super p restricted edge connectivity of line graphs super restricted connectivity and super restricted edge connectivity are more refined network reliability indices than connectivity and edge connectivity in this paper we first introduce the concepts of super restricted connectivity and super restricted edge connectivity and then give a property of graphs with equal p restricted edge connectivity and 1 p restricted connectivity applying this property we show a sufficient condition for a line graph to be super p restricted edge connected and a relationship between super 1 p restricted connected graphs and super p restricted edge connected graphs

normalisers in limit groups let gamma be a limit group s subset gamma a non trivial subgroup and n the normaliser of s if h 1 s mathbb q has finite mathbb q dimension then s is finitely generated and either n s is finite or n is abelian this result has applications to the study of subdirect products of limit groups

matching preclusion for n dimensional torus networks abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has neither perfect matchings nor almost perfect matchings wang et al 13 proved that a class of n dimensional tours networks with even order are super matched later cheng et al 8 further showed that all n dimensional tours networks with even order are super matched in this paper we prove that all n dimensional torus networks with odd order are super matched if n 3 two dimensional torus networks with odd order is maximally matched except for c 3 c 3 our results are complementary to those of wang et al 13 and cheng et al 8

stallings foldings and subgroups of free groups we re cast in a more combinatorial and computational form the topological approach of j stallings to the study of subgroups of free groups

subgroup separability in residually free groups we prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type mathrm fp infty are virtual retracts we describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups

on pro p analogues of limit groups via extensions of centralizers we begin a study of a pro p analogue of limit groups via extensions of centralizers and call mathcal l this new class of pro p groups we show that the pro p groups of mathcal l have finite cohomological dimension type fp and non positive euler characteristic among the group theoretic properties it is proved that they are free by torsion free nilpotent and if non abelian do not have a finitely generated non trivial normal subgroup of infinite index furthermore it is shown that every 2 generated pro p group in the class mathcal l is either free pro p or abelian

recursive circulants and their embeddings among hypercubes we propose an interconnection structure for multicomputer networks called recursive circulant recursive circulant g n d is defined to be a circulant graph with n nodes and jumps of powers of d g n d is node symmetric and has some strong hamiltonian properties g n d has a recursive structure when n cdm 1 c d we develop a shortest path routing algorithm in g cdm d and analyze various network metrics of g cdm d such as connectivity diameter mean internode distance and visit ratio g 2m 4 whose degree is m compares favorably to the hypercube qm g 2m 4 has the maximum possible connectivity and its diameter is 3m 1 4 recursive circulants have interesting relationship with hypercubes in terms of embedding we present expansion one embeddings among recursive circulants and hypercubes and analyze the costs associated with each embedding the earlier version of this paper appeared in park and chwa proc internat symp parallel architectures algorithms and networks ispan 94 kanazawa japan december 1994 pp 73 80

statistically boundedness and statistical core of double sequences the concept of statistical convergence was presented by steinhaus in 1951 this concept was extended to the double sequences by mursaleen and edely in 2003 throughout this paper we will present multidimensional analogues of the results presented by fridy and orhan in 1997 to achieve this goal multidimensional analogues of the definition for bounded statistically sequences statistical inferior and statistical superior will be presented in addition to these results we will investigate statistical core for double sequences and study an inequality related to the statistical and p cores of bounded double sequences

strong matching preclusion for n dimensional torus networks the strong matching preclusion number of a graph is the minimum number of edges and or vertices whose deletion results in the remaining graph that has neither perfect matchings nor almost perfect matchings in 14 wang et al proved that c k c k is super strongly matched where k 3 is odd in this paper we show that c k 1 c k 2 c k n is super strongly matched where n 3 is an integer and k i 3 is an odd integer for each i 1 n our studies generalize the results of wang et al 14

matching preclusion number in product graphs abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has no perfect matchings or almost perfect matchings for many interconnection networks the matching preclusion number is equal to the minimum degree of a vertex in the network and the sets of edges attaining the minimum are precisely those incident to a single vertex of minimum degree we say such networks are super matched in this paper we derive upper and lower bounds for the matching preclusion number for networks constructed using a variety of binary graph operations and give sufficient conditions for such networks to be super matched

strong matching preclusion of n k star graphs the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings this is an extension of the matching preclusion problem that was introduced by park and ihm the generalized n k star graph was introduced to address scaling issues of the star graph and it has many desirable properties in this paper the goal is to find the strong matching preclusion number of n k star graphs and to categorize all optimal strong matching preclusion sets of these graphs since bipartite graphs generally have low strong matching preclusion numbers we assume that k n 2

fraisse limits of limit groups abstract we modify the notion of a fraisse class and show that various interesting classes of groups notably the class of nonabelian limit groups and the class of finitely generated elementary free groups admit fraisse limits furthermore we rediscover lyndon s z t exponential completions of countable torsion free csa groups as fraisse limits with respect to extensions of centralizers

the weak commutativity construction for lie algebras abstract we study the analogue of sidki s weak commutativity construction defined originally for groups in the category of lie algebras this is the quotient g of the lie algebra freely generated by two isomorphic copies g and g of a fixed lie algebra by the ideal generated by the brackets x x for all x we exhibit an abelian ideal of g whose associated quotient is a subdirect sum in g g g and we give conditions for this ideal to be finite dimensional we show that g has a subquotient that is isomorphic to the schur multiplier of g we prove that g is finitely presentable or of homological type f p 2 if and only if g has the same property but f is not of type f p 3 if f is a non abelian free lie algebra

on subdirect products of free pro p groups and demushkin groups of infinite depth abstract we study subdirect products of free and demushkin pro p groups of depth developing theory similar to the abstract case see bridson et al 2002 4 furthermore we classify when a subdirect product has homological type fp m for some m 2 a problem still open for abstract groups for m 3

strong matching preclusion number of graphs abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings the strong matching preclusion number or simply smp number s m p g of a graph g is the minimum number of vertices and or edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings this is an extension of the matching preclusion problem and has been introduced by park and ihm in this paper we first study the smp number of some special graph classes and give some sharp upper and lower bounds of smp number next graphs with large and small smp number are characterized respectively in the end we investigate the nordhaus gaddum type relations on smp number

strong matching preclusion for k composition networks abstract let g 1 v 1 e 1 g 2 v 2 e 2 g k v k e k be k graphs and let f 1 v 1 v 2 f 2 v 2 v 3 f k 1 v k 1 v k f k v k v 1 be k bijections the k composition networks g induced by g 1 g 2 g k is the graph with v g t 1 k v g t and e g t 1 k e g t a t f t a t a t v g t and 1 t k many interconnection networks such as n dimensional torus networks recursive circulant graphs and cayley graphs on abelian groups generated by minimal generating sets are special k composition networks the strong matching preclusion number of a graph is the minimum number of edges and or vertices whose deletion results in the remaining graph has neither perfect matchings nor almost perfect matchings in this paper we study the strong matching preclusion number and strong matching preclusion sets for k composition networks with odd order our results generalize the main conclusion in 12

on the finite presentation of subdirect products and the nature of residually free groups we establish it virtual surjection to pairs vsp as a general criterion for the finite presentability of subdirect products of groups if gamma 1 ldots gamma n are finitely presented and s we use the vsp criterion to characterize the finitely presented residually free groups we prove that the class of such groups is recursively enumerable we describe an algorithm that given a finite presentation of a residually free group constructs a canonical embedding into a direct product of finitely many limit groups we solve the multiple conjugacy problem and membership problem for finitely presentable subgroups of residually free groups we also prove that there is an algorithm that given a finite generating set for such a subgroup will construct a finite presentation new families of subdirect products of free groups are constructed including the first examples of finitely presented subgroups that are neither rm fp infty nor of stallings bieri type

many to many disjoint path covers in the presence of faulty elements a many to many k disjoint path cover k dpc of a graph g is a set of k disjoint paths joining k sources and k sinks in which each vertex of g is covered by a path it is called a paired many to many disjoint path cover when each source should be joined to a specific sink and it is called an unpaired many to many disjoint path cover when each source can be joined to an arbitrary sink in this paper we discuss about paired and unpaired many to many disjoint path covers including their relationships application to strong hamiltonicity and necessary conditions and then we give a construction scheme for paired many to many disjoint path covers in the graph h0 oplus h1 obtained from connecting two graphs h0 and h1 with v h0 v h1 by v h1 pairwise nonadjacent edges joining vertices in h0 and vertices in h1 where h0 g0 oplus g1 and h1 g2 oplus g3 for some graphs gj using the construction we show that every m dimensional restricted hl graph and recursive circulant g 2m 4 with f or less faulty elements have a paired k dpc for any f and k ges 2 with f 2k les m

implicit function theorem over free groups abstract we introduce the notion of a regular quadratic equation and a regular ntq system over a free group we prove the results that can be described as implicit function theorems for algebraic varieties corresponding to regular quadratic and ntq systems we will also show that the implicit function theorem is true only for these varieties in algebraic geometry such results would be described as lifting solutions of equations into generic points from the model theoretic view point we claim the existence of simple skolem functions for particular formulas over free groups proving these theorems we describe in details a new version of the makanin razborov process for solving equations in free groups we also prove a weak version of the implicit function theorem for ntq systems which is one of the key results in the solution of the tarski s problems about the elementary theory of a free group we call it the parametrization theorem

conditional edge fault tolerant edge bipancyclicity of hypercubes the hypercube has been one of the most popular interconnection networks for parallel computer communication systems in this paper we assume that each node is incident with at least two fault free links under this assumption we show that every fault free edge lies on a fault free cycle of every even length from 6 to 2 n inclusive even if it has up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated

on conditional diagnosability of the folded hypercubes the n dimensional folded hypercube fq n a variation of the hypercube proposed by ahmed et al a el amawy s latifi properties and performance of folded hypercubes ieee transactions on parallel and distributed systems 2 3 1991 31 42 is an n 1 regular n 1 connected graph conditional diagnosability a new measure of diagnosability introduced by lai et al pao lien lai jimmy j m tan chien ping chuang lih hsing hsu conditional diagnosability measures for large multiprocessor systems ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of fq n t c fq n is 4n 3 when n 5 or n 8 t c fq 3 3 t c fq 4 7

strong matching preclusion for augmented cubes the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings the concept was introduced by park and son in this paper we study the strong matching preclusion problem for the augmented cube graphs as a result we find smp aqn and classify all optimal solutions

strong matching preclusion for non bipartite torus networks the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in the remaining graph that has neither perfect matchings nor almost perfect matchings the torus network is one of the most popular interconnection network topologies for massively parallel computing systems because of its desirable properties it is known that bipartite torus networks have low strong matching preclusion numbers hu et al 13 proved that non bipartite torus networks with an odd number of vertices have good strong matching preclusion properties to complete the study of strong matching preclusion problem for non bipartite torus networks in this paper we establish the strong matching preclusion number and classify all optimal strong matching preclusion sets for the n dimensional non bipartite torus network with an even number of vertices where n 3

irreducible affine varieties over a free group i irreducibility of quadratic equations and nullstellensatz the object of this paper which consists of two parts is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups we prove that any variety over a free group f can be defined by a finite number of systems of equations s 1 in triangular form where quadratic words play the role of leading terms algebraically irreducible varieties are exactly the varieties whose coordinate groups are fully residually f the crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into lyndon s free z x group f since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety and the group f is fully residually f we obtain a characterization of finitely generated fully residually free groups as subgroups of f the group f and its subgroups have been studied extensively during the last several years in particular every finitely generated subgroup of f hence every finitely generated fully residually free group can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and hnn extensions of the type where amalgamated and associated subgroups are free abelian of finite rank in particular this implies that every finitely generated fully residually free group is finitely presented there are three parts to this paper algebraic geometry over free groups the theory of free exponential groups and makanin razborov s machinery to deal with equations over free groups 11 14 13 the algebraic geometry approach has been shown to be very usefull in dealing with equations over groups it provides necessary topological means and a method to transcribe geometric notions into pure group theoretic language following baumslag myasnikov remeslennikov 1 we use the standard algebraic geometry notions such as variety zariski topology irreducibility of varieties radicals and coordinate groups some of the ideas of the algebraic geometry approach go back to r lyndon 9 e rips j stallings 16

