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In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to h... | https://en.wikipedia.org/wiki/Tucker_decomposition |
In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other b... | https://en.wikipedia.org/wiki/Tucker_decomposition |
In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker. Let T be a triangulation of the closed n-dimensional ball B n {\displaystyle B_{n}} . Assume T is antipodally symmetric on the boundary sphere S n − 1 {\displaystyle S_{n-1}} . That means that the subset o... | https://en.wikipedia.org/wiki/Tucker's_lemma |
Let L: V ( T ) → { + 1 , − 1 , + 2 , − 2 , . . | https://en.wikipedia.org/wiki/Tucker's_lemma |
. , + n , − n } {\displaystyle L:V(T)\to \{+1,-1,+2,-2,...,+n,-n\}} be a labeling of the vertices of T which is an odd function on S n − 1 {\displaystyle S_{n-1}} , i.e, L ( − v ) = − L ( v ) {\displaystyle L(-v)=-L(v)} for every vertex v ∈ S n − 1 {\displaystyle v\in S_{n-1}} . Then Tucker's lemma states that T contai... | https://en.wikipedia.org/wiki/Tucker's_lemma |
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál Turán (1950) (and first published by Szegö (1948)). There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors. If P n ... | https://en.wikipedia.org/wiki/Turán's_inequalities |
{\displaystyle \,\!P_{n}(x)^{2}>P_{n-1}(x)P_{n+1}(x)\ {\text{for}}\ -1 0 , {\displaystyle H_{n}(x)^{2}-H_{n-1}(x)H_{n+1}(x)=(n-1)!\cdot \sum _{i=0}^{n-1}{\frac {2^{n-i}}{i! }}H_{i}(x)^{2}>0,} whilst for Chebyshev polynomials they are T n ( x ) 2 − T n − 1 ( x ) T n + 1 ( x ) = 1 − x 2 > 0 for − 1 < x < 1. {\displaystyl... | https://en.wikipedia.org/wiki/Turán's_inequalities |
In mathematics, Turán's method provides lower bounds for exponential sums and complex power sums. The method has been applied to problems in equidistribution. The method applies to sums of the form s ν = ∑ n = 1 N b n z n ν {\displaystyle s_{\nu }=\sum _{n=1}^{N}b_{n}z_{n}^{\nu }\ } where the b and z are complex number... | https://en.wikipedia.org/wiki/Turán's_method |
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed un... | https://en.wikipedia.org/wiki/Tychonoff_theorem |
(This reference is mentioned in "Topology" by Hocking and Young, Dover Publications, Ind.) Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with Urysohn's lemma). The theorem is also valid for topological spaces based on fuzzy sets. | https://en.wikipedia.org/wiki/Tychonoff_theorem |
In mathematics, Ulugh Beg wrote accurate trigonometric tables of sine and tangent values correct to at least eight decimal places. | https://en.wikipedia.org/wiki/Ulugh_Beg |
In mathematics, Varadhan's lemma is a result from the large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a family of random variables Zε as ε becomes small in terms of a rate function for the variables. | https://en.wikipedia.org/wiki/Varadhan's_lemma |
In mathematics, Veblen's theorem, introduced by Oswald Veblen (1912), states that the set of edges of a finite graph can be written as a union of disjoint simple cycles if and only if every vertex has even degree. Thus, it is closely related to the theorem of Euler (1736) that a finite graph has an Euler tour (a single... | https://en.wikipedia.org/wiki/Veblen's_theorem |
However, Veblen's theorem applies also to disconnected graphs, and can be generalized to infinite graphs in which every vertex has finite degree (Sabidussi 1964). If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite ... | https://en.wikipedia.org/wiki/Veblen's_theorem |
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by Jean-Louis Verdier (1995) as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomo... | https://en.wikipedia.org/wiki/Verdier_duality |
In mathematics, Viennot's geometric construction (named after Xavier Gérard Viennot) gives a diagrammatic interpretation of the Robinson–Schensted correspondence in terms of shadow lines. It has a generalization to the Robinson–Schensted–Knuth correspondence, which is known as the matrix-ball construction. | https://en.wikipedia.org/wiki/Viennot's_geometric_construction |
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta"). | https://en.wikipedia.org/wiki/Vieta's_theorem |
In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech... | https://en.wikipedia.org/wiki/Vinberg's_algorithm |
