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

information theoretic asymptotics of bayes methods in the absence of knowledge of the true density function bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior the authors examine the relative entropy distance d sub n between the true density and the bayesian density and show that the asymptotic distance is d 2 log n c where d is the dimension of the parameter vector therefore the relative entropy rate d sub n n converges to zero at rate log n n the constant c which the authors explicitly identify depends only on the prior density function and the fisher information matrix evaluated at the true parameter value consequences are given for density estimation universal data compression composite hypothesis testing and stock market portfolio selection

fine asymptotics for universal one to one compression of parametric sources universal source coding at short blocklengths is considered for an i i d exponential family of distributions the type size code has previously been shown to be optimal up to the third order rate for universal compression of all memoryless sources over finite alphabets the type size code assigns sequences ordered based on their type class sizes to binary strings ordered lexicographically to generalize this type class approach for parametric sources a natural scheme is to define two sequences to be in the same type class if and only if they are equiprobable under any model in the parametric class this natural approach however is shown to be suboptimal a variation of the type size code is introduced where type classes are defined based on neighborhoods of minimal sufficient statistics the asymptotics of the overflow rate of this variation are derived and a converse result establishes its optimality up to the third order term

optimal lossless data compression non asymptotics and asymptotics this paper provides an extensive study of the behavior of the best achievable rate and other related fundamental limits in variable length strictly lossless compression in the non asymptotic regime the fundamental limits of fixed to variable lossless compression with and without prefix constraints are shown to be tightly coupled several precise quantitative bounds are derived connecting the distribution of the optimal code lengths to the source information spectrum and an exact analysis of the best achievable rate for arbitrary sources is given fine asymptotic results are proved for arbitrary not necessarily prefix compressors on general mixing sources nonasymptotic explicit gaussian approximation bounds are established for the best achievable rate on markov sources the source dispersion and the source varentropy rate are defined and characterized together with the entropy rate the varentropy rate serves to tightly approximate the fundamental nonasymptotic limits of fixed to variable compression for all but very small block lengths

weak variable length source coding given a general source x x sup n sub n 1 sup spl infin source coding is characterized by a pair spl phi sub n spl psi sub n of encoder spl phi sub n and decoder spl psi sub n together with the probability of error spl epsi sub n spl equiv pr spl psi sub n spl phi sub n x sup n spl ne x sup n if the length of the encoder output spl phi sub n x sup n is fixed then it is called fixed length source coding while if the length of the encoder output spl phi sub n x sup n is variable then it is called variable length source coding usually in the context of fixed length source coding the probability of error spl epsi sub n is required to asymptotically vanish i e lim sub n spl rarr spl infin spl epsi sub n 0 whereas in the context of variable length source coding the probability of error spl epsi sub n is required to be exactly zero i e spl epsi sub n 0 spl forall n 1 2 in contrast to these we consider the problem of variable length source coding with asymptotically vanishing probability of error i e lim sub n spl rarr spl infin spl epsi sub n 0 and establish several fundamental theorems on this new subject

new bounds on the expected length of one to one codes we provide new bounds on the expected length l of a binary one to one code for a discrete random variable x with entropy h we prove that l spl ges h log h 1 hlog 1 1 h this bound improves on previous results furthermore we provide upper bounds on the expected length of the best code as function of h and the most likely source letter probability

guessing under source uncertainty this paper considers the problem of guessing the realization of a finite alphabet source when some side information is provided the only knowledge the guesser has about the source and the correlated side information is that the joint source is one among a family a notion of redundancy is first defined and a new divergence quantity that measures this redundancy is identified this divergence quantity shares the pythagorean property with the kullback leibler divergence good guessing strategies that minimize the supremum redundancy over the family are then identified the min sup value measures the richness of the uncertainty set the min sup redundancies for two examples the families of discrete memoryless sources and finite state arbitrarily varying sources are then determined

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

existence of infinitely many homoclinic orbits to aubry sets for positive definite lagrangian systems abstract in this paper we study the problem of homoclinic orbits to aubry sets for time periodic positive definite lagrangian systems we show that there are infinitely many homoclinic orbits to some aubry set under the conditions that the associated mather set is uniquely ergodic and the first relative homology group of the projection of this aubry set is nonzero

source coding using families of universal hash functions this correspondence is concerned with new connections between source coding and two kinds of families of hash functions known as the families of universal hash functions and n strongly universal hash functions where n ges 2 is an integer first it is pointed out that such families contain classes of well known source codes such as bin codes and linear codes next performance of a source coding scheme using either of the two kinds of families is evaluated an upper bound on the expectation of the decoding error probability is obtained for each family the expectation of the decoding error probability is analyzed in detail for the cases of discrete memoryless sources and sources without the memoryless assumption under a certain class of decoders

numerical study of the wrinkling of a stretched thin sheet a thin sheet clamped at opposite ends and stretched develops wrinkles parallel to the direction of the applied tensile strain due to the hindered poisson lateral contraction at the clamps to study this phenomenon a variational model recently proposed by puntel deseri and fried is adopted the relevant energy functional includes bending and membranal contributions and is minimized subject to a constraint on the area of the mid surface of the sheet a fourth order partial differential equation is henceforth obtained and numerically implemented using b splines predictions are obtained concerning the number of wrinkles critical applied stretches and scaling relationships for wrinkle amplitude and wavelength both a linearized version of the boundary value problem based on the small slope approximation and a fully nonlinear one are considered their results are found to be in good agreement for the whole range of applied stretches taken into account comparisons with previous analytical results by puntel deseri and fried who used different boundary conditions and an ansatz on the deflection function are also provided the numerical results substantially confirm the validity of the analytical predictions the present work provides then an alternative numerical method for the study of wrinkling in thin sheets and supports the use of analytical and semi analytical solutions as viable options for specific geometries though further investigation particularly experimental is still needed extensive comparisons of the results with other studies available in the literature provide confirmation for the scaling laws and signal that predicted values of the critical stretches may only be accurate for higher length to width aspect ratios

an algorithm for finite element analysis of partly wrinkled membranes a generalization and numerical implementation is presented of the stein hedgepeth 1961 continuum theory for the analysis of partly wrinkled membranes such as the tensioned membrane surfaces which are found in spacecraft structural components the approach has its basis in experimental observations showing that when wrinkles develop in a membrane parallel to the x direction the associated overall contraction in the y direction exceeds that predicted by the poisson s ratio effect

new bounds on the expected length of optimal one to one codes in this correspondence we consider one to one encodings for a discrete memoryless source which are one shot encodings associating a distinct codeword with each source symbol such encodings could be employed when only a single source symbol rather than a sequence of source symbols needs to be transmitted for example such a situation can arise when the last message must be acknowledged before the next message can be transmitted we consider two slightly different types of one to one encodings depending on whether the empty codeword is used or not and obtain lower and upper bounds on the expected length of optimal one to one codes we first give an extension of a known tight lower bound on the expected length of optimal one to one codes for the case that the the size of the source alphabet is finite and partial information about the source symbol probabilities is available as expected our lower bound is no less than the previously known lower bound obtained without side information about the source symbol probabilities we then consider the case that the source entropy is available and derive arbitrarily tight lower bounds on the expected length of optimal one to one codes we also derive arbitrarily tight lower bounds for the case that the source entropy and the probability of the most likely source symbol are available finally given that the probability of the most likely source symbol is available we obtain an upper bound on the expected length of optimal one to one codes our upper bound is tighter than the best upper bound known in the literature

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

tight lower bounds for optimum code length corresp a new lower bound for the mean code length of all one to one codes for a random variable with n outcomes is derived the bound which is tight improves an earlier one due to leung yan cheong and cover another bound for one to one codes for binary information sources is derived