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

on reliability of the folded hypercubes in this paper we explore the 2 extra connectivity and 2 extra edge connectivity of the folded hypercube fq n we show that k 2 fq n 3n 2 for n 8 and l 2 fq n 3n 1 for n 5 that is for n 8 resp n 5 at least 3n 2 vertices resp 3n 1 edges of fq n are removed to get a disconnected graph that contains no isolated vertices resp edges when the folded hypercube is used to model the topological structure of a large scale parallel processing system these results can provide more accurate measurements for reliability and fault tolerance of the system

conditional matching preclusion for the arrangement graphs the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex recently the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex it is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings in this paper we find this number and classify all optimal sets for the arrangement graphs one of the most popular interconnection networks

ascending chains of finitely generated subgroups abstract we show that a nonempty family of n generated subgroups of a pro p group has a maximal element this suggests that noetherian induction can be used to discover new features of finitely generated subgroups of pro p groups to demonstrate this we show that in various pro p groups e g free pro p groups nonsolvable demushkin groups the commensurator of a finitely generated subgroup h 1 is the greatest subgroup of containing h as an open subgroup we also show that an ascending chain of n generated subgroups of a limit group must terminate this extends the analogous result for free groups proved by takahasi higman and kapovich myasnikov

irreducible affine varieties over a free group ii systems in triangular quasi quadratic form and description of residually free groups abstract we shall prove the conjecture of myasnikov and remeslennikov 4 which states that a finitely generated group is fully residually free every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group if and only if it is embeddable in the lyndon s exponential groupfz x which is thez x completion of the free group herez x is the ring of polynomials of one variable with integer coefficients historically lyndon s attempts to solve tarski s famous problem concerning the elementary equivalence of free groups of different ranks led him to introducefz x an free group is a groupgsuch that the class of formulas true ing is the same as the class of formulas true in a nonabelian free group a finitely generated group is free if and only if it is fully residually free 22 our result gives an algebraic description of free groups we shall give an algorithm to represent a solution set of an arbitrary system of equations overfas a union of finite number of irreducible components in the zariski topology onfn the solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms the possibility of such a decomposition for a solution set was conjectured by razborov in 20 and also by rips we shall give a description of systems of equations determining irreducible components using methods developed in 13 19 it is possible to find some of these methods in 18 we are thankful to e rips for attracting our attention to these techniques

structure and finiteness properties of subdirect products of groups we investigate the structure of subdirect products of groups particularly their finiteness properties we pay special attention to the subdirect products of free groups surface groups and hnn extensions we prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands this leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups we obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup a useful structure theory for subgroups of finite direct products of groups has yet to be developed to begin to study such subgroups it is natural to assume one knows about the subgroups of the direct factors and to concentrate on subdirect products recall that g is termed a subdirect product of the groups a1 an if g a1 an is a subgroup that projects surjectively to each factor work by various authors has exposed the surprisingly rich structure to be found amongst the subdirect products of superficially tame groups for example in contrast to the fact that subdirect products of abelian or nilpotent groups are again in the specified class nonabelian free groups harbour a great diversity of subdirect products including some with unsolvable decision problems 21 this diversity has long been known but it is only as a result of more recent work by baumslag roseblade 5 and bridson howie miller short 9 that it has been understood as a phenomenon that is intimately tied to the failure of various homological finiteness conditions from this background we pick out the three strains of thought to be developed in this article the subtlety of subdirect products in general 1991 mathematics subject classification 20f05 20e07 20f67 20j05 20f10

subgroup properties of fully residually free groups we prove that fully residually free groups have the howson property that is the intersection of any two finitely generated subgroups in such a group is again finitely generated we also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup

subgroups of direct products of limit groups if ti tn are limit groups and s c t x x tn is of type fp q then s contains a subgroup of finite index that is itself a direct product of at most n limit groups this answers a question of sela

covering spaces of 3 orbifolds let o be a compact orientable 3 orbifold with non empty singular locus and a finite volume hyperbolic structure equivalently the interior of o is the quotient of hyperbolic 3 space by a lattice in psl 2 c with torsion then we prove that o has a tower of finite sheeted covers oi with linear growth of p homology for some prime p this means that the dimension of the first homology with mod p coefficients of the fundamental group of oi grows linearly in the covering degree the proof combines techniques from 3 manifold theory with group theoretic methods including the golodshafarevich inequality and results about p adic analytic pro p groups this has several consequences firstly the fundamental group of o has at least exponential subgroup growth secondly the covers oi have positive heegaard gradient thirdly we use it to show that a group theoretic conjecture of lubotzky and zelmanov would imply that o has large fundamental group this implication uses a new theorem of the author which will appear in a forthcoming paper these results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3 orbifold with non empty singular locus has large fundamental group many of the above results apply also to 3 manifolds commensurable with an orientable finite volume hyperbolic 3 orbifold with non empty singular locus this includes all closed orientable hyperbolic 3 manifolds with rank two fundamental group and all arithmetic 3 manifolds

strong matching preclusion for torus networks the torus network is one of the most popular interconnection network topologies for massively parallel computing systems strong matching preclusion that additionally permits more destructive vertex faults in a graph is a more extensive form of the original matching preclusion that assumes only edge faults in this paper we establish the strong matching preclusion number and all minimum strong matching preclusion sets for bipartite torus networks and 2 dimensional nonbipartite torus networks

cubulating malnormal amalgams let g be a group splitting as a graph of groups where each vertex group acts properly on a cat 0 cube complex when g is hyperbolic relative to virtually abelian subgroups we examine conditions producing a proper action of g on a cat 0 cube complex when g is hyperbolic this action is also cocompact

fully residually free pro p groups abstract we prove that the pro p completion of the orientable surface group of genus d is residually free pro p if d is even or p 2 the case when p 2 odd d 1 is still open

automorphisms of the category of free lie algebras we prove that every automorphism of the category of free lie algebras is a semi inner automorphism this solves problem 3 9 from g mashevitzky b plotkin e plotkin electron res announc amer math soc 8 2002 1 10 for lie algebras

statistical convergence of double sequences the idea of statistical convergence was first introduced by fast 1951 but the rapid developments were started after the papers of salat 1980 and fridy 1985 now a days it has become one of the most active area of research in the field of summability in this paper we define and study statistical analogue of convergence and cauchy for double sequences we also establish the relation between statistical convergence and strongly cesaro summable double sequences

conditional matching preclusion for hypercube like interconnection networks the conditional matching preclusion number of a graph with n vertices is the minimum number of edges whose deletion results in a graph without an isolated vertex that does not have a perfect matching if n is even or an almost perfect matching if n is odd we develop some general properties on conditional matching preclusion and then analyze the conditional matching preclusion numbers for some hl graphs hypercube like interconnection networks

the extension of the knopp core theorem to the sequences of fuzzy numbers the concept of the core of a sequence of fuzzy numbers has been introduced by aytar et al in the core of a sequence of fuzzy numbers fuzzy sets and systems 159 24 2008 3369 3379 quite recently some matrix transformations on the classical sets of sequences of fuzzy numbers have been characterized by talo and basar in determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations comput math appl 58 2009 717 733 in this paper we have proved a core theorem for sequences of fuzzy numbers which is analogous to the famous knoop core theorem for real sequences to achieve this goal we have studied some properties of the lim sup and lim inf of sequences of fuzzy numbers also we have characterized a class of positive regular matrices of fuzzy numbers such that these matrices leave the core of any bounded sequence of fuzzy numbers invariant

elementary theory of free non abelian groups we prove that any two non abelian free groups have the same elementary theory and that this theory is decidable these results solve two questions that were raised by tarski in 1945

reliability analysis of twisted cubes connectivity is a vital metric to explore fault tolerance and reliability of network structure based on a graph model let g v e g v e be a connected graph a connected graph g is called supper resp super if every minimum vertex cut edge cut of g is the set of neighbors of some vertex in g let f v f v be a vertex set f is called extra cut if g f g f is not connected and each component of g f g f has more than k vertices the extraconnectivity k g k g is the cardinality of the minimum extra cuts a r component cut of g is a set s of vertices g s g s has at least r components r component connectivity c r g c r g of g is the size of the smallest r component cut the r component edge connectivity c r g c r g can be defined similarly in this paper we determine the r component edge connectivity of twisted cubes tn n t n n for small r and we also prove other properties of tn n t n n

krull dimension of solvable groups in this paper we prove that free solvable groups have finite krull dimension in fact this is true for much wider class of solvable groups termed rigid groups along the way we study the algebraic structure of the limit solvable groups fully residually free solvable groups

groups with free regular length functions in ℤⁿ this is the first paper in a series of three where we take on the unified theory of non archimedean group actions length functions and infinite words our main goal is to show that group actions on zn trees give one a powerful tool to study groups all finitely generated groups acting freely on r trees also act freely on some zn trees but the latter ones form a much larger class the natural effectiveness of all constructions for zn actions which is not the case for r trees comes along with a robust algorithmic theory in this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on zn trees and give necessary and sufficient conditions for such actions

quasi potency and cyclic subgroup separability we study the profinite topology on discrete groups and in particular the property of cyclic subgroup separability we investigate the class of quasi potent cyclic subgroup separable groups producing many examples and showing how it behaves with respect to certain group constructions

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

the diagnosability of the matching composition network under the comparison diagnosis model the classical problem of diagnosability is discussed widely and the diagnosability of many well known networks has been explored we consider the diagnosability of a family of networks called the matching composition network mcn a perfect matching connects two components the diagnosability of mcn under the comparison model is shown to be one larger than that of the component provided some connectivity constraints are satisfied applying our result the diagnosability of the hypercube qn the crossed cube cq sub n the twisted cube tq sub n and the mobius cube mq sub n can all proven to be n for n spl ges 4 in particular we show that the diagnosability of the four dimensional hypercube q sub 4 is 4 which is not previously known

matching preclusion for balanced hypercubes matching preclusion is a measure of robustness in the event of edge failure in interconnection networks the matching preclusion number of a graph g is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching and the conditional matching preclusion number of g is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or an almost perfect matching in this paper we consider balanced hypercubes we obtain that an n dimension balanced hypercube bh n has the matching preclusion number 2n and mainly prove that for the balanced hypercube bh n each matching preclusion set of cardinality 2n is trivial and the conditional matching preclusion number of balanced hypercube is 4n 2 whenever n 2

finding all solutions of equations in free groups and monoids with involution this paper presents a pspace algorithm which yields a finite graph of exponential size that describes the set of all solutions of equations in free groups as well as the set of all solutions of equations with rational constraints in free monoids this became possible due to the recent recompression technique while this technique was successfully applied for pure word equations without involution or rational constraints it could not be used as a black box for free groups actually the presence of an involution and rational constraints complicates the situation and some additional analysis is necessary still the technique is general enough to accommodate both extensions in the end it simplifies proofs that satisfiability of word equations is in pspace and the corresponding result for equations in free groups with rational constraints as a byproduct we can decide in pspace whether the solution set is finite

complexity volumes of splittable groups abstract using graph of groups decompositions of finitely generated groups we define euler characteristic type invariants which are non zero in many interesting classes of hyperbolic limit and csa groups including elementarily free groups and one ended torsion free hyperbolic groups whose jsj decomposition contains a maximal hanging fuchsian vertex group

decision problems and profinite completions of groups abstract we consider pairs of finitely presented residually finite groups p for which the induced map of profinite completions p is an isomorphism we prove that there is no algorithm that given an arbitrary such pair can determine whether or not p is isomorphic to we construct pairs for which the conjugacy problem in can be solved in quadratic time but the conjugacy problem in p is unsolvable let j be the class of super perfect groups that have a compact classifying space and no proper subgroups of finite index we prove that there does not exist an algorithm that given a finite presentation of a group and a guarantee that j can determine whether or not 1 we construct a finitely presented acyclic group h and an integer k such that there is no algorithm that can determine which k generator subgroups of h are perfect

erratum to subgroup properties of fully residually free groups our article 4 contained a number of algorithmic and structural results about finitely generated subgroups of residually free groups while all the arguments and proofs given in 4 are internally consistent our results in the most general form relied on proposition 5 1 of 4 which we attributed to the paper 8 that is currently in preparation and has an additional author v remeslennikov unfortunately it turns out that proposition 5 1 is incorrect as stated by us in 4 as was pointed out to the author by zlil sela mladen bestvina olga kharlampovich alexei myasnikov and inna bumagina after 4 appeared this mistake was due to a failure of communication and misunderstanding on my part of the results of 8 and i take full responsibility for this event all the main results of 4 namely theorems a b c and d are thus incomplete and should be regarded as conjectures for the moment it is clear to the author that these statements are in fact correct but the proofs would require a careful and quite lengthy inductive use of proposition 5 4 of 4 that is due to a myasnikov and v remeslennikov 9 together with the results of r burns 1 2 and d cohen 3 instead of relying on the incorrect proposition 5 1 as a shortcut nevertheless all of the proofs given in 4 are correct and complete for the case of word hyperbolic fully residually free groups since in that case proposition 5 1 was not needed thus what was actually proved in 4 can be summarized in the following

characterizations of the ideal core abstract given an ideal i on and a sequence x in a topological vector space we let the i core of x be the least closed convex set containing x n n i for all i i we show two characterizations of the i core this implies that the i core of a bounded sequence in r k is simply the convex hull of its i cluster points as applications we simplify and extend several results in the context of pringsheim convergence and e convergence of double sequences

algebraic geometry over groups i algebraic sets and ideal theory abstract the object of this paper which is the first in a series of three is to lay the foundations of the theory of ideals and algebraic sets over groups

linearly many faults in dual cube like networks the dual cubes were introduced as a better interconnection network than the hypercubes for large scale distributed memory multiprocessors in this paper we introduce a generalization of these networks called dual cube like networks which preserve the basic structure of dual cubes and retain many of its topological properties we investigate structural properties of these networks beyond simple measures such as connectivity we prove that if up to kn k k 1 2 vertices are deleted from a dual cube like n regular network then the resulting graph will either be connected or will have a large component and small components having at most k 1 vertices in total and this result is sharp for k n as an application we derive additional results such as the cyclic vertex connectivity and the restricted vertex connectivity of these networks

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

a note on generalized matching preclusion in bipartite graphs abstract for a graph with an even number of vertices the matching preclusion number is the minimum number of edges whose deletion results in a graph with no perfect matchings the conditional matching preclusion number introduced as an extension of the matching preclusion number has the additional requirement that the resulting graph have no isolated vertices in bipartite graphs the edge sets that achieve these minimum numbers can be construed in terms of obstruction sets of vertices from this perspective the generalized matching preclusion problem was developed the matching preclusion number of order a is the minimum number of edges whose deletion results in a graph with no perfect matchings and no obstruction sets with fewer than a vertices in this paper we extend known sufficient conditions regarding the matching preclusion and conditional matching preclusion numbers orders 1 and 2 respectively to give sufficient conditions for the matching preclusion problem of order a for any positive integer a

matching preclusion number of graphs abstract the matching preclusion number of a graph g denoted by mp g is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings in this paper we first give some sharp upper and lower bounds of matching preclusion number next graphs with large and small matching preclusion number are characterized respectively in the end we investigate some extremal problems and the nordhaus gaddum type relations on matching preclusion number

on systems of equations over free products of groups using an analogue of the makanin razborov diagrams we give a description of the solution set of systems of equations over an equationally noetherian free product of groups g equivalently we give a parametrisation of the set hom h g of all homomorphisms from a finitely generated group h to g furthermore we show that every algebraic set over g can be decomposed as a union of finitely many images of algebraic sets of ntq systems if the universal horn theory of g the theory of quasi identities is decidable then our constructions are effective 2010 elsevier inc

path embedding in faulty hypercubes abstract there are some interesting results concerning longest paths or even cycles embedding in faulty hypercubes this paper considers the embeddings of paths of all possible lengths between any two fault free vertices in faulty hypercubes let fv respectively fe denote the number of faulty vertices respectively edges in an n dimensional hypercube qn we prove that there exists a fault free path of length l between any two distinct fault free vertices u and v in qn with f v f e n 2 for each l satisfying d q n u v 2 l 2 n 2 f v 1 and 2 l d q n u v the bounds on path length l and faulty set size f v f e for a successful embedding are tight that is the result does not hold if l d q n u v 2 or l 2 n 2 f v 1 or f v f e n 2 moreover our result improves some known results

