conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

analogues of non gibbsianness in joint measures of disordered mean field models it is known that the joint measures on the product of spin space and disorder space are very often non gibbsian measures for lattice systems with quenched disorder at low temperature are there reflections of this non gibbsianness in the corresponding mean field models we study the continuity properties of the conditional probabilities in finite volume of the following mean field models a joint measures of random field ising b joint measures of dilute ising c decimation of ferromagnetic ising the conditional probabilities are functions of the empirical mean of the conditionings so we look at the large volume behavior of these functions to discover non trivial limiting objects for a we find 1 discontinuous dependence for almost any realization and 2 dependence of the conditional probabilities on the phase in contrast to that we see continuous behavior for b and c for almost any realization this is in complete analogy to the behavior of the corresponding lattice models in high dimensions it shows that non gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean field models

on the gibbsian nature of the random field kac model under block averaging we consider the kac ising model in an arbitrary configuration of local magnetic fields left eta kern 1pt i right i quad quad in quad kern 1pt quad mathbb z d in any dimension d at any inverse temperature we investigate the gibbs properties of the renormalized infinite volume measures obtained by block averaging any of the gibbs measures corresponding to fixed with block length small enough compared to the range of the kac interaction we show that these measures are gibbs measures for the same renormalized interaction potential this potential depends locally on the field configuration and decays exponentially uniformly in for which we give explicit bounds the construction of the potential is based on a high temperature type cluster expansion

weak limit and blowup of approximate solutions to h systems abstract let h r 3 r be a continuous function such that h p h 0 r as p fixing a domain in r 2 we study the behaviour of a sequence u n of approximate solutions to the h system u 2 h u u x u y in assuming that sup p r 3 h p h 0 p 1 we show that the weak limit of the sequence u n solves the h system and u n u strongly in h 1 apart from a countable set s made by isolated points moreover if in addition h p h 0 o 1 p as p then in correspondence of each point of s we prove that the sequence u n blows either an h bubble or an h 0 sphere

improved perturbation expansion for disordered systems beating griffiths singularities we introduce a new expansion to prove exponential clustering of connected correlations in a large class of disordered systems our expansion converges for values of the temperature and magnetic field where standard cluster expansions diverge due to the presence of griffiths type singularities it is organized inductively over an infinite sequence of increasing distance scales in each induction step one redefines what is means by the unperturbed system a procedure somewhat reminiscent of k a m theory our techniques may be useful in dealing with the so called large field problem in real space renormalization group schemes

typical configurations for one dimensional random field kac model in this paper we study the typical profiles of a random field kac model we give upper and lower bounds of the space scale where the profiles are constant the results hold almost surely with respect to the realizations of the random field the analysis is based on a block spin construction deviation techniques for the local empirical order parameters and concentration inequalities for the realizations of the random magnetic field for the upper bound we exhibit a scale related to the law of the iterated logarithm where the random field makes an almost sure fluctuation that obliges the system to break its rigidity for the lower bound we prove that on a smaller scale the fluctuations are not strong enough to allow this transition

the regularity for nonlinear parabolic systems of p laplacian type with critical growth abstract we study the holder regularity of weak solutions to the evolutionary p laplacian system with critical growth on the gradient we establish a natural criterion for proving that a small solution and its gradient are locally holder continuous almost everywhere actually our regularity result recovers the classical result in the case p 2 16 and can be applied to study the regularity of the heat flow for m dimensional h systems as well as the m harmonic flow

renormalization group in the uniqueness region weak gibbsianity and convergence we analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature we prove weak gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system the correlation length or the critical length for metastability or both thus to perturbatively treat the problem we have to use a scale adapted expansion moreover such a model below the critical temperature resembles a disordered system in the presence of griffiths singularity then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

the jacobian determinant revisited this paper is devoted to the study of the jacobian determinant of a map g from a smooth bounded open subset of rn into rn n 2 more generally could be a smooth bounded open subset of an n dimensional manifold starting with the seminal work of c b morrey 27 y reshetnyak 32 and j ball 1 it has been known that one can define the distributional jacobian determinant det g under fairly weak assumption on g in particular it is defined for all maps g w 1 n2 n 1 and also for all

large solutions for harmonic maps in two dimensions we seek critical points of the functionale u mathop smallint limits omega u 2 where is the unit disk in 2 andu s2 satisfies the boundary conditionu on we prove that if is not a constant thene has a local minimum which is different from the absolute minimum we discuss in more details the case where x y rx ry sqrt 1 r 2 andr 1

corrections to the critical temperature in 2d ising systems with kac potentials we consider ad 2 ising system with a kac potential whose mean field critical temperature is 1 calling 0 the kac parameter we prove that there existsc 0 so that the true inverse critical temperature cr 1 by2 log 1 for anyb c and correspondingly small we also show that if 0 andb c suitably then the correlation functions normalized and rescaled converge to those of a non gaussian euclidean field theory

relative entropy and variational properties of generalized gibbsian measures we study the relative entropy density for generalized gibbs measures we first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of almost gibbsian measures almost sure continuity of conditional probabilities for quasilocal measures we obtain a full variational principle for the joint measures of the random field ising model we show that the weak gibbs property holds with an almost surely rapidly decaying translation invariant potential for these measures we show that the variational principle fails as soon as the measures lose the almost gibbs property these examples suggest that the class of weakly gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

the dirichlet problem for harmonic maps from riemannian polyhedra to spaces of upper bounded curvature this is a continuation of the cambridge tract harmonic maps between riemannian polyhedra by j eells and the present author the variational solution to the dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary and thereby uniquely determined the domain space is a compact admissible riemannian polyhedron with boundary while the target can be for example a simply connected complete geodesic space of nonpositive alexandrov curvature alternatively the target may have upper bounded curvature provided that the maps have a suitably small range essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain

the rigidity theorems of self shrinkers via gauss maps abstract we study the rigidity results for self shrinkers in euclidean space by restriction of the image under the gauss map the geometric properties of the target manifolds carry into effect in the self shrinking hypersurface situation theorem 3 1 and theorem 3 2 not only improve the previous results but also are optimal in higher codimensional case using geometric properties of the grassmannian manifolds the target manifolds of the gauss map we give a rigidity theorem for self shrinking graphs

generalized autoregressive moving average models a class of generalized autoregressive moving average garma models is developed that extends the univariate gaussian arma time series model to a flexible observation driven model for non gaussian time series data the dependent variable is assumed to have a conditional exponential family distribution given the past history of the process the model estimation is carried out using an iteratively reweighted least squares algorithm properties of the model including stationarity and marginal moments are either derived explicitly or investigated using monte carlo simulation the relationship of the garma model to other models is shown including the autoregressive models of zeger and qaqish the moving average models of li and the reparameterized generalized autoregressive conditional heteroscedastic garch model providing the formula for its fourth marginal moment not previously derived the model is demonstrated by the application of the garma model with a negative binomial conditional distribution to

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

harmonic diffeomorphisms of the hyperbolic plane in this paper we consider the dirichlet problem at infinity for harmonic maps between the poincare model d of the hyperbolic plane h2 and solve this when given boundary data are c4 immersions of d oo the boundary at infinity of d to d oo also we present a construction of nonconformal harmonic diffeomorphisms of d and give a complete description of the boundary behavior including their first derivatives

low temperature dynamics of the curie weiss model periodic orbits multiple histories and loss of gibbsianness we consider the curie weiss model at initial temperature 0 1 in vanishing external field evolving under a glauber spin flip dynamics with temperature 0 1 we study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity bad points we provide a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending earlier work for the case of independent spin flip dynamics

uniqueness of positive solutions to the rotationally symmetric p harmonic map equations abstract let and be positive solutions on 0 to the rotationally symmetric p harmonic map equation on model manifolds m f and m g where f is assumed to be sufficiently large near infinity and g y 0 for y 0 we show that if and have the same limit at infinity then on 0

regularity for the evolution of p harmonic maps abstract this paper presents our study of regularity for p harmonic map heat flows we devise a monotonicity type formula of scaled energy and establish a criterion for a uniform regularity estimate for regular p harmonic map heat flows as application we show the small data global in the time existence of regular p harmonic map heat flow

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

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

unbounded positive entire solutions of rotationally symmetric harmonic map equations abstract we study the existence of unbounded positive entire c 2 solutions of the rotationally symmetric harmonic map equations using the existence result we solve the dirichlet problem at infinity for any nonnegative boundary value at infinity

a new look at time series of counts summary this paper proposes a simple new model for stationary time series of integer counts previous work has focused on thinning methods and classical time series autoregressive moving average difference equations in contrast our methods use a renewal process to generate a correlated sequence of bernoulli trials by superpositioning independent copies of such processes stationary series with binomial poisson geometric or any other discrete marginal distribution can be readily constructed the model class proposed is parsimonious non markov and readily generates series with either short or long memory autocovariances the model can be fitted with linear prediction techniques for stationary series as an example a stationary series with binomial marginal distributions is fitted to the number of rainy days in 210 consecutive weeks at key west florida

weakly gibbsian representations for joint measures of quenched lattice spin models can the joint measures of quenched disordered lattice spin models with finite range on the product of spin space and disorder space be represented as suitably generalized gibbs measures of an annealed system we prove that there is always a potential depending on both spin and disorder variables that converges absolutely on a set of full measure w r t the joint measure weak gibbsianness this positive result is surprising when contrasted with the results of a previous paper k6 where we investigated the measure of the set of discontinuity points of the conditional expectations investigation of a s gibbsianness in particular we gave natural negative examples where this set is even of measure one including the random field ising model further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations we apply them to various examples from this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero our proof uses a martingale argument that allows to cut an infinite volume analogue of the quenched free energy into local pieces along with generalizations of kozlov s constructions

on the blow up of heat flow for conformal 3 harmonic maps using a comparison theorem chang ding and ye 1992 proved a finite time derivative blow up for the heat flow of harmonic maps from d 2 a unit ball in r 2 to s 2 a unit sphere in r 3 under certain initial and boundary conditions we generalize this result to the case of 3 harmonic map heat flow from d 3 to s 3 in contrast to the previous case our governing parabolic equation is quasilinear and degenerate technical issues such as the development of a new comparison theorem have to be resolved

harmonic maps of bounded symmetric domains harmonic maps between complete and non compact manifolds have been studied by schoen yau li tam aviles choi micallef ding wang and others s y l t a c m and d w for further study it might be of interest to consider problems in various concrete manifolds which are more general than hyperbolic space bounded symmetric domains were introduced by e cartan c1 c2 and were systematically studied by hua look and siegel hi h2 l si those are specific cartan hadamard manifolds whose further geometrical and analytical properties should be explored such investigation might also imply more general results for complete and non compact manifolds the purpose of the present paper is to pursue this goal by the work done by y c wong w a bounded symmetric domain can be viewed as a pseudo grassmannian manifold of all the spacelike subspaces of dimension m in pseudo euclidean space r n of index n in sect 2 this interesting point of view is briefly introduced it is well known that the simplest bounded symmetric domain nm 2 can be identified with a product of two hyperbolic planes he in sect 3 we study harmonic maps into nn 2 via harmonic maps into hyperbolic space and obtain an interesting image shrinking property in the striking paper h o s the authors classified all complete constant mean curvature surfaces in r 3 and 4 with certain restricted gauss image one of the analogous properties in ambient minkowski space was already proved by a different approach in the author s previous paper xi also see a by ruh vilms theorem r v we can investigate submanifolds with parallel mean curvature in euclidean space via its harmonic gauss maps in the present

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

a bernstein type theorem for minimal hypersurfaces via gauss maps abstract let m be an n dimensional smooth oriented complete embedded minimal hypersurface in r n 1 with euclidean volume growth we show that if the image under the gauss map of m avoids some neighborhood of a half equator then m must be an affine hyperplane

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

phase transition in the 3d random field ising model we show that the three dimensional ising model coupled to a small random magnetic field is ordered at low temperatures this means that the lower critical dimension dl for the theory isdl 2 settling a long controversy on the subject our proof is based on an exact renormalization group rg analysis of the system this analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of imry and ma the rg acts in the space of ising models and in the space of random field distributions driving the former to zero temperature and the latter to zero variance

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

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

pseudo free energies and large deviations for non gibbsian fkg measures a large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on 0 1 zd is proven in this way a pseudo free energy and pressure is defined for ergodic systems the large deviations property holds with the usual scaling the case of non ergodic systems is also discussed a similar result holds for occupation times the perturbation by an external field is treated

analyzing multiple emotions over time by autoregressive negative multinomial regression models abstract this article presents an autoregressive random coefficient model with overdispersed negative multinomial marginal distributions for the analysis of heterogeneity and serial dependencies in multivariate longitudinal count data the model structure consists of four components that take into account a individual difference effects b random time effects c multiple event categories and d autodependencies the last component is based on a stochastic integer valued autoregressive process proposed by mckenzie the model is applied to analyze count data from a panel diary study about the relationship between personality factors and emotion experiences it is shown that there are large and stable individual personality differences in the incidence and duration of self reported emotional experiences theoretical and clinical implications of this result are discussed

subelliptic harmonic maps we study a nonlinear harmonic map type system of subelliptic pde in particular we solve the dirichlet problem with image contained in a convex ball

continuous spin mean field models limiting kernels and gibbs properties of local transforms we extend the notion of gibbsianness for mean field systems to the setup of general possibly continuous local state spaces we investigate the gibbs properties of systems arising from an initial mean field gibbs measure by application of given local transition kernels this generalizes previous case studies made for spins taking finitely many values to the first step in the direction to a general theory containing the following parts 1 a formula for the limiting conditional probability distributions of the transformed system it holds both in the gibbs and in the non gibbs regime and invokes a minimization problem for a constrained rate function 2 a criterion for gibbsianness of the transformed system for initial lipschitz hamiltonians involving concentration properties of the transition kernels and 3 a continuity estimate for the single site conditional distributions of the transformed system while 2 and 3 have provable lattice counterparts the characterization of 1 is stronger in m

a family of bivariate distributions generated by the bivariate bernoulli distribution abstract if x 1 x 2 is a sequence of independent bernoulli random variables the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution from both the binomial and the negative binomial distributions the poisson distribution is obtainable as a limit moreover gamma distributions integer shape parameters are limits of negative binomial distributions and the normal distribution is a limit of negative binomial poisson and gamma distributions these basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate bernoulli distribution in this article such extensions yielding a family of bivariate distributions are obtained and studied

two connections between random systems and non gibbsian measures in this contribution we discuss the role disordered or random systems have played in the study of non gibbsian measures this role has two main aspects the distinction between which has not always been fully clear 1 from disordered systems disordered systems can be used as a tool analogies with as well as results and methods from the study of random systems can be employed to investigate non gibbsian properties of a variety of measures of physical and mathematical interest 2 of disordered systems non gibbsianness is a property of various joint measures describing quenched disordered systems we discuss and review this distinction and a number of results related to these issues moreover we discuss the mean field version of the non gibbsian property and present some ideas how a kac limit approach might connect the finite range and the mean field non gibbsian properties

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

critical behavior of the two dimensional spin diluted ising model via the equilibrium ensemble approach the equilibrium ensemble approach to disordered systems is used to investigate the critical behavior of the two dimensional ising model in the presence of quenched random site dilution the numerical transfer matrix technique in semi infinite strips of finite width together with phenomenological renormalization and conformal invariance is particularly suited to putting the equilibrium ensemble approach to work a method by which to extract with great precision the critical temperature of the model is proposed and applied a more systematic finite size scaling analysis than in previous numerical studies has been performed a parallel investigation along the lines of the two main scenarios currently under discussion namely the logarithmic corrections scenario with critical exponents fixed in the ising universality class versus the weak universality scenario critical exponents varying with the degree of disorder is carried out in interpreting our data maximum care is constantly taken to be open to both possibilities a critical discussion shows that an unambiguous discrimination between the two scenarios is still not possible on the basis of the available finite size data s1063 651x 99 16110 5

interfaces and typical gibbs configurations for one dimensional kac potentials we consider a one dimensional ising spin system with a ferromagnetic kac potential j r j having compact support we study the system in the limit 0 below the lebowitz penrose critical temperature where there are two distinct thermodynamic phases with different magnetizations we prove that the empirical spin average in blocks of size 1 for any positive converges as 0 to one of the two thermodynamic magnetizations uniformly in the intervals of size p for any given positivep 1 we then show that the intervals where the magnetization is approximately constant have lengths of the order of exp c 1 c 0 and that when normalized they converge to independent variables with exponential distribution we show this by proving large deviation estimates and applying the ventsel and friedlin methods to gibbs random fields finally if the temperature is low enough we characterize the interface namely the typical magnetization pattern in the region connecting the two phases

boundary behavior of harmonic maps on non smooth domains and complete negatively curved manifolds abstract in this paper we develop a theory for harmonic maps which is analogous to the classical theory for bounded harmonic functions in harmonic analysis we solve the dirichlet problem in bounded lipschitz domains and manifolds with negative sectional curvatures generalize the wienner criterion and prove the fatou theorem for harmonic maps into convex balls

the low temperature phase of kac ising models we analyze the low temperature phase of ferromagnetic kax ising models in dimensionsd 2 we show that if the range of interactions is 1 then two disjoint translation invariant gibbs states exist if the inverse temperature satisfies 1 n where d 1 2d 2 d 1 for any e 0 the proof involves the blocking procedure usual for kac models and also a contour representation for the resulting long range almost continuous spin system which is suitable for the use of a variant of the peierls argument

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

on the evolution of harmonic maps in higher dimensions on etablit des resultats de regularite partielle et l existence de solutions regulieres globales pour le probleme d evolution pour des applications harmoniques avec des donnees petites

majority rule at low temperatures on the square and triangular lattices we consider the majority rule renormalization group transformation applied to nearest neighbor ising models for the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low then the transformation is not defined we use the methods of van enter fewrnandez and sokal who proved the renormalized measure is not gibbsian for 7 by 7 blocks if the temperature is too low for the triangular lattice we prove that a zero temperature majorityrule transformation may be defined the resulting renormalized hamiltonian is local with 14 different types of interactions

the regularity of harmonic maps into spheres and applications to bernstein problems we show the regularity of and derive a priori estimates for weakly harmonic maps from a riemannian manifold into a euclidean sphere under the assumption that the image avoids some neighborhood of a half equator the proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems we apply these results to the spherical and euclidean bernstein problems for minimal hypersurfaces obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in euclidean spaces are trivial

spin flip dynamics of the curie weiss model loss of gibbsianness with possibly broken symmetry we study the conditional probabilities of the curie weiss ising model in vanishing external field under a symmetric independent stochastic spin flip dynamics and discuss their set of points of discontinuity bad points we exhibit a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending the results for the corresponding lattice model where only partial answers can be obtained for initial temperature beta 1 geq 1 we prove that the time evolved measure is always gibbsian for frac 2 3 leq beta 1 1 the time evolved measure loses its gibbsian character at a sharp transition time for beta 1 frac 2 3 we observe the new phenomenon of symmetry breaking in the set of points of discontinuity bad points corresponding to non zero spin average appear at a sharp transition time and give rise to biased non gibbsianness of the time evolved measure these bad points become neutral at a later transition time while the measure stays non gibbs in our proof we give a detailed description of the phase diagram of a curie weiss random field ising model with possibly non symmetric random field distribution based on bifurcation analysis

on the ising model with random boundary condition the infinite volume limit behavior of the 2d ising model under possibly strong random boundary conditions is studied the model exhibits chaotic size dependence at low temperatures and we prove that the and phases are the only almost sure limit gibbs measures assuming that the limit is taken along a sparse enough sequence of squares in particular we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces in order to exclude mixtures as possible limit points a detailed multi scale contour analysis is performed

uniqueness and regularity of proper harmonic maps introduction the purpose of this paper is to study some uniqueness and regularity properties of the dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces in general if the metrics of two complete noncompact manifolds mm and n are given locally by ds2 imj gij dxzdxj and dsy en h 3 du du respectively then the energy density function of a c1 map u m n is defined by

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative