solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

heteroclinic connections between periodic orbits in planar restricted circular three body problem part ii we present a method for proving the existence of symmetric periodic heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry as an application we show that the planar restricted circular three body problem pcr3bp corresponding to the sun jupiter oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the lyapunov orbits moreover we show the existence of symbolic dynamics on six symbols for pcr3bp and the possibility of resonance transitions of the comet this extends earlier results by wilczak and zgliczynski 12

periodic orbits in k symmetric dynamical systems abstract a map l is called k symmetric if itskth iterate lk possesses more symmetry than l for some value of k in k symmetric systems there exists a notion of k symmetric orbits this paper deels with k symmetric periodic orbits we derive a relation between orbits that are k symmetric with respect to reversing k symmetries and symmetric orbits of lk with this relation we set out an efficient method for finding systematically all periodic orbits that are k symmetric with respect to reversing k symmetries this k symmetric fixed set iteration fsi method generalizes a celebrated method due to devogelaere that applies to symmetric periodic orbits in reversible dynamical systems we use the fsi method to study k symmetric periodic orbits of a map of the planer2 possessing a crystallographic reversing k symmetry group the explicit findings illustrate a typically k symmetric phenomenon consisting of a nontrivial relation between the symmetry properties of periodic orbits and their periods

finite traces and representations of the group of infinite matrices over a finite field abstract the article is devoted to the representation theory of locally compact infinite dimensional group glb of almost upper triangular infinite matrices over the finite field with q elements this group was defined by s k a v and andrei zelevinsky in 1982 as an adequate n analogue of general linear groups gl n q it serves as an alternative to gl q whose representation theory is poor our most important results are the description of semifinite unipotent traces characters of the group glb via certain probability measures on the borel subgroup b and the construction of the corresponding von neumann factor representations of type ii as a main tool we use the subalgebra a glb of smooth functions in the group algebra l 1 glb this subalgebra is an inductive limit of the finite dimensional group algebras c gl n q under parabolic embeddings as in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups like multiplicativity of indecomposable characters or connections to probabilistic concepts the infinite dimensional iwahori hecke algebra h q plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group s which had been intensively studied in numerous previous papers

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

plancherel representations of u infty and correlated gaussian free fields we study asymptotics of traces of noncommutative monomials formed by images of certain elements of the universal enveloping algebra of the infinitedimensional unitary group in its plancherel representations we prove that they converge to commutative moments of a gaussian process that can be viewed as a collection of simply yet nontrivially correlated two dimensional gaussian free fields the limiting process has previously arisen via the global scaling limit of spectra for submatrices of wigner hermitian random matrices the results of the present work were announced in 4

the kronecker product and stochastic automata networks this paper can be thought of as a companion paper to van loan s the ubiquitous kronecker product paper j comput appl math 123 2000 85 we collect and catalog the most useful properties of the kronecker product and present them in one place we prove several new properties that we discovered in our search for a stochastic automata network preconditioner we conclude by describing one application of the kronecker product omitted from van loan s list of applications namely stochastic automata networks

a quantization of the harmonic analysis on the infinite dimensional unitary group abstract the present work stemmed from the study of the problem of harmonic analysis on the infinite dimensional unitary group u that problem consisted in the decomposition of a certain 4 parameter family of unitary representations which replace the nonexisting two sided regular representation olshanski 31 the required decomposition is governed by certain probability measures on an infinite dimensional space which is a dual object to u a way to describe those measures is to convert them into determinantal point processes on the real line it turned out that their correlation kernels are computable in explicit form they admit a closed expression in terms of the gauss hypergeometric function f 1 2 borodin and olshanski 8 in the present work we describe a nonevident q discretization of the whole construction this leads us to a new family of determinantal point processes we reveal its connection with an exotic finite system of q discrete orthogonal polynomials the so called pseudo big q jacobi polynomials the new point processes live on a double q lattice and we show that their correlation kernels are expressed through the basic hypergeometric function 1 2 a crucial novel ingredient of our approach is an extended version g of the gelfand tsetlin graph the conventional graph describes the gelfand tsetlin branching rule for irreducible representations of unitary groups we find the q boundary of g thus extending previously known results gorin 17

block sor preconditioned projection methods for kronecker structured markovian representations kronecker structured representations are used to cope with the state space explosion problem in markovian modeling and analysis currently an open research problem is that of devising strong preconditioners to be used with projection methods for the computation of the stationary vector of markov chains mcs underlying such representations this paper proposes a block successive overrelaxation bsor preconditioner for hierarchical markovian models hmms throughout the paper the hmm acronym stands for hierarchical markovian models and should not be confused with the hmm that is sometimes used for hidden markov models that are composed of multiple low level models and a high level model that defines the interaction among low level models the kronecker structure of an hmm yields nested block partitionings in its underlying continuous time mc which may be used in the bsor preconditioner the computation of the bsor preconditioned residual in each iteration of a preconditioned projection method becomes the problem of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of the chosen partitioning the proposed bsor preconditioner solves these systems using sparse lu or real schur factors of diagonal blocks the fill in of sparse lu factorized diagonal blocks is reduced using the column approximate minimum degree colamd ordering a set of numerical experiments is presented to show the merits of the proposed bsor preconditioner

reversing symmetry group of and matrices with connections to cat maps and trace maps dynamical systems can have both symmetries and time reversing symmetries together these two types of symmetries form a group called the reversing symmetry group with the symmetries forming a normal subgroup of we give a complete characterization of and hence in the dynamical systems associated with the groups of integral matrices and to do this we use well known methods of number theory such as dirichlet s unit theorem for quadratic fields and gaus results on the equivalence of integer quadratic forms and employ the algebraic structure of the modular group as a free product we show how some recently discussed generalizations of the reversing symmetry group are also nicely illustrated when we consider affine extensions of these matrix groups our results are applicable to hyperbolic toral automorphisms anosov or cat maps pseudo anosov maps and the group of three dimensional 3d trace maps that preserve the fricke vogt invariant

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

area preserving dynamics that is not reversible in this paper we address some global dynamical features in area preserving dynamical systems on the planer2 which obstruct the presence of a time reversal symmetry that is a reflection we show that in the case of flows reversibility and non reversibility are non generic properties in the case of maps we show that reversibility is a non generic property by constructing an example of a persistently non reversible area preserving map

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

a lanczos method for approximating composite functions abstract we seek to approximate a composite function h x g f x with a global polynomial the standard approach chooses points x in the domain of f and computes h x at each point which requires an evaluation of f and an evaluation of g we present a lanczos based procedure that implicitly approximates g with a polynomial of f by constructing a quadrature rule for the density function of f we can approximate h x using many fewer evaluations of g the savings is particularly dramatic when g is much more expensive than f or the dimension of x is large we demonstrate this procedure with two numerical examples i an exponential function composed with a rational function and ii a navier stokes model of fluid flow with a scalar input parameter that depends on multiple physical quantities

markov processes on the path space of the gelfand tsetlin graph and on its boundary we construct a four parameter family of markov processes on infinite gelfand tsetlin schemes that preserve the class of central gibbs measures any process in the family induces a feller markov process on the infinite dimensional boundary of the gelfand tsetlin graph or equivalently the space of extreme characters of the infinite dimensional unitary group u the process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on u posed in olshanski 2003 44 as was shown in borodin and olshanski 2005 11 this measure can also be described as a determinantal point process with a correlation kernel expressed through the gauss hypergeometric function

a posteriori error estimation and adaptivity in stochastic galerkin fem for parametric elliptic pdes beyond the affine case abstract we consider a linear elliptic partial differential equation pde with a generic uniformly bounded parametric coefficient the solution to this pde problem is approximated in the framework of stochastic galerkin finite element methods we perform a posteriori error analysis of galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations practical versions of this error estimate are discussed and tested numerically for a model problem with non affine parametric representation of the coefficient furthermore we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric pde problem the performance of the adaptive algorithm is tested numerically for model problems with two different non affine parametric representations of the coefficient

chaos in the kuramoto sivashinsky equations a computer assisted proof abstract in this paper we present a new technique of proving the existence of an infinite number of symmetric periodic solutions this technique combines the covering relations method introduced by zgliczynski j differential equations 171 2001 173 methods in nonlinear anal 8 1996 169 nonlinearity 10 1997 243 with fixed set iteration method described in lamb reversing symmetries in dynamical systems ph d thesis universiteit van amsterdam 1994 as an example we present a computer assisted proof of symbolic dynamics in the kuramoto sivashinsky equations moreover we prove the existence of an infinite number of symmetric and an infinite number of nonsymmetric periodic solutions

