central configurations of the n body problem via equivariant morse theory in this paper we use the equivariant morse theory to give an estimate of the minimal number of central configurations in the n body problem in 3 in the case of equal masses we prove that the planar central configurations are saddle points for the potential energy from this we deduce the presence of non planar central configurations for every n 4

counterexamples to some conjectures on the number of solutions of nonlinear equations on construit trois contre exemples resolvant par la negative certains problemes d analyse non lineaire

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

equivariant morse theory for flows and an application to the n body problem a l aide de l indice de conley et de la cohomologie equivariante on deduit certaines inegalites de type morse pour un flot equivariant par rapport a l action d un groupe topologique compact

flexible control of chaotic stick slip mechanical systems abstract this paper describes some numerical techniques to locate and control unstable periodic orbits embedded in chaotic attractors of some discontinuous mechanical systems special attention is paid to the case of maps generated by stick slip systems but the techniques presented in the paper are applicable to wider classes of maps

multiplicity results for two kinds of equivariant systems abstract two equivariant problems of the form e u f u are considered where f is a real function which is invariant under the action of a group g and using morse theory for each problem an arbitrarily great number of orbits of solutions is founded choosing e suitably small the first problem is a o 2 equivariant system of two equations which can be seen as a complex ginzburg landau equation while the second one is a system of m equations which is equivariant for the action of a finite group of real orthogonal matrices m m

a dynamic mesh algorithm for curvature dependent evolving interfaces a new finite element method is discussed for approximating evolving interfaces in2nwhose normal velocity equals mean curvature plus a forcing function the method is insensitive to singularity formation and retains the local structure of the limit problem and thus exhibits a computational complexity typical of3n 1without having the drawbacks of front tracking strategies a graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh time stepping is explicit but stability constraints force small time steps only when singularities develop whereas relatively large time steps are allowed before or past singularities when the evolution is smooth the explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step thereby making mesh updating simple and thus practical performance and potentials are fully documented via a number of numerical simulations in 2d 3d 4d and 8d with axial symmetries they include tori and cones for the mean curvature flow minimal and prescribed mean curvature surfaces with given boundary fattening for smooth driving force and volume constraint

asymptotics for a parabolic double obstacle problem we consider a parabolic double obstacle problem which is a version of the allen cahn equation u t delta u epsilon 2 psi prime u in omega times 0 infty where omega is a bounded domain epsilon is a small constant and psi is a double well potential here we take psi such that psi u 1 u 2 when u leq 1 and psi u infty when u 1 we study the asymptotic behaviour as epsilon rightarrow 0 of the solution of the double obstacle problem under some natural restrictions on the initial data we show that after a short time of order epsilon 2 ln epsilon the solution takes value 1 in a region omega t and value 1 in omega t where the region omega backslash omega t cup omega t is a thin strip and is contained in either a o epsilon ln epsilon or o epsilon neighbourhood of a hypersurface gamma t which moves with normal velocity equal to its mean curvature we also study the asymptotic behaviour as t rightarrow infty of the solution in the one dimensional case in particular we prove that the omega limit set consists of a singleton

periodic orbits of a one dimensional non autonomous hamiltonian system in this paper we study the properties of the periodic orbits of x vx t x 0 with x s1 and vx t x a t0 periodic potential called 1t0q the frequency of windings of an orbit in s1 we show that exists an infinite number of periodic solutions with a given we give a lower bound on the number of periodic orbits with a given period and by means of the morse theory

a simple proof of geodesical completeness for compact space times of zero curvature any riemannian metric on a compact manifold is geodesically complete by contrast it is widely known that incomplete compact lorentzian manifolds exist in this article it will be proven that a compact manifold with a smooth lorentz metric must be complete if the metric is flat everywhere this question of completeness which has significant implications for the classification of compact lorentzian space forms has remained unresolved until quite recently when it was finally established by carriere as a special case of an even more general result here a simple alternative proof of this special case of carriere s theorem is given the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space time

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

geometrical concepts and computational techniques of nonlinear dynamics abstract this paper presents a brief overview of some of the geometrical concepts of nonlinear dissipative dynamics computational techniques are described for locating fixed points of poincare maps local bifurcations and homoclinic tangencies of invariant manifolds and following them as parameters vary the cell to cell mapping technique for locating basins of attraction in a computationally efficient way is outlined

periodic solutions of dynamical systems with bounded potential where ve c2 r x rn r v t t x y t x v t x jjbxiw y is bounded and v v t x 0 as 1x1 co for example we prove that if 0 d v r x m and v r x m as 1x1 co there exists at least one t periodic solution of v moreover if v j t 0 0 under suitable assumptions on v y t 0 this solution is not identically zero we also give existence results for the forced system

on the number of minimal two spheres of small area in manifolds with curvature bounded above a beautiful application of the morse theory of closed geodesics is provided by klingenberg s k 1 study of the number of short smooth closed geodesics in sufficiently pinched spheres together with subsequent contributions in this direction by many authors a particularly striking result see ballmann thorbergsson and ziller btz states that a generic metric on a compact simply connected riemannian manifold whose sectional curvatures satisfy 1 4 k a 1

birkhoff minimax principle for minimal surfaces with a free boundary 1 1 x 0 in b 1 2 xu xv 0 xu xv in b 1 3 x c s 1 4 xn w tx w s w c here n is the outer normal to b subscripts denote partial derivatives and are the euclidean norm and scalar product in rn 1 and ty s for y s denotes the tangent space to s at y for brevity problem 1 1 1 4 will be referred to as problem p s a solution of p s is a critical point of the following dirichlet integral

a morse complex for infinite dimensional manifolds part i abstract in this paper and in the forthcoming part ii we introduce a morse complex for a class of functions f defined on an infinite dimensional hilbert manifold m possibly having critical points of infinite morse index and co index the idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of m suitably related with the gradient flow of f this part i deals with the following questions about the intersection w of the unstable manifold of a critical point x and the stable manifold of another critical point y finite dimensionality of w possibility that different components of w have different dimension orientability of w and coherence in the choice of an orientation compactness of the closure of w classification up to topological conjugacy of the gradient flow on the closure of w in the case dim w 2

computational techniques for nonlinear dynamics in multiple friction oscillators this paper presents some numerical techniques used to investigate the nonlinear dynamics of a class of mechanical systems affected by dry friction forces they allow for the exact detection of the points which separate slip phases from stick phases in this way the correct dynamic equations are integrated during the different motion phases finally a method is described to perform the numerical computation of the lyapunov exponents in such non smooth systems

symmetry breaking patterns and extended morse theory an indirect topological method is considered which constrains the possible breaking patterns of a lie group symmetry for higgs landau systems the application of the method and its efficiency are discussed

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

sliding motion on discontinuity surfaces of high co dimension a construction for selecting a filippov vector field in this paper we consider the issue of sliding motion in filippov systems on the intersection of two or more surfaces to this end we propose an extension of the filippov sliding vector field on manifolds of co dimension p with p 2 our model passes through the use of a multivalued sign function reformulation to justify our proposal we will restrict to cases where the sliding manifold is attractive for the case of co dimension p 2 we will distinguish between two types of attractive sliding manifold node like and spiral like the case of node like attractive manifold will be further extended to the case of p 3 finally we compare our model to other existing methodologies on some examples

poincare s closed geodesic on a convex surface we present a new proof for the existence of a simple closed geodesic on a convex surface m this result is due originally to poincare the proof uses the 2k dimensional riemannian manifold m briefly of piecewise geodesic closed curves on m with a fixed number k of corners k chosen sufficiently large in a we consider a submanifold 0 formed by those elements of a which are simple regular and divide m into two parts of equal total curvature 2 the main burden of the proof is to show that the energy integral e restricted to 0 assumes its infimum at the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on m

unstable geodesics and topological field theory a topological field theory is used to study the cohomology of mapping space the cohomology is identified with the becchi rouet stora tyutin cohomology realizing the physical hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua the method is illustrated by a simple example

dynamical systems with effective like potentials on etudie l existence des solutions t periodiques pour des systemes hamiltoniens d ordre 2 de la forme x v x ou v r n 0 r est de classe c 2 et v x quand x 0

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

morse inequalities for r constructible sheaves this note aims at generalizing of classical morse inequalities for betti numbers of compact manifolds cf lo 1 2 in this paper we work with r constructible sheaves instead and encounter the tight relation between morse theory and microlocal analysis of sheaves see witten 111 hellter and sjiistrand s 61 and henniart 7 for another approach to morse inequalities via microlocal analysis and also goresky and macpherson 3 4 who introduced the stratified morse theory this paper may be considered as a variation on kashiwara s index theorem 8 and in fact our proof is a slight modification of his

detectability of critical points of smooth functionals from their finite dimensional approximations abstract given a critical point of a c 2 functional on a separable hilbert space we obtain sufficient conditions for it to be detectable i e visible from finite dimensional rayleigh ritz galerkin rrg approximations while examples show that even nondegenerate critical points are without any further restriction not visible we single out relevant classes of smooth functionals e g the hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations such that their nondegenerate critical points are visible from their rrg approximations

upper bounds on the length of the shortest closed geodesic on simply connected manifolds the subject of this paper is upper bounds on the length of the shortest closed geodesic on simply connected manifolds with non trivial second homology group we will give three estimates the first estimate will explicitly depend on volume and the upper bound for the sectional curvature the second estimate will depend on diameter a positive lower bound for the volume and on the possibly negative lower bound on sectional curvature the third estimate will depend on diameter on a possibly negative lower bound for the sectional curvature and on a lower bound for the simply connectedness radius the technique that we develop in order to obtain the last result will also enable us to estimate the homotopy distance between any two closed curves on compact simply connected manifolds of sectional curvature bounded from below and diameter bounded from above more precisely let c be a constant such that any metric ball of radius leq c is simply connected there exists a homotopy connecting any two closed curves such that the length of the trajectory of the points during this homotopy has an upper bound in terms of the lower bound of the curvature the upper bound of diameter and c

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

on the geodesical connectedness for a class of semi riemannian manifolds we prove a variational principle for geodesics on a semi riemannian manifold m of arbitrary index k and possessing k linearly independent killing vector fields that generate a timelike distribution on m using such a principle and a suitable completeness condition for m we prove some existence and multiplicity results for geodesics joining two fixed points of m

two dimensional map for impact oscillator with drift an impact oscillator with a drift is considered using a simple co ordinates transformation the bounded oscillations are separated from the drift in general the dynamic state of the system is fully described by four variables time relative displacement p and velocity y of the mass and relative displacement of the slider top q however this number can be reduced by two if the beginning of the progression phase is monitored in this paper a new two dimensional numerical map is developed and its dynamics is discussed

topology and phase transitions the case of the short range spherical model we characterize the topology of the phase space of the berlin kac spherical model in the context of the so called topological hypothesis for spins lying in hypercubic lattices of dimension d for zero external field we are able to characterize the topology exactly up to homology we find that even though there is a continuum of changes in the topology of the corresponding manifolds for d 3 there are abrupt discontinuities in some topological functions that could be good candidates to associate with the phase transitions that occur at the thermodynamic level we show however that these changes do not coincide with the phase transitions and conversely that no topological discontinuity can be associated to the points where the phase transitions take place at variance with what happens in the mean field version of this same model we show that these abrupt topological changes are accessible thermodynamically we conclude that even in short range systems the topological mechanism does not seem to be responsible for the triggering of a phase transition we also analyze the case of spins connected to a macroscopic number of but not all neighbors and find that similar to the results found for the fully connected version in this case the topological hypothesis seems to hold the phase transition coincides with an accumulation point of the topological changes present in configuration space the question of the ensemble equivalence in the short range spherical model is also considered

on the numerical computation of poincare maps abstract this note describes a method for finding simply and accurately the intersections of a numerically integrated trajectory with a surface of section

non linear dynamics of multiple friction oscillators this paper deals with the dynamics of multiple friction oscillators in some particular cases when the driving velocity is small the dynamics of the systems can be described by low dimensional discrete maps which allow all existing attractors and their basins of attraction to be detected moreover the maps allow the relevant lyapunov exponents to be computed and the classical path following techniques to be applied the complex dynamics of the systems is clearly illustrated

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

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

length of geodesics on a two dimensional sphere let m be an arbitrary riemannian manifold diffeomorphic to s 2 let x y be two arbitrary points of m we prove that for every k 1 2 3 ldots there exist k distinct geodesics between x and y of length less than or equal to 4k 2 2k 1 d where d denotes the diameter of m to prove this result we demonstrate that for every riemannian metric on s 2 there are two not mutually exclusive possibilities either every two points can be connected by many short geodesics of index 0 or the resulting riemannian sphere can be swept out by short meridians

bumpy metrics and closed parametrized minimal surfaces in riemannian manifolds the purpose of this article is to study conformal harmonic maps f m where is a closed riemann surface and m is a compact riemannian manifold of dimension at least four such maps define parametrized minimal surfaces possibly with branch points we show that when the ambient manifold m is given a generic metric all prime closed parametrized minimal surfaces are free of branch points and are as morse nondegenerate as allowed by the group of automorphisms of e they are morse nondegenerate in the usual sense if has genus at least two lie on two dimensional nondegenerate critical submanifolds if s has genus one and on six dimensional nondegenerate critical submanifolds if has genus zero

a quasi optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem solutions of the so called prescribed curvature problem min a p a a g x g being the curvature field are approximated via a singularly perturbed elliptic pde of bistable type for nondegenerate relative minimizers a we prove an o 2 log 2 error estimate where stands for the perturbation parameter and show that this estimate is quasi optimal the proof is based on the construction of accurate barriers suggested by formal asymptotics this analysis is next extended to a finite element discretization of the pde to prove the same error estimate for discrete minima

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

minimal two spheres of low index in manifolds with positive complex sectional curvature suppose that s n is given a generic riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures this pinching condition is satisfied by manifolds whose real sectional curvatures k r sigma satisfy begin aligned 1 2 k r sigma le 1 end aligned then the number of minimal two spheres of morse index lambda for n 2 le lambda le 2n 5 is at least p 3 lambda n 2 where p 3 k is the number of k cells in the schubert cell decomposition for g 3 mathbb r n 1

singular levels and topological invariants of morse bott integrable systems on surfaces abstract we classify up to homeomorphisms closed curves and eights of saddle points on orientable closed surfaces this classification is applied to morse bott foliations and morse bott integrable systems allowing us to define a complete invariant we state also a realization theorem based in two transformations and one generator the foliation of the sphere with two centers

uniqueness of critical points of solutions to the mean curvature equation with neumann and robin boundary conditions abstract in this paper we investigate the critical points of solutions to the prescribed constant mean curvature equation with neumann and robin boundary conditions respectively in a bounded smooth convex domain of r n n 2 firstly we show the non degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local chen huang s comparison technique and the geometric properties of approximate surfaces at the non degenerate critical points secondly we deduce the uniqueness and non degeneracy of the critical points of solutions in a rotationally symmetric domain of r n n 3 by the projection of higher dimensional space onto two dimensional plane

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow