SYLLABUS Definitions; modular lattices. Application to abstract algebras.
Distributive lattices. Boolean algebras. Applications to logic.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
344 GENERAL ALGEBRA (ADVANCED) Not offered in 1977.
16 lectures. 3 points.
Prerequisite: 334.
SYLLABUS Universal algebras and related topics.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
351 ORDINARY DIFFERENTIAL EQUATIONS (ADVANCED) 24 lectures. 5 points.
Prerequisite: 254 (or good results in 294).
Condition: Students taking this course may not gain credit for 391.
SYLLABUS Existence and uniqueness theorems. Linear systems: funda- mental matrix. Floquet theory. Stability of linear and nonlinear systems.
Plane autonomous systems. Perturbation theory of nonlinear systems.
Comparison theorems and maximum principles for second order equations.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
352 PARTIAL DIFFERENTIAL EQUATIONS (ADVANCED) 24 lectures. 5 points.
Prerequisites: 253, 254.
Condition: Students taking this course may not gain credit for 391.
SYLLABUS Partial differential equations; characteristics, types of equa- tions, boundary conditions. Elliptic equations: maximum principle, mean value theorem, Green's function, variational formulation, elgenfunctlon expansions. Hyperbolic equations: use of characteristics, wave equation.
Heat equation: maximum principle, Green's function and similarity solution.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
353 OPTIMIZATION METHODS (ADVANCED) 16 lectures; 3 points.
Prerequisite: 263 or 264 or 394.
SYLLABUS Constrained and unconstrained minimization problems. Uni- variate search and approximation methods. Multivariate constrained mini- mization by simplex search, conjugate gradient, variable metric methods.
Constrained optimization, Kuhn-Tucker conditions, gradient methods, penalty methods.
121
Books
Prescribed textbook:
Adby P R and Dempster M A H Introduction to Optimization Methods, Chapman and Hall 1974
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
354 FLUID MECHANICS (ADVANCED) 16 lectures; 3 points.
Prerequisite: 262, 371 or 331.
Condition: Students taking this course must take 392 concurrently.
SYLLABUS Conformal mapping and aerodynamics, vorticity and rotating fluids, hydrodynamic stability.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
355 WAVES (ADVANCED) 16 lectures; 3 points.
Prerequisites: 262, 352.
SYLLABUS kinematic waves and shocks. Wave equation and hyperbolic systems. Linear dispersive waves. Applications to acoustics, elasticity, and geophysical fluid mechanics.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
356 VARIATIONAL METHODS 16 lectures; 3 points.
Prerequisite: 263.
SYLLABUS Extrema of functionals: necessary and sufficient conditions.
Minimizing sequences. Direct methods. Eigenvalues. Isoperimetric prob- lems. Applications: variational principles in mechanics (Hamiltonian theory), optics, continuum mechanics, economics, etc.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
357 THERMODYNAMICS AND STATISTICAL MECHANICS 24 lectures. 5 points.
Prerequisite: 261 or 640-224.
SYLLABUS Laws, thermodynamic relations and ideal gas. Elements of kinetic theory. Gibbs ensembles and the thermodynamic limit. Phase transitions and critical points. Model systems. Applications to biology.
Books
Prescribed textbook:
Thompson C J Mathematical Statistical Mechanics, Macmillan 1972 EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
359 SOLID MECHANICS (ADVANCED) Not offered in 1977.
16 lectures. 3 points.
Prerequisites: 262, 371 or 331.
Conditions: Students taking this course are advised to take 352 or 391 concurrently.
SYLLABUS Review of basic equations, energy theorems; integral repre- sentations. Half-space problems. Two dimensional crack problems. Equa- tions of visco-elasticity. Introduction to seismology.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
362 MATHEMATICAL METHODS (ADVANCED) 16 lectures; 3 points.
Prerequisites: 254 and 371 or 331.
Condition: Students taking this course may not gain credit for 391.
SYLLABUS Fourier and Laplace transforms. Asymptotic evaluation of integrals. Asymptotic methods for solving ordinary differential equations.
Applications to the special functions (hypergeometric, Bessel, Legendre).
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
371 COMPLEX ANALYSIS
24 lectures. Practice classes by arrangement.
Prerequisite: 272 or 231.
SYLLABUS Differentiability. Conformal mapping. Power series. Residues.
Contour integration.
BOOKS
Prescribed textbook:
Tall D 0 Functions of a Complex Variable Vols I & 11, Routledge &
kegan Paul 1970
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
372 GEOMETRY
Not offered in 1977.
16 lectures; practice classes by arrangement. 3 points.
123
Prerequisite: At least one of 271, 273 or 274.
Condition: Students taking this course may not gain credit for 338.
SYLLABUS Revision of Euclidean geometry. Cross ratio, harmonic sec.
tion, pole and polar. Geometry of incidence. Axiomatic system. Finite geometries. Real projective plane. Principle of duality. Perspectives, projectivities, Complete quadrangle. Projectivities on a conic. Non-Eucli- dean geometries.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
373 LINEAR ANALYSIS
24 lectures. Practice classes by arrangement.
Prerequisite: 272 or 377.
Condition: Students taking this course may not gain credit for 332.
• SYLLABUS Linear spaces, norms, inner products, Banach and Hilbert spaces. Bessel's and Schwarz's inequalities, Parseval's formula. Sequence spaces, C(I), L2 norm on C(I). Linear functionals and operators, dual spaces, dual of IP Riesz representation theorem. Weierstrass approxima- tion theorem.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
374 RINGS AND MODULES
16 lectures; practice classes by arrangement; 3 points.
Prerequisite: 274.
Condition: Students taking this course may not gain credit for 334.
SYLLABUS Elementary theory of rings including subrings, homomorphisms and polynomial rings. Factorization in integral domains, the Euclidean algorithm and Euclidean domains. Elementary theory of modules including submodules, homomorphisms and direct sums. Decomposition theorem for finitely generated modules over a Euciidean domain and application to the structure of abelian groups.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
375 TOPOLOGY OF SURFACES
16 lectures. Practice classes by arrangement. 3 points.
Prerequisite: 231 or 271.
SYLLABUS Topics selected from: Classification of surfaces, equivalence and orientation, topological polygons, Riemann surfaces.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
377 INTRODUCTION TO METRIC SPACES 16 lectures; 3 points.
Prerequisites: 272 or 231, and 273.
Condition: Students taking this course may not gain credit for 232.
SYLLABUS Metric spaces: convergence in metric; open and closed sets;
Cauchy sequences, continuity, contraction mapping theorem, compactness.
Prescribed textbook:
Pitts C G C Introduction to Metric Spaces, Oliver & Boyd 1972 EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
391 METHODS OF APPLIED MATHEMATICS 32 lectures; practice classes by arrangement; 6 points.
Prerequisite: 294 or 254.
Condition: Students taking this course may not gain credit for 351 or 352.
SYLLABUS A selection from the following topics: Tensor calculus, special functions, Laplace and other integral transforms, Green's functions, asymp- totic expansions, integral equations, differential equations.
BOOKS
Prescribed textbook:
'Spiegel M R Theory and Problems of Laplace Transforms, Schaumm EXAMINATION One 3-hour paper. Prescribed written work may form part of the final assessment.
392 MECHANICS OF CONTINUOUS MEDIA 24 lectures; practice classes by arrangement; 5 points.
Prerequisite: 262.
SYLLABUS Navier-Stokes equations for a viscous fluid, boundary layers, slow viscous flow. Navier equations of elasticity, two-dimensional elasto- static problems, waves in elastic media.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
393 SYSTEMS THEORY
24 lectures; practice classes by arrangement; 5 points.
Prerequisite: 294 or 254. _
SYLLABUS Linear operators, transforms, input, output, feedback, applica- tions: renewal equation, traffic dynamics. Stability. Volterra problem of competing species. Dynamical systems in biology. Elements of optimal control.
125
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
394 LINEAR PROGRAMMING
16 lectures. Practice classes by arrangement. 3 points.
Prerequisite: 273 or 233.
Condition: Students taking this course may not gain credit for 264.
SYLLABUS The linear programming problem. Review of basic algebra.
Simplex and revised simplex methods. Duality. Degeneracy. Transporta- till problem.
BOOKS
Prescribed textbook:
Hadley G Linear Programming, Addison-Wesley 1971
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.
396 OPTIMIZATION
24 lectures; weekly practice classes; 5 points.
Prerequisite: 264 or 394.
SYLLABUS Extensions of linear programming, perturbations, decomposi- tion methods. Transportation and networks. Introduction to dynamic programming. Methods for computing a minimum. The knapsack problem in integer programming.
EXAMINATION One 2-hour paper. Prescribed written work may form part of the final assessment.