note on basic features of large time behaviour of heat kernels large time behaviour of heat semigroups and more gener ally of positive selfadjoint semigroups is studied convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups this framework includes laplacians on manifolds metric graphs and discrete graphs

light structures in infinite planar graphs without the strong isoperimetric property it is shown that the discharging method can be successfully applied on infinite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality the general outline of the method is presented and the following applications are given planar graphs with only finitely many vertices of degree 5 and with subexponential growth contain arbitrarily large finite submaps of the tessellation of the plane or of some tessellation of the cylinder by equilateral triangles every planar graph with isoperimetric number zero and with essential minimum degree 3 has infinitely many edges whose degree sum is at most 15 in particular this holds for all graphs with minimum degree 3 and with subexponential growth the cases without infinitely many edges whose degree sum is 14 or similarly 13 or 12 are also considered several further results are obtained

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

graphical designs and extremal combinatorics abstract a graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the laplacian operator have mean value zero in this paper we show that extremal independent sets make extremal graphical designs that is a design on which the maximum possible number of eigenfunctions have mean value zero we then provide examples of such graphs and sets which arise naturally in extremal combinatorics we also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs and provide examples for them as well we investigate the behavior of graphical designs under the operation of weak graph product in addition we present a family of extremal graphical designs for the hypercube graph

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

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

bayesian image reconstruction in spect using higher order mechanical models as priors while the ml em algorithm for reconstruction for emission tomography is unstable due to the ill posed nature of the problem bayesian reconstruction methods overcome this instability by introducing prior information often in the form of a spatial smoothness regularizer more elaborate forms of smoothness constraints may be used to extend the role of the prior beyond that of a stabilizer in order to capture actual spatial information about the object previously proposed forms of such prior distributions were based on the assumption of a piecewise constant source distribution here the authors propose an extension to a piecewise linear model the weak plate which is more expressive than the piecewise constant model the weak plate prior not only preserves edges but also allows for piecewise ramplike regions in the reconstruction indeed for the authors application in spect such ramplike regions are observed in ground truth source distributions in the form of primate autoradiographs of rcbf radionuclides to incorporate the weak plate prior in a map approach the authors model the prior as a gibbs distribution and use a gem formulation for the optimization they compare quantitative performance of the ml em algorithm a gem algorithm with a prior favoring piecewise constant regions and a gem algorithm with their weak plate prior pointwise and regional bias and variance of ensemble image reconstructions are used as indications of image quality the authors results show that the weak plate and membrane priors exhibit improved bias and variance relative to ml em techniques

every covering of a compact riemann surface of genus greater than one carries a nontrivial l2 harmonic differential this paper contains the proof of the assertion in its title my motivation for considering this problem was the following let m be a compact differentiable manifold and let d be an elliptic differential operator between the spaces of c sections of two bundles on m in 4 m f atiyah proves that if index d 0 then for every galois covering at m the operator induced by d on has nontrivial l 2 solutions u 0 atiyah goes on to ask whether the same is true for coverings which are not galois his guess is that the answer is negative in general but that the counter example will be difficult to construct the simplest situation to consider in this connection is that of a surface and the operator whose index is the euler characteristic for infinite coverings since every l 2 harmonic function would be constant and nonzero constants are not in l 2 atiyah s question reduces to the question of existence of l 2 harmonic forms of degree one it is shown here that a counter example will not be found in this simple setting from now on s will denote an oriented compact surface of genus g g s l equipped with a smooth riemannian metric s s will be an arbitrary usually infinite covering of s and a a will be the laplace operator on s with respect to the pull back metric a differential exterior form of degree one on s is harmonic i e ao 0 and in l 2 if and only if cf 15 theorem 26

the dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality we study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range we assume that the underlying graphg satisfies a strong isoperimetric inequality and that the transition operatorp is strongly reversible uniformly irreducible and satisfies a uniform first moment condition we prove that under these hypotheses the random walk converges almost surely to a random end ofg and that the dirichlet problem forp harmonic functions is solvable with respect to the end compactification if in addition the graph as a metric space is hyperbolic in the sense of gromov then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification the main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality finally it is shown how the same technique can be applied to brownian motion to obtain analogous results for riemannian manifolds satisfying cheeger s isoperimetric inequality in particular in this general context new and simpler proofs of well known results on the dirichlet problem for negatively curved manifolds are obtained

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

isoperimetric inequalities and transient random walks on graphs recurrent we show that any graph satisfying an isoperimetric inequality only slightly stronger than that of z2 is transient more precisely if f k denotes the smallest number of vertices in the boundary of a connected subgraph with k vertices then the graph is transient if the infinite sum e f k 2 converges this can be applied to parabolicity versus hyperbolicity of surfaces 1 introduction let g be a connected graph which is locally finite that is all vertices have finite degree we consider a random walk starting at a vertex v say such that at any vertex u the walk proceeds to a neighbour with probability 1 d u where d u is the degree i e the number of neighbours of u the graph g is recurrent if we revisit v with probability 1 otherwise g is transient it is well known that the three dimensional grid z3 is transient while z2 is recurrent more generally nash williams 13 see also 4 10 proved that any graph with smaller growth rate than z2 is recurrent lyons 10 showed that certain subgraphs of grids are transient provided they grow just a little faster than z2 other results in terms of isoperimetric inequalities supporting the statement that z2 is in a sense an extreme recurrent graph can be derived from work of fernandez 5 grigor yan 7 and varopoulos 15 varopoulos 16 used results of gromov 8 to characterize completely the recurrent cayley graphs we shall carry these results further if v is a vertex set in g then av will denote the boundary of v that is the set of vertices of v having neighbours outside of v let f be a nondecreasing positive real function defined on the natural numbers we say that g satisfies an f isoperimetric inequality if there exists a constant c 0 such that for each finite vertex set v of g

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

performance of map reconstruction for hot lesion detection in whole body pet ct an evaluation with human and numerical observers for positron emission tomography pet imaging different reconstruction methods can be applied including maximum likelihood ml and maximum a posteriori map reconstruction postsmoothed ml images have approximately position and object independent spatial resolution which is advantageous for semi quantitative analysis however the complex object dependent smoothing obtained with map might yield improved noise characteristics beneficial for lesion detection in this contribution map and postsmoothed ml are compared for hot spot detection by human observers and by the channelized hotelling observer cho the study design was based on the ldquomultiple alternative forced choicerdquo approach for the map reconstruction the relative difference prior was used for postsmoothed ml a gaussian smoothing kernel was used both the human observers and the cho performed slightly better on map images than on postsmoothed ml images the average cho performance was similar to the best human performance the cho was then applied to evaluate the performance of priors with reduced penalty for large differences for these priors a poorer detection performance was obtained

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

spectral gap of the discrete laplacian on triangulations our goal in this paper is to find an estimate for the spectral gap of the laplacian on a two simplicial complex consisting on a hypergraph of a complete graph an upper estimate is given by generalizing the cheeger constant the lower estimate is obtained from the first non zero eigenvalue of the discrete laplacian acting on the functions of certain sub graphs

general cheeger inequalities for p laplacians on graphs abstract we prove cheeger inequalities for p laplacians on finite and infinite weighted graphs unlike in previous works we do not impose boundedness of the vertex degree nor do we restrict ourselves to the normalized laplacian and more generally we do not impose any boundedness assumption on the geometry this is achieved by a novel definition of the measure of the boundary which uses the idea of intrinsic metrics for the non normalized case our bounds on the spectral gap of p laplacians are already significantly better for finite graphs and for infinite graphs they yield non trivial bounds even in the case of unbounded vertex degree we furthermore give upper bounds by the cheeger constant and by the exponential volume growth of distance balls

