MATH 100 Commercial Law for Mathematics Students
Contract law. Agency. The law of banking. Real estate law. Insurance law. Partnership law. Company law. The law of intellectual property.
Combinatorics and Optimization (MATH)
197 courses in the 1974–75 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
Notes
Stanford University, Stanford, California Quantum Theory MTechnische Universitat, Miinchen, Germany Functional Equations Stockholm University Stockholm, Sweden Functional Equations Monash University Clayton, Victoria, Australia University of Dundee, Numerical Analysis Dundee, Scotland University of Liverpool, Differential Geometry and Liverpool, England Relativity
Electives for Mathematics Students
The following courses may be counted as non-mathematics electives by mathematics students. Mthel 206 may be taken only by students in the Co-operative Mathematics Teaching Option.
MATH 101 (a) Applications of Mathematics
Mathematics Electives 101, 201, 301 and 401 are a sequence of courses designed to illustrate the representation of a real situation by a mathematical model. The mathematical techniques necessary to formulate the model and to discuss its behaviour are developed as required. Examples will be taken from diverse fields since one of the aims of the course is to formulate models to describe unfamiliar situations. First term: Models using difference equations will be formulated for examples drawn from biology, ecology, economics and psychology. 3 hours lectures, 1 hour tutorial. Fall term
MATH 101 (b) Applications of Mathematics
See preamble to Mthel 101 (a). b) Sets of difference equations, probability and matrix theory will be used to describe models from biology, ecology, economics, psychology and physiology. 3 hours lectures, 1 hour tutorial. Winter term. -
MATH 201 (a) Applications of Mathematics
See preamble to Mthel 101 (a). Ordinary differential equations will be used in models drawn from biology, ecology, economics, physiology and physics. 2 hours lectures, 1 hour tutorial Prerequisite: M101, 120, 130 or consent of instructor
MATH 522 Faculty of Mathematics
MATH 201 (b) Applications of Mathematics
See Mthel 201 (a). Other examples from biology, ecology, economics, physiology and physics are described using sets of differential equations. 2 hours lectures, | hour tutorial Prerequisite: M201 (a)
MATH 206 (a) Introduction to Mathematics Education
A consideration of secondary school Mathematics education under the following headings: Current trends in education, professional practices and administration, the role of the department head, lesson planning, techniques of teaching, evaluation of students, special students, extra-curricular activities, the relationship between elementary and secondary school mathematics, audio-visual materials, current text books. This course is offered to those students in the Co-op Mathematics Teaching option.
MATH 301 (a) Applications of Mathematics
See preamble to Mthel 101 (a). Partial differential equations are applied to diffusion processes, blood flow, and wave phenomena. 2 hours lectures, 1 hour tutorial. Prerequisite: M217 or equivalent, or consent of instructor
MATH 301 (b) Applications of Mathematics
See Mthel 301 (a). Laplace transforms are applied to problems in ecology, physiology and other fields. 2 hours lectures, 1 hour tutorial Prerequisite: M217 or equivalent, or consent of instructor
MATH 302 Applications of Mathematics to the Biological Sciences
Since individual enrolling in Mthel 302 will not usually have a Science background, the course material has been selected with particular reference to some of the fundamentals of Medical Science including Macromolecular processes, Environmental Health, Genetics and Genetic Engineering, aging processes and theories and quantitative models which describe events in these areas, Subjects such as types of distributions, data processing and simple differential equations, as they relate to Biological phenomena, will also form part of the course. 2 hours lectures. Two terms (Cross listed as Health Studies 302* [303*) 302 may not be counted as Mathematics credit but may be counted as an elective credit.
MATH 304 (a) Foundations of Mathematics
An introduction to the problems of the foundations of mathematics. This course will normally be taken in third or fourth year. It will be an elective credit for mathematics students. 3 hours lectures . Prerequisite: Consent of instructor
MATH 304 (b) Foundations of Probability Theory
An introduction to the problems of the foundations of probability theory. This course will normally be taken in third or fourth year. It will be an elective credit for mathematics students. 3 hours lectures Prerequisite: Consent of instructor
Faculty of Mathematics
.
MATH 401 (a) Applications of Mathematics
See preamble to Mthel 101 (a). Integral equations and integral transforms will be applied to systems with memory. 2 hours-lectures, Fall term Prerequisite: Consent of instructor
MATH 401 (b) Applications of Mathematics
See Mthel 401 (a). Students will construct a model and interpret it as a project. 2 hours lectures, Winter term Prerequisite: Consent of instructor
MATH 402 Topics in Mathematical Aspects of Chemistry, Biology, and the Medical Sciences Note 12 Calculus 1 21 Algebra and Vector Geometry 22 Calculus 2 (for Chemical Engineers)
Subjects will be selected from areas such as epidemiology and mathematical models of disease processes including heart disease, cancer of aging. Consideration will be given to the quantitative evaluation of the role of environmental factors important in human disease processes. ‘ 2 hours lectures Prerequisites: M302 or consent of the instructor
Undergraduate Course Descriptions
The number of hours or lectures shown after the course description is an attempt to indicate the “normal”; each instructor determines how often his particular class will meet.
Functions and limits, the derivative, Differentiation formulae, Applications to tangents, rates, extrema. Antiderivatives. The ' definite integral, fundamental theorem of integral calculus. ‘Applications to area, volume, moments, fluid pressure, work, potential. Transcendental functions. Techniques of integration. Parametric and polar equations with applications. Mean value theorem. Introduction to partial derivatives. Introduction to differential equations with applications.‘ Students will work selected problems under supervision. 2 hours lectures, 1-2 hours problems.
Sets, subsets, relations, functions. Mathematical induction. Limits of real sequences. Vector geometry of points, lines, planes and spheres. Arithmetic of complex numbers. Introductory material on systems of linear equations. Systems of linear equations, Gaussian Elimination. Matrices, augumented matrix, row-reduced matrices, elementary matrices, inverses. Vector spaces, spanning sets, bases, dimension. Rowspace, column-space, null-space, rank. Determinants. Linear transformations. Eigenvalues and Eigenvectors. Functions of complex variables. 2 hours lectures, 1-Y2 hours problems
Partial differentiation, the gradient, multiple integrals with applications, line and surface integrals, divergence and curl, Theorems of Green and Stokes. Applications to physical problems. 3 hours lectures, one term
25 Calculus 2 (for Electrical Engineers)
31 Differential Equations
32 Numerical Analysis
Note
35 Advanced Calculus (for Electrical Engineers)
44 Complex Variable
51 Probability and Statistics
52 Applied Probability and Statistics (for Mechanical Engineers)
Note
Faculty of Mathematics
Differential calculus of functions of several variables. Differential Equations. Multiple integrals. Prerequisite: M12 or equivalent
Standard methods of solution. Applications to physical engineering problems. Linear equations with constant coefficients. Systems of differential equations. Solution of differential equations by the Laplace Transformation.
3 hours lectures
A survey of numerical procedures with emphasis upon computer implementation using the FORTRAN IV programming language. In particular, the following topics are introduced: interpolation, curve fitting, solution of non-linear equations, numerical integration, numerical solution of ordinary differential equations, matrix algebra and solution of systems of linear equations, and problems in the solution of partial differential equations.
2 hours lectures, 2 hours problems, one term
Offered in the Fall and Spring terms
Prerequisite: An introductory programming course
Credit will not be granted for both M32 and M240 (a)
Fourier series. Partial Differential Equations; separation of variables, wave equations, heat equation and Laplace’s equation. Fourier integral. Properties of complex analytic functions. Complex integration.
Cauchy-Riemann equations, the Cauchy integral theorems, conformal mapping, the Taylor and Laurent series, contour integration.
2 lectures, 1 hour problems, one term
Frequency distributions; probability; Binomial, Normal, Poisson laws. Sampling theory. Distribution of Chisquared, F, and t. Hypothesis testing, quality control, regression theory. Curve fitting. Analysis of variance, introduction to experimental design. 3 lectures, 2 hours problems, one term
Axiomatic development of probability. Conditional probability, Bayes’ Theorem and independence. Permutations and combinations as related to probability. Discrete and continuous random variables. Special:cases of uniform, binomial, Poisson, normal, etc. distributions. Mathematical expectation and moment generating functions. Derived distributions. Sampling from a normal population. Elements of decision theory. Various methods of estimating parameters and testing hypotheses.
3 hours lectures, 1 hour tutorial, one term
This course is cross listed in Management Science as MS21.
61 Applicable Mathematics
for Systems Design 1 (Also Listed as Systems Design 211)
62 Applicable Mathematics for Systems Design 2 (Also listed as Systems Design 212)
- 72 (a) Introduction to Computer Science in Business and Economics
Note
72 (b) Advanced Topics in Computer Science in Business and Economics
Note
81 Introduction to Functions Analytic Geometry, and Calculus
Faculty of Mathematics 525
First order differential equations, integrating factor, higher order differential equations. Complex variables, forced and free solutions to differential equations, transient and steady state solutions, applications. Fourier series, Fourier transforms and applications.
Laplace transforms. Solution of ordinary linear differential equations by Laplace transforms, applications. Residue calculus and its relation to inverse Laplace transforms. Functions of the complex variable. Difference equations. Z transforms.
The purpose of Math 72 (a) is to introduce the student to the basic concepts of programming electronic digital computers. Students are taught to write computer programmes in two of the most widely used computer languages in the business world, namely COBOL and FORTRAN. The first part of the course deals with the COBOL language and uses this language to write programmes to simulate a large-scale credit card system. Students are shown how computers are employed to write invoices, process customer accounts and mail out credit cards. The second half of the course uses FORTRAN to handle problems involving inventory control, simulation of business problems and plotting.
2 hours lectures, 1 hour tutorial, one term
Offered in the Fall and Winter terms
Credit will only be granted for one of M72, M112, M122 or M132. M72 cannot be counted as a credit towards a B.Math degree..
Topics discussed will include.
a) An introduction to machine hardware — CPU’s, channels, tape drives, disk drives, main memory, slow core, drums, etc. Basic machine configurations will also. be discussed. The relative costs involved will be covered whenever possible.
b) The concept of an operating system, JCL, compilers and system software packages.
c) Some of the basic terminology such as multiprogramming, batch processing, timesharing, telecommunications and data bases. d) The general organization of a computing centre and the position of computer centres in corporate structures.
2 hours lectures, 1 hour tutorial, one term.
Offered in the Winter term
Prerequisite: M72 (a) or the equivalent
Credit will only be granted for one of M72, M112, M122 or M132. M72 cannot be counted as a credit towards a B.Math degree..
