on supersymmetry breaking in the computation of the complexity we study the consequences of supersymmetry breaking in the computation of the number of solutions of the thouless anderson palmer tap equations we show that the kurchan argument that proves the vanishing of the prefactor of the bray and moore saddle point for the total number of solutions can be extended to solutions at any given free energy we also provide a new simple argument for the vanishing of the prefactor and use it to prove that the isolated eigenvalue recently considered by aspelmeier bray and moore is exactly zero in the bm theory because of supersymmetry breaking the behaviour of the eigenvector of the isolated eigenvalue at the lower band edge is also considered

orthonormal dilations of parseval wavelets we prove that any parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the baumslag solitar group bs 1 2 langle u t utu 1 t 2 rangle we give a precise description of this representation in some special cases and show that for wavelet sets it is related to symbolic dynamics theorem 3 14 we prove that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics theorem 3 24 we give concrete examples of parseval wavelets for which we compute the orthonormal dilations in detail we construct parseval wavelet sets which have infinitely many non isomorphic orthonormal dilations

ordinary and modular characters of su 3 p abstract this article deals with the relationship between ordinary and p modular characters of the group su 3 p parallel to the treatment of sl 3 p in j e humphreys ordinary and modular characters of sl 3 p j algebra 72 1981 8 16 here we denote by su 3 q the special unitary subgroup of sl 3 q2 where q is a power of the prime p the notation su 3 q2 is perhaps more standard but we want to emphasize the comparison with sl 3 q our main conclusion is that the decomposition behavior is essentially the same for both groups in a sense to be made more precise below

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

chip firing on dynkin diagrams and mckay quivers this paper establishes new connections between the representation theory of finite groups and sandpile dynamics two classes of avalanche finite matrices and their critical groups integer cokernels are studied from the viewpoint of chip firing sandpile dynamics namely the cartan matrices of finite root systems and the mckay cartan matrices for finite subgroups g of general linear groups in the root system case the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights in the mckay cartan case for finite subgroups of the special linear group the cokernel is related to the abelianization of the subgroup g in the special case of the classical mckay correspondence the critical group and the abelianization are shown to be isomorphic

walks on graphs and their connections with tensor invariants and centralizer algebras abstract the number of walks of k steps from the node 0 to the node on the mckay quiver determined by a finite group g and a g module v is the multiplicity of the irreducible g module g in the tensor power v k and it is also the dimension of the irreducible module labeled by for the centralizer algebra z k g end g v k this paper explores ways to effectively calculate that number using the character theory of g we determine the corresponding poincare series the special case 0 gives the poincare series for the tensor invariants t v g k 0 v k g and a tensor analog of molien s formula for polynomial invariants when g is abelian we show that the exponential generating function for the number of walks is a product of generalized hyperbolic functions many graphs such as circulant graphs can be viewed as mckay quivers and the methods presented here provide efficient ways to compute the number of walks on them

lipschitz equivalence of fractals generated by nested cubes david and semmes fractured fractals and broken dreams vol 7 1997 posed a problem on the lipschitz equivalence among fractals generated by nested cubes according to various rules for this problem we show that there are uncountably many lipschitz equivalence classes if the rules are taken arbitrarily however if the same rule is taken in each step of construction process of fractal under some suitable restriction we show that there is only a unique lipschitz equivalence in particular we obtain the self similar set if only one rule is used in construction process applying the second result to the self similar sets we can obtain the following proposition in xi and xiong c r math acad sci paris 348 15 20 2010 if e a bigcup a in a frac 1 n e a frac a n e b bigcup b in b frac 1 n e b frac b n are totally disconnected self similar sets with a b subset 0 1 ldots n 1 d and a b then ea and eb are lipschitz equivalent