In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients. Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, havin... | https://en.wikipedia.org/wiki/Vincent's_theorem |
In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers. It is an important inequality in analytic number theory, named for I. M. Vinogradov. More specifically, let J s , k ( X ) {\displaystyle J_{s,k}(X)} count the number of solutions to the system of k {\displaystyle k} s... | https://en.wikipedia.org/wiki/Vinogradov's_mean-value_theorem |
{\displaystyle f_{k}(\mathbf {\alpha } ;X)=\sum _{1\leq x\leq X}\exp(2\pi i(\alpha _{1}x+\cdots +\alpha _{k}x^{k})).} Vinogradov's mean-value theorem gives an upper bound on the value of J s , k ( X ) {\displaystyle J_{s,k}(X)} . | https://en.wikipedia.org/wiki/Vinogradov's_mean-value_theorem |
A strong estimate for J s , k ( X ) {\displaystyle J_{s,k}(X)} is an important part of the Hardy-Littlewood method for attacking Waring's problem and also for demonstrating a zero free region for the Riemann zeta-function in the critical strip. Various bounds have been produced for J s , k ( X ) {\displaystyle J_{s,k}(... | https://en.wikipedia.org/wiki/Vinogradov's_mean-value_theorem |
In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets. | https://en.wikipedia.org/wiki/Vitale's_random_Brunn–Minkowski_inequality |
In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles (a diameter) of the sphere (see diagram). Before Viv... | https://en.wikipedia.org/wiki/Viviani's_curve |
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π: It can also be represented as: The formula is named after François Viète, who published it in 1593. As the first formula of European mathematics to represent an infinite... | https://en.wikipedia.org/wiki/Viète's_formula |
The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle formula from trigonometry leads to a generalized formula, discovered by Leonhard Euler, that has Viète's formula as a special case. Many similar ... | https://en.wikipedia.org/wiki/Viète's_formula |
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas o... | https://en.wikipedia.org/wiki/Voigt_notation |
Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off-diagonal. | https://en.wikipedia.org/wiki/Voigt_notation |
Thus it can be expressed as the vector ⟨ x 11 , x 22 , x 12 ⟩ {\displaystyle \langle x_{11},x_{22},x_{12}\rangle } .As another example: The stress tensor (in matrix notation) is given as σ = . {\displaystyle {\boldsymbol {\sigma }}=\left.} In Voigt notation it is simplified to a 6-dimensional vector: σ ~ = ( σ x x , σ... | https://en.wikipedia.org/wiki/Voigt_notation |
{\displaystyle {\tilde {\sigma }}=(\sigma _{xx},\sigma _{yy},\sigma _{zz},\sigma _{yz},\sigma _{xz},\sigma _{xy})\equiv (\sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5},\sigma _{6}).} The strain tensor, similar in nature to the stress tensor—both are symmetric second-order tensors --, is given in matrix for... | https://en.wikipedia.org/wiki/Voigt_notation |
Its representation in Voigt notation is ϵ ~ = ( ϵ x x , ϵ y y , ϵ z z , γ y z , γ x z , γ x y ) ≡ ( ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4 , ϵ 5 , ϵ 6 ) , {\displaystyle {\tilde {\epsilon }}=(\epsilon _{xx},\epsilon _{yy},\epsilon _{zz},\gamma _{yz},\gamma _{xz},\gamma _{xy})\equiv (\epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon _{... | https://en.wikipedia.org/wiki/Voigt_notation |
In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta (1987) about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other ... | https://en.wikipedia.org/wiki/Vojta's_conjecture |
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties: V is differentiable everywhere The derivative V ′ is bounded everywhere The derivative is not Riemann-integrable. | https://en.wikipedia.org/wiki/Volterra's_function |
In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings. Vopěnka's principle was first introduced by Petr V... | https://en.wikipedia.org/wiki/Vopěnka's_principle |
According to Pudlák (2013, p. 204), Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However, before publishing his inconsistency proof he... | https://en.wikipedia.org/wiki/Vopěnka's_principle |
In mathematics, Waldspurger's theorem, introduced by Jean-Loup Waldspurger (1981), is a result that identifies Fourier coefficients of modular forms of half-integral weight k+1/2 with the value of an L-series at s=k/2. | https://en.wikipedia.org/wiki/Waldspurger's_theorem |
In mathematics, Waraszkiewicz spirals are subsets of the plane introduced by Waraszkiewicz (1932). Waraszkiewicz spirals give an example of an uncountable family of pairwise incomparable continua, meaning that there is no continuous map from one onto another. | https://en.wikipedia.org/wiki/Waraszkiewicz_spiral |
In mathematics, Ward's conjecture is the conjecture made by Ward (1985, p. 451) that "many (and perhaps all?) of the ordinary and partial differential equations that are regarded as being integrable or solvable may be obtained from the self-dual gauge field equations (or its generalizations) by reduction". | https://en.wikipedia.org/wiki/Ward's_conjecture |
In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals. | https://en.wikipedia.org/wiki/Watson's_lemma |
In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius b with centers distance 2a apart (taken to be at (±a, 0)). A line segment of length 2c attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as... | https://en.wikipedia.org/wiki/Watt's_curve |
It arose in connection with James Watt's pioneering work on the steam engine. The equation of the curve can be given in polar coordinates as r 2 = b 2 − 2 . {\displaystyle r^{2}=b^{2}-\left^{2}.} | https://en.wikipedia.org/wiki/Watt's_curve |
In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following. Consider two non-singular curves C and C′ having the same genus g > 1. If there is a rational correspondence φ between C and C′, then φ is a birational transformation. | https://en.wikipedia.org/wiki/Weber's_theorem_(Algebraic_curves) |
In mathematics, Wedderburn's little theorem states that every finite division ring is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite alternative division ring is a field. | https://en.wikipedia.org/wiki/Wedderburn_theorem |
In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by Charles Weibel (1980) and proven in full generality by Kerz, Strunk & Tamme (2018) using methods from derived algebraic geometry. Previously partial cases had been proven by Morrow (... | https://en.wikipedia.org/wiki/Weibel's_conjecture |
In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea was formulated first in a 1952 paper. It is based on the explicit formulae of... | https://en.wikipedia.org/wiki/Weil's_criterion |
A single statement thus combines statements on the complex zeroes of all Dirichlet L-functions. Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions (Artin-Hecke L-functions); and to the global function field case. Here the inclusion of Artin L-functions, in ... | https://en.wikipedia.org/wiki/Weil's_criterion |
In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by Weingarten (1978) who found their asymptotic behavior, and named by Collins (2003), who evaluated the... | https://en.wikipedia.org/wiki/Weingarten_function |
In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M × N − {\displaystyle M\times N^{-}} , where the superscript minus means minus the given symplectic form (for example,... | https://en.wikipedia.org/wiki/Symplectic_category |
In mathematics, Weisner's method is a method for finding generating functions for special functions using representation theory of Lie groups and Lie algebras, introduced by Weisner (1955). It includes Truesdell's method as a special case, and is essentially the same as Rainville's method. ... Weisner's group-theoretic... | https://en.wikipedia.org/wiki/Weisner's_method |
In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , and area Δ {\displaystyle \Delta } , the following inequality holds: a 2 + b 2 + c 2 ≥ 4 3 Δ . {\displaystyle a^{2}+b^{2}+c^{2}\geq 4{\sq... | https://en.wikipedia.org/wiki/Weitzenböck's_inequality |
In mathematics, Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space. The bounds are important tools in the design and analysis of certain methods in telecommunication engineering, particularly in coding theory. The bounds were originally publish... | https://en.wikipedia.org/wiki/Welch_bounds |
In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions that are not smooth solutions. Weyl's lemma is a special case of elliptic or hypoelliptic regularity. | https://en.wikipedia.org/wiki/Weyl's_lemma_(Laplace_equation) |
In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid). In purely graph-theoretic terms, this... | https://en.wikipedia.org/wiki/Whitney's_planarity_criterion |
In mathematics, Wiener deconvolution is an application of the Wiener filter to the noise problems inherent in deconvolution. It works in the frequency domain, attempting to minimize the impact of deconvolved noise at frequencies which have a poor signal-to-noise ratio. The Wiener deconvolution method has widespread use... | https://en.wikipedia.org/wiki/Wiener_deconvolution |
In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its atomic part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener. | https://en.wikipedia.org/wiki/Wiener's_lemma |
In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties. | https://en.wikipedia.org/wiki/Wilkie's_theorem |
In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilson (1980) that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by p n ( t 2 ) = ( a + b ) n ( a ... | https://en.wikipedia.org/wiki/Wilson_polynomials |
In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace H 2 {\displaystyle \left.\right.H_{2}} of the simple, unweighted holomor... | https://en.wikipedia.org/wiki/Wirtinger's_representation_and_projection_theorem |
If F ( z ) {\displaystyle \left.\right.\left.F(z)\right.} is of the class L 2 {\displaystyle \left.\right.L^{2}} on | z | < 1 {\displaystyle \left.\right.|z|<1} , i.e. ∬ | z | < 1 | F ( z ) | 2 d S < + ∞ , {\displaystyle \iint _{|z|<1}|F(z)|^{2}\,dS<+\infty ,} where d S {\displaystyle \left.\right.dS} is the area eleme... | https://en.wikipedia.org/wiki/Wirtinger's_representation_and_projection_theorem |
The last formula gives a form for the orthogonal projection from L 2 {\displaystyle \left.\right.L^{2}} to H 2 {\displaystyle \left.\right.H_{2}} . Besides, replacement of F ( ζ ) {\displaystyle \left.\right.F(\zeta )} by f ( ζ ) {\displaystyle \left.\right.f(\zeta )} makes it Wirtinger's representation for all f ( z )... | https://en.wikipedia.org/wiki/Wirtinger's_representation_and_projection_theorem |
Later, after the 1950s, a degree of the Cauchy kernel was called reproducing kernel, and the notation A 0 2 {\displaystyle \left.\right.A_{0}^{2}} became common for the class H 2 {\displaystyle \left.\right.H_{2}} . In 1948 Mkhitar Djrbashian extended Wirtinger's representation and projection to the wider, weighted Hil... | https://en.wikipedia.org/wiki/Wirtinger's_representation_and_projection_theorem |
In mathematics, Witt vector cohomology was an early p-adic cohomology theory for algebraic varieties introduced by Serre (1958). Serre constructed it by defining a sheaf of truncated Witt rings Wn over a variety V and then taking the inverse limit of the sheaf cohomology groups Hi(V, Wn) of these sheaves. Serre observe... | https://en.wikipedia.org/wiki/Witt_vector_cohomology |
In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5 {\displaystyle p\geq 5} , the congruence ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 3 ) {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}} holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one... | https://en.wikipedia.org/wiki/Wolstenholme's_theorem |
In 1819, Charles Babbage showed the same congruence modulo p2, which holds for p ≥ 3 {\displaystyle p\geq 3} . An equivalent formulation is the congruence ( a p b p ) ≡ ( a b ) ( mod p 3 ) {\displaystyle {ap \choose bp}\equiv {a \choose b}{\pmod {p^{3}}}} for p ≥ 5 {\displaystyle p\geq 5} , which is due to Wilhelm Ljun... | https://en.wikipedia.org/wiki/Wolstenholme's_theorem |
A prime that satisfies the congruence modulo p4 is called a Wolstenholme prime (see below). As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers: 1 + 1 2 + 1 3 + ⋯ + 1 p − 1 ≡ 0 ( mod p 2 ) , and {\displaystyle 1+{1 \over 2}+{1 \over 3}+\dots... | https://en.wikipedia.org/wiki/Wolstenholme's_theorem |
In mathematics, Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields, introduced by Yamamoto (1986). | https://en.wikipedia.org/wiki/Yamamoto's_reciprocity_law |
In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks: Is it true that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of S n + 1 {\displaystyle S^{n+1}} is n {\displaystyle n} ? If true, it ... | https://en.wikipedia.org/wiki/Yau's_conjecture_on_the_first_eigenvalue |
Shiing-Shen Chern conjectured that a closed, minimally immersed hypersurface in S n + 1 {\displaystyle S^{n+1}} (1), whose second fundamental form has constant length, is isoparametric. If true, it would have established the Yau's conjecture for the minimal hypersurface whose second fundamental form has constant length... | https://en.wikipedia.org/wiki/Yau's_conjecture_on_the_first_eigenvalue |
In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young. | https://en.wikipedia.org/wiki/Young's_convolution_inequality |
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality. Young's inequality for products can be used to prove Hölder's inequality. It is also widely use... | https://en.wikipedia.org/wiki/Young's_inequality_for_products |
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial ... | https://en.wikipedia.org/wiki/Young's_lattice |
In mathematics, Zahorski's theorem is a theorem of real analysis. It states that a necessary and sufficient condition for a subset of the real line to be the set of points of non-differentiability of a continuous real-valued function, is that it be the union of a Gδ set and a G δ σ {\displaystyle {G_{\delta }}_{\sigma ... | https://en.wikipedia.org/wiki/Zahorski_theorem |
In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers. Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in ... | https://en.wikipedia.org/wiki/Zeckendorf's_theorem |
Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is 64 = 55 + 8 + 1.There are other ways of representing 64 as the sum of Fibonacci numbers 64 = 55 + 5 + 3 + 1 64 = 34 + 21 + 8 + 1 64 = ... | https://en.wikipedia.org/wiki/Zeckendorf's_theorem |
In mathematics, Zolotarev polynomials are polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy of approximation near the origin is of less importance. Zolotarev polynomials differ from the Chebyshev polynomials in that two of the coefficients ar... | https://en.wikipedia.org/wiki/Zolotarev_polynomials |
In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematic... | https://en.wikipedia.org/wiki/Tits_system |
In mathematics, a (classical) gauge field is defined as a connection over a principal G-bundle over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately. A connection on a G-bundle tells you how to glue fibers together at nearby points of M. It starts with a continuous symmetry group G th... | https://en.wikipedia.org/wiki/Quantization_condition |
In physics, the connection is the fundamental physical object. One of the fundamental observations in the theory of characteristic classes in algebraic topology is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over any connection over it. Note that a... | https://en.wikipedia.org/wiki/Quantization_condition |
If spacetime is ℝ4 the space of all possible connections of the G-bundle is connected. But consider what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S2. | https://en.wikipedia.org/wiki/Quantization_condition |
A principal G-bundle over S2 is defined by covering S2 by two charts, each homeomorphic to the open 2-ball such that their intersection is homeomorphic to the strip S1×I. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle S1. So a topological classification of the possible connect... | https://en.wikipedia.org/wiki/Quantization_condition |
The transition function maps the strip to G, and the different ways of mapping a strip into G are given by the first homotopy group of G. So in the G-bundle formulation, a gauge theory admits Dirac monopoles provided G is not simply connected, whenever there are paths that go around the group that cannot be deformed to... | https://en.wikipedia.org/wiki/Quantization_condition |
The total magnetic flux is none other than the first Chern number of the principal bundle, and depends only upon the choice of the principal bundle, and not the specific connection over it. In other words, it is a topological invariant. This argument for monopoles is a restatement of the lasso argument for a pure U(1) ... | https://en.wikipedia.org/wiki/Quantization_condition |
It generalizes to d + 1 dimensions with d ≥ 2 in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension d − 3. Another way is to examine the type of topological singularity at a point with the homotopy group πd−2(G). | https://en.wikipedia.org/wiki/Quantization_condition |
In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every q ∈ M {\displaystyle q\in M} the distance function L q: N → R , L q ( x ) = dist ( x , q ) 2 {\displaystyle L_{q}:N\to \mathbf {R} ,\qquad L_{q}(x)=\operatorname {dist} (x,q)^{2}} is a perfect Mo... | https://en.wikipedia.org/wiki/Taut_submanifold |
In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings. Every left Noetherian ring is left coherent. | https://en.wikipedia.org/wiki/Coherent_ring |
The ring of polynomials in an infinite number of variables over a left Noetherian ring is an example of a left coherent ring that is not left Noetherian. A ring is left coherent if and only if every direct product of flat right modules is flat (Chase 1960), (Anderson & Fuller 1992, p. 229). Compare this to: A ring is l... | https://en.wikipedia.org/wiki/Coherent_ring |
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivati... | https://en.wikipedia.org/wiki/Monge-Ampere_equation |
The most complete results so far have been obtained when the equation is elliptic. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 and later by André-Marie Ampère in 1... | https://en.wikipedia.org/wiki/Monge-Ampere_equation |
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < ... | https://en.wikipedia.org/wiki/Bounded_interval |
The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even ... | https://en.wikipedia.org/wiki/Bounded_interval |
In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product satisfying the Leibniz identity , c ] = ] + , b ] . {\displaystyle ,c]=]+,b].\,} In other words, right multiplic... | https://en.wikipedia.org/wiki/Leibniz_algebra |
Conversely any Lie algebra is obviously a Leibniz algebra. In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature. | https://en.wikipedia.org/wiki/Leibniz_algebra |
For instance, it has been shown that Engel's theorem still holds for Leibniz algebras and that a weaker version of Levi-Malcev theorem also holds.The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that = a 1 ⊗ ⋯ a n ⊗ x for a 1 , … , a n , x ∈ V . {\displaystyle =a_{1}\otimes \cdot... | https://en.wikipedia.org/wiki/Leibniz_algebra |
They attracted interest after Jean-Louis Loday noticed that the classical Chevalley–Eilenberg boundary map in the exterior module of a Lie algebra can be lifted to the tensor module which yields a new chain complex. In fact this complex is well-defined for any Leibniz algebra. The homology HL(L) of this chain complex i... | https://en.wikipedia.org/wiki/Leibniz_algebra |
If L is the Lie algebra of (infinite) matrices over an associative R-algebra A then Leibniz homology of L is the tensor algebra over the Hochschild homology of A. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. It has defining identity: ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) . {\displaystyle (... | https://en.wikipedia.org/wiki/Leibniz_algebra |
In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object ⊥ {\displaystyle \bot } . The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality. | https://en.wikipedia.org/wiki/Dualizing_object |
In mathematics, a *-ring is a ring with a map *: A → A that is an antiautomorphism and an involution. More precisely, * is required to satisfy the following properties: (x + y)* = x* + y* (x y)* = y* x* 1* = 1 (x*)* = xfor all x, y in A. This is also called an involutive ring, involutory ring, and ring with involution.... | https://en.wikipedia.org/wiki/*-algebra |
One can define a sesquilinear form over any *-ring. Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: x ∈ I ⇒ x* ∈ I and so on. *-rings are unrelated to star semirings in the theory of computation. | https://en.wikipedia.org/wiki/*-algebra |
In mathematics, a 2-valued morphism is a homomorphism that sends a Boolean algebra B onto the two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on B, and, in a different way, also the same things as a maximal ideal of B. 2-valued morphisms have also been proposed as a tool for un... | https://en.wikipedia.org/wiki/2-valued_morphism |
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. Thi... | https://en.wikipedia.org/wiki/3-manifold |
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