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

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

hybrid arq in multiple antenna slow fading channels performance limits and optimal linear dispersion code design this paper focuses on studying the fundamental performance limits and linear dispersion code design for the mimo arq slow fading channel the optimal average rate of well known harq protocols is analyzed the optimal design of space time coding for the mimo arq channel is discussed information theoretic measures are used to optimize the rate assignment and derive the optimum design criterion which is then used to evaluate the optimality of existing space time codes a different design criterion which is obtained from the error probability analysis of space time coded mimo harq is presented examples are studied to reveal the gain of arq feedback in space time coded mimo systems

on achievable rates and complexity of ldpc codes over parallel channels bounds and applications a variety of communication scenarios can be modeled by a set of parallel channels upper bounds on the achievable rates under maximum likelihood ml decoding and lower bounds on the decoding complexity per iteration of ensembles of low density parity check ldpc codes are presented the communication of these codes is assumed to take place over statistically independent parallel channels where the component channels are memoryless binary input and output symmetric the bounds are applied to ensembles of punctured ldpc codes where the puncturing patterns are either random or possess some structure our discussion is concluded by a diagram showing interconnections between the new theorems and some previously reported results

the donaldson thomas invariants under blowups and flops using the degeneration formula for doanldson thomas invariants we proved formulae for blowing up a point and simple flops given a smooth projective calabi yau 3 fold x the moduli space of stable sheaves on x has virtual dimension zero donaldson and thomas d t defined the holomorphic casson invariant of x which essentially counts the number of sta ble bundles on x however the moduli space has positive dimension and is singular in general making use of virtual cycle technique see b f and l t thomas showed in thomas that one can define a virtual moduli cycle for some x including calabi yau and fano 3 folds as a consequence one can define donaldson type in variants of x which are deformation invariant donaldson thomas invariants pro vide a new vehicle to study the geometry and other aspects of higher dimensional varieties it is important to understand these invariants much studied gromov witten invariants of x are the counting of stable maps from curves to x in mnop1 mnop2 maulik nekrasov okounkov and pandharipande discovered relations between gromov witten invarints of x and donaldson thomas invariants constructed from moduli spaces of ideal sheaves of curves on x they conjectured that these two invariants can be identified via the equations of partition functions of both theory this suggests that many phe nomena on gromov witten theory have the counterparts in donaldson thomas theory donaldson thomas invariants are deformation independent in the birational geometry of 3 folds we have blowups and flops donaldson thomas invariants couldn t be effective in studying birational geometry unless we understand how invariants change under birational operations li and ruan in l r studied how gromov witten invariants change under a flop for calabi yau 3 fold they proved that one can identify the 3 point functions of x and the flop x f of x up to some transformation of the q variables the same question was aslo studied by liu and yau in l y recently using the j li s degeneration formula from algebraic geometry in hu1 hu2 the first author studied the change of gromov witten

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

reliable channel regions for good binary codes transmitted over parallel channels we study the average error probability performance of binary linear code ensembles when each codeword is divided into j subcodewords with each being transmitted over one of j parallel channels this model is widely accepted for a number of important practical channels and signaling schemes including block fading channels incremental redundancy retransmission schemes and multicarrier communication techniques for frequency selective channels our focus is on ensembles of good codes whose performance in a single channel model is characterized by a threshold behavior e g turbo and low density parity check ldpc codes for a given good code ensemble we investigate reliable channel regions which ensure reliable communications over parallel channels under maximum likelihood ml decoding to construct reliable regions we study a modifed 1961 gallager bound for parallel channels by allowing codeword bits to be randomly assigned to each component channel the average parallel channel gallager bound is simplified to be a function of code weight enumerators and channel assignment rates special cases of this bound average union bhattacharyya ub shulman feder sf simplified sphere ss and modified shulman feder msf parallel channel bounds allow for describing reliable channel regions using simple functions of channel and code spectrum parameters parameters describing the channel are the average parallel channel bhattacharyya noise parameter the average channel mutual information and parallel gaussian channel signal to noise ratios snrs code parameters include the union bhattacharyya noise threshold and the weight spectrum distance to the random binary code ensemble reliable channel regions of repeat accumulate ra codes for parallel binary erasure channels becs and of turbo codes for parallel additive white gaussian noise awgn channels are numerically computed and compared with simulation results based on iterative decoding in addition an examp

on the cohomological dimension of the moduli space of riemann surfaces the moduli space of riemann surfaces of genus g 2 is up to a finite etale cover a complex manifold and so it makes sense to speak of its dolbeault cohomological dimension the conjecturally optimal bound is g 2 this expectation is verified in low genus and supported by harer s computation of its de rham cohomological dimension and by vanishing results in the tautological intersection ring in this paper we prove that such dimension is at most 2g 2 we also prove an analogous bound for the moduli space of riemann surfaces with marked points the key step is to show that the dolbeault cohomological dimension of each stratum of translation surfaces is at most g in order to do that we produce an exhaustion function whose complex hessian has controlled index in the construction of such a function some basic geometric properties of flat surfaces come into play

birational cobordism invariance of uniruled symplectic manifolds 1 department of mathematics zhongshan university guangzhou 510275 p r china e mail stsjxhu mail sysu edu cn 2 school of mathematics university of minnesota minneapolis mn 55455 usa e mail tjli math umn edu 3 department of mathematics university of michigan ann arbor mi 48109 1109 usa e mail ruan umich edu 4 yangtze center of mathematics sichuan university chengdu 610064 p r china

evaluating tautological classes using only hurwitz numbers hurwitz numbers count ramified covers of a riemann surface with prescribed monodromy as such they are purely combinatorial objects tautological classes on the other hand are distinguished classes in the intersection ring of the moduli spaces of riemann surfaces of a given genus and are thus geometric localization computations in gromov witten theory provide non obvious relations between the two this paper makes one such computation and shows how it leads to a master relation theorem 0 1 that reduces the ratios of certain interesting tautological classes to the pure combinatorics of hurwitz numbers as a corollary we obtain a purely combinatorial proof of a theorem of bryan and pandharipande expressing in generating function form classical computations by faber looijenga theorem 0 2

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

the design of rate compatible protograph ldpc codes this paper presents a simple yet effective method for designing nested families of ldpc codes rate compatible codes are essential for many communication applications e g hybrid automatic repeat request harq systems and their design is nontrivial due to the difficulty of simultaneously guaranteeing the quality of several related codes puncturing can be used to generate rate compatible ldpc codes but it produces a gap to capacity that in practice often significantly exceeds the gap of the mother code we propose an alternative method based on successively extending a high rate protograph the resulting codes not only inherit the advantages of protograph codes namely low encoding complexity and efficient decoding algorithms but also cover a wide range of rates and have very good performance with iterative decoding thresholds that are within 0 2 db of their capacity limits

capacity achieving ldpc codes through puncturing the performance of punctured low definition parity check ldpc codes under maximum likelihood ml decoding is studied in this correspondence via deriving and analyzing their average weight distributions awds and the corresponding asymptotic growth rate of the awds in particular it is proved that capacity achieving codes of any rate and for any memoryless binary input output symmetric mbios channel under ml decoding can be constructed by puncturing some original ldpc code with small enough rate moreover it is shown that the gap to capacity of all the punctured codes can be the same as the original code with a small enough rate conditions under which puncturing results in no rate loss with asymptotically high probability are also given in the process these results show high potential for puncturing to be used in designing capacity achieving codes and in rate compatible coding under any mbios channel

weighted blowup correspondence of orbifold gromov witten invariants and applications let mathsf x be a compact symplectic orbifold groupoid with mathsf s being a compact symplectic sub orbifold groupoid and underline mathsf x mathfrak a be the weight mathfrak a blowup of mathsf x along mathsf s with mathsf z being the exceptional divisor we show that there is a weighted blowup correspondence between some certain absolute orbifold gromov witten invariants of mathsf x relative to mathsf s and some certain relative orbifold gromov witten invariants of the pair underline mathsf x mathfrak a mathsf z as an application we prove that the symplectic uniruledness of symplectic orbifold groupoids is a weighted blowup invariant

curves in calabi yau threefolds and topological quantum field theory we continue our study of the local gromov witten invariants of curves in calabi yau threefolds we define relative invariants for local theory which give rise to a 1 1 dimensional topological quantum field theory tqft taking values in the ring mathbb q t the associated frobenius algebra over mathbb q t is semisimple consequently we obtain a structure result for the local invariants as an easy consequence of our structure formula we recover the closed formulas for the local invariants in the case where either the target genus or the degree equals 1

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

capacity achieving ensembles for the binary erasure channel with bounded complexity we present two sequences of ensembles of nonsystematic irregular repeat accumulate ira codes which asymptotically as their block length tends to infinity achieve capacity on the binary erasure channel bec with bounded complexity per information bit this is in contrast to all previous constructions of capacity achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap in rate to capacity the new bounded complexity result is achieved by puncturing bits and allowing in this way a sufficient number of state nodes in the tanner graph representing the codes we derive an information theoretic lower bound on the decoding complexity of randomly punctured codes on graphs the bound holds for every memoryless binary input output symmetric mbios channel and is refined for the binary erasure channel

the class of the locus of intermediate jacobians of cubic threefolds we study the locus of intermediate jacobians of cubic threefolds within the moduli space mathcal a 5 of complex principally polarized abelian fivefolds and its generalization to arbitrary genus the locus of abelian varieties with a singular odd two torsion point on the theta divisor assuming that this locus has expected codimension g which we show to be true for g 5 and conjecturally for any g we compute the class of this locus and of its closure in the perfect cone toroidal compactification mathcal a g mathrm perf in the chow homology and the tautological ring

on the renyi divergence joint range of relative entropies and a channel coding theorem this paper starts by considering the minimization of the renyi divergence subject to a constraint on the total variation distance based on the solution of this optimization problem the exact locus of the points bigl d q p 1 d q p 2 bigr is determined when p 1 p 2 and q are arbitrary probability measures which are mutually absolutely continuous and the total variation distance between p 1 and p 2 is not below a given value it is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet this characterization yields a geometric interpretation of the minimal chernoff information subject to a constraint on the variational distance this paper also derives an exponential upper bound on the performance of binary linear block codes or code ensembles under maximum likelihood decoding its derivation relies on the gallager bounding technique and it reproduces the shulman feder bound as a special case the bound is expressed in terms of the renyi divergence from the normalized distance spectrum of the code or the average distance spectrum of the ensemble to the binomially distributed distance spectrum of the capacity achieving ensemble of random block codes this exponential bound provides a quantitative measure of the degradation in performance of binary linear block codes or code ensembles as a function of the deviation of their distance spectra from the binomial distribution an efficient use of this bound is considered

parity check density versus performance of binary linear block codes over memoryless symmetric channels we derive lower bounds on the density of parity check matrices of binary linear codes which are used over memoryless binary input output symmetric mbios channels the bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit error probability vanishes for every mbios channel we construct a sequence of ensembles of regular low density parity check ldpc codes so that an upper bound on the asymptotic density of their parity check matrices scales similarly to the lower bound the tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right regular ldpc codes which was introduced by shokrollahi and which is known to achieve the capacity of this channel under iterative message passing decoding we show that this sequence of ensembles is asymptotically optimal in a sense to be defined in this paper strengthening a result of shokrollahi finally we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs and study their performance limitations over mbios channels the latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error correction codes

a genus 3 topological recursion relation in this paper we give a new genus 3 topological recursion relation for gromov witten invariants of compact symplectic manifolds this formula also applies to intersection numbers on moduli spaces of spin curves a by product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1 pointed genus 3 stable curves

the zero section of the universal semiabelian variety and the double ramification cycle we study the chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties ppav we describe the subring generated by divisor classes and we compute the class of the partial compactification of the universal zero section which turns out to lie in this subring our formula extends the results for the zero section of the universal uncompactified family the partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of ag our formula provides a first step in a program to understand the chow groups of a g especially of the perfect cone compactification by induction on genus by restricting to the image of mg under the torelli map our results extend the results of hain on the double ramification cycle answering eliashberg s question

double ramification cycles and integrable hierarchies in this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory we conjecture that in the semisimple case our hierarchy is related to the dubrovin zhang hierarchy by a miura transformation and we check it in several examples

on the convergence of infinite products of matrices abstract in the normed metric space of n square complex matrices convergence of i ai implies convergence of i u i a i with u i 1 i if and only if i q u i converges for any integer q the limit of i u i a i may be considered a function u a of the summable sequence a a i the function u is parametrized by u u i the sequences of summable matrices a a i constitute a normed metric space e with norm a e i a i the family of functions u a i u i a i is equicontinuous at a if a e u i

distance verification for classical and quantum ldpc codes the techniques of distance verification known for general linear codes are first applied to the quantum stabilizer codes then these techniques are considered for classical and quantum stabilizer low density parity check ldpc codes new complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of ldpc codes these bounds are expressed in terms of the erasure correcting capacity of the corresponding ensemble we also present a new irreducible cluster technique that can be applied to any ldpc code and takes advantage of parity checks sparsity for both the classical and quantum ldpc codes this technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances which also include all known families of the quantum stabilizer ldpc codes

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

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

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

on the projectivity of the moduli spaces of curves 1 there are basically three algebraic proofs of the projectivity of the moduli spaces of stable curves the original one by knudsen 8 9 the proof by mumford 12 and gieseker 5 which makes heavy use of the machinery of geometric invariant theory and the more recent one by viehweg 15 and kollar 10 which relies instead on the semipositivity of the direct images of powers of the relative dualizing sheaf and at least in kollar s version does not use geometric invariant theory at all in this note combining ideas from the above papers and from 4 we shall outline a proof of projectivity which we believe is simpler and more direct than any of the existing ones however while the proofs by mumford gieseker viehweg and kollar are applicable at least in principle to a wide variety of moduli problems ours uses in an essential way the peculiarities of the problem at hand and it is hard to see how it could be extended to other situations mumford s idea is to use the machinery of geometric invariant theory to directly construct the moduli space of stable curves as a projective quotient of the chow or hilbert scheme of pluricanonically embedded stable curves by the action of the projective linear group to be able to do so one needs to show that the k canonical images of stable curves are stable in the invariant theoretic sense for high k this is quite well understood although not really easy for smooth curves one notices that k canonically embedded stable curves are linearly stable and then proves that for smooth curves linear stability implies invariant theoretic stability in the singular case by contrast both mumford and gieseker have to rely on indirect arguments which are quite long and involved our main point is that one can avoid proving the invariant theoretic stability of singular stable curves provided one is willing to grant that moduli space exists as a complete algebraic space that this is the case incidentally is relatively easy to prove as in 10 we prove projectivity by applying to a suitable invertible sheaf one of the standard numerical ampleness criteria seshadri s criterion in our case the necessary numerical estimates are obtained using the techniques of 4 more precisely the invariant theoretic stability of linearly stable smooth curves is used to prove an inequality for families of stable curves which suitably interpreted says that the hypothesis of seshadri s criterion is satisfied for curves in moduli which are not contained in the boundary the case of curves lying in the boundary is then reduced to the previous one by standard elementary techniques in order to properly carry out this procedure which can be viewed as a sort of induction on the genus it is convenient to deal simultaneously with the moduli spaces of stable n pointed curves as well in a sense then our approach is to use invariant theoretic stability as a means of obtaining numerical inequalities rather than as a step in the construction of quotients by group actions

tropicalizing the space of admissible covers we study the relationship between tropical and classical hurwitz moduli spaces following recent work of abramovich caporaso and payne we outline a tropicalization for the moduli space of generalized hurwitz covers of an arbitrary genus curve our approach is to appeal to the geometry of admissible covers which compactify the hurwitz scheme we study the relationship between a combinatorial moduli space of tropical admissible covers and the skeleton of the berkovich analytification of the classical space of admissible covers we use techniques from non archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization and that the degree of the classical branch map can be recovered from the tropical side as a consequence we obtain a proof at the level of moduli spaces of the equality of classical and tropical hurwitz numbers

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

virasoro constraints for target curves we prove generalized virasoro constraints for the relative gromov witten theories of all nonsingular target curves descendents of the even cohomology classes are studied first by localization degeneration and completed cycle methods descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations as an outcome of our results the relative theories of target curves are completely and explicitly determined

on universal properties of capacity approaching ldpc code ensembles this paper is focused on the derivation of some universal properties of capacity approaching low density parity check ldpc code ensembles whose transmission takes place over memoryless binary input output symmetric mbios channels properties of the degree distributions graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information theoretic bounds these bounds are expressed in terms of the target block bit error probability and the gap in rate to capacity most of the bounds are general for any decoding algorithm and some others are proved under belief propagation bp decoding proving these bounds under a certain decoding algorithm validates them automatically also under any suboptimal decoding algorithm a proper modification of these bounds makes them universal for the set of all mbios channels which exhibit a given capacity bounds on the degree distributions and graphical complexity apply to finite length ldpc codes and to the asymptotic case of an infinite block length the bounds are compared with capacity approaching ldpc code ensembles under bp decoding and they are shown to be informative and are easy to calculate finally some interesting open problems are considered

on the tautological ring of mg we prove that any product of tautological classes of g of degreed in the chow ring of g with rational coefficients vanishes ford g 2 and is proportional to the class of the hyperelliptic locus in degreeg 2

relative virtual localization and vanishing of tautological classes on moduli spaces of curves we prove a localization formula for the moduli space of stable relative maps as an application we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i g 1 genus 0 components as consequences we prove and generalize various conjectures and theorems about various moduli spaces of curves due to getzler ionel faber looijenga pandharipande diaz and others this theorem appears to be the geometric content behind these results the rest is straightforward graph combinatorics the theorem also suggests the importance of the stratification of the moduli space by number of rational components

topological recursive relations in h2g ℳg n we show that any degree at least g monomial in descendant or tautological classes vanishes on g n when g 2 this generalizes a result of looijenga and proves a version of getzler s conjecture the method we use is the study of the relative gromov witten invariants of 1 relative to two points combined with the degeneration formulas of ip1

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

compact global attractors of discrete inclusions abstract the paper is dedicated to the study of the problem of the existence of compact global attractors of discrete inclusions and to the description of its structure we consider a family of continuous mappings of a metric space w into itself and w f i i i is the family of discrete dynamical systems on the metric space w we consider a discrete inclusion 1 u t 1 f u t associated with m f i i i where f u f u f m for all u w we give sufficient conditions the family of maps m is contracting in the extended sense for the existence of a compact global attractor of 1 if the family m consists of a finite number of maps then the corresponding compact global attractor is chaotic we study this problem in the framework of non autonomous dynamical systems cocyles

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

a compactification of the space of maps from curves we construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points it is shown to be a proper deligne mumford stack equipped with a natural virtual fundamental class

stability and paracontractivity of discrete linear inclusions abstract we study stability properties of a finite set of n n matrices such as paracontractivity bv and left convergent products lcps stability and their relations to each other the conjecture on equivalence of paracontractivity and lcp stability is proved moreover we prove the equivalence of the uniform bv stability and the property of vanishing length of steps of any trajectory of

successive maximization for systematic design of universally capacity approaching rate compatible sequences of ldpc code ensembles over binary input output symmetric memoryless channels a systematic construction of capacity achieving low density parity check ldpc code ensemble sequences over the binary erasure channel bec has been proposed by saeedi et al based on a method here referred to as successive maximization sm in sm the fraction of degree i nodes are successively maximized starting from i 2 with the constraint that the ensemble remains convergent over the channel in this paper we propose sm to design universally capacity approaching rate compatible ldpc code ensemble sequences over the general class of binary input output symmetric memoryless biosm channels this is achieved by first generalizing the sm method to other biosm channels to design a sequence of capacity approaching ensembles called the parent sequence the sm principle is then applied to each ensemble within the parent sequence this time to design rate compatible puncturing schemes as part of our results we extend the stability condition which was previously derived for degree 2 variable nodes to other variable node degrees as well as to the case of rate compatible codes consequently we prove that using the sm principle one is able to design universally capacity achieving rate compatible ldpc code ensemble sequences over the bec unlike the previous results in the literature the proposed sm approach is naturally extendable to other biosm channels the performance of the rate compatible schemes designed based on our method is comparable to those designed by optimization

a proof of a conjecture of marino vafa on hodge integrals where i c1 li is the first chern class of li and j cj e is the j th chern class of the hodge bundle hodge integrals arise in the calculations of gromov witten invariants by localization techniques 14 7 the explicit evaluation of hodge integrals is a difficult problem the hodge integrals involving only classes can be computed recursively by witten s conjecture 26 proven by kontsevich 13 algorithms of computing hodge integrals are described in 2 in 23 m marino and c vafa obtained a closed formula for a generating function of certain open gromov witten invariants some of which has been reduced to hodge integrals by localization techniques which are not fully clarified mathematically this leads to a conjectural formula of hodge integrals to state this formula we introduce some notation following 28 let

the gorenstein conjecture fails for the tautological ring of mathcal overline m _ 2 n we prove that for n equal to at least one of the integers 8 12 16 20 the tautological ring r bullet overline mathcal m 2 n is not gorenstein in fact our n equals the smallest integer such that there is a non tautological cohomology class of even degree on overline mathcal m 2 n by work of graber and pandharipande such a class exists on overline mathcal m 2 20 and we present some evidence indicating that n is in fact 20

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

a remark on gromov witten invariants of quintic threefold abstract the purpose of the article is to give a proof of a conjecture of maulik and pandharipande for genus 2 and 3 as a result it gives a way to determine gromov witten invariants of the quintic threefold for genus 2 and 3

absolute and relative gromov witten invariants of very ample hypersurfaces for any smooth complex projective variety x and any smooth very ample hypersurface y subset x we develop the technique of genus zero relative gromov witten invariants of y in x in algebro geometric terms we prove an equality of cycles in the chow groups of the moduli spaces of relative stable maps which relates these relative invariants to the gromov witten invariants of x and y given the gromov witten invariants of x we show that these relations are sufficient to compute all relative invariants as well as all genus zero gromov witten invariants of y whose homology and cohomology classes are induced by x

a short proof of the λg conjecture without gromov witten theory hurwitz theory and the moduli of curves we give a short and direct proof of the g conjecture the approach is through the ekedahl lando shapiro vainshtein theorem which establishes the polynomiality of hurwitz numbers from which we pick off the lowest degree terms the proof is independent of gromovwitten theory we briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

the tropical vertex elements of the tropical vertex group are formal families of symplectomorphisms of the 2 dimensional algebraic torus we prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus zero relative gromov witten invariants of toric surfaces the relative invariants which arise have full tangency to a toric divisor at a single unspecified point the method uses scattering diagrams tropical curve counts degeneration formulas and exact multiple cover calculations in orbifold gromov witten theory

on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

invariant manifolds global attractors almost automorphic and almost periodic solutions of non autonomous differential equations the paper is devoted to the study of non autonomous evolution equations invariant manifolds compact global attractors almost periodic and almost automorphic solutions we study this problem in the framework of general non autonomous cocycle dynamical systems first we prove that under some conditions such systems admit an invariant continuous section an invariant manifold then we obtain the conditions for the existence of a compact global attractor and characterize its structure third we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non autonomous differential equations both odes in finite and infinite spaces and pdes

invariant graphs for forced systems abstract many applications of nonlinear dynamics involve forced systems we consider the case where for a fixed input the driven system is contracting this is for instance the situation in certain classes of filters and in the study of synchronization when such contraction is uniform it is well known that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space if the contraction is sufficiently strong then is smooth we describe the theoretical framework for such results and discuss a number of applications to the filtering of time series to synchronization and to quasiperiodically forced systems we go on to give recent generalizations to the case of non uniform contraction that is contraction measured by lyapunov exponent like quantities we conclude with a number of new results for quasiperiodic forcing a corollary of these is that a strange non chaotic attractor for such a system cannot be the graph of a continuous function nor roughly speaking can it have as open attracting neighbourhood in other words its closure must contain some repelling orbits

hurwitz numbers and intersections on moduli spaces of curves 1 department of mathematics university of stockholm s 10691 stockholm sweden e mail teke matematik su se 2 higher college of mathematics independent university of moscow and institute for system research ras moscow russia e mail lando mccme ru 3 department of mathematics royal institute of technology s 10044 stockholm sweden e mail mshapiro math kth se 4 department of mathematics and department of computer science university of haifa haifa 31905 israel e mail alek mathcs haifa ac il

gromov witten donaldson thomas correspondence for toric 3 folds we prove the equivariant gromov witten theory of a nonsingular toric 3 fold x with primary insertions is equivalent to the equivariant donaldson thomas theory of x as a corollary the topological vertex calculations by agangic klemm marino and vafa of the gromov witten theory of local calabi yau toric 3 folds are proven to be correct in the full 3 leg setting

generating functions for hurwitz hodge integrals we describe explicit generating functions for a large class of hurwitz hodge integrals these are integrals of tautological classes on moduli spaces of admissible covers a stackily smooth compactification of the hurwitz schemes admissible covers and their tautological classes are interesting mathematical objects on their own but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves and the orbifold gromov witten theory of dm stacks

uep constructions of quasi cyclic low density parity check codes via masking in this paper the algebraic constructions of quasi cyclic low density parity check qc ldpc codes with the unequal error protection uep property are considered a criterion for constructing such codes via the masking technique is proposed based on which explicit conditions on the base parity check matrices and masking matrices to achieve uep are provided we also give three specific constructions of uep qc ldpc codes furthermore a sufficient condition to ensure strict uep is presented simulation results demonstrate the superiority of our constructed codes over time sharing schemes the constructed codes also have competitive error performance against randomly constructed equal error protection ldpc codes and irregular uep ldpc codes designed based on the degree distribution

rate compatible ldpc convolutional codes achieving the capacity of the bec in this paper we propose a new family of rate compatible regular low density parity check ldpc convolutional codes the construction is based on graph extension i e the codes of lower rates are generated by successively extending the graph of the base code with the highest rate theoretically the proposed rate compatible family can cover all the rational rates from 0 to 1 in addition the regularity of degree distributions simplifies the code optimization we prove analytically that all the ldpc convolutional codes of different rates in the family are capable of achieving the capacity of the binary erasure channel bec the analysis is extended to the general binary memoryless symmetric channel for which a capacity approaching performance can be achieved analytical thresholds and simulation results for finite check and variable node degrees are provided for both becs and binary input additive white gaussian noise channels the results confirm that the decoding thresholds of the rate compatible codes approach the corresponding shannon limits over both channels

norm conditions for convergence of infinite products abstract we give a finite number of necessary and sufficient conditions on two matrices so that any infinite left product in them is convergent

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

random coding techniques for nonrandom codes this work provides techniques to apply the channel coding theorem and the resulting error exponent which was originally derived for totally random block code ensembles to ensembles of codes with less restrictive randomness demands as an example the random coding technique can even be applied for an ensemble that contains a single code for a specific linear code we get an upper bound for the error probability which equals gallager s 1968 random coding bound up to a factor determined by the maximum ratio between the weight distribution of the code and the expected random weight distribution

chow rings of moduli spaces of curves ii some results on the chow ring of overline mathscr m _4 ring of the moduli space of stable curves of genus 4 these results are not complete we find generators for the chow ring of 4 and for the chow groups in codimension 1 and 2 of w4 for a2 g 4 we find fourteen generators using test surfaces we prove that the dimension of a 2 4 4 is at least 13 and explicitly determine the single relation between the fourteen generators which still can exist finally we have two proofs that this relation does indeed hold so that the dimension of a2 4 4 equals 13 this enables us to determine the chow ring of 4 4 our original proof is based on a rather delicate argument the second proof uses a result of ran see r and is much simpler

results on punctured low density parity check codes and improved iterative decoding techniques this paper first introduces an improved decoding algorithm for low density parity check ldpc codes over binary input output symmetric memoryless channels then some fundamental properties of punctured ldpc codes are presented it is proved that for any ensemble of ldpc codes there exists a puncturing threshold it is then proved that for any rates r1 and r2 satisfying 0 r1 r 2 1 there exists an ensemble of ldpc codes with the following property the ensemble can be punctured from rate r1 to r2 resulting in asymptotically good codes for all rates r1lesrlesr2 specifically this implies that rates arbitrarily close to one are achievable via puncturing bounds on the performance of punctured ldpc codes are also presented it is also shown that punctured ldpc codes are as good as ordinary ldpc codes for bec and arbitrary positive numbers r1 r2 1 the existence of the sequences of punctured ldpc codes that are capacity achieving for all rates r1 lesrlesr2 is shown based on the above observations a method is proposed to design good punctured ldpc codes over a broad range of rates finally it is shown that the results of this paper may be used for the proof of the existence of the capacity achieving ldpc codes over binary input output symmetric memoryless channels

the κ ring of the moduli of curves of compact type the subalgebra of the tautological ring of the moduli of curves of compact type generated by the classes is studied in all genera relations constructed via the virtual geometry of the moduli of stable quotients are used to obtain minimal sets of generators bases and betti numbers of the rings are computed a universality property relating the higher genus rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps the g formula for hodge integrals arises as the simplest consequence