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In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an example of a Drinfeld module.
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https://en.wikipedia.org/wiki/Carlitz_exponential
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In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field Fq of q elements. A polynomial f(x) in Fq of degree d is called exceptional over Fq if every irreducible factor (differing from x − y) or (f(x) − f(y))/(x − y)) over Fq becomes reducible over the algebraic closure of Fq. If q > d4, then f(x) is exceptional if and only if f(x) is a permutation polynomial over Fq.
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https://en.wikipedia.org/wiki/Carlitz-Wan_conjecture
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The Carlitz–Wan conjecture states that there are no exceptional polynomials of degree d over Fq if gcd(d, q − 1) > 1. In the special case that q is odd and d is even, this conjecture was proposed by Leonard Carlitz (1966) and proved by Fried, Guralnick, and Saxl (1993). The general form of the Carlitz–Wan conjecture was proposed by Daqing Wan (1993) and later proved by Hendrik Lenstra (1995) == References ==
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https://en.wikipedia.org/wiki/Carlitz-Wan_conjecture
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In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa. The Carlson elliptic integrals are: Since R C {\displaystyle R_{C}} and R D {\displaystyle R_{D}} are special cases of R F {\displaystyle R_{F}} and R J {\displaystyle R_{J}} , all elliptic integrals can ultimately be evaluated in terms of just R F {\displaystyle R_{F}} and R J {\displaystyle R_{J}} .
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https://en.wikipedia.org/wiki/Carlson_symmetric_form
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The term symmetric refers to the fact that in contrast to the Legendre forms, these functions are unchanged by the exchange of certain subsets of their arguments. The value of R F ( x , y , z ) {\displaystyle R_{F}(x,y,z)} is the same for any permutation of its arguments, and the value of R J ( x , y , z , p ) {\displaystyle R_{J}(x,y,z,p)} is the same for any permutation of its first three arguments. The Carlson elliptic integrals are named after Bille C. Carlson (1924-2013).
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https://en.wikipedia.org/wiki/Carlson_symmetric_form
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In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.
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https://en.wikipedia.org/wiki/Cartan_involution
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In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space.
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https://en.wikipedia.org/wiki/Cartan_model
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In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described as the composition of at most n reflections. The notion of a symmetric bilinear space is a generalization of Euclidean space whose structure is defined by a symmetric bilinear form (which need not be positive definite, so is not necessarily an inner product – for instance, a pseudo-Euclidean space is also a symmetric bilinear space). The orthogonal transformations in the space are those automorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances and angles. These orthogonal transformations form a group under composition, called the orthogonal group.
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https://en.wikipedia.org/wiki/Cartan–Dieudonné_theorem
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For example, in the two-dimensional Euclidean plane, every orthogonal transformation is either a reflection across a line through the origin or a rotation about the origin (which can be written as the composition of two reflections). Any arbitrary composition of such rotations and reflections can be rewritten as a composition of no more than 2 reflections. Similarly, in three-dimensional Euclidean space, every orthogonal transformation can be described as a single reflection, a rotation (2 reflections), or an improper rotation (3 reflections). In four dimensions, double rotations are added that represent 4 reflections.
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https://en.wikipedia.org/wiki/Cartan–Dieudonné_theorem
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In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926. Informally, the conjecture states that negative curvature allows regions with a given perimeter to hold more volume. This phenomenon manifests itself in nature through corrugations on coral reefs, or ripples on a petunia flower, which form some of the simplest examples of non-positively curved spaces.
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https://en.wikipedia.org/wiki/Cartan–Hadamard_conjecture
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In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I {\displaystyle I} . It is named for Élie Cartan and Erich Kähler.
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https://en.wikipedia.org/wiki/Cartan–Kähler_theorem
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In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let X be such a surface, projective and non-singular, and let ω1 and ω2be two differentials of the first kind on X which are linearly independent but with wedge product 0. Then this data can be represented as a pullback of an algebraic curve: there is a non-singular algebraic curve C, a morphism φ: X → C,and differentials of the first kind ω′1 and ω′2 on C such that φ*(ω′1) = ω1 and φ*(ω′2) = ω2.This result is due to Guido Castelnuovo and Michele de Franchis (1875–1946). The converse, that two such pullbacks would have wedge 0, is immediate.
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https://en.wikipedia.org/wiki/Castelnuovo–de_Franchis_theorem
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In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f ( n ) {\displaystyle f(n)} of non-negative real numbers, the series ∑ n = 1 ∞ f ( n ) {\textstyle \sum \limits _{n=1}^{\infty }f(n)} converges if and only if the "condensed" series ∑ n = 0 ∞ 2 n f ( 2 n ) {\textstyle \sum \limits _{n=0}^{\infty }2^{n}f(2^{n})} converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.
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https://en.wikipedia.org/wiki/Cauchy_condensation_test
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In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if f ( z ) {\displaystyle f(z)} is holomorphic in a simply connected domain Ω, then for any simply closed contour C {\displaystyle C} in Ω, that contour integral is zero.
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https://en.wikipedia.org/wiki/Cauchy_integral
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In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the singularity (so the singularity is not covered by the integral).
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https://en.wikipedia.org/wiki/Cauchy_principal_part
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In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis.
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https://en.wikipedia.org/wiki/Cauchy–Hadamard_theorem
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In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy (1842), and the full result by Sofya Kovalevskaya (1874).
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https://en.wikipedia.org/wiki/Cauchy–Kowalevski_theorem
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In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describing the octonions.
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https://en.wikipedia.org/wiki/Cayley_plane
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In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators (Nikolski 1988).
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https://en.wikipedia.org/wiki/Cayley_transform
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In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. The original form states: Assume that two cubics C1 and C2 in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point.A more intrinsic form of the Cayley–Bacharach theorem reads as follows: Every cubic curve C over an algebraically closed field that passes through a given set of eight points P1, ..., P8 also passes through (counting multiplicities) a ninth point P9 which depends only on P1, ..., P8.A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and Isaak Bacharach.
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https://en.wikipedia.org/wiki/Cayley–Bacharach_theorem
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In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics.
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https://en.wikipedia.org/wiki/Cayley–Dickson_construction
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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 this product) is called the norm. The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, associativity of multiplication, and finally alternativity. More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension. : 45 Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed) division algebras (over the real numbers).
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https://en.wikipedia.org/wiki/Cayley–Dickson_construction
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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 subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space. More precisely, it is an adaptation of the Fell topology construction, which itself derives from the Vietoris topology concept.
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https://en.wikipedia.org/wiki/Chabauty_topology
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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 successive integers: C10 = 0.12345678910111213141516… (sequence A033307 in the OEIS).Champernowne constants can also be constructed in other bases, similarly, for example: C2 = 0.11011100101110111… 2 C3 = 0.12101112202122… 3.The Champernowne word or Barbier word is the sequence of digits of C10 obtained by writing it in base 10 and juxtaposing the digits: 12345678910111213141516... (sequence A007376 in the OEIS)More generally, a Champernowne sequence (sometimes also called a Champernowne word) is any sequence of digits obtained by concatenating all finite digit-strings (in any given base) in some recursive order. For instance, the binary Champernowne sequence in shortlex order is 0 1 00 01 10 11 000 001 ... (sequence A076478 in the OEIS)where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated.
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https://en.wikipedia.org/wiki/Champernowne_constant
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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 regular elements σ of order h + 1, in the complex points of the corresponding Chevalley group. He showed that the trace of σ on an irreducible representation is −1, 0, or +1, and if h + 1 is prime then the trace is congruent to the dimension mod h+1. This implies that the dimension of an irreducible representation is always −1, 0, or +1 mod h + 1 whenever h + 1 is prime.
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https://en.wikipedia.org/wiki/Chang_number
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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 a movable singularity that is a natural boundary for its solutions.
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https://en.wikipedia.org/wiki/Chazy_equation
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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.
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https://en.wikipedia.org/wiki/Chazy_equation
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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 logarithm, with the sum extending over all prime numbers p that are less than or equal to x. The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all prime powers not exceeding x ψ ( x ) = ∑ k ∈ N ∑ p k ≤ x log p = ∑ n ≤ x Λ ( n ) = ∑ p ≤ x ⌊ log p x ⌋ log p , {\displaystyle \psi (x)=\sum _{k\in \mathbb {N} }\sum _{p^{k}\leq x}\log p=\sum _{n\leq x}\Lambda (n)=\sum _{p\leq x}\left\lfloor \log _{p}x\right\rfloor \log p,} where Λ is the von Mangoldt function. The Chebyshev functions, especially the second one ψ (x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.
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https://en.wikipedia.org/wiki/Chebyshev_function
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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 this function for different values of w {\displaystyle w} , one obtains every point on a Pareto front, even in the nonconvex parts.
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https://en.wikipedia.org/wiki/Chebyshev_function
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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 functions are named in honour of Pafnuty Chebyshev.
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https://en.wikipedia.org/wiki/Chebyshev_function
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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.
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https://en.wikipedia.org/wiki/Chebyshev_integral
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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({\frac {x-1}{x+1}}\right)} where Tn(x) is a Chebyshev polynomial of the first kind.
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https://en.wikipedia.org/wiki/Chebyshev_rational_functions
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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 ) {\displaystyle K(x,y)} be the transition probability for a reversible Markov chain on X {\displaystyle X} .
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https://en.wikipedia.org/wiki/Cheeger_bound
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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).}
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https://en.wikipedia.org/wiki/Cheeger_bound
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Define the constant Φ {\displaystyle \Phi } as Φ = min S ⊂ X , π ( S ) ≤ 1 2 Q ( S × S c ) π ( S ) . {\displaystyle \Phi =\min _{S\subset X,\pi (S)\leq {\frac {1}{2}}}{\frac {Q(S\times S^{c})}{\pi (S)}}.} The operator K , {\displaystyle K,} acting on the space of functions from | X | {\displaystyle |X|} to | X | {\displaystyle |X|} , defined by ( K ϕ ) ( x ) = ∑ y K ( x , y ) ϕ ( y ) {\displaystyle (K\phi )(x)=\sum _{y}K(x,y)\phi (y)} has eigenvalues λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}} .
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https://en.wikipedia.org/wiki/Cheeger_bound
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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}}.}
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https://en.wikipedia.org/wiki/Cheeger_bound
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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, card shuffling. The graph theoretical notion originated after the Cheeger isoperimetric constant of a compact Riemannian manifold. The Cheeger constant is named after the mathematician Jeff Cheeger.
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https://en.wikipedia.org/wiki/Cheeger_constant_(graph_theory)
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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 Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant). It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry.Riemann–Roch and Atiyah–Singer are other generalizations of the Gauss–Bonnet theorem.
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https://en.wikipedia.org/wiki/Chern–Gauss–Bonnet_formula
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In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.
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https://en.wikipedia.org/wiki/Chern–Simons_3-form
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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 between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} , and let C {\displaystyle \mathbb {C} } denote the algebra of C {\displaystyle \mathbb {C} } -valued polynomials on g {\displaystyle {\mathfrak {g}}} (exactly the same argument works if we used R {\displaystyle \mathbb {R} } instead of C {\displaystyle \mathbb {C} } .)
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https://en.wikipedia.org/wiki/Chern–Weil_homomorphism
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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 {\displaystyle {\mathfrak {g}}} , Given a principal G-bundle P on M, there is an associated homomorphism of C {\displaystyle \mathbb {C} } -algebras, C G → H ∗ ( M ; C ) {\displaystyle \mathbb {C} ^{G}\to H^{*}(M;\mathbb {C} )} ,called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi-simple, then the cohomology ring of the classifying space for G-bundles, B G {\displaystyle BG} , is isomorphic to the algebra C G {\displaystyle \mathbb {C} ^{G}} of invariant polynomials: H ∗ ( B G ; C ) ≅ C G .
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https://en.wikipedia.org/wiki/Chern–Weil_homomorphism
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{\displaystyle H^{*}(BG;\mathbb {C} )\cong \mathbb {C} ^{G}.} (The cohomology ring of BG can still be given in the de Rham sense: H k ( B G ; C ) = lim → ker ( d: Ω k ( B j G ) → Ω k + 1 ( B j G ) ) / im d . {\displaystyle H^{k}(BG;\mathbb {C} )=\varinjlim \operatorname {ker} (d\colon \Omega ^{k}(B_{j}G)\to \Omega ^{k+1}(B_{j}G))/\operatorname {im} d.} when B G = lim → B j G {\displaystyle BG=\varinjlim B_{j}G} and B j G {\displaystyle B_{j}G} are manifolds.)
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https://en.wikipedia.org/wiki/Chern–Weil_homomorphism
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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 (Dieudonné & Carrell 1970, p.53, 1971, p.55). It is named after Claude Chevalley, Nagayoshi Iwahori, and Masayoshi Nagata.
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https://en.wikipedia.org/wiki/Chevalley–Iwahori–Nagata_theorem
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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 theorem was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre.
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https://en.wikipedia.org/wiki/Chevalley–Shephard–Todd_theorem
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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 at 0 with jump 0, and is non-symmetric, but whose support has an infinite number of both positive and negative points.
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https://en.wikipedia.org/wiki/Chihara–Ismail_polynomials
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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 their classification of monoids with growth similar to that of the plactic monoid, and studied in detail by Julien Cassaigne, Marc Espie, Daniel Krob, Jean-Christophe Novelli, and Florent Hivert in 2001.The Chinese monoid has a regular language cross-section a ∗ ( b a ) ∗ b ∗ ( c a ) ∗ ( c b ) ∗ c ∗ ⋯ {\displaystyle a^{*}\ (ba)^{*}b^{*}\ (ca)^{*}(cb)^{*}c^{*}\cdots } and hence polynomial growth of dimension n ( n + 1 ) 2 {\displaystyle {\frac {n(n+1)}{2}}} .The Chinese monoid equivalence class of a permutation is the preimage of an involution under the map w ↦ w ∘ w − 1 {\displaystyle w\mapsto w\circ w^{-1}} where ∘ {\displaystyle \circ } denotes the product in the Iwahori-Hecke algebra with q s = 0 {\displaystyle q_{s}=0} .
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https://en.wikipedia.org/wiki/Chinese_monoid
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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 divisors share a common factor other than 1). For example, if we know that the remainder of n divided by 3 is 2, the remainder of n divided by 5 is 3, and the remainder of n divided by 7 is 2, then without knowing the value of n, we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23. Importantly, this tells us that if n is a natural number less than 105, then 23 is the only possible value of n. The earliest known statement of the theorem is by the Chinese mathematician Sunzi in the Sunzi Suanjing in the 3rd century CE.
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https://en.wikipedia.org/wiki/Chinese_Remainder_Theorem
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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 ideal domain. It has been generalized to any ring, with a formulation involving two-sided ideals.
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https://en.wikipedia.org/wiki/Chinese_Remainder_Theorem
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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 number, χ {\displaystyle \chi } a nontrivial Dirichlet character modulo p {\displaystyle p} , and G ( χ ) = ∑ χ ( a ) ζ a {\displaystyle G(\chi )=\sum \chi (a)\zeta ^{a}} where ζ {\displaystyle \zeta } is a primitive p {\displaystyle p} -th root of unity in the complex numbers, then G ( χ ) | G ( χ ) | {\displaystyle {\frac {G(\chi )}{|G(\chi )|}}} is a root of unity if and only if χ {\displaystyle \chi } is the quadratic residue symbol modulo p {\displaystyle p} . The 'if' part was known to Gauss: the contribution of Chowla and Mordell was the 'only if' direction. The ratio in the theorem occurs in the functional equation of L-functions.
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https://en.wikipedia.org/wiki/Chowla–Mordell_theorem
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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, 1967).
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https://en.wikipedia.org/wiki/Chowla–Selberg_formula
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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 \sum _{j=0}^{n}{\frac {f_{j}(x)f_{j}(y)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}{\frac {f_{n}(y)f_{n+1}(x)-f_{n+1}(y)f_{n}(x)}{x-y}}} where fj(x) is the jth term of a set of orthogonal polynomials of squared norm hj and leading coefficient kj. There is also a "confluent form" of this identity by taking y → x {\displaystyle y\to x} limit:
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https://en.wikipedia.org/wiki/Christoffel–Darboux_formula
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In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev.
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https://en.wikipedia.org/wiki/Christ–Kiselev_maximal_inequality
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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), but in three or more dimensional spaces it will leave to infinity. Specifically, if a position of the particle is described by the vector X n {\displaystyle X_{n}}: where Z 1 , Z 2 , … , Z n {\displaystyle Z_{1},Z_{2},\dots ,Z_{n}} are independent m-dimensional vectors with a given multivariate distribution, then if m = 1 {\displaystyle m=1} , E ( | Z i | ) < ∞ {\displaystyle E(|Z_{i}|)<\infty } and E ( Z i ) = 0 {\displaystyle E(Z_{i})=0} , or if m = 2 {\displaystyle m=2} E ( | Z i 2 | ) < ∞ {\displaystyle E(|Z_{i}^{2}|)<\infty } and E ( Z i ) = 0 {\displaystyle E(Z_{i})=0} , the following holds: However, for m ≥ 3 {\displaystyle m\geq 3} ,
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https://en.wikipedia.org/wiki/Chung–Fuchs_theorem
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In mathematics, the Chvátal–Sankoff constants are mathematical constants that describe the lengths of longest common subsequences of random strings. Although the existence of these constants has been proven, their exact values are unknown. They are named after Václav Chvátal and David Sankoff, who began investigating them in the mid-1970s.There is one Chvátal–Sankoff constant γ k {\displaystyle \gamma _{k}} for each positive integer k, where k is the number of characters in the alphabet from which the random strings are drawn. The sequence of these numbers grows inversely proportionally to the square root of k. However, some authors write "the Chvátal–Sankoff constant" to refer to γ 2 {\displaystyle \gamma _{2}} , the constant defined in this way for the binary alphabet.
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https://en.wikipedia.org/wiki/Chvátal–Sankoff_constants
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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 respect to that path. It is named after the contributions of mathematicians J.M.C. Clark (1970), Daniel Ocone (1984) and U.G. Haussmann (1978).
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https://en.wikipedia.org/wiki/Clark–Ocone_theorem
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In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by Clebsch (1871) and Klein (1873), all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its blowup at the 10 Eckardt points.
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https://en.wikipedia.org/wiki/Clebsch_surface
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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 each associated with the corresponding row/column of A. If entry aji ≠ 0 then there is a directed edge from node i to node j with weight aji. In other words, the Coates graph for matrix A is the one whose adjacency matrix is the transpose of A.
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https://en.wikipedia.org/wiki/Coates_graph
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In mathematics, the Coble variety is the moduli space of ordered sets of 6 points in the projective plane, and can be represented as a double cover of the projective 4-space branched over the Igusa quartic. It is a 4-dimensional variety that was first studied by Arthur Coble.
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https://en.wikipedia.org/wiki/Coble_variety
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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 field. (Equicharacteristic means that the local ring and its residue field have the same characteristic, and is equivalent to the local ring containing a field.) Any complete regular Noetherian local ring that is not equicharacteristic but is unramified is uniquely determined by its residue field and its dimension. Any complete Noetherian local ring is the image of a complete regular Noetherian local ring.
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https://en.wikipedia.org/wiki/Cohen_structure_theorem
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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 = b w {\displaystyle v=bw} (for some b ∈ B {\displaystyle b\in B} and w ∈ V {\displaystyle w\in V} ) whenever lim i ∈ I u i v = v {\displaystyle \displaystyle \lim _{i\in I}u_{i}v=v} . The theorem was introduced by Paul Cohen (1959) and Edwin Hewitt (1964).
