perturbation bound of singular linear systems a classical perturbation bound in numerical linear algebra is extended to more general cases of singular linear algebraic systems and weighted linear least squares problem componentwise perturbation analysis of linear least squares problem is also derived

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

different complex zfs leading to different complex znn models for time varying complex generalized inverse matrices as a special class of recurrent neural network zhang neural network znn has been recently proposed since 2001 for solving various time varying problems and has shown high efficiency and excellent performance for solving the problems in the real domain in this paper to solve online the time varying complex generalized inverse in most cases the pseudoinverse problem in the complex domain a new type of complex valued znn is further proposed and investigated the design of such a complex znn is based on a complex zhang function zf which is indefinite and quite different from the usual error function specially the scalar valued energy function in the studies of conventional algorithms by introducing five different complex zfs five different complex znn models termed complex znn i znn ii znn iii znn iv and znn v models are proposed developed and investigated for the online solution of the time varying complex generalized inverse matrices theoretical results of convergence analysis are presented to show the desirable properties of complex znn models in addition we discover the link between the proposed complex znn models and the getz marsden dynamic system in the complex domain computer simulation results further demonstrate the effectiveness of complex znn models based on different complex zfs for the time varying complex generalized inverse matrices

recurrent neural network for computing the drazin inverse this paper presents a recurrent neural network rnn for computing the drazin inverse of a real matrix in real time this recurrent neural network rnn is composed of n independent parts subnetworks where n is the order of the input matrix these subnetworks can operate concurrently so parallel and distributed processing can be achieved in this way the computational advantages over the existing sequential algorithms can be attained in real time applications the rnn defined in this paper is convenient for an implementation in an electronic circuit the number of neurons in the neural network is the same as the number of elements in the output matrix which represents the drazin inverse the difference between the proposed rnn and the existing ones for the drazin inverse computation lies in their network architecture and dynamics the conditions that ensure the stability of the defined rnn as well as its convergence toward the drazin inverse are considered in addition illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network

generalized sphere packing bounds on the size of codes for combinatorial channels many of the classic problems of the coding theory are highly symmetric which makes it easy to derive the sphere packing upper bounds on the size of codes we discuss the generalizations of the sphere packing bounds to the arbitrary error models these generalizations become especially important when the sizes of the error spheres are nonuniform the best possible sphere packing bounds are the solutions to the linear programs we derive a series of bounds from the approximations to the packing and covering problems and study the relationships and the trade offs between them we show the method to obtain the upper bounds by optimizing across a family of channels that admit the same codes we present a generalization of the local degree bound of kulkarni and kiyavash and use it to improve the best known upper bounds on the sizes of the single deletion correcting codes and the single grain error correcting codes

moment balancing templates for d k constrained codes and run length limited sequences the first order moment of d k constrained codes is investigated in this paper a generalized moment balancing template is proposed to encode a d k sequence into a single insertion or deletion correcting codeword without losing the constraint property by relocating 0 s in moment balancing runs which appear in a pairwise manner of a d k sequence the first order moment of this sequence can be modified to satisfy the varshamov tenengolts construction with a reasonably large base in the modulo system introduced by the varshamov tenengolts construction this generalized moment balancing template can be applied to run length limited sequences the asymptotic bound of the redundancy introduced by the template for d k sequences is of the same order as the universal template for random sequences and therefore the redundancy is small and suitable for long sequences of practical interest

restricted composition deletion correcting codes we investigate deletion correcting codes and restricted composition codes in particular restricted composition codes generalize codes on permutations and multipermutations we use graph theoretic methods to characterize codes establish bounds on code size and describe constructions the substring partial order has a well known property for any string the number of superstrings of a particular length depends only on the length of the original string we generalize this property to take compositions into account for any string the number of superstrings of a particular composition depends only on the composition of the original string we present a bijective proof of this fact we apply this result to obtain a lower bound on the size of restricted composition codes we obtain an upper bound by analyzing deletion errors when the composition of deleted symbols is restricted to a particular worst case composition we construct binary restricted composition single deletion correcting codes and show that they are asymptotically optimal and form an optimal coloring there is a natural distance on compositions that provides a lower bound on deletion distance unrestricted deletion correcting codes can be constructed from the union of restricted composition codes as long as the set of compositions used themselves form a code the nonbinary single deletion correcting codes constructed by tenengolts are a special case of our method

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

neural network approach to computing matrix inversion a neural network for matrix inversion is proposed we show both analytically and by simulations that this network is guaranteed to be stable and to provide results arbitrarily close to the accurate inversion of a matrix within an elapsed time of only a few characteristic time constants of the network the parameters such as the interconnection strengths of this network can be obtained from the known matrix without any computations

the representation and approximation for the weighted moore penrose inverse in hilbert space we present a unified representation theorem of the weighted moore penrose inverse in hilbert space specific expressions and computational procedures for the weighted moore penrose inverse in hilbert space can be uniformly derived

the weighted generalized inverses of a partitioned matrix a constructive proof for the recursive representation of the weighted moore penrose inverse of a partitioned matrix a uv is given using the same thread of reasoning similar results for a 1 3 m and a 1 4 n are derived and all of these results can be represented by an unified form

recurrent neural network for computing outer inverse two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined each of the proposed recurrent neural networks is based on the matrix valued differential equation a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion the moore penrose inversion as well as the drazin inversion under the condition of zero initial state the application of the first approach is conditioned by the properties of the spectrum of a certain matrix the second approach eliminates this drawback though at the cost of increasing the number of matrix operations the cases corresponding to the most common generalized inverses are defined the conditions that ensure stability of the proposed neural network are presented illustrative examples present the results of numerical simulations

solving simultaneous linear equations using recurrent neural networks abstract simultaneous linear algebraic equations can be found in many mathematical model formulations in monitoring and control of dynamic systems there is often a need for solving simultaneous linear algebraic equations in real time in this paper recurrent neural networks for solving simultaneous linear algebraic equations are proposed the asymptotic stability of the proposed neural networks and solvability of simultaneous linear equations by using the neural networks are substantiated a circuit schematic for realizing the neural networks is described the results of numerical simulations are discussed via illustrative examples an extension of the recurrent neural networks for solving quadratic programming problems subject to equality constraints is also discussed

efficient reconstruction of sequences we introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type these erroneous versions are considered as outputs of repeated transmissions over a channel either a combinatorial channel defined by the maximum number of permissible errors of a given type or a discrete memoryless channel we are interested in the smallest n such that n erroneous versions always suffice to reconstruct a sequence of length n either exactly or with a preset accuracy and or with a given probability we are also interested in simple reconstruction algorithms complete solutions for combinatorial channels with some types of errors of interest in coding theory namely substitutions transpositions deletions and insertions of symbols are given for these cases simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found in general for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius a certain sufficient condition for solution of this problem is presented for a discrete memoryless channel the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within hamming distance d with error probability spl epsiv is found when d n and spl epsiv tend to 0 as n spl rarr spl infin a similar result for the continuous channel with discrete time and additive gaussian noise is also obtained

recurrent neural network approach based on the integral representation of the drazin inverse in this letter we present the dynamical equation and corresponding artificial recurrent neural network for computing the drazin inverse for arbitrary square real matrix without any restriction on its eigenvalues conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the drazin inverse are considered several illustrative examples present the results of computer simulations

new bounds and constructions for granular media coding improved lower and upper bounds on the size and the rate of grain correcting codes are presented the lower bound is gilbert varshamov like combined with a construction by gabrys et al and it improves on the previously best known lower bounds on the asymptotic rate of lceil tau n rceil grain correcting codes of length n on the interval 0 0 0668 one of the two newly presented upper bounds improves on the best known upper bounds on the asymptotic rate of lceil tau n rceil grain correcting codes of length n on the interval tau in 0 1 8 and meets the lower bound of 1 2 for tau geq 1 8 moreover in a nonasymptotic regime both upper bounds improve on the previously best known results on the largest size of t grain correcting codes of length n for certain values of n and t constructions of 1 grain correcting codes based on a partitioning technique are presented for lengths up to 18 finally a lower bound of 1 2 log 2 n on the minimum redundancy of infty grain detecting codes of length n is presented

neural network approach to computing outer inverses based on the full rank representation abstract the first intention of this paper is to emphasize equivalence between two well known general representations of outer inverses with prescribed range and null space of a given matrix two dynamical state equations corresponding to particular expressions which are included in these representations are defined within the second objective in this way two gradient based neural networks initiated by introduced dynamical equations are exploited in generating the class of outer inverses the results corresponding to most common generalized inverses are obtained in particular cases numerical results are presented

some topics on weighted moore penrose inverse weighted least squares and weighted regularized tikhonov problems it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems the classical normwise relative condition number formulas are given by higham linear algebra appl 214 1995 193 here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b moreover we consider the nearness to rank deficiency finally we give some decomposition tools and their application in the regularized weighted tikhonov problems

the weighted moore penrose inverse of modified matrices the weighted moore penrose inverse of modified matrix a cb is derived under some range conditions where cb is a full rank factorization the generalized singular value decomposition of a matrix is used to derive systematically the weighted moore penrose inverse for a modified matrix where cb need not be a full rank factorization the results extend the earlier works by various authors

complex neural network models for time varying drazin inverse two complex zhang neural network znn models for computing the drazin inverse of arbitrary time varying complex square matrix are presented the design of these neural networks is based on corresponding matrix valued error functions arising from the limit representations of the drazin inverse two types of activation functions appropriate for handling complex matrices are exploited to develop each of these networks theoretical results of convergence analysis are presented to show the desirable properties of the proposed complex valued znn models numerical results further demonstrate the effectiveness of the proposed models

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

expression for the perturbation of the weighted moore penrose inverse abstract we consider the perturbation formula for the weighted moore penrose inverse of a rectangular matrix and give an explicit expression for the weighted moore penrose inverse of a perturbed matrix under the weakest rank condition this explicit expression extends the earlier work of several authors

lee metric bch codes and their application to constrained and partial response channels shows that each code in a certain class of bch codes over gf p specified by a code length n spl les p sup m 1 and a runlength r spl les p 1 2 of consecutive roots in gf p sup m has minimum lee distance spl ges 2r for the very high rate range these codes approach the sphere packing bound on the minimum lee distance furthermore for a given r the length range of these codes is twice as large as that attainable by berlekamp s 1984 extended negacyclic codes the authors present an efficient decoding procedure based on euclid s algorithm for correcting up to r 1 errors and detecting r errors that is up to the number of lee errors guaranteed by the designed minimum lee distance 2r bounds on the minimum lee distance for r spl ges p 1 2 are provided for the reed solomon case i e when the bch code roots are in gf p the authors present two applications first lee metric bch codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and or deletion of zeros in d k constrained channels second the code construction with its decoding algorithm can be formulated over the integer ring providing an algebraic approach to correcting errors in partial response channels where matched spectral null codes are used

efficient low redundancy codes for correcting multiple deletions we consider the problem of constructing binary codes to recover from k bit deletions with efficient encoding decoding for a fixed k the single deletion case is well understood with the varshamov tenengolts levenshtein code from 1965 giving an asymptotically optimal construction with approx 2 n n codewords of length n i e at most log n bits of redundancy however even for the case of two deletions there was no known explicit construction with redundancy less than n omega 1 for any fixed k we construct a binary code with c k log n redundancy that can be decoded from k deletions in o k n log 4 n time the coefficient c k can be taken to be o k 2 log k which is only quadratically worse than the optimal non constructive bound of o k we also indicate how to modify this code to allow for a combination of up to k insertions and deletions we also note that among linear codes capable of correcting k deletions the k 1 fold repetition code is essentially the best possible

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

interval iterative methods for computing moore penrose inverse in this paper we import interval method to the iteration for computing moore penrose inverse of the full row or column rank matrix through modifying the classical newton iteration by interval method we can get better numerical results the convergence of the interval iteration is proven we also give some numerical examples to compare interval iteration with classical newton iteration

derivation of statistical energy analysis from radiative exchanges abstract this paper is concerned with the derivation of sea equations from structural ray equations rays are assumed to be uncorrelated leading to the additivity of energy inside all subsystems the energy density is the sum of a direct field from driving forces a reflected field from the boundary and a transmitted field from adjacent subsystems assuming a rain on the roof excitation and a compact shape for subsystems actual and fictitious sources on the boundary are found to be constant furthermore if the attenuation of rays during a mean free path normalized attenuation factor is light the field becomes diffuse i e homogeneous and isotropic the net exchanged power between two adjacent subsystems is then proportional to the difference of energy densities and therefore to the difference of modal energies the derived proportionality coefficient is consistent with the well known formula for coupling loss factor in terms of transmission factors these results are illustrated by a numerical simulation for a multi plate system finally the validity domains of sea and the ray theory are discussed and particularly the diffuse field assumption

an improved newton iteration for the weighted moore penrose inverse abstract in this paper we showed that newton iteration may be used to compute the weighted moore penrose inverse of an arbitrary matrix we also take advantage of several new acceleration procedures to reduce the cost of newton iteration which extends the earlier work r schreiber computing generalized inverses and eigenvalues of symmetric matrices using systolic arrays in r glowinski j l lious eds computing methods in applied science and engineering north holland amsterdam 1984 t soderstorm g w stewant on the numerical properties of an iterative method for computing the moore penrose generalized inverse siam j numer anal 11 1974 61 74

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

a note on double insertion deletion correcting codes by using a run length representation of sequences ways to determine suband supersequences are discussed this is then used in determining the number of sub and supersequences of a sequence after double insertions or deletions it is also used in creating subsequence supersequence books that are searched to find new double insertion deletion correcting code books with higher cardinalities than those already known

power flow between two continuous one dimensional subsystems a wave solution abstract the power flow between two point coupled wave bearing subsystems is determined by using a wave approach the subsystems are one dimensional in that each supports just one wave mode and are described by a reflection coefficient from which in certain circumstances the modal overlap and modal density can be defined as continuous functions of frequency the coupling element is described by a reflection coefficient and attention is focused on the case of conservative coupling time harmonic point forces or rain on the roof excitations may act on the subsystems uncertainty exists in the properties of the subsystems which are assumed to be drawn from ensembles of similar subsystems the exact power flow and the ensemble statistics are seen to depend on the magnitudes of the subsystem and coupling reflection coefficients for some particular cases the maximum and minimum power flows are examined together with the ensemble mean and geometric mean the cumulative probability distribution and the probability density function for a given power incident upon the coupling the ensemble mean power flow through the coupling is always less than that expected from a normal sea wave approach due to power re radiation the differences are largest when the damping is light and the coupling strong for a given level of excitation power re radiation effects are also apparent for the ensemble mean power flow from an infinite subsystem to a finite subsystem in the opposite case the ensemble mean power flow from a finite to an infinite subsystem is greater than that between two infinite subsystems due to power re injection the differences are largest for light damping and for weak coupling finally the case where there is uncertainty in one subsystem only is considered

efficient exceedance probability computation for randomly uncertain nonlinear structures with periodic loading abstract a method is developed to compute low level response amplitude exceedance probabilities associated with uncertain nonlinear structures with random parameters and deterministic periodic forcing emphasis is focused on accurate and efficient computation in the tails of the exceedance probability distribution function associated with the largest possible response of one displacement variable for unspecified forcing frequency and normally distributed parameters this gives a measure of system reliability when a large amplitude response exceedance of a specified threshold is designated as the mode of failure the method exploits the first order reliability method form in which the failure surface is constructed via the harmonic balance method hbm this combined approach is tested on a duffing oscillator with harmonic forcing and up to three uncertain parameters for which the frequency of multiple solution maximum amplitude is found directly and the probability computed via the hasofer lind reliability index the accuracy of the proposed hbm form in the tails of the amplitude exceedance probability is shown for the duffing example to be acceptably accurate whereas the efficiency is shown to be around 1000 times faster than direct integration and around 200 times faster than monte carlo simulation

comparison of an integral equation on energy and the ray tracing technique in room acoustics this paper deals with a comparison of two room acoustic models the first one is an integral formulation stemming from power balance and the second is the ray tracing technique with a perfectly diffuse reflection law the common assumptions to both models are the uncorrelated wave hypothesis and the perfectly diffuse reflection law the latter allows the use of these methods for nondiffuse fields beyond the validity domain of sabine s formula comparisons of numerical simulations performed with the softwares rayon and ceres point out that these results are close to each other and finally a formal proof is proposed showing that both methods are actually equivalent

on the number of subsequences when deleting symbols from a string we consider the problem of finding the number of subsequences when deleting symbols from a string we present a framework to find closed form expressions for the number of subsequences and use it to prove the first explicit formulas for and the first formulas for nonbinary alphabets we also present the first efficient algorithm to compute the number of subsequences for any string number of deletions and alphabet size

a multiple insertion deletion correcting code for run length limited sequences a code construction is proposed to add a multiple insertion deletion error correcting capability to a run length limited sequence the codewords of this code are themselves run length limited the insertion deletion correcting capability is achieved by requiring several weighted sums of run lengths in the codewords to satisfy certain congruences modulo primes the construction is similar to the number theoretic code proposed by dolecek and anantharam which can correct multiple repetition errors or equivalently multiple insertions of zeros it is shown that if the codewords in this code are run length limited then the code is capable of correcting both insertions and deletions of zeros and ones an algorithm is proposed for decoding over a multiple insertion deletion channel following the work of dolecek and anantharam a systematic encoding method is also proposed for the codes furthermore it is shown that the proposed construction has a higher rate asymptotically than the helberg code which is unconstrained in terms of run lengths even though our construction has the additional run length constraints the need for run length limited codes that can correct insertion deletion errors is motivated by bit patterned media for magnetic recording

uncertainty propagation in sea for structural acoustic coupled systems with non deterministic parameters abstract considering limited available information on uncertainties in structural acoustic coupled systems two methods namely the vertex method and the legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the statistical energy analysis sea approach for the vertex method an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded which is well suited for an interval input vector in arbitrary dimension instead of calculating the extremum of the response of interest a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the legendre orthogonal polynomial approximation following the theoretical analysis of the accuracy and efficiency of the proposed methods their validation is performed by one numerical example and two applications

pcr algorithm for parallel computing minimum norm t least squares s solution of inconsistent linear equations this paper presents a new highly parallel algorithm for computing the minimum norm t least squares s solution of inconsistent linear equations ax b email protected r m x n r email protected r a by this algorithm the solution x a s t b is obtained in t 1 m 1 log 2m n 6 log 2 n r 1 log 2m log 2n r 1 log 2n steps with p mn processors when m 2 n 1 and with p 2n n 1 processors otherwise

deletion codes in the high noise and high rate regimes the noise model of deletions poses significant challenges in coding theory with basic questions like the capacity of the binary deletion channel still being open in this paper we study the harder model of worst case deletions with a focus on constructing efficiently decodable codes for the two extreme regimes of high noise and high rate specifically we construct polynomial time decodable codes with the following tradeoffs for any varepsilon 0 1 codes that can correct a fraction 1 varepsilon of deletions with rate mathop mathrm poly nolimits varepsilon over an alphabet of size mathop mathrm poly nolimits 1 varepsilon 2 binary codes of rate 1 tilde o sqrt varepsilon that can correct a fraction varepsilon of deletions and 3 binary codes that can be list decoded from a fraction 1 2 varepsilon of deletions with rate mathop mathrm poly nolimits varepsilon this paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet the above mentioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors

fixed length lossy compression in the finite blocklength regime this paper studies the minimum achievable source coding rate as a function of blocklength n and probability that the distortion exceeds a given level d tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength for stationary memoryless sources with separable distortion the minimum rate achievable is shown to be closely approximated by r d v d n q 1 where r d is the rate distortion function v d is the rate dispersion a characteristic of the source which measures its stochastic variability and q 1 is the inverse of the standard gaussian complementary cumulative distribution function

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

list decoding of insertions and deletions list decoding of insertions and deletions in the levenshtein metric is considered the levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other in this paper a johnson like upper bound on the maximum list size when list decoding in the levenshtein metric is derived this bound depends only on the length and minimum levenshtein distance of the code the length of the received word and the alphabet size it shows that polynomial time list decoding beyond half the levenshtein distance is possible for many parameters further we also prove a lower bound on list decoding of deletions with the well known binary varshamov tenengolts codes which shows that the maximum list size grows exponentially with the number of deletions finally an efficient list decoding algorithm for two insertions deletions with vt codes is given this decoder can be modified to a polynomial time list decoder of any constant number of insertions deletions

displacement structure of weighted pseudoinverses estimates for the rank of a m n v ua m n and more general displacement of a m n are presented where a m n is the weighted pseudoinverse of a matrix a the results are applied to the close to toeplitz close to vandermonde and close to cauchy matrices we extend the results due to heinig and hellinger in 1994

an improvement to levenshtein s upper bound on the cardinality of deletion correcting codes we consider deletion correcting codes over a q ary alphabet it is well known that any code capable of correcting s deletions can also correct any combination of s total insertions and deletions to obtain asymptotic upper bounds on code size we apply a packing argument to channels that perform different mixtures of insertions and deletions even though the set of codes is identical for all of these channels the bounds that we obtain vary prior to this paper only the bounds corresponding to the all insertion case and the all deletion case were known we recover these as special cases the bound from the all deletion case due to levenshtein has been the best known for more than 45 years our generalized bound is better than levenshtein s bound whenever the number of deletions to be corrected is larger than the alphabet size

on the computation of weighted moore penrose inverse using a high order matrix method we propose a matrix method for calculating generalized weighted moore penrose inverse numerically it is proved that the scheme is convergent with high order via theoretical analysis some tests are also presented to show the applicability and the superiority of the new method

two finite time convergent zhang neural network models for time varying complex matrix drazin inverse abstract this paper concerns the computation of the drazin inverse of a complex time varying matrix based on two zhang functions constructed from two limit representations of the drazin inverse we present two complex zhang neural network znn models with the li activation function for computing the drazin inverse of a complex time varying square matrix we prove that our znn models globally converge in finite time in addition upper bounds of the convergence time are derived analytically via the lyapunov theory our simulation results verify the theoretical analysis and demonstrate the superiority of our znn models over the gradient based gnn models

recurrent neural network for computing the w weighted drazin inverse abstract two gradient based recurrent neural networks rnns for computing the w weighted drazin inverse of a real rectangular matrix are proposed and considered usage of the first rnn is limited by a specific constraint on the spectrum of a certain matrix the second rnn is usable without restrictions the stability of the recurrent neural networks as well as their convergence are considered numerical examples are given to show the efficiency of the proposed neural networks

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

linear programming based converses for finite blocklength lossy joint source channel coding a linear programming lp based framework is presented for obtaining converses for finite blocklength lossy joint source channel coding problems the framework applies for any loss criterion generalizes certain previously known converses and also extends to multi terminal settings the finite blocklength problem is posed equivalently as a nonconvex optimization problem and using a lift and project like method a close but tractable lp relaxation of this problem is derived lower bounds on the original problem are obtained by the construction of feasible points for the dual of the lp relaxation a particular application of this approach leads to new converses which recover and improve on the converses of kostina and verdu for finite blocklength lossy joint source channel coding and lossy source coding for finite blocklength channel coding the lp relaxation recovers the converse of polyanskiy poor and verdu and leads to a new improvement on the converse of wolfowitz showing thereby that our lp relaxation is asymptotically tight with increasing blocklengths for channel coding lossless source coding and joint source channel coding with the excess distortion probability as the loss criterion using a duality based argument a new converse is derived for finite blocklength joint source channel coding for a class of source channel pairs employing this converse the lp relaxation is also shown to be tight for all blocklengths for the minimization of the expected average symbolwise hamming distortion of a q ary uniform source over a q ary symmetric memoryless channel for any q in mathbb n the optimization formulation and the lift and project method are extended to networked settings and demonstrated by obtaining an improvement on a converse of zhou et al for the successive refinement problem for successively refinable source distortion measure triplets

extension and generalization properties of the weighted minkowski inverse in a minkowski space for an arbitrary matrix the weighted minkowski inverse a m n m n m related to the positive definite matrices m m m and n m n of an arbitrary matrix a m m n including singular and rectangular is one of the important generalized inverses for solving matrix equations in minkowski space in this paper the results are introduced in the following three ways first we establish some new and attractive properties of the weighted minkowski inverse a m n in a minkowski space second new representations and conditions for the continuity of the weighted minkowski inverse a m n in a minkowski space are discussed finally some illustrated counterexamples are also studied to show that some well known properties of the weighted moore penrose inverse a m n in a hilbert space h are not valid in the minkowski inverse a m n in a minkowski space

study on the validity region of energy finite element analysis abstract the validity domain of energy finite element analysis efea is studied in this paper the validity region and criterion of efea are studied theoretically from the formation of reverberant plane wave field one of the main assumptions of efea the studies are acquired by virtue of the equation of radiative energy transfer method a similar wave method that can express the direct field and its conversion relationship with reverberant field exactly the result shows that the sea criterion of diffuse field derived by le bot can be used as a good indicator for the efea validity numerical simulations on a rectangular plate with different physical parameters are applied to validate the criterion the validity region and the diagrams of validity of efea are assessed and discussed some noteworthy conclusions about efea are drawn

finite blocklength converse bounds for quantum channels we derive upper bounds on the rate of transmission of classical information over quantum channels by block codes with a given blocklength and error probability for both entanglement assisted and unassisted codes in terms of a unifying framework of quantum hypothesis testing with restricted measurements our bounds do not depend on any special property of the channel such as memorylessness and generalize both a classical converse of polyanskiy poor and verdu as well as a quantum converse of renner and wang and have a number of desirable properties in particular our bound on entanglement assisted codes is a semidefinite program and for memoryless channels its large blocklength limit is the well known formula for entanglement assisted capacity due to bennett shor smolin and thapliyal

asymmetric lee distance codes for dna based storage we introduce a new family of codes termed asymmetric lee distance ald codes designed to correct errors arising in dna based storage systems and systems with parallel string transmission protocols ald codes are defined over a quaternary alphabet and analyzed in this particular setting but the derived results hold for other alphabet sizes as well our technical contributions are twofold first we derive upper bounds on the size of the codes under the ald metric based on linear programming techniques second we propose a number of code constructions which imply lower bounds

the representation of generalized inverse at s 2 3 and its applications this paper presents the explicit expression for matrix right symmetry factor with prescribed rang and null space moreover the explicit expression for generalized inverse a t s 2 3 which is a 2 3 inverse of a having the prescribed rang t and null space s is derived as an application two numerical examples are given

minimal properties of moore penrose inverses abstract let ax y be consistent let x0 gy be any minimum norm solution satisfying ag ag and let a be the moore penrose inverse of a it is shown that g a for any in a class containing the unitarily invariant matrix norms the conditioning of the system ax y is studied via condition numbers c a g it is shown that c a g c a a for every moreover bounds on c a g are given in terms of singular values parallel results are found when a and g are symmetric with applications to linear models of less than full rank

successive matrix squaring algorithm for parallel computing the weighted generalized inverse amn we derive a successive matrix squaring sms algorithm to approximate the weighted generalized inverse which can be expressed in the form of successive squaring of a composite matrix t given an m by n matrix a with m n we show that the weighted generalized inverse of a can be computed in parallel time ranging from o log n to o log 2n provided that there are enough processors to support matrix multiplication in time o log n

a novel iterative method for computing generalized inverse in this letter we propose a novel iterative method for computing generalized inverse based on a novel kkt formulation the proposed iterative algorithm requires making four matrix and vector multiplications at each iteration and thus has low computational complexity the proposed method is proved to be globally convergent without any condition furthermore for fast computing generalized inverse we present an acceleration scheme based on the proposed iterative method the global convergence of the proposed acceleration algorithm is also proved finally the effectiveness of the proposed iterative algorithm is evaluated numerically

a family of higher order convergent iterative methods for computing the moore penrose inverse abstract in this paper we extend the iterative method for computing the inner inverse of a matrix proposed in li and li w g li z li a family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non square matrix applied mathematics and computation 215 2010 3433 3442 to compute the moore penrose inverse of a matrix and show that the generated sequence converges to the moore penrose inverse of a matrix in a higher order the performance of the method is tested on some randomly generated matrices

construction of fixed length insertion deletion correcting runlength limited codes an algorithm is presented for the construction of fixed length insertion deletion correcting runlength limited rll codes in this construction binary d k constrained codewords are generated by codewords of a q ary lee metric based code it is shown that this new construction always yields better codes than known constructions the use of a q ary lee 1987 metric code q odd is based on the assumption that an error insertion deletion or peak shift has maximal size q 1 2 it is shown that a decoding algorithm for the lee metric code can be extended so that it can also be applied to insertion deletion correcting rll codes furthermore such an extended algorithm can also correct some error patterns containing errors of size more than q 1 2 as a consequence if s denotes the maximal size of an error then in some cases the alphabet size of the generating code can be s 1 instead of 2 spl middot s 1

nonasymptotic upper bounds for deletion correcting codes explicit nonasymptotic upper bounds on the sizes of multiple deletion correcting codes are presented in particular the largest single deletion correcting code for q ary alphabet and string length is shown to be of size at most qn q q 1 n 1 an improved bound on the asymptotic rate function is obtained as a corollary upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length and this idea is demonstrated by application to sets of run length limited strings the problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph this problem is formulated as an integer linear program the upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program the nonasymptotic bounds derived imply the known asymptotic bounds of levenshtein and tenengolts and improve on known nonasymptotic bounds numerical results support the conjecture that in the binary case the varshamov tenengolts codes are the largest single deletion correcting codes

wave derivation of the energetics of driven coupled one dimensional dynamic systems abstract the equation of statistical energy analysis was originally derived on the basis of modal concepts in this paper a derivation of the equation on the basis of wave concepts is outlined although the new formalism is limited at present to the analysis of coupled one dimensional dynamic systems it carries details that were a priori forfeited by the previous formalism the wave approach to sea as developed in the paper reveals a formalism that can be naturally used for the analysis of net power and intensity in coupled dynamic systems this aspect of the formalism is briefly discussed

energy transfer for high frequencies in built up structures this paper is concerned with the prediction of vibrations at high frequencies in built up structures in the high frequency range the dynamical behaviour of structures is driven by energy transfer rather than modal aspects the description of systems in terms of energy is thus well suited and leads to closed form equations of an integral type on energy variables the numerical solution of these equations is achieved with appropriate software called ceres and the results are compared with experimental measurements on a multi plate structure

neural network for computing pseudoinverses and outer inverses of complex valued matrices we propose two continuous time neural networks for computing generalized inverses of complex valued matrices with rank deficient cases the first of them is applicable in the pseudoinverse computation and the second one is applicable in construction of outer inverses the proposed continuous time neural networks have a low complexity of implementation and they are proved to be globally convergent without any condition compared with the existing algorithms for computing the pseudoinverse and outer inverses of matrices the global convergence of the proposed continuous time neural networks is analyzed in the complex domain effectiveness of the proposed continuous time neural networks is evaluated numerically via examples

polynomial time low density parity check codes with rates very close to the capacity of the q ary random deletion channel for large q we demonstrate polynomial time deletion codes based on the verification based decoding paradigm that come arbitrarily close to capacity the random deletion channel takes n symbols from a q ary alphabet and each symbol is deleted independently with probability p taking advantage of recent improvements on the results of luby and mitzenmacher for verification based decoding by shokrollahi and wang we show how to design for any epsi 0 and sufficiently large n and q deletion codes with the following properties the rate is 1 p 1 epsi the failure probability is no 1 epsi2 q and the computational complexity for encoding and decoding is no 1 epsi2 log q we also extend these schemes to obtain the same results even if the undeleted symbols are also transposed arbitrarily and if a sufficiently small number of random symbols are inserted

validity diagrams of statistical energy analysis abstract this paper is concerned with the validity domain of statistical energy analysis sea which is defined in terms of four criteria the mode count n and the modal overlap m must be high the normalized attenuation factor m and the coupling strength must be small the application of dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of sea must be delimited this domain is discussed on the basis of geometry of the surfaces delimiting it the diagrams of validity of sea are introduced and discussed a numerical simulation on a couple of rectangular plates coupled along one edge illustrates the theoretical approach

the representation and approximation for the weighted moore penrose inverse we present a unified representation theorem for the weighted moore penrose inverse specific expressions and computational procedures for the weighted moore penrose inverse can be uniformly derived

a linear program for the finite block length converse of polyanskiy poor verdu via nonsignaling codes motivated by recent work on entanglement assisted codes for sending messages over classical channels the larger easily characterized class of nonsignaling codes is defined analyzing the optimal performance of these codes yields an alternative proof of the finite block length converse of polyanskiy poor and verdu and shows that they achieve this converse this provides an explicit formulation of the converse as a linear program which has some useful features for discrete memoryless channels it is shown that nonsignaling codes attain the channel capacity with zero error probability if and only if the dispersion of the channel is zero

guess check codes for deletions insertions and synchronization we consider the problem of constructing codes that can correct delta deletions occurring in an arbitrary binary string of length n bits varshamov tenengolts vt codes dating back to 1965 are zero error single deletion delta 1 correcting codes and have an asymptotically optimal redundancy finding similar codes for delta geq 2 deletions remains an open problem in this paper we relax the standard zero error i e worst case decoding requirement by assuming that the positions of the delta deletions or insertions are independent of the code word our contribution is a new family of explicit codes that we call guess check gc codes that can correct with high probability up to a constant number of delta deletions or insertions gc codes are systematic and have deterministic polynomial time encoding and decoding algorithms we also describe the application of gc codes to file synchronization

triple reverse order law for weighted generalized inverses in this paper we present a new triple reverse order law of weighted generalized inverses which does not involve the calculation of any generalized inverse our result is an important improvement of the existing reverse order law of weighted generalized inverse

asymptotically good codes correcting insertions deletions and transpositions we present simple polynomial time encodable and decodable codes which are asymptotically good for channels allowing insertions deletions and transpositions as a corollary they achieve exponential error probability in a stochastic model of insertion deletion