on the universality of burnashev s error exponent we consider communication over a time invariant discrete memoryless channel dmc with noiseless and instantaneous feedback we assume that the transmitter and the receiver are not aware of the underlying channel however they know that it belongs to some specific family of dmcs recent results show that for certain families e g binary symmetric channels and z channels there exist coding schemes that universally achieve any rate below capacity while attaining burnashev s error exponent we show that this is not the case in general by deriving an upper bound to the universally achievable error exponent

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

worst case expected capacity loss of slow fading channels for delay limited communication over block fading channels the difference between the ergodic capacity and the maximum achievable expected rate for coding over a finite number of coherent blocks represents a fundamental measure of the penalty incurred by the delay constraint this paper introduces a notion of worst case expected capacity loss focusing on the slow fading scenario one block delay the worst case additive and multiplicative expected capacity losses are precisely characterized for the point to point fading channel extension to the problem of writing on fading paper is also considered where both the ergodic capacity and the additive expected capacity loss over one block delay are characterized to within one bit per channel use

common message broadcast channels with feedback in the nonasymptotic regime full feedback we investigate the maximum coding rate for a given average blocklength and error probability over a k user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable length stop feedback codes for the point to point case polyanskiy et al 2011 demonstrated that variable length coding combined with stop feedback significantly increases the speed of convergence of the maximum coding rate to capacity this speed up manifests itself in the absence of a square root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths i e zero dispersion in this paper we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common message k user discrete memoryless broadcast channel which strengthen and generalize the ones reported in trillingsgaard et al 2015 for the two user case an asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common message broadcast channels e g the binary symmetric broadcast channel furthermore we identify conditions under which our converse and achievability bounds are tight up to the second order through numerical evaluations we illustrate that our second order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point to point case

variable length coding over an unknown channel burnashev in 1976 gave an exact expression for the reliability function of a discrete memoryless channel dmc with noiseless feedback a coding scheme that achieves this exponent needs in general to know the statistics of the channel suppose now that the coding scheme is designed knowing only that the channel belongs to a family q of dmcs is there a coding scheme with noiseless feedback that achieves burnashev s exponent uniformly over q at a nontrivial rate we answer the question in the affirmative for two families of channels binary symmetric and z for these families we show that for any given fraction there is a feedback coding strategy such that for any member of the family i guarantees this fraction of its capacity as rate and ii guarantees the corresponding burnashev s exponent therefore for these families in terms of delay and error probability the knowledge of the channel becomes asymptotically irrelevant in feedback code design there are blind schemes that perform as well as the best coding scheme designed with the foreknowledge of the channel under use however a converse result shows that in general even for families that consist of only two channels such blind schemes do not exist

income distribution an adaptive heterogeneous model abstract in this communication an adaptive process is introduced into a many agent model for closed economic system in order to establish general features of income distribution in this new version agents are able to modify their exchange parameter i of resources through an adaptive process the conclusions indicate that assuming an instantaneous learning behavior of all agents a distribution for income is reproduced while a frozen behavior establishes a pareto s distribution for income with an exponent 0 94 0 02 a third case occurs when a heterogeneous inertia behavior is introduced leading us to a distribution at the low income regime and a power law decay for the large income values with an exponent 2 05 0 05 this method enables investigation of the resources flux in the economic environment and produces also bounding values for the gini index comparable with data evidences

a simple converse of burnashev s reliability function in a remarkable paper published in 1976 burnashev determined the reliability function of variable length block codes over discrete memoryless channels dmcs with feedback subsequently an alternative achievability proof was obtained by yamamoto and itoh via a particularly simple and instructive scheme their idea is to alternate between a communication and a confirmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct we provide a converse that parallels the yamamoto itoh achievability construction besides being simpler than the original the proposed converse suggests that a communication and a confirmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent the proposed converse also makes it intuitively clear why the terms that appear in burnashev s exponent are necessary

recurrent multiple repetition coding for channels with feedback we consider multiple repetition strategies with fixed delay decoding for discrete memoryless channels with noiseless feedback existing binary schemes by schalkwijk and zigangirov are analyzed and their results are extended the general error exponents are computed and presented by elegant expressions in the strictly symmetric case an important class of precoded sequences so called flip sequences is found and their degrading effect on the error exponent is investigated this effect is shown negligible when the repetition parameters are chosen such that the transmission rate is maximized even when signalling at channel capacity the error exponent is shown to be strictly positive

vanishing theorems for f harmonic forms on smooth metric measure spaces abstract in this paper we first establish a monotonicity formula for vector bundle valued f harmonic p forms on a smooth metric measure space provided f r is less that an explicit constant as applications we get some vanishing theorems for l 2 f harmonic forms on concrete geometric models in the second part for a metric measure space with nonnegative bakry emery ricci curvature and with moderate volume growth we prove that any bounded f harmonic 1 form must be parallel moreover some vanishing theorems under nonnegative m bakry emery ricci curvature assumption are also proved finally we consider smooth metric measure spaces with weighted poincare inequality and show some vanishing theorems for l q f harmonic 1 forms

on a 145 32 binary cyclic code we discovered that a 145 32 binary cyclic code generated by the product of an irreducible polynomial of degree 28 with exponent 145 and the polynomial x sup 4 x sup 3 x sup 2 x 1 has a minimum distance of 44

poisson equation on complete manifolds abstract we develop heat kernel and green s function estimates for manifolds with positive bottom spectrum the results are then used to establish existence and sharp estimates of the solution to the poisson equation on such manifolds with ricci curvature bounded below as an application we show that the curvature of a steady gradient ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above

complete manifolds with positive spectrum ii in this paper we continued our investigation of complete manifolds whose spectrum of the laplacian has an optimal positive lower bound in particular we proved a splitting type theorem for n dimensional manifolds that have a finite volume end this can be viewed as a study of the equality case of a theorem of cheng

study of a market model with conservative exchanges on complex networks abstract many models of market dynamics make use of the idea of conservative wealth exchanges among economic agents a few years ago an exchange model using extremal dynamics was developed and a very interesting result was obtained a self generated minimum wealth or poverty line on the other hand the wealth distribution exhibited an exponential shape as a function of the square of the wealth these results have been obtained both considering exchanges between nearest neighbors or in a mean field scheme in the present paper we study the effect of distributing the agents on a complex network we have considered archetypical complex networks erdos renyi random networks and scale free networks the presence of a poverty line with finite wealth is preserved but spatial correlations are important particularly between the degree of the node and the wealth we present a detailed study of the correlations as well as the changes in the gini coefficient that measures the inequality as a function of the type and average degree of the considered networks

optimal feedback communication via posterior matching in this paper we introduce a fundamental principle for optimal communication over general memoryless channels in the presence of noiseless feedback termed posterior matching using this principle we devise a simple sequential generic feedback transmission scheme suitable for a large class of memoryless channels and input distributions achieving any rate below the corresponding mutual information this provides a unified framework for optimal feedback communication in which the horstein scheme bsc and the schalkwijk kailath scheme awgn channel are special cases thus as a corollary we prove that the horstein scheme indeed attains the bsc capacity settling a longstanding conjecture we further provide closed form expressions for the error probability of the scheme over a range of rates and derive the achievable rates in a mismatch setting where the scheme is designed according to the wrong channel model several illustrative examples of the posterior matching scheme for specific channels are given and the corresponding error probability expressions are evaluated the proof techniques employed utilize novel relations between information rates and contraction properties of iterated function systems

the error exponent of variable length codes over markov channels with feedback the error exponent of markov channels with feedback is studied in the variable length block coding setting burnashev s classic result is extended to finite state ergodic markov channels for these channels a single letter characterization of the reliability function is presented under the assumption of full causal output feedback and full causal observation of the channel state both at the transmitter and at the receiver side tools from stochastic control theory are used in order to treat channels with intersymbol interference isi specifically the convex analytic approach to markov decision processes is adopted in order to handle problems with stopping time horizons induced by variable length coding schemes

distortion minimization in gaussian layered broadcast coding with successive refinement a transmitter without channel state information wishes to send a delay limited gaussian source over a slowly fading channel the source is coded in superimposed layers with each layer successively refining the description in the previous one the receiver decodes the layers that are supported by the channel realization and reconstructs the source up to a distortion the expected distortion is minimized by optimally allocating the transmit power among the source layers for two source layers the allocation is optimal when power is first assigned to the higher layer up to a power ceiling that depends only on the channel fading distribution all remaining power if any is allocated to the lower layer for convex distortion cost functions with convex constraints the minimization is formulated as a convex optimization problem in the limit of a continuum of infinite layers the minimum expected distortion is given by the solution to a set of linear differential equations in terms of the density of the fading distribution as the number of channel uses per source symbol tends to zero the power distribution that minimizes expected distortion converges to the one that maximizes expected capacity

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

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

moderate deviation asymptotics for variable length codes with feedback we consider data transmission across discrete memoryless channels dmcs using variable length codes with feedback we consider the family of such codes whose rates are rho n below the channel capacity c where rho n is a positive sequence that tends to zero slower than the reciprocal of the square root of the expectation of the random blocklength n this is known as the moderate deviations regime and we establish the optimal moderate deviations constant we show that in this scenario the error probability decays sub exponentially with speed exp b c n rho n where b is the maximum relative entropy between output distributions of the dmc

communication over individual channels a communication problem in considered where no mathematical model is specified for the channel the achievable rates are determined as a function of the channel input and output sequences known a posteriori without assuming any a priori relation between them for discrete channels the empirical mutual information between the input and output sequences is shown to be achievable while for continuous channels the achievable rate is based on the empirical correlation between the sequences a rate adaptive scheme employing feedback which achieves these rates asymptotically with a guaranteed reliability without prior knowledge of the channel behavior is presented

feedback does not increase the capacity of discrete channels with additive noise we consider discrete time finite alphabet channels with additive random noise we show that output feedback does not increase the capacity of such channels this result holds in the most general case i e for arbitrary additive noise processes

geometry of manifolds with densities abstract we study the geometry of complete riemannian manifolds endowed with a weighted measure where the weight function is of quadratic growth assuming the associated bakry emery curvature is bounded from below we derive a new laplacian comparison theorem and establish various sharp volume upper and lower bounds we also obtain some splitting type results by analyzing the busemann functions in particular we show that a complete manifold with nonnegative bakry emery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions while some of our results are new even for the gradient ricci solitons the novelty here is that only a lower bound of the bakry emery curvature is involved in our analysis

training sequence optimization in mimo systems with colored interference we address the problems of channel estimation and optimal training sequence design for multiple input multiple output systems over flat fading channels in the presence of colored interference in practice knowledge of the unknown channel is often obtained by sending known training symbols to the receiver during the training period we obtain the best linear unbiased estimates of the channel parameters based on the received training block we determine the optimal training sequence set that minimizes the mean square error of the channel estimator under a total transmit power constraint in order to obtain the advantage of the optimal training sequence design long term statistics of the interference correlation are needed at the transmitter hence this information needs to be estimated at the receiver and fed back to the transmitter obviously it is desirable that only a minimal amount of information needs to be fed back from the receiver to gain the advantage in reducing the estimation error of the short term channel fading parameters we develop such a feedback strategy in this paper

error exponents for variable length block codes with feedback and cost constraints variable length block coding schemes are investigated for discrete memoryless channels with ideal feedback under cost constraints upper and lower bounds are found for the minimum achievable probability of decoding error pe min as a function of constraints r p and tau on the transmission rate average cost and average block length respectively for given r and p the lower and upper bounds to the exponent ln pe min tau are asymptotically equal as tau rarr infin the resulting reliability function limtaurarrinfin in pe min tau as a function of r and v is concave in the pair r p and generalizes the linear reliability function of burnashev to include cost constraints the results are generalized to a class of discrete time memoryless channels with arbitrary alphabets including additive gaussian noise channels with amplitude and power constraints

inequalities of wealth distribution in a conservative economy abstract we analyze a conservative market model for the competition among economic agents in a society with conserved total wealth a minimum dynamics ensures that the poorest agent has a chance to improve its economic welfare after a transient the system self organizes into a critical state where the wealth distribution have a minimum threshold with almost no agent below this poverty line also very few extremely rich agents are stable in time above the poverty line the distribution follows an exponential behavior the local solution exhibits a low gini index while the mean field solution of the model generates a wealth distribution similar to welfare states like sweden

elliptic gradient estimates for a weighted heat equation and applications we obtain two elliptic gradient estimates for positive solutions to the f heat equation on a complete smooth metric measure space with only bakry emery ricci tensor bounded below one is a local sharp souplet zhang s type and the other is a global hamilton s type as applications we prove parabolic liouville theorems for ancient solutions satisfying some growth restriction near infinity in particular the liouville results are suitable for the gradient shrinking or steady ricci solitons the estimates of derivation of the f heat kernel are also obtained

variable rate channel capacity this paper introduces the notions of variable to fixed and fixed to variable channel capacity without feedback for channels that satisfy the strong converse these notions coincide with the conventional shannon capacity for channels that do not behave ergodically the conventional fixed rate shannon capacity only depends on least favorable channel conditions while the variable rate capacity notions are able to capture the whole range of channel states and their likelihood even in the absence of any side information about channel state at the transmitter particular emphasis is placed on memoryless channels that are governed by finitely valued states we show that single user variable to fixed channel capacity is intimately connected to the capacity region of broadcast channels with degraded message sets and we give an expression for the fixed to variable capacity

rateless codes for avc models the arbitrarily varying channel avc is a channel model whose state is selected maliciously by an adversary fixed blocklength coding assumes a worst case bound on the adversary s capabilities which leads to pessimistic results this paper defines a variable length perspective on this problem for which achievable rates are shown that depend on the realized actions of the adversary specifically rateless codes are constructed which require a limited amount of common randomness these codes are constructed for two kinds of avc models in the first the channel state cannot depend on the channel input and in the second it can as a by product the randomized coding capacity of the avc with state depending on the transmitted codeword is found and shown to be achievable with a small amount of common randomness the results for this model are proved using a randomized strategy based on list decoding

minimax universal decoding with an erasure option motivated by applications of rateless coding decision feedback and automatic repeat request arq we study the problem of universal decoding for unknown channels in the presence of an erasure option specifically we harness the competitive minimax methodology developed in earlier studies in order to derive a universal version of forney s classical erasure list decoder which in the erasure case optimally trades off between the probability of erasure and the probability of undetected error the proposed universal erasure decoder guarantees universal achievability of a certain fraction xi of the optimum error exponents of these probabilities in a sense to be made precise in the sequel a single letter expression for xi which depends solely on the coding rate and the neyman pearson threshold to be defined is provided the example of the binary symmetric channel is studied in full detail and some conclusions are drawn

opportunistic capacity and error exponent region for the compound channel with feedback variable length communication over a compound channel with feedback is considered traditionally capacity of a compound channel without feedback is defined as the maximum rate that is determined before the start of communication such that communication is reliable this traditional definition is pessimistic in the presence of feedback an opportunistic definition is given capacity is defined as the maximum rate that is determined at the end of communication such that communication is reliable thus the transmission rate can adapt to the realized channel under this definition feedback communication over a compound channel is conceptually similar to multiterminal communication transmission rate is a vector rather than a scalar channel capacity is a region rather than a scalar error exponent is a region rather than a scalar in this paper variable length communication over a compound channel with feedback is formulated its opportunistic capacity region is characterized and lower bounds for its error exponent region are provided

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

erasure list random coding error exponents are not universally achievable we study the problem of universal decoding for unknown discrete memoryless channels in the presence of erasure list option at the decoder in the random coding regime in particular we harness a universal version of forney s classical erasure list decoder developed in earlier studies which is based on the competitive minimax methodology and guarantees universal achievability of a certain fraction of the optimum random coding error exponents in this paper we derive an exact single letter expression for the maximum achievable fraction examples are given in which the maximal achievable fraction is strictly less than unity which imply that in general there is no universal erasure list decoder which achieves the same random coding error exponents as the optimal decoder for a known channel this is in contrast to the situation in ordinary decoding without the erasure list option where optimum exponents are universally achievable as is well known it is also demonstrated that previous lower bounds derived for the maximal achievable fraction are not tight in general we then analyze a generalized random coding ensemble which incorporate a training sequence in conjunction with a suboptimal practical decoder plug in decoder which first estimates the channel using the available training sequence and then decodes the remaining symbols of the codeword using the estimated channel one of the implications of our results is setting the stage for a reasonable criterion of optimal training finally we compare the performance of the plug in decoder and the universal decoder in terms of the achievable error exponents and show that the latter is noticeably better than the former

optimistic shannon coding theorems for arbitrary single user systems the conventional definitions of the source coding rate and of channel capacity require the existence of reliable codes for all sufficiently large block lengths alternatively if it is required that good codes exist for infinitely many block lengths then optimistic definitions of source coding rate and channel capacity are obtained in this work formulas for the optimistic minimum achievable fixed length source coding rate and the minimum spl epsi achievable source coding rate for arbitrary finite alphabet sources are established the expressions for the optimistic capacity and the optimistic spl epsi capacity of arbitrary single user channels are also provided the expressions of the optimistic source coding rate and capacity are examined for the class of information stable sources and channels respectively finally examples for the computation of optimistic capacity are presented

on complete gradient shrinking ricci solitons in this paper we derive optimal growth estimates on the potential functions of complete noncompact shrinking solitons based on this we prove that a complete noncompact gradient shrinking ricci soliton has at most euclidean volume growth this latter result can be viewed as an analog of the well known volume comparison theorem of bishop that a complete noncompact riemannian manifold with nonnegative ricci curvature has at most euclidean volume growth

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

achieving the empirical capacity using feedback memoryless additive models we address the problem of universal communications over an unknown channel with an instantaneous noiseless feedback and show how rates corresponding to the empirical behavior of the channel can be attained although no rate can be guaranteed in advance first we consider a discrete modulo additive channel with alphabet x where the noise sequence zn is arbitrary and unknown and may causally depend on the transmitted and received sequences and on the encoder s message possibly in an adversarial fashion although the classical capacity of this channel is zero we show that rates approaching the empirical capacity log x h emp zn can be universally attained where h emp zn is the empirical entropy of zn for the more general setting where the channel can map its input to an output in an arbitrary unknown fashion subject only to causality we model the empirical channel actions as the modulo addition of a realized noise sequence and show that the same result applies if common randomness is available the results are proved constructively by providing a simple sequential transmission scheme approaching the empirical capacity

on kappa noncollapsed complete noncompact shrinking gradient ricci solitons which split at infinity we discuss some geometric conditions under which a complete noncompact shrinking gradient ricci soliton will split at infinity

leveraging coherent space time codes for noncoherent communication via training training codes are introduced for the multiple antenna noncoherent multiple block rayleigh fading channel in which the fading coefficients which are constant over a fixed number of dimensions coherence interval for each block and then change independently to a new realization are known neither at the transmitter nor at the receiver each codeword of a training code consists of a part known to the receiver used to form a minimum mean squared error mmse estimate of the channel and a part that contains codeword s of a space time block or trellis code designed for the coherent channel in which the receiver has perfect knowledge of the channel the channel estimate is used as if it were error free for decoding the information bearing part of the training codeword training codes are hence easily designed to have high rate and low decoding complexity by choosing the underlying coherent code to have high rate and to be efficiently decodable conditions for which the estimator detector e d receiver is equivalent to the optimal noncoherent receiver are established a key performance analysis result of this paper is that the training codes when decoded with the e d receiver achieve a diversity order of the error probability that is equal to the diversity order of the underlying coherent code in some cases the performance of training codes can be measured relative to coherent reception via training efficiency which is then optimized over the energy allocation between the training and data phases in the limit of increasing block lengths training codes always achieve the performance of coherent reception the examples of training codes provided in this work have polynomial complexity in rate but an error rate comparable to the best performing unitary designs available even though the latter require exponential decoding complexity

feedback capacity of the compound channel in this work we find the capacity of a compound finite state channel fsc with time invariant deterministic feedback we consider the use of fixed length block codes over the compound channel our achievability result includes a proof of the existence of a universal decoder for the family of fscs with feedback as a consequence of our capacity result we show that feedback does not increase the capacity of the compound gilbert elliot channel additionally we show that for a stationary and uniformly ergodic markovian channel if the compound channel capacity is zero without feedback then it is zero with feedback finally we use our result on the fsc to show that the feedback capacity of the memoryless compound channel is given by infthetas maxqx i x y thetas

volume growth and bernstein theorems for translating solitons abstract in this paper we study properties of the potential function of a translating soliton m in r n 1 and the volume growth of the intersection of euclidean balls with m we show that if m is a convex translating soliton then the infimum of the mean curvature function on m is zero i e inf m h 0 we give a condition to obtain the bernstein theorem for the translating solitons we also give an outline of a simple proof of the bernstein theorem due to bombieri de giorgi miranda

heat kernel on smooth metric measure spaces and applications we derive a harnack inequality for positive solutions of the f heat equation and gaussian upper and lower bound estimates for the f heat kernel on complete smooth metric measure spaces with bakry emery ricci curvature bounded below both upper and lower bound estimates are sharp when the bakry emery ricci curvature is nonnegative the main argument is the de giorgi nash moser theory as applications we prove an l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness theorem for f heat equations when f has at most linear growth we also obtain eigenvalues estimates and f green s function estimates for the f laplace operator

the weighted isoperimetric type and sobolev type inequalities for hypersurfaces in carnot groups abstract this paper mainly studies hypersurfaces immersed in a sub riemannian step k carnot group g with a density e f where f is a smooth function on g we give the first variation formula about the weighted horizontal area of hypersurfaces furthermore on the base of some work we derive the weighted isoperimetric type and sobolev type inequalities for hypersurfaces

the capacity of channels with feedback in this paper we introduce a general framework for treating channels with memory and feedback first we prove a general feedback channel coding theorem based on massey s concept of directed information second we present coding results for markov channels this requires determining appropriate sufficient statistics at the encoder and decoder we give a recursive characterization of these sufficient statistics third a dynamic programming framework for computing the capacity of markov channels is presented fourth it is shown that the average cost optimality equation acoe can be viewed as an implicit single letter characterization of the capacity fifth scenarios with simple sufficient statistics are described sixth error exponents for channels with feedback are presented

a broadcast approach for a single user slowly fading mimo channel a broadcast transmission strategy for the slowly fading gaussian multiple input multiple output mimo channel is introduced this broadcast strategy is an extension of the single input single output siso broadcast approach perfect channel state information csi is assumed known at the receiver end only this strategy facilitates to adapt the reliably decoded rate to the actual channel state without having any feedback link to the transmitter transmission of layered coded information is motivated by the theory of majorization we derive the basic equations characterizing achievable rates of the strategy several ad hoc approximations to the achievable region are considered and their performance is compared with the siso setting and the ergodic capacity it has been demonstrated that a single layer outage approach is reasonably efficient in the mimo setting in terms of the average reliably decoded rate a multiple access channel mac broadcast approach is also applied for the mimo case and demonstrated to be relatively efficient

intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds abstract let m g be a complete non compact riemannian manifold together with a function e h which weights the hausdorff measures associated to the riemannian metric in this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of m to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in m centered at a point o m as a consequence we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones a basic tool in our study is the analysis of the weighted laplacian of the distance function from o the technique extends to non compact submanifolds properly immersed in m under certain control on their weighted mean curvature

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

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

streaming data transmission in the moderate deviations and central limit regimes we consider streaming data transmission over a discrete memoryless channel a new message is given to the encoder at the beginning of each block and the decoder decodes each message sequentially after a delay of t blocks in this streaming setup we study the fundamental interplay between the rate and error probability in the central limit and moderate deviations regimes and show that 1 in the moderate deviations regime the moderate deviations constant improves over the block coding or non streaming setup by a factor of t and 2 in the central limit regime the second order coding rate improves by a factor of approximately sqrt t for a wide range of channel parameters for both the regimes we propose coding techniques that incorporate a joint encoding of fresh and previous messages in particular for the central limit regime we propose a coding technique with truncated memory to ensure that a summation of constants which arises as a result of applications of the central limit theorem does not diverge in the error analysis furthermore we explore interesting variants of the basic streaming setup in the moderate deviations regime we first consider a scenario with an erasure option at the decoder i e the decoder can output an erasure symbol instead of a message estimate and show that both the exponents of the total error and the undetected error probabilities improve by factors of t next by utilizing the erasure option we show that the exponent of the total error probability can be improved to that of the undetected error probability in the order sense at the expense of a variable decoding delay

on the use of training sequences for channel estimation suppose q is a family of discrete memoryless channels an unknown member of q will be available with perfect causal output feedback for communication we study a scenario where communication is carried by first testing the channel by means of a training sequence then coding according to the channel estimate we provide an upper bound on the maximum achievable error exponent of any such coding scheme if we consider the binary symmetric and the z families of channels this bound is much lower than burnashev s exponent for example in the case of binary symmetric channels this bound has a slope that vanishes at capacity this is to be compared with our previous result that demonstrates the existence of coding schemes that achieve burnashev s exponent that has a nonzero slope at capacity even though the channel is revealed neither to the transmitter nor to the receiver hence the present result suggests that in terms of error exponent a good universal feedback scheme entangles channel estimation with information delivery rather than separating them

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

a coding theorem for a class of stationary channels with feedback a coding theorem is proved for a class of stationary channels with feedback in which the output yn f xn m n zn m n is the function of the current and past m symbols from the channel input xn and the stationary ergodic channel noise zn in particular it is shown that the feedback capacity is equal to limnrarr infin supp x n y n 1 1 n i xn rarr yn where i xn rarr yn sigmai 1 n i xi yi yi 1 denotes the massey directed information from the channel input to the output and the supremum is taken over all causally conditioned distributions p xn yn 1 pii 1 n p xi xi 1 yi 1 the main ideas of the proof are a classical application of the shannon strategy for coding with side information and a new elementary coding technique for the given channel model without feedback which is in a sense dual to gallager s lossy coding of stationary ergodic sources a similar approach gives a simple alternative proof of coding theorems for finite state channels by yang kavcic tatikonda chen berger and permuter weissman goldsmith

on the redundancy of slepian wolf coding in this paper the redundancy of both variable and fixed rate slepian wolf coding is considered given any jointly memoryless source side information pair xi yi i 1 infin with finite alphabet the redundancy rn isinn of variable rate slepian wolf coding of x1 n with decoder only side information y1 n depends on both the block length n and the decoding block error probability isinn and is defined as the difference between the minimum average compression rate of order n variable rate slepian wolf codes having the decoding block error probability less than or equal to isinn and the conditional entropy h x y where h x y is the conditional entropy rate of the source given the side information the redundancy of fixed rate slepian wolf coding of x1 n with decoder only side information y1 n is defined similarly and denoted by rf n isinn it is proved that under mild assumptions about isinn rn isinn dvradic log isinn n oradic log isinn n and rf n isinn dfradic log isinn n o radic log isinn n where df and dnu are two constants completely determined by the joint distribution of the source side information pair since dv is generally smaller than df our results show that variable rate slepian wolf coding is indeed more efficient than fixed rate slepian wolf coding

extrinsic jensen shannon divergence applications to variable length coding this paper considers the problem of variable length coding over a discrete memoryless channel with noiseless feedback this paper provides a stochastic control view of the problem whose solution is analyzed via a newly proposed symmetrized divergence termed extrinsic jensen shannon ejs divergence it is shown that strictly positive lower bounds on ejs divergence provide nonasymptotic upper bounds on the expected code length this paper presents strictly positive lower bounds on ejs divergence and hence nonasymptotic upper bounds on the expected code length for the following two coding schemes 1 variable length posterior matching and 2 maxejs coding scheme that is based on a greedy maximization of the ejs divergence as an asymptotic corollary of the main results this paper also provides a rate reliability test variable length coding schemes that satisfy the condition s of the test for parameters r and e are guaranteed to achieve a rate r and an error exponent e the results are specialized for posterior matching and maxejs to obtain deterministic one phase coding schemes achieving capacity and optimal error exponent for the special case of symmetric binary input channels simpler deterministic schemes of optimal performance are proposed and analyzed