ricci flow on kahler einstein surfaces in this paper we prove that if m is a kahler einstein surface with positive scalar curvature if the initial metric has nonnegative sectional curvature and the curvature is positive somewhere then the kahler ricci flow converges to a kahler einstein metric with constant bisectional curvature in a subsequent paper 7 we prove the same result for general kahler einstein manifolds in all dimension this gives an affirmative answer to a long standing problem in kahler ricci flow on a compact kahler einstein manifold does the kahler ricci flow converge to a kahler einstein metric if the initial metric has a positive bisectional curvature our main method is to find a set of new functionals which are essentially decreasing under the kahler ricci flow while they have uniform lower bounds this property gives the crucial estimate we need to tackle this problem

on stability and the convergence of the kahler ricci flow assuming uniform bounds for the curvature the exponential convergence of the kahler ricci flow is established under two con ditions which are a form of stability the mabuchi energy is bound ed from below and the dimension of the space of holomorphic vector fields in an orbit of the diffeomorphism group cannot jump up in the limit

sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities the goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and in self similar variables the optimal convergence rates to barenblatt self similar profiles and their generalizations it relies on the identification of the optimal constants in some related hardy poincare inequalities and concludes a long series of papers devoted to generalized entropies functional inequalities and rates for nonlinear diffusion equations

synthesis of positive logic programs for checking a class of definitions with infinite quantification we describe a method based on unfold fold transformations that synthesizes positive logic programs p r with the purpose of checking mechanically definitions of the form d r x r x q y r x y where r is the relation defined by the formula q y r x y x is a set of variables to be instantiated at runtime by ground terms qy is a set of quantified variables on infinite domains q is the quantifier and r x y a quantifier free formula in the language of a first order logic theory this work constitutes a first step towards the construction of a new type of assertion checkers with the ability of handling restricted forms of infinite quantification

phase transitions and symmetry breaking in singular diffusion the long time asymptotics are determined for fast nonlinear diffusion by linearizing otto s gradient descent model at the barenblatt profile the spectrum of the entropy is explicitly determined the dynamics are found to undergo a phase transition in which rotational symmetry is broken as the strength of the nonlinearity is varied

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity ii this article continues the analysis of the preceding article showing that the lyapunov functional introduced by newman can be used to prove the stability of the barenblatt pattle solutions of the porous medium equation

non archimedean analytic curves in abelian varieties one of the main subtleties of non archimedean analysis is that the natural topology that one puts on non archimedean analytic spaces is totally disconnected meaning that there is a base for the topology consisting of sets which are both open and closed this makes it difficult for instance to define a good notion of analytic function so that one has analytic continuation properties of course one can make a sensible definition for what one means by an analytic function and one of the more successful approaches has been the notion of a rigid analytic function this approach was started by tate in ta and a systematic introduction to rigid analysis is given in bgr however rigid analysis is very different from classical analysis in the sense that essentially none of the topological techniques which are commonplace in classical analysis can be used in rigid analysis recently berkovich in ber has come up with a new notion of non archimedean analytic spaces which have nice topological properties for instance berkovich s spaces are locally arc connected locally compact hausdorff spaces using berkovich s theory one is able to add such topological techniques as covering spaces and map liftings to the study of nonarchimedean analysis in his book berkovich proves the following theorem with the aid of the topological techniques his new theory encourages

remarks on the existence problem of positive kahler einstein metrics on introduit un invariant m qui donne un critere pour l existence de metriques de kahler einstein

vanishing cycles for formal schemes let k be a non archimedean field and let x be a formal scheme locally finitely presented over the ring of integers k see 1 in this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre x of x which is a k analytic space to the category of etale sheaves on the closed fibre xs of x which is a scheme over the residue field of k we prove that if x is the formal completion x of a scheme x finitely presented over k along the closed fibre then the vanishing cycles sheaves of x are canonically isomorphic to those of x as defined in sga7 exp xiii in particular the vanishing cycles sheaves of x depend only on x and any morphism x induces a homomorphism from the pullback of the vanishing cycles sheaves of x under s ys xs to those of y furthermore we prove that for each x one can find a nontrivial ideal of k such that if two morphisms x coincide modulo this ideal then the homomorphisms between the vanishing cycles sheaves induced by and coincide these facts were conjectured by p deligne in 1 we associate with a formal scheme x locally finitely presented over k a k analytic space x in the sense of ber1 and ber2 in 2 we find that the morphism y x which is induced by an etale morphism of formal schemes y x possesses a certain property morphisms of k analytic spaces with this property are called quasi etale and they give rise to a quasi etale site xqet of a k analytic space x there is a canonical morphism of sites xqet xet where xet is the etale site introduced in ber2 we show that the inverse image functor identifies the category of etale sheaves x et with a full subcategory of

smooth p adic analytic spaces are locally contractible 0 introduction 293 1 piecewise rs linear spaces 298 2 r colored polysimplicial sets 308 3 r colored polysimplicial sets of length l 313 4 the skeleton of a nondegenerate pluri stable formal scheme 327 5 a colored polysimplicial set associated with a nondegenerate poly stable fibration 336 6 p adic analytic and piecewise linear spaces 346 7 strong local contractibility of smooth analytic spaces 355 8 cohomology with coefficients in the sheaf of constant functions 362

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

best matching barenblatt profiles are delayed the growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self similar solutions however at next order there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self similar barenblatt solutions this distance can be measured in terms of a relative entropy to the best matching barenblatt profile this best matching barenblatt function determines a scale in new variables based on this scale which are given by a self similar change of variables if and only if the initial datum is one of the barenblatt profiles the typical scale is monotone and has a limit coming back to original variables the best matching barenblatt profile is delayed compared to the self similar solution with same initial second moment as the initial datum such a delay is a new phenomenon which has to be taken into account for instance when fitting experimental data

conical kahler ricci flows on fano manifolds abstract in this paper we study the long term behavior of conical kahler ricci flows on fano manifolds first by proving uniform regularities for twisted kahler ricci flows we prove the existence of conical kahler ricci flows by limiting these twisted flows second we obtain uniform perelman s estimates along twisted kahler ricci flows by improving the original proof after that we prove that if there exists a conical kahler einstein metric then conical kahler ricci flow must converge to it

constrained steepest descent in the 2 wasserstein metric we study several constrained variational problems in the 2 wasserstein metric for which the set of probability densities satisfying the constraint is not closed for example given a probability density f0 on d and a time step h 0 we seek to minimize i f hs f w 2 2 f0 f over all of the probability densities f that have the same mean and variance as f0 where s f i sthe entropy of f w eprove existence of minimizers we also analyze the induced geometry of the set of densities satisfying the constraint on the variance and means and we determine all of the geodesics on it from this we determine a criterion for convexity of functionals in the induced geometry it turns out for example that the entropy is uniformly strictly convex on the constrained manifold though not uniformly convex without the constraint the problems solved here arose in a study of a variational approach to constructing and studying solutions of the nonlinear kinetic fokker planck equation which is briefly described here and fully developed in a companion paper

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity i we employ the transformation properties and the integral invariants of the porous medium equation in m dimensions to construct a lyapunov functional which characterizes the evolution of the profile to its self similar solution

the concavity of renyi entropy power we associate to the pth renyi entropy a definition of entropy power which is the natural extension of shannon s entropy power and exhibits a nice behavior along solutions to the p nonlinear heat equation in rn we show that the renyi entropy power of general probability densities solving such equations is always a concave function of time whereas it has a linear behavior in correspondence to the barenblatt source type solutions this result extends costa s concavity inequality for shannon s entropy power to renyi entropies

kahler einstein metrics and the generalized futaki invariant in 1983 futaki introduced his famous invariant this invariant generalizes the obstruction of kazdan warner to prescribing gauss curvature on s 2 cf ful l the futaki invariant is defined for any compact k ihler manifold with positive first chern class that has nontrivial holomorphic vector fields it is a lie algebraic character from the lie algebra of holomorphic vector fields into and its vanishing is a necessary condition for the existence of a k ihler einstein metric on the underlying manifold therefore it can be used to test the existence of k ihlereinstein metrics on a given compact k ihler manifold with positive first chern class an excellent reference on the futaki invariant is futaki s book fu2 until now all known nontrivial obstructions to k ihler einstein metrics come from holomorphic vector fields this suggests the following conjecture

vanishing cycles for non archimedean analytic spaces in this work we develop a formalism of vanishing cycles for non archimedean analytic spaces which is an analog of that for complex analytic spaces from sga7 exp xiv as an application we prove that in the equicharacteristic case the stalks of the vanishing cycles sheaves of a scheme x at a closed point x e x depend only on the formal completion spf ox of x at x in particular any continuous homomorphism ox x y y induces a homorphism from the stalks of the vanishing cycles sheaves of x at x to those of y at y furthermore we prove that given ox x and o y there exists n 1 such that for any pair of continuous homomorphisms ox x oy y that coincide modulo the n th power of the maximal ideal of oy y the induced homomorphisms between the stalks of the vanishing cycles sheaves coincide these facts generalize a result of g laumon from lau see remark 7 6 throughout the paper we fix a non archimedean field k whose valuation is not assumed to be nontrivial in 1 we study etale galois sheaves on k analytic spaces to define the vanishing cycles functor and to work with it we use the language of pro analytic spaces i e pro objects of the category of analytic spaces sga4 exp i examples of such objects are the germs of analytic spaces as in ber2 3 4 another example is considered in 3 in 4 we define the vanishing cycles functor and establish its basic properties in 5 we show that the vanishing cycles sheaves are trivial for smooth morphisms in 6 we prove a comparison theorem for vanishing cycles this theorem is more general than its analog over c from sga7 exp xiv and its proof does not use hironaka s theorem on resolution of singularities in 7 we apply the comparison theorem to prove the properties of the vanishing cycles sheaves of schemes formulated above it is worthwhile to note that this application is obtained by considering non archimedean analytic geometry over fields with trivial valuation like ber3 this work arose from a suggestion of p deligne to apply the etale cohomology theory from ber2 to the study of the vanishing cycles sheaves of schemes i am very grateful to him for useful discussions on the subject i also

determinization of conditional term rewriting systems this paper discusses determinization of conditional term rewriting systems with oriented constructor rules we present a rule based transformation system which transforms a non deterministic one into a deterministic one together with examples of the transformation we prove that the transformation system is simulation sound and simulation complete we also prove that the transformation system is complete for some class by introducing a strategy for the transformation system

special fast diffusion with slow asymptotics entropy method and flow on a riemannian manifold we consider the asymptotic behaviour of positive solutions u t x of the fast diffusion equation u t delta u m m rm div u m 1 nabla u posed for x in mathbb r d t 0 with a precise value for the exponent m d 4 d 2 the space dimension is d 3 so that m 1 and even m 1 for d 3 this case had been left open in the general study blanchet et al in arch rat mech anal 191 347 385 2009 since it requires quite different functional analytic methods due in particular to the absence of a spectral gap for the operator generating the linearized evolution the linearization of this flow is interpreted here as the heat flow of the laplace beltrami operator of a suitable riemannian manifold mathbb r d bf g with a metric g which is conformal to the standard mathbb r d metric studying the pointwise heat kernel behaviour allows to prove suitable gagliardo nirenberg inequalities associated with the generator such inequalities in turn allow one to study the nonlinear evolution as well and to determine its asymptotics which is identical to the one satisfied by the linearization in terms of the rescaled representation which is a nonlinear fokker planck equation the convergence rate turns out to be polynomial in time this result is in contrast with the known exponential decay of such representation for all other values of m

transformations of clp modules we propose a transformation system for clp programs and modules the framework is inspired by the one of tamaki and sato for pure logic programs however the use of clp allows us to introduce some new operations such as splitting and constraint replacement we provide two sets of applicability conditions the first one guarantees that the original and the transformed programs have the same computational behaviour in terms of answer constraints the second set contains more restrictive conditions that ensure compositionality we prove that under these conditions the original and the transformed modules have the same answer constraints also when they are composed with other modules this result is proved by first introducing a new formulation in terms of trees of a resultants semantics for clp as corollaries we obtain the correctness of both the modular and the non modular system w r t the least model semantics

the existence of kahler einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group on utilise la methode de continuite pour demontrer l existence de metriques de kahler einstein pour des varietes de kahler de fibre linge positif anticanonique sous l hypothese additionnelle de l existence d un bon groupe de symetries fini ou compact

the logarithmic sobolev inequalities along geometric flow and applications abstract for some class of geometric flows we obtain the logarithmic sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non collapsing and non inflated properties which generalize the results in the case of ricci flow or list ricci flow or harmonic ricci flow as applications for mean curvature flow in lorentzian space with nonnegative sectional curvature and twisted kahler ricci flow on fano manifolds we get the results above

the generalized kahler ricci flow abstract in this paper we study a generalized kahler ricci flow in which the ricci term ric is replaced by ric where is a closed real semi positive 1 1 form we first introduce the generalized functionals w and and study their monotonicity along the generalized kahler ricci flow then extend some of perelman s estimates to the generalized kahler ricci flow we also study the asymptotic behavior of the generalized ricci potential

ricci flow on kahler einstein manifolds this is the continuation of our earlier article 10 for any kahler einstein surfaces with positive scalar curvature if the initial metric has positive bisectional curvature then we have proved see 10 that the kahler ricci flow converges exponentially to a unique kahler einstein metric in the end this partially answers a long standing problem in ricci flow on a compact kahler einstein manifold does the kahler ricci flow converge to a kahler einstein metric if the initial metric has positive bisectional curvature in this article we give a complete affirmative answer to this problem

a semantics for modular general logic programs modular programs are built as a combination of separate modules which may be developed and verified separately therefore in order to reason over such programs compositionality plays a crucial role the semantics of the whole program must be obtainable as a simple function from the semantics of its individual modules in the field of logic programming the need for a compositional semantics has been long recognized however while for definite i e negationfree logic programs a few such semantics have been proposed in the literature of normal logic programs programs which employ the negation operator compositionality has received scarce attention this is mainly due to the fact that normal programs typically have a nonmonotonic behavior which is difficult to fit in a compositional framework here we propose a declarative compositional semantics for general logic programs first a compositional semantics for first order modules is presented and proved correct wrt the set of logical consequences of the module in three valued logic in a second stage the obtained results are applied to modular normal logic programs obtaining a semantics which is correct with respect to the set of logical consequences of the completion of the program and in contrast with the other approaches which is always computable this semantics might be regarded as a compositional counterpart of kunen s semantics finally we discuss and show how these results have to be modified in order to be applied to normal constraint logic programs

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

asymptotics of the fast diffusion equation via entropy estimates we consider non negative solutions of the fast diffusion equation ut um with m 0 1 in the euclidean space mathbb r d d 3 and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t for m mc d 2 d or as t approaches the extinction time when m mc for a class of initial data we prove that the solution converges with a polynomial rate to a self similar solution for t large enough if m mc or close enough to the extinction time if m mc such results are new in the range m mc where previous approaches fail in the range mc m 1 we improve on known results

solution of a model boltzmann equation via steepest descent in the 2 wasserstein metric we study a model boltzmann equation closely related to the bgk equation using a steepest descent method in the wasserstein metric and prove global existence of energy and momentum conserving solutions we also show that the solutions converge to the manifold of local maxwellians in the large time limit and obtain other information on the behavior of the solutions we show how the wasserstein metric is natural for this problem because it is adapted to the study of both the free streaming and the collisions

nonlinear diffusions extremal properties of barenblatt profiles best matching and delays abstract in this paper we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source type solutions to the fast diffusion or porous medium equations that are also known as barenblatt solutions as functions of time these functionals have convexity properties for generic solutions so that their asymptotic slopes are extremal for barenblatt profiles the method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp gagliardo nirenberg sobolev inequalities in scale invariant form the method also gives refined estimates of the growth of the second moment and as a consequence establishes the monotonicity of the delay corresponding to the best matching barenblatt solution compared to the barenblatt solution with same initial second moment here the notion of best matching is defined in terms of a relative entropy

an equivalence preserving first order unfold fold transformation system an unfold fold transformation system for first order programs is presented which comprises an unfolding rule a folding rule and a replacement rule it is intended to work with a first order theory specifying the meaning of primitives on top of which new relations are built by programs it preserves the provability relationship g between a call consistent program and a goal formula g such that is strict with respect to g it also preserves the logical consequence relationship for three valued logic