the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

a kam theorem for one dimensional schrodinger equation with periodic boundary conditions abstract in this paper one dimensional 1 d nonlinear schrodinger equation iu t u xx mu g u u u 0 with periodic boundary conditions is considered m 1 12 z is a real parameter and the nonlinearity g u u j l j l 4 a jl u j u l a jl a lj r a 22 0 is a real analytic function in a neighborhood of the origin the kam machinery is adapted to fit the above equation so as to construct small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system

a kam theorem for hamiltonian partial differential equations in higher dimensional spaces in this paper we give a kam theorem for a class of infinite dimensional nearly integrable hamiltonian systems the theorem can be applied to some hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi periodic solutions and its local birkhoff normal form the applications to the higher dimensional beam equations and the higher dimensional schrodinger equations with nonlocal smooth nonlinearity are also given in this paper

double series representation of bounded signals series representations of the form f t approximately sigma sub n infinity sup infinity sigma sub k infinity sup infinity a sub n k nu t n sub e sup 2 pi kts for bounded signals f t are studied as are conditions on the unit function nu t such that coefficients a sub n k reveal the energy content of f t in the time interval n 1 2

applications of temperley lieb algebras to lorentz lattice gases motivated by the study of motion in a random environment we introduce and investigate a variant of the temperley lieb algebra this algebra is very rich providing us with three classes of solutions of the yang baxter equation this allows us to establish a theoretical framework to study the diffusive behaviour of a lorentz lattice gas exact results for the geometrical scaling behaviour of closed paths are also presented

optimal tight frames and quantum measurement tight frames and rank one quantum measurements are shown to be intimately related in fact the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank one generalized quantum measurements on that space using this relationship frame theoretical analogs of various quantum mechanical concepts and results are developed the analog of a least squares quantum measurement is a tight frame that is closest in a least squares sense to a given set of vectors the least squares tight frame is found for both the case in which the scaling of the frame is specified constrained least squares frame clsf and the case in which the scaling is chosen to minimize the least squares error unconstrained least squares frame ulsf the well known canonical frame is shown to be proportional to the ulsf and to coincide with the clsf with a certain scaling

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

the analysis and design of windowed fourier frame based multiple description source coding schemes in this paper the windowed fourier encoding decoding scheme applied to the multiple description compression problem is analyzed in the general case four window functions are needed to define the encoder and decoder although this number can be reduced to three or two by using time shift or frequency shift division schemes the encoding coefficients are next divided into two groups according to the eveness of either the modulation or translation index the distortion on each channel is analyzed using the zak transform for the optimal windows explicit representation formulas are obtained and nonlocalization results are proved asymptotic formulas of the total distortion and transmission rate are established and the redundancy is shown to trade off between these two

lattice lorentz gas in cellular automaton fluids where binary collision laws admit reflections the stosszahl ansatz and the boltzmann equation are no longer valid at low densities this is proven for a class of lorentz gas models defined on cubic lattices for which the diffusion coefficient is calculated from a self consistent ring kinetic equation or effective medium theory the calculated results are in excellent agreement with results from molecular dynamics simulations over the whole density interval

a tiling of phase space through self convolution a novel tiling for fourier half space is constructed using erlang functions this tiling is constructed by shifts in discrete time and self convolutions in the fourier domain we show that a subframe of this complex wavelet family can be used for the detection and characterization of events step and peaks in one dimensional 1 d discrete time signals

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 kam algorithm for the resonant non linear schrodinger equation abstract we prove through a kam algorithm the existence of large families of stable and unstable quasi periodic solutions for the nls in any number of independent frequencies the main tools are the existence of a non degenerate integrable normal form proved in 26 27 and a generalization of the quasi toplitz functions defined in 31

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

kam for reversible derivative wave equations we prove the existence of cantor families of small amplitude analytic linearly stable quasi periodic solutions of reversible derivative wave equations

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

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

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

on reducibility of schrodinger equations with quasiperiodic in time potentials we prove that a linear d dimensional schrodinger equation with an x periodic and t quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector the reduction is made by means of a non autonomous linear transformation of the space of x periodic functions this transformation is a quasiperiodic function of t

comment on on the complete integrability of the hirota satsuma system the authors show that the aforementioned paper by chowdhury and mukherjee ibid vol 17 p l231 1984 is false

kam theory for the reversible perturbations of 2d linear beam equations in the present paper we prove an infinite dimensional reversible kolmogorov arnold moser kam theorem as an application we study the existence of kam tori for a class of two dimensional 2d non hamiltonian completely resonant beam equations with derivative nonlinearities the birkhoff normal form theory is also used since there are no external parameters in the equations

on the complete integrability of the hirota satsuma system the authors have applied the painleve test to the coupled nonlinear system advocated by hirota and satsuma 1981 they have also made a lie point symmetry analysis of these equations and have shown that the reduced ordinary nonlinear equations are not members of the painleve class also the painleve test itself is seen to fail so that on both counts the equations are not completely integrable in the usual sense

quasi periodic solutions for a completely resonant beam equation with a nonlinear term depending on the time and space variables abstract this article is devoted to the study of a completely resonant beam equation with an x periodic and t quasi periodic nonlinear term it is proved that the equation admits small amplitude linearly stable and quasi periodic solutions for most values of the frequency vector by utilizing the measure estimation of infinitely many small divisors we construct a real analytic symplectic change of coordinates which can transform the hamiltonian into some birkhoff normal form we show an infinite dimensional kam theorem for non autonomous beam equations which is applied to prove the existence of quasi periodic solutions

periodic solutions for a class of nonlinear partial differential equations in higher dimension we prove the existence of periodic solutions in a class of nonlinear partial differential equations including the nonlinear schrodinger equation the nonlinear wave equation and the nonlinear beam equation in higher dimension our result covers cases of completely resonant equations where the bifurcation equation is infinite dimensional such as the nonlinear schrodinger equation with zero mass for which solutions which at leading order are wave packets are shown to exist

long time tails in lattice lorentz gases we consider a lorentz gas on a square lattice with a fraction c of scattering sites the collision laws are deterministic fixed mirror model or stochastic with transmission reflection and deflection probabilities and respectively if all mirrors are parallel the mirror model is exactly solvable for the general case a self consistent ring kinetic equation is used to calculate the longtime tails of the velocity correlation function 0 t and the tensor correlation q 0 q t withq x y both functions showt 2 tails as opposed to the continuous lorentz gas where the tails are respectivelyt 2 andt 3 inclusion of the self consistent ring collisions increases the low density coefficient of the tail in 0 t by 30 100 as compared to the simple ring collisions depending on the model parameters

resilience properties of redundant expansions under additive noise and quantization representing signals using coarsely quantized coefficients of redundant expansions is an interesting source coding paradigm the most important practical case of which is oversampled analog to digital a d conversion signal reconstruction from quantized redundant expansions and the accuracy of such representations are problems which are not well understood and we study them in this paper for uniform scalar quantization in finite dimensional spaces to give a more global perspective we first present an analysis of the resilience of redundant expansions to degradation by additive noise in general and then focus on the effects of uniform scalar quantization the accuracy of signal representations obtained by applying uniform scalar quantization to coefficients of redundant expansions measured as the mean squared euclidean norm of the reconstruction error has been previously shown to be lower bounded by an 1 r sup 2 expression we establish some general conditions under which the 1 r sup 2 accuracy can actually be attained and under those conditions prove a 1 r sup 2 upper error bound for a particular kind of structured expansions which includes many popular frame classes we propose reconstruction algorithms which attain the 1 r sup 2 accuracy at low numerical complexity these structured expansions moreover facilitate efficient encoding of quantized coefficients in a manner which requires only a logarithmic bit rate increase in redundancy resulting in an exponential error decay in the bit rate results presented in this paper are immediately applicable to oversampled a d conversion of periodic bandlimited signals

sigma delta spl sigma spl delta quantization and finite frames the k level sigma delta spl sigma spl delta scheme with step size spl delta is introduced as a technique for quantizing finite frame expansions for spl ropf sup d error estimates for various quantized frame expansions are derived and in particular it is shown that spl sigma spl delta quantization of a unit norm finite frame expansion in spl ropf sup d achieves approximation error where n is the frame size and the frame variation spl sigma f p is a quantity which reflects the dependence of the spl sigma spl delta scheme on the frame here spl par spl middot spl par is the d dimensional euclidean 2 norm lower bounds and refined upper bounds are derived for certain specific cases as a direct consequence of these error bounds one is able to bound the mean squared error mse by an order of 1 n sup 2 when dealing with sufficiently redundant frame expansions this represents a significant improvement over classical pulse code modulation pcm quantization which only has mse of order 1 n under certain nonrigorous statistical assumptions spl sigma spl delta also achieves the optimal mse order for pcm with consistent reconstruction

probability of first return on cayley trees a lorentz lattice gas with a fraction of the scatterers being pure backscatterers the remaining being stochastic right and left rotators is considered the problem at hand is the evaluation of the probability x that the moving particle returns to its original site in the limit where the density of sites occupied by scatterers is small and the lattice becomes a cayley tree in the special case of deterministic collision rules gunn ortuno model on a cayley tree an exact cubic equation for the laplace transform of the distribution function of first return times is derived and its asymptotic form near the threshold value beyond which x 1 is obtained in the general case of stochastic models a mean field approximation is proposed for x and the mean return time the approximation reduces to the exact result in the case of the deterministic model as well as in the stochastic model without backscatterers comparison with monte carlo simulations shows a reasonable agreement in the dependence of x and on the relevance of the results to the development of approximate analytic expressions for the diffusion coefficient is discussed

diffusion and propagation in triangular lorentz lattice gas cellular automata the diffusion process of point particles moving on regular triangular and random lattices randomly occupied with stationary scatterers a lorentz lattice gas cellular automaton is studied for strictly deterministic scattering rules as a function of the concentration of the scatterers in addition to the normal and various kinds of retarded diffusion found before on the regular square lattice straight line propagation through the scatterers is observed

invariant tori of nonlinear schrodinger equation abstract in this paper we consider one dimensional nonlinear schrodinger equation i u t u x x v x u f u 2 u 0 on 0 r under the boundary conditions a 1 u t 0 b 1 u x t 0 0 a 2 u t b 2 u x t 0 a i 2 b i 2 0 for i 1 2 it is proved that for a prescribed and analytic positive potential v x the above equation admits small amplitude quasi periodic solutions corresponding to d dimensional invariant tori of the associated infinite dimensional dynamical system

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

kam for the nonlinear schrodinger equation we consider the d dimensional nonlinear schrodinger equation under periodic boundary conditions i dot u delta u v x u ep frac p f p bar u x u bar u quad u u t x x in t d where v x sum hat v a e i sc a x is an analytic function with hat v real and f is a real analytic function in re u im u and x this equation is a popular model for the real nls equation where instead of the convolution term v u we have the potential term vu for ep 0 the equation is linear and has time quasi periodic solutions u u t x sum a in a hat u a e i a 2 hat v a t e i sc a x quad hat u a 0 where a is any finite subset of z d we shall treat omega a a 2 hat v a a ina as free parameters in some domain u subset r a this is a hamiltonian system in infinite degrees of freedom degenerate but with external parameters and we shall describe a kam theory which under general conditions will have the following consequence if ep is sufficiently small then there is a large subset u of u such that for all omega in u the solution u persists as a time quasi periodic solution which has all lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients

a normal form for beam and non local nonlinear schrodinger equations we discuss normal forms of the completely resonant nonlinear beam equation and nonlinear schrodinger equation we work in n 1 spatial dimensions and study both periodic and dirichlet boundary conditions on cubes we discuss the applications to the problem of finding quasi periodic solutions in the case of periodic boundary and the dimension n 2 we apply kam theory and prove the existence and stability of quasi periodic solutions

periodic solutions for the schrodinger equation with nonlocal smoothing nonlinearities in higher dimension abstract we consider the nonlinear schrodinger equation in higher dimension with dirichlet boundary conditions and with a nonlocal smoothing nonlinearity we prove the existence of small amplitude periodic solutions in the fully resonant case we find solutions which at leading order are wave packets in the sense that they continue linear solutions with an arbitrarily large number of resonant modes the main difficulty in the proof consists in a small divisor problem which we solve by using a renormalisation group approach

noise reduction in oversampled filter banks using predictive quantization we introduce two methods for quantization noise reduction in oversampled filter banks these methods are based on predictive quantization noise shaping or linear prediction it is demonstrated that oversampled noise shaping or linear predictive subband coders are well suited for subband coding applications where for technological or other reasons low resolution quantizers have to be used in this case oversampling combined with noise shaping or linear prediction improves the effective resolution of the subband coder at the expense of increased rate simulation results are provided to assess the achievable quantization noise reduction and resolution enhancement and to investigate the rate distortion properties of the proposed methods

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

noise reduction in tight weyl heisenberg frames the functions g that give rise to tight weyl heisenberg frames based on time and frequency discretization steps q sub 0 and p sub 0 with p sub 0 q sub 0 2 pi k k in n are characterized and the noise reducing properties of such frames are examined it is shown that when the frame coefficients c sub m n f of a function f are subject to stationary noise n the average squared l sup 2 norm of the corresponding noise contribution to the reconstruction of f is down by a factor proportional to k sup 2 for a fixed number of sample points also a measure of local error of the reconstruction is introduced as opposed to the global l sup 2 error for a given noise n formulae are derived for the optimal choice or choices of the basis function g in the sense of minimizing this local error