second order and moderate deviations asymptotics for successive refinement we derive the optimal second order coding region and moderate deviations constant for successive refinement source coding with a joint excess distortion probability constraint we consider two scenarios 1 a discrete memoryless source dms and arbitrary distortion measures at the decoders and 2 a gaussian memoryless source gms and quadratic distortion measures at the decoders for a dms with arbitrary distortion measures we prove an achievable second order coding region using type covering lemmas by kanlis and narayan and by no ingber and weissman we prove the converse using the perturbation approach by gu and effros when the dms is successively refinable the expressions for the second order coding region and the moderate deviations constant are simplified and easily computable for this case we also obtain new insights on the second order behavior compared with the scenario where separate excess distortion proabilities are considered for example we describe a dms for which the optimal second order region transitions from being characterizable by a bivariate gaussian to a univariate gaussian as the distortion levels are varied we then consider a gms with quadratic distortion measures to prove the direct part we make use of the sphere covering theorem by verger gaugry together with appropriately defined gaussian type classes to prove the converse we generalize kostina and verdu s one shot converse bound for point to point lossy source coding we remark that this proof is applicable to general successively refinable sources in the proofs of the moderate deviations results for both scenarios we follow a strategy similar to that for the second order asymptotics and use the moderate deviations principle

asymmetric evaluations of erasure and undetected error probabilities the problem of channel coding with the erasure option is revisited for discrete memoryless channels the interplay between the code rate the undetected and total error probabilities is characterized using the information spectrum method a sequence of codes of increasing blocklengths n is designed to illustrate this tradeoff furthermore for additive discrete memoryless channels with uniform input distribution we establish that our analysis is tight with respect to the ensemble average this is done by analyzing the ensemble performance in terms of a tradeoff between the code rate the undetected and total errors this tradeoff is parameterized by the threshold in a generalized likelihood ratio test two asymptotic regimes are studied first the code rate tends to the capacity of the channel at a rate slower than n 1 2 corresponding to the moderate deviations regime in this case both error probabilities decay subexponentially and asymmetrically the precise decay rates are characterized second the code rate tends to capacity at a rate of n 1 2 in this case the total error probability is asymptotically a positive constant while the undetected error probability decays as exp bn 1 2 for some b 0 the proof techniques involve the applications of a modified or shifted version of the gartner ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions

data compression with low distortion and finite blocklength this paper considers lossy source coding of n dimensional memoryless sources and shows an explicit approximation to the minimum source coding rate required to sustain the probability of exceeding distortion d no greater than epsilon which is simpler than known dispersion based approximations our approach takes inspiration in the celebrated classical result stating that the shannon lower bound to rate distortion function becomes tight in the limit d to 0 we formulate an abstract version of the shannon lower bound that recovers both the classical shannon lower bound and the rate distortion function itself as special cases likewise we show that a nonasymptotic version of the abstract shannon lower bound recovers all previously known nonasymptotic converses a necessary and sufficient condition for the shannon lower bound to be attained exactly is presented it is demonstrated that whenever that condition is met the rate dispersion function is given simply by the varentropy of the source remarkably all finite alphabet sources with balanced distortion measures satisfy that condition in the range of low distortions most continuous sources violate that condition still we show that lattice quantizers closely approach the nonasymptotic shannon lower bound provided that the source density is smooth enough and the distortion is low this implies that fine multidimensional lattice coverings are nearly optimal in the rate distortion sense even at finite n the achievability proof technique is based on a new bound on the output entropy of lattice quantizers in terms of the differential entropy of the source the lattice cell size and a smoothness parameter of the source density the technique avoids both the usual random coding argument and the simplifying assumption of the presence of a dither signal

neyman pearson test for zero rate multiterminal hypothesis testing the problem of zero rate multiterminal hypothesis testing is revisited from the perspective of information spectrum approach and finite blocklength analysis a neyman pearson like test is proposed and its non asymptotic performance is clarified for a short block length it is numerically determined that the proposed test is superior to the previously reported hoeffding like test proposed by han kobayashi for a large deviation regime it is shown that our proposed test achieves an optimal trade off between the type i and type ii exponents presented by han kobayashi among the class of symmetric type based testing schemes when the type i error probability is non vanishing the proposed test is optimal up to the second order term of the type ii error exponent the latter term is characterized in terms of the variance of the projected relative entropy density the information geometry method plays an important role in the analysis as well as the construction of the test

minimum expected length of fixed to variable lossless compression without prefix constraints the minimum expected length for fixed to variable length encoding of an n block memoryless source with entropy h grows as nh o 1 where the term o 1 lies between 0 and 1 however this well known performance is obtained under the implicit constraint that the code assigned to the whole n block is a prefix code dropping the prefix constraint which is rarely necessary at the block level we show that the minimum expected length for a finite alphabet memoryless source with known distribution grows as nh 1 2 log n o 1 unless the source is equiprobable we also refine this result up to o 1 for those memoryless sources whose log probabilities do not reside on a lattice

nonasymptotic noisy lossy source coding this paper shows new general nonasymptotic achievability and converse bounds and performs their dispersion analysis for the lossy compression problem in which the compressor observes the source through a noisy channel while this problem is asymptotically equivalent to a noiseless lossy source coding problem with a modified distortion function nonasymptotically there is a noticeable gap in how fast their minimum achievable coding rates approach the common rate distortion function as evidenced both by the refined asymptotic analysis dispersion and the numerical results the size of the gap between the dispersions of the noisy problem and the asymptotically equivalent noiseless problem depends on the stochastic variability of the channel through which the compressor observes the source

refined asymptotics for rate distortion using gaussian codebooks for arbitrary sources the rate distortion saddle point problem considered by lapidoth 1997 consists in finding the minimum rate to compress an arbitrary ergodic source when one is constrained to use a random gaussian codebook and minimum euclidean distance encoding is employed we extend lapidoth s analysis in several directions in this paper first we consider refined asymptotics in particular when the source is stationary and memoryless we establish the second order moderate and large deviation asymptotics of the problem second by random gaussian codebook lapidoth referred to a collection of random codewords each of which is drawn independently and uniformly from the surface of an n dimensional sphere to be more precise we term this as a spherical codebook we also consider i i d gaussian codebooks in which each random codeword is drawn independently from a product gaussian distribution we derive the second order moderate and large deviation asymptotics when i i d gaussian codebooks are employed in contrast to the recent work on the channel coding counterpart by scarlett tan and durisi 2017 the dispersions for spherical and i i d gaussian codebooks are identical the ensemble excess distortion exponents for both spherical and i i d gaussian codebooks are established for all rates furthermore we show that the i i d gaussian codebook has a strictly larger excess distortion exponent than its spherical counterpart for any rate greater than the ensemble rate distortion function derived by lapidoth

nonasymptotic and second order achievability bounds for coding with side information we present a novel nonasymptotic or finite blocklength achievability bounds for three side information problems in network information theory these include 1 the wyner ahlswede korner wak problem of almost lossless source coding with rate limited side information 2 the wyner ziv wz problem of lossy source coding with side information at the decoder and 3 the gel fand pinsker gp problem of channel coding with noncausal state information available at the encoder the bounds are proved using ideas from channel simulation and channel resolvability our bounds for all three problems improve on all previous nonasymptotic bounds on the error probability of the wak wz and gp problems in particular those derived by verdu using our novel nonasymptotic bounds we recover the general formulas for the optimal rates of these side information problems finally we also present achievable second order coding rates by applying the multidimensional berry esseen theorem to our new nonasymptotic bounds numerical results show that the second order coding rates obtained using our nonasymptotic achievability bounds are superior to those obtained using existing finite blocklength bounds

complete controllability and contractibility in multimodal systems abstract the concept of a strictly positive definite set of hermitian matrices is introduced it is shown that a strictly positive definite set is always a positive definite set and conditions are found under which a positive definite set is strictly positive definite we also show that a set of hermitian matrices is strictly positive definite if and only if some nonnegative linear combination of these matrices is a positive definite matrix for state dimension two we use this concept to find necessary and sufficient conditions for a two mode completely controllable irreducible multimodal system to be contractible relative to an elliptic norm for general state dimensions we give necessary and sufficient conditions for a special type two mode completely controllable irreducible system to be contractible relative to a weakly monotone norm applying the above results we show that for state dimension two there exists a completely controllable two mode system which is not contractible relative to either an elliptic or a weakly monotone norm we leave open the question whether or not complete controllability implies contractibility relative to some norm for multimodal systems of two or more modes

an error analysis of the finite element method overcoming corner singularities for the stationary stokes problem in this paper we introduce a mixed finite element method to overcome corner singularities of the stationary stokes system on a non convex polygon the reference choi and kweon 2013 says that the velocity field and the pressure function are composed of singular parts and regular remainders near a non convex vertex and furthermore the regular parts are sufficiently smooth on the polygon we use the corner singularity expansion and propose new extraction formulae for coefficients of singularities for the proposed numerical method we first try to find finite element solutions for the regular parts and then calculate approximations of the coefficients by the derived formulae expressed by the remainders and given functions we show error estimates of the approximations for the regular parts and coefficients and give some numerical experiments to confirm the efficiency and reliability of the proposed method

bayesian m ary hypothesis testing the meta converse and verdu han bounds are tight two alternative exact characterizations of the minimum error probability of bayesian m ary hypothesis testing are derived the first expression corresponds to the error probability of an induced binary hypothesis test and implies the tightness of the meta converse bound by polyanskiy et al the second expression is a function of an information spectrum measure and implies the tightness of a generalized verdu han lower bound the formulas characterize the minimum error probability of several problems in information theory and help to identify the steps where existing converse bounds are loose

successive refinement of abstract sources in successive refinement of information the decoder refines its representation of the source progressively as it receives more encoded bits the rate distortion region of successive refinement describes the minimum rates required to attain the target distortions at each decoding stage in this paper we derive a parametric characterization of the rate distortion region for successive refinement of abstract sources our characterization extends csiszar s result to successive refinement and generalizes a result by tuncel and rose applicable for finite alphabet sources to abstract sources this characterization spawns a family of outer bounds to the rate distortion region it also enables an iterative algorithm for computing the rate distortion region which generalizes blahut s algorithm to successive refinement finally it leads a new nonasymptotic converse bound in all the scenarios where the dispersion is known this bound is second order optimal in our proof technique we avoid karush kuhn tucker conditions of optimality and we use basic tools of probability theory we leverage the donsker varadhan lemma for the minimization of relative entropy on abstract probability spaces

the ziv zakai renyi bound for joint source channel coding shannon s capacity and rate distortion function combined with the separation principle provide tight bounds for the minimum possible distortion in joint source channel coding these bounds however are usually achievable only in the limit of a large block length in their 1973 paper ziv and zakai introduced a family of alternative capacity and rate distortion functions based on functionals satisfying the data processing inequality which potentially give tighter bounds for systems with a small block length there is a considerable freedom as to how to choose those functionals and the ways of finding the best possible functionals yielding the best bounds for a given source channel combination are not specified we examine recently conjectured high snr asymptotic expressions for the ziv zakai bounds based on the renyi divergence functional we derive nonasymptotic bounds on the ziv zakai renyi rate distortion function and capacity for a broad class of sources and additive noise channels which hold for arbitrary snr and prove the conjectured asymptotic expressions in the limit of a small distortion high snr the results lead to new bounds on the best achievable distortion in finite dimensional joint source channel coding examples are presented where the new bounds achieve significant improvement upon shannon s original bounds

a posteriori error estimates for the stokes problem an a posteriors error estimate is derived and analyzed for the mini element discretization of the stokes equations the estimate is based on the solution of a local stokes problem in each element of the finite element mesh using spaces of quadratic bump functions for both velocity and pressure errors this results in solving a 9 times 9 system which reduces to two easily invertible 3 times 3 systems comparisons with other estimates based on a petrov galerkin solution are used in this analysis which shows that it provides a reasonable approximation of the actual discretization error numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures

existence and uniqueness of steady solutions to the magnetohydrodynamic equations of compressible flows abstract we establish the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary under the assumption that the external force field is small

for the stationary compressible viscous navier stokes equations with no slip condition on a convex polygon abstract our concern is with existence and regularity of the stationary compressible viscous navier stokes equations with no slip condition on convex polygonal domains note that u p 0 c c a constant is the eigenpair for the singular value 1 of the stokes problem on the convex sector it is shown that except the pair 0 c the leading order of the corner singularities for the nonlinear equations is the same as that of the stokes problem we split the leading corner singularity from the solution and show an increased regularity for the remainder as a consequence the pressure solution changes the sign at the convex corner and its derivatives blow up

approaching the rate distortion limit with spatial coupling belief propagation and decimation we investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled low density generator matrix codes the degree of the check nodes is regular and the one of code bits is poisson distributed with an average depending on the compression rate the performance of a low complexity belief propagation guided decimation algorithm is excellent the algorithmic rate distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows moreover as the check degree grows both curves approach the ultimate shannon rate distortion limit the belief propagation guided decimation encoder is based on the posterior measure of a binary symmetric test channel this measure can be interpreted as a random gibbs measure at a temperature directly related to the noise level of the test channel we investigate the links between the algorithmic performance of the belief propagation guided decimation encoder and the phase diagram of this gibbs measure the phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds in particular the dynamical and condensation phase transition temperatures equivalently test channel noise thresholds are computed we observe that 1 the dynamical temperature of the spatially coupled construction saturates toward the condensation temperature and 2 for large degrees the condensation temperature approaches the temperature i e noise level related to the information theoretic shannon test channel noise parameter of rate distortion theory this provides heuristic insight into the excellent performance of the belief propagation guided decimation algorithm this paper contains an introduction to the cavity method

singularities of a compressible stokes system in a domain with concave edge in r 3 we study a compressible stokes system in a three dimensional domain with a concave edge we subtract an edge singularity from the velocity and show an increased regularity for remainder after the singular function has been subtracted we give a formula for the edge singularity in terms of an edge flux intensity function using the transport character of the continuity equation we show a propagation of the edge singularity into the domain

a review on some contributions to perturbation theory singular limits and well posedness abstract in the beginning of the 1990s we devoted a sequence of papers to perturbation theory singular limits and well posedness problems in particular the strong well posedness of the initial boundary value problem for the compressible euler equations was demonstrate for the first time our method also allowed singular limit results in the strong norm even under assumptions weaker than the current ones in the literature where the strong norm is not reached it is worth noting that until now the above method and results have not been substantially improved hence an introduction to it still looks timely actually in a forthcoming paper by returning to this method we improve in a very substantial way some important results recently appeared in the literature

existence of solutions of stationary compressible navier stokes equations with large force abstract in this paper we consider the navier stokes equations for isentropic compressible flows of a polytropic gas in a bounded domain the equations to be considered are obtained by scaling to dimensionless form and then replacing the density by e 2 where e is a mach number the existence of solutions has been known only for small forces or large potential forces near a rest state the aim of this paper is to give a proof of the existence of stationary compressible navier stokes equations with large force when the mach number e is small

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

strong converses in source coding relative to a fidelity criterion two strong converses are obtained for an abstract alphabet stationary ergodic source coded relative to an appropriate fidelity criterion it is shown that given a distortion rate point d r that lies below the rate distortion curve 1 block codes that operate at rate level r must encode sample source blocks at a rate exceeding d with probability tending to one as the block length tends to infinity and 2 variable rate codes that operate at distortion level d must encode sample source blocks at a rate exceeding r with probability tending to one as the block length tends to infinity the previously known weak converses guarantee only that the indicated probabilities remain bounded away from zero as block length tends to infinity the proofs of the strong converses involve sample converses in source coding theory

viscous flow in domains with corners numerical artifacts their origin and removal abstract viscous flows in domains with boundaries forming two dimensional corners are considered we examine the case where on each side of the corner the boundary condition for the tangential velocity is formulated in terms of stress it is shown that computing such flows numerically by straightforwardly applying well tested algorithms and numerical codes based on their use such as comsol multiphysics can lead to spurious multivaluedness and mesh dependence in the distribution of the fluid s pressure the origin of this difficulty is that near a corner formed by smooth parts of the boundary in addition to the solution of the formulated inhomogeneous problem there also exists an eigensolution for obtuse corner angles this eigensolution a becomes dominant and b has a singular radial derivative of velocity at the corner despite the bulk pressure in the eigensolution being constant when the derivatives of the velocity are singular numerical errors in the velocities calculation near the corner give rise to pressure spikes whose magnitude increases as the mesh is refined a method is developed that uses the knowledge about the eigensolution to remove the artifacts in the pressure distribution the method is first explained in the simple case of a stokes flow in a corner region and then generalized for the navier stokes equations applied to describe steady and unsteady free surface flows encountered in problems of dynamic wetting

lossy compression via sparse linear regression computationally efficient encoding and decoding we propose computationally efficient encoders and decoders for lossy compression using a sparse regression code the codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix the proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence it is shown to achieve the optimal distortion rate function for independent identically distributed i i d gaussian sources under the squared error distortion criterion for a given rate the parameters of the design matrix can be varied to tradeoff distortion performance with encoding complexity an example of such a tradeoff as a function of the block length n is the following with computational resource space or time per source sample of o n log n 2 for a fixed distortion level above the gaussian distortion rate function the probability of excess distortion decays exponentially in n the sparse regression code is robust in the following sense for any ergodic source the proposed encoder achieves the optimal distortion rate function of an i i d gaussian source with the same variance simulations show that the encoder has good empirical performance especially at low and moderate rates

rateless lossy compression via the extremes we begin by presenting a simple lossy compressor operating at near zero rate the encoder merely describes the indices of the few maximal source components while the decoder s reconstruction is a natural estimate of the source components based on this information this scheme turns out to be near optimal for the memoryless gaussian source in the sense of achieving the zero rate slope of its distortion rate function motivated by this finding we then propose a scheme comprised of iterating the above lossy compressor on an appropriately transformed version of the difference between the source and its reconstruction from the previous iteration the proposed scheme achieves the rate distortion function of the gaussian memoryless source under squared error distortion when employed on any finite variance ergodic source it further possesses desirable properties and we respectively refer to as infinitesimal successive refinability ratelessness and complete separability its storage and computation requirements are of order no more than n 2 log beta n per source symbol for beta 0 at both the encoder and the decoder though the details of its derivation construction and analysis differ considerably we discuss similarities between the proposed scheme and the recently introduced sparse regression codes of venkataramanan et al

pointwise redundancy in lossy data compression and universal lossy data compression we characterize the achievable pointwise redundancy rates for lossy data compression at a fixed distortion level pointwise redundancy refers to the difference between the description length achieved by an nth order block code and the optimal nr d bits for memoryless sources we show that the best achievable redundancy rate is of order o spl radic n in probability this follows from a second order refinement to the classical source coding theorem in the form of a one sided central limit theorem moreover we show that along almost any source realization the description lengths of any sequence of block codes operating at distortion level d exceed nr d by at least as much as c spl radic nloglogn infinitely often corresponding direct coding theorems are also given showing that these rates are essentially achievable the above rates are in sharp contrast with the expected redundancy rates of order o log n reported by various authors our approach is based on showing that the compression performance of an arbitrary sequence of codes is essentially bounded below by the performance of shannon s random code we obtain partial generalizations of the above results for arbitrary sources with memory and we prove lossy analogs of barron s lemma barron 1985

navier stokes equations for compressible fluids global existence and qualitative properties of the solutions in the general case we consider the equations which describe the motion of a viscous compressible fluid taking into consideration the case of inflow and or outflow through the boundary by means of some a priori estimates we prove the existence of a global in time solution moreover as a consequence of a stability result we show that there exist a periodic solution and a stationary solution

lossy joint source channel coding in the finite blocklength regime this paper finds new tight finite blocklength bounds for the best achievable lossy joint source channel code rate and demonstrates that joint source channel code design brings considerable performance advantage over a separate one in the nonasymptotic regime a joint source channel code maps a block of k source symbols onto a length n channel codeword and the fidelity of reproduction at the receiver end is measured by the probability e that the distortion exceeds a given threshold d for memoryless sources and channels it is demonstrated that the parameters of the best joint source channel code must satisfy nc kr d nv k v d q 1 e where c and v are the channel capacity and channel dispersion respectively r d and v d are the source rate distortion and rate dispersion functions and q is the standard gaussian complementary cumulative distribution function symbol by symbol uncoded transmission is known to achieve the shannon limit when the source and channel satisfy a certain probabilistic matching condition in this paper we show that even when this condition is not satisfied symbol by symbol transmission is in some cases the best known strategy in the nonasymptotic regime

moderate deviations in channel coding we consider block codes whose rate converges to the channel capacity with increasing blocklength at a certain speed and examine the best possible decay of the probability of error for discrete memoryless channels we prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes

analysis of a pressure stabilized finite element approximation of the stationary navier stokes equations the purpose of this paper is to analyze a finite element approximation of the stationary navier stokes equations that allows the use of equal velocity pressure interpolation the idea is to introduce as unknown of the discrete problem the projection of the pressure gradient multiplied by suitable algorithmic parameters onto the space of continuous vector fields the difference between these two vectors pressure gradient and projection is introduced in the continuity equation the resulting formulation is shown to be stable and optimally convergent both in a norm associated to the problem and in the l 2 norm for both velocities and pressure this is proved first for the stokes problem and then it is extended to the nonlinear case all the analysis relies on an inf sup condition that is much weaker than for the standard galerkin approximation in spite of the fact that the present method is only a minor modification of this

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

discrete lossy gray wyner revisited second order asymptotics large and moderate deviations in this paper we revisit the discrete lossy gray wyner problem in particular we derive its optimal second order coding rate region its error exponent reliability function and its moderate deviations constant under mild conditions on the source to obtain the second order asymptotics we extend some ideas from watanabe s work in particular we leverage the properties of an appropriate generalization of the conditional distortion tilted information density which was first introduced by kostina and verdu the converse part uses a perturbation argument by gu and effros in their strong converse proof of the discrete gray wyner problem the achievability part uses two novel elements 1 a generalization of various type covering lemmas and 2 the uniform continuity of the conditional rate distortion function in both the source joint distribution and the distortion level to obtain the error exponent for the achievability part we use the same generalized type covering lemma and for the converse we use the strong converse together with a change of measure technique finally to obtain the moderate deviations constant we apply the moderate deviations theorem to probabilities defined in terms of information spectrum quantities

variable length compression allowing errors this paper studies the fundamental limits of the minimum average length of lossless and lossy variable length compression allowing a nonzero error probability epsilon for lossless compression we give nonasymptotic bounds on the minimum average length in terms of erokhin s rate distortion function and we use those bounds to obtain a gaussian approximation on the speed of approach to the limit which is quite accurate for all but small blocklengths 1 epsilon k h mathsf s k v mathsf s 2 pi 1 2 mathop rm exp q 1 left epsilon right 2 2 where q 1 left cdot right is the functional inverse of the standard gaussian complementary cumulative distribution function and v mathsf s is the source dispersion a nonzero error probability thus not only reduces the asymptotically achievable rate by a factor of 1 epsilon but this asymptotic limit is approached from below i e larger source dispersions and shorter blocklengths are beneficial variable length lossy compression under an excess distortion constraint is shown to exhibit similar properties

a finite element method for singular solutions of the navier stokes equations on a non convex polygon it is shown in choi and kweon 2013 that a solution of the navier stokes equations with no slip boundary condition on a non convex polygon can be written as u p c 1 1 1 c 2 2 2 u r p r near each non convex vertex where u r p r h 2 h 1 i i are corner singularity functions for the stokes problem with no slip condition and c i r are coefficients which are called the stress intensity factors we design a finite element method to approximate the coefficients c i and the regular part u r p r show the unique existence of the approximations and derive their error estimates some numerical examples are given confirming convergence rates for the approximations

existence uniqueness and asymptotics of steady jets we consider the three dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition we discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity moreover we show that each solution with a bounded dirichlet integral which has a certain weak additional decay behaves like o r 2 as r x and admits a wide jet region these investigations are based on the solvability properties of the linear stokes system in a half space 3 to investigate the stokes problem in 3 we apply the mellin transform technique and reduce the stokes problem to the determination of the spectrum of the corresponding invariant stokes beltrami operator on the hemisphere

strong successive refinability and rate distortion complexity tradeoff we investigate the second order asymptotics source dispersion of the successive refinement problem similar to the classical definition of a successively refinable source we say that a source is strongly successively refinable if successive refinement coding can achieve the second order optimum rate including the dispersion terms at both decoders we establish a sufficient condition for strong successive refinability we show that any discrete source under hamming distortion and the gaussian source under quadratic distortion are strongly successively refinable we also demonstrate how successive refinement ideas can be used in point to point lossy compression problems in order to reduce complexity we give two examples the binary hamming and gaussian quadratic cases in which a layered code construction results in a low complexity scheme that attains optimal performance for example when the number of layers grows with the block length n we show how to design an o n log n algorithm that asymptotically achieves the rate distortion bound

the rate distortion function and excess distortion exponent of sparse regression codes with optimal encoding this paper studies the performance of sparse regression codes for lossy compression with the squared error distortion criterion in a sparse regression code code words are linear combinations of subsets of columns of a design matrix it is shown that with minimum distance encoding sparse regression codes achieve the shannon rate distortion function for i i d gaussian sources r d as well as the optimal excess distortion exponent this completes a previous result which showed that r d and the optimal exponent were achievable for distortions below a certain threshold the proof of the rate distortion result is based on the second moment method a popular technique to show that a non negative random variable x is strictly positive with high probability in our context x is the number of code words within target distortion d of the source sequence we first identify the reason behind the failure of the standard second moment method for certain distortions and illustrate the different failure modes via a stylized example we then use a refinement of the second moment method to show that r d is achievable for all distortion values finally the refinement technique is applied to suen s correlation inequality to prove the achievability of the optimal gaussian excess distortion exponent

a lower bound analysis of hamming distortion for a binary ceo problem with joint source channel coding a two node binary chief executive officer ceo problem is investigated noise corrupted versions of a binary sequence are forwarded by two nodes to a single destination node over orthogonal additive white gaussian noise awgn channels we first reduce the binary ceo problem to a binary multiterminal source coding problem of which an outer bound for the rate distortion region is derived the distortion function is then established by evaluating the relationship between the binary ceo and multiterminal source coding problems a lower bound approximation on the hamming distortion hd is obtained by minimizing a distortion function subject to constraints obtained based on the source channel separation theorem encoding decoding algorithms using concatenated convolutional codes and a joint decoding scheme are used to verify the lower bound on the hd it is found that the theoretical lower bounds on the hd and the computer simulation based bit error rate performance curves have the same tendencies the differences in the threshold signal to noise ratio between the theoretical lower bounds and those obtained by simulations are around 1 5 db in awgn channel the theoretical lower bound on the hd in block rayleigh fading channel is also evaluated by performing monte carlo simulation

the evolution compressible navier stokes system on polygonal domains abstract the evolution compressible navier stokes system is considered on polygonal domains it is shown that the lowest order of the corner singularity of the system is the same as that of the heat equation in a suitable banach space the velocity is split into singular and regular parts and the coefficient of the singularity is expressed by convolution of some two functions in the time variable by a formula of the pressure we observe propagation of the corner singularity along the characteristic lines emanating from the corners and also unboundedness of derivatives of pressure there an increased regularity for the remainder part is established

smoothing brascamp lieb inequalities and strong converses of coding theorems the brascamp lieb inequality in functional analysis can be viewed as a measure of the uncorrelatedness of a joint probability distribution we define the smooth brascamp lieb bl divergence as the infimum of the best constant in the brascamp lieb inequality under a perturbation of the joint probability distribution an information spectrum upper bound on the smooth bl divergence is proved using properties of the subgradient of a certain convex functional in particular in the i i d setting such an infimum converges to the best constant in a certain mutual information inequality we then derive new single shot converse bounds for the omniscient helper common randomness generation problem and the gray wyner source coding problem in terms of the smooth bl divergence where the proof relies on the functional formulation of the brascamp lieb inequality exact second order rates are thus obtained in the stationary memoryless and nonvanishing error setting these offer rare instances of strong converses second order converses for continuous sources when the rate region involves auxiliary random variables

lp approach to steady flows of viscous compressible fluids in exterior domains we investigate steady compressible flows in three dimensional exterior domains for small data and for both zero and nonzero but constant velocity at infinity we prove existence and uniqueness of solutions in l p spaces p 3 and study their regularity as well as their decay at infinity

one shot lossy quantum data compression we provide a framework for one shot quantum rate distortion coding in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level we obtain a one shot characterization of the minimum qubit compression size for an entanglement assisted quantum rate distortion code in terms of the smooth max information a quantity previously employed in the one shot quantum reverse shannon theorem next we show how this characterization converges to the known expression for the entanglement assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source finally we give a tight finite blocklength characterization for the entanglement assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbolwise distortion constraint

regularity of solutions for the navier stokes system of incompressible flows on a polygon abstract we are concerned with singularities and regularities of solutions for the navier stokes system of incompressible flows on a polygonal domain with a concave vertex we subtract the corner singularities by the stokes operator from the solution velocity and pressure functions of the system it is shown that the stress intensity factors are functions of time variable belong to a fractional sobolev space on the time interval and can be expressed in terms of given data an increased regularity for the remainder is obtained

on the dispersions of three network information theory problems we analyze the dispersions of distributed lossless source coding the slepian wolf problem the multiple access channel and the asymmetric broadcast channel for the two encoder slepian wolf problem we introduce a quantity known as the entropy dispersion matrix which is analogous to the scalar dispersions that have gained interest recently we prove a global dispersion result that can be expressed in terms of this entropy dispersion matrix and provides intuition on the approximate rate losses at a given blocklength and error probability to gain better intuition about the rate at which the nonasymptotic rate region converges to the slepian wolf boundary we define and characterize two operational dispersions 1 the local dispersion and 2 the weighted sum rate dispersion the former represents the rate of convergence to a point on the slepian wolf boundary whereas the latter represents the fastest rate for which a weighted sum of the two rates converges to its asymptotic fundamental limit interestingly when we approach either of the two corner points the local dispersion is characterized not by a univariate gaussian but a bivariate one as well as a subset of off diagonal elements of the aforementioned entropy dispersion matrix finally we demonstrate the versatility of our achievability proof technique by providing inner bounds for the multiple access channel and the asymmetric broadcast channel in terms of dispersion matrices all our proofs are unified by a so called vector rate redundancy theorem which is proved using the multidimensional berry esseen theorem

supercloseness and superconvergence of stabilized low order finite element discretizations of the stokes problem the supercloseness and superconvergence properties of stabilized finite element methods applied to the stokes problem are studied we consider consistent residual based stabilization methods as well as inconsistent local projection type stabilizations moreover we are able to show the supercloseness of the linear part of the mini element solution which has been previously observed in practical computations the results on supercloseness hold on three directional triangular axiparallel rectangular and brick type meshes respectively but extensions to more general meshes are also discussed applying an appropriate postprocess to the computed solution we establish superconvergence results numerical examples illustrate the theoretical predictions

second order region for gray wyner network the coding problem over the gray wyner network is studied from the second order coding rates perspective a tilted information density for this network is introduced in the spirit of kostina verdu and under a certain regularity condition the second order region is characterized in terms of the variance of this tilted information density and the tangent vector of the first order region the second order region is proved by the type method the achievability part is proved by the type covering argument and the converse part is proved by a refinement of the perturbation approach that was used by gu effros to show the strong converse of the gray wyner network this is the first instance that the second order region is characterized for a multi terminal problem where the characterization of the first order region involves an auxiliary random variable

solution methods for the poisson equation with corner singularities numerical results in z cai and s kim siam j numer anal 39 2001 pp 286 299 we developed a new finite element method using singular functions for the poisson equation on a polygonal domain with re entrant angles such a method first computes the regular part of the solution then the stress intensity factor and finally the solution itself this paper studies solution methods for solving the system of linear equations arising from the discretization and focuses on numerical results including the finite element accuracy and the multigrid performance

equivocations exponents and second order coding rates under various renyi information measures we evaluate the asymptotics of equivocations their exponents as well as their second order coding rates under various renyi information measures specifically we consider the effect of applying a hash function on a source and we quantify the level of non uniformity and dependence of the compressed source from another correlated source when the number of copies of the sources is large unlike previous works that use shannon information measures to quantify randomness information or uniformity we define our security measures in terms of a more general class of information measures the renyi information measures and their gallager type counterparts a special case of these renyi information measure is the class of shannon information measures we prove tight asymptotic results for the security measures and their exponential rates of decay we also prove bounds on the second order asymptotics and show that these bounds match when the magnitudes of the second order coding rates are large we do so by establishing new classes non asymptotic bounds on the equivocation and evaluating these bounds using various probabilistic limit theorems asymptotically

on the free boundary problem for stationary compressible navier stokes equations we consider the equations which describe a stationary motion of a viscous compressible barotropic fluid in a bounded domain in 3 with a free boundary determined by the surface tension by means of some a priori estimates we prove the existence of rotationally symmetric solutions in reality with some additional symmetry for a sufficiently small external force and in the case of rotationally symmetric force and domain where also we need more symmetry respectively

fixed length lossy compression in the finite blocklength regime this paper studies the minimum achievable source coding rate as a function of blocklength n and probability that the distortion exceeds a given level d tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength for stationary memoryless sources with separable distortion the minimum rate achievable is shown to be closely approximated by r d v d n q 1 where r d is the rate distortion function v d is the rate dispersion a characteristic of the source which measures its stochastic variability and q 1 is the inverse of the standard gaussian complementary cumulative distribution function

optimal zero delay coding of markov sources stationary and finite memory codes the optimal zero delay coding of a finite state markov source is considered the existence and structure of optimal codes are studied using a stochastic control formulation prior results in the literature established the optimality of deterministic markov walrand varaiya type coding policies for the finite time horizon problem and the optimality of both deterministic nonstationary and randomized stationary policies for the infinite time horizon problem our main result here shows that for any irreducible and aperiodic markov source with a finite alphabet deterministic and stationary markov coding policies are optimal for the infinite horizon problem in addition the finite block length time horizon performance of an optimal stationary and markov coding policy is shown to approach the infinite time horizon optimum at a rate o 1 t the results are extended to systems where zero delay communication takes place across a noisy channel with noiseless feedback

the compressible euler equations in a bounded domain existence of solutions and the incompressible limit a short time existence theorem is proven for the euler equations for nonisentropic compressible fluid flow in a bounded domain and solutions with low mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

penalty finite element method for the navier stokes equations abstract we present an analysis of a penalty formulation of the stationary navier stokes equations for an incompressible fluid subject to restrictions on the viscosity and prescribed body force it is shown that there exists a unique solution to this penalty problem the solution to the penalty problem is shown to converge to the solution of the navier stokes problem as o e where e 0 is the penalty parameter existence uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized navier stokes problem presented here numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases

linear programming based converses for finite blocklength lossy joint source channel coding a linear programming lp based framework is presented for obtaining converses for finite blocklength lossy joint source channel coding problems the framework applies for any loss criterion generalizes certain previously known converses and also extends to multi terminal settings the finite blocklength problem is posed equivalently as a nonconvex optimization problem and using a lift and project like method a close but tractable lp relaxation of this problem is derived lower bounds on the original problem are obtained by the construction of feasible points for the dual of the lp relaxation a particular application of this approach leads to new converses which recover and improve on the converses of kostina and verdu for finite blocklength lossy joint source channel coding and lossy source coding for finite blocklength channel coding the lp relaxation recovers the converse of polyanskiy poor and verdu and leads to a new improvement on the converse of wolfowitz showing thereby that our lp relaxation is asymptotically tight with increasing blocklengths for channel coding lossless source coding and joint source channel coding with the excess distortion probability as the loss criterion using a duality based argument a new converse is derived for finite blocklength joint source channel coding for a class of source channel pairs employing this converse the lp relaxation is also shown to be tight for all blocklengths for the minimization of the expected average symbolwise hamming distortion of a q ary uniform source over a q ary symmetric memoryless channel for any q in mathbb n the optimization formulation and the lift and project method are extended to networked settings and demonstrated by obtaining an improvement on a converse of zhou et al for the successive refinement problem for successively refinable source distortion measure triplets

the structure of the controllable set for multimodal systems this paper deals with the structure of the controllable set of a multimodal system we define a maximal component of the controllable set and we investigate the controllable set as the union of its maximal components we show that for each positive integer k state dimension n 3 and control dimension m n 1 there is a multimodal system whose controllable set s l is the union of exactly k maximal subspaces of rn and this system has k as bound on the number of iterations necessary to reach any state in s l from zero we also show the above holds with k we show that for each state dimension n and each control dimension m there is a completely controllable multimodal system having bound 2n 2m

the dispersion of nearest neighbor decoding for additive non gaussian channels we study the second order asymptotics of information transmission using random gaussian codebooks and nearest neighbor decoding over a power limited stationary memoryless additive non gaussian noise channel we show that the dispersion term depends on the non gaussian noise only through its second and fourth moments thus complementing the capacity result lapidoth 1996 which depends only on the second moment furthermore we characterize the second order asymptotics of point to point codes over k sender interference networks with non gaussian additive noise specifically we assume that each user s codebook is gaussian and that nn decoding is employed i e that interference from the k 1 unintended users gaussian interfering signals is treated as noise at each decoder we show that while the first order term in the asymptotic expansion of the maximum number of messages depends on the power of the interfering codewords only through their sum this does not hold for the second order term

non asymptotic converse bounds and refined asymptotics for two source coding problems in this paper we revisit two multi terminal lossy source coding problems the lossy source coding problem with side information available at the encoder and one of the two decoders which we term as the kaspi problem kaspi 1994 and the multiple description coding problem with one semi deterministic distortion measure which we refer to as the fu yeung problem fu and yeung 2002 for the kaspi problem we first present the properties of optimal test channels subsequently we generalize the notion of the distortion tilted information density for the lossy source coding problem to the kaspi problem and prove a non asymptotic converse bound using the properties of optimal test channels and the well defined distortion tilted information density finally for discrete memoryless sources we derive refined asymptotics which includes the second order large and moderate deviations asymptotics in the converse proof of second order asymptotics we apply the berry esseen theorem to the derived non asymptotic converse bound the achievability proof follows by first proving a type covering lemma tailored to the kaspi problem then properly taylor expanding the well defined distortion tilted information densities and finally applying the berry esseen theorem we then generalize the methods used in the kaspi problem to the fu yeung problem as a result we obtain the properties of optimal test channels for the minimum sum rate function a non asymptotic converse bound and refined asymptotics for discrete memoryless sources since the successive refinement problem is a special case of the fu yeung problem as a by product we obtain a non asymptotic converse bound for the successive refinement problem which is a strict generalization of the non asymptotic converse bound for successively refinable sources zhou tan and motani 2017

on the redundancy of lossy source coding with abstract alphabets the redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered for coding at a fixed rate level it is shown that for any fixed rate r 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that the distortion redundancy of c sub n defined as the difference between the performance of c sub n and the distortion rate function d p r of p is upper bounded by spl part d p r spl part r ln n 2n o ln n n for coding at a fixed distortion level it is demonstrated that for any d 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the fixed distortion d such that the rate redundancy of c sub n defined as the difference between the performance of c sub n and the rate distortion function r p d of p is upper bounded by 7ln n 6n o ln n n these results strengthen the traditional berger s 1968 1971 abstract alphabet source coding theorem and extend the positive redundancy results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 and ibid vol 42 p 803 21 1996 on lossy source coding with finite alphabets and the redundancy result of wyner see ibid vol 43 p 1452 64 1997 on block coding of memoryless gaussian sources

channels with cost constraints strong converse and dispersion this paper shows the strong converse and the dispersion of memoryless channels with cost constraints and performs a refined analysis of the third order term in the asymptotic expansion of the maximum achievable channel coding rate showing that it is equal to 1 2 log n n in most cases of interest the analysis is based on a nonasymptotic converse bound expressed in terms of the distribution of a random variable termed the mathsf b tilted information density which plays a role similar to that of the mathsf d tilted information in lossy source coding we also analyze the fundamental limits of lossy joint source channel coding over channels with cost constraints

on finite block length quantization distortion we investigate the upper and lower bounds on the quantization distortions for independent and identically distributed continuous sources in the finite block length regime we derive a lower bound on the quantization distortion as well as an upper bound on the quantization distortion based on random quantization codebooks moreover we apply the obtained bounds to the continuous gaussian source for the gaussian source we propose a computationally tractable method to numerically compute the upper and lower bounds for both bounded and unbounded quantization codebooks numerical results show that the gap between the upper and lower bounds is small for the block length of several hundreds and the upper and lower bounds are tighter than those proposed in the existing works

sample converses in source coding theory the rate and distortion performance of a sequence of codes along a sample sequence of symbols generated by a stationary ergodic information source are studied two results are obtained 1 the source sample sequence is encoded by an arbitrary sequence of block codes which operate at a fixed rate level r and a sample converse is obtained which states that with probability one the lower limit of the code sample distortions is lower bounded by d r the value of the distortion rate function at r 2 the source sample sequence is encoded by an arbitrary sequence of variable rate codes which operate at a fixed distortion level d and a sample converse is obtained which states that with probability one the lower limit of the code sample rates is lower bounded by r d the value of the rate distortion function at d a novel ergodic theorem is used to obtain both sample converses

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

on the stationary transport equations we find solutions in l1 and l for the general domains for certain transport equations arising in the theory of the compressible navier stokes equations some a priori estimates in l1 and l have been found then the asymptotic behavior of l solutions is found we also consider solutions in h1 and bmo we show the uniqueness of the solutions in these four function spaces

an l p theory for the n dimensional stationary compressible navier stokes equations and the incompressible limit for compressible fluids the equilibrium solutions in this paper we study the system 1 1 1 3 which describes the stationary motion of a given amount of a compressible heat conducting viscous fluid in a bounded domain ofr n n 2 hereu x is the velocity field x is the density of the fluid x is the absolute temperature f x andh x are the assigned external force field and heat sources per unit mass andp is the pressure in the physically significant case one hasg 0 we prove that for small data f g h there exists a unique solution u of problem 1 1 1 3 1 in a neighborhood of 0 m 0 for arbitrarily large data the stationary solution does not exist in general see sect 5 moreover we prove that for barotropic flows the stationary solution of the navier stokes equations 1 8 is the incompressible limit of the stationary solutions of the compressible navier stokes equations 1 7 as the mach number becomes small finally in sect 5 we will study the equilibrium solutions for system 4 1 for a more detailed explanation see the introduction

critical behavior in lossy source coding the following critical phenomenon was recently discovered when a memoryless source is compressed using a variable length fixed distortion code the fastest convergence rate of the pointwise compression ratio to r d is either o spl radic n or o log n we show it is always o spl radic n except for discrete uniformly distributed sources

smooth solution of the compressible navier stokes equations in an unbounded domain with inflow boundary condition abstract the barotropic compressible navier stokes equations in an unbounded domain are studied we prove the unique existence of the solution u p of the system 1 1 in the sobolev space h k 3 h k 2 provided that the derivatives of the data of the problem are sufficiently small where k 0 is any integer the proof follows from an analysis of the linearized problem the solvability of the continuity equation and the schauder fixed point theory similar smoothness results are obtained for a linearized form of 1 1

steady flows of compressible fluids in a rigid container with upper free boundary we consider fluid in a smooth rigid container whose lateral boundary is a piece of vertical cylinder bounded above by a free upper surface as basic flow we consider the non homogeneous rest state in the presence of gravity and of a surface tension under these assumptions we study the existence of a steady free boundary and a steady motion in of an isothermal viscous gas resulting as perturbation to the rest state in correspondence of small non potential perturbations to the large potential gravitational force we linearize the problem by prescribing the unknown domain then we make use of the iterative scheme introduced by heywood and padula our method is based on an iteration between the neumann problem for a non homogeneous stokes system for the velocity the neumann problem for an elliptic problem on for height and a steady transport equation for the perturbation to the density the difference of boundary condition between lateral boundary and free upper surfaces causes a singularity at the intersection contact line to avoid singularities on the contact line we adopt weighted sobolev spaces

joint source channel coding with feedback this paper quantifies the fundamental limits of variable length transmission of a general possibly analog source over a memoryless channel with noiseless feedback under a distortion constraint we consider excess distortion average distortion and guaranteed distortion d semifaithful codes in contrast to the asymptotic fundamental limit a general conclusion is that allowing variable length codes and feedback leads to a sizable improvement in the fundamental delay distortion tradeoff in addition we investigate the minimum energy required to reproduce k source samples with a given fidelity after transmission over a memoryless gaussian channel and we show that the required minimum energy is reduced with feedback and an average rather than maximal power constraint

stability of stationary solution for the compressible viscous magnetohydrodynamic equations with large potential force in bounded domain abstract in this paper we consider the 3 d compressible viscous magnetohydrodynamic mhd equations with some large potential force in bounded rigid vessel we firstly construct the non constant stationary solutions of the compressible viscous mhd equations under suitable constitutive assumptions next a critical energy identity is established to achieve a universal stability criterion of the stationary solution in this case the stationary solution is exponential stable for any large external potential force finally we show the well posedness of the initial boundary value problem for the compressible viscous mhd equations with the large potential force provided that the prescribed initial data is close to the stationary solution it implies that the set satisfying the stability criterion is not empty

the redundancy of source coding with a fidelity criterion 1 known statistics the problem of redundancy of source coding with respect to a fidelity criterion is considered for any fixed rate r 0 and any memoryless source with finite source and reproduction alphabets and a common distribution p the nth order distortion redundancy d sub n r of fixed rate coding is defined as the minimum of the difference between the expected distortion per symbol of any block code with length n and rate r and the distortion rate function d p r of the source p it is demonstrated that for sufficiently large n d sub n r is equal to spl part spl part r d p r ln n 2n o ln n n where spl part spl part r d p r is the partial derivative of d p r evaluated at r and assumed to exist for any fixed distortion level d 0 and any memoryless source p the nth order rate redundancy r sub n d of coding at fixed distortion level d or by using d semifaithful codes is defined as the minimum of the difference between the expected rate per symbol of any d semifaithful code of length n and the rate distortion function r p d of p evaluated at d it is proved that for sufficiently large n r sub n d is upper bounded by ln n n o ln n n and lower bounded by in n 2n o in n n as a by product the lower bound of r sub n d derived in this paper gives a positive answer to a conjecture proposed by yu and speed 1993

compressible stokes problem on nonconvex polygonal domains abstract the compressible stokes system with the inflow boundary condition is studied on the polygonal domain d it is shown that the lowest order corner singularity of the system is the same as that of the laplacian the velocity u is split into a singular and regular part near each concave vertex if the polygon is convex it is shown that u h 2 d 2

a case where interference does not affect the channel dispersion in 1975 carleial presented a special case of an interference channel called the very strong interference regime in which the interference does not reduce the capacity of the constituent point to point gaussian channels in this paper we show that in the strictly very strong interference regime the dispersions are similarly unaffected more precisely in this paper we characterize the second order coding rates of the gaussian interference channel in the strictly very strong interference regime in other words we characterize the speed of convergence of rates of optimal block codes toward a boundary point of the rectangular capacity region these second order coding rates are expressed in terms of the average probability of error and variances of appropriately defined information densities which coincide with the dispersion of the single user gaussian channel this allows us to conclude that the dispersions are unaffected by interference in this channel model