the multiplier of the interval 1 1 for the dunkl transform on the real line abstract we study the boundedness of the multiplier of the interval 1 1 for the dunkl transform of order 1 2 on weighted l p spaces with 1 p in particular we get that it is bounded from l p r x 2 1 d x into itself if and only if 4 1 2 3 p 4 1 2 1

yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

bessel functions of two variables some power series and plots for a finite reflection group g there is a rich theory developed by dunkl heckman and opdam leading to the notion of a commuting set of bessel differential operators these systems play an important role in the study of calogero moser systems and other problems of physical interest when g acts on the line one recovers the usual bessel function with a well known power series expansion at the origin we obtain some such expansions in the case of g a2 acting in the plane and we use these to produce plots of some of these functions

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

convergence of numerical solution to stochastic delay differential equation with poisson jump and markovian switching the main purpose of this paper is to study the convergence of numerical solutions to a class of stochastic delay differential equations with poisson jump and markovian switching a numerical approximation scheme is proposed to approximate the solution to stochastic delay differential equations with poisson jump and markovian switching it is proved that the euler approximation solution converge to the analytic solution in probability under weaker conditions some known results are generalized and improved an example is provided to illustrate our theory

convergence and stability of numerical solutions to sddes with markovian switching a class of linear stochastic delay differential equations with markovian switching is considered the main aim of this paper is to investigate the convergence and stability of the euler method of the equations it is proved that the euler method is convergent with strong order p 1 2 the ms stable properties of the euler scheme are also studied

convergence of numerical solutions to stochastic age dependent population equations with markovian switching in this paper a class of stochastic age dependent population equations with markovian switching is considered the main aim of this paper is to investigate the convergence of the numerical approximation of stochastic age dependent population equations with markovian switching it is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions an example is given for illustration

isogeometric analysis and hierarchical refinement for higher order phase field models abstract while the interest in higher order models in physics and mechanics grows their numerical simulation still poses a challenge especially for arbitrary shaped three dimensional domains this contribution presents the mathematical framework as well as the application to different problems in the field of material science fracture mechanics and diffusion problems all models under consideration require at least c 1 continuity which prevents the application of standard finite element analysis and local mesh refinements introducing isogeometric analysis iga for the discretization in a finite element framework enables us to deal with these requirements moreover a general hierarchical refinement scheme based on a subdivision projection is presented here for one two and three dimensional simulations this technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization using this mathematical framework the improved convergence of a kuramoto sivashinsky model a mesh adapted thermal diffusion simulation and computations of a priori unknown crack propagation in different fracture modes underline the versatility of the presented hierarchical refinement scheme

interpolation by weighted paley wiener spaces associated with the dunkl transform abstract given 1 2 0 and 1 p we study the interpolation problem in the space pw p of entire functions f c c of exponential type for which r f x p x 2 1 d x with nodes of interpolation at s j j z where s j j n is the increasing sequence of all positive roots of the bessel function j 1 z of order 1 and s j s j for all j z we prove that if 4 1 2 3 p 1 p p 2 4 1 2 1 the interpolation problem f pw p and f 1 s j c j for all j z has a unique solution for every sequence c j of complex numbers satisfying j z c j p 1 j 2 1 and that if p p 1 the corresponding interpolation problem may not have a solution and that the solution if exists is unique if and only if p p 2 finally we show that r f x p x 2 1 d x 2 2 j z f s j 1 p 1 j 2 1 with the constant of equivalence depending only on p and holds for all entire functions f of exponential type if and only if p 1 p p 2

convergence of numerical solutions to neutral stochastic delay differential equations with markovian switching recently numerical solutions of stochastic differential equations have received a great deal of attention it is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet in the paper the euler maruyama method for neutral stochastic delay differential equations is developed the key aim is to show that the numerical solutions will converge to the true solutions under the local lipschitz condition

the patch test for mixed formulations this paper presents a simple extension of the patch test to mixed formulations to provide the necessary and sufficient conditions for convergence the general algebraic conditions of babuska and brezzi are given a simple form and a conceptual application of the patch tests serves to point out the instability of several well known formulations for incompressible problems the test is also applied to mixed displacement strain formulations of elasticity and limiting formulations are noted

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

dunkl operators and a family of realizations of osp 1 2 in this paper a family of radial deformations of the realization of the lie superalgebra osp 1 vertical bar 2 in the theory of dunkl operators is obtained this leads to a dirac operator depending on 3 parameters several function theoretical aspects of this operator are studied such as the associated measure the related laguerre polynomials and the related fourier transform for special values of the parameters it is possible to construct the kernel of the fourier transform explicitly as well as the related intertwining operator

convergence of numerical solutions to stochastic differential equations with markovian switching abstract the paper develops strong approximation schemes for solutions of stochastic differential equations with markovian switching sdewmss the convergent orders of existing strong numerical schemes all are 0 5 this paper provides a convergence theorem for the construction of strong approximations of any given order of convergence for sdewmss and then constructs the order 1 scheme the order 1 5 scheme the paper also develops a new way to simulate the markov chain and hence the order 1 scheme finally a numerical example is provided to illustrate the theoretical results

mixed norm estimates for the cesaro means associated with dunkl hermite expansions our main goal in this article is to study mixed norm estimates for the cesaro means associated with dunkl hermite expansions on r d these expansions arise when one considers the dunkl hermite operator or dunkl harmonic oscillator h kappa delta kappa vertical bar x vertical bar 2 where delta kappa stands for the dunkl laplacian it is shown that the desired mixed norm estimates are equivalent to vector valued inequalities for a sequence of cesaro means for laguerre expansions with shifted parameter in order to obtain such vector valued inequalities we develop an argument to extend these laguerre operators for complex values of the parameters involved and apply a version of the three lines lemma

global well posedness and scattering for a class of nonlinear dunkl schrodinger equations in this paper we introduce a class of nonlinear schrodinger equations associated with the dunkl operators in this regard we study local and global well posedness and the scattering theory associated with these equations

inherently energy conserving time finite elements for classical mechanics in this paper we develop a finite element method for the temporal discretization of the equations of motion the continuous galerkin method is based upon a weighted residual statement of hamilton s canonical equations we show that the proposed finite element formulation is energy conserving in a natural sense a family of implicit one step algorithms is generated by specifying the polynomial approximation in conjunction with the quadrature formula used for the evaluation of time integrals the numerical implementation of linear quadratic and cubic time finite elements is treated in detail for the model problem of a circular pendulum in addition to that concerning dynamical systems with several degrees of freedom we address the design of nonstandard quadrature rules which retain the energy conservation property our numerical investigations assess the effect of numerical quadrature in time on the accuracy and energy conservation property of the time stepping schemes

nonlinear generalized dunkl wave equations and applications abstract in this paper we introduce a class of nonlinear wave equations associated with the generalized dunkl laplace operator we study local and global well posedness next we establish the linearization of bounded energy solutions in the spirit of gerard 1996 9 the proof uses strichartz type inequalities and the energy estimate

lp fourier multipliers for the dunkl operator on the real line we consider fourier multipliers for lp associated with the dunkl operator on r and establish a version of hormander s multiplier theorem in applying this version we come up with some results regarding the oscillating multipliers partial sum operators and generalized bessel potentials

fefferman stein inequalities for the mathbb z _2 d dunkl maximal operator in this article we establish the fefferman stein inequalities for the dunkl maximal operator associated with a finite reflection group generated by the sign changes similar results are also given for a large class of operators related to dunkl s analysis

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

radial mollifiers mean value operators and harmonic functions in dunkl theory abstract in this paper we show how to use mollifiers to regularize functions relative to a set of dunkl operators in r d with coxeter weyl group w multiplicity function k and weight function k in particular for a w invariant open subset of r d for d r d a radial function and u l loc 1 k x d x we study the dunkl convolution product u k and the action of the dunkl laplacian and the volume mean operators on these functions the results are then applied to obtain an analog of the weyl lemma for dunkl harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators

variational space time elements for large scale systems abstract in this paper we introduce a new galerkin based formulation for transient continuum problems governed by partial differential equations in space and time therefore we aim at a direct finite element discretization of the space time suitable for massive parallel analysis of the arising large scale problem the proposed formulation is applied to thermal mechanical and fluid systems as well as to a kuramoto sivashinsky problem representing the general class of higher order formulations in material science using nurbs based shape functions we verify whenever possible the conservation properties of the formulation finally a series of examples demonstrate the applicability to all systems presented in this paper

conservation properties of a time fe method part iii mechanical systems with holonomic constraints a galerkin based discretization method for index 3 differential algebraic equations pertaining to finite dimensional mechanical systems with holonomic constraints is proposed in particular the mixed galerkin mg methodis introduced which leads in a natural way to time stepping schemes that inherit major conservation properties of the underlying constrained hamiltonian system namely total energy and angular momentum in addition to that the constraints on the configuration level and on the velocity momentum level are fulfilled exactly the application of the mg method to specific mechanical systems such as the pendulum rigid body dynamics and the coupled motion of rigid and flexible bodies is presented related numerical examples are investigated to evaluate the numerical performance of the mg 1 and mg 2 method

stationary dynamical and chaotic states of the two dimensional damped kuramoto sivashinsky equation the long time behavior of the two dimensional damped kuramoto sivashinsky equation is studied numerically in the large aspect ratio limit the equation is shown to lead to three states depending on the magnitude of a damping parameter ensuremath alpha at large intermediate and small values of ensuremath alpha the equation leads respectively to hexagonal breathing hexagonal and disordered states the disordered phase is chaotic in both space and time consistent with spatiotemporal chaos the transitions between these states are examined using standard statistical measures

a continuous space time finite element method for the wave equation we consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time thereby giving a unified treatment of the spatial and temporal discretizations our analysis uses primarily energy arguments which are quite common for spatial discretizations but not for time we present a priori nodal in time superconvergence error estimates without any special time step restrictions our method is based on tensor product spaces for the full discretization

mean convergence of fourier dunkl series abstract in the context of the dunkl transform a complete orthogonal system arises in a very natural way this paper studies the weighted norm convergence of the fourier series expansion associated to this system we establish conditions on the weights in terms of the a p classes of muckenhoupt which ensure the convergence necessary conditions are also proved which for a wide class of weights coincide with the sufficient conditions

riesz transform and riesz potentials for dunkl transform analogues of riesz potentials and riesz transforms are defined and studied for the dunkl transform associated with a family of weight functions that are invariant under a reflection group the lp boundedness of these operators is established in certain cases

enabling local time stepping in the parallel implicit solution of reaction diffusion equations via space time finite elements on shallow tree meshes for many applications local time stepping offers an interesting and worthwhile alternative to the by now well established global time step control in fact local time stepping can allow for a highly detailed resolution of localized features of the solution with strongly reduced computational cost when compared to global time step control however local time stepping is not applicable in a straight forward manner in the context of fully implicit time discretizations here we present a method for the efficient parallel adaptive solution of non linear partial differential equations in particular reaction diffusion equations using spatially adapted time step sizes in the context of a fully implicit solution strategy our proposed method uses a discontinuous galerkin method in time approach within a full space time approach moreover it is designed from scratch for efficient parallel computation we employ shallow tree based mesh data structures in order to ensure a low memory footprint of the adaptive meshes by solving the time dependent partial differential equation on a d 1 dimensional non conforming mesh space time adaptivity is naturally achieved in combination with a discontinuous galerkin method in time the size of the arising systems can be precisely controlled we additionally introduce and discuss a stabilization scheme for space time mortar element methods that also has a highly positive impact on the efficiency of preconditioning techniques for the arising systems of equations we present results from extensive numerical experiments that address the question of convergence and efficiency linear and non linear solver performance parallel scalability up to 2048 cores as well as accuracy for the linear heat equation and a real world non linear reaction diffusion equation from the field of computational electrocardiology

multivariable continuous hahn and wilson polynomials related to integrable difference systems multivariable generalizations of the continuous hahn and wilson polynomials are introduced as eigenfunctions of rational ruijsenaars type difference systems with an external field

bubble functions prompt unusual stabilized finite element methods a second order linear scalar differential equation including a zero th order term is approximated using first the standard galerkin method enriched with bubble functions static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations the method is generalized to allow for a convection operator in the equation this work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena flows with chemical reactions and other important problems

a space time multigrid method for parabolic partial differential equations we consider the solution of parabolic partial differential equations pdes in standard time stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step by contrast the method presented in this paper treats the whole of the space time problem simultaneously thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space time variables of a given grid level the method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil numerical results for the one and two dimensional heat equations are presented and are shown to agree closely with predictions from fourier mode analysis

entropy elasticity of isotropic rubber like solids at finite strains abstract this paper presents a macroscopic continuum formulation and a numerical analysis of constitutive equations for stress entropy and heat flux describing the thermoelastic behavior of amorphous cross linked polymers above the glass transition temperature in which a specimen typically snap back with rubbery characteristics the introduced coupled thermomechanical functional extends the classical strain energy function proposed by ogden volume changes due to thermal expansion are regarded for a large temperature domain and play a remarkable role in thermoelastic materials the non linear thermoelastic problem is solved within a staggered fractional step method which is based on a two phase isentropic operator split each of the two symmetric sub problems retains the characteristic dissipative structure of the implicit monolithic scheme which consequently results to an unconditionally stable product formula algorithm the numerical analysis side shows in particular the capability of the theoretical framework reproducing the realistic physical stress deformation temperature relations of rubber distinctive attention is paid to the thermoelastic inversion phenomena a remarkable property governing the class of rubber like materials

space time discontinuous galerkin method for the compressible navier stokes equations a space time discontinuous galerkin finite element method for the compressible navier stokes equations is presented we explain the space time setting derive the weak formulation and discuss our choices for the numerical fluxes the resulting numerical method allows local grid adaptation as well as moving and deforming boundaries which we illustrate by computing the flow around a 3d delta wing on an adapted mesh and by simulating the dynamic stall phenomenon of a 2d airfoil in rapid pitch up maneuver

convergence of numerical solutions to stochastic age structured population equations with diffusions and markovian switching in this paper it is considered for a class of stochastic age structured population equations with diffusions and markovian switching most kind of equations are nonlinear and cannot be solved explicitly so the construction of efficient computational methods is of great importance the main aim of this paper is to develop a numerical scheme and investigate the convergence of numerical approximation an example is given for illustration

numerical solutions of stochastic differential delay equations under local lipschitz condition in this paper a variant of the euler maruyama method is used to define the numerical solutions for stochastic differential delay equations sddes with variable delay the key aim is to show that the numerical solutions will converge to the true solutions of the sddes under the local lipschitz condition

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

helicopter rotor dynamics by finite element time approximation abstract starting from a variational formulation based on hamilton s principle the paper exploits the finite element technique in the time domain in order to generate by an automated procedure a numerical approximation of the nonlinear periodic equations peculiar to the dynamics of helicopter rotors with this method a unified approach is developed which can be used to solve both the response and stability problem two simple examples are presented that show how the method can be used to study the trimmed response of a fully articulated rigid blade and the mechanical instability analysis of a four bladed rotor

analysis of a new space time parallel multigrid algorithm for parabolic problems we present and analyze a new space time parallel multigrid method for parabolic equations the method is based on arbitrarily high order discontinuous galerkin discretizations in time and a finite element discretization in space the key ingredient of the new algorithm is a block jacobi smoother we present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters a precise criterion for semi coarsening in time or full coarsening and give an asymptotic two grid contraction factor estimate we then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties

convergence of numerical solutions to stochastic delay differential equations with jumps this paper studies a class of stochastic delay differential equations with jumps sddejs explicit solutions can hardly be obtained for the sddejs appropriate numerical approximation schemes such as the euler scheme are needed to apply sddejs in practice or to study their properties in this paper it is proved that the euler approximation solutions converge to the analytic solution for sddejs under weaker conditions than the linear growth condition and global lipschitz condition an example is given for illustration

a numerical approach to space time finite elements for the wave equation we study a space time finite element approach for the nonhomogeneous wave equation using a continuous time galerkin method we present fully implicit examples in 1 1 2 1 and 3 1 dimensions using linear quadrilateral hexahedral and tesseractic elements krylov solvers with additive schwarz preconditioning are used for solving the linear system we introduce a time decomposition strategy in preconditioning which significantly improves performance when compared with unpreconditioned cases

conservation properties of a time fe method part ii time stepping schemes for non linear elastodynamics in the present paper one step implicit integration algorithms for non linear elastodynamics are developed the discretization process rests on galerkin methods in space and time in particular the continuous galerkin method applied to the hamiltonian formulation of semidiscrete non linear elastodynamics lies at the heart of the time stepping schemes algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals we newly introduce the assumed strain method in time which enables the design of energy momentum conserving schemes and which can be interpreted as temporal counterpart of the well established assumed strain method for finite elements in space the numerical examples deal with quasi rigid motion as well as large strain motion copyright 2001 john wiley sons ltd

littlewood paley operators associated with the dunkl operator on r we consider a vector version of l p multipliers for the dunkl transform on r and we prove l p inequalities for the littlewood paley operators

riesz transforms and fractional integration for orthogonal expansions on spheres balls and simplices abstract this paper studies the hardy littlewood sobolev hls inequality and the riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres balls and simplexes with weights being invariant under a general finite reflection group on r d the sharp index for the validity of the hls inequality is determined and the l p boundedness of the riesz transforms is established in particular our results extend a classical inequality of muckenhoupt and stein on conjugate ultraspherical polynomial expansions our idea is based on a new decomposition of the dunkl laplace beltrami operator on the sphere and some sharp asymptotic estimates of the weighted christoffel functions

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

continuous finite elements in space and time for the heat equation on analyse une nouvelle methode variationnelle pour l approximation de l equation de la chaleur a l aide des elements finis continus dans l espace et le temps on presente les estimations d erreurs et les resultats d experiences numeriques

a parallel approach to the variational transfer of discrete fields between arbitrarily distributed unstructured finite element meshes an essential ingredient for the discretization and numerical solution of coupled multiphysics or multiscale problems is stable and efficient techniques for the transfer of discrete fields between nonmatching volume or surface meshes here we present and investigate a new and completely parallel approach it allows for the transfer of discrete fields between unstructured volume and surface meshes which can be arbitrarily distributed among different processors no a priori information on the relation between the different meshes is required our inherently parallel approach is general in the sense that it can deal with both classical interpolation and variational transfer operators e g the l 2 projection and the pseudo l 2 projection it includes a parallel search strategy output dependent load balancing and the computation of element intersections as well as the parallel assembling of the algebraic representation of the respective transfer operator we describe our algorithmic framework and its

stability analysis for stochastic volterra levin equations with poisson jumps fixed point approach this paper is devoted to investigate a class of stochastic volterra levin equations with poisson jumps to the best of the authors knowledge till now the stability problem for this class of new systems has not yet been solved since poisson jumps are considered the main objective of this paper is to fill the gap by using the fixed point theory we first study the existence and uniqueness of the solution as well as the pth moment exponential stability for the considered system then based on the well known borel cantelli lemma we prove that the solution is almost surely pth moment exponentially stable our results improve and generalize those given in the previous literature finally two simple examples are provided to illustrate the effectiveness of the obtained results

comparison principle and stability of ito stochastic differential delay equations with poisson jump and markovian switching abstract in this paper the comparison principle for the nonlinear ito stochastic differential delay equations with poisson jump and markovian switching is established later using this comparison principle we obtain some stability criteria including stability in probability asymptotic stability in probability stability in the p th mean asymptotic stability in the p th mean and the p th moment exponential stability of such equations some known results are generalized and improved

numerical simulation of asymptotic states of the damped kuramoto sivashinsky equation the damped kuramoto sivashinsky equation has emerged as a fundamental tool for the understanding of the onset and evolution of secondary instabilities in a wide range of physical phenomena most existing studies about this equation deal with its asymptotic states on one dimensional settings or on periodic square domains we utilize a large scale numerical simulation to investigate the asymptotic states of the damped kuramoto sivashinsky equation on annular two dimensional geometries and three dimensional domains to this end we propose an accurate efficient and robust algorithm based on a recently introduced numerical methodology namely isogeometric analysis we compared our two dimensional results with several experiments of directed percolation on square and annular geometries and found qualitative agreement