successive refinement with decoder cooperation and its channel coding duals we study cooperation in multiterminal source coding models involving successive refinement specifically we study the case of a single encoder and two decoders where the encoder provides a common description to both the decoders and a private description to only one of the decoders the decoders cooperate via cribbing i e the decoder with access only to the common description is allowed to observe in addition a deterministic function of the reconstruction symbols produced by the other we characterize the fundamental performance limits in the respective settings of noncausal strictly causal and causal cribbing we use a coding scheme referred to as forward encoding and block markov decoding which builds on one recently used by cuff and zhao for coordination via implicit communication finally we use the insight gained to introduce and solve some dual channel coding scenarios involving multiple access channels with cribbing

on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

on adjunctions between lim sl top and sl lim abstract consider l 1 be a commutative strictly two sided quantale with the underlying lattice l being meet continuous two adjunctions one is between limit spaces and stratified l limit spaces and the other is between stratified l limit spaces and stratified l topological spaces are established the first adjunction can be viewed as an extension of lowen s adjunction between the category of topological spaces and stratified 0 1 topological spaces the second is an extension of an adjunction between limit spaces and stratified l topological spaces established in u hohle and t kubiak hohle kubiak semigroup forum 2007

explicit inverse of a generalized vandermonde matrix in this paper the author gives an explicit closed form expression for the nxn inverse matrix v g k n 1 of the generalized nxn vandermonde matrix v g k n by using the elementary symmetric functions symbolic and numerical results are presented

duality between channel capacity and rate distortion with two sided state information we show that the duality between channel capacity and data compression is retained when state information is available to the sender to the receiver to both or to neither we present a unified theory for eight special cases of channel capacity and rate distortion with state information which also extends existing results to arbitrary pairs of independent and identically distributed i i d correlated state information s sub 1 s sub 2 available at the sender and at the receiver respectively in particular the resulting general formula for channel capacity c max sub p u x s sub 1 i u s sub 2 y i u s sub 1 assumes the same form as the generalized wyner ziv 1976 rate distortion function r d min sub p u x s sub 1 p x spl i cap u s sub 2 i u s sub 1 x 1 u s sub 2

lattice valued spaces fuzzy convergence abstract an alternative set of axioms is given for the study of lattice valued convergence spaces these axioms extend the notion of a probabilistic convergence space in the special case when the lattice is l 0 1 the category described here is shown to be topological cartesian closed and extensional

the categorical topology approach to fuzzy topology and fuzzy convergence it is the purpose of this paper to have a look at fuzzy topology from the point of view of categorical topology we first look at interesting subcategories of fts and determine whether they are bireflective and or coreflective we also describe the set of all subcategories of fts which are at the same time bireflective and coreflective due to the fact that top is one of these subcategories of fts ffs itself can be neither cartesian closed nor extensional we therefore secondly construct a supercategory of bts which is a topological quasitopos the construction of this quasitopos is based on the theory of convergence in fts

coordination capacity we develop elements of a theory of cooperation and coordination in networks rather than considering a communication network as a means of distributing information or of reconstructing random processes at remote nodes we ask what dependence can be established among the nodes given the communication constraints specifically in a network with communication rates ri j between the nodes we ask what is the set of all achievable joint distributions p x1 xm of actions at the nodes of the network several networks are solved including arbitrarily large cascade networks distributed cooperation can be the solution to many problems such as distributed games distributed control and establishing mutual information bounds on the influence of one part of a physical system on another

numerical differentiation from a viewpoint of regularization theory in this paper we discuss the classical ill posed problem of numerical differentiation assuming that the smoothness of the function to be differentiated is unknown using recent results on adaptive regularization of general ill posed problems we propose new rules for the choice of the stepsize in the finite difference methods and for the regularization parameter choice in numerical differentiation regularized by the iterated tikhonov method these methods are shown to be effective for the differentiation of noisy functions and the order optimal convergence results for them are proved

closed form expressions for the finite approximations of first and higher derivatives based on taylor series numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series in this paper we have presented closed form expressions of these approximations of arbitrary order for first and higher derivatives a comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses the comparison reveals that the central difference approximations can be used as digital differentiators because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator it is also observed that central difference approximations are in fact the same as maximally flat digital differentiators in the appendix a computer program written in mathematica is presented which can give the approximation of any order to the derivative of a function at a certain mesh point

duality between source coding and channel coding and its extension to the side information case we explore the information theoretic duality between source coding with side information at the decoder and channel coding with side information at the encoder we begin with a mathematical characterization of the functional duality between classical source and channel coding formulating the precise conditions under which the optimal encoder for one problem is functionally identical to the optimal decoder for the other problem we then extend this functional duality to the case of coding with side information by invoking this duality we are able to generalize the result of wyner and ziv 1976 relating to no rate loss for source coding with side information from gaussian to more arbitrary distributions we consider several examples corresponding to both discrete and continuous valued cases to illustrate our formulation for the gaussian cases of coding with side information we invoke geometric arguments to provide further insights into their duality our geometric treatment inspires the construction and dual use of practical coset codes for a large class of emerging applications for coding with side information such as distributed sensor networks watermarking and information hiding communication systems

pretopological and topological lattice valued convergence spaces we show that the classical axiom which characterizes pretopological convergence spaces splits into two axioms in the general heyting algebra valued case furthermore we present a generalization of kowalski s diagonal condition to the lattice valued case

mathematical proof of closed form expressions for finite difference approximations based on taylor series taylor series based finite difference approximations of derivatives of a function have already been presented in closed forms with explicit formulas for their coefficients however those formulas were not derived mathematically and were based on observation of numerical results in this paper we provide a mathematical proof of those formulas by deriving them mathematically from the taylor series

lattice valued convergence spaces and regularity we define a regularity axiom for lattice valued convergence spaces where the lattice is a complete heyting algebra to this end we generalize the characterization of regularity by a dual form of a diagonal condition we show that our axiom ensures that a regular t1 space is separated and that regularity is preserved under initial constructions further we present an extension theorem for a continuous mapping from a subspace to a regular space a characterization in the restricted case that the lattice is a complete boolean algebra in terms of the closure of an l filter is given

extrapolation methods and derivatives of limits of sequences let s m be an infinite sequence whose limit or antilimit s can be approximated very efficiently by applying a suitable extrapolation method e 0 to s m assume that the s m and hence also s are differentiable functions of some parameter d d s being the limit or antilimit of d d s m and that we need to approximate d d s a direct way of achieving this would be by applying again a suitable extrapolation method e 1 to the sequence d d s m and this approach has often been used efficiently in various problems of practical importance unfortunately as has been observed at least in some important cases when d d s m and s m have essentially different asymptotic behaviors as m the approximations to d d s produced by this approach despite the fact that they are good do not converge as quickly as those obtained for s and this is puzzling in this paper we first give a rigorous mathematical explanation of this phenomenon for the cases in which e 0 is the richardson extrapolation process and e 1 is a generalization of it thus showing that the phenomenon has very little to do with numerics following that we propose a procedure that amounts to first applying the extrapolation method e 0 to s m and then differentiating the resulting approximations to s and we provide a thorough convergence and stability analysis in conjunction with the richardson extrapolation process it follows from this analysis that the new procedure for d d s has practically the same convergence properties as e 0 for s we show that a very efficient way of implementing the new procedure is by actually differentiating the recursion relations satisfied by the extrapolation method used and we derive the necessary algorithm for the richardson extrapolation process we demonstrate the effectiveness of the new approach with numerical examples that also support the theory we discuss the application of this approach to numerical integration in the presence of endpoint singularities we also discuss briefly its application in conjunction with other extrapolation methods

multiple access channel with partial and controlled cribbing encoders in this paper we consider a multiple access channel mac with partial cribbing encoders this means that each of the two encoders obtains a deterministic function of the output of the other encoder with or without delay the partial cribbing scheme is especially motivated by the additive noise gaussian mac where perfect cribbing results in the degenerated case of full cooperation between the encoders and requires an infinite entropy link we derive a single letter characterization of the capacity of the mac with partial cribbing for the cases of causal and strictly causal cribbing several numerical examples such as those of quantized cribbing are presented we further consider and derive the capacity region where the cribbing depends on actions that are functions of the previous cribbed observations in particular we consider a scenario where the action is taken to decide to crib or not to crib and show that a naive time sharing strategy is not optimal

the discrete memoryless multiple access channel with partially cooperating encoders corresp we introduce the communication situation in which the encoders of a multiple access channel are partially cooperating these encoders are connected by communication links with finite capacities which permit both encoders to communicate with each other first we give a general definition of such a communication process conference then by proving a converse and giving an achievability proof we establish the capacity region of the multiple access channel with partially cooperating encoders it turns out that the optimal conference is very simple

diagonal conditions for lattice valued uniform convergence spaces we define diagonal conditions for lattice valued uniform convergence spaces and show that these conditions are preserved under initial constructions further the forgetful functor from the category of stratified lattice valued uniform convergence spaces into the category of stratified lattice valued limit spaces maps these conditions to corresponding conditions we especially define uniform regularity and characterize this condition by certain closures of lattice valued filters as an application we generalize an extension theorem for uniformly continuous mappings

the lower and upper p topological p regular modifications for lattice valued convergence spaces in this paper the notions of lower and upper topological regular modifications with respect to a stratified l generalized convergence structure and those with respect to a stratified l strong generalized convergence structure are defined and studied the basic properties and characterizations are presented in particular it is proved that the lower modifications behave reasonably well relative to final structures whereas the upper modifications exhibit comparable behavior relative to initial structures

cascade triangular and two way source coding with degraded side information at the second user in this paper we consider the cascade and triangular rate distortion problems where the same side information is available at the source node and user 1 and the side information available at user 2 is a degraded version of the side information at the source node and user 1 we characterize the rate distortion region for these problems for the cascade setup we show that at user 1 decoding and rebinning the codeword sent by the source node for user 2 is optimum we then extend our results to the two way cascade and triangular setting where the source node is interested in lossy reconstruction of the side information at user 2 via a rate limited link from user 2 to the source node we characterize the rate distortion regions for these settings complete explicit characterizations for all settings are given in the quadratic gaussian case we conclude with two further extensions a triangular source coding problem with a helper and an extension of our two way cascade setting in the quadratic gaussian case

source coding theory for a triangular communication system a rate distortion problem is considered for a triangular communication system tcs which has one encoder f and two decoders g sub x and g sub y the encoder f maps correlated source outputs x sup k y sup k to two codewords w sub x and w sub y which are sent to g sub x and g sub y respectively the decoders g sub x and g sub y can communicate with each other via rate constrained channels as many times as they need and g sub x reproduces x spl circ sup k while g sub y reproduces y spl circ sup k the admissible rate distortion region is determined for this tcs furthermore the relations between the tcs and the gray wyner 1974 system wyner s 1974 common information yamamoto s cascade communication system 1981 and the successive refinement system are discussed

new finite difference formulas for numerical differentiation conventional numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series and closed form expressions of these finite difference formulas have already been presented in this paper we present new finite difference formulas which are more accurate than the available ones especially for the oscillating functions having frequency components near the nyquist frequency closed form expressions of the new formulas are given for arbitrary order a comparison of the previously available three types of approximations is given with the presented formulas a computer program written in mathematica based on new formulas is given in the appendix for numerical differentiation of a function at a specified mesh point

achievable rates for multiple descriptions with feed forward the two channel multiple descriptions problem for an independent and identically distributed i i d source with feed forward to one or both side decoders is considered a single letter achievable rate region is derived it enlarges the best known rate region for multiple descriptions without feed forward the proof of the result uses a block markov superposition source coding strategy in point to point source coding feed forward does not decrease the rate distortion function of an i i d source in contrast an example is provided to show that the derived region can be strictly larger than the optimal multiple description rate distortion region without feed forward

p topologicalness and p regularity for lattice valued convergence spaces in this paper p topologicalness and p regularity for jager s stratified l generalized convergence spaces and those for boustique et al s stratified l convergence spaces are studied through two different methods one method generalizes the well known fischer s dual diagonal condition the other method uses the neighborhood closure of stratified l filter in addition the p topologicalness and p regularity w r t initial constructions and final constructions are also discussed

general explicit difference formulas for numerical differentiation through introducing the generalized vandermonde determinant the linear algebraic system of a kind of vandermonde equations is solved analytically by use of the basic properties of this determinant and then we present general explicit finite difference formulas with arbitrary order accuracy for approximating first and higher derivatives which are applicable to unequally or equally spaced data comparing with other finite difference formulas the new explicit difference formulas have some important advantages basic computer algorithms for the new formulas are given and numerical results show that the new explicit difference formulas are quite effective for estimating first and higher derivatives of equally and unequally spaced data

cascade and triangular source coding with side information at the first two nodes we consider the cascade and triangular rate distortion problem where side information is known to the source encoder and to the first user but not to the second user we characterize the rate distortion region for these problems as well as some of their extensions for the quadratic gaussian case we show that it is sufficient to consider jointly gaussian distributions which leads to an explicit solution

simplified analytical expressions for numerical differentiation via cycle index based on the lagrange interpolation to the function f x 0 for arbitrarily chosen x 0 and logarithmic differentiation we give a simple approach to analytical expressions for numerical differentiation using cycle index a detailed analysis for the remainder is also included

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

lattice valued convergence associated with cns spaces abstract in this paper in the context of meet continuous quantale we present a notion of conical topological lattice valued convergence space which provides the theory of lattice valued convergence associated with such conical neighborhood spaces

taylor series based finite difference approximations of higher degree derivatives a new type of taylor series based finite difference approximations of higher degree derivatives of a function are presented in closed forms with their coefficients given by explicit formulas for arbitrary orders characteristics and accuracies of presented approximations and already presented central difference higher degree approximations are investigated by performing example numerical differentiations it is shown that the presented approximations are more accurate than the central difference approximations especially for odd degrees however for even degrees central difference approximations become attractive as the coefficients of the presented approximations of even degrees do not correspond to equispaced input samples

stratified lmn convergence tower spaces we develop a general theory of convergence for lattice valued spaces based on the concept of s stratified lm filters for different choices of the frames l and m different kinds of filters arise and suitable choices of the lattice n allow to view many existing fuzzy and probabilistic convergence spaces as special examples of our stratified lmn convergence tower spaces the stratification requires a certain relation between the frames l and m and we use a so called stratification mapping to this end the stratification condition is used to show that the resulting category is fiber small and hence cartesian closedness is equivalent to the existence of natural function spaces we give several examples for our spaces for different choices of the lattices l m and n with a special focus on enriched lm fuzzy topological spaces finally we study diagonal axioms and regularity

on many valued stratified l fuzzy convergence spaces based on complete residuated lattices the relations between stratified l fuzzy convergence spaces and enriched l fuzzy topological spaces are studied it is shown that as a reflective subcategory the category of enriched l fuzzy topological spaces can embedded in the category of stratified l fuzzy convergence spaces also the specialization l preorder of stratified l fuzzy convergence spaces is investigated

lattice valued convergence spaces extending the lattice context we define a category of stratified l generalized convergence spaces for the case where the lattice is an enriched cl premonoid we then investigate some of its categorical properties and those of its subcategories in particular the stratified l principal convergence spaces and the stratified l topological convergence spaces for some results we need to introduce a new condition on the lattice which is always true in the case where the lattice is a frame but not always true in the more general case as examples where we may apply the more general lattice context we examine the stratified l topological spaces and probabilistic limit spaces we show that the category of stratified l topological spaces is a reflective subcategory of our category and that the category of probabilistic limit spaces under a t norm is both a reflective and a coreflective subcategory of our category if we choose the lattice context appropriately

derivative kernels numerics and applications a generalized framework for numerical differentiation nd is proposed for constructing a finite impulse response fir filter in closed form the framework regulates the frequency response of nd filters for arbitrary derivative order and cutoff frequency selected parameters relying on interpolating power polynomials and maximally flat design techniques compared with the state of the art solutions such as gaussian kernels the proposed nd filter is sharply localized in the fourier domain with ripple free artifacts here we construct 2d maxflat kernels for image directional differentiation to calculate image differentials for arbitrary derivative order cutoff level and steering angle the resulted kernel library renders a new solution capable of delivering discrete approximation of gradients hessian and higher order tensors in numerous applications we tested the utility of this library on three different imaging applications with main focus on the unsharp masking the reported results highlight the high efficiency of the 2d maxflat kernel and its versatility with respect to robustness and parameter control accuracy

compactification of lattice valued convergence spaces we define compactness for stratified lattice valued convergence spaces and show that a tychonoff theorem is true further a generalization of the classical richardson compactification is given this compactification has a universal property

on diagonal completion of lattice valued diagonal cauchy spaces we define and study diagonal axioms for lattice valued cauchy spaces a completion of a weakly diagonal lattice valued cauchy space is constructed which is weakly diagonal and the coarsest among such completions it is at the same time at least as fine as any weakly regular completion

modifications lattice valued structures several important topological concepts such as regularity local compactness and local boundedness are investigated in the category of lattice valued convergence spaces these definitions coincide with the usual ones in familiar categories

duality of a source coding problem and the semi deterministic broadcast channel with rate limited cooperation the wyner ahlswede korner wak empirical coordination problem where the encoders cooperate via a finite capacity one sided link is considered the coordination capacity region is derived by combining several source coding techniques such as wyner ziv coding binning and superposition coding furthermore a semi deterministic sd broadcast channel bc with one sided decoder cooperation is considered duality principles relating the two problems are presented and the capacity region for the sd bc setting is derived the direct part follows from an achievable region for a general bc that is tight for the sd scenario a converse is established by using telescoping identities the sd bc is shown to be operationally equivalent to a class of relay bcs and the correspondence between their capacity regions is established the capacity region of the sd bc is transformed into an equivalent region that is shown to be dual to the admissible region of the wak problem in the sense that the information measures defining the corner points of both regions coincide achievability and converse proofs for the equivalent region are provided for the converse we use a probabilistic construction of auxiliary random variables that depends on the distribution induced by the codebook several examples illustrate the results

largest and smallest t2 compactifications of lattice valued convergence spaces we give necessary and sufficient conditions for a lattice valued convergence space to possess a largest or a smallest t2 compactification

a common framework for lattice valued uniform spaces and probabilistic uniform limit spaces we study a category of lattice valued uniform convergence spaces where the lattice is enriched by two algebraic operations this general setting allows us to view the category of lattice valued uniform spaces as a reflective subcategory of our category and the category of probabilistic uniform limit spaces as a coreflective subcategory

a note on regularity an alternative characterization of gahler s definition of regularity of a lattice valued convergence space is given in particular a space is regular exactly whenever convergence of filters remains invariant with respect to taking closures verification employs a slight modification of the work by jager

a recursive algorithm for optimizing differentiation in this paper a recursive algorithm will be introduced to improve the differentiation method proposed by hasan et al 2012 their algorithm is based on the undetermined coefficient method and uses the taylor series expansion and vandermonde matrix inverse to calculate weighting coefficients with a complexity of o n 4 our method reduces time complexity from o n 4 to o n 3 moreover we obtain a substantial optimality when the required degree and the order of accuracy increase finally the implementation of the proposed method follows and experimentations prove the validity of the algorithm and compare previous and new methods in terms of execution time besides the combination of our optimization and parallel processing provides valuable results in real world applications particularly qrs detection in ecg signal processing has been considered and we have shown that how features of our method could be used in order to increase accuracy and speed for this application

information embedding on actions the problem of optimal actuation for channel and source coding was recently formulated and solved in a number of relevant scenarios in this class of models actions are taken at encoders or decoders either to acquire side information in an efficient way or to control or probe effectively the channel state in this paper the problem of embedding information on the actions is studied for both the source and the channel coding setups in both cases a decoder is present that observes only a function of the actions taken by an encoder or a decoder of an action dependent point to point link for the source coding model this decoder wishes to reconstruct a lossy version of the source being transmitted over the point to point link while for the channel coding problem the decoder wishes to retrieve a portion of the message conveyed over the link for the problem of source coding with actions taken at the decoder a single letter characterization of the set of all achievable tuples of rate distortions at the two decoders and action cost is derived under the assumption that the mentioned decoder observes a function of the actions noncausally strictly causally or causally a special case of the problem in which the actions are taken by the encoder is also solved a single letter characterization of the achievable capacity cost region is then obtained for the channel coding setup with actions examples are provided that shed light into the effect of information embedding on the actions for the action dependent source and channel coding problems

an algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy in this paper we introduce an algorithm and a computer code for numerical differentiation of discrete functions the algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points the algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas in particular when the derivatives are required with high approximation accuracy moreover the given matlab computer code can be implemented to solve boundary value ordinary and partial differential equations with high numerical accuracy the numerical technique is based on the undetermined coefficient method in conjunction with taylor s expansion to avoid the difficulty of solving a system of linear equations an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions this is done by using an explicit closed formula for the vandermonde matrix inverse moreover the code is designed to give a unified approximation order throughout the given domain a numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code it is found that the method and the code work properly for any degree of derivative and any order of accuracy

representation for the lagrangian numerical differentiation formula involving elementary symmetric functions by using elementary symmetric functions this paper presents an explicit representation for the lagrangian numerical differentiation formula as well as the error estimate for local approximation and we also point out that the numerical differentiation formula constructed by li j p li general explicit difference formulas for numerical differentiation j comput appl math 183 2005 29 52 is actually a special case of the lagrangian numerical differentiation formula to approximate the values of the derivatives at the nodes

regularity lattice valued cauchy spaces a category of lattice valued cauchy spaces is defined and its properties are investigated the relationship between this category and the recent work by jager is presented the notion of regularity in the cauchy space context is the primary emphasis here in particular the category consisting of the regular cauchy spaces is shown to be bireflective in the category of all lattice valued cauchy spaces having cauchy continuous maps as its morphisms moreover completions are also investigated

lattice valued convergence diagonal axioms diagonal axioms are defined in the category whose objects consist of all the lattice valued convergence spaces when the latter is linearly ordered a diagonal condition is given which characterizes those objects in the category that are determined by a probabilistic convergence space which is topological