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{{Short description, none Numerous things are named after the French mathematician
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. Hermi ...
(1822–1901):


Hermite

*
Cubic Hermite spline In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the correspondi ...
, a type of third-degree spline *
Gauss–Hermite quadrature In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind: :\int_^ e^ f(x)\,dx. In this case :\int_^ e^ f(x)\,dx \approx \sum_^n w_i f(x_i) where ''n'' is ...
, an extension of
Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more ...
method * Hermite class * Hermite differential equation *
Hermite distribution In probability theory and statistics, the Hermite distribution, named after Charles Hermite, is a discrete probability distribution used to model ''count data'' with more than one parameter. This distribution is flexible in terms of its ability to ...
, a parametrized family of discrete probability distributions * Hermite–Lindemann theorem, theorem about transcendental numbers *
Hermite constant In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be. The constant ''γn'' for integers ''n'' > 0 is defined as follows. For a lattice ''L'' in Euclidean space ...
, a constant related to the geometry of certain lattices * Hermite-Gaussian modes * The
Hermite–Hadamard inequality In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ :  'a'', ''b''nbsp;→ R is convex function, c ...
on convex functions and their integrals *
Hermite interpolation In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than that takes the s ...
, 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 In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R''n'', the H ...
, a form of row-reduced matrices *
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 ...
s, integers related to the Hermite polynomials * Hermite polynomials, a sequence of polynomials orthogonal with respect to the
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 ...
**
Continuous q-Hermite polynomials In mathematics, the continuous ''q''-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic h ...
**
Continuous big q-Hermite polynomials In mathematics, the continuous big ''q''-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic ...
** Discrete q-Hermite polynomials ** Wiener–Hermite expansion *
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 ...
, a
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 irr ...
concerning covariants of binary forms *
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, the ...
, a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
over which every stably free module is free of unique rank * Hermite-Sobolev spaces


Hermite's

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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&nd ...
, a trigonometric identity * Hermite's criterion *
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_^\ ...
, an identity on fractional parts of integer multiples of real numbers * Hermite's problem, an unsolved problem on certain ways of expressing real numbers * Hermite's theorem, that there are only finitely many
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s of
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
less than a given magnitude


Hermitian

* Einstein–Hermitian vector bundle ** Deformed Hermitian Yang–Mills equation *
Hermitian adjoint In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
*
Hermitian connection In mathematics, a Hermitian connection \nabla is a connection on a Hermitian vector bundle E over a smooth manifold M which is compatible with the Hermitian metric \langle \cdot, \cdot \rangle on E, meaning that : v \langle s,t\rangle = \langle \n ...
, the unique connection on a Hermitian manifold that satisfies specific conditions *
Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
, a specific sesquilinear form *
Hermitian function In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: :f^*(x) = f(-x) (where the ^* indicates the complex conjugate) ...
, a complex function whose complex conjugate is equal to the original function with the variable changed in sign *
Hermitian manifold In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...
/structure **
Hermitian metric In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...
, is a smoothly varying positive-definite Hermitian form on each fiber of a complex vector bundle *
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
, a square matrix with complex entries that is equal to its own conjugate transpose **
Skew-Hermitian matrix __NOTOC__ In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relation ...
*''
Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to it ...
'', an operator (sometimes a symmetric operator, sometimes a symmetric densely defined operator, sometimes a
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
) *
Hermitian 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 ...
, a classical orthogonal polynomial sequence that arise in probability *
Hermitian symmetric space In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian s ...
, a Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space *
Hermitian transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
, the transpose of a matrix and with the complex conjugate of each entry *
Hermitian variety In geometry, a unital is a set of ''n''3 + 1 points arranged into subsets of size ''n'' + 1 so that every pair of distinct points of the set are contained in exactly one subset. This is equivalent to saying that a unital is a 2-(''n''3 + 1, ''n'' ...
, a generalisation of quadrics *
Hermitian wavelet Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The n^\textrm Hermitian wavelet is defined as the n^\textrm derivative of a Gaussian distribution: \Psi_(t)=(2n)^c_He_\left(t\right)e^ where He_ ...
, a family of continuous wavelets *
Non-Hermitian quantum mechanics PT symmetry was initially studied as a specific system in non-Hermitian quantum mechanics, where Hamiltonians are not Hermitian. In 1998, physicist Carl Bender and former graduate student Stefan Boettcher published in ''Physical Review Letters'' ...


Astronomical objects

* 24998 Hermite, a
main-belt asteroid The asteroid belt is a torus-shaped region in the Solar System, located roughly between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies, of many sizes, but much smaller than planets, called ...
*
Hermite (crater) 240px, Lunar Orbiter 4 image of Hermite and surrounding craters Hermite is a lunar impact crater located along the northern lunar limb, close to the north pole of the Moon. Named for Charles Hermite, the crater was formed roughly 3.91 billion y ...
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. Hermi ...