reconstruction of closely spaced small inclusions in this paper we establish an explicit asymptotic formula for the steady state voltage perturbations caused by closely spaced small conductivity inhomogeneities based on this new formula we design a very effective numerical method to identify the location and some geometric features of these inhomogeneities from a finite number of boundary measurements the viability of our approach is documented by numerical examples

optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions if stiff inclusions are closely located then the stress which is the gradient of the solution may become arbitrarily large as the distance between two inclusions tends to zero in this paper we investigate the asymptotic behavior of the stress concentration factor which is the normalized magnitude of the stress concentration as the distance between two inclusions tends to zero for that purpose we show that the gradient of the solution to the case when two inclusions are touching decays exponentially fast near the touching point we also prove a similar result when two inclusions are closely located and there is no potential difference on boundaries of two inclusions we then use these facts to show that the stress concentration factor converges to a certain integral of the solution to the touching case as the distance between two inclusions tends to zero we then present an efficient way to compute this integral

on labeled birooted tree languages with an aim to developing expressive language theoretical tools applicable to inverse semigroup languages that is subsets of inverse semigroups this paper explores the language theory of finite labeled birooted trees munn s birooted trees extended with vertex labeling to this purpose we define a notion of finite state birooted tree automata that simply extends finite state word automata semantics this notion is shown to capture the class of languages that are definable in monadic second order logic and upward closed with respect to the natural order then we derive from these automata the notion of quasi recognizable languages that is languages recognizable by means of adequate premorphisms into finite adequately ordered monoids this notion is shown to capture finite boolean combinations of languages as above a class that contains classical regular languages of finite mono rooted trees this contrasts with the known collapse of classical algebraic tools when applied to inverse semigroups

gradient estimates for solutions to the conductivity problem in this paper we derive very precise gradient estimates for solutions to the conductivity problem in the case where two circular conductivity inclusions are very close but not touching the novelty of these estimates is that they give very specific information about the blow up of the gradient as the conductivities of the inclusions degenerate

an optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions abstract we consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions when inclusions with an extreme conductivity insulators or perfect conductors are closely located the gradient can be concentrated in between inclusions and then becomes arbitrarily large as the distance between inclusions approaches zero the gradient estimate in between insulators in three dimensions has been regarded as a challenging problem while the optimal blow up rates in terms of the distance were successfully obtained for the other extreme conductivity problems in two and three dimensions and are attained on the shortest line segment between inclusions in this paper we establish upper and lower bounds of gradients on the shortest line segment between two insulating unit spheres in three dimensions these bounds present the optimal blow up rate of gradient on the line segment which is substantially different from the rates in the other problems

on groups that have normal forms computable in logspace abstract we consider the class of finitely generated groups which have a normal form computable in logspace we prove that the class of such groups is closed under passing to finite index subgroups direct products wreath products and certain free products and infinite extensions and includes the solvable baumslag solitar groups as well as non residually finite and hence non linear examples we define a group to be logspace embeddable if it embeds in a group with normal forms computable in logspace we prove that finitely generated nilpotent groups are logspace embeddable it follows that all groups of polynomial growth are logspace embeddable

the word and geodesic problems in free solvable groups we study the computational complexity of the word problem wp in free solvable groups s r d where r 2 is the rank and d 2 is the solvability class of the group it is known that the magnus embedding of s r d into matrices provides a polynomial time decision algorithm for wp in a fixed group s r d unfortunately the degree of the polynomial grows together with d so the uniform algorithm is not polynomial in d in this paper we show that wp has time complexity o rn log 2 n in s r 2 and o n 3 rd in s r d for d 3 however it turns out that a seemingly close problem of computing the geodesic length of elements in s r 2 is np complete we prove also that one can compute fox derivatives of elements from s r d in time o n 3 rd in particular one can use efficiently the magnus embedding in computations with free solvable groups our approach is based on such classical tools as the magnus embedding and fox calculus as well as on relatively new geometric ideas in particular we establish a direct link between fox derivatives and geometric flows on cayley graphs

on drift and entropy growth for random walks on groups in this paper we consider drift and entropy growth for symmetric finitary random walks on finitely generated groups we construct examples of various intermediate asymptotics of the drift for such random walks we establish general inequalities which connect drift entropy and exponential growth rate of groups then we apply these inequalities to get estimates for entropy in particular examples

