Littlewood's One Circle Problem

It is shown that Littlewood's one circle problem has a negativeanswer, that is, there exists a continuous bounded functionf on the unit disk U such that f is not harmonic, but neverthelessfor every x U the equality holds for some r(x) with 0 < r(x) < 1 – ||x||.     

Complemented Invariant Subspaces in Bergman Spaces

It is shown that the invariant subspace of the Bergman spaceA^p of the unit disk, generated by either an A^p-interpolatingsequence or a singular inner function with a single point masson the unit circle, is complemented in A^p.     

Error bounds in the isometric Arnoldi process

Error bounds for the eigenvalues computed in the isometric Arnoldi method are derived. The Arnoldi method applied to a unitary matrix U successively computes a sequence of unitary upper Hessenberg matrices H_k, k = 1,2,… The eigenvalues of the H_k's are increasingly better approximations to eigenvalues of U. An upper bound for the distance of the spectrum of H_k from the spectrum of U, and an upper bound for the distance between each individual eigenvalue of H_k and one of U are given. Between two eigenvalues of H_k on the unit circle, there is guaranteed to lie an eigenvalue of U. The results are applied to a problem in signal processing.     

A Counterexample to Uniqueness in the Riemann Mapping Theorem for Univalent Harmonic Mappings

Let f be an orientation-preserving univalent harmonic mappingof the unit disk U. Then , where h and g are analytic in U. Furthermore, f satisfies theequation in U, where a(z)=g'(z)/h'(z), and |a(z)| < 1 in U. The functiona(z) is the analytic dilatation of f. In [2], Hengartner and Schober proved the following versionof the Riemann mapping theorem for univalent harmonic mappings.1991 Mathematics Subject Classification 31A05, 31A20.     

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.     

Algebras with Collapsing Monomials

A semigroup S is called collapsing if there exists a positiveinteger n such that for every subset of n elements in S, atleast two distinct words of length n on these letters are equalin S. In particular, S is collapsing whenever it satisfies alaw. Let U(A) denote the group of units of a unitary associativealgebra A over a field k of characteristic zero. If A is generatedby its nilpotent elements, then the following conditions areequivalent: U(A) is collapsing; U(A) satisfies some semigrouplaw; U(A) satisfies the Engel condition; U(A) is nilpotent;A is nilpotent when considered as a Lie algebra.     

On prime ends and local connectivity

Let U be a simply connected domain whose complement K = \ U contains more than one point. We show that the impressionof a prime end of U contains at most two points at which K islocally connected. This is achieved by establishing a characterizationof local connectivity of K at a point z₀ U in terms of theprime ends of U whose impressions contain z₀, and then invokinga result of Ursell and Young.     

The Coradical Filtration for Quantized Enveloping Algebras

We determine the coradical filtration on a quantized envelopingalgebra U_q(g) = U defined over Q(q). As applications, we determinethe biideals of U and describe the group of Hopf algebra automorphismsof U.     

On a Question of Herman, Baker and Rippon Concerning Siegel Disks

Consider the family of exponential maps E_k(z) = exp(z)+k. Thispaper shows that any unbounded Siegel disk U of E_k containsthe singular value k on its boundary. By a result of Herman,this implies that k U if the rotation number is diophantine.2000 Mathematics Subject Classification 37F10, 30D05.     

Composition Preserves Rigidity

Say that f in H^p (1 p ) of the unit disk is rigid if it isdetermined in H^p by its argument modulo 2 on the unit circle.We show that if f is rigid and u is an inner function, thenthe composition f o u is rigid. The proof uses the disintegrationof Lebesgue measure with respect to u to compute the adjointof the operator of composition with u. This result is a generalization of the work of Younis, who provedthe special case p = 1 using operator theoretic methods.     

Formulas for Liapunov Functions for Systems of Linear Difference Equations

Explicit quadratic Liapunov functions that provide necessaryand sufficient conditions for the asymptotic stability of thesystem of linear difference equations x (t + 1) = Ax(t) areconstructed by transforming the original systems to y (t + 1)= Gy(t), where G is a companion matrix associated with the characteristicpolynomial of A. A necessary and sufficient condition for allroots of the characteristic polynomial to lie in the unit circle|z| < 1 on the complex plane is also derived. 2000 MathematicalsSubject Classification 39A11, 93D05.     

