singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

exclusion and persistence in deterministic and stochastic chemostat models abstract we first introduce and analyze a variant of the deterministic single substrate chemostat model in this model microbe removal and growth rates depend on biomass concentration with removal terms increasing faster than growth terms using a comparison principle we show that persistence of all species is possible in this scenario then we turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation in particular we show that random effects may lead to extinction in scenarios where the deterministic model predicts persistence on the other hand we also establish some stochastic persistence results

morse inequalities for fourier components of kohn rossi cohomology of cr manifolds with s 1 action let x be a compact connected cr manifold of dimension 2n 1 n ge 2 with a transversal cr s 1 action on x we study the fourier components of the kohn rossi cohomology with respect to the s 1 action by studying the szego kernel of the fourier components we establish the morse inequalities on x using the morse inequalities we have established on x we prove that there are abundant cr functions on x when x is weakly pseudoconvex and strongly pseudoconvex at a point

scale sensitive dimensions uniform convergence and learnability learnability in valiant s pac learning model has been shown to be strongly related to the existence of uniform laws of large numbers these laws define a distribution free convergence property of means to expectations uniformly over classes of random variables classes of real valued functions enjoying such a property are also known as uniform glivenko cantelli classes in this paper we prove through a generalization of sauer s lemma that may be interesting in its own right a new characterization of uniform glivenko cantelli classes our characterization yields dudley gine and zinn s previous characterization as a corollary furthermore it is the first based on a gine and zinn s previous characterization as a corollary furthermore it is the first based on a simple combinatorial quantity generalizing the vapnik chervonenkis dimension we apply this result to obtain the weakest combinatorial condition known to imply pac learnability in the statistical regression or agnostic framework furthermore we find a characterization of learnability in the probabilistic concept model solving an open problem posed by kearns and schapire these results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner s hypothesis class

bergman kernels and local holomorphic morse inequalities let x be a hermitian manifold and let lk be a high power of a hermitian holomorphic line bundle over x local versions of demailly s holomorphic morse inequalities that give bounds on the dimension of the dolbeault cohomology groups associated to lk are presented after integration they give the usual holomorphic morse inequalities the local weak inequalities hold on any hermitian manifold x regardless of compactness and completeness the proofs which are elementary are based on a new approach to pointwise bergman kernel estimates where the kernels are estimated by a model kernel in open image in new window

isoperimetry of waists and local versus global asymptotic convex geometries existence of nicely bounded sections of two symmetric convex bodies k and l implies that the intersection of random rotations of k and l is nicely bounded for l subspace this main result immediately yields the unexpected phenomenon if k has one nicely bounded section then most sections of k are nicely bounded this existence implies randomness consequence was proved independently in giannopoulos milman and tsolomitis the main result represents a new connection between the local asymptotic convex geometry study of sections of convex bodies and the global asymptotic convex geometry study of convex bodies as a whole the method relies on the new isoperimetry of waists on the sphere due to gromov

a kollar type vanishing theorem let f x rightarrow y be a smooth fibration between two complex manifolds x and y and let l be a pseudo effective line bundle on x we obtain a sufficient condition for r q f k x y otimes l to be reflexive and hence derive a kollar type vanishing theorem

averages of norms and quasi norms we compute the number of summands in q averages of norms needed to approximate an euclidean norm it turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the euclidean norm particular attention is given to the case q in which the average is replaced with the maxima this is closely connected with the behavior of certain families of projective caps on the sphere

big vector bundles and complex manifolds with semi positive tangent bundles we classify compact kahler manifolds with semi positive holomorphic bisectional and big tangent bundles we also classify compact complex surfaces with semi positive tangent bundles and compact complex 3 folds of the form mathbb p t x whose tangent bundles are nef moreover we show that if x is a fano manifold such that mathbb p t x has nef tangent bundle then x cong mathbb p n

integer cells in convex sets abstract every convex body k in r n has a coordinate projection pk that contains at least vol 1 6 k cells of the integer lattice p z n provided this volume is at least one our proof of this counterpart of minkowski s theorem is based on an extension of the combinatorial density theorem of sauer shelah and vapnik chervonenkis to z n this leads to a new approach to sections of convex bodies in particular fundamental results of the asymptotic convex geometry such as the volume ratio theorem and milman s duality of the diameters admit natural versions for coordinate sections

spectral theory for tensor products of hermitian holomorphic line bundles let l be a quasipositive line bundle over a compact complex hermitian manifold it has been conjectured that the smallest positive eigenvalue of the laplacian acting on the k th tensor power of l is uniformly bounded independent of k various special cases of the conjecture are proved but a counterexample is given to the general conjecture

entropy and the combinatorial dimension we solve talagrand s entropy problem the l2 covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension this extends dudley s theorem on classes of 0 1 valued functions for which the shattering dimension is the vapnik chervonenkis dimension in convex geometry the solution means that the entropy of a convex body k is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of k this has a number of consequences including the optimal elton s theorem and estimates on the uniform central limit theorem in the real valued case

dynamics of stochastic seis epidemic model with varying population size abstract we introduce the stochasticity into a deterministic model which has state variables susceptible exposed infected with varying population size in this paper the infected individuals could return into susceptible compartment after recovering we show that the stochastic model possesses a unique global solution under building up a suitable lyapunov function and using generalized ito s formula the densities of the exposed and infected tend to extinction when some conditions are being valid moreover the conditions of persistence to a global solution are derived when the parameters are subject to some simple criteria the stochastic model admits a stationary distribution around the endemic equilibrium which means that the disease will prevail to check the validity of the main results numerical simulations are demonstrated as end of this contribution

persistence and distribution of a stochastic susceptible infected removed epidemic model with varying population size abstract in this paper the dynamics of a stochastic susceptible infected removed model in a population with varying size is investigated we firstly show that the stochastic epidemic model has a unique global positive solution with any positive initial value then we verify that random perturbations lead to extinction when some conditions are being valid moreover we prove that the solution of the stochastic epidemic model is persistent in the mean by building up a suitable lyapunov function and using generalized ito s formula further the stochastic epidemic model admits a stationary distribution around the endemic equilibrium when parameters satisfy some sufficient conditions to end this contribution and to check the validity of the main results numerical simulations are separately carried out to illustrate these results

stochastic permanence of an siqs epidemic model with saturated incidence and independent random perturbations abstract this article discusses a stochastic siqs epidemic model with saturated incidence we assume that random perturbations always fluctuate at the endemic equilibrium the existence of a global positive solution is obtained by constructing a suitable lyapunov function under some suitable conditions we derive the stochastic boundedness and stochastic permanence of the solutions of a stochastic siqs model some numerical simulations are carried out to check our results

mean width and diameter of proportional sections of a symmetric convex body let k be a symmetric convex body in r the purpose of this paper is to provide upper and lower bounds for the diameter of a random n dimensional central section of k where the proportion 0 1 is arbitrary but fixed there are several aspects of our approach to this question that should be clarified right away 1 we are interested in bounds expressed in terms of average parameters of the body k which can be efficiently computed in a simple way therefore being useful from the computational geometry point of view 2 the dimension n is understood to tend to infinity then we say that our bounds hold for a random n dimensional section of k if they are satisfied by all k where is in a subset of the appropriate grassmannian with haar probability measure greater than 1 h n and h n 0 as n tends to infinity 3 we say that our estimates are tight for a class of bodies and a fixed proportion if the ratio of our upper and lower bounds depends only on it is clear that one cannot obtain tight bounds for the class of all symmetric convex bodies it is not hard to describe almost degenerated bodies in r for example an ellipsoid with highly incomparable semiaxes for which the diameter of n sections does not concentrate around some value so it is an important question to see under what conditions on k the estimates obtained by a method are tight

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

type infratype and the elton pajor theorem consider a sequence x i i n of norm one vectors in a banach space for a subsetj of 1 n consider the equivalence constant j between x i i j and the l1 basis and consider k min j cardj k we give a near optimal relationship between the rate of decay of k and the averagee of left sum nolimits i leqq n varepsilon i x i right over all choices of signs in particular we show that one can choosek such that for some universal constantk k e 2 kn and xi k geqq left frac kn e 2 k right 1 2 left log left frac kn e 2 k right right k this is optimal within the logarithmic term we also prove that forp 2 the notions of type and infratype coincide

dynamics of a stochastic sir epidemic model with saturated incidence in this paper the dynamics of a stochastic sir epidemic model with saturated incidence is investigated firstly we prove that the system has a unique global positive solution with any positive initial value then we verify that random effect may lead the disease to extinction under a simple condition thirdly we establish a sufficient condition for persistence in the mean of the disease moreover we show that there is a stationary distribution to the stochastic system under certain parametric restrictions finally some numerical simulations are carried out to confirm the analytical results

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

diameters of sections and coverings of convex bodies abstract we study the diameters of sections of convex bodies in r n determined by a random n n matrix either as kernels of or as images of entries of are independent random variables satisfying some boundedness conditions and typical examples are matrices with gaussian or bernoulli random variables we show that if a symmetric convex body k in r n has one well bounded k codimensional section then for any m ck random sections of k of codimension m are also well bounded where c 1 is an absolute constant it is noteworthy that in the gaussian case when determines randomness in sense of the haar measure on the grassmann manifold we can take c 1

vapnik chervonenkis type conditions and uniform donsker classes of functions given a bounded class of functions we introduce a combinatorial quantity related to the idea of vapnik chervonenkis classes that is much more explicit than the koltchinskii pollard entropy but is proved to be essentially of the same order

a transcendental approach to kollar s injectivity theorem ii we treat kollar s injectivity theorem from the analytic or differential geometric viewpoint more precisely we give a curvature condition which implies kollar type cohomology injectivity theorems our main theorem is formulated for a compact kahler manifold but the proof uses the space of harmonic forms on a zariski open set with a suitable complete kahler metric we need neither covering tricks desingularizations nor leray s spectral sequence

asymptotic formulas for the diameter of sections of symmetric convex bodies sharpening work of the first two authors for every proportion 0 1 we provide exact quantitative relations between global parameters of n dimensional symmetric convex bodies and the diameter of their random n dimensional sections using recent results of gromov and vershynin we obtain an asymptotic formula for the diameter of random proportional sections