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

nonuniversal critical behaviour in planar ising models with extended radially symmetric defects using conformal mapping and finite size scaling the authors determine the local magnetic exponent near a defect characterized by couplings of the form j r j 1 a r in accordance with phenomenological considerations they find that it depends continuously on the amplitude a both for a defect in the interior and on the boundary

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

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

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 recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

spectral optimization for the stekloff laplacian the stability issue abstract we consider the problem of minimizing the first non trivial stekloff eigenvalue of the laplacian among sets with given measure we prove that the brock weinstock inequality asserting that optimal shapes for this spectral optimization problem are balls can be improved by means of a sharp quantitative stability estimate this result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in betta et al 1999 2 we provide some new sharp quantitative versions of these achieved by means of a suitable calibration technique

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

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

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

capacity approaching low rate ldgm codes we study the ability of low rate serially concatenated low density generator matrix scldgm codes to approach theoretical limits although two layer scldgm codes approach capacity for medium rates 0 25 r 0 9 they do not perform adequately for lower rates in this case substantial performance improvements are obtained using three layers

the topological asymptotic for the helmholtz equation with dirichlet condition on the boundary of an arbitrarily shaped hole the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in the domain in this paper such an expansion is obtained for the helmholtz equation in two and three space dimensions with a dirichlet condition on the boundary of an arbitrarily shaped hole in this case the main difficulty is related to the nonhomogeneous symbol of the helmholtz operator in the numerical part of this work we will show that the topological sensitivity method is very promising for solving shape inverse problems in electromagnetic applications

semi infinite two dimensional ising model with marginally inhomogeneous couplings and conformal invariance the authors consider a semi infinite two dimensional ising model with nearest neighbour couplings that deviate from the bulk critical couplings by al 2 where l is the distance from the surface the surface critical exponents of this system are non universal under a conformal mapping onto a strip of width l the al 1 inhomogeneity transforms into a l pi sin pi l l 1 for the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit the analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance

local critical behaviour at aperiodic surface extended perturbation in the ising quantum chain the surface critical behaviour of the semi infinite one dimensional quantum ising model in a transverse field is studied in the presence of an aperiodic surface extended modulation the perturbed couplings are distributed according to a generalized fredholm sequence leading to a marginal perturbation and varying surface exponents the surface magnetic exponents are calculated exactly whereas the expression of the surface energy density exponent is conjectured from a finite size scaling study the system displays surface order at the bulk critical point above a critical value of the modulation amplitude it may be considered as a discrete realization of the hilhorst van leeuwen model

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

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

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

design of repeat accumulate codes for iterative detection and decoding extrinsic information transfer exit charts are used to design systematic and nonsystematic repeat accumulate ra codes for iterative detection and decoding the convergence problems of nonsystematic ra codes are solved by introducing a biregular or doped layer of check nodes as examples such nonsystematic codes are designed for multi input multi output mimo fading channels and are shown to operate close to capacity

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

an isoperimetric inequality for fundamental tones of free plates we establish an isoperimetric inequality for the fundamental tone first nonzero eigenvalue of the free plate of a given area proving the ball is maximal given 0 the free plate eigenvalues and eigenfunctions u are determined by the equation u u u together with certain natural boundary conditions the boundary conditions are complicated but arise naturally from the plate rayleigh quotient which contains a hessian squared term d 2 u 2

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

eigenvalues of the wentzell laplace operator and of the fourth order steklov problems abstract we prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the wentzell laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a euclidean space we study some fourth order steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them we also find all the eigenvalues and eigenfunctions for two kind of fourth order steklov problems on a euclidean ball

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

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

the computational modeling of problems on domains with small holes abstract the modeling challenges arising when the problem domain has small supported holes in it are considered through a representative membrane problem such problems are sometimes modeled intuitively in engineering practice by taking the limiting case of holes with zero radius this intuitive model is incorrect since it has no mathematical solution it is demonstrated however that finite element approximations based on it can still satisfy verification tests and appear to converge leading to erroneous recovery of quantities of interest this points to the need for an alternate approach where the holes of finite radius are properly incorporated in the modeling and robustness with respect to the radius is maintained to this end a computational method is presented which combines analytic knowledge of the solution singularities with finite element approximation of its smooth components theoretical and numerical results are provided establishing the efficacy and robustness of the method in extracting quantities of interest the method converges both with respect to the size of the holes and the mesh discretization parameter and provides a more accurate alternative to using the asymptotic limit

real analytic families of harmonic functions in a domain with a small hole abstract let n 3 let i and o be open bounded connected subsets of r n containing the origin let 0 0 be such that o contains the closure of i for all 0 0 then for a fixed 0 0 0 we consider a dirichlet problem for the laplace operator in the perforated domain o i we denote by u the corresponding solution if p o and p 0 then we know that under suitable regularity assumptions there exist p 0 and a real analytic operator u p from p p to r such that u p u p for all 0 p thus it is natural to ask what happens to the equality u p u p for negative we show a general result on continuation properties of some particular real analytic families of harmonic functions in domains with a small hole and we prove that the validity of the equality u p u p for negative depends on the parity of the dimension n

radial fredholm perturbation in the two dimensional ising model and gap exponent relation we consider concentric circular defects in the two dimensional ising model which are distributed according to a generalized fredholm sequence i e at exponentially increasing radii this type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation the critical exponent of the local magnetization is obtained through finite size scaling using a corner transfer matrix approach in the extreme anisotropic limit it varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial hilhorst van leeuwen model through a conformal mapping of the system onto a strip the gap exponent relation is shown to remain valid for such an aperiodic defect

simple map decoding of first order reed muller and hamming codes a maximum a posteriori map probability decoder of a block code minimizes the probability of error for each transmitted symbol separately the standard way of implementing map decoding of a linear code is the bahl cocke jelinek raviv bcjr algorithm which is based on a trellis representation of the code the complexity of the bcjr algorithm for the first order reed muller rm 1 codes and hamming codes is proportional to n sup 2 where n is the code s length in this correspondence we present new map decoding algorithms for binary and nonbinary rm 1 and hamming codes the proposed algorithms have complexities proportional to q sup 2 n log sub q n where q is the alphabet size in particular for the binary codes this yields complexity of order n log n

concatenated tree codes a low complexity high performance approach this paper is concerned with a family of concatenated tree ct codes ct codes are special low density parity check ldpc codes consisting of several trees with large spans they can also be regarded as special turbo codes with hybrid recursive nonrecursive parts and multiple constituent codes ct codes are decodable by the belief propagation algorithm they combine many advantages of ldpc and turbo codes such as low decoding cost fast convergence speed and good performance

a few shape optimization results for a biharmonic steklov problem abstract we derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary namely a steklov problem for the biharmonic operator we provide hadamard type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint finally we prove an isoperimetric inequality for the first positive eigenvalue

design of low density parity check codes for modulation and detection a coding and modulation technique is studied where the coded bits of an irregular low density parity check ldpc code are passed directly to a modulator at the receiver the variable nodes of the ldpc decoder graph are connected to detector nodes and iterative decoding is accomplished by viewing the variable and detector nodes as one decoder the code is optimized by performing a curve fitting on extrinsic information transfer charts design examples are given for additive white gaussian noise channels as well as multiple input multiple output mimo fading channels where the receiver but not the transmitter knows the channel for the mimo channels the technique operates within 1 25 db of capacity for various antenna configurations and thereby outperforms a scheme employing a parallel concatenated turbo code by wide margins when there are more transmit than receive antennas

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

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