shape and morse theory of attractors we study the shape in the sense of borsuk of attractors of continuous semi dynamical systems on general metric spaces we show in particular that the natural inclusion of the global attractor into the state space is a shape equivalence this and other results of the paper are used to develop an elementary morse theory of an attractor 2000 john wiley sons inc

dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

unstable dimension variability a source of nonhyperbolicity in chaotic systems the hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set a familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible respectively noninvertible maps of dimension 3 respectively 2 and higher in particular we investigate a situation first considered by abraham and smale in 1970 for different purposes in which the dimension of the unstable and stable tangent spaces are not constant over the chaotic set we call this unstable dimension variability a simple two dimensional map that displays behavior typical of this phenomenon is presented and analyzed

hierarchical preconditioning for the stochastic galerkin method one of the popular methods for numerical solution of partial differential equations with uncertain data is the stochastic galerkin method where function spaces used for discretized problems are tensor products of finite element spaces of spatial variables and of sets of polynomials of random variables related systems of linear equations are thus usually huge studying relations between subspaces of the solution spaces is important for obtaining efficient preconditioning for a hierarchy of polynomials of random variables we introduce upper bounds to the strengthened cauchy bunyakowsky schwarz cbs constants with respect to the scalar product defined by the operator of the weak form of the underlying equation small values of the cbs constant indicate that certain additive or multiplicative two by two block preconditioners reduce enough the condition number of the system moreover we show that a recursive multiplicative two by two block preconditioning can be used resulting in the algebraic multilevel iterative amli method we present the conditions under which the amli preconditioning is of an optimal order numerical experiments confirm the introduced theoretical estimates

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

a note on recent lie parasuperstructures the authors discuss two recently published triple products seen as generalizations in parasupersymmetric quantum mechanics of the usual anticommutator between odd operators in the supersymmetric context they lead to lie parasuperalgebras which are not isomorphic in general

a note on parasupersymmetric quantum mechanics of arbitrary order we revisit parasupersymmetric quantum mechanics of arbitrary order and present a set of nontrivial relations which characterizes the most general multilinear part of the associated parasupersymmetric algebra we then show that the formulation of multilinear relations leads immediately to a polynomial of parasupersymmetric hamiltonian in terms of the corresponding parasupercharges the deduction of higher derivative supersymmetric quantum mechanics directly via this parasupersymmetric formulation is discussed the complete degenerate structure of the energy spectrum for parasupersymmetric quantum mechanics of order p is systematically analyzed finally the notion of cyclic symmetry is introduced and the algebra of cyclic charge operators of arbitrary order is developed based on the parasupersymmetric formalism

intertwined basins of attraction of dynamical systems in this paper we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results

wada basin boundaries and basin cells in dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is nonempty we consider a two dimensional diffeomorphism f that is f is an invertible map and both f and its inverse are differentiable with continuous derivatives which has at least three basins fractal basin boundaries contain infinitely many periodic points generally only finitely many of these periodic points are outermost on the basin boundary that is accessible from a basin for many systems all accessible points lie on stable manifolds of periodic points a point x on the basic boundary is a wada point if every open neighborhood of x has a nonempty intersection with at least three different basins we call the boundary of a basin a wada basin boundary if all its points are wada points our main goal is to have definitions and hypotheses for wada basin boundaries that can be verified by computer the basic notion basin cell will play a fundamental role in our results for numerical verifications assuming each accessible point on the boundary of a basin b is on the stable manifold of some periodic orbit we show that partial derivative b over bar is a wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins in addition we find conditions for basins b 1 b 2 b n n greater than or equal to 3 to satisfy partial derivative b over bar 1 partial derivative b over bar 2 partial derivative b over bar n our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a wada basin boundary our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of wada basin boundaries

a multigrid solver for two dimensional stochastic diffusion equations steady and unsteady diffusion equations with stochastic diffusivity coefficient and forcing term are modeled in two dimensions by means of stochastic spectral representations problem data and solution variables are expanded using the polynomial chaos system the approach leads to a set of coupled problems for the stochastic modes spatial finite difference discretization of these coupled problems results in a large system of equations whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time to accelerate the convergence of the iterative technique a multigrid method based on spatial coarsening is implemented numerical experiments show good scaling properties of the method both with respect to the number of spatial grid points and the stochastic resolution level

a new perspective on the solution of uncertainty quantification and reliability analysis of large scale problems abstract this work revisits the computational performance of non intrusive monte carlo versus intrusive galerkin methods of large scale stochastic systems in the framework of high performance computing environments the purpose of this work is to perform an assessment of the range of the relative superiority of these approaches with regard to a variety of stochastic parameters in both approaches the solution of the resulting algebraic equations is performed with a combination of primal and dual domain decomposition methods implementing specifically tailored preconditioners the solution of repeated simulations of the monte carlo method is accelerated with an a orthogonalization procedure aiming at reducing the iterations of subsequent simulations while the solution of the augmented equations of the stochastic galerkin method is enhanced with preconditioners which combine the block diagonal features of the resulting matrices as well as the sparsity pattern of the off block diagonal terms numerical results are presented demonstrating the efficiency of the proposed implementations on a large scale 3d problem with different stochastic characteristics and useful conclusions are derived regarding the ranges of stochastic parameters in which non intrusive solvers have a superior performance compared to intrusive ones and vice versa

the classification of topologically expansive lorenz maps we show that topologically expansive lorenz maps can be described up to topological conjugacy by their kneading invariants we also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive lorenz map a simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

strange attractors of uniform flows on considere des attracteurs orbitalement stables de ces flots sur le tore solide ouvert d 2 s 1 qui ont une vitesse dans la direction s 1 on trouve qu un tel attracteur est la frontiere d une suite strictement emboitee de tores solides ouverts positivement invariants

wada dye boundaries in open hydrodynamical flows abstract dyes of different colors advected by two dimensional flows which are asymptotically simple can form a fractal boundary that coincides with a chaotic saddle s unstable manifold we show that such dye boundaries can have the wada property every boundary point of a given color on this fractal set is on the boundary of at least two other colors the condition for this is the nonempty intersection of the saddle s stable manifold with at least three differently colored domains in the asymptotic inflow region

countable sections for free actions of groups let the group g act on the set x we will call the set y a countable section if each gx n y is countable p forrest showed in 2 that if a locally compact separable metrizable group g acts freely on a standard measurable space x with quasi invariant probability measure u then there is a countable section y which is in and such that the set gy which is the image of g x y by a measurable function and therefore p measurable has pmeasure 1 in the terminology of 11 y is a complete measurable countable section we prove the following converse to forrest s result

multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification in this paper we study multiscale finite element methods for stochastic porous media flow equations as well as applications to uncertainty quantification we assume that the permeability field the diffusion coefficient is stochastic and can be described in a finite dimensional stochastic space this is common in applications where the coefficients are expanded using chaos approximations the proposed multiscale method constructs multiscale basis functions corresponding to sparse realizations and these basis functions are used to approximate the solution on the coarse grid for any realization furthermore we apply our coarse scale model to uncertainty quantification problem where the goal is to sample the porous media properties given an integrated response such as production data our algorithm employs pre computed posterior response surface obtained via the proposed coarse scale model using fast analytical computations of the gradients of this posterior we propose approximate langevin samples these samples are further screened through the coarse scale simulation and finally used as a proposal in metropolis hasting markov chain monte carlo method numerical results are presented which demonstrate the efficiency of the proposed approach

bddc and feti dp without matrices or vectors abstract the connection between the bddc balancing domain decomposition by constraints algorithm and the feti dp finite element tearing and interconnecting dual primal algorithm is discussed in the language of function spaces operators and linear functionals

unpredictable behavior in the duffing oscillator wada basins this paper describes some numerical experiments giving evidence of wada basin boundaries for the duffing oscillator we suggest some mechanisms by which this fractal property of the boundaries appears and discuss the difficulties that the wada property presents for predicting the behavior of dynamical systems

morse equation of attractors for nonsmooth dynamical systems abstract this paper is concerned with a morse theory of attractors for finite dimensional nonsmooth dynamical systems described by differential inclusions with upper semi continuous righthand sides we first show that all open attractor neighborhoods of an attractor share the same homotopy type then based on this basic fact we introduce the concept of homology index for morse sets and establish morse inequalities and morse equation by using smooth morse lyapunov functions

a reduced spectral function approach for the stochastic finite element analysis the stochastic finite element analysis of elliptic type partial differential equations with non gaussian random fields are considered a novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated it is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors these functions called as the spectral functions can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation based on the projection in a reduced orthonormal vector basis a galerkin error minimization approach is proposed the constants appearing in the galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation a hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution the method is illustrated using the stochastic nanomechanics of a zinc oxide zno nanowire deflected under the atomic force microscope afm tip the results are compared with the results obtained using direct monte carlo simulation classical neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness

on the convergence of a dual primal substructuring method in the dual primal feti method introduced by farhat et al 1999 the domain is decomposed into non overlapping subdomains but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint the continuity of the remaining degrees of freedom on subdomain interfaces is enforced by lagrange multipliers and all degrees of freedom are eliminated the resulting dual problem is solved by preconditioned conjugate gradients we give an algebraic bound on the condition number assuming only a single inequality in discrete norms and use the algebraic bound to show that the condition number is bounded by c 1 log 2 h h for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements as well as for a wide class of finite elements for the reissner mindlin plate model

unstable attractors in manifolds assume that k is a compact attractor with basin of attraction a k for some continuous flow phi in a space m stable attractors are very well known but otherwise without the stability assumption the situation can be extremely wild in this paper we consider the class of attractors with no external explosions where a mild form of instability is allowed after obtaining a simple description of the trajectories in a k k we study how k sits in a k by performing an analysis of the poincare polynomial of the pair a k k in case m is a surface we obtain a nice geometric characterization of attractors with no external explosions as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence finally we explore the strong relations which exist between the shape in the sense of borsuk of k and the shape in the intuitive sense of the whole phase space m much in the spirit of the morse conley theory

scalable domain decomposition solvers for stochastic pdes in high performance computing abstract stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non linear problems with billions of unknowns for stochastic modeling it is therefore essential to design robust parallel and scalable algorithms that can efficiently utilize high performance computing to tackle such large scale systems domain decomposition based iterative solvers can handle such systems although these algorithms exhibit excellent scalabilities significant algorithmic and implementational challenges exist to extend them to solve extreme scale stochastic systems using emerging computing platforms intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in house implementation sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi level iterative solvers parallel sparse matrix vector operations are used to reduce the floating point operations and memory requirements numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non gaussian stochastic process scalability of the solvers with respect to the number of random variables is also investigated

an intrinsic description of shape we give in this paper a description of the shape category of compacta in terms of multivalued maps we introduce the notion of a multi net and prove that the shape category of compacta is isomorphic to the category hn whose objects are metric compacta and whose morphisms are homotopy classes of multi nets this description is intrinsic in the sense that it does not make use of external elements such as anr expansions or embeddings in appropriate ar spaces we present many applications of this new formulation of shape

addressing the curse of dimensionality in ssfem using the dependence of eigenvalues in kl expansion on domain size this work is aimed at reducing the dimensionality in the spectral stochastic finite element method ssfem thus the computational cost through a domain decomposition dd method this reduction hinges on some new mathematical results on domain size dependence of the karhunen loeve kl expansion it has been reported in the literature that for few covariance kernels a lower domain size leads to a faster convergence of the kl eigenvalues this observation leads to an early truncation of the kl expansion and this reduction in stochastic dimensionality brings down the total computational cost in this work first we mathematically show the generalization of this faster convergence that is for any arbitrary covariance kernel this is achieved via developing a bound on eigenvalues as a function of the domain size then we prove that for a chosen number of terms in the kl expansion with any kernel for a one dimensional process the approximation error in the trace norm reduces with the domain size based on this domain size dependence we propose an algorithm for solving a stochastic elliptic equation in a dd framework the computational cost gain is demonstrated by a numerical study and is observed that the serial implementation of the proposed algorithm is about an order of magnitude faster compared to the existing method the cost saving increases with the stochastic dimensionality in the global domain the structure of this algorithm provides scope for parallelization which would help in efficiently solving large scale problems the sharpness of the proposed eigenvalue bounds is also tested for gaussian and exponential kernels the generalization opens avenues for developing further dd based ssfem solvers c 2016 elsevier b v all rights reserved

diophantine conditions imply critical points on the boundaries of siegel disks of polynomials let f be a polynomial map of the riemann sphere of degree at least two we prove that if f has a siegel disk g on which the rotation number satisfies a diophantine condition then either the boundary b of g contains a critical point or b is a lakes of wada indecomposable continuum with one of the lakes containing a critical point consequently if the boundary b of g has only 2 complementary domains then b contains a critical point we also show without any assumption on the rotation number that each proper nondegenerate subcontinuum of the boundary b of g is tree like and any other bounded complementary domain of b is a preperiodic component of the grand orbit of g finally we establish some conditions under which b contains no periodic point

schwarz preconditioners for stochastic elliptic pdes abstract increasingly the spectral stochastic finite element method ssfem has become a popular computational tool for uncertainty quantification in numerous practical engineering problems for large scale problems however the computational cost associated with solving the arising linear system in the ssfem still poses a significant challenge the development of efficient and robust preconditioned iterative solvers for the ssfem linear system is thus of paramount importance for uncertainty quantification of large scale industrially relevant problems in the context of high performance computing the preconditioner must scale to a large number of processors therefore in this paper a two level additive schwarz preconditioner is described for the iterative solution of the ssfem linear system the proposed preconditioner can be viewed as a generalization of the mean based block diagonal preconditioner commonly used in the literature for the numerical illustrations two dimensional steady state diffusion and elasticity problems with spatially varying random coefficients are considered the performance of the algorithm is investigated with respect to the geometric parameters strength of randomness dimension and order of the stochastic expansion

pseudocircles in dynamical systems we construct an example of a coo map on a 3 manifold which has an invariant set with an uncountable number of components each of which is a pseudocircle furthermore any map which is sufficiently close in the c1metric to the constructed map has a similar set

chaos periodicity and snakelike continua the results of this paper relate the dynamics of a continuous map f of the interval and the topology of the inverse limit space with bonding map f these inverse limit spaces have been studied by many authors and are examples of what bing has called snakelike continua roughly speaking we show that when the dynamics of f are complicated the inverse limit space contains indecomposable subcontinua we also establish a partial converse

on the global structure of invariant regions of flows with asymptotically stable attractors several authors have pointed out the usefulness of shape theory which was introduced by k borsuk in 1968 as a tool in the study of dynamical systems in particular bogatyi and gutsu 4 garay 10 gunther and segal 11 hastings 12 robbin and salamon 16 rogers 17 and tezer 24 have obtained interesting results in dynamical systems using shape theoretical techniques one of the authors of the present article has given another connection between shape and topological dynamics 20 based on the approach to shape developed in 18 and 19 see also 21 for applications of shape theory to the study of uniform attractors most of the mentioned papers study properties of flows although 16 and 24 are devoted to discrete dynamical systems in particular a shape conley index is constructed in 16 for discrete dynamical systems and some shape theoretical ideas are used in 24 to study properties of topological conjugacy of shifts and homotopical shift equivalence for maps of polyhedra in the papers 4 10 11 12 20 21 some topological properties of the attractors of flows are studied for instance hastings established in 12 a higher dimensional poincare bendixson theorem proving the existence of attractors of certain flows in the interior of submanifolds of the euclidean space and comparing the shape of the attractor with that of the manifold other properties studied in the above papers are related to strong cellularity of the global attractors 10 and polyhedral shape of asymptotically stable compacta 4 11 20 a remarkable fact proved by

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd

intertwined basins of attraction generated by the stable manifold of a saddle point this article deals with the intertwined basins of attraction we give a necessary and sufficient condition for the existence of intertwined basins of attraction

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

realizing rotation numbers on annular continua an annular continuum is a compact connected set k which separates a closed annulus a into exactly two connected components one containing each boundary component the topology of such continua can be very intricate for instance non locally connected we adapt a result proved by handel in the case where k a showing that if k is an invariant annular continuum of a homeomorphism of a isotopic to the identity then the rotation set in k is closed moreover every element of the rotation set is realized by an ergodic measure supported in k and by a periodic orbit if the rotation number is rational and most elements are realized by a compact invariant set our second result shows that if the continuum k is minimal with the property of being annular what we call a circloid then every rational number between the extrema of the rotation set in k is realized by a periodic orbit in k as a consequence the rotation set is a closed interval and every number in this interval rational or not is realized by an orbit moreover by an ergodic measure in k this improves a previous result of barge and gillette

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

on the intertwined basins of attraction for planar flows in this paper we discuss the intertwined basins of attraction one of the simplest cases of obstruction to predictability some sufficient conditions to guarantee the existence of intertwined basins of attraction are given

connected simple systems and the conley index of isolated invariant sets the object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows simple explicit formulae are derived for the maps which play a central role in the theory the presentation is self contained

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

stochastic reduced basis methods stochastic reduced basis methods for solving large scale linear random algebraic systems of equations such as those obtained by discretizing linear stochastic partial differential equations in space time and the random dimension are introduced the fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients or random functions we present a theoretical justification for employing basis vectors spanning the preconditioned stochastic krylov subspace to approximate the response process subsequently variants of the bubnov galerkin scheme are employed to compute the undetermined coefficients which allow explicit expressions for the response quantities to be derived we also examine some theoretical properties of the projection scheme and procedures for computing the response statistics numerical studies are presented for static and dynamic analysis of stochastic structural systems we demonstrate that significant improvements over the neumann expansion scheme as well as other relevant techniques in the literature can be achieved

the coset equivalence relation and topologies on subgroups the paper studies the structure of the homogeneous space g h for g a polish group and h g a borel not necessarily closed subgroup of g from the point of view of the theory of definable equivalence relations it makes a connection between the complexity of the natural it coset equivalence relation associated with g h and it polishability of h that is the possibility of introducing a polish group topology on h respecting its borel structure in particular it is proved that if h is an abelian borel subgroup of a polish group g then either h is polishable or bbb e 1 continuously embeds into the coset equivalence relation induced by h on g the same conclusion is shown to hold if h is an increasing union of a sequence of polishable subgroups of g

numerical solution of spectral stochastic finite element systems abstract this paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element ssfem formulation two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems a brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure the differences between these matrices from their deterministic finite element counterparts are illustrated an iterative solution scheme is proposed which utilizes their specific properties for efficient memory management and enhanced convergence behavior results from numerical tests are presented comparisons with standard algorithms illustrate the efficiency of the proposed algorithm the second solution procedure presented in this paper is based on hierarchical basis concepts results from numerical tests are again provided and the limitations of this approach are assessed the performance of both proposed algorithms indicates that the linear algebraic systems from the underlying ssfem formulation can be solved with considerably less effort in memory and computation time than their size suggests furthermore the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic fem

vaught s conjecture and the glimm effros property for polish transformation groups we extend the original glimm effros theorem for locally compact groups to a class of polish groups including the nilpotent ones and those with an invariant metric for this class we thereby obtain the topological vaught conjecture

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

prey predator systems with intertwined basins of attraction in this article we show the existence of intertwined basins of attraction for a class of prey predator systems which improves the extant results by deleting some crucial and hard testing conditions

shape of global attractors in topological spaces in this paper we apply the notion and properties of compactly generated shape to study attractors in topological spaces

unstable manifold conley index and fixed points of flows we study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows we show that some parts of the unstable manifold admit sections carrying a considerable amount of information these sections enable the construction of parallelizable structures which facilitate the study of the flow from this fact many nice consequences are derived specially in the case of plane continua for instance we give an easy method of calculation of the conley index provided we have some knowledge of the unstable manifold and as a consequence a relation between the brouwer degree and the unstable manifold is established for smooth vector fields we study the dynamics of non saddle sets properties of existence or non existence of fixed points of flows and conditions under which attractors are fixed points morse decompositions preservation of topological properties by continuation and classify the bifurcations taking place at a critical point

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

basins of wada abstract we describe situations in which there are several regions more than two with the wada property namely that each point that is on the boundary of one region is on the boundary of all we argue that such situations arise even in studies of the forced damped pendulum where it is possible to have three attractor regions coexisting and the three basins of attraction have the wada property

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

the mosaic of high performance domain decomposition methods for structural mechanics formulation interrelation and numerical efficiency of primal and dual methods a multitude of domain decomposition methods ddm for structural mechanics is available in the literature today a unified framework for formulating primal and dual ddm is thus presented in this paper aiming at providing a mathematical platform for a uniform treatment of high performance ddm in structural mechanics a novel approach for developing new ddm from existing methods is also proposed and is applied to dual and primal methods in the field of the feti methods this approach leads to a new category of methods derived from existing feti variants furthermore two alternative formulations of the balancing domain decomposition method are described while interrelations between the introduced and existing methods are established finally comparative numerical tests demonstrate the differences in the computational performance of the methods in question

hierarchy of the parasupersymmetric hamiltonians in quantum mechanics the simplest parasupersymmetric model found in one dimensional quantum mechanics is considered it contains one bosonic and one parafermionic degree of freedom by successive restriction of the superpotentials the author shows how the algebra of symmetry is simplified some discussion of the general parasupersymmetry algebras is presented

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

global topological properties of the hopf bifurcation we study the homotopical and homological properties of the attractors evolving from a generalized hopf bifurcation we consider the lorenz equations for parameter values near the hopf bifurcation and study a natural morse decomposition of the global attractor calculating the cech homotopy type of the lorenz attractor the shape indexes of the morse sets and the morse equation of the decomposition

asynchronous space time domain decomposition method with localized uncertainty quantification abstract the computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest in this case a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response we present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions in particular we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions as a result computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required to verify the numerical implementation we consider elastic wave propagation in an axially loaded beam moreover we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain a projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method

hall littlewood symmetric functions and the bkp equation the connection between solutions of the bkp equation and hall littlewood symmetric functions is utilised in a unified approach to soliton and polynomial solutions this is analogous to the wronskian formulation of the solution of the kp equation as a by product two novel expressions for certain hall littlewood functions in terms of pfaffians are derived

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

a complete classification of homogeneous plane continua we show that the only compact and connected subsets i e continua x of the plane mathbb r 2 which contain more than one point and are homogeneous in the sense that the group of homeomorphisms of x acts transitively on x are up to homeomorphism the circle mathbb s 1 the pseudo arc and the circle of pseudo arcs these latter two spaces are fractal like objects which do not contain any arcs it follows that any compact and homogeneous space in the plane has the form x z where x is either a point or one of the three homogeneous continua above and z is either a finite set or the cantor set the main technical result in this paper is a new characterization of the pseudo arc following lelek we say that a continuum x has span zero provided for every continuum c and every pair of maps f g colon c to x such that f c subset g c there exists c 0 in c so that f c 0 g c 0 we show that a continuum is homeomorphic to the pseudo arc if and only if it is hereditarily indecomposable i e every subcontinuum is indecomposable and has span zero

