entrywise relative perturbation bounds for exponentials of essentially non negative matrices a real square matrix is said to be essentially non negative if all of its off diagonal entries are non negative we establish entrywise relative perturbation bounds for the exponential of an essentially non negative matrix our bounds are sharp and contain a condition number that is intrinsic to the exponential function as an application we study sensitivity of continuous time markov chains

time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

the randomization technique as a modeling tool and solution procedure for transient markov processes we present a randomization procedure for computing transient solutions to discrete state space continuous time markov processes this procedure computes transient state probabilities it is based on a construction relating a continuous time markov process to a discrete time markov chain modifications and extensions of the randomization method allow for computation of distributions of first passage times and sojourn times in markov processes and also the computation of expected cumulative occupancy times and expected number of events occurring during a time interval several implementations of the randomization procedure are discussed in particular we present an implementation for a general class of markov processes that can be described in terms of state space s event set e rate vectors r and target vectors t abbreviated as sert this general approach can handle systems whose state spaces are quite large if they have sparse generators

the pade method for computing the matrix exponential we analyze the pade method for computing the exponential of a real matrix more precisely we study the roundoff error introduced by the method in the general case and in three special cases 1 normal matrices 2 essentially nonnegative matrices aij 0 i j 3 matrices a such that a d 1 bd with d diagonal and b essentially nonnegative for these special matrices it turns out that the pade method is stable finally we compare the ward upper bound with our results and show that our bounds are generally tighter

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

computing exponentials of essentially non negative matrices entrywise to high relative accuracy a real square matrix is said to be essentially non negative if all of its off diagonal entries are non negative it has recently been shown that the exponential of an essentially non negative matrix is determined entrywise to high relative accuracy by its entries up to a condition number intrinsic to the exponential function numer math 110 2008 pp 393 403 thus the smaller entries of the exponential may be computed to the same relative accuracy as the bigger entries this paper develops algorithms to compute exponentials of essentially non negative matrices entrywise to high relative accuracy

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

wavelets generated by using discrete singular convolution kernels this paper explores the connection between wavelet methods and an efficient computational algorithm the discrete singular convolution dsc many new dsc kernels are constructed and they are identified as wavelet scaling functions two approaches are proposed to generate wavelets from dsc kernels two well known examples the canny filter and the mexican hat wavelet are found to be special cases of the present dsc kernel generated wavelets a family of wavelet generators proposed in this paper are found to form an infinite dimensional lie group which has an invariant subgroup of translation and dilation if dsc kernels form an orthogonal system they are found to span a wavelet subspace in a multiresolution analysis

petrov galerkin methods for linear volterra integro differential equations in this paper we study a class of petrov galerkin solutions that have global optimal convergence rates for linear volterra integro differential equations these solutions also possess certain local and global superconvergence asymptotic expansions of the errors in these solutions are established which can be used to form higher order approximations by richardson extrapolation and defect corrections several postprocessing techniques are introduced to enhance these solutions as by products these higher order numerical approximations can be used to generate a posteriori error estimators representative numerical results are also provided

componentwise accurate fluid queue computations using doubling algorithms markov modulated fluid queues are popular stochastic processes frequently used for modelling real life applications an important performance measure to evaluate in these applications is their steady state behaviour which is determined by the stationary density computing it requires solving a nonsymmetric m matrix algebraic riccati equation and indeed computing the stationary density is the most important application of this class of equations xue et al numer math 120 671 700 2012 provided a componentwise first order perturbation analysis of this equation proving that the solution can be computed to high relative accuracy even in the smallest entries and suggested several algorithms for computing it an important step in all proposed algorithms is using so called triplet representations which are special representations for m matrices that allow for a high accuracy variant of gaussian elimination the gth like algorithm however triplet representations for all the m matrices needed in the algorithm were not found explicitly this can lead to an accuracy loss that prevents the algorithms from converging in the componentwise sense in this paper we focus on the structured doubling algorithm the most efficient among the proposed methods in xue et al and build upon their results providing i explicit and cancellation free expressions for the needed triplet representations allowing the algorithm to be performed in a really cancellation free fashion ii an algorithm to evaluate the final part of the computation to obtain the stationary density and iii a componentwise error analysis for the resulting algorithm the first explicit one for this class of algorithms we also present numerical results to illustrate the accuracy advantage of our method over standard normwise accurate algorithms

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

discrete singular convolution and its application to the analysis of plates with internal supports part 1 theory and algorithm this paper presents a novel computational approach the discrete singular convolution dsc algorithm for analysing plate structures the basic philosophy behind the dsc algorithm for the approximation of functions and their derivatives is studied approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels unified features of the dsc algorithm for solving differential equations are explored it is demonstrated that different methods of implementation for the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point the use of the algorithm for the vibration analysis of plates with internal supports is discussed detailed formulation is given to the treatment of different plate boundary conditions including simply supported elastically supported and clamped edges this work paves the way for applying the dsc approach in the following paper to plates with complex support conditions which have not been fully addressed in the literature yet

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

communicability in complex networks we propose a new measure of the communicability of a complex network which is a broad generalization of the concept of the shortest path according to the new measure most of the real world networks display the largest communicability between the most connected popular nodes of the network assortative communicability there are also several networks with the disassortative communicability where the most popular nodes communicate very poorly to each other using this information we classify a diverse set of real world complex systems into a small number of universality classes based on their structure dynamic correlation in addition the new communicability measure is able to distinguish finer structures of networks such as communities into which a network is divided a community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network

pade approximation for the exponential of a block triangular matrix abstract in this work we obtain improved error bounds for pade approximations to e a when a is block triangular as a result improved scaling strategies ensue which avoid some common overscaling difficulties

non smooth initial data error estimates with the weight norms for the linear finite element method of parabolic partial differential equations we analyze the galerkin approximation of a parabolic equation for the second order continuous time method with initial data only in l2 we prove an optimal order error estimate with weight norm our analysis uses elementary energy techniques and spectral representations and weight function techniques the same techniques are shown to give an analysis of the galerkin method time discretization

quadrature rule based bounds for functions of adjacency matrices bounds for entries of matrix functions based on gauss type quadrature rules are applied to adjacency matrices associated with graphs this technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities estrada index subgraph centrality communicability that describe properties of networks

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

communicability betweenness in complex networks abstract betweenness measures provide quantitative tools to pick out fine details from the massive amount of interaction data that is available from large complex networks they allow us to study the extent to which a node takes part when information is passed around the network nodes with high betweenness may be regarded as key players that have a highly active role at one extreme betweenness has been defined by considering information passing only through the shortest paths between pairs of nodes at the other extreme an alternative type of betweenness has been defined by considering all possible walks of any length in this work we propose a betweenness measure that lies between these two opposing viewpoints we allow information to pass through all possible routes but introduce a scaling so that longer walks carry less importance this new definition shares a similar philosophy to that of communicability for pairs of nodes in a network which was introduced by estrada and hatano e estrada n hatano phys rev e 77 2008 036111 having defined this new communicability betweenness measure we show that it can be characterized neatly in terms of the exponential of the adjacency matrix we also show that this measure is closely related to a frechet derivative of the matrix exponential this allows us to conclude that it also describes network sensitivity when the edges of a given node are subject to infinitesimally small perturbations using illustrative synthetic and real life networks we show that the new betweenness measure behaves differently to existing versions and in particular we show that it recovers meaningful biological information from a protein protein interaction network

superconvergence of numerical solutions to weakly singular volterra integro differential equations we discuss the application of a class of spline collocation methods to first order volterra integro differential equations vides which contain a weakly singular kernel t s with 0 1 it will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes the grading exponents of graded meshes used are not greater thanm the polynomial degree which is independent of this is in contrast to the theories of spline collocation methods for volterra or fredholm integral equation of the second kind numerical examples are given to illustrate the theoretical results

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

chaotic motion of a parametrically forced simple pendulum abstract this paper describes the behavior of a simple pendulum with support in a steady and rather slow circular motion in a vertical plane the governing equation of this model is nonlinear and its motion is about the direction of gravity in the equation of motion there are three parameters which are related to the radius and angular velocity of the circular motion the length of the pendulum and the damping constant for some combinations of these parameters this model is found to have chaotic motion by numerical integration this paper not only draws the time series and poincare map to show the characteristics of regular and chaotic motions but also explores the feasibility of using the amplitude probability distribution by numerical method to diagnose whether the motion is chaotic the effect of geometry and damping constant on the motion of the system is investigated furthermore whether the poincare map and amplitude probability distribution depend sensitively on the initial conditions is also studied

lagrange distributed approximating functionals lagrange distributed approximating functionals ldafs are proposed as the basis for a new collocation type method for accurately approximating functions and their derivatives both on and off discrete grids example applications are presented to illustrate the use of ldafs for solving the schr umlt o dinger equation and fokker planck equation ldafs are constructed by combining the daf concept with the lagrange interpolation scheme copyright ital 1997 ital the american physical society

quasi wavelet based numerical method for a class of partial integro differential equation abstract in this paper we study the numerical solution of initial boundary problem for a class of partial integro differential equations the quasi wavelet method is proposed to handle the spatial derivatives while the forward euler method is used to discretize the temporal derivatives detailed discrete schemes are given and some numerical experiments are included to demonstrate the effectiveness of the discrete technique the comparisons of the present numerical results with the exact analytical solutions show that the quasi wavelet based numerical method has distinctive local property and can achieve accurate results

on the naturality of the iteration and recursive specifications abstract in 5 the naturality of level of sentence control structures concatenation conditional and iterative recursion is given in the present article we emphasize from a categorical semantics point of view following 11 the iteration and some of its relations with general recursion at the unit level

accurate computation of the smallest eigenvalue of a diagonally dominant m matrix if each off diagonal entry and the sum of each row of a diagonally dominant m matrix are known to certain relative accuracy then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers in this paper we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms

computing the exponential of large block triangular block toeplitz matrices encountered in fluid queues the erlangian approximation of markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block triangular block toeplitz structure to this end we propose some algorithms which exploit the toeplitz structure and the properties of generators such algorithms allow us to compute the exponential of very large matrices which would otherwise be untreatable with standard methods we also prove interesting decay properties of the exponential of a generator having a block triangular block toeplitz structure