some recent variations on the expected number of distinct sites visited by an n step random walk abstract asymptotic forms for the expected number of distinct sites visited by an n step random walk being calculable for many random walks have been used in a number of analyses of physical models we describe three recent extensions of the problem the first replacing the single random walker by n random walkers the second to the study of a random walk in the presence of a trapping site and the third to a random walk in the presence of a trapping hyperplane

relating p adic eigenvalues and the local smith normal form abstract conditions are established under which the p adic valuations of the invariant factors diagonal entries of the smith form of an integer matrix are equal to the p adic valuations of the eigenvalues it is then shown that this correspondence is the typical case for most matrices density counts are given for when this property holds as well as easy transformations to this typical case

pirock a swiss knife partitioned implicit explicit orthogonal runge kutta chebyshev integrator for stiff diffusion advection reaction problems with or without noise a partitioned implicit explicit orthogonal runge kutta method pirock is proposed for the time integration of diffusion advection reaction problems with possibly severely stiff reaction terms and stiff stochastic terms the diffusion terms are solved by the explicit second order orthogonal chebyshev method rock2 while the stiff reaction terms solved implicitly and the advection and noise terms solved explicitly are integrated in the algorithm as finishing procedures it is shown that the various coupling between diffusion reaction advection and noise can be stabilized in the pirock method the method implemented in a single black box code that is fully adaptive provides error estimators for the various terms present in the problem and requires from the user solely the right hand side of the differential equation numerical experiments and comparisons with existing chebyshev methods imex methods and partitioned methods show the efficiency and flexibility of our new algorithm

eigenvalues and the smith normal form abstract we compare the smith normal form snf over the integers of an integral nonsingular matrix with its spectrum when its eigenvalues are integers our results include tight bounds on the size of the largest element of the snf when the matrix is diagonalizable with nonzero integer eigenvalues with no assumptions on the diagonalizing matrices

wavelet based iterative methods for a class of 2d partial integro differential equations abstract in this paper an iterative method based on quasilinearization is presented to solve a class of two dimensional partial integro differential equations that arise in nuclear reactor models and population models two different approaches based on haar and legendre wavelets are studied to develop numerical methods in the first approach time domain is approximated with the help of forward finite difference approach in the second approach both time as well as space domains are approximated by wavelets appropriate examples are solved using these methods and the obtained results are compared with the methods available in the recent literature

convergence properties of the runge kutta chebyshev method the runge kutta chebyshev method is ans stage runge kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations method of lines the method possesses an extended real stability interval with a length proportional tos 2 the method can be applied withs arbitrarily large which is an attractive feature due to the proportionality of withs 2 the involved stability property here is internal stability internal stability has to do with the propagation of errors over the stages within one single integration step this internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space time grid refinement for a model class of linear problems we prove convergence under the sole condition that the necessary time step restriction for stability is satisfied these error bounds are valid for anys and independent of the stiffness of the problem numerical examples are given to illustrate the theoretical results

multiparticle trapping problem in the half line abstract a variation of rosenstock s trapping model in which n independent random walkers are all initially placed upon a site of a one dimensional lattice in the presence of a one sided random distribution with probability c of absorbing traps is investigated the probability survival probability n t that no random walker is trapped by time t for n 1 is calculated by using the extended rosenstock approximation this requires the evaluation of the moments of the number sn t of distinct sites visited in a given direction up to time t by n independent random walkers the rosenstock approximation improves when n increases working well in the range dt ln 2 1 c ln n d being the diffusion constant the moments of the time lifetime before any trapping event occurs are calculated asymptotically too the agreement with numerical results is excellent

volume of the domain visited by n spherical brownian particles the average value and variance of the volume of the domain visited in timet byn spherical brownian particles starting initially at the same point are presented as quadratures of the solutions of simple diffusion problems of the survival of a point brownian particle in the presence of one and two spherical traps as an illustration explicit time dependences are obtained for the average volume in one and three dimensions

a note on lattice random walks with an excluded point we derive some asymptotic results for the moments of displacement of a random walk on a lattice with a single absorbing point in one dimension we are able to find an asymptotic expression for the probability distribution of random walks that have not been trapped

order statistics of rosenstock s trapping problem in disordered media the distribution of times t j n elapsed until the first j independent random walkers from a set of n ensuremath gg 1 all starting from the same site are trapped by a quenched configuration of traps randomly placed on a disordered lattice is investigated in doing so the cumulants of the distribution of the territory explored by n independent random walkers s n t and the probability ensuremath phi n t that no particle of an initial set of n is trapped by time t are considered simulation results for the two dimensional incipient percolation aggregate show that the ratio between the nth cumulant and the nth moment of s n t is for large n i very large in comparison with the same ratio in euclidean media and ii almost constant the first property implies that in contrast with euclidean media approximations of the order higher than the standard zeroth order rosenstock approximation are required to provide a reasonable description of the trapping order statistics fortunately the second property which has a geometric origin can be exploited to build these higher order rosenstock approximations simulation results for the two dimensional incipient percolation aggregate confirm the predictions of our approach

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

matrices over rings of algebraic integers abstract for matrices with algebraic integer entries this paper studies i completions with prescribed determinant ii the hermite triangular form iii the schur triangular form iv the smith diagonal form and v links connecting eigenvalues and smith invariants

smith forms for adjacency matrices of circulant graphs we calculate the smith normal form of the adjacency matrix of each of the following graphs or their complements or both complete graph cycle graph square of the cycle power graph of the cycle distance matrix graph of cycle andrasfai graph doob graph cocktail party graph crown graph prism graph mobius ladder the proofs operate by finding the abelianization of a cyclically presented group whose relation matrix is column equivalent to the required adjacency matrix

solving parabolic integro differential equations by an explicit integration method vasudeva murthy a s and j g verwer solving parabolic integro differential equations by an explicit integration method journal of computational and applied mathematics 39 1992 121 132 the subject of this note is the numerical integration in time of nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain the integral terms may appear in the boundary conditions and or in the governing partial differential equation itself these terms generally complicate the application of standard methods our purpose here is to draw attention to an explicit method the runge kutta chebyshev method whose application remains straightforward in many cases of practical interest we discuss two numerical examples to illustrate this a comparison is made respectively with the implicit bdf code dassl and with the keller box scheme

application of the adomian decomposition method for two dimensional parabolic equation subject to nonstandard boundary specifications in this work we are concerned with the solution of a nonlocal boundary value problem an approach is presented for solving the two dimensional parabolic partial differential equation subject to integral boundary specifications the main objective is to propose an alternative method of solution one not based on finite difference methods or finite element schemes or spectral techniques the aim of the present paper is to investigate the application of the adomian decomposition method for solving the two dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions the adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques the introduction of this idea as will be discussed not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume the solutions will be handle more easily quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions in this approach the solution is found in the form of a convergent power series with easily computed components the adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization to give a clear overview of the methodology we have selected illustrative example

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

random walks on lattices the problem of visits to a set of points revisited a general method is outlined for calculating the statistical properties of the number of visits to a set of points in a random walk in illustrative examples known results and new results are easily derived

numerical procedures for a boundary value problem with a non linear boundary condition numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial boundary value problem for the two dimensional parabolic partial differential equation pde with a non linear boundary condition this class of parabolic pdes plays a very important role in many branches of science and engineering the non linear condition is in the form of a double integral giving the specification of mass in the solution domain not only the problem has both neumann and dirichlet boundary conditions but the dirichlet boundary condition is in a non standard form while sharing some common features with the one dimensional models the solution of two dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of one dimensional case due to the structure of the boundary conditions a reduced one dimensional parabolic equation for the new unknown v y t email protected 1 0u x y t dx can be formulated the resulting problem has a non local boundary condition the new two dimensional parabolic pde with neumann s boundary conditions will be solved numerically by using the method of lines semi discretization approach the space derivatives in the pde are approximated by finite difference replacements the solution of the resulting system of first order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function the accuracy in time is controlled by choosing several subdiagonal pade approximants to replace this matrix exponential term numerical techniques are developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem from the literature the article concludes with the results of some numerical experiments

the spectral methods for parabolic volterra integro differential equations in this paper we study the numerical solutions to parabolic volterra integro differential equations in one dimensional bounded and unbounded spatial domains in a bounded domain the given parabolic volterra integro differential equation is converted to two equivalent equations then a legendre collocation method is used to solve them and finally a linear algebraic system is obtained for an unbounded case we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain in both cases some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

constructably laplacian integral graphs a graph is laplacian integral if the spectrum of its laplacian matrix consists entirely of integers we consider the class of constructably laplacian integral graphs those graphs that be constructed from an empty graph by adding a sequence of edges in such a way that each time a new edge is added the resulting graph is laplacian integral we characterize the constructably laplacian integral graphs in terms of certain forbidden vertex induced subgraphs and consider the number of nonisomorphic laplacian integral graphs that can be constructed by adding a suitable edge to a constructably laplacian integral graph we also discuss the eigenvalues of constructably laplacian integral graphs and identify families of isospectral nonisomorphic graphs within the class