subharmonic oscillations of forced pendulum type equations where f is periodic with minimal period t and mean value zero we have in mind as a particular case the pendulum equation where g x a sin x first results on the existence of subharmonic orbits in a neighborhood of a given periodic motion were obtained by birkhoff and lewis cf 3 and 143 by perturbation type techniques rabinowitz 151 was able to prove the existence of subharmonic solutions for hamiltonian systems by the use of variational methods his approach is not of local type like the one in 3 and enables one to obtain a sequence of solutions whose minimal period tends toward infinity in the case when the hamiltonian function has subquadratic or superquadratic growth these results have been extended in various directions cf 2 5 6 8 13 16 181 local results on subharmonies for the forced pendulum equation can be found in 19 hamiltonian systems with periodic nonlinearity were studied by conley and zehnder 6 they proved the existence of subharmonic solutions under some assumptions on the nondegenerateness of the solutions by the use of morse conley theory in this paper we will prove the existence of subharmonic oscillations of a pendulum type equation by the use of classical morse theory together with an iteration formula for the index due to bott 4 and developed in 7 and l

dual morse index estimates for periodic solutions of hamiltonian systems in some nonconvex superquadratic cases in l ekeland and hofer stated a very general result for the existence of periodic solutions with any prescribed minimal period for hamiltonian systems assuming that the hamiltonian function h is strictly convex with superquadratic behaviour at the origin and at infinity the arguments given in l are based on an estimate for the morse index of a critical point of mountain pass type for a functional which is associated with the hamiltonian system via the duality principle of clarke and ekeland 2 unfortunately the functional is only c and not c then the result is achieved only through a finite dimensional reduction and as the authors themselves say through an inordinate amount of technicalities in this paper we give a direct definition of the morse index of a periodic solution which does not use the duality principle indeed it is based on a study of the morse indexes of approximating solutions obtained by a galerkin scheme such as the one introduced by rabinowitz in 3 taking into account some results about linking point theory recently stated in 4 71 we get the same index estimates as in l but in a simple way moreover some generalizations are obtained in the sense that the convexity assumption on h can be replaced either by a local convexity condition or by a condition on the second derivative of h plus eventually a symmetry condition see also 8 91 for some other results obtained in this framework in particular in the locally convex case the index estimates yield direct information on the minimal period of the corresponding solution finally let us remark that this definition of the morse index can be adapted to other nonconvex cases and one can still get some index estimates of the same type

on the distribution of the effective field for a spin glass on a cayley tree at zero temperature an ansatz for the probability distribution of the effective field for a spin glass on a cayley tree at zero temperature is proposed the continuous part of the distribution is not differentiable at the origin in contrast to an ansatz by katsura and in qualitative agreement with numerical results of de oliveira possible implications are discussed

monte carlo simulations of the ising spin glass on lattices with finite connectivity the authors have simulated the ising spin glass model on a random lattice with a finite average coordination number and also on the bethe lattice with various different boundary conditions in particular they have calculated the overlap function p q for two independent samples for the random lattice the results are consistent with a spin glass transition above which p q converges to a dirac delta function for large n number of sites and below which p q has in addition a long tail similar to previous results obtained for the infinite range model for the bethe lattice they obtain results in the interior by discarding the two outer shells of the cayley tree when calculating the thermal averages for fixed uncorrelated boundary conditions p q seems to converge to a delta function even below the spin glass transition whereas on a closed lattice correlated boundary conditions p q has a long tail similar to its behaviour in the random lattice case

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

the order parameter in a spin glass various possible precise definitions of an edwards anderson type of order parameter for an ising model spin glass are considered using boundary conditions for a finite system states of an infinite system and a duplicate system approach several of these definitions are shown to yield identical results

stochastic conservation laws weak in time formulation and strong entropy condition abstract this article is an attempt to complement some recent developments on conservation laws with stochastic forcing in a pioneering development feng and nualart 8 have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition however the authors formulation of entropy inequalities are weak in space but strong in time in the absence of a priori path continuity for the solutions we take a critical outlook towards this formulation and offer an entropy formulation which is weak in time and weak in space

the replica symmetric solution for potts models on d regular graphs we establish an explicit formula for the limiting free energy density log partition function divided by the number of vertices for ferromagnetic potts models on uniformly sparse graph sequences converging locally to the d regular tree for d even covering all temperature regimes this formula coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d regular tree the so called replica symmetric solution for uniformly random d regular graphs we further show that the replica symmetric bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions

multiple solutions of perturbed superquadratic second order hamiltonian systems abstract in this paper the author proves the existence of infinitely many distinct t periodic solutions of the perturbed second order hamiltonian systems q v q f t under the condition that v is c and superquadratic via minimax methods we also obtain similar results for general nonautonomous second order hamiltonian systems and perturbed lagrangian systems keywords invariance asymmetry a priori growth estimates multiple periodic solutions

ising spin glass in the bethe approximation at zero temperature abstract we study the properties of the ising spin glass in the bethe approximation at zero temperature for the case of a j distribution of bonds we show that there is an infinite number of solutions for the probability distribution function of the effective field each one is a sum of 2n 1 delta functions located at jn n n 0 1 2 n in the limit n we obtain a solution with a continous part in addition to delta functions located at 0 and j for each case we calculate the spin glass order parameter the energy and entropy

convergence of a flux splitting finite volume scheme for conservation laws driven by levy noise abstract we explore numerical approximation of multidimensional stochastic balance laws driven by multiplicative levy noise via flux splitting finite volume method the convergence of the approximations is proved towards the unique entropy solution of the underlying problem

on the absence of breakdown of symmetry for the plane rotator model with long range random interaction we study the plane rotator model with hamiltonian frac 1 2 sum limits x ne y frac jxy cos mathcal o x mathcal o y left x y right 3 mathcal e where jxy for different pair x y are independent symmetric unbounded random variables it is proved that for almost all j all gibbs states p j are invariant by rotation

spin glass order parameter on a bethe lattice an order parameter function for the spin glass state on a bethe lattice of coordination number z is calculated by the use of hierarchical boundary conditions the order parameter is discrete for finite z and becomes a continuous function in the limit z to infinity numerical calculations are performed at zero temperature and zero external field for several values of z

on the mean field ising model in a random external field we use a method developed by van hemmen to obtain the free energy of the mean field ising model in a random external magnetic field some results of previous mean field calculations are confirmed and generalized the tricritical point in the global phase diagram is discussed in detail we also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point

periodic solutions of superquadratic hamiltonian systems with bounded forcing terms on etudie l existence et la multiplicite des solutions periodiques pour le systeme hamiltonien z z t z ou z r r 2n z dz dt j o j j o j etant la matrice identite dans r n

periodic solutions of large norm of hamiltonian systems abstract hamiltonian systems of ordinary differential equations are studied the only assumption made on the hamiltonian is appropriately rapid growth at infinity it is proved that for any given period there is an unbounded sequence of periodic solutions of the system having the given period

one dimensional stochastic burgers equation driven by levy processes abstract in this paper we proved the global existence and uniqueness of the strong weak and mild solutions for one dimensional burgers equation perturbed by a poisson form process a poisson form and q wiener process with the dirichlet bounded condition we also proved the existence of the invariant measure of these models

glassy behaviour in the ferromagnetic ising model on a cayley tree we present a detailed study of the nearest neighbour ferromagnetic ising model on a cayley tree in the limit of zero field the system displays glassy behaviour below a crossover temperature that scales inversely with the logarithm of the number of generations thus is inversely proportional to the logarithm of the logarithm of the number of sites non gaussian magnetization distributions are observed for reminiscent of that associated with the central spin of the edwards anderson model on the same tree furthermore a dynamical study indicates metastability long relaxation times and ageing consistent with the development of glassy behaviour for a finite but macroscopic number of sites

phase ordering after a deep quench the stochastic ising and hard core gas models on a tree consider a low temperature stochastic ising model in the phase coexistence regime with markov semigroup p t a fundamental and still largely open problem is the understanding of the long time behavior of p t when the initial configuration is sampled from a highly disordered state e g a product bernoulli measure or a high temperature gibbs measure exploiting recent progresses in the analysis of the mixing time of monte carlo markov chains for discrete spin models on a regular b ary tree open image in new window we study the above problem for the ising and hard core gas independent sets models on open image in new window if is a biased product bernoulli law then under various assumptions on the bias and on the thermodynamic parameters we prove almost sure weak convergence of p t to an extremal gibbs measure pure phase and show that the limit is approached at least as fast as a stretched exponential of the time t in the context of randomized algorithms and if one considers the glauber dynamics on a large finite tree our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution

bethe lattice spin glass the effects of a ferromagnetic bias and external fields i bifurcation analysis we present a rigorous analysis of the j ising spin glass model on the bethe lattice with fixed uncorrelated boundary conditions phase diagrams are derived as a function of temperature vs concentration of ferromagnetic bonds and for a symmetric distribution of bonds external field vs temperature in this part we characterize the bulk ordered phases using bifurcation theory we prove the existence of a distribution of single site magnetizations far inside the lattice which is stable with respect to changes in the boundary conditions

the stability of subharmonic solutions for hamiltonian systems abstract in this paper by using the dual morse index theory we study the stability of subharmonic solutions of the non autonomous hamiltonian systems we obtain a infinite sequence of geometrically distinct periodic solutions such that every element has at most one direction of instability i e it has at least 2 n 2 floquet multipliers lying on the unit circle in the complex plane if the periodic solution is non degenerate or it is elliptic all its 2 n floquet multipliers are lying on the unit circle if the periodic solution is degenerate

the tightness of the kesten stigum reconstruction bound of symmetric model with multiple mutations it is well known that reconstruction problems as the interdisciplinary subject have been studied in numerous contexts including statistical physics information theory and computational biology to name a few we consider a 2q state symmetric model with two categories of q states in each category and 3 transition probabilities the probability to remain in the same state the probability to change states but remain in the same category and the probability to change categories we construct a nonlinear second order dynamical system based on this model and show that the kesten stigum reconstruction bound is not tight when q ge 4

second order phase transitions and tricritical points in the random field ising models abstract a method that applies to a wide class of the random field ising models is used to investigate the influence of random field on the critical phenomena the 3d random field ising models with nearest neighbour interactions exhibit tricritical behavior the obtained results express rather large deviations from the mean field theory for particular cases the formulation gives some well known exact results

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

subharmonics for convex nonautonomous hamiltonian systems on etudie des solutions periodiques du systeme hamiltonien jx h t x ou j est la matrice symplectique standard et h c 2 r r 2n r est t periodique en la premiere variable pour tout t x h t x note le gradient partiel par rapport a la seconde variable

glauber dynamics on trees boundary conditions and mixing time we give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the glauber dynamics in the so called bethe approximation specifically we show that the spectral gap and the log sobolev constant of the glauber dynamics for the ising model on an n vertex regular tree with boundary are bounded below by a constant independent of n at all temperatures and all external fields this implies that the mixing time is o logn in contrast to the free boundary case where it is not bounded by any fixed polynomial at low temperatures in addition our methods yield simpler proofs and stronger results for the spectral gap and log sobolev constant in the regime where the mixing time is insensitive to the boundary condition our techniques also apply to a much wider class of models including those with hard core constraints like the antiferromagnetic potts model at zero temperature proper colorings and the hard core lattice gas independent sets

large amplitude subharmonic oscillations for scalar second order differential equations with asymmetric nonlinearities abstract we provide sufficient conditions for the existence of subharmonic solutions for equations whose nonlinearity may have a different qualitative behaviour to the right and to the left of the origin the proofs are based on variational methods

subharmonic solutions of second order subquadratic hamiltonian systems with potential changing sign abstract with the aid of some symplectic transformations the existence of subharmonic solutions of the second order hamiltonian system v x t x is considered where the potential v is subquadratic in x as x goes to infinity and can change sign with respect to t

weak versus strong uniqueness of gibbs measures a regular short range example we provide an example of a nearest neighbour random model on a regular lattice which almost certainly has a unique disordered gibbs state for every boundary condition although by choosing interaction dependent boundary conditions one can obtain different gibbs states

the random cluster model on a homogeneous tree the random cluster model on a homogeneous tree is defined and studied it is shown that for 1 q 2 the percolation probability in the maximal random cluster measure is continuous inp while forq 2 it has a discontinuity at the critical valuep p c q it is also shown that forq 2 there is nonuniqueness of random cluster measures for an entire interval of values ofp the latter result is in sharp contrast to what happens on the integer lattice z d

potts models on feynman diagrams we investigate numerically and analytically potts models on thin random graphs generic feynman diagrams using the idea that such models may be expressed as the limit of a matrix model the thin random graphs in this limit are locally tree like in distinction to the fat random graphs that appear in the planar feynman diagram limit more familiar from discretized models of two dimensional gravity the interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree like structures such as the bethe lattice q state potts models display a first order transition in the mean field for q 2 so the thin graph potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation such discontinuous transitions also appear in multiple ising models on thin graphs and may have implications for the use of the replica trick in spin glass models on random graphs

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

on mean field equations for spin glasses in this paper we study rigorously the random ising model on a cayley tree in the limit of infinite coordination numberz 8 an iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice in this way we obtain local properties of the model in the thermodynamic limit of an infinite number of shells when the coupling constants are independent gaussian random variables the sk expressions emerge as stable fixed points of our scheme and provide a valid local mean field theory of spin glasses in which negative local entropy at low temperatures while perfectly possible mathematically may still perhaps be physically undesirable finally we examine the tap equations and show that if the average over bond disorder and the limitz 8 are actually performed one recovers our iterative scheme and hence the sk equations also in the thermodynamic limit

on nonlinear stochastic balance laws we are concerned with multidimensional stochastic balance laws we identify a class of nonlinear balance laws for which uniform spatial bv bound for vanishing viscosity approximations can be achieved moreover we establish temporal equicontinuity in l1 of the approximations uniformly in the viscosity coefficient using these estimates we supply a multidimensional existence theory of stochastic entropy solutions in addition we establish an error estimate for the stochastic viscosity method as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions various further generalizations of the results are discussed

origin magnetisation distribution of the site diluted ising model on a rooted cayley tree a stochastic iterative technique is used to obtain the origin magnetisation distribution of the site diluted ising ferromagnet on a rooted cayley tree with coordination number z 3 this distribution shows marked morphological changes with temperature crossing over from a broad distribution near the ferromagnetic phase boundary to a highly structured shape at lower temperatures the large scale structure of the intermediate and low temperature distributions a superposition of similarly shaped but scaled peaks can be understood on the basis of the contributions of clusters distinguished by their structure close to the origin

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

continuous dependence on the nonlinearities of solutions of degenerate parabolic equations explicit estimates for the continuous dependence inl 0 t l1 rd of solutions of the equation lv v v in 0 rdwith initial conditionv 0 h with respect to the nonlinear continuously differential functions and are established

stochastic scalar conservation laws we introduce a notion of stochastic entropic solution a la kruzkov but with ito s calculus replacing deterministic calculus this results in a rich family of stochastic inequalities defining what we mean by a solution a uniqueness theory is then developed following a stochastic generalization of l1 contraction estimate an existence theory is also developed by adapting compensated compactness arguments to stochastic setting we use approximating models of vanishing viscosity solution type for the construction while the uniqueness result applies to any spatial dimensions the existence result in the absence of special structural assumptions is restricted to one spatial dimension only

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

weighted averages and order parameters for the infinite range ising spin glass the sherrington kirkpatrick spin glass model 1975 is studied by replicas and by analysing the mean field equations of thouless anderson and palmer tap 1977 the authors show that the standard order parameter defined by statistical mechanics is given by the average of all off diagonal components of the matrix qalpha beta there is consequently no violation of the fluctuation dissipation theorem they describe how to weight the different solutions of the tap equations and argue that there is no entropy from solution degeneracy this assumption is shown to be internally consistent the square of a spin expectation value for a solution averaged over solutions is shown to be q 1 if they make the parisi sompolinsky ansatz for qalpha beta

inclusion of short range order in a mean field theory for the disordered magnetic lattice gas the disordered magnetic lattice gas dmlg as a unifying description of many simpler random spin systems has been investigated in an attempt to devise a mean field theory which goes beyond the infinitely long ranged model by incorporating short range order sro the authors have shown rigorously that the local thermodynamic properties of the dmlg on a cayley tree of finite coordination number z are identical to the thermodynamic properties of the dmlg in a pair approximation obtained by using the method of the distribution function further a modified pair approximation for the dmlg is presented which is exactly solvable it is formulated for general random bond distribution functions and is then examined for the special case of gaussian distributions

subharmonic solutions for second order differential systems ink this paper we consider the existence of subharmonic solutions for the problem utt g u t where g rn r is not necessarily convex and r rn is periodic with minimal period t 0

mean field theory for percolation models of the ising type the q 2 random cluster model is studied in the context of two mean field models the bethe lattice and the complete graph for these systems the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high density side which vindicates the results of earlier studies in particular the exponent which characterizes the divergence of the average size of finite clusters is 1 2 and the exponent associated with the length scale of finite clusters is 1 4 the full collection of exponents indicates an upper critical dimension of 6 the standard mean field exponents of the ising system are also present in this model 1 2 1 which implies in particular the presence of two diverging length scales furthermore the finite cluster exponents are stable to the addition of disorder which near the upper critical dimension may have interesting implications concerning the generality of the disordered system correlation length bounds

the dirichlet problem for a conservation law with a multiplicative stochastic perturbation abstract in this paper we are interested in the dirichlet boundary value problem for a multi dimensional nonlinear conservation law with a multiplicative stochastic perturbation using the concept of measure valued solutions and kruzhkov s semi entropy formulations a result of existence and uniqueness of the entropy solution is proved

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

existence of periodic solutions of a particular type of super quadratic hamiltonian systems abstract this paper considers the hamiltonian systems with new super quadratic conditions covering the case h t z p 1 q 1 ln 1 z where 1 with 1 1 1 and 2 1 using the mini max principle we obtain the existence of the periodic solutions

replica symmetry breaking in finite connectivity systems a large connectivity expansion at finite and zero temperature parisi replica symmetry breaking is extended to random bond systems with finite fixed connectivity m 1 the free energy f is explicitly calculated for large m within the first stage of symmetry breaking at finite temperature t the 1 m expansion is found to diverge as t to o indeed direct evaluation at zero temperature shows that the large m expansion is in powers of 1 m introduction of symmetry breaking brings the value of f close approximately 1 for 10 m 20 to numerical estimates of banavar et al the same techniques apply to systems with average finite connectivity

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

homoclinic orbits in a first order superquadratic hamiltonian system convergence of subharmonic orbits abstract we consider the existence of homoclinic orbits for a first order hamiltonian system z jh z t z we assume h t z is of form h t z 1 2 az z w t z where a is a symmetric matrix with ja i r and w t z is 2 periodic in t and has superquadratic growth in z we prove the existence of a nontrivial homoclinic solution z t and subharmonic solutions zt t t n i e 2 t periodic solutions of hs such that zt t z t in cloc1 r r 2n as t

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

explicit isoperimetric constants and phase transitions in the random cluster model the random cluster model is a dependent percolation model that has applications in the study of ising and potts models in this paper several new results are obtained for the random cluster model on nonamenable graphs with cluster parameter q geq 1 among these the main ones are the absence of percolation for the free random cluster measure at the critical value and examples of planar regular graphs with regular dual where p mathrm c mathrm free q p mathrm u mathrm wired q for q large enough the latter follows from considerations of isoperimetric constants and we give the first nontrivial explicit calculations of such constants such considerations are also used to prove nonrobust phase transition for the potts model on nonamenable regular graphs

continuous dependence estimate for conservation laws with levy noise abstract we are concerned with multidimensional stochastic balance laws driven by levy processes using bounded variation bv estimates for vanishing viscosity approximations we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that levy noise only depends on the solution this result is used to show the error estimate for the stochastic vanishing viscosity method in addition we establish fractional bv estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable

on the minimal periodic solutions of nonconvex superlinear hamiltonian systems abstract this paper proves a multiplicity result for the minimal periodic solutions of hamiltonian system x t j h x t under a superquadratic hypothesis on h weaker than the ambrosetti rabinowitz type condition we also define a class of homogeneous functions that are more general than the classical ones and prove the rabinowitz s conjecture in the case of h satisfying such new homogeneity

a remark on the notion of robust phase transitions we point out that the high qpotts model on a regular lattice at its transition temperature provides an example of a nonrobust in the sense recently proposed by pemantle and steif phase transition

scalar conservation laws with stochastic forcing we show that the cauchy problem for a randomly forced periodic multi dimensional scalar first order conservation law with additive or multiplicative noise is well posed it admits a unique solution characterized by a kinetic formulation of the problem which is the limit of the solution of the stochastic parabolic approximation

modulated structures of an ising model with competing nearest neighbour interactions the authors use a two dimensional mapping to analyse an ising model with competing ferromagnetic and antiferromagnetic nearest neighbour interactions on a cayley tree the phase diagram which had some resemblance to the case of an ising spin glass displays some tricritical points and many modulated phases

stiffness exponent number of pure states and almeida thouless line in spin glasses we show how an inequality for the stiffness exponent for spin glass models proposed by fisher and huse could be violated we analyze their derivation and point out that their scaling arguments do not apply to investigations of the difference between systems with periodic and antiperiodic boundary conditions as a consequence the possibility remains open that in sufficiently high dimensions an infinite number of pure states exists and that an almeida thouless line spin glass transition in a field occurs as is predicted in competing theories