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

the spectrum of a coxeter transformation affine coxeter transformations and the defect map abstract the spectrum of the coxeter transformation r in a reflection group defined by a matrix a without odd cycles is determined in terms of spec a for affine a r determines the defect map which is given a simple formulation in terms of null roots the affine exponents determined by spec r are related to the coefficients of the fundamental co weight corresponding to the fork node of the coxeter dynkin diagram

trees parking functions syzygies and deformations of monomial ideals for a graph g we construct two algebras whose dimensions are both equal to the number of spanning trees of g one of these algebras is the quotient of the polynomial ring modulo certain monomial ideal while the other is the quotient of the polynomial ring modulo certain powers of linear forms we describe the set of monomials that forms a linear basis in each of these two algebras the basis elements correspond to g parking functions that naturally came up in the abelian sandpile model these ideals are instances of the general class of monotone monomial ideals and their deformations we show that the hilbert series of a monotone monomial ideal is always bounded by the hilbert series of its deformation then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals in some cases these resolutions coincide with scarf resolutions we prove several formulas for hilbert series of monotone monomial ideals and investigate when they are equal to hilbert series of deformations in the appendix we discuss the abelian sandpile model

the partition algebra and the kronecker coefficients we propose a new approach to study the kronecker coefficients by using the schur weyl duality between the symmetric group and the partition algebra we explain the limiting behaviour and associated bounds in the context of the partition algebra our analysis leads to a uniform description of the reduced kronecker coefficients when one of the indexing partitions is a hook or a two part partition

on finite subgroups of su 2 simple lie algebras and the mckay correspondence the question is considered of how the restriction n where n is irreducible for all n and is a finite subgroup of su 2 decomposes into irreducibles for any arbitrary n unk it is announced that using the mckay correspondence the problem has an elegant solution in terms of the coxeter element for the associated lie algebra and that the numbers involved come in a beautiful way from the root structure

enumerating linear systems on graphs the divisor theory of graphs views a finite connected graph g as a discrete version of a riemann surface divisors on g are formal integral combinations of the vertices of g and linear equivalence of divisors is determined by the discrete laplacian operator for g as in the case of riemann surfaces we are interested in the complete linear system d of a divisor d the collection of nonnegative divisors linearly equivalent to d unlike the case of riemann surfaces the complete linear system of a divisor on a graph is always finite we compute generating functions encoding the sizes of all complete linear systems on g and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the dual of the jacobian group of g if g is a cycle graph our results lead to a bijection between complete linear systems and binary necklaces our results also apply to a model in which the laplacian is replaced by an invertible integral m matrix

lipschitz equivalence of fractal triangles abstract given an integer number n 2 and two digit sets a k 1 k 2 k 1 k 2 n 1 and k 1 k 2 n 0 b k 1 k 2 2 k 1 k 2 n and k 1 k 2 n where 1 0 1 2 3 2 there is a self similar set t t a b r 2 satisfying the set equation t t a b t n we call such t a fractal triangle by examining deeper the types of connected components in each step of constructing such fractal triangles we in this paper successfully characterize the lipschitz equivalence of two classes of totally disconnected fractal triangles in it through the number of basic triangles in the first step of construction

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

the heat semigroup and brownian motion on strip complexes abstract we introduce the notion of strip complex a strip complex is a special type of complex obtained by gluing strips along their natural boundaries according to a given graph structure the most familiar example is the one dimensional complex classically associated with a graph in which case the strips are simply copies of the unit interval our setup actually allows for variable edge length a leading key example is treebolic space a geometric object studied in a number of recent articles which arises as a horocyclic product of a metric tree with the hyperbolic plane in this case the graph is a regular tree the strips are 0 1 r and each strip is equipped with the hyperbolic geometry of a specific strip in upper half plane we consider natural families of dirichlet forms on a general strip complex and show that the associated heat kernels and harmonic functions have very strong smoothness properties we study questions such as essential self adjointness of the underlying differential operator acting on a suitable space of smooth functions satisfying a kirchhoff type condition at points where the strip complex bifurcates compatibility with projections that arise from proper group actions is also considered

static spherically symmetric solutions of the so 5 einstein yang mills equations globally regular ie asymptotically flat and regular interior spher ically symmetric and localised particle like solutions of the coupled ein stein yang mills eym equations with gauge group su 2 have been known for more than 20 years yet their properties are still not well understood spher ically symmetric yang mills fields are classified by a choice of isotropy genera tor and so 5 is distinguished as the simplest model with a non abelian resid ual little group su 2 u 1 and which admits globally regular particle like solutions we exhibit an algebraic gauge condition which normalises the resid ual gauge freedom to a finite number of discrete symmetries this generalises the well known reduction to the real magnetic potential w r t in the original su 2 ym model reformulating using gauge invariant polynomials dramat ically simplifies the system and makes numerical search techniques feasible we find three families of embedded su 2 eym equations within the so 5 system one of which was first detected only within the gauge invariant polyno mial reduced system numerical solutions representing mixtures of the three su 2 sub systems are found classified by a pair of positive integers

on tensor categories of lie type en n 9 let v be the minuscule representation of the lie algebra g en n 6 7 we show that the centralizer of the action of the quantum group uqg en on v n is generated by the r matrices and one additional element appearing in the n 1 st tensor power a similar description can be given for the classical limit an analogous statement is true for a certain direct summand of v n for uqg en with n 5 n 9 here v is the module whose highest weight is the fundamental weight labeled by the vertex furthest from the triple point of the graph en moreover we obtain a 2 parameter family of braid representations generalizing a q version of the brauer centralizer algebras

replica trick to calculate means of absolute values applications to stochastic equations a version of the replica trick can be used to evaluate means of the absolute values of functions that are not positive definite and with some modifications the mean logarithm of the absolute value this trick can be applied in the context of stochastic equations where the absolute value of the jacobian of the stochastic function in each solution must be computed the calculation of the average number of solutions of the naive mean field equations for the sk spin glass is discussed an extension of the considerations to other mean field equations such as tap is straightforward the brst supersymmetry is found to provide a useful tool in uncovering some puzzling aspects in these calculations the trick presented provides a possible solution to these problems

geometric analysis on cantor sets and trees dusing uniformization cantor type sets can be regarded as boundaries of rooted trees in this setting we show that the trace of a first order sobolev space on the boundary of a regular rooted tre

killing and the coxeter transformation of kac moody algebras on etudie le spectre de la transformation de coxeter associee a une algebre de kac moody arbitraire et plus particulierement a une algebre affine

invariant polynomials and molien functions the molien function associated with a finite dimensional representation of a compact lie group is a useful tool in representation theory because it acts as a generating function for counting the number of invariant polynomials on the representation space the main purpose of this paper is to introduce a more general and apparently new generating function which in the same sense counts not only the number of invariant real polynomials in a real representation or the number of invariant complex polynomials in a complex representation as a function of their degree but encodes the number of invariant real polynomials in a complex representation as a function of their bidegree the first and second component of this bidegree being the number of variables in which the polynomial is holomorphic and antiholomorphic respectively this is obviously an additional and non trivial piece of information for representations which are truly complex i e not self conjugate or are pseudo real but it provides ad

stein s method and random character ratios stein s method is used to prove limit theorems for random character ratios tools are developed for four types of structures finite groups gelfand pairs twisted gelfand pairs and association schemes as one example an error term is obtained for a central limit theorem of kerov on the spectrum of the cayley graph of the symmetric group generated by i cycles other main examples include an error term for a central limit theorem of ivanov on character ratios of random projective representations of the symmetric group and a new central limit theorem for the spectrum of certain random walks on perfect matchings the results are obtained with very little information a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk the limit theorems stated in this paper are for normal approximation but many of the tools developed are applicable for arbitrary distributional approximation

tap complexity the cavity method and supersymmetry we compute the bray and moore bm tap complexity for the sherrington kirkpatrick model through the cavity method showing that some essential modifications are needed with respect to the standard formulation of the method this allows us to understand various features recently discovered and to unveil at last the physical meaning of the parameters of the bm theory we also reconsider the supersymmetric susy formulation of the problem finding that the bm solution satisfies some proper susy ward identities that are different from the standard ones the susy relationships encode the physical meaning of the parameters obtained through the cavity method the problem of the vanishing prefactor is addressed showing how it can be avoided

quantum random walk on the dual of su n we study a quantum random walk ona su n the von neumann algebra of su n obtained by tensoring the basic representation of su n two classical markov chains are derived from this quantum random walk by restriction to commutative subalgebras ofa su n and the main result of the paper states that these two markov chains are related by means of doob sh processes

norm convergence of random walks on compact hypergroups where ig is the order of g and p is the nth convolut ion power of p they show that this is a natural tool for studying strong uniform times for r a n d o m walks on g and they show in theorem 7 of 1 stated as theorem 5 in 7 chap 4c that if i p ml o as n then s n o as n at a rate that depends on how fast l ip m converges to 0 but does not depend on the size of the group g in particular if j p m is small then s 2n is small

the farrell jones conjecture for the solvable baumslag solitar groups in this paper we prove the farrell jones conjecture for the solvable baumslag solitar groups with coefficients in an additive category we also extend our results to groups of the form z 1 p semidirect product with any virtually cyclic group where p is a prime number

invariants equivariants and characters in symmetric bifurcation theory in the analysis of stability in bifurcation problems it is often assumed that the appropriate reduced equations are in normal form in the presence of symmetry the truncated normal form is an equivariant polynomial map therefore the determination of invariants and equivariants of the group of symmetries of the problem is an important step in general these are hard problems of invariant theory and in most cases they are tractable only through symbolic computer programs nevertheless it is desirable to obtain some of the information about invariants and equivariants without actually computing them for example the number of linearly independent homogeneous invariants or equivariants of a certain degree generating functions for these dimensions are generally known as molien functions we obtain formulae for the number of linearly independent homogeneous invariants or equivariants for hopf bifurcation in terms of characters we also show how to construct molien functions for invariants and equivariants for hopf bifurcation our results are then applied to the computation of the number of invariants and equivariants for hopf bifurcation for several finite groups and the continuous group o 3

conditioned random walks from kac moody root systems random paths are time continuous interpolations of random walks by using littelmann path model we associate to each irreducible highest weight module of a kac moody algebra g a random path w under suitable hypotheses we make explicit the probability of the event e w never exits the weyl chamber of g we then give the law of the random walk defined by w conditioned by the event e and proves this law can be recovered by applying to w the generalized pitmann transform introduced by biane bougerol and o connell this generalizes the main results of 10 and 16 to kac moody root systems and arbitrary highest weight modules moreover we use here a completely new approach by exploiting the symmetry of our construction under the action of the weyl group of g rather than renewal theory and doob s theorem on martin kernels

algebraic model for the evolution of the genetic code a search for symmetries based on the compact simple lie algebras is performed to verify to what extent the genetic code is a manifestation of some underlying symmetry an exact continuous symmetry group cannot be found to reproduce the present genetic code however a unique approximate symmetry group is compatible with codon assignment for the fundamental amino acids and the termination codon in order to obtain the actual genetic code the symmetry must be slightly broken

stein s method and characters of compact lie groups stein s method is used to study the trace of a random element from a compact lie group or symmetric space central limit theorems are proved using very little information character values on a single element and the decomposition of the square of the trace into irreducible components this is illustrated for lie groups of classical type and dyson s circular ensembles the approach in this paper will be useful for the study of higher dimensional characters where normal approximations need not hold

tensor product markov chains abstract we analyze families of markov chains that arise from decomposing tensor products of irreducible representations this illuminates the burnside brauer theorem for building irreducible representations the mckay correspondence and pitman s 2 m x theorem the chains are explicitly diagonalizable and we use the eigenvalues eigenvectors to give sharp rates of convergence for the associated random walks for modular representations the chains are not reversible and the analytical details are surprisingly intricate in the quantum group case the chains fail to be diagonalizable but a novel analysis using generalized eigenvectors proves successful

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

on exactness and isoperimetric profiles of discrete groups abstract we introduce a quasi isometry invariant related to property a and explore its connections to various other invariants of finitely generated groups this allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other we apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory

asymmetric abeiian sandpile models in the abelian sandpile models introduced by dhar long time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system dhar proposed a simple procedure theburning algorithm as a possible test of whether a configuration is recurrent and later with majumdar verified the correctness of this test when the toppling rules of the sandpile are symmetric we observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles we also obtain necessary and sufficient conditions for the validity of the original test the results are applied to a family of deterministic one dimensional sandpile models originally studied by lee liang and tzeng

the partial order of dominant weights abstract the weight lattice of a crystallographic root system is partially ordered by the rule that if is a nonnegative integer linear combination of positive roots in this paper we study the subposet formed by the dominant weights in particular we prove that covers in this partial order only if belongs to a distinguished subset of the positive roots also if the root system is irreducible we prove that the mobius function of the partial order takes on only the values 0 1 2

a characterization of cocompact hyperbolic and finite volume hyperbolic groups in dimension three we show that a cocompact hyperbolic group in dimension 3 is char acterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space we prove a similar theorem for finite volume hyperbolic groups in dimension 3 a group g is called cocompact hyperbolic of dimension n if and only if it can act isometrically properly discontinuously and cocompactly on hyperbolic space h proper discontinuity implies that the action has at most finite kernel our main result states that a group g is cocompact hyperbolic of dimension 3 if and only if it has a finite generating set c and associated cayley graph f r cr c such that the word metric on t is comparable in the large quasi isometric with the standard hyperbolic metric on h3 we have a similar result for finite volume hyperbolic groups in dimension 3 we believe that our results generalize to all dimensions at least in the cocompact case our result for groups of finite volume employs a different metric space instead of the cayley graph with word metric we use an augmented cayley graph associated with a group g and a finite family of euclidean subgroups of g we prove that the group acts isometrically and properly discontinuously on h3 with orbit space of finite volume if and only if the augmented cayley graph is quasi isometric with h3 precise definitions of all of these terms will appear in 2 the proof of the cocompact case appears in 3 the proof of the finite volume case appears in 4 our results are similar to a result of gromov gl but without the added assumption that the group be the fundamental group of a convex path space there are other results on the classification of groups up to quasi isometry in g2 in particular a group is quasi isometric to e if and only if there is a subgroup z of finite index it would be interesting to know which groups are quasi isometric to the 3 dimensional geometries h2 x e1 solv and psl 2 r gromov s work on groups with polynomial growth g3 bk implies that a group is quasi isometric to nil if and only if it has a subgroup of finite index which is the fundamental group of a compact nil 3 orbifold notice that all compact

bilipschitz equivalence is not equivalent to quasi isometric equivalence for finitely generated groups we show that certain lamplighter groups that are quasi isometric to each other are not bilipschitz equivalent this gives a positive answer to a question of de la harpe

convergence condition of the tap equation for the infinite ranged ising spin glass model it is shown that the power expansion of the gibbs potential of the sk model up to second order in the exchange couplings leads to the tap equation this result remains valid for the general including a ferromagnetic exchange sk model theorems of power expansions and resolvent techniques are employed to solve the convergence problem the convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations

critical groups of group representations abstract this paper investigates the critical group of a faithful representation of a finite group it computes the order of the critical group in terms of the character values and gives some restrictions on its subgroup structure it also computes the precise structure of the critical group both for the regular representation of any finite group and for the reflection representation of s n

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

group splittings and asymptotic topology it is a consequence of the theorem of stallings on groups with many ends that splittings over finite groups are preserved by quasi isometries in this paper we use asymptotic topology to show that group splittings are preserved by quasi isometries in many cases roughly speaking we show that splittings are preserved under quasi isometries when the vertex groups are fundamental groups of aspherical manifolds or more generally coarse pd n groups and the edge groups are smaller than the vertex groups

mckay centralizer algebras for a finite subgroup g of the special unitary group su2 we study the centralizer algebra zk g endg v k of g acting on the k fold tensor product of its defining representation v c2 the mckay corre spondence relates the representation theory of these groups to an associated affine dynkin diagram and we use this connection to study the structure and representation theory of zk g via the combinatorics of the dynkin diagram when g equals the binary tetrahedral octahedral or icosahedral group we exhibit remarkable connections between zk g and the martin jones set partition algebras

coarse differentiation of quasi isometries i spaces not quasi isometric to cayley graphs in this paper we prove that certain spaces are not quasi isometric to cayley graphs of nitely generated groups in particular we answer a question of woess and prove a conjecture of diestel and leader by showing that certain homogeneous graphs are not quasi isometric to a cayley graph of a nitely generated group this paper is the rst in a sequence of papers proving results announced in our 2007 article quasi isometries and rigidity of solvable groups in particular this paper contains many steps in the proofs of quasi isometric rigidity of lattices in sol and of the quasi isometry classication of lamplighter groups the proofs of those results are completed in coarse dierentiation of quasi isometries ii rigidity for lattices in sol and lamplighter groups the method used here is based on the idea of coarse dierentiation introduced in our 2007 article

quasisymmetric maps on the boundary of a negatively curved solvable lie group we describe all the self quasisymmetric maps on the ideal boundary of a particular negatively curved solvable lie group as applications we derive some rigidity properties for quasiisometries of the solvable lie group

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

on the asymptotic geometry of abelian by cyclic groups gromov s polynomial growth theorem gro81 states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups gromov s theorem inspired the more general problem see e g gdlh91 of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure in short are solvable groups quasi isometrically rigid in general they aren t very recently a dioubina dio99 has found a solvable group which is quasi isometric to a group which is not virtually solvable these groups are finitely generated but not finitely presentable in the opposite direction first steps in identifying quasi isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples the solvable baumslagsolitar groups in fm98 and fm99b the goal of the present paper is to show that a much broader class of solvable groups the class of finitely presented nonpolycyclic abelian bycyclic groups is characterized among all finitely generated groups by its quasi isometry type we also give a complete quasi isometry classification of the groups in this class such a classification for nilpotent groups remains a major open question motivated by these results we offer a conjectural picture of quasi isometric classification and rigidity for polycyclic abelianby cyclic groups in 10 1

characters of algebras containing a jones basic construction the temperley lieb okada brauer and birman wenzl algebras abstract we begin by determining in a general form the characters of irreducible representations of a jones basic construction and use this result to compute the characters of the temperley lieb algebras and the okada algebras in the case of the birman wenzl algebra some of the characters are determined by the general theorem and the others are computed by using the duality between the birman wenzl algebras and the drinfeld jimbo quantum groups of types b and c the computations involve certain characters of the quantum group these are polynomials invariant under the weyl group of type b we are able to decompose these weyl group symmetric functions and obtain a combinatorial rule for computing the irreducible characters of the birman wenzl algebras the combinatorial rules for computing the irreducible characters of the iwahori hecke algebras and the combinatorial rule for computing the irreducible characters of the brauer algebras are both special cases of our rule for the birman wenzl algebras

differential invariants for symplectic lie algebras realized by boson operators the study of algebraic symplectic models in high energy physics and even in molecular genetics has received much attention in the past decades in this paper we first survey one known bosonic realization with n types of bosons operators in a 2j 1 dimensional space with j being the semi integer then using the general theory of differential invariants we determine two sets of sp 2j 1 differential invariants in one set when n 2j 1 all independent invariants of any order are obtained in a simple closed form when n 2j 1 we could present only particular matrix forms in the latter case the differential invariants up to the second order are explicitly presented for the symplectic algebras sp 2 sp 4 and sp 6 realized by up to three types of bosons as applications we present the simplest sp 2 invariant lagrangians null lagrangians new solutions to the helmholtz inverse problem and evolution equations which will be of fundamental importance in the construction of dynamical systems invariant under the symplectic group

poincare series exponents of affine lie algebras and mckay slodowy correspondence abstract let n be a normal subgroup of a finite group g and v a fixed finite dimensional g module the poincare series for the multiplicities of induced modules and restriction modules in the tensor algebra t v k 0 v k are studied in connection with the mckay slodowy correspondence in particular it is shown that the closed formulas for the poincare series associated with the distinguished pairs of subgroups of su 2 give rise to the exponents of all untwisted and twisted affine lie algebras except a 2 n 1

poincare series for tensor invariants and the mckay correspondence abstract for a finite group g and a finite dimensional g module v we prove a general result on the poincare series for the g invariants in the tensor algebra t v k 0 v k we apply this result to the finite subgroups g of the 2 2 special unitary matrices and their natural module v of 2 1 column vectors because these subgroups are in one to one correspondence with the simply laced affine dynkin diagrams by the mckay correspondence the poincare series obtained are the generating functions for the number of walks on the simply laced affine dynkin diagrams

root system chip firing i interval firing jim propp recently introduced a variant of chip firing on a line where the chips are given distinct integer labels hopkins mcconville and propp showed that this process is confluent from some but not all initial configurations of chips we recast their set up in terms of root systems labeled chip firing can be seen as a root firing process which allows the moves for alpha in phi whenever langle lambda alpha vee rangle 0 where phi is the set of positive roots of a root system of type a and lambda is a weight of this root system we are thus motivated to study the exact same root firing process for an arbitrary root system actually this central root firing process is the subject of a sequel to this paper in the present paper we instead study the interval root firing processes determined by for alpha in phi whenever langle lambda alpha vee rangle in k 1 k 1 or langle lambda alpha vee rangle in k k 1 for any k ge 0 we prove that these interval firing processes are always confluent from any initial weight we also show that there is a natural way to consistently label the stable points of these interval firing processes across all values of k so that the number of weights with given stabilization is a polynomial in k we conjecture that these ehrhart like polynomials have nonnegative integer coefficients