numerical study of fisher s reaction diffusion equation by the sinc collocation method fisher s equation which describes a balance between linear diffusion and nonlinear reaction or multiplication is studied numerically by the sinc collocation method the derivatives and integrals are replaced by the necessary matrices and a system of algebraic equations is obtained to approximate solution of the problem the error in the approximation of the solution is shown to converge at an exponential rate numerical examples are given to illustrate the accuracy and the implementation of the method the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large both the limiting wave fronts and the limiting speed are independent of the initial values

exponential error bounds for random codes in the arbitrarily varying channel random codes for the arbitrarily varying channel are investigated the code ensemble is restricted to increase only exponentially with codeword length a quantity called key rate is used as a measure of the rate of increase the reliability function for such codes is related to the reliability function for codes with unlimited key rate explicit results are obtained in some examples of practical interest

why do block length and delay behave differently if feedback is present for output symmetric discrete memoryless channels dmcs at even moderately high rates fixed block length communication systems show no improvements in their error exponents with feedback this paper studies systems with fixed end to end delay and shows that feedback generally provides dramatic gains in the error exponents a new upper bound the uncertainty focusing bound is given on the probability of symbol error in a fixed delay communication system with feedback this bound turns out to have a form similar to viterbi s bound used for the block error probability of convolutional codes as a function of the fixed constraint length the uncertainty focusing bound is shown to be asymptotically achievable with noiseless feedback for erasure channels as well as for any output symmetric dmc that has strictly positive zero error capacity furthermore it can be achieved in a delay universal anytime fashion even if the feedback itself is delayed by a small amount finally it is shown that for end to end delay it is generally possible at high rates to beat the sphere packing bound for general dmcs thereby providing a counterexample to a conjecture of pinsker

common randomness in information theory and cryptography i secret sharing as the first part of a study of problems involving common randomness at distance locations information theoretic models of secret sharing generating a common random key at two terminals without letting an eavesdropper obtain information about this key are considered the concept of key capacity is defined single letter formulas of key capacity are obtained for several models and bounds to key capacity are derived for other models

numerical solution of fisher s equation using a moving mesh method the paper investigates the viability of using moving mesh methods to simulate travelling wave solutions of fisher s equation results are presented that illustrate the weaknesses in moving mesh methods based on equidistribution of some popular monitor functions it is shown that knowledge of the differential equation and the travelling wave solution may be used to construct a monitor function that yields accurate results with suitably chosen moving mesh methods a comparison is made between a moving mesh partial differential equation and a moving mesh differential algebraic equation for the evolution in time

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

rings quadratic forms and complete degeneracy for a subclass of highly overmoded rectangular waveguides in this article eigenvalue degeneracy formulas for a subclass of highly overmoded rectangular waveguides are determined this subclass consists of those rectangular geometries with side length ratio a b equal to the square root of a rational integer z and contains an unusual amount of degeneracy even when the side length ratio is irrational the complete degeneracy for members of this subclass 10 is determined by interpreting its normalized eigenvalue formula as a the norm of an associated ring of algebraic integers and b a special case of a positive definite binary integral quadratic form

variable length coding over an unknown channel burnashev in 1976 gave an exact expression for the reliability function of a discrete memoryless channel dmc with noiseless feedback a coding scheme that achieves this exponent needs in general to know the statistics of the channel suppose now that the coding scheme is designed knowing only that the channel belongs to a family q of dmcs is there a coding scheme with noiseless feedback that achieves burnashev s exponent uniformly over q at a nontrivial rate we answer the question in the affirmative for two families of channels binary symmetric and z for these families we show that for any given fraction there is a feedback coding strategy such that for any member of the family i guarantees this fraction of its capacity as rate and ii guarantees the corresponding burnashev s exponent therefore for these families in terms of delay and error probability the knowledge of the channel becomes asymptotically irrelevant in feedback code design there are blind schemes that perform as well as the best coding scheme designed with the foreknowledge of the channel under use however a converse result shows that in general even for families that consist of only two channels such blind schemes do not exist

arbitrarily varying channels with constrained inputs and states random coding theorems are proved for discrete memoryless arbitrarily varying channels avcs with constraints on the transmitted codewords and channel state sequences two types of constraints are considered peak i e required for each n length sequence almost surely and average over the message set or over an ensemble for peak constraints on the codewords and on the channel state sequences the avc is shown to have a strong random coding capacity if the codewords and or the channel state sequences are constrained in the average sense the avcs do not possess strong capacities only epsilon capacities are shown to exist

exponential error bounds for random codes on gaussian arbitrarily varying channels the main objective is to develop exponential bounds to the best error probability achievable with random coding on the gaussian arbitrarily varying channel gavc in the one case where a strong capacity exists i e with peak time averaged power constraints on both the transmitter and interference the gavc models a channel corrupted by thermal noise and by an unknown interfering signal of bounded power the upper and lower bounds to the best error probability achievable on this channel with random coding are presented the asymptotic exponents of these bounds agree in a range of rates near capacity the exponents are universally larger than the corresponding exponents for the discrete time gaussian channel with the same capacity it is further shown that the decoder can be taken to be the minimum euclidean distance rule at all rates less than capacity

stability of moving mesh systems of partial differential equations moving mesh methods based on the equidistribution principle ep are studied from the viewpoint of stability of the moving mesh system of differential equations for fine spatial grids the moving mesh system inherits the stability of the original discretized partial differential equation pde unfortunately for some pdes the moving mesh methods require so many spatial grid points that they no longer appear to be practical failures and successes of the moving mesh method applied to three reaction diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh pde

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

capacity of the gaussian arbitrarily varying channel the gaussian arbitrarily varying channel with input constraint gamma and state constraint lambda admits input sequences x x sub 1 x sub n of real numbers with sigma x sub i sup 2

a unified approach for the solution of the fokker planck equation this paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the fokker planck equation the unified features of the discrete singular convolution algorithm are discussed it is demonstrated that different implementations of the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point three benchmark stochastic systems the repulsive wong process the black scholes equation and a genuine nonlinear model are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the fokker planck equation via a time dependent method an additional example the incompressible euler equation is used to further validate the present approach for more difficult problems numerical results indicate that the present unified approach is robust and accurate for solving the fokker planck equation

a comparison of differential quadrature methods for the solution of partial differential equations abstract discrete ordinate methods for the solution of partial differential equations are examined and compared a novel approach for the discrete representation of differential operators based on split range polynomial expansions is introduced the utility of the method is demonstrated for the case of differentiation of functions involving steep gradients the solution of burgers equation is presented to illustrate the effectiveness of the technique for the solution of nonlinear partial differential equations exhibiting nearly discontinuous solutions comparisons are made with other differential quadrature methods

rateless coding for gaussian channels a rateless code i e a rate compatible family of codes has the property that codewords of the higher rate codes are prefixes of those of the lower rate ones a perfect family of such codes is one in which each of the codes in the family is capacity achieving we show by construction that perfect rateless codes with low complexity decoding algorithms exist for additive white gaussian noise channels our construction involves the use of layered encoding and successive decoding together with repetition using time varying layer weights as an illustration of our framework we design a practical three rate code family we further construct rich sets of near perfect rateless codes within our architecture that require either significantly fewer layers or lower complexity than their perfect counterparts variations of the basic construction are also developed including one for time varying channels in which there is no a priori stochastic model

the maximal error capacity of arbitrarily varying channels for constant list sizes the capacity of an arbitrarily varying channel avc for list codes of arbitrarily small list rate under the maximal error criterion has previously been determined the authors prove that for an avc a any rate r below the list code capacity c sub 1 a is achievable with a particular list size dependent on a r and y where y is the output alphabet

roots of certain transcendental equations for elastic angular regions certain important transcendental equations occur in the case of elastic angular regions while analyzing these regions for flexure vibration and buckling previously these transcendental equations have been solved for roots and the data has been tabulated for different boundary conditions as the angle of the region is varied the purpose of this paper is to demonstrate that once we solve for the roots at a specific angular region the roots for angular regions with angles ranging from 0 to 2 can be obtained via forward integration

on error exponents for arbitrarily varying channels the minimum probability of error achievable by random codes on the arbitrarily varying channel avc is investigated new exponential error bounds are found and applied to the avc with and without input and state constraints also considered is a simple subclass of random codes called randomly modulated codes in which encoding and decoding operations are separate from code randomization a universal coding theorem is proved which shows the existence of randomly modulated codes that achieve the same error bounds as fully random codes for all avcs

free vibration analysis of arbitrarily shaped polygonal plates with simply supported edges using a sub domain method as an extension of the ndif method developed by the authors a practical analytical method for the free vibration analysis of a simply supported polygonal plate with arbitrary shape is proposed especially the method is more effective for plates highly concave shapes because it employs a sub domain method dividing the plate of interest with two sub plates the approximate solution of each sub plate is assumed by linearly superposing plane waves propagated from edges of the sub plate sub system matrix equations for the two sub plates are extracted by applying the simply supported boundary condition to the edges of each sub plate excepting the common interface of the two sub plates finally the sub system matrix equations is merged into a single system matrix equation for the entire plate by considering the compatibility condition that the two sub plates have the same displacement and slope at the common interface the eigenvalues and mode shapes of the single plate are obtained from the determinant of a system matrix extracted from the entire system matrix equation it is shown by several case studies that the proposed method has a good convergence characteristics and yields accurate eigenvalues and mode shapes compared with another analytical method ndif method and fem nastran

bounds to eigenvalues of the laplacian on l shaped domain by variational methods in this study the bounds for eigenvalues of the laplacian operator on an l shaped domain are determined by adopting some special functions in goerisch method for lower bounds and in traditional rayleigh ritz method for upper bounds very accurate bounds to eigenvalues for the problem are obtained numerical results show that these functions can also be successfully used to solve the problem on the region with other reentrant angle

the unexplained accuracy of the lagrange mesh method the lagrange mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a gauss quadrature in order to analyze its accuracy four different lagrange mesh calculations based on the zeros of laguerre polynomials are compared with exact variational calculations based on the corresponding laguerre basis the comparison is performed for three solvable radial potentials the morse harmonic oscillator and coulomb potentials the results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy even in the presence of the centrifugal singularity the same property holds for the approximate wave functions this striking accuracy remains unexplained

the capacity of a vector gaussian arbitrarily varying channel the random coding capacity of a vector gaussian arbitrarily varying channel vgavc is determined along with a simple general method for computing this capacity the vgavc is a discrete time memoryless vector channel with an input power constraint and additive gaussian noise that is further corrupted by an additive jamming signal the statistics of this jamming signal are unknown and can be arbitrary subject only to a power constraint

a new numerical scheme for the fisher equation in the context of the one dimensional fisher equation 1937 the authors study a new numerical scheme for reaction diffusion equations proposed by oono and puri 1987 they find that this scheme enables a reasonable simulation of the fisher equation at high values of the time increment where conventional schemes are not applicable for lower values of the time increment the new scheme compares favourably with conventional schemes

comparison of the nodal integral method and nonstandard finite difference schemes for the fisher equation the relationship between the so called nonstandard finite difference schemes and the nodal integral method nim is investigated using the fisher equation as a model problem exact and best finite difference schemes are reviewed first next the nim for the fisher equation is developed it is shown that the nim leads to a nonstandard evaluation of the derivatives moreover the resulting scheme possesses the desirable characteristics of the nonstandard finite difference schemes such as the nonlocal evaluation of the nonlinear terms thus the nim provides a systematic framework to obtain schemes similar to the best finite difference schemes numerical results for a propagating front problem show that the nim can capture the shape and speed of the front very accurately results also show that the best finite difference scheme is stable for large grid sizes but only at the cost of inaccuracy in the front propagation speed additional results are obtained using the nim for symmetric and asymmetric initial conditions these describe the interaction of two fronts of advantageous genes that approach each other and merge it is noted that the traveling fronts evolving from certain asymmetric initial conditions become indistinguishable from those that evolve from symmetric initial conditions

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

on traveling wave solutions of fisher s equation in two spatial dimensions it is shown for the quadratic fisher equation in two spatial dimensions that along with a plane wave there exist several other traveling waves with nontrivial front geometry some of the solutions are found in explicit form others are constructed approximately the dispersion relationship and velocity curvature dependence generated by these solutions are studied

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

a chebyshev collocation method for computing the eigenvalues of the laplacian chebyshev collocation techniques are developed in this paper to compute the eigenvalues of the laplacian based on a boundary integral formulation for two dimensional domains with piecewise smooth boundaries unlike the traditional domain methods for example the finite element method which discretizes the eigenfunctions on the two dimensional domain only a one dimensional function defined on the boundary is discretized global expansions in terms of chebyshev polynomials are used in each smooth piece of the boundary to solve the integral equation comparing with the boundary element method this method obtains higher accuracy for a smaller discretized matrix finally an efficient algorithm for generating the discretized matrix say n n is developed that requires only o n2 log n operations

high accuracy approximation techniques for analytic functions a generalization of the familiar mesh point technique for numerical approximation of functions is presented high accuracy and very rapid convergence may be obtained by thoughtful choice of the reference function chosen for interpolation between the mesh points in particular derivative operators are represented by highly nonlocal matrices but this is no drawback when one has computing machines to perform the algebraic manipulations some examples are given from familiar quantum mechanical problems

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

the capacity of the arbitrarily varying channel revisited positivity constraints a well known result of r ahlswede 1970 asserts that the deterministic code capacity of an arbitrarily varying channel avc under the average error probability criterion either equals its random code capacity or else is zero a necessary and sufficient condition is identified for deciding between these alternative namely the capacity is zero if and only if the avc is symmetrizable the capacity of the avc is determined with constraints on the transmitted codewords as well as on the channel state sequences and it is demonstrated that it may be positive but less than the corresponding random code capacity a special case of the results resolves a weakened version of a fundamental problem of coding theory

vibration analysis of arbitrarily shaped membranes using local radial basis function based differential quadrature method in this study a recently developed local radial basis function based differential quadrature lrbfdq method is applied for the vibration analysis of arbitrarily shaped membranes lrbfdq method combines the good features of differential quadrature dq approximation of derivatives and mesh free nature of the radial basis functions rbfs in a local region the derivative at a reference point is approximated as a linear weighted sum of functional values at a set of scattered points in the local supporting region of the reference point the helmholtz equation governing membrane vibration is directly discretized into algebraic equations from which the wavenumbers natural frequencies and mode shapes of freely vibrating membranes are easily calculated owing to the properties of mesh free and local approximation of the lrbfdq method the problems with arbitrarily shaped domains can be solved readily and accurately in particular for highly concave shaped membranes and multi connected membranes with a hole very accurate numerical results can be easily obtained without the use of any domain decomposition technique it is also shown that the lrbfdq method can produce more accurate solutions than fem when the two methods use nearly the same number of points in a domain

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

free vibration of regular polygonal plates with simply supported edges the free vibration of regular polygonal plates with simply supported edges is studied by the dynamical analogy with membranes a regular polygonal membrane is formed on a rectangular membrane by fixing several segments with the reaction forces acting on all edges of an actual polygonal membrane regarded as unknown harmonic loads the stationary response of the membrane to these loads is expressed by the eigenfunctions of the extended rectangular membrane without internal supports the force distributions along the edges are expanded into fourier sine series with unknown coefficients and the homogeneous equations for the coefficients are derived by restraint conditions on the edges the natural frequencies and the mode shapes of the actual membrane are determined by calculating the eigenvalues and eigenvectors of the equations the method is applied to an equilateral triangular through a regular decagonal membrane the natural frequencies and mode shapes are calculated numerically and the effect of the shape of membrane is discussed the numerical values obtained for polygonal membranes are immediately converted into those of simply supported polygonal plates

arbitrarily shaped hollow waveguide analysis by the alpha interpolation method an efficient numerical method named the alpha interpolation method aim has been developed for solving hollow waveguide problems the aim is a simpler and more systematic procedure than the finite element method and when the optimum interpolation parameter is chosen for any number of linear triangular elements the cutoff wavenumbers or the exact eigenvalues may be estimated by the aim numerical results and comparisons are given

robust and efficient solution of the drum problem via nystrom approximation of the fredholm determinant the drum problem finding the eigenvalues and eigenfunctions of the laplacian with dirichlet boundary condition has many applications yet remains challenging for general domains when high accuracy or high frequency is needed boundary integral equations are appealing for large scale problems yet certain difficulties have limited their use we introduce the following two ideas to remedy this 1 we solve the resulting nonlinear eigenvalue problem using boyd s method for analytic root finding applied to the fredholm determinant and we show that this is many times faster than the usual iterative minimization of a singular value 2 we fix the problem of spurious exterior resonances via a combined field representation this also provides the first robust boundary integral eigenvalue method for non simply connected domains we implement the new method in two dimensions using spectrally accurate nystrom product quadrature we prove exponential convergence of the determinant at roots for domains with

a direct spectral collocation poisson solver in polar and cylindrical coordinates in this paper we present a direct spectral collocation method for the solution of the poisson equation in polar and cylindrical coordinates the solver is applied to the poisson equations for several different domains including a part of a disk an annulus a unit disk and a cylinder unlike other poisson solvers for geometries such as unit disks and cylinders no pole condition is involved for the present solver the method is easy to implement fast and gives spectral accuracy we also use the weighted interpolation technique and nonclassical collocation points to improve the convergence

delay induced traveling wave fronts in reaction diffusion equations of kpp fisher type this paper explores the impact of delay on the existence of traveling wave fronts in reaction diffusion equations of kpp fisher type for two such equations one being local and the other with nonlocal effect arising from the age structure of the population we show that delay can induce slow traveling wave fronts

solution of evolutionary partial differential equations using adaptive finite differences with pseudospectral post processing a coordinate transformation approach is described that enables pseudospectral methods to be applied efficiently to unsteady differential problems with steep solutions the work is an extension of a method presented by mulholland huang and sloan for the adaptive pseudospectral solution of steady problems a coarse grid is generated by a moving mesh finite difference method that is based on equidistribution and this grid is used to construct a time dependent coordinate transformation a sequence of spatial transformations may be generated at discrete points in time or a single transformation may be generated as a continuous function of space and time the differential problem is transformed by the coordinate transformation and then solved using a method that combines pseudospectral discretisation in space with a suitable integrator in time numerical results are presented for unsteady problems in one space dimension

a pseudospectral method of solution of fisher s equation in this paper we develop an accurate and efficient pseudospectral solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations the collocation method employed is based on chebyshev gauss lobatto quadrature points we compare results for a single domain as well as for a subdivision of the main domain into subintervals instabilities that occur in the numerical solution for a single domain analogous to those found by others are attributed to round off errors arising from numerical features of the discrete second derivative matrix operator however accurate stable solutions of fisher s equation are obtained with a multidomain pseudospectral method a detailed comparison of the present approach with the use of the sine interpolation is also carried out

comment on the discrete ordinate method in the kinetic theory of gases abstract the discrete ordinate do method for solving the linear integro differential equations of kinetic theory is investigated and other numerical methods for solving this type of problem are reviewed it is pointed out that the do method can be regarded as a special case of the method of weighted residuals an efficient method for calculating the matrix d ij representing the differential operator is described and it is shown that the method does not necessarily require the use of the nodes of a gaussian quadrature

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

rateless codes for avc models the arbitrarily varying channel avc is a channel model whose state is selected maliciously by an adversary fixed blocklength coding assumes a worst case bound on the adversary s capabilities which leads to pessimistic results this paper defines a variable length perspective on this problem for which achievable rates are shown that depend on the realized actions of the adversary specifically rateless codes are constructed which require a limited amount of common randomness these codes are constructed for two kinds of avc models in the first the channel state cannot depend on the channel input and in the second it can as a by product the randomized coding capacity of the avc with state depending on the transmitted codeword is found and shown to be achievable with a small amount of common randomness the results for this model are proved using a randomized strategy based on list decoding

chaos free numerical solutions of reaction diffusion equations two numerical methods which do not bring contrived chaos into the solution are proposed for the solution of the riccati logistic equation though implicit in nature with the resulting improvements in stability the methods are applied explicitly when extended to the numerical solution of fisher s equation in which the quadratic polynomial representing the derivative in the riccati equation appears as the reaction term the solution is found by solving a linear system of algebraic equations at each time step as opposed to solving a nonlinear system which frequently happens when solving nonlinear partial differential equations the approaches adopted are extended to an ordinary differential equation in which the derivative is expressed as a cubic polynomial in the dependent variable the solution of this initial value problem is not available in closed form for finite values of the independent variable t under the conditions stated numerical solutions are seen to converge to the correct steady state solution a nonlinear partial differential equation which governs the conduction of electrical impulses along a nerve axon and which has the aforementioned cubic polynomial as its reaction term is solved by applying the numerical methods developed for solving the ordinary differential equation the solution to this nonlinear reaction diffusion equation is determined by solving a linear algebraic system at each time step

a multidomain spectral method for supersonic reactive flows this paper has a dual purpose it presents a multidomain chebyshev method for the solution of the two dimensional reactive compressible navier stokes equations and it reports the results of the application of this code to the numerical simulations of high mach number reactive flows in recessed cavity the computational method utilizes a newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations the results of the simulations are relevant to recessed cavity flame holders

haar wavelet method for solving fisher s equation in this paper we develop an accurate and efficient haar wavelet solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations moreover the use of haar wavelets is found to be accurate simple fast flexible convenient small computation costs and computationally attractive

exact free vibration analysis for membrane assemblies with general classical boundary conditions abstract this paper proposes exact dynamic stiffness formulations for membranes and their assemblies under any arbitrary classical boundary conditions first by taking exact solutions in one direction satisfying all possible opposite edge supports we can derive exact general solutions of the helmholtz equation for membrane vibration then generic force and displacement boundary conditions in the other direction are expressed in terms of the general solutions finally the dynamic stiffness matrices of rectangular membrane elements are formulated which can be assembled directly and allows applications of arbitrary boundary conditions as an accurate and efficient modal solution technique the wittrick williams ww algorithm is applied onto the global dynamic stiffness matrix of the final structure the most important issue of the ww algorithm j 0 count has been resolved with the analytical expressions derived for all possible cases the proposed dynamic stiffness method dsm is then applied to several examples including individual membranes and their assemblies high accuracy and exactness of the proposed method within the whole frequency range is demonstrated by comparison with the finite element method besides interesting findings have been observed on repeated eigenvalues with distinct mode shapes corresponding to certain aspect ratio and tension ratio where physical and mathematical understanding has been provided

list decoding for the arbitrarily varying channel under state constraints list decoding for arbitrarily varying channels avcs under state constraints is investigated it is shown that rates within of the randomized coding capacity of avcs with input dependent state can be achieved under maximal error with list decoding using lists of size o 1 under the average error criterion an achievable rate and converse bound are given for lists of size l these bounds are based on two different notions of symmetrizability and do not coincide in general an example is given which shows that for list size l the capacity may be positive but strictly smaller than the randomized coding capacity in contrast to the situation without constraints

upper bounds on the capacity of binary channels with causal adversaries in this paper we consider the communication of information in the presence of a causal adversarial jammer in the setting under study a sender wishes to communicate a message to a receiver by transmitting a codeword x x1 xn bit by bit over a communication channel the sender and the receiver do not share common randomness the adversarial jammer can view the transmitted bits xi one at a time and can change up to a p fraction of them however the decisions of the jammer must be made in a causal manner namely for each bit xi the jammer s decision on whether to corrupt it or not must depend only on xj for j i this is in contrast to the classical adversarial jamming situations in which the jammer has no knowledge of x or knows x completely in this study we present upper bounds that hold under both the average and maximal probability of error criteria on the capacity which hold for both deterministic and stochastic encoding schemes

fundamental frequency of clamped plates with circularly periodic boundaries abstract a boundary perturbation method is developed to determine the fundamental frequency of vibrating plates the method is then applied to wavy star shape and polygonal plates with clamped boundary conditions approximate analytical solutions of the fundamental frequency are obtained with an accuracy of o e4 where e is the deviation from the unit circle

achieving the empirical capacity using feedback memoryless additive models we address the problem of universal communications over an unknown channel with an instantaneous noiseless feedback and show how rates corresponding to the empirical behavior of the channel can be attained although no rate can be guaranteed in advance first we consider a discrete modulo additive channel with alphabet x where the noise sequence zn is arbitrary and unknown and may causally depend on the transmitted and received sequences and on the encoder s message possibly in an adversarial fashion although the classical capacity of this channel is zero we show that rates approaching the empirical capacity log x h emp zn can be universally attained where h emp zn is the empirical entropy of zn for the more general setting where the channel can map its input to an output in an arbitrary unknown fashion subject only to causality we model the empirical channel actions as the modulo addition of a realized noise sequence and show that the same result applies if common randomness is available the results are proved constructively by providing a simple sequential transmission scheme approaching the empirical capacity

two sided bounds for eigenvalues of differential operators with applications to friedrichs poincare trace and similar constants we present a general numerical method for computing guaranteed two sided bounds for principal eigenvalues of symmetric linear elliptic differential operators the approach is based on the galerkin method on the method of a priori a posteriori inequalities and on a complementarity technique the two sided bounds are formulated in a general hilbert space setting and as a byproduct we prove an abstract inequality of friedrichs poincare type the abstract results are then applied to friedrichs poincare and trace inequalities and fully computable two sided bounds on the optimal constants in these inequalities are obtained accuracy of the method is illustrated in numerical examples

computational solution of two dimensional unsteady pdes using moving mesh methods numerical experiments are described which illustrate some important features of the performance of moving mesh methods for solving two dimensional partial differential equations pdes here we are concerned with algorithms based on moving mesh methods proposed by w huang and r d russell siam j sci comput 20 998 1999 we show that the accuracy of the computations is strongly influenced by the choice of monitor function and we present a monitor function which yields a higher rate of convergence than those that are commonly used in an earlier paper g beckett j a mackenzie a ramage and d m sloan j comput phys 167 372 2001 we demonstrated a robust and efficient algorithm for problems in one space dimension in which the mesh equation is decoupled from the physical pde and the time step is controlled automatically the present work extends this algorithm to deal with problems in two space dimensions

chebyshev pseudospectral multi domain technique for viscous flow calculation abstract a novel chebyshev pseudospectral multi domain technique is introduced for the numerical solution of the navier stokes equations in the primitive variable formulation careful consideration is given to the proper interface condition in the multi domain method which decomposes the solution domain into subdomains by overlapping one grid point a finite difference approximation is used for the time discretization which transforms the system to a set of coupled equations which is then solved by an efficient computational boundary cb method in the cb method the pressure neumann boundary conditions are simply obtained by re arranging the discrete algebraic equations resulting from chebyshev pseudospectral multi domain approximation for the space variables to demonstrate the effectiveness of this technique we present some accurate computational results for calculating the driven cavity flows with re up to 10 000 and for the flow past a circular cylinder with re up to 100 the results for both applications are in good agreement with those of previous researchers

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

the uniform distribution as a universal prior in this correspondence we discuss the properties of the uniform prior as a universal prior i e a prior that induces a mutual information that is simultaneously close to the capacity for all channels we determine bounds on the amount of the mutual information loss in using the uniform prior instead of the capacity achieving prior specifically for the class of binary input channels with any output alphabet we show that the z channel has the minimal mutual information with uniform prior out of all channels with a given capacity from this we conclude that the degradation of the mutual information with respect to the capacity is at most 0 011 bit and as was shown previously at most 6 a related result is that the capacity achieving prior for any channel is not far from uniform some of these results are extended to channels with nonbinary input

exact vibration solutions for some irregularly shaped membranes and simply supported plates abstract a procedure is discussed for realizing some exact analytical eigenfunctions and corresponding eigenfrequencies for vibrating membranes with fixed rectilinear edges delineable on a grid of squares and their diagonals by the membrane plate analogy the corresponding eigenmodes and frequencies of simply supported vibrating plates of the same shapes may be obtained the fundamental modes of certain shapes with curved boundaries are also considered

eigenvalues and musical instraments most musical instruments are built from physical systems that oscillate at certain natural frequencies the frequencies are the imaginary parts of the eigenvalues of a linear operator and the decay rates are the negatives of the real parts so it ought to be possible to give an approximate idea of the sound of a musical instrument by a single plot of points in the complex plane nevertheless the authors are unaware of any such picture that has ever appeared in print this paper attempts to fill that gap by plotting eigenvalues for simple models of a guitar string a flute a clarinet a kettledrum and a musical bell for the drum and the bell simple idealized models have eigenvalues that are irrationally related but as the actual instruments have evolved over the generations the leading five or six eigenvalues have moved around the complex plane so that their relative positions are musically pleasing

gaussian arbitrarily varying channels the em arbitrarily varying channel avc can be interpreted as a model of a channel jammed by an intelligent and unpredictable adversary we investigate the asymptotic reliability of optimal random block codes on gaussian arbitrarily varying channels gavc s a gavc is a discrete time memoryless gaussian channel with input power constraint p t and noise power n e which is further corrupted by an additive jamming signal the statistics of this signal are unknown and may be arbitrary except that they are subject to a power constraint p j we distinguish between two types of power constraints em peak and em average for peak constraints on the input power and the jamming power we show that the gavc has a random coding capacity for the remaining cases in which either the transmitter or the jammer or both are subject to average power constraints no capacities exist and only lambda capacities are found the asymptotic error probability suffered by optimal random codes in these cases is determined our results suggest that if the jammer is subject only to an average power constraint reliable communication is impossible at any positive code rate

the steady state response of a damped circular plate driven at a diameter abstract the steady state response of a damped circular plate transversely or rotationally driven at a diameter is determined by the series type method with the reaction force or moment acting along a diameter of a circular plate regarded as an unknown harmonic load the response of the plate to the load is expressed by the green function the force or moment distribution along the diameter is expanded into fourier series with unknown coefficients and the simultaneous linear equations for the coefficients are derived by the restraint condition on the diameter the dynamic response of the plate is determined by solving the equations the method is applied to circular plates driven by a sinusoidally varying transverse deflection or angular rotation at a diameter and the steady state response of the plates is studied

spectral differencing with a twist spectral collocation methods have become very useful in providing highly accurate solutions to differential equations a straightforward implementation of these methods involves the use of spectral differentiation matrices to obtain optimal accuracy these matrices must be computed carefully we demonstrate that naive algorithms for computing these matrices suffer from severe loss of accuracy due to roundoff errors several improvements are analyzed and compared a number of numerical examples are provided demonstrating significant differences between the sensitivity of the forward problem and the inverse problem

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

the errors in calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points abstract we discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points we propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices

fisher equation for anisotropic diffusion simulating south american human dispersals the fisher equation is commonly used to model population dynamics this equation allows describing reaction diffusion processes considering both population growth and diffusion mechanism some results have been reported about modeling human dispersion always assuming isotropic diffusion nevertheless it is well known that dispersion depends not only on the characteristics of the habitats where individuals are but also on the properties of the places where they intend to move then isotropic approaches cannot adequately reproduce the evolution of the wave of advance of populations solutions to a fisher equation are difficult to obtain for complex geometries moreover when anisotropy has to be considered and so few studies have been conducted in this direction with this scope in mind we present in this paper a solution for a fisher equation introducing anisotropy we apply a finite difference method using the crank nicholson approximation and analyze the results as a function of the characteristic parameters finally this methodology is applied to model south american human dispersal

higher order time stepping methods for time dependent reaction diffusion equations arising in biology abstract this paper demonstrates the use of higher order methods to solve some time dependent stiff pdes in the past the most popular numerical methods for solving system of reaction diffusion equations was based on the combination of low order finite difference method with low order time stepping method we extend in this report the compatibility of fourth order finite difference scheme in space coupled with fourth order time stepping methods such as imexlm4 imexpc4 imexrk4 and etdrk4 b in time for direct integration of reaction diffusion equations in one space dimension some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio temporal dynamical structures

comparison of the discrete singular convolution and three other numerical schemes for solving fisher s equation in this paper a discrete singular convolution dsc algorithm is introduced to solve fisher s equation to which obtaining an accurate and reliable traveling wave solution is a challenging numerical problem two novel numerical treatments a moving frame scheme and a new asymptotic scheme are designed to overcome the subtle difficulties involved in the numerical solution of fisher s equation the moving frame scheme is proposed to provide accurate and efficient spatial resolution of the problem the new asymptotic scheme is introduced to facilitate the correct prediction of the wave speed after long time integrations the resulting dsc algorithm is able to correctly predict long time traveling wave behavior the performance of the present algorithm is demonstrated through extensive numerical experiments as well as through a comparison with three other standard approaches i e methods of the fourier pseudospectral the accurate spatial derivatives and the crank nicolson schemes the spatial and temporal accuracies of all four methods are analyzed and numerically verified

free vibration analysis of arbitrarily shaped plates with clamped edges using wave type functions the basic idea related to the non dimensional dynamic influence green s function defined in an infinite membrane has been applied to the free vibration analysis of arbitrarily shaped plates with clamped edges another non dimensional dynamic influence function newly defined in the paper exactly satisfies the governing homogeneous differential equations of plates and is regular in the entire domain the proposed method uses the boundary discretization scheme similar to that of the boundary element method but does not require any interpolation function between the nodes distributed along the boundary as a result the method has a significant advantage in its simplicity and accuracy although the method uses no interpolation function it gives very accurate eigenvalues and mode shapes compared with the exact solutions or fem results it should be also noted that particular attention is given to reduce the size of the system matrix whose determinant equation yields eigenvalues and to overcome the functional dependence problem of the non dimensional dynamic influence functions

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

vibration of non homogeneous rectangular membranes with arbitrary interfaces abstract the study deals with the generalized solution of the title problem the free vibration problem of a rectangular membrane with partial domains each of uniform density and arbitrary interface is tackled previous studies by other researchers include straight parallel to the borders and oblique interfaces and bent ones the solution is found by means of a direct variational method with a series composed with a complete set of functions two alternative sets are explored trigonometric and power series such series are uniformly convergent to the exact solution the approach is straightforward and very efficient from the computational viewpoint a determinant factorization method is employed to automatically eliminate eventual spurious frequencies the well known analogy between plates and membranes does not hold in this problem and a demonstration is included diverse illustrations are worked out as the cases of an oblique straight line interface and an open curve line which divides the membrane in two domains each of different density are first presented also a rectangular membrane with an interior closed domain is stated and the numerical example of a circular interior zone is included comparison between the two alternatives and with other authors results show excellent agreement in all cases the computational cost is very low