weyl manifolds and connections a study is made of weyl connections and their associated holonomy groups the extent to which a weyl connection determines its associated metric and 1 form structures is also discussed finally the related concept of a locally metric connection is introduced and some relations between metric locally metric and weyl connections are established

mean square stability of second order runge kutta methods for stochastic differential equations in a previous paper we proposed the stochastic generalization of classical second order two stage explicit runge kutta rk methods the obtained stochastic schemes have second order in the weak sense in this paper the numerical stability of these rk schemes is studied the study focuses on stability with respect to the second moment ms stability figures of the stability domains of the numerical schemes are shown numerical examples that confirm the theoretical results are also presented

higher order implicit strong numerical schemes for stochastic differential equations higher order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed these are based on a stochastic taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero the regions of absolute stability of these implicit and related explicit methods are also examined

the composite euler method for stiff stochastic differential equations in this paper we present the composite euler method for the strong solution of stochastic differential equations driven by d dimensional wiener processes this method is a combination of the semi implicit euler method and the implicit euler method at each step either the semi implicit euler method or the implicit euler method is used in order to obtain better stability properties we give criteria for selecting the semi implicit euler method or the implicit euler method for the linear test equation the convergence properties of the composite euler method depend on the criteria for selecting the methods numerical results suggest that the convergence properties of the composite euler method applied to nonlinear sdes is the same as those applied to linear equations the stability properties of the composite euler method are shown to be far superior to those of the euler methods and numerical results show that the composite euler method is a very promising method

representation of lie algebra t3 and 2 variable 2 parameter bessel functions abstract in this paper we deal with the problem of framing 2 variable 2 parameter bessel functions 2v2pbf into the context of the representation q w m 0 of the 3 dimensional lie algebra t 3 further we derive generating relations and identities involving 2v2pbf furthermore we deduce some new and known relations involving various other forms of generalized bessel functions gbf and bessel functions bf as applications of these results

infinite variable bessel functions in two dimensional wannier stark systems we discuss the spectrum and the eigenfunctions of a 2d wannier stark system in both the tight binding and single band approximations we show that the unfamiliar infinite variable bessel functions play a crucial role in these considerations furthermore a closed formula for the eigenfunction for an arbitrary tight binding ansatz in terms of an expansion in wannier states is derived finally we give a closed form solution for the wannier stark states in the scope of the single band approximation

the αth moment stability for the stochastic pantograph equation in this paper we investigate the ath moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the razumikhin technique especially the linear stochastic pantograph equations and the semi implicit euler method applying them are considered the convergence result of the semi implicit euler method is obtained the stability conditions of the analytic solution of those equations and the numerical method are given finally some experiments are given

recursive algorithm for arrays of generalized bessel functions numerical access to dirac volkov solutions in the relativistic and the nonrelativistic theoretical treatment of moderate and high power laser matter interaction the generalized bessel function occurs naturally when a schrodinger volkov and dirac volkov solution is expanded into plane waves for the evaluation of cross sections of quantum electrodynamic processes in a linearly polarized laser field it is often necessary to evaluate large arrays of generalized bessel functions of arbitrary index but with fixed arguments we show that the generalized bessel function can be evaluated in a numerically stable way by utilizing a recurrence relation and a normalization condition only without having to compute any initial value we demonstrate the utility of the method by illustrating the quantum classical correspondence of the dirac volkov solutions via numerical calculations

conditions for a symmetric connection to be a metric connection a recent result in general circumstances for a four dimensional space time giving algebraic conditions on a curvature tensor of a symmetric connection so that the connection be metric is shown to be a special case of a more general result both of these results are shown formally to be of a generic nature in this new result the conditions are imposed on a tensor of a more general character than the curvature tensor in addition it is shown that once the symmetric connection is known to be metric the metric is uniquely defined up to a constant conformal factor for those special curvature tensors which are excluded from the original result supplementary conditions are suggested which alongside the original conditions are sufficient to ensure that most of these excluded curvature tensors are also riemann tensors

generating functions of multivariable generalized bessel functions and jacobi elliptic functions it is pointed out that the jacobi elliptic functions are the natural basis to get generating functions of the multivariable generalized bessel functions analytical and numerical results are given of interest for applications

generalized bessel functions and kapteyn series kaptey n series axe introduced for generaliq ed bessel functions in full analogy with or dinary bessel functions apart from the relevance that kapteyn series have in astronomical problems and in electromagnetic radiation theory interest in this type of series is justified by the possibility of expanding analytic functions of a complex variable in series of bessel functions with specific fea tures accordingly in this paper it is shown that it is possible to obtain expansions of m vaxiable analytic functions in series of gbf s which reproduce the structure of kapteyn series of ordinary bessel functions the possible field of applications is also suggested

on the compatibility of lorentz metrics with linear connections on four dimensional manifolds this paper considers four dimensional manifolds upon which there is a lorentz metric h and a symmetric connection which are originally assumed unrelated it then derives sufficient conditions on h and expressed through the curvature tensor of for to be the levi civita connection of some local lorentz metric g and calculates the relationship between g and h some examples are provided which help to assess the strength of the sufficient conditions derived

theory of two index bessel functions and applications to physical problems in this article the theory of two index bessel functions is presented their generating function series expansion and integral representations are discussed their usefulness in physical problems is also discussed in the context of analysis of radiation emitted by relativistic electrons in two frequency undulators finally the theoretical analysis proving addition and multiplication theorems for two index bessel functions are completed and their modified forms are introduced

convergence and stability of the split step θ method for stochastic differential equations in this paper we construct a new split step method for solving stochastic differential equations namely the split step q method under lipschitz and linear growth conditions we establish a mean square convergence theory of split step q approximate solutions moreover the mean square stability of the method for a linear test equation with real parameters is considered and the real mean square stability region is plotted finally numerical results are presented to demonstrate the efficiency of the split step q method

approximation of upper lyapunov exponents of bilinear stochastic differential systems a bilinear stochastic differential system is considered whose solution starting from x is denoted by x t x it is supposed that its upper lyapunov exponent lambda exists the purpose of the paper is to propose an efficient algorithm to approximate lambda it is shown that for markov chains bar x p h x p in mathbb n defined by several approximation schemes the lyapunov exponents bar lambda h are well defined and can be computed estimates are given in terms of the discretization step h of the theoretical error lambda bar lambda h then these results are applied to an engineering problem the stability of the motion of helicopter rotor blades in a turbulent wind

affine collineations in space time the existence of affine collineations in space time is discussed and the types of space time admitting proper affine collineations is displayed the close connection between such space times and their holonomy structure and local decomposability is established affine collineations with fixed points are also considered as is the problem of extending local affine collineations to the whole of space time

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

purely magnetic space times purely magnetic space times in which the riemann tensor satisfies rabcdubud 0 for some unit timelike vector ua are studied the algebraic consequences for the weyl and ricci tensors are examined in detail and consideration given to the uniqueness of ua some remarks concerning the nature of the congruence associated with ua are made

on the algebra of parallel endomorphisms of a pseudo riemannian metric on a pseudo riemannian manifold m g some fields of endomorphisms i e sections of end tm may be parallel for g they form an associative algebra e which is also the commutant of the holonomy group of g as any associative algebra e is the sum of its radical and of a semi simple algebra s this s may be of eight different types see 8 then for any self adjoint nilpotent element n of the commutant of such an s in end tm the set of germs of metrics such that e s n is non empty we parametrize it generically the holonomy algebra of those metrics is the full commutant o g s n and then apart from some degenerate cases e s n where n is the ideal spanned by n to prove it we introduce an analogy with complex differential calculus the ring r x x n replacing the field c this treats the case where the radical of e is principal and consists of self adjoint elements we add a glimpse in the case where this radical is not principal we build germs of pseudo riemannian metric admitting a parallel nilpotent endomorphism n end tm together with a semisimple associative algebra of parallel endomorphisms s 6 n possibly s rid motivation let m g be a pseudo riemannian manifold h its

a unified point of view on the theory of generalized bessel functions abstract bessel functions have been generalized in a number of ways and many of these generalizations have been proved to be important tools in applications in this paper we present a unified treatment thus proving that many of the seemingly different generalizations may be viewed as particular cases of a two variable function of the type introduced by miller during the sixties

normal form of a metric admitting a parallel field of planes local coordinate normal forms for two classes of metrics are obtained the first class is comprised of those metrics that have a field of degenerate but not necessarily null planes thereby extending a result of a g walker the second class of metrics are artinian metrics that possess a pair of parallel complementary null distributions this latter class is shown to be equivalent to a certain class of symplectic forms

covariantly constant tensors and holonomy structure in general relativity this paper attempts to give a comprehensive discussion of space times that admit vector fields or second order symmetric or skew symmetric tensor fields which are covariantly constant or recurrent the relationship between such space times m and their holonomy group is given

delay dependent stability analysis of numerical methods for stochastic delay differential equations this paper is concerned with the numerical solution of stochastic delay differential equations the focus is on the delay dependent stability of numerical methods for a linear scalar test equation with real coefficients by using the so called root locus technique the full asymptotic stability region in mean square of stochastic theta methods is obtained which is characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients as well as time stepsize and method parameter theta then this condition is compared with the analytical stability condition it is proved that the backward euler method completely preserves the asymptotic mean square stability of the underlying system and the euler maruyama method preserves the instability of the system our investigation also shows that not all theta methods with q 12 preserve this delay dependent stability some numerical examples are presented to confirm the theoretical results

mean square stability of two classes of theta methods for numerical computation and simulation of delayed stochastic hopfield neural networks abstract recently the investigation on the stability of the numerical solutions to delayed stochastic differential equations has received an increasing attention but there has been little work on the stability analysis of the numerical solutions to delayed stochastic hopfield neural networks dshnns so far the aim in this paper is to study the mean square exponential stability of the split step theta sst method and the stochastic linear theta slt method for the underlying model it is proved that for any 0 1 2 there exists a constant 0 depending on such that the numerical schemes produced by the sst method and the slt method are mean square exponentially stable for 0 under the same assumptions as those to guarantee the mean square exponential stability of the underlying continuous model for the case 1 2 1 we show the same stability conclusion for all 0 to carry out the required conclusion a novel technique for the stability analysis of discrete numerical schemes with multi time delays namely the weighted sum lyapunov functional method is proposed finally a numerical example is given to illustrate the application of the suggested methods and to verify the stability conclusions obtained

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

non normal drift structures and linear stability analysis of numerical methods for systems of stochastic differential equations we investigate mean square asymptotic stability of equilibria in linear systems of stochastic differential equations with non normal drift coefficients with particular emphasis on the role of interactions between the drift and diffusion structures that act along orthogonally to and laterally to the flow hence we construct test systems with non normal drift coefficients and characteristic diffusion structures for the purposes of a linear stability analysis of the q maruyama method next we discretise these test systems and examine the mean square asymptotic stability of equilibria of the resulting systems of stochastic difference equations finally we indicate how this approach may help to shed light on numerical discretisations of stochastic partial differential equations with multiplicative space time perturbations

double warped space times an invariant characterization of double warped space times is given in terms of newman penrose formalism and a classification scheme is proposed a detailed study of the conformal algebra of these space times is also carried out and some remarks are made on certain classes of exact solutions

an analysis of stability of milstein method for stochastic differential equations with delay this paper deals with the adapted milstein method for solving linear stochastic delay differential equations it is proved that the numerical method is mean square ms stable under suitable conditions the obtained result shows that the method preserves the stability property of a class of linear constant coefficient problems this is also verified by several numerical examples

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

the holonomy lie algebras of neutral metrics in dimension four the lie algebra isomorphism between su 1 1 su 1 1 and o 2 2 is used to obtain a list of subalgebras of the latter the resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the lie algebra of the holonomy group of a neutral metric in four dimensions the conclusions taken in conjunction with previously known results furnish a classification of such lie subalgebras of o 2 2 with only one case remaining unresolved

generalized bessel functions in tunnelling ionization we develop two new approximations for the generalized bessel function that frequently arises in the analytical treatment of strong field processes especially in non perturbative multiphoton ionization theories both these new forms are applicable to the tunnelling environment in atomic ionization and are analytically much simpler than the currently used low frequency asymptotic approximation for the generalized bessel function the second of the new forms is an approximation to the first and it is the second new form that exhibits the well known tunnelling exponential

on two dimensional bessel functions the general properties of two dimensional generalized bessel functions are discussed various asymptotic approximations are derived and applied to analyse the basic structure of the two dimensional bessel functions as well as their nodal lines

strong convergence and exponential stability of stochastic differential equations with piecewise continuous arguments for non globally lipschitz continuous coefficients abstract the paper deals with a split step method for stochastic differential equations with piecewise continuous arguments sepcas the strong convergence of the method is proved under non globally lipschitz conditions the exponential stability of the exact and numerical solutions is obtained some experiments are given to illustrate the conclusions

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

exponential mean square stability of numerical methods for systems of stochastic differential equations this paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split step theta method for stochastic systems first we consider linear autonomous systems under a sufficient and necessary condition for exponential mean square stability of the exact solution it is proved that the two classes of theta methods with q 0 5 are exponentially mean square stable for all positive step sizes and the methods with q 0 5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does not have this property finally we consider stochastic differential equations with jumps some similar results are derived

numerical integration of multiplicative noise stochastic differential equations a two stage runge kutta algorithm for vector ito and by transform also stratonovich stochastic differential equations with multiplicative noise has been developed the method is second order accurate but for vanishing drift the algorithm yields a martingale independent of step size several examples are included to illustrate our method a discussion of errors shows that sample size can be as important as truncation

t stability of the semi implicit euler method for delay differential equations with multiplicative noise the paper deals with the t stability of the semi implicit euler method for delay differential equations with multiplicative noise a difference equation is obtained by applying the numerical method to a linear test equation in which the wiener increment is approximated by a discrete random variable with two point distribution the conditions under which the method is t stable are considered and the numerical experiments are given

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

comment on generalized bessel functions in tunnelling ionization in the recent paper reiss and krainov 2003 j phys a math gen 36 5575 developed two new approximations for the generalized bessel function that arises in the analytical treatment of strong field multiphoton ionization theories we show that their tunnelling approximation 1 systematically underestimates the theoretical ionization rate by a factor of 2 or more for typical laser frequencies more important still only a few first ati peaks of the lowest energy may be reproduced quite well using this approximation

a spinor approach to walker geometry a four dimensional walker geometry is a four dimensional manifold m with a neutral metric g and a parallel distribution of totally null two planes this distribution has a natural characterization as a projective spinor field subject to a certain constraint spinors therefore provide a natural tool for studying walker geometry which we exploit to draw together several themes in recent explicit studies of walker geometry and in other work of dunajski 11 and plebanski 30 in which walker geometry is implicit in addition to studying local walker geometry we address a global question raised by the use of spinors

numerical approximations to the stationary solutions of stochastic differential equations in this paper we investigate the possibility of approximating the stationary solution of a stochastic differential equation sde we start with the random dynamical system generated by the sde with the multiplicative noise we prove that the pullback flow has a stationary point however the stationary point is not constructible explicitly therefore we look at the numerical approximation we prove that the discrete time random dynamical system also has a stationary point finally we prove mean square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes as well as almost surely convergence when the time step is rational

holonomy groups in general relativity the infinitesimal holonomy group structure of space time is discussed and related to the petrov type of the weyl tensor and the algebraic segre type of the energy momentum tensor the number of covariant derivatives of the curvature tensor required to determine the infinitesimal holonomy group is determined in each case and the complete classification scheme is tabulated some special cases of physical interest are investigated in more detail a geometrical approach is followed throughout

two dimensional bloch oscillations a lie algebraic approach a lie algebraic approach successfully used to describe one dimensional bloch oscillations in a tight binding approximation is extended to two dimensions this extension has the same algebraic structure as the one dimensional case while the dynamics shows a much richer behaviour the bloch oscillations are discussed using analytical expressions for expectation values and widths of the operators of the algebra it is shown under which conditions the oscillations survive in two dimensions and the centre of mass of a wavepacket shows a lissajous like motion in contrast to the one dimensional case a wavepacket shows systematic dispersion that depends on the direction of the field and the dispersion relation of the field free system

de rham wu decomposition of holomorphic riemannian manifolds it is shown that a de rham wu decomposition for holomorphic riemannian manifolds is obtained easily from the underlying real geometry of holomorphic riemannian manifolds by appealing to h wu s extension to pseudo riemannian manifolds of de rham s decomposition theorem

curvature structure in general relativity this paper investigates the extent to which the curvature structure of space time determines the metric stucture it continues the work of earlier papers by prescribing the curvature structure and the curvature covariant derivatives up to certain orders it is shown that with the exception of the so called generalized pp waves the curvature and its first and second covariant derivatives essentially determine the metric up to coordinate transformations

stability of analytical and numerical solutions of nonlinear stochastic delay differential equations this paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations sddes we derive sufficient conditions for the stability contractivity and asymptotic contractivity in mean square of the solutions for nonlinear sddes the results provide a unified theoretical treatment for sddes with constant delay and variable delay including bounded and unbounded variable delays then the stability contractivity and asymptotic contractivity in mean square are investigated for the backward euler method it is shown that the backward euler method preserves the properties of the underlying sddes the main results obtained in this work are different from those of razumikhin type theorems indeed our results hold without the necessity of constructing or finding an appropriate lyapunov functional

affine collineations in general relativity and their fixed point structure a general discussion of affine vector fields collineations in space time is given following a previous article by hall and da costa some general remarks are presented on the zeros of affine vector fields and then the general theory of such zeros is established some preliminary results regarding homothetic symmetries in two and three dimensional space times are also given the impossibility of reducing an affine algebra containing proper affine vector fields to homothetic or killing vector fields by means of a change of compatible metric is demonstrated

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

spacetimes admitting a vector field whose inner product with the riemann tensor is zero the equation v r 0 arises in various places in general relativity in particular as the integrability conditions of the equations lvg 2 g where is a constant these are the equations of a homothetic vector field v with a zero homothetic bivector dv 0 in some space time with metric g and r are the components of the riemann tensor of that metric in some frame in this paper the equation v r 0 is examined and the components of the riemann tensor for the spacetimes which admit nonzero solutions v of this equation are given the petrov types of the weyl tensors of these spacetimes are listed and as a result a correction is then made to a theorem in a paper by collinson and fugere about the petrov types of spacetimes which admit the type of separation that they require of the hamilton jacobi equation for these spacetimes

walker s theorem without coordinates we provide a coordinate free version of the local classification due to walker q j math 1 69 1950 of null parallel distributions on pseudo riemannian manifolds the underlying manifold is realized locally as the total space of a fiber bundle each fiber of which is an affine principal bundle over a pseudo riemannian manifold all structures just named are naturally determined by the distribution and the metric in contrast with the noncanonical choice of coordinates in the usual formulation of walker s theorem

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

curvature collineations in general relativity ii this paper is the second of a set of two papers on curvature collineations in general relativity the first paper presented the mathematical basis of curvature collineations and a possible approach to their study this paper continues from the first one by investigating in detail many of the cases where curvature collineations can occur in space time it is based on a classification of the curvature tensor which is discussed in the first paper and reviewed briefly here this together with the geometrical approach favored in this paper leads to a rather general discussion of the problem which it is hoped does not obscure those physical aspects of the situation that are important in einstein s theory

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations we develop two classes of test equations for the linear stability analysis of numerical methods applied to systems of stochastic ordinary differential equations of ito type sodes motivated by the theory of stochastic stabilization and destabilization these test equations capture certain fundamental effects of stochastic perturbation in systems of sodes while remaining amenable to analysis before and after discretization we then carry out a linear stability analysis of the theta maruyama method applied to these test equations investigating mean square and almost sure asymptotic stability of the test equilibria we discuss the implications of our work for the notion of a stability of the theta maruyama method and use numerical simulation to suggest extensions of our results to test systems with nonnormal drift coefficients

convergence and stability of the split step theta method for stochastic differential equations with piecewise continuous arguments abstract in this paper we consider the stochastic differential equations with piecewise continuous arguments sdepcas in which both the drift and the diffusion coefficients do not satisfy the global lipschitz and linear growth conditions especially the diffusion coefficients are highly non linear growing it is proved that the split step theta sst method with 1 2 1 is strongly convergent to sdepcas under the local lipschitz monotone and one sided lipschitz conditions it is also obtained that the sst method with 1 2 1 preserves the exponential mean square stability of sdepcas under the monotone condition and some condition on the step size without any restriction on the step size there exists 1 2 1 such that the sst method with 1 is exponentially stable in mean square moreover for sufficiently small step size the rate constant can be reproduced some numerical simulations are presented to illustrate the analytical theory

on a novel iterative method to compute polynomial approximations to bessel functions of the first kind and its connection to the solution of fractional diffusion diffusion wave problems we present an iterative method to obtain approximations to bessel functions of the first kind jp x p 1 via the repeated application of an integral operator to an initial seed function f0 x the class of seed functions f0 x leading to sets of increasingly accurate approximations fn x is considerably large and includes any polynomial when the operator is applied once to a polynomial of degree s it yields a polynomial of degree s 2 and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree we focus on the set of polynomial approximations generated from the seed function f0 x 1 this set of polynomials is useful not only for the computation of jp x but also from a physical point of view as it describes the long time decay modes of certain fractional diffusion and diffusion wave problems

quantum transport and localization in biased periodic structures under bi and polychromatic driving we consider the dynamics of a quantum particle in a one dimensional periodic potential lattice under the action of a static and time periodic field the analysis is based on a nearest neighbour tight binding model which allows a convenient closed form description of the transport properties in terms of generalized bessel functions the case of bichromatic driving is analysed in detail and the intricate transport and localization phenomena depending on the communicability of the two excitation frequencies and the bloch frequency are discussed the case of polychromatic driving is also discussed in particular for flipped static fields i e rectangular pulses which can support an almost dispersionless transport with a velocity independent of the field amplitude