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wiki_6400_chunk_0 | Vitali–Carathéodory theorem | In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the conditions stated below, integrable functions can be approximated in L1 from above and below by lower- and upper-semicontinuous functions, respectively. It is named after Giuseppe Vitali and Constantin Carathéodory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6401_chunk_0 | Vitali–Hahn–Saks theorem | In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali (1907), Hahn (1922), and Saks (1933), proves that under some conditions a sequence of measures converging point-wise does so uniformly and the limit is also a measure. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6402_chunk_0 | Vogel plane | In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of P2/S3, the projective plane P2 divided out by the symmetric group S3 of permutations of coordinates. It was introd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6403_chunk_0 | Volterra integral equation | In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is f ( t ) = ∫ a t K ( t , s ) x ( s ) d s {\displaystyle f(t)=\int _{a}^{t}K(t,s)\,x(s)\,ds} where f is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6404_chunk_0 | Volterra integral equation | {\displaystyle x(t)=f(t)+\int _{a}^{t}K(t,s)x(s)\,ds.} In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6405_chunk_0 | Volterra integral equation | A linear Volterra integral equation is a convolution equation if x ( t ) = f ( t ) + ∫ t 0 t K ( t − s ) x ( s ) d s . {\displaystyle x(t)=f(t)+\int _{t_{0}}^{t}K(t-s)x(s)\,ds.} The function K {\displaystyle K} in the integral is called the kernel. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6406_chunk_0 | Volterra integral equation | Such equations can be analyzed and solved by means of Laplace transform techniques. For a weakly singular kernel of the form K ( t , s ) = ( t 2 − s 2 ) − α {\displaystyle K(t,s)=(t^{2}-s^{2})^{-\alpha }} with 0 < α < 1 {\displaystyle 0<\alpha <1} , Volterra integral equation of the first kind can conveniently be trans... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6407_chunk_0 | Volterra lattice | In mathematics, the Volterra lattice, also known as the discrete KdV equation, the Kac–van Moerbeke lattice, and the Langmuir lattice, is a system of ordinary differential equations with variables indexed by some of the points of a 1-dimensional lattice. It was introduced by Marc Kac and Pierre van Moerbeke (1975) and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6408_chunk_0 | Volterra lattice | The Volterra lattice also behaves like a discrete version of the KdV equation. The Volterra lattice is an integrable system, and is related to the Toda lattice. It is also used as a model for Langmuir waves in plasmas. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6409_chunk_0 | Voorhoeve index | In mathematics, the Voorhoeve index is a non-negative real number associated with certain functions on the complex numbers, named after Marc Voorhoeve. It may be used to extend Rolle's theorem from real functions to complex functions, taking the role that for real functions is played by the number of zeros of the funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6410_chunk_0 | Vámos matroid | In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be represented as a matrix over any field. It is named after English mathematician Peter Vámos, who first described it in an unpublished manuscript in 1968. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6411_chunk_0 | Walter's theorem | In mathematics, the Walter theorem, proved by John H. Walter (1967, 1969), describes the finite groups whose Sylow 2-subgroup is abelian. Bender (1970) used Bender's method to give a simpler proof. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6412_chunk_0 | Kantorovich metric | In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space M {\displaystyle M} . It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on M {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6413_chunk_0 | Weber modular function | In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6414_chunk_0 | Weierstrass M-test | In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6415_chunk_0 | Weierstrass elliptic function | In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6416_chunk_0 | Weierstrass function | In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6417_chunk_0 | Weierstrass function | The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply alm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6418_chunk_0 | Weierstrass sigma function | In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and ℘ {\displaystyle \wp } functions is analogous to that between the sine, cotangent, and squared cos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6419_chunk_0 | Weierstrass preparation theorem | In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic, and whose coefficients of lower d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6420_chunk_0 | Weierstrass product inequality | In mathematics, the Weierstrass product inequality states that for any real numbers 0 ≤ x1, ..., xn ≤ 1 we have ( 1 − x 1 ) ( 1 − x 2 ) ( 1 − x 3 ) ( 1 − x 4 ) . . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6421_chunk_0 | Weierstrass product inequality | ( 1 − x n ) ≥ 1 − S n , {\displaystyle (1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})....(1-x_{n})\geq 1-S_{n},} ( 1 + x 1 ) ( 1 + x 2 ) ( 1 + x 3 ) ( 1 + x 4 ) . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6422_chunk_0 | Weierstrass product inequality | . ( 1 + x n ) ≥ 1 + S n , {\displaystyle (1+x_{1})(1+x_{2})(1+x_{3})(1+x_{4})....(1+x_{n})\geq 1+S_{n},} where S n = x 1 + x 2 + x 3 + x 4 + . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6423_chunk_0 | Weierstrass product inequality | . . + x n . {\displaystyle S_{n}=x_{1}+x_{2}+x_{3}+x_{4}+....+x_{n}.} The inequality is named after the German mathematician Karl Weierstrass. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6424_chunk_0 | Weierstrass transform | In mathematics, the Weierstrass transform of a function f: R → R, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x. Specifically, it is the function F defined by F ( x ) = 1 4 π ∫ − ∞ ∞ f ( y ) e − ( x − y ) 2 4 d y = 1 4 π ∫ − ∞... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6425_chunk_0 | Weierstrass transform | Instead of F(x) one also writes W(x). Note that F(x) need not exist for every real number x, when the defining integral fails to converge. The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function f describes the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6426_chunk_0 | Weierstrass–Enneper parameterization | In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863. Let f {\displaystyle f} and g {\displaystyle g} be functions on either the entire complex plane or the unit disk, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6427_chunk_0 | Weil conjectures | In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. The conjectures concern the generating functions (known as l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6428_chunk_0 | Weil conjectures | The generating function has coefficients derived from the numbers Nk of points over the extension field with qk elements. Weil conjectured that such zeta functions for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6429_chunk_0 | Conjecture | In mathematics, the Weil conjectures were some highly influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. A variety V over a finite field with q elements has a finite number of ration... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6430_chunk_0 | Conjecture | Weil conjectured that such zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis. The rationality was proved by Dwork (1960), the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6431_chunk_0 | Weil pairing | In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6432_chunk_0 | Weil reciprocity | In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then f((g)) = g((f))where the notation has this meaning: (h) is the divisor of the function h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6433_chunk_0 | Weil reciprocity | When f and g both take the values 0 or ∞ at P, the definition is essentially in limiting or removable singularity terms, by considering (up to sign) fagbwith a and b such that the function has neither a zero nor a pole at P. This is achieved by taking a to be the multiplicity of g at P, and −b the multiplicity of f at ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6434_chunk_0 | Weinstein conjecture | In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the conjecture claims that on a compact contact manifold, its Reeb vector field should carry at least one periodic orbit. By definition, a level set of contact type ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6435_chunk_0 | Weinstein conjecture | It is a fact that any contact manifold (M,α) can be embedded into a canonical symplectic manifold, called the symplectization of M, such that M is a contact type level set (of a canonically defined Hamiltonian) and the Reeb vector field is a Hamiltonian flow. That is, any contact manifold can be made to satisfy the req... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6436_chunk_0 | Weinstein conjecture | In several cases, the existence of a periodic orbit was known. For instance, Rabinowitz showed that on star-shaped level sets of a Hamiltonian function on a symplectic manifold, there were always periodic orbits (Weinstein independently proved the special case of convex level sets). Weinstein observed that the hypothes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6437_chunk_0 | Weinstein conjecture | (Weinstein's original conjecture included the condition that the first de Rham cohomology group of the level set is trivial; this hypothesis turned out to be unnecessary). The Weinstein conjecture was first proved for contact hypersurfaces in R 2 n {\displaystyle \mathbb {R} ^{2n}} in 1986 by Viterbo, then extended to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6438_chunk_0 | Weinstein conjecture | All these cases dealt with the situation where the contact manifold is a contact submanifold of a symplectic manifold. A new approach without this assumption was discovered in dimension 3 by Hofer and is at the origin of contact homology.The Weinstein conjecture has now been proven for all closed 3-dimensional manifold... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6439_chunk_0 | Sylvester's determinant theorem | In mathematics, the Weinstein–Aronszajn identity states that if A {\displaystyle A} and B {\displaystyle B} are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided A B {\displaystyle AB} (and hence, also B A {\displaystyle BA} ) is of trace class, det ( I m + A B ) = d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6440_chunk_0 | Weyl character formula | In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl (1925, 1926a, 1926b). There is a closely related formula for the character of an irreducible representation o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6441_chunk_0 | Weyl character formula | In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimensi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6442_chunk_0 | Weyl character formula | The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. Knowledge of the character χ {\displaystyle \chi } of π {\displaystyle \pi } gives a lot of information about π {\displaystyle \pi } itself. Weyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6443_chunk_0 | Weyl integration formula | In mathematics, the Weyl integration formula, introduced by Hermann Weyl, is an integration formula for a compact connected Lie group G in terms of a maximal torus T. Precisely, it says there exists a real-valued continuous function u on T such that for every class function f on G: ∫ G f ( g ) d g = ∫ T f ( t ) u ( t )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6444_chunk_0 | Weyl–von Neumann theorem | In mathematics, the Weyl–von Neumann theorem is a result in operator theory due to Hermann Weyl and John von Neumann. It states that, after the addition of a compact operator (Weyl (1909)) or Hilbert–Schmidt operator (von Neumann (1935)) of arbitrarily small norm, a bounded self-adjoint operator or unitary operator on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6445_chunk_0 | Weyl–von Neumann theorem | The theorem and its generalizations were one of the starting points of operator K-homology, developed first by Lawrence G. Brown, Ronald Douglas and Peter Fillmore and, in greater generality, by Gennadi Kasparov. In 1958 Kuroda showed that the Weyl–von Neumann theorem is also true if the Hilbert–Schmidt class is replac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6446_chunk_0 | Whitehead continuum | In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R 3 . {\displaystyle \mathbb {R} ^{3}.} J. H. C. Whitehead (1935) discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead (1934, theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6447_chunk_0 | Whitehead continuum | One can ask whether all contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the Riemann mapping theorem. Dimension 3 presents the first counterexample: the Whitehead manifold. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6448_chunk_0 | Whitehead product | In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in (Whitehead 1941). The relevant MSC code is: 55Q15, Whitehead products and generalizations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6449_chunk_0 | Whitney inequality | In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of smoothness. It was first proved by Hassler Whitney in 1957, and is an important tool in the field of approximation theory for obtaining upper estimates on the errors of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6450_chunk_0 | Wielandt theorem | In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers z {\displaystyle z} for which R e z > 0 {\displaystyle \mathrm {Re} \,z>0} by Γ ( z ) = ∫ 0 + ∞ t z − 1 e − t d t , {\displaystyle \Gamma (z)=\int _{0}^{+\infty }t^{z-1}\mathrm {e} ^{-t}\,\mathrm {d} t,} as the only f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6451_chunk_0 | Wiener algebra | In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by A(T), is the space of absolutely convergent Fourier series. Here T denotes the circle group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6452_chunk_0 | Wiener integral | In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6453_chunk_0 | Wiener integral | In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6454_chunk_0 | Wiener integral | It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6455_chunk_0 | Wiener integral | In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6456_chunk_0 | Wiener series | In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite polynomial expansion has to a power series. For... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6457_chunk_0 | Wiener series | The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise. This property allows the terms to be identified in applications by the Lee–Schetzen method. The Wiener series is important in nonlinear system identification. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6458_chunk_0 | Wiener series | In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience. The name Wiener series is almost exclusively used in system theory. In the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6459_chunk_0 | Wiener–Wintner theorem | In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by Wiener and Wintner (1941). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6460_chunk_0 | Witten zeta function | In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group. These zeta functions were introduced by Don Zagier who named them after Edward Witten's study of their special values (among other things). Note... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6461_chunk_0 | Wright Omega function | In mathematics, the Wright omega function or Wright function, denoted ω, is defined in terms of the Lambert W function as: ω ( z ) = W ⌈ I m ( z ) − π 2 π ⌉ ( e z ) . {\displaystyle \omega (z)=W_{{\big \lceil }{\frac {\mathrm {Im} (z)-\pi }{2\pi }}{\big \rceil }}(e^{z}).} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6462_chunk_0 | Wythoff array | In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6463_chunk_0 | X-ray transform | In mathematics, the X-ray transform (also called ray transform or John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6464_chunk_0 | X-ray transform | The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ƒ represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ƒ. Inversion of the X-ray transform is therefore of practical impor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6465_chunk_0 | Yang–Mills–Higgs equations | In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are D A ∗ F A + = 0 , D A ∗ D A Φ = 0 {\displaystyle {\begin{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6466_chunk_0 | Yoneda lemma | In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6467_chunk_0 | Yoneda lemma | It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6468_chunk_0 | Young–Deruyts development | In mathematics, the Young–Deruyts development is a method of writing invariants of an action of a group on an n-dimensional vector space V in terms of invariants depending on at most n–1 vectors (Dieudonné & Carrell 1970, 1971, p.36, 39). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6469_chunk_0 | Z function | In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function. It can be defined in t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6470_chunk_0 | Z function | It follows from the functional equation of the Riemann zeta function that the Z function is real for real values of t. It is an even function, and real analytic for real values. It follows from the fact that the Riemann-Siegel theta function and the Riemann zeta function are both holomorphic in the critical strip, wher... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6471_chunk_0 | Zahlbericht | In mathematics, the Zahlbericht (number report) was a report on algebraic number theory by Hilbert (1897, 1998, (English translation)). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6472_chunk_0 | Zak transform | In mathematics, the Zak transform (also known as the Gelfand mapping) is a certain operation which takes as input a function of one variable and produces as output a function of two variables. The output function is called the Zak transform of the input function. The transform is defined as an infinite series in which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6473_chunk_0 | Zassenhaus algorithm | In mathematics, the Zassenhaus algorithm is a method to calculate a basis for the intersection and sum of two subspaces of a vector space. It is named after Hans Zassenhaus, but no publication of this algorithm by him is known. It is used in computer algebra systems. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6474_chunk_0 | Zernike polynomial | In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play important roles in various optics branches such as beam optics and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6475_chunk_0 | Modulus of a complex number | In mathematics, the absolute value or modulus of a real number x {\displaystyle x} , denoted | x | {\displaystyle |x|} , is the non-negative value of x {\displaystyle x} without regard to its sign. Namely, | x | = x {\displaystyle |x|=x} if x {\displaystyle x} is a positive number, and | x | = − x {\displaystyle |x|=-x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6476_chunk_0 | Modulus of a complex number | The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6477_chunk_0 | Abstract additive Schwarz method | In mathematics, the abstract additive Schwarz method, named after Hermann Schwarz, is an abstract version of the additive Schwarz method for boundary value problems on partial differential equations, formulated only in terms of linear algebra without reference to domains, subdomains, etc. Many if not all domain decompo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6478_chunk_0 | Actuarial polynomials | In mathematics, the actuarial polynomials a(β)n(x) are polynomials studied by Toscano (1950) given by the generating function ∑ n a n ( β ) ( x ) n ! t n = exp ( β t + x ( 1 − e t ) ) {\displaystyle \displaystyle \sum _{n}{\frac {a_{n}^{(\beta )}(x)}{n! }}t^{n}=\exp(\beta t+x(1-e^{t}))} (Roman 1984, 4.3.4), Boas & Bu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6479_chunk_0 | Additive Schwarz method | In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6480_chunk_0 | Additive polynomial | In mathematics, the additive polynomials are an important topic in classical algebraic number theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6481_chunk_0 | Valuation vector | In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring. An adele derive... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6482_chunk_0 | Valuation vector | The word stands for 'ideal element' (abbreviated: id.el.). Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6483_chunk_0 | Valuation vector | The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that G {\displaystyle G} -bundles on an algebraic curve over a finite field can be described in terms o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6484_chunk_0 | Noetherian | Noetherian relation, a binary relation that satisfies the ascending chain condition on its elements. Noetherian topological space, a topological space that satisfies the descending chain condition on closed sets. Noetherian induction, also called well-founded induction, a proof method for binary relations that satisfy ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6485_chunk_0 | Triviality (mathematics) | In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6486_chunk_0 | Triviality (mathematics) | The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove.The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for so... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6487_chunk_0 | Adjoint endomorphism | In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L ( n , R ) {\displaystyle GL(n,\mathbb {R} )} , the Lie group of real n-by-n inve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6488_chunk_0 | Affine general linear group | In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6489_chunk_0 | Affine span | In mathematics, the affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S, or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace. The affine hull aff(S) of S is the set of all affine com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6490_chunk_0 | Upper-extended real line | In mathematics, the affinely extended real number system is obtained from the real number system R {\displaystyle \mathbb {R} } by adding two infinity elements: + ∞ {\displaystyle +\infty } and − ∞ , {\displaystyle -\infty ,} where the infinities are treated as actual numbers. It is useful in describing the algebra on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6491_chunk_0 | Algebra of sets | In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6492_chunk_0 | Butterfly curve (algebraic) | In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation x 6 + y 6 = x 2 . {\displaystyle x^{6}+y^{6}=x^{2}.} The butterfly curve has a single singularity with delta invariant three, which means it is a curve of genus seven. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6493_chunk_0 | Butterfly curve (algebraic) | The only plane curves of genus seven are singular, since seven is not a triangular number, and the minimum degree for such a curve is six. The butterfly curve has branching number and multiplicity two, and hence the singularity link has two components, pictured at right. The area of the algebraic butterfly curve is giv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6494_chunk_0 | Algebraic topology (object) | In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every g in G. This terminology is often used in the case of the algebraic topology on the set of discrete, faithful... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6495_chunk_0 | Amoeba order | In mathematics, the amoeba order is the partial order of open subsets of 2ω of measure less than 1/2, ordered by reverse inclusion. Amoeba forcing is forcing with the amoeba order; it adds a measure 1 set of random reals. There are several variations, where 2ω is replaced by the real numbers or a real vector space or t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6496_chunk_0 | Amplitwist | In mathematics, the amplitwist is a concept created by Tristan Needham in the book Visual Complex Analysis (1997) to represent the derivative of a complex function visually. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6497_chunk_0 | Analytic Fredholm theorem | In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and the Hilbert–Schmidt theorem. The result is named after the Swedish mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6498_chunk_0 | Analytic subgroup theorem | In mathematics, the analytic subgroup theorem is a significant result in modern transcendental number theory. It may be seen as a generalisation of Baker's theorem on linear forms in logarithms. Gisbert Wüstholz proved it in the 1980s. It marked a breakthrough in the theory of transcendental numbers. Many longstanding ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6499_chunk_0 | Annihilator (ring theory) | In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S. Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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