generalized two step maruyama methods for stochastic differential equations abstract in this paper we propose generalized two step maruyama methods for solving ito stochastic differential equations numerical analysis concerning consistency convergence and numerical stability in the mean square sense is presented we derive sufficient and necessary conditions for linear mean square stability of the generalized two step maruyama methods we compare the stability region of the generalized two step maruyama methods of adams type with that of the corresponding two step maruyama methods of adams type and show that our proposed methods have better linear mean square stability a numerical example is given to confirm our theoretical results

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

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

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

determining the best decision in the presence of imprecise information abstract this paper addresses general aspects of decision making in an imprecise environment here the task of choosing a decision among the available alternatives with imprecisely defined outcomes is analyzed new definitions for the concept of best decision in such an environment are introduced these definitions are based on fuzzy set theoretic representations of decision outcomes

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

mean square stability of second order runge kutta methods for multi dimensional linear stochastic differential systems in this paper the mean square stability of second order runge kutta schemes for multi dimensional linear stochastic differential systems is studied motivated by the work of tocino mean square stability of second order runge kutta methods for stochastic differential equations j comput appl math 175 2005 355 367 and saito and mitsui mean square stability of numerical schemes for stochastic differential systems in international conference on scientific computation and differential equations july 29 august 3 2001 vancouver british columbia canada we investigate the mean square stability of second order runge kutta schemes for multi dimensional linear stochastic differential systems with one multiplicative noise stability criteria are established and numerical examples that confirm the theoretical results are also presented

numerical solution of stochastic differential problems in the biosciences stochastic differential equations sdes models play a prominent role in many application areas including biology epidemiology and population dynamics mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do so suitable numerical methods must be introduced to simulate the solutions of the resulting stochastic differential systems in this work we take into account both euler taylor expansion and runge kutta type methods for stochastic ordinary differential equations sodes and the euler maruyama method for stochastic delay differential equations sddes focusing on the most relevant implementation issues the corresponding matlab codes for both sodes and sddes problems are tested on mathematical models arising in the biosciences math comput simulation numerical analysis of a campbell like stochastic delay model preprint

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

asymptotic mean square stability of two step maruyama schemes for stochastic differential equations the mean square stability for two step schemes applied to scalar stochastic differential equations is studied necessary and sufficient conditions in terms of the parameters of the schemes guaranteeing their ms stability are derived particular members of the studied family are considered their stability regions are plotted and compared with the stability region of the linear test equation it is proved that the stochastic two step bdf scheme is unconditionally ms stable numerical experiments that confirm the theoretical results are shown

the lusternik schnirelmann category of a lie groupoid we propose a new homotopy invariant for lie groupoids which generalizes the classical lusternik schnirelmann category for topological spaces we use a bicategorical approach to develop a notion of contraction in this context we propose a notion of homotopy between generalized maps given by the 2 arrows in a certain bicategory of fractions this notion is invariant under morita equivalence thus when the groupoid defines an orbifold we have a well defined ls category for orbifolds we prove an orbifold version of the classical lusternik schnirelmann theorem for critical points

convergence and asymptotic stability of the explicit steklov method for stochastic differential equations in this paper we develop a new numerical method with asymptotic stability properties for solving stochastic differential equations sdes the foundations for the new solver are the steklov mean and an exact discretization for the deterministic version of the sdes strong consistency and convergence properties are demonstrated for the proposed method moreover a rigorous linear and nonlinear asymptotic stability analysis is carried out for the multiplicative case in a mean square sense and for the additive case in a path wise sense using the pullback limit in order to emphasize the characteristics of the steklov discretization we use as benchmarks the stochastic logistic equation and the langevin equation with a nonlinear potential of the brownian dynamics we show that the steklov method has mild stability requirements and allows long time simulations in several applications

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

ranking of fuzzy sets based on the concept of existence abstract various approaches have been proposed for the comparison or ranking of fuzzy sets however due to the complexity of the problem a general method which can be used for any situation still does not exist this paper formalizes the concept of existence for the ranking of fuzzy sets many of the existing fuzzy ranking methods are shown to be some application of this concept an improved fuzzy ranking method is then introduced based on this concept this newly introduced method is expanded for treating both normal and nonnormal convex and nonconvex fuzzy sets emphasis is placed on the use of the subjectivity of the decision maker such as the optimistic or the pessimistic view points an improved procedure for obtaining linguistic conclusions is also developed finally some numerical examples are given to illustrate the approach

split step forward methods for stochastic differential equations in this paper we discuss split step forward methods for solving ito stochastic differential equations sdes eight fully explicit methods the drifting split step euler drsse method the diffused split step euler disse method and the three stage milstein tsm 1a tsm 1f methods are constructed based on euler maruyama method and milstein method respectively in this paper their order of strong convergence is proved the analysis of stability shows that the mean square stability properties of the methods derived in this paper are improved on the original methods the numerical results show the effectiveness of these methods in the pathwise approximation of ito sdes

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

fuzzy optimal search methods abstract this paper gives analytical approaches and algorithms for decision tree search techniques the decision trees are assumed to have a fixed number of stages and predefined possible states at every stage the costs of traversing the tree are characterized as approximate and defined as fuzzy numbers two search methods each drawing from an existing non fuzzy search algorithm are described the first method is a dynamic programming search where the principle of optimality with fuzzy costs is addressed the second method is an a search for which the notion of a lower bound estimate of costs is utilized to increase the efficiency of the search algorithms for each method are included theorems and proofs which complete the analytical development of these techniques are also included finally a numerical example illustrating the procedure is given

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

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

van kampen theorems for toposes abstract in this paper we introduce the notion of an extensive 2 category to be thought of as a 2 category of generalized spaces we consider an extensive 2 category k equipped with a binary product preserving pseudofunctor c k op cat which we think of as specifying the coverings of our generalized spaces we prove in this context a van kampen theorem which generalizes and refines one of brown and janelidze the local properties required in this theorem are stated in terms of morphisms of effective descent for the pseudofunctor c we specialize the general van kampen theorem to the 2 category top s of toposes bounded over an elementary topos s and to its full sub 2 category ltop s determined by the locally connected toposes after showing both of these 2 categories to be extensive we then consider three particular notions of coverings on toposes corresponding respectively to local homeomorphisms covering projections and unramified morphisms in each case we deduce a suitable version of a van kampen theorem in terms of coverings and under further hypotheses also one in terms of fundamental groupoids an application is also given to knot groupoids and branched coverings along the way we are led to investigate locally constant objects in a topos bounded over an arbitrary base topos s and to establish some new facts about them

delay dependent stability of stochastic split step θ methods for stochastic delay differential equations abstract in this paper the delay dependent asymptotic mean square stability of the stochastic split step method for a scalar linear stochastic delay differential equation with real coefficients is studied the full stability region of this method is given by using root locus technique the necessary and sufficient condition with respect to the equation coefficients time stepsize and method parameter is derived it is proved that the stochastic split step backward euler can preserve the asymptotic mean square stability of the underlying system completely in addition the numerical stability regions of the stochastic split step method and the stochastic method are compared with each other at last we validate our conclusions by numerical experiments

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

a shift splitting hierarchical identification method for solving lyapunov matrix equations in the present paper we propose a hierarchical identification method sshi for solving lyapunov matrix equations which is based on the symmetry and skew symmetry splitting of the coefficient matrix we prove that the iterative algorithm consistently converges to the true solution for any initial values with some conditions and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factors appropriately furthermore we show that the method adopted can be easily extended to study iterative solutions of other matrix equations such as sylvester matrix equations finally we test the algorithms and show their effectiveness using numerical examples

diagonally drift implicit runge kutta methods of strong order one for stiff stochastic differential systems the class of strong stochastic runge kutta methods for stochastic differential equations sdes due to ro sler is considered based on the order conditions of this class we design a family of strong diagonally drift implicit stochastic runge kutta ddisrk methods of order one for solving ito sde systems with an m dimensional standard wiener process we then explicitly provide the structure of the mean square stability matrix of the new ddisrk methods for the general form of linear sde systems with some specific values of the coefficients for this family particular ddisrk methods that have good stability properties are proposed in order to check their convergence order and stability properties some numerical experiments are performed

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

etale stacks as prolongations abstract we give a complete and categorical characterization of etale stacks generalized orbifolds in various geometric contexts including differentiable stacks and topological stacks this characterization states that etale stacks are precisely those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms and makes no mention of an atlas or groupoid presentation moreover we show that the bicategory of etale differentiable stacks and local diffeomorphisms is equivalent to the bicategory of stacks on the site of smooth manifolds and local diffeomorphisms an analogous statement holds for other flavors of manifolds topological c k complex super and topological spaces locally homeomorphic to a given space x a slight modification of this result also holds in an even more general context including all etale topological stacks and zariski etale stacks which are analogues of deligne mumford stacks for the zariski topology and we also sketch a proof of an analogous characterization of classical deligne mumford algebraic stacks which we prove in full in 7 by different methods we go on to characterize effective etale stacks e g reduced orbifolds as precisely those stacks arising as the prolongations of sheaves of sets rather than stacks of groupoids it follows that etale stacks and in particular orbifolds induce a small gerbe over their effective part and all gerbes over effective etale stacks arise in this way we show that well known lie groupoids from foliation theory literature can be reproduced from this framework as natural presentations for moduli stacks of certain geometric structures for example there exists a classifying stack for riemannian metrics presented by haefliger s groupoid r 13 and submersions into this stack classify riemannian foliations and similarly for symplectic structures with the role of r replaced with s p we also prove some unexpected results for example the category of smooth n manifolds and local diffeomorphisms has binary products

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

runge kutta methods for numerical solution of stochastic differential equations the way to obtain deterministic runge kutta methods from taylor approximations is generalized for stochastic differential equations now by means of stochastic truncated expansions about a point for sufficiently smooth functions of an ito process a class of explicit runge kutta schemes of second order in the weak sense for systems of stochastic differential equations with multiplicative noise is developed also two runge kutta schemes of third order have been obtained for scalar equations with constant diffusion coefficients numerical examples that compare the proposed schemes to standard ones are presented

stability analysis of high order runge kutta methods for index 1 stochastic differential algebraic equations with scalar noise abstract in this paper a new class of implicit stochastic runge kutta srk methods is constructed for numerically solving systems of index 1 stochastic differential algebraic equations sdaes with scalar multiplicative noise by applying rooted tree theory analysis the family of coefficients of the proposed methods of order 1 5 are calculated in the mean square sense in particular we derive some four stage stiffly accurate semi implicit srk methods for approximating index 1 sdaes with scalar noise for these methods first ms stability functions applied to a scalar linear test equation with multiplicative noise are calculated then their regions of ms stability are compared with the corresponding ms stability region of the original sde accordingly we illustrate that the proposed schemes seems to have good stability properties numerical results will be presented to check the convergence order and computational efficiency of the new methods

numerical algorithms for solving discrete lyapunov tensor equation abstract in this paper we present several numerical algorithms to solve the discrete lyapunov tensor equation which is the generalized form of discrete lyapunov matrix equation based on the structure of the tensor equation we firstly propose a simple iterative si method to consider the numerical solution of the discrete lyapunov tensor equation then we extend the gradient based iterative gbi method and the residual norm steepest descent rnsd iterative method to study the tensor equation respectively furthermore we develop a residual norm conjugate gradient rncg iterative method to accelerate the convergence speed of the rnsd method convergence analysis shows that the proposed methods converge to an exact solution for any initial value finally some numerical examples are provided to illustrate the efficiency and validity of these methods proposed

mean square stability and almost sure exponential stability of two step maruyama methods of stochastic delay hopfield neural networks abstract in this paper the two step maruyama methods of stochastic delay hopfield neural networks are studied we have found that under what choices of step size the two step maruyama methods of stochastic delay hopfield networks maintain the stability of exact solutions the mean square stability of two step maruyama methods of stochastic delay hopfield neural networks is investigated under suitable conditions also the almost sure exponential stability of two step maruyama methods of stochastic delay hopfield networks is proved using the semi martingale convergence theorem further the comparisons of stability conditions to the previous results in liu and zhu 2015 rathinasamy 2012 and ronghua et al 2010 are given numerical experiments are provided to illustrate our theoretical results

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

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

a derivative free explicit method with order 1 0 for solving stochastic delay differential equations a new explicit stochastic scheme of order 1 is proposed for solving stochastic delay differential equations sddes with sufficiently smooth drift and diffusion coefficients and a scalar wiener process the method is derivative free and is shown to be stable in mean square a stability theorem for the continuous strong approximation of the solution of a linear test equation by the milstein method is also proved which shows the stepsize restriction for stability is larger than those given previously in the literature the case of linear sddes is further investigated in order to compare the stepsize restrictions for stability of these two methods numerical experiments are given to illustrate the obtained stability properties

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

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

mean square stability properties of an adaptive time stepping sde solver we consider stability properties of a class of adaptive time stepping schemes based upon the milstein method for stochastic differential equations with a single scalar forcing in particular we focus upon mean square stability for a class of linear test problems with multiplicative noise we demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls one for the drift term and one for the diffusion

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

strongly graded groupoids and strongly graded steinberg algebras abstract we study strongly graded groupoids which are topological groupoids g equipped with a continuous surjective functor g to a discrete group such that 1 1 1 for all we introduce the category of graded g sheaves and prove an analogue of dade s theorem g is strongly graded if and only if every graded g sheaf is induced by a g e sheaf the steinberg algebra of a graded ample groupoid is graded and we prove that the algebra is strongly graded if and only if the groupoid is applying this result we obtain a complete graphical characterisation of strongly graded leavitt path and kumjian pask algebras

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

the w transform in stability analysis for stochastic linear functional difference equations abstract we apply the method of integral transforms to study stability properties of rather general linear functional difference equations of ito type this approach called the w method is based on integral transforms coming from auxiliary equations that already possess the desired asymptotic properties some sufficient conditions for different kinds of stability are found in this way a general framework is offered as well that can be used to study asymptotic properties of other classes of stochastic difference equations

an error corrected euler maruyama method for stiff stochastic differential equations in this paper we propose an error corrected euler maruyama method which is constructed by adding an error correction term to the euler maruyama scheme the correction term is derived from an approximation of the difference between the exact solution of stochastic differential equations and the euler maruyama s continuous time extension the method is proved to be mean square convergent with order 1 2 and is as easy to implement as standard explicit schemes but much more efficient for solving stiff stochastic problems for a linear scalar test equation with a scalar noise term it is shown that the mean square stability domain of the method is much bigger than that of the euler maruyama method it is proved the method preserves the mean square stability and asymptotic stability of the linear scalar equation without any constraint on the numerical step size finally numerical examples are reported to show the accuracy and effectiveness of the method

on mean square stability properties of a new adaptive stochastic runge kutta method we analyze the mean square ms stability properties of a newly introduced adaptive time stepping stochastic runge kutta method which relies on two local error estimators based on drift and diffusion terms of the equation a foroush bastani s m hosseini a new adaptive runge kutta method for stochastic differential equations j comput appl math 206 2007 631 644 in the same spirit as h lamba t seaman mean square stability properties of an adaptive time stepping sde solver j comput appl math 194 2006 245 254 and with applying our adaptive scheme to a standard linear multiplicative noise test problem we show that the ms stability region of the adaptive method strictly contains that of the underlying stochastic differential equation some numerical experiments confirms the validity of the theoretical results

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

choice of θ and mean square exponential stability in the stochastic theta method of stochastic differential equations this paper examines the relationship of choice of and mean square exponential stability in the stochastic theta method stm of stochastic differential equations sdes and mainly includes the following three results i under the linear growth condition for the drift term when 0 1 2 the stm may preserve the mean square exponential stability of the exact solution but the counterexample shows that the stm cannot reproduce this stability without this linear growth condition ii when 1 2 1 without the linear growth condition for the drift term the stm may reproduce the mean square exponential stability of the exact solution but the bound of the lyapunov exponent cannot be preserved iii when 1 this stm is called as the backward euler maruyama bem method the stm can reproduce not only the mean square exponential stability but also the bound of the lyapunov exponent this paper also gives the sufficient and necessary conditions of the mean square exponential stability of the stm for the linear sde when 0 1 2 and 1 2 1 respectively and the simulations also illustrate these theoretical results

on the discrete lyapunov matrix equation some bounds for the arithmetic and the geometric means of the characteristic roots of the positive semidefinite solution to the discrete lyapunov matrix equation are derived

the sylvester equation and approximate balanced reduction abstract the purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems the resulting approach is completely automatic once an error tolerance is specified and also yields an error bound this is to be contrasted with existing projection methods namely pade via lanczos pvl and rational krylov which do not satisfy these properties our approach is based on the computation and approximation of the cross gramian of the system the cross gramian is the solution of a sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system it is well suited to large scale problems because there are no matrix factorizations of the large sparse system matrix only matrix vector products are required

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

exponential mean square stability of the theta approximations for neutral stochastic differential delay equations in this paper a split step theta sst method is introduced and analyzed for neutral stochastic differential delay equations nsddes it is proved that the sst method with 0 1 2 can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient but for 1 2 1 the sst can reproduce the exponential mean square stability unconditionally then based on the stability results of sst scheme we examine the exponential mean square stability of the stochastic linear theta slt approximation for nsddes and obtain the similar stability results to that of the sst method moreover for sufficiently small stepsize we show that the decay rate as measured by the lyapunov exponent can be reproduced arbitrarily accurately these results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for nsddes

localic metric spaces and the localic gelfand duality abstract in this paper we prove as conjectured by b banachewski and c j mulvey that the constructive gelfand duality can be extended into a duality between compact regular locales and unital abelian localic c algebras in order to do so we develop a constructive theory of localic metric spaces and localic banach spaces we study the notion of localic completion of such objects and the behavior of these constructions with respect to pull back along geometric morphisms

stability of the drift implicit and double implicit milstein schemes for nonlinear sdes abstract this paper examines the stability of numerical solutions of nonlinear stochastic differential equations sdes with non global lipschitz continuous coefficients two implicit milstein schemes called drift implicit milstein scheme and double implicit milstein scheme are considered to simulate the underlying sdes it is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems

convergence and stability of the semi tamed euler scheme for stochastic differential equations with non lipschitz continuous coefficients recently explicit tamed schemes were proposed to approximate the sdes with the non lipschitz continuous coefficients this work proposes a semi tamed euler scheme which is also explicit to solve the sdes with the drift coefficient equipped with the lipschitz continuous part and non lipschitz continuous part it is shown that the semi tamed euler converges strongly with the standard order one half to the exact solution of the sde we also investigate the stability inheritance of the semi tamed euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution numerical experiments confirm the theoretical analysis

toposes as homotopy groupoids the purpose of this paper is to deepen our understanding of toposes as generalized spaces we prove that for any grothendieck topos 8 there exists a space x and a collection p of paths in x so that 6 is equivalent to the category of sheaves on x which are constant along the paths in p furthermore by taking homotopy classes of paths in p we obtain a continuous groupoid g 3 x so that the sheaves on x which are acted upon by g are exactly those sheaves which are constant along the paths in p as a consequence we thus deduce one of the main results proved in jtj showing that any topos can be described as the category of equivariant sheaves for some continuous groupoid our construction has the following additional properties the covering map cp x 6 is connected and locally connected and tiberwise contractible this implies that cp induces a full embedding of a into the category of sheaves on a which preserves exponentials in addition to tirst order logic moreover it implies that cp induces an isomorphism between the etale cohomology groups of the topos 8 and the cohomology of the space x this last aspect will be discussed in a different paper

time interval model reduction of bilinear systems in this paper a new method for model reduction of bilinear systems is presented the proposed technique is from the family of gramian based model reduction methods the method uses time interval generalised gramians in the reduction procedure rather than the ordinary generalised gramians and in such a way that it improves the accuracy of the approximation within the time interval for which the method is applied the time interval generalised gramians are the solutions to the generalised time interval lyapunov equations the conditions for these equations to be solvable are derived and an algorithm is proposed to solve these equations iteratively the method is further illustrated with the help of two illustrative examples the numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

high weak order methods for stochastic differential equations based on modified equations inspired by recent advances in the theory of modified differential equations we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations this approach is illustrated with the constructions of new methods of weak order two in particular semi implicit integrators well suited for stiff mean square stable stochastic problems and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system numerical examples confirm the theoretical results and show the versatility of our methodology

iterative solutions of coupled discrete markovian jump lyapunov equations in this paper two iterative methods are given to solve coupled discrete markovian jump lyapunov equations arising from the stability analysis of discrete time markovian jump systems when such equations have unique positive definite solutions a sufficient condition for the convergence of iterates for one iterative method is derived when such equations only have unique solutions but not necessarily positive definite the above iterative method may fail and an alterative iterative method is provided by using the solutions of discrete time lyapunov equations as its intermediate steps a sufficient condition is also presented for the convergence of the iterates

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

on the numerical behavior of matrix splitting iteration methods for solving linear systems we study the numerical behavior of stationary one step or two step matrix splitting iteration methods for solving large sparse systems of linear equations we show that inexact solutions of inner linear systems associated with the matrix splittings may considerably influence the accuracy of the approximate solutions computed in finite precision arithmetic for a general stationary matrix splitting iteration method we analyze two mathematically equivalent implementations and discuss the conditions when they are componentwise or normwise forward or backward stable we distinguish two different forms of matrix splitting iteration methods and prove that one of them is significantly more accurate than the other when employing inexact inner solves the theoretical results are illustrated by numerical experiments with an inexact one step and an inexact two step splitting iteration method

ms stability of nonnormal stochastic differential systems abstract mean square stability of 2 dimensional stochastic differential systems with nonnormal drift coefficient is investigated using routh hurwitz criterion explicit necessary and sufficient conditions in terms of the parameters of the stochastic system as well as geometrical representation of the stability regions are given in addition using jury s test a similar ms stability analysis is carried out for the numerical recurrence obtained applying an stochastic method to solve the system analytical comparison of the stability regions stochastic differential system vs stochastic numerical difference system is done concluding that stochastic methods are a stable in mean square for 1 2

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

asymptotic mean square stability of weak second order balanced stochastic runge kutta methods for multi dimensional ito stochastic differential systems abstract in this paper the linear asymptotic mean square stability of the weak second order stochastic runge kutta methods for multi dimensional ito stochastic differential equations due to rosler 2009 and tang and xiao 2017 are obtained further we have developed the stochastic runge kutta methods for multi dimensional ito stochastic differential equations to the balanced stochastic runge kutta methods by using the control functions we carry out a linear stability analysis of the class of stochastic runge kutta methods for the linear test equations with multiplicative noise thereby providing an explicit structure of stability matrices some comparisons and illustrations shows that there is an improvement in the stability and error analysis of these new balanced stochastic runge kutta methods comparatively to stochastic runge kutta methods and thus conforming the obtained theoretical results

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

almost sure and moment exponential stability of euler maruyama discretizations for hybrid stochastic differential equations positive results are derived concerning the long time dynamics of numerical simulations of stochastic differential equation systems with markovian switching euler maruyama discretizations are shown to capture almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

the classifying topos of a continuous groupoid i we investigate some properties of the functor b which asso ciates to any continuous groupoid g its classifying topos bg of equivariant g sheaves in particular it will be shown that the category of toposes can be obtained as a localization of a category of continuous groupoids if g is a group the category of g sets sets equipped with a right g action is a topos bg which classifies principal g bundles for instance if a is a topological space there is an equivalence between topos morphisms sheaves a bg and principal g bundles over x the construction of bg also applies to the case where g is a topological group or more generally a topological groupoid it is a rather surprising result that this essentially exhausts the range of toposes joyal and tierney 1984 have shown that any topos is equivalent to one of the form bg for a topological group g provided one works with a slightly generalized notion of topological space by taking the lattice of open sets as the primitive notion rather than the set of points one sometimes speaks of pointless spaces the continuous groupoids of this paper are the groupoid objects in this category of generalized spaces the aim of this paper is threefold first g bg is a functor and we wish to investigate how the properties of the topos bg depend on those of the contin uous groupoid g and more generally how the properties of a geometric morphism bg bh depend on those of the map of continuous groupoids g h the second aim is to extend the joyal tierney result and not only represent toposes in terms of continuous groupoids but also the geometric morphisms from one topos to another there are several possible solutions to this problem in this paper i present one approach and show that the category of toposes can be obtained as a category of fractions from a category of continuous groupoids another approach somewhat similar in spirit to the morita theorems for categories of modules will be presented elsewhere the third aim of this paper is of a more methodological nature in presenting many arguments concerning generalized pointless spaces i have tried to convey the idea that by using change of base techniques and exploit ing the internal logic of a grothendieck topos point set arguments are perfectly suitable for dealing with pointless spaces at least as long as one stays within the stable part of the theory although the general underlying idea is very clear see e g the discussion in 5 3 below it is a challenging open problem to express this