limits of hypercyclic and supercyclic operators abstract an operator t on a complex separable infinite dimensional hilbert space h is hypercyclic if there is a vector y in h such that the orbit y ty t2y is dense in h it is shown that if t is hypercyclic then t does not have any eigenvalue and that the union of the spectrum of t and the unit circle is a connected set this result is used to obtain a spectral characterization of the norm closure of the class hc h of all hypercyclic operators acting on h as well as a formula for the distance from a given operator to hc h analogous results are obtained for the closely related class sc h of all supercyclic operators this paper includes some information about the structure of the set of all hypercyclic supercyclic vectors of a given hypercyclic supercyclic resp operator t

supercyclic subspaces spectral theory and weighted shifts abstract a vector x in a banach space b is called hypercyclic for a bounded operator t if the orbit t n x n 0 is dense in b if the scalar multiples of the elements in the orbit are dense then the vector x is called supercyclic we give a general sufficient condition for a bounded operator on a banach space to have an infinite dimensional closed subspace of supercyclic vectors as a consequence we also obtain a spectral sufficient condition for the existence of such a subspace for an operator these results allow us to characterize unilateral and bilateral weighted shifts that have an infinite dimensional closed subspace of supercyclic vectors surprisingly there are weighted shift operators that have supercyclic vectors but in which all closed subspaces of supercyclic vectors are finite dimensional our results complement recent work on hypercyclic subspaces and supercyclic subspaces they also suggest that the problem of the existence of infinite dimensional closed subspaces of supercyclic vectors is not only different but also more difficult to handle than the corresponding problem for hypercyclic vectors

a new adaptive runge kutta method for stochastic differential equations in this paper we will present a new adaptive time stepping algorithm for strong approximation of stochastic ordinary differential equations we will employ two different error estimation criteria for drift and diffusion terms of the equation both of them based on forward and backward moves along the same time step we will use step size selection mechanisms suitable for each of the two main regimes in the solution behavior which correspond to domination of the drift based local error estimator or diffusion based one numerical experiments will show the effectiveness of this approach in the pathwise approximation of several standard test problems

dispersion and wave propagation in discrete and continuous models for granular materials abstract a generalised continuum model for granular media is derived by direct homogenisation of the discrete equations of motion in contrast to previous works on this topic continuum concepts such as stress and moment stress are introduced after homogenisation first a very simple one dimensional model is considered and the continuum version for this model is derived by replacing the difference quotients of the discrete model by differential quotients the dispersion relations of the discrete and the continuous model are derived and compared variational boundary conditions for the continuous model are deduced from the stationarity of the corresponding lagrangian the three dimensional case is treated in an essentially similar fashion the resulting continuum theory is a combination of a cosserat continuum and a higher order deformation gradient continuum the salient features of the theory are illustrated by means of the dispersion relations for planar wave propagation

the role of coefficients of a general spde on the stability and convergence of a finite difference method in this paper for the approximate solution of stochastic partial differential equations spdes of ito type the stability and application of a class of finite difference method with regard to the coefficients in the equations is analyzed the finite difference methods discussed here will be either explicit or implicit and a comparison between them will be reported we prove the consistency and stability of these methods and investigate the influence of the multiplier particularly multiplier of the random noise in mean square stability from stochastic version of lax richtmyer the convergence of these methods under some conditions are established numerical experiments are included to show the efficiency of the methods

a general implicit splitting for stabilizing numerical simulations of ito stochastic differential equations in this paper we describe a general 2nd order accurate weak sense procedure for stabilizing monte carlo simulations of ito stochastic differential equations the splitting procedure includes explicit runge kutta heun methods semi implicit methods and the trapezoidal rule we prove the semi implicit method of ottinger stochastic processes in polymeric fluids springer verlag berlin 1996 for stabilizing simulations in the presence of nonlinear drift coefficients

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

a physical perspective of the length scales in gradient elasticity through the prism of wave dispersion abstract the goal of this study is to understand the physical meaning and evaluate the intrinsic length scale parameters featured in the theories of gradient elasticity by deploying the analytical treatment and experimental measurements of the dispersion of elastic waves the developments are focused on examining the propagation of longitudinal waves in an aluminum rod with periodically varying cross section first the analytical solution for the dispersion relationship based on the periodic cell analysis of a bi layered laminate and bloch theorem is compared to two competing models of gradient elasticity it is shown that the customary gradient elastic model with two length scale parameters is able to capture the dispersion accurately up to the beginning of the first band gap on the other hand the gradient elastic model with an additional length scale affiliated with the fourth order time derivative in the field equation is shown to capture not only the first dispersion branch before the band gap but also the band gap itself and the preponderance of the second branch closed form relations between the microstructure parameters and the intrinsic length scales are obtained for both gradient elasticity models by way of the asymptotic treatment in the limit of a weak contrast between the laminae a clear physical meaning and scaling of the length scale parameters was established in terms of i the microstructure given by the size of the unit cell and the contrast between the laminae and ii thus induced dispersion relationship characterized by the location and the width of the band gap the analysis is verified through an experimental observation of wave dispersion and wave attenuation within the band gap a comparison between the analytical treatment the gradient elastic model with three intrinsic length scales and experimental measurements demonstrates a good agreement over the range of frequencies considered

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

semi fredholm theory hypercyclic and supercyclic subspaces a vector x in a banach space cal b is called hypercyclic for a bounded linear operator t cal b rightarrow cal b if the orbit t n x n geq 1 is dense in cal b we prove that if t is hereditarily hypercyclic and its essential spectrum meets the closed unit disk then there is an infinite dimensional closed subspace consisting except for zero entirely of hypercyclic vectors for t the converse is also true even if t is not hereditarily hypercyclic in this way we improve and extend to banach spaces a recent result for hilbert spaces we also investigate the corresponding problem for supercyclic operators in this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite dimensional closed subspace of hypercyclic vectors moreover for certain kinds of supercyclic operators we can characterize when they have an infinite dimensional closed subspace of supercyclic vectors 1991 mathematics subject classification 47b38 47b99

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

gradient elasticity in statics and dynamics an overview of formulations length scale identification procedures finite element implementations and new results in this paper we discuss various formats of gradient elasticity and their performance in static and dynamic applications gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher order spatial derivatives of strains stresses and or accelerations we focus on the versatile class of gradient elasticity theories whereby the higher order terms are the laplacian of the corresponding lower order terms one of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum we start with discussing the general mindlin theory that in its most general form has 903 constitutive elastic parameters but which were reduced by mindlin to three independent material length scales further simplifications are often possible in particular the aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures we also address how this can be extended to dynamics an overview of length scale identification and quantification procedures is given finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions details are provided for particular formats of gradient elasticity that can be implemented with simple linear finite element shape functions new numerical results show the removal of singularities in statics and dynamics as well as the size dependent mechanical response predicted by gradient elasticity

s linear and convex structures in function families abstract in this paper the notion of s lineability originally coined by vladimir i gurariy is introduced and developed in a general abstract setting this new notion is then applied to specific situations as for instance classes of differentiable nowhere monotone functions as well as families of vectors having dense orbit with respect to an operator large convex structures are also shown to exist inside the family of topologically mixing continuous selfmaps of a real compact interval

hereditarily hypercyclic operators abstract we show that a continuous linear operator t on a frechet space satisfies the so called hypercyclicity criterion if and only if it is hereditarily hypercyclic and if and only if the direct sum t t is hypercyclic in particular hypercyclic operators with either a dense generalized kernel or a dense set of periodic points i e chaotic in the sense of r l devaney 1989 an introduction to chaotic dynamical systems addison wesley reading ma must satisfy the criterion finally we provide a characterization of those weighted shifts t that are hereditarily hyper cyclic with respect to a given sequence n k of positive integers as well as conditions under which t and t n k k 1 share the same set of hypercyclic vectors

hypercyclic and cyclic vectors abstract let x denote an arbitrary separable banach space over the field of complex numbers and b x the banach algebra of all bounded linear operators on x we prove the following results 1 an element of the space x is hypercyclic supercyclic for all positive powers tn of an operator t in b x if it is hypercyclic supercyclic for t 2 under some condition on the spectrum of the adjoint of a cyclic operator the set of all cyclic vectors of the operator is dense this result extends to any cyclic commutative subset of b x 3 under a mild condition on the spectrum of a cyclic operator t the set of all separating vectors for the commutant t of t is dense this also extends to any cyclic commutative subset of b x 4 a slightly stronger version of a theorem of k f clancey and d d rogers on cyclic vectors finally we define and discuss hereditarily hypercyclic operators

elastic field of a spherical inclusion with non uniform eigenfields in second strain gradient elasticity in this paper within the complete form of mindlin s second strain gradient theory the elastic field of an isolated spherical inclusion embedded in an infinitely extended homogeneous isotropic medium due to a non uniform distribution of eigenfields is determined these eigenfields in addition to eigenstrain comprise eigen double and eigen triple strains after the derivation of a closed form expression for green s function associated with the problem two different cases of non uniform distribution of the eigenfields are considered as follows i radial distribution i e the distributions of the eigenfields are functions of only the radial distance of points from the centre of inclusion and ii polynomial distribution i e the distributions of the eigenfields are polynomial functions in the cartesian coordinates of points while the obtained solution for the elastic field of the latter case takes the form of an infinite series the solution to the former case is represented in a closed form moreover eshelby s tensors associated with the two mentioned cases are obtained

existence of common and upper frequently hypercyclic subspaces abstract we provide criteria for the existence of upper frequently hypercyclic subspaces and for common hypercyclic subspaces which include the following consequences there exist frequently hypercyclic operators with upper frequently hypercyclic subspaces and no frequently hypercyclic subspace on the space of entire functions each differentiation operator induced by a non constant polynomial supports an upper frequently hypercyclic subspace and the family of its non zero scalar multiples has a common hypercyclic subspace a question of costakis and sambarino on the existence of a common hypercyclic subspace for a certain uncountable family of weighted shift operators is also answered

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

non local dispersive model for wave propagation in heterogeneous media one dimensional case non local dispersive model for wave propagation in heterogeneous media is derived from the higher order mathematical homogenization theory with multiple spatial and temporal scales in addition to the usual space time co ordinates a fast spatial scale and a slow temporal scale are introduced to account for rapid spatial fluctuations of material properties as well as to capture the long term behaviour of the homogenized solution by combining various order homogenized equations of motion the slow time dependence is eliminated giving rise to the fourth order differential equation also known as a bad boussinesq problem regularization procedures are then introduced to construct the so called good boussinesq problem where the need for c1 continuity is eliminated numerical examples are presented to validate the present formulation copyright 2002 john wiley sons ltd

on causality of the gradient elasticity models abstract this paper is concerned with causality of the gradient elasticity models of heterogeneous materials as a rule these models are not strictly causal since they allow an infinite speed of energy transfer by means of either propagating or transient evanescent waves a discussion is presented in this paper of both physical and mathematical implications of this fact this discussion is carried out employing one dimensional 1d second order gradient models a phenomenological enhancement is proposed which makes these models causal the main idea behind this enhancement is that a partial differential equation that governs dynamic behaviour of a causal gradient elasticity model must be of the same order with respect to spatial coordinate and with respect to time the validity of this idea is confirmed in this paper by deriving a second order 1d continuum model for concrete a brief comparison is provided in conclusion of the equations of motion of the 1d second gradient elasticity models and those used in dynamics of thin bars it is shown that the proposed causal model corresponds to the most advanced dynamic models of such bars namely to the timoshenko model for the bending motion and the mindlin model for the longitudinal motion of a bar

hypercyclic operators on non normable frechet spaces abstract every infinite dimensional separable non normable frechet space admits a continuous hypercyclic operator a large class of separable countable inductive limits of banach spaces with the same property is given but an example of a separable complete inductive limit of banach spaces which admits no hypercyclic operator is provided it is also proved that no compact operator on a locally convex space is hypercyclic

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

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

asymptotic equivalence of homogenisation procedures and fine tuning of continuum theories abstract long wave models obtained in the process of asymptotic homogenisation of structures with a characteristic length scale are known to be non unique the term non uniqueness is used here in the sense that various homogenisation strategies may lead to distinct governing equations that usually for a given order of the governing equation approximate the original problem with the same asymptotic accuracy a constructive procedure presented in this paper generates a class of asymptotically equivalent long wave models from an original homogenised theory the described non uniqueness manifests itself in the occurrence of additional parameters characterising the model a simple problem of long wave propagation in a regular one dimensional lattice structure is used to illustrate important criteria for selecting these parameters the procedure is then applied to derive a class of continuum theories for a two dimensional square array of particles applications to asymptotic structural theories are also discussed in particular we demonstrate how to improve the governing equation for the rayleigh love rod and explain the reasons for the well known numerical accuracy of the mindlin plate theory

universality of sequences of operators related to taylor series abstract in this note the universality of a sequence of operators associated to the partial sums of the taylor series of a holomorphic function is investigated the emphasis is put on the fact that the taylor series are evaluated at a prescribed point and the variable is the center of the expansion the dynamics of the sequence of operators linked to the partial sums of a power series that is not generated by an entire function is also studied

gradient elastic stress analysis near cylindrical holes in a plane under bi axial tension fields abstract this paper is devoted to a gradient elastic stress analysis of an infinite plate weakened by a cylindrical hole and subjected to two perpendicular and independent uni axial tensions at infinity the problem setting can be considered as an extension and generalization of the well known kirsch problem of the classical elasticity theory which is here extended with respect to the external loadings and generalized with respect to the continuum framework a closed form solution in terms of displacements is derived for the problem within the strain gradient elasticity theory on plane stress strain assumptions the main characters of the total and cauchy stress fields are analyzed near the circumference of the hole for different combinations of bi axial tensions and for different parameter values for the original kirsch problem concerning a uni axially stretched plate the analytical solution fields for stresses and strains are compared to numerical results these results are shown to be in a full agreement with each other and in particular they reveal a set of new qualitative findings about the scale dependence of the stresses and strains provided by the gradient theory not common to the classical theory based on these findings we finally consider the physicalness of the concepts total and cauchy stress appearing in the strain gradient model

linear structure of hypercyclic vectors abstract a vector x in a banach space b is called hypercyclic for a bounded linear operator t b b if the orbit t n x n 1 is dense in b our main result states that if t is a compact perturbation of an operator of norm 1 and satisfies an appropiate extra hypothesis then there is an infinite dimensional closed subspace consisting except for zero entirely of hypercyclic vectors for t in particular the result applies to compact perturbations of the identity we also include applications to some weighted backward shifts and compact perturbations of the identity by weighted backward shifts this last result in combination with a recent one that states that every banach space admits an operator with a hypercyclic vector proves that in all banach space there is an operator t with an infinite dimensional closed subspace consisting except for zero of hypercyclic vectors the main result also applies to the differentiation operator and the translation operator t f z f z 1 on certain hilbert spaces consisting of entire functions

boundedness of derivatives and anti derivatives of holomorphic functions as a rare phenomenon abstract in this article we prove a general result which in particular suggests that on a simply connected domain in c all the derivatives and anti derivatives of the generic holomorphic function are unbounded a similar result holds for the operator t n of partial sums of the taylor expansion with center at z 0 seen as functions of the center we also discuss a universality result of these operators t n

universality of taylor series as a generic property of holomorphic functions for every pair of lebesgue measurable functions h g 0 2 there exists an increasing sequence n 1 2 3 such that re s n e h and im s n e g as n almost everywhere such trigonometric series are called universal trigonometric series 17 2 in 1970 14 luh proved the existence of universal taylor series with respect to summability matrices for f0 z 0 a z we write s z k 0 akz luh considers triangular matrices a ank n k 0 ank c ank 0 for k n satisfying the two conditions

hypercyclic subspaces for frechet space operators abstract a continuous linear operator t x x is hypercyclic if there is an x x such that the orbit t n x is dense and such a vector x is said to be hypercyclic for t recent progress show that it is possible to characterize banach space operators that have a hypercyclic subspace i e an infinite dimensional closed subspace h x of except for zero hypercyclic vectors the following is known to hold a banach space operator t has a hypercyclic subspace if there is a sequence n i and an infinite dimensional closed subspace e x such that t is hereditarily hypercyclic for n i and t n i 0 pointwise on e in this note we extend this result to the setting of frechet spaces that admit a continuous norm and study some applications for important function spaces as an application we also prove that any infinite dimensional separable frechet space with a continuous norm admits an operator with a hypercyclic subspace

hypercyclic subspaces and weighted shifts abstract we first generalize the results of leon saavedra and muller 2006 10 on hypercyclic subspaces to sequences of operators on frechet spaces with a continuous norm then we study the particular case of iterates of an operator t and show a simple criterion for having no hypercyclic subspace finally we deduce from this criterion a characterization of weighted shifts with hypercyclic subspaces on the spaces l p or c 0 on the space of entire functions and on certain kothe sequence spaces we also prove that if p is a non constant polynomial and d is the differentiation operator on the space of entire functions then p d possesses a hypercyclic subspace

operators with dense invariant cyclic vector manifolds abstract we study a class of banach space operators patterned after the weighted backward shifts on hilbert space and show that any non scalar operator in the commutant of one of these generalized backward shifts has a dense invariant linear manifold whose non zero members are cyclic vectors under appropriate extra hypotheses on the commuting operator stronger forms of cyclicity are possible the most extreme being hypercyclicity density of an orbit motivated by these results we examine the cyclic behavior of two seemingly unrelated classes of operators adjoint multiplications on hilbert spaces of holomorphic functions and differential operators on the frechet space of entire functions we show that each of these operators other than the scalar multiples of the identity possesses a dense invariant linear submanifold each of whose non zero elements is hypercyclic finally we explore some connections with dynamics many of the hypercyclic operators discussed here are in at least one of the commonly accepted senses of the word chaotic

hypercyclic subspaces in omega abstract we show that any countable family of operators of the form p b where p is a non constant polynomial and b is the backward shift operator on the countably infinite product of lines has a common hypercyclic subspace

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

spectral theory and hypercyclic subspaces a vector x in a hilbert space h is called hypercyclic for a bounded operator t h h if the orbit tnx n 1 is dense in h our main result states that if t satisfies the hypercyclicity criterion and the essential spectrum intersects the closed unit disk then there is an infinite dimensional closed subspace consisting except for zero entirely of hypercyclic vectors for t the converse is true even if t is a hypercyclic operator which does not satisfy the hypercyclicity criterion as a consequence other characterizations are obtained for an operator t to have an infinite dimensional closed subspace of hypercyclic vectors these results apply to most of the hypercyclic operators that have appeared in the literature in particular they apply to bilateral and backward weighted shifts perturbations of the identity by backward weighted shifts multiplication operators and composition operators the main result also applies to the differentiation operator and the translation operator t f z f z 1 defined on certain hilbert spaces consisting of entire functions we also obtain a spectral characterization of the norm closure of the class of hypercyclic operators which have an infinite dimensional closed subspace of hypercyclic vectors

second order homogenization of boundary and transmission conditions for one dimensional waves in periodic media abstract we consider the homogenized boundary and transmission conditions governing the mean field approximations of 1d waves in finite periodic media within the framework of two scale analysis we establish the homogenization ansatz up to the second order of approximation for both types of problems by obtaining the relevant boundary correctors and exposing the enriched boundary and transmission conditions as those of robin type rigorous asymptotic analysis is performed for boundary conditions while the applicability to transmission conditions is demonstrated via numerical simulations within this framework we also propose an optimized second order model of the homogenized wave equation for 1d periodic media that follows more accurately the exact dispersion relationship and generally enhances the performance of second order approximation the proposed analysis is applied toward the long wavelength approximation of waves in finite periodic bilaminates subject to both boundary and transmission conditions a set of numerical simulations is included to support the mathematical analysis and illustrate the effectiveness of the homogenization scheme

universal and chaotic multipliers on spaces of operators we use tensor product techniques to study universality hypercyclicity and chaos of multipliers defined on operator ideals and of multiplication operators on the space of all continuous and linear operators thus continuing the work of kit chan we also obtain the first examples of outer multipliers on a banach algebra which are chaotic in the sense of devaney and prove sufficient conditions for the existence of closed subspaces of universal vectors for operators between frechet spaces

an adaptive timestepping algorithm for stochastic differential equations we introduce a variable timestepping procedure using local error control for the pathwise strong numerical integration of a system of stochastic differential equations forced by a single wiener process the milstein method is used to advance the numerical solution and the stepsizes are determined via two local error estimates that roughly correspond to leading order deterministic and stochastic local error components one advantage of using two error controls is an increased flexibility that allows for the treatment of both drift and diffusion dominated regimes in a consistent manner numerical results are presented and the generalization of this approach to wider classes of problems and methods is discussed

some results and open questions on spaceability in function spaces a subset m of a topological vector space x is called lineable respectively spaceable in x if there exists an infinite dimensional linear space respectively an infinite dimensional closed linear space y subset of m boolean or 0 in this article we prove that for every infinite dimensional closed subspace x of c 0 1 the set of functions in x having infinitely many zeros in 0 1 is spaceable in x we discuss problems related to these concepts for certain subsets of some important classes of banach spaces such as c 0 1 or muntz spaces we also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of c 0 1 as well as oscillating and annulling properties of subspaces of c 0 1