brownian motion with restoring drift micro canonical ensemble and the thermodynamic limit we take up the old problem of micro canonical conditioning in the context of diffusion starting with a potential the schrodinger operator with ground state is carried by a conjugation into the diffusion generator with invariant density the latter motion is made micro canonical by first conditioning the path to be periodic and then further conditioning on the empirical mean square or particle number the thermodynamics are then studied by taking while d remains fixed the problem in this form owes its inception to mckean vaninsky cite mv2 who obtained the following result for with they showed the same type of diffusion appears in the thermodynamic limit but with drift arising from the shifted potential being such that the limiting mean square equals d their method of proof predicts the same outcome for so long as d is smaller than the canonical mean square while if the matter was unresolved the purpose of this note is to show a type of phase transition takes place in this case the conditioning is overcome in the limit and one sees the original stationary diffusion on the line the proof employs an entropy inequality due to csiszar 1

random tree type partitions as a model for acyclic polymerization holtsmark 3 2 stable distribution of the supercritical gel we study the asymptotic behavior in the thermodynamic limit as n of a particular probability distribution on the set of all forests of trees on a set of n elements monomers we establish the existence of three stages of polymerization dependent upon the ration of association and dissociation rates of monomers and we analyse the near critical and supercritical stages in the supercritical stage we discover that the molecular weight of the largest connected component gel has the holtsmark distribution

integrable dynamics of toda type on square and triangular lattices in a recent paper we constructed an integrable generalization of the toda law on the square lattice we construct other examples of integrable dynamics of toda type on the square lattice as well as on the triangular lattice as nonlinear symmetries of the discrete laplace equations on square and triangular lattices we also construct the ensuremath tau function formulations and darboux b acklund transformations of these dynamics

two integrable differential difference equations derived from the discrete bkp equation and their related equations in this paper two integrable differential difference equations derived from the discrete bkp equation and their related equations are studied the corresponding bilinear backlund transformations bts are presented starting from the bilinear bts soliton solutions of these equations are generated and the corresponding lax pairs are obtained

periodic wave solutions of the boussinesq equation the boussinesq equation usually arises in a physical problem as a long wave equation the present work extends the search of periodic wave solutions for it the hirota bilinear method and riemann theta function are employed in the process we also analyse the asymptotic property of periodic waves in detail furthermore it is of interest to note that well known soliton solutions can be reduced from the periodic wave solutions

a note on blow up of solutions of a quasilinear heat equation with vanishing initial energy abstract in this work we consider an initial boundary value problem related to the equation u t div u m 2 u f u and prove under suitable conditions on f a blow up result for solutions with vanishing or negative initial energy

entropy barriers and slow relaxation in some random walk models we study the zero temperature limit of a simple model of slow relaxation without energy barriers proposed by ritort 1995 as well as two other closely related models with a much faster relaxation these models can be mapped onto random walk problems which allows for their analytic study we analyse in particular a specific aspect of the former model namely the existence of a bias leading to entropy barriers and to a very slow relaxation

exact periodic wave solutions for the hkdv equation abstract in this paper by using the hirota bilinear method and the jacobian theta functions for the higher order kdv equation the existence of periodic wave solutions with one and two period are obtained the asymptotic properties of the periodic wave solutions are analyzed in detail it is shown that the well known soliton solutions can be reduced from the periodic wave solutions

bounds for blow up time for the heat equation under nonlinear boundary conditions a differential inequality technique is used to determine a lower bound on the blow up time for solutions to the heat equation subject to a nonlinear boundary condition when blow up of the solution does occur in addition a sufficient condition which implies that blow up does occur is determined

clustering in coagulation fragmentation processes random combinatorial structures and additive number systems asymptotic formulae and limiting laws we develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper in the case when the parametric function of the models is regularly varying with positive exponent the approach is based on khintchine s probabilistic method that grew out of the darwin fowler method in statistical physics our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters components as the total size of a structure number of particles goes to infinity we discover that n1 l 1 is the threshold for the limiting distribution of the largest cluster as a byproduct of our study we prove the independence of the numbers of groups of fixed sizes as n this is in accordance with the general principle of asymptotic independence of sites in mean field models the latter principle is commonly accepted in statistical physics but not rigorously proved

the structure of random partitions of large integers random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability the focus is on limit theorems as the number being partitioned approaches the limiting probability distribution of the appropriately normalized number of parts of some small size is exponential the large parts are described by a particular markov chain a central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes the major tool is a simple construction of random partitions that treats the number being partitioned as a random variable the same technique is useful when some restriction is placed on partitions such as the requirement that all parts must be distinct

lower current large deviations for zero range processes on a ring we study lower large deviations for the current of totally asymmetric zero range processes on a ring with concave current density relation we use an approach by jensen and varadhan which has previously been applied to exclusion processes to realize current fluctuations by travelling wave density profiles corresponding to non entropic weak solutions of the hyperbolic scaling limit of the process we further establish a dynamic transition where large deviations of the current below a certain value are no longer typically attained by non entropic weak solutions but by condensed profiles where a non zero fraction of all the particles accumulates on a single fixed lattice site this leads to a general characterization of the rate function which is illustrated by providing detailed results for four generic examples of jump rates including constant rates decreasing rates unbounded sublinear rates and asymptotically linear rates our results on the dynamic transition are supported by numerical simulations using a cloning algorithm

structure of the condensed phase in the inclusion process we establish a complete picture of condensation in the inclusion process in the thermodynamic limit with vanishing diffusion covering all scaling regimes of the diffusion parameter and including large deviation results for the maximum occupation number we make use of size biased sampling to study the structure of the condensed phase which can extend over more than one lattice site and exhibit an interesting hierarchical structure characterized by the poisson dirichlet distribution while this approach is established in other areas including population genetics or random permutations we show that it also provides a powerful tool to analyse homogeneous condensation in stochastic particle systems with stationary product distributions we discuss the main mechanisms beyond inclusion processes that lead to the interesting structure of the condensed phase and the connection to other generic particle systems our results are exact and we present monte carlo simulation data and recursive numerics for partition functions to illustrate the main points

matrix product representation of the stationary state of the open zero range process many one dimensional lattice particle models with open boundaries like the paradigmatic asymmetric simple exclusion process asep have their stationary states represented in the form of a matrix product with matrices that do not explicitly depend on the lattice site in contrast the stationary state of the open 1d zero range process zrp takes an inhomogeneous factorized form with site dependent probability weights we show that in spite of the absence of correlations the stationary state of the open zrp can also be represented in a matrix product form where the matrices are site independent non commuting and determined from algebraic relations resulting from the master equation we recover the known distribution of the open zrp in two different ways first using an explicit representation of the matrices and boundary vectors second from the sole knowledge of the algebraic relations satisfied by these matrices and vectors finally an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden markov chains

a direct method for dromion solutions of the davey stewartson equations and their asymptotic properties a new direct approach to determining a class of solutions including the dromion solutions of the davey stewartson equations is presented the form of solution obtained allows a detailed asymptotic analysis of the dromion solutions and compact expression for the phase shifts and changes of amplitude as a result of interaction of the dromions to be determined a class of solutions describing dromion scattering is discussed and computer plots of the interactions of such solutions are presented

laplace approximations for sums of independent random vectors we consider expressions of the form zn e exp nf sn n where sn is the sum of n i i d random vectors with values in a banach space and f is a smooth real valued function by results of donsker varadhan and bahadur zabell one knows that lim 1 n log zn supxf x h x where h is the so called entropy function in an earlier paper a more precise evaluation of zn is given in the case where there was a unique point maximizing f h and the curvature at the maximum was nonvanishing the present paper treats the more delicate problem where these conditions fail to hold

independent process approximations for random combinatorial structures abstract many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables conditioned on the value of a weighted sum of the variables it is interesting to compare the combinatorial structure directly to the independent discrete process without renormalizing the quality of approximation can often be conveniently quantified in terms of total variation distance for functionals which observe part but not all of the combinatorial and independent processes among the examples are combinatorial assemblies e g permutations random mapping functions and partitions of a set multisets e g polynomials over a finite field mapping patterns and partitions of an integer and selections e g partitions of an integer into distinct parts and square free polynomials over finite fields we consider issues common to all the above examples including equalities and upper bounds for total variation distances existence of limiting processes heuristics for good approximations the relation to standard generating functions moment formulas and recursions for computing densities refinement to the process which counts the number of parts of each possible type the effect of further conditioning on events of moderate probability large deviation theory and nonuniform measures on combinatorial objects and the possibility of getting useful results by overpowering the conditioning

integrable equations in 2 1 dimensions associated with symmetric and homogeneous spaces generalizations of the n wave davey stewartson and kadomtsev petviashvili equations associated with homogeneous and symmetric spaces are presented these equations are 2 1 dimensional generalizations of those presented by fordy and kulish commun math phys 89 427 1983 and athorne and fordy j phys a 20 1377 1987 examples are explicitly presented that are associated with the simplest spaces in particular a new single component 2 1 dimensional generalization of the kdv equation is presented

galton watson trees with vanishing martingale limit we show that an infinite galton watson tree conditioned on its martingale limit being smaller than varepsilon agrees up to generation k with a regular mu ary tree where mu is the essential minimum of the offspring distribution and the random variable k is strongly concentrated near an explicit deterministic function growing like a multiple of log 1 varepsilon more precisely we show that if mu ge 2 then with high probability as varepsilon downarrow 0 k takes exactly one or two values this shows in particular that the conditioned trees converge to the regular mu ary tree providing an example of entropic repulsion where the limit has vanishing entropy our proofs are based on recent results on the left tail behaviour of the martingale limit obtained by fleischmann and wachtel 11

refinements of the gibbs conditioning principle refinements of sanov s large deviations theorem lead via csiszar s information theoretic identity to refinements of the gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence sharp bounds on the growth of the block length with the length of the conditioning sequence are derived

domain growth in a one dimensional driven diffusive system the low temperature coarsening dynamics of a one dimensional ising model with conserved magnetisation and subject to a small external driving force is studied analytically in the limit where the volume fraction mu of the minority phase is small and numerically for general mu the mean domain size l t grows as t 1 2 in all cases and the domain size distribution for domains of one sign is very well described by the form p l l propto l l 3 exp lambda mu l 2 l 2 which is exact for small mu and possibly for all mu the persistence exponent for the minority phase has the value 3 2 for mu to 0

a new condition for the concavity method of blow up solutions to p laplacian parabolic equations abstract in this paper we consider an initial boundary value problem of the p laplacian parabolic equation u t x t div u x t p 2 u x t f u x t x t 0 u x t 0 x t 0 u x 0 u 0 0 x where p 2 and is a bounded domain of r n n 1 with smooth boundary the main contribution of this work is to introduce a new condition c p 0 u f s d s u f u u p u 0 for some 0 with 0 p 1 p p where 1 p is the first eigenvalue of p laplacian p and we use the concavity method to obtain the blow up solutions to the above equations in fact it will be seen that the condition c p improves the conditions ever known so far

quasi periodic waves and an asymptotic property for the asymmetrical nizhnik novikov veselov equation based on a multi dimensional riemann theta function the hirota bilinear method is extended to explicitly construct multi periodic quasi periodic wave solutions for the asymmetrical nizhnik novikov veselov equation among these periodic waves two periodic waves are a direct generalization of well known cnoidal waves their surface pattern is two dimensional the main physical result is the description of the behavior of nonlinear waves in shallow water a limiting procedure is presented to analyze asymptotic properties of the two periodic waves in details relations between the periodic wave solutions and the well known soliton solutions are established it is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit

domain number distribution in the nonequilibrium ising model we study domain distributions in the one dimensional ising model subject to zero temperature glauber and kawasaki dynamics the survival probability of a domain s t t and an unreacted domain q1 t t are characterized by two independent nontrivial exponents we develop an independent interval approximation that provides close estimates for many characteristics of the domain length and number distributions including the scaling exponents

finite size effects in dynamics of zero range processes the finite size effects prominent in zero range processes exhibiting a condensation transition are studied by using continuous time monte carlo simulations we observe that well above the thermodynamic critical point both static and dynamic properties display fluidlike behavior up to a density c l which is the finite size counterpart of the critical density c c l we determine this density from the crossover behavior of the average size of the largest cluster we then show that several dynamical characteristics undergo a qualitative change at this density in particular the size distribution of the largest cluster at the moment of relocation the persistence properties of the largest cluster and the correlations in its motion are studied

integrable generalization of the toda law to the square lattice we generalize the toda lattice or toda chain equation to the square lattice i e we construct an integrable nonlinear equation for a scalar field taking values on the square lattice and depending on a continuous time variable characterized by an exponential law of interaction in both discrete directions of the square lattice we construct the darboux b acklund transformations for such lattice and the corresponding formulas describing their superposition we finally use these darboux b acklund transformations to generate examples of explicit solutions of exponential and rational type the exponential solutions describe the evolution of one and two smooth two dimensional shock waves on the square lattice

on bilinear representations and infinite conservation laws of a nonlinear variable coefficient equation the kdv equation for non uniform media with relaxation effects has extensive applications in mathematics and physics satsuma 1976 20 and hirota 1979 21 once investigated its algebraic properties in the paper we shall generalize it a more general integrable equation with variable coefficients and further discuss the corresponding bilinear representation backlund transformation lax pair and infinite conservation laws by making use of the bell polynomials

subset source coding this paper studies the fundamental limits of storage for structured data where statistics and structure are both critical to the application accordingly a framework is proposed for optimal lossless and lossy compression of subsets of the possible realizations of a discrete memoryless source dms for the lossless subset compression problem it turns out that the optimal source code may not index the conventional source typical sequences but rather index certain subset typical sequences consistent with the subset of interest building upon an achievability and a strong converse an analytic expression is given based on the shannon entropy relative entropy and subset entropy which identifies such subset typical sequences for a broad class of subsets of a dms intuitively subset typical sequences belong to those typical sets which highly intersect the subset of interest but are still closest to the source distribution in the sense of relative entropy for the lossy subset compression problem an upper bound is derived on the subset rate distortion function in terms of the subset mutual information optimized over the set of conditional distributions that satisfy the expected distortion constraint with respect to the subset typical distribution and over a set of certain auxiliary subsets by proving a strong converse result this upper bound is shown to be tight for a class of symmetric subsets as shown in our numerical examples more often than not one achieves a gain in the fundamental limits in that the optimal compression rate for the subset in both the lossless and lossy settings can be strictly smaller than the source entropy and the source rate distortion function respectively although exceptions are also possible

hydrodynamic limit of mean zero condensing zero range processes with sub critical initial profiles condensing zero range processes are interacting particle systems with zero range interaction exhibiting phase separation at densities above a finite critical density we prove the hydrodynamic limit of mean zero condensing zero range processes with bounded local jump rate for sub critical initial profiles i e for initial profiles that are everywhere strictly below the critical density the proof is based on h t yau s relative entropy method and is made possible by a generalisation of the one block estimate

bayesian t optimal discriminating designs the problem of constructing bayesian optimal discriminating designs for a class of regression models with respect to the t optimality criterion introduced by atkinson and fedorov 1975a is considered it is demonstrated that the discretization of the integral with respect to the prior distribution leads to locally t optimal discriminating design problems with a large number of model comparisons current methodology for the numerical construction of discrimination designs can only deal with a few comparisons but the discretization of the bayesian prior easily yields to discrimination design problems for more than 100 competing models a new efficient method is developed to deal with problems of this type it combines some features of the classical exchange type algorithm with the gradient methods convergence is proved and it is demonstrated that the new method can find bayesian optimal discriminating designs in situations where all currently available procedures fail

condensation in perturbed zero range processes condensation phenomena in zero range processes have attracted significant attention recently here we consider zero range processes in which n sites are occupied by m particles that jump between sites according to a jump rate function given by g let z z1 z2 zn denote the steady state of a zero range process and let z n be the size of the maximum cluster a non complete condensation is defined as an event in which z n n converges to a positive constant less than 1 as n tends to infinity in this work we provide evidence that condensation depends on the detailed form of the jump rates by showing that non complete condensation occurs when g is given by where mk are bounded constants satisfying suitable conditions note that this choice of g is a perturbation of which exhibits perfect condensation i e n z n converges to 0 as n tends to infinity

laplace approximations for sums of independent random vectors ii degenerate maxima and manifolds of maxima we consider expressions of the form z n e exp nf s n n where s n is the sum of n i i d random vectors with values in a banach space and f is a smooth real valued function by results of donsker varadhan and bahadur zabell one knows that lim 1 n log z n sup x f x h x where h is the so called entropy function in an earlier paper a more precise evaluation of z n is given in the case where there was a unique point maximizing f h and the curvature at the maximum was nonvanishing the present paper treats the more delicate problem where these conditions fail to hold

an integrable symmetric 2 1 dimensional lotka volterra equation and a family of its solutions a symmetric 2 1 dimensional lotka volterra equation is proposed by means of a dependent variable transformation the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables a bilinear backlund transformation is found and then the corresponding lax pair is derived explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric 2 1 dimensional lotka volterra equation are given as a special case of the pfaffian solutions we obtain soliton solutions and dromions

discontinuous condensation transition and nonequivalence of ensembles in a zero range process we study a zero range process where the jump rates do not only depend on the local particle configuration but also on the size of the system rigorous results on the equivalence of ensembles are presented characterizing the occurrence of a condensation transition in contrast to previous results the phase transition is discontinuous and the system exhibits ergodicity breaking and metastable phases this leads to a richer phase diagram including nonequivalence of ensembles in certain phase regions the paper is motivated by results from granular clustering where these features have been observed experimentally

condensation of non reversible zero range processes in this article we investigate the condensation phenomena for a class of non reversible zero range processes on a fixed finite set by establishing a novel inequality bounding the capacity between two sets and by developing a robust framework to perform quantitative analysis on the metastability of non reversible processes we prove that the condensed site of the corresponding zero range processes approximately behaves as a markov chain on the underlying graph whose jump rate is proportional to the capacity with respect to the underlying random walk the results presented in the current paper complete the generalization of the work of beltran and landim probab theory relat fields 152 781 807 2012 on reversible zero range processes and that of landim commun math phys 330 1 32 2014 on totally asymmetric zero range processes on a one dimensional discrete torus

the random cluster model on the complete graph the random cluster model of fortuin and kasteleyn contains as special cases the percolation ising and potts models of statistical physics when the underlying graph is the complete graph onn vertices then the associated processes are called mean field in this study of the mean field random cluster model with parametersp n andq we show that its properties for any value ofq 0 may be derived from those of an erd s renyi random graph in this way we calculate the critical point c q of the model and show that the associated phase transition is continuous if and only ifq 2 exact formulae are given for c q the density of the largest component the density of edges of the model and the free energy this work generalizes earlier results valid for the potts model whereq is an integer satisfyingq 2 equivalent results are obtained for a fixed edge number random cluster model as a consequence of the results of this paper one obtains large deviation theorems for the number of components in the classical random graph models whereq 1

bell polynomial approach and n soliton solution for the extended lotka volterra equation in plasmas symbolically investigated in this paper is the extended lotka volterra elv equation which can govern the kinetics of the discrete peaks of the weak langmuir turbulence in plasmas without the linear damping and random noise binary bell polynomials are applied to the bilinearization of the discrete system bilinear backlund transformation of the elv equation is constructed n soliton solution in terms of the extended casorati determinant is also presented and verified propagation and interaction behaviors of the langmuir turbulence are analyzed it is demonstrated that the number of the interacting langmuir waves can influence the soliton velocity and amplitude as well as the collision phase shift graphic illustrations of the solitonic collisions show that the repulsion effects and nonlinear interactions are also associated with the number of the interacting langmuir waves

a martingale problem for an absorbed diffusion the nucleation phase of condensing zero range processes we prove uniqueness of a martingale problem with boundary conditions on a simplex associated to a differential operator with an unbounded drift we show that the solution of the martingale problem remains absorbed at the boundary once it attains it and that after hitting the boundary it performs a diffusion on a lower dimensional simplex similar to the original one we also prove that in the diffusive time scale condensing zero range processes evolve as this absorbed diffusion

negative order korteweg de vries equations in this paper based on the regular korteweg de vries kdv system we study negative order kdv nkdv equations particularly their hamiltonian structures lax pairs conservation laws and explicit multisoliton and multikink wave solutions thorough bilinear backlund transformations the nkdv equations studied in our paper are differential and actually derived from the first member in the negative order kdv hierarchy the nkdv equations are not only gauge equivalent to the camassa holm equation through reciprocal transformations but also closely related to the ermakov pinney systems and the kupershmidt deformation the bi hamiltonian structures and a darboux transformation of the nkdv equations are constructed with the aid of trace identity and their lax pairs respectively the single and double kink wave and bell soliton solutions are given in an explicit formula through the darboux transformation the one kink wave solution is expressed in the form of tanh while the one bell soliton is in the form of sech and both forms are very standard the collisions of two kink wave and two bell soliton solutions are analyzed in detail and this singular interaction differs from the regular kdv equation multidimensional binary bell polynomials are employed to find bilinear formulation and backlund transformations which produce n soliton solutions a direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the nkdv equations furthermore the relations between quasiperiodic wave solutions and soliton solutions are clearly described finally we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions

size of the largest cluster under zero range invariant measures we study the finite zero range process with occupancy dependent rate function g under the invariant measure which can be written explicitly in terms of g particles are distributed over sites and we regard all particles at a fixed site as a cluster in the density one case that is equal numbers of particles and sites we determine asymptotically the size of the largest cluster as the number of particles tends to infinity and determine its dependence on the rate function

absorption time in certain urn models we show that the average absorption time in a certain m urn model recently introduced by ritort in the context of glassy dynamics has a lower bound where n is the number of balls a monte carlo scheme which accurately estimates the absorption time even when indicates that for large n we have

quasi periodic waves of the n 1 supersymmetric modified korteweg de vries equation abstract based on the hirota bilinear method and the riemann theta function a straightforward way is shown to construct quasi periodic wave solutions of supersymmetric equations the resulting theory is applied to the supersymmetric modified korteweg de vries equation further we analyze the asymptotic properties of the solutions and give their asymptotic relations between the quasi periodic wave solutions and the soliton solutions

blow up and global solutions for a class of nonlinear reaction diffusion equations under dirichlet boundary conditions abstract in this paper we consider the following reaction diffusion equations under dirichlet boundary condition g u t u 2 u k t f u in 0 t u x t 0 on 0 t u x 0 u 0 x 0 in where is a bounded domain of r n n 2 with smooth boundary by constructing some appropriate auxiliary functions and using a first order differential inequality technique the upper and lower bounds on blow up time when blow up occurs are presented moreover conditions on the data to ensure that the solution exists globally or blows up at some finite time are also derived

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

optimal designs for the identification of the order of a fourier regression for the fourier regression model we determine optimal designs for identifying the order of periodicity it is shown that the optimal design problem for trigonometric regression models is equivalent to the problem of optimal design for discriminating between certain homo and heteroscedastic polynomial regression models these optimization problems are then solved using the theory of canonical moments and the optimal discriminating designs for the fourier regression model can be found explicitly in contrast to many other optimality criteria for the trigonometric regression model the optimal discriminating designs are not uniformly distributed on equidistant points

conditional limit theorems under markov conditioning let x 1 x 2 cdots be independent identically distributed random variables taking values in a finite set x and consider the conditional joint distribution of the first m elements of the sample x 1 cdots x n on the condition that x 1 x 1 and the sliding block sample average of a function h cdot cdot defined on x 2 exceeds a threshold alpha eh x 1 x 2 for m fixed and n rightarrow infty this conditional joint distribution is shown to converge m the m step joint distribution of a markov chain started in x 1 which is closest to x l x 2 cdots in kullback leibler information divergence among all markov chains whose two dimensional stationary distribution p cdot cdot satisfies sum p x y h x y geq alpha provided some distribution p on x 2 having equal marginals does satisfy this constraint with strict inequality similar conditional limit theorems are obtained when x 1 x 2 cdots is an arbitrary finite order markov chain and more general conditioning is allowed

optimal discriminating designs for several competing regression models the problem of constructing optimal designs for a class of regression models is considered we investigate a version of the tp optimality criterion as introduced by atkinson and fedorov 1975b and demonstrate that optimal designs with respect to this type of criteria can be obtained by solving nonlinear vector valued approximation problems we provide a characterization of the best approximations in this context and use these results to develop an efficient algorithm for the determination of the optimal discriminating designs the results are illustrated by fnumerical examples

metastability in a condensing zero range process in the thermodynamic limit zero range processes with decreasing jump rates are known to exhibit condensation where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value we study such a process on a one dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed super critical particle density we show that the process exhibits metastability with respect to the condensate location i e the suitably accelerated process of the rescaled location converges to a limiting markov process on the unit torus this process has stationary independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice our result extends previous work for fixed lattices and diverging density in beltran and landim probab theory relat fields 152 3 4 781 807 2012 and we follow the martingale approach developed there and in subsequent publications besides additional technical difficulties in estimating error bounds for transition rates the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells corresponding to a typical set of configurations with a particular condensate location the total exit rates from individual wells turn out to diverge in the limit which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells which is of a general nature and can be adapted to other models

on the critical set for fujita type blow up of solutions to the discrete laplacian parabolic equations with nonlinear source on networks abstract in this paper we are interested in long time behaviors of solutions to the discrete laplacian parabolic equations u t u f u with nonnegative and non trivial initial data in particular we assume that the function f is convex only on a short interval and f s f f s for 0 1 s 0 and we present a critical set depending on the function in the sense that if f is in the critical set then solutions blow up in finite time for any initial data and if not then solutions are global or blow up according to the size of initial data

construction of soliton cellular automaton from the vertex model the discrete 2d toda equation and the bogoyavlensky lattice we study the soliton cellular automaton sca in 2 1 dimensions from the viewpoint of the integrable vertex model as in our previous paper we relate the sca the so called box ball system to an integrable vertex model associated with the bogoyavlensky lattice we extend this framework and introduce the 2 1 dimensional sca which can be interpreted as the ultradiscretization of the 2d toda equation we also construct the n soliton solutions for this system

a graph valued markov process as rings allowed polymerization model subcritical behavior in previous papers random tree type partitions of a finite set as a stochastic model for acyclic polymerization were studied in this paper the model is extended one important step further by allowing rings formation the main result is that in the subcritical domain the asymptotic behavior of the process remains essentially the same as in the acyclic case despite the fact that cycles are not excluded

dynamics of the condensate in zero range processes for stochastic processes leading to condensation the condensate once it is formed performs an ergodic stationary state motion over the system we analyse this motion and especially its characteristic time for zero range processes the characteristic time is found to grow with the system size much faster than the diffusive time scale but not exponentially fast this holds both in the mean field geometry and on finite dimensional lattices in the generic situation where the critical mass distribution follows a power law the characteristic time grows as a power of the system size

two dimensional schro dinger operator inverse scattering transform and evolutional equations abstract the inverse problem for the two dimensional schrodinger operator on the data from one energy level is solved in a special class of finite zone or algebraic operators this class seems to be dense among all smooth periodic schrodinger operators evolutional equations associated with this problem are constructed

t optimal designs for discrimination between two polynomial models the paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree n 2 and n in a fundamental paper atkinson and fedorov 1975a proposed the t optimality criterion for this purpose recently atkinson 2010 determined t optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide a half of the circle into equal parts if the coefficient of xn 1 in the polynomial of larger degree vanishes in the present paper we give a strong justification of the conjecture and determine all t optimal designs explicitly for any degree n n in particular we show that there exists a one dimensional class of t optimal designs moreover we also present a generalization to the case when the ratio between the coefficients of xn 1 and xn is smaller than a certain critical value because of the complexity of the optimization problem t optimal designs have only been determined numerically so far and this paper provides the first explicit solution of the t optimal design problem since its introduction by atkinson and fedorov 1975a finally for the remaining cases where the ratio of coefficients is larger than the critical value we propose a numerical procedure to calculate the t optimal designs the results are also illustrated in an example ams subject classification 62k05

thermodynamic limit for the invariant measures in supercritical zero range processes we prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles it is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system we show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density this improves the result of groskinsky schutz and spohn where convergence is obtained for the finite dimensional marginals we obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

quasi periodic wave solutions for the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation abstract the hirota bilinear method is implemented to directly construct quasi periodic wave solutions in terms of theta functions for a hirota bilinear equation the result is applied to the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation thereby yielding their quasi periodic wave solutions under the backlund transformation u y 3 l n f x y t x

condensation for a fixed number of independent random variables a family of m independent identically distributed random variables indexed by a chemical potential 0 represents piles of particles as increases to the mean number of particles per site converges to a maximal density c the distribution of particles conditioned on the total number of particles equal to n does not depend on canonical ensemble for fixed m as n goes to infinity the canonical ensemble measure behave as follows removing the site with the maximal number of particles the distribution of particles in the remaining sites converges to the grand canonical measure with density c the remaining particles concentrate condensate on a single site

localization for a random walk in slowly decreasing random potential we consider a continuous time random walk x in a random environment on such that its potential can be approximated by the function v given by v x sigma w x frac b 1 alpha x 1 alpha where w a brownian motion with diffusion coefficient 0 and parameters b are such that b 0 and 0 1 2 we show that p a s where p is the averaged law lim t to infty frac x t c ln ln t 1 ln t frac 1 alpha 1 with c frac 2 alpha b sigma 2 1 2 alpha in fact we prove that by showing that there is a trap located around c ln ln t 1 ln t frac 1 alpha with corrections of smaller order where the particle typically stays up to time t this is in sharp contrast to what happens in the pure sinai s regime where the location of this trap is random on the scale ln2t

metastability of reversible condensed zero range processes on a finite set let r s x s r be the jump rates of an irreducible random walk on a finite set s reversible with respect to some probability measure m for alpha 1 let g n r be given by g 0 0 g 1 1 g k k k 1 alpha k 2 consider a zero range process on s in which a particle jumps from a site x occupied by k particles to a site y at rate g k r x y let n stand for the total number of particles in the stationary state as n up arrow infinity all particles but a finite number accumulate on one single site we show in this article that in the time scale n1 alpha the site which concentrates almost all particles evolves as a random walk on s whose transition rates are proportional to the capacities of the underlying random walk

on an integrable modified 2 1 dimensional lotka volterra equation in this paper an integrable 2 1 dimensional modified lotka volterra equation is considered this is a semi discrete system having discrete space and continuous time which has a known 1 1 dimensional self dual network equation as a reduction its bilinear backlund transformation and lax pair are also presented and explicit solutions including soliton solutions expressed in terms of pfaffians are obtained finally the reduction of these solutions to 1 1 dimensions is considered

an efficient design for model discrimination and parameter estimation in linear models summary we consider experimental designs in a regression set up where the unknown regression function belongs to a known family of nested linear models the objective of our design is to select the correct model from the family of nested models as well as to estimate efficiently the parameters associated with that model we show that our proposed design is able to choose the true model with probability tending to one as the number of trials grows to infinity we also establish that our selected design converges to the optimal design distribution for the true linear model ensuring asymptotic efficiency of least squares estimators of model parameters

phase transition for perfect condensation and instability under the perturbations on jump rates of the zero range process let z z1 z2 zn represent the steady state of a zero range process in which n sites are occupied by m particles with a jump rate between sites given by g if m n a particle density of 1 and z n is the maximum cluster size perfect condensation occurs if n z n converges to 0 in probability as n tends to infinity in this paper we improve the description of the conditions for perfect condensation first introduced by jeon et al 2000 ann prob 28 1162 and jeon and march 2000 stochastic models proc int conf on stochastic models in honor of professor donald a dawson ottawa canada 10 13 june 1998 p 233 applying the results to a few special cases we demonstrate the existence of an interesting phase transition and conclude that the maximum cluster size in a zero range process is unstable with respect to fluctuations in the jump rate g

binary bell polynomial manipulations on the integrability of a generalized 2 1 dimensional korteweg de vries equation abstract this paper investigates the integrability of a generalized 2 1 dimensional korteweg de vries equation with the aid of binary bell polynomials its bilinear formalism bilinear backlund transformation lax pair and darboux covariant lax pair are succinctly constructed which can be reduced to the ones of several integrable equations such as the korteweg de vries equation and the calogero bogoyavlenskii schiff equation moreover the infinite conservation laws of the generalized 2 1 dimensional korteweg de vries equation are found by virtue of binary bell polynomials all conserved densities and fluxes are given with explicit recursion formulas

two periodic waves and asymptotic property for generalized 2d toda lattice equation abstract in this paper two periodic wave solutions are constructed for the 2 1 dimensional generalized toda lattice equation by using hirota bilinear method and riemann theta function at the same time we analyze in details asymptotic properties of the two periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions

condensation in stochastic particle systems with stationary product measures we study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities we review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit this leads to strengthened results on weak convergence for subcritical systems and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions extending previous results which focused on attractive and finite systems we use relative entropy techniques which provide simple proofs making use of general versions of local limit theorems for independent random variables

quasi periodic wave solution and asymptotic behavior for the 2 1 dimensional toda lattice equation abstract one and two periodic wave solutions for the 2 1 dimensional toda lattice equation are presented based on the hirota bilinear method and the riemann theta function the asymptotic behaviors of these two solutions are considered and the rigorous proof is given that the periodic wave solutions tend to the soliton solutions in an appropriate limiting procedure

robust t optimal discriminating designs this paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the t optimality criterion introduced by atkinson and fedorov 1975a t optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification as a solution of this problem we propose a bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the t optimality criterion it is shown that the corresponding bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria for the problem of discriminating between two polynomial regression models which differ in the degree by two the robust t optimal discriminating designs can be found explicitly the results are illustrated in several examples ams subject classification 62k05

phase transitions on markovian bipartite graphs an application of the zero range process we analyze the existence and the size of the giant component in the stationary state of a markovian model for bipartite multigraphs in which the movement of the edge ends on one set of vertices of the bipartite graph is a zero range process the degrees being static on the other set the analysis is based on approximations by independent variables and on the results of molloy and reed for graphs with prescribed degree sequences the possible types of phase diagrams are identified by studying the behavior below the zero range condensation point as a specific example we consider the so called evans interaction in particular we examine the values of a critical exponent describing the growth of the giant component as the value of the dilution parameter controlling the connectivity is increased above the critical threshold rigorous analysis spans a large portion of the parameter space of the model exactly at the point of zero range condensation these results supplemented with conjectures supported by monte carlo simulations suggest that the phenomenological landau theory for percolation on graphs is not broken by the fluctuations

relaxation at late stages in an entropy barrier model for glassy systems the ground state dynamics of an entropy barrier model proposed recently for describing relaxation of glassy systems is considered at late stages of evolution the dynamics can be described by a simple variant of the ehrenfest urn model analytical expressions for the relaxation times from arbitrary initial states to the ground state are derived upper and lower bounds for the relaxation times as a function of system size are obtained