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

quasi periodic stability of normally resonant tori we study quasi periodic tori under a normal internal resonance possibly with multiple eigenvalues two non degeneracy conditions play a role the first of these generalizes invertibility of the floquet matrix and prevents drift of the lower dimensional torus the second condition involves a kolmogorov like variation of the internal frequencies and simultaneously versality of the floquet matrix unfolding we focus on the reversible setting but our results carry over to the hamiltonian and dissipative contexts

the q gelfand tsetlin graph gibbs measures and q toeplitz matrices abstract the problem of the description of finite factor representations of the infinite dimensional unitary group investigated by voiculescu in 1976 is equivalent to the description of all totally positive toeplitz matrices vershik kerov showed that this problem is also equivalent to the description of the simplex of central i e possessing a certain gibbs property measures on paths in the gelfand tsetlin graph we study a quantum version of the latter problem we introduce a notion of a q centrality and describe the simplex of all q central measures on paths in the gelfand tsetlin graph conjecturally q central measures are related to representations of the quantized universal enveloping algebra u gl we also define a class of q toeplitz matrices and show that extreme q central measures correspond to q toeplitz matrices with non negative minors finally our results can be viewed as a classification theorem for certain gibbs measures on rhombus tilings of the halfplane we use a class of q interpolation polynomials related to schur functions one of the key ingredients of our proofs is the binomial formula for these polynomials proved by okounkov

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

the boundary of the gelfand tsetlin graph a new approach the gelfand tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups the boundary of the gelfand tsetlin graph has at least three incarnations as a discrete potential theory boundary as the set of finite indecomposable characters of the infinite dimensional unitary group and as the set of doubly infinite totally positive sequences an old deep result due to albert edrei and dan voiculescu provides an explicit description of the boundary it can be realized as a region in an infinite dimensional coordinate space the paper contains a novel approach to the edrei voiculescu theorem it is based on a new explicit formula for the number of semi standard young tableaux of a given skew shape or of gelfand tsetlin schemes of trapezoidal shape the formula is obtained via the theory of symmetric functions and new schur like symmetric functions play a key role in the derivation

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

trace maps as 3d reversible dynamical systems with an invariant one link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so called trace maps a subclass of them are mappings on a one parameter family of 2d surfaces that foliate 3 and also 3 they are derived from transfer matrix approaches to properties of 1d quasicrystals in this article we consider various dynamical properties of trace maps we first discuss the fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant 2d surfaces we highlight a particular surface where the motion is integrable and semiconjugate to an anosov system i e the mapping acts as a pseudo anosov map we identify properties of symmetry and reversibility time reversal symmetry in the fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits we show that a conservative period boubling sequence can be identified when moving through the one parameter family of 2d surfaces by using generator trace maps in terms of which all trace maps obtained from invertible two letter substitution rules can be expressed we show that many features of the fibonacci trace map hold in general the role of the fricke character hat i x y z x 2 y 2 z 2 2xyz 1 its symmetry group and reversibility for the nielsen trace maps are described algebraically finally we outline possible higher dimensional generalizations

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

efficient descriptor vector multiplications in stochastic automata networks this paper examines numerical issues in computing solutions to networks of stochastic automata it is well known that when the matrices that represent the automata contain only constant values the cost of performing the operation basic to all iterative solution methods that of matrix vector multiply is given by r n i 1 n n i i 1 n n i where n i is the number of states in the i th automaton and n is the number of automata in the network we introduce the concept of a generalized tensor product and prove a number of lemmas concerning this product the result of these lemmas allows us to show that this relatively small number of operations is sufficient in many practical cases of interest in which the automata contain functional and not simply constant transitions furthermore we show how the automata should be ordered to achieve this

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

the finite dimensional reversible kam theory abstract the main results of the theory of quasi periodic motions in finite dimensional reversible dynamical systems are surveyed lower dimensional invariant tori and excitation of elliptic normal modes are discussed in detail the case where the reversing involution is perturbed is considered several fundamental differences between the reversible and hamiltonian theories are listed

asymptotics of random lozenge tilings via gelfand tsetlin schemes a gelfand tsetlin scheme of depth n is a triangular array with m integers at level m m 1 ldots n subject to certain interlacing constraints we study the ensemble of uniformly random gelfand tsetlin schemes with arbitrary fixed n th row we obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble and also of its q deformation this provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice or equivalently of random stepped surfaces we work with a class of polygons which allows arbitrarily large number of sides we show that the local limit behavior of random tilings as all dimensions of the polygon grow is directed by ergodic translation invariant gibbs measures the slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by kenyon and okounkov acta math 199 2 263 302 2007 we also prove that at the edge of the limit shape the asymptotic behavior of random tilings is given by the airy process in particular our results cover the most investigated case of random boxed plane partitions when the polygon is a hexagon

reversible relative periodic orbits abstract we study the bundle structure near reversible relative periodic orbits in reversible equivariant systems in particular we show that the vector field on the bundle forms a skew product system by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves we also discuss possibilities for drifts along group orbits our results extend those recently obtained in the equivariant context by b sandstede et al 1999 j nonlinear sci 9 439 478 and c wulff et al 2001 ergodic theory dynam systems 21 605 635

aggregation of stochastic automata networks with replicas we present techniques for computing the solution of large markov chain models whose generators can be represented in the form of a generalized tensor algebra such as stochastic automata networks san many large systems include a number of replications of identical components this paper exploits replication by aggregating similar components this leads to a state space reduction based on lumpability we define san with replicas and we show how such san models can be strongly aggregated taking functional rates into account a tensor representation of the matrix of the aggregated markov chain is proposed allowing to store this chain in a compact manner and to handle larger models with replicas more efficiently examples and numerical results are presented to illustrate the reduction in state space and consequently the memory and processing time gains

k integrals and k lie symmetries in discrete dynamical systems abstract we generalize the concept of symplectic maps to that of k symplectic maps maps whose kth iterates are symplectic similarly k symmetries and k integrals are symmetries resp integrals of the kth iterate of the map it is shown that k symmetries and k integrals are related by the k symplectic structure as in the k 1 continuous case noether s theorem examples are given of k integrals and their related k symmetries for k 1 4

a factorization of the spectral galerkin system for parameterized matrix equations derivation and applications recent work has explored solver strategies for the linear system of equations arising from a spectral galerkin approximation of the solution of pdes with parameterized or stochastic inputs we consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters e g a pde with stochastic inputs semidiscretized in space and examine the linear system arising from a similar galerkin approximation of the solution we derive a useful factorization of this system of equations which yields bounds on the eigenvalues clues to preconditioning and a flexible implementation method for a wide array of problems we complement this analysis with i a numerical study of preconditioners on a standard elliptic pde test problem and ii a fluids application using existing cfd codes the matlab codes used in the numerical studies are available online

bifurcation from discrete rotating waves discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries we present a systematic approach to the study of local bifurcation from discrete rotating waves the approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group actions in the domain and the range our results are valid for dynamical systems with finite symmetry group and more generally for bifurcations from isolated discrete rotating waves in dynamical systems with compact symmetry group

symmetric singularities of reversible vector fields in dimension three abstract for a large class of reversible vector fields on the three dimensional space we present all the topological types and their respective normal forms of the symmetric singularities of codimension 0 1

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

reversing symmetries in dynamical systems dynamical systems may possess in addition to symmetries that leave the equations of motion invariant reversing symmetries that invert the equations of motion such dynamical systems are called weakly reversible some consequences of the existence of reversing symmetries for dynamical systems with discrete time mappings are discussed a reversing symmetry group is introduced and it is shown that every weakly reversible mapping l can be decomposed into two mappings k0 and k1 of the same order 2l limit l to infinity included such that k02 k12 i some applications are discussed briefly

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

log concavity of multiplicities with application to characters ofu the log concavity of the reduction multiplicities for the classical groups of typean bn cnis proved moreover the skew schur functionss are shown to be log concave coefficient by coefficient the results are applied to the calculation of the characters of the infinite dimensional classical groups the log concavity of density of the push forward of the liouville measure on coadjoint orbits under moment map is proved

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

crystallographic symmetries of stochastic webs in studies of the dynamics of a kicked pendulum webs along which stochastic diffusion takes place have been observed to occur at resonance the author deals with stochastic webs with crystallographic symmetries via an analysis of the reversing symmetries of the equations of motion symmetry properties of stochastic webs are revealed furthermore the threshold for stochastic diffusion is discussed in relation to the reversing symmetries of the equations of motion

on vector kronecker product multiplication with rectangular factors the infinitesimal generator matrix underlying a multidimensional markov chain can be represented compactly by using sums of kronecker products of small rectangular matrices for such compact representations analysis methods based on vector kronecker product multiplication need to be employed when the factors in the kronecker product terms are relatively dense vector kronecker product multiplication can be performed efficiently by the shuffle algorithm when the factors are relatively sparse it may be more efficient to obtain nonzero elements of the generator matrix in kronecker form on the fly and multiply them with corresponding elements of the vector this work proposes a modification to the shuffle algorithm that multiplies relevant elements of the vector with submatrices of factors in which zero rows and columns are omitted this approach avoids unnecessary floating point operations that evaluate to zero during the course of the multiplication and possibly reduces the amount of memory used numeri

on symmetric attractors in reversible dynamical systems abstract let o n be a finite group acting orthogonally on r n we say that is a reversing symmetry group of a homeomorphism diffeomorphism or flow f t r n r n t z or t r if has an index two subgroup whose elements commute with ft and for all elements and all t ft x f t x we give necessary group and representation theoretic conditions for subgroups of reversing symmetry groups to occur as symmetry groups of attractors lyapunov stable limit sets these conditions arise due to topological obstructions in dimensions 1 and 2 we present a complete description of possible symmetry groups of asymptotically stable attractors for homeomorphisms and diffeomorphisms these attractors cannot possess reversing symmetries we also have a fairly complete description in the context of subgroups which contain reversing symmetries for all dimensions n we present complete results on the possible symmetry groups of connected asymptotically stable attractors in addition we summarize our results in the context of reversible equivariant flows which are complete in dimensions 1 and 2 as well as for subgroups of o n in any dimension n a survey of the corresponding results in the equivariant context is also given

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

a natural class of generalized fibonacci chains in this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typid properties with the quasiperiodic fibonacci chain for a subclass we explicitly con vct the atomic surface moreover scaling properties of the energy specmm are discussed in relation to the dynamics of trace maps

the problem of harmonic analysis on the infinite dimensional unitary group abstract the goal of harmonic analysis on a noncommutative group is to decompose the most natural unitary representations of this group like the regular representation on irreducible ones the infinite dimensional unitary group u is one of the basic examples of big groups whose irreducible representations depend on infinitely many parameters our aim is to explain what the harmonic analysis on u consists of we deal with unitary representations of a reasonable class which are in 1 1 correspondence with characters central positive definite normalized functions on u the decomposition of any representation of this class is described by a probability measure called spectral measure on the space of indecomposable characters the indecomposable characters were found by dan voiculescu in 1976 the main result of the present paper consists in explicitly constructing a 4 parameter family of natural representations and computing their characters we view these representations as a substitute of the nonexisting regular representation of u we state the problem of harmonic analysis on u as the problem of computing the spectral measures for these natural representations a solution to this problem is given in the next paper harmonic analysis on the infinite dimensional unitary group and determinantal point processes math 0109194 to appear in ann math joint with alexei borodin we also prove a few auxiliary general results in particular it is proved that the spectral measure of any character of u can be approximated by a sequence of discrete spectral measures for the restrictions of the character to the compact unitary groups u n this fact is a starting point for computing spectral measures

iterative methods based on splittings for stochastic automata networks abstract this paper presents iterative methods based on splittings jacobi gauss seidel successive over relaxation and their block versions for stochastic automata networks sans these methods prove to be better than the power method that has been used to solve sans until recently with the help of three examples we show that the time it takes to solve a system modeled as a san is still substantial and it does not seem to be possible to solve systems with tens of millions of states on standard desktop workstations with the current state of technology however the san methodology enables one to solve much larger models than those could be solved by explicitly storing the global generator in the core of a target architecture especially if the generator is reasonably dense

block sor for kronecker structured representations the kronecker structure of a hierarchical markovian model hmm induces nested block partitionings in the transition matrix of its underlying markov chain this paper shows how sparse real schur factors of certain diagonal blocks of a given partitioning induced by the kronecker structure can be constructed from smaller component matrices and their real schur factors furthermore it shows how the column approximate minimum degree colamd ordering algorithm can be used to reduce fill in of the remaining diagonal blocks that are sparse lu factorized combining these ideas the paper proposes three level block successive over relaxation bsor as a competitive steady state solver for hmms finally on a set of numerical experiments it demonstrates how these ideas reduce storage required by the factors of the diagonal blocks and improve solution time compared to an all lu factorization implementation of the bsor solver

on the symmetry lines of the standard mapping abstract the taylor and chirikov standard mapping is used as a model to represent the dynamics of non integrable systems by means of the poincare section of the trajectories intersecting a fixed plane of the phase space this study is based on a systematic use of the decomposition of the standard mapping in a product of two involutions as was birkhoff s in the restricted three body problem an infinite family of involutions is generated by these two involutions forming a group including the iterates of the mapping and their inverses the symmetry lines of the family of involutions determine the position of the periodic points and explain the onset of new chains of periodic points produced by the winding of the symmetry lines around the stable periodic points

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

kronecker products and shuffle algebra relates three classical concepts viz mixed radix number system kronecker product of matrices and perfect shuffle it presents an algebra which describes the hardware organization of the computation of a product mv where m is a matrix in kronecker product form and v is a vector the algebraic formalism describes both the blockwise structure of the computation and the various possible connection patterns

the reversible umbilic bifurcation abstract hamiltonian systems with several degrees of freedom regularly lead to the investigation of bifurcating equilibria in reduced one degree of freedom systems this paper concerns equilibria with vanishing linearisation a co dimension two phenomenon in the reversible context under appropriate transversality conditions such equilibria have versal unfoldings related to the elliptic and hyperbolic umbilic catastrophes this has applications to gyrostat motion and also helps to explain the dynamics defined by the normal form of the henon heiles system the occurring unfoldings turn out to be versal even in the general reversible context of not necessarily hamiltonian systems

the numerical solution of stochastic automata networks abstract stochastic automata networks san s have recently received attention in the literature as an efficient means of modelling parallel systems such as communicating processes concurrent processors shared memory etc the advantage that the san approach has over generalized stochastic petri nets and indeed over any markovian analysis that requires the generation of a transition matrix is that its representation remains compact even as the number of states in the underlying markov chain begins to explode our concern in this paper is with the numerical issues that are involved in solving san networks we introduce stochastic automata and consider the numerical difficulties that result from their interaction we examine how the product of a vector with a compact san descriptor may be formed for this operation is the basis of all iterative solution methods we describe possible solution methods including the power method the method of arnoldi and gmres and show that the two latter methods greatly out perform the power method the method usually used in conjunction with stochastic automata networks finally we consider one possible means of preconditioning but conclude that further research is needed

lumpable continuous time stochastic automata networks abstract the generator matrix of a continuous time stochastic automata network san is a sum of tensor products of smaller matrices which may have entries that are functions of the global state space this paper specifies easy to check conditions for a class of ordinarily lumpable partitionings of the generator of a continuous time san in which aggregation is performed automaton by automaton when there exists a lumpable partitioning induced by the tensor representation of the generator it is shown that an efficient aggregation iterative disaggregation algorithm may be employed to compute the steady state distribution the results of experiments with two san models show that the proposed algorithm performs better than the highly competitive block gauss seidel in terms of both the number of iterations and the time to converge to the solution

versal deformations and normal forms for reversible and hamiltonian linear systems the problem of this article is the characterization of equivalence classes and their versal deformations for reversible and reversible hamiltonian matrices in both cases the admissible transformations form a subgroup g of gl m therefore the gl m orbits of a given matrix may split into several g orbits these orbits are characterized by signs for each sign we have a normal form and a corresponding versal deformation the main tool in the characterization is reduction to the semi simple case c 1996 academic press inc

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data we introduce a stochastic galerkin mixed formulation of the steady state diffusion equation and focus on the efficient iterative solution of the saddle point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions so called mean based preconditioners based on fast solvers for scalar diffusion problems are introduced for use with the minimum residual method we derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters

singularities of reversible vector fields abstract for a large class of reversible vector fields on the plane we present all the topological types and their respective normal forms of the symmetric singularities of codimensions 0 1 and 2

reversing k symmetries in dynamical systems we generalize the concept of reversing symmetries of a dynamical system i e we study dynamical systems that possess symmetry properties only if considered on a proper time scale in particular considering dynamical systems with discrete time the kth iterate of a map may possess more reversing symmetries than the map itself in this way the concepts of reversing symmetries and reversing symmetry groups are generalized to reversing k symmetries and reversing k symmetry groups furthermore a method is studied for finding orbits that are k symmetric with respect to reversing k symmetries firstly an existing method for finding orbits that are symmetric with respect to one reversing symmetry is extended to the case of more than one reversing symmetry and secondly a generalization of this method to the case of reversing k symmetries is introduced some physically relevant examples of dynamical systems possessing reversing k symmetries are discussed briefly

asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory we develop a new method for studying the asymptotics of symmetric polynomials of representation theoretic origin as the number of variables tends to infinity several applications of our method are presented we prove a number of theorems concerning characters of infinite dimensional unitary group and their q deformations we study the behavior of uniformly random lozenge tilings of large polygonal domains and find the gue eigenvalues distribution in the limit we also investigate similar behavior for alternating sign matrices equivalently six vertex model with domain wall boundary conditions finally we compute the asymptotic expansion of certain observables in theo n 1 dense loop model

reversibility in polynomial systems of ode s abstract for a given family of real planar polynomial systems of ordinary differential equations depending on parameters we consider the problem of how to find the systems in the family which become time reversible after some affine transformation we first propose a general computational approach to solve this problem and then demonstrate its usage for the case of the family of quadratic systems

fluctuations of particle systems determined by schur generating functions abstract we develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems we introduce and study the notion of the schur generating function of a random discrete configuration our main result provides a central limit theorem clt for such a configuration given certain conditions on the schur generating function as applications of this approach we prove clt s for several probabilistic models coming from asymptotic representation theory and statistical physics including random lozenge and domino tilings non intersecting random walks decompositions of tensor products of representations of unitary groups

efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering in this study we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions we show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution we also show how to generate a variety of statistical quantities from the computed solution finally we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering

the representation ring of the unitary groups and markov processes of algebraic origin abstract the paper consists of two parts the first part introduces the representation ring for the family of compact unitary groups u 1 u 2 this novel object is a commutative graded algebra r with infinite dimensional homogeneous components it plays the role of the algebra of symmetric functions which serves as the representation ring for the family of finite symmetric groups the purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper the second part deals with a family of markov processes on the dual object to the infinite dimensional unitary group u these processes were defined in a joint work with alexei borodin 2012 5 the main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators it is shown that the generators are implemented by certain second order partial differential operators with countably many variables initially defined as operators on r

a two directional arnoldi process and its application to parametric model order reduction we consider a two directional krylov subspace k k a j b j where besides the dimensionality k of the subspace increases the matrix a j and vector b j which induce the subspace may also augment specifically we consider the case where the matrix a j and the vector b j are augmented by block triangular bordering we present a two directional arnoldi process to efficiently generate a sequence of orthonormal bases q k j of the krylov subspaces the concept of a two directional krylov subspace and an arnoldi process is triggered by the need of a multiparameter moment matching based model order reduction technique for parameterized linear dynamical systems numerical examples illustrate computational efficiency and flexibility of the proposed two directional arnoldi process

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