common hypercyclic vectors for multiples of backward shift we prove that l 2 contains a dense set of vectors which are hypercyclic simultaneously for all multiples of the backward shift oper ator by constants of absolute value greater than 1

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

on some new contiguous relations for the gauss hypergeometric function with applications contiguous relations for hypergeometric series contain an enormous amount of hidden information applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series in this paper a new set of contiguous function relations are established applications of such relations to hypergeometric summation formulas and the theory of jacobi polynomials are presented

complete systems of partial derivatives of entire functions and frequently hypercyclic operators abstract we find some sufficient conditions for a system of partial derivatives of an entire function to be complete in the space h c d of all entire functions of d variables as an application of this result we describe new classes of frequently hypercyclic operators on h c d

a generalized preconditioned parameterized inexact uzawa method for singular saddle point problems in this short paper we introduce and analyze a generalized preconditioned parameterized inexact uzawa method for solving singular saddle point problems this method is a generalization of the ppiu method see gao and kong 2010 then we give the semi convergent conditions of this method which are more practical for the applications

distributional chaos for operators on banach spaces abstract four notions of distributional chaos namely dc1 dc2 dc 2 1 2 and dc3 are studied within the framework of operators on banach spaces it is known that for general dynamical systems dc1 dc2 dc 2 1 2 dc3 we show that dc1 and dc2 coincide in our context which answers a natural question in contrast there exist dc 2 1 2 operators which are not dc2 under the condition that there exists a dense set x 0 x such that t n x 0 for any x x 0 dc3 operators are shown to be dc1 moreover we prove that any upper frequently hypercyclic operator is dc 2 1 2 finally several examples are provided to distinguish between different notions of distributional chaos li yorke chaos and irregularity

a class of accelerated uzawa algorithms for saddle point problems in this paper we establish a class of accelerated uzawa au algorithms for solving the large sparse nonsingular saddle point problems by making use of the extrapolation technique this extrapolation technique is based on the eigenvalues of the iterative matrix these au algorithms involve two iteration parameters whose special choices can cover the known classical uzawa method as well as yield new ones firstly the accelerated model for the uzawa algorithm is established and the detail algorithm description of au method is presented then the convergence analyse of the au method is given moreover theoretical analyses show that the au algorithm converges faster than some uzawa type methods the uzawa method is also included in when the eigenvalues of the iterative matrix and the parameter satisfy some conditions numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the au method

the numerical solution of equality constrained quadratic programming problems this paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem the constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints among this class of convergent iterative schemes are generalizations of the relaxed jacobi gauss seidel and symmetric gauss seidel schemes

generalized watson s summation formula for 3 f 2 1 abstract a summation formula is given for 3 f 2 a b c 1 2 a b i 1 2c j 1 with fixed j and arbitrary i i j z this result generalizes the classical watson s theorem which deals with the case i j 0 extensions to the cases of 3f2 a 1 i j a c e 1 i 2c e 1 and 3f2 a b c 1 i a b 1 i j a c 1 are given notice that the case i j 0 corresponds to the classical theorems due to whipple and dixon respectively

hypercyclic weighted shifts an operator t acting on a hilbert space is hypercyclic if for some vector x in the space the orbit tnx n 0 is dense in this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic as a consequence we show within the class of weighted shifts that multi hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic for general hypercyclic operators the corresponding questions were posed by d a herrero and they still remain open using a different technique we prove that i t is hypercycic whenever t is a unilateral backward weighted shift thus answering in more generality a question recently posed by k c chan and j h shapiro

on semi convergence of ult iterative method for the singular saddle point problems zheng and ma recently proposed an efficient upper and lower triangular ult splitting iterative method for solving the large sparse nonsingular saddle point problems see zheng and ma 2016 in this paper we further prove the semi convergence of this method when it is applied to solve the large sparse singular saddle point problems under suitable conditions the characteristic of eigenvalues of the iteration matrix of the ult method is analyzed also the pseudo optimal iteration parameters and the corresponding pseudo optimal semi convergence factor for some special cases of the ult method are determined in addition numerical experiments are used to show the feasibility and effectiveness of the ult iterative method for solving singular saddle point problems

a study of new hypergeometric transformations the aim of this research paper is to establish a quite general transformation involving hypergeometric functions by the method of elementary manipulation of the series representations certain known as well as new hypergeometric transformations and identities not previously recorded in the literature are then deduced by means of the generalized kummer and dixon theorems obtained earlier by lavoie et al an application of a newly obtained identity is also given

algebraic genericity of frequently universal harmonic functions on trees abstract we show that the set of frequently universal harmonic functions on a tree t contains a vector space except 0 which is dense in the space of harmonic functions on t seen as subset of c t in order to prove this we replace the complex plane c by any separable frechet space e and we repeat all the theory

genericity of wild holomorphic functions and common hypercyclic vectors abstract let t be the translation operator by in the space of entire functions h c defined by t f z f z we prove that there is a residual set g of entire functions such that for every f g and every c 0 the sequence t n f is dense in h c that is g is a residual set of common hypercyclic vectors functions for the family t c 0 also we prove similar results for many families of operators as multiples of differential operator multiples of backward shift weighted backward shifts

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

some hypergeometric and other evaluations of ζ 2 and allied series numerous interesting solutions of the problem of evaluating the riemann z 2 k 1 k 2 which was of vital importance to euler and the bernoulli brothers james and john bernoulli have appeared in the mathematical literature ever since euler first solved this problem in 1736 the main object of this note is to present yet another evaluation of z 2 and related sums by applying the theory of hypergeometric series some other approaches to this problem are also indicated

strong transitivity properties for operators abstract given a furstenberg family f of subsets of n an operator t on a topological vector space x is called f transitive provided for each non empty open subsets u v of x the set n z t n u v belongs to f we classify the topologically transitive operators with a hierarchy of f transitive subclasses by considering families f that are determined by various notions of largeness and density in z

hypercyclic operators that commute with the bergman backward shift the backward shift b on the bergman space of the unit disc is known to be hypercyclic meaning it has a dense orbit here we ask which operators that commute with b inherit its hypercyclicity we show that the problem reduces to the study of operators of the form 9p b where so is a holomorphic self map of the unit disc that multiplies the dirichlet space into itself and that the question of hypercyclicity for such an operator depends on how freely 9p z is allowed to approach the unit circle as lz s 1

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

an efficient iterative method for the generalized stokes problem the generalized stokes problem which arises frequently in the simulation of time dependent navier stokes equations for incompressible fluid flow gives rise to symmetric linear systems of equations these systems are indefinite due to a set of linear constraints on the velocity causing difficulty for most preconditioners and iterative methods this paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method this new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite numerical experiments indicating superior convergence compared to existing methods are presented a natural extension of this method to elliptic problems is also proposed along with theoretical bounds on the rate of convergence and results of experiments demonstrating robust and effective preconditioning

on parameterized generalized skew hermitian triangular splitting iteration method for singular and nonsingular saddle point problems recently krukier et al 2014 and dou et al 2014 have studied the generalized skew hermitian triangular splitting gsts iteration method for nonsingular and singular saddle point problems respectively in this paper we further extend the gsts method to a parameterized gsts pgsts method for solving non hermitian nonsingular and singular saddle point problems by singular value decomposition technique we derive conditions of the new iterative method for guaranteeing the convergence for non hermitian nonsingular saddle point problems and its semi convergence for singular saddle point problems respectively in addition the choice of the acceleration parameters in a practical manner is studied numerical experiments are provided which further confirm our theoretical results and show the new method is feasible and effective for non hermitian nonsingular or singular saddle point problems

on semi convergence of parameterized uzawa methods for singular saddle point problems abstract bai et al recently proposed an efficient parameterized uzawa method for solving the nonsingular saddle point problems z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 in this paper we further prove the semi convergence of this method when it is applied to solve the singular saddle point problems under suitable restrictions on the involved iteration parameters the optimal iteration parameters and the corresponding optimal semi convergence factor for the parameterized uzawa method are determined in addition numerical experiments are used to show the feasibility and effectiveness of the parameterized uzawa method for solving singular saddle point problems

reduction formulae for generalised hypergeometric functions of one variable series of gamma functions can usually be expressed as generalised hypergeometric functions of one or more variables providing a tool for classifying series transformations while most studies of these functions concern the multivariable type the authors have derived new transformations and extended known results for single variable functions the derivations of the new identities are straightforward and the results are presented in their most general form

hypercyclic orbits intersect subspaces in wild ways abstract let t x x be a hypercyclic operator of an infinite dimensional separable banach space x by modifying a construction of grivaux we will show that the t orbit of a hypercyclic vector can intersect certain closed vector subspaces of x in many strange ways moreover the set of visiting times of the orbit to the subspace can also be quite exotic especially when the operator satisfies a stronger form of hypercyclicity as a by product we will improve and or supply new proofs to some of the recent results about subspace hypercyclicity

a new sor like method for the saddle point problems abstract in previous years golub wu and yuan presented a generalized successive over relaxation sor like method for solving the saddle point problems in this paper we present a new sor like nsor like method which has three parameters our new method can be applied to the nonsingular saddle point problems as well as the singular cases the characteristic of eigenvalues of the iteration matrix of this nsor like method is analyzed then we give the convergence semi convergence theorem of the new iterative method by giving the restrictions imposed on the parameter moreover that convergence semi convergence theorem is applied to some special cases to give the convergence region for the parameters we can see that nsor like method has a wider convergence semi convergence region for and than the parameterized uzawa method which covers preconditioned uzawa method and uzawa method in addition the optimal iteration parameters and the corresponding convergence semi convergence factor for the uzawa method are presented

mean li yorke chaos in banach spaces abstract we investigate the notion of mean li yorke chaos for operators on banach spaces we show that it differs from the notion of distributional chaos of type 2 contrary to what happens in the context of topological dynamics on compact metric spaces we prove that an operator is mean li yorke chaotic if and only if it has an absolutely mean irregular vector as a consequence absolutely cesaro bounded operators are never mean li yorke chaotic dense mean li yorke chaos is shown to be equivalent to the existence of a dense or residual set of absolutely mean irregular vectors as a consequence every mean li yorke chaotic operator is densely mean li yorke chaotic on some infinite dimensional closed invariant subspace a dense mean li yorke chaos criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained moreover we construct an example of an invertible hypercyclic operator t such that every nonzero vector is absolutely mean irregular for both t and t 1 several other examples are also presented finally mean li yorke chaos is also investigated for c 0 semigroups of operators on banach spaces

topologically transitive extensions of bounded operators let x be any banach space and t a bounded operator on x an extensionopen image in new window of the pair x t consists of a banach space open image in new window in which x embeds isometrically through an isometry i and a bounded operatoropen image in new window on open image in new window such that open image in new window when x is separable it is additionally required that open image in new window be separable we say that open image in new window is a topologically transitive extension of x t when open image in new window is topologically transitive on open image in new window i e for every pair open image in new window of non empty open subsets of open image in new window there exists an integer n such that open image in new window is non empty we show that any such pair x t admits a topologically transitive extension open image in new window and that when h is a hilbert space h t admits a topologically transitive extension open image in new window where open image in new window is also a hilbert space we show that these extensions are indeed chaotic

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

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

universal properties of harmonic functions on trees abstract we consider an infinite locally finite tree t equipped with nearest neighbor transition coefficients giving rise to a space of harmonic functions we show that except for trivial cases the generic harmonic function on t has dense range in c by looking at forward only transition coefficients we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of t we also study algebraic genericity spaceability and frequent universality of these phenomena

some summation formulas for the series 3f2 1 summation formulas contiguous to watson s and whipple s theorems in the theory of the generalized hypergeometric series are obtained certain limiting cases of these results are given 1 two results contiguous to watson s theorem the two following summation formulas for series 3f2 1 are useful interesting easily established and probably new they are

constraint preconditioners for solving singular saddle point problems we study constraint preconditioners for solving singular saddle point problems we analyze properties of the preconditioned matrices in particular their eigenvalue distributions and prove that for solving singular saddle point problems by preconditioned gmres methods with constraint preconditioners gmres will determine the least squares solutions at breakdown in addition we present some numerical examples to demonstrate the convergence behavior of preconditioned gmres for solving singular saddle point problems

generalizations of dixon s theorem on the sum of a 3 f 2 twenty three formulas closely related to dixon s theorem in the theory of the generalized hypergeometric series are obtained twenty six lim iting cases are also deduced

existence of hypercyclic operators on topological vector spaces the present paper introduces a very simple but very useful notion of the so called quasi extension ofl1 operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators in particular it answers in the affirmative a question of s rolewicz posed in 1969 whether or not every infinite dimensional separable banach space admits a continuous hypercyclic operator

common hypercyclic functions for translation operators with large gaps abstract let h c be the set of entire functions endowed with the topology of local uniform convergence fix a sequence of non zero complex numbers n n which satisfies the following property for every m 0 there exists a subsequence n of n such that i n 1 n m for every n 1 2 and ii n 1 1 n we prove that there exists a residual set g h c such that for every f g and every non zero complex number a the set f z n a n 1 2 is dense in h c this answers in the affirmative question 1 in 29 and it also provides an extension of a theorem due to costakis and sambarino in 32

the role of the angle in supercyclic behavior abstract a bounded operator t acting on a hilbert space h is said to be supercyclic if there is a vector f h such that the projective orbit t n f n 0 and c is dense in h we use a new method based on a very simple geometric idea that allows us to decide whether an operator is supercyclic or not the method is applied to obtain the following result a composition operator acting on the hardy space whose inducing symbol is a parabolic linear fractional map of the disk onto a proper subdisk is not supercyclic this result finishes the characterization of the supercyclic behavior of composition operators induced by linear fractional maps and thus completes previous work of bourdon and shapiro

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

a class of triangular splitting methods for saddle point problems in this paper we study a class of efficient iterative algorithms for the large sparse nonsingular saddle point problems based on the upper and lower triangular ult splitting of the coefficient matrix we call these algorithms ult methods first the ult algorithm is established and the characteristic of eigenvalues of the iteration matrix of these new methods is analyzed then we give the sufficient and necessary conditions for the convergence of these ult methods moreover the optimal iteration parameters and the corresponding convergence factors for some special cases of the ult methods are presented numerical experiments on a few model problems are presented to support the theoretical results and examine the numerical effectiveness of these new methods

invariant gaussian measures for operators on banach spaces and linear dynamics we give conditions for an operator t on a complex separable banach space x with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non degenerate invariant gaussian measure with respect to which it is weak mixing the existence of such a measure depends on the geometry of the banach space and on the possibility of parametrizing the mathbb t eigenvector fields of t in a regular way we also investigate the connection with frequent hypercyclicity

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

the clausenian hypergeometric function 3 f 2 with unit argument and negative integral parameter differences new explicit as well as manifestly symmetric three term summation formulas are derived for the clausenian hypergeometric series 3 f 2 1 with negative integral parameter differences our results generalize and naturally extend several similar relations published in recent years by many authors an appropriate and useful connection is established with the quite underestimated 1974 paper by p w karlsson

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

sums of hypercyclic operators abstract every bounded operator on a complex infinite dimensional separable hilbert space can be written as the sum of two hypercyclic operators and also as the sum of two chaotic operators

preconditioned ahss pu alternating splitting iterative methods for saddle point problems in order to solve large sparse saddle point problems spp quickly and efficiently wang and zhang recently studied the preconditioned accelerated hermitian and skew hermitian splitting pahss methods through accelerating the pahss iteration algorithms by using parameterized uzawa pu method a preconditioned ahss pu alternating splitting iterative method pahss pu method for solving saddle point problems is proposed in this paper the convergence results of this new method are given under some suitable conditions moreover we can obtain that if the parameters are suitable selected then the pahss pu algorithm will outperform the pahss algorithm and some uzawa type methods in the same precision condition numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the pahss pu method

inverse of u frequently hypercyclic operators abstract we show that there exists an invertible u frequently hypercyclic operator on l p n 1 p whose inverse is not u frequently hypercyclic

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

frequently hypercyclic operators with irregularly visiting orbits abstract we prove that a bounded operator t on a separable banach space x satisfying a strong form of the frequent hypercyclicity criterion which implies in particular that the operator is universal in the sense of glasner and weiss admits frequently hypercyclic vectors with irregularly visiting orbits i e vectors x x such that the set n t x u n 1 t n x u of return times of x into u under the action of t has positive lower density for every non empty open set u x but there exists a non empty open set u 0 x such that n t x u 0 has no density

two remarks on frequent hypercyclicity abstract we show that if t x x is a continuous linear operator on an f space x 0 then the set of frequently hypercyclic vectors of t is of first category in x and this answers a question of a bonilla and k g grosse erdmann we also show that if t x x is a bounded linear operator on a banach space x 0 and if t is frequently hypercyclic or more generally syndetically transitive then the t orbit of every non zero element of x is bounded away from 0 and in particular t is not hypercyclic

hypercyclicity and unimodular point spectrum abstract we show that an operator on a separable complex banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic we apply this result to construct hypercyclic operators with prescribed k unimodular point spectrum we show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators and prove that the family of composition operators c on h 2 d where is a disk automorphism having 1 as attractive fixed point has a residual set of common hypercyclic vectors

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

semi convergence analysis of uzawa methods for singular saddle point problems recently zheng bai and yang studied the parameterized uzawa method for solving singular saddle point problems b zheng z z bai x yang on semi convergence of parameterized uzawa methods for singular saddle point problems linear algebra appl 431 2009 808 817 in this paper we discuss the inexact uzawa method which covers the uzawa method the preconditioned uzawa method and the parameterized uzawa method to solve the singular saddle point problems we prove the semi convergence result under restrictions by verifying two necessary and sufficient conditions that is all elementary divisors associated with the eigenvalue 1 of its iterative matrix are linear and the pseudo spectral radius of the iterative matrix is less than 1 sufficient conditions for the semi convergence of several uzawa type methods are also provided in addition numerical examples are given to demonstrate the semi convergence of uzawa type methods

a new generalized parameterized inexact uzawa method for solving saddle point problems recently bai parlett and wang presented a class of parameterized inexact uzawa piu methods for solving saddle point problems bai et al 2005 in this paper we develop a new generalized piu method for solving both nonsingular and singular saddle point problems the necessary and sufficient conditions of the convergence semi convergence for solving nonsingular singular saddle point problems are derived meanwhile the characteristic of eigenvalues of the iteration matrix corresponding to the above iteration method is discussed we further show that the generalized piu type method proposed in this paper has a wider convergence semi convergence region than some classical uzawa methods such as the inexact uzawa method the sor like method the gsor method and so on finally numerical examples are given to illustrate the feasibility and efficiency of this method

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

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

optimal growth of entire functions frequently hypercyclic for the differentiation operator we solve a problem posed by bonilla and grosse erdmann 2007 7 by constructing an entire function f which is frequently hypercyclic with respect to the differentiation operator and satisfies mf r cerr 1 4 where c 0 may be chosen arbitrarily small this growth rate is sharp we also obtain optimal results for minimal growth in terms of average lp norms among other tools the proof uses the rudin shapiro polynomials and heat kernel estimates

modified accelerated parameterized inexact uzawa method for singular and nonsingular saddle point problems abstract recently bai and wang 2008 9 presented an efficient accelerated parameterized inexact uzawa apiu method for solving both symmetric and nonsymmetric nonsingular saddle point problems in this work we propose a modified apiu mapiu method for solving saddle point problems which is an extension of the apiu iteration method and covers a series of existing iterative methods the mapiu method can be applied not only to the nonsingular case but also to the singular case the convergence properties for the nonsingular saddle point problems and the semi convergence properties for the singular one are carefully discussed under suitable restrictions we prove that the mapiu method has a wider convergence or semi convergence region than the apiu method

operators with the specification property abstract we study a version of the specification property for linear dynamics operators having the specification property are investigated and relationships with other well known dynamical notions such as mixing devaney chaos and frequent hypercyclicity are obtained

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

comparison of performance of iterative methods for singular and nonsingular saddle point linear systems arising from navier stokes equations abstract we compare the convergence performance of iterative methods for solving singular and nonsingular linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of singularity and nonsingularity on convergence we will show that the convergence performance for solving a singular linear system is better than that for solving a nonsingular linear system

a bridge between u frequent hypercyclicity and frequent hypercyclicity abstract given a the family of weights a a n n decreasing to 0 such that the series n 0 a n diverges we show that the supremum on a of lower weighted densities coincides with the unweighted upper density and that the infimum on a of upper weighted densities coincides with the unweighted lower density we then investigate the notions of u frequent hypercyclicity and frequent hypercyclicity associated to these weighted densities we show that there exists an operator which is u frequently hypercyclic for each weight in a but not frequently hypercyclic although the set of frequently hypercyclic vectors always coincides with the intersection of sets of u frequently hypercyclic vectors for each weight in a

invertible composition operators on hp abstract the weakly closed algebras generated by certain sets of composition operators are shown to be reflexive a structure theorem for invertible composition operators on h2 is obtained and used to show that such operators are reflexive the structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal thus the general invariant subspace problem for hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators

on the convergence of iterative methods for solving singular linear systems in this paper we first show that for the stationary iterative methods for solving consistent singular linear systems the convergence and the quotient convergence are equivalent then we use this assertion to analyze the convergence of the multisplitting algorithms for the solution of linear systems when the coefficient matrices of the linear systems are singular hermitian positive semidefinite and the linear systems are consistent

strong mixing measures for linear operators and frequent hypercyclicity abstract we construct strongly mixing invariant measures with full support for operators on f spaces which satisfy the frequent hypercyclicity criterion for unilateral backward shifts on sequence spaces a slight modification shows that one can even obtain exact invariant measures

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

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

generalizations of whipple s theorem on the sum of a 3 f 2 thirty eight summation closely related to whipple s theorem in the theory of the generalized hypergeometric series are obtained some limiting cases are also considered

the principal measure of a quantum harmonic oscillator at the end of his paper oprocha 2006 j phys a math gen 39 14 559 65 oprocha said that it is difficult and interesting to know the exact value of the principal measure of an annihilation operator of the unforced quantum harmonic oscillator motivated by this in this paper we prove that this exact value is equal to 1 besides we point out that exhibits distributional chaos for any 0 2

a generalization of the inexact parameterized uzawa methods for saddle point problems abstract for large sparse saddle point problems bai and wang recently studied a class of parameterized inexact uzawa methods see z z bai z q wang on paramaterized inexact uzawa methods for generalized saddle point problems linear algebra appl 428 2008 2900 2932 in this paper we generalize these methods and propose a class of generalized inexact parameterized iterative schemes for solving the saddle point problems we derive conditions for guaranteeing the convergence of these iterative methods with different choices of the parameter matrices the generalized iterative methods lead to a series of existing and new iterative methods including the classical uzawa method the inexact uzawa method the gsor method and the giaor method

extended shift splitting preconditioners for saddle point problems abstract in this paper we consider to solve the linear systems of the saddle point problems by preconditioned krylov subspace methods the preconditioners are based on a special splitting of the saddle point matrix the convergence theory of this class of the extended shift splitting preconditioned iteration methods is established the spectral properties of the preconditioned matrices are analyzed numerical implementations show that the resulting preconditioners lead to fast convergence when they are used to precondition krylov subspace iteration methods such as gmres

a single step iteration method for non hermitian positive definite linear systems abstract an efficient single step iteration method is presented for solving the large sparse non hermitian positive definite linear systems we theoretically prove that this method converges to the unique solution of the system of linear equations under suitable restrictions moreover we derive an upper bound for the spectral radius of the new iteration matrix furthermore we consider acceleration of the new iteration by krylov subspace methods and some special properties of the new preconditioned matrix are proposed numerical experiments on a few model problems are presented to further examine the effectiveness of our new method