Computer Science (MATH)

MATH 131 Algebra and Solid Geometry

The real and complex number systems. Polynomial functions. Theory of equations. Determinants. Vector and matrix notation. Elementary solid geometry, planes and lines, the standard quadric surfaces. Co-ordinate systems. 2 lectures. one hour problems.

MATH 132 Introductionto Computer Science

A thorough introduction to algorithms, stored-programme computers and programming languages. Concept and properties of an algorithm, language and notation for describing algorithms. Analysis of computational problems and development of algorithms for their solution. A procedure-oriented language (FORTRAN IV) and machine and assembly languages are used to implement algorithms on the computer. 2 hours lectures, 2 hours problems.

Scanned page 263 MATH 229 today

MATH 229 Linear Algebra

Vector spaces, matrices, groups of transformations. 3 hours lectures.

MATH 231 Differential Calculus

Real numbers, sequences. Limits. Continuity. Derivatives. Functions of several variables. Implicit functions. Jacobians. Taylor’s series with geometric applications. Power series with complex terms. The elementary functions of a complex variable. 2 lectures.

MATH 232 Integral Calculus

Methods of formal integration. Differential equations. Fourier series. Riemann integration and applications. Calculus of variations. Line integrals. Stokes’ and Green’s theorems. 3 lectures.

MATH 233 Probability and Statistics

Frequency distributions. Probability. Binomial. Normal, and Poisson laws. Sampling theory. Curve fitting. Distributions of Chi-squared, F. and t. Hypothesis testing. Quality control. Regression theory. Analysis of variance. Introduction to experimental design. 2 lectures. I hour problems.

MATH 234 Introduction to Applied Mathematics

Differential equations, including first order and constant coefficients. The Laplace transform, Applications of differential equations to problems in the Physical Sciences. Vector analysis. Newtonian mechanics. Central forces. Transformation of co-ordinate systems. 2 hours lectures.

Scanned page 263 MATH 235 today

MATH 235 (a) Actuarial Mathematics Mathematics of Investment(b) Basic Life Insurance Mathematics Differential

The theory of rates of interest and discount, annuities and sinking funds; application to financial problems, including the determination of mortgage payments and the price and yield of bonds. 2 hours lectures. Fall term.

Applications of probability to problems of life and death. The determination of single and annual premiums for assurances and annuities. Reserves. Company expenses and their incorporation into premium and cash value calculations. 2 hours lectures. Winter term.

MATH 236 Elementary Equations Blas.

An introduction to first order differential equations. Applications to erowth, decay, reactions. Linear equations with constant coefficients. Systems of equations; vector and matric methods. Solutions in series. Introduction to partial differential equations. 2 lectures.

As 2 4L oat AIT * =..

Faculty of Mathematics

Scanned page 264 MATH 237 today

MATH 237 Differential and Integral Calculus

Real numbers, sequences, limits, continuity, derivatives. Functions of several variables, implicit, functions, Jacobians, Taylor’s series. Methods of formal integration; differential equations. Fourier series. Riemann integration; line integrals. 3 lectures.

MATH 239 Introduction to Combinatorics and Optimization

A selection of topics from discrete mathematics including an introduction to graph theory, difference equations, generating functions. convex sets, finite geometries and partially ordered sets. The ideas are illustrajed with examples from modern applications to such topics as network theory, queueing theory, random walks, linear programming, design of experiments, ranking problems and PERT. 2 hours lectures.

Scanned page 264 MATH 239 today

MATH 240 Applications in Computer Science

Some numerical methods are introduced in the first term and programmed for the computer using FORTRAN IV. Concepts of Numerical Errors. Methods in Interpolation, numerical integration, solution of nonlinear equations, linear systems of equations. In the second term non-numeric computing is introduced including simulation, the concept of a list and elementary list processing techniques, sorting, symbol manipulation. 2 hours lectures. 1 hour problems

MATH 243 Statistics for the Scleoces

The topics of Mathematics 233 with particular emphasis on the analysis and interpretation of experimental data and the design of experiments in the Sciences. The more difficult mathematical techniques associated with these problems will be omitted and many examples from Physics, Chemistry, Biology and other natural Sciences will be considered. 2 hours lectures, | hour laborotory.

MATH 300 Discovery and Invention in Mathematics

The art and technique of problem solving from the point of view of discovery and invention (heuristic); induction, analogy, pattern generalization, analysis, similarity and plausible reasoning. Problems will be se.- lected from various areas of mathematics including Geometry (Euclid to Hilbert), Algebraic Structure (Groups in Algebra and Geometry), Set Theory (point sets, Boolean Algebra) and Logic. 2 hours lectures.

MATH 329 Abstract Algebra

Groups; permutation groups, Lagrange’s theorem, normal subgroups, homomorphisms. Fields. Rings and ideals. Integral domains. 2 hours lectures.

MATH 330 (8) Projective Geometry(b) Non-Euclidean Geometry

Finite geometries,Distance geometry, convex sets. 3 lectures.

MATH 332 Theory of Functions

The real number system. Uniform convergence. Theory of Fourier series. The complex number system. Analytic functions. The Cauchy theorems. Contour integration. Singularities. Conformal mapping. 3 lectures.

MATH 333 Differential Equations

First order differential equations; existence theorems; singular solutions; solution by Laplace Transforms; numerical methods of solutions; solution in series; hypergeometric, Bessel and Legendre functions; introduction to partial differential equations and integral equations.

MATH 262 Faculty of Mathematics

MATH 334 Numerical Methods

Finite differences. Solution of equations. Computation with series and integrals. Linear systems and matric methods. Difference equations and relaxation methods. Numerical solution of partial differential equations. Methods employing electronic computers. 2 lectures. 2 hours laboratory.

MATH 335 Differential amd Difference Equations.

Differential equations with constant coefficients, systems of equations, D-operator methods, variation of parameters, solution in series, an introduction to the Laplace transform. Finite difference calculus, summation, numerical integration, finite difference equations. 2 hours lectures.

MATH 336 Life Contingencies

(a) An advanced course on problems with single lives. 3 hours lectures. 1 term. (b) An advanced course on problems with multiple lives; population and multiple’ decrement theory. 3 lectures. / term.

MATH 338 Mathematical Statistics

An advanced mathematical course including the multivariate normal distribution and related distributions. Normal Regression Theory. Application to hypothesis testing and analysis of variance. Maximum likelihood and large sample theory. Prerequisite: Mathematics 233 or equivalent. 2 hours lectures.

MATH 340 Computer Systems

A discussion of the hardware and software required in a computer system. Overlapped channels, interrupt facilities, memory protection, buffers, input-output control systems, macro-programming, monitor systems. Relocation schemes, multi-programming, multi-processing, dynamic memory allocation, time-sharing. Special purpose computer systems and simulation of computer systems. Peripheral equipment. Introduction to the theory of sequential machines and the logical design of computers. 2 lectures.

MATH 341 Abstract Algebra

Lattices; groups, Lagrange’s Theorem, normal subgroups, homomorphisms, permutation groups; rings, ideals and integral domains; modules, tensor prodcts; fields and extensions. 3 hours lectures.

MATH 342 Real Variable Theory

Real number system; metric space setting for ideas of continuity, uniform continuity, compactness, convergence, open sets, closed sets, connected sets. Cantor ternary set. Stone-Weierstrass theorem, uniform convergence, equicontinuity. Ascoli’s theorem, fixed point theorems and applications, tangent spaces, differential forms, exterior differentiation, Stokes’ Theorem. Introduction to Lebesgue integration. 2 hours lectures.

MATH 343 Complex Variable Theory

Complex numbers; continuity and differentiability, Cauchy-Riemann equations; solution of Laplace equation; bilinear transformations and conformal mapping; power series; integral theorems; Taylor and Laurent expansions; calculus of residues; series solution of differential equations; theory of Laplace transforms; solution of differential equations by integrals. 2 hours lectures.

MATH 344 Point Set Topology

Topological spaces. Cartesian products, continuous mappings, separation axioms, connectedness. Compact and locally compact spaces, me- &ability.

Faculty of Mathematics

MATH 345 Topics in Pure Mathematics for Combinatorial Mathematicians

Elementary introduction to cardinal and ordinal numbers; convexity and its applications to optimization; introduction to the main concepts of projective geometry, metric and topological spaces.

MATH 351 Introduction to Graph Theorynod Combinatorical Analysis

Graphs, paths, arcs, circuits, connected graphs, trees, spanning trees, digraphs, Euler paths, tournaments, complete paths in tournaments, factors of bipartite graphs, Menger’s Theorem and flows in networks, theory of enumeration, generating functions, Ramsey’s Theorem and applications, systems of distinct representatives, Latin squares, balanced incomplete block designs and (v.k.) configurations. Applications to practical problems and other areas of mathematics.

MATH 352 Mathematical Operations Research

Review of mathematical techniques frequently used in Operations Research; including difference-differential equations, Lagrange Multipliers, generating functions, quadratic forms, minimax algorithms in directed graphs. Introduction to Linear programming, Dynamic programming, transport network theory. Applications to production planning, inventory problems, growth and survival models, scheduling problems, replacement policies and reliability optimization of complex systems. 2 hours lectures.

MATH 360 (a) Tensor Analysis(b) Differential Geometry

Vector analysis in curvilinear coordinates; gradient, divergence, curl. Transformation laws of covariant, contravariant, relative tensors. Covariant and intrinsic differentiation. The Riemann-Christoffel tensors, Bianchi identities, Ricci and Einstein tensors. Integrability theorems. Applications to classical mechanics and elasticity. Introduction to general relativity. 2 hours lectures. Fall term.

The metric tensor, and elementary differential geometry of surfaces, Introduction to Riemannian geometry; Riemannian curvature, Gaussian curvature and Einstien curvature tensors. Spaces of affine connection and Finsler spaces with application to electromagnetic theory. 2 hours lectures. Winter term.

MATH 361 (a) Calculus of Variations(b) Mechanics

Euler-Lagrange Equations, Legendre and Jacobi condition, the E-function. Fields of extremals and sufficient conditions; Hilbert’s invariant integral. Parameter invariant single and multiple integral problems; Hamilton-Jacobi theory. Introduction to optimal control problems; introduction to direct methods and partial differential equations. 2 hours lectures. Fall term.

2 Hamilton’s principle. Lagrangian and canonical Equations of motion. Relativistic mechanics. 2 hours lectures. Winter term.

MATH 363 Differential Equations

Existence of solutions. Second-order linear equations. Green’s functions. Series solutions. Laplace transforms. Sturm-Louisville boundary-value problems. Fourier series. Partial differential equations. 2 hours lectures.

MATH 399 Reading in M athematics

MATH 410 Rings and Modules

Projective and injective modules. Baer’s theorem. Types of rings. Fundamental Theorem of Abelian groups. Artin-Wedderburn theorem. Krull-Schmidt Theory. Hilbert’s Theorem. Algebras. Tensor and extenw .

MATH 264 Faculty of Mathematics

MATH 411 Group Theory

Elementary properties of Abelian groups, solvable groups, nilpotent groups. Direct products and semi-direct products. Free groups and free products. Elements of group representations. 2 hours lectures.

MATH 412 Theory of Fields

Algebraic Extensions. Separable and purely inseparable extensions. Normal extensions. Algebraic closure. Finite fields. Galois extensions. Fundamental theorem of Galois theory. Cyclotomic fields. Solvability in radicals. Trace and norm. Cyclic extensions. Hilbert’s Theorem 90. Infinite Galois theory. Transcendental extensions. 2 hours lectures.

MATH 413 Non-linear Differential Equations

Riccati equation, conflict and pursuit problems, elliptic integrals and functions, phase plane, non-linear mechanics. Non-linear integral equations. 2 hours lectures.

MATH 425 Introduction to the Theory of Numbers

2 hours lectures.

MATH 426 Topology

Continuation of general topology. Topological manifolds, a weak simpliembedding theorem, classification of |-manifolds. Triangulation, cial complexes, simplicial homology. Classification and homology groups of closed (triangulated) 2-manifolds. The fundamental group covering spaces, homotopy groups, the Hurewicz isomorphism theorem. 2 hours lectures.

MATH 427 Graph Theory

Graphs and subgraphs. Connection modulo a subgraph. Trees, arcs, and Separation. 3-connection. The structure of 2-connected polygons. graphs. Planar graphs. Kuratowski’s theorem. 2 hours lectures.

MATH 428 Lattice Theory

Ordered sets, lattices, Galois connections. Special attention is given to geometric lattices (matroid theory). 2 hours lectures.

MATH 429 Combinatorial Topology

Homology theory of complexes. Theorems of invariance, covering, and duality. 2 hour lectures.

MATH 430 Finite Projective Geometries

Representations by fields, groups, and algebras. Finite Euclidean geom- Sub-geometries. Collineation groups. Tactical configuration. etries. 2 lectures.

MATH 432 Functions of a Complex Variable

meromorphic func- Analytic continuation. The Riemann surface. Entire, tions. Function spaces. Linear operators. 2 hours lectures.

MATH 433 Measure and Integration

Measure in Euclidean n-space, the Lebesque integral, convergence theorems, the Fubini theorem, differentiability, absolute continuity. A study of Banach spaces. 2 lectures.

MATH 434 Differential Equations of Mathematical Physics

Green’s functions. Cauchy problem. Classification of partial differential equations. Orthogonal functions. Bessel functions. Legendre polynomials. Fourier Integral and Laplace transform. Variational properties of eiaenvalues and _ eieenfunctions.

Faculty of Mathematics

MATH 435 Laboratory

Numerical problems arising in actuarial science and statistics. 2 hours laboratory.

MATH 436 Mathematical Logic

An informal introduction to the logic of sentences and predicates, with emphasis on analogies with familiar mathematical structures. Syllogisms and algebra of sets related to predicate logic. Simplification, logical deduction, duality, consistency and completeness. The concepts of constant, variable, function, and set. Axiomatics. 2 lectures.

MATH 437 Construction of Contimgemcy Tables

(a) Accidental death and disability benefits. Pension fund mathematics. Methods of analysing data to produce raw rates for mortality tables. 2 hours lectures. 1 term.

(b) Methods of graduating mortality tables. 2 hours lectures. I term.

MATH 438 Estimation and Hypothesis Testing

The mathematics and logic of estimation and hypothesis testing. Consistency, efficiency and sufficiency of estimates. Fiducial and confidence intervals, likelihood, Bayes’ theorem, tests of significance. 2 hours lectures.

MATH 439 Theory of Experimental Design 4% Probability

The logic of experimental design with reference to randomization, experimental and sampling errors. Construction and analysis of various kinds of designs. 2 hours lectures.

Classical probability and introduction to stochastic processes with applications such as branching processes, epidemics, etc. 2 hours lectures.

MATH 441 Quantum Mechanics

The Schroedinger wave equation, interpretation of wave functions, harmonic oscillator in wave mechanics. The hydrogen atom. Perturbation theory. The variation methods. The W.K.B. approximation. 2 lectures.

MATH 442 Theory of Relativity

Covariant and contravariant tensors. Riemannian space. Gaussian curvature. The Lorentz transformation and Einstein’s special theory. Introduction to the general theory of relativity. 2 lectures. Winter term.

MATH 443 Electromagnetism

Applications of the Maxwell equations. Reflection and refraction. Introduction to wave guides and antennae. 2 lectures, Fall term.

MATH 444 Elasticity

Problems in stress and strain. Isotropic bodies. Equilibrium equations. Torsion and flexure, 2 lectures, Winter term.

MATH 445 Hydrodynamics

Kinematics and dynamics of flow. Equations of motion. Inviscid twodimensional motion. Applications of conformal transformation to fluid flow. Viscous motion and similarity. Oseen’s approximation. 2 hours lectures.

MATH 446 History of Mathematics

The development of Mathematics from ancient to modern times, including study of prominent mathematicians and their works. Problems will hea warlad neoinng hath manadaca and hiatacinnl « amennn ban

MATH 266 Faculty of Mathematics

MATH 447 Statistical Mechanics

Applications of probability theory to theoretical Physics. 2 lectures. Fall term.

MATH 448 Differential Geometry

Curves in space. Gaussian curvature. Tensorial notation. Introduction to differentiable manifolds. 2 lectures.

MATH 449 Experimental Design

Similar to Mathematics 439 but with more accent on the logic and methods than on the mathematics. Primarily for students in the Sciences. 2 hours lectures.

MATH 450 Applied Analysis Applied Mathematics for Science majors. Ordinary and partial differen-

tial equations. Laplace and Fourier transforms. Special functions. Fourier series. Vector and matric methods. Calculus of variations. Other topics in advanced calculus. 2 lectures.

MATH 451 Optimization

Discrete functions; continuous functions of n variables. Quadratic forms; convex functions. Equality constraints; Lagrange multipliers and generalizations. Inequality constraints; Kuhn-Tucker theorem. Steepest descent and gradient methods. Fibonnaci search, successive approximations, simplex method and other techniques. Applications to the solution of equations and non-linear regression. 2 hours lectures.

MATH 452 Linear Programming

Mathematical foundations. Simplex method and revised simplex method. Duality. Computational methods. Assignment and _ transportation problems. Integar Programming. Applications to game theory, industrial problems and economic theory. 2 hours lectures.

MATH 453 Queueing Theory

Input and Service-time distributions; Poisson process. Erlangian distribution. Single channel queue; differential-difference equations, integral equations. Complex queueing models. Monte Carlo techniques. 2 hours lectures.

MATH 454 Game Theory

Classification of games. Zero-sum matrix games. Infinite zero-sum games. n-person co-operative and non-co-operative games. 2 hours lectures.

MATH 455 Mathematical Programming

An introduction to the methods and applications of integer programming, non-linear programming and dynamic programming. 2 hours lectures.

Combinatorial

MATH 457 Applied Mathematics

May include any subset of the following topics: applications of combinatorial mathematics to optimization, selected topics from the Beckenbach book, structural models, graphs and electrical networks, graph theory and statistical mechanics, the Ising problem, percolation processes, the dimer problem, random walk, applications to chemistry, automata. 2 hours lectures.

MATH 458 Topological Graph Theory

Selected topics within the general area of graphs and maps in the plane and on other surfaces, the four colour problem, connectivity and separation in graphs. thickness and genus of sranhs ete

Faculty of Mathematics

MATH 459 Algebraic Graph Theory

Graphs and matroids. Applications of algebraic techniques (e.g. matroids, matrices, chain groups) to graph theory or related combinatorial topics. Applications of graph theory to algebra. 2 hours lectures.

MATH 460 Combinatorics

Enumerative mathematics. Combinatorial properties of various mathematical structures (e.g. graphs, geometries, etc.) 2 hours lectures.

MATH 461 (8) Advamced Actuarial Theory(a) Risk Theory.(b) Actuarial Laboratory

Individual and collective risk theory. Ruin theory. Stop-loss reinsurance premiums. Monte Carlo methods. 2 hours lectures,] term.

A tutorial course for the advanced actuarial student. 2 hours problems, I term.

MATH 442 (a) Measure an4 Imtegration(b) Fourier Series amd Orthogonal Functions

The theory of measure and the Lebesgue integral. 2 hours lectures. Fall term.

Linear spaces, orthogonal functions, Fourier series, Legendre and Bessel functions, harmonic analysis. 2 hours lectures. Winter term.

MATH 463 Control Theory

Dynamical systems, control ability and observability. Minimization of functions and functionals. Optimal control. Pontryagm’s Maximum Principle. 2 hours lectures, Fail Term.

MATH 464 Topics in Mathematical Physics

A selection of topics given by various members of the Applied Mathematics.Department. 2 hours lectures. 2 terms.

MATH 470 Numerical Solution of Ordinary and Partial Differential Equations47a Switching Circuits

Existence and uniqueness of solutions of ordinary differential equations, one-step methods for scalar and systems of differential equations, multistep methods, linear boundary value problems. Discrete approximations to partial differential equations, convergence, stability and convergence rate estimates, generalization to non-linear equations. 2 hours lectures.

