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

maximum likelihood erasure decoding of ldpc codes pivoting algorithms and code design this paper investigates efficient maximum likelihood ml decoding of low density parity check ldpc codes over erasure channels a set of algorithms referred to as pivoting algorithms is developed the aim is to limit the average number of pivots or reference variables from which all the other erased symbols are recovered iteratively the suggested algorithms exhibit different trade offs between complexity of the pivoting phase and average number of pivots moreover a systematic procedure to design ldpc code ensembles for efficient ml decoding is proposed numerical results illustrate that the designed ldpc codes achieve a near optimum performance very close to the singleton bound at least down to a codeword error rate level 10 8 with an affordable decoding complexity for one of the presented codes and algorithms a software implementation has been developed which is capable to provide data rates above 1 5 gbps on a commercial computing platform

structural optimization under uncertain loads and nodal locations this paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes the second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements we first review the discrete form of the uncertain loads problem which has been previously solved using a weighted average of multiple load patterns with minor modifications we extend this solution to include loads described by continuous joint probability density functions we then proceed to the main contribution of this paper structural optimization under uncertainty in the nodal locations this optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix it is shown however that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads by expressing these equivalent loads in terms of continuous random variables we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper the optimization algorithms are developed in the context of minimum compliance maximum stiffness design simple examples are presented the results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design for structures containing thin substructures under axial loads it is shown that these uncertainties a are of first order significance influencing the linear elastic response quantities and b can affect designs by avoiding unrealistically optimistic and potentially unstable structures the additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem

weight distributions of non binary multi edge type ldpc code ensembles analysis and efficient evaluation non binary multi edge type ensembles of low density parity check codes are analyzed in terms of non binary codeword weight distribution and its growth rate in particular an exact expression of the growth rate for small weights is developed as a side result the stopping set distributions of these ensembles are developed examples of weight distributions are provided showing that the derived closed form expressions can be easily evaluated the obtained results can thus be exploited to analyze and design non binary low density parity check codes that fall within the multi edge type framework such as but not limited to protograph based codes

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

complicated dynamics of scalar reaction diffusion equations with a nonlocal term we consider the dynamics of scalar equations ut uxx f x u c x u 0 x l where denotes some weighted spatial average and dinchlet boundary conditions are assumed prescribing f c appropriately it is shown that complicated dynamics can occur specifically linearisations at equilibria can have any number of purely imaginary eigenvalues moreover the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily up to any finite order these results are in marked contrast with the case 0 where bounded solutions are known to converge to equilibrium

on ensembles of low density parity check codes asymptotic distance distributions we derive expressions for the average distance distributions in several ensembles of regular low density parity check codes ldpc among these ensembles are the standard one defined by matrices having given column and row sums ensembles defined by matrices with given column sums or given row sums and an ensemble defined by bipartite graphs

on inhomogeneous eigenvalue problems i abstract in studying stability of solutions of linear differential equations one naturally encounters liapunov equations in a suitable setting they can be interpreted as equations for the normalized directions of these solutions when applying discretizations to the liapunov equations one is led to a problem which in its most elementary form can be stated as given a matrix a and a vector b determine a vector x with xtx 1 and a scalar such that ax b x here is called the inhomogeneous eigenvalue we consider the question of how many solution pairs x of this problem exist we also give some numerical methods to compute such a pair they are based on generalizations of shifted power iterations finally we consider the case that b is small so that the inhomogeneous eigenvalues can be viewed as perturbations of the homogeneous ones i e b 0

on finite length performance of polar codes stopping sets error floor and concatenated design this paper investigates properties of polar codes that can be potentially useful in real world applications we start with analyzing the performance of finite length polar codes over the binary erasure channel bec while assuming belief propagation as the decoding method we provide a stopping set analysis for the factor graph of polar codes where we find the size of the minimum stopping set we also find the girth of the graph for polar codes our analysis along with bit error rate ber simulations demonstrate that finite length polar codes show superior error floor performance compared to the conventional capacity approaching coding techniques in order to take advantage from this property while avoiding the shortcomings of polar codes we consider the idea of combining polar codes with other coding schemes we propose a polar code based concatenated scheme to be used in optical transport networks otns as a potential real world application comparing against conventional concatenation techniques for otns we show that the proposed scheme outperforms the existing methods by closing the gap to the capacity while avoiding error floor and maintaining a low complexity at the same time

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

bifurcation of steady state solutions in predator prey and competition systems we discuss steady state solutions of systems of semilinear reaction diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region it is easy to find solutions to the systems such that either u or v is identically zero such solutions correspond to the case where one of the species is extinct by using decoupling and global bifurcation theory techniques we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co exist

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

weight distribution of low density parity check codes we derive the average weight distribution function and its asymptotic growth rate for low density parity check ldpc code ensembles we show that the growth rate of the minimum distance of ldpc codes depends only on the degree distribution pair it turns out that capacity achieving sequences of standard unstructured ldpc codes under iterative decoding over the binary erasure channel bec known to date have sublinearly growing minimum distance in the block length

bounds on the decoding error probability of binary linear codes via their spectra bounds on the error probability of maximum likelihood decoding of a binary linear code are considered the bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel the bounds derived are applied to a code with an average spectrum and the result is compared to the random coding exponent the author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound for the case of awgn channel the author shows that berlekamp s 1980 tangential bound can be improved but even this improved bound does not coincide with the random coding bound although it can be very close to it

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

distance distributions in ensembles of irregular low density parity check codes we derive asymptotic expressions for the average distance distributions in ensembles of irregular low density parity check ldpc codes the ensembles are defined by matrices with given profiles of column and row sums

tree structured expectation propagation for ldpc decoding over bms channels in this paper we put forward the tree structured expectation propagation tep algorithm for decoding block and convolutional low density parity check codes over any binary channel we have already shown that tep improves belief propagation bp over the binary erasure channel bec by imposing marginal constraints over a set of pairs of variables that form a tree or a forest the tep decoder is a message passing algorithm that sequentially builds a tree forest of erased variables to capture additional information disregarded by the standard bp decoder which leads to a noticeable reduction of the error rate for finite length codes in this paper we show how the tep can be extended to any channel specifically to binary memoryless symmetric bms channels we particularly focus on how the tep algorithm can be adapted for any channel model and more importantly how to choose the tree forest to keep the gains observed for block and convolutional ldpc codes over the bec

design of irregular ldpc codes with optimized performance complexity tradeoff the optimal performance complexity tradeoff for error correcting codes at rates strictly below the shannon limit is a central question in coding theory this paper proposes a numerical approach for the minimization of decoding complexity for long block length irregular low density parity check ldpc codes the proposed design methodology is applicable to any binary input memoryless symmetric channel and any iterative message passing decoding algorithm with a parallel update schedule a key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence this paper shows that the proposed complexity measure can be accurately estimated from a density evolution and extrinsic information transfer chart analysis of the code a sufficient condition is presented for convexity of the complexity measure in the variable edge degree distribution when it is not satisfied numerical experiments nevertheless suggest that the local minimum is unique the results presented herein show that when the decoding complexity is constrained the complexity optimized codes significantly outperform threshold optimized codes at long block lengths within the ensemble of irregular codes

on the asymptotic input output weight distributions and thresholds of convolutional and turbo like encoders we present a general method for computing the asymptotic input output weight distribution of convolutional encoders in some instances one can derive explicit analytic expressions in general though to determine the growth rate of the input output weight distribution for a particular normalized input weight spl kappa and output weight spl omega a system of polynomial equations has to be solved this method is then used to determine the asymptotic weight distribution of various concatenated code ensembles and to derive lower bounds on the thresholds of these ensembles under maximum likelihood ml decoding

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

robustness analysis of trusses with separable load and structural uncertainties this paper discusses evaluation techniques of the robustness function of trusses which is regarded as one of measures of structural robustness under the uncertainties of member stiffnesses and external forces by using quadratic embedding of the uncertainty and the s procedure we formulate a quasiconvex optimization problem which provides lower bounds of the robustness functions a bisection method is proposed where we solve a finite number of semidefinite programming problems in order to obtain a global optimal solution to the proposed quasiconvex optimization problem the lower bounds of the robustness functions are computed for various trusses under several uncertainty circumstances

topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system abstract this paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load the design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors the idea of aggregation used in the field of control is introduced to assess the value of the robust compliance the aggregation solution technique provides the direct relationship between the uncertain input load and output displacement as a small linear system composed of these vectors and the reduced size of a symmetric matrix in the context of a discretized linear elasticity problem using the finite element method introducing the constraint that the euclidean norm of the uncertain load set is fixed the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the rayleigh ritz theorem for symmetric matrices moreover the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix the proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes mma the numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures

quasi cyclic ldpc codes based on pre lifted protographs quasi cyclic low density parity check qc ldpc codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting however these restrictions impose undesirable fixed upper limits on important code parameters such as minimum distance and girth in this paper we consider an approach to constructing qc ldpc codes that uses a two step lifting procedure based on a protograph and by following this method instead of the usual one step procedure we obtain improved minimum distance and girth properties we also present two new design rules for constructing good qc ldpc codes using this two step lifting procedure and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared with one step circulant based liftings the expected performance improvement is verified by simulation results

tree structure expectation propagation for ldpc decoding over the bec we present the tree structure expectation propagation tree ep algorithm to decode low density parity check ldpc codes over discrete memoryless channels dmcs expectation propagation generalizes belief propagation bp in two ways first it can be used with any exponential family distribution over the cliques in the graph second it can impose additional constraints on the marginal distributions we use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the ldpc code s tanner graph thanks to these additional constraints the tree ep marginal estimates for each variable in the graph are more accurate than those provided by bp we also reformulate the tree ep algorithm for the binary erasure channel bec as a peeling type algorithm tep and we show that the algorithm has the same computational complexity as bp and it decodes a higher fraction of errors we describe the tep decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters the solution of these equations is used to predict the tep decoder performance in both the asymptotic regime and the finite length regimes over the bec while the asymptotic threshold of the tep decoder is the same as the bp decoder for regular and optimized codes we propose a scaling law for finite length ldpc codes which accurately approximates the tep improved performance and facilitates its optimization

topology optimization of continuum structures with local stress constraints we introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria we first consider relevant stress criteria for porous composite materials initially by studying the stress states of the so called rank 2 layered materials then on the basis of the theoretical study of the rank 2 microstructures we propose an empirical model that extends the power penalized stiffness model also called simp for solid isotropic microstructure with penalization for inter mediate densities in a second part solution aspects of topology problems are considered to deal with the so called singularity phenomenon of stress constraints in topology design an constraint relaxation of the stress constraints is used we describe the mathematical programming approach that is used to solve the numerical optimization problems and show results for a number of example applications 1998 john wiley sons ltd

capacity achieving codes with bounded graphical complexity and maximum likelihood decoding in this paper the existence of capacity achieving codes for memoryless binary input output symmetric mbios channels under maximum likelihood ml decoding with bounded graphical complexity is investigated graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding irregular repeat accumulate ira codes are studied first utilizing the asymptotic average weight distribution aawd of these codes and invoking divsalar s bound on the binary input additive white gaussian noise biawgn channel it is shown that simple nonsystematic ira ensembles outperform systematic ira and regular low density parity check ldpc ensembles with the same graphical complexity and are at most 0 124 db away from the shannon limit however a conclusive result as to whether these nonsystematic ira codes can really achieve capacity cannot be reached motivated by this inconclusive result a new family of codes is proposed called low density parity check and generator matrix ldpc gm codes which are serially concatenated codes with an outer ldpc code and an inner low density generator matrix ldgm code it is shown that these codes can achieve capacity on any mbios channel using ml decoding and also achieve capacity on any bec using belief propagation bp decoding both with bounded graphical complexity moreover it is shown that under certain conditions these capacity achieving codes have linearly increasing minimum distances and achieve the asymptotic gilbert varshamov bound for all rates

effects of single cycle structure on iterative decoding of low density parity check codes we consider communication over the binary erasure channel bec using low density parity check ldpc codes and belief propagation bp decoding for fixed numbers of bp iterations the bit error probability approaches a limit as the blocklength tends to infinity and the limit is obtained via density evolution the finite blocklength correction behaves like e t n n 2 as the blocklength n tends to infinity where e t denotes a specific constant determined by the code ensemble considered the number t of iterations and the erasure probability e of the bec in this paper we derive a set of recursive formulas which allows the evaluation of the constant e t for standard irregular ensembles the dominant difference e t n can be considered as effects of cycle free and single cycle structures of local graphs furthermore it is confirmed via numerical simulations that estimation of the bit error probability using e t is accurate even for small blocklengths

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

new nonasymptotic channel coding theorems for structured codes new nonasymptotic random coding theorems with error probability epsilon and finite block length n based on gallager parity check ensemble and general parity check ensembles are derived in this paper the resulting nonasymptotic achievability bounds when combined with nonasymptotic equipartition properties developed in this paper can be easily computed analytically these nonasymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as n goes to infinity with either constant or subexponentially decreasing epsilon in the case of gallager parity check ensemble and to imply that low density parity check ldpc codes be capacity achieving in the case of ldpc ensembles numerically they are also compared favorably for finite n and epsilon of practical interest with existing nonasymptotic achievability bounds in the literature

designing a good low rate sparse graph code this paper deals with the design of low rate sparse graph codes having a linear minimum distance d min in the blocklength n its main contributions are a a necessary condition on a general family of sparse graph codes with linear d min b a justification of having degree 1 bits in the low rate code structure c a new efficient ensemble of low rate sparse graph codes with bits of degree 1 designed so that the necessary condition a is satisfied

topology optimization of continuum structures with stress constraints and uncertainties in loading summary topology optimization using stress constraints and considering uncertainties is a serious challenge since a reliability problem has to be solved for each stress constraint for each element in the mesh in this paper an alternative way of solving this problem is employed where uncertainty quantification is performed through the first order perturbation approach with proper validation by monte carlo simulation uncertainties are considered in the loading magnitude and direction the minimum volume problem subjected to local stress constraints is formulated as a robust problem where the stress constraints are written as a weighted average between their expected value and standard deviation the augmented lagrangian method is used for handling the large set of local stress constraints whereas a gradient based algorithm is used for handling the bounding constraints it is shown that even in the presence of small uncertainties in loading direction different topologies are obtained when compared to a deterministic approach the effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied where the main observed result is loss of symmetry in optimal topologies this article is protected by copyright all rights reserved

coding for parallel channels gallager bounds and applications to turbo like codes the transmission of coded communication systems is widely modeled to take place over a set of parallel channels this model is used for transmission over block fading channels rate compatible puncturing of turbo like codes multicarrier signaling multilevel coding etc new upper bounds on the maximum likelihood ml decoding error probability are derived in the parallel channel setting we focus on the generalization of the gallager type bounds and discuss the connections between some versions of these bounds the tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes repeat accumulate ra codes and some of their recent variations e g punctured accumulate repeat accumulate codes the bounds on the decoding error probability of an ml decoder are compared to computer simulations of iterative decoding the new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels this improvement is exemplified for relatively short block lengths and it is pronounced when the block length is increased in the asymptotic case where we let the block length tend to infinity inner bounds on the attainable channel regions of modern coding techniques under ml decoding are obtained based solely on the asymptotic growth rates of the average distance spectra of these code ensembles

a nonlocal sturm liouville eigenvalue problem a nonlocal eigenvalue problem of the form u a x u bu u with homogeneous dirichlet boundary conditions is considered where b is a rank one bounded linear operator and x belongs to some bounded interval on the real line the behaviour of the eigenvalues is studied using methods of linear perturbation theory in particular some results are given which ensure that the spectrum remains real a sturm type comparison result is obtained finally these results are applied to the study of some nonlocal reaction diffusion equations

construction and application of an ellipsoidal convex model using a semi definite programming formulation from measured data abstract as a set theory based convex model the ellipsoidal model provides an attractive framework for treating uncertain but bounded variations in the structural reliability analysis and design optimization however improper modeling of the uncertainties may give rise to misleading non probabilistic reliability analysis thus result in either unsafe or over conservative designs this paper presents a systematic study on the mathematical formulation for constructing the minimum volume ellipsoidal convex model using a given set of sample data and shows its application in existing methods of non probabilistic reliability analysis and design optimization of structures with bounded uncertainties in this method the uncertain parameters are first divided into groups according to their sources for each individual group of uncertainties the minimum volume ellipsoid problem is reformulated into a semi definite programming sdp problem and thus can be efficiently solved to its global optimum further a linear transformation based on the eigenvalue analysis is employed to map the ellipsoidal model into a standard uncertainty space this uncertainty modeling technique enables a compact and differentiable bound description of the parameter variations moreover it has another useful property the affine invariance which is shown to be necessary for meaningful definition of a non probabilistic reliability index the effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources

weight distributions of regular low density parity check codes over finite fields the average weight distribution of a regular low density parity check ldpc code ensemble over a finite field is thoroughly analyzed in particular a precise asymptotic approximation of the average weight distribution is derived for the small weight case and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved based on this analysis a general result including all previous results as special cases is established for the minimum distance of individual codes in a regular ldpc code ensemble

robust topology optimization under multiple independent unknown but bounded loads abstract the optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable or even lead to catastrophic failures a two level optimization formulation is often used in the robust topology optimization rto under uncertain loads various approaches have been reported to identify the critical loads associated with the worst structure responses because convex model approaches apply convex approximations to the original non convex model at the lower level the optimal designs obtained by these methods are greatly dependent on the quality of the approximation in this paper a new formulation based on the wolfe duality for the rto problems with multiple independent unknown but bounded loads is proposed following the two level formulation the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the wolfe duality both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non convexity associated with the lower level formulation numerical examples are also presented to demonstrate the proposed approach

interactive encoding and decoding based on binary ldpc codes with syndrome accumulation interactive encoding and decoding based on binary low density parity check codes with syndrome accumulation sa ldpc ied is proposed and investigated assume that the source alphabet is gf 2 and the side information alphabet is finite it is first demonstrated how to convert any classical universal lossless code cn with block length n and side information available to both the encoder and decoder into a universal sa ldpc ied scheme it is then shown that with the word error probability approaching 0 subexponentially with n the compression rate including both the forward and backward rates of the resulting sa ldpc ied scheme is upper bounded by a functional of that of cn which in turn approaches the compression rate of cn for each and every individual sequence pair xn yn and the conditional entropy rate h x y for any stationary ergodic source and side information x y as the average variable node degree l of the underlying ldpc code increases without bound when applied to the class of binary source and side information x y correlated through a binary symmetrical channel with crossover probability unknown to both the encoder and decoder the resulting sa ldpc ied scheme can be further simplified yielding even improved rate performance versus the bit error probability when l is not large simulation results coupled with linear time belief propagation decoding on binary source side information pairs confirm the theoretic analysis and further show that the sa ldpc ied scheme consistently outperforms the slepian wolf coding scheme based on the same underlying ldpc code as a by product probability bounds involving ldpc established in the course are also interesting on their own and expected to have implications on the performance of ldpc for channel coding as well

doubly generalized ldpc codes stability bound over the bec the iterative decoding threshold of low density parity check ldpc codes over the binary erasure channel bec fulfills an upper bound depending only on the variable and check nodes with minimum distance 2 this bound is a consequence of the stability condition and is here referred to as stability bound in this paper a stability bound over the bec is developed for doubly generalized ldpc codes where variable and check nodes can be generic linear block codes assuming maximum a posteriori erasure correction at each node it is proved that also in this generalized context the bound depends only on the variable and check component codes with minimum distance 2 a condition is also developed namely the derivative matching condition under which the bound is achieved with equality the stability bound leads to consider single parity check codes used as variable nodes as an appealing option to overcome common problems created by generalized check nodes

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

generalized and doubly generalized ldpc codes with random component codes for the binary erasure channel in this paper a method for the asymptotic analysis of generalized low density parity check gldpc codes and doubly generalized low density parity check d gldpc codes over the binary erasure channel bec based on extrinsic information transfer exit chart is described this method overcomes the problem consisting of the impossibility to evaluate the exit function for the check or variable component codes in situations where the information functions or split information functions for component codes are unknown according to the proposed technique gldpc codes and d gldpc codes where the generalized check and variable component codes are random codes with minimum distance at least 2 are considered a technique is then developed which finds the exit chart for the overall gldpc or d gldpc code by evaluating the expected exit function for each check and variable component code this technique is finally combined with the differential evolution algorithm in order to generate some good gldpc and d gldpc edge distributions numerical results of long random codes are presented which confirm the effectiveness of the proposed approach they also reveal that d gldpc codes can outperform standard ldpc codes and gldpc codes in terms of both waterfall performance and error floor

on the error probability of signals in additive white gaussian noise a new upper bound to the probability of error in detecting one of m equally probable signals in additive white gaussian noise is presented this bound is easy to calculate can be applied to any signal set it is always better than the union and minimum distance bounds examples demonstrate the use of the bound

spectral shape of doubly generalized ldpc codes efficient and exact evaluation this paper analyzes the asymptotic exponent of the weight spectrum for irregular doubly generalized ldpc d gldpc codes in the process an efficient numerical technique for its evaluation is presented involving the solution of a 4 4 system of polynomial equations the expression is consistent with previous results including the case where the normalized weight or stopping set size tends to zero the spectral shape is shown to admit a particularly simple form in the special case where all variable nodes are repetition codes of the same degree a case which includes tanner codes for this case it is also shown how certain symmetry properties of the local weight distribution at the cns induce a symmetry in the overall weight spectral shape function finally using these new results weight and stopping set size spectral shapes are evaluated for some example generalized and doubly generalized ldpc code ensembles

tight bounds for ldpc and ldgm codes under map decoding a new method for analyzing low density parity check ldpc codes and low density generator matrix ldgm codes under bit maximum a posteriori probability map decoding is introduced the method is based on a rigorous approach to spin glasses developed by francesco guerra it allows one to construct lower bounds on the entropy of the transmitted message conditional to the received one based on heuristic statistical mechanics calculations we conjecture such bounds to be tight the result holds for standard irregular ensembles when used over binary input output symmetric bios channels the method is first developed for tanner graph ensembles with poisson left degree distribution it is then generalized to multi poisson graphs and by a completion procedure to arbitrary degree distribution

robust topology optimization minimization of expected and variance of compliance robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies this paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude using exact formulations and analytically derived sensitivities it shows that only a few additional load cases are required which scales in polynomial time with the number of uncertain load cases the approach is implemented using the level set topology optimization method shape sensitivities are derived using the adjoint method several examples are used to investigate the effect of including variance in robust compliance optimization

a general framework for robust topology optimization under load uncertainty including stress constraints we present a worst case approach for topology optimization under load uncertainty based on a general problem formulation involving maxima of quadratic functions of an uncertain load vector as both

average coset weight distribution of combined ldpc matrix ensembles in this paper the average coset weight distribution acwd of structured ensembles of low density parity check ldpc matrices which are called combined ensembles is discussed a combined ensemble is composed of a set of simpler ensembles such as a regular bipartite ensemble two classes of combined ensembles have prime importance a stacked ensemble and a concatenated ensemble the acwd formulas of these ensembles are shown in this paper such formulas play a key role to evaluate the average weight distribution of some classes of combined ensembles

how to solve a semi infinite optimization problem abstract after an introduction to main ideas of semi infinite optimization this article surveys recent developments in theory and numerical methods for standard and generalized semi infinite optimization problems particular attention is paid to connections with mathematical programs with complementarity constraints lower level wolfe duality semi smooth approaches as well as branch and bound techniques in adaptive convexification procedures a section on recent genericity results includes a discussion of the symmetry effect in generalized semi infinite optimization

on logistic diffusion equations with nonlocal interaction terms abstract this paper is concerned with logistic diffusion equations with nonlocal interaction terms appearing in population biology we intend to study effects of nonlocal terms and discuss the similarity and difference between local problems and nonlocal problems mainly the stationary problem is investigated for a certain class of nonlocal terms a constructive approach is proposed to look for positive stationary solutions and the unique existence of such a positive solution is established the analysis of stationary solutions depends on the spectrum for the linearized operator around the stationary solution however the linearized operator contains a nonlocal term which makes the spectral analysis delicate and difficult putting some additional assumptions we will derive the asymptotic stability of the unique positive solution and furthermore its global attractivity finally it will be seen that some arguments are valid to show the unique existence of a positive stationary solution for a considerably general class of nonlocal terms

second order weight distributions a fundamental property of codes the second order weight distribution is proposed to solve the problems such as computing second moments of weight distributions of linear code ensembles a series of results parallel to those for weight distributions is established for second order weight distributions in particular an analogue of macwilliams identities is proved the second order weight distributions of regular ldpc code ensembles are then computed as easy consequences the second moments of weight distributions of regular ldpc code ensembles are obtained furthermore the application of second order weight distributions in random coding approach is discussed the second order weight distributions of the ensembles generated by a so called 2 good random generator or parity check matrix are computed where a 2 good random matrix is a kind of generalization of the uniformly distributed random matrix over a finite filed and is very useful for solving problems that involve pairwise or triple wise properties of sequences it is shown that the 2 good property is reflected in the second order weight distribution which thus plays a fundamental role in some well known problems in coding theory and combinatorics an example of linear intersecting codes is finally provided to illustrate this fact

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices abstract this paper proposes an efficient approach to solving robust topology optimization problem of structures under loading uncertainty the objective is to minimize a weighted sum of the mean and standard deviation of structural compliance loading uncertainties can be in either concentrated loads or uniformly distributed loads by exploiting of the linear elastic nature of structure monte carlo sampling is completely separated from the topology optimization process thus accurate calculation of objective function becomes possible efficient sensitivity analysis method is developed and its computational cost is only linearly proportional to the number of uncertain loads the sensitivity analysis is also integrated into the density based topology optimization approach to solve the robust topology optimization problems the numerical examples demonstrate the effectiveness of the proposed approach the effect of uncertainty level probability distribution of uncertainty and different influence of loading magnitude and directional uncertainty on the robust designs are also shown by the numerical examples

performance of ldpc codes under faulty iterative decoding departing from traditional communication theory where decoding algorithms are assumed to perform without error a system where noise perturbs both computational devices and communication channels is considered here this paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low density parity check ldpc codes concentration of decoding performance around its average is shown to hold when noise is introduced into message passing and local computation density evolution equations for simple faulty iterative decoders are derived in one model computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases but arbitrarily small probability of error is not achievable probability of error may be driven to zero in another system model the decoding threshold again decreases smoothly with decoder noise as an application of the methods developed an achievability result for reliable memory systems constructed from unreliable components is provided

upper bound on error exponent of regular ldpc codes transmitted over the bec the error performance of the ensemble of typical low definition parity check ldpc codes transmitted over the binary erasure channel bec is analyzed in the past lower bounds on the error exponents were derived in this paper a probabilistic upper bound on this error exponent is derived this bound holds with some confidence level

maxwell construction the hidden bridge between iterative and maximum a posteriori decoding there is a fundamental relationship between belief propagation and maximum a posteriori decoding a decoding algorithm which is called the maxwell decoder is introduced and provides a constructive description of this relationship both the algorithm itself and the analysis of the new decoder are reminiscent of the maxwell construction in thermodynamics this paper investigates in detail the case of transmission over the binary erasure channel while the extension to general binary memoryless channels is discussed in a companion paper

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

the maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems abstract in this work we deal with the problem of the existence and uniqueness of principal eigenvalues for some linear weighted boundary value problems associated to a general second order uniformly elliptic operator for a large class of sign definited weights we characterize whether the boundary value problem admits a principal eigenvalue or not

lower bounds on the error rate of ldpc code ensembles the ensemble of regular low definition parity check ldpc codes is considered using concentration results on the weight distribution lower bounds on the error rate of a random code in the ensemble are derived these bounds hold with some confidence level combining these results with known lower bounds on the error exponent confidence intervals on the error exponent under maximum likelihood ml decoding are obtained over a large range of channel parameter and transmission rate values when the graph connectivity is sufficiently large the upper bound of the interval approaches the lower bound and the probability that the error exponent is within the interval can be arbitrarily close to one in fact in this case the true error exponent approaches the maximum between the random coding and the expurgated random coding exponents with probability that approaches one

global stability and pattern formation in a nonlocal diffusive lotka volterra competition model abstract a diffusive lotka volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed the nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species it is shown that when there is no nonlocal intraspecific competition the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive lotka volterra competition model regardless of the strength of nonlocal interspecific competition global stability of nonnegative constant equilibria are proved using lyapunov or upper lower solution methods on the other hand strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity for the weak competition case the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients

protograph based raptor like ldpc codes this paper proposes protograph based raptor like pbrl codes as a class of rate compatible low density parity check codes for binary input awgn channels as with the raptor codes exclusive or operations on precoded bits produce additional parity bits providing extensive rate compatibility unlike raptor codes each additional parity bit in the protograph is explicitly designed to optimize the density evolution threshold during the lifting process approximate cycle extrinsic message degree ace and circulant progressive edge growth cpeg constraints are used to avoid undesirable graphical structures some density evolution performance is sacrificed to obtain lower error floors particularly at short blocklengths simulation results are shown for information block sizes of k 1032 hbox and 16 384 for a target frame error rate of 10 5 at each rate the k 1032 hbox and 16 384 code families perform within 1 db and 0 4 db of both the gallager bound and the normal approximation respectively the 16 384 code family outperforms the best known standardized code family namely the ar4ja codes the pbrl codes also outperform dvb s2 codes that have the advantages of longer blocklengths and outer bch codes performance is similar to rc code families designed by nguyen et al that do not constrain codes to have the pbrl structure and involve simulation in the optimization process at each rate

shape optimization under uncertainty a stochastic programming perspective we present an algorithm for shape optimization under stochastic loading and representative numerical results our strategy builds upon a combination of techniques from two stage stochastic programming and level set based shape optimization in particular usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces which often is much smaller than the number of different realizations of the stochastic forces numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions

further spectral properties of uniformly elliptic operators that include a non local term in this paper we consider the eigenvalues of a class of non local differential operators the real eigenvalues of certain examples of these operators have previously been plotted as functions of a fundamental parameter in the operator using appropriate computer software here we show how the complex eigenvalues of such operators may also be plotted we use the plots of both the real and the complex eigenvalues to motivate a number of new theoretical results regarding the structure of the spectrum which we subsequently prove under suitable hypotheses

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