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wiki_5600_chunk_0 | Cayley–Dickson construction | The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with itself, with multiplication defined in a specific way (different from the componentwise multiplication) and an involution known as conjugation. The product of an element and its conjugate (or sometimes the square root of thi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5601_chunk_0 | Chabauty topology | In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G. The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5602_chunk_0 | Champernowne constant | In mathematics, the Champernowne constant C10 is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933.For base 10, the number is defined by concatenating representations of successi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5603_chunk_0 | Chang number | In mathematics, the Chang number of an irreducible representation of a simple complex Lie algebra is its dimension modulo 1 + h, where h is the Coxeter number. Chang numbers are named after Chang (1982), who rediscovered an element of order h + 1 found by Kac (1981). Kac (1981) showed that there is a unique class of re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5604_chunk_0 | Chazy equation | In mathematics, the Chazy equation is the differential equation d 3 y d x 3 = 2 y d 2 y d x 2 − 3 ( d y d x ) 2 . {\displaystyle {\frac {d^{3}y}{dx^{3}}}=2y{\frac {d^{2}y}{dx^{2}}}-3\left({\frac {dy}{dx}}\right)^{2}.} It was introduced by Jean Chazy (1909, 1911) as an example of a third-order differential equation with... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5605_chunk_0 | Chazy equation | One solution is given by the Eisenstein series E 2 ( τ ) = 1 − 24 ∑ σ 1 ( n ) q n = 1 − 24 q − 72 q 2 − ⋯ . {\displaystyle E_{2}(\tau )=1-24\sum \sigma _{1}(n)q^{n}=1-24q-72q^{2}-\cdots .} Acting on this solution by the group SL2 gives a 3-parameter family of solutions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5606_chunk_0 | Chebyshev function | In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ (x) or θ (x) is given by ϑ ( x ) = ∑ p ≤ x log p {\displaystyle \vartheta (x)=\sum _{p\leq x}\log p} where log {\displaystyle \log } denotes the natural logari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5607_chunk_0 | Chebyshev function | Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function: f T c h b ( x , w ) = max i w i f i ( x ) . {\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}f_{i}(x).} By minimizing th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5608_chunk_0 | Chebyshev function | Often the functions to be minimized are not f i {\displaystyle f_{i}} but | f i − z i ∗ | {\displaystyle |f_{i}-z_{i}^{*}|} for some scalars z i ∗ {\displaystyle z_{i}^{*}} . Then f T c h b ( x , w ) = max i w i | f i ( x ) − z i ∗ | . {\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}|f_{i}(x)-z_{i}^{*}|.} All three function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5609_chunk_0 | Chebyshev integral | In mathematics, the Chebyshev integral, named after Pafnuty Chebyshev, is ∫ x p ( 1 − x ) q d x = B ( x ; 1 + p , 1 + q ) , {\displaystyle \int x^{p}(1-x)^{q}\,dx=B(x;1+p,1+q),} where B ( x ; a , b ) {\displaystyle B(x;a,b)} is an incomplete beta function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5610_chunk_0 | Chebyshev rational functions | In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree n is defined as: R n ( x ) = d e f T n ( x − 1 x + 1 ) {\displaystyle R_{n}(x)\ {\stackrel {\mathrm {def} }{=}}\ T_{n}\left... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5611_chunk_0 | Cheeger bound | In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs. Let X {\displaystyle X} be a finite set and let K ( x , y ) {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5612_chunk_0 | Cheeger bound | Assume this chain has stationary distribution π {\displaystyle \pi } . Define Q ( x , y ) = π ( x ) K ( x , y ) {\displaystyle Q(x,y)=\pi (x)K(x,y)} and for A , B ⊂ X {\displaystyle A,B\subset X} define Q ( A × B ) = ∑ x ∈ A , y ∈ B Q ( x , y ) . {\displaystyle Q(A\times B)=\sum _{x\in A,y\in B}Q(x,y).} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5613_chunk_0 | Cheeger bound | It is known that λ 1 = 1 {\displaystyle \lambda _{1}=1} . The Cheeger bound is a bound on the second largest eigenvalue λ 2 {\displaystyle \lambda _{2}} . Theorem (Cheeger bound): 1 − 2 Φ ≤ λ 2 ≤ 1 − Φ 2 2 . {\displaystyle 1-2\Phi \leq \lambda _{2}\leq 1-{\frac {\Phi ^{2}}{2}}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5614_chunk_0 | Cheeger constant (graph theory) | In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers, c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5615_chunk_0 | Chern–Gauss–Bonnet formula | In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Ri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5616_chunk_0 | Chern–Weil homomorphism | In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5617_chunk_0 | Chern–Weil homomorphism | Let C G {\displaystyle \mathbb {C} ^{G}} be the subalgebra of fixed points in C {\displaystyle \mathbb {C} } under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that f ( Ad g x ) = f ( x ) {\displaystyle f(\operatorname {Ad} _{g}x)=f(x)} , for all g in G and x in g {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5618_chunk_0 | Chevalley–Iwahori–Nagata theorem | In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomials is an isomorphism if this ring is finitely generated and all orbits of G on V are closed (D... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5619_chunk_0 | Chevalley–Shephard–Todd theorem | In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex general linear group the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5620_chunk_0 | Chihara–Ismail polynomials | In mathematics, the Chihara–Ismail polynomials are a family of orthogonal polynomials introduced by Chihara and Ismail (1982), generalizing the van Doorn polynomials introduced by van Doorn (1981) and the Karlin–McGregor polynomials. They have a rather unusual measure, which is discrete except for a single limit point ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5621_chunk_0 | Chinese monoid | In mathematics, the Chinese monoid is a monoid generated by a totally ordered alphabet with the relations cba = cab = bca for every a ≤ b ≤ c. An algorithm similar to Schensted's algorithm yields characterisation of the equivalence classes and a cross-section theorem. It was discovered by Duchamp & Krob (1994) during t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5622_chunk_0 | Chinese Remainder Theorem | In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two div... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5623_chunk_0 | Chinese Remainder Theorem | The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers. The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5624_chunk_0 | Chowla–Mordell theorem | In mathematics, the Chowla–Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and published independently by Sarvadaman Chowla and Louis Mordell, around 1951. In detail, if p {\displaystyle p} is a prime num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5625_chunk_0 | Chowla–Selberg formula | In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers. The result was essentially found by Lerch (1897) and rediscovered by Chowla and Selberg (1949, 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5626_chunk_0 | Christoffel–Darboux formula | In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878). It states that ∑ j = 0 n f j ( x ) f j ( y ) h j = k n h n k n + 1 f n ( y ) f n + 1 ( x ) − f n + 1 ( y ) f n ( x ) x − y {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5627_chunk_0 | Christ–Kiselev maximal inequality | In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5628_chunk_0 | Chung–Fuchs theorem | In mathematics, the Chung–Fuchs theorem, named after Chung Kai-lai and Wolfgang Heinrich Johannes Fuchs, states that for a particle undergoing a random walk in m-dimensions, it is certain to come back infinitely often to any neighborhood of the origin on a one-dimensional line (m = 1) or two-dimensional plane (m = 2), ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5629_chunk_0 | Clark–Ocone theorem | In mathematics, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itô integral with respe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5630_chunk_0 | Coates graph | In mathematics, the Coates graph or Coates flow graph, named after C.L. Coates, is a graph associated with the Coates' method for the solution of a system of linear equations.The Coates graph Gc(A) associated with an n × n matrix A is an n-node, weighted, labeled, directed graph. The nodes, labeled 1 through n, are eac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5631_chunk_0 | Cohen structure theorem | In mathematics, the Cohen structure theorem, introduced by Cohen (1946), describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures of Krull: Any complete regular equicharacteristic Noetherian local ring is a ring of formal power series over a fi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5632_chunk_0 | Cohen–Hewitt factorization theorem | In mathematics, the Cohen–Hewitt factorization theorem states that if V {\displaystyle V} is a left module over a Banach algebra B {\displaystyle B} with a left approximate unit ( u i ) i ∈ I {\displaystyle (u_{i})_{i\in I}} , then an element v {\displaystyle v} of V {\displaystyle V} can be factorized as a product v =... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5633_chunk_0 | Conley–Zehnder theorem | In mathematics, the Conley–Zehnder theorem, named after Charles C. Conley and Eduard Zehnder, provides a lower bound for the number of fixed points of Hamiltonian diffeomorphisms of standard symplectic tori in terms of the topology of the underlying tori. The lower bound is one plus the cup-length of the torus (thus 2n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5634_chunk_0 | Conway polynomial (finite fields) | In mathematics, the Conway polynomial Cp,n for the finite field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define a standard representation of Fpn as a splitting field of Cp,n. Conway polynomials were named after John H. Conway by Richard A. Parker, who was the first to define th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5635_chunk_0 | Courant minimax principle | In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix. It is named after Richard Courant. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5636_chunk_0 | Second Cousin problem | In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by Pierre Cousin in 1895. They are now posed, and solved, for any complex manifold M, in terms of conditi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5637_chunk_0 | Coxeter–Todd lattice | In mathematics, the Coxeter–Todd lattice K12, discovered by Coxeter and Todd (1953), is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is analogous to the Barnes–Wall lattice. The automorphism gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5638_chunk_0 | Cramér–Wold theorem | In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on R k {\displaystyle \mathbb {R} ^{k}} is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5639_chunk_0 | Crofton formula | In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5640_chunk_0 | Cuntz algebra | In mathematics, the Cuntz algebra O n {\displaystyle {\mathcal {O}}_{n}} , named after Joachim Cuntz, is the universal C*-algebra generated by n {\displaystyle n} isometries of an infinite-dimensional Hilbert space H {\displaystyle {\mathcal {H}}} satisfying certain relations. These algebras were introduced as the firs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5641_chunk_0 | Davenport constant | In mathematics, the Davenport constant D(G ) is an invariant of a group studied in additive combinatorics, quantifying the size of nonunique factorizations. Given a finite abelian group G, D(G ) is defined as the smallest number such that every sequence of elements of that length contains a non-empty subsequence adding... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5642_chunk_0 | Dawson function | In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5643_chunk_0 | Dawson–Gärtner theorem | In mathematics, the Dawson–Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller” topological space to a “larger” one. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5644_chunk_0 | Dedekind eta function | In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5645_chunk_0 | Dedekind zeta functions | In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5646_chunk_0 | Dehn-Somerville equations | In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927. The Dehn–Sommervi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5647_chunk_0 | Denjoy's theorem on rotation number | In mathematics, the Denjoy theorem gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. Denjoy (1932) proved the theorem in the course of his topological classification of homeomorphisms of the circle. He also... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5648_chunk_0 | Denjoy–Koksma inequality | In mathematics, the Denjoy–Koksma inequality, introduced by Herman (1979, p.73) as a combination of work of Arnaud Denjoy and the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums ∑ k = 0 m − 1 f ( x + k ω ) {\displaystyle \sum _{k=0}^{m-1}f(x+k\omega )} of functions f of bounded variation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5649_chunk_0 | Denjoy–Luzin theorem | In mathematics, the Denjoy–Luzin theorem, introduced independently by Denjoy (1912) and Luzin (1912) states that if a trigonometric series converges absolutely on a set of positive measure, then the sum of its coefficients converges absolutely, and in particular the trigonometric series converges absolutely everywhere. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5650_chunk_0 | Denjoy–Luzin–Saks theorem | In mathematics, the Denjoy–Luzin–Saks theorem states that a function of generalized bounded variation in the restricted sense has a derivative almost everywhere, and gives further conditions of the set of values of the function where the derivative does not exist. N. N. Luzin and A. Denjoy proved a weaker form of the t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5651_chunk_0 | Denjoy–Wolff theorem | In mathematics, the Denjoy–Wolff theorem is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers into itself. The result was proved independently in 1926 by the French mathematician Arnaud Denjoy and the Dutch mathemat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5652_chunk_0 | Denjoy–Young–Saks theorem | In mathematics, the Denjoy–Young–Saks theorem gives some possibilities for the Dini derivatives of a function that hold almost everywhere. Denjoy (1915) proved the theorem for continuous functions, Young (1917) extended it to measurable functions, and Saks (1924) extended it to arbitrary functions. Saks (1937, Chapter ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5653_chunk_0 | Deuring–Heilbronn phenomenon | In mathematics, the Deuring–Heilbronn phenomenon, discovered by Deuring (1933) and Heilbronn (1934), states that a counterexample to the generalized Riemann hypothesis for one Dirichlet L-function affects the location of the zeros of other Dirichlet L-functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5654_chunk_0 | Dickson polynomial | In mathematics, the Dickson polynomials, denoted Dn(x,α), form a polynomial sequence introduced by L. E. Dickson (1897). They were rediscovered by Brewer (1961) in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials. Over the complex numbers, Dickson polynomials are essen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5655_chunk_0 | Dini test | In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5656_chunk_0 | Dini–Lipschitz criterion | In mathematics, the Dini–Lipschitz criterion is a sufficient condition for the Fourier series of a periodic function to converge uniformly at all real numbers. It was introduced by Ulisse Dini (1872), as a strengthening of a weaker criterion introduced by Rudolf Lipschitz (1864). The criterion states that the Fourier s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5657_chunk_0 | Catalan beta function | In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5658_chunk_0 | Dirichlet convolution | In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5659_chunk_0 | Dirichlet eigenvalue | In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5660_chunk_0 | Dirichlet eigenvalue | {\displaystyle \Delta u={\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}.} The boundary value problem (1) is the Dirichlet problem for the Helmholtz equation, and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5661_chunk_0 | Dirichlet eigenvalue | More generally, in spectral geometry one considers (1) on a manifold with boundary Ω. Then Δ is taken to be the Laplace–Beltrami operator, also with Dirichlet boundary conditions. It can be shown, using the spectral theorem for compact self-adjoint operators that the eigenspaces are finite-dimensional and that the Diri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5662_chunk_0 | Dirichlet eigenvalue | This operator is invertible, and its inverse is compact and self-adjoint so that the usual spectral theorem can be applied to obtain the eigenspaces of Δ and the reciprocals 1/λ of its eigenvalues. One of the primary tools in the study of the Dirichlet eigenvalues is the max-min principle: the first eigenvalue λ1 minim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5663_chunk_0 | Dirichlet energy | In mathematics, the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space H1. The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5664_chunk_0 | Dirichlet function | In mathematics, the Dirichlet function is the indicator function 1Q or 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q, i.e. 1Q(x) = 1 if x is a rational number and 1Q(x) = 0 if x is not a rational number (i.e. an irrational number). It is named after the mathematician Peter Gustav Leje... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5665_chunk_0 | Dirichlet space | In mathematics, the Dirichlet space on the domain Ω ⊆ C , D ( Ω ) {\displaystyle \Omega \subseteq \mathbb {C} ,\,{\mathcal {D}}(\Omega )} (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space H 2 ( Ω ) {\displaystyle H^{2}(\Omega... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5666_chunk_0 | Dirichlet space | It is not a norm in general, since D ( f ) = 0 {\displaystyle {\mathcal {D}}(f)=0} whenever f is a constant function. For f , g ∈ D ( Ω ) {\displaystyle f,\,g\in {\mathcal {D}}(\Omega )} , we define D ( f , g ) := 1 π ∬ Ω f ′ ( z ) g ′ ( z ) ¯ d A ( z ) . {\displaystyle {\mathcal {D}}(f,\,g):={1 \over \pi }\iint _{\Ome... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5667_chunk_0 | Dirichlet space | The Dirichlet space is not an algebra, but the space D ( Ω ) ∩ H ∞ ( Ω ) {\displaystyle {\mathcal {D}}(\Omega )\cap H^{\infty }(\Omega )} is a Banach algebra, with respect to the norm ‖ f ‖ D ( Ω ) ∩ H ∞ ( Ω ) := ‖ f ‖ H ∞ ( Ω ) + D ( f ) 1 / 2 ( f ∈ D ( Ω ) ∩ H ∞ ( Ω ) ) . {\displaystyle \|f\|_{{\mathcal {D}}(\Omega )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5668_chunk_0 | Dirichlet conditions | In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the disco... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5669_chunk_0 | Dixmier mapping | In mathematics, the Dixmier mapping describes the space Prim(U(g)) of primitive ideals of the universal enveloping algebra U(g) of a finite-dimensional solvable Lie algebra g over an algebraically closed field of characteristic 0 in terms of coadjoint orbits. More precisely, it is a homeomorphism from the space of orbi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5670_chunk_0 | Dixmier trace | In mathematics, the Dixmier trace, introduced by Jacques Dixmier (1966), is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces. Some applications of Dixmier traces to noncommutative geometry are described in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5671_chunk_0 | Dottie number | In mathematics, the Dottie number is a constant that is the unique real root of the equation cos x = x {\displaystyle \cos x=x} ,where the argument of cos {\displaystyle \cos } is in radians. The decimal expansion of the Dottie number is 0.739085... {\displaystyle 0.739085...} .Since cos ( x ) − x {\displaystyle \c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5672_chunk_0 | Double extension set theory | In mathematics, the Double extension set theory (DEST) is an axiomatic set theory proposed by Andrzej Kisielewicz consisting of two separate membership relations on the universe of sets, denoted here by ∈ {\displaystyle \in } and ε {\displaystyle \varepsilon } , and a set of axioms relating the two. The intention behin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5673_chunk_0 | Double extension set theory | Then, the axioms of DEST posit a set A = { x | ϕ ( x ) } {\displaystyle A=\{x|\phi (x)\}} such that x ε A ⟺ ϕ ( x ) {\displaystyle x\varepsilon A\iff \phi (x)} . For instance, x ∉ x {\displaystyle x\notin x} is a formula involving only ∈ {\displaystyle \in } , and thus DEST posits the Russell set R = { x | x ∉ x } {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5674_chunk_0 | Double extension set theory | Since the membership relations are different, we thus avoid the Russell's paradox. The focus in DEST is on regular sets, which are sets whose extensions under the two membership relations coincide, i.e., sets A {\displaystyle A} for which it holds that ∀ x . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5675_chunk_0 | Double extension set theory | x ∈ A ⟺ x ε A {\displaystyle \forall x.x\in A\iff x\varepsilon A} . The preceding discussion suggests that the Russell set R = { x | x ∉ x } {\displaystyle R=\{x|x\notin x\}} cannot be regular, as otherwise it leads to the Russell's paradox. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5676_chunk_0 | Drazin inverse | In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix. Let A be a square matrix. The index of A is the least nonnegative integer k such that rank(Ak+1) = rank(Ak). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5677_chunk_0 | Drazin inverse | The Drazin inverse of A is the unique matrix AD that satisfies A k + 1 A D = A k , A D A A D = A D , A A D = A D A . {\displaystyle A^{k+1}A^{\text{D}}=A^{k},\quad A^{\text{D}}AA^{\text{D}}=A^{\text{D}},\quad AA^{\text{D}}=A^{\text{D}}A.} It's not a generalized inverse in the classical sense, since A A D A ≠ A {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5678_chunk_0 | Drazin inverse | If A is invertible with inverse A − 1 {\displaystyle A^{-1}} , then A D = A − 1 {\displaystyle A^{\text{D}}=A^{-1}} . If A is a block diagonal matrix A = {\displaystyle A={\begin{bmatrix}B&0\\0&N\end{bmatrix}}} where B {\displaystyle B} is invertible with inverse B − 1 {\displaystyle B^{-1}} and N {\displaystyle N} is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5679_chunk_0 | Drazin inverse | The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or {1,2,5}-inverse and denoted A#. The group inverse can be defined, equivalently, by the properties AA#A = A, A#AA# = A#, and AA# = A#A. A projection matrix P, defined as a matrix such that P2 = P, has index 1 (or 0) and has Drazin inverse PD =... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5680_chunk_0 | Drinfeld upper half plane | In mathematics, the Drinfeld upper half plane is a rigid analytic space analogous to the usual upper half plane for function fields, introduced by Drinfeld (1976). It is defined to be P1(C)\P1(F∞), where F is a function field of a curve over a finite field, F∞ its completion at ∞, and C the completion of the algebraic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5681_chunk_0 | Duflo isomorphism | In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. It was introduced by Michel Duflo (1977) and later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich. The P... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5682_chunk_0 | Duflo isomorphism | It is equivariant with respect to the natural representation of g {\displaystyle {\mathfrak {g}}} on these spaces, so it restricts to a vector space isomorphism F: S ( g ) g → U ( g ) g {\displaystyle F\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}} where the superscript indicates the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5683_chunk_0 | Duflo isomorphism | {\displaystyle F\circ G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}.} Later, using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5684_chunk_0 | Duflo isomorphism | Call this element a d ∈ S ( g ∗ ) ⊗ E n d ( g ) {\displaystyle \mathrm {ad} \in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})} Both S ( g ∗ ) {\displaystyle S({\mathfrak {g}}^{\ast })} and E n d ( g ) {\displaystyle \mathrm {End} ({\mathfrak {g}})} are algebras so their tensor product is as well. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5685_chunk_0 | Duflo isomorphism | As a result, the algebra S ( g ∗ ) {\displaystyle S({\mathfrak {g}}^{\ast })} acts on as differential operators on S ( g ) {\displaystyle S({\mathfrak {g}})} , and this extends to an action of S ( g ) {\displaystyle S({\mathfrak {g}})} on S ( g ) {\displaystyle S({\mathfrak {g}})} . We can thus define a linear map G: S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5686_chunk_0 | Dushnik–Miller theorem | In mathematics, the Dushnik–Miller theorem is a result in order theory stating that every infinite linear order has a non-identity order embedding into itself. It is named for Ben Dushnik and E. W. Miller, who published this theorem for countable linear orders in 1940. More strongly, they showed that in the countable c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5687_chunk_0 | Dwork conjecture on unit root zeta functions | In mathematics, the Dwork unit root zeta function, named after Bernard Dwork, is the L-function attached to the p-adic Galois representation arising from the p-adic etale cohomology of an algebraic variety defined over a global function field of characteristic p. The Dwork conjecture (1973) states that his unit root ze... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5688_chunk_0 | Dynkin index | In mathematics, the Dynkin index I ( λ ) {\displaystyle I({\lambda })} of a finite-dimensional highest-weight representation of a compact simple Lie algebra g {\displaystyle {\mathfrak {g}}} with highest weight λ {\displaystyle \lambda } is defined by where V 0 {\displaystyle V_{0}} is the 'defining representation', wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5689_chunk_0 | Dynkin index | Since the trace forms are bilinear forms, we can take traces to obtain I ( λ ) = dim V λ 2 dim g ( λ , λ + 2 ρ ) {\displaystyle I(\lambda )={\frac {\dim V_{\lambda }}{2\dim {\mathfrak {g}}}}(\lambda ,\lambda +2\rho )} where the Weyl vector ρ = 1 2 ∑ α ∈ Δ + α {\displaystyle \rho ={\frac {1}{2}}\sum _{\alpha \in \De... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5690_chunk_0 | Dyson conjecture | In mathematics, the Dyson conjecture (Freeman Dyson 1962) is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it to the q-Dyson conjecture, proved by Zeilberger and Bressoud and sometimes called the Zeilberger–Bressoud theorem. M... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5691_chunk_0 | E8 lattice | In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E8 root system. The norm of the E8 lattice (divided by 2) is a positive definite even unimodular quadr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5692_chunk_0 | ELSV formula | In mathematics, the ELSV formula, named after its four authors Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves. Several fundamental results in the intersection theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5693_chunk_0 | Earle–Hamilton fixed-point theorem | In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point. The result was proved in 1968 by Clifford Earle and Richard S. Hamilton by showing that, w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5694_chunk_0 | Eckmann–Hilton argument | In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two unital magma structures on a set where one is a homomorphism for the other. Given this, the structures are the same, and the resulting magma is a commutative monoid. This can then be used to prov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5695_chunk_0 | Ehrenpreis conjecture | In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any K greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers which are K-quasiconformal: that is, the covers are arbitrarily close in the Teichmüller metric. A proof was announced by Jeremy Kahn and Vladi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5696_chunk_0 | Eisenstein ideal | In mathematics, the Eisenstein ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of the Hecke algebra of Hecke operators that annihilate the Eisenstein series. It was introduced by Barry Mazur (1977), in studying the rational points of modular curves. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5697_chunk_0 | Erdős–Ko–Rado theorem | In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central resu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5698_chunk_0 | Erdős–Ko–Rado theorem | One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when n {\displaystyle n} is large enough for the problem to be non... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5699_chunk_0 | Erdős–Ko–Rado theorem | When n = 2 r {\displaystyle n=2r} there are other equally-large families, but for larger values of n {\displaystyle n} only the families constructed in this way can be largest. The Erdős–Ko–Rado theorem can also be described in terms of hypergraphs or independent sets in Kneser graphs. Several analogous theorems apply ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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