the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

persistence properties of a system of coagulating and annihilating random walkers we study a d dimensional system of diffusing particles that on contact either annihilate with probability 1 q ensuremath 1 or coagulate with probability q ensuremath 2 q ensuremath 1 in one dimension the system models the zero temperature glauber dynamics of domain walls in the q state potts model we calculate overline p m t the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly m ensuremath 1 coagulations using perturbative renormalization group analysis for dl2 we show that if the number of coagulations m is much less than the typical number m t then overline p m t ensuremath sim m ensuremath zeta d t ensuremath ensuremath theta with ensuremath theta dq q q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 ensuremath zeta 2q ensuremath 1 ensuremath epsilon 2q ensuremath 1 q ensuremath 1 1 2 aq ensuremath epsilon 2 o ensuremath epsilon 3 where q q ensuremath 1 q ensuremath epsilon 2 ensuremath d and a ensuremath 0 006 dots m t is shown to scale as m t ensuremath sim t d 2 ensuremath ensuremath delta where ensuremath delta d 1 ensuremath q q ensuremath 1 q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 in two dimensions we show that overline p m t ensuremath sim mathrm ln t q 3 ensuremath 2q mathrm ln m 2q ensuremath 1 2 t ensuremath 2q for m ensuremath ll t 2q ensuremath 1 we also derive an exact nonperturbative relation between the exponents namely ensuremath delta q ensuremath theta 1 ensuremath q the one dimensional results corresponding to ensuremath epsilon 1 are compared with results from monte carlo simulations

dynamical critical behaviours of the ising spin chain swendsen wang and wolff algorithms we study the zero temperature ising chain evolving according to the swendsen wang dynamics we determine analytically the domain length distribution and various historical characteristics e g the density of unreacted domains is shown to scale with the average domain length as l with 3 2 for the q state potts model 1 q 1 we also compute the domain length distribution for the ising chain endowed with the zero temperature wolff dynamics

ageing in one dimensional coagulation diffusion processes and the fredrickson andersen model we analyse the ageing dynamics of the one dimensional fredrickson andersen fa model in the nonequilibrium regime following a low temperature quench relaxation then effectively proceeds via diffusion limited pair coagulation dlpc of mobility excitations by employing a familiar stochastic similarity transformation we map exact results from the free fermion case of diffusion limited pair annihilation to dlpc crucially we are able to adapt the mapping technique to averages involving multiple time quantities this relies on knowledge of the explicit form of the evolution operators involved exact results are obtained for two time correlation and response functions in the free fermion dlpc process the corresponding long time scaling forms apply to a wider class of dlpc processes including the fa model we are thus able to exactly characterize the violations of the fluctuation dissipation theorem fdt in the ageing regime of the fa model we find nontrivial scaling forms for the fluctuation dissipation ratio fdr x x tw t but with a negative asymptotic value x 3 6 16 3 307 while this prevents a thermodynamic interpretation in terms of an effective temperature it is a direct consequence of probing fdt with observables that couple to activated dynamics the existence of negative fdrs should therefore be a widespread feature in non mean field systems

a new conformal invariant on 3 dimensional manifolds abstract by improving the analysis developed in the study of k yamabe problem we prove in this paper the following new inequality for any 3 dimensional riemannian manifold m 3 g of non negative scalar curvature m 1 g d v g 2 3 vol g m 2 g d v g equality holds if and only if m 3 g is a space form

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

statistical description of contact interacting brownian walkers on the line the distribution of interval lengths between brownian walkers on the line is investigated the walkers are independent until collision at collision the left walker disappears and the right walker survives with probability p this problem arises in the context of diffusion limited reactions and also in the scaling limit of the voter model a systematic expansion in correlation between neighbor intervals gives a series of approximations of increasing accuracy for the probability density functions of interval lengths the first approximation beyond mere statistical independence between successive intervals already gives excellent results as established by comparison with direct numerical simulations

naming games in two dimensional and small world connected random geometric networks we investigate a prototypical agent based model the naming game on two dimensional random geometric networks the naming game a baronchelli et al j stat mech theory exp 2006 p06014 is a minimal model employing local communications that captures the emergence of shared communication schemes languages in a population of autonomous semiotic agents implementing the naming games with local broadcasts on random geometric graphs serves as a model for agreement dynamics in large scale autonomously operating wireless sensor networks further it captures essential features of the scaling properties of the agreement process for spatiallyembedded autonomous agents among the relevant observables capturing the temporal properties of the agreement process we investigate the cluster size distribution and the distribution of the agreement times both exhibiting dynamic scaling we also present results for the case when a small density of long range communication links are added on top of the random geometric graph resulting in a small world like network and yielding a significantly reduced time to reach global agreement we construct a finite size scaling analysis for the agreement times in this case

wigner surmise for domain systems in random matrix theory the spacing distribution functions p n s are well fitted by the wigner surmise and its generalizations in this approximation the spacing functions are completely described by the behavior of the exact functions in the limits s 0 and s most non equilibrium systems do not have analytical solutions for the spacing distribution and correlation functions because of that we explore the possibility to use the wigner surmise approximation in these systems we found that this approximation provides a first approach to the statistical behavior of complex systems in particular we use it to find an analytical approximation to the nearest neighbor distribution of the annihilation random walk

mean field approximation for spacing distribution functions in classical systems we propose a mean field method to calculate approximately the spacing distribution functions p n s in one dimensional classical many particle systems we compare our method with two other commonly used methods the independent interval approximation and the extended wigner surmise in our mean field approach p n s is calculated from a set of langevin equations which are decoupled by using a mean field approximation we find that in spite of its simplicity the mean field approximation provides good results in several systems we offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system the physical interpretation of each method is also discussed

survival probability of a diffusing particle in the presence of poisson distributed mobile traps the problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d we consider the case where the traps are initially randomly distributed in space with uniform density ensuremath rho and derive upper and lower bounds for the probability q t averaged over all particle and trap trajectories that the particle survives up to time t we show that for 1l dl 2 the bounds converge asymptotically to give q t ensuremath sim mathrm exp ensuremath ensuremath lambda d t d 2 for 1l dl2 where ensuremath lambda d 2 ensuremath pi d mathrm sin ensuremath pi d 2 4 ensuremath pi d d 2 ensuremath rho and d is the diffusion constant of the traps and that q t ensuremath sim mathrm exp ensuremath 4 ensuremath pi ensuremath rho dt mathrm ln t for d 2 for dg2 bounds can still be derived but they no longer converge for large t for 1l dl 2 these asymptotic forms are independent of the diffusion constant of the particle the results are compared with simulation results obtained using a new algorithm v mehra and p grassberger phys rev e 65 050101 2002 which is described in detail deviations from the predicted asymptotic forms are found to be large even for very small values of q t indicating slowly decaying corrections whose form is consistent with the bounds we also present results in d 1 for the case where the trap densities on either side of the particle are different for this case we can still obtain exact bounds but they no longer converge

estimates of the conformal scalar curvature equation via the method of moving planes in this paper we derive a local estimate of a positive singular solution u near its singular set z of the conformal equation where k x is a positive continuous function z is a compact subset of and g satisfies that is nonincreasing for t 0 assuming that the order of flatness at critical points of k on z is no less than we prove that through the application of the method of moving planes the inequality holds for any solution of 0 1 with cap z 0 by the same method we also derive a harnack type inequality for smooth positive solutions let u satisfy assume that the order of flatness at critical points of k is no less than n 2 then the inequality holds for r 1 we also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality 0 4 1997 john wiley sons inc

on the definition and the properties of the principal eigenvalue of some nonlocal operators in this article we study some spectral properties of the linear operator mathcal l omega a defined on the space c bar omega by mathcal l omega varphi a varphi int omega k x y varphi y dy a x varphi x where omega subset mathbb r n is a domain possibly unbounded a is a continuous bounded function and k is a continuous non negative kernel satisfying an integrability condition we focus our analysis on the properties of the generalised principal eigenvalue lambda p mathcal l omega a defined by lambda p mathcal l omega a sup lambda in mathbb r exists varphi in c bar omega varphi 0 textit such that mathcal l omega varphi a varphi lambda varphi le 0 text in omega we establish some new properties of this generalised principal eigenvalue lambda p namely we prove the equivalence of different definitions of the principal eigenvalue we also study the behaviour of lambda p mathcal l omega a with respect to some scaling of k for kernels k of the type k x y j x y with j a compactly supported probability density we also establish some asymptotic properties of lambda p left mathcal l sigma m omega frac 1 sigma m a right where mathcal l sigma m omega is defined by displaystyle mathcal l sigma m omega varphi frac 1 sigma 2 n int omega j left frac x y sigma right varphi y dy in particular we prove that lim sigma to 0 lambda p left mathcal l sigma 2 omega frac 1 sigma 2 a right lambda 1 left frac d 2 j 2n delta a right where d 2 j int mathbb r n j z z 2 dz and lambda 1 denotes the dirichlet principal eigenvalue of the elliptic operator in addition we obtain some convergence results for the corresponding eigenfunction varphi p sigma

a fully nonlinear conformal flow on locally conformally flat manifolds we study a fully nonlinear flow for conformal metrics the long time existence and the sequential convergence of flow will be established for locally conformally flat manifolds as an application we solve the k yamabe problem for locally conformal flat manifolds when k n 2

efficient numerical solution of stochastic differential equations using exponential timestepping we present an exact timestepping method for brownian motion that does not require gaussian random variables to be generated time is incremented in steps that are exponentially distributed random variables boundaries can be explicitly accounted for at each timestep the method is illustrated by numerical solution of a system of diffusing particles

general solutions for multispin two time correlation and response functions in the glauber ising chain the kinetic glauber ising spin chain is one of the very few exactly solvable models of non equilibrium statistical mechanics nevertheless existing solutions do not yield tractable expressions for two time correlation and response functions of observables involving products of more than one or two spins we use a new approach to solve explicitly the full hierarchy of differential equations for the correlation and response functions from this general solution follow closed expressions for arbitrary multispin two time correlation and response functions for the case where the system is quenched from equilibrium at ti 0 to some arbitrary t 0 by way of application we give the results for two and four spin two time correlation and response functions from the standard mapping these also imply new exact results for two time particle correlation and response functions in one dimensional diffusion limited annihilation

a conformally invariant gap theorem characterizing mathbb cp 2 via the ricci flow we extend the sphere theorem of chang et al publ math inst ht etudes sci 98 105 434 2003 to give a conformally invariant characterization of mathbb cp 2 g fs in particular we introduce a conformal invariant beta m 4 g ge 0 defined on conformal four manifolds satisfying a positivity condition it follows from chang et al 2003 that if 0 le beta m 4 g 0 and 4 le beta m 4 g 0 small enough then m 4 is diffeomorphic to mathbb cp 2 the ricci flow is used in a crucial way to pass from the bounds on beta to pointwise curvature information

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

statistical behavior of domain systems we study the statistical behavior of two out of equilibrium systems the first one is a quasi one dimensional gas with two species of particles under the action of an external field which drives each species in opposite directions the second one is a one dimensional spin system with nearest neighbor interactions also under the influence of an external driving force both systems show a dynamical scaling with domain formation the statistical behavior of these domains is compared with models based on the coalescing random walk and the interacting random walk we find that the scaling domain size distribution of the gas and the spin systems is well fitted by the wigner surmise which lead us to explore a possible connection between these systems and the circular orthogonal ensemble of random matrices however the study of the correlation function of the domain edges shows that the statistical behavior of the domains in both gas and spin systems is not completely well described by a circular orthogonal ensemble nor it is by other models proposed such as the coalescing random walk and the interacting random walk nevertheless we find that a simple model of independent intervals describes more closely the statistical behavior of the domains formed in these systems

