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Ore Algebra
In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. The concept is named after Øystein Ore. Definition Let K be a (commutative) field and A = K _1, \ldots, x_s/math> be a commutative polynomial ring (with A = K when s = 0). The iterated skew polynomial ring A partial_1; \sigma_1, \delta_1\cdots partial_r; \sigma_r, \delta_r/math> is called an Ore algebra when the \sigma_i and \delta_j commute for i \neq j, and satisfy \sigma_i(\partial_j) = \partial_j, \delta_i(\partial_j) = 0 for i > j. Properties Ore algebras satisfy the Ore condition, and thus can be embedded in a (skew) field of fractions. The constraint of commutation in the definition makes Ore algebras have a non-commutative generalization theory of Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and comput ... [...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] |
Ore Extension
In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups. Definition Suppose that ''R'' is a (not necessarily commutative) ring, \sigma \colon R \to R is a ring homomorphism, and \delta\colon R\to R is a ''σ''-derivation of ''R'', which means that \delta is a homomorphism of abelian groups satisfying : \delta(r_1 r_2) = \sigma(r_1)\delta(r_2)+\delta(r_1)r_2. Then the Ore extension R ;\sigma,\delta/math>, also called a skew polynomial ring, is the noncommutative ring obtained by giving the ring of polynomials R /math> a new multiplication, subject ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics. Life Ore graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled ''Zur Theorie der algebraischen Körper'', supervised by Thoralf Skolem. Ore also studied at Göttingen University, where he learned Emmy Noether's new approach to abstract algebra. He was also a fellow at the Mittag-Leffler Institute in Sweden, and spent some time at the University of Paris. In 1925, he was appointed research assistant at the University of Oslo. Yale University’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928, then to full professor in 1929. In 1931, he became a Sterling Prof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ore Condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The ''right Ore condition'' for a multiplicative subset ''S'' of a ring ''R'' is that for and , the intersection . A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly. General idea The goal is to construct the right ring of fractions ''R'' 'S''−1with respect to a multiplicative subset ''S''. In other words, we want to work with elements of the form ''as''−1 and have a ring structure on the set ''R'' 'S''−1 The problem is that there is no obvious interpretation of the product (''as''−1)(''bt''−1); indeed, we need a method to "move" ''s''−1 past ''b''. This means that we need t ... [...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] |
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] |