dissonant points and the region of influence of non saddle sets abstract the aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non saddle set we see in particular that some topological conditions are sufficient to guarantee that these sets are attractors or repellers we study in detail the existence of dissonant points of the flow which play a key role in the description of the region of influence of a non saddle set these points are responsible for much of the dynamical and topological complexity of the system we also study non saddle sets from the point of view of the conley index theory and consider among other things the case of flows on manifolds with trivial first cohomology group for flows on these manifolds dynamical robustness is equivalent to topological robustness we carry out a particular study of 2 dimensional flows and give a topological condition which detects the existence of dissonant points for flows on surfaces we also prove that isolated invariant continua of planar flows with global region of influence are necessarily attractors or repellers

partial hyperbolicity and robust transitivity throughout this paper m denotes a three dimensional boundaryless compact manifold and diff m the space of gl diffeomorphisms defined on m endowed with the usual cl topology a invariant set a is transitive if a w x for some xea here w x is the forward limit set of x the accumulation points of the positive orbit of x the maximal invariant set of in an open set u denoted by a u is the set of points whose whole orbit is contained in u i e a u i e z i u the set a u is robustly transitive if ar is transitive for every diffeomorphism r clclose to a diffeomorphism ediff m is transitive if m w x for some xem i e if a m m is transitive analogously is robustly transitive if every r glclose to also is transitive i e if a m m is robustly transitive in this paper we focus our attention on forms of hyperbolicity uniform partial and strong partial of a maximal invariant set a u derived from its robust transitivity observe that u can be equal to m and then is robustly transitive on one hand in dimension one there do not exist robustly transitive diffeomorphisms the diffeomorphisms with finitely many hyperbolic periodic points morse smale are open and dense in diff s1 on the other hand for two dimensional diffeomorphisms every robustly transitive set a u is a basic set i e a u is hyperbolic transitive and the periodic points of are dense in a u in particular every robustly transitive surface diffeomorphism is anosov and the unique surface which supports such diffeomorphisms is the torus t 2 these assertions follow from m3 and m4 in dimension bigger than or equal to three besides anosov hyperbolic diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type as far as we know three types of such diffeomorphisms have been constructed skew products

orbits of the pseudocircle the following theorem is proved theorem the pseudocircle has uncountably many orbits under the action of its homeomorphism group each orbit is the union of uncountably many composants a pseudocircle is a circularly chainable hereditarily indecomposable separating plane continuum r h bing 1 constructed a pseudocircle in 1951 and asked two questions are any two pseudocircles homeomorphic is a pseudocircle homogeneous in 1968 l fearnley 6 proved that the answer to the first question is yes and fearnley 5 and j t rogers jr 14 independently proved that the answer to the second question is no since the advent of the effros theorem several elegant proofs of the nonhomogeneity of the pseudocircle have appeared 7 10 and 13 in 1968 after seeing the proof that the pseudocircle is not homogeneous f b jones asked the second author how many orbits the pseudocircle had under the action of its homeomorphism group but the matter was not pursued further recently however the question has arisen again and wayne lewis 3 has asked if the pseudocircle has uncountably many orbits the purpose of this paper is to prove the following theorem which gives an affirmative answer to this question theorem the pseudocircle has uncountably many orbits under the action of its homeomorphism group each of these orbits is the union of uncountably many composants an interesting sidelight of the proof is the construction of two homeomorphic sets one being an open set of the pseudocircle and the other being an open set of the pseudoarc during the proof we construct an uncountable abelian subgroup of the homeomorphism group of the pseudocircle the homeomorphism constructed by handel 8 might be an element of this group each orbit of the pseudocircle is a borel set 4 lewis has announced that no orbit can be a g6 in the course of our argument we prove this but we do not determine further restrictions on the type of the borel set for the orbit question what classes of borel sets occur as the orbits of the pseudocircle in particular are each two orbits of the same class received by the editors may 1 1984 and in revised form august 13 1985 1980 mathematics subject classification primary 54f20 54f50

on the fine structure of the global attractor of a uniformly persistent flow abstract we study the internal structure of the global attractor of a uniformly persistent flow we show that the restriction of the flow to the global attractor has duality properties which can be expressed in terms of certain attractor repeller decompositions we also study some natural morse decompositions of the flow and calculate their morse equations these equations provide necessary and sufficient conditions for the existence of attractors with the shape of s 1 or such that their suspension has spherical shape

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

on the creation of wada basins in interval maps through fixed point tangent bifurcation basin boundaries play an important role in the study of dynamics of nonlinear models in a variety of disciplines such as biology chemistry economics engineering and physics one of the goals of nonlinear dynamics is to determine the global structure of the system such as boundaries of basins a basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins is called a wada basin and its boundary is called a wada basin boundary here we consider maps on the interval we present a sufficient and necessary condition guaranteeing that three wada basins are emerging from a tangent bifurcation for certain one dimensional maps having negative schwarzian derivative two fixed point attractors on one side of the tangent bifurcation and three fixed point attractors on the other side of the tangent bifurcation all the conditions involved are numerically verifiable

on the conjecture of wood and projective homogeneity abstract in 2005 kawamura and rambla independently constructed a metric counterexample to wood s conjecture from 1982 we exhibit a new nonmetric counterexample of a space l such that c 0 l c is almost transitive and show that it is distinct from a nonmetric space whose existence follows from the work of greim and rajagopalan in 1997 up to our knowledge this is only the third known counterexample to wood s conjecture we also show that contrary to what was expected if a one point compactification of a space x is r h bing s pseudo circle then c 0 x c is not almost transitive for a generic choice of points finally we point out close relation of these results on wood s conjecture to a work of irwin and solecki on projective fraisse limits and projective homogeneity of the pseudo arc and addressing their conjecture we show that the pseudo circle is not approximately projectively homogeneous

a kronecker product preconditioner for stochastic galerkin finite element discretizations the discretization of linear partial differential equations with random data by means of the stochastic galerkin finite element method results in general in a large coupled linear system of equations using the stochastic diffusion equation as a model problem we introduce and study a symmetric positive definite kronecker product preconditioner for the galerkin matrix we compare the popular mean based preconditioner with the proposed preconditioner which in contrast to the mean based construction makes use of the entire information contained in the galerkin matrix we report on results of test problems where the random diffusion coefficient is given in terms of a truncated karhunen loeve expansion or is a lognormal random field

domain decomposition of stochastic pdes theoretical formulations we present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations the methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space the probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes the spatial decomposition is achieved through a schur complement based domain decomposition the methodology aims to exploit the full potential of high performance computing platforms by reducing discretization errors with high resolution numerical model in conjunction to giving due regards to uncertainty in the system the mathematical formulation is numerically validated with an example of waves in random media copyright 2008 john wiley sons ltd

a conley index study of the evolution of the lorenz strange set abstract in this paper we study the lorenz equations using the perspective of the conley index theory more specifically we examine the evolution of the strange set that these equations posses throughout the different values of the parameter we also analyze some natural morse decompositions of the global attractor of the system and the role of the strange set in these decompositions we calculate the corresponding morse equations and study their change along the successive bifurcations we see how the main features of the evolution of the lorenz system are explained by properties of the dynamics of the global attractor in addition we formulate and prove some theorems which are applicable in more general situations these theorems refer to poincare andronov hopf bifurcations of arbitrary codimension bifurcations with two homoclinic loops and a study of the role of the traveling repellers in the transformation of repeller attractor pairs into attractor repeller ones

uniform persistence and hopf bifurcations in r n we consider parameterized families of flows in locally compact metrizable spaces and give a characterization of those parameterized families of flows for which uniform persistence continues on the other hand we study the generalized poincare andronov hopf bifurcations of parameterized families of flows at boundary points of r n or more generally of an n dimensional manifold and show that this kind of bifurcations produce a whole family of attractors evolving from the bifurcation point and having interesting topological properties in particular in some cases the bifurcation transforms a system with extreme non permanence properties into a uniformly persistent one we study in the paper when this phenomenon happens and provide an example constructed by combining a holling type interaction with a pitchfork bifurcation