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A Condition for Associativity of Univariate Resultants

Examples and Counterexamples.  Published online on 2 May 2023. 4 pages.

DOI: https://doi.org/10.1016/j.exco.2023.100113

Factoring Variants of Chebyshev Polynomials with Minimal Polynomials of cos(2 π / d)

Bulletin of the Australian Mathematical Society.  Published online on 21 March 2022. 10 pages.

DOI: https://doi.org/10.1017/S0004972722000235
 

Factoring Variants of Chebyshev Polynomials of the First and Second Kinds with Minimal Polynomials of cos(2 π / d)

The American Mathematical Monthly.  129(2) (2022) 172-176, and published online on 6 January 2022. 5 pages. 

DOI: https://doi.org/10.1080/00029890.2022.2005391

Below are some publications from when I was a lecturer at ANU.

Electronic Notes in Theoretical Computer Science,

Volume 31 (Elsevier, Amsterdam, 2000) pages 1-184.

Guest Editor, Proceedings of Computing: the Australasian Theory Symposium, 2000. CATS 2000 was held at ANU and I was Chair of the Programme Committee. The invited speakers were Rod Burstall (Edinburgh), Mariangiola Dezani-Ciancaglini (Turin), Lance Fortnow (NEC, Princeton) and Emo Welzl, (ETH, Zürich).

Complexity of nilpotent unification and matching problems

with Qing Guo and Paliath Narendran

Information and Computation  162 , 3-23 (October 2000) 

Abstract

We consider the complexity of equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x, x)=0 where 0 is a constant. We show that nilpotent unification and matching are NP-complete. In the presence of associativity and commutativity, the problems still remain NP-complete. However, when 0 is also assumed to be the unity for the function f, the problems are solvable in polynomial time. We also show that the problem remains in P even when a homomorphism is added. An application of this result to a subclass of set constraints is illustrated. Second-order matching modulo nilpotence is shown to be undecidable.

A formula for the general solution of a constant-coefficient difference equation

Journal of Symbolic Computation 29, 1 (2000) 79-82.

This paper continues some ideas in the one below on generalized Fibonacci recurrences, and provides a formula for their solution.

Abstract

We give a formula for the general solution of a d th-order linear difference equation with constant coefficients in terms of one of the solutions of its associated homogeneous equation. The formula neither uses the roots of the characteristic equation nor their multiplicities. It can be readily generalized to the case where the domain of the difference equation is the real numbers, and the initial values are given by a function defined on the interval [0,d ). In both cases, we express the general solution of the difference equation in terms of a single solution of its associated homogeneous equation at integer arguments.

Solving generalized Fibonacci recurrences

These sequences are defined by the sum of the previous n values where n > 1 and also generalised by using initial functions instead of initial values.

Lemma 3.6 in this paper gives bounds on the roots of the characteristic equation of generalised Fibonacci sequences. This Lemma has been applied widely in other research. The bound for the positive root of the characteristic equation was sharpened by Fan Chung, Persi Diaconis and Ron Graham, Advances in Applied Mathematics 126 (2021) 101916, 50 pp.

The paper gives the solution in radicals for the positive root of the characteristic equation when n = 4, the Tetranacci constant, and also a series for the positive root for any n > 0. These results were summarised in a Wikipedia article on Generalizations of Fibonacci numbers.

An appraisal of INTERNIST-I

Artificial Intelligence in Medicine 7, 2 (1995) 93-116.

Abstract

"INTERNIST-I was an expert system designed in the early 1970's to diagnose multiple diseases in internal medicine by modelling the behaviour of clinicians. Its form and operation are described, and evaluations of the system are surveyed. The major result of the project was its knowledge base which has been used in successor systems for medical education and clinical use. We also survey the effects of the project through these systems, and conclude that the most successful of them in the near future is likely to be Quick Medical Reference (QMR) when used as an “electronic textbook” of medicine."

This paper has various citations including one of many in U.S. patent 7230529, for “System, method, and computer program for interfacing an expert system to a clinical information system" that was awarded to Theradoc  in 2007.