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Hermitian
{{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 * Hermi ... [...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 (mathematical constant), e'', the base of natural logarithms, is a transcendental number. His methods were used later by Ferdinand von Lindemann to prove that pi, π is transcendental. Life Hermite was born in Dieuze, Moselle (department), 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 care ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Number
In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials. Formal definition The numbers ''H''n = ''H''n(0), where ''H''n(''x'') is a Hermite polynomial of order ''n'', may be called Hermite numbers.Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html The first Hermite numbers are: :H_0 = 1\, :H_1 = 0\, :H_2 = -2\, :H_3 = 0\, :H_4 = +12\, :H_5 = 0\, :H_6 = -120\, :H_7 = 0\, :H_8 = +1680\, :H_9 =0\, :H_ = -30240\, Recursion relations Are obtained from recursion relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...s of Hermitian polynomials for ''x'' = 0: :H_ = -2(n-1)H_.\,\! Since ''H''0 = 1 and ''H''1 = 0 one c ... [...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 ... [...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 Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal d ... [...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 ca ..., 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 ... [...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 numbers, 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 In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ..., 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] |
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 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, 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, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Ring
In Abstract_algebra, algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects. According to (p. 465), a ring (mathematics), 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 module, 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) inver ... [...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 re ... [...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] |
