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Topic: Word problem for groups

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	Distributional Word Problem for Groups
		A finite presentation of a group consists of a set of generators (abstract symbols) and a set of relations that relate the freely generated words.
		Wang [Wan95a] studied the following distributional word problem for finitely presented groups, which is a slight modification of the original word problem of Dehn and Thue.
		The distributional word problem for finitely presented groups was shown to be average-case NP-complete by Wang [Wan95a].
	www.uncg.edu /mat/avg/avgnp/node14.html (342 words)

	Word problem for groups - Wikipedia, the free encyclopedia
		Although it is common to speak of the word problem for the group G stricly speaking it is a presentation of the group that does or does not have solvable word problem.
		The related but different uniform word problem for a class K of recursively presented groups is the algorithmic problem of deciding, given as input a presentation P for a group G in the class K and two words in the generators of G, whether the words represent the same element of G.
		The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra.
	en.wikipedia.org /wiki/Word_problem_for_groups (2231 words)

	Groups with Word Problem in NP, and Higman Embeddings
		Nevertheless not every finitely presented group with polynomial Dehn function has a simply connected asymptotic cone because if the cone is symply connected then the group has a linear isodiametric function, and Theorem 3 allows one to construct lots of groups with polynomial Dehn functions which cannot have linear isodiametric functions.
		So perhaps the class of groups with word problem in NP (which by Theorem 14 is the class of all subgroups of finitely presented groups with polynomial Dehn functions) can be considered as the class of ``tame" groups.
		The class of finitely presented groups with polynomial Dehn functions is, by Theorem 14, the ``universal" subclass of the class of all groups with word problem in NP.
	www.math.vanderbilt.edu /~msapir/Talk1/node6.html (1879 words)

	Groups that Work
		The key is that the group should be viewed as an important resource whose maintenance must be managed just like any other resource and that this management should be undertaken by the group itself so that it forms a normal part of the group's activities.
		Groups are particularly good at combining talents and providing innovative solutions to possible unfamiliar problems; in cases where there is no well established approach/procedure, the wider skill and knowledge set of the group has a distinct advantage over that of the individual.
		Finally, a word about the much vaunted "recognition of the worth of the individual" which is often given as the reason for delegating responsibility to groups of subordinates.
	www.ee.ed.ac.uk /~gerard/Management/art0.html (2553 words)

	Distributional Word Problem for Groups
		is dominated by the default uniform distribution of the word problem with respect to the reduction.
		We will first construct an intermediate word problem for Thue systems.
		Group operations can be applied on this representation in a natural way.
	www.uncg.edu /mat/avg/avgnp/node27.html (874 words)

	80.07.10: Math Is Everywhere: A Problem Solving Teaching Unit
		Problem solving offers the opportunity for students to make the connection between the mathematical concepts that are taught in school and the real world Therefore; it becomes necessary to make the learning process as concrete as possible.
		There are basically two types of word problems which are frequently assigned to students; those which require the direct translation of the verbal languageto mathematical language, and the subsequent application of the math skills (s)he possesses, and those problems which require the student to apply mathematical skills and concepts in a discovery manner.
		Both types of problems require the student to draw upon present knowledge and past experiences, but it is the latter which offers the greatest challenge to the student, and which has the greatest potential for helping a student to understand the many ways in which math relates to the other fields of study.
	www.yale.edu /ynhti/curriculum/units/1980/7/80.07.10.x.html (4850 words)

	BACKGROUND
		Note that the growth rate of the free metabelian group of rank 2 should be strictly less than 3 (which is the growth rate of the free group of rank 2) since a free metabelian group is not free.
		This is the case, in particular, with the word as well as the conjugacy problem for a given finitely presented group and with the problem of isomorphism to a given finitely presented group.
		The word problem in Aut(F_n) has a straightforward solution (by just acting on the generators of F_n), but this algorithm is exponential with respect to the length of the input.
	www.sci.ccny.cuny.edu /~shpil/gworld/problems/Back2.html (4913 words)

	BACKGROUND
		We note that the word problem for groups admitting finitely many defining relations in the variety of all solvable groups of a given derived length >2, is, in general, unsolvable - see [O. Kharlampovich, A finitely presented solvable group with unsolvable word problem.
		14 (1973), 1351--1355, 1368] settled this problem in the affirmative for metabelian groups.
		For a general setup that motivated this problem, we refer to a recent preprint [V. Nekrashevych, S. Sidki, Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms], which is available here.
	zebra.sci.ccny.cuny.edu /web/nygtc/problems/Back2.html (3180 words)

	Research (Site not responding. Last check: 2007-10-28)
		The first construction of the braid group was extended into larger classes of groups such as Artin groups and Garside groups, and the braid groups became immensely important in many fields, such as group theory, combinatorial group theory, topology, knot theory, computer science and algorithms, cryptography, algebraic geometry and physics.
		The first and probably the most important problem, when given a group presented by generators and relations, is to be able to determine if two products of the generators and their inverse represent the same element in the group.
		The generalized algorithms for computing braid monodromy together with the algorithms for solving the braid word problem may be used together as an automated method to compute the fundamental group of the complement of a given curve.
	www.cs.biu.ac.il /~kaplansh/research.htm (1233 words)

	Introduction (Site not responding. Last check: 2007-10-28)
		This chapter deals with another, quite specialised, class of finitely presented groups for which the word problem is solvable, the category of braid groups.
		The word problem in braid groups is solvable, that is, there is a normal form for elements of a braid group and elements can be compared.
		In Magma, the elements of a braid group are represented as words a_(i_1)^(e_1)...
	www.umich.edu /~gpcc/scs/magma/text448.htm (551 words)

	Word:mac - VBA: MacWord vs. WinWord
		VBA in Word 97 for Windows is also based on VB5, but later versions of Word for Windows use VB6 as the basis for VBA.
		This lack is not usually a problem unless you use an Excel worksheet as a data source, but you should be aware of it.
		You can, of course, copy-and-paste the code into a Word document and save it as text file with the ‘.bas’ extension, and it is possible, using additional code, to export and import forms and modules, but these workarounds are not as convenient as using menu commands, as can be done in Windows.
	word.mvps.org /Mac/MacWordVBA.html (1061 words)

	ChangeLog
		Existing genetic algorithms for finitely presented groups (word problem, triviality problem, "is a group abelian ?" problem) and for solving equations in a free group were modified to show better performance.
		A word which is trivial in the given group is re-expressed as a product of conjugates of the defining relator (and its inverse).
		A word which represents an element of finite order is re-expressed as a conjugate of a power of r, where here r^n is the given defining relator.
	www.msri.org /about/computing/docs/magnus/ChangeLog.html (542 words)

	Lectures Abstracts (Site not responding. Last check: 2007-10-28)
		While many of the problems involving braid group factorizations remain open to this date (in particular the "Hurwitz problem", whose solution would in principle give a classification of symplectic manifolds), the braid monodromy data associated to a plane curve gives access to useful information, such as the fundamental group of the complement.
		The first condition for using a group in cryptography is to be able to specify its elements, hence, for a presented group, to have an efficient solution to the word problem.
		Braid groups have been proposed to be a good choice for these cryptosystems, so we will study the conjugacy problem in these groups.
	www.cs.biu.ac.il /~eni/Eilat2005Abstracts.htm (424 words)

	Power-conjugate Presentations (Site not responding. Last check: 2007-10-28)
		However, the class of groups defined by means of a polycyclic presentation has a soluble word problem.
		Since soluble groups are often given in a form other than by a pc-presentation, it is usually necessary to explicitly construct such a presentation.
		If G is a finitely presented group known to be a p-group, then the p-quotient algorithm may be used to construct a pc-presentation for G.
	www.math.ufl.edu /help/magma/text237.html (446 words)

	COMETS WORKSHOP: TEACHING MATHEMATICS WORD PROBLEM SOLVING TO DEAF STUDENTS
		Pairs then share with the large group what they believe the real core problem to be; through large-group discussion, consensus is reached on the core problem, and it is written clearly by the leader for the group to see.
		Divide the larger group into three sub-groups or teams, and each group then tries to carry out the actual solution according to one solution path; thus each of the teams has a different solution path which they are testing (Executing the Plan).
		At this stage, it is VERY important that the groups not only carry out their respective plans, but also make notes about the processes used; appointing one member of each group as a recorder will facilitate this process.
	www.rit.edu /~comets/pages/workshops/problemsolvingwkshop.html (1169 words)

	Word Groups (Site not responding. Last check: 2007-10-28)
		Word groups with these properties have been referred to as finite projective geometries, a term which reflects the underlying mathematics but says nothing about the word relationships.
		If one waives the requirement that each word should have exactly m letters in common with each other word, but retains the requirement that each possible pair of letters appears in exactly n words, one is led to a large number of interesting word groups.
		Possible combinations for which no word groups have yet been found include a group of 15 six-letter words with two-letter overlap, a group of 14 seven-letter words with three-letter overlap, a group of 14 eight-letter words with four-letter overlap, and a group of 12 eight-letter words with five-letter overlap.
	www.wordways.com /groups.htm (1934 words)

	[No title]
		Date: Fri, 18 Feb 2000 15:21:46 GMT Newsgroups: sci.math,sci.logic Summary: [missing] The word problem for groups is often given as: "given a finite group presentation and relators, decide whether a given word reduces to the identity element." This problem, for general groups, is known to be unsolvable by any algorithm.
		My impression is yes, because if the set were recursive, then an algorithm would exist to decide the word problem for this group; though perhaps one could not prove (in ZFC, say) that the algorithm always gives the right answer...
		Does the reducibility question amount to: "does this word reduce to e following some fixed rules based on the relators" or is it: "is the _value_ of this word the identity element e?" I read an introduction to this problem (group presentations, etc) in the Feb. 1997 issue of the American Math.
	www.math.niu.edu /~rusin/known-math/00_incoming/wordprob (266 words)

	Math 415
		Braid groups and mapping class groups are important in the study of both topology and group theory.
		An algorithm for the word problem in braid groups, by
		A new approach to the word and conjugacy problems in the braid group by Birman, Ko,, Lee
	www.math.uiuc.edu /~kwhittle/math415.html (311 words)

	ChangeLog
		Now it is possible to store examples of groups and their descriptions in the Database as well as save and rstore objects in the Workspace.
		Whitehead reduction for tuples of words in a free group.
		Andrews-Curtis reduction for tuples of words in a free group.
	zebra.sci.ccny.cuny.edu /web/nygtc/software/ChangeLog.html (694 words)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: 2007-10-28)
		Planar graphs, duality, and the geometry of the word problem for groups
		This has arisen in work in Geometric Group Theory and has applications concerning the geometry of the word problem for finite presentations of groups.
		Also the graph theory facilitates geometric inequalities that constrain the space complexity measure of the word problem called filling length.
	www.math.unl.edu /pi/colloquia/abstract-20031211.txt (93 words)

	Literature-Math Crossover Lesson (Site not responding. Last check: 2007-10-28)
		I think that they have the necessary skills of word comprehension and knowledge of subtraction to work a word problem.
		I will ask that the problems be short (2-3 sentences) and that the problem be something that can be answered by the class.
		The problems should be no more than 2-3 sentences long and be something that the class can answer." I will collect all of the problems (and their answers -- the groups will provide the answers to the problems) when they are finished.
	www.halcyon.com /marcs/litmath.html (654 words)

	WordStuff #2
		The complication here is that, in addition to single words, we want to generate 2 and 3 word phrases that use all of the given letters.
		I sort the encrypted words from longest to shortest on the (untested) theory that there are fewer long words than short words in the dictionary, thus we should prune invalid combinations quicker.
		Capitalized words are now saved as capitalized and the characters ",C" are appended to match the ",A" and ",F" used to flag abbreviations and foreign words.
	www.delphiforfun.org /Programs/wordstuff2.htm (1606 words)

	Boone biography
		Boone's doctoral supervisor at Princeton was Church and his thesis was entitled Several Simple, Unsolvable Problems of Group Theory Related to the Word Problem.
		In 1950 Turing gave an example of a cancellative semigroup with insoluble word problem (having at one stage believed incorrectly that he could solve the group problem).
		Boone proved in 1959 that many other decision problems for groups were insoluble.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Boone.html (309 words)

	algebra.help -- Basic Word Problems (Site not responding. Last check: 2007-10-28)
		A word problem in algebra is the equivalent of a story problem in math.
		When you solved story problems in your math class you had to decide what information you had and what you needed to find out.
		Word problems are solved by separating information from the problems into two equal groups, one for each side of an equation.
	www.algebrahelp.com /lessons/wordproblems/basics (215 words)

	Word problem - Wikipedia, the free encyclopedia
		For a type of textbook problem designed to help students apply abstract mathematical concepts to "real-world" situations, see Word problem (mathematics education).
		For the word problem for encodings of sets in mathematics and computer science, see word problem (mathematics).
		For the word problem for groups, see word problem for groups.
	en.wikipedia.org /wiki/Word_problem (113 words)

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	www.uniquecontentpro.com (1427 words)

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