random walks for image segmentation a novel method is proposed for performing multilabel interactive image segmentation given a small number of pixels with user defined or predefined labels one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels by assigning each pixel to the label for which the greatest probability is calculated a high quality image segmentation may be obtained theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits this algorithm is formulated in discrete space i e on a graph using combinatorial analogues of standard operators and principles from continuous potential theory allowing it to be applied in arbitrary dimension on arbitrary graphs

vector generation functions q spectral functions of hyperbolic geometry and vertex operators for quantum affine algebras we investigate the concept of q replicated argument in symmetric functions with its connection to spectral functions of hyperbolic geometry this construction suffices for vector generation functions in the form of q series and string theory we hope that the mathematical side of the construction can be enriched by ideas coming from physics

expansion in sl d z q z q arbitrary let s be a fixed finite symmetric subset of sl d z and assume that it generates a zariski dense subgroup g we show that the cayley graphs of q g with respect to the generating set q s form a family of expanders where q is the projection map z z q z

mathematical representation of clausius and kelvin s statements of the second law and irreversibility we provide a stochastic mathematical representation for clausius and kelvin planck s statements of the second law of thermodynamics in terms of the entropy productions of a finite compact driven markov system and its lift a surjective map is rigorously established through the lift when the state space is either a discrete graph or a continuous n dimensional torus mathbb t n the corresponding lifted processes have detailed balance thus a natural potential function but no stationary probability we show that in the long time limit the entropy production of the finite driven system precisely equals the potential energy decrease in the lifted system this theorem provides a dynamic foundation for the two equivalent statements of second law of thermodynamics a la kelvin s and clausius it suggests a modernized combined statement a mesoscopic engine that works in completing irreversible internal cycles statistically has necessarily an external effect that lowering a weight accompanied by passing heat from a warmer to a colder body

boundary and entropy of space homogeneous markov chains we study the poisson boundary equiv representation of bounded harmonic functions of markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations this automorphism group is locally compact but not necessarily discrete or unimodular the main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups while however the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other various new examples are given as applications including a description of the poisson boundary for random walks on vertex transitive graphs with infinitely many ends and on the diestel leader graphs

vertex operators and composite supersymmetric s functions the s function basis for the fock space of vertex operator constructions is considered where states are associated with partitions mod lambda the authors show that the matrix element lambda mod valpha z mod mu is essentially given by the composite s function slambda mu z where the argument z identical to z 1 is replicated mod alpha mod times matrix elements of products of vertex operators lambda mod valpha z v beta w mod mu are given up to functions of z w independent of lambda and mu by the composite supersymmetric s function slambda mu z w where the z w are replicated mod alpha mod mod beta mod times the supersymmetric arguments being those associated with negative roots

two constructions of affine lie algebra representations and boson fermion correspondence in quantum field theory we establish an isomorphism between the vertex and spinor representations of affine lie algebras for types dl 1 and dl 1 2 we also study decomposition of spinor representations using the infinite family of casimir operators and prove that they are either irreducible or have two irreducible components we show that the vertex and spinor constructions of the representations can be reformulated in the language of two dimensional quantum field theory in this physical context the two constructions yield the generalized sine gordon and thirring models respectively already in renormalized form the isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson fermion correspondence

cohomology of noncompact surfaces this paper is motivated by the question of whether nonzero l2 harmonic differentials exist on coverings of a riemann surface of genus 2 our approach will be via an analogue of the de rham theorem some results concerning the invariance of l2 homology and the intersection number of lu cycles are demonstrated a growth estimate for triangulations of planar coverings of the two holed torus is derived finally the equivalence between the existence of l2 harmonic one cycles and the transience of random walks on a planar surface is shown

a sears type self adjointness result for discrete magnetic schrodinger operators abstract in the context of a weighted graph with vertex set v and bounded vertex degree we give a sufficient condition for the essential self adjointness of the operator w where is the magnetic laplacian and w v r is a function satisfying w x q x for all x v with q v 1 the condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph the main result is a discrete analogue of the results of i oleinik and m a shubin in the setting of non compact riemannian manifolds

the essential spectrum of the laplacian on rapidly branching tessellations in this paper we characterize absence of the essential spectrum of the laplacian under a hyperbolicity assumption for general graphs moreover we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

a new convex edge preserving median prior with applications to tomography in a bayesian tomographic maximum a posteriori map reconstruction an estimate of the object f is computed by iteratively minimizing an objective function that typically comprises the sum of a log likelihood data consistency term and prior or penalty term the prior can be used to stabilize the solution and to also impose spatial properties on the solution one such property preservation of edges and locally monotonic regions is captured by the well known median root prior mrp an empirical method that has been applied to emission and transmission tomography we propose an entirely new class of convex priors that depends on f and also on m an auxiliary field in register with f we specialize this class to our median prior mp the approximate action of the median prior is to draw at each iteration an object voxel toward its own local median this action is similar to that of mrp and results in solutions that impose the same sorts of object properties as does mrp our map method is not empirical since the problem is stated completely as the minimization of a joint on f and m objective we propose an alternating algorithm to compute the joint map solution and apply this to emission tomography showing that the reconstructions are qualitatively similar to those obtained using mrp

the heat flow on metric random walk spaces abstract in this paper we study the heat flow on metric random walk spaces which unifies into a broad framework the heat flow on locally finite weighted connected graphs the heat flow determined by finite markov chains and some nonlocal evolution problems we give different characterizations of the ergodicity and prove that a metric random walk space with positive ollivier ricci curvature is ergodic furthermore we prove a cheeger inequality and as a consequence we show that a poincare inequality holds if and only if an isoperimetric inequality holds we also study the bakry emery curvature dimension condition and its relation with functional inequalities like the poincare inequality and the transport information inequalities

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

amenability via random walks we use random walks to show that the basilica group is amenable answering an open question of grigorchuk and zuk our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits these classes are separated even within the realm of finitely presented groups

martin and end compactifications for non locally finite graphs we consider a connected graph having countably infinite vertex set x which is permitted to have vertices of infinite degree for a transient irreducible transition matrix p corresponding to a nearest neighbor random walk on x we study the associated harmonic functions on x and in particular the martin compactification we also study the end compactification of the graph when the graph is a tree we show that these compactifications coincide they are a disjoint union of x the set of ends and the set of improper vertices new points associated with vertices of infinite degree other results proved include a solution of the dirichlet problem in the context of the end compactification of a general graph applications are given to e g the cayley graph of a free group on infinitely many generators

spatial resolution properties of penalized likelihood image reconstruction space invariant tomographs this paper examines the spatial resolution properties of penalized likelihood image reconstruction methods by analyzing the local impulse response the analysis shows that standard regularization penalties induce space variant local impulse response functions even for space invariant tomographic systems paradoxically for emission image reconstruction the local resolution is generally poorest in high count regions we show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections this analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution the modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized likelihood methods

asymptotic entropy and green speed for random walks on countable groups we study asymptotic properties of the green metric associated with transient random walks on countable groups we prove that the rate of escape of the random walk computed in the green metric equals its asymptotic entropy the proof relies on integral representations of both quantities with the extended martin kernel in the case of finitely generated groups where this result is known benjamini and peres probab theory related fields 98 1994 91 112 we give an alternative proof relying on a version of the so called fundamental inequality relating the rate of escape the entropy and the logarithmic volume growth extended to random walks with unbounded support

asymptotic behavior of the transition probability of a random walk on an infinite graph ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition in the course of our argument we employ perturbation theory for the maximal eigenvalues of twisted transition operator as a result an asymptotic of the probabilityp n x y that a particle starting atxreachesyat timenasngoes to infinity is established

new branching rules induced by plethysm we derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor t of young symmetry type where is an integer partition the branchings and fixing a vector vi a symmetric tensor gij gji and an antisymmetric tensor fij fji respectively are obtained as special cases all new branchings are governed by schur function series obtained from plethysms of the schur function s by the basic m series of complete symmetric functions and the l m 1 series of elementary symmetric functions our main technical tool is that of hopf algebras and our main result is the derivation of a coproduct for any schur function series obtained by plethysm from another such series therefrom one easily obtains generalized newell littlewood formulae and the algebra of the formal group characters of these subgroups is established concrete examples and extensive tabulations are displayed for and showing their involved and nontrivial representation theory the nature of the subgroups is shown to be in general affine and in some instances non reductive we discuss the complexity of the coproduct formula and give a graphical notation to cope with it we also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2 cocycles the algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters modification rules are beyond the scope of the present paper but are briefly discussed

isoperimetric inequalities and random walks on quotients of graphs and buildings let be a euclidean or hyperbolic building and let g aut be a locally compact unimodular group which acts strongly transitively on we use graphs quasi isometric to to study asymptotic properties of quotients where is a discrete subgroup of g if g has kazhdan s property t we show that such quotients satisfy strong isoperimetric inequalities this yields new examples of graphs with positive cheeger constant such graphs cannot be bi lipschitz embedded into hilbert space moreover simple random walks on such quotients are shown to be recurrent if and only if is a uniform lattice in g

transient random walks on graphs and metric spaces with applications to hyperbolic surfaces we introduce an r net 0 2r r of a metric space m as a maximal graph whose vertices are elements in m of pairwise distance at least r such that any two vertices of distance at most r are adjacent we show that for a large class of metric spaces including many riemannian manifolds the property of transience of a net and the property of the net carrying a non constant harmonic function of bounded energy is independent of the choice of the net we give a new necessary and sufficient condition for a graph with bounded degrees and satisfying an isoperimetric inequality to have no non constant harmonic functions for this purpose we develop equivalent analytic conditions for graphs satisfying an isoperimetric inequality some of these results have been discovered recently by others in more general settings but our treatment here is specific and self contained we use graph transience to prove that scherk s surface is hyperbolic a problem posed by osserman in 1965

sur la dynamique unidimensionnelle en regularite intermediaire using probabilistic methods we prove new rigidity results for groups and pseudo groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class i e between c 1 and c 2 in particular we show some generalizations of denjoy theorem and the classical kopell lemma for abelian groups these techniques are also applied to the study of codimension 1 foliations for instance we obtain several generalized versions of sacksteder theorem in class c 1 we conclude with some remarks about the stationary measure

the direction of an automorphism of a totally disconnected locally compact group we describe the asymptotic behavior of automorphisms of totally disconnected locally compact groups in terms of a set of directions which comes equipped with a natural pseudo metric the structure at infinity obtained by completing the induced metric quotient space of the set of directions recovers familiar objects such as the set of ends of the tree for the group of inner automorphisms of the group of isometries of a regular locally finite tree and the spherical bruhat tits building for the group of inner automorphisms of the set of rational points of a semisimple group over a local field

multi way dual cheeger constants and spectral bounds of graphs we introduce a set of multi way dual cheeger constants and prove universal higher order dual cheeger inequalities for eigenvalues of normalized laplace operators on weighted finite graphs our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces we further extend those results to a general reversible markov operator and find applications in characterizing its essential spectrum

the poisson formula for groups with hyperbolic properties the poisson boundary of a group g with a probability measure is the space of ergodic components of the time shift in the path space of the associated random walk via a generalization of the classical poisson formula it gives an integral representation of bounded harmonic functions on g in this paper we develop a new method of identifying the poisson boundary based on entropy estimates for conditional random walks it leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the poisson boundary with natural topological boundaries for several classes of groups word hyperbolic groups and discontinuous groups of isometries of gromov hyperbolic spaces groups with inflnitely many ends cocompact lattices in cartan hadamard manifolds discrete subgroups of semisimple lie groups

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given