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wiki_2100_chunk_0 | Cauchy determinant | In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form a i j = 1 x i − y j ; x i − y j ≠ 0 , 1 ≤ i ≤ m , 1 ≤ j ≤ n {\displaystyle a_{ij}={\frac {1}{x_{i}-y_{j}}};\quad x_{i}-y_{j}\neq 0,\quad 1\leq i\leq m,\quad 1\leq j\leq n} where x i {\displaystyle x_{i}} a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2101_chunk_0 | Cauchy sequence | As a result, no matter how far one goes, the remaining terms of the sequence never get close to each other; hence the sequence is not Cauchy. The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2102_chunk_0 | Cauchy-continuous function | In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the Cauchy completion of their domain. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2103_chunk_0 | Cayley graph | In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2104_chunk_0 | Cayley metric | In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2105_chunk_0 | Chevalley basis | In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2106_chunk_0 | Clifford multiplication | In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2107_chunk_0 | Clifford bundle | In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold M which is called the Clifford bundle of M. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2108_chunk_0 | Clifford module | In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined. The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2109_chunk_0 | Cohen–Macaulay ring | In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2110_chunk_0 | Colombeau algebra | In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Such a multiplication of distributions has long ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2111_chunk_0 | Colombeau algebra | As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microloc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2112_chunk_0 | Coons surface | In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements. Coons patches a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2113_chunk_0 | Costas array | In mathematics, a Costas array can be regarded geometrically as a set of n points, each at the center of a square in an n×n square tiling such that each row or column contains only one point, and all of the n(n − 1)/2 displacement vectors between each pair of dots are distinct. This results in an ideal "thumbtack" auto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2114_chunk_0 | Costas array | Costas arrays are named after John P. Costas, who first wrote about them in a 1965 technical report. Independently, Edgar Gilbert also wrote about them in the same year, publishing what is now known as the logarithmic Welch method of constructing Costas arrays. The general enumeration of Costas arrays is an open proble... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2115_chunk_0 | Coulomb functions | In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2116_chunk_0 | D-module | In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations. Since around 1970, D-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2117_chunk_0 | D-module | The methods of D-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. The approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2118_chunk_0 | Kleene algebra (with involution) | In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that: (A, ∨, ∧, 0, 1) is a bounded distributive lattice, and ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an involution that additionally satisfies ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2119_chunk_0 | Kleene algebra (with involution) | Thus ¬ is a dual automorphism of (A, ∨, ∧, 0, 1). If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2120_chunk_0 | Kleene algebra (with involution) | (i-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of Antonio Monteiro.De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic. The standard fuzzy algebra F = (, max(x, y), min(x, y), 0, 1, 1 − x) is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2121_chunk_0 | Delzant's theorem | In mathematics, a Delzant polytope is a convex polytope in R n {\displaystyle \mathbb {R} ^{n}} such for each vertex v {\displaystyle v} , exactly n {\displaystyle n} edges meet at v {\displaystyle v} , and these edges form a collection of vectors that form a Z {\displaystyle \mathbb {Z} } -basis of Z n {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2122_chunk_0 | Demazure module | In mathematics, a Demazure module, introduced by Demazure (1974a, 1974b), is a submodule of a finite-dimensional representation generated by an extremal weight space under the action of a Borel subalgebra. The Demazure character formula, introduced by Demazure (1974b, theorem 2), gives the characters of Demazure module... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2123_chunk_0 | Matiyasevich's theorem | In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns. A Diophantine set is a subset S of N j {\displaystyle \mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2124_chunk_0 | Matiyasevich's theorem | The use of natural numbers both in S and the existential quantification merely reflects the usual applications in computability and model theory. It does not matter whether natural numbers refer to the set of nonnegative integers or positive integers since the two definitions for Diophantine set are equivalent. We can ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2125_chunk_0 | Matiyasevich's theorem | Also it is sufficient to assume P is a polynomial over Q {\displaystyle \mathbb {Q} } and multiply P by the appropriate denominators to yield integer coefficients. However, whether quantification over rationals can also be substituted for quantification over the integers is a notoriously hard open problem.The MRDP theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2126_chunk_0 | Matiyasevich's theorem | This means that the concept of general Diophantine set, apparently belonging to number theory, can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work. Matiyasevich's completion of the MRDP theorem settled Hilbert's tenth pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2127_chunk_0 | Diophantine equation | In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponentia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2128_chunk_0 | Diophantine equation | Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called Diophantine geometry. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2129_chunk_0 | Dirac comb | In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula for some given period T {\displaystyle T} . Here t is a real variable and the sum extends over all integers k. The Dirac delta function δ {\displaystyle \delta } and the Dirac comb are t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2130_chunk_0 | Dirac comb | This implies Because the Dirac comb function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel: The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling and aliasing, in a single framework of continuous Fourier analysis on tempered distr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2131_chunk_0 | Dirac measure | In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2132_chunk_0 | Dirichlet L-function | In mathematics, a Dirichlet L-series is a function of the form L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n s . {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.} where χ {\displaystyle \chi } is a Dirichlet character and s a complex variable with real part greater than 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2133_chunk_0 | Dirichlet L-function | It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2134_chunk_0 | Dirichlet L-function | In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1. Otherwise, the L-function is entire. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2135_chunk_0 | Dirichlet algebra | In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space X. It is a closed subalgebra of C(X), the uniform algebra of bounded continuous functions on X, whose real parts are dense in the algebra of bounded continuous real functions on X. The concept was introduced by A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2136_chunk_0 | Dirichlet's problem | In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Lapl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2137_chunk_0 | Formal Dirichlet series | In mathematics, a Dirichlet series is any series of the form where s is complex, and a n {\displaystyle a_{n}} is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2138_chunk_0 | Ditkin set | In mathematics, a Ditkin set, introduced by (Ditkin 1939), is a closed subset of the circle such that a function f vanishing on the set can be approximated by functions φnf with φ vanishing in a neighborhood of the set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2139_chunk_0 | Drinfeld module | In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2140_chunk_0 | Drinfeld module | He later invented shtukas and used shtukas of rank 2 to prove the remaining cases of the Langlands conjectures for GL2. Laurent Lafforgue proved the Langlands conjectures for GLn of a function field by studying the moduli stack of shtukas of rank n. "Shtuka" is a Russian word штука meaning "a single copy", which comes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2141_chunk_0 | Dupin cyclide | 1802 by (and named after) Charles Dupin, while he was still a student at the École polytechnique following Gaspard Monge's lectures. The key property of a Dupin cyclide is that it is a channel surface (envelope of a one-parameter family of spheres) in two different ways. This property means that Dupin cyclides are natu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2142_chunk_0 | Dupin cyclide | Dupin cyclides are often simply known as cyclides, but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions. Dupin cyclides were investigated not only by Dupin, but also by A. Cayley, J... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2143_chunk_0 | Euclidean distance matrix | In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space. For points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} in k-dimensional space ℝk, the elements of their Euclidean distance matrix A are given by squares of distances between t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2144_chunk_0 | Euclidean distance matrix | However, in the Euclidean case, squares of distances are used to avoid computing square roots and to simplify relevant theorems and algorithms. Euclidean distance matrices are closely related to Gram matrices (matrices of dot products, describing norms of vectors and angles between them). The latter are easily analyzed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2145_chunk_0 | Euclidean distance matrix | This allows to characterize Euclidean distance matrices and recover the points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} that realize it. A realization, if it exists, is unique up to rigid transformations, i.e. distance-preserving transformations of Euclidean space (rotations, reflections, translati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2146_chunk_0 | Euclidean distance matrix | The goal may be to visualize such data by points in Euclidean space whose distance matrix approximates a given dissimilarity matrix as well as possible — this is known as multidimensional scaling. Alternatively, given two sets of data already represented by points in Euclidean space, one may ask how similar they are in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2147_chunk_0 | Fatou–Bieberbach domain | In mathematics, a Fatou–Bieberbach domain is a proper subdomain of C n {\displaystyle \mathbb {C} ^{n}} , biholomorphically equivalent to C n {\displaystyle \mathbb {C} ^{n}} . That is, an open set Ω ⊊ C n {\displaystyle \Omega \subsetneq \mathbb {C} ^{n}} is called a Fatou–Bieberbach domain if there exists a bijective... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2148_chunk_0 | Fedosov manifold | In mathematics, a Fedosov manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (M, ω, ∇), where (M, ω) is a symplectic manifold (that is, ω {\displaystyle \omega } is a symplectic form, a non-degenerate closed exterior 2-form, on a C ∞ {\displaystyle C^{\infty }} -manifold M), ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2149_chunk_0 | Fedosov manifold | In other words, the symplectic form is parallel with respect to the connection, i.e., its covariant derivative vanishes.) Note that every symplectic manifold admits a symplectic torsion-free connection. Cover the manifold with Darboux charts and on each chart define a connection ∇ with Christoffel symbol Γ j k i = 0 {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2150_chunk_0 | Fekete polynomial | In mathematics, a Fekete polynomial is a polynomial f p ( t ) := ∑ a = 0 p − 1 ( a p ) t a {\displaystyle f_{p}(t):=\sum _{a=0}^{p-1}\left({\frac {a}{p}}\right)t^{a}\,} where ( ⋅ p ) {\displaystyle \left({\frac {\cdot }{p}}\right)\,} is the Legendre symbol modulo some integer p > 1. These polynomials were known in nine... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2151_chunk_0 | Fermat quintic threefold | In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation V 5 + W 5 + X 5 + Y 5 + Z 5 = 0 {\displaystyle V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0} .This threefold, so named after Pierre de Fermat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2152_chunk_0 | Fredholm kernel | In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory. Fredholm kernels are named in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2153_chunk_0 | Frey curve | In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve y 2 = x ( x − a ℓ ) ( x + b ℓ ) {\displaystyle y^{2}=x(x-a^{\ell })(x+b^{\ell })} associated with a (hypothetical) solution of Fermat's equation a ℓ + b ℓ = c ℓ . {\displaystyle a^{\ell }+b^{\ell }=c^{\ell }.} The curve is named after Gerhard... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2154_chunk_0 | Fuchsian group | In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the disc model of hyperbolic geometry). General Fuchsian groups were first studied by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2155_chunk_0 | G-measure | In mathematics, a G-measure is a measure μ {\displaystyle \mu } that can be represented as the weak-∗ limit of a sequence of measurable functions G = ( G n ) n = 1 ∞ {\displaystyle G=\left(G_{n}\right)_{n=1}^{\infty }} . A classic example is the Riesz product G n ( t ) = ∏ k = 1 n ( 1 + r cos ( 2 π m k t ) ) {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2156_chunk_0 | GCD domain | In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM).A GCD domain gen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2157_chunk_0 | Galois field extension | In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2158_chunk_0 | Normal integral basis | In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2159_chunk_0 | Error curve | In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form and with parametric extension for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2160_chunk_0 | Gelfand pair | In mathematics, a Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2161_chunk_0 | Generalized Clifford algebra | In mathematics, a Generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882), and organized by Cartan (1898) and Schwinger.Clock and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2162_chunk_0 | Goldman domain | In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A. They are named after Oscar Goldman. An overring (i.e., an intermediate ring lying between the ring and its field of fractions) of a Goldman domain is again a Goldman domain. There exists... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2163_chunk_0 | Golomb ruler | The Golomb ruler was named for Solomon W. Golomb and discovered independently by Sidon (1932) and Babcock (1953). Sophie Piccard also published early research on these sets, in 1939, stating as a theorem the claim that two Golomb rulers with the same distance set must be congruent. This turned out to be false for six-p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2164_chunk_0 | Golomb ruler | Distributed.net has completed distributed massively parallel searches for optimal order-24 through order-28 Golomb rulers, each time confirming the suspected candidate ruler.Currently, the complexity of finding optimal Golomb rulers (OGRs) of arbitrary order n (where n is given in unary) is unknown. In the past there w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2165_chunk_0 | Grassmann–Cayley algebra | In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product. It is the most general structure in which projective properties are expressed in a coordinate-free way. The technique is based on work by German mathematician... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2166_chunk_0 | Green’s function | In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L {\displaystyle \operatorname {L} } is the linear differential operator, then the Green's function G {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2167_chunk_0 | Gregory number | In mathematics, a Gregory number, named after James Gregory, is a real number of the form: G x = ∑ i = 0 ∞ ( − 1 ) i 1 ( 2 i + 1 ) x 2 i + 1 {\displaystyle G_{x}=\sum _{i=0}^{\infty }(-1)^{i}{\frac {1}{(2i+1)x^{2i+1}}}} where x is any rational number greater or equal to 1. Considering the power series expansion for arc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2168_chunk_0 | Grothendieck category | In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2169_chunk_0 | Grothendieck universe | (In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.). Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algeb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2170_chunk_0 | Grothendieck universe | The existence of a nontrivial Grothendieck universe goes beyond the usual axioms of Zermelo–Fraenkel set theory; in particular it would imply the existence of strongly inaccessible cardinals. Tarski–Grothendieck set theory is an axiomatic treatment of set theory, used in some automatic proof systems, in which every set... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2171_chunk_0 | Data encoding | In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem. Here, the idea was to map mathematical notation to a natural number (using a Gödel numbering). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2172_chunk_0 | Gödel numbering for sequences | In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2173_chunk_0 | Hadamard manifold | In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} that is complete and simply connected and has everywhere non-positive sectional curvature. By Cartan–Hadamard theorem all Cartan–... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2174_chunk_0 | Hadamard matrix | In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2175_chunk_0 | Hadamard matrix | The n-dimensional parallelotope spanned by the rows of an n×n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2176_chunk_0 | Haken hierarchy | This conjecture was proven by Ian Agol.Haken manifolds were introduced by Wolfgang Haken (1961). Haken (1962) proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2177_chunk_0 | Hamiltonian matrix | In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix J = {\displaystyle J={\begin{bmatrix}0_{n}&I_{n}\\-I_{n}&0_{n}\\\end{bmatrix}}} and In is the n-by-n identity matrix. In other words, A is Hamiltonian if and only if (JA)T = JA where ()T denotes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2178_chunk_0 | Hankel contour | The contour is traversed in the positively-oriented sense, meaning that the circle around the origin is traversed counter-clockwise. Use of Hankel contours is one of the methods of contour integration. This type of path for contour integrals was first used by Hermann Hankel in his investigations of the Gamma function. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2179_chunk_0 | Hardy field | In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2180_chunk_0 | Fixed-point space | In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function f: X → X {\displaystyle f:X\rightarrow X} has a fixed point. For example, any closed interval in R {\displaystyle \mathbb {R} } is a fixed point space, and it can be proved from the intermediate value property of real contin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2181_chunk_0 | Fixed-point space | To see it, consider the function f ( x ) = a + 1 b − a ⋅ ( x − a ) 2 {\displaystyle f(x)=a+{\frac {1}{b-a}}\cdot (x-a)^{2}} , for example. Any linearly ordered space that is connected and has a top and a bottom element is a fixed point space. Note that, in the definition, we could easily have disposed of the condition ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2182_chunk_0 | Hecke algebra (disambiguation) | In mathematics, a Hecke algebra is classically the algebra of Hecke operators studied by Erich Hecke. It may also refer to one of several algebras (some of which are related to the classical Hecke algebra): Iwahori–Hecke algebra of a Coxeter group. Hecke algebra of a pair (g,K) where g is the Lie algebra of a Lie group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2183_chunk_0 | Hecke algebra of a locally compact group | In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2184_chunk_0 | Hermitian connection | In mathematics, a Hermitian connection ∇ {\displaystyle \nabla } is a connection on a Hermitian vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle M} which is compatible with the Hermitian metric ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } on E {\displaystyle E} , meaning that v ⟨ s ,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2185_chunk_0 | Hermitian matrices | In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: or in matrix form: Hermitian... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2186_chunk_0 | Hermitian symmetric domain | The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2187_chunk_0 | Hermitian symmetric domain | In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C). Irreducible compact Hermitian symmetric spaces are exactly the homog... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2188_chunk_0 | Heyting algebra | In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that (c ∧ a) ≤ b is equivalent to c ≤ (a → b). From a logical standpoin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2189_chunk_0 | Heyting algebra | As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when a → b is defined as ¬a ∨ b, as is every complete distributive lattice satisfying a one-sided infinite distributive law when a → b is taken to be the supremum of the set of all c for which c ∧ a ≤ b. In the finite case, every... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2190_chunk_0 | Heyting algebra | Although the negation operation ¬a is not part of the definition, it is definable as a → 0. The intuitive content of ¬a is the proposition that to assume a would lead to a contradiction. The definition implies that a ∧ ¬a = 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2191_chunk_0 | Heyting algebra | It can further be shown that a ≤ ¬¬a, although the converse, ¬¬a ≤ a, is not true in general, that is, double negation elimination does not hold in general in a Heyting algebra. Heyting algebras generalize Boolean algebras in the sense that Boolean algebras are precisely the Heyting algebras satisfying a ∨ ¬a = 1 (excl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2192_chunk_0 | Heyting algebra | The internal logic of an elementary topos is based on the Heyting algebra of subobjects of the terminal object 1 ordered by inclusion, equivalently the morphisms from 1 to the subobject classifier Ω. The open sets of any topological space form a complete Heyting algebra. Complete Heyting algebras thus become a central ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2193_chunk_0 | Heyting algebra | It follows that even among the finite Heyting algebras there exist infinitely many that are subdirectly irreducible, no two of which have the same equational theory. Hence no finite set of finite Heyting algebras can supply all the counterexamples to non-laws of Heyting algebra. This is in sharp contrast to Boolean alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2194_chunk_0 | Higgs bundle | In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle \varphi } , a holomorphic 1-form taking values in the bundle of endomorphisms of E such that φ ∧ φ = 0 {\displaystyle \varphi \wedge \varphi =0} . Such pairs wer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2195_chunk_0 | Hilbert modular form | In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m-fold product of upper half-planes H {\displaystyle {\mathcal {H}}} satisfying a certain kind of functional equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2196_chunk_0 | Hilbert–Schmidt integral operator | In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k: Ω × Ω → C with ∫ Ω ∫ Ω | k ( x , y ) | 2 d x d y < ∞ {\displaystyle \int _{\Omega }\int _{\Omega ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2197_chunk_0 | Hilbert–Schmidt integral operator | {\displaystyle \Vert K\Vert _{\mathrm {HS} }=\Vert k\Vert _{L^{2}}.} Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators). The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2198_chunk_0 | Hilbert–Schmidt integral operator | Specifically, let X be a locally compact Hausdorff space equipped with a positive Borel measure. Suppose further that L2(X) is a separable Hilbert space. The above condition on the kernel k on Rn can be interpreted as demanding k belong to L2(X × X). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2199_chunk_0 | Hilbert–Schmidt integral operator | Then the operator ( K f ) ( x ) = ∫ X k ( x , y ) f ( y ) d y {\displaystyle (Kf)(x)=\int _{X}k(x,y)f(y)\,dy} is compact. If k ( x , y ) = k ( y , x ) ¯ {\displaystyle k(x,y)={\overline {k(y,x)}}} then K is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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