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Hermitian Yang–Mills Equations
{{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature method * Hermite class * Hermite differential equation * Hermite distribution, a parametrized family of discrete probability distributions * Hermite–Lindemann theorem, theorem about transcendental numbers * Hermite constant, a constant related to the geometry of certain lattices * Hermite-Gaussian modes * The Hermite–Hadamard inequality on convex functions and their integrals * Hermite interpolation, a method of interpolating data points by a polynomial * Hermite–Kronecker–Brioschi characterization * The Hermite–Minkowski theorem, stating that only finitely many number fields have small discriminants * Hermite normal form, a form of row-reduced matrices * Hermite numbers, integers related to the Hermite polynomials * Hermite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré. He was the first to prove that '' e'', the base of natural logarithms, is a transcendental number. His methods were used later by Ferdinand von Lindemann to prove that π is transcendental. Life Hermite was born in Dieuze, Moselle, on 24 December 1822, with a deformity in his right foot that would impair his gait throughout his life. He was the sixth of seven children of Ferdinand Hermite and his wife, Madeleine née Lallemand. Ferdinand worked in the drapery business of Madeleine's family while also pursuing a career as an artist. The drapery business relocate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite's Identity
In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number ''x'' and for every positive integer ''n'' the following identity holds:. : \sum_^\left\lfloor x+\frac\right\rfloor=\lfloor nx\rfloor . Proof Split x into its integer part and fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ..., x=\lfloor x\rfloor+\. There is exactly one k'\in\ with :\lfloor x\rfloor=\left\lfloor x+\frac\right\rfloor\le x<\left\lfloor x+\frac\right\rfloor=\lfloor x\rfloor+1. By subtracting the same integer from inside the floor operations on the left and right sides of this inequality, it may be rewritten as : |
Permutation Polynomial
In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x \mapsto g(x) is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function, so in particular every permutation of the elements of that field, can be written as a polynomial function. In the case of finite rings Z/''n''Z, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms. Single variable permutation polynomials over finite fields Let be the finite field of characteristic , that is, the field having elements where for some prime . A polynomial with coefficients in (symbolically written as ) is a ''permutation polynomial'' of if the function from to itself defined by c \mapsto f(c) is a permutation of . Due to the finiteness of , thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite's Cotangent Identity
In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite.Warren P. Johnson, "Trigonometric Identities à la Hermite", ''American Mathematical Monthly'', volume 117, number 4, April 2010, pages 311–327 Suppose ''a''1, ..., ''a''''n'' are complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...s, no two of which differ by an integer multiple of . Let : A_ = \prod_ \cot(a_k - a_j) (in particular, ''A''1,1, being an empty product, is 1). Then : \cot(z - a_1)\cdots\cot(z - a_n) = \cos\frac + \sum_^n A_ \cot(z - a_k). The simplest non-trivial example is the case ''n'' = 2: : \cot(z - a_1)\cot(z - a_2) = -1 + \cot(a_1 - a_2)\cot(z - a_1) + \cot(a_2 - a_1)\cot(z - a_2). \, Notes a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oscillator Representation
In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself. The contraction operators, determined only up to a sign, have kernels that are Gaussian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (algebra)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a Set (mathematics), set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, function (mathematics), functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is Associative property, associative, is Distributive property, distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term "rng (algebra), " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Ring
In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects. According to (p. 465), a ring is right Hermite if, for every two elements ''a'' and ''b'' of the ring, there is an element ''d'' of the ring and an invertible 2 by 2 matrix ''M'' over the ring such that ''(a b)M=(d 0)''. (The term left Hermite is defined similarly.) Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix (, p. 468.) (appendix to §I.4) calls this property K-Hermite, using ''Hermite'' instead in the sense given below. According to (§I.4, p. 26), a ring is right Hermite if any finitely generated stably free right module over the ring is free. This is equivalent to requiring that any row vector ''(b1,...,bn)'' of elements of the ring which generate it as a right module (i.e., ''b1R+...+bnR=R'') can be completed to a (not necessarily square) invertible matrix by adding some number of rows. (The criteri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocity Law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial f(x) = x^2 + ax + b splits into linear terms when reduced mod p. That is, it determines for which prime numbers the relationf(x) \equiv f_p(x) = (x-n_p)(x-m_p) \text (\text p)holds. For a general reciprocity lawpg 3, it is defined as the rule determining which primes p the polynomial f_p splits into linear factors, denoted \text\. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (''p''/''q'') generalizing the quadratic reciprocity symbol, that describes when a prime number is an ''n''th power residue modulo another prime, and gave a relation between (''p''/''q'') and (''q''/''p''). Hilbert refo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Reciprocity
In mathematics, Hermite's law of reciprocity, introduced by , states that the degree ''m'' covariants of a binary form of degree ''n'' correspond to the degree ''n'' covariants of a binary form of degree ''m''. In terms of representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ... it states that the representations ''S''''m'' ''S''''n'' C2 and ''S''''n'' ''S''''m'' C2 of ''GL''2 are isomorphic. References *{{Citation , last1=Hermite , first1=Charles , title=Sur la theorie des fonctions homogenes à deux indéterminées , authorlink=Charles Hermite , url=http://resolver.sub.uni-goettingen.de/purl?PPN600493962_0009 , year=1854 , journal=Cambridge and Dublin Mathematical Journal , volume=9 , pages=172–217 Invariant theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Series
In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite polynomial expansion has to a power series. For this reason it is also known as the Wiener–Hermite expansion. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise. This property allows the terms to be identified in applications by the ''Lee–Schetzen method''. The Wiener series is important in nonlinear system identification. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience. The name Wiener series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Q-Hermite Polynomials
In mathematics, the discrete ''q''-Hermite polynomials are two closely related families ''h''''n''(''x'';''q'') and ''ĥ''''n''(''x'';''q'') of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. ''h''''n''(''x'';''q'') is also called discrete q-Hermite I polynomials and ''ĥ''''n''(''x'';''q'') is also called discrete q-Hermite II polynomials. Definition The discrete ''q''-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials In mathematics, Al-Salam–Carlitz polynomials ''U''(''x'';''q'') and ''V''(''x'';''q'') are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. Definit ... by :\displaystyle h_n(x;q)=q^_2\phi_1(q^,x^;0;q,-qx) = x^n_2\phi_0(q^,q^;;q^2,q^/x^2) = U_n^(x;q) :\displaystyle \hat h_n(x;q)=i^q^_2\phi_0(q^,ix;;q,-q^n) = x^n_2\phi_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
