cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

infinite dimensional diffusion processes as gibbs measures on c 0 1 z d an infinite lattice system of interacting diffusion processes is characterized as a gibbs distribution on c 0 1 z d with continuous local conditional probabilities using estimates for the vasserstein metric onc 0 1 dobrushin s contraction technique is applied in order to obtain information about macroscopic properties of the entire diffusion process

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

gibbs non gibbs transitions via large deviations computable examples we give new and explicitly computable examples of gibbs non gibbs transitions of mean field type using the large deviation approach introduced in van enter et al in mosc math j 10 687 711 2010 these examples include brownian motion with small variance and related diffusion processes such as the ornstein uhlenbeck process as well as birth and death processes we show for a large class of initial measures and diffusive dynamics both short time conservation of gibbsianness and dynamical gibbs non gibbs transitions

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

on asymptotically equivalent shallow water wave equations abstract the integrable third order korteweg de vries kdv equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves however at quadratic order this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other under a group of nonlinear non local normal form transformations introduced by kodama in combination with the application of the helmholtz operator these kodama helmholtz kh transformations are used to present connections between shallow water waves the integrable fifth order kdv equation and a generalization of the camassa holm ch equation that contains an additional integrable case the dispersion relation of the full water wave problem and any equation in this family agree to fifth order the travelling wave solutions of the ch equation are shown to agree to fifth order with the exact solution

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

coupled camassa holm equations n peakons and infinitely many conservation laws abstract three coupled camassa holm equations are proposed which are associated with a 4 4 matrix spectral problem with two potentials with the aid of the zero curvature equation we derive a hierarchy of new nonlinear evolution equations and establish their bi hamiltonian structures the three coupled camassa holm equations are exactly negative flows in the hierarchy and exact solutions with n peakons and infinite sequences of conserved quantities are discussed

distances defined by neighborhood sequences abstract this paper investigates general properties of distance functions defined over digitized space we assume that a distance between two points is defined as the length of a shortest path connecting them in the underlying graph which is defined by a given neighborhood sequence many typical distance functions can be described in this form but there are cases in which given neighborhood sequence do not define distance functions we first derive a necessary and sufficient condition for a neighborhood sequence to define a distance function we then discuss another important problem of estimating how tight such distances can approximate the euclid distance from the view point of relative error and absolute error

distance functions defined by variable neighborhood sequences abstract in the field of pattern recognition many researchers adopt the definition that distance between two points x and y on digitized space is the length of the shortest path from x to y determined by a specific sequence of neighborhood forms the diamond distance and various octagonal distances are typical examples of these kinds of distances however not necessarily every sequence of neighborhood forms does define a distance function thus in this paper we present a necessary and sufficient condition for a sequence of neighborhood forms to define a distance function two applications of this condition are also presented

a simple inductive approach to the problem of convergence of cluster expansions of polymer models we explain a simple inductive method for the analysis of the convergence of cluster expansions taylor expansions mayer expansions for the partition functions of polymer models we give a very simple proof of the dobrushin koteck preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used

some remarks on ultracontractivity we study the semi group t t generated by the operator 12 u on the lebesgue space l2 rd with the measure e u we prove via different methods using probabilistic techniques or pde arguments that t t is ultracontractive i e for t 0 it maps l1 into l when the function u satisfies a growth condition at infinity which is essentially for instance when the dimension d 1 the integrability of 1 u at infinity also we consider the analogous properties of the semi group generated by the fractional powers of the above operator

propagation of gibbsianness for infinite dimensional gradient brownian diffusions we study the strong gibbsian character on mathbb r mathbb z d of the law at time t of an infinite imensional gradient brownian diffusion when the initial distribution is gibbsian

digital distance functions on three dimensional grids in this paper we examine five different three dimensional grids suited for image processing digital distance functions are defined on the cubic face centered cubic body centered cubic honeycomb and diamond grids we give the parameters that minimize an error function that favors distance functions with low rotational dependency we also give an algorithm for computing the distance transform the tool by which these distance functions can be applied in image processing applications

global specifications and nonquasilocality of projections of gibbs measures reference gr pf article 1997 001doi 10 1214 aop 1024404514view record in web of science record created on 2008 11 27 modified on 2017 05 12

robustness of the non gibbsian property some examples we discuss some examples of measures on lattice systems which lack the property of being a gibbs measure in a rather strong sense

une approche gibbsienne des diffusions browniennes infini dimensionnelles nous montrons qu il y a correspondance biunivoque entre les lois de diffusions browniennes infini dimensionnelles et les etats de gibbs sur l espace des trajectoires omega c 0 1 mathbb z d ce resultat est ensuite applique aux problemes de transition de phases du retournement du temps et a l etude des mesures reversibles le principal outil est une caracterisation des etats de gibbs par une version infini dimensionnelle de la formule d integration par parties du calcul de malliavin

loss without recovery of gibbsianness during diffusion of continuous spins we consider a specific continuous spin gibbs distribution t 0 for a double well potential that allows for ferromagnetic ordering we study the time evolution of this initial measure under independent diffusions

the associated camassa holm equation and the kdv equation recently the associated camassa holm ach equation related to the fuchssteiner fokas camassa holm equation by a reciprocal transformation was introduced by schiff who derived backlund transformations by a loop group technique and used these to obtain some simple soliton and rational solutions we show how the ach equation is related to schrodinger operators and kdv and describe how to construct solutions of ach from tau functions of the kdv hierarchy rational n soliton and elliptic solutions are considered as well as exact solutions given by a particular case of the third painleve transcendent

distance functions in digital geometry abstract an analysis of paths and distances in n dimensions is carried out using variable neighborhood sequences a symbolic expression for the distance function between any two points in this quantized space is derived an algorithm for finding the shortest path is presented the necessary and sufficient condition for such distance functions to satisfy the properties of a metric has been derived certain practical and efficient methods to check for metric properties are also presented

variational description of gibbs non gibbs dynamical transitions for the curie weiss model we perform a detailed study of gibbs non gibbs transitions for the curie weiss model subject to independent spin flip dynamics infinite temperature dynamics we show that in this setup the program outlined in van enter fernandez den hollander and redig 3 can be fully completed namely that gibbs non gibbs transitions are equivalent to bifurcations in the set of global minima of the large deviation rate function for the trajectories of the magnetization conditioned on their endpoint as a consequence we show that the time evolved model is non gibbs if and only if this set is not a singleton for some value of the final magnetization a detailed description of the possible scenarios of bifurcation is given leading to a full characterization of passages from gibbs to non gibbs and vice versa with sharp transition times under the dynamics gibbsianness can be lost and can be recovered our analysis expands the work of ermolaev and kulske 7 who considered zero mag netic field and finite temperature spin flip dynamics we consider both zero and non zero magnetic field but restricted to infinite temperature spin flip dynamics our results re veal an interesting dependence on the interaction parameters including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in the optimal trajectories the numerical plots provided are obtained with the help of mathematica

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

a reciprocal transformation for the geng xue equation in this paper we construct a reciprocal transformation for the geng xue equation and show that with help of this transformation we relate the first negative flow of the modified boussinesq hierarchy to the geng xue equation furthermore we analyze the construction of conserved quantities and present new ones

on reciprocal transformation of a 3 component camassa holm type system abstract we construct a reciprocal transformation to connect a 3 component camassa holm type system proposed by geng and xue with the first negative flow in a generalized mkdv hierarchy we discuss the hamiltonian pair and infinitely many conserved quantities for the generalized mkdv hierarchy

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries