morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

a simple smooth backfitting method for additive models in this paper a new smooth backfitting estimate is proposed for additive regression models the estimate has the simple structure of nadaraya watson smooth backfitting but at the same time achieves the oracle property of local linear smooth backfitting each component is estimated with the same asymptotic accuracy as if the other components were known

asymptotics for in sample density forecasting this paper generalizes recent proposals of density forecasting models and it develops theory for this class of models in density forecasting the density of observations is estimated in regions where the density is not observed identification of the density in such regions is guaranteed by structural assumptions on the density that allows exact extrapolation in this paper the structural assumption is made that the density is a product of one dimensional functions the theory is quite general in assuming the shape of the region where the density is observed such models naturally arise when the time point of an observation can be written as the sum of two terms e g onset and incubation period of a disease the developed theory also allows for a multiplicative factor of seasonal effects seasonal effects are present in many actuarial biostatistical econometric and statistical studies smoothing estimators are proposed that are based on backfitting full asymptotic theory is derived for them a practical example from the insurance business is given producing a within year budget of reported insurance claims a small sample study supports the theoretical results

second quantization of the elliptic calogero sutherland model we construct a quantum field theory model of anyons on a circle and at finite temperature we find an anyon hamiltonian providing a second quantization of the elliptic calogero sutherland model this allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic calogero sutherland hamiltonian

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

on a conjecture of anosov abstract in this paper we prove that for every bumpy finsler n sphere s n f with reversibility and flag curvature k satisfying 1 2 k 1 there exist 2 n 1 2 prime closed geodesics this gives a confirmed answer to a conjecture of anosov 1 in 1974 for a generic case

subharmonic solutions of hamiltonian equations on tori let the torus t2n be equipped with the standard symplectic structure and a periodic hamiltonian dt e c3 sx x t2n r we look for periodic orbits of the hamiltonian flow u t j vdt t a subharmonic solution is a periodic orbit with minimal period an integral multiple m of the period of dt with m 1 we prove that if the hamiltonian flow has only finitely many orbits with the same period as dt then there are subharmonic solutions with arbitrarily high minimal period thus there are always infinitely many distinct periodic orbits the results proved here were proved in the nondegenerate case by conley and zehnder and in the case n 1 by le calvez

multiple closed geodesics on bumpy finsler n spheres abstract in this paper we prove that for every bumpy finsler metric f on every rationally homological n dimensional sphere s n with n 2 there exist always at least two distinct prime closed geodesics

on the average indices of closed geodesics on positively curved finsler spheres in this paper we prove that on every finsler n sphere s n f for n 6 with reversibility and flag curvature k satisfying frac lambda lambda 1 2 k le 1 either there exist infinitely many prime closed geodesics or there exist frac n 2 2 closed geodesics possessing irrational average indices if in addition the metric is bumpy then there exist n 3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite

iteration inequalities of the maslov type index theory with applications abstract in this paper by using the index theory introduced by y long in 1999 pacific j math 187 113 149 in particular the splitting numbers maslov type mean index and the homotopy component of symplectic matrix we establish various inequalities of the maslov type index theory for iterations of symplectic paths starting from the identity as an application these results are used to study rabinowitz conjecture on the prescribed minimal period solution problem of nonlinear hamiltonian systems

the index growth and multiplicity of closed geodesics abstract in the recent paper 31 of long and duan 2009 we classified closed geodesics on finsler manifolds into rational and irrational two families and gave a complete understanding on the index growth properties of iterates of rational closed geodesics this study yields that a rational closed geodesic cannot be the only closed geodesic on every irreversible or reversible including riemannian finsler sphere and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible including riemannian finsler 3 dimensional manifold in this paper we study the index growth properties of irrational closed geodesics on finsler manifolds this study allows us to extend results in 31 of long and duan 2009 on rational and in 12 of duan and long 2007 39 of rademacher 2010 and 40 of rademacher 2008 on completely non degenerate closed geodesics on spheres and cp 2 to every compact simply connected finsler manifold then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible including riemannian finsler 4 dimensional manifold

maslov type indices and linear stability of elliptic euler solutions of the three body problem in this paper we use the central configuration coordinate decomposition to study the linearized hamiltonian system near the 3 body elliptic euler solutions then using the maslov type omega index theory of symplectic paths and the theory of linear operators we compute the omega indices and obtain certain properties of linear stability of the euler elliptic solutions of the classical three body problem

the minimal period problem of periodic solutions for autonomous superquadratic second order hamiltonian systems abstract in this paper we study the existence of periodic solutions with prescribed minimal period for superquadratic autonomous second order hamiltonian systems defined on r n with no convexity assumptions we use the direct variational approach for this problem on a w 1 2 space of even functions and prove new iteration inequalities on morse indices using these tools and the saddle point theorem we obtain results under precisely rabinowitz superquadratic condition on potential functions we show that for every t 0 the above mentioned system possesses a t periodic even solution with minimal period not smaller than t n 2

closed characteristics on partially symmetric compact convex hypersurfaces in r2n abstract in this paper we study the multiplicity of closed characteristics on partially symmetric convex compact hypersurfaces in r 2n the main ingredient of the proof is a new p index function and its iteration theory for symplectic matrix paths and some symplectic orthogonal matrix p this theory yields estimates on this multiplicity via index estimates for a new type of iterations of any symplectic matrix path 0 sp 2n defined by t p t p for any t 0

multiple closed geodesics on 3 spheres abstract this paper is devoted to a study on closed geodesics on finsler and riemannian spheres we call a prime closed geodesic on a finsler manifold rational if the basic normal form decomposition cf y long bott formula of the maslov type index theory pacific j math 187 1999 113 149 of its linearized poincare map contains no 2 2 rotation matrix with rotation angle which is an irrational multiple of or irrational otherwise we prove that if there exists only one prime closed geodesic on a d dimensional irreversible finsler sphere with d 2 it cannot be rational then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible finsler 3 dimensional sphere our method yields also at least two geometrically distinct closed geodesics on every reversible finsler as well as riemannian 3 dimensional sphere we prove also such results hold for all compact simply connected 3 dimensional manifolds with irreversible or reversible finsler as well as riemannian metrics

two elliptic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f with reversibility and flag curvature k satisfying 1 2 k 1 either there exist infinitely many closed geodesics or there exist at least two elliptic closed geodesics and each linearized poincare map has at least one eigenvalue of the form e 1 with being an irrational multiple of

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

time varying additive models for longitudinal data the additive model is an effective dimension reduction approach that also provides flexibility in modeling the relation between a response variable and key covariates the literature is largely developed to scalar response and vector covariates in this article more complex data are of interest where both the response and the covariates are functions we propose a functional additive model together with a new backfitting algorithm to estimate the unknown regression functions whose components are time dependent additive functions of the covariates such functional data may not be completely observed since measurements may only be collected intermittently at discrete time points we develop a unified platform and an efficient approach that can cover both dense and sparse functional data and the needed theory for statistical inference we also establish the oracle properties of the proposed estimators of the component functions

fitting a bivariate additive model by local polynomial regression while the additive model is a popular nonparametric regression method many of its theoretical properties are not well understood especially when the backfitting algorithm is used for computation of the estimators this article explores those properties when the additive model is fitted by local polynomial regression sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given asymptotic approximations to the bias and the variance of a homoscedastic bivariate additive model with local polynomial terms of odd and even degree are computed this model is shown to have the same rate of convergence as that of univariate local polynomial regression

bandwidth selection for smooth backfitting in additive models the smooth backfitting introduced by mammen linton and nielsen ann statist 27 1999 1443 1490 is a promising technique to fit additive regression models and is known to achieve the oracle efficiency bound in this paper we propose and discuss three fully automated bandwidth selection methods for smooth backfitting in additive models the first one is a penalized least squares approach which is based on higher order stochastic expansions for the residual sums of squares of the smooth backfitting estimates the other two are plug in bandwidth selectors which rely on approximations of the average squared errors and whose utility is restricted to local linear fitting the large sample properties of these bandwidth selection methods are given their finite sample properties are also compared through simulation experiments

multiple periodic points of the poincare map of lagrangian systems on tori in this paper suppose l t x p 1 over 2 a x p cdot p v t x a is positive definite and symmetric and both a and v are c 3 and 1 periodic in all of their variables we prove that the poincare map i e the time 1 solution map of the lagrangian system

trace formula for linear hamiltonian systems with its applications to elliptic lagrangian solutions in the present paper we build up trace formulas for both the linear hamiltonian systems and sturm liouville systems the formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the hessian of the action functional a natural application is to study the non degeneracy of linear hamiltonian systems precisely by the trace formula we can give an estimation for the upper bound such that the non degeneracy preserves moreover we could estimate the relative morse index by the trace formula consequently a series of new stability criteria for the symmetric periodic orbits is given as a concrete application the trace formula is used to study the linear stability of elliptic lagrangian solutions of the classical planar three body problem which depends on the mass parameter beta in 0 9 and the eccentricity e in 0 1 based on the trace formula we estimate the stable region and hyperbolic region of the elliptic lagrangian solutions

the second closed geodesic on finsler spheres of dimension n 2 we show the existence of at least two geometrically distinct closed geodesics on an n dimensional sphere with a bumpy and non reversible finsler metric for n 2 2000 msc classification 53c22 53c60 58e10

closed characteristics on compact convex hypersurfaces in r 2n for any given compact c 2 hypersurface in r 2n bounding a strictly convex set with nonempty interior in this paper an invariant n is defined and satisfies n n 2 1 where a denotes the greatest integer which is not greater than a r the following results are proved in this paper there always exist at least n geometrically distinct closed characteristics on if all the geometrically distinct closed characteristics on are nondegenerate then n n if the total number of geometrically distinct closed characteristics on is finite there exists at least an elliptic one among them and there exist at least n 1 of them possessing irrational mean indices if this total number is at most 2 n 2 there exist at least two elliptic ones among them

a new obstruction to embedding lagrangian tori n let 2n be endowed with the canonical symplectic form a i 1 dxi dy i and consider a lagrangian embedding of a compact manifold j l r 2 a a well known result due to g r o m o v g states that the cohomology class of a in h z e 2 l is non zero we identify l with its image in r2 an equivalent assertion is that the liouville form 2 y 7 1 xldy i pulls back to a closed form j 2 on l whose cohomology class does not vanish g r o m o v s proof uses subtle properties of holomorphic maps into c he shows that there is always a holomorphic disk a in c with boundary in l on such a disk a is positive hence sa a soa 2 0 in this paper we shall only consider the case l t and give an elementary proof of gromov s result elementary should be understood in the sense of number theory that is without using holomorphic functions in fact our result is somewhat more precise and can be stated as follows

irrationally elliptic closed characteristics on compact convex hypersurfaces in r6 abstract in this paper let r 6 be a compact convex hypersurface which carries exactly three geometrically distinct closed characteristics we prove that at least two of them must be irrationally elliptic

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

multiple brake orbits in bounded convex symmetric domains abstract in this paper we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in r n for n 2

multiplicity of closed characteristics on symmetric convex hypersurfaces in mathbb r 2n let sigma be a compact c 2 hypersurface in mathbb r 2n bounding a convex set with non empty interior in this paper it is proved that there always exist at least n geometrically distinct closed characteristics on sigma if sigma is symmetric with respect to the origin

a bandwidth selector for local linear density estimators local linear density estimators achieve automatic boundary corrections and enjoy some typical optimal properties proper choice of the smoothing parameters is crucial for their performance a data based bandwidth selector is developed in the spirit of plug in rules consistency and asymptotic normality of the selected bandwidth are demonstrated the bandwidth is very efficient regardless of whether there are non smooth boundaries in the support of the density or not

a sphere theorem for non reversible finsler metrics for a non reversible finsler metric f on a compact smooth manifold m we introduce the reversibility max f x f x 1 1 we prove the following generalization of the classical sphere theorem in riemannian geometry a simply connected and compact finsler manifold of dimension n 3 with reversibility and with flag curvature left 1 frac 1 1 lambda right 2 k le 1 is homotopy equivalent to the n sphere

stable closed characteristics on partially symmetric convex hypersurfaces abstract let n be a positive integer and p diag in i in i for some integer 0 n in this paper we prove that for any convex compact smooth hypersurface in r2n with n 2 there always exists at least one closed characteristic on which possesses at least 2n 4 floquet multipliers on the unit circle of the complex plane provided is p symmetric i e x implies px

geodesics on s2 and periodic points of annulus homeomorphisms we show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points a consequence is the theorem that every smooth riemannian metric ons2 with positive gaussian curvature has infinitely many distinct closed geodesics

resonance identity and multiplicity of non contractible closed geodesics on finsler rpn abstract in this paper we establish first the resonance identity for non contractible homologically visible prime closed geodesics on finsler n dimensional real projective space r p n f when there exist only finitely many distinct non contractible closed geodesics on r p n f where the integer n 2 then as an application of this resonance identity we prove the existence of at least two distinct non contractible closed geodesics on r p n with a bumpy and irreversible finsler metric together with two previous results on bumpy and reversible finsler metrics in 14 and 39 it yields that every r p n with a bumpy finsler metric possesses at least two distinct non contractible closed geodesics

non hyperbolic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f n 3 with reversibility and flag curvature k satisfying 1 2 k 1 there exist at least three distinct closed geodesics and at least two of them are elliptic if the number of prime closed geodesics is finite when n 6 these three distinct closed geodesics are non hyperbolic

existence of closed geodesics on positively curved manifolds on applique la theorie de lusternik schnirelmann pour obtenir l existence de plusieurs geodesiques fermees entre certains niveaux de longueur on etudie les proprietes de stabilite de ces geodesiques fermees

equivariant morse theory for starshaped hamiltonian systems soit une hypersurface en forme d etoile dans r 2n le probleme de trouver des caracteristiques fermees de peut se reduire classiquement a un probleme variationnel ceci conduit a etudier une fonctionnelle s 1 equivariante sur un espace de hilbert

infinitely many periodic solution orbits of autonomous lagrangian systems on tori we prove that for any 0 the autonomous lagrangian system ddtlx x x lx x x 0 on the standard n dimensional torus possesses infinitely many geometrically distinct contractible periodic solution orbits whose periods are integer multiples of provided that the lagrangian function satisfies l x p 12ap p v x where a is a positive definite symmetric real matrix and v is c3 and 1 periodic in all of its variables

resonance identity for symmetric closed characteristics on symmetric convex hamiltonian energy hypersurfaces and its applications in this paper we establish a new resonance identity for symmetric closed characteristics on symmetric compact convex hypersurface in r2n when there exist only finitely many geometrically distinct symmetric closed characteristics as its applications some interesting results about the stability and multiplicity of symmetric closed characteristics are obtained and also we prove that if is c generic it carries infinitely many symmetric closed characteristics

the existence of two closed geodesics on every finsler 2 sphere in this paper we prove that for every finsler metric on s2 there exist at least two distinct prime closed geodesics

topological structure of non contractible loop space and closed geodesics on real projective spaces with odd dimensions abstract in this paper we use chas sullivan theory on loop homology and leray serre spectral sequence to investigate the topological structure of the non contractible component of the free loop space on the real projective spaces with odd dimensions then we apply the result to get the resonance identity of non contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite

index iteration theories for periodic orbits old and new abstract in this paper we give a survey of the index iteration theories developed for periodic solution orbits of hamiltonian systems in treating various problems from 1950s up to now including some recent developments and some of their applications to periodic solution problems

the conley conjecture we prove the conley conjecture for a closed symplectically aspherical symplectic manifold a hamiltonian diffeomorphism of such a manifold has infinitely many periodic points more precisely we show that a hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period this theorem generalizes for instance a recent result of hingston establishing the conley conjecture for tori

an optimization interpretation of integration and back fitting estimators for separable nonparametric models we provide an optimization interpretation of both back fitting and integration estimators for additive nonparametric regression we find that the integration estimator is a projection with respect to a product measure we also provide further understanding of the back fitting method

identification of the age period cohort model and the extended chain ladder model we consider the identification problem that arises in the age period cohort models as well as in the extended chain ladder model we propose a canonical parameterization based on the accelerations of the trends in the three factors this parameterization is exactly identified and eases interpretation estimation and forecasting the canonical parameterization is applied to a class of index sets which have trapezoidal shapes including various lexis diagrams and the insurance reserving triangles

convex hamiltonian energy surfaces and their periodic trajectories in this paper we introduce symplectic invariants for convex hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations in particular we show that they can be used to prove multiplicity results for the number of periodic trajectories

lusternik schnirelmann theory and closed reeb orbits we develop a variant of lusternik schnirelmann theory for the shift operator in equivariant floer and symplectic homology our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated as an application we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere the unit cotangent bundle to the sphere and some other contact manifolds we also show that the lower conley zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of long and zhu this is the second combinatorial ingredient in the proof of the multiplicity results

on the average indices of closed geodesics on definit la notion d indices moyens de geodesiques fermees on demontre une relation entre ces indices pour une variete compacte simplement connexe dotee d une metrique admissible

resonance identities for closed characteristics on compact star shaped hypersurfaces in r2n resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding hamiltonian systems in this paper we establish two new resonance identities for closed characteristics on every compact star shaped hypersurface in r2n when the number of geometrically distinct closed characteristics on is finite which extend those identities established by c viterbo in 1989 for star shaped hypersurfaces assuming in addition that all the closed characteristics and their iterates are non degenerate and that by w wang x hu and y long in 2007 for strictly convex hypersurfaces in r2n

seifert conjecture in the even convex case in this paper we prove that there exist at least n geometrically distinct brake orbits on every c2 compact convex symmetric hypersurface in 2n satisfying the reversible condition n with n diag in in as a consequence we show that if the hamiltonian function is convex and even then seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integern 2014 wiley periodicals inc

hyperbolic closed characteristics on compact convex smooth hypersurfaces inr2n in this paper we prove that on every compactc2hypersurface inr2nbounding a convex set with non empty interior either there exists a sequence of variationally visible hyperbolic closed characteristcs with their minimal periods tending to infinity or there exists at least one variationally visible nonhyperbolic closed characteristic

iteration theory of l index and multiplicity of brake orbits abstract in this paper we first establish the bott type iteration formulas and some abstract precise iteration formulas of the maslov type index theory associated with a lagrangian subspace for symplectic paths as an application we prove that there exist at least n 2 1 geometrically distinct brake orbits on every c 2 compact convex symmetric hypersurface in r 2 n satisfying the reversible condition n furthermore if all brake orbits on this hypersurface are nondegenerate then there are at least n geometrically distinct brake orbits on it as a consequence we show that there exist at least n 2 1 geometrically distinct brake orbits in every bounded convex symmetric domain in r n furthermore if all brake orbits in this domain are nondegenerate then there are at least n geometrically distinct brake orbits in it in the symmetric case we give a positive answer to the seifert conjecture of 1948 under a generic condition

stability of closed geodesics on finsler 2 spheres abstract in this paper we prove that on every finsler 2 dimensional sphere either there exist infinitely many prime closed geodesics or there exist at least two irrationally elliptic prime closed geodesics

elliptic relative equilibria in the n body problem a planar central configuration of the n body problem gives rise to a solution where each particle moves on a specific keplerian orbit while the totality of the particles move on a homothety motion if the keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution an elliptic relative equilibrium the totality of such solutions forms a four dimensional symplectic subspace and we give a symplectic coordinate system which is adapted to this subspace and its symplectic complement in our coordinate system the linear variational equations of such a solution decouple into three subsystems one subsystem simply gives the motion of the center of mass another is kepler s problem and the third determines the nontrivial characteristic multipliers using these coordinates we study the linear stability of the elliptic relative equilibrium defined by the equilateral triangular central configuration of the three body problem we reproduce the analytic studies of g roberts we also study the linear stability of the four and five body problem where three or four bodies of unit mass are at the vertices of a equilateral triangle or

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation