stability for quasi periodically perturbed hill s equations we consider a perturbed hill s equation of the form open image in new window p0 t p1 t 0 where p0 is real analytic and periodic p1 is real analytic and quasi periodic and is small assuming diophantine conditions on the frequencies of the decoupled system i e the frequencies of the external potentials p0 and p1 and the proper frequency of the unperturbed 0 hill s equation but without making any assumptions on the perturbing potential p1 other than analyticity we prove that quasi periodic solutions of the unperturbed equation can be continued into quasi periodic solutions if lies in a cantor set of relatively large measure in open image in new window where 0 is small enough our method is based on a resummation procedure of a formal lindstedt series obtained as a solution of a generalized riccati equation associated to hill s problem

unsupervised learning of human motion an unsupervised learning algorithm that can obtain a probabilistic model of an object composed of a collection of parts a moving human body in our examples automatically from unlabeled training data is presented the training data include both useful foreground features as well as features that arise from irrelevant background clutter the correspondence between parts and detected features is unknown the joint probability density function of the parts is represented by a mixture of decomposable triangulated graphs which allow for fast detection to learn the model structure as well as model parameters an em like algorithm is developed where the labeling of the data part assignments is treated as hidden variables the unsupervised learning technique is not limited to decomposable triangulated graphs the efficiency and effectiveness of our algorithm is demonstrated by applying it to generate models of human motion automatically from unlabeled image sequences and testing the learned models on a variety of sequences

unsupervised learning of human action categories using spatial temporal words we present a novel unsupervised learning method for human action categories a video sequence is represented as a collection of spatial temporal words by extracting space time interest points the algorithm automatically learns the probability distributions of the spatial temporal words and the intermediate topics corresponding to human action categories this is achieved by using latent topic models such as the probabilistic latent semantic analysis plsa model and latent dirichlet allocation lda our approach can handle noisy feature points arisen from dynamic background and moving cameras due to the application of the probabilistic models given a novel video sequence the algorithm can categorize and localize the human action s contained in the video we test our algorithm on three challenging datasets the kth human motion dataset the weizmann human action dataset and a recent dataset of figure skating actions our results reflect the promise of such a simple approach in addition our algorithm can recognize and localize multiple actions in long and complex video sequences containing multiple motions

boosted multi class semi supervised learning for human action recognition human action recognition is a challenging task due to significant intra class variations occlusion and background clutter most of the existing work use the action models based on statistic learning algorithms for classification to achieve good performance on recognition a large amount of the labeled samples are therefore required to train the sophisticated action models however collecting labeled samples is labor intensive to tackle this problem we propose a boosted multi class semi supervised learning algorithm in which the co em algorithm is adopted to leverage the information from unlabeled data three key issues are addressed in this paper firstly we formulate the action recognition in a multi class semi supervised learning problem to deal with the insufficient labeled data and high computational expense secondly boosted co em is employed for the semi supervised model construction to overcome the high dimensional feature space weighted multiple discriminant analysis wmda is used to project the features into low dimensional subspaces in which the gaussian mixture models gmm are trained and boosting scheme is used to integrate the subspace models thirdly we present the upper bound of the training error in multi class framework which is able to guide the novel classifier construction in theory the proposed solution is proved to minimize this upper error bound experimental results have shown good performance on public datasets

quasi periodic solutions for two level systems we consider the schrodinger equation for a class of two level atoms in a quasi periodic external field in the case in which the spacing 2 between the two unperturbed energy levels is small and we study the problem of finding quasi periodic solutions of a related generalized riccati equation we prove the existence of quasi periodic solutions of the latter equation for a cantor set of values of around the origin which is of positive lebesgue measure such solutions can be obtained from the formal power series by a suitable resummation procedure the set can be characterized by requesting infinitely many diophantine conditions of mel nikov type

double wells perturbation series summable to the eigenvalues and directly computable approximations we give a rigorous proof of the analyticity of the eigenvalues of the double well schrodinger operators and of the associated resonances we specialize the rayleigh schrodinger perturbation theory to such problems obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method by an approximation we obtain new series expansions directly computable still summable which in the case of the herbst simon model can be given in an explicit form

the distributional borel summability and the large coupling φ4 lattice fields we complete and correct some proofs of an earlier paper on distributional borel summability and we add an application which can be useful in the discussion of semiclassical problems

renormalization group and the melnikov problem for pde s we give a new proof of persistence of quasi periodic low dimensional elliptic tori in infinite dimensional systems the proof is based on a renormalization group iteration that was developed recently in bgk to address the standard kam problem namely persistence of invariant tori of maximal dimension in finite dimensional near integrable systems our result covers situations in which the so called normal frequencies are multiple in particular it provides a new proof of the existence of small amplitude quasi periodic solutions of nonlinear wave equations with periodic boundary conditions

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

a kam theorem with applications to partial differential equations of higher dimensions the existence of lower dimensional kam tori is shown for a class of nearly integrable hamiltonian systems of infinite dimensions where the second melnikov s conditions are completely eliminated and the algebraic structure of the normal frequencies are not needed as a consequence it is proved that there exist many invariant tori and thus quasi periodic solutions for nonlinear wave equations schrodinger equations and other equations of any spatial dimensions

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

quasiperiodic hamiltonian motions scale invariance harmonic oscillators the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point

human motion tracking with a kinematic parameterization of extremal contours this paper addresses the problem of human motion tracking from multiple image sequences the human body is described by five articulated mechanical chains and human body parts are described by volumetric primitives with curved surfaces if such a surface is observed with a camera an extremal contour appears in the image whenever the surface turns smoothly away from the viewer we describe a method that recovers human motion through a kinematic parameterization of these extremal contours the method exploits the fact that the observed image motion of these contours is a function of both the rigid displacement of the surface and of the relative position and orientation between the viewer and the curved surface first we describe a parameterization of an extremal contour point velocity for the case of developable surfaces second we use the zero reference kinematic representation and we derive an explicit formula that links extremal contour velocities to the angular velocities associated with the kinematic model third we show how the chamfer distance may be used to measure the discrepancy between predicted extremal contours and observed image contours moreover we show how the chamfer distance can be used as a differentiable multi valued function and how the tracker based on this distance can be cast into a continuous non linear optimization framework fourth we describe implementation issues associated with a practical human body tracker that may use an arbitrary number of cameras one great methodological and practical advantage of our method is that it relies neither on model to image nor on image to image point matches in practice we model people with 5 kinematic chains 19 volumetric primitives and 54 degrees of freedom we observe silhouettes in images gathered with several synchronized and calibrated cameras the tracker has been successfully applied to several complex motions gathered at 30 frames second

construction of coarse grained order parameters in nonequilibrium systems we develop a renormalization group rg procedure that includes important system specific features the key ingredient is to systematize the coarse graining procedure that generates the rg flow the coarse graining technology comes from the control and operator theoretic model reduction the resulting generalized rg is a consistent generalization of the wilsonian rg we apply the procedure to a deterministic nonlinear wave equation nlwe with probabilistic initial conditions we derive the form of the projection operator from the dynamics of the nlwe and then use it to generate the rg flow for the distribution of initial conditions the probability density of the initial conditions is chosen to be a boltzmann weight that is quartic in the field variables in our calculation we find that in contrast to conventional implementations of the rg naive power counting breaks down we also show that the resulting rg equations are different from those derived from the conventional rg

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

a renormalization approach to lower dimensional tori with brjuno frequency vectors abstract we extend the renormalization scheme for vector fields on t d r m in order to construct lower dimensional invariant tori with brjuno frequency vectors for near integrable hamiltonian flows for every brjuno frequency vector r d and every vector r d satisfying a diophantine condition with respect to there exists an analytic manifold w of infinitely renormalizable hamiltonian vector fields each vector field on w is shown to have an analytic invariant torus with frequency vector

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

degenerate elliptic resonances quasi periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations the parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations here we prove that for a class of elliptic tori a resummation algorithm can be devised and proved to be convergent thus extending to such lower dimensional invariant tori the methods employed to prove convergence of the lindstedt series either for the maximal i e kam tori or for the hyperbolic lower dimensional invariant tori

fractional lindstedt series the parametric equations of the surfaces on which highly resonant quasiperiodic motions develop lower dimensional tori cannot be analytically continued in general in the perturbation parameter e i e they are not analytic functions of e however rather generally quasiperiodic motions whose frequencies satisfy only one rational relation resonances of order 1 admit formal perturbation expansions in terms of a fractional power of e depending on the degeneration of the resonance we find conditions for this to happen and in such a case we prove that the formal expansion is convergent after suitable resummation

l2 index formulae for perturbed dirac operators the callias index theorem is generalized from the euclidean case to certain spin manifolds with warped ends making use of certain index preserving deformations

hyperbolic low dimensional invariant tori and summations of divergent series we consider a class of a priori stable quasi integrable analytic hamiltonian systems and study the regularity of low dimensional hyperbolic invariant tori as functions of the perturbation parameter we show that under natural nonresonance conditions such tori exist and can be identified through the maxima or minima of a suitable potential they are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin the invariant tori admit an asymptotic series at the origin with taylor coefficients that grow at most as a power of a factorial and a remainder that to any order n is bounded by the n 1 st power of the argument times a power of n we show the existence of a summation criterion of the generically divergent series in powers of the perturbation size that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

borel summability and lindstedt series resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation such series may be sometimes summed via suitable sum rules defining c functions of the perturbation strength here we find sufficient conditions for the borel summability of their sums in the case of two dimensional rotation vectors with diophantine exponent 1 e g with ratio of the two independent frequencies equal to the golden mean

summation of divergent series and borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms we consider a class of second order ordinary differential equations describing one dimensional systems with a quasiperiodic analytic forcing term and in the presence of damping as a physical application one can think of a resistor inductor varactor circuit with a periodic or quasiperiodic forcing function even if the range of applicability of the theory is much wider in the limit of large damping we look for quasiperiodic solutions which have the same frequency vector of the forcing term and we study their analyticity properties in the inverse of the damping coefficient we find that even the case of periodic forcing terms is nontrivial as the solution is not analytic in a neighborhood of the origin it turns out to be borel summable in the case of quasiperiodic forcing terms we need renormalization group techniques in order to control the small divisors arising in the perturbation series we show the existence of a summation criterion of the series in this case also however this cannot be interpreted as borel summability

persistence of hyperbolic tori in hamiltonian systems abstract we generalize the well known result of graff and zehnder on the persistence of hyperbolic invariant tori in hamiltonian systems by considering non floquet frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies the preservation of part or full frequency components associated to the degree of non degeneracy is considered as applications we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non integrable hamiltonian system

resummation of perturbation series and reducibility for bryuno skew product flows we consider skew product systems on d l sl 2 for bryuno base flows close to constant coefficients depending on a parameter in any dimension d and we prove reducibility for a large measure set of values of the parameter the proof is based on a resummation procedure of the formal power series for the conjugation and uses techniques of renormalisation group in quantum field theory

periodic and quasi periodic orbits for the standard map we consider both periodic and quasi periodic solutions for the standard map and we study the corresponding conjugating functions i e the functions conjugating the motions to trivial rotations we compare the invariant curves with rotation numbers satisfying the bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents n p n q n we prove the following results for n 1 for rotation numbers n n we study the radius of convergence of the conjugating functions and we find lower bounds on them which tend to a limit which is a lower bound on the corresponding quantity for 2 the periodic orbits consist of points which are more and more close to the invariant curve with rotation number 3 such orbits lie on analytical curves which tend uniformly to the invariant curve

double wells nevanlinna analyticity distributional borel sum and asymptotics we consider the energy levels of a stark family in the parameterj of quartic double wells with perturbation parameterg h g j p 2 x 2 1 gx 2 j left gx frac 1 2 right for non evenj and for the symmetric casej 0 we prove analyticity in the full nevanlinna disk g 2 r 1 of theg2 plane as predicted by crutchfield by means of an approximation we give a heuristic estimate of the asymptotic smallg behaviour showing the relation between the avoided crossings and the failure of the usual perturbation series

an equivariant index for proper actions i abstract equivariant indices taking values in group theoretic objects have previously been defined in cases where either the group acting or the orbit space of the action is compact in this paper we define an equivariant index without assuming the group or the orbit space to be compact this allows us to generalise an index of deformed dirac operators defined for compact groups by braverman in parts ii and iii of this series we explore some properties and applications of this index

bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit in this paper we investigate bifurcation phenomena such as those of the periodic solutions for the unperturbed nonlinear system g x cbxz 0 with g x z ac x 2 1c x 2 and ao1 bo0 when we add the two competing termskf t cg x with f t a time periodic analytic forcing function and go0 the dissipative parameter the resulting differential equation g x xcbxcg xkf t z 0 describes approximately an electronic system known as the saturating inductor circuit for any periodic orbit of the unperturbed system we provide conditions which give rise to the appearance of subharmonic solutions furthermore we show that other bifurcation phenomena arise as there is a periodic solution with the same period as the forcing function f t which branches off the origin when the perturbation is switched on we also show that such a solution which encircles the origin attracts the entire phase space when the dissipative parameter becomes large enough we then compute numerically the basins of attraction of the attractive periodic solutions by choosing specific examples of the forcing function f t which are dictated by experience we provide evidence showing that all the dynamics of the saturating inductor circuit is organized by the persistent subharmonic solutions and by the periodic solution around the origin

oscillator synchronisation under arbitrary quasi periodic forcing we study the problem of existence of response solutions for a real analytic one dimensional system consisting of a rotator subject to a small quasi periodic forcing with bryuno frequency vector we prove that at least one response solution always exists without any assumption on the forcing besides smallness and analyticity this strengthens the results available in the literature where generic non degeneracy conditions are assumed the proof is based on a diagrammatic formalism and relies on renormalisation group techniques which exploit the formal analogy with problems of quantum field theory a crucial role is played by remarkable identities between classes of diagrams

graphical templates for model registration a new method of model registration is proposed using graphical templates a graph of landmarks is chosen in the template image all possible candidates for these landmarks are found in the data image using local operators a dynamic programming algorithm on decomposable subgraphs of the template graph finds the optimal match to a subset of the candidate points in polynomial time this combination of local operators to describe points of interest landmarks and a graph to describe their geometric orientation in the plane yields fast and precise matches of the model to the data with no initialization required

multidimensional continued fractions dynamical renormalization and kam theory the disadvantage of traditional multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors following ideas of dani lagarias and kleinbock margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space sl d mathbb z backslash sl d mathbb r the space of lattices of covolume one that indeed yields best possible approximations to any irrational vector the algorithm is ideally suited for a number of dynamical applications that involve small divisor problems as an example we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension

distributional borel summability of odd anharmonic oscillators it is proved that the divergent rayleigh schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are borel summable in the distributional sense to the resonances naturally associated with the system

kam theorem and quantum field theory we give a new proof of the kam theorem for analytic hamiltonians the proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale the crucial cancellations of resonances are shown to follow from the ward identities expressing the translation invariance of the corresponding field theory

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

perturbative analysis of dynamical localization in this paper we present a mathematical investigation of the phenomenon of dynamical localization in a class of quasi periodically and periodically time dependent two level systems our results are based on a sort of renormalization procedure which is developed here in a systematic way in order to adapt it to the case of dynamical localization in the quasi periodic case this procedure leads to a formal perturbative expansion free of secular terms in the periodic case a convergent perturbative solution is obtained and in particular a convergent perturbative expansion for the secular frequency is presented the case of ac dc fields is discussed in some detail leading to the conclusion that the phenomenon of dynamical localization is not exact in that situation

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

quasi periodic solutions of completely resonant nonlinear wave equations it is shown that there are many elliptic invariant tori and thus quasi periodic solutions for the completely resonant nonlinear wave equation subject to periodic boundary conditions via kam theory

compensations in small divisor problems a general direct method alternative to kam theory apt to deal with small divisor problems in the real analytic category is presented and tested on several small divisor problems including the construction of maximal quasi periodic solutions for nearly integrable non degenerate hamiltonian or lagrangian systems and the construction of lower dimensional resonant tori for nearly integrable hamiltonian systems the method is based on an explicit graph theoretical representation of the formal power series solutions which allows to prove compensations among the monomials forming such representation

learning and matching of dynamic shape manifolds for human action recognition in this paper we learn explicit representations for dynamic shape manifolds of moving humans for the task of action recognition we exploit locality preserving projections lpp for dimensionality reduction leading to a low dimensional embedding of human movements given a sequence of moving silhouettes associated to an action video by lpp we project them into a low dimensional space to characterize the spatiotemporal property of the action as well as to preserve much of the geometric structure to match the embedded action trajectories the median hausdorff distance or normalized spatiotemporal correlation is used for similarity measures action classification is then achieved in a nearest neighbor framework to evaluate the proposed method extensive experiments have been carried out on a recent dataset including ten actions performed by nine different subjects the experimental results show that the proposed method is able to not only recognize human actions effectively but also considerably tolerate some challenging conditions e g partial occlusion low quality videos changes in viewpoints scales and clothes within class variations caused by different subjects with different physical build styles of motion etc

the hyperbolic invariant tori of symplectic mappings abstract in this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings under a russmann type non degenerate condition by introducing a modified kam iteration scheme we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough

a normal form theorem for brjuno skew systems through renormalization abstract we develop a dynamical renormalization method for the problem of local reducibility of analytic linear quasi periodic skew product flows on t 2 sl 2 r this approach is based on the continued fraction expansion of the linear base flow and deals with small divisors by turning them into large divisors we use this method to give a new proof of a normal form theorem for a brjuno base flow