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

on the error exponents of improved tangential sphere bounds the performance of maximum likelihood ml decoded binary linear block codes over the additive white gaussian noise awgn channel is addressed via the tangential sphere bound tsb and two of its recent improved versions the correspondence is focused on the derivation of the error exponents of these bounds although it was shown that some recent improvements of the tsb tighten this bound for finite length codes it is demonstrated in this correspondence that their error exponents coincide for an arbitrary ensemble of binary linear block codes the common value of these error exponents is explicitly expressed in terms of the asymptotic growth rate of the average distance spectrum

variational computation of homoclinic orbits for twist maps abstract variational methods have been used extensively in the study and computation of monotone twist maps of the annulus dimension d 1 and in their higher dimensional generalizations in particular periodic orbits can be characterized and computed as critical points of certain real valued functions constructed from the generating function of the map in this paper we propose a similar variational method for the computation of orbits homoclinic or heteroclinic in some cases to periodic orbits this again consists in finding critical points of a certain function constructed with knowledge of the generating function of the map and of the stable and unstable manifolds to the periodic orbits near the periodic points we also prove that under relatively simple conditions when d 1 there exist heteroclinic connections between fixed points

condition numbers of hankel matrices for exponential weights abstract we obtain the rate of growth of the largest eigenvalues and euclidean condition numbers of the hankel matrices i t j k w 2 t d t j k 0 n for a general class of even exponential weights w 2 exp 2 q on an interval i as particular examples we discuss q x x on i r and q x d 2 x 2 on i d d

the condition number of real vandermonde krylov and positive definite hankel matrices we show that the euclidean condition number of any positive definite hankel matrix of order n geq 3 may be bounded from below by gamma n 1 16n with gamma exp 4 cdot it catalan pi approx 3 210 and that this bound may be improved at most by a factor 8 gamma n similar estimates are given for the class of real vandermonde matrices the class of row scaled real vandermonde matrices and the class of krylov matrices with hermitian argument improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval our findings confirm that all such matrices including for instance the famous hilbert matrix are ill conditioned already for moderate order as application we describe implications of our results for the numerical condition of various tasks in numerical analysis such as polynomial and rational i nterpolation at real nodes determination of real roots of polynomials computation of coefficients of orthogonal polynomials or the iterative solution of linear systems of equations

study of a class of subharmonic motions using a non smooth temporal transformation nstt abstract a nonlinear boundary value problem nbvp formulation for computing strongly nonlinear subharmonic orbits of a class of harmonically forced conservative systems is presented the formulation is based on a non smooth temporal transformation nstt to replace the independent temporal variable of the problem with two new piecewise linear periodic temporal variables as a result the problem of computing the subharmonic motions is reduced to a set of nbvps with homogeneous boundary conditions some numerical and analytical solutions of the reduced nbvps are given and the asymptotic behavior in the limit of large period of the derived analytical approximations is discussed in particular the formulation shows a definite connection between vibro impact oscillations considered in mechanics and strongly nonlinear oscillations close to a heteroclinic orbit an interesting feature of a family of subharmonic solutions analyzed in this work is that in the limit of large periods it degenerates to the perturbed stable and unstable manifolds of an unstable periodic orbit of the system

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

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

existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems abstract in this paper we study a generalization of arnold s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori that has subsequently been referred to as arnold diffusion in case when the number of degrees of freedom n 3 namely we consider a widely studied model of a pendulum weakly coupled with n 1 rotors with the degeneracies in the unperturbed hamiltonian corresponding to different timescales existing in the problem using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation thus improving the estimate of gallavotti 1994 which he calls quasiflat and generalizing the analogous recent estimate of delshams et al 1996 for the rapidly quasiperiodically forced pendulum to a much larger class of hamiltonian systems in particular the exponentially small upper bound for the transversality measure of the splitting applies when the hamiltonian has extra degeneracies namely when the frequencies on a torus become near resonant in fact we show that in such a case the quantity in question becomes smaller which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger nekhoroshev exponent nekhoroshev exponent nevertheless we emphasize that getting uniform estimates for an arbitrary n 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector although recent developments show that the first order of canonical perturbation theory given by melnikov integrals generally cannot be accepted as the leading order answer for the splitting distance for the case of more than two degrees of freedom including the rapidly quasiperiodically forced pendulum problem we suggest an analytic perturbation the majority of whose fourier components are strictly nondashzero for which melnikov integrals can be vindicated as the leading order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions this allows us to bound the splitting distance from below furthermore having such a perturbation for the case of three degrees of freedom we use a simple number theoretical argument to find the asymptotics of the fourier series with exponentially small coefficients involved this enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure and by proving the domineering contribution of the first order of perturbation theory for the transversality measure to suggest a leading order answer for this quantity which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies we elucidate the numerous arithmetic issues that obstruct getting a compact leading order approximation for the splitting size most of which can be overcome in the case of three degrees of freedom as our example shows these obstacles can be also possibly avoided in the same fashion for an arbitrary n 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n 3 or n 2

small eigenvalues of large hankel matrices in this paper we investigate the smallest eigenvalue denoted as n of a n 1 n 1 hankel or moments matrix associated with the weight w x exp x x 0 0 in the large n limit using a previous result the asymptotics for the polynomials pn z z 0 orthonormal with respect to w which are required in the determination of n are found adopting an argument of szegothe asymptotic behaviour of n for where the related moment problem is determinate is derived this generalizes the result given by szegofor 1 it is shown that for the smallest eigenvalue of the infinite hankel matrix is zero while for 0 it is greater then a positive constant this shows a phase transition in the corresponding hermitian random matrix model as the parameter varies with identified as the critical point the smallest eigenvalue at this point is conjectured

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

non existence of truly solitary waves in water with small surface tension this paper considers permanent capillary gravity waves on the free surface of a two dimensional incompressible inviscid fluid of finite depth it is shown that there are no solitary wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals the proof is based upon an estimate of a constant that is related to the approximation of the solution if it exists near its singularity the approximation satisfies a fourth order nonlinear ordinary differential equation when the solution is extended to the complex plane then the non existence of truly solitary waves is obtained by using a contradiction on this constant

identical capacity channel decomposition for design of universal ldpc codes design of low density parity check ldpc codes suitable for all channels which exhibit a given capacity c is investigated such codes are referred to as universal ldpc codes first based on numerous observations a conjecture is put forth that a code working on n equal capacity channels also works on any convex combination of these n channels as a supporting evidence we prove that a code satisfying the stability condition on n channels also satisfies the stability condition on the convex hull of these n channels then a channel decomposition method is suggested which spans any given channel with capacity c in terms of a number of identical capacity basis channels we expect codes that work on the basis channels to be suitable for any convex combination of the bases i e all channels with capacity c such codes are found over a wide range of rates an upper bound on the achievable rate of universal ldpc codes is suggested through examples it is shown that our codes achieve rates extremely close to this upper bound in comparison with existing ldpc codes designed for a given channel significant performance gain is reported when codes are used over various channels of equal capacity

exponentially small splitting of separatrices in a weakly hyperbolic case abstract we validate the poincare melnikov method in the singular case of high frequency periodic perturbations of the hamiltonian h 0 x y 1 2 y 2 x 3 x 4 under appropriate conditions which among other things imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one the splitting is exponentially small

periodic solutions of lagrangian systems on a compact manifold abstract let m be a smooth n dimensional manifold and let tm be its tangent bundle we consider a time periodic lagrangian of period t l t tm r and we seek t periodic solutions of the lagrange equations which in local coordinates are d dt l q t q q l q t q q 0 i 1 n our main result states that if the fundamental group of m is finite then has infinitely many t periodic solutions provided that l t satisfies certain physically reasonable assumptions

parity check density versus performance of binary linear block codes new bounds and applications the moderate complexity of low density parity check ldpc codes under iterative decoding is attributed to the sparseness of their parity check matrices it is therefore of interest to consider how sparse parity check matrices of binary linear block codes can be a function of the gap between their achievable rates and the channel capacity this issue was addressed by sason and urbanke and it is revisited in this paper the remarkable performance of ldpc codes under practical and suboptimal decoding algorithms motivates the assessment of the inherent loss in performance which is attributed to the structure of the code or ensemble under maximum likelihood ml decoding and the additional loss which is imposed by the suboptimality of the decoder these issues are addressed by obtaining upper bounds on the achievable rates of binary linear block codes and lower bounds on the asymptotic density of their parity check matrices as a function of the gap between their achievable rates and the channel capacity these bounds are valid under ml decoding and hence they are valid for any suboptimal decoding algorithm the new bounds improve on previously reported results by burshtein and by sason and urbanke and they hold for the case where the transmission takes place over an arbitrary memoryless binary input output symmetric mbios channel the significance of these information theoretic bounds is in assessing the tradeoff between the asymptotic performance of ldpc codes and their decoding complexity per iteration under message passing decoding they are also helpful in studying the potential achievable rates of ensembles of ldpc codes under optimal decoding by comparing these thresholds with those calculated by the density evolution technique one obtains a measure for the asymptotic suboptimality of iterative decoding algorithms

interaction of sine gordon kinks with defects the two bounce resonance a model of soliton defect interactions in the sine gordon equations is studied using singular perturbation theory melnikov theory is used to derive a critical velocity for strong interactions which is shown to be exponentially small for weak defects matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem which are then used to derive analytical formulas for the locations of the well known two and three bounce resonance windows as well as several other phenomena seen in numerical simulations

a geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of 2 we give a proof based in geometric perturbation theory of a result proved by j n mather using variational methods namely the existence of orbits with unbounded energy in perturbations of a generic geodesic flow in 2 by a generic periodic potential

ghost circles for twist maps abstract completely ordered invariant circles are found for the gradient of the energy flow in the state space containing the critical sets corresponding to the birkhoff orbits of all rotation number in particular these ghost circles contain the aubry mather sets and map invariant circles as completely critical sets when these exist we give a criterion for a sequence of rational ghost circles to converge to a completely critical one

smallest eigenvalue of large hankel matrices at critical point comparing conjecture with parallelised computation abstract we propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill conditioned hankel matrices it is based on the ldlt decomposition and involves finding a k k sub matrix of the inverse of the original n n hankel matrix h n 1 the computation involves extremely high precision arithmetic message passing interface and shared memory parallelisation we demonstrate that this approach achieves good scalability on a high performance computing cluster hpcc which constitutes a major improvement of the earlier approaches we use this method to study a family of hankel matrices generated by the weight w x e x supported on 0 and 0 such weight generates a hankel determinant a fundamental object in random matrix theory in the situation where 1 2 the smallest eigenvalue tends to 0 exponentially fast if 1 2 it is conjectured that the smallest eigenvalue tends to 0 algebraically with a precise exponent the algorithm run on the hpcc producing a fantastic match between the theoretical value of 2 and the numerical result

exponentially small splitting of separatrices under fast quasiperiodic forcing we consider fast quasiperiodic perturbations with two frequencies 1 epsiv of a pendulum where is the golden mean number the complete system has a two dimensional invariant torus in a neighbourhood of the saddle point we study the splitting of the three dimensional invariant manifolds associated to this torus provided that the perturbation amplitude is small enough with respect to and some of its fourier coefficients the ones associated to fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the melnikov function

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

extremes of information combining extreme densities for information combining are found for two important channel models the binary input symmetric parallel broadcast channel and the parity constrained input symmetric parallel channels following upper and lower mutual information thresholds are stated for per bit maximum a posteriori probability map decoding and low density parity check ldpc code ensembles

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

on the role of high order resonances in normal forms and in separatrix splitting we discuss the role of high order resonances in the construction of normal forms for e close to integrable hamiltonian systems by heuristic considerations based on standard estimates we show that the remainder of normal forms is dominated by the terms corresponding to the main high order resonances and we provide a general argument to show that the size of such leading terms is exponentially small in 1e we apply this method to the problem of estimating the splitting of separatrices in resonant perturbed systems

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

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

mel nikov s function for two dimensional mappings this paper calculates explicitly the mel nikov function for general perturbations of two dimensional maps possessing a saddle connection this result may be considered as the discrete analogue of ziglin s calculation of the mel nikov function for time periodic perturbations of two dimensional continuous flows key words integrable planar maps perturbations homoclinic orbits mel nikov s function poisson summation formula

melnikov method and exponentially small splitting of separatrices abstract we study the splitting of separatrices for high frequency perturbations of a planar hamiltonian system it is well known that in the analytic case the splitting of separatrices is exponentially small with respect to the perturbation period 2 e and that the constant in the exponent is related to the position of singularities of the unperturbed separatrix in the complex time plane on the other hand in many examples the melnikov function can be calculated explicitly and is exponentially small with an exponent very close but larger than the exponent from the upper estimates so the melnikov function appears to be smaller than the upper estimate as well as the standard estimate of the approximation error we found a class of hamiltonians such that the melnikov function gives actual asymptotics for the splitting provided the perturbation amplitude is less than 0ep for some p 0

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

oscillatory motions for the restricted planar circular three body problem the restricted three body problem models the motion of a massless body under the influence of the newtonian gravitational force caused by two other bodies called the primaries when they move along circular keplerian orbits and the third body moves in the same plane one has the restricted planar circular three body problem rpc3bp in suitable coordinates it is a hamiltonian system of two degrees of freedom the conserved energy is usually called the jacobi constant llibre and simo math ann 248 2 153 184 1980 proved the existence of oscillatory motions for this system that is orbits which leave every bounded region but which return infinitely often to some fixed bounded region to prove their existence they had to assume the ratio between the masses of the primaries to be small enough in this paper we prove the existence of such motions for any value of the mass ratio mu closing the problem of existence of oscillatory motions in the rpc3bp to obtain such motions we restrict ourselves to the level sets of the jacobi constant we show that for any value of the mass ratio and for large values of the jacobi constant there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets these transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits the main achievement is to prove the existence of these orbits without assuming the mass ratio mu small when mu is not small this transversality can not be checked by means of classical perturbation theory since our method is valid for all values of mu we are able to detect a curve in the parameter space formed by mu and the jacobi constant where cubic homoclinic tangencies between the invariant manifolds of infinity appear

a study on universal codes with finite block lengths based on random codes and typical set decoding an alternative proof of root and varaiya s compound channel coding theorem for linear gaussian channels is presented the performance limit of codes with finite block length under a compound channel is studied through error bounds and simulation although the theorem promises uniform convergence of the probability of error as the block length approaches infinity with short block lengths the performance can differ considerably for individual channels simulation results show that universal performance can be a practical goal as the block lengths become large

width of stochastic layers in near integrable two dimensional symplectic maps abstract width of stochastic layers around separatrix eight is studied for two dimensional symplectic maps close to integrable we prove that under rather general assumptions the relation w d 1 log holds where w is the width of the stochastic layer d is the width of a lobe domain and is a multiplier at the corresponding hyperbolic point

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

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

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

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

beyond the melnikov method ii multidimensional setting abstract we present a melnikov type approach for establishing transversal intersections of stable unstable manifolds of perturbed normally hyperbolic invariant manifolds we do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation we also do not need to compute any integrals along such homoclinics all needed bounds are established using rigorous computer assisted numerics lastly and most importantly the method establishes intersections for an explicit range of parameters and not only for perturbations that are small enough as is the case in the classical melnikov approach

an analysis of the block error probability performance of iterative decoding asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes n and the number of iterations i go to infinity three classes of codes are considered these are gallager s regular low density parity check ldpc codes tanner s generalized ldpc gldpc codes and the turbo codes due to berrou et al it is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability p sub b but also the block frame error probability p sub b goes to zero as n and i go to infinity

bounds on the belief propagation threshold of non binary ldpc codes we consider low density parity check ldpc code ensembles over non binary galois fields when used for transmission over arbitrary discrete memoryless channels belief propagation decoding for these codes has been shown to achieve excellent results however computing the decoding threshold using density evolution is usually impractical since one needs to propagate multi dimensional probability distributions and monte carlo simulations are required instead by considering the evolution of the message bhattacharyya parameter and the message expected value parameter we derive a simple lower bound on the performance of the algorithm this bound applies for both regular and irregular non binary ldpc ensembles

non binary ldpc code design for the poisson ppm channel this paper investigates the design of non binary protograph low density parity check codes for the poisson channel with m ary pulse position modulation the field order over which the code is constructed is matched to the pulse position modulation order yielding a coded modulation scheme the optimization of the low density parity check code structure is performed via protograph density evolution on a surrogate m ary erasure channel the surrogate design is illustrated to be not only accurate but also robust for a range of practical values of channel background noise and various modulation orders as a result the proposed codes show excellent performance over the poisson channel with pulse position modulation outperforming competing schemes as a side product of this paper finite length benchmarks on the block error probability are provided together with a union bound to characterize the code performance in the error floor region

the smallest eigenvalue of large hankel matrices generated by a singularly perturbed laguerre weight an asymptotic expression of the orthonormal polynomials pn z as n associated with the singularly perturbed laguerre weight w x t x e x tx x 0 1 t 0 is derived based on this we establish the asymptotic behavior of the smallest eigenvalue n of the hankel matrix generated by the weight w x t

periodic orbits for hamiltonian systems in cotangent bundles we prove the existence of at least cl m periodic orbits for certain time dependent hamiltonian systems on the cotangent bundle of an arbitrary compact manifold m these hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a riemannian metric on m we discretize the variational problem by decomposing the time i map into a product of symplectic twist maps a second theorem deals with homotopically non trivial orbits of negative curvature

beyond the melnikov method a computer assisted approach abstract we present a melnikov type approach for establishing transversal intersections of stable unstable manifolds of perturbed normally hyperbolic invariant manifolds nhims the method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem which establishes the existence of a nhim together with its associated invariant manifolds and bounds on their first and second derivatives we do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation we also do not need to compute any integrals along such homoclinics all needed bounds are established using rigorous computer assisted numerics lastly and most importantly the method establishes intersections for an explicit range of parameters and not only for perturbations that are small enough as is the case in the classical melnikov approach

melnikov potential for exact symplectic maps the splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered the non degenerate critical points of a real valued function called the melnikov potential are associated to transverse homo clinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given moreover if the unperturbed invariant manifolds are completely dou bled it is shown that there exist in general at least 4 primary homoclinic orbits 4 n in antisymmetric maps both lower bounds are optimal two examples are presented a 2n dimensional central standard like map and the hamiltonian map associated to a magnetized spherical pendulum several topics are studied about these examples existence of splitting explicit computations of melnikov potentials transverse homoclinic orbits exponentially small splitting etc

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

solving hankel matrix approximation problem using semidefinite programming positive semidefinite hankel matrices arise in many important applications some of their properties may be lost due to rounding or truncation errors incurred during evaluation the problem is to find the nearest matrix to a given matrix to retrieve these properties the problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second order cone problem the duality and optimality conditions are obtained and the primal dual algorithm is outlined explicit expressions for a diagonal preconditioned and crossover criteria have been presented computational results are presented a possibility for further improvement is indicated

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

total destruction of lagrangian tori for an integrable tonelli hamiltonian with d d 2 degrees of freedom we show that all of the lagrangian tori can be destroyed by analytic perturbations which are arbitrarily small in the cd topology

perturbations of elliptic billiards abstract the dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied in the case of an elliptic billiard table the corresponding twist map is integrable and has a saddle connection between two hyperbolic period two points using a discrete analog to the melnikov method we are able to show that this saddle connection can be deformed into a transversal heteroclinic connection under certain analytic perturbations of the table from the formulas that we get we can show that the splitting of the separatrices is exponentially small as a function of the eccentricity of the original unperturbed elliptic table in addition we also include a characterization of the period two periodic points for any billiard table

infinite order cross validated local polynomial regression many practical problems require nonparametric estimates of regression func tions and local polynomial regression has emerged as a leading approach in applied settings practitioners often adopt either the local constant or local linear variants or choose the order of the local polynomial to be slightly greater than the order of the maximum derivative esti mate required but such ad hoc determination of the polynomial order may not be optimal in general while the joint determination of the polynomial order and bandwidth presents some interesting theoretical and practical challenges in this paper we propose a data driven approach towards the joint determination of the polynomial order and bandwidth provide theoretical underpinnings and demonstrate that improvements in both nite sample e ciency and rates of convergence can thereby be obtained in the case where the true data generating process dgp is in fact a polynomial whose order does not depend on the sample size our method is capable of attaining the p n rate often associated with correctly speci ed parametric models while the estimator is shown to be uniformly consistent for a much larger class of dgps theoretical underpinnings are provided and nite sample properties are examined

the generalized area theorem and some of its consequences there is a fundamental relationship between belief propagation bp and maximum a posteriori decoding the case of transmission over the binary erasure channel was investigated in detail in a companion paper c meacuteasson a montanari and r urbanke maxwell s construction the hidden bridge between iterative and maximum a posteriori decoding ieee transactions on information theory submitted for publication this paper investigates the extension to general memoryless channels paying special attention to the binary case an area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed we show that this area theorem gives rise to an upper bound on the maximum a posteriori threshold for sparse graph codes in situations where this bound is tight the extrinsic soft bit estimates delivered by the bp decoder coincide with the correct a posteriori probabilities above the maximum a posteriori threshold more generally it is conjectured that the fundamental relationship between the maximum a posteriori probability map and the bp decoder which was observed for transmission over the binary erasure channel carries over to the general case we finally demonstrate that in order for the design rate of an ensemble to approach the capacity under bp decoding the component codes have to be perfectly matched a statement which is well known for the special case of transmission over the binary erasure channel

finite dimensional bounds on bbz _m and binary ldpc codes with belief propagation decoders this paper focuses on finite dimensional upper and lower bounds on decodable thresholds of zopfm and binary low density parity check ldpc codes assuming belief propagation decoding on memoryless channels a concrete framework is presented admitting systematic searches for new bounds two noise measures are considered the bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability map decoder on the uncoded channel for zopf m ldpc codes an iterative m dimensional bound is derived for m ary input symmetric output channels which gives a sufficient stability condition for zopfm ldpc codes and is complemented by a matched necessary stability condition introduced herein applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed for binary codes two new lower bounds are provided for symmetric channels including a two dimensional iterative bound and a one dimensional noniterative bound the latter of which is the best known bound that is tight for binary symmetric channels bscs and is a strict improvement over the existing bound derived by the channel degradation argument by adopting the reverse channel perspective upper and lower bounds on the decodable bhattacharyya noise parameter are derived for nonsymmetric channels which coincides with the existing bound for symmetric channels

an approach to mel nikov theory in celestial mechanics using a completely analytic procedure based on a suitable extension of a classical method we discuss an approach to the poincare mel nikov theory which can be conveniently applied also to the case of nonhyperbolic critical points and even if the critical point is located at the infinity in this paper we concentrate our attention on the latter case and precisely on problems described by kepler type potentials in one or two degrees of freedom in the presence of general time dependent perturbations we show that the appearance of chaos possibly including arnol d diffusion can be proved quite easily and in a direct way without resorting to singular coordinate transformations such as the mcgehee or blowing up transformations natural examples are provided by the classical gylden problem originally proposed in celestial mechanics but also of interest in different fields and by the general three body problem in classical mechanics

poincare birkhoff periodic orbits for mechanical hamiltonian systems on t tn here a version of the arnol d conjecture first studied by conley and zehnder giving a generalization of the poincare birkhoff last geometrical theorem is proved inside viterbo s framework of the generating functions quadratic at infinity we give brief overviews of some tools that are often utilized in symplectic topology

periodic solutions of spatially periodic hamiltonian systems abstract this work is concerned with the study of existence and multiplicity of periodic solutions of hamiltonian systems of ordinary differential equations z j hz z t f t when the hamiltonian h z t h p q t is periodic in the variable q and superlinear in the variable p by imposing a growth condition on the derivative of h we obtain the existence of at least n 1 periodic solutions where n is the dimension of the system the existence of periodic solutions is obtained by using a saddle point theorem recently proved by lui we consider a functional over e x m where e is a hilbert space and m is a compact manifold satisfying a saddle point condition on e uniformly on m we present a proof of this saddle point theorem using standard minimax techniques based on the cup length of m

bandwidth efficient and rate matched low density parity check coded modulation a new coded modulation scheme is proposed at the transmitter the concatenation of a distribution matcher and a systematic binary encoder performs probabilistic signal shaping and channel coding at the receiver the output of a bitwise demapper is fed to a binary decoder no iterative demapping is performed rate adaption is achieved by adjusting the input distribution and the transmission power the scheme is applied to bipolar amplitude shift keying ask constellations with equidistant signal points and it is directly applicable to two dimensional quadrature amplitude modulation qam the scheme is implemented by using the dvb s2 low density parity check ldpc codes at a frame error rate of 10 3 the new scheme operates within less than 1 1 db of the awgn capacity frac 1 2 log 2 1 mathsf snr at any spectral efficiency between 1 and 5 bits s hz by using only 5 modes i e 4 ask with code rate 2 3 8 ask with 3 4 16 ask and 32 ask with 5 6 and 64 ask with 9 10

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

transversality of homoclinic orbits to hyperbolic equilibria in a hamiltonian system via the hamilton jacobi equation abstract we consider a hamiltonian system with 2 degrees of freedom with a hyperbolic equilibrium point having a loop or homoclinic orbit or alternatively two hyperbolic equilibrium points connected by a heteroclinic orbit as a step towards understanding the behavior of nearly integrable hamiltonians near double resonances we provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop inside their common energy level for the system considered we establish a necessary and sufficient condition for the transversality in terms of a riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop the key point of our approach is to write the invariant manifolds in terms of generating functions which are solutions of the hamilton jacobi equation in some examples we show that it is enough to analyze the phase portrait of the riccati equation without solving it explicitly finally we consider an analogous problem in a perturbative situation if the invariant manifolds of the unperturbed loop coincide we have a problem of splitting of separatrices in this case the riccati equation is replaced by a melnikov potential defined as an integral providing a condition for the existence of a perturbed loop and its transversality this is also illustrated with a concrete example

on universal ldpc code ensembles over memoryless symmetric channels a design of robust error correcting codes that achieve reliable communication over various channels is of great theoretical and practical interest such codes are termed universal this paper considers the universality of low density parity check ldpc code ensembles over families of memoryless binary input output symmetric mbios channels universality is considered both under belief propagation bp and maximum likelihood ml decoding for the bp decoding case we derive a density evolution based analytical method for designing ldpc code ensembles that are universal over various families of mbios channels we also derive a necessary condition for universality of ldpc code ensembles under bp decoding this condition is used to provide bounds on the universally achievable fraction of capacity these results enable us to provide conditions for reliable unreliable communications under bp decoding that are based on the bhattacharyya parameter of the channel for the ml decoding case we prove that properly selected regular ldpc code ensembles are universally capacity achieving for the set of equi capacity mbios channels and extend this result to punctured regular ldpc code ensembles

orbits of unbounded energy in quasi periodic perturbations of geodesic flows abstract we show that certain mechanical systems including a geodesic flow in any dimension plus a quasi periodic perturbation by a potential have orbits of unbounded energy the assumptions we make in the case of geodesic flows are a the metric and the external perturbation are smooth enough b the geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection c the frequency of the external perturbation is diophantine d the external potential satisfies a generic condition depending on the periodic orbit considered in b the assumptions on the metric are c 2 open and are known to be dense on many manifolds the assumptions on the potential fail only in infinite codimension spaces of potentials the proof is based on geometric considerations of invariant manifolds and their intersections the main tools include the scattering map of normally hyperbolic invariant manifolds as well as standard perturbation theories averaging kam and melnikov techniques we do not need to assume that the metric is riemannian and we obtain results for finsler or lorentz metrics indeed there is a formulation for hamiltonian systems satisfying scaling hypotheses we do not need to make assumptions on the global topology of the manifold nor on its dimension

geometric properties of the scattering map of a normally hyperbolic invariant manifold given a normally hyperbolic invariant manifold for a map f whose sta ble and unstable invariant manifolds intersect transversally we consider its associated scattering map that is the map that given an asymptotic orbit in the past gives the asymptotic orbit in the future we show that when f and are symplectic resp exact symplectic then the scattering map is symplectic resp exact symplectic furthermore we show that in the exact symplectic case there are extremely easy formulas for the primitive function which have a variational interpretation as dierence of actions we use this geometric information to obtain ecien t perturbative calculations of the scattering map using deformation theory this perturbation theory generalizes and extends several results already obtained using the melnikov method analogous results are true for hamiltonian o ws the proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems hence the results can be used to obtain intersection properties of objects of any type we also reexamine the calculation of the scattering map in a geodesic o w perturbed by a quasi periodic potential we show that the geometric theory reproduces the results obtained in dls06b using methods of fast slow systems moreover the geometric theory allows to compute perturbatively the dependence on the slow variables which does not seem to be accessible to the previous methods

a practical use of the melnikov homoclinic method using cutoff functions and periodic extensions we prove that the melnikov homoclinic method gives a criterium to show that for a finite time interval t t with t arbitrarily large the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions the method is applied to specific perturbations of the pendulum and of the gylden s problem

does the performance of ldpc codes depend on the channel in this letter we discuss the performance of low density parity check ldpc codes on memoryless channels using a recently proposed analysis technique based on extrinsic information transfer exit charts we present an interpretation of the known fact that the bit error rate ber performance of an ensemble of ldpc codes shows little dependence on the specific memoryless channel this result has been partially observed in the literature for symmetric channels and is here extended to asymmetric channels we conjecture and demonstrate that the performance of an ensemble of ldpc codes depends primarily and solely on the mutual information mi between the input and the output of the channel as a validation of this conjecture we compare the performance of a few ldpc codes with various rates for five representative memoryless both symmetric and asymmetric channels obtaining results in excellent agreement with the exit chart based prediction

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

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

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each

constrained information combining theory and applications for ldpc coded systems this paper tightens previous information combining bounds on the performance of iterative decoding of binary low density parity check ldpc codes over binary input symmetric output channels by tracking the probability of erroneous bit in conjunction with mutual information evaluation of the new bounds as well as of other known bounds on different ldpc ensembles demonstrates sensitivity of the finite dimensional iterative bounds to lambda2 the fraction of edges connected to degree 2 variable nodes

a geometric criterion for positive topological entropy we prove that a diffeomorphism possessing a homoclinic point with a topological crossing possibly with infinite order contact has positive topological entropy along with an analogous statement for heteroclinic points we apply these results to study area preserving perturbations of area preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections in the heteroclinic case the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection in the homoclinic case the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection these results significantly simplify the application of the poincare arnold melnikov sotomayor method the results apply even when the contraction and expansion at the fixed point is subexponential

smallest eigenvalues of hankel matrices for exponential weights abstract we obtain the rate of decay of the smallest eigenvalue of the hankel matrices i t j k w 2 t dt j k 0 n for a general class of even exponential weights w2 exp 2q on an interval i more precise asymptotics for more special weights have been obtained by many authors