spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

inequalities for numerical invariants of sets of matrices we prove three inequalities relating some invariants of sets of matrices such as the joint spectral radius one of the inequalities in which proof we use geometric invariant theory has the generalized spectral radius theorem of berger and wang as an immediate corollary

on the zero stability of variable stepsize multistep methods the spectral radius approach in this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients in particular we deal with the zero stability of the 3 step bdf method on grids with variable stepsize for the numerical solution of ivps for odes our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results which give a slight improvement to the best results already known from the literature the success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

least squares finite element methods for optimization and control problems for the stokes equations the approximate solution of optimization and control problems for systems governedby the stokes equations is considered modern computational techniques for such problems are predominantly based on the application of the lagrange multiplier rule while penalty formulations even though widely used in other settings have not enjoyed the same level of popularity for this class of problems a discussion is provided that explains why naively defined penalty methods may not be practical then practical penalty methods are defined using methodologies associated with modern least squares finite element methods the advantages with respect to efficiency of penalty leasts squares methods for optimal control problems compared to methods based on lagrange multipliers are highlighted a tracking problem for the stokes system is used for illustrative purposes

on the asymptotic properties of a family of matrices in this paper we consider bounded families f of complex n n matrices after introducing the concept of asymptotic order we investigate how the norm of products of matrices behaves as the number of factors goes to infinity in the case of defective families f u sing the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so called generalized spectral radius with reference to the well known finiteness conjecture for finite families we also introduce the concepts of spectrum maximizing product and limit spectrum maximizing product showing that for finite families of 2 2matrices defectivity is equivalent to the existence of defective such limit products 2001 elsevier science inc all rights reserved

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

duality results for the joint spectral radius and transient behavior for linear inclusions in discrete or continuous time several quantities characterizing the growth behavior of the corresponding semigroup are analyzed these quantities are the joint spectral radius the initial growth rate and for bounded semigroups the transient bound it is discussed how these constants relate to one another and how they are characterized by various norms a complete duality theory is developed in this framework relating semigroups and dual semigroups and extremal or transient norms with their respective dual norms

some criteria for spectral finiteness of a finite subset of the real matrix space rd d abstract in this paper we present some checkable criteria for the spectral finiteness of a finite subset of the real d d matrix space r d d where 2 d

extremal and barabanov semi norms of a semigroup generated by a bounded family of matrices abstract let s s i i i be an arbitrary family of complex n by n matrices where 1 n let s denote the joint spectral radius of s defined as s lim sup l sup i 1 i l i l s i 1 s i l 1 l which is independent of the norm used here a semi norm on c n is called extremal of s if it satisfies x 0 and x s i s x x x 1 x n c n and i i in this paper using an elementary analytical approach we show that if s is bounded in c n n then there always exists for s an extremal semi norm on c n if additionally s is compact in c n n this extremal semi norm has the barabanov type property i e to any x c n one can find an infinite sequence i n i with x s i 1 s i k s k x for each k 1 as a common starting point this directly implies the fundamental results barabanov s norm theorem berger wang s formula and elsner s reduction theorem

on the complexity of computing the capacity of codes that avoid forbidden difference patterns some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed the maximal number of n bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n the coefficient of the exponent is the capacity of the forbidden patterns in this paper new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy the computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given subsequently a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be np hard when the sets of forbidden patterns are defined over an extended set of symbols finally the existence of extremal norms is proved for any set of matrices arising in the capacity computation based on this result a second capacity approximating algorithm is proposed the usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately

a gel fand type spectral radius formula and stability of linear constrained switching systems abstract using ergodic theory in this paper we present a gel fand type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup s restricted to a subset that need not carry the algebraic structure of s this generalizes the berger wang formula using it as a tool we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one sided markov shift associated to s

rank one characterization of joint spectral radius of finite matrix family abstract in this paper we study the joint generalized spectral radius of a finite set of matrices in terms of its rank one approximation by singular value decomposition in the first part of the paper we show that any finite set of matrices with at most one element s rank being greater than one satisfies the finiteness property under the framework of invariant extremal norm formula for the computation of joint generalized spectral radius for this class of matrix family is derived based on that in the second part we further study the joint generalized spectral radius of finite sets of general matrices through constructing rank one approximations in terms of singular value decomposition and some new characterizations of joint generalized spectral radius are obtained several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

the finite step realizability of the joint spectral radius of a pair of d d matrices one of which being rank one abstract we study the finite step realizability of the joint generalized spectral radius of a pair of real square matrices s 1 and s 2 one of which has rank 1 where 2 d let a denote the spectral radius of a square matrix a then we prove that there always exists a finite length word i 1 i m 1 2 m for some finite m 1 such that s i 1 s i m m sup n 1 max i 1 i n 1 2 n s i 1 s i n n in other words there holds the spectral finiteness property for s 1 s 2 explicit formula for computation of the joint spectral radius is derived this implies that the stability of the switched system induced by s 1 s 2 is algorithmically decidable in this case

criteria for the stability of the finiteness property and for the uniqueness of barabanov norms a set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product the extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research in this article we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set but also for all sufficiently nearby sets of equal cardinality we also prove a theorem giving conditions under which the barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar and show that in certain cases these conditions are also persistent under small perturbations

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

a rapidly converging lower bound for the joint spectral radius via multiplicative ergodic theory abstract we use ergodic theory to prove a quantitative version of a theorem of m a berger and y wang which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices the proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded

on the finiteness property for rational matrices in this chapter we analyze a recent conjecture stating that the finiteness property holds for pairs of binary matrices we show that the finiteness property holds for all pairs of binary matrices if and only if it holds for all sets of nonnegative rationalmatrices we provide a similar result for matrices with positive and negative entries we finally prove the conjecture for 2 2 matrices

finiteness property of pairs of 2 2 sign matrices via real extremal polytope norms this paper deals with the joint spectral radius of a finite set of matrices we s ay that a set of matrices has the finiteness property if the maximal rate of growth in the multiplicative semigroup it generates is given by the powers of a finite product here we address the problem of establishing the finiteness properties of p airs of 2 2 sign matrices such problem is related to the conjecture that pairs of sign ma trices fulfil the finiteness property for any dimension this would imply by a recent result by j ungers and blondel that finite sets of rational matrices fulfil the finiteness property wh ich would be very important property in terms of the computation of the joint spectral radius the finiteness property of n n sign matrices remains open but the knowledge of its validity in the 2 2 case gives the hope to solve it by using some induction argument on the dimension as a main tool of our proof we make use of a procedure to compute a so called real polytope extremal norm for the set in particular we present an algorithm which under some suitable assumptions is able to check if a certain product in the multiplicative semig roup is spectrum maximizing for pairs of sign matrices we develop the computations exactly and hence are able to prove analytically the finiteness property on the other hand the algorithm c an be used in a floating point arithmetic and provide a general tool for approximating the join t spectral radius of a set of matrices

a space time coupled p version least squares finite element formulation for unsteady two dimensional navier stokes equations this paper presents a p version least squares finite element formulation for two dimensional unsteady fluid flow described by navier stokes equations where the effects of space and time are coupled the dimensionless form of the navier stokes equations are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c0 element approximation the element properties are derived by utilizing the p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure the application of least squares minimization to the set of coupled first order partial differential equations results in finding a solution vector which makes gradient of error functional with respect to a null vector this is accomplished by using newton s method with a line search a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach the generality success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements

hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non negative matrices abstract recently blondel nesterov and protasov proved 1 2 that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non negative matrices with independent row column uncertainty we show that this result can be obtained as a simple consequence of the so called hourglass alternative used in 3 by the author and his companions to analyze the minimax relations between the spectral radii of matrix products axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property which includes the sets of non negative matrices with independent row uncertainty this class of matrices supplemented by the zero and identity matrices forms a semiring with the minkowski operations of addition and multiplication of matrix sets which gives means to construct new sets of non negative matrices possessing the finiteness property for the generalized and the lower spectral radii

optimal disturbance attenuation for discrete time switched and markovian jump linear systems an exact condition for uniform stabilization and disturbance attenuation for switched linear systems is given in the discrete time domain via the union of an increasing family of linear matrix inequality conditions associated with each markovian jump linear system is a switched linear system so we obtain a necessary and sufficient condition for almost sure uniform stabilization and disturbance attenuation for markovian jump linear systems as well the results lead to semidefinite programming based controller synthesis techniques from which optimal finite path dependent linear dynamic output feedback controllers arise naturally in particular under the notion of path by path optimal disturbance attenuation finite path dependent controllers can outperform the usual mode dependent ones

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

brief paper periodically switched stability induces exponential stability of discrete time linear switched systems in the sense of markovian probabilities the conjecture that periodically switched stability implies absolute asymptotic stability of random infinite products of a finite set of square matrices has recently been disproved under the guise of the finiteness conjecture in this paper we show that this conjecture holds in terms of markovian probabilities more specifically let s k c n x n 1 k k be arbitrarily given k matrices and s k k j j 1 1 k j k for each j 1 where n k 2 then we study the exponential stability of the following discrete time switched dynamics s x j s k j s k 1x 0 j 1 and x 0 c n where k j j 1 s k can be an arbitrary switching sequence for a probability row vector p p 1 p k r k and an irreducible markov transition matrix p r k x k with pp p we denote by m p p the markovian probability on s k corresponding to p p by using symbolic dynamics and ergodic theoretic approaches we show that if s possesses the periodically switched stability then i it is exponentially stable m p p almost surely ii the set of stable switching sequences k j j 1 s k has the same hausdorff dimension as s k thus the periodically switched stability of a discrete time linear switched dynamics implies that the system is exponentially stable for almost all switching sequences

a discontinuous petrov galerkin methodology for adaptive solutions to the incompressible navier stokes equations the discontinuous petrov galerkin methodology with optimal test functions dpg of demkowicz and gopalakrishnan 18 20 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes whereas bubnov galerkin methods use identical trial and test spaces petrov galerkin methods allow these function spaces to differ in dpg test functions are computed on the fly and are chosen to realize the supremum in the inf sup condition the method is equivalent to a minimum residual method for well posed problems with sufficiently regular solutions dpg can be shown to converge at optimal rates the inf sup constants governing the convergence are mesh independent and of the same order as those governing the continuous problem 48 dpg also provides an accurate mechanism for measuring the error and this can be used to drive adaptive mesh refinements we employ dpg to solve the steady incompressible navier stokes equations in two dimensions building on previous work on the stokes equations and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh we apply our approach to a manufactured solution due to kovasznay as well as the lid driven cavity flow backward facing step and flow past a cylinder problems

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

characterization of joint spectral radius via trace the joint spectral radius for a bounded collection of the square matrices with complex entries and of the same size is characterized by the trace of matrices this characterization allows us to give some estimates concerning the computation of the joint spectral radius

an algorithm for finding extremal polytope norms of matrix families in this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered we present an algorithm which under some suitable assumptions is able to check if a certain product in the multiplicative semigroup is spectrum maximizing the algorithm proceeds by attempting to construct a suitable extremal norm for the family namely a complex polytope norm as examples for testing our technique we first consider the set of two 2 dimensional matrices recently analyzed by blondel nesterov and theys to disprove the finiteness conjecture and then a set of 3 dimensional matrices arising in the zero stability analysis of the 4 step bdf formula for ordinary differential equations

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

asymptotic stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

an efficient lower bound for the generalized spectral radius of a set of matrices the generalized spectral radius gsr is a fundamental concept in studying the regularity of compactly supported wavelets here we describe an efficient method for estimating a lower bound for the gsr let mq be the set of all q q matrices with complex entries suppose a0 am 1 is a collection of m matrices in mq let ln be the set of all products of length n of the elements of define n maxa ln a in where a is the spectral radius of a the generalized spectral radius of is then lim supn n the standard method for estimating from below and at level n is to calculate the spectral radii of all mn products in ln and select the largest here we use three elementary theorems from linear algebra combinatorics and number theory to show that the same result can be obtained with no more than mnn matrix calculations

a collocation penalty least squares finite element formulation for incompressible flows abstract we present a penalty least squares finite element formulation for incompressible flows the pressure degree of freedom is eliminated using the classical penalty approach and the least squares model is formed in terms of velocity vorticity and dilatation an iterative penalization is implemented which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix the h convergence is verified using the exact solution of kovasznay s flow numerical results are presented for a number of benchmark problems e g steady flow past a large circular cylinder in a channel transient flow over a backward facing step unsteady flow past a circular cylinder and buoyancy driven natural convection in a square cavity for all numerical examples the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters 10 100 these problems are also solved on coarse meshes to show that good mass conservation is achieved

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

spectral hp penalty least squares finite element formulation for the steady incompressible navier stokes equations in this paper we present spectral hp penalty least squares finite element formulation for the numerical solution of steady incompressible navier stokes equations the continuity equation is treated as a constraint on velocity field and the constraint is enforced using the penalty method which is equivalent to replacing the pressure variable p with p n p n email protected u where u is the velocity field c is the penalty parameter and superscript n is non linear iteration number the modified navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity dilatation or stresses as additional independent variables and the least squares method is used to develop the finite element model high order basis functions are used to construct the discrete model unlike other penalty finite element formulations equal order integration is used for all terms of the coefficient matrix schur complement method is used and all interior element degrees of freedom are condensed out and interface degrees of freedom are solved either by direct solver banded cholesky factorization or iterative solver preconditioned conjugate gradient present formulation produces very accurate results for even very low penalty parameters 10 40 in this way present formulation and implementation fixes perennial problem of bad conditioning of coefficient matrix associated with penalty formulations proposed formulation is verified using the exact solution of kovasznay flow numerical results are also presented for a number of two dimensional benchmark problems e g 2d lid driven cavity flow flow over a backward facing step steady flow past a circular cylinder and 2d flow past a large circular cylinder in a channel finally the proposed formulation is extended to coupled velocity temperature problems and buoyant flow inside a square enclosure is solved to test the penalty least squares finite element model developed herein

computing the joint spectral radius this paper presents algorithms for finding an arbitrarily small interval that contains the joint spectral radius of a finite set of matrices it also presents a numerical criterion for verifying in certain cases that the joint spectral radius is the maximum of the spectral radii of the given matrices error bounds are derived for the case where calculations are done with finite precision and the matrices are not known exactly the algorithms are implemented and applied to estimate holder exponents of the orthonormal wavelets n constructed by daubechies for 3 n 8

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

the generalized spectral radius theorem an analytic geometric proof abstract let be a bounded set of complex matrices m a 1 a m a i the generalized spectral radius theorem states that where and are defined as follows lim sup m 1 m where m sup a a m with a the spectral radius lim sup m 1 m where m sup a a m with any matrix norm we give an elementary proof based on analytic and geometric tools which is in some ways simpler than the first proof by berger and wang

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

the generalized spectral radius is strictly increasing using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense as an application some consequences of this property in the area of time varying stability radii are discussed in particular using the implicit function theorem sufficient conditions for lipschitz continuity are derived an example is presented of a linearly increasing family of matrix polytopes for which the proximal subgradient of the generalized spectral radius at a certain polytope contains 0 so that the implicit function theorem is not applicable in all cases

remote source coding under gaussian noise dueling roles of power and entropy power the distributed remote source coding the so called ceo problem is studied in the case where the underlying source not necessarily gaussian has finite differential entropy and the observation noise is gaussian the main result is a new lower bound for the sum rate distortion function under arbitrary distortion measures when specialized to the case of mean squared error it is shown that the bound exactly mirrors a corresponding upper bound except that the upper bound has the source power variance whereas the lower bound has the source entropy power bounds exhibiting this pleasing duality of power and entropy power have been well known for direct and centralized source coding since shannon s work while the bounds hold generally their value is most pronounced when interpreted as a function of the number of agents in the ceo problem

on accuracy of approximation of the spectral radius by the gelfand formula abstract the famous gelfand formula a limsup n a n 1 n for the spectral radius of a matrix is of great importance in various mathematical constructions unfortunately the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities a n 1 n to a in the paper this deficiency is made up to some extent by using the bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities a n 1 n to a the obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

least squares spectral element solution of incompressible navier stokes equations with adaptive refinement least squares spectral element solution of steady two dimensional incompressible flows are obtained by approximating velocity pressure and vorticity variable set on gauss lobatto legendre nodes constrained approximation method is used for h and p type nonconforming interfaces of quadrilateral elements adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient effective use of p refinement to overcome poor mass conservation drawback of least squares formulation and successful use of h and p refinement together to solve problems with geometric singularities are demonstrated capabilities and limitations of the developed code are presented using kovasznay flow flow past a circular cylinder in a channel and backward facing step flow

weakly birkhoff recurrent switching signals almost sure and partial stability of linear switched dynamical systems let a1 ak cd d be arbitrary k matrices where k and d both 2 for any 0 we denote by l pc r k the set of all switching sequences u t n 1 k r satisfying tj tj 1 and 0 t0 t1 tj 1 tj with tj differently from the classical weak topology and l1 norm we equip l pc r k with the topology so that the one sided markov type shift l pc r k l pc r k defined by u j tj j 1 u j 1 tj 1 t1 j 1 is continuous which is different from and simpler than the classical continuous time translation we study the stability of the linear switched dynamics a x t au t x t x 0 cd and t 0 where u t j if tj 1 0 then for any ergodic probability p on l pc r k either limj 1jlog u tj 0for p a s u j tj j 1 or j u tj k tj 1 k j 0for p a s u j tj j 1 some applications are presented including i equivalence of various stabilities ii almost sure exponential stability of periodically switched stable systems iii partial stability and iv how to approach arbitrarily the stable manifold by that of periodically switched signals and how to select a stable switching signal for any initial data

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

an explicit counterexample to the lagarias wang finiteness conjecture abstract the joint spectral radius of a finite set of real d d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set a set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth j c lagarias and y wang conjectured in 1995 that every finite set of real d d matrices satisfies the finiteness property however t bousch and j mairesse proved in 2002 that counterexamples to the finiteness conjecture exist showing in particular that there exists a family of pairs of 2 2 matrices which contains a counterexample similar results were subsequently given by v d blondel j theys and a a vladimirov and by v s kozyakin but no explicit counterexample to the finiteness conjecture has so far been given the purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the lagarias wang finiteness conjecture namely for the set a 1 1 0 1 1 0 1 1 we give an explicit value of 0 749326546330367557943961948091344672091327370236064317358024 such that a does not satisfy the finiteness property