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Derivative Of The Exponential Map
In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group into . In case is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted , is analytic and has as such a derivative , where is a path in the Lie algebra, and a closely related differential . Appendix on analytic functions. The formula for was first proved by Friedrich Schur (1891). It was later elaborated by Henri Poincaré (1899) in the context of the problem of expressing Lie group multiplication using Lie algebraic terms. It is also sometimes known as Duhamel's formula. The formula is important both in pure and applied mathematics. It enters into proofs of theorems such as the Baker–Campbell–Hausdorff formula, and it is used frequently in physics for example in quantum field theory, as in the Magnus expansion in perturbation theory, and in lattice gauge theory. Throughout, the notations and will be used intercha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation Of A Lie Algebra
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sides Of An Equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. definition and example of abbreviation More generally, these terms may apply to an or inequality; the right-hand side is everything on the right side of a < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mercator Series
In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac+\frac-\frac+\cdots In summation notation, :\ln(1+x)=\sum_^\infty \frac x^n. The series converges to the natural logarithm (shifted by 1) whenever -1 History The series was discovered independently by Johannes Hudde and . It was first published by , in his 1668 treatise ''Logarithmotechnia''.Derivation The series can be obtained from |
Eugene Dynkin 2003
Eugene may refer to: People and fictional characters * Eugene (given name), including a list of people and fictional characters with the given name * Eugene (actress) (born 1981), Kim Yoo-jin, South Korean actress and former member of the singing group S.E.S. * Eugene (wrestler), professional wrestler Nick Dinsmore * Franklin Eugene (producer), American film producer * Gene Eugene, stage name of Canadian born actor, record producer, engineer, composer and musician Gene Andrusco (1961–2000) * Wendell Eugene (1923–2017), American jazz musician Places Canada * Mount Eugene, in Nunavut; the highest mountain of the United States Range on Ellesmere Island United States * Eugene, Oregon, a city ** Eugene, OR Metropolitan Statistical Area ** Eugene (Amtrak station) * Eugene Apartments, NRHP-listed apartment complex in Portland, Oregon * Eugene, Indiana, an unincorporated town * Eugene, Missouri, an unincorporated town Business * Eugene Green Energy Standard, an in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Function Theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at the point''. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth. The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem. Statements For functions of a single varia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Correspondence
A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be interpreted as deliberately false statements or misleading statements. Lies may also serve a variety of instrumental, interpersonal, or psychological functions for the individuals who use them. Generally, the term "lie" carries a negative connotation, and depending on the context a person who communicates a lie may be subject to social, legal, religious, or criminal sanctions; for instance, perjury, or the act of lying under oath, can result in criminal and civil charges being pressed against the perjurer. Although people in many cultures believe that deception can be detected by observing nonverbal behaviors (e.g. not making eye contact, fidgeting, stuttering) research indicates that people overestimate both the significance of such cues an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
