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Integrable Algorithm
Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems. Background The theory of integrable systems has advanced with the connection between numerical analysis. For example, the discovery of solitons came from the numerical experiments to the KdV equation by Norman Zabusky and Martin David Kruskal. Today, various relations between numerical analysis and integrable systems have been found (Toda lattice and numerical linear algebra, discrete soliton equations and series acceleration), and studies to apply integrable systems to numerical computation are rapidly advancing. Integrable difference schemes Generally, it is hard to accurately compute the solutions of nonlinear differential equations due to its non-linearity. In order to overcome this difficulty, R. Hirota has made discrete versions of integrable systems with the viewpoint of "Preserve mathematical structures of integrable systems in the discrete versions". ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable Systems
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax Pair
In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems. Definition A Lax pair is a pair of matrices or operators L(t), P(t) dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation: :\frac= ,L/math> where ,LPL-LP is the commutator. Often, as in the example below, P depends on L in a prescribed way, so this is a nonlinear equation for L as a function of t. Isospectral property It can then be shown that the eigenvalues and more generally the spectrum of ''L'' are independent of ''t''. The matrices/operators ''L'' are said to be ''isospectral'' as t varies. The core observation is that the matrices L(t) are all similar by virtue of :L(t)=U(t,s) L( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Science
Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems. * Algorithms ( numerical and non-numerical): mathematical models, computational models, and computer simulations developed to solve science (e.g., biological, physical, and social), engineering, and humanities problems * Computer hardware that develops and optimizes the advanced system hardware, firmware, networking, and data management components needed to solve computationally demanding problems * The computing infrastructure that supports both the science and engineering problem solving and the developmental computer and information science In practical use, it is typically the application of computer simulation and other fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". Definition A single, consensus definition of a soliton is difficult to find. ascribe three properties to solitons: # They are of permanent form; # They are localized within a region; # They can interact with other solitons, and emerge from the collision unchanged, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Society For Industrial And Applied Mathematics
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics. SIAM is one of the four member organizations of the Joint Policy Board for Mathematics. Membership Membership is open to both individuals and organizations. By the end of its first full year of operation, SIAM had 130 memb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark J
Mark may refer to: Currency * Bosnia and Herzegovina convertible mark, the currency of Bosnia and Herzegovina * East German mark, the currency of the German Democratic Republic * Estonian mark, the currency of Estonia between 1918 and 1927 * Finnish markka ( sv, finsk mark, links=no), the currency of Finland from 1860 until 28 February 2002 * Mark (currency), a currency or unit of account in many nations * Polish mark ( pl, marka polska, links=no), the currency of the Kingdom of Poland and of the Republic of Poland between 1917 and 1924 German * Deutsche Mark, the official currency of West Germany from 1948 until 1990 and later the unified Germany from 1990 until 2002 * German gold mark, the currency used in the German Empire from 1873 to 1914 * German Papiermark, the German currency from 4 August 1914 * German rentenmark, a currency issued on 15 November 1923 to stop the hyperinflation of 1922 and 1923 in Weimar Germany * Lodz Ghetto mark, a special currency for Lodz Ghetto. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities. Definition Given a sequence :S=\_ having a limit :\lim_ s_n = \ell, an accelerated series is a second sequence :S'=\_ which converges faster to \ell than the original sequence, in the sense that :\lim_ \frac = 0. If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit \ell. The mappings from the original to the transformed series may be linear (as defined in the article sequence transfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Linear Algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible. Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
