its ''Edge of Stability'' (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)-the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.

Reduction in effective spacetime dimensionality can occur in field-theory models more general than the widely studied dimensional reductions based on technically consistent truncations. Situations where wavefunction factors depend nontrivially on coordinates transverse to the effective lower dimension can give rise to unusual patterns of gauge symmetry breaking. Leading-order gauge modes can be left massless, but naturally occurring Stueckelberg modes can couple importantly at quartic order and higher, thus generating a "covert" pattern of gauge symmetry breaking. Such a situation is illustrated in a five-dimensional model of scalar electrodynamics in which one spatial dimension is taken to be an interval with Dirichlet/Robin boundary conditions on opposing ends. This simple model illuminates a mechanism which also has been found in gravitational braneworld scenarios.

According to a scalar theory of gravity with a preferred frame, electromagnetism in the presence of a gravitational field implies that there is an additional energy tensor, which might contribute to dark matter. The expression of this tensor is determined by a mere scalar p, that depends on the EM field and (for a weak field) on the Newtonian gravitational field. We briefly recall why this tensor arises and how the EM field in a galaxy can be calculated. The data fields that enter the PDE for the scalar field p oscillate very quickly in space and time, as does the EM field. This prevents integration of that PDE at the relevant galactic scale. Therefore, a homogenization of that PDE has to be operated. We discuss in some detail three possible ways of applying the homogenization theory to that PDE: time, space, or spacetime homogenization. The second and third ways may lead to feasible, albeit heavy calculations.

We show how to construct perturbative effective field theories for strong phase transitions, where the standard tools of high-temperature dimensional reduction fail. By identifying the relevant power counting relations for the couplings, masses and background fields, we construct a controlled perturbative expansion adapted to this regime. We find that the order parameter field remains light at the transition, so that there is a controlled derivative expansion, even though large external fields need to be resummed. The resulting effective field theories for strong phase transitions live in three dimensions and have nonpolynomial field dependence. Applying our framework to a two-scalar model, we verify renormalisation scale cancellation through next-to-next-to-leading order and find decreasing corrections to the latent heat as the perturbative order is increased. Extending to include fermions and gauge fields, we identify a broad class of models in which strong-transition thermodynamics are systematically improvable.

We construct high-accuracy semi-analytic mode functions for scalar cosmological perturbations in mLQC-I within the hybrid approach to study ultraviolet regularity and gravitational particle creation. The evolution comprises an asymptotic pre-de Sitter branch, a pre-inflationary region containing the quantum bounce, and inflation. In the intermediate region, we construct the complete third-order uniform asymptotic solution and introduce adaptive defect correction. The third-order approximation achieves sub-percent complex-mode accuracy for representative higher-wavenumber modes; defect-corrected solutions agree with numerical evolution at the 0.00001% level over the tested low-wavenumber range. Retaining amplitudes and phases provides a controlled basis for future precision calculations of primordial power spectra and non-Gaussian correlations. We compare the de Sitter vacuum selected in the remote contracting branch with a fourth-order adiabatic state imposed at the bounce. Finite-wavenumber matching and the strict ultraviolet limit require different asymptotic treatments: extrapolating the leading connection to infinite wavenumber yields a spurious constant negative-frequency coefficient and an apparent ultraviolet divergence. Instead, the large-wavenumber de Sitter mode agrees with the fourth-order adiabatic expansion through the order required for energy-momentum tensor renormalization, and smooth exact evolution preserves this ultraviolet structure. Finite and intermediate modes can nevertheless undergo nonadiabatic Bogoliubov mixing and particle creation. Finite-mode particle production thus coexists with ultraviolet regularity. The ultraviolet argument also extends to the dressed-metric approach in its corresponding large-wavenumber regime, while quantitative mode functions must be recalculated using its effective mass.

We study numerically the 6D (2,0) superconformal bootstrap using the soft-Actor-Critic (SAC) algorithm as a stochastic optimizer. We focus on the four-point functions of scalar superconformal primaries in the energy-momentum multiplet. Starting from the supergravity limit, we perform searches for adiabatically varied central charges and derive two curves for a collection of 80 CFT data (70 of these data correspond to unprotected long multiplets and 10 to protected short multiplets). We conjecture that the two curves capture the A- and D-series (2,0) theories. Our results are competitive when compared to the existing bounds coming from standard numerical bootstrap methods, and data obtained using the OPE inversion formula. With this paper we are also releasing our Python implementation of the SAC algorithm, BootSTOP. The paper discusses the main functionality features of this package.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

We study scalaron quasinormal modes and quasi-bound states of Schwarzschild black holes directly within the Bondi-Sachs formulation of metric f(R) gravity, combining frequency-domain analyses with characteristic time evolution. Starting from the linearised characteristic equations, we obtain the massive scalar master equation and show that the tensor master equation takes the same form as in general relativity. For the analytic models considered here, the linear scalar spectra depend on the f(R) theory only through the dimensionless combination μM. We compute both quasinormal and quasi-bound spectral branches using Leaver's continued-fraction method and compare them with independent literature benchmarks. To complement the frequency domain analysis, we evolve the scalaron perturbation equations using a fourth order convergent characteristic scheme. Applying matrix-pencil stability scans to the resulting waveforms, we independently recover fundamental quasinormal frequencies consistent with the continued-fraction results within the measured extraction-stability scales. Finally, for a selected finite-mass case, we demonstrate the extraction of both oscillation frequencies and damping rates of the fundamental quasinormal mode and quasi-bound state from separate time windows of the same waveform at one extraction radius, without imposing either spectral branch at the finite outer boundary.

We study a one dimensional nonlinear Klein-Gordon equation with a potential, a localized quadratic nonlinearity, and a non-localized quartic nonlinearity. We assume that the linearized operator has a single discrete eigenvalue below the continuous spectrum, corresponding to an ''internal mode''. This eigenvalue generates localized, time-periodic solutions for the linear equation. Assuming the natural Fermi Golden Rule, we give a global description of the dynamics of the amplitude of the internal mode through the radiation damping mechanism, and prove scattering for the radiation. Our approach is based on the distorted Fourier transform and a collection of refined dispersive decay and smoothing estimates. The corresponding cubic problem is closely related to the question of global-in-space dispersive asymptotics for perturbations of kink solutions in the classical ϕ^4 model, which remains a challenging open question. From the perspective of nonlinear estimates, a quartic (non-localized) interaction is essentially sharp for our analysis, and closing the estimates with such a low degree nonlinear term requires a careful and delicate treatment. As a direct application of our analysis, we prove a result on the asymptotic stability of kinks for classes of scalar field models that can be seen as perturbations of the ϕ^4 model.

Ridgelet transform has been a fundamental mathematical tool in the theoretical studies of neural networks. However, the practical applicability of ridgelet transform to conducting learning tasks was limited since its numerical implementation by conventional classical computation requires an exponential runtime exp(O(D)) as data dimension D increases. To address this problem, we develop a quantum ridgelet transform (QRT), which implements the ridgelet transform of a quantum state within a linear runtime O(D) of quantum computation. As an application, we also show that one can use QRT as a fundamental subroutine for quantum machine learning (QML) to efficiently find a sparse trainable subnetwork of large shallow wide neural networks without conducting large-scale optimization of the original network. This application discovers an efficient way in this regime to demonstrate the lottery ticket hypothesis on finding such a sparse trainable neural network. These results open an avenue of QML for accelerating learning tasks with commonly used classical neural networks.

Muon is an increasingly widely used optimizer that replaces a gradient G=USV^top with its polar factor UV^top, thereby flattening the singular spectrum. However, full flattening discards singular-value information that may matter for adaptation. We introduce Muon^p, a Muon-style optimizer that instead uses fractional spectral-power updates US^pV^top for rational pin(0,1), interpolating between Muon and gradient descent. To make it practical, we prove that fractional spectral powers cannot be computed by any fixed univariate polynomial iteration, and furthermore derive low-degree odd bivariate recurrences that approximate US^pV^top using only matrix multiplications, preserving Muon's matrix-multiplication-only structure and compute complexity. We show that Muon^p maximizes the linear improvement in loss under the Schatten q-norm for q=1+1{p}. Empirically, Muon^p is especially effective for finetuning: on billion-scale models, Muon^p improves validation perplexity and downstream task performance. We further analyze when Muon^p is less suitable, through the lens of spectral geometry. Our results reveal important insights on when preserving the singular spectrum can bring significant gains, and introduce a principled way to achieve them.

The term GreenAI refers to a novel approach to Deep Learning, that is more aware of the ecological impact and the computational efficiency of its methods. The promoters of GreenAI suggested the use of Floating Point Operations (FLOPs) as a measure of the computational cost of Neural Networks; however, that measure does not correlate well with the energy consumption of hardware equipped with massively parallel processing units like GPUs or TPUs. In this article, we propose a simple refinement of the formula used to compute floating point operations for convolutional layers, called {\alpha}-FLOPs, explaining and correcting the traditional discrepancy with respect to different layers, and closer to reality. The notion of {\alpha}-FLOPs relies on the crucial insight that, in case of inputs with multiple dimensions, there is no reason to believe that the speedup offered by parallelism will be uniform along all different axes.

In the 3-3-1 model with right-handed neutrinos, three triplets of scalars engender the correct sequence of symmetry breaking, SU(3)_C times SU(3)_L times U(1)_X rightarrow SU(3)_C times SU(2)_L times U(1)_Y rightarrow SU(3)_C times U(1)_{EM}, generating mass for all fermions, except neutrinos. Tiny neutrino masses may be achieved by adding one sextet of scalars to the original scalar content. As consequence, it emerges a very complex scalar sector, involving terms that violate lepton number explicitly, too. The main obstacle to the development of the phenomenology of such scenario is the knowledge of its spectrum of scalars since, now, there are 15 massive scalar particles on it. The proposal of this work is to do an exhaustive analysis of such scalar sector with lepton number being explicitly violated at low, electroweak and high energy scales by means of trilinear terms in the potential. The first case can be addressed analytically and, as a nice result, we have observed that the scalar content of such case is split into two categories: One belonging to the 331 energy scale and the other belonging to the EWSB energy scale, with the last recovering the well known THDM+triplet. For the other cases, the scalar sector can be addressed only numerically. Hence, we proposed a very general approach for the numerical study of the potential, avoiding simplifications that can make us reach conclusions without foundation. We show that, in the case of lepton number being explicitly violated at electroweak scale, it is possible to recover the same physics of the THDM+triplet, as the previous case. Among all the possibilities, we call the attention to one special case which generates the 3HDM+triplet scenario. For the last case, when lepton number is violated at high energy scale, the sextet become very massive and decouples from the original scalar content of the 3-3-1 model.

Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by the eigenvalue density. We further probe the origin of these results from the viewpoint of integrability, which is taken here to mean wider classes of recursions and differential equations, and give extensions. Prominent in our study are first order linear matrix differential equations. One application given is to the derivation of the third order scalar equation for the density. Another is to obtain the explicit rational function formula for the variance of the purity statistic in the β generalised fixed trace Laguerre ensemble. In the original case (β= 2), the purity cumulants are expressed in terms of the large argument expansion of a particular σ-Painlevé IV transcendent. In a different but related direction, the exact computation of the two-point correlation for the fixed determinant circular unitary ensemble SU(N) is given the Appendix.

We present an escape rate theory for current-induced chemical reactions. We use Keldysh nonequilibrium Green's functions to derive a Langevin equation for the reaction coordinate. Due to the out of equilibrium electronic degrees of freedom, the friction, noise, and effective temperature in the Langevin equation depend locally on the reaction coordinate. As an example, we consider the dissociation of diatomic molecules induced by the electronic current from a scanning tunnelling microscope tip. In the resonant tunnelling regime, the molecular dissociation involves two processes which are intricately interconnected: a modification of the potential energy barrier and heating of the molecule. The decrease of the molecular barrier (i.e. the current induced catalytic reduction of the barrier) accompanied by the appearance of the effective, reaction-coordinate-dependent temperature is an alternative mechanism for current-induced chemical reactions, which is distinctly different from the usual paradigm of pumping vibrational degrees of freedom.

We develop a novel systematic method for constructing exact, static, and asymptotically flat black-hole solutions in shift-symmetric and Z_2-symmetric beyond Horndeski theories. In contrast to previous studies, our approach does not rely on the restrictive reciprocal condition g_{tt}g_{rr}=-1, allowing for more general geometries characterized by two independent metric functions. The resulting solution space contains two qualitatively distinct types of hair originating from different branches of the theory. One branch is characterized by khronometric hair, while the other by genuine scalar hair. In both, the scalar profile defines a nontrivial preferred foliation, but the shift-symmetry Noether current and the associated propagating degree of freedom are trivial for the former and nonvanishing for the latter. Focusing mainly on the second branch, we derive several distinct families of exact black-hole solutions. In general, these solutions are singular and exhibit primary scalar hair associated with an independent scalar charge. Nevertheless, we demonstrate that they can be regularized to yield regular black-hole or solitonic configurations, for which the scalar hair becomes secondary. We further investigate the stability properties of these solutions under linear perturbations by analyzing the axial sector and the monopole (ell=0) mode of the even-parity sector. For representative solutions with scalar hair, the perturbative analysis provides evidence for linear stability under the modes considered.

High-performance computing (HPC) is increasingly important for scalable quantum chemistry workflows that couple classical generative models, quantum circuit simulation, and selected configuration interaction postprocessing. We present the generative quantum-inspired Kolmogorov-Arnold eigensolver (GQKAE), a parameter-efficient extension of the generative quantum eigensolver (GQE) for quantum chemistry. GQKAE replaces the parameter-heavy feed-forward network components in GPT-style generative eigensolvers with hybrid quantum-inspired Kolmogorov-Arnold network modules, forming a compact HQKANsformer backbone. The method preserves autoregressive operator selection and the quantum-selected configuration interaction evaluation pipeline, while using single-qubit DatA Re-Uploading ActivatioN modules to provide expressive nonlinear mappings. Numerical benchmarks on H4, N2, LiH, C2H6, H2O, and the H2O dimer show that GQKAE achieves chemical accuracy comparable to the GPT-based GQE architecture, while reducing trainable parameters and memory by approximately 66% and improving wall-time performance. For strongly correlated systems such as N2 and LiH, GQKAE also improves convergence behavior and final energy errors. These results indicate that quantum-inspired Kolmogorov-Arnold networks can reduce classical-side overhead while preserving circuit-generation quality, offering a scalable route for HPC-quantum co-design on near-term quantum platforms.

Multi-Resolution Hash Encoding (MHE), the foundational technique behind Instant Neural Graphics Primitives, provides a powerful parameterization for neural fields. However, its spatial behavior lacks rigorous understanding from a physical systems perspective, leading to reliance on heuristics for hyperparameter selection. This work introduces a novel analytical approach that characterizes MHE by examining its Point Spread Function (PSF), which is analogous to the Green's function of the system. This methodology enables a quantification of the encoding's spatial resolution and fidelity. We derive a closed-form approximation for the collision-free PSF, uncovering inherent grid-induced anisotropy and a logarithmic spatial profile. We establish that the idealized spatial bandwidth, specifically the Full Width at Half Maximum (FWHM), is determined by the average resolution, N_{avg}. This leads to a counterintuitive finding: the effective resolution of the model is governed by the broadened empirical FWHM (and therefore N_{avg}), rather than the finest resolution N_{max}, a broadening effect we demonstrate arises from optimization dynamics. Furthermore, we analyze the impact of finite hash capacity, demonstrating how collisions introduce speckle noise and degrade the Signal-to-Noise Ratio (SNR). Leveraging these theoretical insights, we propose Rotated MHE (R-MHE), an architecture that applies distinct rotations to the input coordinates at each resolution level. R-MHE mitigates anisotropy while maintaining the efficiency and parameter count of the original MHE. This study establishes a methodology based on physical principles that moves beyond heuristics to characterize and optimize MHE.

We study the parity-perturbed Hofstadter Q-recursion $ widetilde Q(1)=widetilde Q(2)=1,qquad widetilde Q(n)=widetilde Q(n-widetilde Q(n-1)) +widetilde Q(n-widetilde Q(n-2))+(-1)^n, and the associated two-variable Dirichlet series Z_{widetilde Q}(s,t)=sum_{nge1}n^{-s}widetilde Q(n)^{-t}. The estimate \widetilde Q(n)=n/2+O(n/\log n) gives the exact domain of absolute convergence Re(s+t)>1. With w=s+t, we separate the universal term 2^tζ(w) and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns t=-1. For E(n)=2\widetilde Q(n)-n and A(X)=\sum_{n\le X}E(n), the binary-arch clock yields A(X)=Xlog_2X+XΩ!left(log_23X{32}right) +O!left(X{log X}right), where Ω is an explicit continuous periodic function. This continues the normalized correction to Rew>0 and yields a boundary resonance lattice: a double resonance at w=0 and simple resonances at 2πi m/\log2. After subtracting the full-slice order-X skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal n/\log n pointwise scale with an explicit signed constant. The negative-arch mass satisfies A_r=512{92π}16^r{sqrt r} left(1-13{16r}+O(r^{-2})right). We do not claim a full-slice continuation across Rew=0$.

The asymptotic safety conjecture is examined for quantum gravity in four dimensions. Using the renormalisation group, we find evidence for an interacting UV fixed point for polynomial actions up to the 34th power in the Ricci scalar. The extrapolation to infinite polynomial order is given, and the self-consistency of the fixed point is established using a bootstrap test. All details of our analysis are provided. We also clarify further aspects such as stability, convergence, the role of boundary conditions, and a partial degeneracy of eigenvalues. Within this setting we find strong support for the conjecture.

We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Hilbert spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distributions in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3% observation, our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg-Witten curves can be systematically studied via the Nekrasov-Shatashvili functions. In this paper, we explore another aspect of the relation between N=2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlev\'e equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg-Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an O(2) matrix model. We then show that these eigenfunctions are computed by surface defects in SU(2) super Yang-Mills in the self-dual phase of the Omega-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

An accurate description of the scalar potential at finite temperature is crucial for studying cosmological first-order phase transitions (FOPT) in the early Universe. At finite temperatures, a precise treatment of thermal resummations is essential, as bosonic fields encounter significant infrared issues that can compromise standard perturbative approaches. The Partial Dressing (or the tadpole resummation) method provides a self consistent resummation of higher order corrections, allowing the computation of thermal masses and the effective potential including the proper Boltzmann suppression factors and without relying on any high-temperature approximation. We systematically compare the Partial dressing resummation scheme results with the Parwani and Arnold Espinosa (AE) ones to investigate the thermal phase transition dynamics in the Two-Higgs-Doublet Model (2HDM). Our findings reveal that different resummation prescriptions can significantly alter the nature of the phase transition within the same region of parameter space, confirming the differences that have already been noticed between the Parwani and AE schemes. Notably, the more refined resummation prescription, the Partial Dressing scheme, does not support symmetry non-restoration in 2HDM at high temperatures observed using the AE prescription. Furthermore, we quantify the uncertainties in the stochastic gravitational wave (GW) spectrum from an FOPT due to variations in resummation methods, illustrating their role in shaping theoretical predictions for upcoming GW experiments. Finally, we discuss the capability of the High-Luminosity LHC and proposed GW experiments to probe the FOEWPT-favored region of the parameter space.

The precise equivalence between discretized Euclidean field theories and a certain class of probabilistic graphical models, namely the mathematical framework of Markov random fields, opens up the opportunity to investigate machine learning from the perspective of quantum field theory. In this contribution we will demonstrate, through the Hammersley-Clifford theorem, that the φ^{4} scalar field theory on a square lattice satisfies the local Markov property and can therefore be recast as a Markov random field. We will then derive from the φ^{4} theory machine learning algorithms and neural networks which can be viewed as generalizations of conventional neural network architectures. Finally, we will conclude by presenting applications based on the minimization of an asymmetric distance between the probability distribution of the φ^{4} machine learning algorithms and target probability distributions.

In metrology, Fisher information is an important metric that quantifies the precision that can be achieved in a measurement. For optical measurements using coherent light it has been shown that Fisher information can be expressed simply using the scattering matrix of the system. Fisher information can be maximized over the input modes to achieve maximum information states, which produce optimally precise estimates for a parameter when the system is limited by photon noise. Here, we extend this approach to multiparameter estimation, in which case Fisher information takes the form of a matrix. We consider several scalar functions of the Fisher matrix to optimize the precision in multiple parameters at the same time. We also consider strategies for dealing with nuisance parameters, which can degrade the achievable precision of other parameters but are not of interest to measure. We corroborate our findings numerically using a scattering system of 2D coupled dipoles.

We consider beta-Dyson Brownian motion, with β>= 1, started from a deterministic configuration with uniformly bounded support. Let μ_t be the semicircular free-convolution flow issued from the initial empirical measure, and set S_t = supp(μ_t). For every fixed T, ε> 0, with probability at least 1 - C exp(-(log n)^2), every particle remains within an epsilon-neighborhood of S_t for all 0 <= t <= T. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an o((n η)^(-1)) comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.

Estimating the symbolic or analytical form of probability density functions (PDFs) from observed samples is a fundamental challenge in statistical and computational modelling. This process is critical for deriving interpretable and generalizable relationships characterizing the underlying phenomenon. Traditionally, this estimation depends strongly on domain expertise and prior field-specific knowledge, with experts selecting appropriate functional forms or parametric families based on empirical evidence and theoretical understanding. The coefficients of these forms are then typically determined through parameter estimation. In this paper, we develop a framework for estimating symbolic expressions of unnormalized distributions from observed samples using domain-specific prior knowledge, such as the range of interactions and a predefined set of primitive functions. We integrate deep generative models with symbolic regression (SR), incorporating inductive biases, such as factorizing large distributions, to keep the problem tractable. The deep generative models we examine include likelihood-based models, viz., flow models, and score-based models. Experiments show the effectiveness of the proposed framework for estimating density functions for multivariate toy distributions as well as lattices from computational physics, namely, XY model and ϕ^4 theory. When applied to the renormalization problem in ϕ^4 theory, the proposed framework estimates compact symbolic approximations of the hamiltonian function at different scales directly from samples, yielding expressions that may be challenging to derive using traditional perturbative or analytic approaches in nonperturbative settings.

Deriving simple and accurate three-flavor neutrino oscillation probability formulas in matter is critical for understanding the phenomenology of upcoming experiments. In this article, we extend the Jacobi diagonalization method for deriving effective parameter mappings for neutrino oscillations in matter. The key insight behind the extension is to purposefully parameterize the mixing matrix, U_{PMNS}, using particular arrangements of Euler rotations. This novel extension allows for deriving simple and accurate effective mixing parameter mappings for exotic types of matter potentials. First, we revisit the Jacobi diagonalization method for neutrino oscillations with standard matter effects and use this to demonstrate our generalization. Subsequently, we apply the Jacobi diagonalization method to scalar non-standard interactions (SNSI) and derive effective parameter mappings for all three single diagonal coupling cases. The resulting formulas reveal that all diagonal SNSI couplings could give rise to a scalar MSW (SMSW) effect, that is an energy independent resonant enhancement analogous to the MSW effect. Finally, we study the accuracy of the derived SNSI formulas for various contemporary and upcoming experiments. For δm_{ee}, the accuracy is excellent for all considered experiments. For δm_{μμ} and δm_{ττ} the accuracy is worse since A_{CC} must be neglected, and first order perturbative corrections are applied to amend this.

We consider the question of learning Q-function in a sample efficient manner for reinforcement learning with continuous state and action spaces under a generative model. If Q-function is Lipschitz continuous, then the minimal sample complexity for estimating ε-optimal Q-function is known to scale as Ω(1{ε^{d_1+d_2 +2}}) per classical non-parametric learning theory, where d_1 and d_2 denote the dimensions of the state and action spaces respectively. The Q-function, when viewed as a kernel, induces a Hilbert-Schmidt operator and hence possesses square-summable spectrum. This motivates us to consider a parametric class of Q-functions parameterized by its "rank" r, which contains all Lipschitz Q-functions as r to infty. As our key contribution, we develop a simple, iterative learning algorithm that finds ε-optimal Q-function with sample complexity of O(1{ε^{max(d_1, d_2)+2}}) when the optimal Q-function has low rank r and the discounting factor γ is below a certain threshold. Thus, this provides an exponential improvement in sample complexity. To enable our result, we develop a novel Matrix Estimation algorithm that faithfully estimates an unknown low-rank matrix in the ell_infty sense even in the presence of arbitrary bounded noise, which might be of interest in its own right. Empirical results on several stochastic control tasks confirm the efficacy of our "low-rank" algorithms.

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires O(N) operations on a dataset composed of N data points. Therefore, a direct evaluation of ECDFs at N evaluation points requires a quadratic O(N^2) operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed and compared. The first one is based on fast summation in lexicographical order, with a O(N{log}N) complexity and requires the evaluation points to lie on a regular grid. The second one is based on the divide-and-conquer principle, with a O(Nlog(N)^{(d-1){vee}1}) complexity and requires the evaluation points to coincide with the input points. The two fast algorithms are described and detailed in the general d-dimensional case, and numerical experiments validate their speed and accuracy. Secondly, the paper establishes a direct connection between cumulative distribution functions and kernel density estimation (KDE) for a large class of kernels. This connection paves the way for fast exact algorithms for multivariate kernel density estimation and kernel regression. Numerical tests with the Laplacian kernel validate the speed and accuracy of the proposed algorithms. A broad range of large-scale multivariate density estimation, cumulative distribution estimation, survival function estimation and regression problems can benefit from the proposed numerical methods.

We implement physics-informed neural networks (PINNs) to solve the time-independent Schr\"odinger equation for three canonical one-dimensional quantum potentials: an infinite square well, a finite square well, and a finite barrier. The PINN models incorporate trial wavefunctions that exactly satisfy boundary conditions (Dirichlet zeros at domain boundaries), and they optimize a loss functional combining the PDE residual with a normalization constraint. For the infinite well, the ground-state energy is known (E = pi^2 in dimensionless units) and held fixed in training, whereas for the finite well and barrier, the eigenenergy is treated as a trainable parameter. We use fully-connected neural networks with smooth activation functions to represent the wavefunction and demonstrate that PINNs can learn the ground-state eigenfunctions and eigenvalues for these quantum systems. The results show that the PINN-predicted wavefunctions closely match analytical solutions or expected behaviors, and the learned eigenenergies converge to known values. We present training logs and convergence of the energy parameter, as well as figures comparing the PINN solutions to exact results. The discussion addresses the performance of PINNs relative to traditional numerical methods, highlighting challenges such as convergence to the correct eigenvalue, sensitivity to initialization, and the difficulty of modeling discontinuous potentials. We also discuss the importance of the normalization term to resolve the scaling ambiguity of the wavefunction. Finally, we conclude that PINNs are a viable approach for quantum eigenvalue problems, and we outline future directions including extensions to higher-dimensional and time-dependent Schr\"odinger equations.

The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap Δ encountered at quantum phase transitions. We investigate the scaling of Δ with system size N for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as N increases. This indicates that the unfavorable super-algebraic scaling of Δ, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to Δpropto N^{-1/3}. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.

We present a high-order boundary integral equation (BIE) method for the frequency-domain acoustic scattering of a point source by a singly-periodic, infinite, corrugated boundary. We apply it to the accurate numerical study of acoustic radiation in the neighborhood of a sound-hard two-dimensional staircase modeled after the El Castillo pyramid. Such staircases support trapped waves which travel along the surface and decay exponentially away from it. We use the array scanning method (Floquet--Bloch transform) to recover the scattered field as an integral over the family of quasiperiodic solutions parameterized by their on-surface wavenumber. Each such BIE solution requires the quasiperiodic Green's function, which we evaluate using an efficient integral representation of lattice sum coefficients. We avoid the singularities and branch cuts present in the array scanning integral by complex contour deformation. For each frequency, this enables a solution accurate to around 10 digits in a couple of seconds. We propose a residue method to extract the limiting powers carried by trapped modes far from the source. Finally, by computing the trapped mode dispersion relation, we use a simple ray model to explain an observed acoustic "raindrop" effect (chirp-like time-domain response).

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon with three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). By rebalancing the momentum matrix before finite-step Newton--Schulz orthogonalization, {\method} improves the geometry seen by orthogonalization. We show that finite-step orthogonalization is governed by the input spectrum, especially stable rank and condition number, and that row/column normalization acts as a zeroth-order surrogate for whitening. For hidden matrix weights, R is the default variant. Theoretically, {\method} (R) retains the standard mathcal O(T^{-1/4}) Muon-type nonconvex stationarity guarantee with decoupled weight decay and a horizon-free diminishing learning-rate schedule, and extends it to finite-step NS5 up to an explicit inexactness constant. In LLaMA2 pretraining on C4, {\method} (R) consistently outperforms Muon on 130M, 350M, and 1B models, with faster convergence and lower validation perplexity. The code is available at the https://github.com/MaeChd/muon-eq{MuonEq codebase}.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

Many machine learning models based on neural networks exhibit scaling laws: their performance scales as power laws with respect to the sizes of the model and training data set. We use large-N field theory methods to solve a model recently proposed by Maloney, Roberts and Sully which provides a simplified setting to study neural scaling laws. Our solution extends the result in this latter paper to general nonzero values of the ridge parameter, which are essential to regularize the behavior of the model. In addition to obtaining new and more precise scaling laws, we also uncover a duality transformation at the diagrams level which explains the symmetry between model and training data set sizes. The same duality underlies recent efforts to design neural networks to simulate quantum field theories.