entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

real quadrics in cn complex manifolds and convex polytopes in this paper we investigate the topology of a class of non kahler compact complex manifolds generalizing that of hopf and calabi eckmann manifolds these manifolds are diffeomorphic to special systems of real quadrics cn which are invariant with respect to the natural action of the real torus s1 n onto cn the quotient space is a simple convex polytope the problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes we prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich we also resolve an associated wall crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope finally as a nice consequence we obtain that affine non kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups contrasting with the kahler situation

totally geodesic foliations on 4 manifolds we give a rather detailed description of the behavior of 2 dimensional totally geodesic foliations on compact riemannian 4 manifolds in particular we obtain a complete characterization in the simply connected case

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

kuranishi type moduli spaces for proper cr submersions fibering over the circle kuranishi s fundamental result 1962 associates to any compact complex manifold x 0 sub a finite dimensional analytic space which has to be thought of as a local moduli space of complex structures close to x 0 sub in this paper we give an analogous statement for levi flat cr manifolds fibering properly over the circle by describing explicitely an infinite dimensional kuranishi type local moduli space of levi flat cr structures we interpret this result in terms of kodaira spencer deformation theory making clear the likenesses as well as the differences with the classical case the article ends with applications and examples

local models and integrability of certain almost kahler 4 manifolds we classify up to a local isometry all non kahler almost kahler 4 manifolds for which the fundamental 2 form is an eigenform of the weyl tensor and whose ricci tensor is invariant with respect to the almost complex structure equivalently such almost kahler 4 manifolds satisfy the third curvature condition of a gray we use our local classification to show that in the compact case the third curvature condition of gray is equivalent to the integrability of the corresponding almost complex structure

linearly positive histories probabilities for a robust family of sequences of quantum events non negative probabilities that obey the sum rules may be assigned to a much wider family of sets of histories than decohering histories the resulting linearly positive histories avoid the highly restrictive decoherence conditions and yet give the same probabilities when those conditions apply thus linearly positive histories are a broad extension of decohering histories moreover the resulting theory is manifestly time reversal invariant

consistent histories and the interpretation of quantum mechanics the usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times called consistent histories through a criterion which ensures that within limits which are explicitly defined within the formalism classical rules for probabilities are satisfied the interpretive scheme which results is applicable to closed isolated quantum systems is explicitly independent of the sense of time i e past and future can be interchanged has no need for wave function collapse makes no reference to processes of measurement though it can be used to analyze such processes and can be applied to sequences of microscopic or macroscopic events or both as long as the mathematical condition of consistency is satisfied when applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus

counting real zeros of analytic functions satisfying linear ordinary differential equations abstract we suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients if the equation formula has no singular points in a small neighborhooduof a real segmentk all the coefficientsaj t have absolute value aonuanda0 t 1 then any solution of this equation may have no more than a zeros onk where u k is a geometric constant depending only onkandu if the principal coefficienta0 t is nonconstant but its modulus is at leasta 0 somewhere onk then the number of real zeros onkof any solution analytic inu does not exceed a a with some u k

the transverse dynamics of a holomorphic flow on considere un champ vectoriel polynomial homogene p z dans c n 1 les solutions du systeme d equations differentielles dz i dt p i z t c i 0 n feuillette c n 1 par des courbes holomorphes ayant des singularites le long du lieu commun p i z 0 on definit la notion de dynamique transverse du feuilletage

the arrival time distribution the extended probability approaches we present a calculation of the arrival time probability distribution based on the extended probability as proposed by hartle 2008 phys rev a 78 012108 we show that two cases of the arrival time probability distribution can be used first we use the path decomposition expansion to discuss a fine grained history for which a particle has a first crossing x 0 at time the arrival time probability related to the probability current is obtained second we consider that the arrival time distribution is modeled by the first photon emitted from a two level atom the basic atom laser model the partition dynamic of this system is discussed each history is in the form of a coarse grained history we confirm that the arrival time probability distribution computed by this method agrees with that computed via the probability of no photon detection crucially the resulting arrival time distributions can be obtained without the restriction of the consistency condition

holomorphic foliations in ruled surfaces we analyse the universal families of holomorphic foliations with singularities in a ruled surface in terms of chern classes we determine the general and the special families we also classify all nonsingular foliations a holomorphic foliation with singularities by curves in a complex manifold m may be defined locally as the integral curves of a nonidentically zero holomorphic vector field and globally by a holomorphic bundle map from a line bundle on m to the tangent bundle of m x l tm uniquely defined up to multiplication by a never vanishing holomorphic function the set of all holomorphic foliations by curves in a compact manifold m has a natural structure of a complex analytic variety usually with infinitely many connected components and in case m is a hodge manifold then each connected component is compact 5 the objective of this paper is to describe these components for the case of a ruled surface s we organise the components according to the chern classes of the defining bundle l and we determine those elements in h 2 s z that appear as chern classes of l using the notion of an ample direction b or of the ample cone in h 2 s z we observe a generic behaviour of the components 2m m for m large enough and a special behaviour for m small the pattern observed resembles the pattern obtained in the theory of special divisors on a riemann surface 7 we also determine all nonsingular foliations in ruled surfaces and give topological and analytic descriptions of them we finish by determining which foliations in ruled surfaces accept a change of complex structure on the leaves keeping the transversal structure fixed 6 the paper is organised as follows in 1 we describe the results of ruled surfaces i e holomorphic cpl bundles over a riemann surface b that we will need in 2 we will obtain restrictions on the chern classes c l e h 2 s z of holomorphic foliations x l ts and describe the nonsingular foliations if the genus g of b is distinct from 1 in 3 we analyse the case g 1 in 4 we describe the components as well as the subvariety of that parametrizes foliations that have an algebraic leaf of fixed fundamental class finally in 5 we apply the variational method of 6 to the ruled surface case received by the editors october 15 1987 1980 mathematics subject classification 1985 revision primary 58f18 14j26 supported by conacyt and university of warwick 1989 american mathematical society 0002 9947 89 1 00 25 per page

on the number of singularities in meromorphic foliations on compact complex surfaces here we study meromorphic foliations with singularities on a smooth compact complex surface aim study the locations of the singularities using vector bundle techniques and techniques introduced in the cohomological study of projective geometry roughly speaking a meromorphic foliation with singularities on a complex connected manifold s is described by a subsheaf f of the tangent bundle ts with f closed under lie brackets see for instance gl g2 g3 g4 g5 gk gm or for the case dim 5 2 here in 1 8 1 the singularities of the foliations are the points of s at which f is not a subbundle of ts with this definition the singularities may occur only in codimension at least 2 but a meromorphic foliation without singularities is not a holomorphic foliation in the classical sense note that any rank one subsheaf of ts is involutive and it is contained in a line bundle contained in ts in general not as subbundle thus a rank one meromorphic foliation g with singularities on s is given by a line bundle m on s and a nonzero map m ts which induces an inclusion as sheaves of m in ts up to a scalar factor of course one imposes that s o has no hypersurface as component i e that m is a saturated subsheaf of ts equivalent g is given by m and s e h s ts m with s 0 up to a scalar factor the singularities of g are the zero locus s o of s such a foliation is called also a foliation by curves thus for fixed m e pic s we are looking at a linear problem many papers are concerned with this case see e g g2 g3 g4 gk from different points of view and with several different kinds of problems in mind when dim 5 2 as always in this paper every foliation is a foliation by curves for the case dim 5 2 relevant references are among others g2 and g3 from now on we assume dim 5 2 it happens very seldom that s has a meromorphic foliation without singularities by definition this happens if and only if ts is an extension of two line bundles here we are interested essentially on the possible location for various m e pic is of the singularities associated to some s e h s ts m very often we will be interested in a general such s the main problems considered here are a given a zero dimensional scheme z c s find if there is seh s ts m 5 0

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

on the relationship between complex potentials and strings of projection operators it is of interest in a variety of contexts and in particular in the arrival time problem to consider the quantum state obtained through unitary evolution of an initial state regularly interspersed with periodic projections onto the positive x axis pulsed measurements echanobe del campo and muga have given a compelling but heuristic argument that the state thus obtained is approximately equivalent to the state obtained by evolving in the presence of a certain complex potential of step function form in this paper with the help of the path decomposition expansion of the associated propagators we give a detailed derivation of this approximate equivalence the propagator for the complex potential is known so the bulk of the derivation consists of an approximate evaluation of the propagator for the free particle interspersed with periodic position projections this approximate equivalence may be used to show that to produce significant reflection the projections must act at time spacing less than e where e is the energy scale of the initial state

a new approach to quantum backflow we derive some rigorous results concerning the backflow operator introduced by bracken and melloy we show that it is linear bounded self adjoint and non compact thus the question is underlined whether the backflow constant is an eigenvalue of the backflow operator from the position representation of the backflow operator we obtain a more efficient method to determine the backflow constant finally detailed position probability flow properties of a numerical approximation to the perhaps improper wavefunction of maximal backflow are displayed

pulling back fixed points if x y are g spaces spaces on which a compact lie group g acts and f x y is a g map then the image of a fixed point in x is fixed in i1 when does the inverse image of a fixed point contain a fixed point in particular if x y are g manifolds and f has degree prime to gi order of g in unpublished work of the author1 this is shown for elementary abelian p groups with x y more general spaces n near manifolds in this paper we show that this is true for abelian p groups g acting smoothly on x with p odd or with p 2 with an additional complex hypothesis we give a counterexample for nonabelian g the method is to prove a generalization of a theorem of bredon 1973 for 7lip actions which says that maps of degree prime to p induce injections on the mod p cohomology of the fixed set we generalize this theorem to abelian p groups where the action on x is smooth and p 4 2 or other hypothesis this seems to be new even in the case of elementary abelian p groups of rank 1 we actually prove a stronger theorem for a generalization of the notion of degree which may be non zero for maps between manifolds of different dimension the proof involves complex linear equivariant k theory in an essential way which requires smoothness and p 2 or a complexity hypothesis on x in order to produce the appropriate complex linear bundles among the corollaries we show that an abelian p group p odd cannot act smoothly on a closed manifold with exactly one isolated fixed point generalizing an old result of atiyah and bott 1964 for g z p this was found independently by ewing and stong 1986 by a different argument i am indebted to ib madsen for his comments in particular for discussions on equivariant k theory thanks are due also to michael davis for carefully reading the manuscript and pointing out several difficulties i also wish to thank the referee for his comments

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles