A continued fraction algorithm

Summary Two existing algorithms for the evaluation of a finite sequence of convergents of a continued fraction are considered. Each method has a drawback concerning numerical stability or computational effort. A third algorithm is presented which requires less computations than the first method, and generally is more stable than the second one. The results are illustrated by numerical examples. The connection with Mikloko's algorithm is shown.     

An open mapping theorem using lower semicontinuous functions

We give a simple and direct proof, using lower semicontinuous functions, of a generalization of the open mapping theorem for metrizable topological vector spaces (that are not necessarily locally convex) and operators with complete graph. Our result is in a form more applicable to applied (convex) analysis.     

Estimates for norms of matrix-valued and operator-valued functions and some of their applications

A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators.From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved.From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.     

An algebraic structure for Faber polynomials

Let Σ be the set of functions, convergent for all |z|>1, with a Laurent series of the form f(z)=z+∑_n?0a_nz^-n. In this paper, we prove that the set of Faber polynomial sequences over Σ and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the Bell^(k) subgroup of the Riordan group, respectively. Further, a relationship between such Faber polynomial sequences and Lucas and Sheffer polynomial sequences is derived.     

J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations

This is the first of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted largely to foundational material (much of which is of independent interest) on the theory of assorted classes of meromorphic matrix valued functions. Particular attention is paid to the structure of J-inner functions and connections with bitangential interpolation problems and reproducing kernel Hilbert spaces. Some new characterizations of regular, singular and strongly regular J-inner functions in terms of the associated reproducing kernel Hilbert spaces are presented.D. Z. Arov wishes to thank the Weizmann Institute of Science for hospitality and support; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research.     

An algorithm for evolutionary surfaces

Summary An Algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a series of surfaces. This gives a new method to solve Plateau's problem forH-surfaces.     

A Constructive Inversion Framework for Twisted Convolution

In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be interpreted as another twisted convolution on a finite cyclic group. In analogy with the twisted convolution of finite discrete signals, we derive an anti-homomorphism between the sequence space and a suitable matrix algebra which preserves the algebraic structure. In this way, the problem reduces to the analysis of finite matrices whose entries are sequences supported on corresponding cosets. The invertibility condition then follows from Cramer’s rule and Wiener’s lemma for this special class of matrices. The problem results from a well known approach of studying the invertibility properties of the Gabor frame operator in the rational case. The presented approach gives further insights into Gabor frames. In particular, it can be applied for both the continuous (on ) and the finite discrete setting. In the latter case, we obtain algorithmic schemes for directly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature.     

Intersections of Sets Starshaped Via Paths of Bounded Length

 Let S be a nonempty closed, simply connected set in the plane. For α > 0, let ℳ denote the family of all maximal subsets of S which are starshaped via paths of length at most α. Then ⋂{M : M in ℳ} is either starshaped via α-paths or empty. The result fails without the simple connectedness condition. However, even with a simple connectedness requirement, there is no Helly theorem for intersections of sets which are starshaped via α-paths. Received November 19, 2001; in revised form April 25, 2002 Published online November 18, 2002     

Spectral Properties of Interval Exchange Maps

 In [7], Nogueira and Rudolph proved that for irreducible permutations not of rotation class almost every (a.e.) interval exchange transformation (i.e.t.) is topological weak mixing. It is conjectured that the claim holds if topological weak mixing is replaced by weak mixing. Here we study the behaviour of eigenfunctions of i.e.t. Our analysis gives alternative proofs of results due to Katok and Stepin [4] and Veech [10]: for certain permutations a.e. i.e.t. is weak mixing and for irreducible permutations a.e. i.e.t. is totally ergodic. (Received 1 February 2001)     

A characterization of Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.     

A potential method for the biharmonic equation

Summary A new formulation of the Dirichlet problem for the biharmonic operator is presented. This gives rise to a simple numerical method to solve the above problem. Convergence is proved in the unidimensional case. Numerical results in one and two dimensional test problems are presented.     

On complete partially ordered sets and compatible topologies

We describe two complete partially ordered sets which are the intersection of complete linear orderings but which have no compatible Hausdorff topology. One is two-dimensional, while the second is countable, and leads to an example of a countable, compact, T ₁ space with a countable base which is not the continuous image of any compact Hausdorff space.     

Sets of Locally Maximal Visibility and Finite Unions of Starshaped Sets

 Let , where is an open connected subset of some linear topological space, such that S contains all triangular regions whose (relative) boundaries lie in S. If some finite subset T of S has locally maximal visibility in S, then . Hence S is a finite union of starshaped sets whose kernels are determined by T. An analogous result holds for S open. Moreover, counterexamples show that neither the requirement on triangular regions nor the restriction to a finite set T can be deleted. (Received 7 September 1998; in revised form 25 October 1999)     

Regular closure operators

In an E,M-categoryX for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms inM to factor through the lattice of all closure operators onM, and to factor through certain sublattices. This leads to the notion ofregular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün-Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class ofX-objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.Dedicated to Professor Dieter Pumplün, on his 60th birthdayResearch partially supported by the Faculty of Arts and Sciences, University of Puerto Rico, Mayagüez Campus during a sabbatical visit at Kansas State University.     

The pullback closure operator and generalisations of perfectness

A categorical closure operator induced via pullback by a pointed endofunctor is introduced. Various notions of a perfect morphism relative to a pointed endofunctor and the induced closure are then considered. The main result explores how these notions are interrelated, linking also with earlier notions of perfectness.The author acknowledges financial support from the University of Cape Town, from the Foundation for Research Development through the Categorical Topology Research Group at the University of Cape Town, and from the University of L'Aquila.     

Almost distributive sublattices and congruence heredity

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary. Presented by E. W. Kiss. Received July 18, 2005; accepted in final form April 2, 2006.     

Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

Invariance Principles for Energy Functionals on Spheres

We study a generalised version of the g-energy functionals defined by Damelin and Grabner. We comment on invariance principles for finite energies and use these principles to obtain expansions of these latter energies in terms of cap discrepancies for a subclass of g. This allows for discrepancy estimates knowing bounds on the energy and vice versa. We are, in particular, able to carefully analyse the case when g gives a Riesz kernel g^R_s when 0<sd or a logarithmic kernel g^L₀ in the limits when 0⁺.The author is supported by the START project Y96-MAT of the Austrian Science Fund.     

Riordan arrays associated with Laurent series and generalized Sheffer-type groups

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples.