modulated lattice vector quantization how to make quantization index modulation an efficient variable rate source coder the design of a variable rate joint watermarking and compression jwc scheme is examined in this paper the proposed approach called modulated lattice vector quantization mlvq is based on dither modulation quantization index modulation dm qim which allows for embedding information while maintaining good coding performance in the first part we propose a specific indexing method to make mlvq with a multidimensional codebook feasible furthermore a quantization parameter estimation was designed to ensure the invertibility of the embedding the limitations of the compression performance of jwc schemes are studied in the second part we show the existence of a coding rate lower bound which depends mainly on the codebook characteristics and dramatically decreases coding performance to circumvent this drawback two improved mlvq schemes are proposed in the first one called arbitrary mlvq the embedding is performed on part of the signal to ensure a low embedding coding ratio in the second one called deadzone mlvq the coding efficiency is further improved by maintaining the sparsity of the quantized host signal it consists in excluding the sparse signal components from the embedding process then thresholding them these schemes both applying wavelet coding demonstrate their efficiency as variable rate coders

nearest neighbor algorithm for spherical codes from the leech lattice the leech lattice is a regular arrangement of points in 24 dimensional euclidean space that yields an extremely dense packing when equal spheres are centered at these points a subset of the leech lattice can be used as a signal set for the gaussian channel or as representative vectors for a vector quantizer of particular interest are the spherical codes or code books that consist of the points of the leech lattice which lie on a sphere centered at the origin the code points do not have to be stored because they can be obtained from a very small set of basic vectors using permutations of the components in a manner dictated by the words of the extended golay code a nearest neighbor algorithm that works on this is developed to determine the point in the code closet to some arbitrary vector in r sup 24 the performance of this approach when quantizing independent identically distributed gaussian samples is reported

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

an l bound for solutions of the cahn hilliard equation in this paper we consider the equation partial t u varepsilon delta u varepsilon sum limits i j 1 n partial ij f ij u varepsilon x varepsilon 2 delta 2 u varepsilon with fij constant when u e is large and we obtain l bounds independent of 0

optimal decay rate of solutions for cahn hilliard equation with inertial term in multi dimensions abstract the cauchy problem of the cahn hilliard equation with inertial term is considered based on green s function method together with energy estimates we get the global in time existence and optimal decay rate of solutions furthermore the viscous case is investigated in the last section

pyramidal lattice vector quantization for multiscale image coding introduces a new image coding scheme using lattice vector quantization the proposed method involves two steps biorthogonal wavelet transform of the image and lattice vector quantization of wavelet coefficients in order to obtain a compromise between minimum distortion and bit rate we must truncate and scale the lattice suitably to meet this goal we need to know how many lattice points lie within the truncated area we investigate the case of laplacian sources where surfaces of equal probability are spheres for the l sup 1 metric pyramids for arbitrary lattices we give explicit generating functions for the codebook sizes for the most useful lattices like z sup n d sub n e sub s spl and sub 16

vector quantization by companding a union of z lattices an encoding scheme is presented based on random coding theory in conjunction with companding techniques this combination provides an easy design and a low storage quantizer the groundwork of the study is gaussian sources the motivation is the attractive properties of gaussian and gaussian mixtures gms and their applications in data compression here codebooks are generated by companding a union of randomly translated cubic lattices the compander functions operate scalar wise providing an affordable encoding complexity that is rate independent for such a codebook an optimal search and the indexing of the codevectors are addressed performance close to finite dimensional random coding is achieved while the complexity is not far from scalar coding the structure of the codebook balances between complexity and performance in comparison to some other methods the advantage of the proposed scheme is more distinct for correlated sources

global existence and asymptotics of solutions of the cahn hilliard equation abstract this paper is concerned with the cauchy problem of the cahn hilliard equation u t u 2 u 0 x r n t 0 u t 0 u 0 x x r n first we construct a local smooth solution u t x to the above cauchy problem then by combining some a priori estimates sobolev s embedding theorem and the continuity argument the local smooth solution u t x is extended step by step to all t 0 provided that the smooth nonlinear function u satisfies a certain local growth condition at some fixed point u r and that u 0 x u l 1 r n is suitably small secondly we show that the global smooth solution u t x satisfies the following temporal decay estimates d k u t x u l p r n c 1 t k 4 n 4 1 1 p t 0 k 0 1 here p 1 c 0 is a constant depending on and 0 is any positive constant which can be chosen sufficiently small at last we show that under a strong assumption on the growth of the nonlinear function u at u u the asymptotics of solutions of the above cauchy problem is described by u 0 t n 4 g x t 4 here 0 r n u 0 x u d x g x r n exp 4 i x d

exponential attractors and inertial manifolds for singular perturbations of the cahn hilliard equations abstract we consider the singular perturbations of two boundary value problems concerning respectively the viscous and the nonviscous cahn hilliard equations in one dimension of space we show that the dynamical systems generated by these two problems admit exponential attractors and inertial manifolds in the phase space h01 0 h 1 0

diffuse interface model for rapid phase transformations in nonequilibrium systems a thermodynamic approach to rapid phase transformations within a diffuse interface in a binary system is developed assuming an extended set of independent thermodynamic variables formed by the union of the classic set of slow variables and the space of fast variables we introduce finiteness of the heat and solute diffusive propagation at the finite speed of the interface advancing to describe transformations within the diffuse interface we use the phase field model which allows us to follow steep but smooth changes of phase within the width of the diffuse interface governing equations of the phase field model are derived for the hyperbolic model a model with memory and a model of nonlinear evolution of transformation within the diffuse interface the consistency of the model is proved by the verification of the validity of the condition of positive entropy production and by outcomes of the fluctuation dissipation theorem a comparison with existing sharp interface and diffuse interface versions of the model is given

global attractors for singular perturbations of the cahn hilliard equations abstract we consider the singular perturbations of two boundary value problems concerning respectively the viscous and the nonviscous cahn hilliard equations in one dimension of space we show that the dynamical systems generated by these two problems admit global attractors in the phase space h 0 1 0 h 1 0 and that these global attractors are at least upper semicontinuous with respect to the vanishing of the perturbation parameter

quasi uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions abstract we introduce several quasi uniformities on the hyperspace of a topological space in order to study convergence of nets of semicontinuous multifunctions the interplay between conjugate quasi uniformities yields results on both the topological structure of hyperspaces and the convergence of semicontinuous multifunctions

the lp lq estimates of solutions to one dimensional damped wave equations and their application to the compressible flow through porous media we first obtain the lp lq estimates of solutions to the cauchy problem for one dimensional damped wave equation vtt vxx vt 0 v vt t 0 v0 v1 x x t r r corresponding to that for the parabolic equation t xx 0 t 0 v0 v1 x the estimates are shown by v t e t 2v0 t v0 t 2lp ct 121q 1p 1 v0 v1 lq t 1 etc for 1 q p to show the explicit formula of the damped wave equation will be used to apply the estimates to nonlinear problems is the second aim we will treat the system of a compressible flow through porous media the solution is expected to behave as the diffusion wave which is the solution to the porous media equation due to the darcy law when the initial data has the same constant state at a sharp lp convergence rate for p 2 has been recently obtained by nishihara proc roy soc edinburgh sect a 133a 2003 1 20 by choosing a suitably located diffusion wave we will show the l1 convergence applying

entropy coded lattice vector quantization dedicated to the block mixture densities entropy coded lattice vector quantization eclvq with codebooks dedicated to independent identically distributed i i d generalized gaussian sources have proven their high coding performances in the wavelet domain it is well known that wavelet coefficients with high magnitude corresponding to edges and textures tend to be clustered in a few amount of vectors in this paper we first show that this property has a major influence on the performances of eclvq schemes since this clustering property cannot be taken into account by the classical i i d assumption our first proposal is to model the joint distribution of vectors by a multidimensional mixture of generalized gaussian mmgg densities the main outcome of this mmgg model is to provide a theoretical framework to simply derive from i i d models the corresponding mmgg models in a second part a new codebook better suited to wavelet coding is proposed the so called dead zone lattice vector quantizers dzlvq it consists of generalizing the scalar dead zone to vector quantization by thresholding vectors according to their energy we show that dzlvq improves the rate distortion tradeoff experimental results are provided for the pyramidal lvq scheme under the assumption of a multidimensional mixture of laplacian mml densities results performed on a set of real life images show the precision of the analytical curves and the efficiency of the dzlvq scheme

enumeration encoding and decoding algorithms for pyramid cubic lattice and trellis codes a pyramid source code is a code that assigns equal length binary strings to all reproduction codevectors of equal weighted spl epsiv sub 1 norm a pyramid source encoding is partitioned into two concatenated mappings the first from source word to reproduction codeword within a codebook the second from the reproduction codevector to a binary string the first mapping allows distortion and is accomplished using lattice quantization or trellis coded quantization the second mapping is noiseless and is denoted as enumeration efficient pyramid enumeration encoding and decoding algorithms are presented for use with fixed rate or variable rate pyramid lattice and trellis codes

pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions abstract in this paper we study the pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions by the green function method furthermore we obtain the l p convergence rate of the solution

distortion rate models for entropy coded lattice vector quantization the increasing demand for real time applications requires the use of variable rate quantizers having good performance in the low bit rate domain in order to minimize the complexity of quantization as well as maintaining a reasonably high psnr ratio we propose to use an entropy coded lattice vector quantizer eclvq these quantizers have proven to outperform the well known ezw algorithm s performance in terms of rate distortion tradeoff we focus our attention on the modeling of the mean squared error mse distortion and the prefix code rate for eclvq first we generalize the distortion model of jeong and gibson 1993 on fixed rate cubic quantizers to lattices under a high rate assumption second we derive new rate models for eclvq efficient at low bit rates without any high rate assumptions simulation results prove the precision of our models

topological existence and stability for min sup problems resume tout d abord nous donnons des conditions de caractere minimal qui garantissent l existence d un point min sup pour une fonctionnelle f ensuite nous considerons un probleme de min sup parametrise et nous determinons une classe minimale pour f de maniere a obtenir la stabilite du probleme parametrise

a conserved phase field system with memory and relaxed chemical potential abstract in this paper we propose a conserved phase field model with memory and relaxed chemical potential our model could be considered also in the frame of cahn hilliard s theory of spinodal decomposition of a binary mixture where there is a coupling to a slowly relaxing variable e g in phase separation of polymer mixtures near a glass transition the global well posedness in an l p setting is obtained under suitable relaxation functions and under homogeneous conditions

on the hyperbolic relaxation of the one dimensional cahn hilliard equation abstract we consider the one dimensional cahn hilliard equation with an inertial term u t t for 0 this equation endowed with proper boundary conditions generates a strongly continuous semigroup s t which acts on a suitable phase space and possesses a global attractor our main result is the construction of a robust family of exponential attractors m whose common basins of attraction are the whole phase space

the pointwise estimates of solutions for semilinear dissipative wave equation in multi dimensions abstract in this paper we study the global in time existence and the pointwise estimates of solutions to the cauchy problem for the dissipative wave equation in multi dimensions using the fixed point theorem we obtain the global existence of the solution in addition the pointwise estimates of the solution are obtained by the method of the green function furthermore we obtain the l p 1 p convergence rate of the solution

low complexity indexing method for z sup n and d sub n lattice quantizers code vector indexing is a key problem in lattice quantization applications in order to solve this problem we propose a method based on the idea of a coding table encompassing a set of points as small as possible our method works for both spherical and pyramidal code books it provides a good tradeoff between computational complexity and storage requirements

large time behavior and lp lq estimate of solutions of 2 dimensional nonlinear damped wave equations abstract we show the asymptotic behavior of the solution to the cauchy problem of the two dimensional damped wave equation it is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the lp lq type estimate this is a two dimensional generalization of the three dimensional result due to nishihara math z 244 2003 631 to show this we use the fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations by using the lp lq estimate we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity u u our result covers the whole super critical case 1 where the 1 is well known as the fujita exponent when n 2

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

lattice codebook enumeration for generalized gaussian source the goal of this correspondence is to propose a low complexity enumeration algorithm for lattice vectors based on a geometrical interpretation and valid for different source distributions i e for different l sub p norms in the range 0 p spl les 2 as a particular case we obtain the laplacian enumeration formula of fischer this point of view offers various advantages and particularly it enables one to make the link with the generalized theta series and to reduce the algorithm to the calculation of a few convolutional products in the special cases p 1 and p 2 using a dedicated digital signal processing dsp architecture convolutional products are easy to implement and require few arithmetic operations our algorithm developed for the z sup n lattice can be generalized to other lattices like the d sub n

on structure of upper semicontinuity abstract the refinement of a choquet theorem on strong upper semi continuity and its relation to the vainstein lemma are dealt with here relevance of subcontinuity is discussed consequently an improvement of a characterization theorem of dolecki and rolewicz is achieved

continuous convergence and preservation of convergences of sets abstract a systematic study of convergence of images of families of relation is undertaken structurally these are questions from the theory of continuous convergence

decay properties of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u p 1 u 0 t x r r n the behavior of u as t is expected to be the same as that for the corresponding heat equation t p 1 0 t x r r n which has the similarity solution w a t x with the form t 1 p 1 f x t depending on a lim x x 2 p 1 f x 0 provided that p is less than the fujita exponent p c n 1 2 n in this paper as a first step if 1 p p c n and the data u 0 u 1 x decays exponentially as x without smallness condition the solution is shown to decay with rates as t u t l 2 u t l p 1 u t l 2 o t 1 p 1 n 4 t 1 p 1 n 2 p 1 t 1 p 1 1 2 n 4 those of which seem to be reasonable because the similarity solution w a t x has the same decay rates as for the proof the weighted l 2 energy method will be employed with suitable weight similar to that in todorova and yordanov y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489