Numerical analysis of a quadratic matrix equation

The quadratic matrix equation AX²+ BX + C = 0in n x nmatricesarises in applications and is of intrinsic interest as oneof the simplest nonlinear matrix equations. We give a completecharacterization of solutions in terms of the generalized Schurdecomposition and describe and compare various numerical solutiontechniques. In particular, we give a thorough treatment offunctional iteration methods based on Bernoulli’s method.Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions.We show that functional iteration applied to the quadraticmatrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem (²A + B + C)x = 0.     

A paired queueing system arising in multimedia synchronization

One of the most important and distinguishing features of multimedia applications is the integration of multiple media streams that have to be presented in a synchronized fashion. In this paper, a queueing system with a special service mechanism arising in multimedia synchronization is considered. The system is characterized by arrival of two types of customers (media streams), and servicing of customers (processing of packets) in pairs with one customer from each type for a pair. The exact transient system size probabilities are obtained as the stationary solutions do not exist for this system and these are illustrated numerically. The density function of the first return to the origin for the queueing system is also obtained.     

On the Padé table for e x and the simple continued fractions for e and e L/M

A recently observed connection between some Padé approximants for the exponential series and the convergents of the simple continued fraction for e is established, leading to an alternative proof of the latter. Similar results for the simple continued fraction e ²,e ^1/M and e ^2/M , when M is a natural number greater than one, are derived.      

Solution of the Semi-Infinite Toda Lattice for Unbounded Sequences

The semi-infinite Toda lattice is the system of differential equations d _n (t)/dt = _n (t)(b _n+1(t) – b _n (t)), db _n (t)/dt = 2( _n ²(t) – _n–1 ²(t)), n = 1, 2, ..., t > 0. The solution of this system (if it exists) is a pair of real sequences _n (t), b _n (t) which satisfy the conditions _n (0) = _n ,, b _n (0) = b _n , where _n > 0 and b _n are given sequences of real numbers. It is well known that the system has a unique solution provided that both sequences _n and b _n are bounded. When at least one of the known sequences _n and b _n is unbounded, many difficulties arise and, to the best of our knowledge, there are few results concerning the solution of the system. In this letter we find a class of unbounded sequences _n and b _n such that the system has a unique solution. The results are illustrated with a typical example where the sequences _i (t), b _i (t), i = 1, 2, ... can be exactly determined. The connection of the Toda lattice with the semi-infinite differential-difference equation d²/dt ² log h _n = h _n+1 + h _n–1 – 2h _n , n = 1, 2, ... is also discussed and the above results are translated to analogous results for the last equation.     

On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities

The notion of equivalence of multidimensional continued fractions is introduced. We consider some properties and state some conjectures related to the structure of the family of equivalence classes of two-dimensional periodic continued fractions. Our approach to the study of the family of equivalence classes of two-dimensional periodic continued fractions leads to revealing special subfamilies of continued fractions for which the triangulations of the torus (i.e., the combinatorics of their fundamental domains) are subjected to clear rules. Some of these subfamilies are studied in detail; the way to construct other similar subfamilies is indicated.     

The distribution of special subsets of the Farey sequence

We will examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are not divisible by a fixed prime p. In particular, we will provide an asymptotic result on the distribution of H-tuples of consecutive fractions in , as Q→∞.     

Units in some families of algebraic number fields

Multi-dimensional continued fractions associated with are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature (2,1).

     

Continued fractions in local fields, II

The present paper is a continuation of an earlier work by the author. We propose some new definitions of -adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every if then has a periodic continued fraction expansion. The same is not true in for some larger values of