non trivial exponents in coarsening phenomena abstract one of the simplest examples of stochastic automata is the glauber dynamics of ferromagnetic spin models such as ising or potts models at zero temperature if the initial condition is random one observes a pattern of growing domains with a characteristic size which increases with time like t 1 2 in this self similar regime the fraction of spins which never flip up to time t decreases like t where the exponent is non trivial and depends both on the number q of states of the potts model and on the dimension of space this exponent can be calculated exactly in one dimension similar non trivial exponents are also present in even simpler models of coarsening where the dynamical rule is deterministic

some properties of the schouten tensor and applications to conformal geometry the riemannian curvature tensor decomposes into a conformally invariant part the weyl tensor and a non conformally invariant part the schouten tensor a study of the kth elementary symmetric function of the eigenvalues of the schouten tensor was initiated in an earlier paper by the second author and a natural condition to impose is that the eigenvalues of the schouten tensor are in a certain cone k we prove that this eigenvalue condition for k n 2 implies that the ricci curvature is positive we then consider some applications to the locally conformally flat case in particular to extremal metrics of k curvature functionals and conformal quermassintegral inequalities using the results of the first and third authors

hardy littlewood sobolev inequalities on compact riemannian manifolds and applications abstract in this paper we extend hardy littlewood sobolev inequalities on compact riemannian manifolds for dimension n 2 as one application we solve a generalized yamabe problem on locally conformally flat manifolds via a new designed energy functional and a new variational approach even for the classic yamabe problem on locally conformally flat manifolds our approach provides a new and relatively simpler solution

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds

distribution of domain sizes in the zero temperature glauber dynamics of the one dimensional potts model for the zero temperature glauber dynamics of the q state potts model we calculate the exact distribution of domain sizes by mapping the problem on an exactly soluble one species coagulation model a1a a in the long time limit this distribution is universal and from its complicated exact expression we extract its behavior in various regimes our results are tested in a simulation and compared to the predictions of a simple approximation proposed recently considering the dynamics of domain walls as a reaction diffusion model a1a a with probability q22 q21 and a1a f with probability 1 q21 we calculate the pair correlation function in the long time regime s1063 651x 96 12809 9

rigidite conforme des hemispheres rm s 4_ et rm s 6_ conformal rigidity of the hemispheres rm s 4_ and rm s 6_ let m g be a four or six dimensional compact riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical in this note we prove that if the euler characteristic of m is equal to 1 and if its yamabe invariant is positive then m g is conformally isometric to the standard hemisphere as an application and using a result of hang and wang comm anal geom 14 1 91 106 2006 we prove a rigidity result for these hemispheres regarding the min oo conjecture

classification of singular radial solutions to the σk yamabe equation on annular domains the study of the kth elementary symmetric function of the weyl schouten curvature tensor of a riemannian metric the so called k curvature has produced many fruitful results in conformal geometry in recent years in these studies in conformal geometry the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic pde important advances have been made in recent years in the understanding of the analytic behavior of solutions of the pde however the singular behavior of these solutions which is important in describing many important questions in conformal geometry is little understood this note classifies all possible radial solutions in particular the singular solutions of the k yamabe equation which describes conformal metrics whose k curvature equals a constant although the analysis involved is of elementary nature these results should provide useful guidance in studying the behavior of singular solutions in the general situation

conformal deformations of the smallest eigenvalue of the ricci tensor we consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the ricci tensor is a constant it is related to the notion of minimal volumes in comparison geometry such a metric with the smallest eigenvalue of the ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class the problem is reduced to solve a pucci type equation with respect to the schouten tensor we establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations combining it with the theory of fully nonlinear equations we establish the existence of solutions for this equation

conformally invariant monge ampere equations global solutions in this paper we will examine a class of fully nonlinear partial differential equations which are invariant under the conformal group so n 1 1 these equations are elliptic and variational using this structure and the conformal invariance we will prove a global uniqueness theorem for solutions in rn with a quadratic growth condition at infinity