joint source channel coding with side information using hybrid digital analog codes we study the joint source channel coding problem of transmitting a gaussian source over a gaussian channel in two cases i the presence of interference known only to the transmitter and ii in the presence of side information about the source known only to the receiver we introduce hybrid digital analog forms of the costa and wyner ziv coding schemes we present the random coding counterpart of schemes based on lattices proposed by kochman and zamir then we discuss applications of the hybrid digital analog schemes in the case of channel signal to noise ratio mismatch and for lossy multicasting of a common source with bandwidth compression

lattice coding and decoding achieve the optimal diversity multiplexing tradeoff of mimo channels this paper considers communication over coherent multiple input multiple output mimo flat fading channels where the channel is only known at the receiver for this setting we introduce the class of lattice space time last codes we show that these codes achieve the optimal diversity multiplexing tradeoff defined by zheng and tse under generalized minimum euclidean distance lattice decoding our scheme is based on a generalization of erez and zamir mod lambda scheme to the mimo case in our construction the scalar scaling of erez zamir and costa gaussian dirty paper schemes is replaced by the minimum mean square error generalized decision feedback equalizer mmse gdfe this result settles the open problem posed by zheng and tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity multiplexing tradeoff moreover our results shed more light on the structure of optimal coding decoding techniques in delay limited mimo channels and hence open the door for novel approaches for space time code constructions in particular 1 we show that mmse gdfe plays a fundamental role in approaching the limits of delay limited mimo channels in the high signal to noise ratio snr regime unlike the additive white gaussian noise awgn channel case and 2 our random coding arguments represent a major departure from traditional space time code designs based on the rank and or mutual information design criteria

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

channel state feedback schemes for multiuser mimo ofdm downlink channel state feedback schemes for the mimo broadcast downlink have been widely studied in the frequency flat case this work focuses on the more relevant frequency selective case where some important new aspects emerge we consider a mimo ofdm broadcast channel and compare achievable ergodic rates under three channel state feedback schemes analog feedback direction quantized feedback and time domain channel quantized feedback the first two schemes are direct extensions of previously proposed schemes the third scheme is novel and it is directly inspired by rate distortion theory of gaussian correlated sources for each scheme we derive the conditions under which the system achieves full multiplexing gain the key difference with respect to the widely treated frequency flat case is that in mimo ofdm the frequency domain channel transfer function is a gaussian correlated source the new time domain quantization scheme takes advantage of the channel frequency correlation structure and outperforms the other schemes furthermore it is by far simpler to implement than complicated spherical vector quantization in particular we observe that no structured codebook design and vector quantization is actually needed for efficient channel state information feedback

joint wyner ziv dirty paper coding by modulo lattice modulation the combination of source coding with decoder side information the wyner ziv problem and channel coding with encoder side information the gel fand pinsker problem can be optimally solved using the separation principle in this work we show an alternative scheme for the quadratic gaussian case which merges source and channel coding this scheme achieves the optimal performance by applying a modulo lattice modulation to the analog source thus it saves the complexity of quantization and channel decoding and remains with the task of ldquoshapingrdquo only furthermore for high signal to noise ratio snr the scheme approaches the optimal performance using an snr independent encoder thus it proves for this special case the feasibility of universal joint source channel coding

the golden code a 2 spl times 2 full rate space time code with nonvanishing determinants in this paper the golden code for a 2 spl times 2 multiple input multiple output mimo system is presented this is a full rate 2 spl times 2 linear dispersion algebraic space time code with unprecedented performance based on the golden number 1 spl radic 5 2

channel state feedback over the mimo mac in order to exploit the full multiplexing gain of multi antenna multi user downlink schemes accurate channel state information at the transmitter i e at the base station is required we consider the design of a closed loop channel state information feedback scheme where user terminals feed back their channel state information simultaneously to a multi antenna base station the underlying information theoretic problem consists of lossy source channel coding of multiple independent analog sources i e the users channel coefficients over a gaussian multiple input multiple output multi access channel mimo mac unlike the classical source channel coding setting this application requires low latency otherwise the channel state information would be outdated hence source channel codewords can span only a single fading state of the uplink feedback channel furthermore the transmitters are ignorant of the realization of the uplink channel coefficients in this scenario the scaling of the maximum of the estimated downlink channel mean square errors with the snr dominates the performance of the multiuser downlink this scaling is described by the distortion snr exponent previously introduced in a single user mimo setting this paper analyzes the max min distortion snr exponent of the mimo mac for both separated source channel coding and a particular hybrid digital analog joint source channel coding scheme for the case of single antenna users we prove that the distortion snr exponent of separated source channel coding can be achieved by the concatenation of scalar quantization and uncoded quadrature amplitude modulation qam transmission with lattice decoding at the base station receiver the resulting scheme has very low encoding latency only a few symbols of the uplink slot and generally outperforms currently proposed channel state feedback schemes based on analog unquantized transmission or vector quantization with fixed codebooks

fliess series a generalization of the ree s theorem and an algebraic approach to a homogeneous approximation problem in this paper an algebraic technique originated by m fliess is developed and applied to a homogeneous approximation problem for nonlinear affine control systems the central role is played by a generalization of the r ree s theorem on a connection of lie elements and the shuffle product operation in a free associative algebra

on the distortion snr exponent of some layered transmission schemes we consider the problem of joint source channel coding for transmitting k samples of a complex gaussian source overt bk uses of a block fading multiple input multiple output mimo channel with m transmit and n receive antennas we consider the case when we are allowed to code over l blocks the channel gain is assumed to be constant over a block and channel gains for different blocks are assumed to be independent the performance measure of interest is the rate of decay of the expected mean squared error with the signal to noise ratio snr called the distortion snr exponent we first show that using a broadcast strategy similar to that of gunduz and erkip but with a different power and rate allocation policy the optimal distortion snr exponent can be achieved for 0 les b les n m 1 min m n and for b mnl2 this is the first time the optimal exponent is characterized for 1 min m n b n m 1 min m n then we propose a digital layered transmission scheme that uses both time layering and superposition the new scheme is at least as good as currently known schemes for the entire range of bandwidth expansion factors b whereas at least for some m n and b it is strictly better than the currently known schemes

on the support of graph lie algebras we deal with the support of graph lie algebras and we characterize the set of independence alphabets for which the noncommutative result can naturally be extended we also provide an algorithm which decides if an independence alphabet fulfills the preceding condition

comments on approximate characterizations for the gaussian source broadcast distortion region recently tian et al 1 considered joint source channel coding of transmitting a gaussian source over k user gaussian broadcast channel and derived an outer bound on the admissible distortion region in 1 they stated due to its nonlinear form it appears difficult to determine whether it is always looser than the trivial outer bound in all distortion regimes with bandwidth compression however in this correspondence we solve this problem and prove that for the bandwidth expansion case with k geq 2 this outer bound is strictly tighter than the trivial outer bound with each user being optimal in the point to point setting while for the bandwidth compression or bandwidth match case this outer bound actually degenerates to the trivial outer bound therefore our results imply that on one hand the outer bound given in 1 is nontrivial only for gaussian broadcast communication k geq 2 with bandwidth expansion on the other hand unfortunately no nontrivial outer bound exists so far for gaussian broadcast communication k geq 2 with bandwidth compression

a note on the optimality of variant i permutation modulation codes in this correspondence the optimality of variant i permutation codes initially proposed by slepian see proc ieee vol 53 no 3 p 228 236 mar 1965 is shown in a simple way

on achievable distortion exponents for a gaussian source transmitted over parallel gaussian channels with correlated fading and asymmetric snrs this paper considers the end to end mean squared error distortion in reconstructing a memoryless proper complex gaussian source transmitted over a set of parallel block fading gaussian noise channels where the fading gains are modeled as correlated rayleigh distributed random variables with different average powers thus resulting in asymmetric average received signal noise ratios snrs the distortion exponent i e how fast the average distortion decays to zero as the average received snr increases of several coding strategies based on separate source and channel coding is characterized the definition of distortion exponent commonly used in the literature for snr symmetric channels is generalized to the case of snr asymmetric channels it is shown that fading correlation degrades the achievable mean squared error distortion but does not affect the distortion exponent in the analyzed achievable schemes the logarithm of the determinant of the fading correlation matrix is found to be a proxy for measuring the performance degradation due to correlation as compared with the case of independent fading the proposed framework allows one to study any number of correlated parallel channels contrary to the most of the literature that restricts attention to two channels only with the same received snr and with independent fading in particular a scheme based on the multiple description coding with more than two descriptions is analyzed it is shown that determining the distortion exponent in this setting reduces to solving a linear program which can be done numerically very efficiently the proposed methodology relies on combining the ideas from linear innovation sequences and properties of determinant of sub matrices interestingly it is found that snr asymmetry is beneficial for multiple description coding when the total average received snr in decibel is held constant even in the snr symmetric case asymmetry in the compression rates is shown to lead to a larger distortion exponent than symmetric rates

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

the extension problem of the mean curvature flow i we show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in mathbb r 3

uniqueness of blowups and łojasiewicz inequalities once one knows that singularities occur one naturally wonders what the singu larities are like for minimal varieties the first answer already known to federer fleming in 1959 is that they weakly resemble cones 1 for mean curvature flow by the combined work of huisken ilmanen and white singularities weakly resemble shrinkers unfortunately the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup but under higher magnification it might as far as anyone knows resemble a completely different blowup whether this ever happens is perhaps the most fundamental question about singularities it is this long stand ing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities

uniqueness of convex ancient solutions to mean curvature flow in mathbb r 3 r 3 a well known question of perelman concerns the classification of noncompact ancient solutions to the ricci flow in dimension 3 which have positive sectional curvature and are kappa noncollapsed in this paper we solve the analogous problem for mean curvature flow in mathbb r 3 and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in mathbb r 3 which is strictly convex and noncollapsed

on estimation error outage for scalar gauss markov signals sent over fading channels measurements of a scalar linear gauss markov process are sent over a fading channel the fading channel is modeled as independent and identically distributed random variables with known realization at the receiver the optimal estimator at the receiver is the kalman filter in contrast to the classical kalman filter theory given a random channel the kalman gain and the error covariance become random then the probability distribution function of expected estimation error and its outage probability can be chosen for estimation quality assessment in this paper and in order to get the estimation error outage we provide means to characterize the stationary probability density function of the random expected estimation error furthermore and for the particular case of the i i d rayleigh fading channels upper and lower bounds for the outage probability are derived which provide insight and simpler means for design purposes we also show that the bounds are tight for the high snr regime and that the outage probability decreases linearly with the inverse of the average channel snr

tcm ssma communication systems with cascaded sequences and pam qam signal sets an expression in matrix form for the multiple access interference mai in an asynchronous direct sequence spread spectrum multiple access ds ssma communication system with cascaded sequences cvs arbitrary chip waveforms and trellis coded modulation tcm with a pulse amplitude modulation pam or quadrature amplitude modulation qam signal set is obtained tcm provides significant coding gain while the cvs decrease the correlation between the mai of adjacent data intervals the expression is used to calculate arbitrarily accurate probability density functions pdfs of the mai in the tcm system and to derive an accurate approximation of the mai variance it also helps illustrate some properties of the mai by separating contributing parameters into different matrices we derive an approximation of the upper union bound on the bit error probability and investigate its applicability the results show that cv schemes can greatly reduce the pairwise error probabilities peps until the length of the cv becomes greater than that of the error weight sequence ews under consideration

distortion outage analysis for joint space time coding and kalman filtering in this paper we consider the scenario of transmitting a first order gauss markov vector signal over a multi input multi output mimo rayleigh nonfrequency selective fading channel the signal is reconstructed at the receiver side with the help of a kalman filter in order to minimize the mean squared error orthogonal space time codes are utilized in order to increase the quality of estimation and mitigate the destructing effects of the fading channel as a criterion for estimation quality assessment we use the distortion outage probability we first obtain upper and lower bounds for the outage probability as a function of system parameters we then perform high snr analysis of the bounds through which we prove the achievability of the maximum diversity order for a n times k mimo fading channel in addition we obtain upper and lower bounds for the coding gain of the distortion outage probability in the high snr regime and outline the relation between system parameters and the coding gain

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

a linear encoding approach to index assignment in lossy source channel coding we present a general scheme for the lossy transmission of a source with arbitrary statistics through a noisy channel under the mean square error fidelity criterion our approach is based on transform coding scalar quantization of the transform coefficients and linear encoding of the quantization indices entropy coding and channel coding are merged into a single linear encoding function such that the catastrophic behavior of conventional entropy coding is avoided and the full power of modern coding techniques and iterative belief propagation decoding can be exploited we show that this approach is asymptotically optimal in the limit of large block length for arbitrary source statistics and binary input output symmetric channel in the practical regime of finite block length and low decoding complexity we show through the explicit construction of codes for the lossy transmission of digital images over a binary symmetric channel that our approach yields significant improvements with respect to previously proposed channel optimized quantization schemes and also with respect to the conventional concatenation of state of the art image coding with state of the art channel coding although our constructive example focuses on a special case the approach is general and can be applied to other classes of sources of practical interest

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

joint source channel codes for mimo block fading channels we consider transmission of a continuous amplitude source over an l block rayleigh fading mt x mr multiple input multiple output mimo channel when the channel state information is only available at the receiver since the channel is not ergodic shannon s source channel separation theorem becomes obsolete and the optimal performance requires a joint source channel approach our goal is to minimize the expected end to end distortion particularly in the high signal to noise ratio snr regime the figure of merit is the distortion exponent defined as the exponential decay rate of the expected distortion with increasing snr we provide an upper bound and lower bounds for the distortion exponent with respect to the bandwidth ratio among the channel and source bandwidths for the lower bounds we analyze three different strategies based on layered source coding concatenated with progressive superposition or hybrid digital analog transmission in each case by adjusting the system parameters we optimize the distortion exponent as a function of the bandwidth ratio we prove that the distortion exponent upper bound can be achieved when the channel has only one degree of freedom that is l 1 and min mt mr 1 when we have more degrees of freedom our achievable distortion exponents meet the upper bound for only certain ranges of the bandwidth ratio we demonstrate that our results which were derived for a complex gaussian source can be extended to more general source distributions as well

diversity effects in kalman filtering over rayleigh fading channels a scalar first order gauss markov signal is sent over d parallel i i d rayleigh fading channels using uncoded transmission assuming perfect channel estimation at the receiver the optimal mmse estimator is the kalman filter with random instantaneous estimation error variance statistical characteristics of the instantaneous estimation error variance are studied to characterize the estimation quality one such characteristic is the estimation error outage which probabilistically depicts how often the value of the instantaneous estimation error variance exceeds a certain threshold in this paper we study the effect of channel diversity on the estimation error outage probability we show that for a certain range of outage thresholds the estimation error outage probability decreases with inverse of d th power of the average signal to noise ratio at the receiver side this means that analogous to the context of digital communication over fading channels our result implies a diversity order of d in estimating the input process

hybrid digital analog hda joint source channel codes for broadcasting and robust communications we consider the problem of transmitting a band limited gaussian source on an additive band limited gaussian noise channel the well known threshold effect dictates that the more powerful a code is the more sensitive it is to the exact knowledge of the channel noise a code is said to be robust if it is asymptotically optimal for a wide range of channel noise thus robust codes have a graceful degradation characteristic and are free of the threshold effect it is demonstrated that robust codes exist whenever the source and channel bandwidths are equal in the unequal bandwidth case a collection of nearly robust joint source channel codes is constructed using a hybrid digital analog hda coding technique for designing nearly robust codes a matched tandem code whose channel encoder s output is partially fully matched to its input is proposed and the existence of an asymptotically optimal matched tandem code is shown the nearly robust codes achieve the shannon limit theoretically optimum distortion and have a less severe threshold effect finally for the case of two different noise conditions the distortion regions of these codes are determined

analog matching of colored sources to colored channels analog uncoded transmission provides a simple and robust scheme for communicating a gaussian source over a gaussian channel under the mean squared error mse distortion measure unfortunately its performance is usually inferior to the all digital separation based source channel coding solution which requires exact knowledge of the channel at the encoder the loss comes from the fact that except for very special cases e g white source and channel of matching bandwidth bw it is impossible to achieve perfect matching of source to channel and channel to source by linear means we show that by combining prediction and modulo lattice operations it is possible to match any colored gaussian source to any colored gaussian noise channel of possibly different bw hence achieve shannon s optimum attainable performance r d c furthermore when the source and channel bws are equal but otherwise their spectra are arbitrary this scheme is asymptotically robust in the sense that for high signal to noise ratio snr a single encoder independent of the noise variance achieves the optimum performance the derivation is based upon a recent modulo lattice modulation scheme for transmitting a wyner ziv source over a dirty paper channel

orthogonal projection onto the free lie algebra abstract as an answer to a question of garsia we give a universal formula in a normal form for the orthogonal projection of free associative algebra onto free lie algebra

analog multiple description joint source channel coding based on lattice scaling joint source channel coding schemes based on analog mappings for point to point channels have recently gained attention for their simplicity and low delay in this paper these schemes are extended either to scenarios with or without side information at the decoders to transmit multiple descriptions of a gaussian source over independent parallel channels they are based on a lattice scaling approach together with bandwidth reduction analog mappings adapted for this multiple description scenario the rationale behind lattice scaling is to improve performance through bandwidth expansion another important contribution of this paper is the proof of the separation theorem for the communication scenario of multiple description with side information at the decoders this proof allows us to compare the performance of the proposed schemes with the theoretical limits when considering awgn channels finally when the channels are parallel slow fading rayleigh the proposed schemes are optimal in terms of distortion exponent

multiple description analog joint source channel coding to exploit the diversity in parallel channels the complexity and delay introduced by efficient digital coding strategies may be a barrier in some real time communications in this sense these last years joint source channel coding schemes based on analog mappings have gained prominence precisely for their simplicity and their implicit low delay in this work analog mappings originally designed for point to point communications are adapted to the case of parallel channels by following the multiple description strategy traditionally used in source coding in principle the coding scheme is designed to transmit over parallel awgn on off channels which are characterized by the possibility of having failures we also show that our scheme performs satisfactorily over slow rayleigh fading parallel channels

the shuffle algebra and its derivations abstract the shuffle algebra on a partially commutative alphabet and its derivations are studied the derivation of the shuffle algebra is characterized in terms of admissible mappings the module of all derivations of the shuffle algebra is proved to be isomorphic to the free partially commutative power series lie algebra with the coefficient ring extended the orthogonality between the lie elements and the proper shuffles is established

minimal 2 spheres in 3 spheres we prove that any manifold diffeomorphic to s3 and endowed with a generic metric contains at least two embedded minimal 2 spheres the existence of at least one minimal 2 sphere was obtained by simon and smith in 1983 our approach combines ideas from min max theory and mean curvature flow we also establish the existence of smooth mean convex foliations in 3 manifolds we apply our methods to solve a problem posed by s t yau in 1987 on whether the planar 2 spheres are the only minimal spheres in ellipsoids centered about the origin in r4 finally considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity 1 conjecture and the equidistribution of widths conjecture are in a certain sense sharp

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

the nature of singularities in mean curvature flow of mean convex sets let k be a compact subset of r or more generally of an n 1 dimensional riemannian manifold we suppose that k is mean convex if the boundary of k is smooth and connected this means that the mean curvature of k is everywhere nonnegative with respect to the inward unit normal and is not identically 0 more generally it means that ft k is contained in the interior of k for t 0 where ft k is the set obtained by letting k evolve for time t under the level set mean curvature flow as k evolves it traces out a closed set k of spacetime k x t r r x ft k also there is associated to k a brakke flow m t 7 mt of rectifiable varifolds we call the pair m k a mean convex flow let x x t be a point in spacetime with t 0 suppose xi ti is a sequence of points converging to x and i is a sequence of positive numbers tending to infinity translate the pair m and k in spacetime by y 7 y xi ti and then dilate parabolically by y 7 iy i to get new flows mi and ki the sequence mi ki is called a blow up sequence at x general compactness theorems guarantee that this sequence will have subsequential limits a subsequential limit m k is called a limit flow here m t 7 m t

systematic lossy source channel coding the fundamental limits of systematic communication are analyzed in systematic transmission the decoder has access to a noisy version of the uncoded raw data analog or digital the coded version of the data is used to reduce the average reproduced distortion d below that provided by the uncoded systematic link and or increase the rate of information transmission unlike the case of arbitrarily reliable error correction d spl rarr 0 for symmetric sources channels where systematic codes are known to do as well as nonsystematic codes we demonstrate that the systematic structure may degrade the performance for nonvanishing d we characterize the achievable average distortion and we find necessary and sufficient conditions under which systematic communication does not incur loss of optimality the wyner ziv 1976 rate distortion theorem plays a fundamental role in our setting the general result is applied to several scenarios for a gaussian bandlimited source and a gaussian channel the invariance of the bandwidth signal to noise ratio snr in decibels product is established and the optimality of systematic transmission is demonstrated bernoulli sources transmitted over binary symmetric channels and over certain gaussian channels are also analyzed it is shown that if nonnegligible bit error rate is tolerated systematic encoding is strictly suboptimal

the streaming dmt of fading channels we study sequential transmission of a stream of messages over a block fading multi input multi output mimo channel a new message arrives in each coherence block and the decoder is required to output each message after a delay of t coherence blocks we establish the optimal diversity multiplexing tradeoff dmt and show that it can be achieved using a simple interleaving of messages the converse is based on an outage amplification technique which appears to be new we also discuss another coding scheme based on a sequential tree code this coding scheme only requires the knowledge of delay at the decoder and yet realizes the optimal dmt we finally discuss some extensions when multiple messages at uniform intervals arrive within each coherence period

on robustness of hybrid digital analog source channel coding with bandwidth mismatch for transmission of memoryless gaussian sources over channels with additive white gaussian noise the tradeoff between the distortion when the channel quality is good versus bad is investigated under the constraint that the distortion is optimal when the channel has the targeted median quality the problem was proposed by tian and shamai who subsequently proved remarkable achievability results for bandwidth expansion ratios kappa which are integers or unit fractions i e kappa 2 3 4 ldots or kappa 1 2 1 3 1 4 ldots their result is extended here to all kappa geq 2 and kappa leq 1 2 by generalizing the hybrid digital analog hda scheme of wilson et al to the case of bandwidth mismatch finally novel schemes are proposed for 1 2 leq kappa and 1 achieving nontrivial tradeoffs outperforming all known schemes these latter schemes rely on another extension of the hda scheme by wilson et al namely the relaxation of the independence of source and channel input sequences

mean curvature flow with surgery of mean convex surfaces in mathbb r 3 we define a notion of mean curvature flow with surgery for two dimensional surfaces in mathbb r 3 with positive mean curvature our construction relies on the earlier work of huisken and sinestrari in the higher dimensional case one of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author invent math 2015

single sample robust joint source channel coding achieving asymptotically optimum scaling of sdr versus snr in this paper we consider the problem of zero delay encoding a single source sample robust joint source channel coding over an additive white gaussian noise channel we propose a new scheme that unlike previously known coding schemes achieves the optimal scaling of the source signal to distortion ratio sdr versus channel signal to noise ratio snr also we propose a family of robust codes which together maintain a bounded gap with the optimum sdr curve in terms of decibel to show the importance of this result we derive some theoretical bounds on the asymptotic performance of a widely used class of delay limited hybrid digital analog hda coding schemes based on superposition of analog and digital components we show that unlike the delay unlimited case for this class of delay limited hda codes the asymptotic performance loss is unbounded in terms of decibels although the main focus of this paper is on uniform sources it is also shown that the results are also valid for a more general class of well behaved distributions

a lower bound on the expected distortion of joint source channel coding we consider the classic joint source channel coding problem of transmitting a memoryless source over a memoryless channel the focus of this work is on the long standing open problem of finding the rate of convergence of the smallest attainable expected distortion to its asymptotic value as a function of the blocklength n our main result is that in general the convergence rate is not faster than n 1 2 in particular we show that for the problem of transmitting i i d uniform bits over a binary symmetric channels with hamming distortion the smallest attainable distortion bit error rate is at least omega n 1 2 above the asymptotic value if the bandwidth expansion ratio is above 1

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

on source transmission over mimo channels with limited feedback the problem of source channel coding over a multiple antenna mimo channel with quantized channel state information at the transmitter csit is considered upper bounds on the distortion exponents achieved with partial csit under a long term power constraint are developed it is shown that the distortion exponent with perfect csit grows unbounded as the ratio between the channel and source bandwidth increases while the exponent achieved with any feedback link of fixed finite resolution is bounded above by a polynomial of the product between the number of transmit and number of receive antennas the resolution of the feedback link should grow with the bandwidth ratio to make the distortion exponent scale as fast as that in the perfect csit case we show that in order to achieve the optimal scaling the csit feedback resolution must grow logarithmically with the bandwidth ratio for mimo channels and faster than linear for the single input single output channel the achievable distortion exponent of some hybrid schemes with heavily quantized feedback is also derived the results demonstrate that dramatic performance improvement over the case of no csit can be achieved by combining simple schemes with a very coarse csit feedback

hybrid digital analog coding with bandwidth compression for gaussian source channel pairs three hybrid digital analog hda systems denoted by hda i hda and hda ii for the coding of a memoryless discrete time gaussian source over a discrete time additive memoryless gaussian channel under bandwidth compression are studied the systems employ simple linear coding in their analog component and superimpose their analog and digital signals before channel transmission information theoretic upper bounds on the asymptotically optimal mean squared error distortion of the systems are obtained under both matched and mismatched channel conditions allocation schemes for distributing the channel input power between the analog and the digital signals are also examined it is shown that systems hda and hda ii can asymptotically achieve the optimal shannon limit performance under matched channel conditions low complexity and low delay versions of systems hda i and hda ii are next designed and implemented without the use of error correcting codes the parameters of these hda systems which employ vector quantization in conjunction with binary phase shift keying modulation in their digital part are optimized via an iterative algorithm similar to the design algorithm for channel optimized vector quantizers both systems have low complexity and low delay and guarantee graceful performance improvements for high csnrs for memoryless gaussian sources the designed hda ii system is shown to be superior to the hda i designed system when applied to a gauss markov source under karhunen loeve processing the hda i system is shown to provide considerably better performance

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

on the distortion snr exponent of hybrid digital analog space time coding we consider the transmission of a real independent and identically distributed i i d analog source over a quasi static m input n output multiple input multiple output mimo block fading channel the relevant performance criterion is end to end average quadratic distortion d versus channel signal to noise ratio snr for given spectral efficiency eta defined as the ratio of the source bandwidth over the channel bandwidth in the limit of high snr we define the distortion snr exponent a eta as the largest a such that d esdot snr a over all possible source channel coding schemes of spectral efficiency eta we find a simple upper bound on a eta an achievable lower bound asep eta achievable by separated tandem source channel coding and a tighter lower bound ahybrid eta achievable by new hybrid digital analog space time coding schemes as a corollary we have that a eta is completely determined for the scalar case m n 1 and for the bandwidth compression case eta ges 2 min m n expiicit and simple construction of hybrid space time codes achieving ahybrid eta are also given

mean curvature flow of mean convex hypersurfaces in the last 15 years white and huisken sinestrari developed a far reaching structure theory for the mean curvature flow of mean convex hypersurfaces their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow in the present paper we give a new treatment of the theory of mean convex and k convex flows this includes 1 an estimate for derivatives of curvatures 2 a convexity estimate 3 a cylindrical estimate 4 a global convergence theorem 5 a structure theorem for ancient solutions and 6 a partial regularity theorem our new proofs are both more elementary and substantially shorter than the original arguments our estimates are local and universal a key ingredient in our new approach is the new noncollapsing result of andrews 2 some parts are also inspired by the work of perelman 32 33 in a forthcoming paper 17 we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper note added in may 2015 since the first version of this paper was posted on arxiv in april 2013 the estimates have been used to construct mean convex flow with surgery in 3 by brendle and huisken 5 in september 2013 and in another paper by the authors in april 2014 2016 wiley periodicals inc

distortion bounds for broadcasting with bandwidth expansion we consider the problem of broadcasting a single gaussian source to two listeners over a gaussian broadcast channel with rho channel uses per source sample where rho 1 a distortion pair d1 d2 is said to be achievable if one can simultaneously achieve a mean squared error mse d1 at receiver 1 and d2 at receiver 2 the main result of this correspondence is an outer bound for the set of all achievable distortion pairs that is we find necessary conditions under which d1 d2 is achievable we then apply this result to the problem of point to point transmission over a gaussian channel with unknown signal to noise ratio snr and rho 1 we show that if a system must be optimal at a certain snrmin then asymptotically the system distortion cannot decay faster than o 1 snr as for achievability we show that a previously reported scheme due to mittal and phamdo 2002 is optimal at high snr we introduce two new schemes for broadcasting with bandwidth expansion combining digital and analog transmissions we finally show how a system with a partial feedback returning from the bad receiver to the transmitter and to the good receiver achieves a distortion pair that lies on the outer bound derived here

a monotonicity formula for mean curvature flow with surgery abstract we prove a monotonicity formula for mean curvature flow with surgery this formula differs from huisken s monotonicity formula by an extra term involving the mean curvature as a consequence we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is in fact free of surgeries this result is used in the analysis of mean curvature flow with surgery in riemannian three manifolds cf 5

joint source channel coding of a gaussian mixture source over the gaussian broadcast channel suppose that we want to send a description of a single source to two listeners through a gaussian broadcast channel where the channel is used once per source sample the problem of joint source channel coding is to design a communication system to minimize the distortion d sub 1 at receiver 1 and at the same time minimize the distortion d sub 2 at receiver 2 if the source is gaussian the optimal solution is well known and it is achieved by an uncoded analog scheme we consider a gaussian mixture source we derive inner and outer bounds for the distortion region of all d sub 1 d sub 2 pairs that are simultaneously achievable the outer bound is based on the entropy power inequality while the inner bound is attained by a digital over analog encoding scheme which we present we also show that if the modes of the gaussian mixture are highly separated our bounds are tight and hence our scheme attains the entire distortion region this optimal region exceeds the region attained by separating source and channel coding although it does not contain the ideal point d sub 1 d sub 2 r sup 1 c sub 1 r sup 1 c sub 2

mean curvature flow with surgeries of two convex hypersurfaces we consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow it is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded the purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure compared with the notions of weak solutions existing in the literature the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold our construction is inspired by the procedure originally introduced by hamilton for the ricci flow and then employed by perelman in the proof of thurston s geometrization conjecture in this paper we consider initial hypersurfaces which have dimension at least three and are two convex that is such that the sum of the two smallest principal curvatures is nonnegative everywhere under these assumptions we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology as a corollary we obtain a classification up to diffeomorphism of the hypersurfaces under consideration

the achievable distortion region of sending a bivariate gaussian source on the gaussian broadcast channel we provide a complete characterization of the achievable distortion region for the problem of sending a bivariate gaussian source over bandwidth matched gaussian broadcast channels where each receiver is interested in only one component of the source this setting naturally generalizes the simple single gaussian source bandwidth matched broadcast problem for which the uncoded scheme is known to be optimal we show that a hybrid scheme can achieve the optimum for the bivariate case but neither an uncoded scheme alone nor a separation based scheme alone is sufficient we further show that in this joint source channel coding setting the gaussian scenario is the worst scenario among the sources and channel noises with the same covariances

mean curvature flow with surgery we give a new proof for the existence of mean curvature flow with surgery of 2 convex hypersurfaces in r as announced in hk13a our proof works for all n 3 including mean convex surfaces in r we also derive a priori estimates for a more general class of flows in a local and flexible setting

a sharp bound for the inscribed radius under mean curvature flow we consider a family of embedded mean convex hypersurfaces which evolve by the mean curvature flow it follows from general results of white that the inscribed radius at each point on the hypersurface is at least frac c h where c is a positive constant that depends only on the initial data andrews recently gave a new proof of that fact using the maximum principle in this paper we show that the inscribed radius is at least frac 1 1 delta h at each point where the curvature is sufficiently large