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96,000 | It also includes tables for integral transforms. Another advantage of Gradshteyn and Ryzhik compared to computer algebra systems is the fact that all special functions and constants used in the evaluation of the integrals are listed in a registry as well, thereby allowing reverse lookup of integrals based on special fu... | Gradshteyn and Ryzhik | 0.778457 |
96,001 | Its main function was basic nuclear and particle physics research, with a focus on proton-proton and proton-neutron scattering.Comparisons were frequently drawn between the second cyclotron at the Harvard Cyclotron Laboratory and the Harwell Synchrocyclotron, and in 1974 clinicians from Oxford's Radcliffe Infirmary led... | Harwell Synchrocyclotron | 0.778457 |
96,002 | The InterMine web application allows creation of custom bioinformatics queries, includes template queries (web forms to run 'canned' queries). Users can upload and operate on lists of data. It is possible to configure/create widgets to analyse lists with graphs and enrichment statistics. An admin user can publish new t... | InterMine | 0.778456 |
96,003 | In some cases the two methods may yield essentially the same results. The basis functions are typically found by computing the eigenvectors of the covariance matrix of the data set. A more advanced technique is to form a kernel out of the data, using a fixed kernel. The basis functions from the eigenvectors of the kern... | Empirical orthogonal function | 0.778456 |
96,004 | In statistics and signal processing, the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. The term is also interchangeable with the geographically weighted Principal components analysis in geophys... | Empirical orthogonal function | 0.778456 |
96,005 | For a commutative ring R and R-modules A and B, TorRi(A, B) is an R-module (using that A ⊗R B is an R-module in this case). For a non-commutative ring R, TorRi(A, B) is only an abelian group, in general. If R is an algebra over a ring S (which means in particular that S is commutative), then TorRi(A, B) is at least an ... | Tor functor | 0.778456 |
96,006 | In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book Homological Al... | Tor functor | 0.778456 |
96,007 | In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie alge... | Tor functor | 0.778456 |
96,008 | The reason: every abelian group A has a free resolution of length 1, since every subgroup of a free abelian group is free abelian. For any ring R, Tor preserves direct sums (possibly infinite) and filtered colimits in each variable. For example, in the first variable, this says that Flat base change: for a commutative ... | Tor functor | 0.778455 |
96,009 | For example, if R is the polynomial ring k over a field k, then Tor ∗ R ( k , k ) {\displaystyle \operatorname {Tor} _{*}^{R}(k,k)} is the exterior algebra over k on n generators in Tor1. Tor i Z ( A , B ) = 0 {\displaystyle \operatorname {Tor} _{i}^{\mathbb {Z} }(A,B)=0} for all i ≥ 2. | Tor functor | 0.778455 |
96,010 | For a set S ∈ Σ, define Then ν is a Lebesgue measure on (Ω, Σ). Monotone convergence theorem: Suppose {fk}k ∈ N is a sequence of non-negative measurable functions such that Then, the pointwise limit f of fk is Lebesgue measurable and The value of any of the integrals is allowed to be infinite. Fatou's lemma: If {fk}k ∈... | Lebesgue integration | 0.778455 |
96,011 | Denote B R ≥ 0 {\textstyle \operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}}} the σ-algebra of Borel sets on . (By definition, B R ≥ 0 {\textstyle \operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}}} contains the set {+∞} and all Borel subsets of R ≥ 0 {\displaystyle \mathbb {R} _{\geq 0}} .) Consider a ( Σ , B R... | Lebesgue integration | 0.778455 |
96,012 | As shown above, the indicator function 1Q on the rationals is not Riemann integrable. In particular, the Monotone convergence theorem fails. To see why, let {ak} be an enumeration of all the rational numbers in (they are countable so this can be done). | Lebesgue integration | 0.778455 |
96,013 | A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to an orientation: Generalizing this to higher dimensions yields integration of differential forms. B... | Lebesgue integration | 0.778455 |
96,014 | Hence ∫ has a unique extension to all of L1. This integral is precisely the Lebesgue integral. More generally, when the measure space on which the functions are defined is also a locally compact topological space (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon ... | Lebesgue integration | 0.778455 |
96,015 | More precisely, the compactly supported functions form a vector space that carries a natural topology, and a (Radon) measure is defined as a continuous linear functional on this space. The value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to ... | Lebesgue integration | 0.778455 |
96,016 | In 2009, NASA's Kepler spacecraft was observing stars on its photometer, the instrument it uses to detect transit events, in which a planet crosses in front of and dims its host star for a brief and roughly regular period of time. In its last test, Kepler observed 50000 stars in the Kepler Input Catalog, including Kepl... | Kepler-90i | 0.778455 |
96,017 | Kepler-90i (also known by its Kepler Object of Interest designation KOI-351.08) is a super-Earth exoplanet with a radius 1.32 times that of Earth, orbiting the early G-type main sequence star Kepler-90 every 14.45 days, discovered by NASA's Kepler spacecraft. It is located about 2,840 light-years (870 parsecs, or nearl... | Kepler-90i | 0.778455 |
96,018 | Topological maps are a method of environment representation which capture the connectivity (i.e., topology) of the environment rather than creating a geometrically accurate map. Topological SLAM approaches have been used to enforce global consistency in metric SLAM algorithms.In contrast, grid maps use arrays (typicall... | Simultaneous localization and mapping | 0.778454 |
96,019 | For applications in mobile robotics (ex. drones, service robots), it is valuable to use low-power, lightweight equipment such as monocular cameras, or microelectronic microphone arrays. Audio-Visual SLAM can also allow for complimentary function of such sensors, by compensating the narrow field-of-view, feature occlusi... | Simultaneous localization and mapping | 0.778454 |
96,020 | SLAM algorithms are based on concepts in computational geometry and computer vision, and are used in robot navigation, robotic mapping and odometry for virtual reality or augmented reality. SLAM algorithms are tailored to the available resources and are not aimed at perfection but at operational compliance. Published a... | Simultaneous localization and mapping | 0.778454 |
96,021 | Recent approaches apply quasi-optical wireless ranging for multi-lateration (real-time locating system (RTLS)) or multi-angulation in conjunction with SLAM as a tribute to erratic wireless measures. A kind of SLAM for human pedestrians uses a shoe mounted inertial measurement unit as the main sensor and relies on the f... | Simultaneous localization and mapping | 0.778454 |
96,022 | In robotics, GraphSLAM is a SLAM algorithm which uses sparse information matrices produced by generating a factor graph of observation interdependencies (two observations are related if they contain data about the same landmark). | Simultaneous localization and mapping | 0.778454 |
96,023 | In robotics, EKF SLAM is a class of algorithms which uses the extended Kalman filter (EKF) for SLAM. Typically, EKF SLAM algorithms are feature based, and use the maximum likelihood algorithm for data association. In the 1990s and 2000s, EKF SLAM had been the de facto method for SLAM, until the introduction of FastSLAM... | Simultaneous localization and mapping | 0.778454 |
96,024 | In neuroscience, the hippocampus appears to be involved in SLAM-like computations, giving rise to place cells, and has formed the basis for bio-inspired SLAM systems such as RatSLAM. | Simultaneous localization and mapping | 0.778454 |
96,025 | Resonance Raman spectroscopy with ultraviolet excitation can be used to examine the chemistry, structure, and intermolecular interactions of nucleic acids, specifically the bases. Interactions between nucleic acids and DNA-binding compounds such as drugs can be examined by selectively exciting either the nucleobases or... | Resonance Raman spectroscopy | 0.778454 |
96,026 | The real rank of a unital C*-algebra A is the smallest non-negative integer n, denoted RR(A), such that for every (n + 1)-tuple (x0, x1, ... ,xn) of self-adjoint elements of A and every ε > 0, there exists an (n + 1)-tuple (y0, y1, ... ,yn) of self-adjoint elements of A such that ∑ i = 0 n y i 2 {\displaystyle \sum _{i... | Real rank (C*-algebras) | 0.778453 |
96,027 | Formulas that hold for dimension may not generalize for real rank. For example, Brown and Pedersen conjectured that RR(A ⊗ B) ≤ RR(A) + RR(B), since it is true that dim(X × Y) ≤ dim(X) + dim(Y). They proved a special case that if A is AF and B has real rank zero, then A ⊗ B has real rank zero. But in general their conj... | Real rank (C*-algebras) | 0.778453 |
96,028 | If X is a locally compact Hausdorff space, then RR(C0(X)) = dim(X), where dim is the Lebesgue covering dimension of X. As a result, real rank is considered a noncommutative generalization of dimension, but real rank can be rather different when compared to dimension. For example, most noncommutative tori have real rank... | Real rank (C*-algebras) | 0.778453 |
96,029 | Since simple graph C*-algebras are either AF or purely infinite, every simple graph C*-algebra has real rank zero. Having real rank zero is a property closed under taking direct limits, hereditary C*-subalgebras, and strong Morita equivalence. In particular, if A has real rank zero, then Mn(A), the algebra of n × n mat... | Real rank (C*-algebras) | 0.778453 |
96,030 | C*-algebras with real rank zero are of particular interest. By definition, a unital C*-algebra has real rank zero if and only if the invertible self-adjoint elements of A are dense in the self-adjoint elements of A. This condition is equivalent to the previously studied conditions: (FS) The self-adjoint elements of A w... | Real rank (C*-algebras) | 0.778453 |
96,031 | In mathematics, the real rank of a C*-algebra is a noncommutative analogue of Lebesgue covering dimension. The notion was first introduced by Lawrence G. Brown and Gert K. Pedersen. | Real rank (C*-algebras) | 0.778453 |
96,032 | PicSat is a French observatory nanosatellite, designed to measure the transit of Beta Pictoris b, an exoplanet which orbits the star Beta Pictoris. PicSat was designed and built by a team of scientists led by Dr. Sylvestre Lacour, astrophysicist and instrumentalist at the High Angular Resolution in Astrophysics group i... | PicSat | 0.778453 |
96,033 | The revision DTLS 1.2 based on TLS 1.2 was published in January 2012.TLS 1.3 (2018) specified in RFC 8446 includes major optimizations and security improvements. QUIC (2021) specified in RFC 9000 and DTLS 1.3 (2022) specified in RFC 9147 builds on TLS 1.3. The publishing of TLS 1.3 and DTLS 1.3 obsoleted TLS 1.2 and DT... | Comparison of TLS implementations | 0.778452 |
96,034 | Linchpin analysis proceeds from information that is certain, or with a high probability of being certain. In mathematics and physics, a similar problem formation, which constrains the solution by certain known or impossible conditions, is the boundary value condition. By starting from knowns (and impossibilities), the ... | Intelligence analyst | 0.778449 |
96,035 | gbXML is the underlying architecture of Autodesk's Green Building Studio commercial on-line energy analysis product, and is the main export option for energy analysis from their modeling products. It is often used for geometry data transformation, but the quality of exported models is not good. Lighting systems, Heatin... | Green Building XML | 0.778448 |
96,036 | The Kintex-7 family is the first Xilinx mid-range FPGA family that the company claims delivers Virtex-6 family performance at less than half the price while consuming 50 percent less power. The Kintex family includes high-performance 12.5 Gbit/s or lower-cost optimized 6.5 Gbit/s serial connectivity, memory, and logic ... | Xilinx Kintex | 0.778445 |
96,037 | This campaign enables AWS Marketplace's Amazon Machine Images (AMIs) with associated Amazon FPGA Instances created by partners. The two companies released software development tools to simplify the creation of FPGA technology. The tools create and manage the machine images created and sold by partners.In July 2018, Xil... | Xilinx Kintex | 0.778445 |
96,038 | International Federation of Gynecology and Obstetrics (FIGO) staging is done at the time of surgery: Stage 0: Carcinoma in situ Stage I: Growth limited to fallopian tubes Stage II: Growth involving one or both fallopian tubes with extension to pelvis Stage III: Tumor involving one or both fallopian tubes with spread ou... | Fallopian tube cancer | 0.778445 |
96,039 | Rothblatt envisions the creation of "mindfiles" – collections of data from all kinds of sources, including the photos we upload to Facebook, the discussions and opinions we share on forums or blogs, and other social media interactions that reflect our life experiences and our unique self.Richard Grandmorin summarized t... | Digital immortality | 0.778445 |
96,040 | In the novel / Netflix series Altered Carbon, a person's memories and consciousness can be stored in a disk-shaped device called a cortical stack, which is implanted into the cervical vertebrae. In Frictional Games' SOMA, the story revolves around the problem of existing as a digital personality scan taken from a physi... | Digital immortality | 0.778445 |
96,041 | Defining the avatar to be alive allows it to communicate with the future in the sense that it continues to learn, evolve and interact with people, if they still exist. Technically, the operation exists to implement an artificial intelligence system to the avatar. This artificial intelligence system is then assumed to t... | Digital immortality | 0.778445 |
96,042 | The National Science Foundation has awarded a half-million-dollar grant to the universities of Central Florida at Orlando and Illinois at Chicago to explore how researchers might use artificial intelligence, archiving, and computer imaging to create convincing, digital versions of real people, a possible first step tow... | Digital immortality | 0.778445 |
96,043 | Several miniature double-focusing mass analyzers have been fabricated. A non-scanning Mattauch–Herzog geometry sector was developed using new materials to construct a lighter magnet. Under the collaboration of University of Minnesota and Universidad de Costa Rica, a miniature double-focusing sector was produced under s... | Miniature mass spectrometer | 0.778443 |
96,044 | These mass analyzers are used for supporting the pharmaceutical process chemistry.Several other MMS instruments have been also fabricated using ion trap mass analyzers, including Tridion-9 GCMS from TorionInc, now part of Perkin Elmer (AmericanFork, Utah), GC/QIT from the Jet Propulsion Laboratory, Chemsense 600 from G... | Miniature mass spectrometer | 0.778443 |
96,045 | Ion traps include quadrupole ion traps or Paul trap, Fourier transform ion cyclotron resonance or Penning trap and newly developed orbitrap. However, Paul trap receives a great focus from researchers for a MMS because of its distinct advantages over other mass analyzers for building MMS. One of the benefits is that ion... | Miniature mass spectrometer | 0.778443 |
96,046 | The privacy calculus model posits that two factors determine privacy behavior, namely privacy concerns (or perceived risks) and expected benefits. By now, the privacy calculus has been supported by several studies. | Information dissemination | 0.77844 |
96,047 | Similarly to actions which reduce privacy, there are multiple angles of privacy and multiple techniques to improve them to varying extents. When actions are done at an organizational level, they may be referred to as cybersecurity. | Information dissemination | 0.77844 |
96,048 | In software engineering, CI/CD or CICD is the combined practices of continuous integration (CI) and continuous delivery (CD) or, less often, continuous deployment. They are sometimes referred to collectively as continuous development or continuous software development. | CI/CD | 0.77844 |
96,049 | Given a *-ring, there is also the map −*: x ↦ −x*. It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as 1 ↦ −1, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where x ... | Involutive ring | 0.77844 |
96,050 | The *-operation on a *-ring is analogous to complex conjugation on the complex numbers. The *-operation on a *-algebra is analogous to taking adjoints in complex matrix algebras. | Involutive ring | 0.77844 |
96,051 | Not every algebra admits an involution: Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra: Any nontrivial antiautomorphism necessarily has the form: for any complex number z ∈ C {\displaystyle z\in \mathbb {C} } . It follows that any nontrivial antiautomorphism fails to be idempotent: ... | Involutive ring | 0.77844 |
96,052 | It consists of two parts: The first, larger part contains peer-reviewed scientific articles from the various fields of chemistry and chemical engineering, written in English and accompanied by abstracts in Macedonian. The second part, written in Macedonian or in English, contains society and related news. The journal w... | Macedonian Journal of Chemistry and Chemical Engineering | 0.77844 |
96,053 | The Macedonian Journal of Chemistry and Chemical Engineering is a peer-reviewed scientific journal of chemistry established in 1974 by the Society of Chemists and Technologists of Macedonia. Since 2022 it is co-published with the Ss. Cyril and Methodius University of Skopje. | Macedonian Journal of Chemistry and Chemical Engineering | 0.77844 |
96,054 | The reaction centers around an imine - lithium enolate cycloaddition. In order to ensure the correct stereochemistry (the phenyl group and the silyl ether must adopt a cis configuration) in the β-lactam a chiral auxiliary is used in the enolate synthesis. The enolate synthesis starts from glycolic acid. | Ojima lactam | 0.778438 |
96,055 | In the 1980s the aperture synthesis interferometric imaging technique was extended to visible light and infrared astronomy by the Cavendish Astrophysics Group, providing the first very high resolution images of nearby stars. In 1995 this technique was demonstrated on an array of separate optical telescopes for the firs... | Radar interferometry | 0.778437 |
96,056 | One of the first uses of optical interferometry was applied by the Michelson stellar interferometer on the Mount Wilson Observatory's reflector telescope to measure the diameters of stars. The red giant star Betelgeuse was the first to have its diameter determined in this way on December 13, 1920. In the 1940s radio in... | Radar interferometry | 0.778437 |
96,057 | Environmental Health Perspectives Environmental Toxicology Environmental Toxicology & Pharmacology Journal of Environmental Science and Health Journal of Toxicology and Environmental Health Toxicological & Environmental Chemistry | Environmental toxicology | 0.778436 |
96,058 | American College of Toxicology (ACT) Society of Environmental Toxicology and Chemistry (SETAC) Society of Toxicology (SOT) | Environmental toxicology | 0.778436 |
96,059 | When humans and animals are exposed to high radiation levels, they can develop cancer, congenital disabilities, or skin burns. Plants also face problems when exposed to large levels of radiation. After the Chernobyl disaster in 1986, the nuclear radiation damaged the surrounding plants' reproductive tissues, and it too... | Environmental toxicology | 0.778436 |
96,060 | A computational simulation shows that nucleotide concentration of nucleotide catalysis of "the energy currency pathway is favored, as energy is limiting; favoring this pathway feeds forward into a greater nucleotide synthesis". Fast nucleotide catalysis of CO2 fixation lowers nucleotide concentration as protocell growt... | Hydrothermal vent | 0.778433 |
96,061 | The chemical and thermal dynamics in hydrothermal vents makes such environments highly suitable thermodynamically for chemical evolution processes to take place. Therefore, thermal energy flux is a permanent agent and is hypothesized to have contributed to the evolution of the planet, including prebiotic chemistry.Günt... | Hydrothermal vent | 0.778433 |
96,062 | The primary concern of automated synthesis is job displacement. Other concerns are high initial investment and maintenance costs, privacy concerns, and an over-dependence on technology. There are also ethical concerns, regarding the use of artificial intelligence and robotics. See Ethics of artificial intelligence, Rob... | Fully automated synthesis | 0.778432 |
96,063 | Parts of procedures and techniques were automated throughout the 19th and 20th centuries, using simple circuit boards. The first fully automatic synthesis was a peptide synthesis by Robert Merrifield and John Stewart in 1966. Applications of artificial intelligence to organic synthesis also started in the 1960s with th... | Fully automated synthesis | 0.778432 |
96,064 | Over the years, multiple synthesizers have been developed to assist with automated synthesis, including the Chemspeed Accelerator (SLT106, SLT II, ASW2000, SwingSLT, Autoplant A100, and SLT100), the Symyx system, and Freeslate ScPPR. Recently, researchers have investigated the optimization of these methods for controll... | Fully automated synthesis | 0.778432 |
96,065 | Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module. | Artinian ring | 0.778432 |
96,066 | For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Wedderburn–Artin theorem characterizes every simple Artinian ring as a ring of matrices over a division ring. | Artinian ring | 0.778432 |
96,067 | In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain con... | Artinian ring | 0.778432 |
96,068 | For each n ≥ 1 {\displaystyle n\geq 1} , the full matrix ring M n ( R ) {\displaystyle M_{n}(R)} over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian).The following two are examples of non-Artinian rings. If R is any ring, then the polynomial ring R is not Artinian, since the idea... | Artinian ring | 0.778432 |
96,069 | Then k / ( t n ) {\displaystyle k/(t^{n})} is Artinian for every positive integer n. Similarly, k / ( x 2 , y 3 , x y 2 ) = k ⊕ k ⋅ x ⊕ k ⋅ y ⊕ k ⋅ x y ⊕ k ⋅ y 2 {\displaystyle k/(x^{2},y^{3},xy^{2})=k\oplus k\cdot x\oplus k\cdot y\oplus k\cdot xy\oplus k\cdot y^{2}} is an Artinian ring with maximal ideal ( x , y ) {... | Artinian ring | 0.778432 |
96,070 | Let M be a left module over a left Artinian ring. Then the following are equivalent (Hopkins' theorem): (i) M is finitely generated, (ii) M has finite length (i.e., has composition series), (iii) M is Noetherian, (iv) M is Artinian. | Artinian ring | 0.778432 |
96,071 | In 2010, the Astro2010 Decadal Report was released and did not recommend that NASA continue the development of the SIM Lite Astrometric Observatory. This prompted NASA Astronomy and Physics Director, Jon Morse, to issue a letter on 24 September 2010 to the SIM Lite project manager, informing him that NASA was discontin... | Space Interferometry Mission | 0.778431 |
96,072 | However, the preliminary budget for NASA for 2008 included zero dollars for SIM.In 2007, the Congress restored funding for fiscal year 2008 as part of an omnibus appropriations bill which the President later signed. At the same time the Congress directed NASA to move the mission forward to the development phase. In 200... | Space Interferometry Mission | 0.778431 |
96,073 | After an additional $51.9 million decrease in FY 2009, the program was reduced to $6 million in FY 2010 supplemented by substantial carryover from the previous year while awaiting the results of the Astronomy and Astrophysics Decadal Survey, Astro2010.By February 2007 many of the budget cuts outlined in the FY 2007 bud... | Space Interferometry Mission | 0.778431 |
96,074 | An isosceles triangle also has two angles of the same measure, namely the angles opposite to the two sides of the same length. This fact is the content of the isosceles triangle theorem, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an... | Scalene triangle | 0.778431 |
96,075 | So the sum of the angles in this triangle is 90° + 90° + 90° = 270°. The triangle encloses 1/4 of the northern hemisphere (90°/360° as viewed from the North Pole) and therefore 1/8 of the Earth's surface, so in the formula f = 1/8; thus the formula correctly gives the sum of the triangle's angles as 270°. From the abov... | Scalene triangle | 0.778431 |
96,076 | A non-planar triangle is a triangle which is not contained in a Euclidean plane. Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. While the measures of the internal angles in planar triangles always sum to 18... | Scalene triangle | 0.778431 |
96,077 | NGC 6751, also known as the Glowing Eye Nebula or the Dandelion Puffball Nebula, is a planetary nebula in the constellation Aquila. It is estimated to be about 6,500 light-years (2.0 kiloparsecs) away.NGC 6751 was discovered by the astronomer Albert Marth on 20 July 1863. John Louis Emil Dreyer, the compiler of the New... | Glowing Eye | 0.778427 |
96,078 | It consists of elementary transformations that represent a generalized inversion into a four-dimensional hypersphere: x ′ = k 2 x x 2 + y 2 + z 2 + u 2 , z ′ = k 2 z x 2 + y 2 + z 2 + u 2 , y ′ = k 2 y x 2 + y 2 + z 2 + u 2 , u ′ = k 2 u x 2 + y 2 + z 2 + u 2 , {\displaystyle {\begin{aligned}x'&={\frac {k^{2}x}{x^{2}+y... | Spherical wave transformation | 0.778426 |
96,079 | This group of transformations has been discussed by S. Lie; it is the group of transformations which transform lines of curvature on a surface enveloped by spherical waves into lines of curvature on the surface enveloped by the corresponding spherical waves. Depending on λ {\displaystyle \lambda } they can be different... | Spherical wave transformation | 0.778426 |
96,080 | Bateman, Harry (1912) . "Some geometrical theorems connected with Laplace's equation and the equation of wave motion". American Journal of Mathematics. | Spherical wave transformation | 0.778426 |
96,081 | 54 (14): 1–13. Bateman, Harry (1910b). "The Relation between Electromagnetism and Geometry". | Spherical wave transformation | 0.778426 |
96,082 | It was recently pointed out: A Laguerre transformation (L-transform) is a mapping which is bijective on the sets of oriented planes and oriented spheres, respectively, and preserves tangency between plane and sphere. L-transforms are more easily understood if we use the so-called cyclographic model of Laguerre geometry... | Spherical wave transformation | 0.778426 |
96,083 | On the other hand, Laguerre once introduced into geometry a group of transformations that change the spheres into spheres. These two groups are isomorphic, so that mathematically these two theories, one physical, the other one geometric, show no essential difference. | Spherical wave transformation | 0.778426 |
96,084 | After the first comparison of the Lorentz transformation and the Laguerre inversion by Bateman (1910) as mentioned above, the equivalence of both groups was pointed out by Cartan in 1912 and 1914, and he expanded upon it in 1915 (published 1955) in the French version of Klein's encyclopedia. Also Poincaré (1912, publis... | Spherical wave transformation | 0.778426 |
96,085 | Another method for solving such sphere problems was to write down the coordinates together with the sphere's radius. This was employed by Lie (1871) in the context of Lie sphere geometry which represents a general framework of sphere-transformations (being a special case of contact transformations) conserving lines of ... | Spherical wave transformation | 0.778426 |
96,086 | doi:10.1016/0039-3681(92)90015-X. Warwick, Andrew (2003). Masters of Theory: Cambridge and the Rise of Mathematical Physics. | Spherical wave transformation | 0.778426 |
96,087 | To appear in Einstein Studies, D. Rowe, ed., Basel: Birkhäuser. Warwick, Andrew (1992). "Cambridge mathematics and Cavendish physics: Cunningham, Campbell and Einstein's relativity 1905–1911 Part I: The uses of theory". | Spherical wave transformation | 0.778426 |
96,088 | "Erlangen program and higher geometry – Laguerre geometry". In Heinrich Behnke (ed.). Fundamentals of Mathematics: Geometry. | Spherical wave transformation | 0.778426 |
96,089 | "On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics". Annalen der Physik. 520 (9–10): 631–690. | Spherical wave transformation | 0.778426 |
96,090 | Cecil, Thomas E. (2008) , "Laguerre geometry", Lie sphere geometry, Springer, pp. 37–46, ISBN 978-0387746555 Coolidge, Julian (1916). | Spherical wave transformation | 0.778426 |
96,091 | Classical Geometries in Modern Contexts: Geometry of Real Inner Product Spaces Third Edition. Springer. pp. | Spherical wave transformation | 0.778426 |
96,092 | However, only the Lorentz/Poincaré groups represent symmetries of all laws of nature including mechanics, whereas the conformal group is related to certain areas such as electrodynamics. In addition, it can be shown that the conformal group of the plane (corresponding to the Möbius group of the extended complex plane) ... | Spherical wave transformation | 0.778426 |
96,093 | Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman giving the transformation its name. They correspond to the conformal group o... | Spherical wave transformation | 0.778426 |
96,094 | doi:10.1007/bf01446331. S2CID 122317672. English translation by David Delphenich: On complexes - in particular, line and sphere complexes - with applications to the theory of partial differential equations Lie, Sophus; Scheffers, Georg (1896). | Spherical wave transformation | 0.778426 |
96,095 | It was also noted that the Laguerre group is part of a wider group conserving tangential distances, called the "equilong group" by Scheffers (1905).In R2 the Laguerre group leaves invariant the relation d x 2 + d y 2 − d r 2 {\displaystyle dx^{2}+dy^{2}-dr^{2}} , which can be extended to arbitrary Rn as well. For insta... | Spherical wave transformation | 0.778426 |
96,096 | As shown above, the group of conformal point transformations in Rn (composed of motions, similarities, and inversions) can be related by minimal projection to the group of contact transformations in Rn-1 transforming circles or spheres into other circles or spheres. In addition, Lie (1871, 1896) pointed out that in R3 ... | Spherical wave transformation | 0.778426 |
96,097 | Following Müller's paper, Smith (1900) discussed Laguerre's transformation and the related "group of the geometry of reciprocal directions". Alluding to Klein's (1893) treatment of minimal projection, he pointed out that this group "is simply isomorphic with the group of all displacements and symmetry transformations i... | Spherical wave transformation | 0.778426 |
96,098 | Above, the connection of conformal transformations with coordinates including the radius of spheres within Lie sphere geometry was mentioned. The special case λ = 1 {\displaystyle \lambda =1} corresponds to a sphere transformation given by Edmond Laguerre (1880-1885), who called it the "transformation by reciprocal dir... | Spherical wave transformation | 0.778426 |
96,099 | One of the axioms of a Weil theory is the so-called hard Lefschetz theorem (or axiom): Begin with a fixed smooth hyperplane section W = H ∩ X,where X is a given smooth projective variety in the ambient projective space P N and H is a hyperplane. Then for i ≤ n = dim(X), the Lefschetz operator L: H i(X) → H i+2(X),which... | Standard conjectures on algebraic cycles | 0.778425 |
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