|
Magma (computer Algebra System)
Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. It runs on Unix-like operating systems, as well as Windows. Introduction Magma is produced and distributed by thComputational Algebra Groupwithin the Sydney School of Mathematics and Statistics at the University of Sydney. In late 2006, the booDiscovering Mathematics with Magmawas published by Springer as volume 19 of the Algorithms and Computations in Mathematics series. The Magma system is used extensively within pure mathematics. The Computational Algebra Group maintain a list of publications that cite Magma, and as of 2010 there are about 2600 citations, mostly in pure mathematics, but also including papers from areas as diverse as economics and geophysics. History The predecessor of the Magma system was named Cayley (1982–1993), after Arthur Cayley. Magma was officially released in August 1993 (version 1.0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Sydney School Of Mathematics And Statistics
The School of Mathematics and Statistics is a constituent body of the Faculty of Science at the University of Sydney, Australia. It was established in its present form in 1991. As of 29 August 2022, the Head of School is Professor Dingxuan Zhou, and the Deputy Head of School is Professor Mary Myerscough. The Magma computer algebra system Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. It runs on Unix-like operating systems, as well as Windows. Introduction Magma ... is produced and distributed by the Computational Algebra Group within the School. History Mathematics has been taught at the university since its establishment. The first Professor of Mathematics was Morris Pell, one of the university's three foundation professors. Pell gave the first lecture in mathematics on 13 October 1852, two days after the university's inauguration, to all 24 students at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Simons Foundation
The Simons Foundation is an American private foundation established in 1994 by Marilyn and James Harris Simons, Jim Simons with offices in New York City. As one of the largest charitable organizations in the United States with assets of over $5 billion in 2022, the foundation's mission is to advance the frontiers of research in mathematics and basic sciences. The foundation supports science by making grants to individual researchers and their projects. In 2021, Marilyn Simons stepped down as president after 26 years at the helm, and Astrophysics, astrophysicist David Spergel was appointed president. The Flatiron Institute In 2016, the foundation launched the Flatiron Institute, its in-house multidisciplinary research institute focused on computational science. The Flatiron Institute hosts centers for computational science in five areas: Funding areas The foundation makes grants in four program areas: Simons Investigators awardees Among other programs, the Simons Foundat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Straight-line Program
In computer science, a straight-line program is, informally, a program that does not contain any loop or any test, and is formed by a sequence of steps that apply each an operation to previously computed elements. This article is devoted to the case where the allowed operations are the operations of a group, that is multiplication and inversion. More specifically a straight-line program (SLP) for a finite group ''G'' = ⟨''S''⟩ is a finite sequence ''L'' of elements of ''G'' such that every element of ''L'' either belongs to ''S'', is the inverse of a preceding element, or the product of two preceding elements. An SLP ''L'' is said to ''compute'' a group element ''g'' ∈ ''G'' if ''g'' ∈ ''L'', where ''g'' is encoded by a word in ''S'' and its inverses. Intuitively, an SLP computing some ''g'' ∈ ''G'' is an ''efficient'' way of storing ''g'' as a group word over ''S''; observe that if ''g'' is constructed in ''i'' steps, the word l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Braid Group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Emil Artin, Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see ); and in monodromy invariants of algebraic geometry. Introduction In this introduction let ; the generalization to other values of will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Polycyclic Group
In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational point of view. Terminology Equivalently, a group ''G'' is polycyclic if and only if it admits a subnormal series with cyclic factors, that is a finite set of subgroups, let's say ''G''0, ..., ''G''''n'' such that * ''G''''n'' coincides with ''G'' * ''G''0 is the trivial subgroup * ''G''''i'' is a normal subgroup of ''G''''i''+1 (for every ''i'' between 0 and ''n'' - 1) * and the quotient group ''G''''i''+1 / ''G''''i'' is a cyclic group (for every ''i'' between 0 and ''n'' - 1) A metacyclic group is a polycyclic group with ''n'' ≤ 2, or in other words an extension of a cyclic group by a cyclic group. Examples Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Solvable Group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Motivation Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equations. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial f \in F /math> there is a tower of field extensionsF = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=Ksuch that # F_i = F_ alpha_i/math> where \alpha_i^ \in F_, so \alpha_i is a solution to the equation x^ - a where a \in F_ # F_m contains a splitting field for f(x) Example The smallest Galois field extension of \mathbb containing the elementa = \sqr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Finitely Presented Group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Permutation Group
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to itself). The group of ''all'' permutations of a set ''M'' is the symmetric group of ''M'', often written as Sym(''M''). The term ''permutation group'' thus means a subgroup of the symmetric group. If then Sym(''M'') is usually denoted by S''n'', and may be called the ''symmetric group on n letters''. By Cayley's theorem, every group is isomorphic to some permutation group. The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry. Basic properties and terminology A ''permutation group'' is a subgroup of a symmetric group; that is, its elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |