design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

a class of convex fuzzy mappings based on the concept of ordering defined by goetschel and voxman fuzzy sets and systems 18 1996 31 in this paper we present the concepts of epigraph and convexity of the fuzzy mappings and describe characteristics of the convex fuzzy mappings and quasi convex fuzzy mappings then we discuss the properties of convex fuzzy optimizations

rate compatible puncturing of low density parity check codes in this correspondence we consider puncturing of low density parity check ldpc codes for additive white gaussian noise awgn channels we show that good puncturing patterns exist and that the puncturing can be performed in a rate compatible fashion furthermore rate compatible puncturing results in a small loss of performance with respect to threshold namely the punctured code is good in terms of threshold across a range of rates when compared with the optimal codes for each rate this allows one to implement a single mother encoder and decoder that is good across a wide range of rates

on integration of fuzzy mappings abstract in this paper we introduce the concept of an integral of a fuzzy mapping this generalises the integral introduced by matloka 10 and also extends riemann stieltjes integral over a closed interval to fuzzy mappings

invex and generalized convex fuzzy mappings in this paper we introduce the concepts of pseudo convexity invexity and pseudo invexity for fuzzy mappings of one variable based on the notion of differentiability proposed by goetschel and voxman 4 and investigate the relationship between convex fuzzy mappings preinvex fuzzy mappings and these classes of fuzzy mappings we shall prove that pseudo convex fuzzy mappings and invex fuzzy mappings are pseudo invex and that a differentiable convex resp preinvex fuzzy mapping is pseudo convex resp invex in addition sufficient optimality conditions are obtained for pseudo convex invex and pseudo invex fuzzy mappings

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

a note on generalized convexity for fuzzy mappings through a linear ordering in de campos ibanez and gonzalez munoz fuzzy sets syst 29 145 154 1989 6 goestschel and voxman fuzzy sets syst 18 31 43 1986 7 the authors considered a linear ordering on the space of fuzzy intervals for each fuzzy mapping fuzzy interval valued mapping f based on the aforementioned linear ordering they introduced a real valued function t f on the domain of the fuzzy mapping f recently chalco cano et al fuzzy sets syst 231 70 83 2013 4 have studied the relationship between the generalized hukuhara differentiability of a fuzzy mapping f g differentiability for short and the differentiability of t f and some properties of local global minima this paper studies such properties for fuzzy mappings using new concepts which generalize the existing ones

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

convex fuzzy mapping with differentiability and its application in fuzzy optimization abstract the concepts of differentiability convexity generalized convexity and minimization of a fuzzy mapping are known in the literature the purpose of this present paper is to extend and generalize these concepts to fuzzy mappings of several variables using buckley feuring approach for fuzzy differentiation and derive karush kuhn tucker condition for the constrained fuzzy minimization problem

nonconvex fuzzy mappings and the fuzzy pre variational inequality in this paper we introduce the concepts of fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping from r n to the set of fuzzy numbers and put forward the concepts of fuzzy extended directional differential subgradient and subdifferential some properties are obtained for fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping relations are discussed between the fuzzy pre variational inequality and the fuzzy vector optimization problem

duality theory in fuzzy mathematical programming problems with fuzzy coefficients in this paper the notions of subgradient subdifferential and differential with respect to convex fuzzy mappings are investigated which provides the basis for the fuzzy extremum problem theory we consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and or sufficient optimality conditions furthermore the concept of saddle points and minimax theorems under fuzzy environment is discussed the results obtained are used to formulate the lagrangian dual of fuzzy programming under certain fuzzy convexity assumptions kkt conditions for fuzzy programming are derived and the perturbed convex fuzzy programming is considered finally these results are applied to fuzzy linear programming and fuzzy quadratic programming

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

evaluate fuzzy optimization problems based on biobjective programming problems abstract the embedding theorem shows that the set of fuzzy numbers can be embedded into a banach space inspired by this embedding theorem we propose a solution concept of the fuzzy optimization problem by solving a biobjective optimization problem which is transformed from the original fuzzy optimization problem using the embedding theorem we then also consider the fuzzy optimization problem with fuzzy coefficients i e the coefficients are assumed to be fuzzy numbers under a setting of core values of fuzzy numbers we show that the optimal solution of its corresponding crisp optimization problem the usual optimization problem is also an optimal solution of the original fuzzy optimization problem

the universal operation of ldpc codes over scalar fading channels root and varaiya proved the existence of a code that can communicate reliably over any linear gaussian channel for which the channel mutual information level exceeds the transmitted rate this paper provides several examples of scalar single input single output fading channels and shows that on these channels the performance of low density parity check ldpc codes lies in close proximity to the performance limits identified by root and varaiya specifically we consider periodic fading channels and partial band jamming pbj channels a special case of periodic fading is the variation of signal to noise ratio across orthogonal frequency division modulation subchannels the robustness of ldpc codes to periodic fading and pbj across parameterizations of these different channels is demonstrated through the consistency of the required mutual information to provide a specified bit error rate for the periodic fading case the gaussian approximation to density evolution has been adapted such that asymptotic threshold measures can be compared to simulated code performance in various periodic fading scenarios

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

selective avoidance of cycles in irregular ldpc code construction this letter explains the effect of graph connectivity on error floor performance of low density parity check ldpc codes under message passing decoding a new metric called extrinsic message degree emd measures cycle connectivity in bipartite graphs of ldpc codes using an easily computed estimate of emd we propose a viterbi like algorithm that selectively avoids small cycle clusters that are isolated from the rest of the graph this algorithm is different from conventional girth conditioning by emphasizing the connectivity as well as the length of cycles the algorithm yields codes with error floors that are orders of magnitude below those of random codes with very small degradation in capacity approaching capability

trellis codes for periodic erasures this paper describes techniques for the design and analysis of trellis codes that provide reliable communication over every channel in a specified set of possible channels where each channel is characterized by additive white gaussian noise with a distinct periodic variation in signal to noise ratio an important practical application for such trellis codes is the periodic erasure channel produced by partial band interference dispersed by a block interleaver we present trellis codes that provide reliable communication over all periodic erasure patterns of a given period for which the number of unerased coded bits per period is at least equal to the number of information bits per period

a subjective approach for ranking fuzzy numbers abstract starting from a subjective assignation of weights related to the relative importance of the different level sets we define a new method of comparison of fuzzy numbers this procedure is an extension of some well known indices adamo tsumura et al yager some properties of our index are studied in this paper as well as its behaviour on several particular cases

duality theorems in fuzzy mathematical programming problems based on the concept of necessity the fuzzy valued lagrangian function for the fuzzy mathematical programming problem via the concept of the fuzzy scalar inner product is proposed a solution concept of fuzzy optimization problems which is essentially similar to the notion of pareto solution in multiobjective programming problems is also introduced by ranking the fuzzy numbers using the necessity indices under these settings the weak and strong duality theorems can be elicited we show that the primal and dual fuzzy mathematical programming problems have no duality gap under suitable convexity assumptions for fuzzy valued functions

universal trellis codes for diagonally layered space time systems foschini s diagonally layered space time transmission system known as d blast is an architecture designed for a rayleigh fading environment using multiple element antenna arrays at both the transmit and receive sites to achieve very high spectral efficiencies in this paper we propose a simple coding technique for d blast that uses a single trellis code with finite traceback viterbi decoding we examine the performance of universal trellis codes that are designed to have a distance structure that is matched to the periodic signal to noise ratio snr variation of the channel created by d blast under the assumption that the channel is static during one burst but may change from burst to burst we show that a universal 64 state trellis code on a 2 spl times 2 d blast system with long enough block lengths displays universal behavior working on almost every 2 spl times 2 channel with at least the mutual information required by a standard 64 state awgn trellis code the only 2 spl times 2 channel where more mutual information is required is a certain rotation of the zero eigenvalue channel we also present 4 spl times 4 and 8 spl times 8 examples

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

generalized derivative and π derivative for set valued functions in this paper we study the generalized derivative and the p derivative for interval valued functions we show the connections between these derivatives some illustrative examples and applications to interval differential equations and fuzzy functions are presented

convex fuzzy mappings abstract in this note the concept of convexity and logarithmic convexity for fuzzy mappings are introduced and some applications to optimization are discussed

minimum memory for generating rare events we classify the rare events of structured memoryful stochastic processes and use this to analyze sequential and parallel generators for these events given a stochastic process we introduce a method to construct a process whose typical realizations are a given process rare events this leads to an expression for the minimum memory required to generate rare events we then show that the recently discovered classical quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations