sur le theoreme d atiyah singer on donne une version simplifiee de la demonstration probabiliste du theoreme de l indice pour l operateur de dirac donnee par j m bismut au lieu d utiliser la construction du mouvement brownien au moyen du fibre des reperes on utilise la construction de schwartz en plongeant la variete ambiante dans un espace vectoriel de dimension plus grande on evite aussi l utilisation de la decomposition de l espace de wiener en deux utilisee par j m bismut le calcul des variations stochastiques etant notre outil principal de ce fait on perd la relation avec la cohomologie de l espace des lacets

li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

the basic principles of irreversible thermodynamics in the context of an informational statistical approach abstract we consider on the light of information statistical thermodynamics the basic principles of phenomenological thermodynamic theories namely the principles of equipresence objectivity and memory as well as the characteristics of entropy production informational statistical thermodynamics arises as a particular application of maximum information entropy inference we discuss how in this perspective together with the property of contraction of description based on bogoliubov s hierarchy of relaxation times it is possible to account for the above mentioned principles particular analyses are carried out on the basis of a model system moreover selecting the basic set of thermodynamical variables in a closed and in truncated approaches we assess the validity of the approximations in particular we derive the limiting conditions under which one recovers the constitutive equations of linear irreversible thermodynamics the associated equations of evolution for the basic macrovariables and a particular h theorem are described and discussed

on the heat kernel of the bergmann metric on algebraic varieties the restriction of the standard fubini study metric of ycpn l to m m is an incomplete kahler metric g called the bergmann metric let us consider the operators d and 3 defined on c1 functions and c1 1 forms on m m respectively we define the domain 9y d of d to be the set of c1 functions 2 f defined on m m such that both f and df are in l similarly we define the domain 92 6 of to be the set of c1 1 forms co such that both co and 2 6 o are in l we then define the laplacian a with respect to the bergmann metric by a d with domain 9 a given by the set of c2 functions f such that f e 9 d and df e 6 the main purpose of this paper is to show that

remarks on differential harnack inequalities abstract in this paper we give a generalization of global and local differential harnack inequalities for heat equations obtained by li and xu j f li x j xu differential harnack inequalities on riemannian manifolds i linear heat equation adv math 226 5 2011 4456 4491 and baudoin and garofalo f baudoin n garofalo perelman s entropy and doubling property on riemannian manifolds j geom anal 21 2011 1119 1131 from this we can derive new harnack inequalities and new lower bounds for the associated heat kernel also we provide some new entropy formulas with monotonicity

li yau and harnack type inequalities in rcd k n metric measure spaces abstract metric measure spaces satisfying the reduced curvature dimension condition cd k n and where the heat flow is linear are called rcd k n spaces this class of non smooth spaces contains gromov hausdorff limits of riemannian manifolds with ricci curvature bounded below by k and dimension bounded above by n we prove that in rcd k n spaces the following properties of the heat flow hold true a li yau type inequality a bakry qian inequality the harnack inequality

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

hamilton s gradient estimates for fast diffusion equations under the ricci flow abstract let m be a complete noncompact riemannian manifold of dimension n we derive a hamilton s gradient estimate for positive solutions of the fast diffusion equations u t u m 1 4 n 8 m 1 on m 0 under the ricci flow

linear response for systems far off equilibrium abstract the kubo theory of linear response is extended with respect to reference states being excited far off equilibrium using the time dependent projector technique generalized response functions are introduced the structure of the corresponding evolution equations is mainly determined by the observable algebra of the relevant subspace generalizations of retarded green s functions may be introduced in the limit of two well separated time scales of evolution but generally the energy representation becomes unsuitable in systems far off equilibrium after a reformulation of the evolution superoperator a transparent example of a two level system is considered where the exciting laser pulse has the shape of a gaussian envelope

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

gradient estimates and harnack inequalities of a parabolic equation under geometric flow abstract in this paper we consider a manifold evolving by a general geometric flow and study parabolic equation q x t t u x t a u x t x t m 0 t we establish space time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation by integrating the gradient estimates we derive the corresponding harnack inequalities finally as applications we give gradient estimates of some specific parabolic equations

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

gradient estimates and liouville theorems for nonlinear parabolic equations on noncompact riemannian manifolds abstract let m be a complete noncompact riemannian manifold in this paper we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation t u x t x t u x t 0 on m 0 we also obtain a theorem of liouville type for positive solutions of this nonlinear equation this paper extends the result of souplet and zhang p souplet qi s zhang sharp gradient estimate and yau s liouville theorem for the heat equation on noncompact manifolds bull lond math soc 38 2006 1045 1053

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

superlinear elliptic inequalities on manifolds abstract let m be a complete non compact riemannian manifold and let be a radon measure on m we study the problem of existence or non existence of positive solutions to a semilinear elliptic inequality u u q in m where q 1 we obtain necessary and sufficient criteria for existence of positive solutions in terms of green function of in particular explicit necessary and sufficient conditions are given when m has nonnegative ricci curvature everywhere in m or more generally when green s function satisfies the 3g inequality

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

a cameron martin type quasi invariance theorem for pinned brownian motion on a compact riemannian manifold the results in driver 13 for quasi invariance of wiener measure on the path space of a compact riemannian manifold m are extended to the case of pinned wiener measure to be more explicit let h 0 1 tom be a c1 function where m is a compact riemannian manifold o e m is a base point and tom is the tangent space to m at o e m let w m be the space of continuous paths from 0 1 into m v be wiener measure on w m concentrated on paths starting at o e m and hs w denote the stochastic parallel translation operator along a path to e w m up to time s note hs w is only well defined up to v equivalence for c e w m let xh w denote the vector field along a given by xsh c hs cw h s for each s e 0 1 one should interpret xh as a vector field on w m the vector field xh induces a flow sh t w m w m which leaves wiener measure v quasi invariant see driver 13 it is shown in this paper that the same result is valid if h l 0 and the wiener measure v is replaced by a pinned wiener measure ive the measure ve is proportional to the measure v conditioned on the set of paths which start at 0 e m and end at a fixed end point e e m also as in 13 one gets an integration by parts formula for the vector fields xh defined above

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

heat kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

statistical mechanics of nonlinear hydrodynamic fluctuations abstract the paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics the generalized fokker planck equation for the distribution function of coarse grained densities of conserved quantities is derived from the liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix we have obtained the functional differential fokker planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse grained fluctuating fields an outline of the derivation of fokker planck equations containing the burnett terms is also given the explicit coordinate representation for the hydrodynamic fokker planck equation is discussed in the case of one component simple fluid the general scheme of a change of coarse grained functional variables is developed for hydrodynamic fokker planck equations the corresponding transformation rules are found for drift terms diffusion coefficients and thermodynamic forces the dynamical equations and stationary conditions for averages of functions functionals of hydrodynamic fields are discussed by using the fokker planck operators acting on such functions the explicit form of these operators are found for various sets of fluctuating fields as an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state

morse inequalities a function space integral approach abstract in this paper we prove the morse inequalities in the non degenerate and degenerate cases like the approach of j m bismut ours is based on the idea suggested by witten in fact if anything our approach is closer to witten s original idea than bismut s

the witten complex and the degenerate morse inequalities on utilise le complexe introduit par witten pour demontrer des inegalites de morse dans le cas ou les points critiques de la fonction de morse sont isoles et aussi les inegalites de morse degenerees de bott

fractional cauchy problems with almost sectorial operators this paper concerns the abstract cauchy problem of fractional evolution equations with almost sectorial operators the suitable mild solutions of evolution equations with riemann liouville derivative and caputo derivative are introduced respectively the existence theorems of mild solutions of cauchy problems are established the results obtained here improve and generalize some known results finally an example is given for demonstration

path integral formulae for heat kernels and their derivatives the heat kernel and its derivatives of a vector laplacian on the sections of a bundle over a compact riemannian manifold are expressed as products of the scalar heat kernel of the manifold and path integrals over the brownian bridge the small time asymptotics of these integrals are computed

on recent developments in the theory of abstract differential equations with fractional derivatives abstract this note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives we show that the existence results in agarwal et al 2009 1 belmekki and benchohra 2010 2 darwish et al 2009 3 hu et al 2009 4 mophou and n guerekata 2009 6 7 mophou 2010 8 9 muslim 2009 10 pandey et al 2009 11 rashid and el qaderi 2009 12 and tai and wang 2009 13 are incorrect since the considered variation of constant formulas is not appropriate in this note we also consider a different approach to treat a general class of abstract fractional differential equations

a harnack type inequality for non symmetric markov semigroups abstract for a strong feller and irreducible markov semigroup on a locally compact polish space the harnack type inequality 1 1 holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive moreover new sufficient conditions for this inequality to hold as well as upper bound estimates of the underlying constant are presented for diffusion semigroups on riemannian manifolds

on the burnett and higher order equations of hydrodynamics abstract several derivations of the burnett and higher order in the gradients equation of classical hydrodynamics are critically examined it is clearly pointed out that the two available phenomenological derivations and grad s kinetic derivation are compatible with the local equilibrium assumption and the entropy balance equation furthermore the failure to fulfill these requirements when such equations are obtained from the chapman enskog method for solving boltzmann s equation is accounted for a comparison with other more microscopic methods to obtain these equations is also provided

a stochastic approach to the poincare hopf theorem in this paper the poincare hopf theorem hopf 8 is proved by a probabilistic method the proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution we use the malliavian calculus to estimate the fundametal solution our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

feedback and memory in kinetic equations of highly excited semiconductors abstract the memory kernels of kinetic equations in highly excited semiconductors in ultra short laser pump experiments are investigated with respect to a many body treatment at each point on the time scale this method results in a shift and a decay of the time dependent mean field tdmf states involving these effects a generalized form of kinetic equations is derived utilizing the internal 2n u 2 symmetry of the highly excited semiconductor

self interacting diffusions this paper is concerned with a general class of self interacting diffusions x t t 0 living on a compact riemannian manifold m these are solutions to stochastic differential equations of the form dx t brownian increments drift term depending on x t and t the normalized occupation measure of the process it is proved that the asymptotic behavior of t can be precisely related to the asymptotic behavior of a deterministic dynamical semi flow t t 0 defined on the space of the borel probability measures on m in particular the limit sets of t are proved to be almost surely attractor free sets for these results are applied to several examples of self attracting repelling diffusions on the n sphere for instance in the case of self attracting diffusions our results apply to prove that t can either converge toward the normalized riemannian measure or to a gaussian measure depending on the value of a parameter measuring the strength of the attraction

a semilinear parabolic volterra integrodifferential equation abstract we consider the semilinear volterra integrodifferential equation u t a t u t t 0 t a t s g s u s ds t u t t t 0 u t 0 u 0 we assume that for each t 0 the linear operator a t is the negative generator of an analytic semigroup in a banach space x the nonlinear operator g t is lipschitz continuous on the domain of a 0 into x and the forcing term t u satisfies a holder condition in both variables we prove existence of solutions in the banach space x and uniqueness of solutions when x is a hilbert space we also consider an example where a t a a t s a t s and show that the corresponding solutions u t x converge as t to a solution u x of a related semilinear elliptic boundary value problem

linear response theory for a highly excited system far from equilibrium abstract in this paper a generalization of kubo s linear response theory is treated for systems which are driven far off equilibrium by external fields the reduced density operator is calculated by a time dependent projector technique in a fast moving reference frame the response of the system with respect to a second retarded probe pulse is considered which reflects the higher complexity of the corresponding equations in comparison with kubo s linear response theory the response of a virtually excited exciton system to a white probe pulse is discussed for a highly excited semiconductor in the athermal stage and in the low density limit the poles of the two particle green s functions contain the differences of time dependent shifted quasi particle energies this behavior reflects the dynamic renormalization of the band structure as the laser induced gap creation or the dynamic stark effect

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

an l 2 theory for differential forms on path spaces i an l 2 theory of differential forms is proposed for the banach manifold of continuous paths on a riemannian manifold m furnished with its brownian motion measure differentiation must be restricted to certain hilbert space directions the h tangent vectors to obtain a closed exterior differential operator the relevant spaces of differential forms the h forms are perturbed by the curvature of m a hodge decomposition is given for l 2 h one forms and the structure of h two forms is described the dual operator d is analysed in terms of a natural connection on the h tangent spaces malliavin calculus is a basic tool c 2007 elsevier inc all rights reserved

recurrence and transience of branching diffusion processes on riemannian manifolds we relate the recurrence and transience of a branching diffusion process on a riemannian manifold m to some properties of a linear elliptic operator onm including spectral properties there is a trade off between the tendency of the transient brownian motion to escape and the birth process of the new particles if the latter has a high enough intensity then it may override the transience of the brownian motion leading to the recurrence of the branching process and vice versa in the case of a spherically symmetric manifold the critical intensity of the population growth can be found explicitly

holomorphic morse inequalities for orbifolds we prove that demailly s holomorphic morse inequalities hold true for complex orbifolds by using a heat kernel method then we introduce the class of moishezon orbifolds and as an application of our inequalties we give a geometric criterion for a compact connected orbifold to be a moishezon orbifolds thus generalizing siu s and demailly s answers to the grauert riemenschneider conjecture to the orbifold case

lifschitz tail on hyperbolic space neumann conditions on considere un espace hyperbolique h de dimension n et l operateur laplacien hyperbolique h etant donne une distribution aleatoire de points a l aide d un processus ponctuel de poisson tel que chaque point se situe dans une boule hyperbolique de rayons on etudie la question suivante si est un petit nombre reel combien y a t il de valeurs propres inferieures a pour l operateur 1 2 h dans une boule b de centre o avec des conditions de neumann sur la frontiere et avec des conditions aux limites de dirichlet sur les boules aleatoires formant des intersections avec b si le rayon de b est suffisamment grand

morse type inequalities for dynamical systems and the witten laplacian we provide an analytic proof of morse type inequalities for vector flelds determining a morse decomposition with normally hyperbolic dynamics in the demonstration we reduce the problem to the gradient case using a morse bott lyapunov function and

sur la l 2 cohomologie des varietes a courbure negative we give a topological interpretation of the space of l2 harmonic forms on finite volume manifolds with sufficiently pinched negative curvature we give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants the method consists first in comparing l2 cohomology with weighted l2 cohomology thanks to previuos works done by t ohsawa and then in identifying these weighted spaces

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

morse homotopy and chern simons perturbation theory we define and invariant of a three manifold equipped with a flat bundle with vanishing homology the construction is based on morse theory using several morse functions simultaneously and is regarded as a higher loop analogue of various product operations in algebraic topology there is a heuristic argument that this invariant is related to perturbative chern simons gauge theory by axelrod singer etc there is also a theorem which gives a relation of the construction to open string theory on the cotangent bundle

demailly s asymptotic morse inequalities a heat equation proof on donne une demonstration qui s appuie sur l equation de la chaleur pour les inegalites asymptotiques de demailly pour le complexe

optimal mild solutions and weighted pseudo almost periodic classical solutions of fractional integro differential equations abstract in this paper we investigate a class of fractional integro differential equations given by d x t d t a x t f t x t g x t our main results concern the existence uniqueness of optimal mild solutions and weighted pseudo almost periodic classical solutions an example is given to illustrate our results

compact manifolds with a little negative curvature on etend le theoreme de bochner et le theoreme de myers pour des p formes en donnant des obstructions topologiques pour des varietes admettant des metriques avec une petite courbure negative

on estimation of the logarithmic sobolev constant and gradient estimates of heat semigroups this paper presents some explicit lower bound estimates of logarithmic sobolev constant for diffusion processes on a compact riemannian manifold with negative ricci curvature let ric k for some k 0 and d d be respectively the dimension and the diameter of the manifold if the boundary of the manifold is either empty or convex then the logarithmic sobolev constant for brownian motion is not less than

kirwan novikov inequalities on a manifold with boundary we extend the novikov morse type inequalities for closed 1 forms in 2 directions first we consider manifolds with boundary second we allow a very degenerate structure of the critical set of the form assuming only that the form is non degenerated in the sense of kirwan in particular we obtain a generalization of a result of floer about the usual morse inequalities on a manifold with boundary we also obtain an equivariant version of our inequalities our proof is based on an application of the witten deformation technique the main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end this leads to a considerable simplification of the local analysis in particular we obtain a new analytic proof of the morse bott inequalities on a closed manifold

harnack estimates for nonlinear backward heat equations in geometric flows let m be a closed riemannian manifold with a family of riemannian metrics gij t evolving by a geometric flow tgij 2sij where sij t is a family of smooth symmetric two tensors we derive several differential harnack estimates for positive solutions to the nonlinear backward heat type equation f t f flog f asf where a and are constants and s gijsij is the trace of sij our abstract formulation provides a unified framework for some known results proved by various authors and moreover leads to new harnack inequalities for a variety of geometric flows

the method of green s functions for a time dependent hamiltonian in systems far off equilibrium abstract a green function method is outlined for open systems being used if the hamiltonian depends sensitively on time the method is applied to the quantum mechanical treatment of the photon field for the problem of highly excited semiconductors as a result we obtain the creation of laser induced n dependent gaps where n 0 1 2 are the eigenvalues of the photon number operator of the laser field

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

derivation in a nonequilibrium ensemble formalism of a far reaching generalization of a quantum boltzmann theory abstract within the framework of the nonequilibrium statistical ensemble formalism provided by the nonequilibrium statistical operator method we derive a quantum boltzmann style transport theory of a broad scope this is done by choosing the single and two particle dynamical density operators as the basic informational statistical variables the equations of evolution for their average values over the nonequilibrium ensemble the nonequilibrium reduced dirac landau bogoliubov type density matrices are obtained from the resulting generalized nonlinear quantum transport theory after resorting to perturbative like expansions a far reaching generalization of boltzmann equation for the single particle distribution function is derived a type of traditional boltzmann equation follows after using stringent approximations whose limits of validity are evaluated

semilinear fractional integro differential equations with compact semigroup abstract in this paper we study the local and global existence of mild solutions to a class of fractional integro differential equations in an arbitrary banach space associated with operators generating compact semigroup on the banach space

dimension dependent hypercontractivity for gaussian kernels we derive sharp local and dimension dependent hypercontractive bounds on the markov kernel of a large class of diffusion semigroups unlike the dimension free ones they capture refined properties of markov kernels such as trace estimates they imply classical bounds on the ornstein uhlenbeck semigroup and a dimensional and refined transportation talagrand inequality when applied to the hamilton jacobi equation hypercontractive bounds on the ornstein uhlenbeck semigroup driven by a non diffusive levy semigroup are also investigated curvature dimension criteria are the main tool in the analysis

the foundations of informational statistical thermodynamics revisited abstract the aim of this paper is to review the fundamental ideas of the underlying method behind informational statistical thermodynamics this method is set forth to deal with phenomena that occur in nonequilibrium systems the most significant aspects of this analysis are 1 i to show that abel s theorem guarantees that in the asymptotic limit the nonequilibrium statistical operator nso obtained by maxent generates a stationary solution of the liouville equation this fact is consistent with the experimental behavior of an equilibrium system 2 ii further it is also shown how a liouville equation with sources can be obtained by the nso determined by maxent whose formal solution proves that the general interpretation of abel s theorem leading to memory effects is incorrect rather this theorem introduces a time smoothing function in a time interval t 0 t t 1 t 1 initial time of an observation which is to be understood as one that connects an adiabatic perturbation for t t 1 in fact the memory effects appear in the evolution equations for the average values of the dynamical variables obtained by the nso when these evolution equations are calculated up to second order in the perturbation hamiltonian 3 iii also some criticisms that have been presented against maxent formalism are discussed and it is shown that they are inapplicable such criticisms are related to the memory effects and with the inconsistency of evolution equations for macrovariables with respect to the total energy conservation equation

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

generalized kinetic equation for far from equilibrium many body systems a kinetic equation for the single particle distribution function in an open many body system when in far away from equilibrium conditions is derived in the context of a non equilibrium thermo statistics of ample scope it consists of a generalization of traditional kinetic equations in that no restrictions are imposed on the characteristics of the nonequilibrium thermodynamic state of the system this kinetic equation do contain some contributions that become relevant in systems with a nonlinear kinetics when driven sufficiently far from equilibrium certain complex systems moreover the handling of the kinetic equation in a multiple moment approach provides a generalized nonlinear higher order thermo hydrodynamics

steady state of periodically driven systems in quantum statistics abstract a formal theory for the statistical operator of the steady state in periodically driven matter is given for this purpose the periodic fields are changed to have an amplitude increasing with e t where is taken to be zero at the end this procedure allows for a correct description of energy denominators the results following from the steady state operator are compared with the long time behaviour obtained from n th order response and are shown to agree thus the proposed statistical operator has all properties required

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

remarks on li yau inequality on graphs abstract in this paper we study li yau gradient estimates for the solutions u to the heat equation t u u on graphs under the curvature condition c d n k which is introduced by bauer et al in 5 as applications we derive harnack inequalities and heat kernel estimates on graphs also we present a type of hamilton gradient estimates

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

a nonlinear quantum transport theory abstract a nonlinear quantum transport theory for many body systems arbitrarily far away from equilibrium based on the nonequilibrium statistical operator method is discussed an iterative process is described that allows for the calculation of the collision operator in a series of instantaneous in time partial contributions of ever increasing power in the interaction strengths these partial collision operators are shown to be composed of three contributions to the dissipative processes one is a direct result of the collisions another is accounted for in the internal state variables and the third arises from memory effects in the lowest order the so called linear theory of relaxation the equations become markovian