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

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

revised adomian decomposition method for solving systems of ordinary and fractional differential equations a modification of the adomian decomposition method applied to systems of linear nonlinear ordinary and fractional differential equations which yields a series solution with accelerated convergence has been presented illustrative examples have been given

smoothing of the stokes phenomenon using mellin barnes integrals abstract recently berry 1989 has put forward a formal theory of the smoothing of the stokes phenomenon based upon optimal truncation of asymptotic expansions and borel summation of the resulting exponentially small remainder terms for certain classes of functions defined by laplace and stieltjes transforms olver jones and boyd 1990 have obtained rigorous uniform exponentially improved asymptotic expansions which describe the smooth transition of a stokes multiplier across a stokes line in this paper we present an alternative theory for functions defined by mellin barnes integrals this approach enables uniform exponentially improved expansions to be established in a direct manner including estimates for the remainder term in the exponentially small series the theory is illustrated by application to the confluent hypergeometric functions considered by olver 1990 1991

solution of the system of ordinary differential equations by adomian decomposition method in this article we use adomian decomposition method which is a well known method for solving functional equations now a days to solve systems of differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one theoretical considerations are being discussed and convergence of the method for theses systems is addressed some examples are presented to show the ability of the method for linear and non linear systems of differential equations

overlapping stokes smoothings survival of the error function and canonical catastrophe integrals we derive doubly uniform approximations for the remainder in the optimally truncated saddle point expansion for an integral containing a large parameter double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate as described by catastrophe theory and while the subdominant exponentials they contribute switch on and off as described by the error function smoothing of the stokes phenomenon two sorts of asymptotic singularity are thereby united in a common framework the formula for the remainder incorporates both the stokes error function and the canonical catastrophe integrals a numerical illustration is given in which the distant cluster contains two saddles the asymptotic theory gives an accurate description of the details of the fractional remainder even when this is of order exp 36

an adaptive n body algorithm of optimal order picard iteration is normally considered a theoretical tool whose primary utility is to establish the existence and uniqueness of solutions to first order systems of ordinary differential equations odes however in 1996 parker and sochacki neural parallel sci comput 4 1996 published a practical numerical method for a certain class of odes based upon modified picard iteration that generates the maclaurin series of the solution to arbitrarily high order the applicable class of odes consists of first order autonomous systems whose right hand side functions generators are projectively polynomial that is they can be written as polynomials in the unknowns the class is wider than might be expected the method is ideally suited to the classical n body problem which is projectively polynomial here we recast the n body problem in polynomial form and develop a picard based algorithm for its solution the algorithm is highly accurate parameter free and simultaneously adaptive in time and order test cases for both benign and chaotic n body systems reveal that optimal order is dynamic that is in addition to dependency upon n and the desired accuracy optimal order depends upon the configuration of the bodies at any instant

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

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

a reliable method for the numerical solution of the kinetics problems this paper deals with the implementation of the adomian s decomposition method in chemical applications one often encounters systems of ordinary differential equations this method is intended to solve the approximate solutions of the linear and nonlinear kinetic problems without usual restrictive assumptions a new approach to differential problems is particularly valuable as a tool for scientists and applied mathematicians because it provides immediate and visible symbolic terms of analytic solution as well as its numerical approximate solution to both linear and nonlinear problems without linearization adomian solving frontier problems of physics the decomposition method kluwer academic publishers boston 1994 j math anal appl 135 1988 501 it does also not require discretization and consequently massive computation we also give a comparison between a reliable decomposition and a conventional methods such as the taylor series and the fourth order runge kutta method for system of ordinary differential equations the numerical results demonstrate that the new method is quite accurate and readily implemented

a note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points for the uniform asymptotic expansion of certain types of contour integrals one of whose critical points is an endpoint of the interval of integration a method alternative to bleistein s is introduced and numerically tested by way of a non trivial physical example

a double integral containing the modified bessel function asymptotics and computation on considere une integrale bidimensionnelle comportant exp u t io 2 ut io z est la fonction de bessel modifiee et l integrale est prise sur le rectangle o u x o t y on presente des algorithmes efficaces de calcul

exponentially improved asymptotics for the gamma function by expressing the error term in truncation of the asymptotic expansion in terms of a mellin barnes integral we obtain an exponentially improved asymptotic expansion for z as z in arg z we describe the need for such an expansion arising in applications the result is compared with formal results of dingle obtained via borel summation and rigorous methods of olver using laplace integrals for a different class of functions

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

an innovative low density parity check code design with near shannon limit performance and simple implementation a novel parity check matrix design for low density parity check ldpc codes is described by eliminating the routing problem associated with ldpc codes the design results in a small implementation area and the codes have outstanding error rate performance close to the shannon limit for a wide range of code rates from 1 4 to 9 10 and for various modulation schemes such as binary phase shift keying psk quaternary psk 8 psk 16 amplitude psk apsk and 32 apsk as a result ldpc codes designed with this method have been standardized for next generation digital video broadcasting

asymptotics beyond all orders asymptotology and borel summation asymptotics superasymptotics hyperasymptotics m berry the geometric model of crystal growth numerical analysis of the geometric model of crystal growth j m hammersley dendritic crystal growth an experimental assessment of continuum models of dendritic growth j p gollub directional solidification of solids directional growth of dilute mixtures and lamellar eutectics k kassner et al flow in a hele shaw cell viscous displacement in a hele shaw cell s tanveer the rapidly forced pendulum exponentially small estimate for separatrix splittings j scheurle et al ordinary differential equations singular perturbations of solitons y pomeau solitary water waves in the presence of small surface tension solitary waves with ripples beyond all orders j t beale problems in optics potpourri 23 additional articles index

uniform asymptotic expansions of integrals a selection of problems abstract on the occasion of the conference we mention examples of stieltjes work on asymptotics of special functions the remaining part of the paper gives a selection of asymptotic methods for integrals in particular on uniform approximations we discuss several standard problems and examples in which known special functions error functions airy functions bessel functions etc are needed to construct uniform approximations finally we discuss the recent interest and new insights in the stokes phenomenon an extensive bibliography on uniform asymptotic methods for integrals is given together with references to recent papers on the stokes phenomenon for integrals and related topics

uniform asymptotic expansions of a class of integrals with finite endpoints of integration on the same path of steepest descent and with nearby saddle points abstract for the frequently required uniform asymptotic expansion of a certain class of integrals that have both their finite endpoints on the same path of steepest descent with a number of nearby saddle points a modification of an earlier procedure is given over bleistein s general method it offers the convenience of completely separating the processes of determining the expansion coefficients and of repeatedly integrating by parts this is of practical relevance with a view to delegating the otherwise very tedious calculation of higher expansion coefficients to one of the available computer codes for the algebraic analytic manipulation of given expressions

selective avoidance of cycles in irregular ldpc code construction this letter explains the effect of graph connectivity on error floor performance of low density parity check ldpc codes under message passing decoding a new metric called extrinsic message degree emd measures cycle connectivity in bipartite graphs of ldpc codes using an easily computed estimate of emd we propose a viterbi like algorithm that selectively avoids small cycle clusters that are isolated from the rest of the graph this algorithm is different from conventional girth conditioning by emphasizing the connectivity as well as the length of cycles the algorithm yields codes with error floors that are orders of magnitude below those of random codes with very small degradation in capacity approaching capability

a new class of multiple rate codes based on block markov superposition transmission the hadamard transform ht over the binary field provides a natural way to implement multiple rate codes referred to as ht coset codes where the code length n 2 p is fixed but the code dimension k can be varied from 1 to n 1 by adjusting the set of frozen bits the ht coset codes including reed muller rm codes and polar codes as typical examples can share the same fundamental encoder decoder architecture with the implementation complexity of order o n log n however to guarantee that all codes with designated rates perform well ht coset coding usually requires a sufficiently large code length which in turn causes difficulties in the determination of which bits are better for being frozen in this paper we propose to transmit short ht coset codes in the so called block markov superposition transmission bmst manner at the transmitter signals are spatially coupled via superposition resulting in long codes at the receiver these coupled signals are recovered by a sliding window iterative soft successive cancellation decoding algorithm most importantly the performance around or below the bit error rate ber of 10 5 can be predicted by a simple genie aided lower bound both these bounds and simulation results show that the bmst of short ht coset codes performs well within one db away from the corresponding shannon limits in a wide range of code rates

high orders of the weyl expansion for quantum billiards resurgence of periodic orbits and the stokes phenomenon a formalism is developed for calculating high coefficients c r of the weyl high energy expansion for the trace of the resolvent of the laplace operator in a domain b with smooth boundary b the c r are used to test the following conjectures a the sequence of c r diverges factorially controlled by the shortest accessible real or complex periodic geodesic b if this is a 2 bounce orbit it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the mobius strip which is the space of chords of b c this orbit gives an exponential contribution to the remainder when the weyl series truncated at its least term is subtracted from the resolvent the exponential switches on smoothly according to an error function where it is smallest that is across the negative energy axis stokes line these conjectures are motivated by recent results in asymptotics they survive tests for the circle billiard and for a family of curves with 2 and 3 bulges where the dominant orbit is not always the shortest and is sometimes complex for some systems which are not smooth billiards e g a particle on a ring or in a billiard where b is a polygon the weyl series terminates and so no geodesics are accessible for a particle on a compact surface of constant negative curvature only the complex geodesics are accessible from the weyl series

a simple explanation of the stokes phenomenon the stokes phenomenon is known to be a pervasive feature of asymptotics but its explanation in the literature is obscured by intricate and lengthy technicalities this article presents a simpler approach to its understanding and treatment as a natural aspect of a well motivated characterization of functions by approximands of different multivaluedness

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

robust rate compatible punctured ldpc convolutional codes a family of robust rate compatible rc punctured low density parity check convolutional codes ldpc ccs is derived from a time invariant ldpc cc mother code by periodically puncturing encoded bits variable nodes with respect to several criteria 1 ensuring the recoverability of punctured variable nodes 2 minimizing the number of completely punctured cycle trapping sets cpctss and 3 minimizing the number of punctured variable nodes involved in short cycles the influence of 1 and 3 on iterative decoding performance is felt most strongly in the waterfall region of the bit error rate ber curve while 2 has a larger effect in the error floor or high signal to noise ratio snr region we show that the length of the puncturing period is an important parameter when designing high rate punctured codes and moreover that extending the puncturing period can improve the decoding performance and extend the range of compatible rates as examples we obtain families of rc ldpc ccs from several time invariant ldpc cc mother codes with monomial and binomial entries in their polynomial syndrome former matrices

multiple rate low density parity check codes with constant blocklength this paper describes and analyzes low density parity check code families that support variety of different rates while maintaining the same fundamental decoder architecture such families facilitate the decoding hardware design and implementation for applications that require communication at different rates for example to adapt to changing channel conditions combining rows of the lowest rate parity check matrix produces the parity check matrices for higher rates an important advantage of this approach is that all effective code rates have the same blocklength this approach is compatible with well known techniques that allow low complexity encoding and parallel decoding of these ldpc codes this technique also allows the design of programmable analog ldpc decoders the proposed design method maintains good graphical properties and hence low error floors for all rates

picard iteration algorithm combined with gauss seidel technique for initial value problems abstract in this paper we introduce and implement a relatively new improvement for the well known picard s method for studying linear and nonlinear systems of ordinary differential equations in this method the solution takes the form of a rapidly convergent series with easily computable components a mathematica computer algebra system mathematica 5 1 is used to introduce a code which generates the series different illustrative examples have been considered as well as different comparison criterion between the proposed method and modern reliable and conventional methods the deeply mathematical roots of the method fixed point theorem and the correspondence with the theory of algebraic systems makes it is very interesting theoretical considerations are being discussed and convergence of the method is addressed other extensions and applications for further work are mentioned

uniform and quasi uniform asymptotic expansions of incomplete diffraction integrals two types of incomplete diffraction integrals defined on semi infinite contours are considered the integrals are characterized by three critical points a simple saddle point a simple pole and a finite endpoint of integration uniform asymptotic expansions of these integrals valid to the order of lambda 1 are constructed the expansions obtained are then simplified to quasi uniform and nonuniform ones

a technique for the numerical verification of asymptotic expansions a technique is presented that uses numerical solutions to verify the order of the accuracy of an asymptotic expansion for several types of problems this technique may be introduced in any beginning course on asymptotic or perturbation methods but is equally suited to verifying advanced asymptotic results

the uniform asymptotic swallowtail approximation practical methods for oscillating integrals with four coalescing saddle points practical methods for the numerical implementation of the uniform swallowtail approximation have been developed this approximation arises in the uniform asymptotic evaluation of oscillating integrals with four coalescing saddle points a complex contour quadrature technique has been used to evaluate the swallowtail canonical integral s x y z and its partial derivatives delta s delta x delta s delta y delta s delta z this method has the advantage that it is straightforward to implement on a computer and results of high accuracy are readily obtained a comparison is made with other methods that have been reported in the literature for the evaluation of s x y z isometric plots of mod s x y z mod mod delta s delta x mod mod delta s delta y mod mod delta s delta z mod are presented and some properties of the zeros of s 0 y z that lie on the line y 0 are also discussed two methods for the evaluation of the mapping parameters x y z are described an iterative method that is valid when x y z is not close to the swallowtail caustic and an algebraic method valid for x y z on the caustic and for y 0 symbolic algebraic computer programs have been used to carry out the necessary algebraic manipulations in practice both methods for determining x y z are complementary an application of the uniform swallowtail approximation to the butterfly canonical integral has been made the uniform asymptotic swallowtail approximation can now be regarded as a practical tool for the evaluation of oscillating integrals with four coalescing saddle points

on the expansion of confluent hypergeometric functions in terms of bessel functions abstract for the confluent hypergeometric functions u a b z and m a b z asymptotic expansions are given for a the expansions contain modified bessel functions for real values of the parameters rigorous error bounds are given

hyperasymptotics for integrals with finite endpoints berry howls 1991 hereinafter called bh refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp kf z along doubly infinite contours in the complex plane passing over saddlepoints of f z here we derive analogous results for integrals with integrands of a similar form but whose local expansions in powers of 1 k are made about the finite endpoints of semi infinite contours of integration we treat endpoints where f z behaves locally linearly or quadratically generically local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f z we derive resurgence relations which express the original integral exactly as a truncated endpoint expansion plus a remainder involving the global saddle structure of f z via integrals through certain adjacent saddles the saddles adjacent to the endpoint are determined by a topological rule if the least term of the endpoint expansion is the n 0 k th summing to here calculates the endpoint integral up to an error of approximately exp n 0 k we develop a scheme involving iteration of the new resurgence relations with a similar one derived in bh which can reduce this error down to exp 2 386n 0 k this hyperasymptotic formalism parallels that of bh and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane provided that it does not coincide with other saddles we illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc x and a constructed incomplete airy function

an iterative method for solving nonlinear functional equations an iterative method for solving nonlinear functional equations viz nonlinear volterra integral equations algebraic equations and systems of ordinary differential equation nonlinear algebraic equations and fractional differential equations has been discussed

on the asymptotics of some pearcey type integrals the author discusses the asymptotic behaviour of the pearcey type integrali alpha x y 2 sup approximately 0 ualpha 1 exp i u4 xu2 jalpha yu du for 1 alpha 5 2 where jalpha is a bessel function as x to or infinity y fixed as y to infinity x fixed and as y rho 2 3 mod x mod 32 x to infinity rho fixed the case alpha 1 2 gives the classical pearcey integral whose asymptotics has been investigated recently by kaminski 1989 and paris 1991 in the case alpha 0 i alpha x y as a function of y or 0 represents the radial part of the impulse response function describing the image formation in high resolution electron microscopes at normalized defocus x he uses the approach of paris by representing i alpha x y in terms of weber parabolic cylinder functions and he augments this approach by invoking the chester friedman ursell method 1957 to obtain the leading asymptotics of i alpha x y around the caustic y2 2 3 mod x mod 3 x to infinity

large girth nonbinary qc ldpc codes of various lengths in this paper we construct nonbinary quasi cyclic low density parity check qc ldpc codes whose parity check matrices consist of an array of square sub matrices which are either zero matrices or circulant permutation matrices we propose a novel method to design the shift offset values of the circulant permutation sub matrices so that the code length can vary while maintaining a large girth extensive monte carlo simulations demonstrate that the obtained codes of a wide range of rates from 1 2 to 8 9 with length from 1000 to 10000 bits have very good performance over both awgn and rayleigh fading channels furthermore the proposed method is extended to design multiple nonbinary qc ldpc codes simultaneously where each individual code can achieve large girth with variable lengths the proposed codes are appealing to practical adaptive systems where the block length and code rate need to be adaptively adjusted depending on traffic characteristics and channel conditions

on the numerical solution of stiff systems abstract in this paper we use power series method to solve stiff ordinary differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one theoretical considerations has been discussed and some examples were presented to show the ability of the method for linear and nonlinear systems of differential equations we use maple computer algebra systems for numerical calculations g frank maple v crc press inc florida 1996

infinitely many stokes smoothings in the gamma function the stokes lines for gamma z are the positive and negative imaginary axes where all terms in the divergent asymptotic expansion for ln gamma z have the same phase on crossing these lines from the right to the left half plane infinitely many subdominant exponentials appear rather than the usual one the exponentials increase in magnitude towards the negative real axis anti stokes line where they add to produce the poles of gamma z corresponding to each small exponential is a separate component asymptotic series in the expansion for ln gamma z if each is truncated near its least term its exponential switches on smoothly across the stokes lines according to the universal error function law by appropriate subtractions from ln gamma z the switching on of successively smaller exponentials can be revealed the procedure is illustrated by numerical computations

design of efficiently encodable moderate length high rate irregular ldpc codes this paper presents a new class of irregular low density parity check ldpc codes of moderate length 10 sup 3 spl les n spl les 10 sup 4 and high rate r spl ges 3 4 codes in this class admit low complexity encoding and have lower error rate floors than other irregular ldpc code design approaches it is also shown that this class of ldpc codes is equivalent to a class of systematic serial turbo codes and is an extension of irregular repeat accumulate codes a code design algorithm based on the combination of density evolution and differential evolution optimization with a modified cost function is presented moderate length high rate codes with no error rate floors down to a bit error rate of 10 sup 9 are presented although our focus is on moderate length high rate codes the proposed coding scheme is applicable to irregular ldpc codes with other lengths and rates

picard iteration and pade approximations for stiff fractional point kinetics equations a model of stiff point kinetics equations is one of the important models in the nuclear reactor dynamics this model describes the neutron density and the precursor concentrations of delayed neutrons into nuclear reactors in this work a fractional model of the stiff point kinetics equations is studied to describe the neutron density behavior by the fractional order picard iteration and pade approximations are presented to solve the stiff fractional point kinetics equations with multi group of delayed neutrons the validity of the fractional model is tested for different cases of step ramp and sinusoidal reactivities the numerical results of picard iteration and pade approximations are computed for various fractional order the results of pade11 approximation are in good agreement with the results of picard iteration than pade01 approximation in addition the numerical results confirm that the neutron density for a positive negative reactivity is increasing decreasing quicker with decreases the fractional order