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Janet Basis
In mathematics, a Janet basis is a normal form for systems of linear homogeneous partial differential equations (PDEs) that removes the inherent arbitrariness of any such system. It was introduced in 1920 by Maurice Janet.M. JanetLes systèmes d'équations aux dérivées partielles Journal de mathématiques pures et appliquées 8 ser., t. 3 (1920), pages 65–123. It was first called the Janet basis by Fritz Schwarz in 1998.F. Schwarz"Janet Bases for Symmetry Groups" in: ''Gröbner Bases and Applications; Lecture Notes Series'' 251, London Mathematical Society, pages 221–234 (1998); B. Buchberger and F. Winkler, Edts. The left hand sides of such systems of equations may be considered as differential polynomials of a ring, and Janet's normal form as a special basis of the ideal that they generate. By abuse of language, this terminology will be applied both to the original system and the ideal of differential polynomials generated by the left hand sides. A Janet basis is the pred ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and which allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Janet
Maurice Janet (1888–1983) was a French mathematician. Education and career In 1912 as a student he visited the University of Göttingen. He was a professor at the University of Caen. He was an Invited Speaker of the International Congress of Mathematicians in 1924 in Toronto, in 1932 in Zürich, and in 1936 in Oslo. Named in his honor are Janet bases, Janet sequences and a related algorithm in the theory of systems of partial differential equations. In 1926 he proved results that were later generalized by John Forbes Nash Jr. in his embedding theorem. In 1948 Janet was the president of the Société Mathématique de France. He was a close friend of the mathematician Ernest Vessiot Ernest Vessiot (; 8 March 1865 – 17 October 1952) was a French mathematician. He was born in Marseille, France, and died in La Bauche, Savoie, France. He entered the École Normale Supérieure in 1884. He was Maître de Conférences at Lille .... Selected publications ArticlesLes système ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gröbner Basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger's algorithm), by Bruno Buchberger in his Ph.D. thesis. He named them after h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruno Buchberger
Bruno Buchberger (born 22 October 1942) is Professor of Computer Mathematics at Johannes Kepler University Linz, Johannes Kepler University in Linz, Austria. In his 1965 Ph.D. thesis, he created the theory of Gröbner basis, Gröbner bases, and has developed this theory throughout his career. He named these objects after his advisor Wolfgang Gröbner. Since 1995, he has been active in the Theorema project at the University of Linz. Career In 1987 Buchberger founded and chaired the Research Institute for Symbolic Computation (RISC) at Johannes Kepler University. In 1985 he started the Journal of Symbolic Computation, which has now become the premier publication in the field of computer algebra. Buchberger also conceived Softwarepark Hagenberg in 1989 and since then has been directing the expansion of this Austrian technology park for software. In 2014 he became a member of the ''Global Digital Mathematical Library Working Group'' of the International Mathematical Union, IMU. Awa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loewy Decomposition
In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced by Alfred Loewy. Solving differential equations is one of the most important subfields in mathematics. Of particular interest are solutions in closed form. Breaking ODEs into largest irreducible components, reduces the process of solving the original equation to solving irreducible equations of lowest possible order. This procedure is algorithmic, so that the best possible answer for solving a reducible equation is guaranteed. A detailed discussion may be found in., F.Schwarz, Loewy Decomposition of Linear Differential Equations, Springer, 2012 Loewy's results have been extended to linear partial differential equations (PDEs) in two independent variables. In this way, algorithmic methods for solving large classes of linear PDEs have become available. Decomposing linear ordinary dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
