machine calculations in weyl groups a geometrical description of the minimal automaton recognizing lexicographically minimal reduced expressions for elements of a weyl group is given this complements the construction of a related automaton in a recent preprint of brink and howlett by going on to present a practical way to construct it a transducer is described that will output in reverse the word corresponding to right multiplication by a generato r on such a word in a right to left scan the question is raised whether or not this transducer is finite

rigidity of coxeter groups let w be a coxeter group acting properly discontinuously and cocompactly on manifolds n and m 9m 0 such that the fixed point sets of finite subgroups are contractible let f n n mx dk m xsk1 be a w homotopy equivalence which restricts to a w homeomorphism on the boundary under an assumption on the three dimensional fixed point sets we show that then f is w homotopic to a w homeomorphism

coxeter groups and hyperbolic manifolds the theory of coxeter groups is used to provide an algebraic construction of finite volume hyperbolic manifolds combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds three examples in 4 5 and 6 dimensions are given each of very small volume and in one case of smallest possible volume

gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type abstract we study the gevrey solvability of a class of complex vector fields defined on s 1 given by l t a x i b x x b 0 near the characteristic set 0 s 1 we show that the interplay between the order of vanishing of the functions a and b at x 0 plays a role in the gevrey solvability

automata reduced words and garside shadows in coxeter groups abstract in this article we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated coxeter system w s the definition of these automata is straightforward as it only requires the notion of weak order on w s and the related notion of garside shadows in w s an analog of the notion of a garside family then we discuss the relations between this class of automata and the canonical automaton built from brink and howlett s small roots we end this article by providing partial positive answers to two conjectures 1 the automata associated to the smallest garside shadow is minimal 2 the canonical automaton is minimal if and only if the support of all small roots is spherical i e the corresponding root system is finite

a global theory of internal solitary waves in two fluid systems abstract the study of single crested progressing gravity waves was initiated over a century ago with the observations by russell of what he termed solitary waves which progressed without change of form over a considerable distance on the glasgow edinburgh canal the mathematical analysis of this wave motion on the surface of water begun in the nineteenth century has undergone a rapid development in the last three decades due to the scattering theory for the korteweg de vries equation which models the motion of long waves due to the development of techniques in nonlinear analysis allowing for the analysis of finite amplitude motions the work on surfce waves has many parallels in the study of waves in fluids with variable density in the case of a heterogeneous fluid with a free upper surface gravity waves still occur in analogy with surface waves in a fluid of constant density what is distinctive about a fluid with density stratification however is the presence of waves which are predominantly due to the stratification and not to the free surface these waves called internal waves exist in a heterogeneous fluid even when it is confined between horizontal boundaries a configuration which precludes gravity waves in a fluid of constant density this paper is concerned with progressing solitary gravity waves in a system consisting of two fluids of differing densities confined in a channel of unit depth and infinite horizontal extent

automorphisms of coxeter groups of rank 3 with infinite bonds abstract if w is a rank 3 coxeter group whose coxeter diagram contains at least one infinite bond then the automorphism group of w is larger than the group generated by the inner automorphisms and the automorphisms induced from automorphisms of the coxeter diagram in each case the automorphism group of w is described

small internal waves in two fluid systems this paper treats travelling waves in a heterogeneous inviscid non diffusive fluid bounded between two horizontal boundaries the fluid has two incompressible components of different but constant density and is acted on by gravity the flow is steady when viewed in a moving reference frame and gives rise to a quasilinear elliptic problem with an eigenvalue parameter related to the wave speed the small amplitude solutions are analyzed using a dynamical systems approach a center manifold reduction in combination with a conserved quantity for the flow is used to parametrise all small solutions of the full elliptic system in terms of solutions of an autonomous first order ordinary differential equation for a principal component of the wave amplitude the result is a characterization of all small waves irrotational in each fluid near the critical speed for the system they are solitary waves surges connecting distinct conjugate flows at extreme ends of the channel conjugate flows and periodic waves

solvability for perturbations of a class of real vector fields on the two torus abstract let l t a x x be a real vector field defined on the two dimensional torus t 2 where a is a real valued and smooth function on t 1 we deal with the global solvability of equations in the form l u p u f where p f c t 2 solvability to the equation l u f is well understood we show that a perturbation of zero order may affect the global solvability of l we may maintain gain or lose solvability by adding a perturbation this phenomenon is linked to the order of vanishing of the coefficient a of l we obtained results in the class of smooth functions on t 2 and also in the space of schwartz distributions d t 2

automorphisms of coxeter groups we compute aut w for any even coxeter group whose coxeter diagram is connected contains no edges labeled 2 and cannot be separated into more than 2 connected components by removing a single vertex the description is given explicitly in terms of the given presentation for the coxeter group and admits an easy characterization of those groups w for which out w is finite

some new biautomatic coxeter groups abstract a coxeter system is called two dimensional if every parabolic subgroup of rank at least 3 is infinite we apply results of niblo and reeves and of b t williams to show that coxeter groups with two dimensional systems are biautomatic if the corresponding presentation contains no visible euclidean triangle groups this provides a partial positive answer to a question posed by niblo and reeves in g niblo l reeves coxeter groups act on cat 0 cube complexes j group theory 6 3 2003 399 413

rationality and reciprocity for the greedy normal form of a coxeter group we show that the characteristic series for the greedy normal form of a coxeter group is always a rational series and prove a reciprocity formula for this series when the group is right angled and the nerve is eulerian as corollaries we obtain many of the known rationality and reciprocity results for the growth series of coxeter groups as well as some new ones

cells in coxeter groups i abstract the purpose of this article is to shed new light on the combinatorial structure of kazhdan lusztig cells in infinite coxeter groups w our main focus is the set d of distinguished involutions in w which was introduced by lusztig in one of his first papers on cells in affine weyl groups we conjecture that the set d has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite coxeter groups moreover to each element of d we assign an explicitly defined set of equivalence relations on w that altogether conjecturally determine the partition of w into left right cells we are able to prove these conjectures only in a special case but even from these partial results we can deduce some interesting corollaries

on the existence of small amplitude solitary waves with strong surface tension an existence theory for small amplitude solitary waves with surface tension effects included is developed for large values of the surface tension parameter 13 using ideas of beale the nash moser implicit function theorem is applied to justify the well known approximation of korteweg and devries some of the recent results of amick and kirchgassner are thereby achieved more directly and additional insight obtained for the open case 13

asymptotic entropy of random walks on fuchsian buildings and kac moody groups in this article we prove existence of the asymptotic entropy for isotropic random walks on regular fuchsian buildings moreover we give formulae for the asymptotic entropy and prove that it is equal to the rate of escape of the random walk with respect to the green distance when the building arises from a fuchsian kac moody group our results imply results for random walks induced by bi invariant measures on these groups however our results are proven in the general setting without the assumption of any group acting on the building the main idea is to consider the retraction of the isotropic random walk onto an apartment of the building to prove existence of the asymptotic entropy for this retracted walk and to lift this in order to deduce the existence of the entropy for the random walk on the building

nonexistence of global solutions for a class of complex vector fields on two torus abstract the goal of this paper is study the global solvability of a class of complex vector fields of the special form l t a i b x x a b c s 1 r defined on two torus t 2 r 2 2 z 2 the kernel of transpose operator l t is described and the solvability near the characteristic set is also studied

embeddings of root systems ii permutation root systems abstract this paper studies a class of embeddings of permutation root systems with properties abstracting those of examples of embeddings of linear root systems over commutative rings as considered in the first paper in this series the root system of one coxeter group w regarded as a permutation representation is realized with its roots as sets of roots of another coxeter group w with action induced by an embedding of w in a completion of w

global s solvability and global s hypoellipticity for certain perturbations of zero order of systems of constant real vector fields abstract the main purpose of this paper is to show in the two dimensional torus a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s solvable we are also interested in studying its global s hypoellipticity we present connections between these global concepts and a priori estimates we also present two applications of our results for systems of operators with variable coefficients

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

bipolar coxeter groups we consider the class of those coxeter groups for which removing from the cayley graph any tubular neighbourhood of any wall leaves exactly two connected components we call these coxeter groups bipolar they include the virtually poincare duality coxeter groups the pseudo manifold coxeter groups and the infinite irreducible 2 spherical ones we show in a geometric way that a bipolar coxeter group admits a unique conjugacy class of coxeter generating sets moreover we provide a characterisation of bipolar coxeter groups in terms of the associated coxeter diagram

closedness of the range for vector fields on the torus abstract we consider vector fields of the form l t a x x on the 2 torus and study the closedness of the range of l in spaces of functions and distributions when a vanishes somewhere but not everywhere we show that the range is closed if and only if all zeros of a are of finite order in the most interesting case one obtains a finiteness condition on the speed of accumulation of orbits similar to what happens in the known tube case