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

connections on naturally reductive spaces their dirac operator and homogeneous models in string theory given a reductive homogeneous space m g h endowed with a naturally reductive metric we study the one parameter family of connections t joining the canonical and the levi civita connection t 0 1 2 we show that the dirac operator dt corresponding to t 1 3 is the so called cubic dirac operator recently introduced by b kostant and derive the formula for its square for any t thus generalizing the classical parthasarathy formula on symmetric spaces applications include the existence of a new g invariant first order differential operator on spinors and an eigenvalue estimate for the first eigenvalue of d1 3 this geometric situation can be used for constructing riemannian manifolds which are ricci flat and admit a parallel spinor with respect to some metric connection whose torsion t 0 is a 3 form the geometric model for the common sector of string theories we present some results about solutions to the string equations and a detailed discussion of a 5 dimensional example

metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

composition algebras with large derivation algebras abstract the finite dimensional flexible composition algebras include the hurwitz algebras composition algebras with unit element but also other interesting classes of algebras the para hurwitz and the okubo algebras the above mentioned algebras present many symmetries and this is reflected in their large derivation algebras in the present paper we study the opposite question what can be said about the composition algebras if we have some information about their derivation algebras our main result is the classification of all the composition algebras with such large derivation algebras

young tableau methods for kronecker products of representations of the classical groups diagrammatic methods for decomposing kronecker products of arbitrary representations of any of the classical groups are presented for convenience efficient ways of computing the dimensions and quadratic casimir s c2 r are also given these methods seem more useful for hand calculations than the method of schur functions or characteristic polynomials an appendix presents the kronecker products for any two representations of dimension 100

deriving good ldpc convolutional codes from ldpc block codes low density parity check ldpc convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity in this paper we discuss several graph cover based methods for deriving families of time invariant and time varying ldpc convolutional codes from ldpc block codes and show how earlier proposed ldpc convolutional code constructions can be presented within this framework some of the constructed convolutional codes significantly outperform the underlying ldpc block codes we investigate some possible reasons for this convolutional gain and we also discuss the mostly moderate decoder cost increase that is incurred by going from ldpc block to ldpc convolutional codes

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

new simple lie superalgebras in characteristic 3 abstract symplectic respectively orthogonal triple systems provide constructions of lie algebras respectively superalgebras however in characteristic 3 it is shown that this role can be interchanged and that lie superalgebras respectively algebras can be built out of symplectic triple systems respectively orthogonal triple systems with a different construction as a consequence new simple finite dimensional lie superalgebras as well as new models of some nonclassical simple lie algebras over fields of characteristic 3 will be obtained

a class of nonbinary ldpc codes with fast encoding and decoding algorithms this letter is concerned with a class of nonbinary low density parity check ldpc codes referred to as column scaled ldpc cs ldpc codes whose parity check matrices have a property that each column is a scaled binary vector the cs ldpc codes which include algebraically constructed nonbinary ldpc codes as subclasses admit fast encoding and decoding algorithms specifically for a code over the finite field f2p the encoder can be implemented with p parallel binary ldpc encoders followed by a series of bijective mappers while the decoder can be implemented with an iterative decoder in which no message permutations are required during the iterations in addition there exist low complexity iterative multistage decoders that can be utilized to trade off the performance against the complexity simulation results show that the performance degradation caused by the iterative multistage decoding algorithms is relevant to the code structure

focal sets and real hypersurfaces in complex projective space let m be a real submanifold of cpm and let j denote the complex structure we begin by finding a formula for the location of the focal points of m in terms of its second fundamental form this takes a particularly tractable form when m is a complex submanifold or a real hypersurface on which ji is a principal vector for each unit normal to m the rank of the focal map onto a sheet of the focal set of m is also computed in terms of the second fundamental form in the case of a real hypersurface on which je is principal with corresponding principal curvature u if the map onto a sheet of the focal set corresponding to u has constant rank then that sheet is a complex submanifold over which m is a tube of constant radius theorem 1 the other sheets of the focal set of such a hypersurface are given a real manifold structure in theorem 2 these results are then employed as major tools in obtaining two classifications of real hypersurfaces in cpm first there are no totally umbilic real hypersurfaces in cpm but we show theorem 3 let m be a connected real hypersurface in cpm m 3 with at most two distinct principal curvatures at each point then m is an open subset of a geodesic hypersphere secondly we show that there are no einstein real hypersurfaces in cpm and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the ricci tensor in theorem 4 one of the first results in the geometry of submanifolds is that an umbilic hypersurface m in euclidean space must be an open subset of a hyperplane or sphere the proof goes as follows assume that the shape operator is a scalar multiple of the identity a ax and use the codazzi equation to show that x is constant then either x 0 in which case m lies on a hyperplane or the focal points fx x x l x t the unit normal all coincide and m lies on the sphere of radius 1 x centered at the unique focal point this simple idea suggests a plan of attack for classifying hypersurfaces in terms of the nature of the principal curvatures under fairly general conditions the set of focal points corresponding to a principal curvature x can be given a differentiable received by the editors december 19 1980 presented to the society at its annual meeting in san francisco january 10 1981 1980 mathematics subject classificatiom primary 53b25 53c40

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

invariant affine connections on lie groups the space of bi invariant affine connections is determined for arbitrary compact lie groups in particular there is a surprising new family of such connections on su n

an extended freudenthal magic square in characteristic 3 abstract freudenthal s magic square which in characteristic 0 contains the exceptional lie algebras other than g 2 is extended over fields of characteristic 3 through the use of symmetric composition superalgebras to a larger square that contains both lie algebras and superalgebras with one exception the simple lie superalgebras that appear have no counterpart in characteristic 0

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

distance bounds for periodically time varying and tail biting ldpc convolutional codes existence type lower bounds on the free distance of periodically time varying ldpc convolutional codes and on the minimum distance of tail biting ldpc convolutional codes are derived it is demonstrated that the bound on free distance of periodically time varying ldpc convolutional codes approaches the bound on free distance of general nonperiodic time varying ldpc convolutional codes as the period increases the proof of the bound is based on lower bounding the minimum distance of corresponding tail biting ldpc convolutional codes which is of interest in its own right

performance comparison of ldpc block and spatially coupled codes over gf q in this paper we compare the finite length performance of protograph based spatially coupled low density paritycheck sc ldpc codes and ldpc block codes ldpc bcs over gf q to reduce computational complexity and latency a sliding window decoder with a stopping rule based on a soft belief propagation bp estimate is used for the q ary sc ldpc codes two regimes are considered one when the constraint length of q ary sc ldpc codes is equal to the block length of q ary ldpc bcs and the other when the two decoding latencies are equal simulation results confirm that in both regimes 3 6 3 9 and 3 12 regular non binary sc ldpc codes can significantly outperform both binary and non binary ldpc bcs and binary sc ldpc codes finally we present a computational complexity comparison of q ary sc ldpc codes and q ary ldpc bcs under equal decoding latency and equal decoding performance assumptions

low complexity x ems algorithms for nonbinary ldpc codes the extended min sum ems algorithm is redescribed as a reduced search trellis algorithm called m ems algorithm two variants of the m ems algorithm called t ems algorithm and d ems algorithm are presented simulation results show that these three algorithms referred to as x ems algorithms for convenience combined with factor correction techniques perform almost as well as the q ary sum product algorithm qspa but with a much lower complexity

non binary protograph based ldpc codes enumerators analysis and designs this paper provides a comprehensive analysis of nonbinary low density parity check ldpc codes built out of protographs we consider both random and constrained edge weight labeling and refer to the former as the unconstrained nonbinary protograph based ldpc codes u nbpb codes and to the latter as the constrained nonbinary protograph based ldpc codes c nbpb codes equipped with combinatorial definitions extended to the nonbinary domain ensemble enumerators of codewords trapping sets stopping sets and pseudocodewords are calculated the exact enumerators are presented in the finite length regime and the corresponding growth rates are calculated in the asymptotic regime an exit chart tool for computing the iterative decoding thresholds of protograph based ldpc codes is presented followed by several examples of finite length u nbpb and c nbpb codes with high performance throughout this paper we provide accompanying examples which demonstrate the advantage of nonbinary protograph based ldpc codes over their binary counterparts and over random constructions the results presented in this paper advance the analytical toolbox of nonbinary graph based codes

on some algebras related to simple lie triple systems abstract the set hom der t t f t t is determined for any simple finite dimensional lie triple system t over a field f of characteristic zero it turns out that it contains nontrivial elements if and only if t is related to a simple jordan algebra in particular this provides a new proof of the determination by laquer of the invariant affine connections in the simply connected compact irreducible riemannian symmetric spaces

structural analysis of array based non binary ldpc codes structural properties of array based non binary low density parity check nbldpc codes are studied in this paper first we characterize graphical substructures induced by codewords of symbol weight six in array based nbldpc codes defined by parity check matrices with column weight three we also reveal necessary conditions for these graphical substructures to incur weight 6 codewords such conditions can be used to select nonzero elements for avoiding weight 6 codewords or reducing the multiplicity of weight 6 codewords second we show that there exist weight 7 codewords in array based nbldpc codes defined by parity check matrices with column weight three as a byproduct we find that the graphical substructure induced by a weight 7 codeword takes the graphical substructure induced by the related weight 6 codewords as a subgraph third we show that there may exist codewords with symbol weight four six and seven in array based nbldpc codes defined by parity check matrices with column weight two these results enrich the structural analysis of array based ldpc codes in addition simulation results show the performance advantage of array based nbldpc codes

near shannon limit quasi cyclic low density parity check codes this letter presents two classes of quasi cyclic low density parity check codes that perform close to the shannon limit

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

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

pseudocodeword performance analysis for ldpc convolutional codes message passing iterative decoders for low density parity check ldpc block codes are known to be subject to decoding failures due to so called pseudocodewords these failures can cause the large signal to noise ratio snr performance of message passing iterative decoding to be worse than that predicted by the maximum likelihood ml decoding union bound in this paper we address the pseudocodeword problem from the convolutional code perspective in particular we compare the performance of ldpc convolutional codes with that of their wrapped quasi cyclic block versions and we show that the minimum pseudoweight of an ldpc convolutional code is at least as large as the minimum pseudoweight of an underlying quasi cyclic code this result which parallels a well known relationship between the minimum hamming weight of convolutional codes and the minimum hamming weight of their quasi cyclic counterparts is due to the fact that every pseudocodeword in the convolutional code induces a pseudocodeword in the block code with pseudoweight no larger than that of the convolutional code s pseudocodeword this difference in the weight spectra leads to improved performance at low to moderate snrs for the convolutional code a conclusion supported by simulation results

convexity of the cost functional in an optimal control problem for a class of positive switched systems abstract this paper deals with the optimal control of a class of positive switched systems the main feature of this class is that switching alters only the diagonal entries of the dynamic matrix the control input is represented by the switching signal itself and the optimal control problem is that of minimizing a positive linear combination of the final state variable first the switched system is embedded in the class of bilinear systems with control variables living in a simplex for each time point the main result is that the cost is convex with respect to the control variables this ensures that any pontryagin solution is optimal algorithms to find the optimal solution are then presented and an example taken from a simplified model for hiv mutation mitigation is discussed

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

threshold saturation for nonbinary sc ldpc codes on the binary erasure channel we analyze the asymptotic performance of nonbinary spatially coupled low density parity check sc ldpc code ensembles defined over the general linear group on the binary erasure channel in particular we prove the threshold saturation of belief propagation decoding to the so called potential threshold using the proof technique based on potential functions introduced by yedla et al assuming that the potential function exists we rewrite the density evolution of nonbinary sc ldpc codes in an equivalent vector recursion form which is suited for the use of the potential function we then discuss the existence of the potential function for the general case of vector recursions defined by multivariate polynomials and give a method to construct it we define a potential function in a slightly more general form than the one by yedla et al in order to make the technique based on potential functions applicable to the case of nonbinary ldpc codes we show that the potential function exists if a solution to a carefully designed system of linear equations exists furthermore we numerically show the existence of a solution to the system of linear equations for a large number of nonbinary ldpc code ensembles which allows us to define their potential function and thus prove threshold saturation

optimal control of switching systems this paper considers an optimal control problem for a switching system for solving this problem we do not make any assumptions about the number of switches nor about the mode sequence they are determined by the solution of the problem the switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter it is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system the relationship between the two sets of trajectories 1 motivates the shift of focus from the original problem to the more general one and 2 underlies the engineering relevance of the study of the second problem sufficient and necessary conditions for optimality are formulated for the second optimization problem if they exist bang bang type solutions of the embedded optimal control problem are solutions of the original problem otherwise suboptimal solutions are obtained via the chattering lemma

7 dimensional compact riemannian manifolds with killing spinors using a link between einstein sasakian structures and killing spinors we prove a general construction principle of odd dimensional riemannian manifolds with real killing spinors in dimensionn 7 we classify all compact riemannian manifolds with two or three killing spinors finally we classify nonflat 7 dimensional riemannian manifolds with parallel spinor fields

design of spatially coupled ldpc codes over gf q for windowed decoding in this paper we study spatially coupled low density parity check sc ldpc codes over finite fields gf q q geq 2 and develop design rules for q ary sc ldpc code ensembles based on their iterative belief propagation decoding thresholds with particular emphasis on low latency windowed decoding wd we consider transmission over both the binary erasure channel bec and the binary input additive white gaussian noise channel biawgnc and present results for a variety of j k regular sc ldpc code ensembles constructed over gf q using protographs thresholds are calculated using the protograph versions of q ary density evolution for the bec and the q ary extrinsic information transfer analysis for the biawgnc we show that the wd of q ary sc ldpc codes provides significant threshold gains compared with corresponding uncoupled q ary ldpc block code ldpc bc ensembles when the window size w is large enough and that these gains increase as the finite field size q 2 m increases moreover we demonstrate that the new design rules provide wd thresholds that are close to capacity even when both m and w are relatively small thereby reducing decoding complexity and latency the analysis further shows that compared with standard flooding schedule decoding the wd of q ary sc ldpc code ensembles results in significant reductions in both the decoding complexity and the decoding latency and that these reductions increase as m increases for the applications with a near threshold performance requirement and a constraint on decoding latency we show that using q ary sc ldpc code ensembles with moderate q 2 instead of their binary counterparts results in reduced decoding complexity

optimized design of finite length separable circulant based spatially coupled codes an absorbing set based analysis in this paper we characterize the finite length performance of separable circulant based spatially coupled scb sc ldpc codes for transmission over the additive white gaussian noise channel for a general class of finite length graph based codes it is known that the existence of small absorbing sets causes a performance degradation in the error floor regime we first present the mathematical conditions for the existence of absorbing sets in binary scb sc codes this analysis enables us to find the exact number of absorbing sets as a function of the design parameters in particular our results show that the choice of the cutting vector affects the number of absorbing sets and therefore the error floor performance of the code for a fixed column weight we find provably optimal cutting vectors that result in the least number of absorbing sets furthermore we extend our analysis to nonbinary scb sc codes where we show that the choice of the cutting vector is not as critical as in the binary case we provide an algorithm which provably removes the problematic nonbinary absorbing sets from nonbinary scb sc codes by informed selection of edge labels our simulation results show the superior error floor performance of our designed binary and nonbinary scb sc codes compared with binary unstructured and nonbinary quasi cyclic sc codes available in the open literature

quasi cyclic ldpc codes an algebraic construction this paper presents two new large classes of qc ldpc codes one binary and one non binary codes in these two classes are constructed by array dispersions of row distance constrained matrices formed based on additive subgroups of finite fields experimental results show that codes constructed perform very well over the awgn channel with iterative decoding based on belief propagation codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry ldpc codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low complexity reliability based iterative decoding algorithms such as binary message passing decoding algorithms non binary codes decoded with a fast fourier transform based sum product algorithm achieve significantly large coding gains over reed solomon codes of the same lengths and rates decoded with either the hard decision berlekamp massey algorithm or the algebraic soft decision kotter vardy algorithm they have potential to replace reed solomon codes in some communication or storage systems where combinations of random and bursts of errors or erasures occur

on the holonomy of connections with skew symmetric torsion we investigate the holonomy group of a linear metric connection with skew symmetric torsion in case of the euclidian space and a constant torsion form this group is always semisimple it does not preserve any non degenerated 2 form or any spinor suitable integral formulas allow us to prove similar properties in case of a compact riemannian manifold equipped with a metric connection of skew symmetric torsion on the aloff wallach space n 1 1 we construct families of connections admitting parallel spinors furthermore we investigate the geometry of these connections as well as the geometry of the underlying riemannian metric finally we prove that any 7 dimensional 3 sasakian manifold admits 2 parameter families of linear metric connections and spinorial connections defined by 4 forms with parallel spinors

iterative decoding threshold analysis for ldpc convolutional codes an iterative decoding threshold analysis for terminated regular ldpc convolutional ldpcc codes is presented using density evolution techniques the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the awgn channel with binary inputs it is shown that for a terminated ldpcc code ensemble the thresholds are better than for corresponding regular and irregular ldpc block codes

on lie admissible algebras whose commutator lie algebras are lie subalgebras of prime associative algebras abstract we describe third power associative multiplications on noncentral lie ideals of prime algebras and skew elements of prime algebras with involution provided that x y y x x y for all x y and the prime algebras in question do not satisfy polynomial identities of low degree we also obtain necessary and sufficient conditions for these multiplications to be fourth power associative or flexible

distance bounds for an ensemble of ldpc convolutional codes an ensemble of j k regular low density parity check ldpc convolutional codes is introduced and existence type lower bounds on the minimum distance dl of code segments of finite length l and on the free distance dfree are derived for sufficiently large constraint lengths v the distances are shown to grow linearly with v and the ratio dl v approaches the ratio dfee v for large l moreover the ratio of free distance to constraint length is several times larger than the ratio of minimum distance to block length for gallager s ensemble of j k regular ldpc block codes

a new proof of a theorem on optimal control of switched systems abstract recently sorin c bengea and raymond a decarlo sorin c bengea raymond a decarlo optimal control of switching systems automatica j ifac 41 2005 11 27 have offered a key result that the set of trajectories of the two switched system is dense in the set of trajectories of the embedded system this result was proven by means of relaxed controls and the chattering lemma in this paper we use the lyapunov theorem to give a new simple proof

branching rules for irreducible representations of e8 into d8 the branching rules for irreducible representations of e8 into d8 are calculated using the knowledge of the kronecker products for those two algebras tables of kronecker products for both e8 and d8 algebras are also included

minimum distance and trapping set analysis of protograph based ldpc convolutional codes low density parity check ldpc convolutional codes have been shown to be capable of achieving capacity approaching performance with iterative message passing decoding in the first part of this paper using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio we show that several ensembles of regular and irregular ldpc convolutional codes derived from protograph based ldpc block codes have the property that the free distance grows linearly with respect to the constraint length i e the ensembles are asymptotically good in particular we show that the free distance to constraint length ratio of the ldpc convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding ldpc block code ensembles a large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal to noise ratios under maximum likelihood decoding when suboptimal decoding methods are employed there are many factors that affect the performance of a code recently it has been shown that so called trapping sets are a significant factor affecting decoding failures of ldpc codes over the additive white gaussian noise channel with iterative message passing decoding in the second part of this paper we study the trapping sets of the asymptotically good protograph based ldpc convolutional codes considered earlier by extending the theory presented in part one and using similar bounding techniques we show that the size of the smallest non empty trapping set grows linearly with the constraint length for these ensembles

affine connections on 3 sasakian and manifolds the space of invariant affine connections on every 3 sasakian homogeneous manifold of dimension at least seven is described in particular the subspace of invariant affine metric connections and the subclass with skew torsion are also determined to this aim an explicit construction of all 3 sasakian homogeneous manifolds is exhibited it is shown that the 3 sasakian homogeneous manifolds which admit nontrivial einstein with skew torsion invariant affine connections are those of dimension seven that is mathbb s 7 mathbb r p 7 and the aloff wallach space mathfrak w 7 1 1 on mathbb s 7 and mathbb r p 7 the set of such connections is bijective to two copies of the conformal linear transformation group of the euclidean space while it is strictly bigger on mathfrak w 7 1 1 the set of invariant connections with skew torsion whose ricci tensor satisfies that its eigenspaces are the canonical vertical and horizontal distributions is fully described on 3 sasakian homogeneous manifolds an affine connection satisfying these conditions is distinguished by parallelizing all the reeb vector fields associated with the 3 sasakian structure which is also einstein with skew torsion on the 7 dimensional examples the invariant metric affine connections on 3 sasakian homogeneous manifolds with parallel skew torsion have been found finally some results have been adapted to the non homogeneous setting

efficient encoding of quasi cyclic low density parity check codes efficient encoding of quasi cyclic low density parity check codes quasi cyclic qc low density parity check ldpc codes form an important subclass of ldpc codes these codes have encoding advantage over other types of ldpc codes this paper addresses the issue of efficient encoding of qc ldpc codes two methods are presented to find the generator matrices of qc ldpc codes in systematic circulant form from their parity check matrices given in circulant form based on the systematic circulation form of the generator matrix of a qc ldpc code various types of encoding circuits using simple shift registers are devised it is shown that the encoding complexity of a qc ldpc code is linearly proportional to the number of parity bits of the code for serial encoding and to the length of the code for high speed parallel encoding

decoding algorithms for nonbinary ldpc codes over gf q in this letter we address the problem of decoding nonbinary low density parity check ldpc codes over finite fields gf q with reasonable complexity and good performance in the first part of the letter we recall the original belief propagation bp decoding algorithm and its fourier domain implementation we show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding in the second part of the letter we introduce a simplified decoder which is inspired by the min sum decoder for binary ldpc codes we called this decoder extended min sum ems we show that it is possible to greatly reduce the computational complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message by choosing appropriate correction factors or offsets we show that the ems decoder performance is quite good and in some cases better than the regular bp decoder the optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution our simulations on ultra sparse codes over very high order fields show that nonbinary ldpc codes are promising for applications which require low frame error rates for small or moderate codeword lengths the ems decoder is a good candidate for practical hardware implementations of such codes

transactions papers constructions of nonbinary quasi cyclic ldpc codes a finite field approach this paper is concerned with construction of efficiently encodable nonbinary quasi cyclic ldpc codes based on finite fields four classes of nonbinary quasi cyclic ldpc codes are constructed experimental results show that codes constructed perform well with iterative decoding using a fast fourier transform based q ary sum product algorithm and they achieve significant coding gains over reed solomon codes of the same lengths and rates decoded with either algebraic hard decision berlekamp massey algorithm or algebraic soft decision kotter vardy algorithm

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

a new young diagrammatic method for kronecker products of and a new simple young diagrammatic method for kronecker products of o n and sp 2m is proposed based on the representation theory of brauer algebras a general procedure for the decomposition of the tensor products of representations for o n and sp 2m is outlined which is similar to that for u n known as the littlewood rules together with trace contractions from a brauer algebra and some modification rules given by king

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

endomorphism valued cohomology and gauge neutral matter for several physically interesting calabi yau manifolds we count and parametrize gauge neutral matter particles occurring in corresponding superstring compactifications to this end we use the technique of exact and spectral sequences and then describe and discuss our results in the more familiar tensor notation

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

moment invariants for the vlasov equation moment invariants functions of the moments of a vlasov distribution that are invariant under sp 6 are classified using young diagrams the connection between the moment invariants and the poincare invariants is established an application using the moment invariants as phase space coordinates is considered for a matching section in a particle beam accelerator and a lie poisson numerical integration algorithm for the moment dynamics is proposed

quasi cyclic ldpc codes for fast encoding in this correspondence we present a special class of quasi cyclic low density parity check qc ldpc codes called block type ldpc b ldpc codes which have an efficient encoding algorithm due to the simple structure of their parity check matrices since the parity check matrix of a qc ldpc code consists of circulant permutation matrices or the zero matrix the required memory for storing it can be significantly reduced as compared with randomly constructed ldpc codes we show that the girth of a qc ldpc code is upper bounded by a certain number which is determined by the positions of circulant permutation matrices the b ldpc codes are constructed as irregular qc ldpc codes with parity check matrices of an almost lower triangular form so that they have an efficient encoding algorithm good noise threshold and low error floor their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices

threshold saturation via spatial coupling why convolutional ldpc ensembles perform so well over the bec convolutional low density parity check ldpc ensembles introduced by felstrom and zigangirov have excellent thresholds and these thresholds are rapidly increasing functions of the average degree several variations on the basic theme have been proposed to date all of which share the good performance characteristics of convolutional ldpc ensembles we describe the fundamental mechanism that explains why convolutional like or spatially coupled codes perform so well in essence the spatial coupling of individual codes increases the belief propagation bp threshold of the new ensemble to its maximum possible value namely the maximum a posteriori map threshold of the underlying ensemble for this reason we call this phenomenon threshold saturation this gives an entirely new way of approaching capacity one significant advantage of this construction is that one can create capacity approaching ensembles with an error correcting radius that is increasing in the blocklength although we prove the threshold saturation only for a specific ensemble and for the binary erasure channel bec empirically the phenomenon occurs for a wide class of ensembles and channels more generally we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly this might give rise to improved algorithms and new techniques for analysis

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

spatially coupled ldpc codes constructed from protographs in this paper we construct protograph based spatially coupled low density parity check ldpc codes by coupling together a series of l disjoint or uncoupled ldpc code tanner graphs into a single coupled chain by varying l we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary l we demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design capacity approaching iterative belief propagation bp decoding thresholds and linear growth of minimum distance with block length in particular we show that for sufficiently large l the bp thresholds on both the binary erasure channel and the binary input additive white gaussian noise channel saturate to a particular value significantly better than the bp decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled ldpc code when all variable nodes in the coupled chain have degree greater than two asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good ldpc codes with fast convergence rates and bp thresholds close to the shannon limit further the gap to capacity decreases as the density of the graph increases opening up a new way to construct capacity achieving codes on memoryless binary input symmetric output channels with low complexity bp decoding

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

low complexity decoding for non binary ldpc codes in high order fields in this paper we propose a new implementation of the extended min sum ems decoder for non binary ldpc codes a particularity of the new algorithm is that it takes into accounts the memory problem of the non binary ldpc decoders together with a significant complexity reduction per decoding iteration the key feature of our decoder is to truncate the vector messages of the decoder to a limited number nm of values in order to reduce the memory requirements using the truncated messages we propose an efficient implementation of the ems decoder which reduces the order of complexity to nm log2 nm this complexity starts to be reasonable enough to compete with binary decoders the performance of the low complexity algorithm with proper compensation is quite good with respect to the important complexity reduction which is shown both with a simulated density evolution approach and actual simulations

color algebras and affine connections on s6 the classification of algebras having a given lie group as their automorphism group has recently surged as an important problem in the study of general structure theory of algebras particle physics differential geometry and invariant system theory 2 4 11 14 17 191 since the automorphism group aut a of a real or complex algebra a is a lie group whose lie algebra is the derivation algebra der a of a zl 1 this is equivalent to the problem of classifying algebras having a given lie algebra as their derivation algebra symmetries that frequently appear in physics are represented by automorphisms of certain nonassociative algebras a where the largeness of aut a or der a reflects the symmetries such algebras with the symmetry group x 3 c su 3 or the compact exceptional lie group g2 have extensively been investigated in particle physics geometry of affrne connections on a reductive homogeneous space and for the study of nite dimensional real division algebras z 4 14 191 the color algebra of domokos and kiivesi domokos 4 o onion and para octonion algebras and the complex or real okubo algebra s are examples of such algebras a well known theorem of f adams asserts that if the sphere s admits

conformally parallel g2 structures on a class of solvmanifolds starting from a 6 dimensional nilpotent lie group n endowed with an invariant su 3 structure we construct a homogeneous conformally parallel g 2 metric on an associated solvmanifold we classify all half flat su 3 structures that endow the rank one solvable extension of n with a conformally parallel g 2 structure by suitably deforming the su 3 structures obtained we are able to describe the corresponding non homogeneous ricci flat metrics with holonomy contained in g 2 in the process we also find a new metric with exceptional holonomy

on algebras satisfying x 2x 2 n x x the commutative algebras satisfying the adjoint identity x 2x 2 n x x where n is a cubic form are shown to be related to a class of generically algebraic jordan algebras of degree at most 4 and to the pseudo composition algebras they are classified under a nondegeneracy condition

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