metric structures and probabilistic computation continuous first order logic is used to apply model theoretic analysis to analytic structures e g hilbert spaces banach spaces probability spaces etc classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first order logic the present paper shows that probabilistic computation sometimes called randomized computation and continuous logic stand in a similar close relationship the main result of this paper is an effective completeness theorem showing that every decidable continuous first order theory has a probabilistically decidable model we also show that probabilistically computable structures give rise to a model of aca 0in a natural way and describe a connection with complexity theory

growth series of some wreath products the growth series of certain finitely generated groups which are wreath products are investigated these growth series are intimately related to the traveling salesman problem on certain graphs a large class of these growth series is shown to consist of irrational algebraic functions n 0 since only one generating set will be associated with each group below the generating set associated with fr x will be obvious let h be a group with a finite generating set sh these will be fixed for the rest of the paper the cayley graph of the pair h sh is as usual the directed graph whose vertices are the elements of h and there is an edge from a vertex hx xoa vertex h2 if and only if h2 hxh for some h in sh us 1 in particular the edge from hx to h2 has an opposite edge from h2 to hx let c be the graph gotten from the cayley graph of h sh simply by identifying opposite edges in other words c might be called the undirected cayley graph of h sh hex k also be a group with a finite generating set sk these will also be fixed for the rest of the paper it is possible to form what might be called a restricted direct product group p which consists of all functions p from the vertices of c to k such that there are only finitely many vertices v in c with p v t 1 this is a subgroup of the direct product group whose elements consist of all functions from the vertex set of c to k the group p admits h as a group of automorphisms by means of the action of h on c the resulting semidirect product p x h is the restricted wreath product k h

isoperimetric inequalities and the homology of groups an isoperimetric function for a finitely presented group g bounds the number of defining relations needed to prove that a word w g 1 in terms of the length of w suppose g f r is a finitely presented group where f is a finitely generated free group and r is the normal closure of a finite set of relators for each w r we define the area w to be the least number of uses of the given relators needed to prove that w g 1 the isoperimetric function g n is defined to be the maximum of the areas of all the words in r of length n one compares the size of such functions using a suitable order 1 on functions different presentations give rise to equivalent isoperimetric functions where is the equivalence relation determined by 1 interest in such functions was stimulated by gromov 13 who characterized word hyperbolic groups as those with linear isoperimetric function that is g n n further information was obtained by cannon et al 8 who showed that automatic groups satisfy g 1 n and that asynchronously automatic groups satisfy g 1 2 they also showed the free nilpotent group of class 2 and rank 2 has g n n gersten 10 used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups he exploited the fact that for a torsion free one relator groupg the usual 2 dimensional complex having fundamental group isomorphic to g is aspherical the general technique for giving upper bounds for g n is to describe a solution to the word problem for g and analyze its use of defining relations it seems quite difficult to establish lower bounds for g n in this manner in this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for g instead of trying to analyze r directly we consider some of its quotient groups for which additional machinery is available for instance we consider for each w r its image w r f in r r f and define a suitable centralized isoperimetric function g n such that g n 1 g n using the hopf formula it can be shown that r r f is a direct sum of the second homology group h2 g z with a free abelian

sublinear deviation between geodesics and sample paths we give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic like properties as an application we prove sublinear tracking in teichmuller distance for random walks on mapping class groups and on cayley graphs of a large class of finitely generated groups

rational and transcendental growth series for the higher heisenberg groups this paper considers growth series of 2 step nilpotent groups with infinite cyclic derived subgroup every such group g has a subgroup of finite index of the form h n m where h n is the discrete heisenberg group of length 2n 1 we call n the heisenberg rank of g we show that every group of this type has some finite generating set such that the corresponding growth series is rational on the other hand we prove that if g has heisenberg rank n 2 then g possesses a finite generating set such that the corresponding growth series is a transcendental power series

rank 2 arithmetically cohen macaulay bundles on a nonsingular cubic surface abstract rank 2 indecomposable arithmetically cohen macaulay bundles e on a nonsingular cubic surface x in p 3 are classified by means of the possible forms taken by the minimal graded free resolution of e over p 3 the admissible values of the chern classes of e are listed and the vanishing locus of a general section of e is studied properties of e such as slope semi stability and simplicity are investigated the number of relevant families is computed together with their dimension

on the spectrum of the poincare variational problem for two close to touching inclusions in 2d we study the spectrum of the poincare variational problem for two close to touching inclusions in r2 we derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero

extensions and submonoids of automatic monoids in this paper submonoids and extensions of automatic and p automatic monoids are studied the concept of a p automatic monoid is a variant on the usual concept of an automatic monoid designed to allow a geometric characterization analogous to the group case in the case of right cancellative monoids the two concepts coincide here we study rational submonoids of p automatic monoids being able to show in many cases that p automaticity is inherited our sharpest results concern rational subgroups also closure properties are established for various notions of extensions of p automatic monoids including different types of products ideal extensions and rees matrix constructions

a strong geometric hyperbolicity property for directed graphs and monoids abstract we introduce and study a strong thin triangle condition for directed graphs which generalises the usual notion of hyperbolicity for a metric space we prove that finitely generated left cancellative monoids whose right cayley graphs satisfy this condition must be finitely presented with polynomial dehn functions and hence word problems in np under the additional assumption of right cancellativity or in some cases the weaker condition of bounded indegree they also admit algorithms for more fundamentally semigroup theoretic decision problems such as green s relations l r j d and the corresponding pre orders in contrast we exhibit a right cancellative but not left cancellative finitely generated monoid in fact an infinite class of them whose cayley graph is essentially a tree hence hyperbolic in our sense and probably any reasonable sense but which is not even recursively presentable this seems to be strong evidence that no geometric notion of hyperbolicity will be strong enough to yield much information about finitely generated monoids in absolute generality

commutativity in free inverse monoids abstract this article characterizes the solutions of the commutativity equation xy yx in free inverse monoids the main result implies the following interesting property that is the natural generalization to free inverse monoids of the solutions of the same equation in free monoids let x and y be non idempotent elements of a free inverse monoid such that xy yx then there exist some elements and z such that x and y are conjugate by to some positive powers of z namely x zn and y zm with n m 1 we also show that the centralizer of a given non idempotent element is a rational non recognizable subset of the free inverse monoid

the geometry of profinite graphs with applications to free groups and finite monoids we initiate the study of the class of profinite graphs defined by the following geometric property for any two vertices v and w of there is a unique smallest connected profinite subgraph of containing them such graphs are called tree like profinite trees in the sense of gildenhuys and ribes are tree like but the converse is not true a profinite group is then said to be dendral if it has a tree like cayley graph with respect to some generating set a bass serre type characterization of dendral groups is provided also such groups including free profinite groups are shown to enjoy a certain small cancellation condition we define a pseudovariety of groups h to be arboreous if all finitely generated free pro h groups are dendral with respect to a free generating set our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids we prove for arboreous pseudovarieties h a pro h analog of the ribes and zalesskii product theorem for the profinite topology on a free group also arboreous pseudovarieties are characterized as precisely the solutions h to the much studied pseudovariety equation j h j h

characterization of electric fields between two spherical perfect conductors with general radii in 3d abstract in composite materials the inclusions are frequently spaced very closely the electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large in this paper we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions an explicit blowup factor relying on radii is obtained which also involves the digamma function and euler mascheroni constant and so the role of inclusions radii played in such blowup analysis is identified

strong influence of a small fiber on shear stress in fiber reinforced composites abstract in stiff fiber reinforced composites it has been known that the shear stress increases at the rate of 1 as the distance between adjacent fibers approaches 0 this paper reveals a strong influence of a combination of a triple fiber as well as the distance between a pair of fibers on the blow up so that the stress concentration can be significantly accelerated by adding a small fiber in between fibers specifically if a fiber f 2 with a small diameter is located in between fibers f 1 and f 3 1 dist f 1 f 2 and 2 dist f 2 f 3 then the stress blows up at the exact rates of 1 1 and 1 2 between f 1 and f 2 and between f 2 and f 3 respectively this estimate still holds even when a part of f 2 overlaps with f 3 the magnification factor 1 yields the enormous increase in the stress that greatly surpasses the expectancy by previous methods

gorenstein biliaison and acm sheaves abstract let x be a normal arithmetically gorenstein scheme in p n we give a criterion for all codimension two acm subschemes of x to be in the same gorenstein biliaison class on x in terms of the category of acm sheaves on x these are sheaves that correspond to the graded maximal cohen macaulay modules on the homogeneous coordinate ring of x using known results on mcm modules we are able to determine the gorenstein biliaison classes of codimension two subschemes of certain varieties including the nonsingular quadric surface in p 3 and the cone over it in p 4 as an application we obtain a new proof of some theorems of lesperance about curves in p 4 and answer some questions he raised

asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities abstract we consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities when two inclusions get closer and their conductivities degenerate to zero or infinity the gradient of the solution can be arbitrary large in this paper we derive an asymptotic formula of the solution which characterizes the gradient blow up of the solution in terms of conductivities of inclusions as well as the distance between inclusions the asymptotic formula is expressed in bipolar coordinates in terms of the lerch transcendent function and it is valid for inclusions with arbitrary constant conductivities we illustrate our results with numerical calculations

pushdown processes a pushdown game is a two player perfect information infinite game on a transition graph of a pushdown automaton a winning condition in such a game is defined in terms of states appearing infinitely often in the play it is shown that if there is a winning strategy in a pushdown game then there is a winning strategy realized by a pushdown automaton an exptime procedure for finding a winner in a pushdown game is presented the procedure is then used to solve the model checking problem for the pushdown processes and the propositional calculus the problem is shown to be dexptime complete

gradient estimates for solutions of the lame system with partially infinite coefficients we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

a constructive version of the ribes zalesski product theorem for any given finitely generated subgroups h1 hn of a free group f and any element w of f not contained in the product h1 hn a finite quotient of f is explicitly constructed which separates the element w from the set h1 hn this provides a constructive version of the product theorem stating that h1 hn is closed in the profinite topology of f the method of proof also applies to other profinite topologies it is efficient for the profinite topology as well as for the pro p topology of f the main tools used are universal p extensions and inverse automata

inverse monoids trees and context free languages this paper is concerned with a study of inverse monoids presented by a set x subject to relations of the form e i f i i i where e i and f i are dyck words i e idempotents of the free inverse monoid on x some general results of stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the cayley graph of the free group on x in the finitely presented case the word problem is solved by using rabin s theorem on the second order monadic logic of the infinite binary tree some connections with the theory of rational subsets of the free group and the theory of context free languages are explored

the word problem for inverse monoids presented by one idempotent relator abstract we study inverse monoids presented by a finite set of generators and one relation e 1 where e is a word representing an idempotent in the free inverse monoid and 1 is the empty word we show that 1 the word problem is solvable by a polynomial time algorithm 2 every congruence class in the free monoid with respect to such a presentation is a deterministic context free language such congruence classes can be viewed as generalizations of parenthesis languages and 3 the word problem is solvable by a linear time algorithm in the more special case where e is a positively labeled idempotent

gradient estimates for the perfect conductivity problem we establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect stiff conductors are closely spaced inside an open bounded domain and away from the boundary these results give the optimal blow up rates of the stress for conductors with arbitrary shape and in all dimensions

vector spaces of skew symmetric matrices of constant rank abstract we study the orbits of vector spaces of skew symmetric matrices of constant rank 2 r and type n 1 n 1 under the natural action of sl n 1 over an algebraically closed field of characteristic zero we give a complete description of the orbits for vector spaces of dimension 2 relating them to some 1 generic matrices of linear forms we also show that for each rank two vector bundle on p 2 defining a triple veronese embedding of p 2 in g 1 7 there exists a vector space of 8 8 skew symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle

decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions abstract when two inclusions get closer and their conductivities degenerate to zero or infinity the gradient of the solution to the conductivity equation blows up in general in this paper we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form one of them has a bounded gradient and the gradient of the other part blows up using the decomposition we derive the best possible estimates for the blow up of the gradient we then consider the case when the inclusions have positive permittivities we show quantitatively that in this case the size of the blow up is reduced

the orthogonal lie algebra of operators ideals and derivations abstract we study in this paper the infinite dimensional orthogonal lie algebra o c which consists of all bounded linear operators t on a separable infinite dimensional complex hilbert space h satisfying c t c t where c is a conjugation on h by employing results from the theory of complex symmetric operators and skew symmetric operators we determine the lie ideals of o c and their dual spaces we study derivations of o c and determine their spectra these results complete some results of p de la harpe and provide interesting contrasts between o c and the algebra b h of all bounded linear operators on h

solvable groups with polynomial dehn functions given a finitely presented group h finitely generated subgroup b of h and a monomorphism b h we obtain an upper bound of the dehn function of the corresponding hnn extension g hh t j t i1 bt b i in terms of the dehn function of h and the distortion of b in g using such a bound we construct first examples of non polycyclic solvable groups with polynomial dehn functions the constructed groups are metabelian and contain the solvable baumslag solitar groups in particular this answers a question posed by birget ol shanskii rips and sapir

gradient estimates for solutions of the lame system with partially infinite coefficients in dimensions greater than two abstract we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

isoperimetric functions of groups and computational complexity of the word problem we prove that the word problem of a finitely generated group g is in np solvable in polynomial time by a nondeterministic turing machine if and only if this group is a subgroup of a finitely presented group h with polynomial isoperimetric function the embedding can be chosen in such a way that g has bounded distortion in h this completes the work started in 6 and 25

gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients in this paper we derive global w1 and piecewise c1 estimates for solutions to divergence form elliptic equations with piecewise holder continuous coefficients the novelty of these estimates is that even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients they are independent of the distance between these surfaces

spectral analysis of the neumann poincare operator and characterization of the stress concentration in anti plane elasticity when holes or hard elastic inclusions are closely located stress which is the gradient of the solution to the anti plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero it is important to precisely characterize the blow up of the gradient of such an equation in this paper we show that the blow up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1 2 of a neumann poincare type operator defined on the boundaries of the inclusions by comparing the singular function with the one corresponding to two disks osculating to the inclusions we quantitatively characterize the blow up of the gradient in terms of explicit functions in electrostatics our results apply to the electric field which is the gradient of the solution to the conductivity equation in the case where perfectly conducting or insulating inclusions are closely located

optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross sections abstract we consider high stresses in stiff fiber reinforced materials which increase rapidly as fibers approximate to one another this paper presents the optimal blow up rate of the stresses with respect to the distance between a pair of stiff fibers in r 3 the blow up result plays an important role in our understanding of low strengths of fiber reinforced composites referring to a problem of anti plane shear the stresses can be interpreted as the electric fields outside closely spaced perfect conductors in r 2 under the action of applied electric field h it has been shown by ammari kang et al that in the particular case of circular inclusions the electric field blows up at the optimal rate 1 2 as 0 where is the distance between conductors recently yun has extended the blow up result to pairs of conductors associated with a large class of shapes whose complements can be transformed conformally to the outside of a circle with c 2 mapping however it presented a suboptimal result that only for a special uniform field h 1 0 the electric fields blow up at the exact rate 1 2 in this paper an upper bound with the rate 1 2 of electric field for any harmonic function h is established this yields the optimal blow up rate 1 2 for the inclusions in the same class of shapes as yun

inverse monoids decidability and complexity of algebraic questions this paper investigates the word problem for inverse monoids generated by a set c subject to relations of the form e f where e and f are both idempotents in the free inverse monoid generated by c it is shown that for every fixed monoid of this form the word problem can be solved both in linear time on a ram as well as in deterministic logarithmic space which solves an open problem of margolis and meakin for the uniform word problem where the presentation is part of the input exptime completeness is shown for the cayley graphs of these monoids it is shown that the first order theory with regular path predicates is decidable regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language as a corollary the decidability of the generalized word problem is deduced

small cancellation labellings of some infinite graphs and applications we construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree we use them to define infinite graphical small cancellation presentations of groups this technique allows us to provide examples of groups with exotic properties we construct the first examples of finitely generated coarsely non amenable groups that is groups without guoliang yu s property a that are coarsely embeddable into a hilbert space moreover our groups act properly on cat 0 cubical complexes we construct the first examples of finitely generated groups with expanders embedded isometrically into their cayley graphs in contrast in the case of the gromov monster expanders are not even coarsely embedded we present further applications