Linear resolvent growth of rank one perturbation of a unitary operator does not imply its similarity to a normal operator

The main result of this paper is that for any unitary (selfadjoint) operatorU with non-trivial absolutely continuous part of the spectrum, there exists a rank-one perturbationK = ba* = (a)b such that the operatorT = U + K satisfies the Linear Resolvent Growth condition (LRG),

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank (S : U) of a subsemigroup U of a semigroupS is the minimum size of a set V S such that U together withV generates the whole of S. As a consequence of a result ofSierpiski, it follows that for U T_X, the monoid of all self-mapsof an infinite set X, rank(T_X : U) is either 0, 1 or 2, or uncountable.In this paper, the relative ranks rank(T_X : O_X) are considered,where X is a countably infinite partially ordered set and O_Xis the endomorphism monoid of X. We show that rank(T_X : O_X) 2 if and only if either: there exists at least one elementin X which is greater than, or less than, an infinite numberof elements of X; or X has |X| connected components. Four examplesare given of posets where the minimum number of members of T_Xthat need to be adjoined to O_X to form a generating set is,respectively, 0, 1, 2 and uncountable. 2000 Mathematics SubjectClassification 08A35 (primary), 06A07, 20M20 (secondary).     

Bounded Local Resolvents for Operators with Single-Valued Extension Property

Let T be a bounded operator with (SVEP) on its localizable spectrum \(\sigma _\mathrm{loc}(T)\). We show that for every open subset U of \(\sigma _\mathrm{loc}(T)\), there exists a unit vector x whose local spectrum coincides with the closure of U, and such that its local resolvent function is bounded. This result answers positively to an open question stated by several authors, and extends the both cases of operators with trivial divisible subspace and operators whose point spectrum has empty interior.     

Squared Givens Rotation

Given a matrix A, let U be the upper triangular matrix thatresults from the application of Gaussian elimination to A^TA.We develop a fast and square-root-free Givens rotation admittingthe direct calculation of U from A.     

UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS

Let P²ⁿ⁺¹ be a two-cell complex which is formed by attachinga (2n + 1)-cell to a 2m-sphere by a suspension map. We constructa universal space U for P²ⁿ⁺¹ in the category of homotopy associative,homotopy commutative H-spaces. By universal, we mean that Uis homotopy associative, homotopy commutative and has the propertythat any map f: P²ⁿ⁺¹ Y to a homotopy associative, homotopycommutative H-space Y extends to a uniquely determined H-map: U Y. We then prove upper and lowerbounds of the H-homotopy exponent of U. In the case of a modp^r, Moore space U is the homotopy fibre S²ⁿ⁺¹{p^r} of the p^r-powermap on S²ⁿ⁺¹, and we reproduce Neisendorfer's result that S²ⁿ⁺¹{p^r}is homotopy associative, homotopy commutative and that the p^r-powermap on S²ⁿ⁺¹{p^r} is null homotopic.     

A Geometric Invariant for Metabelian Pro-p Groups

In [2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see [6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of F_p and U be the formal powerseries algebra F[T], with group of units U^x. Let Q be a finitelygenerated abelian pro-p group. We write Z_p[Q] for the completedgroup algebra of Q over Z_p. Let T(Q) be the abelian group Hom(Q,U^x) of continuous homomorphisms from Q to U^x. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Z_p[Q]to U.     

Conformal Images of Borel Sets

For any holomorphic map in the unit disk, the set of radiallimits at a Borel set on the unit circle is a Suslin-analyticset. Here it is proved that, for a conformal map, this set is,in fact, Borel. As a consequence, the sets of accessible boundarypoints, of cut points and of transition points are Borel. Inaddition, it is shown that the set of end points is a G-set.2000 Mathematics Subject Classification 28A05, 30C35.     

Estimation of the parameters for unstable AR models

ＥＳＴＩＭＡＴＩＯＮＯＦＴＨＥＰＡＲＡＭＥＴＥＲＳＦＯＲＵＮＳＴＡＢＬＥＡＲＭＯＤＥＬＳＡＮＨｏＮＧＺＨＩ（安鸿志）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ,ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅ,Ｂｅｉｊｉｎｇ１０...