Keith Taylor's Mathematical Chemistry Page

Mathematical Chemistry Research Unit

My interest in mathematical questions from chemistry began in the early 80s from work on the symmetry groups of crystals. Through that work I established scholarly contact with chemists and became invoved with the formation of the Mathematical Chemistry Research Unit at the University of Saskatchewan. The unit is co-directed by Prof. Paul Mezey from the Department of Chemistry and myself. Its purpose is to facilitate collaborative work between mathematicians and chemists.

Asymptotic Properties

One of the current projects of the Mathematical Chemistry Research Unit that is of particular interest to me involves certain asymptotic linearity phenomena observed in the laboratory setting with families of hydrocarbons. The driving force behind this project is Prof. Shigeru Arimoto who has been studying these and related phenomena since 1978 and gave the first proof of the Asymptotic Linearity Theorem in 1987 which is fundamental to elucidate the mechanism of additivity phenomena in physico-chemical network systems. He has formulated the mathematical notion of a Repeat Space which he and collaborators used to explain the observed asymptotic linearities. These results suggested a multitude of other questions which we are investigating. I will sketch the mathematical setting below. Let q and r be fixed natural numbers.

Let Q_-r, ^..., Q_-1, Q₀, Q₁, ^..., Q_r be fixed q by q complex matrices which satisfy Q_-j* = Q_j, for -r < j < r. For each natural number n define the qn by qn block Toeplitz matrix T_n by

T_n=

Q₀. Q_-1 . . . Q_-r .0 . . . . . . . . . . . . . 0
Q₁. Q₀. Q_-1 . . . Q_-r 0 . . . . . . . . . . .0
.
Q_r. Q_r-1
. 0 . Q_r . Q_r-1 . . . . . . . . . . Q_-r 0 . . 0
.
.
. 0 . . . . . . . . . . . . . 0 Q_r . . . Q₁. Q₀

Let k be a multiple of q.....(I'll have to finish this later)

Back to Keith Taylor's homepage.

Last update: July 8, 1998