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

periodic shadowing sensitivity analysis of chaotic systems abstract the sensitivity of long time averages of a hyperbolic chaotic system to parameter perturbations can be determined using the shadowing direction the uniformly bounded in time solution of the sensitivity equations although its existence is formally guaranteed for certain systems methods to determine it are hardly available one practical approach is the least squares shadowing lss algorithm wang 2014 18 whereby the shadowing direction is approximated by the solution of the sensitivity equations with the least square average norm here we present an alternative potentially simpler shadowing based algorithm termed periodic shadowing the key idea is to obtain a bounded solution of the sensitivity equations by complementing it with periodic boundary conditions in time we show that this is not only justifiable when the reference trajectory is itself periodic but also possible and effective for chaotic trajectories our error analysis shows that periodic shadowing has the same convergence rates as lss when the time span t is increased the sensitivity error first decays as 1 t and then asymptotically as 1 t we demonstrate the approach on the lorenz equations and also show that as t tends to infinity periodic shadowing sensitivities converge to the same value obtained from long unstable periodic orbits lasagna 2018 14 for which there is no shadowing error finally finite difference approximations of the sensitivity are also examined and we show that subtle non hyperbolicity features of the lorenz system introduce a small yet systematic bias

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

tilting modules and tilting torsion theories abstract we generalize basic results about classical tilting modules and partial tilting modules to the infinite dimensional case over an arbitrary ring r the methods employed combine classical techniques of representation theory of finite dimensional algebras with new techniques of the theory of modules using a generalization of the bongartz lemma we characterize tilting torsion theories in mod r i e torsion theories induced by infinitely generated tilting modules we investigate lattices gen p p of torsion classes induced by partial tilting modules p applying our results to tilting torsion classes we prove a version of the brenner butler theorem and a generalization of the assem smalo theorem to the case when r is artinian

a class of quasi cyclic ldpc codes over gf 2 m low density parity check ldpc codes over gf 2m are an extension of binary ldpc codes performances of gf 2m ldpc codes have been shown to be higher than binary ldpc codes but the complexity of the encoders decoders increases hence there is a substantial lack of implementations for ldpc codes over gf 2m codes this paper presents a class of quasi cyclic ldpc codes over gf 2m these codes can alleviate the encoding decoding complexity without excessive loss of performances it is shown how from a performance point of view such codes are better than m times bigger binary codes and as good as 2m longer binary codes

weighted nonbinary repeat accumulate codes repeat accumulate ra codes are random like codes having remarkably good performance over an additive white gaussian noise awgn channel like turbo and low density parity check ldpc codes in this correspondence we introduce an ensemble of random codes called weighted nonbinary repeat accumulate wnra codes whose encoder consists of a nonbinary repeater a weighter a pseudorandom symbol interleaver and an accumulator over a finite field gf q they can be decoded in a simple way by applying the sum product algorithm to their factor graphs over gf q simulation results show that wnra codes with proper weighting values over gf 4 or gf 8 are superior to binary ra codes on awgn channels

self compensation technique for simplified belief propagation algorithm the min sum algorithm is the most common method to simplify the belief propagation algorithm for decoding low density parity check ldpc codes however there exists a performance gap between the min sum and belief propagation algorithms due to nonlinear approximation in this paper a self compensation technique using dynamic normalization is thus proposed to improve the approximation accuracy the proposed scheme scales the min sum algorithm by a dynamic factor that can be derived theoretically from order statistics moreover applying the proposed technique to several ldpc codes for dvb s2 system the average signal to noise ratio degradation which results from approximation inaccuracy and quantization error is reduced to 0 2 db not only does it enhance the error correcting capability of the min sum algorithm but the proposed self compensation technique also preserves a modest hardware cost after realized with 0 13 mum standard cell library the dynamic normalization requires about 100 additional gates for each check node unit in the min sum algorithm

performance analysis of ldpc coded diversity combining on rayleigh fading channels with impulsive noise spatial diversity is an effective method to mitigate the effects of fading and when used in conjunction with low density parity check ldpc codes it can achieve excellent error correcting performance noise added at each branch of the diversity combiner is generally assumed to be additive white gaussian noise but there are many applications where the received signal is impaired by noise with a non gaussian distribution in this paper we derive the exact bit error probability of different linear combining techniques on rayleigh fading channels with impulsive noise which is modeled using symmetric alpha stable distributions the relationship for the signal to noise ratios of these linear combiners is derived and then different non linear detectors are presented a detector based on the bi parameter cauchy gaussian mixture model is used and shows near optimal performance with a significant reduction in complexity when compared with the optimal detector furthermore the threshold signal to noise ratio of ldpc codes for different combining techniques on these channels is derived using density evolution and an estimation of the waterfall performance of ldpc codes is derived that reduces the gap between simulated and asymptotic performance

the lindstedt poincare technique as an algorithm for computing periodic orbits the lindstedt poincare technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations it uses a nearby periodic orbit of the unperturbed differential equation as the first approximation we derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems the algorithm unlike the lindstedt poincare technique does not require the dynamical system to be a small perturbation of a solvable differential equation this makes it more broadly applicable the algorithm is quadratically convergent it works with equal facility as examples show irrespective of whether the periodic orbit is attracting or repelling or a saddle one of the examples presents what is possibly the most accurate computation of hill s orbit of lunation since its justly celebrated discovery in 1878

some results and problems on quasi weakly almost periodic points in this note we mainly present some results and problems on quasi weakly almost periodic points in discrete dynamical systems

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

a preconditioned multiple shooting shadowing algorithm for the sensitivity analysis of chaotic systems abstract we propose a preconditioner that can accelerate the rate of convergence of the multiple shooting shadowing mss method 1 this recently proposed method can be used to compute derivatives of time averaged objectives also known as sensitivities to system parameter s for chaotic systems we propose a block diagonal preconditioner which is based on a partial singular value decomposition of the mss constraint matrix the preconditioner can be computed using matrix vector products only i e it is matrix free and is fully parallelised in the time domain two chaotic systems are considered the lorenz system and the 1d kuramoto sivashinsky equation combination of the preconditioner with a regularisation method leads to tight bracketing of the eigenvalues to a narrow range this combination results in a significant reduction in the number of iterations and renders the convergence rate almost independent of the number of degrees of freedom of the system and the length of the trajectory that is used to compute the time averaged objective this can potentially allow the method to be used for large chaotic systems such as turbulent flows and optimal control applications the singular value decomposition of the constraint matrix can also be used to quantify the effect of the system condition on the accuracy of the sensitivities in fact neglecting the singular modes affected by noise we recover the correct values of sensitivity that match closely with those obtained with finite differences for the kuramoto sivashinsky equation in the light turbulent regime we notice a similar improvement for the lorenz system as well

matrix free continuation of limit cycles for bifurcation analysis of large thermoacoustic systems in order to define the nonlinear behaviour of a thermoacoustic system it is important to find the regions of parameter space where limit cycles exist continuation methods find limit cycles numerically in the time domain with no additional assumptions other than those used to form the governing equations once the limit cycles are found these continuation methods track them as the operating condition of the system changes most continuation methods are impractical for finding limit cycles in large thermoacoustic systems because the methods require too much computational time and memory in the literature there are therefore only a few applications of continuation methods to thermoacoustics all with low order models matrix free shooting methods efficiently calculate the limit cycles of dissipative systems and have been demonstrated recently in fluid dynamics but are as yet unused in thermoacoustics these matrix free methods are shown to converge quickly to limit cycles by implicitly using a reduced order model property this is because the methods preferentially use the influential bulk motions of the system whilst ignoring the features that are quickly dissipated in time the matrix free methods are demonstrated on a model of a ducted 2d diffusion flame and the stability limits are calculated as a function of the peclet number and the heat release parameter both subcritical and supercritical hopf bifurcations are found physical information about the flame acoustic interaction is found from the limit cycles and floquet modes invariant subspace preconditioning higher order prediction techniques and multiple shooting techniques are all shown to reduce the time required to generate bifurcation surfaces two types of shooting are compared and two types of matrix free evaluation are compared

construction of nonbinary cyclic quasi cyclic and regular ldpc codes a finite geometry approach this paper presents five methods for constructing nonbinary ldpc codes based on finite geometries these methods result in five classes of nonbinary ldpc codes one class of cyclic ldpc codes three classes of quasi cyclic ldpc codes and one class of structured regular ldpc codes experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the awgn channel and they achieve significant coding gains over reed solomon codes of the same lengths and rates with either algebraic hard decision decoding or kotter vardy algebraic soft decision decoding at the expense of a larger decoding computational complexity

least squares shadowing sensitivity analysis of chaotic limit cycle oscillations abstract the adjoint method among other sensitivity analysis methods can fail in chaotic dynamical systems the result from these methods can be too large often by orders of magnitude when the result is the derivative of a long time averaged quantity this failure is known to be caused by ill conditioned initial value problems this paper overcomes this failure by replacing the initial value problem with the well conditioned least squares shadowing lss problem the lss problem is then linearized in our sensitivity analysis algorithm which computes a derivative that converges to the derivative of the infinitely long time average we demonstrate our algorithm in several dynamical systems exhibiting both periodic and chaotic oscillations

bandwidth efficient modulation codes based on nonbinary irregular repeat accumulate codes using nonbinary low density parity check ldpc codes with random coset mapping bennatan and burshtein constructed bandwidth efficient modulation codes with remarkable performance under belief propagation bp decoding however due to the random nature of ldpc codes most of the good ldpc codes found in the literature do not have a simple encoding structure thus the encoding complexity of those ldpc codes can be as high as o n 2 where n is the codeword length to reduce the encoding complexity in this paper nonbinary irregular repeat accumulate ira codes with time varying characteristic and random coset mapping are proposed for bandwidth efficient modulation schemes the time varying characteristic and random coset mapping result in both permutation invariance and symmetry properties respectively in the densities of decoder messages the permutation invariance and symmetry properties of the proposed codes enable the approximations of densities of decoder messages using gaussian distributions under the gaussian approximation extrinsic information transfer exit charts for nonbinary ira codes are developed and several codes of different spectral efficiencies are designed based on exit charts in addition by proper selection of nonuniform signal constellations the constructed codes are inherently capable of obtaining shaping gains even without separate shaping codes simulation results indicate that the proposed codes not only have simple encoding schemes but also have remarkable performance that is even better than that constructed using nonbinary ldpc codes

approximating the llr distribution for a class of soft output mimo detectors we present approximations of the llr distribution for a class of fixed complexity soft output mimo detectors such as the optimal soft detector and the soft output via partial marginalization detector more specifically in a mimo awgn setting we approximate the llr distribution conditioned on the transmitted signal and the channel matrix with a gaussian mixture model gmm our main results consist of an analytical expression of the gmm model including the number of modes and their corresponding parameters and a proof that in the limit of high snr this llr distribution converges in probability towards a unique gaussian distribution

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

analysis of the iterative decoding of ldpc and product codes using the gaussian approximation we propose a novel density evolution approach to analyze the iterative decoding algorithms of low density parity check ldpc codes and product codes based on gaussian densities namely for these classes of codes we derive a one dimensional 1d map whose iterates directly represent the error probability both for the additive white gaussian noise awgn and the rayleigh fading channel these simple models allow a qualitative analysis of the nonlinear dynamics of the decoding algorithm as an application we compute the decoding thresholds and show that they are consistent with the simulation results available in the literature

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

maximizing the spectral efficiency of coded cdma under successive decoding we investigate the spectral efficiency achievable by random synchronous code division multiple access cdma with quaternary phase shift keying qpsk modulation and binary error control codes in the large system limit where the number of users the spreading factor and the code block length go to infinity for given codes we maximize spectral efficiency assuming a minimum mean square error mmse successive stripping decoder for the cases of equal rate and equal power users in both cases the maximization of spectral efficiency can be formulated as a linear program and admits a simple closed form solution that can be readily interpreted in terms of power and rate control we provide examples of the proposed optimization methods based on off the shelf low density parity check ldpc codes and we investigate by simulation the performance of practical systems with finite code block length

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

sequential message passing decoding of ldpc codes by partitioning check nodes in this paper we analyze the sequential message passing decoding algorithm of low density parity check ldpc codes by partitioning check nodes this decoding algorithm shows better bit error rate ber performance than the conventional message passing decoding algorithm especially for the small number of iterations analytical results indicate that as the number of partitioned subsets of check nodes increases the ber performance is improved we also derive the recursive equations for mean values of messages at check and variable nodes by using density evolution with a gaussian approximation from these equations the mean values are obtained at each iteration of the sequential decoding algorithm and the corresponding ber values are calculated they show that the sequential decoding algorithm converges faster than the conventional one finally the analytical results are confirmed by the simulation results

least squares shadowing method for sensitivity analysis of differential equations for a parameterized hyperbolic system frac du dt f u s the derivative of the ergodic average langle j rangle lim t to infty frac 1 t int 0 t j u t s to the parameter s can be computed via the least squares shadowing lss algorithm we assume that the system is ergodic which means that langle j rangle depends only on s not on the initial condition of the hyperbolic system the algorithm solves a constrained least squares problem and from the solution to this problem computes the desired derivative frac d langle j rangle ds the purpose of this paper is to prove that the value given by the lss algorithm approaches the exact derivative when the timespan used to formulate the least squares problem grows to infinity it then illustrates the convergence result through a numerical example

every module is finitely generated abstract we solve one of the main problems of the theory of representable equivalences of module categories we prove that every module over an arbitrary associative ring r with unit is finitely generated consequently we obtain the equality of the classes of all modules and all quasi progenerators provided that r is commutative

performance analysis of iterative decoding algorithms with memory over memoryless channels density evolution is often used to determine the performance of an ensemble of low density parity check ldpc codes under iterative message passing algorithms conventional density evolution techniques over memoryless channels are based on the assumption that messages at iteration l are only a function of the messages at iteration l 1 and possibly the channel output this assumption is valid for many algorithms such as standard belief propagation bp and min sum ms algorithms however there are other important iterative algorithms such as successive relaxation sr versions of bp and ms and differential decoding with binary message passing dd bmp algorithm of mobini et al for which this assumption is not valid the reason is the introduction of memory in these algorithms in this work we propose a model for iterative decoding algorithms with memory which covers sr and dd bmp algorithms as special cases based on this model we derive a bayesian network for iterative algorithms with memory over memoryless channels and use this representation to analyze the performance of the algorithms using density evolution the density evolution technique is developed based on truncating the memory of the decoding process and approximating it with a finite order markov process and can be implemented efficiently as an example we apply our technique to analyze the performance of dd bmp on regular ldpc code ensembles and make a number of interesting observations with regard to the performance complexity tradeoff of dd bmp in comparison with bp and ms algorithms the model presented in this paper is based on certain simplifying assumptions about the memory structure of iterative algorithms such as the existence of memory only at the output of variable nodes in the code s tanner graph rather than at both outputs of variable and check nodes the bayesian network framework introduced here however can still be used to analyze the more general scenarios

sensitivity and transitivity of fuzzified dynamical systems abstract in this paper it is proved that a set valued dynamical system is sensitively dependent on initial conditions resp f sensitive multi sensitive if and only if its g fuzzification is sensitively dependent on initial conditions resp f sensitive multi sensitive where f is a furstenberg family as an application it is shown that there exists a sensitive dynamical system whose g fuzzification does not have such sensitive dependence for any g in a certain domain moreover a sufficient condition is derived for ensuring that the g fuzzification of every nontrivial dynamical system is not transitive

design of regular 2 d sub c ldpc codes over gf q using their binary images in this paper a method to design regular 2 dc ldpc codes over gf q with both good waterfall and error floor properties is presented based on the algebraic properties of their binary image first the algebraic properties of rows of the parity check matrix h associated with a code are characterized and optimized to improve the waterfall then the algebraic properties of cycles and stopping sets associated with the underlying tanner graph are studied and linked to the global binary minimum distance of the code finally simulations are presented to illustrate the excellent performance of the designed codes

sensitivity of iterated function systems abstract the present work is concerned with the equicontinuity and sensitivity of iterated function systems ifss here we consider more general case of ifss i e the ifss generated by a family of relations we generalize the concepts of transitivity sensitivity and equicontinuity to these kinds of systems this note investigates the relationships between these concepts then several sufficient conditions for sensitivity of ifss are presented we introduce the notion of weak topologically exact for ifss generated by a family of relations it is proved that non minimal weak topologically exact ifss are sensitive that yields to different examples of non minimal sensitive systems which are not an m system moreover some interesting examples are given which provide some facts about the sensitive property of ifss

probability density adjoint for sensitivity analysis of the mean of chaos sensitivity analysis especially adjoint based sensitivity analysis is a powerful tool for engineering design which allows for the efficient computation of sensitivities with respect to many parameters however these methods break down when used to compute sensitivities of long time averaged quantities in chaotic dynamical systems this paper presents a new method for sensitivity analysis of ergodic chaotic dynamical systems the density adjoint method the method involves solving the governing equations for the system s invariant measure and its adjoint on the system s attractor manifold rather than in phase space this new approach is derived for and demonstrated on one dimensional chaotic maps and the three dimensional lorenz system it is found that the density adjoint computes very finely detailed adjoint distributions and accurate sensitivities but suffers from large computational costs

morita like equivalence of infinite matrix subrings abstract the concept of morita equivalence of matrix subrings with identity and of categories of modules is extended to morita like equivalence this enables the generalizations of the theorems of morita garcia and xu from the module of n n matrices over a ring r to the module of matrices where is an arbitrary indexing set

forward and adjoint sensitivity computation of chaotic dynamical systems this paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems such as the lorenz attractor the algorithms compute the derivative of long time averaged statistical quantities to infinitesimal perturbations of the system parameters the algorithms are demonstrated on the lorenz attractor we show that sensitivity derivatives of statistical quantities can be accurately estimated using a single short trajectory over a time interval of 20 on the lorenz attractor

chaotic properties of mappings on a probability space sensitive dependence on initial conditions is widely understood as being the central idea of chaos we first give sufficient conditions both topological and ergodic on an endomorphism to ensure the sensitivity property then a strong sensitivity concept is introduced sufficient conditions on a transformation implying strong sensitivity are given we also provide bounds for the strong sensitivity constant

approaching v blast capacity with adaptive modulation and ldpc encoding this paper presents a practical implementation of the vertical bell laboratories layered space time v blast type system in which the multiple input multiple output mimo open loop capacity can be approached with conventional scalar coding using adaptive modulation with appropriate channel codes e g low density parity check ldpc codes and optimum successive detection osd the density evolution technique is employed to determine the maximal achievable rate of an ldpc code for each transmit antenna for a given channel realization at a given signal to noise ratio numerical results will show that the average sum rate of the adaptively modulated ldpc encoded system is quite close to the v blast capacity with both rate and power adaptations considering the performance degradation caused by error propagation due to the imperfect feedback and relatively long decoding delay in the osd we use parallel interference cancellation followed by minimum mean square error filtering in the bit error rate ber performance simulation simulation results will show that a target ber of 10 5 can be achieved by the optimally designed ldpc codes to simplify the code design we replace the ldpc codes optimally designed for each channel realization with rate compatible punctured ldpc codes at the cost of a slight sum rate loss if the fading process is nonergodic the outage capacity corresponding to a given outage probability is used to measure the channel performance as an example we design the ldpc codes for an adaptively modulated 2times2 v blast system to approach its outage capacity for a given outage probability

ldpc coded mimo multiple access with iterative joint decoding an efficient scheme for the multiple access multiple input multiple output mimo channel is proposed which operates well also in the single user regime as well as in a direct sequence spread spectrum ds cdma setting the design features scalability and is of limited complexity the system employs optimized low density parity check ldpc codes and an efficient iterative belief propagation bp detection which combines linear minimum mean square error lmmse detection and iterative interference cancellation ic this combination is found to be necessary for efficient operation in high system loads spl alpha 1 an asymptotic density evolution de is used to optimize the degree polynomials of the underlining ldpc code and thresholds as close as 0 77 db to the channel capacity are evident for a system load of 2 replacing the lmmse with the complex individually optimal multiuser detector io mud further improves the performance up to 0 14 db from the capacity comparing the thresholds of a good single user ldpc code to the multiuser optimized ldpc code both over the above multiuser channel reveals a surprising 8 db difference emphasizing thus the necessity of optimizing the code the asymptotic analysis of the proposed scheme is verified by simulations of finite systems which reveal meaningful differences between the performances of mimo systems with single and multiple users and demonstrate performance similar to previously reported techniques but with higher system loads and significantly lower receiver complexity

nonbinary hybrid ldpc codes in this paper a new class of low density parity check ldpc codes named hybrid ldpc codes is introduced hybrid ldpc codes are characterized by an irregular connectivity profile and heterogeneous orders of the symbols in the codeword it is shown in particular that the class of hybrid ldpc codes can be asymptotically characterized and optimized using density evolution de framework and a technique to maximize the minimum distance of the code is presented numerical assessment of hybrid ldpc code performances is provided by comparing them to protograph based and multiedge type met ldpc codes hybrid ldpc codes are shown to allow to achieve an interesting tradeoff between good error floor performance and good waterfall region with nonbinary coding techniques

convergence of the least squares shadowing method for computing derivative of ergodic averages for a parameterized hyperbolic system u i 1 f u i s the derivative of an ergodic average langle j rangle lim n rightarrow infty frac1n sum 1 n j u i s to the parameter s can be computed via the least squares shadowing method this method solves a constrained least squares problem and computes an approximation to the desired derivative frac d langle j rangle ds from the solution this paper proves that as the size of the least squares problem approaches infinity the computed approximation converges to the true derivative

waterfall performance analysis of finite length ldpc codes on symmetric channels an efficient method for analyzing the performance of finite length low density parity check ldpc codes in the waterfall region when transmission takes place on a memoryless binary input output symmetric channel is proposed this method is based on studying the variations of the channel quality around its expected value when observed during the transmission of a finite length codeword we model these variations with a single parameter this parameter is then viewed as a random variable and its probability distribution function is obtained assuming that a decoding failure is the result of an observed channel worse than the codeiquests decoding threshold the block error probability of finite length ldpc codes under different decoding algorithms is estimated using an extrinsic information transfer chart analysis the bit error probability is obtained from the block error probability different parameters can be used for modeling the channel variations in this work two of such parameters are studied through examples it is shown that this method can closely predict the performance of ldpc codes of a few thousand bits or longer in the waterfall region

characterizing dynamics with covariant lyapunov vectors a general method to determine covariant lyapunov vectors in both discrete and continuous time dynamical systems is introduced this allows us to address fundamental questions such as the degree of hyperbolicity which can be quantified in terms of the transversality of these intrinsic vectors for spatially extended systems the covariant lyapunov vectors have localization properties and spatial fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the lyapunov exponents

sensitivity theory for nonlinear systems i nonlinear functional analysis approach concepts of nonlinear functional analysis are employed to investigate the mathematical foundations underlying sensitivity theory this makes it possible not only to ascertain the limitations inherent in existing analytical approaches to sensitivity analysis but also to rigorously formulate a considerably more general sensitivity theory for physical problems characterized by systems of nonlinear equations and by nonlinear functionals as responses two alternative formalisms labeled the forward sensitivity formalism and the adjoint sensitivity formalism are developed in order to evaluate the sensitivity of the response to variations in the system parameters the forward sensitivity formalism is formulated in normed linear spaces and the existence of the g teaux differentials of the operators appearing in the problem is shown to be both necessary and sufficient for its validity this formalism is conceptually straightforward and can be advantageously used to assess the effects of relatively few parameter alterations on many responses on the other hand for problems involving many parameter alterations or a large data base and comparatively few functional type responses the alternative adjoint sensitivity formalism is computationally more economical however it is shown that this formalism can be developed only under conditions that are more restrictive than those underlying the validity of the forward sensitivity formalism in particular the requirement that operators acting on the state vector and on the system parameters must admit densely defined g teaux derivatives is shown to be of fundamental importance for the validity of this formalism the present analysis significantly extends the scope of sensitivity theory and provides a basis for still further generalizations

multiple shooting shadowing for sensitivity analysis of chaotic dynamical systems abstract sensitivity analysis methods are important tools for research and design with simulations many important simulations exhibit chaotic dynamics including scale resolving turbulent fluid flow simulations unfortunately conventional sensitivity analysis methods are unable to compute useful gradient information for long time averaged quantities in chaotic dynamical systems sensitivity analysis with least squares shadowing lss can compute useful gradient information for a number of chaotic systems including simulations of chaotic vortex shedding and homogeneous isotropic turbulence however this gradient information comes at a very high computational cost this paper presents multiple shooting shadowing mss a more computationally efficient shadowing approach than the original lss approach through an analysis of the convergence rate of mss it is shown that mss can have lower memory usage and run time than lss

weak mixing implies sensitive dependence abstract let x be a metric space b the algebra of borel subsets of x and a probability measure on x b in this note for a measure preserving map t respectively a measure preserving semi flow on x b we prove that if supp x and t respectively is weak mixing then t respectively has sensitive dependence

on the decomposition method for linear programming decoding of ldpc codes in this paper we focus on solving the linear programming lp problem that arises in the decoding of low density parity check ldpc codes by means of the revised simplex method in order to take advantage of the structure of the lp problem we reformulate the dual lp and apply the idea of dantzig wolfe d w decomposition method to solve the problem each subproblem in the d w decomposition method is an lp over a convex polyhedral cone we define the convex polyhedral cone as local parity check cone and discuss its special structures then we enumerate its extreme rays and use these extreme rays to design an efficient method for the general lp decoding problem the proposed method exhibits advantages in reducing both the storage and computational requirements

linear response of the lorenz system the present numerical study provides strong evidence that at standard parameters the response of the lorenz system to small perturbations of the control parameter r is linear this evidence is obtained not only directly by determining the response in the observable a x z but also indirectly by validating various implications of the assumption of a linear response like a quadratic response at twice the perturbation frequency a vanishing response in a x x the kramers kronig relations and relations between different response functions since the lorenz system is nonhyperbolic the present results indicate that in contrast to a recent speculation the large system limit thermodynamic limit need not be invoked to obtain a linear response for chaotic systems of this type

sensitivity of long periodic orbits of chaotic systems the properties of long numerically determined periodic orbits of two low dimensional chaotic systems the lorenz equations and the kuramoto sivashinsky system in a minimal domain configuration are examined the primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite amplitude parameter perturbations to address this question an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed the expectation of period averages floquet exponents and sensitivities over such set is then obtained it is shown that all these quantities converge to a limiting value as the orbit period is increased however while period averages and floquet exponents appear to converge to analogous quantities computed from chaotic trajectories the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state similar to observations made with other shadowing algorithms

generalized low density parity check codes based on hadamard constraints in this paper we consider the design and analysis of generalized low density parity check gldpc codes in awgn channels the gldpc codes are specified by a bipartite tanner graph as with standard ldpc codes but with the single parity check constraints replaced by general coding constraints in particular we consider imposing hadamard code constraints at the check nodes for a low rate approach termed ldpc hadamard codes we introduce a low complexity message passing based iterative soft input soft output siso decoding algorithm which employs the a posteriori probability app fast hadamard transform fht for decoding the hadamard check codes at each decoding iteration the achievable capacity with the gldpc codes is then discussed a modified ldpc hadamard code graph is also proposed we then optimize the ldpc hadamard code ensemble using a low complexity optimization method based on approximating the density evolution by a one dimensional dynamic system represented by an extrinsic mutual information transfer exit chart simulation results show that the optimized ldpc hadamard codes offer better performance in the low rate region than low rate turbo hadamard codes but also enjoy a fast convergence rate a rate 0 003 ldpc hadamard code with large block length can achieve a bit error rate ber performance of 10 5 at 1 44 db which is only 0 15 db away from the ultimate shannon limit 1 592 db and 0 24 db better than the best performing low rate turbo hadamard codes

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

systematic design of low density parity check code ensembles for binary erasure channels we propose a systematic method to design irregular low density parity check ldpc codes for binary erasure channels bec compared to the existing methods which are based on the application of asymptotic analysis tools such as density evolution or extrinsic information transfer exit charts in an optimization process the proposed method is much simpler and faster through a number of examples we demonstrate that the codes designed by the proposed method perform very closely to the best codes designed by optimization an important property of the proposed designs is the flexibility to select the number of constituent variable node degrees p the proposed designs include existing systematic designs as a special case with p n 1 where n is the maximum variable node degree compared to the existing systematic designs for a given rate and a given 0 the designed ensembles can have a threshold in neighborhood of the capacity upper bound with smaller values of p and n they can also achieve the capacity of the bec as n and correspondingly p and the maximum check node degree tend to infinity

tilting modules of finite projective dimension tilting modules are well known and are useful in the representation theory of artinian algebras 2 4 6 11 the purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results a unital left a module a t over a ring a is said to be a tilting module of projective dimension 0 is an integer if it satisfies the following three conditions 1 a t has a projective resolution

lp decoding of regular ldpc codes in memoryless channels we study error bounds for linear programming decoding of regular low density parity check ldpc codes for memoryless binary input output symmetric channels we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph for memoryless binary input awgn channel we prove lower bounds on the threshold for regular ldpc codes whose factor graphs have logarithmic girth under lp decoding specifically we prove a lower bound of 0 735 upper bound of eb n0 2 67 db on the threshold of 3 6 regular ldpc codes whose factor graphs have logarithmic girth our proof is an extension of a recent paper of arora daskalakis and steurer stoc 2009 who presented a novel probabilistic analysis of lp decoding over a binary symmetric channel their analysis is based on the primal lp representation and has an explicit connection to message passing algorithms we extend this analysis to any mbios channel

hybrid optical rf channels characterization and performance study using low density parity check codes a hybrid channel consisting of a free space optical fso link and a parallel radio frequency rf link is considered the fso link carries most of the transmitted data while leaving only a small fraction of the data to be carried by the rf link it is first shown that the capacity of the proposed hybrid structure is much higher than the capacity of a single fso link next the performance of the hybrid channel is studied using a low density parity check code a density evolution strategy for the hybrid channel is derived and a gaussian approximation technique is presented using these techniques the conditions for the convergence of the message passing algorithm in terms of minimum data carrying rate through the rf link are derived finally numerical results showing the benefits of the hybrid channel under various channel conditions are presented

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

raptor codes in the low snr regime in this paper we revisit the design of raptor codes for binary input additive white gaussian noise channels where we are interested in very low signal to noise ratios snrs a linear programming degree distribution optimization problem is defined for raptor codes in the low snr regime through several approximations we also provide an exact expression for the polynomial representation of the degree distribution with infinite maximum output node degree in the low snr regime which enables us to calculate the exact value of the fractions of output nodes of small degrees a more practical degree distribution design is also proposed for raptor codes in the low snr regime where we include the rate efficiency and the decoding complexity in the optimization problem and an upper bound on the maximum rate efficiency is derived for given design parameters simulation results show that the raptor code with the designed degree distributions can approach rate efficiencies larger than 0 95 in the low snr regime

polar coded modulation a framework is proposed that allows for a joint description and optimization of both binary polar coding and 2m ary digital pulse amplitude modulation pam schemes for the latter the multilevel coding mlc approach as well as bit interleaved coded modulation bicm are considered the conceptual equivalence of polar coding and multilevel coding is covered in detail based on an alternative characterization of the channel polarization phenomenon rules for the optimum choice of the labeling in coded modulation schemes employing polar codes are developed simulation results regarding the error performance of the proposed schemes on the awgn channel are included

efficient sensitivity analysis method for chaotic dynamical systems the direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems the key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method an additional constraint based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations the application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the lagrange multipliers the lu factorization technique to calculate the lagrange multipliers leads to a better performance for the convergence problem and the computational expense numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods the numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method

iterative multiuser joint decoding optimal power allocation and low complexity implementation we consider a canonical model for coded code division multiple access cdma with random spreading where the receiver makes use of iterative belief propagation bp joint decoding we provide simple density evolution analysis in the large system limit large number of users of the performance of the bp decoder and of some suboptimal approximations based on interference cancellation ic based on this analysis we optimize the received user signal to noise ratio snr distribution in order to maximize the system spectral efficiency for given user channel codes channel load users per chip and target user bit error rate ber the optimization of the received snr distribution is obtained by solving a simple linear program and can be easily incorporated into practical power control algorithms remarkably under the optimized snr assignment the suboptimal minimum mean square error mmse ic based decoder performs almost as well as the more complex bp decoder moreover for a large class of commonly used convolutional codes we observe that the optimized snr distribution consists of a finite number of discrete snr levels based on this observation we provide a low complexity approximation of the mmse ic decoder that suffers from very small performance degradation while attaining considerable savings in complexity as by products of this work we obtain a closed form expression of the multiuser efficiency me of power mismatched mmse filters in the large system limit and we extend the analysis of the symbol by symbol maximum a posteriori probability map multiuser detector in the large system limit to the case of nonconstant user powers and nonuniform symbol prior probabilities

explosions of chaotic sets abstract large scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied these bifurcations include sudden changes in the size and or type of the set an explosion is a bifurcation in which new recurrent points suddenly appear at a non zero distance from any pre existing recurrent points we discuss the following in a generic one parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur 1 a saddle node bifurcation isolated from other recurrent points 2 a saddle node bifurcation embedded in the set of recurrent points 3 outer homoclinic tangencies and 4 outer heteroclinic tangencies the term outer tangency refers to a particular configuration of the stable and unstable manifolds at tangency in particular we examine different types of tangencies of stable and unstable manifolds from orbits of pre existing invariant sets this leads to a general theory that unites phenomena such as crises basin boundary metamorphoses explosions of chaotic saddles etc we illustrate this theory with numerical examples

on the heart of a faithful torsion theory abstract in r colpi k r fuller tilting objects in abelian categories and quasitilted rings trans amer math soc in press tilting objects in an arbitrary abelian category h are introduced and are shown to yield a version of the classical tilting theorem between h and the category of modules over their endomorphism rings moreover it is shown that given any faithful torsion theory x y in mod r for a ring r the corresponding heart h x y is an abelian category admitting a tilting object which yields a tilting theorem between the heart and mod r in this paper we first prove that h x y is a prototype for any abelian category h admitting a tilting object which tilts to x y in mod r then we study ab type properties of the heart and commutations with direct limits this allows us to show for instance that any abelian category h with a tilting object is ab4 and to find necessary and sufficient conditions which guarantee that h is a grothendieck or even a module category as particular situations we examine two main cases when x y is hereditary cotilting proving that h x y is grothendieck and when x y is tilting proving that h x y is a module category

maximum likelihood laplacian correlation channel estimation in layered wyner ziv coding the distributed source coding theory teaches that efficient compression can be achieved by exploiting the correlation between encoded sources at the decoder only a critical issue in this coding paradigm is how to accurately estimate the correlation statistics at the decoder this paper proposes a novel correlation channel estimation method designed for generic layered wyner ziv wz coding driven by real life applications i e video coding the correlation noise is assumed to be memoryless zero mean laplacian with a block stationary behavior unlike existing methods the proposed technique derives a novel maximum likelihood estimate of the noise scaling parameter per stationarity block adhering to layered coding the derived estimate is successively refined as more stages are decoded the cramer rao lower bound crlb for our estimator is derived and the mean squared error performance of the proposed successive refinement algorithm is evaluated through simulations apart from achieving an accuracy close to the derived crlb the proposed algorithm is shown to yield a wz rate distortion performance approaching the ideal but unrealistic scenario where the decoder knows a priori the correlation statistics moreover the proposed technique has been integrated into our latest wz video codec for wireless capsule endoscopy experimentation reveals that the proposed method delivers improved compression performance compared to state of the art techniques while simultaneously reducing the overall decoding complexity

successive refinement for the wyner ziv problem and layered code design we examine successive refinement for the wyner ziv problem described in a recent paper by steinberg and merhav where the authors showed that if the side information for all stages is identical then the jointly gaussian source with squared error distortion measure is successively refinable we first extend this result to the case where the difference between the source and the side information is gaussian and independent of the side information as a byproduct we give an alternative proof that the wyner ziv problem for these sources has no rate loss a result that was recently shown by pradhan et al through invoking the duality between the gaussian wyner ziv problem and the costa problem we then perform layered wyner ziv code design for this general type of source based on nested scalar quantization bit plane coding and low density parity check ldpc code based slepian wolf coding source coding with side information we show that density evolution can be used to analyze the slepian wolf code performance provided that certain symmetry conditions which have been dubbed dual symmetry are satisfied by the hypothetical channel between the source and the side information we also show that the dual symmetry condition is indeed satisfied by the hypothetical channel in our slepian wolf coding setup this justifies the use of density evolution in our ldpc code based slepian wolf code design for wyner ziv coding for the jointly gaussian source our layered coder performs 1 29 to 3 45 db from the wyner ziv bound for rates ranging from 0 47 to 4 68 bits per sample when the side information is laplacian and the source equals the side information plus an independent gaussian noise term our layered coder performs 1 33 to 3 90 db from the wyner ziv bound for rates ranging from 0 48 to 4 64 bits per sample

construction of structured regular ldpc codes a design theoretic approach a new combinatorial technique for constructing girth 6 structured binary regular low density parity check ldpc codes based on special types of t designs is given a very large number of well known t designs can be used by this method for code construction based on this method a t v k design d x b can be exploited for code construction if it satisfies the following three conditions 1 b1 b2 t for any two blocks b1 b2 b and b1 b2 2 1 and 3 k t though the technique works for any t design satisfying these conditions we focus only on the utilization of simple triple systems super simple bibds steiner systems and large sets lss of t designs we also construct binary and non binary girth 6 qc ldpc codes from the t designs satisfying these conditions by using matrix dispersion method experimental results show that the constructed non binary qc ldpc codes can provide good practical performance under iterative decoding using the fast fourier transform based q ary sum product algorithm fft qspa and they can achieve acceptable coding gains over random like codes of comparable parameters decoded with sum product algorithm spa

f sensitivity and multi sensitivity of hyperspatial dynamical systems abstract let f be a continuous self map defined on a compact metric space x and f be a continuous self map naturally induced by f on the hyperspace k x of all nonempty compact subsets of x endowed with a hausdorff metric firstly a few sufficient and necessary conditions to ensure a dynamical system be f sensitive or multi sensitive are obtained then the following results are proved 1 if x f is a non minimal m system then k x f has f s sensitive pairs almost everywhere 2 if k x f or k y g is f sensitive then k x y f g is f sensitive 3 k x y f g is multi sensitive if and only if k x f or k y g is multi sensitive if and only if x f or y g is multi sensitive moreover it is proved that f g is multi sensitive if and only if f or g is multi sensitive this is a positive answer to a question posed in r li and x zhou 2013 18

an mse based transfer chart for analyzing iterative decoding schemes using a gaussian approximation an alternative to extrinsic information transfer exit charts called mean square error mse charts that use a measure related to the mse instead of mutual information is proposed using the relationship between mutual information and minimum mean square error mmse for the additive white gaussian noise awgn channel a relationship between the rate of any code and the area under a plot of mmse versus signal to noise ratio snr is obtained when the a priori log likelihood ratio llr is from a binary input gaussian channel using this result a justification is provided for designing concatenated codes by matching the exit curves of the inner and outer decoder when the llrs are assumed to be gaussian which is also the typical assumption used for code design using exit charts even though the gaussian assumption is almost never true the results presented in this paper represent a step toward the analysis of iterative decoding schemes using a single parameter finally for the special case of awgn channel it is shown that any capacity achieving code has an exit curve that is a step function

spatially coupled ensembles universally achieve capacity under belief propagation we investigate spatially coupled code ensembles for transmission over the binary erasure channel it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble this explains why convolutional ldpc ensembles originally introduced by felstrom and zigangirov perform so well over this channel we show that the equivalent result holds true for transmission over general binary input memoryless output symmetric channels more precisely given a desired error probability and a gap to capacity we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding in fact most codes in this ensemble have this property the quantifier universal refers to the single ensemble code that is good for all channels but we assume that the channel is known at the receiver the key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble we conclude by discussing some interesting open problems

on lyapunov exponent and sensitivity sensitive dependence on initial conditions is widely understood as being the central idea of chaos for a large class of transformations of the interval we prove that positiveness of the lyapunov exponent implies the sensitivity property we also provide bounds for the sensitivity constant

on pairwise sensitivity we define the concept of pairwise sensitivity with respect to initial conditions for the endomorphisms on lebesgue metric spaces the idea is that the orbits of almost every pair of nearby initial points for the product of the invariant measure of a pairwise sensitive map may diverge from a positive quantity independent of the initial points we study the relationships between pairwise sensitivity and weak mixing pairwise sensitivity and positiveness of metric entropy and we compute the largest sensitivity constant

topological morita equivalences induced by ideals generated by dense idempotents abstract given two complete right linearly topologized rings r and s and a bimodulerbsendowed with a complete topology in such a way that bs clt s and there be a continuous ring homomorphism r cendus b we define a functor rb clt r clt s which is left adjoint to the functor chomus b clt s clt r then we consider the particular case in which s erewith its induced topology whereeis a dense idempotent ofr that is reris dense in r under these hypotheses we show that the pair of functors r re clt r clt s and ser clt s clt r is an equivalence of categories as an application of this result we re obtain a theorem of xu shum and turner smith on similarities between infinite matrix subrings

low density parity check codes over gaussian channels with erasures we consider low density parity check code ldpcc design for additive white gaussian noise awgn channels with erasures this model for example represents a common situation in magnetic and optical recording where defects or thermal asperities in the system are detected and presented to the decoder as erasures we give thresholds of regular and irregular ldpccs and discuss practical code design over the mixed gaussian erasures channel the analysis is an extension of the gaussian approximation work of chung et al in the two limiting cases of no erasures and large signal to noise ratio snr the analysis tends to the results of chung et al see ibid vol 47 p 657 670 feb 2001 and luby et al 1997 respectively giving a general tool for a class of mixed channels we derive a steady state equation which gives a graphical interpretation of decoder convergence this allows one to estimate the maximum erasure capability on the mixture channel or conversely to estimate the additional signal power required to compensate for the loss due to erasures we see that a good capacity approaching ldpcc over an awgn channel is also good over the mixed channel up to a moderate erasure probability we also investigate practical issues such as the maximum number of iterations of message passing decoders the coded block length and types of erasure patterns random block erasures finally we design an optimized ldpcc for the mixed channel which shows better performance if the erasure probability is larger than a certain value 0 1 in our simulation at the expense of performance degradation at unerased awgn channel and lower erasure probability regions less than 0 1 in our simulation

capacity approaching bandwidth efficient coded modulation schemes based on low density parity check codes we design multilevel coding mlc and bit interleaved coded modulation bicm schemes based on low density parity check ldpc codes the analysis and optimization of the ldpc component codes for the mlc and bicm schemes are complicated because in general the equivalent binary input component channels are not necessarily symmetric to overcome this obstacle we deploy two different approaches one based on independent and identically distributed i i d channel adapters and the other based on coset codes by incorporating i i d channel adapters we can force the symmetry of each binary input component channel by considering coset codes we extend the concentration theorem based on previous work by richardson et al see ibid vol 47 p 599 618 feb 2001 and kavc spl caron ic spl acute et al see ibid vol 49 p 1636 52 july 2003 we also discuss the relation between the systems based on the two approaches and show that they indeed have the same expected decoder behavior next we jointly optimize the code rates and degree distribution pairs of the ldpc component codes for the mlc scheme the optimized irregular ldpc codes at each level of mlc with multistage decoding msd are able to perform well at signal to noise ratios snr very close to the capacity of the additive white gaussian noise awgn channel we also show that the optimized bicm scheme can approach the parallel independent decoding pid capacity as closely as does the mlc pid scheme simulations with very large codeword length verify the accuracy of the analytical results finally we compare the simulated performance of these coded modulation schemes at finite codeword lengths and consider the results from the perspective of a random coding exponent analysis

a unified approach to the construction of binary and nonbinary quasi cyclic ldpc codes based on finite fields a unified approach for constructing binary and nonbinary quasi cyclic ldpc codes under a single framework is presented six classes of binary and nonbinary quasi cyclic ldpc codes are constructed based on primitive elements additive subgroups and cyclic subgroups of finite fields numerical results show that the codes constructed perform well over the awgn channel with iterative decoding

several sufficient conditions for sensitive dependence on initial conditions this paper is concerned with sensitive dependence on initial conditions for maps and semi flows several sufficient conditions for sensitive dependence are given where it is not required that maps and semi flows are continuous and spaces are compact in particular it is shown that if a measure preserving map resp a measure preserving semi flow on a metric probability space with a fully supported measure is topologically strongly ergodic then it has sensitive dependence these results relax and extend the conditions of some existing results

some stronger forms of topological transitivity and sensitivity for a sequence of uniformly convergent continuous maps abstract let h d be a metric space f be a furstenberg family and g m m z be a sequence of continuous map on h which converges uniformly to a map g on h in this paper under the condition lim m d g m m g m 0 a necessary and sufficient condition for g to be f mixing is established moreover let t z be an infinite set and lim m t m d g m m g m 0 then some necessary and sufficient conditions or sufficient conditions for g to be some stronger forms of topological transitivity sensitivity ergodic or mixing are obtained

an analysis of belief propagation on the turbo decoding graph with gaussian densities motivated by its success in decoding turbo codes we provide an analysis of the belief propagation algorithm on the turbo decoding graph with gaussian densities in this context we are able to show that under certain conditions the algorithm converges and that somewhat surprisingly though the density generated by belief propagation may differ significantly from the desired posterior density the means of these two densities coincide since computation of posterior distributions is tractable when densities are gaussian use of belief propagation in such a setting may appear unwarranted indeed our primary motivation for studying belief propagation in this context stems from a desire to enhance our understanding of the algorithm s dynamics in a non gaussian setting and to gain insights into its excellent performance in turbo codes nevertheless even when the densities are gaussian belief propagation may sometimes provide a more efficient alternative to traditional inference methods