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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, that its motion is confined to a submanifold of much smaller dimensionality than that of 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 ''generic'' dynamical systems, which are more typically chaotic syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax Pair
A lax is a salmon. LAX as an acronym most commonly refers to Los Angeles International Airport in Southern California, United States. LAX or Lax may also refer to: Places Within Los Angeles * Union Station (Los Angeles), Los Angeles' main train depot, whose Amtrak station code is "LAX" * The Port of Los Angeles, whose port identifier code is "LAX" Other * Lax, Switzerland, a municipality of the canton of Valais * Lax Lake (other) * La Crosse, Wisconsin, a city on the Mississippi River Sports * Los Angeles Xtreme, a former American football team * Lacrosse, a sport * The Latin American Xchange, a professional wrestling stable Media and entertainment * ''LAX'' (album), the third studio album from rapper The Game * ''LAX'' (TV series), a 2004–05 television series set in Los Angeles International Airport * " LA X", the two-part sixth season 2010 premiere of the television show ''Lost'' * LAX, a night club at Luxor Las Vegas * "LAX", a song by the rapper X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the profession, 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 Mathematical analysis, applied analysis, most notably differential equations; approximation theory (broadly construed, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Science
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science, and more specifically the Computer Sciences, which uses advanced computing capabilities to understand and solve complex physical problems. While this typically extends into computational specializations, this field of study includes: * Algorithms ( numerical and non-numerical): mathematical models, computational models, and computer simulations developed to solve sciences (e.g, physical, biological, 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 compu ... [...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 to find 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 like economics, 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 simulati ... [...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, that its motion is confined to a submanifold of much smaller dimensionality than that of 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 ''generic'' dynamical systems, which are more typically chaotic syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a balanced 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 were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell 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". The Korteweg–de Vries equation was later formulated to model such waves, and the term "soliton" was coined by Zabu ... [...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 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark J
Mark may refer to: In the Bible * Mark the Evangelist (5–68), traditionally ascribed author of the Gospel of Mark * Gospel of Mark, one of the four canonical gospels and one of the three synoptic gospels Currencies * Mark (currency), a currency or unit of account in many nations * 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 1928 * Finnish markka (), the currency of Finland from 1860 until 28 February 2002 * Polish mark (), 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 iss ... [...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 to find 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 like economics, 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 simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Acceleration
Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in serialism including tone rows * Harmonic series (music) * Serialism, including the twelve-tone technique Types of series in arts, entertainment, and media * Anime series * Book series * Comic book series * Film series * Manga series * Podcast series * Radio series * Television series * "Television series", the Australian, British, and a number of others countries' equivalent term for the North American " television season", a set of episodes produced by a television serial * Video game series * Web series Mathematics and science * Series (botany), a taxonomic rank between genus and species * Series (mathematics), the sum of a sequence of terms * Series (stratigraphy), a stratigraphic unit deposited during a certain i ... [...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 scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
