Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

We present some general results concerning so-called biorthogonal polynomials of R_II type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel–Darboux formula is obtained.     

Precise (m,k)-Zipf diagram analysis of mathematical and financial time series when m=6, k=2

For studying short-range time correlations in financial signals, we have envisaged to combine the Zipf method and the i-variability diagrams (VD) as useful tools. The 2-VD describes the local curvature short-range correlations. We have resulted into ranking the 2-VD data according to their frequency of occurrence. After having tested the ideas and estimated the error bars on a Brownian motion signal, we have examined two stocks, i.e. SGP and OXHP closing price and volume of transaction long series. A precise (m,k)-Zipf diagram analysis when m=6, k=2 has been shown to lead to a non-immediate information on the signal behavior, even taking into account error bars. The set of curvatures (translated into “words”) indicates a Brownian motion-like set for the closing price local curvature of such signals over a 6 day span. Moreover, it has been shown that the conjecture about a simple relationship between the Hurst exponent H and the ζ exponent of Zipf plots does not seem to be substantiated here.     

Swelling of dextran gel and osmotic pressure of soluble dextran in the presence of salts

The swelling of dextran gels (Sephadex) in salt solutions with a water activity of 0.937, compared with the swelling in pure water, exhibited anion specificity as evidenced by an increased swelling ratio in the following order: Na₂SO₄ < H₂O < NaCl < NaSCN. The swelling ratio showed a good linear correlation with the osmotic pressure of dextran (500 kD) in these solutions. The salt‐concentration difference (imbalance) between the polymer‐solution side of the membrane and the polymer‐free permeate side during the osmotic‐pressure measurements positively correlated with the effect of the salt on the polymer osmotic pressure. These phenomena conform to Hofmeister‐type (or lyotropic) behavior. The diminishing augmentation of dextran osmotic pressure and the change in the salt‐concentration imbalance with rising NaSCN concentration imply a positive preferential interaction and adsorption of the salt onto the polymer. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2740–2750, 2001     

Stability in the presence of degeneracy and error estimation

Received February 10, 1997 / Revised version received June 6, 1998 Published online October 9, 1998     

A Performance Analysis of Appel's Algorithm for Performing Pairwise Calculations in a Many Particle System

Appel's algorithm for efficiently performing pairwise potential energy calculations is analyzed with regard to both computing time required and error incurred. Four factors are identified which determine the practical value of Appel's method: the total number of particles in the system, the tolerance for error in the calculations, the complexity of the pairwise energy calculation, and the spacial distribution of the particles.     

A Proof of Krzyż's Conjecture for the Fifth Coefficient

In this article some problems related to the Krzy? conjecture are discussed. In particular, we prove the Krzy? conjecture for the fifth coefficient.     

Robust Bounds for Krylov Methods

In this paper we give bounds for polynomials of operators. These bounds are robust in low rank perturbations. This problem is encountered in the study of the convergence of Krylov methods. The central idea here is to view the resolvent as a meromorphic function.     

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H _p (G), 1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ _n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L _p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H _p (G), Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ _n,p , 1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b \frac{\log(d μ/ d x)}{\sqrt{(x-a)(b-x)}} dx >- ∈fty. July 27, 2000. Final version received: May 19, 2001.     

Criteria for Feller transition functions

This paper presents some conditions for the minimal Q-function to be a Feller transition function, for a given q-matrix Q. We derive a sufficient condition that is stated explicitly in terms of the transition rates. Furthermore, some necessary and sufficient conditions are derived of a more implicit nature, namely in terms of properties of a system of equations (or inequalities) and in terms of the operator induced by the q-matrix. The criteria lead to some perturbation results. These results are applied to birth-death processes with killing, yielding some sufficient and some necessary conditions for the Feller property directly in terms of the rates. An essential step in the analysis is the idea of associating the Feller property with individual states.     

A probabilistic property of Katsuura's continuous nowhere differentiable function

Katsuura's example of a continuous nowhere differentiable function is based on a fixed-point method. In this paper we use probability theory to study the increments of Katsuura's function.