gaussian multiple description coding with low density generator matrix codes it is shown that the coding problem for an arbitrary point on the dominant face of an l description el gamal cover egc region can be converted to that for a vertex of a k description egc region for some k 2l 1 where the latter problem can be solved via successive coding a practical scheme is proposed for the quadratic gaussian case by reducing each step in successive coding to a gaussian quantization operation and implementing such an operation using low density generator matrix codes the effectiveness of this scheme is verified through extensive simulation experiments

smoothing effect for some schrodinger equations abstract we study the cauchy problem for the following schrodinger equation in r n n n i t u 1 2 u v 1 u v 2 u 2 u t x r r n u 0 x r n where v 1 v 1 x 1 x 1 v 2 v 2 x k 2 3 k x k k r 1 k 3 0 1 min 2 n 2 0 2 3 min 2 n we prove the existence uniqueness and smoothing effect of global solutions of with not necessarily in the sobolev space h1 rn

dithered lattice based quantizers for multiple descriptions multiple description md source coding is aimed at achieving graceful degradation in reconstruction with respect to losing portions of the code with the cost of some redundancy we examine md schemes which use entropy coded dithered lattice quantizers ecdq we propose two techniques one based on successive refinement sr and the other a dithered and periodic version of the mid scalar quantizer mdsq with distributed cells proposed by vaishampayan 1993 1994 similarly to the single description case both techniques are universal in nature and are equivalent to additive noise channels this allows one to derive analytical expressions for the rate distortion performance for general sources and to compare them to the optimal rate regions at both high and low resolutions among other results we establish that while the dithered mdsq scheme loses only the space filling loss of the scalar lattice at any resolution the sr based scheme loses an additional 0 5 bit at any lattice dimension possible improvements such as refinement time sharing and dependent dithering are discussed

on solutions of the initial value problem for the nonlinear schrodinger equations on considere le probleme aux valeurs initiales pour l equation de schrodinger non lineaire de la forme iu t u f u 2 u t x r r n u 0 x x x r n on etudie la regularite des solutions quand les fonctions initiales ne sont pas regulieres

optimal quantizer design for noisy channels an approach to combined source channel coding we present an analysis of the zero memory quantization of memoryless sources when the quantizer output is to be encoded and transmitted across a noisy channel necessary conditions for the joint optimization of the quantizer and the encoder decoder pair are presented and an iterative algorithm for obtaining a locally optimum system is developed the performance of this locally optimal system obtained for the class of generalized gaussian distributions and the binary symmetric channel is compared against the optimum performance theoretically attainable using rate distortion theoretic arguments as well as against the performance of lloyd max quantizers encoded using the natural binary code and the folded binary code it is shown that this optimal design could result in substantial performance improvements the performance improvements are more noticeable at high bit rates and for broad tailed densities

on the derivative nonlinear schro dinger equation abstract in this paper we discuss the cauchy problem for the derivative nonlinear schrodinger equation i t 2i x 2 0 0 x x where 0 under an explicit smallness condition of the initial data we prove the unique global existence of solutions to this problem in the usual sobolev spaces in the weighted sobolev spaces and in the schwartz class we describe the smoothing effect in detail furthermore for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction

on the design of symmetric entropy constrained multiple description scalar quantizer with linear joint decoders this paper addresses the design of symmetric entropy constrained multiple description scalar quantizers ec mdsq with linear joint decoders i e where some of the decoders compute the reconstruction by averaging the reconstructions of individual descriptions thus the use of linear decoders reduces the space complexity at the decoder since only a subset of the codebooks needs to be stored the proposed design algorithm locally minimizes the lagrangian which is a weighted sum of the expected distortion and of the side quantizers rates the algorithm is inspired by the ec mdsq design algorithm proposed by vaishampayan and domaszewicz and it is adapted from two to k descriptions differently from the aforementioned work the optimization of the reconstruction values can no longer be performed separately at the decoder optimization step interestingly we show that the problem is a convex quadratic optimization problem which can be efficiently solved moreover the generalization of the encoder optimization step from two to k descriptions increases drastically the amount of computations we show how to exploit the special form of the cost function conferred by the linear joint decoders to significantly reduce the time complexity at this step we compare the performance of the proposed design with multiple description lattice vector quantizers mdlvq and with the multiple description scheme based on successive refinement and unequal erasure protection uep our experiments show that the proposed approach outperforms mdlvq with dimension 1 quantization as expected additionally when more codebooks are added our scheme even beats mdlvq with quantization dimension approaching infty for rates sufficiently high furthermore the proposed approach is also superior to uep with dimension 1 quantization when the rates are low

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

universal multiple description scalar quantization analysis and design this paper introduces a new high rate analysis of the multiple description scalar quantizer mdsq with balanced descriptions the analysis provides insight into the structure of the mdsq suggesting the nonoptimality of uniform central quantizer cell lengths as well as a method to approximate optimal cell lengths for both level constrained and entropy constrained mdsq new upper bounds on the granular distortion for sources with smooth probability density functions pdfs are derived under the mean squared error measure which are 0 4 db lower than previous results based on the insights a universal multiple description scalar quantizer umdsq is proposed which at high rate can achieve nearly the same performance as the fully optimized entropy constrained mdsq ecmdsq without requiring extensive training the proposed umdsq has only two control parameters and a continuum of tradeoff points between the central and side distortions can be achieved as the two parameters are varied

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

the rate distortion region for multiple descriptions without excess rate during recent years there has been strong interest in a certain source coding problem which some authors call the problem of multiple descriptions old and new wringing techniques enable us to establish a single letter characterization of the rate distortion region in the case of no excess rate for the joint description

optimality and suboptimality of multiple description vector quantization with a lattice codebook the asymptotic analysis of multiple description vector quantization mdvq with a lattice codebook for sources with smooth probability density functions pdfs is considered in this correspondence goyal et al 2002 observed that as the side distortion decreases and the central distortion correspondingly increases the quantizer cells farther away from the coarse lattice points shrink in a spatially periodic pattern in this correspondence two special classes of index assignments are used along strategic groupings of central quantizer cells to derive a straightforward asymptotic analysis which provides an analytical explanation for the aforementioned observation mdvq with a lattice codebook was shown earlier to be asymptotically optimal in high dimensions with a curious converging property that the side quantizers achieve the space filling advantage of an n dimensional sphere instead of an n dimensional optimal polytope the analysis presented here explains this behavior readily while central quantizer cells on a uniform lattice are asymptotically optimal in high dimensions the present authors have shown that by using nonuniform rather than uniform central quantizer cells the central side distortion product in an mdsq can be reduced by 0 4 db at asymptotically high rate the asymptotic analysis derived here partially unifies these previous results in the same framework though a complete characterization is still beyond reach

bit error resilient index assignment for multiple description scalar quantizers this paper addresses the problem of increasing the robustness to bit errors for two description scalar quantizers our approach is to start with an m diagonal index assignment and further apply a permutation to the indexes of each description to increase the minimum hamming distance d rm min of the set of valid index pairs in particular we show how to construct linear permutation pairs achieving d rm min geq 3 and establish a lower bound in terms of the description rate r for the highest value of m for which such permutations exist

universal quantization for separate encodings and joint decoding of correlated sources we consider the multi user lossy source coding problem for continuous alphabet sources in a previous work ziv proposed a single user universal coding scheme which uses uniform quantization with dither followed by a lossless source encoder entropy coder in this paper we generalize ziv s scheme to the multi user setting for this generalized universal scheme upper bounds are derived on the redundancies defined as the differences between the actual rates and the closest corresponding rates on the boundary of the rate region for the mean square error distortion measure it is shown that this scheme can achieve redundancies of no more than 0 754 b per sample for each user these bounds are obtained without the knowledge of the multi user rate region which is an open problem in general as a direct consequence of these results the inner and outer bounds on the rate distortion achievable region are obtained

monotonicity and uniqueness results in identifying an unknown coefficient in a nonlinear diffusion equation we consider the problem of identifying an unknown diffusion coefficient in a nonlinear diffusion equation from overspecified data measured on the boundary conditions ensuring compatibility of the data are derived and it is shown that the mapping which carries the diffusion coefficient into the data is isotonic this result is then used to derive a uniqueness result for the inverse problem and for an associated identification problem some results of numerical experiments are presented to illustrate the various results

smoothing effects and dispersion of singularities for the schrodinger evolution group we describe smoothing effects and dispersion of singularities for the schrodinger evolution group in the weighted sobolev spaces under a fairly general assumption on the potential it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay we measure the regularity gained by the wavefunction by the decay property of the initial state no assumptions on the regularity of the initial state are imposed throughout the paper

alphabet constrained data compression the optimal data compression problem is posed in terms of an alphabet constraint rather than an entropy constraint solving the optimal alphabet constrained data compression problem yields explicit source encoder decoder designs which is in sharp contrast to other approaches the alphabet constrained approach is shown to have the additional advantages that 1 classical waveform encoding schemes such as pulse code modulation pcm differential pulse code modulation dpcm and delta modulation dm as well as rate distortion theory motivated tree trellis coders fit within this theory 2 the concept of preposterior analysis in data compression is introduced yielding a rich new class of coders and 3 it provides a conceptual framework for the design of joint source channel coders for noisy channel applications examples are presented of single path differential encoding delayed or tree encoding preposterior analysis and source coding over noisy channels

space time behavior of propagators for schrodinger evolution equations with stark effect abstract we give a detailed description of the space time behavior of the unitary propagator u t s t s r determined by the schrodinger evolution equation i t u 1 2 u e xu v t x u t r x r n where e is a homogeneous electric field and v is a time dependent potential

optimum quantizer performance for a class of non gaussian memoryless sources the performance of optimum quantizers subject to an entropy constraint is studied for a wide class of memoryless sources for a general distortion criterion necessary conditions are developed for optimality and a recursive algorithm is described for obtaining the optimum quantizer under a mean square error criterion the performance of entropy encoded uniform quantization of memoryless gaussian sources is well known to be within 0 255 bits sample of the rate distortion bound at relatively high rates despite claims to the contrary it is demonstrated that similar performance can be expected for a wide range of memoryless sources indeed for the cases considered the worst case performance is observed to be less than 0 3 bits sample from the rate distortion bound and in most cases this disparity is less at iow rates

minimum breakdown degradation in binary source encoding a memoryless binary equiprobable source produces one letter per second two people each are provided separately with private information about the source data at a rate of 1 2 bit per second suppose that by pooling their information they can produce a long mn reconstruction of the source output that has arbitrarily small error frequency we prove that then the least common asymptotic error frequency d that each can achieve without the other s help is sqrt 2 1 2 0 207 since it had been shown previously that 0 200 leq d leq 0 207 our result closes the so called 007 gap new analytical techniques introduced to effect the proof are of broader significance in multiuser information theory

smoothness and non smoothness of the fundamental solution of time dependent schrodinger equations the fundamental solutione t s x y of time dependent schrodinger equationsi u t 1 2 u v t x u is studied it is shown that e t s x y is smooth and bounded fort s if the potential is sub quadratic in the sense thatv t x o x 2 at infinity in one dimension ifv t x v x is time independent and super quadratic in the sense thatv x c 1 x 2 e at infinity c 0 and e 0 thene t s x y is nowherec1

commutator methods and a smoothing property of the schrodinger evolution group on montre que le groupe d evolution de schrodinger unitaire a une propriete de lissage dans certains espaces de sobolev ponderes

a method for solving an inverse biharmonic problem this paper deals with the problem of determining of an unknown coefficient in an inverse boundary value problem using a nonconstant overspecified data it has been shown that the solution to this inverse problem exists and is unique

new results in binary multiple descriptions an encoder whose input is a binary equiprobable memoryless source produces one output of rate r 1 and another of rate r 2 let d 1 d 2 and d 0 respectively denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate r l only the encoder output of rate r 2 only and both encoder outputs the two descriptions problem is to determine the region r of all quintuples r 1 r 2 d 1 d 2 d 0 that are achievable in thc usual shannon sense let r d 1 d log 2 d 1 d log 2 1 d denote the error frequency rate distortion function of the source the no excess rate case prevails when r 1 r 2 r d 0 and the excess rate case when r 1 r 2 r d 0 denote the section of r at r 1 r 2 d 0 by d r 1 r 2 d 0 d 1 d 2 r 1 r 2 d 1 d 2 d 0 in r in the no excess rate case we show that a portion of the boundary of d r 1 r 2 d 0 coincides with the curve frac 1 2 d 1 2d 0 frac 1 2 d 2 2d 0 frac 1 2 1 2d 0 2 this curve is an extension of witsenhausen s hyperbola bound to the case d 0 0 it follows that the projection of r onto the d 1 d 2 plane at fixed d 0 consists of all d 1 geq d 0 and d 2 geq d 0 that lie on or above this hyperbola in the excess rate case we show by counterexample that the achievable region of el gamal and cover is not tight

optimal index assignment for multiple description scalar quantization with translated lattice codebooks we design a k description scalar quantizer whose construction is based on a structure of translated scalar lattices and a lattice in k 1 dimensional space the use of translated lattices provides a performance advantage by exploiting a so called staggering gain the use of the k 1 dimensional lattice facilitates analytic insight into the performance and significantly speeds up the computation of the index assignment compared to state of the art methods using a common decoding method the proposed index assignment is proven to be optimal for the k description case it is shown that the optimal index assignment is not unique this is illustrated for the two description case where a periodic index assignment is selected from possible optimal assignments and described in detail the performance of the proposed quantizer accurately matches theoretic analysis over the full range of operational redundancies moreover the quantizer outperforms the state of the art md scheme as the redundancy among the description increases

design of entropy constrained multiple description scalar quantizers the problem of entropy constrained multiple description scalar quantizer design is posed as an optimization problem necessary conditions for optimality are derived and an iterative design algorithm is presented performance results are presented for a gaussian source along with comparisons to the multiple description rate distortion bound and a reference system

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

multiple description wavelet based image coding we consider the problem of coding images for transmission over error prone channels the impairments we target are transient channel shutdowns as would occur in a packet network when a packet is lost or in a wireless system during a deep fade when data is delivered it is assumed to be error free but some of the data may never reach the receiver the proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most efficient subband coders a given image is encoded into multiple independent packets of roughly equal length when packets are lost the quality of the approximation computed at the receiver depends only on the number of packets received but does not depend on exactly which packets are actually received when compared with previously reported results on the performance of robust image coders based on multiple descriptions on standard test images our coders attain similar psnr values using typically about 50 60 of the bit rate required by these other state of the art coders while at the same time providing significantly more freedom in the mechanism for allocation of redundancy among descriptions

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

multiuser successive refinement and multiple description coding in this correspondence we consider the multiuser successive refinement msr problem where the users are connected to a central server via links with different noiseless capacities and each user wishes to reconstruct in a successive refinement fashion an achievable region is given for the two user two layer case and it provides the complete rate distortion region for the gaussian source under the mse distortion measure the key observation is that this problem includes the multiple description md problem with two descriptions as a subsystem and the techniques useful in the md problem can be extended to this case it is shown that the coding scheme based on the universality of random binning is suboptimal because multiple gaussian side informations only at the decoders do incur performance loss in contrast to the case of single side information at the decoder it is further shown that unlike the single user case when there are multiple users the loss of performance by a multistage coding approach can be unbounded for the gaussian source the result suggests that in such a setting the benefit of using successive refinement is not likely to justify the accompanying performance loss the msr problem is also related to the source coding problem where each decoder has its individual side information while the encoder has the complete set of the side informations the msr problem further includes several variations of the md problem for which the specialization of the general result is investigated and the implication is discussed

asymptotic performance of multiple description transform codes the multiple description transform coder is introduced for sources with memory and an asymptotic analysis is presented for the squared error distortion for stationary gaussian sources the optimal transform and the optimal bit allocation for the multiple description coder are identical to those for the single description coder

multiple description coding by dithered delta sigma quantization we address the connection between the multiple description md problem and delta sigma quantization the inherent redundancy due to oversampling in delta sigma quantization and the simple linear additive noise model resulting from dithered lattice quantization allow us to construct a symmetric and time invariant md coding scheme we show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion asymptotically as the dimension of the lattice vector quantizer and order of the noise shaping filter approach infinity the entropy rate of the dithered delta sigma quantization scheme approaches the symmetric two channel md rate distortion function for a memoryless gaussian source and mean square error mse fidelity criterion at any side to central distortion ratio and any resolution in the optimal scheme the infinite order noise shaping filter must be minimum phase and have a piecewise flat power spectrum with a single jump discontinuity an important advantage of the proposed design is that it is symmetric in rate and distortion by construction so the coding rates of the descriptions are identical and there is therefore no need for source splitting

a numerical solution technique for a one dimensional inverse nonlinear parabolic problem abstract this study is intended to provide a new numerical algorithm for solving a one dimensional inverse nonlinear parabolic problem at the beginning of the study taylor s series expansion is employed to linearize nonlinear terms and then finite difference method is used to discretize the problem domain the present approach is to rearrange the matrix forms of the differential governing equations and estimate unknown functions the least squares method is adopted to find the solution it is assumed that no prior information is available on the functional form of the unknown coefficient in the present study thus it is classified as the function estimation results show that an excellent estimation on the unknown functions of the inverse problem which can be obtained within a couple of minutes cpu time at pentium iv 2 4 ghz pc

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

quantization as histogram segmentation optimal scalar quantizer design in network systems an algorithm for scalar quantizer design on discrete alphabet sources is proposed the proposed algorithm can be used to design fixed rate and entropy constrained conventional scalar quantizers multiresolution scalar quantizers multiple description scalar quantizers and wyner ziv scalar quantizers the algorithm guarantees globally optimal solutions for conventional fixed rate scalar quantizers and entropy constrained scalar quantizers for the other coding scenarios the algorithm yields the best code among all codes that meet a given convexity constraint in all cases the algorithm run time is polynomial in the size of the source alphabet the algorithm derivation arises from a demonstration of the connection between scalar quantization histogram segmentation and the shortest path problem in a certain directed acyclic graph

multiple description quantization via gram schmidt orthogonalization the multiple description md problem has received considerable attention as a model of information transmission over unreliable channels a general framework for designing efficient md quantization schemes is proposed in this paper we provide a systematic treatment of the el gamal cover egc achievable md rate distortion region and show it can be decomposed into a simplified egc segc region and a superimposed refinement operation furthermore any point in the segc region can be achieved via a successive quantization scheme along with quantization splitting for the quadratic gaussian case the proposed scheme has an intrinsic connection with the gram schmidt orthogonalization which implies that the whole gaussian md rate distortion region is achievable with a sequential dithered lattice based quantization scheme as the dimension of the optimal lattice quantizers becomes large moreover this scheme is shown to be universal for all independent and identically distributed i i d smooth sources with performance no worse than that for an i i d gaussian source with the same variance and asymptotically optimal at high resolution a class of md scalar quantizers in the proposed general framework is also constructed and is illustrated geometrically the performance is analyzed in the high resolution regime which exhibits a noticeable improvement over the existing md scalar quantization schemes

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

near sufficiency of random coding for two descriptions we give a single letter outer bound for the two descriptions problem for independent and identically distributed i i d sources that is universally close to the el gamal and cover egc inner bound the gaps for the sum and individual rates using a quadratic distortion measure are upper bounded by 1 5 and 0 5 bits sample respectively and are universal with respect to the source being encoded and the desired distortion levels variants of our basic ideas are presented including upper and lower bounds on the second channel s rate when the first channel s rate is arbitrarily close to the rate distortion function these bounds differ in the limit as the code block length goes to infinity by not more than 2 bits sample an interesting aspect of our methodology is the manner in which the matching single letter outer bound is obtained as we eschew common techniques for constructing single letter bounds in favor of new ideas in the field of rate loss bounds we expect these techniques to be generally applicable to other settings of interest

a numerical method for solving of a nonlinear inverse diffusion problem in this paper we will first study the existence and uniqueness of the solution for a one dimensional nonlinear inverse diffusion problem via an auxiliary problem and the schauder fixed point theorem furthermore a numerical algorithm based on the finite difference method and the least squares scheme for solving a nonlinear inverse problem is proposed at the beginning of the numerical algorithm taylor s series expansion is employed to linearize nonlinear terms and then the finite difference method is used to discretize the problem domain the present approach is to rearrange the matrix forms of the differential governing equations and estimate the unknown diffusion coefficient the least squares method is adopted to find the solution it is assumed that no prior information is available on the functional form of the unknown diffusion coefficient in the present study and thus it is classified as the function estimation in inverse calculation results show that an excellent estimation on the diffusion coefficient can be obtained within a couple of minutes of cpu time on a pentium iv 2 4ghz pc