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

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

guessing subject to distortion we investigate the problem of guessing a random vector x within distortion level d our aim is to characterize the best attainable performance in the sense of minimizing in some probabilistic sense the number of required guesses g x until the error falls below d the underlying motivation is that g x is the number of candidate codewords to be examined by a rate distortion block encoder until a satisfactory codeword is found in particular for memoryless sources we provide a single letter characterization of the least achievable exponential growth rate of the spl rho th moment of g x as the dimension of the random vector x grows without bound in this context we propose an asymptotically optimal guessing scheme that is universal both with respect to the information source and the value of spl rho we then study some properties of the exponent function e d spl rho along with its relation to the source coding exponents finally we provide extensions of our main results to the gaussian case guessing with side information and sources with memory

optimum overflow thresholds in variable length source coding allowing non vanishing error probability the variable length source coding problem allowing the error probability up to some constant is considered for general sources in this problem the optimum mean codeword length of variable length codes has already been determined on the other hand in this paper we focus on the overflow or excess codeword length probability instead of the mean codeword length the infimum of overflow thresholds under the constraint that both of the error probability and the overflow probability are smaller than or equal to some constant is called the optimum overflow threshold in this paper we first derive finite length upper and lower bounds on these probabilities so as to analyze the optimum overflow thresholds then by using these bounds we determine the general formula of the optimum overflow thresholds in both of the first order and second order forms next we consider another expression of the derived general formula so as to reveal the relationship with the optimum coding rate in the fixed length source coding problem finally we apply the general formula derived in this paper to the case of stationary memoryless sources

weak variable length slepian wolf coding with linked encoders for mixed sources coding problems for correlated information sources were first investigated by slepian and wolf they considered the data compression system called the sw system where two sequences emitted from correlated sources are separately encoded to codewords and sent to a single decoder which has to output the original sequence pairs with a small probability or error in this paper we investigate the coding problem of a modified sw system allowing two encoders to communicate with zero rate first we consider the fixed length coding and clarify that the admissible rate region for general sources is equal to that of the original sw system next we investigate the variable length coding having the asymptotically vanishing probability of error we clarify the admissible rate region for mixed sources characterized by two ergodic sources and show that this region is strictly wider than that for fixed length codes further we investigate the universal coding problem for memoryless sources in the system and show that the sw system with linked encoders has much more flexibility than the original sw system

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

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

asymptotics and non asymptotics for universal fixed to variable source coding universal fixed to variable lossless source coding for memoryless sources is studied in the finite blocklength and higher order asymptotic regimes optimal three term fixed error asymptotic expressions are derived for general fixed to variable codes and for prefix codes it is shown that the non prefix type size code in which codeword lengths are chosen in ascending order of type class size achieves the optimal third order term and outperforms classical two stage codes converse results are proved making use of a result on the distribution of the empirical entropy and laplace s approximation finally the fixed to variable coding problem without a prefix constraint is shown to be essentially the same as the universal guessing problem

a one to one code and its anti redundancy one to one codes are ldquoone shotrdquo codes that assign a distinct codeword to source symbols and are not necessarily prefix codes more generally uniquely decodable interestingly as wyner proved in 1972 for such codes the average code length can be smaller than the source entropy by how much we call this difference the anti redundancy various authors over the years have shown that the anti redundancy can be as big as minus the logarithm of the source entropy however to the best of our knowledge precise estimates do not exist in this note we consider a block code of length n generated for a binary memoryless source and prove that the average anti redundancy is 1 2 log2 n c f n o 1 where c is a constant and either f n 0 if log2 1 p p is irrational where p is the probability of generating a ldquo0rdquo or f n is a fluctuating function as the code length increases this relatively simple finding requires a combination of analytic tools such as precise evaluation of bernoulli sums the saddle point method and theory of distribution of sequences modulo 1

minimum guesswork with an unreliable oracle we study a guessing game where alice holds a discrete random variable x and bob tries to sequentially guess its value before the game begins bob can obtain side information about x by asking an oracle carole any binary question of his choosing carole s answer is however unreliable and is incorrect with probability epsilon we show that bob should always ask carole whether the index of x is odd or even with respect to a descending order of probabilities this question simultaneously minimizes all the guessing moments for any value of epsilon in particular this result settles a conjecture of burin and shayevitz we further consider a more general setup where bob can ask a multiple choice m ary question and then observe carole s answer through a noisy channel when the channel is completely symmetric i e when carole decides whether to lie regardless of bob s question and has no preference when she lies a similar question about the ordered index of x modulo m is optimal interestingly however the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels we provide supporting evidence for this difficulty by showing that a core property required in our proofs becomes np hard to test in the general m ary case we establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution which we prove to be np hard for m geq 3

a c0 discontinuous galerkin method for the stationary quasi geostrophic equations of the ocean abstract this work concerns the development of a finite element algorithm for the stationary quasi geostrophic equations to treat the large scale wind driven ocean circulation the algorithm is developed based on the streamfunction formulation involving fourth order gradients of the streamfunction here we examine the adaptation of a relatively inexpensive nonconforming method based on c 0 basis functions we develop the variational form of the method and establish consistency the method weakly enforces continuity of the normal flux across interelement boundaries and stabilization is achieved via nitsche s method explicit expression for the choice of the stabilization parameter is derived moreover the theoretical error estimate for the linear stommel munk model is performed numerical results from several benchmark problems on rectangular and curved domains are provided to demonstrate the accuracy and robustness of the method we also provide the mediterranean sea example to demonstrate the capability of the approach to the wind driven ocean circulation simulation

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

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

second order noiseless source coding theorems shannon s celebrated source coding theorem can be viewed as a one sided law of large numbers we formulate second order noiseless source coding theorems for the deviation of the codeword lengths from the entropy for a class of sources that includes markov chains we prove a one sided central limit theorem and a law of the iterated logarithm

feedback in the non asymptotic regime without feedback the backoff from capacity due to non asymptotic blocklength can be quite substantial for blocklengths and error probabilities of interest in many practical applications in this paper novel achievability bounds are used to demonstrate that in the non asymptotic regime the maximal achievable rate improves dramatically thanks to variable length coding and feedback for example for the binary symmetric channel with capacity 1 2 the blocklength required to achieve 90 of the capacity is smaller than 200 compared to at least 3100 for the best fixed blocklength code even with noiseless feedback virtually all the advantages of noiseless feedback are shown to be achievable even if the feedback link is used only to send a single signal informing the encoder to terminate the transmission stop feedback it is demonstrated that the non asymptotic behavior of the fundamental limit depends crucially on the particular model chosen for the end of packet control signal fixed blocklength codes and related questions concerning communicating with a guaranteed delay are discussed in which situation feedback is demonstrated to be almost useless even non asymptotically

variable length source dispersions differ under maximum and average error criteria variable length compression without prefix free constraints and with side information available at both encoder and decoder is considered instead of requiring the code to be error free we allow for it to have a non vanishing error probability we derive one shot bounds on the optimal average codeword length by proposing two new information quantities namely the conditional and unconditional varepsilon cutoff entropies using these one shot bounds we obtain the second order asymptotics of the problem under two different formalisms the average and maximum probabilities of error over the realization of the side information while the first order terms in the asymptotic expansions for both formalisms are identical we find that the source dispersion under the average error formalism is in most cases strictly smaller than its maximum error counterpart applications to a certain class of guessing problems previously studied by kuzuoka 2020 are also discussed

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

second and third order asymptotics of the continuous time poisson channel the paper derives the optimal second order coding rate for the continuous time poisson channel we also obtain bounds on the third order coding rate this is the first instance of a second order result for a continuous time channel the converse proof hinges on a novel construction of an output distribution induced by wyner s discretized channel and the construction of an appropriate epsilon net of the input probability simplex while the achievability proof follows the general program to prove the third order term for non singular discrete memoryless channels put forth by polyanskiy several non standard techniques such as new definitions and bounds on the probabilities of typical sets using logarithmic sobolev inequalities are employed to handle the continuous nature of the channel

spline based finite element method for the stationary quasi geostrophic equations on arbitrary shaped coastal boundaries abstract this work concerns a b spline based finite element algorithm for the stationary quasi geostrophic equations to treat the large scale wind driven ocean circulation on arbitrary shaped domains the algorithm models arbitrary shaped coastal boundaries on intra element or embedded boundaries dirichlet boundary conditions on the embedded boundaries are weakly imposed and stabilization is achieved via nitsche s method we employ a hierarchical local refinement approach to improve the geometrical representation of curved boundaries results from several benchmark problems on rectangular and curved domains are provided to demonstrate the accuracy and robustness of the method we also provide the mediterranean sea example that illustrates the effectiveness of the approach in the wind driven ocean circulation simulation

universal coding with minimum probability of codeword length overflow lossless block to variable length source coding is studied for finite state finite alphabet sources the aim is to minimize the probability that the normalized length of the codeword will exceed a given threshold b subject to the kraft inequality it is shown that the lempel ziv algorithm 1978 asymptotically attains the optimal performance in the sense just defined independently of the source and the value of b for the subclass of unifilar markov sources faster convergence to the asymptotic optimum performance can be accomplished by using the minimum description length universal code for this subclass it is demonstrated that these universal codes are also nearly optimal in the sense of minimizing buffer overflow probability and asymptotically optimal in a competitive sense

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

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

dynamics near an isolated m semi static homoclinic orbit abstract we show that there are infinitely many periodic orbits in any neighborhood of an isolated m semi static orbit homoclinic to an aubry set for time periodic positive lagrangian systems

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

second order asymptotics in fixed length source coding and intrinsic randomness there is a difference between the optimal rates of fixed length source coding and intrinsic randomness when we care about the second order asymptotics we prove this difference for general information sources and then investigate independent and identically distributed i i d random variables and markovian variables as examples the difference is demonstrated through an investigation of the second order asymptotic behavior of the rates a universal fixed length source code attaining the second order optimal rate is also proposed the difference between the rates of fixed length source coding and intrinsic randomness proves that the outputs of fixed length source codes are not uniformly distributed

differential equation for the amplitude of wrinkles introduction t he renewed interest in the phenomenonof wrinkling of membranes fueled by applications to the aerospace industry has spurred considerable activity on both the theoretical and computational aspectsof the subject althoughthe wrinklingphenomenonis known to be a case of shell buckling it seems unwise to utilize this knowledge in actual computations because treating the membrane as a shell with a vanishinglysmall thickness leads to the frustrating effects of numerical instability there exist on the other hand techniques based on themembrane theory alone that permit us to obtain with relative ease and numerical robustness not only the extent of thewrinkledzones but also the eldof directionsof thewrinkles 1i4 all that is needed therefore is a means for calculating the amplitude and wavelength of the wrinkles in an already known wrinkled domain in this note we derive an ordinary differential equation whose solution delivers such information for clarity of the presentation the equation is obtained under a number of simplifying assumptions mostly involving the smallness of the deformations however the method is straightforwardlyextendable to the general case our guiding principle has been to keep only those aspects of the model that are clearly determinant of the result while leaving all other aspects for future re nement we believe that this philosophy will serve to highlight the essential features rather than the technicalities of the proposedequation note that even under a geometrically andmaterially linear regime the resultingdifferentialequation is highly nonlinear the physical reason for this nonlinearityis to be foundin that becausewrinklingentailsno stress in thedirectionperpendicular to the wrinkles there exists a severe constraint between the wavelength and amplitude of the wrinkles this can be phrased intuitively as follows for a given amount of wrinkling strain the larger the number of wrinkles the smaller their amplitude to the best of our knowledge the only attempt in a direction similar to the one proposed herein is due to wong and pellegrino in their work these authors present a model basedmainly on the particular case of shearing of a rectangular panel their results are very valuable and will be used here as a benchmark for validation purposes

error estimates of b spline based finite element methods for the stationary quasi geostrophic equations of the ocean abstract this paper presents theoretical error estimates of b spline based finite element methods for the streamfunction formulation of the stationary quasi geostrophic equations which describe the large scale wind driven ocean circulation we introduce variational formulations of the streamfunction formulation inspired by the interior penalty discontinuous galerkin method dirichlet boundary conditions are weakly enforced in the formulations and stabilizations are achieved via nitsche s method existence and uniqueness of the approximation are proved and optimal error estimates in the energy norm are demonstrated under a small data assumption numerical experiments are performed to verify the theoretical error estimates on rectangular and l shape geometries

minimax redundancy for the class of memoryless sources let x sup n x sub 1 x sub n be a memoryless source with unknown distribution on a finite alphabet of size k we identify the asymptotic minimax coding redundancy for this class of sources and provide a sequence of asymptotically minimax codes equivalently we determine the limiting behavior of the minimax relative entropy min sub qxn max sub pxn d p sub xn spl par q sub xn where the maximum is over all independent and identically distributed i i d source distributions and the minimum is over all joint distributions we show in this paper that the minimax redundancy minus k 1 2 log n 2 spl pi e converges to log spl int spl radic det i spl theta d spl theta log spl gamma 1 2 sup k spl gamma k 2 where i spl theta is the fisher information and the integral is over the whole probability simplex the bayes strategy using jeffreys prior is shown to be asymptotically maximin but not asymptotically minimax in our setting the boundary risk using jeffreys prior is higher than that of interior points we provide a sequence of modifications of jeffreys prior that put some prior mass near the boundaries of the probability simplex to pull down that risk to the asymptotic minimax level in the limit

b spline based finite element method for the stationary quasi geostrophic equations of the ocean abstract we present a b spline based conforming finite element method of the streamfunction formulation of the stationary one layer quasi geostrophic equations for the study of the large scale wind driven ocean circulation the method encompasses standard simplifications of these equations in particular the linear stommel and stommel munk models a variational form of the method is developed and its consistency is established in this formulation dirichlet boundary conditions are enforced only weakly and stabilization is achieved via nitsche s method stability parameters are evaluated by solving a local generalized eigenvalue problem on the dirichlet boundary and by monitoring the condition number of the resulting linear systems comparisons of the results from our simulations with previously published results and convergence studies lead us to conclude that our finite element discretization is accurate

tension field theory a general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations the principal contribution is a partial differential equation describing a geometrical property of tension trajectories this is one of a system of two equations which describes the state of stress independently of the deformation this system is strongly elliptic at any stable solution whereas the deformation is described by a system of parabolic type controllable solutions i e those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone are obtained the general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples existing small strain theories are shown to correspond to a singular limit of the general theory at which the underlying system changes from elliptic to parabolic type

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

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

asymptotic properties on codeword lengths of an optimal fv code for general sources this correspondence is concerned with asymptotic properties on the codeword length of a fixed to variable length code fv code for a general source x sup n sub n 1 sup spl infin with a finite or countably infinite alphabet suppose that for each n spl ges 1 x sup n is encoded to a binary codeword spl phi sub n x sup n of length l spl phi sub n x sup n letting spl epsiv sub n denote the decoding error probability we consider the following two criteria on fv codes i spl epsiv sub n 0 for all n spl ges 1 and ii lim sup sub n spl rarr spl infin spl epsiv sub n spl les spl epsiv for an arbitrarily given spl epsiv spl isin 0 1 under criterion i we show that if x sup n is encoded by an arbitrary prefix free fv code asymptotically achieving the entropy 1 nl spl phi sub n x sup n 1 nlog sub 2 1 px sup n x sup n spl rarr 0 in probability as n spl rarr spl infin under a certain condition where p sub x sup n denotes the probability distribution of x sup n under criterion ii we first determine the minimum rate achieved by fv codes next we show that 1 nl spl phi sub n x sup n of an arbitrary fv code achieving the minimum rate in a certain sense has a property similar to the lossless case

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

on m semi static homoclinic orbits abstract we study m semi static homoclinic orbits to aubry sets in positive definite lagrangian systems

an achievable bound for optimal noiseless coding of a random variable corresp for a discrete n valued random variable n possibly denumerably infinite leung yan cheong and cover have given bounds for the minimal expected length of a one to one not necessarily uniquely decodable code l 1 1 sum i 1 n p i log left frac 1 2 1 right it is shown that the best possible case occurs for certain denumerably infinite sets of nonzero probabilities this absolute bound is related to the shannon entropy h of the distribution by h cdot is the binary entropy function