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https://en.wikipedia.org/wiki/Cohen–Hewitt_factorization_theorem
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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+1, where 2n is the dimension of the considered torus), and it can be strengthen to the rank of the homology of the torus (which is 22n) provided all the fixed points are non-degenerate, this latter condition being generic in the C1-topology. The theorem was conjectured by Vladimir Arnold, and it was known as the Arnold conjecture on fixed points of symplectomorphisms. Its validity was later extended to more general closed symplectic manifolds by Andreas Floer and several others.
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https://en.wikipedia.org/wiki/Conley–Zehnder_theorem
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In mathematics, the Contou-Carrère symbol 〈a,b〉 is a Steinberg symbol defined on pairs of invertible elements of the ring of Laurent power series over an Artinian ring k, taking values in the group of units of k. It was introduced by Contou-Carrère (1994).
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https://en.wikipedia.org/wiki/Contou-Carrère_symbol
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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 them and compute examples. Conway polynomials satisfy a certain compatibility condition that had been proposed by Conway between the representation of a field and the representations of its subfields. They are important in computer algebra where they provide portability among different mathematical databases and computer algebra systems. Since Conway polynomials are expensive to compute, they must be stored to be used in practice. Databases of Conway polynomials are available in the computer algebra systems GAP, Macaulay2, Magma, SageMath, and at the web site of Frank Lübeck.
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https://en.wikipedia.org/wiki/Conway_polynomial_(finite_fields)
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In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix. It is named after Richard Courant.
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https://en.wikipedia.org/wiki/Courant_minimax_principle
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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 conditions on M. For both problems, an open cover of M by sets Ui is given, along with a meromorphic function fi on each Ui.
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https://en.wikipedia.org/wiki/Second_Cousin_problem
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In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; they form the apartments of a building.
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https://en.wikipedia.org/wiki/Tits_cone
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
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https://en.wikipedia.org/wiki/Coxeter_number
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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 group of the Coxeter–Todd lattice has order 210·37·5·7=78382080, and there are 756 vectors in this lattice of norm 4 (the shortest nonzero vectors in this lattice).
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https://en.wikipedia.org/wiki/Coxeter–Todd_lattice
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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 Herman Ole Andreas Wold. Let X n = ( X n 1 , … , X n k ) {\displaystyle {X}_{n}=(X_{n1},\dots ,X_{nk})} and X = ( X 1 , … , X k ) {\displaystyle \;{X}=(X_{1},\dots ,X_{k})} be random vectors of dimension k. Then X n {\displaystyle {X}_{n}} converges in distribution to X {\displaystyle {X}} if and only if: ∑ i = 1 k t i X n i → n → ∞ D ∑ i = 1 k t i X i . {\displaystyle \sum _{i=1}^{k}t_{i}X_{ni}{\overset {D}{\underset {n\rightarrow \infty }{\rightarrow }}}\sum _{i=1}^{k}t_{i}X_{i}.} for each ( t 1 , … , t k ) ∈ R k {\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} ^{k}} , that is, if every fixed linear combination of the coordinates of X n {\displaystyle {X}_{n}} converges in distribution to the correspondent linear combination of coordinates of X {\displaystyle {X}} .If X n {\displaystyle {X}_{n}} takes values in R + k {\displaystyle \mathbb {R} _{+}^{k}} , then the statement is also true with ( t 1 , … , t k ) ∈ R + k {\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} _{+}^{k}} .
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https://en.wikipedia.org/wiki/Cramér–Wold_theorem
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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.
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https://en.wikipedia.org/wiki/Crofton_formula
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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 first concrete examples of a separable infinite simple C*-algebra, meaning as a Hilbert space, O n {\displaystyle {\mathcal {O}}_{n}} is isometric to the sequence space l 2 ( N ) {\displaystyle l^{2}(\mathbb {N} )} and it has no nontrivial closed ideals. These algebras are fundamental to the study of simple infinite C*-algebras since any such algebra contains, for any given n, a subalgebra that has O n {\displaystyle {\mathcal {O}}_{n}} as quotient.
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https://en.wikipedia.org/wiki/Cuntz_algebra
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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 up to 0. In symbols, this is D ( G ) = min { N: ∀ ( { g n } n = 1 N ∈ G N ) ( ∃ { n k } k = 1 K: ∑ k = 1 K g n k = 0 ) } . {\displaystyle D(G)=\min \left\{N:\forall \left(\{g_{n}\}_{n=1}^{N}\in G^{N}\right)\left(\exists \{n_{k}\}_{k=1}^{K}:\sum _{k=1}^{K}{g_{n_{k}}}=0\right)\right\}.}
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https://en.wikipedia.org/wiki/Davenport_constant
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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.
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https://en.wikipedia.org/wiki/Dawson_function
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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.
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https://en.wikipedia.org/wiki/Dawson–Gärtner_theorem
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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.
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https://en.wikipedia.org/wiki/Dedekind_eta_function
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In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) is the number of monotone boolean functions of n variables. Equivalently, it is the number of antichains of subsets of an n-element set, the number of elements in a free distributive lattice with n generators, and one more than the number of abstract simplicial complexes on a set with n elements. Accurate asymptotic estimates of M(n) and an exact expression as a summation are known. However Dedekind's problem of computing the values of M(n) remains difficult: no closed-form expression for M(n) is known, and exact values of M(n) have been found only for n ≤ 9 (sequence A000372 in the OEIS).
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https://en.wikipedia.org/wiki/Dedekind_number
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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 functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2. The Dedekind zeta function is named for Richard Dedekind who introduced it in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.
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https://en.wikipedia.org/wiki/Dedekind_zeta_functions
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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–Sommerville equations can be restated as a symmetry condition for the h-vector of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes.
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https://en.wikipedia.org/wiki/Dehn-Somerville_equations
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In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by Demazure (1974, p. 83). The conjecture implies that many of the results of his paper can be extended from complex algebraic groups to algebraic groups over fields of other characteristics or over the integers. V. Lakshmibai, C. Musili, and C. S. Seshadri (1979) showed that Demazure's conjecture (for classical groups) follows from their work on standard monomial theory, and Peter Littelmann extended this to all reductive algebraic groups.
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https://en.wikipedia.org/wiki/Demazure_conjecture
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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 gave an example of a C1 diffeomorphism with an irrational rotation number that is not conjugate to a rotation.
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https://en.wikipedia.org/wiki/Denjoy's_theorem_on_rotation_number
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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.
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https://en.wikipedia.org/wiki/Denjoy–Koksma_inequality
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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.
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https://en.wikipedia.org/wiki/Denjoy–Luzin_theorem
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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 theorem, and Saks (1937, theorem 7.2, page 230) later strengthened their theorem.
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https://en.wikipedia.org/wiki/Denjoy–Luzin–Saks_theorem
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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 mathematician Julius Wolff.
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https://en.wikipedia.org/wiki/Denjoy–Wolff_theorem
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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 IX, section 4) and Bruckner (1978, chapter IV, theorem 4.4) give historical accounts of the theorem.
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https://en.wikipedia.org/wiki/Denjoy–Young–Saks_theorem
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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.
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https://en.wikipedia.org/wiki/Deuring–Heilbronn_phenomenon
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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 essentially equivalent to Chebyshev polynomials with a change of variable, and, in fact, Dickson polynomials are sometimes called Chebyshev polynomials. Dickson polynomials are generally studied over finite fields, where they sometimes may not be equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed α, they give many examples of permutation polynomials; polynomials acting as permutations of finite fields.
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https://en.wikipedia.org/wiki/Dickson_polynomial
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In mathematics, the Dieudonné plank is a specific topological space introduced by Dieudonné (1944). It is an example of a metacompact space that is not paracompact. The notion has since been generalized (by Barr et al.) to that of an absolute CR-epic space.
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https://en.wikipedia.org/wiki/Dieudonné_plank
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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.
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https://en.wikipedia.org/wiki/Dini_test
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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 series of a periodic function f converges uniformly on the real line if lim δ → 0 + ω ( δ , f ) log ( δ ) = 0 {\displaystyle \lim _{\delta \rightarrow 0^{+}}\omega (\delta ,f)\log(\delta )=0} where ω {\displaystyle \omega } is the modulus of continuity of f with respect to δ {\displaystyle \delta } .
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https://en.wikipedia.org/wiki/Dini–Lipschitz_criterion
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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.
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https://en.wikipedia.org/wiki/Catalan_beta_function
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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.
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https://en.wikipedia.org/wiki/Dirichlet_convolution
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In mathematics, the Dirichlet density (or analytic density) of a set of primes, named after Peter Gustav Lejeune Dirichlet, is a measure of the size of the set that is easier to use than the natural density.
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https://en.wikipedia.org/wiki/Dirichlet_density
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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 Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function u ≠ 0 and eigenvalue λ Here Δ is the Laplacian, which is given in xy-coordinates by Δ u = ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 .
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https://en.wikipedia.org/wiki/Dirichlet_eigenvalue
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{\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 for the corresponding Neumann problem. The Laplace operator Δ appearing in (1) is often known as the Dirichlet Laplacian when it is considered as accepting only functions u satisfying the Dirichlet boundary condition.
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https://en.wikipedia.org/wiki/Dirichlet_eigenvalue
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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 Dirichlet eigenvalues λ are real, positive, and have no limit point.
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https://en.wikipedia.org/wiki/Dirichlet_eigenvalue
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Thus they can be arranged in increasing order: 0 < λ 1 ≤ λ 2 ≤ ⋯ , λ n → ∞ , {\displaystyle 0<\lambda _{1}\leq \lambda _{2}\leq \cdots ,\quad \lambda _{n}\to \infty ,} where each eigenvalue is counted according to its geometric multiplicity. The eigenspaces are orthogonal in the space of square-integrable functions, and consist of smooth functions. In fact, the Dirichlet Laplacian has a continuous extension to an operator from the Sobolev space H 0 2 ( Ω ) {\displaystyle H_{0}^{2}(\Omega )} into L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} .
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https://en.wikipedia.org/wiki/Dirichlet_eigenvalue
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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 minimizes the Dirichlet energy. To wit, λ 1 = inf u ≠ 0 ∫ Ω | ∇ u | 2 ∫ Ω | u | 2 , {\displaystyle \lambda _{1}=\inf _{u\not =0}{\frac {\int _{\Omega }|\nabla u|^{2}}{\int _{\Omega }|u|^{2}}},} the infimum is taken over all u of compact support that do not vanish identically in Ω. By a density argument, this infimum agrees with that taken over nonzero u ∈ H 0 1 ( Ω ) {\displaystyle u\in H_{0}^{1}(\Omega )} . Moreover, using results from the calculus of variations analogous to the Lax–Milgram theorem, one can show that a minimizer exists in H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} . More generally, one has λ k = sup inf ∫ Ω | ∇ u | 2 ∫ Ω | u | 2 {\displaystyle \lambda _{k}=\sup \inf {\frac {\int _{\Omega }|\nabla u|^{2}}{\int _{\Omega }|u|^{2}}}} where the supremum is taken over all (k−1)-tuples ϕ 1 , … , ϕ k − 1 ∈ H 0 1 ( Ω ) {\displaystyle \phi _{1},\dots ,\phi _{k-1}\in H_{0}^{1}(\Omega )} and the infimum over all u orthogonal to the ϕ i {\displaystyle \phi _{i}} .
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https://en.wikipedia.org/wiki/Dirichlet_eigenvalue
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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.
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https://en.wikipedia.org/wiki/Dirichlet_energy
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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 Lejeune Dirichlet. It is an example of pathological function which provides counterexamples to many situations.
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https://en.wikipedia.org/wiki/Dirichlet_function
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