unicity in an inverse problem for an unknown reaction term in a reaction diffusion equation in this paper we consider an inverse problem in which we seek to deter mine an unknown source or reaction term in a reaction diffusion equation from overspecified data measured on the boundary of the spatial region where the equation applies the analysis is based on the observation that the overspecified data depends monotonically on the unknown source term in the equation here we use this monotonicity to establish a type of unicity result for the inverse problem the attractiveness of the monotonicity methods illustrated here lies in their versatility surveying the rapidly expanding literature devoted to the topic of inverse problems one finds a considerable variety of ad hoc approaches methods which apply to a general class of problems seem to be few monotonicity methods however apply to a variety of inverse problems involving partial differential equations of parabolic or elliptic type this paper is organized as follows in section 1 we formulate the direct initial boundary value problem ibvp and state assumptions on the data and a priori assumptions on the unknown source term several properties of the solution to the direct problem are derived in section 2 the principal monotonicity estimates are established and in section 3 these are applied to the inverse problem to prove a type of unicity result for the solution to the inverse problem in particular it is a corollary of theorem 3 2 that in the class of analytic source terms the solution of the inverse problem is unique the problem of identifying an unknown source term in a heat equation 155

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

multiple description distribution preserving quantization the notion of a multiple description quantizer mdq includes providing multiple distortion levels if a human observer is involved the design of mdq requires a suitable distortion measure to achieve a graceful quality degradation in the case of description losses while the mean squared error is a ubiquitous distortion measure for many classic mdq schemes it is known to be perceptually relevant only at low distortions we propose a new mdq designed according to an unconventional distortion criterion that combines the mean squared error with a constraint on the probability distribution of the reconstructed signal the performance of the new mdq is shown to approach that of the classic mdq asymptotically as rate increases however once applied in the context of transform audio coding the new mdq significantly outperforms a classic mdq in perceptual tests the new scheme is suitable for a wide range of distortions and renders a seamless transition between coding that preserves signal features and coding of a waveform

asymptotic analysis of multiple description quantizers a high rate analysis of multiple description quantizers is presented for rth power distortions and general source densities both fixed length and variable length encoding of the quantizer indices are considered optimal companding functions are shown to be the same as for single channel quantizers as compared to the bound of ozarow 1980 a gap of 8 69 db and 3 07 db exists between the entropy constrained and level constrained cases respectively for a memoryless gaussian source and r 2

multiple description vector quantization with lattice codebooks design and analysis the problem of designing a multiple description vector quantizer with lattice codebook spl lambda is considered a general solution is given to a labeling problem which plays a crucial role in the design of such quantizers numerical performance results are obtained for quantizers based on the lattices a sub 2 and z sup i i 1 2 4 8 that make use of this labeling algorithm the high rate squared error distortions for this family of l dimensional vector quantizers are then analyzed for a memoryless source with probability density function pdf p and differential entropy h p spl infin for any a spl epsiv 0 1 and rate pair r r it is shown that the two channel distortion d sub o and the channel 1 or channel 2 distortion d sub s satisfy lim sub r spl rarr spl infin d sub o 2 sup 2r 1 a 1 4 g spl lambda 2 sup 2h p and lim sub r spl rarr spl infin d sub s 2 sup 2r 1 a g s sub l 2 sup 2h p where g spl lambda is the normalized second moment of a voronoi cell of the lattice spl lambda and g s sub l is the normalized second moment of a sphere in l dimensions

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

time decay of solutions to the cauchy problem for time dependent schrodinger hartree equations we consider the time dependent schrodinger hartree equation iu t delta u left frac 1 r u 2 right u lambda frac u r t x in mathbb r times mathbb r 3 1 u 0 x phi x in sigma 2 2 x in mathbb r 3 2 where 0 and sigma 2 2 g in l 2 parallel g parallel sigma 2 2 2 sum limits a leqq 2 parallel d a g parallel 2 2 sum limits beta leqq 2 parallel x beta g parallel 2 2 infty

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

time decay for some schrodinger equations on considere le probleme de cauchy pour l equation de schrodinger dans r n n n i t u u 2 v 1 u v 2 u 2 u t x r r n u o x x x r n ou v 1 v 1 x 1 x u 1 1 0 0 1 min 2 n 2 v 2 v 2 x k 2 3 k x u k k 0 0 k min 2 n on etudie les proprietes de decroissance temporelles des solutions

quasi solution approach for a two dimensional nonlinear inverse diffusion problem this paper is related to a class of inverse problems for identification of the coefficient in a square porous medium the unknown coefficient depends on the solution and belongs the class of admissible coefficients using continuous dependence of solutions on the coefficient convergence in the corresponding direct problems the existence of the quasi solution of the inverse problem is obtained in the appropriate class of admissible coefficients

on solutions of the initial value problem for the nonlinear schrodinger equations in one space dimension t 0 xe has a regularizing property that the solutions become smooth instantly for t 0 even if their initial data are not smooth for example in l n our aim of this paper is to show that the nls equation 1 1 1 2 has a similar regularizing effect more precisely all solutions of finite energy to 1 1 1 2 are smooth for tq 0 provided the initial functions in hi n or in

local smoothing and convergence properties of schrodinger type equations abstract the initial value problem for the schrodinger type equation u t ip d u u 0 x u 0 x x r n is considered here p is a principal type polynomial of order m i e p c 1 m 1 r or a certain type of other real symbol such as p m m 1 the following results are proved a if u 0 l 2 r n then for any fixed r 0 write u 0 u 1 u 2 where u 1 and u 2 are square integrable and the fourier transform of u 1 vanishes for r and that of u 2 vanishes for r if u 1 t x e itp d u 1 x u 2 t x e itp d u 2 x then u 1 is real analytic in r r n and satisfies for every multi index sup t x d u 1 t x c u 0 l 2 while 1 x 2 1 2 u 2 t x l 2 r t h m 1 2 r x n b if u 0 l 2 s r n s 0 then for every t 0 u t h loc m 1 s r n c if u 0 h s r n s 1 2 then for a e x r n u t x u 0 x as t 0

combinatorial message sharing and a new achievable region for multiple descriptions this paper presents a new achievable rate distortion region for the general l channel multiple descriptions mds problem a well known general region for this problem is due to venkataramani kramer and goyal vkg their encoding scheme is an extension of the el gamal cover ec and zhang berger zb coding schemes to the l channel case and includes a combinatorial number of refinement codebooks one for each subset of the descriptions as in zb the scheme also allows for a single common codeword to be shared by all descriptions this paper proposes a novel encoding technique involving combinatorial message sharing cms where every subset of the descriptions may share a distinct common message this introduces a combinatorial number of shared codebooks along with the refinement codebooks of these shared codebooks provide a more flexible framework to tradeoff redundancy across the messages for resilience to descriptions loss we derive an achievable rate distortion region for the proposed technique and show that it subsumes the vkg region for general sources and distortion measures we further show that cms provides a strict improvement of the achievable region for any source and distortion measures for which some two description subset is such that zb achieves points outside the ec region we then show a more surprising result cms outperforms vkg for a general class of sources and distortion measures including scenarios where the zb and ec regions coincide for all two description subsets in particular we show that cms strictly improves on vkg for the l channel quadratic gaussian md problem for all l geq 3 despite the fact that the ec region is complete for the corresponding two descriptions problem consequently the correlated quantization scheme an extreme special case of vkg that has been proven to be optimal for several cross sections of the l channel quadratic gaussian md problem is strictly suboptimal in general using the encoding principles derived we show that the cms scheme achieves the complete rate distortion region for several asymmetric cross sections of the l channel quadratic gaussian md problem

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

n channel symmetric multiple descriptions part i n k source channel erasure codes in this two part paper we present a new achievable rate region for the general n channel symmetric multiple descriptions problem in part i inspired by the concept of maximum distance separable mds erasure channel codes we consider a special case of this rate region where the source is encoded into n descriptions each with rate r these descriptions are transmitted over n bandwidth constrained and errorless channels during transmission a subset of these channels can break down thus erasing the corresponding descriptions the decoder is interested in recovering the source with the reception of at least k descriptions thus the encoder is allowed to sample only one realization of this breakdown process during the entire transmission for gaussian sources we have the following interesting result when any k descriptions arrive the achievable distortion exactly matches the optimal distortion rate performance corresponding to a source rate of kr bits with the reception of any m k descriptions the source reconstruction quality is strictly better the improvement being nearly linear in the number of descriptions received

multiple description coding for stationary gaussian sources we consider the problem of multiple description coding for stationary gaussian sources under the squared error distortion measure the rate region is characterized for the 2 description case it is shown that each supporting line of the rate region is achievable with a transform lattice quantization scheme we show the optimal coding scheme has a natural spectral domain coding interpretation which yields a reverse water filling solution with a frequency dependent water level instead of the flat water level as in the conventional single description case

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

existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem abstract the problem of identifying the coefficient in a square porous medium is considered it is shown that under certain conditions of data f g and for a properly specified class a of admissible coefficients there exists at least one a a such that a u is a solution of the corresponding inverse problem