Faculty of Mathematic Handbook,1987 - University of Newcastle

PART IV MATHEMATICS TOPICS
664179 History of Analysis to Around 1900 - R.F. Berghout
Prerequisite
Hours
Examination
Content
Nil
About 27 lecture hours
One 2·hour paper
A course of 26 lectures on the history of mathematics with emphasis on analysis. Other
branches of mathematics will be referred to putting the analysis into context. Where
feasible, use will be made of original material, in translation. The course will be
assessed by essays and a final 2-hour examination.
Topics to be covered include: pre-Greek concepts of exactness and approximation; Greek
concepts of continuity, irrationality, infinity, infinitesimal, magnitude, ratio, proportion
and their treatment in Elements V, XU and the works of Archimedes; developments of
number systems and their equivalents; schOlastic mathematics; virtual motion;
Renaissance quadrature/cubature by infinitesimals and by "geometry"; Cartesian
geometry; 17th and 18th century calculus; rigorization of analysis in the 19th century
with stress on the developments of number systems, continuity, function concept,
differentiability, integrability.
Text
References
Nil
664151 Radicals & Annihilators - R.F. Berghout
Prerequisite
Hours
Examination
Content
Lists will be presented during the course.
Topics T or X
About 27 lecture hours
One 2-hour paper
This topic will briefly outline the classical theory of finite dimensional algebras and the
emergence of the concepts of radical, idempotence, ring, chain conditions, etc. Hopefully
thus set in perspective, the next part will deal with the Artin-Hopkins-Jacobson ring
theory and the significance of other radicals when finiteness conditions are dropped. The
relations between various radicals, noetherian rings, left and right annihilators and the
Goldie-Small theorems will end the topic.
Text
References
Cohn, P.
Divinsky, N.
Herstein, IN.
Kaplansky, 1
McCoy, N.
664166 Symmetry - W. Brisley
Prerequisites
Hours
Examination
Nil
Algebra Vol. 2 (Wiley 1977)
Rings and Radicals (Allen-Unwin 1964)
Non-commutative Rings (Wiley 1968)
Fields and Rings (Chicago 1969)
The Theory of Rings (McMillan 1965)
Topics D and K
About 27 lecture hours
One 2-hour paper
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Content
This course studies various aspects of symmetry. Matters discussed may include:
invariance of lattices, crystals and associated functions and equations; permutation
groups; finite geometries; regular and strongly-regular graphs; designs; tactical
configurations, "classical" simple groups, Matrix groups, representations, characters.
Text Nil
References
Biggs, N. Finite Groups of AUlonwrphisms (Cambridge 1971)
Cannichael, R.D. Groups of Finite Order (Dover reprint)
Harris, D.C. & Bertolucci, M.D. Symmetry and Spectroscopy (Oxford 1978)
Rosen, J. Symmetry Discovered (Cambridge 1975)
Shubnikov, A.V. & Koptsik, V.A. Symmetry in Science and Art (plenum Press 1974)
Weyl, H. Symmetry (princeton 1973)
White, A.T. Graphs, Groups and Surfaces (North-Holland 1973)
664106 Combinatorics - W. Brisley
Prerequisite
Hours
Examination
Content
Topic K
About 27 lecture hours
One 2-hour paper
Permutations and combinations, inclusion-exclusion and generating functions. Poly a's
theorem and its application to counting various kinds of structures and graphs will be
discussed. Also asymptotic analysis of many of the exact results.
Text
References
Beckenback, E.F. (ed.)
Hall,M.
Harary, F. & Palmer, E.M.
Liu, C.L.
Riordan, J.
Nil
664169 Nonlinear Oscillations - J.G. Couper
Prerequisite
Hours
Examinatio'n
Content
Applied Combinatorial Mathematics (Wiley 1964)
Combinatorial Theory (Blaisdell 1967)
Graphical Enumeration (Academic 1974)
Introduction to Combinatorial Mathematics
(McGraw Hill 1968)
Combinatorial Analysis (Wiley 1958)
Topic P
About 27 lecture hours
One 2-hour paper
Physical problems often give rise to ordinary differential equations which have oscillatory
solutions. This course will be concerned with the existence and stability of periodic
solutions of such differential equations, and will cover the following subjects; twodimensional
autonomous systems, limit sets, and the Poincare-Bendixson theorem.
Brouwer's. fixed point theorem and its use in finding periodic solutions. Non-critical
linear systems and their perturbations. The method of averaging. Frequency locking,
jump phenomenon, and subharmonics. Bifurcation of periodic solutions. Attention will
be paid to applications throughout the course.
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