almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks stability of numerical solutions to stochastic delay differential equations have received an increasing attention but there has been so far little work on the stability analysis of numerical solutions to stochastic delay hopfield neural networks the aim of this paper is to study the almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks by using two approaches the euler method and the backward euler method under some reasonable conditions both the euler scheme and the backward euler scheme are proved to be almost sure exponential stability in particular the euler method and the backward euler method are mainly based on the semimartingale convergence theorem

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

m dimensional lattice sums of generalized hypergeometric functions representations and convergence criteria for infinite m dimensional lattice sums of generalized hypergeometric functions pfp 1 are deduced by appealing to the principle of mathematical induction in particular we show that such a lattice sum may be expressed essentially as a finite sum of mellin transforms of products of bessel functions of order m 2 and the functions pfp 1 in the lattice sum in addition a direct derivation for the three dimensional case is provided moreover we construct a countably infinite class of null functions on increasingly larger open intervals which are parametrically independent of the functions pfp 1 generating them

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

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

almost sure asymptotic stability of drift implicit θ methods for bilinear ordinary stochastic differential equations in r 1 global almost sure asymptotic stability of stochastic methods with nonrandom variable step sizes when applied to bilinear nonautonomous homogeneous test systems of ordinary stochastic differential equations sdes is investigated sufficient conditions for almost sure asymptotic stability are proved for both analytical and numerical solutions in r1 the results of saito and mitsui world sci ser appl math 2 1993 333 siam j numer anal 33 1996 2254 higham siam j numer anal 38 2001 753 and schurz stochastic anal appl 14 1996 313 handbook of stochastic analysis and applications 2002 for the constant step sizes are carried over to the case with variable step sizes and nonautonomous linear test equations the investigations indicate that methods with variable step sizes or variable parameter governed by certain conditions can successfully be used to guarantee almost sure asymptotic stability while discretizing nonautonomous sdes

lattice sums of generalized hypergeometric functions representations and convergence criteria for one dimensional 1d and two dimensional 2d lattice sums of generalized hypergeometric functions pfp 1 are derived by using apparently new integral representations for the latter functions and 1d and 2d forms of the poisson summation formula these lattice sum representations concisely unify and generalize various specializations given previously by several authors and should prove useful in analysing finite size effects in systems subjected to non periodic boundary conditions

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

an auslander type result for gorenstein projective modules an artin algebra a is said to be cm finite if there are only finitely many up to isomorphisms indecomposable finitely generated gorenstein projective a modules we prove that for a gorenstein artin algebra it is cm finite if and only if every its gorenstein projective module is a direct sum of finitely generated gorenstein projective modules this is an analogue of auslander s theorem on algebras of finite representation type m auslander a functorial approach to representation theory in representations of algebras workshop notes of the third internat conference in lecture notes in math vol 944 springer verlag berlin 1982 pp 105 179 m auslander representation theory of artin algebras ii comm algebra 1974 269 310

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

s rock chebyshev methods for stiff stochastic differential equations we present and analyze a new class of numerical methods for the solution of stiff stochastic differential equations sdes these methods called s rock for stochastic orthogonal runge kutta chebyshev are explicit and of strong order 1 and possess large stability domains in the mean square sense for mean square stable stiff sdes they are much more efficient than the standard explicit methods proposed so far for stochastic problems and give significant speed improvement the explicitness of the s rock methods allows one to handle large systems without linear algebra problems usually encountered with implicit methods numerical results and comparisons with existing methods are reported

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

almost sure exponential stability of numerical solutions for stochastic delay differential equations using techniques based on the continuous and discrete semimartingale convergence theorems this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations sddes the important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of sddes directly this is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the chebyshev inequality and the borel cantelli lemma

numerical solutions of stochastic differential equations implementation and stability issues stochastic differential equations sdes arise from physical systems where the parameters describing the system can only be estimated or are subject to noise there has been much work done recently on developing numerical methods for solving sdes this paper will focus on stability issues and variable stepsize implementation techniques for numerically solving sdes effectively

periodic complexes and group actions in this paper we show that the cohomology of a connected cw complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional as applications we characterize those discrete groups that act freely and properly on in x sm we construct nonstandard free actions of rank two simple groups on finite complexes y sn x sm and we prove that a finite p group p acts freely on such a complex if and only if it does not contain a subgroup isomorphic to z p 3

relative homological algebra in the category of quasi coherent sheaves abstract in this paper we prove the existence of a flat cover and of a cotorsion envelope for any quasi coherent sheaf over a scheme x o x indeed we prove something more general we define what it is understood by the category of quasi coherent r modules where r is a representation by rings of a quiver q and we prove the existence of a flat cover and a cotorsion envelope for quasi coherent r modules then we use the fact that the category of quasi coherent sheaves on x o x is equivalent to the category of quasi coherent r modules for some q and r to get our result

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

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

a transformation formula for the generalized hypergeometric series 4f3 1 for positive a and b by employing representations for the mellin transform of the product of two generalized hypergeometric functions 1f2 a2x2 1f2 b2x2 a new transformation formula for the series 4f3 1 is deduced

almost sure exponential stability of the backward euler maruyama scheme for stochastic delay differential equations with monotone type condition this paper is a continuation of our previous paper in which the second author with mao and szpruch examined the almost sure stability of the euler maruyama em and the backward euler maruyama bem methods for stochastic delay differential equations sddes in the previous results although the drift coefficient may defy the linear growth condition the diffusion coefficient is required to satisfy the linear growth condition in this paper we want to further relax the condition under monotone type condition this paper will give the almost sure stability of the bem for sddes whose both drift and diffusion coefficients may defy the linear condition this improves the existing results considerably

totally acyclic complexes over noetherian schemes abstract we define a notion of total acyclicity for complexes of flat quasi coherent sheaves on a semi separated noetherian scheme and study these complexes using the pure derived category of flat quasi coherent sheaves we prove that a scheme is gorenstein if and only if every acyclic complex of flat quasi coherent sheaves is totally acyclic our formalism also removes the need for a dualising complex in several known results for rings including jorgensen s proof of the existence of gorenstein projective precovers

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

irreducible recurrences and representation theorems for 3f2 1 abstract by examining the irreducibility of a certain recurrence we show that the hypergeometric function of the title cannot be represented by gamma functions

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

asymptotic moment boundedness of the numerical solutions of stochastic differential equations few papers look at the asymptotic boundedness of numerical solutions of stochastic differential equations sdes one of the open questions is whether numerical approximations can reproduce the boundedness property of the underlying sdes in this paper we give positive answer to this question firstly we discuss the asymptotic moment upper bound of the ito type sdes and show that the euler maruyama em method is capable to preserve the boundedness property for sdes with the linear growth condition on both drift and diffusion coefficients but under the weaker assumption the one sided lipschitz on the drift coefficient the em method fails to work we then show that the backward em method can work in this situation

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

on the mellin transform of products of bessel and generalized hypergeometric functions we deduce in an elementary way representations for the mellin transform of a product of bessel functions 0f1 a2x2 and generalized hypergeometric functions pfp 1 b2x2 for a b 0 as a corollary we obtain a transformation formula for p 1fp 1 which was discovered by wimp in 1987 by using bailey s method for the specialization 3f2 1

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

cohomology theories based on flats abstract let a be an associative ring with identity k flat a the homotopy category of flat modules and k p flat a the full subcategory of pure complexes the quotient category k flat a k p flat a called here the pure derived category of flats was introduced by neeman in this category flat resolutions are unique up to homotopy and so can be used to compute cohomology we develop theories of tate and complete cohomology in the pure derived category of flats these theories extend naturally to sheaves over semi separated noetherian schemes where there are not always enough projectives but we do have enough flats as applications we characterize rings with finite sfli and schemes which are locally gorenstein

the homotopy category of complexes of projective modules abstract the homotopy category of complexes of projective left modules over any reasonably nice ring is proved to be a compactly generated triangulated category and a duality is given between its subcategory of compact objects and the finite derived category of right modules

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

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

cohomology theories based on gorenstein injective modules in this paper we study relative and tate cohomology of modules of finite gorenstein injective dimension using these cohomology theories we present variations of grothendieck local cohomology modules namely gorenstein and tate local cohomology modules by applying a sort of avramov martsinkovsky exact sequence we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by j herzog we discuss some properties of these modules and give some results concerning their vanishing and non vanishing

gorenstein projective dimension for complexes we define and study a notion of gorenstein projective dimension for complexes of left modules over associative rings for complexes of finite gorenstein projective dimension we define and study a tate cohomology theory tate cohomology groups have a natural transformation to classical ext groups in the case of module arguments we show that these maps fit into a long exact sequence where every third term is a relative cohomology group defined for left modules of finite gorenstein projective dimension

some adjoints in homotopy categories let r be a ring in a previous paper 11 we found a new description for the category k r proj it is equivalent to the verdier quotient k r flat s for some suitable s c k r flat in this article we show that the quotient map from k r flat to k r flat s always has a right adjoint this gives a new fully faithful embedding of k r proj into k r flat its virtue is that it generalizes to nonaffine schemes

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

flat covers of complexes abstract in this article we define and study flat complexes over any ring also we prove that any complex over a commutative noetherian ring with finite krull dimension has a flat cover and a dg flat cover

an integral homological characterization of finite groups abstract we show that a group g is finite if and only if every injective z g module has projective length one

sonine gegenbauer type integrals abstract representations for complete and incomplete sonine gegenbauer type integrals are given in terms of generalized hypergeometric functions in two and three variables and in terms of neumann type series certain interesting reductions of these integrals are also given in addition an application to a problem in diffraction theory is provided

the derived category of quasi coherent sheaves and axiomatic stable homotopy we prove in this paper that for a quasi compact and semi separated nonnecessarily noetherian scheme x the derived category of quasi coherent sheaves over x is a stable homotopy category in the sense of hovey palmieri and strickland answering a question posed by strickland moreover we show that it is unital and algebraic we also prove that for a noetherian semi separated formal scheme its derived category of sheaves of modules with quasi coherent torsion homologies is a stable homotopy category it is algebraic but if the formal scheme is not a usual scheme it is not unital therefore its abstract nature differs essentially from that of the derived category which is equivalent to in the case of a usual scheme

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

an alternative derivation of closed form representations for multidimensional lattice sums of generalized hypergeometric functions some representations for infinite m dimensional lattice sums of generalized hypergeometric functions which were deduced previously by the first named author are rederived here by appealing to an alternative more direct method in particular it is shown that these lattice sums are proportional to a finite sum of meijer s g functions

on certain cohomological invariants of groups abstract let r be a left and right 0 noetherian ring we show that if all projective left and all projective right r modules have finite injective dimension then all injective left and all injective right r modules have finite projective dimension using this result we prove that the invariants silp z g and spli z g which were introduced by gedrich and gruenberg 1987 15 are equal for any group g as an application of the latter equality we show that a group g is finite if and only if cd g 0 where cd is the generalized cohomological dimension of groups introduced by ikenaga 1984 21

the homotopy category of flat modules and grothendieck duality let r be a ring we prove that the homotopy category k r proj is always aleph 1 compactly generated and depending on the ring r it may or may not be compactly generated we use this to give a description of k r proj as a quotient of k r flat the remarkable fact is that this new description of k r proj generalizes to non affine schemes this will appear in murfet s thesis

a class of generalized hypergeometric summations abstract summations over the positive integers n of the generalized hypergeometric expressions 1 n p f p 1 n 2 x 2 x 0 are derived in closed form the specialization p 0 for example reduces to known results for schlomilch series in addition we record the apparently not readily available sine and cosine transforms of p f p 1 b 2 x 2 b 0 the latter of which is used together with a form of the poisson summation formula to deduce the aforementioned results

asymptotic expansion of a bessel function integral using hypergeometric functions abstract generalized hypergeometric functions are used to extend simplify and complete the analysis of stoyanov and farrel math comput 49 1987 275 279 and of wong math comput 50 1998 229 234 as well as putting their considerations within a wider framework the integral 0 2 sin a cos b j sin j sin d where j is the bessel function of order is analyzed in terms of generalized hypergeometric functions and a complete asymptotic expansion is given for 0 2 j sin j sin d

exponential stability of the exact solutions and θ em approximations to neutral sddes with markov switching exponential stability of the exact solutions and em 1 2 1 approximations to neutral stochastic differential delay equations with markov switching are investigated in this paper sufficient conditions are obtained to ensure the p th moment p 1 and almost sure exponential stability of the exact solutions as well as em approximations p 2 an example is provided to support our conclusions

on the critical case of the weber schafheitlin integral and a certain generalization abstract when the orders of the product of two bessel functions differ by an odd integer we show that the critical case defined in the sequel as the supercritical case of the discontinuous integral of weber and schafheitlin is proportional to a ratio of products of gamma functions the derivation we give is elementary in the sense that it avoids the use of contour integration and the calculus of residues we consider also a generalization of the later integral by essentially replacing one of the bessel functions by the hypergeometric function 1 f 2 as a byproduct of this investigation we deduce whipple s transformation formula for a well poised 6 f 5 1 and allied results