an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

efficient reconstruction of band limited sequences from nonuniformly decimated versions by use of polyphase filter banks an efficient polyphase structure for the reconstruction of a band limited sequence from a nonuniformly decimated version is developed theoretically the reconstruction involves the implementation of a bank of multilevel filters and it is shown that how all these reconstruction filters can be obtained at the cost of one mth band low pass filter and a constant matrix multiplier the resulting structure is therefore more general than previous schemes in addition the method offers a direct means of controlling the overall reconstruction distortion t z by appropriate design of a low pass prototype filter p z extension of these results to multiband band limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized along with generalizations to the multidimensional case design examples are included to demonstrate the theory and the complexity of the new method is seen to be much lower than earlier ones

nonuniform sampling of periodic bandlimited signals digital processing techniques are based on representing a continuous time signal by a discrete set of samples this paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples in practical applications only a finite number of values are given extending the samples periodically across the boundaries and assuming that the underlying continuous time signal is bandlimited provides a simple way to deal with reconstruction from finitely many samples two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed in the first the reconstruction functions constitute a basis while in the second they form a frame so that there are more samples than needed for perfect reconstruction the advantages and disadvantages of each method are analyzed specifically it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments next we use the theory of finite dimensional frames to characterize the stability of our algorithms we then consider two special distributions of sampling points uniform and recurrent nonuniform and show that for these cases the reconstruction formulas as well as the stability analysis are simplified significantly the advantage of our methods over conventional approaches is demonstrated by numerical experiments

efficient recovery of sub nyquist sampled sparse multi band signals using reconfigurable multi channel analysis and modulated synthesis filter banks sub nyquist cyclic nonuniform sampling cnus of a sparse multi band signal generates a nonuniformly sampled signal assuming that the corresponding uniformly sampled signal satisfies the nyquist sampling criterion the sequence obtained via cnus can be passed through a reconstructor to recover the missing uniform grid samples at present these reconstructors have very high design and implementation complexity that offsets the gains obtained due to sub nyquist sampling in this paper we propose a scheme that reduces the design and implementation complexity of the reconstructor in contrast to the existing reconstructors which use only a multi channel synthesis filter bank fb the proposed reconstructor utilizes both analysis and synthesis fbs which makes it feasitble to achieve an order of magnitude reduction of the complexity the analysis filters are implemented using polyphase networks whose branches are allpass filters with distinct fractional delays and phase shifts in order to reduce both the design and the implementation complexity of the synthesis fb the synthesis filters are implemented using a cosine modulated fb in addition to the reduced complexity of the reconstructor the proposed multi channel recovery scheme also supports online reconfigurability which is required in flexible multi mode systems where the user subband locations vary with time

exact interpolation and iterative subdivision schemes we examine the circumstances under which a discrete time signal can be exactly interpolated given only every mth sample after pointing out the connection between designing an m fold interpolator and the construction of an m channel perfect reconstruction filter bank we derive necessary and sufficient conditions on the signal under which exact interpolation is possible bandlimited signals are one obvious example but numerous others exist we examine these and show how the interpolators may be constructed a main application is to iterative interpolation schemes used for the efficient generation of smooth curves we show that conventional bandlimited interpolators are not suitable in this context a better criterion is to use interpolators that are exact for polynomial functions we demonstrate that these interpolators converge when iterated and show how these may be designed for any polynomial degree n and any interpolation factor m this makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium

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

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

energy estimates and local well posedness of 3d interfacial hydroelastic waves between two incompressible fluids abstract in the current paper we are concerned with waves propagating through the deformation of a thin elastic sheet between two incompressible and inviscid fluids which are usually called hydroelastic waves in the literature to model deformable sheets interacting with surrounding fluids the main purpose of the present study is to solve a basic question on the theoretical side i e the local well posedness issue the problem is formulated by the full euler equations without the assumption of irrotationality for fluids combined with the plotnikov toland model for the elastic sheet based on geometric considerations we derive energy estimates and prove the local existence and uniqueness of solution for this system in n 2 dimensions even if velocity fields are rotational

periodic solutions of large norm of hamiltonian systems abstract hamiltonian systems of ordinary differential equations are studied the only assumption made on the hamiltonian is appropriately rapid growth at infinity it is proved that for any given period there is an unbounded sequence of periodic solutions of the system having the given period

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

filter design for mimo sampling and reconstruction we address the problem of finite impulse response fir filter design for uniform multiple input multiple output mimo sampling this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases the input signals are modeled as either continuous time or discrete time multiband input signals with different band structures we present conditions on the channel and the sampling rate that allow perfect inversion of the channel additionally we provide a stronger set of conditions under which the reconstruction filters can be chosen to have frequency responses that are continuous we also provide conditions for the existence of fir perfect reconstruction filters and when such do not exist we address the optimal approximation of the ideal filters using fir filters and a minmax l sub 2 end to end distortion criterion the design problem is then reduced to a standard semi infinite linear program an example design of fir reconstruction filters is given

large amplitude time periodic solutions of a semilinear wave equation abstract this paper studies the existence of periodic solutions for a family of semilinear wave equations where the restoring force is independent of time monotone and grows at a more rapid rate than linear near infinity with appropriate technical assumptions it is shown that there is an unbounded sequence of such free vibrations i e there are solutions of arbitrarily large amplitude if the restoring force is independent of x the monotonicity assumption can be omitted

a nekhoroshev type theorem for hamiltonian systems with infinitely many degrees of freedom we study the propagation of lattice vibrations in models of disorder ed classical anharmonic crystals using classical perturbation theory with an optimally chosen remainder term i e a nekhoroshev type scheme we are able to show that vibrations corresponding to localized initial conditions do essentially not propagate through the crystal up to times larger than any inverse power of the strength of the anharmonic couplings

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

existence of infinitely many periodic solutions for the radially symmetric wave equation with resonance abstract in this paper we consider the periodic dirichlet problem for a forced nonlinear wave equation with resonance u t t u u a t x u p 1 u in a n dimensional ball under some suitable assumptions on p and a t x we prove the existence of infinitely many radially symmetric time periodic solutions for the problem by variational methods

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

sampling theorems for uniform and periodic nonuniform mimo sampling of multiband signals we examine a multiple input multiple output mimo sampling scheme for a linear time invariant continuous time mimo channel the input signals are modeled as multiband signals with different spectral supports and the channel outputs are sampled on either uniform or periodic nonuniform sampling sets with possibly different but commensurate intervals on the different outputs this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases we derive necessary and sufficient conditions on the channel and the sampling rate that allow stable perfect reconstruction of the inputs or equivalently perfect inversion of the channel from an implementation viewpoint we note that it is desirable that the reconstruction filters have continuous frequency responses we derive necessary and sufficient conditions that guarantee this continuity property the frequency responses of the reconstruction filters are specified as solutions to a system of linear equations finally we demonstrate that perfect reconstruction may be possible even when the channel outputs are sampled at an average rate that does not allow the reconstruction of any output from its samples alone in certain instances this average rate can achieve the recently presented fundamental bounds on mimo sampling density

periodic solutions of symmetric wave equations this paper concerns the existence of solutions to the following nonautonomous wave equation gu uttu f 2 f x t is nondecreasing in and f x t 0 if and only if 0 our first result deals with the case where f is superlinear at theorem 1 let f 1 and f 2 be satisfied assume moreover that f satisfies f 3 there are r 0 and 2 such that

fast reconstruction methods for bandlimited functions from periodic nonuniform sampling a well known generalization of shannon s sampling theorem states that a bandlimited function can be reconstructed from its periodic nonuniformly spaced samples if the effective sampling rate is at least the nyquist rate analogous to shannon s sampling theorem this generalization requires that an infinite number of samples be available which however is never the case in practice most existing reconstruction methods for periodic nonuniform sampling yield very low order often not even first order accuracy when only a finite number of samples is given in this paper we propose a fast numerically robust root exponential accurate reconstruction method the efficiency and accuracy of the algorithm is obtained by fully exploiting the sampling structure and utilizing localized fourier analysis we discuss applications in analog to digital conversion where nonuniform periodic sampling arises in various situations finally we demonstrate the performance of our algorithm by numerical examples

infinitely many periodic solutions for a semilinear wave equation in a ball in rn abstract in this paper we study the wave equation with sub linearity in a ball with periodic dirichlet conditions combining the minimax principle with an approximation argument we prove the existence of infinitely many time periodic solutions for the problem the proof also relies on the structure of the spectrum of the linearized problem

higher order sampling and recovering of lowpass signals a higher order sampling function applied to lowpass signals is described to recover the original signal from its periodic nonuniform samples a formulation of the sampling function is introduced based on the occurrence of local gaps in the sampled signal spectrum as a result the interpolant is greatly simplified

cyclic lti systems in digital signal processing cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer l in such cases the frequency domain is defined as a uniform discrete grid as in l point dft this offers more freedom in theoretical as well as design aspects while circular convolution has been the centerpiece of many algorithms in signal processing for decades such freedom especially from the viewpoint of linear system theory has not been studied in the past in this paper we introduce the fundamentals of cyclic multirate systems and filter banks presenting several important differences between the cyclic and noncyclic cases cyclic systems with allpass and paraunitary properties are studied the paraunitary interpolation problem is introduced and it is shown that the interpolation does not always succeed state space descriptions of cyclic lti systems are introduced and the notions of reachability and observability of state equations are revisited it is shown that unlike in traditional linear systems these two notions are not related to the system minimality in a simple way throughout the paper a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

mmse design of interpolation and downsampling fir filters in the context of periodic nonuniform sampling the generalized sampling theorem states that any analog signal whose spectrum is limited to 1 t can be exactly recovered from n sequences of samples taken at a rate 2 nt and all having a different sampling phase when n 2 the exact interpolation formula can be derived quite easily the ideal interpolation filters have infinite impulse responses iirs this paper addresses first the question of recovering from the two initial sequences any other sequence taken at the same rate 1 t and with a different sampling phase the design problem is dealt with for finite length filters and the criterion is the minimization of the mean squared interpolation error next the problem of computing from the two initial sequences a third one at a lower rate is addressed fir decimation filters are also designed for an mmse criterion these problems are illustrated for two typical covariance functions

global solutions of the gravity capillary water wave system in three dimensions in this paper we prove global regularity for the full water wave system in three dimensions for small data under the influence of both gravity and surface tension this problem presents essential difficulties which were absent in all of the earlier global regularity results for other water wave models to construct global solutions we use a combination of energy estimates and matching dispersive estimates there is a significant new difficulty in proving energy estimates in our problem namely the combination of slow pointwise decay of solutions no better than lvert t rvert 5 6 and the presence of a large codimension 1 set of quadratic time resonances to deal with such a situation we propose here a new mechanism which exploits a non degeneracy property of the time resonant hypersurfaces and some special structure of the quadratic part of the non linearity connected to the conserved energy of the system the dispersive estimates rely on analysis of the duhamel formula in the fourier space the main contributions come from the set of space time resonances which is a large set of dimension 1 to control the corresponding bilinear interactions we use harmonic analysis techniques such as orthogonality arguments in the fourier space and atomic decompositions of functions most importantly we construct and use a refined norm which is well adapted to the geometry of the problem

minimum mean squared error reconstruction for generalized undersampling of cyclostationary processes the generalized sampling problem is considered using a filter bank model with a stochastic process framework a signal reconstruction solution minimizing the time averaged mean squared error is found using a filter bank optimization technique for cyclostationary signals the use of a stochastic model approach to find a minimized error solution rather than a perfect reconstruction solution allows consideration of sampling problems that deterministic approaches cannot solve for instance a particular sampling density is not required allowing reconstruction optimization in cases of undersampling an extension is also presented for optimal reconstruction in the presence of additive noise and interference the result is a generalized sampling model which can be used for an extensive variety of scenarios combining aspects of undersampling multiple input multiple output mimo sampling and periodic nonuniform sampling

a property of exponential stability in nonlinear wave equations near the fundamental linear mode abstract we consider the nonlinear wave equation utt c2uxx u u t 0 u t 0 where is an analytic function satisfying 0 0 and 00 0 for each value of the harmonic energy e i e energy when 0 we consider the closed curve e which is the phase space trajectory of the linear mode with lowest frequency if the perturbation parameter and the harmonic energy e are small enough then near e we construct by means of a suitable simplified equation a closed curve e with the following property provided the distance between the initial datum and such a curve is small enough the corresponding solution remains close to e up to times exponentially long with 1 eve cannot expect the curves e to be trajectories of solutions of such curves depend smoothly on the parameters e and the above result is deduced from a general theorem on generic hamiltonian perturbations of completely resonant linear systems

perfect reconstruction formulas and bounds on aliasing error in sub nyquist nonuniform sampling of multiband signals we examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples this sampling scheme which has been studied previously has an interesting optimality property that uniform sampling lacks one can sample and reconstruct the class spl bscr spl fscr of multiband signals with spectral support spl fscr at rates arbitrarily close to the landau 1969 minimum rate equal to the lebesgue measure of spl fscr even when spl fscr does not tile r under translation using the conditions for exact reconstruction we derive an explicit reconstruction formula we compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band limited to the span of spl fscr the smallest interval containing spl fscr which is a bigger class than the valid signals spl bscr spl fscr band limited to spl fscr we also examine the performance of the reconstruction system when the input contains additive sample noise

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

optimal sub nyquist nonuniform sampling and reconstruction for multiband signals we study the problem of optimal sub nyquist sampling for perfect reconstruction of multiband signals the signals are assumed to have a known spectral support spl fscr that does not tile under translation such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching landau s 1967 lower bound equal to the measure of spl fscr for signals with sparse spl fscr this rate can be much smaller than the nyquist rate unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity in a previous study we derived bounds on the error due to mismodeling and sample additive noise adopting these bounds as performance measures we consider the problems of optimizing the reconstruction sections of the system choosing the optimal base sampling rate and designing the nonuniform sampling pattern we find that optimizing these parameters can improve system performance significantly furthermore uniform sampling is optimal for signals with spl fscr that tiles under translation for signals with nontiling spl fscr which are not amenable to efficient uniform sampling the results reveal increased error sensitivities with sub nyquist sampling however these can be controlled by optimal design demonstrating the potential for practical multifold reductions in sampling rate

reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters this paper considers the problem of reconstructing a class of nonuniformly sampled bandlimited signals of which a special case occurs in e g time interleaved analog to digital converter adc systems due to time skew errors to this end we propose a synthesis system composed or digital fractional delay filters the overall system i e nonuniform sampling and the proposed synthesis system can be viewed as a generalization of time interleaved adc systems to which the former reduces as a special case compared with existing reconstruction techniques our method has major advantages from an implementation point of view to be precise 1 we can perform the reconstruction as well as desired in a certain sense by properly designing the digital fractional delay filters and 2 if properly implemented the fractional delay filters need not be redesigned in case the time skews are changed the price to pay for these attractive features is that we need to use a slight oversampling it should be stressed however that the oversampling factor is less than two as compared with the nyquist rate the paper includes error and quantization noise analysis the former is useful in the analysis of the quantization noise and when designing practical fractional delay filters approximating the ideal filters

periodic solutions of a dirac equation with concave and convex nonlinearities abstract this paper is concerned with multiplicity of periodic solutions of a dirac equation with concave and convex nonlinearities we establish a variational setting and obtain two sequences of periodic solutions with large and small energies respectively for the problem by variational method

free and forced vibrations of nonlinear wave equations in ball first we shall deal with the free vibrations of a nonlinear radially symmetric wave equation t2 u f r u in n dimensional ball ba with center at the origin and radius a where f is smooth monotone decreasing in u and satisfies f r 0 0 f r u has asymptotic properties f r u o u3 u 0andu for n 1 3 we shall show the existence of infinitely many radially symmetric time periodic solutions with different periods of irrational multiple of a second we shall deal with bvp for a forced nonlinear wave equation t2 u eg r t u where g is t periodic in t and e is a small parameter under some diophantine condition on a t we shall show the existence of time periodic solutions of the bvp for 1 n 5 we shall construct infinitely many t satisfying the above diophantine inequality using asymptotic expansions of the zero points of the bessel functions

a nekhoroshev type theorem for the nonlinear wave equation abstract we prove a nekhoroshev type theorem for the nonlinear wave equation nlw 0 1 u t t u x x m u f u under dirichlet boundary conditions where f is an analytic function more precisely we prove the subexponential long time stability result of the origin for equation 0 1 by using birkhoff normal form technique for infinite dimensional hamiltonian systems and the so called tame property of the nonlinearity in gevrey or analytic space which is inspired by the work 6 where the authors proved the polynomial long time stability result in sobolev space for a class of hamiltonian pdes

borsuk ulam theorems for arbitrary s1 actions and applications abstract an s to the first power version of the borsuk ulam theorem is proved for a situation where fix s to the first power may be nontrivial the proof is accomplished with the aid of a new relative index theory applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an s to the first power symmetry minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by hamilton s equations in these arguments one obtains the solutions of the differential equations as critical points of an associated lagrangian by minimaxing the lagrangian over appropriate classes of sets intersection theorems such as are proved in this paper play a crucial role in this process in addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an s to the first power symmetry group

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

long time stability of kam tori for nonlinear wave equation abstract it is proved that the kam tori thus quasi periodic solutions are long time stable for infinite dimensional hamiltonian systems generated by nonlinear wave equation by constructing a partial normal form of higher order around the kam torus and showing that p tame property persists under kam iterative procedure and normal form iterative procedure

realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory part ii the so called problem of the realization of the holonomic constraints of classical mechanics is here revisited in the light of nekhoroshev like classical perturbation theory precisely if constraints are physically represented by very steep potential wells with associated high frequency transversal vibrations then one shows that within suitable assumptions the vibrational energy and the energy associated to the constrained motion are separately almost constant for a very long time scale growing exponentially with the frequency i e with the rigidity of the constraint one aims to realize this result can also be applied to microscopic physics providing a possible entirely classical mechanism for the freezing of the high frequency degrees of freedom in terms of non equilibrium statistical mechanics according to some ideas expressed by boltzmann and jeans at the turn of the century in this part i we introduce the problem and prove a first theorem concerning the realization of a single constraint within a system of any number of degrees of freedom the problem of the realization of many constraints will be considered in a forthcoming part ii

existence of forced oscillations for some nonlinear differential equations abstract this article studies the existence of t periodic solutions for systems of nonlinear second order ordinary differential equations assuming function v to be superquadratic it is shown that this system possesses infinitely many t periodic solutions the proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial some more general results concerning systems of another type are also presented

effective stability for a hamiltonian system near an elliptic equilibrium point with an application to the restricted three body problem abstract we consider an n degrees of freedom hamiltonian system near an elliptic equilibrium point the system is put in normal form up to an arbitrary order and with respect to some resonance module and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form a bound to the rate of diffusion is thus found and by optimizing the order of normalization exponential estimates of nekhoroshev s type are obtained this provides explicit estimates for the stability properties of the elliptic point and leads in some cases to effective stability i e stability up to finite but long times an application to the stability of the triangular libration points in the spatial restricted three body is also given

minimum rate sampling and reconstruction of signals with arbitrary frequency support we examine the question of reconstruction of signals from periodic nonuniform samples this involves discarding samples from a uniformly sampled signal in some periodic fashion we give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed we give an implicit characterization of the reconstruction system and a design method by which the ideal reconstruction filters may be approximated we demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing as in earlier work previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made or where the minimum rate was approached asymptotically we show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered when approaching the minimum rate this freedom allows us in certain cases to have a far less complex reconstruction system

the rabinowitz floer homology for a class of semilinear problems and applications abstract in this paper we construct a rabinowitz floer type homology for a class of non linear problems having a starshaped potential we consider some equivariant cases as well we give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model equations