how to detect integrability in cellular automata ultra discrete equations are generalized cellular automata in the sense that the dependent and independent variables take only integer values we present a new method for identifying integrable ultra discrete equations which is the equivalent of the singularity confinement property for difference equations and the painleve property for differential equations using this criterion we find integrable ultra discrete equations which include the ultra discrete painleve equations

discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

a bk inequality for randomly drawn subsets of fixed size the bk inequality van den berg and kesten in j appl probab 22 556 569 1985 says that for product measures on 0 1 n the probability that two increasing events a and b occur disjointly is at most the product of the two individual probabilities the conjecture in van den berg and kesten 1985 that this holds for all events was proved by reimer combin probab comput 9 27 32 2000 several other problems in this area remained open for instance although it is easy to see that non product measures cannot satisfy the above inequality for all events there are several such measures which intuitively should satisfy the inequality for all increasing events one of the most natural candidates is the measure assigning equal probabilities to all configurations with exactly k 1 s and probability 0 to all other configurations the main contribution of this paper is a proof for these measures we also point out how our result extends to weighted versions of these measures and to products of such measures

bk type inequalities and generalized random cluster representations recently van den berg and jonasson gave the first substantial extension of the bk inequality for non product measures they proved that for k out of n measures the probability that two increasing events occur disjointly is at most the product of the two individual probabilities we show several other extensions and modifications of the bk inequality in particular we prove that the antiferromagnetic ising curie weiss model satisfies the bk inequality for all increasing events we prove that this also holds for the curie weiss model with three body interactions under the so called negative lattice condition for the ferromagnetic ising model we show that the probability that two events occur cluster disjointly is at most the product of the two individual probabilities and we give a more abstract form of this result for arbitrary gibbs measures the above cases are derived from a general abstract theorem whose proof is based on an extension of the fortuin kasteleyn random cluster representation for all probability distributions and on a folding procedure which generalizes an argument of reimer

when is negativity not a problem for the ultradiscrete limit the ultradiscrete limit has provided a link between integrable difference equations and cellular automata displaying soliton like solutions in particular this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the max operator although it certainly is the case that dropping these positivity conditions creates potential difficulties it is still possible for solutions to persist under the ultradiscrete limit even in their absence to recognize when this will occur one must consider whether a certain expression involving a measure of the rates of convergence of different terms in the difference equation and their coefficients is equal to zero applications discussed include the solution of elementary ordinary difference equations a discretization of the hirota bilinear difference equation and the identification of integrals of motion for ultradiscrete equations

multiprocess parallel antithetic coupling for backward and forward markov chain monte carlo antithetic coupling is a general stratification strategy for reducing monte carlo variance without increasing the simulation size the use of the antithetic principle in the monte carlo literature typically employs two strata via antithetic quantile coupling we demonstrate here that further stratification obtained by using k 2 e g k 3 10 antithetically coupled variates can offer substantial additional gain in monte carlo efficiency in terms of both variance and bias the reason for reduced bias is that antithetically coupled chains can provide a more dispersed search of the state space than multiple independent chains the emerging area of perfect simulation provides a perfect setting for implementing the k process parallel antithetic coupling for mcmc because without antithetic coupling this class of methods delivers genuine independent draws furthermore antithetic backward coupling provides a very convenient theoretical tool for investigating antithetic forward coupling however the generation of k 2 antithetic variates that are negatively associated that is they preserve negative correlation under monotone transformations and extremely antithetic that is they are as negatively correlated as possible is more complicated compared to the case with k 2 in this paper we establish a theoretical framework for investigating such issues among the generating methods that we compare latin hypercube sampling and its iterative extension appear to be generalpurpose choices making another direct link between monte carlo and quasi monte carlo

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

integrable systems without the painleve property we examine whether the painleve property is a necessary condition for the integrability of nonlinear ordinary differential equations we show that for a large class of linearizable systems this is not the case in the discrete domain we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non painleve continuous linearizable systems we find that while these discrete systems are themselves linearizable they possess non confined singularities

some concepts of negative dependence abstract the theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial dirichlet and the multivariate hypergeometric some conditions of negative dependence that are satisfied by these distributions and which have practical meaning are introduced preservation results for some of these concepts are derived useful inequalities for some widely used distributions are obtained results of mallows 1969 that apply to the multinomial distributions are extended to more distributions examples are listed author

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

linearisable ultradiscrete systems with sign variables and the confinement of singularities we present the singularity analysis of the ultradiscrete analogue of linearisable mappings of quispel roberts thompson qrt type the ultradiscretisation method used here is one which keeps track of signs and thus can be applied without the positivity restrictions of the classical ultradiscretisation approach we show that in all cases the mappings possess confined singularities the same is true for two non autonomous equations which are equally linearisable in the discrete case we construct explicitly the solutions of the ultradiscrete mappings analysed here

the gambier mapping abstract we propose a discrete form for an equation due to gambier and which belongs to the class of the fifty second order equations that possess the painleve property in the continuous case the solutions of the gambier equation is obtained through a system of riccati equations the same holds true in the discrete case also we use the singularity confinement criterion in order to study the integrability of this new mapping

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

discrete versions of the transport equation and the shepp olkin conjecture we introduce a framework to consider transport problems for integer valued random variables we introduce weighting coecients which allow us to characterise transport problems in a gradient ow setting and form the basis of our introduction of a discrete version of the benamou brenier formula further we use these coecients to state a new form of weighted log concavity these results are applied to prove the monotone case of the shepp olkin entropy concavity conjecture msc2000 primary 60e15 secondary 94a17 60d99

symplectic balancing of hamiltonian matrices we discuss the balancing of hamiltonian matrices by structure preserving similarity transformations the method is closely related to balancing nonsymmetric matrices for eigenvalue computations as proposed by osborne j acm 7 1960 pp 338 345 and parlett and reinsch numer math 13 1969 pp 296 304 and implemented in most linear algebra software packages it is shown that isolated eigenvalues can be deflated using similarity transformations with symplectic permutation matrices balancing is then based on equilibrating row and column norms of the hamiltonian matrix using symplectic scaling matrices due to the given structure it is sufficient to deal with the leading half rows and columns of the matrix numerical examples show that the method improves eigenvalue calculations of hamiltonian matrices as well as numerical methods for solving continuous time algebraic riccati equations

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

balancing sparse matrices for computing eigenvalues abstract applying a permuted diagonal similarity transform dpap t d 1 to a matrix a before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed this is often called balancing this paper describes several balancing algorithms for sparse matrices and compares them against each other and the traditional dense algorithm we first discuss our sparse implementation of the dense algorithm our code is faster than the dense algorithm when the density of the matrix is no more than approximately 5 and is much faster for large sparse matrices we next describe a set of randomized balancing algorithms for matrices that are not given explicitly i e given a vector x we can compute only ax and perhaps a t x we motivate these krylov based algorithms using perron frobenius theory results are given comparing the krylov based algorithms to each other and to the sparse and dense direct balancing algorithms looking at norm reduction running times and the accuracy of eigenvalues computed after a matrix is balanced we conclude that sparse balancing algorithms are efficient preconditioners for eigensolvers

balancing sparse hamiltonian eigenproblems balancing a matrix by a simple and accurate similarity transformation can improve the speed and accuracy of numerical methods for computing eigenvalues we describe balancing strategies for a large and sparse hamiltonian matrix h it is first shown how to permute h to irreducible form while retaining its structure this form can be used to decompose the hamiltonian eigenproblem into smaller sized problems next we discuss the computation of a symplectic scaling matrix d so that the norm of d 1 h d is reduced the considered scaling algorithm is solely based on matrix vector products and thus particularly suitable if the elements of h are not explicitly given the merits of balancing for eigenvalue computations are illustrated by several practically relevant examples c 2004 elsevier inc all rights reserved

negative association of random variables with applications random variables x 1 cdots x k are said to be negatively associated na if for every pair of disjoint subsets a 1 a 2 of 1 2 cdots k operatorname cov lbrack f x 1 i in a 1 g x j j in a 2 rbrack leq 0 for all nondecreasing functions f g the basic properties of negative association are derived especially useful is the property that nondecreasing functions of mutually exclusive subsets of na random variables are na this property is shown not to hold for several other types of negative dependence recently proposed one consequence is the inequality p x i leq x i i 1 cdots k leq prod k 1p x i leq x i for na random variables x 1 cdots x k and the dual inequality resulting from reversing the inequalities inside the square brackets in another application it is shown that negatively correlated normal random variables are na other na distributions are the a multinomial b convolution of unlike multinomials c multivariate hypergeometric d dirichlet and e dirichlet compound multinomial negative association is shown to arise in situations where the probability measure is permutation invariant applications of this are considered for sampling without replacement as well as for certain multiple ranking and selection procedures in a somewhat striking example na and positive association representing quite strong opposing types of dependence are shown to exist side by side in models of categorical data analysis

on the tail of the waiting time in a markov modulated m g 1 queue we show that the exponential decay parameter of the waiting time in a markov modulatedm g 1 queue is no larger than that of the correspondingm g 1 queue with averaged parameters and we give a necessary and sufficient condition for equality we also explore the effect of speeding up the modulation process a key tool is a markov modulated fluid model

towards a theory of negative dependence the fkg theorem says that the positive lattice condition an easily checkable hypothesis which holds for many natural families of events implies positive association a very useful property thus there is a natural and useful theory of positively dependent events there is as yet no corresponding theory of negatively dependent events there is however a need for such a theory this paper unfortunately contains no substantial theorems its purpose is to present examples that motivate a need for such a theory give plausibility arguments for the existence of such a theory outline a few possible directions such a theory might take and state a number of specific conjectures which pertain to the examples and to a wish list of theorems

singularity confinement test for ultradiscrete equations with parity variables we present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign we construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations

stochastic monotonicity and realizable monotonicity we explore and relate two notions of monotonicity stochastic and realizable for a system of probability measures on a common finite partially ordered set poset f when the measures are indexed by another poset a we give counterexamples to show that the two notions are not always equivalent but for various large classes of f we also present conditions on the poset a that are necessary and sufficient for equivalence when a f the condition that the cover graph of f have no cycles is necessary and sufficient for equivalence this case arises in comparing applicability of the perfect sampling algorithms of propp and wilson and the first author of the present paper

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

on a combinatorial conjecture concerning disjoint occurrences of events recently van den berg and kesten have obtained a correlation like inequality for bernoulli sequences this inequality which goes in the opposite direction of the fkg inequality states that the probability that two monotone i e increasing or decreasing events occur disjointly is smaller than the product of the individual probabilities they conjecture that the monotonicity condition is immaterial i e that the inequality holds for all events in the present paper we try to make clear the intuitive meaning of the conjecture and prove some nontrivial special cases one of which a pure correlation inequality is an extension of harris fkg inequality

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

minimal spanning forests minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs these limits can be taken with free or wired boundary conditions and are denoted fmsf free minimal spanning forest and wmsf wired minimal spanning forest respectively the wmsf is also the union of the trees that arise from invasion percolation started at all vertices we show that on any cayley graph where critical percolation has no infinite clusters all the component trees in the wmsf have one end a s in z d this was proved by alexander ann probab 23 1995 87 104 but a different method is needed for the nonamenable case we also prove that the wmsf components are thin in a different sense namely on any graph each component tree in the wmsf has p c 1 a s where p c denotes the critical probability for having an infinite cluster in bernoulli percolation on the other hand the fmsf is shown to be thick on any connected graph the union of the fmsf and independent bernoulli percolation with arbitrarily small parameter is a s connected in conjunction with a recent result of gaboriau this implies that in any cayley graph the expected degree of the fmsf is at least the expected degree of the fsf the weak limit of uniform spanning trees we also show that the number of infinite clusters for bernoulli p u percolation is at most the number of components of the fmsf where p u denotes the critical probability for having a unique infinite cluster finally an example is given to show that the minimal spanning tree measure does not have negative associations

on the entropy of compound distributions on nonnegative integers some entropy comparison results are presented concerning compound distributions on nonnegative integers the main result shows that under a log concavity assumption two compound distributions are ordered in terms of shannon entropy if both the ldquonumbers of claimsrdquo and the ldquoclaim sizesrdquo are ordered accordingly in the convex order several maximum minimum entropy theorems follow as a consequence most importantly two recent results of johnson 2008 on maximum entropy characterizations of compound poisson and compound binomial distributions are proved under fewer assumptions and with simpler arguments

monotonicity thinning and discrete versions of the entropy power inequality we consider the entropy of sums of independent discrete random variables in analogy with shannon s entropy power inequality where equality holds for normals in our case infinite divisibility suggests that equality should hold for poisson variables we show that some natural analogues of the epi do not in fact hold but propose an alternative formulation which does always hold the key to many proofs of shannon s epi is the behavior of entropy on scaling of continuous random variables we believe that renyi s operation of thinning discrete random variables plays a similar role to scaling and give a sharp bound on how the entropy of ultra log concave random variables behaves on thinning in the spirit of the monotonicity results established by artstein ball barthe and naor we prove a stronger version of concavity of entropy which implies a strengthened form of our discrete epi

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

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

do ultradiscrete systems with parity variables satisfy the singularity confinement criterion ultradiscrete singularity confinement test which is an integrability detector for ultradiscrete equations with parity variables is applied to various ultradiscrete equations the ultradiscrete equations exhibit singularity structures analogous to those of the discrete counterparts exact solutions to linearisable ultradiscrete equations are also constructed to explain the singularity structures

the law of the iterated logarithm for positively dependent random variables the levy s type maximal inequality is a key to establish the law of the iterated logarithm for associated random variables unfortunately this type inequality cannot be obtained for a generalization of association i e linear positive quadrant dependence because of their special dependence structure the purpose of this paper is to provide a different approach to obtain a law of the iterated logarithm for a sequence of linear positive quadrant dependent random variables

percolation beyond ℤd the contributions of oded schramm oded schramm 1961 2008 influenced greatly the development of percolation theory beyond the usual d setting in particular the case of nonamenable lattices here we review some of his work in this field

g majorization group induced cone orderings and reflection groups a vector y is g majorized by a vector x if y is an element of the convex hull of the orbit of x under the action of a group g it is known that if g is a finite reflection group then g majorization is equivalent to a group induced cone ordering in this paper it is established that if for a finite subgroup g of the orthogonal group g majorization is equivalent to a group induced cone ordering then g must be a reflection group

long range properties of spanning trees we compute some large scale properties of the uniform spanning tree process on bounded regions in z2 in particular we compute the distribution of the meeting point of the branches of the tree issued from three boundary points we also compute the crossing probabilities of branches of the tree on rectangular and annular regions as well as the winding number of the branches of the tree

indistinguishability of trees in uniform spanning forests we prove that in both the free and the wired uniform spanning forest fusf and wusf of any unimodular random rooted network in particular of any cayley graph it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely this confirms a conjecture of benjamini et al ann probab 29 1 1 65 2001 we also answer positively two additional questions of benjamini et al ann probab 29 1 1 65 2001 under the assumption of unimodularity we prove that on any unimodular random rooted network the fusf is either connected or has infinitely many connected components almost surely and if the fusf and wusf are distinct then every component of the fusf is transient and infinitely ended almost surely all of these results are new even for cayley graphs

indistinguishability of the components of random spanning forests we prove that the infinite components of the free uniform spanning forest fusf of a cayley graph are indistinguishable by any invariant property given that the forest is different from its wired counterpart similar result is obtained for the free minimal spanning forest fmsf we also show that with the above assumptions there can only be 0 1 or infinitely many components which solves the problem for the fusf of caylay graphs completely these answer questions by benjamini lyons peres and schramm for cayley graphs which have been open up to now our methods apply to a more general class of percolations those satisfying weak insertion tolerance and work beyond cayley graphs in the more general setting of unimodular random graphs

subsequential scaling limits of simple random walk on the two dimensional uniform spanning tree the first main result of this paper is that the law of the rescaled two dimensional uniform spanning tree is tight in a space whose elements are measured rooted real trees continuously embedded into euclidean space various properties of the intrinsic metrics measures and embeddings of the subsequential limits in this space are obtained with it being proved in particular that the hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8 5 in addition the tightness result is applied to deduce that the annealed law of the simple random walk on the two dimensional uniform spanning tree is tight under a suitable rescaling for the limiting processes which are diffusions on random real trees embedded into euclidean space detailed transition density estimates are derived

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

harmonic models and spanning forests of residually finite groups abstract we prove a number of identities relating the sofic entropy of a certain class of non expansive algebraic dynamical systems the sofic entropy of the wired spanning forest and the tree entropy of cayley graphs of residually finite groups we also show that homoclinic points and periodic points in harmonic models are dense under general conditions

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

majorization polyhedra and statistical testing problems abstract there are important connections between majorization and convex polyhedra both weak majorization and majorization are preorders related to certain simple convex cones we investigate the facial structure of a polyhedral cone c associated with a layered directed graph a generalization of weak majorization based on c is introduced it defines a preorder of matrices an application in statistical testing theory is discussed in some detail

transfer current and pattern fields in spanning trees when a simply connected domain d subset mathbb r d d ge 2 is approximated in a good way by embedded connected weighted graphs we prove that the transfer current matrix defined on the edges of the graph viewed as an electrical network converges up to a local weight factor to the differential of green s function on d this observation implies that properly rescaled correlations of the spanning tree model and correlations of minimal subconfigurations in the abelian sandpile model have a universal and conformally covariant limit we further show that on a periodic approximation of the domain all pattern fields of the spanning tree model as well as the minimal pattern e g zero height fields of the sandpile converge weakly in distribution to gaussian white noise

negative dependence and the geometry of polynomials we introduce the class of strongly rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter this class covers important models such as determinantal measures e g product measures and uniform random spanning tree measures and distributions for symmetric exclusion processes we show that strongly rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to liggett pemantle and wagner respectively moreover we extend lyons recent results on determinantal measures and we construct counterexamples to several conjectures of pemantle and wagner on negative dependence and ultra log concave rank sequences

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

order restricted statistical tests on multinomial and poisson parameters the starshaped restriction abstract likelihood ratio statistics for i testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped cf shaked ann statist 1979 and for ii testing the null hypothesis that this parameter vector is starshaped are considered for both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi bar square distribution analogous tests on a collection of poisson means are also found to have asymptotic chi bar square distributions author

do integrable cellular automata have the confinement property we analyse a criterion introduced by joshi and lafortune for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure we show that while this criterion can be used in order to single out integrable ultradiscrete systems there do exist cases where the system is nonintegrable and still the criterion is satisfied conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied we investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be finally we comment on the growth properties of ultradiscrete systems

the shift inverted j lanczos algorithm for the numerical solutions of large sparse algebraic riccati equations abstract the goal of solving an algebraic riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left half plane the purpose of this paper is to propose a structure preserving lanczos type algorithm incorporated with shift and invert techniques named shift inverted j lanczos algorithm for computing the stable invariant subspace for large sparse hamiltonian matrices the algorithm is based on the j tridiagonalization procedure of a hamiltonian matrix using symplectic similarity transformations we give a detailed analysis on the convergence behavior of the j lanczos algorithm and present error bound analysis and paige type theorem numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high speed vehicles are reported

again linearizable mappings abstract we examine a family of three point mappings that include mappings solvable through linearization the different origins of mappings of this type are examined projective equations and gambier systems the integrable cases are obtained through the application of the singularity confinement criterion and are explicitly integrated

the component graph of the uniform spanning forest transitions in dimensions 9 10 11 ldots 9 10 11 we prove that the uniform spanning forests of mathbb z d and mathbb z ell have qualitatively different connectivity properties whenever ell d ge 4 in particular we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex which we call the component graph we introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8 to separate dimensions 5 6 7 and 8 we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph our result sharpens a theorem of benjamini kesten peres and schramm who proved that the diameter of the component graph increases by one every time the dimension increases by four

on the maximum entropy properties of the binomial distribution it is shown that the binomial n p distribution maximizes the entropy in the class of ultra log concave distributions of order n with fixed mean np this result which extends a theorem of shepp and olkin 1981 is analogous to that of johnson 2007 who considers the poisson case the proof constructs a markov chain whose limiting distribution is binomial n p and shows that the entropy never decreases along the iterations of this markov chain

ultradiscretization without positivity we present a new ultradiscretization approach which can be applied to discrete systems the solutions of which are not positive definite this was made possible thanks to an ansatz involving the hyperbolic sine function we apply this new procedure to simple mappings for the linear and homographic mappings we obtain ultradiscrete forms and explicitly construct their solutions two discrete painleve ii equations are also analysed and ultradiscretized we show how to construct the ultradiscrete analogues of their rational and airy type solutions