a matrix convergence problem

564 PROBLEMS AND SOLUTIONS Problem 77-13", A Property o]" the First Erlang Function, by ILIA KAUFMAN (Bell Canada, Ottawa, Ontario, Canada). Let where f(x) (x +c)[B(x +clx- 1)-B(x + clx)], B(ylx) y e-’ dt" x->l, c=>O, The function B, or its restriction to integral values of x, B(yln) y.,=o y,/k !, is called the firstErlangfunction. It is easy to prove that for any fixed value of c, lim f(x)= 2/7r. Determine or numerically estimate A= c_->0inf SxU_>__ IL(x)-2/rl. Our own numerical results suggest that for c=0.75, supx_>_l If(x)-2/rrl is sufficiently small that it may be neglected in some practical calculations. This problem arises in queuing theory and has applications in telephone plant engineering. Problem 77-14", A Matrix Convergence Problem, by G. K. KRISTIANSEN (Research Establishment RISO, Roskilde, Denmark). Let P {p,,} be a symmetric matrix having (i) p, 0 for Ir sl > 1 and p,, > 0 T otherwise, (ii) spectral radius 1, and (iii) p_l, +p+x,, _-< 1 for all s. Denote by e the 1 x n matrix with all entries 1, and let I {6} be the n x n unit matrix. Let c be a nonnegative n x i matrix with e TC 1. Prove or disprove that the matrix F (I-c e T)p has spectral radius at most equal to 1. If a counterexample is found, try to minimize the order n. The problem arose in an investigation of methods for solution of the neutron diffusion equation in reactor physics. Problem 77-15", A Conjectured Minimum Valuation Tree, by I. CAHIT (Turkish Telecommunications, Nicosia, Cyprus). Let T denote a tree on n vertices. Each vertex of the tree is labeled with distinct integers from the set 1, 2, , n. The weight of an edge of T is defined as the absolute value of the difference between the vertex numbers at its endpoints. If S denotes the sum of all the edge weights of T with respect to a given labeling, it Downloaded 11/16/14 to 130.207.50.37. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

Upload: g-k

Post on 22-Mar-2017

217 views

Category:

Documents

4 download

Report

Download

Embed Size (px):

TRANSCRIPT

564 PROBLEMS AND SOLUTIONS

Problem 77-13", A Property o]" the First Erlang Function, by ILIA KAUFMAN (BellCanada, Ottawa, Ontario, Canada).

Let

where

f(x) (x +c)[B(x +clx- 1)-B(x + clx)],

B(ylx) y e-’ dt"

x->l, c=>O,

The function B, or its restriction to integral values of x,

B(yln) y.,=o y,/k !,

is called the firstErlangfunction. It is easy to prove that for any fixed value of c,

lim f(x)= 2/7r.

Determine or numerically estimate

A= c_->0inf SxU_>__ IL(x)-2/rl.

Our own numerical results suggest that for c=0.75, supx_>_l If(x)-2/rrl issufficiently small that it may be neglected in some practical calculations. Thisproblem arises in queuing theory and has applications in telephone plantengineering.

Problem 77-14", A Matrix Convergence Problem, by G. K. KRISTIANSEN(Research Establishment RISO, Roskilde, Denmark).Let P {p,,} be a symmetric matrix having (i) p, 0 for Ir sl > 1 and p,, > 0

Totherwise, (ii) spectral radius 1, and (iii) p_l, +p+x,, _-< 1 for all s. Denote by ethe 1 x n matrix with all entries 1, and let I {6} be the n x n unit matrix. Let c bea nonnegative n x i matrix with e TC 1. Prove or disprove that the matrix

F (I-c e T)phas spectral radius at most equal to 1. If a counterexample is found, try tominimize the order n. The problem arose in an investigation of methods forsolution of the neutron diffusion equation in reactor physics.

Problem 77-15", A Conjectured Minimum Valuation Tree, by I. CAHIT (TurkishTelecommunications, Nicosia, Cyprus).Let T denote a tree on n vertices. Each vertex of the tree is labeled with

distinct integers from the set 1, 2, , n. The weight of an edge of T is defined asthe absolute value of the difference between the vertex numbers at its endpoints.If S denotes the sum of all the edge weights of T with respect to a given labeling, it

Dow

nloa

ded

11/1

6/14

to 1

30.2

07.5

0.37

. Red

istr

ibut