Uniform Stability for Time-Varying Infinite-Dimensional Discrete Linear Systems

Stability for time-varying discrete linear systems in a Banachspace is investigated. On the one hand is established a fairlycomplete collection of necessary and sufficient conditions foruniform asymptotic equistability for input-free systems. Thisincludes uniform and strong power equistability, and uniformand strong l_p-equistability, among other technical conditionswhich also play an essential role in stability theory. On theother hand, it is shown that uniform asymptotic equistabilityfor input-free systems is equivalent to each of the followingconcepts of uniform stability for forced systems: l_p-input l_p-state,e_o-input e_o-state, bounded-input bounded-state, l_p-input bounded-state(with p>1), e_o-input bounded-state, and convergent-inputbounded-state; these are also equivalent to their nonuniformcounterparts. For time-varying convergent systems, the aboveis also equivalent to convergent-input convergent-state stability.The proofs presented here are all ‘lementary’ inthe sense that they are based essentially only on the Banach–Steinhaustheorem.     

(k, l)-Algebraic Stability of Gauss Methods

It is well known that the s-stage Gauss Runge-Kutta methodsof order 2s are algebraically stable, or equivalently (1, 0)-algebraicallystable. In this paper, we show that there exists some l_s >0 such that the Gauss methods are (k, l) algebraically stablefor l [0, l_s) with k(l)=e^2l+O(l^p+1, where p=2s if s=1 or s=2,and p=2 if s>3.     

Mahler Measure of Alexander Polynomials

Let l be an oriented link of d components in a homology 3-sphere.For any nonnegative integer q, let l(q) be the link of d–1components obtained from l by performing 1/q surgery on itsdth component l_d. The Mahler measure of the multivariable Alexanderpolynomial _l(q) converges to the Mahler measure of _l as q goesto infinity, provided that l_d has nonzero linking number withsome other component. If l_d has zero linking number with eachof the other components, then the Mahler measure of _l(q) hasa well defined but different limiting behavior. Examples aregiven of links l such that the Mahler measure of _l is small.Possible connections with hyperbolic volume are discussed.     

Wandering Domains in Non-Archimedean Polynomial Dynamics

We extend a recent result on the existence of wandering domainsof polynomial functions defined over the p-adic field C_p toany algebraically closed complete non-archimedean field C_K withresidue characteristic p > 0. We also prove that polynomialswith wandering domains form a dense subset of a certain one-dimensionalfamily of degree p + 1 polynomials in C_K[Z]. 2000 MathematicsSubject Classification 12J25 (primary), 37F99 (secondary).     

On a Question of Haskell P. Rosenthal Concerning a Characterization of c0 and lp

The following property of a normalized basis in a Banach spaceis considered: any normalized block sequence of the basis hasa subsequence equivalent to the basis. Under uniformity or othernatural assumptions, a basis with this property is equivalentto the unit vector basis of c₀ or l_p. An analogous problem concerningspreading models is also addressed. 2000 Mathematics SubjectClassification 46B20 (primary), 46B15 (secondary).     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

Plain Representations of Lie Algebras

In this paper we study representations of finite dimensionalLie algebras. In this case representations are not necessarilycompletely reducible. As the general problem is known to beof enormous complexity, we restrict ourselves to representationsthat behave particularly well on Levi subalgebras. We call suchrepresentations plain (Definition 1.1). Informally, we showthat the theory of plain representations of a given Lie algebraL is equivalent to representation theory of finitely many finitedimensional associative algebras, also non-semisimple. The senseof this is to distinguish representations of Lie algebras thatare of complexity comparable with that of representations ofassociative algebras. Non-plain representations are intrinsicallymuch more complex than plain ones. We view our work as a steptoward understanding this complexity phenomenon. We restrict ourselves also to perfect Lie algebras L, that is,such that L = [L, L]. In our main results we assume that L isperfect and sl₂-free (which means that L has no quotient isomorphicto sl₂). The ground field F is always assumed to be algebraicallyclosed and of characteristic 0.     

Universal Operator Algebras Associated to Contractive Sequences of Non-Commuting Operators

We characterize the minimal isometric dilation of a non-commutativecontractive sequence of operators as a universal object forcertain diagrams of completely positive maps. A non-spatialconstruction of the minimal isometric dilation is also given,using Hilbert modules over C*-algebras. It is shown that the non-commutative disc algebras A_n (n2) arethe universal algebras generated by contractive sequences ofoperators and the identity, and C*(S₁, ..., S_n) (n2), the extensionthrough compact operators of the Cuntz algebra O_n, is the universalC*-algebra generated by a contractive sequence of isometries.It is also shown that the algebras A_n and C*(S₁, ..., S_n) arecompletely isometrically isomorphic to some free operator algebrasconsidered by D. Blecher. In particular, the universal operatoralgebra of a row (respectively column) contraction is identifiedwith a subalgebra of C*(S₁, ..., S_n). The internal characterizationof the matrix norm on a universal algebra leads to some factorizationtheorems.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

Reduced-order filtering with energy-to-peak performance for discrete-time Markovian jumping systems

In this paper, the l₂–l (energy-to-peak) performanceof the discrete-time Markovian jump linear system is investigated.The jump parameters are modelled by a discrete-time Markov process.Furthermore, we study the l₂–l reduced-order filteringproblem for the Markovian jump linear system. A reduced-orderfilter with the same randomly jumping parameters is proposedwhich can make the error systems with Markovian jump parametersstochastically stable with a prescribed l₂–lperformance.Sufficient conditions in terms of linear matrix inequalities(LMIs) and a coupling non-convex rank constraint are derivedfor the existence of a solution to the reduced-order filteringproblems. A numerical example is given to illustrate the designprocedures.     

Bounds on Norms of Compound Matrices and on Products of Eigenvalues

An upper bound on operator norms of compound matrices is presented,and special cases that involve the l₁, l₂ and l norms are investigated.The results are then used to obtain bounds on products of thelargest or smallest eigenvalues of a matrix. 1991 MathematicsSubject Classification 15A15, 15A18, 15A42.     

On Ritt's Factorization of Polynomials

Ritt has shown that any complex polynomial p can be writtenas the composition of polynomials p₁,...,p_m, where each p_j isprime in the sense that it cannot be written as a non-trivialcomposition of polynomials. The factors p_j are not unique butthe number m of them is, as is the set of the degrees of thep_j. The paper extends Ritt's theory and, in particular, a thirdinvariant of the decomposition is introduced.     

A Sharp Lp Inequality for Dyadic A1 Weights in Rn

The exact best possible range of p is determined such that anydyadic A₁ weight w on Rⁿ satisfies a reverse Hölder inequalityfor p, which depends on the dimension n and the correspondingA₁ constant of w. The proof is based on an effective linearizationof the dyadic maximal operator applied to dyadic step functions.2000 Mathematics Subject Classification 42B25.     

On the Existence of WKB-Type Asymptotics for the Generalized Eigenvectors of Discrete String Operators

Let J be a Jacobi real symmetric matrix on l₂ with zero diagonaland non-diagonal entries of the form {1 + p_n}. If p_n–1± p_n = O(n^–) with some 2/3, then the existenceof bounded solutions of Ju = u is proved for almost every (–2, 2) with the WKB-type asymptotic behavior. 2000 MathematicsSubject Classification 47B36, 47B37, 47B39.     

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_q[G] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_q[G] is Noetherian recovers aresult due to Joseph [10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.     

An additive Schwarz method for the h-p version of the boundary element method for hypersingular integral equations in R3

We study a preconditioner for the h-p version of the boundaryelement method for hypersingular integral equations in threedimensions. The preconditioner is based on a three-level decompositionof the underlying ansatz space, the levels being piecewise bilinearfunctions on a coarse grid, piecewise bilinear functions ona fine grid, and piecewise polynomials of high degree on thefine grid. We prove that the condition number of the preconditionedlinear system is bounded by max_j (1 + log H_jp_j/h_j)² where H_jis the diameter of an element _j of the coarse grid, h_j is thesize of the elements of the fine grid on _j, and p_j is the maximumof the polynomial degrees used in _j. Numerical results supportingour theory are reported. Received 9 March 1999. Accepted 19 July 1999.     

l1 solution of linear inequalities

The numerical solution of a possible inconsistent system oflinear inequalities in the l₁ sense is considered. The non-differentiablel₁ norm minimization problem is approximated by a piecewisequadratic Huber smooth function. A continuation algorithm isdesigned to find an l₁ solution of the inequality system. Inthe case where the linear inequality system is consistent, asolution is obtained by solving any smoothed problem. Otherwise,the algorithm is shown to terminate in a finite number of iterations.We also consider an alternative smoothing scheme which sharessimilar properties with the first one, but results in an improvedcomputational performance of the continuation algorithm on inconsistentsystems. Numerical experiments are conducted to test the efficiencyof the algorithm.     

One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

On the Location of Zeros of Certain Classes of Polynomials with Applications to Numerical Analysis

A polynomial is said to be of type (p₁, p₂, p₃) relative tothe unit circle if it has p₁ zeros interior to, p₂ on, and p₃exterior to the unit circle. Stability criteria frequently arisewhere a polynomial or a family of polynomials must be shownto be of type (p₁, p₂, 0) or of type (p₁, 0, 0). Here we reconsiderthe practical problem of showing that a polynomial is of oneor other of these types, and we show that the testing of a polynomialof degree n may always be reduced to the testing of one of degreen–1. The simplicity of the method is illustrated by itsapplication to several well known difference schemes for partialdifferential equations.     

On the Poles of Igusa's Local Zeta Function for Algebraic Sets

Let K be a p-adic field, let Z (s, f), sC, with Re(s) > 0,be the Igusa local zeta function associated to f(x) = (f₁(x),..., f_l(x)) [K (x₁, ..., x_n)]^l, and let be a Schwartz–Bruhatfunction. The aim of this paper is to describe explicitly thepoles of the meromorphic continuation of Z (s, f). Using resolutionof singularities it is possible to express Z (s, f) as a finitesum of p-adic monomial integrals. These monomial integrals arecomputed explicitly by using techniques of toroidal geometry.In this way, an explicit list of the candidates for poles ofZ (s, f) is obtained. 2000 Mathematics Subject Classification11S40, 14M25, 11D79.