electric field concentration in the presence of an inclusion with eccentric core shell geometry in this paper we analyze the gradient blow up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core shell geometry assuming that the core and shell have circular boundaries that are nearly touching we derive an asymptotic formula for the solution in terms of the single and double layer potentials with image line charges we also deduce an integral formula with image line charges for the problem relating to two nearly touching separated conductors

on degeneracy schemes of maps of vector bundles and applications to holomorphic foliations in this paper we provide sufficient conditions for maps of vector bundles on smooth projective varieties to be uniquely determined by their degeneracy schemes we then specialize to holomorphic distributions and foliations in particular we provide sufficient conditions for foliations of arbitrary rank on mathbb p n to be uniquely determined by their singular schemes

quantitative characterization of stress concentration in the presence of closely spaced hard inclusions in two dimensional linear elasticity in the region between close to touching hard inclusions stress may be arbitrarily large as the inclusions get closer this stress is represented by the gradient of a solution to the lame system of linear elasticity we consider the problem of characterizing the gradient blow up of the solution in the narrow region between two inclusions and estimating its magnitude we introduce singular functions which are constructed in terms of nuclei of strain and hence are solutions of the lame system and then show that the singular behavior of the gradient in the narrow region can be precisely captured by singular functions as a consequence of the characterization we are able to regain the existing upper bound on the blow up rate of the gradient namely 1 2 where is the distance between two inclusions we then show that it is in fact an optimal bound by showing that there are cases where 1 2 is also a lower bound this work is the first to completely reveal the singular nature of the gradient blow up and to obtain the optimal blow up rate in the context of the lame system with hard inclusions the singular functions introduced in this paper play essential roles in overcoming the difficulties in applying the methods of previous works the main tools of this paper are the layer potential techniques and the variational principle the variational principle can be applied because the singular functions of this paper are solutions of the lame system

quasi actions on trees i bounded valence given a bounded valence bushy tree t we prove that any cobounded quasi action of a group g on t is quasiconjugate to an action of g on another bounded valence bushy tree t this theorem has many applications quasi isometric rigidity for fundamental groups of finite bushy graphs of coarse pd n groups for each fixed n a generalization to actions on cantor sets of sullivan s theorem about uniformly quasiconformal actions on the 2 sphere and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice finally we give the first examples of two finitely generated groups which are quasi isometric and yet which cannot act on the same proper geodesic metric space properly discontinuously and cocompactly by isometries

constructing divisions into power groups the result due to henckell margolis pin and rhodes modulo ash s solution to the pointlike conjecture that every finite block group divides a power group has long been considered to be one of the deepest results in finite semigroup and algebraic automata theory however the proof is not constructive solving a long standing problem we provide in this paper an explicit construction of such a division we also generalize the result to a large class of pseudovariefies of groups local group pseudovarieties are also considered generalizing and making constructive results of margolis and the second author some applications to language theory are mentioned

growth and ergodicity of context free languages a language l over a finite alphabet e is called growth sensitive if forbidding any set of subwords f yields a sublanguage l f whose exponential growth rate is smaller than that of l it is shown that every ergodic unambiguous nonlinear context free language is growth sensitive ergodic means for a context free grammar and language that its dependency di graph is strongly connected the same result as above holds for the larger class of essentially ergodic context free languages and if growth is considered with respect to the ambiguity degrees then the assumption of unambiguity may be dropped the methods combine a construction of grammars for 2 block languages with a generating function technique regarding systems of algebraic equations

the theory of ends pushdown automata and second order logic abstract a class of edge labeled graphs called context free are defined according to their behavior at infinity such graphs are generalizations of cayley graphs of context free groups they are also shown to be definable in a very natural way in terms of push down automata using rabin s theorem on the monadic second order theory of the finite binary tree these graphs are also shown to have a decidable monadic second order theory questions about tiling systems and cellular automata operating on these graphs are decidable even when the analogous questions are not when the systems operate on a two dimensional grid

isoperimetric and isodiametric functions of groups this is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity in particular we show how to construct a finitely presented group with np complete word problem one of the main results of this paper states that if a real number cx 4 is computable in time 0 then na is equivalent big o to the dehn function of a finitely presented group the smallest isodiametric function of this group is n3 8 4 on the other hand if na is equivalent to the dehn function of a finitely presented group then oe is computable in time 4 are equivalent to dehn functions of some finitely presented groups and that n1r and na for all rational numbers oe 3 are equivalent to the smallest isodiametric functions of finitely presented groups moreover we describe all dehn functions of finitely presented groups f n4 as time functions of turing machines modulo two conjectures

the word problem of the brin thompson group is conp complete abstract we prove that the word problem of the brin thompson group nv over a finite generating set is conp complete for every n 2 it is known that n v n 1 is an infinite family of infinite finitely presented simple groups we also prove that the word problem of the thompson group v over a certain infinite set of generators related to boolean circuits is conp complete

pfaffian bundles on cubic surfaces and configurations of planes we construct a canonical birational map between the moduli space of pfaffian vector bundles on a cubic surface and the space of complete pentahedra inscribed in the cubic surface the universal situation is also considered and we obtain a rationality result as a by product we provide an explicit normal form for five general lines in mathbb p 5 applications to the geometry of palatini threefolds and debarre voisin hyper kahler manifolds are also discussed

algorithmic theory of free solvable groups randomized computations abstract we design new deterministic and randomized algorithms for computational problems in free solvable groups in particular we prove that the word problem and the power problem can be solved in quasi linear time and the conjugacy problem can be solved in quasi quartic time by monte carlo type algorithms

optimal estimates for the conductivity problem by green s function method we study a class of second order elliptic equations of divergence form with discontinuous coefficients and data which models the conductivity problem in composite materials we establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions these extend the known results in the literature and answer open problem b proposed by li and vogelius 2000 for the isotropic conductivity problem we also obtain more interesting higher order derivative estimates which answers open problem c of li and vogelius 2000 it is worth pointing out that the equations under consideration in this paper are non homogeneous

random walks on free solvable groups for any finitely generated group g let n mapsto phi g n be the function that describes the rough asymptotic behavior of the probability of return to the identity element at time 2n of a symmetric simple random walk on g this is an invariant of quasi isometry we determine this function when g is the free solvable group mathbf s d r of derived length d on r generators and some related groups

on the definition of word hyperbolic groups formal languages based on multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups

skew symmetric matrices and palatini scrolls we prove that for m greater than 3 and k greater than m 2 the grassmannian of m dimensional subspaces of the space of skew symmetric forms over a vector space of dimension 2k is birational to the hilbert scheme of palatini scrolls in p2k 1 for m 3 and k 3 this grassmannian is proved to be birational to the set of pairs mathcal e y where y is a smooth plane curve of degree k and mathcal e is a stable rank 2 bundle on y whose determinant is fancyscript o y k 1

proper actions of wreath products and generalizations we study stability properties of the haagerup property and of coarse embeddability in a hilbert space under certain semidirect products in particular we prove that they are stable under taking standard wreath products our construction also allows for a characterization of subsets with relative property t in a standard wreath product

optimal boundary gradient estimates for lame systems with partially infinite coefficients abstract in this paper we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the lame systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary when the distance tends to zero the optimal blow up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions

groups finitely presented in burnside varieties abstract for all sufficiently large odd integers n the following version of higman s embedding theorem is proved in the variety b n of all groups satisfying the identity x n 1 a finitely generated group g from b n has a presentation g a r with a finite set of generators a and a recursively enumerable set r of defining relations if and only if it is a subgroup of a group h finitely presented in the variety b n it follows that there is a universal 2 generated finitely presented in b n group containing isomorphic copies of all finitely presented in b n groups as subgroups

c graph automatic groups abstract we generalize the notion of a graph automatic group introduced by kharlampovich khoussainov and miasnikov by replacing the regular languages in their definition with more powerful language classes for a fixed language class c we call the resulting groups c graph automatic we prove that the class of c graph automatic groups is closed under change of generating set direct and free product for certain classes c we show that for quasi realtime counter graph automatic groups where normal forms have length that is linear in the geodesic length there is an algorithm to compute normal forms and therefore solve the word problem in polynomial time the class of quasi realtime counter graph automatic groups includes all baumslag solitar groups and the free group of countably infinite rank context sensitive graph automatic groups are shown to be a very large class which encompasses for example groups with unsolvable conjugacy problem the grigorchuk group and thompson s groups f t and v