entropy and source coding for integer dimensional singular random variables entropy and differential entropy are important quantities in information theory a tractable extension to singular random variables which are neither discrete nor continuous has not been available so far here we present such an extension for the practically relevant class of integer dimensional singular random variables the proposed entropy definition contains the entropy of discrete random variables and the differential entropy of continuous random variables as special cases we show that it transforms in a natural manner under lipschitz functions and that it is invariant under unitary transformations we define joint entropy and conditional entropy for integer dimensional singular random variables and we show that the proposed entropy conveys useful expressions of the mutual information as first applications of our entropy definition we present a result on the minimal expected codeword length of quantized integer dimensional singular sources and a shannon lower bound for integer dimensional singular sources

optimum sequence multisets for synchronous code division multiple access channels it is shown that the sum capacity of the symbol synchronous code division multiple access channel with equal average input energy constraints is maximized precisely by those spreading sequence multisets that meet welch s lower bound on total squared correlation it is further shown that the symmetric capacity of the channel determined by these same sequence multisets is equal to the sum capacity

optimal phase transitions in compressed sensing compressed sensing deals with efficient recovery of analog signals from linear encodings this paper presents a statistical study of compressed sensing by modeling the input signal as an i i d process with known distribution three classes of encoders are considered namely optimal nonlinear optimal linear and random linear encoders focusing on optimal decoders we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise respectively the optimal phase transition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms in particular we show that gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding our results also provide a rigorous justification of previous results based on replica heuristics in the weak noise regime

channel capacity under sub nyquist nonuniform sampling this paper investigates the effect of sub nyquist sampling upon the capacity of an analog channel the channel is assumed to be a linear time invariant gaussian channel where perfect channel knowledge is available at both the transmitter and the receiver we consider a general class of right invertible time preserving sampling methods which includes irregular nonuniform sampling and characterize in closed form the channel capacity achievable by this class of sampling methods under a sampling rate and power constraint our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal to noise ratio among all spectral sets of measure equal to the sampling rate this can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates or through a single branch of modulation and filtering followed by uniform sampling these results reveal that for a large class of channels employing irregular nonuniform sampling sets while are typically complicated to realize in practice does not provide capacity gain over uniform sampling sets with appropriate preprocessing our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario which is in contrast to the benefits obtained from random mixing in spectrum blind compressive sampling schemes

the pinching theorem for hypersurfaces of minkowski spaces let mn f be a compact finsler hypersurface of a minkowski space vn 1 f with constant mean curvature h in this paper using the gauss formula of chern connection for finsler submanifolds we prove that if the norm square s of the second fundamental form of m satisfies s n2h2n 1 then m with the induced metric is isometric to the standard euclidean sphere which was obtained by kim 1997 1 for the riemannian case

signals on graphs uncertainty principle and sampling in many applications the observations can be represented as a signal defined over the vertices of a graph the analysis of such signals requires the extension of standard signal processing tools in this paper first we provide a class of graph signals that are maximally concentrated on the graph domain and on its dual then building on this framework we derive an uncertainty principle for graph signals and illustrate the conditions for the recovery of band limited signals from a subset of samples we show an interesting link between uncertainty principle and sampling and propose alternative signal recovery algorithms including a generalization to frame based reconstruction methods after showing that the performance of signal recovery algorithms is significantly affected by the location of samples we suggest and compare a few alternative sampling strategies finally we provide the conditions for perfect recovery of a useful signal corrupted by sparse noise showing that this problem is also intrinsically related to vertex frequency localization properties

multiterminal source coding with high resolution we consider separate encoding and joint decoding of correlated continuous information sources subject to a difference distortion measure we first derive a multiterminal extension of the shannon lower bound for the rate region then we show that this shannon outer bound is asymptotically tight for small distortions these results imply that the loss in the sum of the coding rates due to the separation of the encoders vanishes in the limit of high resolution furthermore lattice quantizers followed by slepian wolf lossless encoding are asymptotically optimal we also investigate the high resolution rate region in the remote coding case where the encoders observe only noisy versions of the sources for the quadratic gaussian case we establish a separation result to the effect that multiterminal coding aimed at reconstructing the noisy sources subject to the rate constraints followed by estimation of the remote sources from these reconstructions is optimal under certain regularity conditions on the structure of the coding scheme

convergence of double fourier series and w classes the double fourier series of functions of the generalized bounded variation class n ln n 1 bv are shown to be pringsheim convergent everywhere in a certain sense this result cannot be improved in general functions of class a bv defined here have quadrant limits at every point and for f e a bv there exist at most countable sets p and q such that for x p and y q f is continuous at x y it is shown that the previously studied class abv contains essentially discontinuous functions unless the sequence a satisfies a strong condition

eigenvalues and eigenfunctions of metric measure manifolds in this paper we study the eigenvalues and eigenfunctions of metric measure manifolds we prove that any eigenfunction is c 1 alpha at its critical points and c infty elsewhere moreover the eigenfunction corresponding to the first eigenvalue in the dirichlet problem does not change sign we also discuss the first eigenvalue the sobolev constants and their relationship with the isoperimetric constants 2000 mathematics subject classification 47j05 47j10 53c60 58e05 58c40

universal compressed sensing for almost lossless recovery in this paper the problem of developing universal algorithms for noiseless compressed sensing of stochastic processes is studied first renyi s notion of information dimension id is generalized to continuous valued discrete time stationary processes this provides a measure of complexity for such processes and is connected to the rate of measurement sampling rate required for their accurate recovery then based on occam s razor a minimum entropy pursuit mep optimization approach for universal compressed sensing is proposed it is proven that for any stationary process satisfying certain mixing conditions if the sampling rate is larger than the id of the source process mep optimization can reliably recover the source vector almost losslessly without having any prior information about its distribution then a lagrangian type relaxation of mep optimization problem referred to as lagrangian mep is studied it is shown that lagrangian mep is identical to an implementable algorithm proposed by baron and coauthors and for the right choice of parameters has the same asymptotic performance as mep optimization finally it is proven that lagrangian mep is robust to small measurement noise

near optimal sensor placement for linear inverse problems a classic problem is the estimation of a set of parameters from measurements collected by only a few sensors the number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates unfortunately the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest we propose framesense a greedy algorithm for the selection of optimal sensor locations the core cost function of the algorithm is the frame potential a scalar property of matrices that measures the orthogonality of its rows notably framesense is the first algorithm that is near optimal in terms of mean square error meaning that its solution is always guaranteed to be close to the optimal one moreover we show with an extensive set of numerical experiments that framesense achieves state of the art performance while having the lowest computational cost when compared to other greedy methods

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

on the boundary integral formulation of the plane theory of elasticity with applications analytical aspects abstract a formulation of the plane strain problem of the theory of elasticity in stresses for simply connected domains is carried out in terms of real functions within the frame of what is known as the boundary integral method special attention is devoted to the problem of determination of the arbitrary constants appearing in the solution in view of work in progress where numerical techniques are used relying on some mathematical results formulated in the appendix simple applications concerning the first and the second fundamental problems for the circle and for the ellipse are given which show the correctness of the formulation and the necessity of recurring to numerical techniques once the geometry of the problem or the type of boundary conditions deviates from being simple following parts of the present work are devoted to the numerical treatment of the obtained system of equations as well as to the theories of thermoelasticity and thermo electromagneto elasticity

on the information dimension of stochastic processes in 1959 renyi proposed the information dimension and the d dimensional entropy to measure the information content of general random variables this paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly quantized stochastic process divided by minus the logarithm of the quantizer step size 1 m in the limit as m to infty it is demonstrated that the information dimension rate coincides with the rate distortion dimension defined as twice the rate distortion function r d of the stochastic process divided by log d in the limit as d downarrow 0 it is further shown that among all multivariate stationary processes with a given matrix valued spectral distribution function sdf the gaussian process has the largest information dimension rate and the information dimension rate of multivariate stationary gaussian processes is given by the average rank of the derivative of the sdf the presented results reveal that the fundamental limits of almost zero distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general

comparison theorems in finsler geometry and their applications we prove hessian comparison theorems laplacian comparison theorems and volume comparison theorems for finsler manifolds under various curvature conditions as applications we derive mckean type theorems for the first eigenvalue of finsler manifolds as well as generalize to finsler manifolds a result on fundamental groups due to milnor

on the boundary integral formulation of the plane theory of elasticity computational aspects a numerical scheme is proposed for the solution of the system of field equations and boundary conditions of the static linear plane strain problem of the theory of elasticity in stresses the work is based on a previous analytical approach by the authors within the boundary integral method for a homogeneous and isotropic elastic material occupying a simply connected region the differential and integral operators appearing in the resulting system of integro differential equations for the determination of the boundary values of four unknown harmonic functions are discretized and the problem at this stage is reduced to finding the solution of a linear system of algebraic equations formulae are provided for the determination of the displacements and stresses at the boundary as an illustration the proposed numerical scheme is applied to find the solution of the following problems i a nearly circular boundary subject to a uniform pressure or to a uniform normal displacement ii a square subject to a uniform pressure or to different prescribed normal displacements this choice of the problems has allowed to assess the efficiency of the method through comparison with known analytical solutions and to test the accuracy of the results under different smoothness properties of the boundary and boundary conditions whenever new the obtained results were thoroughly discussed future work in this field will be devoted to the extension of the method to include multiply connected regions and mixed boundary conditions and to investigate the prediction and isolation of the singularities in the solution caused by boundary irregularities

rate region of the quadratic gaussian two encoder source coding problem we determine the rate region of the quadratic gaussian two encoder source coding problem this rate region is achieved by a simple architecture that separates the analog and digital aspects of the compression furthermore this architecture requires higher rates to send a gaussian source than it does to send any other source with the same covariance our techniques can also be used to determine the sum rate of some generalizations of this classical problem our approach involves coupling the problem to a quadratic gaussian ldquoceo problem rdquo

a variant of trefftz s method by boundary fourier expansion for solving regular and singular plane boundary value problems the main objective of the present work is to introduce a variant of trefftz s method for finding approximate solutions to regular or singular two dimensional boundary value problems for laplace s equation after expressing the solution as a finite linear combination of trial functions the method foresees the enforcement of the boundary condition by using a boundary fourier expansion technique instead of the usual pointwise approach the procedure ultimately produces a rectangular set of linear algebraic equations the method is used to find approximate solutions to five prototype problems mainly in rectangular regions using polar harmonics as trial functions more complicated cases could be envisaged the results are discussed and compared with those obtained by the boundary collocation method for the same set of trial functions it appears that the proposed method yields better results in the case of a singular behaviour of the solution at the boundary for which the singularity could be isolated we have compared the results obtained by the proposed method with isolation and without isolation of the singularity the proposed method may be easily extended to deal with other boundary value problems involving more complicated differential operators boundary geometries and boundary conditions in two or in three dimensions

a bernstein type theorem on a randers space we consider finsler spaces with a randers metric f on the three dimensional real vector space where is the euclidean metric and is a 1 form with norm b 0 b 1 by using the notion of mean curvature for immersions in finsler spaces introduced by z shen we obtain the partial differential equation that characterizes the minimal surfaces which are graphs of functions for each b 0 b 1 open image in new window we prove that it is an elliptic equation of mean curvature type then the bernstein type theorem and other properties such as the nonexistence of isolated singularities of the solutions of this equation follow from the theory developped by l simon for b 1 open image in new window the differential equation is not elliptic moreover for every b 1 open image in new window b 1 we provide solutions which describe minimal cones with an isolated singularity at the origin

uncoded transmission is exactly optimal for a simple gaussian sensor network a single memoryless gaussian source is observed by many terminals subject to independent gaussian observation noises the terminals are linked to a fusion center via a standard gaussian multiple access channel the fusion center needs to recover the underlying gaussian source with respect to mean squared error in this correspondence a theorem of witsenhausen is shown to imply that an optimal communication strategy is uncoded transmission i e each terminal s channel input is merely a scaled version of its noisy observation

estimation in gaussian noise properties of the minimum mean square error consider the minimum mean square error mmse of estimating an arbitrary random variable from its observation contaminated by gaussian noise the mmse can be regarded as a function of the signal to noise ratio snr as well as a functional of the input distribution of the random variable to be estimated it is shown that the mmse is concave in the input distribution at any given snr for a given input distribution the mmse is found to be infinitely differentiable at all positive snr and in fact a real analytic function in snr under mild conditions the key to these regularity results is that the posterior distribution conditioned on the observation through gaussian channels always decays at least as quickly as some gaussian density furthermore simple expressions for the first three derivatives of the mmse with respect to the snr are obtained it is also shown that as functions of the snr the curves for the mmse of a gaussian input and that of a non gaussian input cross at most once over all snrs these properties lead to simple proofs of the facts that gaussian inputs achieve both the secrecy capacity of scalar gaussian wiretap channels and the capacity of scalar gaussian broadcast channels as well as a simple proof of the entropy power inequality in the special case where one of the variables is gaussian

sample distortion for compressed imaging we propose the notion of a sample distortion sd function for independent and identically distributed i i d compressive distributions to fundamentally quantify the achievable reconstruction performance of compressed sensing for certain encoder decoder pairs at a given sampling ratio two lower bounds on the achievable performance and the intrinsic convexity property is derived a zeroing procedure is then introduced to improve non convex sd functions the sd framework is then applied to analyse compressed imaging with a multi resolution statistical image model using both the generalized gaussian distribution and the two state gaussian mixture distribution we subsequently focus on the gaussian encoder bayesian optimal approximate message passing amp decoder pair whose theoretical sd function is provided by the rigorous analysis of the amp algorithm given the image statistics analytic bandwise sample allocation for bandwise independent model is derived as a reverse water filling scheme som and schniter s turbo message passing approach is further deployed to integrate the bandwise sampling with the exploitation of the hidden markov tree structure of wavelet coefficients natural image simulations confirm that with oracle image statistics the sd function associated with the optimized sample allocation can accurately predict the possible compressed sensing gains finally a general sample allocation profile based on average image statistics not only illustrates preferable performance but also makes the scheme practical

volume comparison and its applications in riemann finsler geometry abstract we introduce a new geometric quantity the mean covariation for finsler metrics and establish a volume comparison theorem as an application we obtain some precompactness and finiteness theorems for finsler manifolds

distortion rate function of sub nyquist sampled gaussian sources the amount of information lost in sub nyquist sampling of a continuous time gaussian stationary process is quantified we consider a combined source coding and sub nyquist reconstruction problem in which the input to the encoder is a noisy sub nyquist sampled version of the analog source we first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate limited version of its samples this expression is a function of the sampling frequency and the average number of bits describing each sample it is given as the sum of two terms minimum mean square error in estimating the source from its noisy but otherwise fully observed sub nyquist samples and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source we extend this result to multi branch uniform sampling where the samples are available through a set of parallel channels with a uniform sampler and a pre sampling filter in each branch further optimization to reduce distortion is then performed over the pre sampling filters and an optimal set of pre sampling filters associated with the statistics of the input signal and the sampling frequency is found this results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering these results thus unify the shannon whittaker kotelnikov sampling theorem and shannon rate distortion theory for gaussian sources

reconstructing gaussian sources by spatial sampling consider a gaussian memoryless multiple source gmms with m components with joint probability distribution known only to lie in a given class of distributions a subset of k leq m components is sampled and compressed with the objective of reconstructing all the m components within a specified level of distortion under a mean squared error criterion in bayesian and nonbayesian settings the notion of universal sampling rate distortion function for gaussian sources is introduced to capture the optimal tradeoffs among sampling compression rate and distortion level single letter characterizations are provided for the universal sampling rate distortion function our achievability proofs highlight the following structural property it is optimal to compress and reconstruct first the sampled components of the gmms alone and then form estimates for the unsampled components based on the former

sensor placement by maximal projection on minimum eigenspace for linear inverse problems this paper presents two new greedy sensor placement algorithms named minimum nonzero eigenvalue pursuit mnep and maximal projection on minimum eigenspace mpme for linear inverse problems with greater emphasis on the mpme algorithm for performance comparison with existing approaches in both mnep and mpme we select the sensing locations one by one in this way the least number of required sensor nodes can be determined by checking whether the estimation accuracy is satisfied after each sensing location is determined for the mpme algorithm the minimum eigenspace is defined as the eigenspace associated with the minimum eigenvalue of the dual observation matrix for each sensing location the projection of its observation vector onto the minimum eigenspace is shown to be monotonically decreasing w r t the worst case error variance wcev of the estimated parameters we select the sensing location whose observation vector has the maximum projection onto the minimum eigenspace of the current dual observation matrix the proposed mpme is shown to be one of the most computationally efficient algorithms our monte carlo simulations showed that mpme outperforms the convex relaxation method the sparsense method and the framesense method in terms of wcev and the mean square error mse of the estimated parameters especially when the number of available sensor nodes is very limited

on stability of approximation methods for the muskhelishvili equation we study approximation methods for the muskhelishvili integral equations on curves with comer points and establish necessary and sufficient conditions for their stability

minimal surfaces in a randers sphere with the rotational killing vector field abstract the minimal surfaces in finsler geometry with respect to the busemann hausdorff measure and the holmes thompson measure are called bh minimal and ht minimal surfaces respectively let p 1 p 2 p 3 p 4 be the coordinates of r 4 and s 3 f be a randers sphere of flag curvature k 1 with the navigation data h w where h is the standard sphere metric and w e 0 0 p 4 p 3 0 e 1 is a killing vector field in this paper we study the rotationally invariant minimal surface in s 3 f generated by rotating the curve x s y s z s 0 in the upper half sphere of s 2 around the p 1 p 2 plane s r we first show that such a rotational bh minimal surface in s 3 f is either a great 2 sphere or the catenoid in s 3 h then we give a classification of the rotational ht minimal surfaces where we use the angle data to analyze the solutions of the system of ode that characterizes the ht minimality and prove that such a rotational ht minimal surface must be a great 2 sphere an ht minimal torus or a rotational surface of unduloid type as a special case we obtain a distinguished embedded compact ht minimal torus depending on e the completeness of these surfaces is also studied

support recovery with sparsely sampled free random matrices consider a bernoulli gaussian complex n vector whose components are vi xibi with xi c n 0 px and binary bi mutually independent and iid across i this random q sparse vector is multiplied by a square random matrix u and a randomly chosen subset of average size n p p 0 1 of the resulting vector components is then observed in additive gaussian noise we extend the scope of conventional noisy compressive sampling models where u is typically a matrix with iid components to allow u satisfying a certain freeness condition this class of matrices encompasses haar matrices and other unitarily invariant matrices we use the replica method and the decoupling principle of guo and verdu as well as a number of information theoretic bounds to study the input output mutual information and the support recovery error rate in the limit of n we also extend the scope of the large deviation approach of rangan and characterize the performance of a class of estimators encompassing thresholded linear mmse and l1 relaxation

recovery from linear measurements with complexity matching universal signal estimation we study the compressed sensing cs signal estimation problem where an input signal is measured via a linear matrix multiplication under additive noise while this setup usually assumes sparsity or compressibility in the input signal during recovery the signal structure that can be leveraged is often not known a priori in this paper we consider universal cs recovery where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself inspired by kolmogorov complexity and minimum description length we focus on a maximum a posteriori map estimation framework that leverages universal priors to match the complexity of the source our framework can also be applied to general linear inverse problems where more measurements than in cs might be needed we provide theoretical results that support the algorithmic feasibility of universal map estimation using a markov chain monte carlo implementation which is computationally challenging we incorporate some techniques to accelerate the algorithm while providing comparable and in many cases better reconstruction quality than existing algorithms experimental results show the promise of universality in cs particularly for low complexity sources that do not exhibit standard sparsity or compressibility

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

sampling rate distortion consider a discrete memoryless multiple source with m components of which k leq m possibly different sources are sampled at each time instant and jointly compressed in order to reconstruct all the m sources under a given distortion criterion a new notion of sampling rate distortion function is introduced and is characterized first for the case of fixed set sampling next for independent random sampling performed without knowledge of the source outputs it is shown that the sampling rate distortion function is the same regardless of whether or not the decoder is informed of the sequence of sampled sets furthermore memoryless random sampling is considered with the sampler depending on the source outputs and with an informed decoder it is shown that deterministic sampling characterized by a conditional point mass is optimal and suffices to achieve the sampling rate distortion function for memoryless random sampling with an uninformed decoder an upper bound for the sampling rate distortion function is seen to possess a similar property of conditional point mass optimality it is shown by example that memoryless sampling with an informed decoder can outperform strictly any independent random sampler and that memoryless sampling can do strictly better with an informed decoder than without

intersecting random translates of invariant cantor sets given two cantor setsx andy in 0 1 invariant under the mapx b x mod 1 the hausdorff dimension of x t y is constant almost everywhere whenx y are defined by admissible digits in baseb and more generally by sofic systems we compute this dimension in terms of the largest lyapunov exponent of a random product of matrices the results are extended to higher dimensions and multiple intersections

a boundary integral formulation and 2d fundamental solutions for piezoelectric media abstract first based on the basic equations of piezoelectricity the boundary integral formulation is derived by utilizing the reciprocal work theorem second for the plane problem of piezoelectric media one general solution in terms of harmonic functions is derived and the fundamental solutions are therefore obtained numerical calculations by boundary element method bem are performed to give the stress concentration coefficient of an infinite piezoelectric plane with a circular hole as well as the stress and electric intensity factors of a central crack locating on an infinite plate results are found to agree well with the exact solutions

a rigorous approach to high resolution entropy constrained vector quantization the nonnegativity of relative entropy implies that the differential entropy of a random vector x with probability density function pdf f is upper bounded by mathsf e left log g text x right for any arbitrary pdf g using this inequality with a cleverly chosen g we derive a lower bound on the asymptotic excess rate of entropy constrained vector quantization for d dimensional sources and r th power distortion where the asymptotic excess rate is defined as the difference between the smallest output entropy of a vector quantizer satisfying the distortion constraint and the rate distortion function in the limit as the distortion tends to zero specialized to the one dimensional case this lower bound coincides with the asymptotic excess rate achieved by a uniform quantizer thereby recovering the result by gish and pierce that uniform quantizers are asymptotically optimal as the allowed distortion tends to zero furthermore in the one dimensional case the derivation of the lower bound reveals a necessary condition for a sequence of quantizers to be asymptotically optimal this condition implies that any sequence of asymptotically optimal almost regular quantizers must converge to a uniform quantizer as the distortion tends to zero while the obtained lower bound itself is not novel to the best of our knowledge we present the first rigorous derivation that follows the direct approach by gish and pierce without resorting to heuristic high resolution approximations commonly found in the quantization literature furthermore our derivation holds for all d dimensional sources having finite differential entropy and whose integer part has finite entropy in contrast to gish and pierce we do not require additional constraints on the continuity or decay of the source pdf

functional properties of minimum mean square error and mutual information in addition to exploring its various regularity properties we show that the minimum mean square error mmse is a concave functional of the input output joint distribution in the case of additive gaussian noise the mmse is shown to be weakly continuous in the input distribution and lipschitz continuous with respect to the quadratic wasserstein distance for peak limited inputs regularity properties of mutual information are also obtained several applications to information theory and the central limit theorem are discussed

analysis and design of optimum sparse array configurations for adaptive beamforming in this paper we analyze the effect of nonuniform array configurations on adaptive beamforming for enhanced signal to interference plus noise ratio sinr the array is configured using a given number of antennas or through a selection of subset of antennas from a larger available set leading to a sparse array in both cases the bounds on the highest achievable sinr for a given number of antennas are formulated and used to offer new insights into open loop adaptive beamforming the upper and lower bounds underscore the role of array configurations in optimizing performance for interference free and interference active environments respectively this paper considers the general case of multiple sources and interferences in the field of view we formulate three angles namely eigenspace angle conventional angle and minimum canonical angle for characterizing spatial separation between the source and interference subspaces which represent performance loss incurred by interference nulling three sparse array design methods incorporating these angles are proposed simulation examples confirm the role of subspace angles in optimum beamforming and validate the utility of antenna selection algorithms for sparse array design

fundamental distortion limits of analog to digital compression representing a continuous time signal by a set of samples is a classical problem in signal processing we study this problem under the additional constraint that the samples are quantized or compressed in a lossy manner under a limited bitrate budget to this end we consider a combined sampling and source coding problem in which an analog stationary gaussian signal is reconstructed from its encoded samples these samples are obtained by a set of bounded linear functionals of the continuous time path with a limitation on the average number of samples per unit time given in this setting we provide a full characterization of the minimal distortion in terms of the sampling frequency the bitrate and the signal s spectrum assuming that the signal s energy is not uniformly distributed over its spectral support we show that for each compression bitrate there exists a critical sampling frequency smaller than the nyquist rate such that the distortion in signal reconstruction when sampling at this frequency is minimal our results can be seen as an extension of the classical sampling theorem for bandlimited random processes in the sense that they describe the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples and provide the minimal sampling frequency required in order to achieve this distortion finally we compare the fundamental limits in the combined source coding and sampling problem to the performance of pulse code modulation where each sample is quantized by a scalar quantizer using a fixed number of bits

the plenacoustic function and its sampling the spatialization of the sound field in a room is studied in particular the evolution of room impulse responses as a function of their spatial positions it was observed that the multidimensional spectrum of the solution of the wave equation has an almost bandlimited character therefore sampling and interpolation can easily be applied using signals on an array the decay of the spectrum is studied on both temporal and spatial frequency axes the influence of the decay on the performance of the interpolation is analyzed based on the support of the spectrum the number and the spacing between the microphones is determined for the reconstruction of the sound pressure field up to a certain temporal frequency and with a certain reconstruction quality the optimal sampling pattern for the microphone positions is given for the linear planar and three dimensional case existing techniques usually make use of room models to recreate the sound field present at some point in the space the presented technique simply starts from the measurements of the sound pressure field in a finite number of positions and with this information the sound pressure field can be recreated at any spatial position finally simulations and experimental results are presented and compared with the theory

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

polyhedral finsler spaces with locally unique geodesics abstract we study finsler pl spaces that is simplicial complexes glued out of simplices cut off from some normed spaces we are interested in the class of finsler pl spaces featuring local uniqueness of geodesics for complexes made of euclidean simplices this property is equivalent to local cat 0 though non euclidean normed spaces never satisfy cat 0 it turns out that they share many common features in particular a globalization theorem holds in a simply connected finsler pl space local uniqueness of geodesics implies the global one however the situation is more delicate here some basic convexity properties do not extend to the pl finsler case

on sampling and coding for distributed acoustic sensing the issue of how to efficiently represent the data collected by a network of microphones recording spatio temporal acoustic wave fields is addressed each sensor node in the network samples the sound field quantizes the samples and transmits the encoded samples to some central unit which computes an estimate of the original sound field based on the information received from all the microphones our analysis is based on the spectral properties of the sound field which are induced by the physics of wave propagation and have a significant impact on the efficiency of the chosen sampling lattice and coding scheme as field acquisition by a sensor network typically implies spatio temporal sampling of the field a multidimensional sampling theorem for homogeneous random fields with compactly supported spectral measures is proved to assess the loss of information implied by source coding rate distortion functions for various coding schemes and sampling lattices are determined in particular centralized coding independent coding and some multiterminal schemes are compared under the assumption of spectral whiteness of the sound field it is shown that sampling with a quincunx lattice followed by independent coding is optimal as it achieves the lower bound given by centralized coding

an improved outer bound for multiterminal source coding we prove a new outer bound on the rate distortion region for the multiterminal source coding problem this bound subsumes the best outer bound in the literature and improves upon it strictly in some cases the improved bound enables us to obtain a new conclusive result for the binary erasure version of the ceo problem the bound recovers many of the converse results that have been established for special cases of the problem including the recent one for the gaussian two encoder problem

randomly spread cdma asymptotics via statistical physics this paper studies randomly spread code division multiple access cdma and multiuser detection in the large system limit using the replica method developed in statistical physics arbitrary input distributions and flat fading are considered a generic multiuser detector in the form of the posterior mean estimator is applied before single user decoding the generic detector can be particularized to the matched filter decorrelator linear minimum mean square error mmse detector the jointly or the individually optimal detector and others it is found that the detection output for each user although in general asymptotically non gaussian conditioned on the transmitted symbol converges as the number of users go to infinity to a deterministic function of a hidden gaussian statistic independent of the interferers thus the multiuser channel can be decoupled each user experiences an equivalent single user gaussian channel whose signal to noise ratio snr suffers a degradation due to the multiple access interference mai the uncoded error performance e g symbol error rate and the mutual information can then be fully characterized using the degradation factor also known as the multiuser efficiency which can be obtained by solving a pair of coupled fixed point equations identified in this paper based on a general linear vector channel model the results are also applicable to multiple input multiple output mimo channels such as in multiantenna systems

on the rate distortion function of random vectors and stationary sources with mixed distributions the asymptotic small distortion behavior of the rate distortion function of an n dimensional source vector with mixed distribution is derived the source distribution is a finite mixture of components such that under each component distribution a certain subset of the coordinates have a discrete distribution while the remaining coordinates have a joint density the expected number of coordinates with a joint density is shown to equal the rate distortion dimension of the source vector also the exact small distortion asymptotic behavior of the rate distortion function of a special but interesting class of stationary information sources is determined

multidimensional manhattan sampling and reconstruction this paper introduces manhattan sampling in two and higher dimensions and proves sampling theorems for them in 2 d manhattan sampling which takes samples densely along a manhattan grid of lines can be viewed as sampling on the union of two rectangular lattices one dense horizontally and the other vertically with the coarse spacing of each being a multiple of the fine spacing of the other the sampling theorem shows that the images bandlimited to the union of the nyquist regions of the two rectangular lattices can be recovered from their manhattan samples and an efficient procedure for doing so is given such recovery is possible even though there is an overlap among the spectral replicas induced by manhattan sampling in three and higher dimensions there are many possible configurations for manhattan sampling each consisting of the union of special rectangular lattices called bi step lattices this paper identifies them proves a sampling theorem showing that the images bandlimited to the union of the nyquist regions of the bi step rectangular lattices are recoverable from manhattan samples presents an efficient onion peeling procedure for doing so and shows that the union of nyquist regions is as large as any bandlimited region such that all images supported by such can be stably reconstructed from samples taken at the rate of the manhattan sampling it also develops a special representation for the bi step lattices with a number of useful properties while most of this paper deals with continuous space images manhattan sampling of discrete space images is also considered for infinite as well as finite support images

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

implementation of trefftz method for the solution of some elliptic boundary value problems the purpose of this paper is to implement the well known trefftz method for the approximate solution of some elliptic boundary value problems for the two dimensional laplace s and the biharmonic operators through the use of polar harmonics as trial functions four applications treating four different situations are presented the dirichlet problem for the steady state temperature distribution in an infinite elliptic cylinder giving rise to poisson s or laplace s equations with discontinuous boundary function for the complete and the truncated ellipses cases 1 2 and the first and the second fundamental problems of the plane strain theory of elastostatics for an infinite cylinder with rectangular normal cross section cases 3 4 for two of the examples cases 1 3 the results could be checked against known exact solutions to show in particular that the method can sometimes give the exact solution while for the other two cases 2 4 the method has provided solutions for new problems it is shown that the use of polar harmonics possibly together with other harmonics belonging to different coordinate systems can handle many situations in an efficient way i e by using few trial functions

band limited signal extrapolation by truncated bernstein polynomials abstract an algorithm is given for everywhere extrapolating a band limited signal known only on an interval of arbitrary finite length the scheme utilizes a finite number of equally spaced samples of the given function and provides a time limited polynomial approximation the approximation functions are shown to converge everywhere pointwise and uniformly in any compact interval to the band limited signal when the original band limited signal is also lebesgue integrable it is also established that the fourier transform of the approximating signal converges uniformly to the fourier transform of the original signal

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

smallest singular value of a random rectangular matrix we prove an optimal estimate of the smallest singular value of a random sub gaussian matrix valid for all dimensions for an nn matrix a with inde pendent and identically distributed sub gaussian entries the smallest singular value of a is at least of the order p np n 1 with high probability a sharp

minimal surfaces in some 4 dimensional general α β spaces abstract in this paper we study minimal submanifolds in some general spaces we firstly establish the equations for minimal submanifolds in a general space then we construct some kinds of minimal surfaces in some 4 dimensional general space

metric mean dimension and analog compression wu and verdu developed a theory of almost lossless analog compression where one imposes various regularity conditions on the compressor and the decompressor with the input signal being modelled by a typically infinite entropy stationary stochastic process in this work we consider all stationary stochastic processes with trajectories in a prescribed set of bi infinite sequences and find uniform lower and upper bounds for certain compression rates in terms of metric mean dimension and mean box dimension an essential tool is the recent lindenstrauss tsukamoto variational principle expressing metric mean dimension in terms of rate distortion functions we obtain also lower bounds on compression rates for a fixed stationary process in terms of the rate distortion dimension rates and study several examples

a statistical mechanics approach to large system analysis of cdma multiuser detectors we present a theory based on statistical mechanics to evaluate analytically the performance of uncoded fully synchronous randomly spread code division multiple access cdma multiuser detectors with additive white gaussian noise awgn channel under perfect power control and in the large system limit application of the replica method a tool developed in the literature of statistical mechanics allows us to derive analytical expressions for the bit error rate as well as the multiuser efficiency of the individually optimum io and jointly optimum jo multiuser detectors over the whole range of noise levels the information theoretic capacity of the randomly spread cdma channel and the performance of decorrelating and linear minimum mean square error mmse detectors are also derived in the same replica formulation thereby demonstrating validity of the statistical mechanical approach

the entropy of a randomly stopped sequence a wald like equation is proved for the entropy of a randomly stopped sequence of independent identically distributed discrete random variables x sub 1 x sub 2 with a nonanticipating stopping time n the authors first define a general stopping time and the associated stopped sequence and then present the two main theorems for the entropy of a stopped sequence the formal proofs of the lemmas necessary for the proof of the theorems are given the randomness in the stopped sequence x sup n is the expected number of calls for x times the entropy per call plus the residual randomness in the stopping time conditioned on the unstopped sequence x sup infinity

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

on classical analogues of free entropy dimension abstract we define a classical probability analogue of voiculescu s free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on r n we show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension first it can be viewed as a kind of fractal dimension second if one extends bochner s inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical bochner s inequalities then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in bochner s inequality for a measure regularized by convolution with the gaussian law as the regularization is removed we introduce a free analogue of the bochner inequality and study the related free entropy dimension quantity we show that it is greater or equal to the non microstates free entropy dimension

sampling theorems for two dimensional isotropic random fields sampling theorems are developed for isotropic random fields and their associated fourier coefficient processes a wave number limited isotropic random field z r is considered whose spectral density function is zero outside a disk of radius b centered at the origin of the wavenumber plane z r can be reconstructed in the mean square sense from its observation on the countable number of circles with radius r sub i i pi b i in n or of radius r sub i a sub i n b i in n where a sub 1 n denotes the ith zero of the nth order bessel function j sub n x and n is arbitrary

degrees of freedom in multiple antenna channels a signal space approach multiple antenna systems that are limited by the area and geometry of antenna arrays are considered given these physical constraints the limit on the available number of spatial degrees of freedom is derived the commonly used statistical multiple input multiple output mimo model is inadequate antenna theory is applied to take into account the area and geometry constraints and to define the spatial signal space so as to interpret experimental channel measurements in an array independent but manageable description of the physical environment based on these modeling strategies for a spherical array of effective aperture a in a physical environment of angular spread spl omega in solid angle the number of spatial degrees of freedom is shown to be a spl omega for uni polarized antennas and 2a spl omega for tri polarized antennas together with the 2wt degrees of freedom for a system of bandwidth w transmitting in an interval t the total degrees of freedom of a multiple antenna channel is therefore 4wta spl omega

the distributed karhunen loeve transform the karhunen loeve transform klt is a key element of many signal processing and communication tasks many recent applications involve distributed signal processing where it is not generally possible to apply the klt to the entire signal rather the klt must be approximated in a distributed fashion this paper investigates such distributed approaches to the klt where several distributed terminals observe disjoint subsets of a random vector we introduce several versions of the distributed klt first a local klt is introduced which is the optimal solution for a given terminal assuming all else is fixed this local klt is different and in general improves upon the marginal klt which simply ignores other terminals both optimal approximation and compression using this local klt are derived two important special cases are studied in detail namely the partial observation klt which has access to a subset of variables but aims at reconstructing them all and the conditional klt which has access to side information at the decoder we focus on the jointly gaussian case with known correlation structure and on approximation and compression problems then the distributed klt is addressed by considering local klts in turn at the various terminals leading to an iterative algorithm which is locally convergent sometimes reaching a global optimum depending on the overall correlation structure for compression it is shown that the classical distributed source coding techniques admit a natural transform coding interpretation the transform being the distributed klt examples throughout illustrate the performance of the proposed distributed klt this distributed transform has potential applications in sensor networks distributed image databases hyper spectral imagery and data fusion

mmse dimension if n is standard gaussian the minimum mean square error mmse of estimating a random variable x based on snr x n vanishes at least as fast as 1 snr as snr we define the mmse dimension of x as the limit as snr of the product of snr and the mmse mmse dimension is also shown to be the asymptotic ratio of nonlinear mmse to linear mmse for discrete absolutely continuous or mixed distribution we show that mmse dimension equals renyi s information dimension however for a class of self similar singular x e g cantor dis tribution we show that the product of snr and mmse oscillates around information dimension periodically in snr db we also show that these results extend considerably beyond gaussian noise under various technical conditions

minimal surfaces of rotation in finsler space with a randers metric we consider finsler spaces with a randers metric f on the three dimensional real vector space where is the euclidean metric and bdx3 is a 1 form with norm b 0 b 1 by using the notion of mean curvature for immersions in finsler spaces introduced by z shen we get the ordinary differential equation that characterizes the minimal surfaces of rotation around the x3 axis we prove that for every b 0 b 1 there exists up to homothety a unique forward complete minimal surface of rotation the surface is embedded symmetric with respect to a plane perpendicular to the rotation axis and it is generated by a concave plane curve moreover for every there are non complete minimal surfaces of rotation which include explicit minimal cones

rate distortion theory of finite point processes we study the compression of data in the case where the useful information is contained in a set rather than a vector i e the ordering of the data points is irrelevant and the number of data points is unknown our analysis is based on rate distortion theory and the theory of finite point processes we introduce fundamental information theoretic concepts and quantities for point processes and present general lower and upper bounds on the rate distortion function to enable a comparison with the vector setting we concretize our bounds for point processes of fixed cardinality in particular we analyze a fixed number of unordered gaussian data points and show that we can significantly reduce the required rates compared to the best possible compression strategy for gaussian vectors as an example of point processes with variable cardinality we study the best possible compression of poisson point processes for the specific case of a poisson point process with uniform intensity on the unit square our lower and upper bounds are separated by only a small gap and thus provide a good characterization of the rate distortion function

a proof of the fisher information inequality via a data processing argument the fisher information j x of a random variable x under a translation parameter appears in information theory in the classical proof of the entropy power inequality epi it enters the proof of the epi via the de bruijn identity where it measures the variation of the differential entropy under a gaussian perturbation and via the convolution inequality j x y sup 1 spl ges j x sup 1 j y sup 1 for independent x and y known as the fisher information inequality fii the fii is proved in the literature directly in a rather involved way we give an alternative derivation of the fii as a simple consequence of a data processing inequality for the cramer rao lower bound on parameter estimation

sampling and reconstruction of spatial fields using mobile sensors spatial sampling is traditionally studied in a static setting where static sensors scattered around space take measurements of the spatial field at their locations in this paper we study the emerging paradigm of sampling and reconstructing spatial fields using sensors that move through space we show that mobile sensing offers some unique advantages over static sensing in sensing bandlimited spatial fields since a moving sensor encounters such a spatial field along its path as a time domain signal a time domain anti aliasing filter can be employed prior to sampling the signal received at the sensor such a filtering procedure when used by a configuration of sensors moving at constant speeds along equispaced parallel lines leads to a complete suppression of spatial aliasing in the direction of motion of the sensors we analytically quantify the advantage of using such a sampling scheme over a static sampling scheme by computing the reduction in sampling noise due to the filter we also analyze the effects of nonuniform sensor speeds on the reconstruction accuracy using simulation examples we demonstrate the advantages of mobile sampling over static sampling in practical problems we extend our analysis to sampling and reconstruction schemes for monitoring time varying bandlimited fields using mobile sensors we demonstrate that in some situations we require a lower density of sensors when using a mobile sensing scheme instead of the conventional static sensing scheme the exact advantage is quantified for a problem of sampling and reconstructing an audio field

rate distortion behavior of sparse sources the rate distortion behavior of sparse memoryless sources is studied these serve as models of sparse signal representations and facilitate the performance analysis of sparsifying transforms like the wavelet transform and nonlinear approximation schemes for strictly sparse binary sources with hamming distortion r d is shown to be almost linear for nonstrictly sparse continuous valued sources termed compressible two measures of compressibility are introduced incomplete moments and geometric mean the former lead to low and high rate upper bounds on mean squared error d r while the latter yields lower and upper bounds on source entropy thereby characterizing asymptotic r d behavior thus the notion of compressibility is quantitatively connected with actual lossy compression these bounding techniques are applied to two source models gaussian mixtures and power laws matching the approximately scale invariant decay of wavelet coefficients the former are versatile models for sparse data which in particular allow to bound high rate compression performance of a scalar mixture compared to a corresponding unmixed transform coding system such a comparison is interesting for transforms with known coefficient decay but unknown coefficient ordering e g when positions of highest variance coefficients are unknown the use of these models and results in distributed coding and compressed sensing scenarios are also discussed

sampling high dimensional bandlimited fields on low dimensional manifolds consider the task of sampling and reconstructing a bandlimited spatial field in r2 using moving sensors that take measurements along their path it is inexpensive to increase the sampling rate along the paths of the sensors but more expensive to increase the total distance traveled by the sensors per unit area which we call the path density in this paper we introduce the problem of designing sensor trajectories that are minimal in path density subject to the condition that the measurements of the field on these trajectories admit perfect reconstruction of bandlimited fields we study various possible designs of sampling trajectories generalizing some ideas from the classical theory of sampling on lattices we obtain necessary and sufficient conditions on the trajectories for perfect reconstruction we show that a single set of equispaced parallel lines has the lowest path density from certain restricted classes of trajectories that admit perfect reconstruction we then generalize some of our results to higher dimensions we first obtain results on designing sampling trajectories in higher dimensional fields further interpreting trajectories as 1 d manifolds we extend some of our ideas to higher dimensional sampling manifolds we formulate the problem of designing dimensional sampling manifolds for d dimensional spatial fields that are minimal in manifold density a natural generalization of the path density we show that our results on sampling trajectories for fields in r2 can be generalized to analogous results on d dimensional sampling manifolds for d dimensional spatial fields

nontrivial minimal surfaces in a class of finsler spheres of vanishing s curvature abstract the minimal surface theory in finsler geometry deserves to be well developed as that in riemannian geometry in this paper we derive the mean curvature of submanifolds in a general manifold by considering the busemann hausdorff measure and holmes thompson measure respectively we then study the rotationally invariant minimal surfaces in the finsler 3 sphere endowed with an metric f k k k k k 1 where is a smooth function s 3 k is the berger sphere e 4 k 1 and k is a killing one form of constant length along the hopf fibers of s 3 we define the energy of the minimal surfaces and by using the volume ratio function introduced by the author and y b shen we give the explicit local expressions of the rotationally invariant bh minimal and ht minimal surfaces in such sphere respectively as a special case we give a detailed study of the rotationally invariant ht minimal surfaces in the 3 sphere with square metric

on source coding for distributed temperature sensing with shift invariant geometries we study the source coding problem in sensor networks deployed to monitor the evolution of spatio temporal temperature distributions the sensors sample the temperature field quantize the samples and transmit the encoded samples through digital channels to some central unit which computes an estimate of the original temperature field our analysis is based on the heat kernel s spectral properties which are induced by the physics of heat diffusion we determine rate distortion functions for various source coding schemes in particular we compare centralized coding independent coding berger tung coding and predictive quantization

block definition approaches to source coding with a fidelity criterion we give a new variable rate and variable distortion coding scheme and show a coding theorem for time discrete stationary sources with abstract alphabets and a single letter fidelity criterion without assuming reference letters the approach used to prove the coding theorem is then specialized to maximum distortion coding and to fixed rate coding and the corresponding coding theorems are proved for stationary sources we also consider the possibility of extending the results to time continuous sources we still need a reference letter for fixed rate coding

the rate distortion dimension of sets and measures data compression of independent samples drawn from a fractal set is considered the asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution this quantity is shown to be identical to renyi s 1959 information dimension for self similar fractal sets this dimension is distribution dependent in sharp contrast with the behavior of absolutely continuous measures the rate distortion dimension of a set is defined as the maximal rate distortion dimension for distributions supported on this set kolmogorov s metric dimension is an upper bound on the rate distortion dimension while the hausdorff dimension is a lower bound examples of sets for which the rate distortion dimension differs from these bounds are provided

the distortion rate function of sampled wiener processes we consider the recovery of a continuous time wiener process from a quantized or a lossy compressed version of its uniform samples under limited bitrate and sampling rate we derive a closed form expression for the optimal tradeoff among sampling rate bitrate and quadratic distortion in this setting this expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite rank operators associated with the optimal estimator of the wiener process from its samples we show that the ratio between this expression and the standard distortion rate function of the wiener process describing the optimal tradeoff between bitrate and distortion without a sampling constraint is only a function of the number of bits per sample we also consider a sub optimal lossy compression scheme in which the continuous time process is estimated from the output of an encoder that is optimal with respect to the discrete time samples we show that the latter is strictly greater than the distortion under optimal encoding but only by at most 3 we therefore conclude that near optimal performance is attained even if the encoder is unaware of the continuous time origin of the samples

a computable fourier condition generating alias free sampling lattices we propose a fourier analytical condition linking alias free sampling with the fourier transform of the indicator function defined on the given frequency support our discussions center around how to develop practical computation algorithms based on the proposed analytical condition we address several issues along this line including the derivation of simple closed form expressions for the fourier transforms of the indicator functions defined on arbitrary polygonal and polyhedral domains a complete and nonredundant enumeration of all quantized sampling lattices via the hermite normal forms of integer matrices and a quantitative analysis of the approximation of the original infinite fourier condition by using finite computations combining these results we propose a computational testing procedure that can efficiently search for the optimal alias free sampling lattices for a given polygonal or polyhedral shaped frequency domain several examples are presented to show the potential of the proposed algorithm in multidimensional filter bank design as well as in applications involving the design of efficient sampling patterns for multidimensional band limited signals