fault tolerance of locally twisted cubes abstract let g v e be a connected graph and p be graph theoretic property a network is often modeled by a graph g v e one fundamental consideration in the design of networks is reliability the connectivity is an important parameter to measure the fault tolerance and reliability of network the conditional connectivity g p or g p is the minimum cardinality of a set of edges or vertices if it exists whose deletion disconnects g and each remaining component has property p let f be a vertex set or edge set of g and p be the property of with at least k components then we have the k component connectivity c k g and the k component edge connectivity c k g in this paper we determine the k component edge connectivity of locally twisted cubes ltqn for small k and we also prove other properties of ltqn

volume gradients and homology in towers of residually free groups we study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups g n g of increasing finite index in a fixed finitely generated group g assuming bigcap n g n 1 we focus in particular on finitely presented residually free groups calculating their ell 2 betti numbers rank gradient and asymptotic deficiency if g is a limit group and k is any field then for all j ge 1 the limit of dim h j g n k g g n as n rightarrow infty exists and is zero except for j 1 where it equals chi g we prove a homotopical version of this theorem in which the dimension of dim h j g n k is replaced by the minimal number of j cells in a k g n 1 this includes a calculation of the rank gradient and the asymptotic deficiency of g both the homological and homotopical versions are special cases of general results about the fundamental groups of graphs of slow groups we prove that if a residually free group g is of type mathrm fp m but not of type mathrm fp infty then there exists an exhausting filtration by normal subgroups of finite index g n so that lim n dim h j g n k g g n 0 hbox for j le m if g is of type mathrm fp infty then the limit exists in all dimensions and we calculate it

matching preclusion and conditional matching preclusion problems for the folded petersen cube the matching preclusion number of an even graph is the minimum number of edges whose deletion results in a graph that has no perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex recently the conditional matching preclusion number of an even graph was introduced to look for obstruction sets beyond those induced by a single vertex it is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices and no perfect matchings in this paper we study this problem for the folded petersen cube fpq n k via some matching preclusion and conditional matching preclusion results of the cartesian products of graphs

fractional matching preclusion of product networks abstract the matching preclusion number of a graph introduced in 2 as a fault analysis is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings as a generalization liu and liu 14 recently introduced the concept of fractional matching preclusion number the fractional matching preclusion number of graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matching if the sets of edges of the graph attaining the minimum are precisely those incident to a single vertex of minimum degree we say such graph is fractional super matched in this paper the upper and lower bounds for the fractional matching preclusion number for cartesian product direct product strong product and lexicographic product are obtained and we give sufficient conditions for such graphs to be fractional super matched

the equation xp yq zr and groups that act freely on l trees let g be a finitely generated group that acts freely on a lambda tree where lambda is an ordered abelian group and let x y z be elements in g we show that if x p y q z r with integers p q r geq 4 then x y and z commute

edge fault tolerance of super edge connectivity for three families of interconnection networks let g v e be a connected graph g is said to be super edge connected or super l for short if every minimum edge cut of g isolates one of the vertex of g a graph g is called m super l if for any edge set s e g with s

conditional matching preclusion sets abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex in this paper we look for obstruction sets beyond these sets we introduce the conditional matching preclusion number of a graph it is the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings we find this number and classify all optimal sets for several basic classes of graphs

cohomological goodness and the profinite completion of bianchi groups the concept of cohomological goodness was introduced by j p serre in his book on galois cohomology 31 this property relates the cohomology groups of a group to those of its profinite completion we develop properties of goodness and establish goodness for certain important groups we prove for example that the bianchi groups i e the groups psl 2 o where o is the ring of integers in an imaginary quadratic number field are good as an application of our improved understanding of goodness we are able to show that certain natural central extensions of fuchsian groups are residually finite this result contrasts with examples of p deligne 5 who shows that the analogous central extensions of sp 4 z do not have this property

strong matching preclusion the matching preclusion problem introduced by brigham et al r c brigham f harary e c violin and j yellen perfect matching preclusion congressus numerantium 174 2005 185 192 studies how to effectively make a graph have neither perfect matchings nor almost perfect matchings by deleting as small a number of edges as possible extending this concept we consider a more general matching preclusion problem called the strong matching preclusion in which deletion of vertices is additionally permitted we establish the strong matching preclusion number and all possible minimum strong matching preclusion sets for various classes of graphs