Switching devices. Binary codes and arithmetic. Boolean algebra and its application to switching circuits. Cononical forms for Boolean functions. Simplification of switching functions. Analysis of sequential circuits. Synchronous and asynchronous operation. Design of sequential circuits. 2 hours lectures. Fall term.

MATH 471B Computer System Organizationand Logic Design

Parr /. Logic Design: Brief review of basic switching theory; Number systems; logic circuits; storage elements; standard computer subsystems; overall logical design of a conventional processor; automated design of processors — computer compilers. Parr 2. System Organization: Definition of total system requirements for batch processing, time-sharing, real-time processing and other applitradeoffs and the impact of large-scale intehardware-software cations; gration; unorthodox hardware organizations evaluation of solutions. Some attention will be paid to design for reliability and fault diagnosis as additional system requirements.

MATH 268 Faculty of Mathematics

MATH 472A Introduction to Automata Theory

Sequential machine models. Reduction of completely and incompletely specified machines. Decomposition of machines. Nondeterministic machines. Finite automata and regular events. Multitape-finite-state nonwriting automata. Prerequisite: Consent of instructor. 2 hours lectures. Fall term.

MATH 472B Introduction to Turing Machines and Computability Theory

. Formalisms or Turing machines. Variants of Turing machines. Universal Turing machines. Unsolvability of the halting problem and related results. Recursive and primitive recursive functions. Models similar to digital computers. Brief introduction to formal languages. Post’s symbol-manipulation systems. Simple bases for computability. Prerequisite: Mathematics 472a or Consent o/instructor. 2 hours lectures. Winter Term.

MATH 480 Functional Equations

Cauchy’s, Pexider’s and similar equations. Equations for polynomials and for trigonometric functions. Reduction to differential equations. General methods and theorems. Iteration. Applications. Further topics, as Equations for functions of several variables, or Equations for analytic functions, or Equations on algebraic structures. 2 hours lectures.

MATH 499 Reading in Mathematics Applied Analysis Faculty

Graduate and Research Programmes

While the faculty offers graduate work over a broad spectrum of the various branches of Mathematics, fields of special interest include: Abstract Algebra, Combinatorial Analysis, Computer Science, Differential Equations, Differential Geometry, Foundations, Functional Analysis, Functional Equations, Graph Theory, Harmonic Analysis, Mathematical Logic, Number Theory, Numerical Analysis, Optimization, Probability, Statistical Inference, Switching and Automata Theory. The University of Waterloo has at present four computer systems, namely an IBM System /360 Model 75, and IBM System /360 Model 44, an IBM 1710, an IBM 1620 and peripheral equipment which includes an IBM 2250 Graphic Display and a CALCOMP plotter. The IBM 1620 computer is available to undergraduate students at all times. The IBM 1710 is used by undergraduates, graduates, faculty and staff. The System /360 Model 75 is run on a closed shop basis but all members of the University can get programmes processed within a reasonably short period of time. The System 360 Model 44 is used in process control applications and provides the University with a centralized process control facility. Extensive financial assistance is available through National Research Council Scholarships and Bursaries and Province of Ontario Graduate Fellowships. In addition, the Faculty of Mathematics offers a number of Teaching Fellowships and some Research Assistantships. For details, correspondence should be addressed to the Graduate Office of the Faculty or Department Chairmen. Candidates for the degrees of M. Math., M. Phil. and Ph.D. are, accepted under the general regulations set forth in the section of the calendar pertaining to the Graduate Studies. Students proceeding to the M. Math. degree are encouraged to present theses. Those proceeding by course work must submit a substantial essay in connection with at least one of the courses.

members of the Annlied Analvsis and Camnuter Science Ne.

Faculty of Mathematics 269

ing: Computer Graphics, Formal Languages, Switching Theory, Theory of Automata, Numerical Analysis, Information Theory, Functional Equations, Compiler Construction, Programming Systems, Machine Organization and Fault Diagnosis, and Functional Equations.

Research in compiler construction has produced the WATFOR fast FORTRAN IV compiler currently being used by over one hundred System /360 Computing Centres throughout the world. Over seventy installations are using a similar compiler developed in 1965 for 7040-7044 Series machines (Communications of the ACM, January, 1967).

Among the problems investigated in the areas of Formal Languages, Automata and Switching Theory are the algebraic theory of languages, regular languages and finite automata, structure theory of sequential machines, design of arithmetic units for residue number systems, and computer-assisted theorem proving.

Topics currently being studied in Computability Theory are Turing machines, complexity classifications of computable functions, abstract models of digital computers and hierarchies of subclasses of recursive functions.

Our Graphic Display unit is promoting research in new techniques of graphic programming, computer-assisted instruction, design procedures and analogue simulation.

In Numerical Analysis, algorithms for two-point boundary problems, stiff systems of differential equations, atomic structure equations, unilateral matrix equations and optimization problems are under investigation at the present time.

Research in the field of Machine Organization and Fault Diagnosis includes the development of software aids to fault diagnosis, the investigation of unorthodox machine organizations from the viewpoints of fault diagnosis and system availability and the selfdiagnosis problem for digital processors. Also under study are system problems arising from time sharing, and the problem of hyper-efficient compiling.

In applied Analysis, among others, functional equations are studied. i.e. the theory of determining functions from elementary equations containing them. It is a field of mathematics with a two century history although the somewhat more general theory has developed only in the last two decades. Functional equations have applications in many modem disciplines including information theory, mathematical psychology and universal algebra.

At the University of Waterloo the research being done in functional equations includes general methods of solution, uniqueness theorems and applications in differential geometry, homological algebra, functional analysis, theories of continuous groups, quasigroups and nets, complex functions, nomography and the theory of distributions.

Applied Mathematics Research in this department is being carried out in a number of areas directly or indirectly applicable to problems in the Physical Sciences. The range of interests of the members of the department reflects the diversity of problems of a mathematical nature which arise in the real world.

The major areas of specialization include Nonlinear Differential

Equations, Spectral Theory of Differential Operators, Optimal Control Theory, Boundary Layer Problems in Fluid Dynamics, Harmonic Analysis, Theory of Relativity, Particle and Quantum Physics, Statistical Mechanics and the Theory of Liquids.

Pure Mathematics Current research in this area includes studies in sheaf theory and cate-

anesian Aiffasaatial manmaten tamalaan af ALDnanesiabd. 2220.14, TT

Faculty of Mathematics

groups, theory of integration and measure, topological vector spaces, Banach algebras, logic and foundations of set theory.

Statistics Research in the Department of Statistics centres on the controversial

Note 1

problem of formulating exact statistical inferences in many diverse situations using generalizations of the approach introduced by the late Sir Ronald Fisher in his foundation of the theory of fiducial inference, the theory of the likelihood function and the theory of estimation in general. The extensive computing facilities available at the University of Waterloo have given rise to investigations into the uses of computers to obtain exact results on the basis of these theories in situations previously treated by asymptotic methods, and also research into exact tests in multivariate analysis.

Closely related to this research, there is active interest in problems of experimental design, with particular reference-to incomplete designs, multivariate analysis, estimation theory, mathematical genetics and stochastic processes.

Members of the department are also interested in problems in combinatorial mathematics, such as the construction of balanced incomplete block designs, which arise in the design of experiments. Other combinatorial questions under consideration include edge-chromatic graphs and the Ramsey numbers.

Increasing contact with the departments of Chemistry. Physics and Psychology provides a variety of new problems and also gives advanced students a valuable opportunity to apply lecture material to useful practical situations. An arrangement with the Statistics Unit of the Department of Agriculture in Ottawa provides potential summer employment for advanced undergraduates and at the same time. a further opportunity for practical experience.

Graduate Courses

An asterisk (*) indicates courses given in 1968-I 969 academic year. Other graduate courses may be given from lime 10 time as demand warrants.

Note 2 All courses with title “Topics in ,. .”’ are indicated by dagger (t) to

mean that the course material is subjectto yearly change.