homology and closure properties of autostackable groups abstract autostackability for finitely presented groups is a topological property of the cayley graph combined with formal language theoretic restrictions that implies solvability of the word problem the class of autostackable groups is known to include all asynchronously automatic groups with respect to a prefix closed normal form set and all groups admitting finite complete rewriting systems although groups in the latter two classes all satisfy the homological finiteness condition f p we show that the class of autostackable groups includes a group that is not of type f p 3 we also show that the class of autostackable groups is closed under graph products and extensions

on the knuth bendix completion for concurrent processes abstract we consider critical pairs for replacement systems over free partially commutative monoids this is done in order to apply the knuth bendix completion procedure to concurrent processes we will see that there are one rule systems which have no finite set of critical pairs therefore we develop a sufficient and computable condition such that finite trace replacement systems have a finite set of critical pairs this condition is always satisfied in the purely free case of semi thue systems or purely commutative case of vector replacement systems in fact it generalizes and unifies both cases we will give examples of how one can use this generalization

automorphisms of graph groups abstract given a graph v e the graph group on is the group generated by the vertex set v with the defining relations being that two adjacent vertices commute in the following the centralizer problem for graph groups is solved and it is shown that for some graph groups aut f is finitely generated by generators which are natural analogues to the nielsen automorphisms for a free group

the geometry of purely loxodromic subgroups of right angled artin groups we prove that finitely generated purely loxodromic subgroups of a right angled artin group a fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups mod s in particular such subgroups are quasiconvex in a in addition we identify a milder condition for a finitely generated subgroup of a that guarantees it is free undistorted and retains finite generation when intersected with a for subgraphs of these results have applications to both the study of convex cocompactness in mod s and the way in which certain groups can embed in right angled artin groups

nielsen realisation for untwisted automorphisms of right angled artin groups we prove nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of raags in the following sense given any graph gamma and any finite group g u0 a gamma out 0 a gamma we find a non positively curved cube complex with fundamental group a gamma on which g acts by isometries realising the action on a gamma

jacobi matrices and transversality the paper deals with smooth nonlinear ode systems in n f x such that the derivative f x has a matrix representation of jacobi type not necessarily symmetric with positive off diagonal entries a discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system f x t y two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system it is also proved that the nonwandering points are critical points a new class of morse smale systems in n is then explicitly constructed

algorithms detecting stability and morseness for finitely generated groups abstract the notions of stable and morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word hyperbolic group for a word hyperbolic group g kapovich 31 provided a partial algorithm which on input a finite set s of g halts if s generates a quasiconvex subgroup of g and runs forever otherwise in this paper we give various detection and decidability algorithms for stability and morseness of a finitely generated subgroup of mapping class groups right angled artin groups toral relatively hyperbolic groups and finitely generated groups discriminated by a locally quasiconvex torsion free hyperbolic group for example ordinary limit groups

acylindrical hyperbolicity of groups acting on trees we provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees in particular we consider amalgamated products and hnn extensions one relator groups automorphism groups of polynomial algebras 3 manifold groups and graph products acylindrical hyperbolicity is then used to obtain some results about the algebraic structure analytic properties and measure equivalence rigidity of groups from these classes

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

sturm attractors for quasilinear parabolic equations abstract the goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations as it was done for the semilinear case by brunovsk and fiedler 1986 8 and generalized by fiedler and rocha 1996 13 in particular we construct heteroclinic connections between hyperbolic equilibria stating necessary and sufficient conditions for heteroclinics to occur such conditions can be computed through a permutation of the equilibria lastly an example is computed yielding the well known chafee infante attractor

self similar blow up in higher order semilinear parabolic equations we study the cauchy problem in re times re for one dimensional 2mth order m 1 semilinear parabolic pdes of the form d x partial partial x u t dx u u p 1 u mbox where p 1 quad mbox and quad u t dx u e u with bounded initial data u0 x specifically we are interested in those solutions that blow up at the origin in a finite time t we show that in contrast to the solutions of the classical second order parabolic equations ut uxx up and ut uxx eu from combustion theory the blow up in their higher order counterparts is asymptotically self similar in particular there exist exact nontrivial self similar blow up solutions u x t t t 1 p 1 f y in the case of the polynomial nonlinearity and u x t ln t t f y for the exponential nonlinearity where y x t t 1 2m is the backward higher order heat kernel variable the profiles f y satisfy related semilinear odes that share the same non self adjoint higher order linear dif

algorithms and topology of cayley graphs for groups autostackability for finitely generated groups is defined via a topological property of the associated cayley graph which can be encoded in a finite state automaton autostackable groups have solvable word problem and an effective inductive procedure for constructing van kampen diagrams with respect to a canonical finite presentation a comparison with automatic groups is given another characterization of autostackability is given in terms of prefix rewriting systems every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix closed set of normal forms is also autostackable as a consequence the fundamental group of every closed 3 manifold with any of the eight possible uniform geometries is autostackable

a permutation characterization of sturm global attractors of hamiltonian type abstract we consider neumann boundary value problems of the form u t u x x f on the interval 0 x for dissipative nonlinearities f f u a permutation characterization for the global attractors of the semiflows generated by these equations is well known even in the much more general case f f x u u x we present a permutation characterization for the global attractors in the restrictive class of nonlinearities f f u in this class the stationary solutions of the parabolic equation satisfy the second order ode v f v 0 and we obtain the permutation characterization from a characterization of the set of 2 periodic orbits of this planar hamiltonian system our results are based on a diligent discussion of this mere pendulum equation

convex cocompactness in mapping class groups via quasiconvexity in right angled artin groups we characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right angled artin groups that is if the right angled artin group g mod s satisfies certain conditions that imply g is quasi isometrically embedded in mod s then a purely pseudo anosov h g is convex cocompact in mod s if and only if it is combinatorially quasiconvex in g we use this criterion to construct convex cocompact subgroups of mod s whose orbit maps into the curve complex have small lipschitz constants

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

separating quasiconvex subgroups of right angled artin groups a graph group or right angled artin group is a group given by a presentation where the only relators are commutators of the generators a graph group presentation corresponds in a natural way to a simplicial graph with each generator corresponding to a vertex and each commutator relator corresponding to an edge suppose that g is a graph group whose corresponding graph is a tree and h is a subgroup of g we show that if h is quasiconvex with respect to either the word metric on g or the cat 0 metric on the universal cover of the standard complex for g then h is separable that is h is the intersection of finite index subgroups of g we also discuss some consequences relating to certain 3 manifold groups

centralizers in graph products of groups abstract a graph product of groups is a construction which generalizes both free products and direct products of groups in this paper we study commuting elements in graph products of groups and find explicit formulas for the centralizer the subgroup consisting of elements which commute with a given element of elements of a graph product of groups in terms of the centralizers of elements in each vertex group and the structure of the graph

realization of meander permutations by boundary value problems abstract we consider neumann boundary value problems of the form uxx f x u ux 0 on the unit interval 0 x 1 for a certain class of dissipative nonlinearities f associated to these problems we have i meanders in the phase space u ux r 2 which are connected oriented simple curves on the plane intersecting a fixed oriented line the u axis in n points corresponding to the solutions and ii meander permutations f s n obtained by ordering the intersection points first along the u axis and then along the meander the meander permutation f is the permutation defined by the braid of solutions in the space x u ux it was recently shown by fiedler and rocha that f determines the global attractor of the dynamical system generated by the semilinear parabolic differential equation ut uxx f x u ux up to c0 orbit equivalence therefore these permutations are of considerable importance in the classification problem of the morse smale attractors for these dynamical systems in this paper we present a purely combinatorial characterization of the set of meander permutations that are realizable by the above boundary value problems

artin groups of large type are shortlex automatic with regular geodesics we prove that any artin group of large type is shortlex automatic with respect to its standard generating set and that the set of all geodesic words over the same generating set satisfies the falsification by fellow traveller property fftp and hence is regular

on a spectrum of blow up patterns for a higher order semilinear parabolic equation we describe a spectrum of blow up patterns for the 2 m th order semilinear parabolic equation u t m u u p x r n t 0 m 1 p 1 this problem is well understood in the second order case m 1 the semilinear heat equation where for p p s n 2 n 2 the inner space time structure of blow up patterns is locally governed by separable hermite polynomials of arbitrary finite order as eigenfunctions of a linear self adjoint differential operator we consider m 1 and describe a principle idea of constructing of a spectrum of blow up patterns which in the inner self similar region are composed of a countable subset of separable kummer9s polynomials which are eigenfunctions of a non self adjoint linear differential operator a matching procedure extends such linearized structures to the nonlinear intermediate region where the asymptotic blow up behaviour is described by a unique self similar solution of the first order hamilton jacobi equation

explicit construction of an inertial manifold for a reaction diffusion equation abstract an inertial manifold is constructed for the scalar reaction diffusion equation u t vu xx u with a cubic nonlinearity uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria this uniformity leads to the global parameterization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium by the introduction of local techniques near each equilibrium we succeed in constructing an inertial manifold of lowest possible dimension

on the two dimensional muskat problem with monotone large initial data we consider the evolution of two incompressible immiscible fluids with different densities in porous media known as the muskat problem 21 which in two dimensions is analogous to the hele shaw cell 24 we establish for a class of large and monotone initial data the global existence of weak solutions the proof is based on a local well posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function 2016 wiley periodicals inc

on the huge lie superalgebra of pseudo superdifferential operators and super kp hierarchies lie superalgebraic methods are used to establish a connection between the huge lie superalgebra of super pseudo differential operators and various super kp hierarchies we show in particular that splits into 5 2 2 1 graded algebras expected to correspond to five classes of super kp hierarchies generalizing the well known manin radul and figueroa mas ramos supersymmetric kp hierarchies

transversality of stable and unstable manifolds for parabolic problems arising in composite materials abstract in this paper we study one dimensional parabolic problems that arise from composite materials we show that the eigenvalues and eigenfunctions of the associated linear unbounded operator have the sturm liouville property and the nonincrease of the lap number along the solutions these results are used to show that the stable and unstable manifolds of equilibrium points are transversal

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

conjugacy growth series and languages in groups in this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group we study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series as well as on regularity of the geodesic conjugacy language and spherical conjugacy language in particular we show that regularity of the geodesic conjugacy language is preserved by the graph product construction and rationality of the geodesic conjugacy growth series is preserved by both direct and free products 2010 mathematics subject classification 20f65 20e45

algorithms and geometry for graph products of groups recent work of gromov epstein cannon thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups see 16 17 and 14 many of these structures are preserved by taking graph products the graph product of groups not to be confused with the fundamental group of a graph of groups is a product mixing direct and free products whether the product between two groups in the graph product is free or direct is determined by a simplicial graph given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

complicated dynamics in scalar semilinear parabolic equations in higher space dimension abstract we study the dynamics of the boundary value problem ut lu g x u u x 1 u 0 2 where l is a second order uniformly elliptic operator and r n is diffeomorphic to the ball in r n n 2 the main result asserts that given any ck vector field v on r n 1 with v 0 0 one can adjust coefficients of l and the function g such that the corresponding problem 1 2 has an n 1 dimensional invariant manifold through the equilibrium u 0 and the taylor expansion at u 0 of the vector field representing the flow on this manifold coincides in appropriate coordinates with the taylor expansion of v up to k th order terms this result implies that a hyperbolic invariant n torus can be found in 1 2 if l and g are appropriately chosen this result also indicates that chaotic dynamics is likely to occur for some choices of l and g

solutions to the nonautonomous bistable equation with specified morse index part i existence we investigate the existence of unstable solutions of specified morse index for the equation e 2 u xx f x u 0 on a finite interval and neumann boundary conditions

generic hyperbolicity for the equilibria of the one dimensional parabolic equation ut a x ux x f u abstract we show for some classes of diffusion coefficients that generically in f all equilibria of the reaction diffusion equation u t a x u x x f u 0 with homogeneous neumann boundary conditions are hyperbolic

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

stable patterns for fourth order parabolic equations we consider fourth order parabolic equations of gradient type for the sake of simplicity the analysis is carried out for the specific equation u

subgroups of graph groups given a graph x we define a group gx as follows gx is generated by the vertices of x with a defining relation xy jt for each pair y 2 of ver tices joined by an edge of x a group is called a graph group if it is isomorphic to gx for some graph x we will not distinguish by notation in what follows between a vertex of x and the corresponding element of gx we remark that the extreme cases of graphs namely the complete and the completely disconnected graphs correspond respectively to free abelian and free groups it is well known that any subgroup of a group of either of these types is again of the same type and it is thus natural to ask to what extent and in what form this is true of the graph groups that lie in between these extreme cases in this article we will prove the following

on the geometry of normal forms in discrete groups we study normal forms in finitely generated groups from the geometric viewpoint of combings we introduce notions of combability considerably weaker than those commonly in use we prove that groups which satisfy these conditions are finitely presented and satisfy isoperimetric and isodiametric inequalities of a controlled nature

metric systolicity and two dimensional artin groups we introduce the notion of metrically systolic simplicial complexes we study geometric and large scale properties of such complexes and of groups acting on them geometrically we show that all two dimensional artin groups act geometrically on metrically systolic complexes as direct corollaries we obtain new results on two dimensional artin groups and all their finitely presented subgroups we prove that the conjugacy problem is solvable and that the dehn function is quadratic we also show several large scale features of finitely presented subgroups of two dimensional artin groups lying background for further studies concerning their quasi isometric rigidity

graph groups coherence and three manifolds 1 introou t n we study groups given by presentations each of whose defining relations is of the form ql yx for some generators y and y to such a presentation we associate a graph x whose vertices are the generators two vertices y and 1 being adjacent in x if and only if xy yx is a defining relation given a graph x we denote by gx the group defined by the presentation associated to x in this way we call gx a graph group these groups have been studied by kim and roush s and by dicks 3 in this paper we prove the following

tits alternatives for graph products we discuss various types of tits alternative for subgroups of graph products of groups and prove that under some natural conditions a graph product of groups satisfies a given form of tits alternative if and only if each vertex group satisfies this alternative as a corollary we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large as another corollary we prove that every non abelian subgroup of a right angled artin group has an epimorphism onto the free group of rank 2 in the course of the paper we develop the theory of parabolic subgroups which allows to describe the structure of subgroups of graph products that contain no non abelian free subgroups we also obtain a number of results regarding the stability of some group properties under taking graph products

reaction diffusion systems coupled at the boundary and the morse smale property abstract we study an one dimensional nonlinear reaction diffusion system coupled on the boundary such system comes from modeling problems of temperature distribution on two bars of same length jointed together with different diffusion coefficients we prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic to this end we write the system as an equation with noncontinuous diffusion coefficient we then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the sturm liouville properties of the solutions of a linear elliptic problem

generalising some results about right angled artin groups to graph products of groups we prove three results about the graph product g g gv v v of groups gv over a graph the first result generalises a result of servatius droms and servatius proved by them for right angled artin groups we prove a necessary and sufficient condition on a finite graph for the kernel of the map from g to the associated direct product to be free one part of this result already follows from a result in s kim s phd thesis the second result generalises a result of hermiller and sunic again from right angled artin groups we prove that for a graph with finite chromatic number g has a series in which every factor is a free product of vertex groups the third result provides an alternative proof of a theorem due to meier which provides necessary and sufficient conditions on a finite graph for g to be hyperbolic