backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

symplectic and multisymplectic lobatto methods for the good boussinesq equation in this paper we construct second order symplectic and multisymplectic integrators for the good boussineq equation using the two stage lobatto iiia iiib partitioned runge kutta method which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative numerical dispersion properties and the stability of both integrators are investigated numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preserving local and global energy and momentum

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

numerical simulation of interaction between schrodinger field and klein gordon field by multisymplectic method abstract we propose a multisymplectic scheme to solve the coupled klein gordon schrodinger system the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law and can simulate the original waves well in a long time this scheme also has discrete quasi norm conservation law numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results

variational adaptive correlation method for flow estimation a variational approach is presented to the estimation of turbulent fluid flow from particle image sequences in experimental fluid mechanics the approach comprises two coupled optimizations for adapting size and shape of a gaussian correlation window at each location and for estimating the flow respectively the method copes with a wide range of particle densities and image noise levels without any data specific parameter tuning based on a careful implementation of a multiscale nonlinear optimization technique we demonstrate robustness of the solution over typical experimental scenarios and highest estimation accuracy for an international benchmark data set piv challenge

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

symplectic and multi symplectic methods for the nonlinear schrodinger equation the hamiltonian and the multi symplectic formulations of the nonlinear schrodinger equation are considered for the multi symplectic formulation a new six point difference scheme which is equivalent to the multi symplectic preissman integrator is derived numerical experiments are also reported c 2002 elsevier science ltd all rights reserved

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

accuracy of classical conservation laws for hamiltonian pdes under runge kutta discretizations we investigate conservative properties of runge kutta methods for hamiltonian partial differential equations it is shown that multi symplecitic runge kutta methods preserve precisely the norm square conservation law based on the study of accuracy of runge kutta methods applied to ordinary and partial differential equations we present some results on the numerical accuracy of conservation laws of energy and momentum for hamiltonian pdes under runge kutta discretizations

stochastic uncertainty models for the luminance consistency assumption in this paper a stochastic formulation of the brightness consistency used in many computer vision problems involving dynamic scenes for instance motion estimation or point tracking is proposed usually this model which assumes that the luminance of a point is constant along its trajectory is expressed in a differential form through the total derivative of the luminance function this differential equation linearly links the point velocity to the spatial and temporal gradients of the luminance function however when dealing with images the available information only holds at discrete time and on a discrete grid in this paper we formalize the image luminance as a continuous function transported by a flow known only up to some uncertainties related to such a discretization process relying on stochastic calculus we define a formulation of the luminance function preservation in which these uncertainties are taken into account from such a framework it can be shown that the usual deterministic optical flow constraint equation corresponds to our stochastic evolution under some strong constraints these constraints can be relaxed by imposing a weaker temporal assumption on the luminance function and also in introducing anisotropic intensity based uncertainties we also show that these uncertainties can be computed at each point of the image grid from the image data and hence provide meaningful information on the reliability of the motion estimates to demonstrate the benefit of such a stochastic formulation of the brightness consistency assumption we have considered a local least squares motion estimator relying on this new constraint this new motion estimator significantly improves the quality of the results

on the preservation of phase space structure under multisymplectic discretization in this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and fourier spectral approximations multisymplectic ms schemes are developed for two benchmark nonlinear wave equations the sine gordon and nonlinear schrodinger equations we examine the implications of preserving the ms structure under discretization on the numerical scheme s ability to preserve phase space structure as measured by the nonlinear spectrum of the governing equation we find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws dynamical invariants and complicated phase space structures

multi symplectic runge kutta nystrom methods for nonlinear schrodinger equations with variable coefficients in this paper we consider runge kutta nystrom rkn methods applied to nonlinear schrodinger equations with variable coefficients nlsevc concatenating symplectic nystrom methods in spatial direction and symplectic runge kutta methods in temporal direction for nlsevc leads to multi symplectic integrators i e to numerical methods which preserve the multi symplectic conservation law mscl we present the corresponding discrete version of mscl it is shown that the multi symplectic rkn methods preserve not only the global symplectic structure in time but also local and global discrete charge conservation laws under periodic boundary conditions we present a 4 order multi symplectic rkn method and use it in numerical simulation of quasi periodically solitary waves for nlsevc and we compare the multi symplectic rkn method with a non multi symplectic rkn method on the errors of numerical solutions the numerical errors of discrete energy discrete momentum and discrete charge the precise conservation of discrete charge under the multi symplectic rkn discretizations is attested numerically some numerical superiorities of the multi symplectic rkn methods are revealed

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

dense estimation and object based segmentation of the optical flow with robust techniques we address the issue of recovering and segmenting the apparent velocity field in sequences of images as for motion estimation we minimize an objective function involving two robust terms the first one cautiously captures the optical flow constraint while the second a priori term incorporates a discontinuity preserving smoothness constraint to cope with the nonconvex minimization problem thus defined we design an efficient deterministic multigrid procedure it converges fast toward estimates of good quality while revealing the large discontinuity structures of flow fields we then propose an extension of the model by attaching to it a flexible object based segmentation device based on deformable closed curves different families of curve equipped with different kinds of prior can be easily supported experimental results on synthetic and natural sequences are presented including an analysis of sensitivity to parameter tuning

some new structure preserving algorithms for general multi symplectic formulations of hamiltonian pdes many partial differential equations pdes can be written as a multi symplectic hamiltonian system which has three local conservation laws namely multi symplectic conservation law local energy conservation law and local momentum conservation law in this paper we give several systematic methods for discretizing general multi symplectic formulations of hamiltonian pdes including a local energy preserving algorithm a class of global energy preserving methods and a local momentum preserving algorithm the methods are illustrated by the nonlinear schrodinger equation and the korteweg de vries equation numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods

multi symplectic runge kutta methods for nonlinear dirac equations in this paper we consider the multi symplectic runge kutta msrk methods applied to the nonlinear dirac equation in relativistic quantum physics based on a discovery of the multi symplecticity of the equation in particular the conservation of energy momentum and charge under msrk discretizations is investigated by means of numerical experiments and numerical comparisons with non msrk methods numerical experiments presented reveal that msrk methods applied to the nonlinear dirac equation preserve exactly conservation laws of charge and momentum and conserve the energy conservation in the corresponding numerical accuracy to the method utilized it is verified numerically that msrk methods are stable and convergent with respect to the conservation laws of energy momentum and charge and msrk methods preserve not only the inner geometric structure of the equation but also some crucial conservative properties in quantum physics a remarkable advantage of msrk methods applied to the nonlinear dirac equation is the precise preservation of charge conservation law

dense estimation of fluid flows in this paper we address the problem of estimating and analyzing the motion of fluids in image sequences due to the great deal of spatial and temporal distortions that intensity patterns exhibit in images of fluids the standard techniques from computer vision originally designed for quasi rigid motions with stable salient features are not well adapted in this context we thus investigate a dedicated minimization based motion estimator the cost function to be minimized includes a novel data term relying on an integrated version of the continuity equation of fluid mechanics which is compatible with large displacements this term is associated with an original second order div curl regularization which prevents the washing out of the salient vorticity and divergence structures the performance of the resulting fluid flow estimator is demonstrated on meteorological satellite images in addition we show how the sequences of dense motion fields we estimate can be reliably used to reconstruct trajectories and to extract the regions of high vorticity and divergence

highly accurate optic flow computation with theoretically justified warping in this paper we suggest a variational model for optic flow computation based on non linearised and higher order constancy assumptions besides the common grey value constancy assumption also gradient constancy as well as the constancy of the hessian and the laplacian are proposed since the model strictly refrains from a linearisation of these assumptions it is also capable to deal with large displacements for the minimisation of the rather complex energy functional we present an efficient numerical scheme employing two nested fixed point iterations following a coarse to fine strategy it turns out that there is a theoretical foundation of so called warping techniques hitherto justified only on an experimental basis since our algorithm consists of the integration of various concepts ranging from different constancy assumptions to numerical implementation issues a detailed account of the effect of each of these concepts is included in the experimental section the superior performance of the proposed method shows up by significantly smaller estimation errors when compared to previous techniques further experiments also confirm excellent robustness under noise and insensitivity to parameter variations

power of the scan statistic for detection of clustering the scan statistic is the maximum number of events in an interval of fixed length w as the subinterval moves over the entire time frame previous research derived the null distribution of the scan statistic under the conditional model which assumed that the total number of events was fixed and under the unconditional model which let the total number of events be a random variable this paper derives approximations for the power of the scan test for a pulse alternative under this alternative the relative risk of disease on a subinterval tau tau w tau unknown is theta fold as high as it is for other subintervals of length w two sets of approximations are given for each model the first approximation gives highly accurate results but requires use of a personal computer the second procedure can be performed on a hand held calculator and appears very accurate for the cases examined

large displacement optical flow descriptor matching in variational motion estimation optical flow estimation is classically marked by the requirement of dense sampling in time while coarse to fine warping schemes have somehow relaxed this constraint there is an inherent dependency between the scale of structures and the velocity that can be estimated this particularly renders the estimation of detailed human motion problematic as small body parts can move very fast in this paper we present a way to approach this problem by integrating rich descriptors into the variational optical flow setting this way we can estimate a dense optical flow field with almost the same high accuracy as known from variational optical flow while reaching out to new domains of motion analysis where the requirement of dense sampling in time is no longer satisfied

observation model based on scale interactions for optical flow estimation in this paper an original observation model for multiresolution optical flow estimation is introduced multiresolution frameworks often based on coarse to fine warping strategies are widely used by state of the art optical flow methods they allow the recovery of large motions by successive estimations of the flow field at several resolution levels although such approaches perform very efficiently and usually lead to faster minimizations they generally consider independent problems at each resolution levels and do not exploit the existing interactions between scales especially the influences of fine scales on larger ones in this paper we tackle this issue by proposing a flexible framework inspired from fluid mechanics able to partly counter these limitations for each resolution level our process filters the equations of interest and decomposes the key variables into resolved i e at a given resolution and unresolved i e at finer resolutions components this enables to derive a new data term that takes into account at coarse resolutions the influence of their unresolved parts from this new term we propose two different estimation strategies depending on whether we explicitly know the type of relations between the different scales as for physical processes or not in order to test the efficiency of this new observation model we have embedded it in a simple multiresolution lucas kanade estimator comparing the usual optical flow constraint equation with this new term in the same motion estimation procedure it clearly appears that the proposed term leads to more consistent estimates and prevents from errors propagation apparition during the estimation in all situations synthetic real physical images or not our new term is able to greatly improve the results compared with usual conservation constraints

numerical methods for nonlinear dirac equation abstract this paper presents a review of the current state of the art of numerical methods for nonlinear dirac nld equation several methods are extendedly proposed for the 1 1 dimensional nld equation with the scalar and vector self interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge energy and linear momentum those methods are the crank nicolson cn schemes the linearized cn schemes the odd even hopscotch scheme the leapfrog scheme a semi implicit finite difference scheme and the exponential operator splitting os schemes the nonlinear subproblems resulted from the os schemes are analytically solved by fully exploiting the local conservation laws of the nld equation the effectiveness of the various numerical methods with special focus on the error growth and the computational cost is illustrated on two numerical experiments compared to two high order accurate runge kutta discontinuous galerkin methods theoretical and numerical comparisons show that the high order accurate os schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency a fourth order accurate os scheme is further applied to investigating the interaction dynamics of the nld solitary waves under the scalar and vector self interaction the results show that the interaction dynamics of two nld solitary waves depend on the exponent power of the self interaction in the nld equation collapse happens after collision of two equal one humped nld solitary waves under the cubic vector self interaction in contrast to no collapse scattering for corresponding quadric case

the multi symplectic fourier pseudospectral method for solving two dimensional hamiltonian pdes in this paper the multi symplectic fourier pseudospectral msfp method is generalized to solve two dimensional hamiltonian pdes with periodic boundary conditions using the fourier pseudospectral method in the space of the two dimensional hamiltonian pde 2d hpde the semi discrete system obtained is proved to have semi discrete multi symplectic conservation laws and a global symplecticity conservation law then the implicit midpoint rule is employed for time integration to obtain the msfp method for the 2d hpde the fully discrete multi symplectic conservation laws are also obtained in addition the proposed method is applied to solve the zakharov kuznetsov zk equation and the kadomtsev petviashvili kp equation numerical experiments on soliton solutions of the zk equation and the kp equation show the high accuracy and effectiveness of the proposed method

effect of relative risk and cluster configuration on the power of the one dimensional scan statistic the scan test for clustering in time is based on the maximum number of events in an interval window of width w as the window moves across the entire time frame power estimates of the scan statistic are simulated for a variety of epidemiologically motivated situations two cluster configurations are used a rectangular pulse and a triangular pulse designed to emulate environmental contamination for a rectangular pulse the relative risk r of disease in the cluster region in r fold as high as it is for the background region the power is strongly influenced by the sample size the relative risk and the width or duration of the cluster region whereas the effect of the cluster configuration is small using a 5 per cent significance level a relative risk of 4 a standardized cluster duration of 0 10 a relative window width of 1 5 and a non random sample size of 50 the simulated power is approximately 80 per cent indicating that the minimum sample size in the cluster region for adequate power is in the 12 32 range for values of the parameters used in this study

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

a note on multisymplectic fourier pseudospectral discretization for the nonlinear schrodinger equation based on the multisymplectic fourier pseudospectral scheme for the nonlinear schrodinger nls equation we investigate some discrete properties corresponding to local conservation laws of the original equation the discrete normal conservation law is proved and the error estimation of local and global energy conservation laws are also obtained numerical experiments for cubic nls equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results

multi symplectic preissmann scheme for a high order wave equation of kdv type the high order wave equation of kdv type which describes many important physical phenomena has been investigated widely in last several decades here the new multi symplectic formulations for the high order wave equation of kdv type are presented and the associated local conservation laws are shown the multi symplectic discretization of each formulation is exemplified by the multi symplectic preissmann scheme the numerical experiments are given and the results verify the efficiency of the multi symplectic preissmann scheme

explicit multi symplectic methods for klein gordon schrodinger equations in this paper we propose explicit multi symplectic schemes for klein gordon schrodinger equation by concatenating suitable symplectic runge kutta type methods and symplectic runge kutta nystrom type methods for discretizing every partial derivative in each sub equation it is further shown that methods constructed in this way are multi symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions in the aim of the commonly practical applications a novel 2 order one parameter family of explicit multi symplectic schemes through such concatenation is constructed and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods furthermore some high order explicit multi symplectic schemes of such category are given as well good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments

multi symplectic spectral discretizations for the zakharov kuznetsov and shallow water equations abstract the time evolution of multi symplectic pdes with periodic boundary conditions in one and two space dimensions is considered we introduce the idea of a multi symplectic fourier transform for systems on periodic domains leading to a semi discretization on fourier space and to the concept of multi symplecticity on fourier space the spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it a discrete wave model a by product of the semi discretization is that it leads automatically to a system of hamiltonian odes in time when truncated we show that the one dimensional shallow water equations and the two dimensional zakharov kuznetsov equation are multi symplectic and derive spectral discretizations for these systems and present numerical experiments

an approximation for the distribution of the scan statistic the scan statistic evaluates whether an apparent cluster of disease in time is due to chance the statistic employs a moving window of length w and finds the maximum number of cases revealed through the window as it scans or slides over the entire time period t computation of the probability of observing a certain size cluster under the hypothesis of a uniform distribution is infeasible when n the total number of events is large and w is of moderate or small size relative to t we give an approximation that is an asymptotic upper bound easy to compute and for the purposes of hypothesis testing more accurate than other approximations presented in the literature the approximation applies both when n is fixed and when n has a poisson distribution we illustrate the procedure on a data set of trisomic spontaneous abortions observed in a two year period in new york city

determining optical flow optical flow cannot be computed locally since only one independent measurement is available from the image sequence at a point while the flow velocity has two components a second constraint is needed a method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image an iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences the algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time it is also insensitive to quantization of brightness levels and additive noise examples are included where the assumption of smoothness is violated at singular points or along lines in the image

the multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this article we consider partitioned runge kutta prk methods for hamiltonian partial differential equations pdes and present some sufficient conditions for multi symplecticity of prk methods of hamiltonian pdes

a multigrid platform for real time motion computation with discontinuity preserving variational methods variational methods are among the most accurate techniques for estimating the optic flow they yield dense flow fields and can be designed such that they preserve discontinuities estimate large displacements correctly and perform well under noise and varying illumination however such adaptations render the minimisation of the underlying energy functional very expensive in terms of computational costs typically one or more large linear or nonlinear equation systems have to be solved in order to obtain the desired solution consequently variational methods are considered to be too slow for real time performance in our paper we address this problem in two ways i we present a numerical framework based on bidirectional multigrid methods for accelerating a broad class of variational optic flow methods with different constancy and smoothness assumptions thereby our work focuses particularly on regularisation strategies that preserve discontinuities ii we show by the examples of five classical and two recent variational techniques that real time performance is possible in all cases even for very complex optic flow models that offer high accuracy experiments show that frame rates up to 63 dense flow fields per second for image sequences of size 160 120 can be achieved on a standard pc compared to classical iterative methods this constitutes a speedup of two to four orders of magnitude

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

quadratic invariants and multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this paper we study the preservation of quadratic conservation laws of runge kutta methods and partitioned runge kutta methods for hamiltonian pdes and establish the relation between multi symplecticity of runge kutta method and its quadratic conservation laws for schrodinger equations and dirac equations it reveals that multi symplectic runge kutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation respectively

the numerical study of a nonlinear one dimensional dirac equation we study the convergence of a difference scheme for a nonlinear one dimensional dirac equation also we show some numerical experiments by using this scheme

approximations and bounds for the distribution of the scan statistic abstract the scan statistic is used in many areas of science to test the null hypothesis of uniformity against a clustering alternative this article derives approximations and bounds for the distribution of scan statistics an extensive simulation study is carried out to compare the approximations and the bounds derived in this article with other known approximations and bounds the new lower and upper bounds that have been derived are very useful in evaluating the approximations and the simulation a quantity of average number of points in the scanning interval of length d d is introduced to study the approximations for the distribution of the scan statistic for low values of d the approximation derived in this article is the most accurate one the importance of the scan statistics arises from their applications in many areas including nuclear physics geology radio optics photography and epidemiology

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed