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

boundary stabilization of linear elastodynamic systems it is proved that an elastic medium which occupies a bounded region in three dimensional euclidean space can be uniformly stabilized by means of traction forces applied on a portion of the boundary

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

persistently expansive geodesic flows we prove thatc1 persistently expansive geodesic flows of compact boundaryless riemannian manifolds have the property that the closure of the set of closed orbits is a hyperbolic set in the case of compact surfaces we deduce that the geodesic flow isc1 persistently expansive if and only if it is an anosov flow

well posedness of systems of linear elasticity with dirichlet boundary control and observation an open loop system of linear elasticity with dirichlet boundary control and collocated observation is considered the main result we obtained states that this system is well posed in the sense of salamon the result deduces the exponential stability of the closed loop system under proportional output feedback hence it answers positively an open question proposed by liu and krstic ima j appl math 65 2000 pp 109 121 moreover the well posedness together with the regularity of the open loop system in the sense of weiss that was obtained in our separate paper s g chai and b z guo feedthrough operator for linear elasticity system with boundary control and observation preprint school of computational and applied mathematics university of the witwatersrand south africa 2008 makes this infinite dimensional system parallel in many ways to a linear finite dimensional system in the framework of well posed and regular linear infinite dimensional systems

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 kahler ricci flow on fano surfaces i suppose m g t 0 t is a kahler ricci flow solution on a fano surface if rm is not uniformly bounded along this flow we can blowup at the maximal curvature points to obtain a limit complete riemannian manifold x we show that x must have certain topological and geometric properties using these properties we are able to prove that rm is uniformly bounded along every kahler ricci flow on toric fano surface whose initial metric has toric symmetry in particular such a kahler ricci flow must converge to a kahler ricci soliton metric therefore we give a new ricci flow proof of the existence of kahler ricci soliton metrics on toric fano surfaces

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

loop products and closed geodesics the critical points of the length function on the free loop space m of a compact riemannian manifold m are the closed geodesics on m the length function gives a filtration of the homology of m and we show that the chas sullivan product hi hj hi j n is compatible with this filtration we obtain a very simple expression for the associated graded homology ring grh m when all geodesics are closed or when all geodesics are nondegenerate we also interpret sullivan s coproduct see su1 su2 on c as a product in cohomology h 0 h 0 hi j n 1 0 where 0 m is the constant loop we show that is also compatible with the length filtration and we obtain a similar expression for the ring grh 0 the nonvanishing of products n and n is shown to be determined by the rate at which the morse index grows when a geodesic is iterated we determine the full ring structure h 0 for spheres m s n 3

decay rates for the three dimensional linear system of thermoelasticity we consider the two and three dimensional system of linear thermoelasticity in a bounded smooth domain with dirichlet boundary conditions we analyze whether the energy of solutions decays exponentially uniformly to zero as t to infty first of all by a decoupling method we reduce the problem to an observability inequality for the lame system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component we show that when the domain is convex the decay rate is never uniform in fact the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary we also show that in three space dimensions the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement in two space dimensions we prove a sufficient and almost necessary condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents we also give an example due to d hulin in which these geometric properties hold in three space dimensions we indicate without proof how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold in two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite dimensional subspace of solutions except when the domain is a ball or an annulus finally we discuss some closely related controllability and spectral issues

feedthrough operator for linear elasticity system with boundary control and observation we are concerned with the feedthrough operator for the open loop system of linear elasticity with dirichlet boundary control and collocated observation this system has been known to be regular in the sense of g weiss in this paper the analytic expression of the corresponding feedthrough operator is presented by means of differential geometry and fourier transform methods

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

hyperbolicity through stable shadowing for generic geodesic flows abstract we prove that the closure of the closed orbits of a generic geodesic flow on a closed riemannian n 2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds c 2 robustly on the metric we obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization

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

iteration inequalities of the maslov p index theory with applications abstract in this paper the p splitting numbers and p mean index for a symplectic path starting from the identity with p s p 2 n are defined first and then by using the maslov p index theory for a symplectic path various inequalities of the maslov p index theory for iterations of symplectic paths starting from the identity are established as an application these results are used to study the minimal period problem of p symmetric periodic solution of nonlinear autonomous hamiltonian systems

a homotopy theoretic realization of string topology let m be a closed oriented manifold of dimension d let lm be the space of smooth loops in m in 2 chas and sullivan defined a product on the homology h lm of degree d they then investigated other structure that this product induces including a batalin vilkovisky structure and a lie algebra structure on the s1 equivariant homology h s 1 lm these algebraic structures as well as others came under the general heading of the string topology of m in this paper we will describe a realization of the chas sullivan loop product in terms of a ring spectrum structure on the thom spectrum of a certain virtual bundle over the loop space we also show that an operad action on the homology of the loop space discovered by voronov has a homotopy theoretic realization on the level of thom spectra this is the cactus operad defined in 6 which is equivalent to operad of framed disks in this operad action realizes the chas sullivan bv structure on h lm we then describe a cosimplicial model of this ring spectrum and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the hochschild cohomology hh c m c m

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

the numerical computation of the confluent hypergeometric functionu a b z an algorithm is given for the computation of the confluent hypergeometric functionu a b z for real values ofa b andz z 0 algol 60 procedures are given the computations are based on a miller algorithm and on asymptotic expansions

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

a geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of 2 we give a proof based in geometric perturbation theory of a result proved by j n mather using variational methods namely the existence of orbits with unbounded energy in perturbations of a generic geodesic flow in 2 by a generic periodic potential

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

the fadell rabinowitz index and multiplicity of non contractible closed geodesics on finsler rpn abstract in this paper we prove that for every irreversible finsler n dimensional real projective space r p n f with reversibility and flag curvature k satisfying 16 9 1 2 k 1 with 3 there exist at least n 1 non contractible closed geodesics in addition if the metric f is bumpy with 64 25 1 2 k 1 and 5 3 then there exist at least 2 n 1 2 non contractible closed geodesics which is the optimal lower bound due to katok s example the main ingredients of the proofs are the fadell rabinowitz index theory of non contractible closed geodesics on r p n f and the s 1 equivariant poincare series of the non contractible component of the free loop space on r p n

existence of closed geodesics on finsler n spheres abstract in this paper we prove that on every finsler n sphere s n f with reversibility satisfying f 2 1 2 g 0 and l s n f 1 1 there always exist at least n prime closed geodesics without self intersections where g 0 is the standard riemannian metric on s n with constant curvature 1 and l s n f is the length of a shortest geodesic loop on s n f we also study the stability of these closed geodesics

on a generic property of geodesic flows in this paper we are concerned with a sufficient condition for a riemannian metric on a compact simply connected manifold to have infinitely many geometrically distinct closed geodesics 1969 gromoll meyer proved in gm that for every riemannian metric on a compact simply connected manifold m there are infinitely many geometrically distinct closed geodesics if the sequence bi m f i of betti numbers of the free loop space m is unbounded for some field f using the theory of minimal models vigue poirrier sullivan proved that the rational cohomology algebra h m i q of m is generated by a single element if and only if the sequence bi m i q i of betti numbers of the free loop space m of m is bounded it is a conjecture that the same statement holds for all fields of prime characteristic partial results are due to mccleary ziller mz and halperin vigue poirrier hv now we turn to the manifolds for which the hypothesis of gromoll meyer s theorem does not hold for example spheres and projective spaces then stability properties of the closed geodesics become important a closed geodesic is hyperbolic if the linearized poincare map has no eigenvalue of norm 1 from the bumpy metrics theorem due to abraham ab and anosov an2 and from a pertubation result due to klingenberg takens kt one can conclude a c4 generic metric on a compact manifold has either a non hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic in the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by moser mo in the second case there are infinitely many closed geodesics if m is simply connected due to results by hingston hi and the author ra1 hence it follows from these results that a c4 generic metric on a compact simply connected manifold has infinitely many geometrically distinct closed geodesics we remark that it is an open question whether there is a riemannian metric on a simply connected compact manifold all of whose closed geodesics are hyperbolic in ra2 we show that the examples of metrics on the 2

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

well posedness of boundary control systems continuity of the input output map for boundary control systems is shown through the system transfer function our approach transforms the question of continuity of the input output map of a boundary control system to uniform boundedness of the solution to a related elliptic problem this is shown for a class of boundary control systems with dirichlet neumann or robin boundary control

well posedness and regularity of naghdi s shell equation under boundary control and observation abstract a system of naghdi s shell equation with dirichlet boundary control and collocated observation is considered results on the associated nonhomogeneous boundary value problem are presented based on these results it is shown that the system is well posed in the sense of d salamon and regular in the sense of g weiss the expression of the corresponding feedthrough operator is explicitly found by means of riemannian geometric method and partial fourier transform these properties make this partial differential control system parallel in many ways finite dimensional ones in the general framework of well posed and regular infinite dimensional systems

dyer lashof operations in the string topology of spheres and projective spaces we compute the 2 primary dyer lashof operations in the string topology of several families of manifolds specifically spheres and a variety of projective spaces these operations while well known in the context of iterated loop spaces give a collection of homotopy invariants of manifolds new to string topology the computations presented here begin an exploration of these invariants

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

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

boundary observability controllability and stabilization of linear elastodynamic systems in 1988 lions obtained observability and exact controllability results for linear homogeneous isotropic elastodynamic systems siam rev 30 1988 pp 1 68 applying some new identities we extend his theorems to nonisotropic systems in 1991 lagnese obtained uniform stabilizability results for two dimensional linear homogeneous isotropic systems by applying a somewhat artificial feedback nonlinear anal 16 1991 pp 35 54 then he asked whether analogous results hold for a natural and physically implementable boundary feedback using some new identities and applying a method introduced in 1987 by zuazua and the second author j math pures appl 69 1990 pp 33 54 we give an affirmative answer to this question in all dimensions and also for nonisotropic systems moreover we obtain good decay estimates finally applying a recent general method of uniform stabilization we construct boundary feedbacks leading to arbitrarily large energy decay rates

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

coulomb and bessel functions of complex arguments and order abstract the coulomb wavefunctions originally constructed for real 0 real and integer 0 are defined for and all complex we examine the complex continuation of a variety of series and continued fraction expansions for the coulomb functions and their logarithmic derivatives and then see how these expansions may be selectively combined to calculate both the regular and irregular functions and their derivatives the resulting algorithm 46 is a complex generalisation of steed s method 6 7 as it appears in the real procedure coulfg 10 complex whittaker confluent hypergeometric and bessel functions can also be calculated

the temme s sum rule for confluent hypergeometric functions revisited the possibility of extending to generalized hypergeometric functions a sum rule for confluent hypergeometric functions found by temme is considered

geodesic flows with positive topological entropy twist maps and hyperbolicity we prove a perturbation lemma for the derivative of geodesic flows in high dimension this implies that a c 2 generic riemannian metric has a nontrivial hyperbolic basic set in its geodesic flow

genericity of geodesic flows with positive topological entropy on s2 we show that the set of c riemannian metrics on s 2 or rp 2 whose geodesic flow has positive topological entropy is open and dense in the c 2 topology the proof is partially based on an analogue of franks lemma for geodesic flows on surfaces

simple closed geodesics on most alexandrov surfaces abstract we study the existence of simple closed geodesics on most in the sense of baire category alexandrov surfaces with curvature bounded below compact and without boundary we show that it depends on both the curvature bound and the topology of the surfaces

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

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

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

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

uniform exponential energy decay of wave equations in a bounded region with l2 0 l2 γ feedback control in the dirichlet boundary conditions etant donne un ouvert borne r n n 2 on considere l equation d onde w tt w dans o avec des conditions initiales w o x w o x l 2 w 1 o x w 1 x h 1 et des conditions aux limites non homogenes de type dirichlet w t u t sur o etant la frontiere de

well posedness of a structural acoustics control model with point observation of the pressure abstract we consider a controlled and observed partial differential equation pde which describes a structural acoustics interaction physically this pde describes an acoustic chamber with a flexible chamber wall the control is applied to this flexible wall and the class of controls under consideration includes those generated by piezoceramic patches the observation we consider is point measurements of acoustic pressure inside the cavity mathematically the model consists of a wave equation coupled through boundary trace terms to a structurally damped plate or beam equation and the point controls and observations for this system are modeled by highly unbounded operators we analyze the map from the control to the observation since the properties of this map are central to any control design which is based upon this observation we also show there exists an appropriate state space x so that if the initial state is in x and the control is in l2 then the state evolves continuously in x and the observation is in l2 the analysis of this system entails a microlocal analysis of the wave component of the system and the use of pseudodifferential machinery

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

closed geodesics on positively curved finsler spheres in this paper we prove that for every finsler n sphere sn f for n3 with reversibility and flag curvature k satisfying either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized poincare map has at least one eigenvalue which is of the form exp pi with an irrational furthermore there always exist three prime closed geodesics on any s3 f satisfying the above pinching 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

implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity abstract uniform exponential decay of solution is established for the elastodynamic system of elasticity using boundary feedback control energy dissipation is introduced via linear velocity feedbacks acting through a portion of the boundary as traction forces two primary goals are achieved first these results are proven without the imposition of strong geometric assumptions on the controlled portion of the boundary thus extending earlier work which required that the domain be star shaped second the feedback is only a function of velocity as opposed to also containing the tangential derivative of the displacement resulting in a physically viable feedback proof is based on the multiplier method and relies critically on sharp trace estimates for the tangential derivative of the displacement on the boundary as well as on unique continuation results for the corresponding static system