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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. | https://en.wikipedia.org/wiki/Carlitz_exponential |
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 al... | https://en.wikipedia.org/wiki/Carlitz-Wan_conjecture |
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 wa... | https://en.wikipedia.org/wiki/Carlitz-Wan_conjecture |
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:... | https://en.wikipedia.org/wiki/Carlson_symmetric_form |
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 ) {\displa... | https://en.wikipedia.org/wiki/Carlson_symmetric_form |
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... | https://en.wikipedia.org/wiki/Cartan_involution |
In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space. | https://en.wikipedia.org/wiki/Cartan_model |
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 ... | https://en.wikipedia.org/wiki/Cartan–Dieudonné_theorem |
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 comp... | https://en.wikipedia.org/wiki/Cartan–Dieudonné_theorem |
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 F... | https://en.wikipedia.org/wiki/Cartan–Hadamard_conjecture |
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. | https://en.wikipedia.org/wiki/Cartan–Kähler_theorem |
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 pull... | https://en.wikipedia.org/wiki/Castelnuovo–de_Franchis_theorem |
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 ... | https://en.wikipedia.org/wiki/Cauchy_condensation_test |
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... | https://en.wikipedia.org/wiki/Cauchy_integral |
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 n... | https://en.wikipedia.org/wiki/Cauchy_principal_part |
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. Hadama... | https://en.wikipedia.org/wiki/Cauchy–Hadamard_theorem |
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 ... | https://en.wikipedia.org/wiki/Cauchy–Kowalevski_theorem |
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. | https://en.wikipedia.org/wiki/Cayley_plane |
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 ... | https://en.wikipedia.org/wiki/Cayley_transform |
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 eve... | https://en.wikipedia.org/wiki/Cayley–Bacharach_theorem |
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 number... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Cayley–Dickson_construction |
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://en.wikipedia.org/wiki/Chabauty_topology |
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://en.wikipedia.org/wiki/Champernowne_constant |
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://en.wikipedia.org/wiki/Chang_number |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Chazy_equation |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Chebyshev_function |
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://en.wikipedia.org/wiki/Chebyshev_integral |
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://en.wikipedia.org/wiki/Chebyshev_rational_functions |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Cheeger_bound |
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 | {\disp... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Cheeger_bound |
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://en.wikipedia.org/wiki/Cheeger_constant_(graph_theory) |
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://en.wikipedia.org/wiki/Chern–Gauss–Bonnet_formula |
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. | https://en.wikipedia.org/wiki/Chern–Simons_3-form |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Chern–Weil_homomorphism |
{\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 \Omeg... | https://en.wikipedia.org/wiki/Chern–Weil_homomorphism |
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://en.wikipedia.org/wiki/Chevalley–Iwahori–Nagata_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://en.wikipedia.org/wiki/Chevalley–Shephard–Todd_theorem |
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://en.wikipedia.org/wiki/Chihara–Ismail_polynomials |
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://en.wikipedia.org/wiki/Chinese_monoid |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Chinese_Remainder_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://en.wikipedia.org/wiki/Chowla–Mordell_theorem |
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://en.wikipedia.org/wiki/Chowla–Selberg_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://en.wikipedia.org/wiki/Christoffel–Darboux_formula |
In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev. | https://en.wikipedia.org/wiki/Christ–Kiselev_maximal_inequality |
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://en.wikipedia.org/wiki/Chung–Fuchs_theorem |
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 t... | https://en.wikipedia.org/wiki/Chvátal–Sankoff_constants |
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://en.wikipedia.org/wiki/Clark–Ocone_theorem |
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 blow... | https://en.wikipedia.org/wiki/Clebsch_surface |
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://en.wikipedia.org/wiki/Coates_graph |
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. | https://en.wikipedia.org/wiki/Coble_variety |
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://en.wikipedia.org/wiki/Cohen_structure_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://en.wikipedia.org/wiki/Cohen–Hewitt_factorization_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://en.wikipedia.org/wiki/Conley–Zehnder_theorem |
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). | https://en.wikipedia.org/wiki/Contou-Carrère_symbol |
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://en.wikipedia.org/wiki/Conway_polynomial_(finite_fields) |
In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix. It is named after Richard Courant. | https://en.wikipedia.org/wiki/Courant_minimax_principle |
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://en.wikipedia.org/wiki/Second_Cousin_problem |
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. | https://en.wikipedia.org/wiki/Tits_cone |
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. | https://en.wikipedia.org/wiki/Coxeter_number |
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://en.wikipedia.org/wiki/Coxeter–Todd_lattice |
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://en.wikipedia.org/wiki/Cramér–Wold_theorem |
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://en.wikipedia.org/wiki/Crofton_formula |
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://en.wikipedia.org/wiki/Cuntz_algebra |
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://en.wikipedia.org/wiki/Davenport_constant |
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://en.wikipedia.org/wiki/Dawson_function |
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://en.wikipedia.org/wiki/Dawson–Gärtner_theorem |
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://en.wikipedia.org/wiki/Dedekind_eta_function |
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... | https://en.wikipedia.org/wiki/Dedekind_number |
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://en.wikipedia.org/wiki/Dedekind_zeta_functions |
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://en.wikipedia.org/wiki/Dehn-Somerville_equations |
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 ... | https://en.wikipedia.org/wiki/Demazure_conjecture |
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://en.wikipedia.org/wiki/Denjoy's_theorem_on_rotation_number |
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://en.wikipedia.org/wiki/Denjoy–Koksma_inequality |
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://en.wikipedia.org/wiki/Denjoy–Luzin_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://en.wikipedia.org/wiki/Denjoy–Luzin–Saks_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://en.wikipedia.org/wiki/Denjoy–Wolff_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://en.wikipedia.org/wiki/Denjoy–Young–Saks_theorem |
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://en.wikipedia.org/wiki/Deuring–Heilbronn_phenomenon |
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://en.wikipedia.org/wiki/Dickson_polynomial |
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. | https://en.wikipedia.org/wiki/Dieudonné_plank |
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://en.wikipedia.org/wiki/Dini_test |
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://en.wikipedia.org/wiki/Dini–Lipschitz_criterion |
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://en.wikipedia.org/wiki/Catalan_beta_function |
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://en.wikipedia.org/wiki/Dirichlet_convolution |
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. | https://en.wikipedia.org/wiki/Dirichlet_density |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Dirichlet_eigenvalue |
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, an... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Dirichlet_eigenvalue |
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://en.wikipedia.org/wiki/Dirichlet_energy |
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://en.wikipedia.org/wiki/Dirichlet_function |
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