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Whitehead's Algorithm
Whitehead's algorithm is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group ''Fn''. The algorithm is based on a classic 1936 paper of J. H. C. Whitehead. J. H. C. Whitehead, ''On equivalent sets of elements in a free group'', Ann. of Math. (2) 37:4 (1936), 782–800. It is still unknown (except for the case ''n'' = 2) if Whitehead's algorithm has polynomial time complexity. Statement of the problem Let F_n=F(x_1,\dots, x_n) be a free group of rank n\ge 2 with a free basis X=\. The automorphism problem, or the automorphic equivalence problem for F_n asks, given two freely reduced words w, w'\in F_n whether there exists an automorphism \varphi\in \operatorname(F_n) such that \varphi(w)=w'. Thus the automorphism problem asks, for w, w'\in F_n whether \operatorname(F_n)w=\operatorname(F_n)w'. For w, w'\in F_n one has \operatorname(F_n)w=\operatorname(F_n)w' if and only if \operatorname(F_n) \operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Culler
Marc Edward Culler (born November 22, 1953) is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at University of California, Berkeley, Berkeley where he graduated in 1978. He is now at the University of Illinois at Chicago. Culler is the son of Glen Culler, Glen Jacob Culler who was an important early innovator in the development of the Internet. Work Culler specializes in group theory, low dimensional topology, 3-manifolds, and hyperbolic geometry. Culler frequently collaborates with Peter Shalen and they have co-authored many papers. Culler and Shalen did joint work that related properties of representation varieties of hyperbolic 3-manifold groups to decompositions of 3-manifolds. In particular, Culler and Shalen used the Bass–Serre theory, applied to the function field of the SL(2,C)-Character variety of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jakob Nielsen (mathematician)
Jakob Nielsen (15 October 1890 in Mjels, Als – 3 August 1959 in Helsingør) was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North Schleswig, in modern-day Denmark. His mother died when he was 3, and in 1900 he went to live with his aunt and was enrolled in the Realgymnasium. In 1907 he was expelled for membership to an illicit student club. Nevertheless, he matriculated at the University of Kiel in 1908. Nielsen completed his doctoral dissertation in 1913. Soon thereafter, he was drafted into the German Imperial Navy. He was assigned to coastal defense. In 1915 he was sent to Constantinople as a military adviser to the Turkish Government. After the war, in the spring of 1919, Nielsen married Carola von Pieverling, a German medical doctor. In 1920 Nielsen took a position at the Technical University of Breslau. The next year he published a paper in Mathematisk Tidsskrift in which he pr ... [...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. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The Edinburgh Mathematical Society
In academia and librarianship, conference proceedings is a collection of academic papers published in the context of an academic conference or workshop. Conference proceedings typically contain the contributions made by researchers at the conference. They are the written record of the work that is presented to fellow researchers. In many fields, they are published as supplements to academic journals; in some, they are considered the main dissemination route; in others they may be considered grey literature. They are usually distributed in printed or electronic volumes, either before the conference opens or after it has closed. A less common, broader :wikt:proceedings, meaning of proceedings are the acts and happenings of an discipline (academia), academic field, a learned society. For example, the title of the ''Acta Crystallographica'' journals is New Latin for "Proceedings in Crystallography"; the ''Proceedings of the National Academy of Sciences of the United States of America' ... [...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] |
EXPTIME
In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2''p''(''n'')) time, where ''p''(''n'') is a polynomial function of ''n''. EXPTIME is one intuitive class in an exponential hierarchy of complexity classes with increasingly more complex oracles or quantifier alternations. For example, the class 2-EXPTIME is defined similarly to EXPTIME but with a doubly exponential time bound. This can be generalized to higher and higher time bounds. EXPTIME can also be reformulated as the space class APSPACE, the set of all problems that can be solved by an alternating Turing machine in polynomial space. EXPTIME relates to the other basic time and space complexity classes in the following way: P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE. Furthemore, by the time hierarchy theorem and the space hierarchy the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Journal Of Algebra And Computation
The ''International Journal of Algebra and Computation'' is published by World Scientific, and contains articles on general mathematics, as well as: * Combinatorial group theory and semigroup theory * Universal algebra * Algorithmic and computational problems in algebra * Theory of automata * Formal language theory * Theory of computation * Theoretical computer science According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.719. Abstracting and indexing The journal is indexed in: * ISI Alerting Services * CompuMath Citation Index * Science Citation Index * Current Contents/Physical, Chemical and Earth Sciences * Mathematical Reviews * INSPEC * Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ... * Computer Abstracts Mathematics j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Breadth-first Search
Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next depth level. Extra memory, usually a queue, is needed to keep track of the child nodes that were encountered but not yet explored. For example, in a chess endgame a chess engine may build the game tree from the current position by applying all possible moves, and use breadth-first search to find a win position for white. Implicit trees (such as game trees or other problem-solving trees) may be of infinite size; breadth-first search is guaranteed to find a solution node if one exists. In contrast, (plain) depth-first search, which explores the node branch as far as possible before backtracking and expanding other nodes, may get lost in an infinite branch and never make it to the solution node. Iterative deepening depth-first search avoids ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pacific Journal Of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation, and the University of California, Berkeley. It was founded in 1951 by František Wolf and Edwin F. Beckenbach and has been published continuously since, with five two-issue volumes per year and 12 issues per year. Full-text PDF versions of all journal articles are available on-line via the journal's website with a subscription. The journal is incorporated as a 501(c)(3) organization. References Mathematics journals Publications established in 1951 Mathematical Sciences Publishers academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group. Definition If is a group and is an element of (alternatively, if is a ring, and is a unit), then the function :\begin \varphi_g\colon G&\to G \\ \varphi_g(x)&:= g^xg \end is called (right) conjugation by (see also conjugacy class). This function is an endomorphism of : for all x_1,x_2\in G, :\varphi_g(x_1 x_2) = g^ x_1 x_2g = \left(g^ x_1 g\right)\left(g^ x_2 g\right) = \varphi_g(x_1)\varphi_g(x_2), where the second equality is given by the insertion of the identity between x_1 and x_2. Furthermore, it has a left and ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Out(Fn)
In mathematics, Out(''Fn'') is the outer automorphism group of a free group on ''n'' generators. These groups play an important role in geometric group theory. Outer space Out(''Fn'') acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles. Definition A point of the outer space is essentially an \R-graph ''X'' homotopy equivalent to a bouquet of ''n'' circles together with a certain choice of a free homotopy class of a homotopy equivalence from ''X'' to the bouquet of ''n'' circles. An \R-graph is just a weighted graph with weights in \R. The sum of all weights should be 1 and all weights should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 3. A more descriptive view avoiding the homotopy equivalence ''f'' is the following. We may fix an identification of the fundamental gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
