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Alekseev–Gröbner Formula
The Alekseev–Gröbner formula, or nonlinear variation-of-constants formula, is a generalization of the linear variation of constants formula which was proven independently by Wolfgang Gröbner in 1960 and Vladimir Mikhailovich Alekseev in 1961. It expresses the global error of a perturbation in terms of the local error and has many applications for studying perturbations of ordinary differential equations. Formulation Let d \in \mathbb N be a natural number, let T \in (0, \infty) be a positive real number, and let \mu \colon , T\times \mathbb^ \to \mathbb^ \in C^(, T\times \mathbb^) be a function which is continuous on the time interval , T/math> and continuously differentiable on the d-dimensional space \mathbb^. Let X \colon , T \times \mathbb^ \to \mathbb^, (s, t, x) \mapsto X_^ be a continuous solution of the integral equation X_^ = x + \int_^ \mu(r, X_^) dr. Furthermore, let Y \in C^(, T The comma is a punctuation mark that appears in several variants in different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variation Of Constants
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations. Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Gröbner
Wolfgang Gröbner (11 February 1899 – 20 August 1980) was an Austrian mathematician. His name is best known for the Gröbner basis, used for computations in algebraic geometry. However, the theory of Gröbner bases for polynomial rings was developed by his student Bruno Buchberger in 1965, who named them for Gröbner. Gröbner is also known for the Alekseev-Gröbner formula, which is actually proven by him. Early life Gröbner was born in Gossensaß, which at that time was in part of the County of Tyrol of the Austro-Hungarian Empire and is now part of Italy. Gröbner first studied engineering at the University of Technology in Graz, Austria, but switched in 1929 to mathematics. Career He wrote his dissertation ''Ein Beitrag zum Problem der Minimalbasen'' in 1932 at the University of Vienna; his advisor was Phillip Furtwängler. After his promotion, he did further studies at the University of Göttingen under Emmy Noether, in what is now known as commutative algebra. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Mikhailovich Alekseev
Vladimir Mikhailovich Alekseev (Владимир Михайлович Алексеев, sometimes transliterated as "Alexeyev" or "Alexeev", 17 June 1932, Bykovo, Ramensky District, Moscow Oblast – 1 December 1980) was a Russian mathematician who specialized in celestial mechanics and dynamical systems.D. Anosov, V. Arnold, A. N. Kolmogorov, Y. Sinai et al., (Obituary in Russian) Mathematical Surveys, vol. 36, 1981, pp. 201–206Russian on mathnet.ru/ref> He attended secondary school in Moscow at one of the special schools of mathematics affiliated with Moscow State University and participated in several mathematical olympiads. From 1950 he studied at the Faculty of Mathematics and Mechanics at the Moscow State University, where he worked as a student of Andrei Kolmogorov on the asymptotic behavior in the three-body problem of celestial mechanics. Already as an undergraduate, Alekseev proved significant new results on quasi-random motion associated with the three-body problem. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Algebra
Nonlinear algebra is the nonlinear analogue to linear algebra, generalizing notions of spaces and transformations coming from the linear setting. Algebraic geometry is one of the main areas of mathematical research supporting nonlinear algebra, while major components coming from computational mathematics support the development of the area into maturity. The topological setting for nonlinear algebra is typically the Zariski topology, where closed sets are the algebraic sets. Related areas in mathematics are tropical geometry, commutative algebra, and optimization. Algebraic geometry Nonlinear algebra is closely related to algebraic geometry, where the main objects of study include algebraic equations, algebraic varieties, and schemes. Computational nonlinear algebra Current methods in computational nonlinear algebra can be broadly broken into two domains: symbolic and numerical. Symbolic methods often rely on the computation of Gröbner bases and resultants. On the othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |