a characterization of the number of subsequences obtained via the deletion channel motivated by the study of deletion channels this paper presents improved bounds on the number of subsequences obtained from a binary string x of length n under t deletions it is known that the number of subsequences in this setting strongly depends on the number of runs in the string x where a run is a maximal substring of the same character our improved bounds are obtained by a structural analysis of the family of r run strings x an analysis in which we identify the extremal strings with respect to the number of subsequences specifically for every r we present r run strings with the minimum respectively maximum number of subsequences under any t deletions we perform an exact analysis of the number of subsequences of these extremal strings and show that this number can be calculated in polynomial time

generalized sphere packing bound kulkarni and kiyavash recently introduced a new method to establish upper bounds on the size of deletion correcting codes this method is based upon tools from hypergraph theory the deletion channel is represented by a hypergraph whose edges are the deletion balls or spheres so that a deletion correcting code becomes a matching in this hypergraph consequently a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear programming problem the problem of finding fractional transversals the method by kulkarni and kiyavash can be applied not only for the deletion channel but also for other error channels this paper studies this method in its most general setup first it is shown that if the error channel is regular and symmetric then the upper bound by this method coincides with the well known sphere packing bound and thus is called here the generalized sphere packing bound even though this bound is explicitly given by a linear programming problem finding its exact value may still be a challenging task the art of finding the exact upper bound or slightly weaker ones is the assignment of weights to the hypergraph s vertices in a way that they satisfy the constraints in the linear programming problem in order to simplify the complexity of the linear programming we present a technique based upon graph automorphisms that in many cases significantly reduces the number of variables and constraints in the problem we then apply this method on specific examples of error channels we start with the z channel and show how to exactly find the generalized sphere packing bound for this setup next studied is the nonbinary limited magnitude channel both for symmetric and asymmetric errors where we focus on the single error case we follow up on the deletion channel which was the original motivation of the work by kulkarni and kiyavash and show how to improve upon their upper bounds for single deletion correcting codes since the deletion and grain error channels have a similar structure for a single error we also improve upon the existing upper bounds on single grain error correcting codes finally we apply this method for projective spaces and find its generalized sphere packing bound for the single error case

scaling of lyapunov exponents of coupled chaotic systems we develop a statistical theory of the coupling sensitivity of chaos the effect was first described by daido prog theor phys 72 853 1984 it appears as a logarithmic singularity in the lyapunov exponent in coupled chaotic systems at very small couplings using a continuous time stochastic model for the coupled systems we derive a scaling relation for the largest lyapunov exponent the singularity is shown to depend on the coupling and the systems mismatch generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered too the analytical results are confirmed by numerical simulations

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

new lower bounds for asymmetric and unidirectional codes new lower bounds on the sizes of asymmetric codes and unidirectional codes are presented various methods are used three of them of special interest the first is a partitioning method that is a modification of a method used to construct constant weight codes the second is a combining codes method that is used to obtain a new code from a few others the third method is shortening by weights that is applied on symmetric codes or on codes generated by the combining codes method tables for the sizes of codes of length n

more on modifications and improvements of classical iterative schemes for m matrices in the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an m matrix in order to improve on the convergence rates of the classical iterative schemes jacobi gauss seidel etc the present work is a contribution towards the generalization of the most common preconditioners used so far

strongly coupled ising chain under a weak random field abstract we present here an analytical method based on the transfer matrix m to study quenched disorder in one dimensional spin systems in the limit of strong couplings and weak disorder the procedure is formulated for the random field ising chain of finite length l and its properties represented as functions of m satisfy a differential equation of the fokker planck type this equation describes evolution with l and a central limit theorem of a novel kind provides the equation and its solutions with universal character we obtain analytical expressions for the moments of the magnetization of the infinite length chain and study the approach to the infinite coupling limit we find that the random field free energy f h and the edwards anderson order parameter m 2 satisfy a simple relation we discuss our results in connection to previous work by luck and nieuwenhuizen

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

greedy and heuristic algorithms for codes and colorings many of the fundamental coding problems can be represented as graph problems these problems are often intrinsically difficult and unsolved even if the code length is relatively small with the motivation to improve lower bounds on the sizes of constant weight codes and asymmetric codes we suggest a few greedy algorithms and other heuristic methods which result in new record breaking codes some of the heuristics used are based on tabu search and evolutionary algorithms tables of new codes are presented

the aor iterative method for new preconditioned linear systems 461 abstract in this paper we apply the aor method to preconditioned linear systems different from those considered in evans and martins internat j comput math 5 1995 69 76 gunawardena et al linear algebra appl 154 156 1991 123 143 and li and evans technical report no 901 department of computer studies university of loughborough 1994 our results show that some improvements in the convergence rate of this iterative method can be obtained

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

almost perfect lattices the covering radius problem and applications to ajtai s connection factor lattices have received considerable attention as a potential source of computational hardness to be used in cryptography after a breakthrough result of ajtai in proceedings of the 28th annual acm symposium on theory of computing philadelphia pa 1996 pp 99 108 connecting the average case and worst case complexity of various lattice problems the purpose of this paper is twofold on the expository side we present a rigorous self contained proof of results along the lines of ajtai s seminal work at the same time we explore to what extent ajtai s original results can be quantitatively improved as a by product we define a random class of lattices such that computing short nonzero vectors in the class with nonnegligible probability is at least as hard as approximating the length of the shortest nonzero vector in any n dimensional lattice within worst case approximation factors gamma n n 3 omega sqrt log n log log n this improves previously known best connection factor gamma n n 4 epsilon j y cai and a p nerurkar in proceedings of the 38th annual ieee symposium on foundations of computer science miami beach fl 1997 pp 468 477 we also show how our reduction implies the existence of collision resistant cryptographic hash functions based on the worst case inapproximability of the shortest vector problem within the same factors gamma n n 3 omega sqrt log n log log n in the process we distill various new lattice problems that might be of independent interest related to the covering radius the bounded distance decoding problem approximate counting of lattice points inside convex bodies and the efficient construction of lattices with good geometric and algorithmic decoding properties we also show how further investigation of these new lattice problems might lead to even stronger connections between the average case and worst case complexity of the shortest vector problem possibly leading to connection factors as low as gamma n n 1 5 omega log n

expected length of the longest common subsequence for large alphabets we consider the length l of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k ary alphabet subadditivity arguments yield that e l n converges to a constant k we prove a conjecture of sankoff and mainville from the early 80 s claiming that gamma kappa sqrt k longrightarrow 2 as k longrightarrow infty

multiscaling of noise induced parametric instability we describe the statistical properties of growth rates of a linear oscillator driven by a parametric noise we show that in general the fluctuations of local lyapunov exponents are non gaussian and demonstrate multiscaling analytical calculations of the generalized lyapunov exponents are complemented with approximative and numerical results this allows us to identify the parameter range where the deviations from the gaussian statistics become important

some new preconditioned generalized aor methods for generalized least squares problems in this paper we present some new preconditioned gaor methods for solving generalized least squares problems and their comparison results comparison results show that the convergence rates of the new preconditioned gaor methods are better than those of the preconditioned gaor methods presented by zhou et al 2009 19 and wang et al 2013 18 whenever these methods are convergent lastly numerical experiments are provided to confirm the theoretical results

further results on the covering radius of codes a number of upper and lower bounds are obtained for k n r the minimal number of codewords in any binary code of length n and covering radius r several new constructions are used to derive the upper bounds including an amalgamated direct sum construction for nonlinear codes this construction works best when applied to normal codes and we give some new and stronger conditions which imply that a linear code is normal an upper bound is given for the density of a covering code over any alphabet and it is shown that k n 2 r 1 leq k n r holds for sufficiently large n

accelerated iterative method for z matrices it has recently been reported that the convergence of the preconditioned gauss seidel method which uses a matrix of the type i u as a preconditioner is faster than the basic iterative method in this paper we generalize the preconditioner to the type i bu where b is a positive real number after discussing convergence of the method applied to z matrices we propose an algorithm for estimating the optimum b numerical examples are also given which show the effectiveness of our algorithm

bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6 upper bounds to the maximum number of codewords in a binary code of length n and asymmetric distance delta are derived for some values of n and delta a method is given in which a code of length n m and asymmetric distance at least t 1 is constructed by expurgating and puncturing a code of length n and hamming distance at least 2t 1 novel asymmetric error correcting codes are constructed by applying this method to some celebrated symmetric error correcting codes a table is presented on the size of optimal asymmetric error correcting codes of length less than 24 and asymmetric distance less than 6

upper bounds for constant weight codes let a n d w denote the maximum possible number of codewords in an n d w constant weight binary code we improve upon the best known upper bounds on a n d w in numerous instances for n spl les 24 and d spl les 12 which is the parameter range of existing tables most improvements occur for d 8 10 where we reduce the upper bounds in more than half of the unresolved cases we also extend the existing tables up to n spl les 28 and d spl les 14 to obtain these results we develop new techniques and introduce new classes of codes we derive a number of general bounds on a n d w by means of mapping constant weight codes into euclidean space this approach produces among other results a bound on a n d w that is tighter than the johnson bound a similar improvement over the best known bounds for doubly constant weight codes studied by johnson and levenshtein is obtained in the same way furthermore we introduce the concept of doubly bounded weight codes which may be thought of as a generalization of the doubly constant weight codes subsequently a class of euclidean space codes called zonal codes is introduced and a bound on the size of such codes is established this is used to derive bounds for doubly bounded weight codes which are in turn used to derive bounds on a n d w we also develop a universal method to establish constraints that augment the delsarte inequalities for constant weight codes used in the linear programming bound in addition we present a detailed survey of known upper bounds for constant weight codes and sharpen these bounds in several cases all these bounds along with all known dependencies among them are then combined in a coherent framework that is amenable to analysis by computer this improves the bounds on a n d w even further for a large number of instances of n d and w

unidirectional covering codes a code c spl sube z sup n sub 2 where z sub 2 0 1 has unidirectional covering radius r if r is the smallest integer so that any word in z sup n sub 2 can be obtained from at least one codeword c spl isin c by replacing either 1s by 0s in at most r coordinates or 0s by 1s in at most r coordinates the minimum cardinality of such a code is denoted by e n r upper bounds on this function are here obtained by constructing codes using tabu search lower bounds on the other hand are mainly obtained by integer programming and exhaustive search best known bounds on e n r for n spl les 13 and r spl les 6 are tabulated

improving the modified gauss seidel method for z matrices abstract in 1991 a d gunawardena et al reported that the convergence rate of the gauss seidel method with a preconditioning matrix i s is superior to that of the basic iterative method in this paper we use the preconditioning matrix i s if a coefficient matrix a is an irreducibly diagonally dominant z matrix then i s a is also a strictly diagonally dominant z matrix it is shown that the proposed method is also superior to other iterative methods

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

covering codes with improved density we prove a general recursive inequality concerning spl mu sup r the asymptotic least density of the best binary covering codes of radius r in particular this inequality implies that spl mu sup r spl les e spl middot rlogr logr loglogr 2 which significantly improves the best known density 2 sup r r sup r r 1 r our inequality also holds for covering codes over arbitrary alphabets

preconditioned aor iterative methods for m matrices linear systems with m matrices often appear in a wide variety of areas in this paper we give general preconditioners for solving the systems with nonsingular m matrix we show that our preconditioners increase the convergence rate of aor iterative methods numerical results for corresponding preconditioned gmres methods are also given

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

a unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations abstract a unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods suitable to the simd and mimd multiprocessor systems respectively is presented and its convergence theory is established under rather weak conditions these afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting in addition how the known parallel matrix multisplitting iterative methods can be classified into this new framework and what novel ones can be generated by it are shown in detail

a new method based on generalized taylor expansion for computing a series solution of the linear systems in this paper based on the generalized taylor expansion and using the iteration matrix g of the iterative methods we introduce a new method for computing a series solution of the linear systems this method can be used to accelerate the convergence of the basic iterative methods in addition we show that by applying the new method to a divergent iterative scheme it is possible to construct a convergent series solution and to find the convergence intervals of control parameter for special cases numerical experiments are given to show the efficiency of the new method

improved bounds on the average length of longest common subsequences it has long been known lchvatal and sankoff 1975r that the average length of the longest common subsequence of two random strings of length n over an alphabet of size k is asymptotic to kn for some constant k depending on k the value of these constants remains unknown and a number of papers have proved upper and lower bounds on them we discuss techniques involving numerical calculations with recurrences on many variables for determining lower and upper bounds on these constants to our knowledge the previous best known lower and upper bounds for 2 were those of dancik and paterson approximately 0 773911 and 0 837623 ldancik 1994 dancik and paterson 1995r we improve these to 0 788071 and 0 826280 this upper bound is less than the 2 given by steele s old conjecture see steele l1997 page 3r that 2 e 2 1 s2 a 0 828427 as steele points out experimental evidence had already suggested that this conjectured value was too high finally we show that the upper bound technique described here could be used to produce for any k a sequence of upper bounds converging to k though the computation time grows very quickly as better bounds are guaranteed

modified iterative methods for nonnegative matrices and m matrices linear systems the purpose of this paper is to present new preconditioning techniques for solving nonnegative matrices linear system and m matrices linear system ax b based on the i s a type preconditioning matrices provided by hadjidimos et al 1 and evans et al 2 convergence analysis of the proposed methods are given numerical results are presented which show the improvements on the convergence rate of the jacobi type and gauss seidel type preconditioned iterative methods

a general preconditioner for linear complementarity problem with an m matrix abstract in this paper we first present a general preconditioner p for solving linear complementarity problem lcp associated with an m matrix a and a vector f and prove that the lcp a f is equivalent to the lcp p a p f then based on this general preconditioner p two preconditioned ssor methods for solving the linear complementarity problems are proposed we show that this general preconditioner p accelerates the convergence of two ssor methods under the assumption that p a is a z matrix in addition we also give a practically concrete choice for the preconditioner p satisfying aforementioned assumption numerical examples are used to illustrate the theoretical results obtained

comparison results of the preconditioned aor methods for l matrices abstract in this paper we first provide comparison results of several types of the preconditioned aor paor methods for solving a linear system whose coefficient matrix is an l matrix satisfying some weaker conditions than those used in the recent literature next we propose an application of paor method to a preconditioner of krylov subspace method lastly numerical results are provided to show that krylov subspace method with the paor preconditioner performs quite well as compared with the ilu 0 preconditioner

continuous approach for the random field ising chain we study the random field ising chain in the limit of strong exchange coupling in order to calculate the free energy we apply a continuous langevin type approach this continuous model can be solved exactly whereupon we are able to locate the crossover between an exponential and a power law decay of the free energy with increasing coupling strength in terms of magnetization this crossover restricts the validity of the linear scaling the known analytical results for the free energy are recovered in the corresponding limits the outcomes of numerical computations for the free energy are presented which confirm the results of the continuous approach we also discuss the validity of the replica method which we then utilize to investigate the sample to sample fluctuations of the finite size free energy

a preconditioned two step modulus based matrix splitting iteration method for linear complementarity problem abstract in this paper a preconditioned two step modulus based matrix splitting iteration method for linear complementarity problems associated with an m matrix is proposed the convergence analysis of the presented method is given in particular we provide a comparison theorem between preconditioned two step modulus based gauss seidel ptmgs iteration method and two step modulus based gauss seidel tmgs iteration method which shows that ptmgs method improves the convergence rate of original tmgs method for linear complementarity problem numerical tested examples are used to illustrate the theoretical analysis

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

a note on the preconditioned gauss seidal gs method for linear systems in this note recent comparison results for preconditioned gauss seidel gs methods are discussed a new strict comparison result between two different preconditioned gs methods is given some errors in a recent article by niki et al j comput appl math 164 165 2004 587 are pointed out and a new proof for the corresponding results in niki et al is presented

modified gauss seidel type methods and jacobi type methods for z matrices abstract in this paper we present the convergence analysis for some modified gauss seidel and jacobi type iterative methods and provide a comparison of spectral radius among the gauss seidel iterative method and these modified methods some recent results are improved

convergence analysis of the two preconditioned iterative methods for m matrix linear systems in this paper we present the two preconditioners i s and i s r for solving m matrix linear systems and discuss the convergence of the two preconditioned iterative methods meanwhile we obtain comparison theorems between the two preconditioned iterative methods and consider the solution of m matrix linear systems by preconditioned krylov subspace methods numerical experiments are given to validate the performance of the preconditioners