embeddings of submanifolds and normal bundles this paper studies the embeddings of a complex submanifold s inside a complex manifold m in particular we are interested in comparing the embedding of s in m with the embedding of s as the zero section in the total space of the normal bundle ns of s in m we explicitly describe some cohomological classes allowing to measure the difference between the two embeddings in the spirit of the work by grauert griffiths and camacho movasati and sad we are also able to explain the geometrical meaning of the separate vanishing of these classes our results hold for any codimension but even for curves in a surface we generalize previous results due to laufert and camacho movasati and sad

positive solutions of sub superlinear sturm liouville problems in this paper we introduce the notion of a strict lower upper solution to nonlinear sturm liouville boundary value problems based on the maximum principles we establish a result of leray schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions applying the result and the fixed point index theory in cones we obtain the global existence results of positive solutions for sub superlinear sturm liouville problems

residual indices of holomorphic maps relative to singular curves of fixed points on surfaces let m be a two dimensional complex manifold and let f m to m be a holomorphic map that fixes pointwise a possibly singular compact reduced and globally irreducible curve c subset m we give a notion of degeneracy of f at a point of c it turns out that f is non degenerate at one point if and only if it is non degenerate at every point of c when f is non degenerate on c we define a residual index for f at each point of c then we prove that the sum of the indices is equal to the self intersection number of c

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

multiple positive solutions for nonlinear periodic problems abstract we consider a nonlinear periodic equation driven by the scalar p laplacian and with a caratheodory asymptotically p 1 linear nonlinearity using variational methods coupled with suitable truncation techniques we show that the problem has at least two positive solutions for the semilinear case p 2 using morse theory we show that the problem has three distinct positive solutions

germs of holomorphic vector fields in c3 without a separatrix oblatum 28 xii 1988 26 iii 1990 10 x 1990 13 vi 1991 30 ix 1991 supported by universite de toulouse conacyt cnrs and caicyt 87 336 fulltext preview

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

hypersurfaces quasi invariant by codimension one foliations we present a variant of the classical darboux jouanolou theorem our main result provides a characterization of foliations which are pull backs of foliations on surfaces by rational maps as an application we provide a structure theorem for foliations on 3 folds admitting an infinite number of extremal rays

on codimension one foliations with prescribed cuspidal separatrix abstract in this paper we will construct a pre normal form for germs of codimension one holomorphic foliation having a particular type of separatrix of cuspidal type as an application we will explain how this form could be used in order to study the analytic classification of the singularities via the projective holonomy in the generalized surface case we will also give an application to the analytic classification of singularities and a sufficient condition in the quasi homogeneous three dimensional case for these foliations to be of generalized surface type

darboux integrating factors inverse problems abstract we discuss planar polynomial vector fields with prescribed darboux integrating factors in a nondegenerate affine geometric setting we establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems and show that the space of vector fields with a given integrating factor modulo a subspace of explicitly known standard vector fields has finite dimension for several classes of examples we determine this space explicitly

generalized ambrosetti prodi conditions for nonlinear two point boundary value problems 1 under the assumptions that g is of class c2 strictly convex and such that o lim g u l lim g u l 2 u ro it 0 where 1 a2 are the first two eigenvalues of the dirichlet problem on 52 for a they proved that the holder space c a was split into two open sets 0 and o2 by a c manifold m such that i if k e oo 1 has no solution ii if 12 e m 1 has exactly one solution iii if h e 02 1 has exactly two solutions 422

extendable cohomologies for complex analytic varieties we introduce a cohomology called extendable cohomology for abstract complex singular varieties based on suitable differential forms beside a study of the general properties of such a cohomology we show that given a complex vector bundle one can compute its topological chern classes using the extendable chern classes defined via a chern weil type theory we also prove that the localizations of the extendable chern classes represent the localizations of the respective topological chern classes thus obtaining an abstract residue theorem for compact singular complex analytic varieties as an application of our theory we prove a camacho sad type index theorem for holomorphic foliations of singular complex varieties

on the quasi ordinary cuspidal foliations in c 3 0 abstract in this paper we study a class of singularities of codimension 1 holomorphic germs of foliations in c 3 0 namely those ones having only one separatrix that is a quasi ordinary surface and whose reduction of singularities agrees with the combinatorial desingularization of the separatrix we show that the analytic classification of these germs can be read in the holonomy of a certain component of the exceptional divisor of the desingularization

index theorems for holomorphic self maps let m be a complex manifold and s subset m a possibly singular subvariety of m let f colon m to m be a holomorphic map such that f restricted to s is the identity we show that one can associate to f a holomorphic section x f of a sheaf related to the embedding of s in m and that such a section reads the dynamical behavior of f along s in particular we prove that under generic hypotheses the canonical section x f induces a holomorphic action in the sense of bott on the normal bundle of the regular part of s in m and this allows to obtain for holomorphic self maps with non isolated fixed points index theorems similar to camacho sad baum bott and variation index theorems for holomorphic foliations finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self maps of surfaces with a compact curve of fixed points

size biased sampling of poisson point processes and excursions some general formulae are obtained for size biased sampling from a poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function these formulae explain why in certain cases gamma and stable the size biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking residual allocation scheme defined by independent beta random variables an application is made to length biased sampling of excursions of a markov process away from a recurrent point of its statespace with emphasis on the brownian and bessel cases when the associated inverse local time is a stable subordinator results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by brownian motion

sur les integrales premieres dans la classe de nilsson d equations differentielles holomorphes we study two classes of holomorphic differential equations the first one is constitued by elements admitting solutions defined in an algebraic way the so called liouville class and the second of elements admitting solutions defined in an analytic way the nilsson class we build up links between these two classes using special properties of the holonomy and its results on the monodromy

universal statistics of longest lasting records of random walks and levy flights we study the record statistics of random walks after n steps x0 x1 xn with arbitrary symmetric and continuous distribution p of the jumps i xi xi 1 we consider the age of the records i e the time up to which a record survives depending on how the age of the current last record is defined we propose three distinct sequences of ages indexed by i ii iii associated to a given sequence of records we then focus on the longest lasting record which is the longest element among this sequence of ages to characterize the statistics of these longest lasting records we compute i the probability that the record of the longest age is broken at step n denoted by q n which we call the probability of record breaking and ii the duration of the longest lasting record we show that both q n and the full statistics of are universal i e independent of the jump distribution p we compute exactly the large n asymptotic behaviors of q n as well as when it exists and show that each case gives rise to a different universal constant associated to random walks including levy flights while two of them appeared before in the excursion theory of brownian motion for which we provide here a simpler derivation the third case gives rise to a non trivial new constant ciii 0 241 749 associated to the records of random walks other observables characterizing the ages of the records exhibiting an interesting universal behavior are also discussed

meromorphic vector fields with single valued solutions on complex surfaces abstract we study ordinary differential equations in the complex domain given by meromorphic vector fields on compact complex kahler surfaces we prove that if such an equation has a maximal single valued solution in particular an entire one whose image is not contained in a proper analytic subset then up to a bimeromorphic transformation either the vector field is holomorphic or it preserves a fibration

complete algebraic vector fields on affine surfaces part ii in this work we study the singularities at infinity of algebraic vector fields in dimension 2 these singularities will be classified under a mild assumption the general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in part ii our main results are local and therefore can be carried over more general surfaces whereas we deal with complete vector fields the results also apply to complete ones thanks to a theorem of forstneric 7

a class of nonlinear sturm liouville problems with infinitely many solutions on etudie l existence des solutions des problemes de sturm liouville non lineaires dont la partie lineaire est a la resonance on montre que de tels problemes peuvent avoir une infinite de solutions si les perturbations non lineaires sont periodiques

ages of records in random walks we consider random walks with continuous and symmetric step distributions we prove universal asymptotics for the average proportion of the age of the kth longest lasting record for k 1 2 ldots and for the probability that the record of the kth longest age is broken at step n due to the relation to the chinese restaurant process the ranked sequence of proportions of ages converges to the poisson dirichlet distribution

positive solutions for nonlinear periodic problems with concave terms abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a parametric concave term and a caratheodory perturbation whose potential primitive exhibits a p superlinear growth near without satisfying the usual in such cases ambrosetti rabinowitz condition using critical point theory and truncation techniques we prove a bifurcation type theorem describing the nonexistence existence and multiplicity of positive solutions as the parameter varies

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

arcsine laws and interval partitions derived from a stable subordinator levy discovered that the fraction of time a standard one dimensional brownian motion b spends positive before time t has arcsine distribution both for a fixed time when b t 0 almost surely and for t an inverse local time when b t 0 almost surely this identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of b away from 0 similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator for instance a bessel process of dimension d for 0 d 2

multiple positive solutions for a superlinear problem a topological approach we study the multiplicity of positive solutions for a two point boundary value problem associated to the nonlinear second order equation u f x u 0 we allow x f x s to change its sign in order to cover the case of scalar equations with indefinite weight roughly speaking our main assumptions require that f x s s is below 1 as s 0 and above 1 as s in particular we can deal with the situation in which f x s has a superlinear growth at zero and at infinity we propose a new approach based on topological degree which provides the multiplicity of solutions applications are given for u a x g u 0 where we prove the existence of 2n 1 positive solutions when a x has n positive humps and a x is sufficiently large

longest excursion of fractional brownian motion numerical evidence of non markovian effects we study using exact numerical simulations the statistics of the longest excursion l text max t up to time t for the fractional brownian motion with hurst exponent 0lhl1 we show that in the large t limit l text max t ensuremath propto q ensuremath infty t where q ensuremath infty ensuremath equiv q ensuremath infty h depends continuously on h these results are compared with exact analytical results for a renewal process with an associated persistence exponent ensuremath theta 1 ensuremath h this comparison shows that q ensuremath infty h carries the clear signature of non markovian effects for h ensuremath ne 1 2 the preasymptotic behavior of l text max t is also discussed

liouvillian integration and bernoulli foliations analytic foliations in the 2 dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable it is shown that such a condition leads to the existence of a liouville type first integral and under generic extra conditions it is proven that these foliations can be defined by bernoulli equations

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

record statistics of a strongly correlated time series random walks and levy flights we review recent advances on the record statistics of strongly correlated time series whose entries denote the positions of a random walk or a l evy flight on a line after a brief survey of the theory of records for independent and identically distributed random variables we focus on random walks during the last few years it was indeed realized that random walks are a very useful laboratory to test the effects of correlations on the record statistics we start with the simple one dimensional random walk with symmetric jumps both continuous and discrete and discuss in detail the statistics of the number of records as well as of the ages of the records i e the lapses of time between two successive record breaking events then we review the results that were obtained for a wide variety of random walk models including random walks with a linear drift continuous time random walks constrained random walks like the random walk bridge and the case of multiple independent random walkers finally we discuss further observables related to records like the record increments as well as some questions raised by physical applications of record statistics like the effects of measurement error and noise

on invariance of plurigenera for foliations on surfaces let x be a smooth algebraic surface a foliation f on x is roughly speaking a subline bundle t f of the tangent bundle of x the dual of t f is called the canonical bundle of the foliation k f in the last few years birational methods have been successfully used in order to study foliations more precisely geometric properties of the foliation are translated into properties of the canonical bundle of the foliation one of the most important invariants describing the properties of a line bundle l is its kodaira dimension kappa l which measures the growth of the global sections of l and its tensor powers the kodaira dimension of a foliation f is defined as the kodaira dimension of its canonical bundle kappa k f in their fundamental works brunella and mcquillan give a classfication of foliations on surfaces on the model of enriques kodaira classification of surfaces the next step is the study of the behaviour of families of foliations brunella proves that for a family of foliations x t f t of dimension one on surfaces satisfying certain hypotheses of regularity the kodaira dimension of the foliation does not depend on t by analogy with siu s invariance of plurigenera it is natural to ask whether for a family of foliations x t f t the dimensions of global sections of the canonical bundle and its powers depend on t in this talk we will discuss to which extent an invariance of plurigenera for foliations is true and under which hypotheses on the family of foliations it holds

exchangeable and partially exchangeable random partitions call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn1 nk the probability that the partition breaks the firstn1 nk integers intok particular classes of sizesn1 nk in order of their first elements has the same valuep n1 nk for every possible choice of classes subject to the sizes constraint a random partition is exchangeable iff it is partially exchangeable for a symmetric functionp n1 nk a representation is given for partially exchangeable random partitions which provides a useful variation of kingman s representation in the exchangeable case results are illustrated by the two parameter generalization of ewens partition structure

embedding smooth and formal diffeomorphisms through the jordan chevalley decomposition abstract in xiang zhang the embedding flows of c hyperbolic diffeomorphisms j differential equations 250 5 2011 2283 2298 zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field we present a new and more conceptual proof of such result using the jordan chevalley decomposition in algebraic groups and the properties of the exponential operator we characterize the hyperbolic smooth resp formal diffeomorphisms that are embedded in a smooth resp formal flow we introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable this solves a conjecture of zhang the criterion is optimal we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions