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Automatic Group
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator. More precisely, let ''G'' be a group and ''A'' be a finite set of generators. Then an ''automatic structure'' of ''G'' with respect to ''A'' is a set of finite-state automata: * the ''word-acceptor'', which accepts for every element of ''G'' at least one word in A^\ast representing it; *''multipliers'', one for each a \in A \cup \, which accept a pair (''w''1, ''w''2), for words ''w''''i'' accepted by the word-acceptor, precisely when w_1 a = w_2 in ''G''. The property of being automatic does not depend on the set of generators. Properties Automatic groups have word problem solvable in quadratic time. More strongly, a given word can actual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. Definition The definition uses the idea of a central series for a group. The followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of Groups
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object *Material properties, properties by which the benefits of one material versus another can be assessed *Chemical property, a material's properties that becomes evident during a chemical reaction *Physical property, any property that is measurable whose value describes a state of a physical system *Semantic property *Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances *Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Properties File to store program settings as name-value p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automatic Semigroup
In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed ... over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator. Formally, let S be a semigroup and A be a finite set of generators. Then an ''automatic structure'' for S with respect to A consists of a regular language L over A such that every element of S has at least one representative in L and such that for each a \in A \cup \, the relation consisting of pairs (u,v) with ua = v is regular, viewed as a subset of (''A''# × ''A''#)*. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Group Of Finite Type
Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun, an Armenian given name * 15378 Artin, a main-belt asteroid See also {{disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ) chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others. Definition Let G be a finitely g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baumslag–Solitar Group
In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. They are given by the group presentation : \left \langle a, b \ : \ b a^m b^ = a^n \right \rangle. For each integer and , the Baumslag–Solitar group is denoted . The relation in the presentation is called the Baumslag–Solitar relation. Some of the various are well-known groups. is the free abelian group on two generators, and is the fundamental group of the Klein bottle. The groups were defined by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non-Hopfian group In mathematics, a Hopfian group is a group ''G'' for which every epimorphism :''G'' → ''G'' is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group ''G'' is co-Hopfia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A subgroup of a finitely generated group need not be finitely generated. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrically Finite Group
In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ... is called geometrically finite if it can be described in terms of geometrically finite groups. Geometrically finite polyhedra A convex polyhedron ''C'' in hyperbolic space is called geometrically finite if its closure in the conformal compactification of hyperbolic space has the following property: *For each point ''x'' in , there is a neighborhood ''U'' of ''x'' such that all faces of meeting ''U'' also pass through ''x'' . For example, every polyhedron with a finite number of faces is geometrically finite. In hyperbolic space of dimension at most 2, every geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