Polynomial functions. Inverse of a function. Rational functions. Exponential functions and identities. The straight line. Conic sections. Problems solved by analytic means. Simple transformation. The limit concept and an introduction to Calculus including the derivative and its application to graph sketching.
2 hours lectures ‘
\
526 § 82 Introduction to Algebra 83 Introductory Mathematics for Sociology : Students 84 Algebra 85 Mathematics for Environmental Studies
Faculty of Mathematics
Properties of real numbers including inequalities and absolute value. Systems of equations in two and three variables including linear, quadratic, and cubic equations. Complex numbers and polar representation. Exponents. Permutations and combinations. The Binomial Theorem. Sequences and series. 2 hours lectures
Set Theory, Permutations and Combinations, Vectors and Matrices, Probability Theory, Solution of Linear Equations, Game Theory, Linear Programming. 2 hours lectures, 2 hours problems lab (alternate weeks) 2 terms
A special course designed for students in Psychology and related subjects. Polynomial, rational, exponential and trigonometric functions. Elementary analytical geometry. Polar co-ordinates. Fundamental concepts in the Calculus. Linear and quadratic systems of equations. Vectors, complex numbers. Mathematical Induction Permutations and combinations, the. Binomial theorem. 2 hours lectures, 2 hours problems lab (alternate weeks) Winter term only
Quantitative analysis in environmental research. Elementary concepts in Algebra, notation, terminology, operations. Introductory matrix algebra. Probability Theory. Permutations and Combinations, approaches to probability, dependent and independent events, conditional probability, Markov processes, distribution functions, including the Binomial, Poisson and Normal distributions, with applications to problems in Environmental Studies. 2 hours lectures, one term Winter term only
MATH 100 Fundamental Concepts of Modern Mathematics
A course for non-mathematicians to provide some insight into the many aspects of modern mathematics. The “human interest” point of view will be stressed in order to reveal mathematics as an endeavor holding a strong place in man’s culture. Emphasis will be on twentieth century ideas. To integrate the traditional with the modern, material will be organized in a conceptual rather than a chronological manner.
Faculty of Mathematics
MATH 112 (a) Introduction to Computers Note
The purpose of Mathematics 112 is to provide students in programmes of study which do not emphasize Mathematics with an appreciation of the capabilities and limitations of machine computing, together with a reasonable capability for programming in one or more programming languages. Topics covered in Mathematics 112a will include: concept of an algorithm, representation of information, programming in a higher level language, concept of a compiler, applications in business, the social sciences and other areas. 2 hours lectures, 1 hour tutorial, Fall term Prerequisite: None although Grade 12 Mathematics is recommended
Credit will only be granted for one of M72, M112, M122 or M132. M112 cannot be counted as a credit towards a B.Math Honours degree.
MATH 112 (b) Applications and Implications of Computers Note
A continuation of the concepts introduced in Mathematics 112a — the applications of computers to several fields and the impact (past, present and future) of computing on society. Topics discussed will include: computer applications, various programming languages (their assets and liabilities), the concepts of compilers, assemblers, interpreters, interactive systems and text editors, computer simulation, computers and society (including data banks, privacy, employment, etc.) artificial intelligence, and other directions of computer science, the limitations of . computing devices. 2 hours lectures, I hour tutorial, Winter term Prerequisite: M112 (a)
Credit will only be granted for one of M72, M112, M122 or M132. M112 cannot be counted as a credit towards a B.Math Honours degree
MATH 119 Algebra and Geometry Note
Functions and permutations, Elementary number theory. Real and. complex number systems. Polynomial functions. Linear geometry and algebra in 2 and 3 space. 3 hours lectures, 1 hour tutorial
M119 is not a course for Honours Mathematics students
MATH 120 Calculus Note
Functions and limits. The derivative. Differentiation of algebraic functions. Applications to tangents, rates, minima and maxima. Integration as the limit of a sum. Fundamental theorem of integral calculus with applications to areas, and moments. Power series. Partial differentiation. 3 hours lectures, 1 hour tutorial
M120 is not a course for Honours Mathematics students.
MATH 528 Faculty of Mathematics
MATH 122 (a) Introduction to Computer Programming
An introduction to algorithms with emphasis on the solution of problems of a scientific nature. Concept and properties of an algorithm, language and notation for describing algorithms. Analysis of scientific problems, development of algorithms for their solution and their implementation in a procedure oriented language (FORTRAN IV). Other topics will include: enumerative and iterative algorithms, nature of errors and uncertainty, use of existing library routines for manual proccdures and graph plotting. 2 hours lectures, 2 hours tutorial Offered in the Fall and Winter terms Prerequisite: Grade 13 Mathematics is recommended
Only one of M72 (a), 112 (a), 122 (a) and 132 (a) can be taken for credit.
MATH 129 Algebra and Geometry
Functions and permutations. Elementary number theory. Real and complex number systems. Polynomial functions. Linear geometry and algebra in 2 and 3 space. Algebraic systems. 3 hours lectures, 1 hour tutorial
MATH 130 Calculus
Functions and limits, the derivative, differentiation of algebraic and other functions, applications of the derivative. The integral, theorems on the integral, applications of the integral. Sequences, convergence of series, power series. Some elementary differential equations. 3 hours lectures, I hour tutorial
MATH 131 (a) Algebra and Solid Geometry
The real and complex number system, Mathematical Induction, the Binomial Theorem, Monotone sequences and the Cauchy criterion, polynomial functions, Theory of equations. 3 hours lectures
MATH 131 (b) Algebra and Solid Geometry
Determinants Vector and Matrix notation, Elementary Solid Geometry, Linear transformations, Eigenvalues and Eigenvectors. 3 hours lectures
MATH 132 (a) Introduction to Computer Science — Programming Note
A thorough introduction to the use of computers. Concept and properties of an algorithm, language and notation for describing algorithms. Analysis of problems of both a scientific and business nature, and development of algorithms for their solution. Introduction to procedure-oriented languages (COBOL and FORT- RAN IV). The preparation, implementation, and debugging of programs in these languages. Students are expected to spend considerable time running programmes on the computer. 2 hours lectures, 2 hours tutorial (laboratory), one term Normally offered in the Fall term Prerequisite: Grade 13 mathematics
Credit will only be granted for one of M72, M112, M122 or M132.
Faculty of Mathematics
MATH 132 (b) Introduction to Computer Science — Characteristics of Computers Note
A general introduction to the nature and concepts of Computer Science. Basic machine architecture. Introduction to machine and assembly-language programming. The representation of data in a computer. Address modification, indexing and indirection. Character manipulation, floating point operations and subroutine linkage. Characteristics of peripheral devices. The logical design of basic machine components. A brief survey of software which assists user programmes: Compilers, loaders, input-output routines, operating systems. Applications of a computer in various segments of society. 3 hours lectures, 1 hour tutorial (laboratory), one term. Normally offered in the Winter and Spring terms Prerequisite: Grade 13 mathematics
Credit will only be granted for one of M72, M112 or M132.
MATH 217 Advanced Calculus Note
Differential calculus of functions of several variables. Multiple integrals. Line and Surface integrals; Green’s Theorem, Divergence Theorem, Stokes’ Theorem. Infinite Series. 3 hours lectures, I hour tutorial
M217 is not a course for Honours Mathematics students
MATH 219 Linear Algebra Note
Vector spaces, linear maps. Inner products. Bilinear and quadratic forms. Matrix theory. 3 hours lectures
M219 is not a course for Honours Mathematics students
MATH 223 (a) Introduction to Statistical Methods (Descriptive)
Descriptive statistics, graphical methods, model fitting; correlation, regression and the method of least squares; an introduction to time series and forecasting. 3 hours lectures Prerequisites: M120. or M130; one of M112 (a), M122 (a), M132 (a)..
MATH 223 (b) Introduction to Statistical Methods (Inferential) Notes
Probability theory, discrete and continuous random variables; tests of significance, maximum likelihood estimation and large sample theory; estimation and testing in the normal distribution. 3 hours lectures Prerequisite: M223 (a)
M223 is not a course for Honours Mathematics students. Credit will be given for only one of M223, M233 and M243.
MATH 228 Introduction to Pure Mathematics
Examples and results in modern geometry, number theory, and algebra. Logical foundations of Mathematics (Hilbert’s proof theory and Brouwer’s intuitionism). 2 hours lectures
MATH 229 Linear Algebra
Real and complex vector spaces. Linear transformations. Matrices. Determinants. Diagonalization of matrices. Inner product spaces. Orthogonal and unitary matrices. The spectral theorem for Hermitian and real. symmetric matrices. ; 3 hours lectures
MATH 530 Faculty of Mathematics
MATH 233 Probability and Statistics
An introduction to probability theory and Statistics, with emphasis on applications to practical problems, and On the logical principles involved. Several examples to be discussed require use of the computer. Topics from probability theory include the laws of probability; discrete and continuous random variables; expectation; and the central limit theorem. Topics from statistics include the use of relative likelihood in estimation; sufficiency; tests of significance; contingency tables; normal distribution theory; simple linear regression. 3 hours lectures, 2 hours tutorial Prerequisites: M120 or M130; M122 (a) or M132 (a).
MATH 234 Introduction to Applied Mathematics
MATH 234 (a) Differential Equations
First order equations; second order equations with constant coefficients. Applications to problems in the physical and biological sciences. Solution of differential equations by series. 2 hours lectures, one term Prerequisite: M237 (may be taken concurrently)
MATH 234 (b) Mathematical Modelling Note Mathematics
Mathematical models from various disciplines. Introduction to Newtonian mechanics will also be included in the course. 2 hours lectures, one term Prerequisites: First year Physics or Consent of Instructor M31 or M234 (a) or M 236
M 234 (a) and M 236 may not both be taken for credit
The theory of rates of interest and discount, annuities and sinking
MATH 235 (a) Actuarial Mathematics of Investment
funds; application to financial problems, including the determination of mortgage payments and the price and yield of bonds. 2 hours lectures, I hour problems
MATH 235 (b) Basic Life Insurance : Mathematics
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, I hour problems
MATH 236 Elementary Differential Equations Note
An introduction to first order differential equations. Applications to growth, decay, reactions. Linear equations with constant coefficients. Systems of equations; vector and matric methods. Solutions in series. Introduction to partial differential equations. 2 hours lectures
M236 is not a course for Honours Mathematics students
Faculty of Mathematics
MATH 237 Advanced Calculus
Sequences and series of functions: uniform convergence, power series. Introduction to ordinary differential equations: first and second order linear equations, power series solution method, applications. Differential calculus of functions of several variables: continuity, partial derivatives, differentials, chain rule, Taylor expansion, extreme values, mappings and Jacobians. Integral calculus of function of several variables: multiple integrals, change of variable, line integrals, Green’s theorem and independence of path. 3 hours lectures, 2 hour tutorial Prerequisite: First year algebra (M119] 129)
MATH 239 Introduction to Combinatorics and Optimization
MATH 239 (a) An Introduction to Combinatorics
Existence and non-existence. Enumeration. Combinations and Permutations. Inclusion-exclusion principle. Linear equations with unit coefficients. Recursion. Generating functions. Chromatic polynomials. Applications to geometry designs. Recommended for students wishing an introduction to discrete mathematics, and for students intending to pursue a course of study in combinatorics and optimization. 2 hours lectures, 2 hours tutorial Prerequisite: First year algebra (M119[129)
MATH 239 (b) An Introduction to Optimization
Convex sets. Linear programming. Directed graphs. Applications of Boolean Algebra to Combinatorial Circuits. Probability; Markov chains; random walks. Theory of Games. Recommended for students wishing an introduction to optimization, and for students intending to pursue a course of study - in combinatorics or optimization. 2 hours lectures, 2 hours tutorial, one term Prerequisites: First year algebra (M129); Note that 239 (a) is not a prerequisite, but is highly recommended.
MATH 240 (a) Numerical Applications in Computer Science
An introduction to Numerical Procedures with emphasis upon computer implementation using the FORTRAN IV programming language. In particular, the following topics are introduced: concept of numerical errors, interpolation, curve fitting, solution of non-linear equations, numerical integration, matrix operations and solution of systems of linear equations, numerical solution of ordinary differential equations. Offered in the Fall, Winter and Spring terms. Prerequisite: M132 (a) or equivalent, and M120 or M130 and M119 or M129. There is a special section for non-mathematics students.
MATH 240 (b) Principles of Programming Languages and Data Structures
This course is intended to cover a number of basic principles of programming languages and data structures. The emphasis will be on basic principles with motivation for programming languages arising from practical examples. The following programming will be covered: ALGOL, SNOBOL, L6, PL/1. 2 hours lectures, 2 hours tutorial. Offered in the Fall and Winter terms. Prerequisite: M132(a) and M132(b)
MATH 532 Faculty of Mathematics
MATH 243 Statistics for the Sciences Note:
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, 1 hour laboratory.
M243 and 233 can not both be taken for credit. Honours Mathematics students must take M233. ‘
MATH 300 (a) Mathematical Discovery and Invention
A study of about 100 challenging problems taken from many areas of elementary mathematics, number theory, combinatorics, geometry, probability, logic. Emphasis is placed on student participation. This course was designed for those planning to go into high school teaching. However, the problems have universal appeal to all who like mathematics. This is a “do” course, and is not recommended for passive students. 2 hours lecture, one term. Prerequisite: No formal prerequisites are demanded. However, the broader and deeper one’s mathematical experience the better. This course focusses on applying one’s knowledge and ingenuity.
MATH 300 (b) Mathematical Discovery and Invention-B
A study of about 100 challenging problems taken from many areas of elementary mathematics — number theory, combinatorics, geometry, probability, logic. Emphasis is placed on student. participation. This course was designed for those planning to go into high school teaching. However, the problems have universal appeal to all who like mathematics. This is a “do” course, and is not recommended for passive students. 2 hours lectures, one term. Prerequisites: No formal prerequisites are demanded. However, the broader and deeper one’s mathematical experience the better. This course focusses on applying one’s knowledge and ingenuity.. Note: 300(a) is not a prerequisite to 300(b).
MATH 307 Combinatorial Geometry
A study of combinatorial properties of plane figures, covering problems, addition of figures, maxima and minima problems, curves of constant breadth and delta-curves. This course is designed for the student who particularly likes geometry and for the serious student of combinatorics. Occasionally brief contact is made with the subject of analysis. Thus a concurrent course in real variables would be helpful, but not essential. 2 hours lectures. Prerequisite: First year calculus (M120] 130).
Faculty of Mathematics
MATH 312 (a) Elements of Real Variable Theory Note Note
Metric properties of Rn, functions from Rn to R; differentials; Rieman integrals; and one of a) an introduction to the Lebesque integral; b) an introduction to metric spaces; c) orthogonal systems. 3 hours lectures. Emphasis will be on applications rather than theory. M312(a) is not a course for Honours Mathematics students.
MATH 312 (b) An Introduction to Complex Variable Theory Note Note
Complex numbers: continuity, differentiability, analyticity of functions; the Cauchy-Riemann equations; solutions of Laplace’s equation; conformal mapping by elementary functions, and applications; the Cauchy and allied theorems; Taylor and Laurent expansions; uniform convergence and power series; the residue calculus, and applications. 3 hours lectures. Emphasis will be on applications rather than theory. M312(b) is not a course for Honours Mathematics students.
MATH 319 Abstract Algebra Note Note
Rings, integral domains, fields, groups. Quotient groups and quotient rings. .
Emphasis will be on examples rather than proofs of theorems. M319 is not a course for Honours Mathematics students
MATH 329 Abstract Algebra
Relations and operations in sets. Rings, integral domains, fields, groups. Quotient structures. Linear maps of inner-product spaces.
MATH 330 (a) Projective Geometry
Projective spaces over fields, collineations and correlations, quadric curves and surfaces. Reference to non-euclidean geometries. 3 hours lectures, one term. Prerequisite: M229
MATH 330 (b) Geometry of the Complex Numbers
The plane of complex numbers. The group of circle preserving mappings and its subgroups. Connections with non-euclidean geometries. Other number systems and their geometries (Laguerre, Minkowski). 3 hours lectures, one term. Prerequisite: M229
MATH 330 (c) Euclidean Geometry
Concurrent lines, collinear points, the Euler line, the Simson line, the nine point circle. Cross ratio, projection, Harmonic range, the quadrilateral and the quadrangle. Properties of circles. 3 hours lectures, one term.
MATH 534 Faculty of Mathematics
MATH 332 (a) Real Variables
Description of real numbers by means of a few fundamental properties (l.u.b. axiom, principle of Archimedes, etc.); proof of some theorems on continuous functions using l.u.b. axiom.
_ functions of bounded variation and the Riemann Stieltjes integral.
Cesaro summability of series; derivation, discussion and applications of Cauchy-Schwartz and Minkowski inequalities for series and integrals; inner product spaces and normed linear spaces with examples; orthonormal systems with examples including Legendre, Tschebyscheff, and trigonometric systems; Fourier series; ordinary differential equations and Picard existence Theorem (by means of fixed point theorem). 3 hours lectures.
MATH 332 (b) An Introduction to Complex Variable Theory
Complex numbers; continuity, differentiability, analyticity of functions; the Cauchy-Riemann equations; solutions of Laplace’s equation; conformal mapping by elementary functions, and applications; the Cauchy and allied theorems; Taylor and Laurent expansions; uniform convergence and power series; the residue calculus, and applications. ‘ 3 hours lectures.
MATH 334 (a) Numerical Algebra
Zero finding: real functions of a single real variable, polynomials, systems of non-linear equations. Linear Algebra: linear systems of equations, linear least squares, eigenvalue problems. 3 hours lectures, 2 hours problems, one term. Offered in the Spring and Fall terms. Prerequisite: M240(a). Also recommended are M219 or 229 and M217 or 237.
MATH 334 (b) Numerical Approximation
Interpolatory, least squares, and minimax approximation of functions by polynomials, trigonometric series, rational functions, splines and exponentials. Inverting Laplace transforms. Acceleration techniques. Numerical approximation of derivatives. Quadrature. 3 hours lectures, 2 hours tutorial. Winter term. Prerequisite: M240(a). Also recommended are M219 or 229 and M217 or 237.
MATH 335 (a) Finite Differences
A course in the calculus of finite difference, to include: summation, numerical. integration and differentiation, relation between integration and summation; and error theory. 2 hours lectures.
MATH 335 (b) Graduation of Tables
Applications of finite differences to actuarial problems in graduation of statistical tables. 2 hours lectures. Prerequisites: M235(a), M235(b).
MATH 336 (a) Life Contingencies
An advanced course on problems with single lives, including population theory. 3 hours lectures, one term. Prerequisite: M235(b).
Faculty of Mathematics
MATH 336 (b) Life Contingencies
An advanced course on problems with multiple lives; multiple decrement theory; accidental death and disability benefits. 3 hours lectures, one term. Prerequisite: M235(b).
MATH 338 (a) Mathematical Statistics
Continuous random variables; moments and moment generating function; distribution of t, chi-squared, and F, and their applications. Large sample theory. ; 3 hours lectures Prerequisites: M233, M237 ©
MATH 338 (b) Mathematical Statistics ' 339 (a) Probability and Stochastic Processes
The multivariate normal distribution, its properties and uses; quadratic forms and Cochran’s theorem; multiple regression; introduction to the analysis of variance. 3 hours lectures Prerequisites: M229, M338 (a)
Probability distributions, joint variation, central limit theorem. Introduction to continued time stochastic processes. Poisson process and application to simple congestion situations. 3 hours lectures, Fall or Spring term ; Prerequisite: One of M223, M233 or M243
MATH 339 (b) Probability and Stochastic Processes
Discrete probability models. Random walks. Introduction to theory of Markov chains. Applications to biological and social processes. 3 hours lectures, Winter term Prerequisite: One of M223, M233 or M243
Parts (a) and (b) may be taken in either order
MATH 340 (a) Machine, Data and Programme Structures
Representation of numbers and strings. Addressing schemes. Compound data structures. Programme structure. 3 hours lectures, one term Offered in the Fall, Winter and Spring terms Prerequisite: M240 (b)
MATH 340 (b) Implementation Schemes for Programming Structures
Searching and sorting. Run time representation of a high level language. Dynamic storage management. Text manipulation. 3 hours lectures, one term Offered in the Fall and Winter terms Prerequisite: M340 (a)
MATH 341 Algebra
Fundamentals of group, ring, and field theory. 3 hours lectures.
Corequisite: M229
MATH 342 Real Analysis 1
Theory of functions of real variables. The notions of compactness, connectedness, and uniformity are used in a study of continuity, differentiation, and integration. 2 hours lectures Prerequisite: M237 .
MATH 536 Faculty of Mathematics
MATH 343 Complex Analysis 1
Analysis of complex numbers; fundamental theorems of holomorphic functions; meromorphic functions. 2 hours lectures Prerequisite: M237
MATH 344 Topology (a) ‘ ;
Intuitive set theory, metric spaces, point set topology. 3 hours lectures ,one term Consent of instructor
MATH 345 (a) Topics in Pure Mathematics for Combinatorial Mathematicians ~- A
Not offered in 1974–75.
MATH 345 (b) Topics in Pure Mathematics for Combinatorial Mathematicians —- B
Not offered in 1974–75.
MATH 349 (a) Applied Statistical Aanalysis
Review of Normal, t, Chi squared and F distributions and their applications. Introduction to the design and analysis of experiments and surveys. Analysis of variance, multiple regression. 3 hours lectures, Spring and Fall terms Prerequisite: One of M223, M233 or M243
MATH 349 (b) Statistical Decision Theory Note
The decision problem; utilities and subjective probabilities; decision rules; decision trees, normal and extensive analyses; conjugate prior distributions and associated distribution theory; applications to business decisions. Loss functions, risk functions, admissability, minimax rules, tests of hypotheses. An introduction to statistical quality control and control charts. 3 hours lectures, Winter term Prerequisites: M223, M233, or M243; M217 or M237
M349 (a) is not a prerequisite for M349 (b). M349 and M338 may not both be taken for credit
MATH 351 Introduction to Graph Theory and Combinatorial Analysis
MATH 351 (a) Introduction to Graph Theory
Basic concepts; Konigsberg bridge problem, Euler trails and Hamiltonian cycles; marriage problem, maximum matchings, Hall’s theorem; planar graphs, four colour problem, chromatic number; Ramsey’s theorem. Stress will be on problem solving. Recommended for students interested in learning the basics of graph theory and for those intending to take higher level courses in graph theory. 2 hours lectures, one term Prerequisites: No formal prerequisites. M239 recommended
MATH 351 (b) Applied Graph Theory
Applications of graph theory to communication networks, timetabling, ranking, and other combinatorial problems. 2 hours lectures, one term Prerequisite: M351 (a)
: Faculty of Mathematics
MATH 352 (a) Fundamentals of ; Optimization
Introductory study of optimization techniques common to a wide variety of operations research and engineering problems, and. of fundamental theoretical interest. Linear programming, simplex method, duality; integer and non-linear programming; convexity; branch and bound, network programming, optimal paths and trees, transportation problems, flows, PERT, replacement policies, knapsack problems. Recommended for students desiring mathematical background for. operations research, industrial engineering, computer science and for those wishing to pursue advanced courses in mathematical programming and combinatorics. 2 hours lectures, one term Prerequisite: Completion of M229, 237, 239 or consent of instructor
MATH 352 (b) Fundamentals of Optimization
Introductory study of optimization techniques complementary to those studied in 352 (a). Extrema of functions of n-variables, implicit funtcions and Lagrange multipliers, inventory problems; difference methods; queuing, recurrent events and replacement policies; game;s job- sequencing in problems; dynamic programming. : Recommended for students desiring mathematical background for operations research and industrial engineering and for those wishing to pursue advanced courses in mathematical programming and queuing theory. 2 hours lectures, one term Prerequisites: Same as M352 (a)
MATH 353 Applications of Combinatorics
MATH 353 (a) Combinatorial Mathematics in Information Retrieval
Brief survey of information retrieval and its applications; discrete models; representation problems; semantic adequacy; matching problems; -partitioning of aggregates of elements; generalized distance measures; application of classification . techniques; graph theoretic techniques; hierarchical techniques; minimizaticn techniques; local and global methods; retrieval performance; measures of effectiveness; problems of retrieval relevance judgements. 2 hours, lectures, One term
MATH 353 (b) Modelling Techniques in Optimization
Theory of mathematical modelling, deterministic and stochastic. Methods for optimization including standard analysis, steepest descent, probabilistic methods. Applications to business problems such as inventory theory, capital budgeting, reliability, queueing,, etc..
2 hours lectures, one term
MATH 360 (a) Tensor Analysis
Elementary introduction to classical differential geometry and tensor analysis. Curves and surfaces in Euclidean spaces, coordinate transformation, invariance of length, introduction of tensors and metric, Riemannian spaces, covariant differentiation, geodesics, curvature and Einstein tensors. 2 hours lectures, + 1 hour tutorial, Fall term Prerequisites: M237, consent of instructor
MATH 538 Faculty of Mathematics
MATH 360 (b) Differential ‘Geometry
Intermediate level course on Riemannian spaces (continuation of 360a). Conformal spaces, spaces with corresponding geodesics, theory of subspaces. 2 hours lectures + 1 hour tutorial, Winter term Prerequisites: M237, M360 (a) or consent of instructor
MATH 361 (a) Calculus of Variations
Euler-Lagrange equations for constrained and unconstrained single and double integral variational problems. Parameterinvariant single integrals. General variational formula. The canonical formalism. Hilbert’s independent integral. Hamilton-Jacobi equation and the Carathéodory complete figure. Fields and the Legendre and Weierstrass sufficient conditions. 2 hours lectures + 1 hour tutorial, Fall term Prerequisites: M237 or consent of instructor
MATH 361 (b) Mechanics
Three-dimensional rigid body motion. Lagrange’s equations of motion for holonomic and non-holonomic systems. Ignorable cordinates. Hamilton’s principle for holonomic systems and its counterparts for non-holonomic systems. Noether’s theorem and conservation laws. Canonical equations of motion. Canonical transformations. Lagrange and Poisson bracket relations. Integral invariants. 2 hours lectures + 1 hour tutorial, Winter term Prerequisites: M237, M361 (a) or consent of instructor
MATH 362 (a) Introduction to Continuum Mechanics
Motion of a system of particles, methods of continuum mechanics. Stress and strain tensors; analysis of stress and strain. Lagrangian and Eulerian methods for describing flow. Equations of continuity, motion and energy. Navier-Stokes equation. 2 hours lectures, I hour tutorial, Winter term Prerequisites: M234, M237 or consent of instructor
MATH 362 (b) Partial Differential Equations of Applied Mathematics
Vector Integral Calculus, derivation of the equations of heat flow, the vibrating membrane and the electro-magnetic field equations by various elementary techniques and illustrations of the use of the calculus of variations in the derivation of equations; solutions of the above equations by separation of variables; Fourier analysis and Fourier Integral. 2 hours lectures, 1 hour tutorial, Fall term Prerequisites: M234, M237 or consent of instructor
MATH 363 Differential Equations
Existence of solutions. Second-order linear equations. Green’s functions. Series solution. Laplace transforms. Sturm-Liouville boundary-value problems. Fourier series. Partial differential equations. 2 hours lectures, + 1 hour tutorial Prerequisites: M234, M237 or equivalent, M332 (Real Variables) may be taken concurrently
Faculty of Mathematics
MATH 371 (a) Switching Circuits
Switching devices. Binary codes and arithmetic. Boolean algebra and its application to switching circuits. Canonical forms for Boolean functions. Simplification of switching functions. Analysis of sequential circuits. Synchronous and asynchronous operation. Design of sequential circuits. 3 hours lectures, one term Offered in the Fall, Winter and Spring terms Prerequisite: M132. Recommended Corequisite: M319 or M329
MATH 371 (b) Architecture of Computers
This course is intended to prepare the students to choose a suitable computer for a given application. Review of combinatorial and sequential logic circuits.. Discussion of modules or “building blocks” — central processing units, stores, input/output systems, and bus structures. Case studies of machines. 3 hours lectures Offered in the Fall and Winter terms Prerequisite: M371 (a)
MATH 372 (b) 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. 3 hours lectures, Winter term Prerequisite: M371 (a) or consent of the instructor Recommended Corequisite: M319 or M329
MATH 380 Introduction to Information Theory with Applications
Finite probabilistic schemes (experiments). Shannon’s Entropy as a measure of uncertainty and expected information. Fundamental properties of Shannon’s Entropy. Encoding and decoding. Optimal coding elementary problems in communication theory, programming and classification. Maximizing expected information; applications to problem solving in logic, linguistics representations of questionnaires. Applications to diagnostics, economics and social sciences.
398 Undergraduate Seminar
2 hours lectures
399 Reading in Mathematics
406 Linear Algebra 2
Continuation of linear algebra. Main topics; representations of endomorphisms, structures of bilinear forms, multilinear products. 2 hours lectures Prerequisite: M329 or M341
MATH 407 Algebraic Geometry
An introduction to the theory of algebraic varieties. Special topics such. as the Theorem of Riemann-Roch. 2 hours lectures Prerequisite: M329 or M341
MATH 408 Mathematical Logic
First order languages and theories. A treatment of at least one of the following: set theory, model theory, undecidability. This course is more specialized and at a more advanced level than Mathematics 436. 2 hours lectures
MATH 540 Faculty of Mathematics
MATH 409 Foundations of Geometry
An axiomatic treatment of geometry. Geometrical structures, such as projective planes and inversive planes. 2 hours lectures.
MATH 410 Ring Theory
Continuation of the theory of rings and modules. ; 2 hours lectures Prerequisite: M329 or M341
MATH 411 Group Theory
Continuation of group theory. 2 hours lectures Corequisite: M329 or M341
MATH 412 Field Theory
Field extensions and Galois theory. 2 hours lectures Prerequisite: M329 or M341 Non-linear mechanics, stability, quasi-linear and strongly non-
MATH 413 Non-Linear Differential Equations
linear systems, linear periodic systems, non-linear integral equations. 2 hours lectures Prerequisite: M363 or consent of instructor
MATH 417 (a) Optimizational Combinatorics-A
Algorithms and theory relating to both combinatorics and optimization. Brief review of linear programming duality theory. Optimum paths and flows in networks. Efficiency of algorithms. Convex polyhedra. Combinatorial max-min relations. Totally _ unimodular matrices. Integer programming. Optimum matchings and constrained subgraphs. Recommended for students interested in theoretically advanced study in Combinatorics and Optimization or Computer Science. 2 hours lectures, one term Prerequisite: M352 (a) or consent of instructor
MATH 417 (b) Optimizational Combinatorics-B
MATH 418 (a) Combinatorial Applications of Computers — A \
Branchings in networks. Elementary matroid theory illustrated by forests, transversals, algebraic dependence, and geometry. Matroid partitions and intersections. Submodular functions and polymatroids. Recommended for students interested in theoretically advanced study in Combinatorics and Optimization or Computer Science. 2 hours lectures, one term Prerequisite: M417 (a) or consent of instructor
General topics: methods of data storage for combinatorial problems, representation of sets, etc. Algorithms for permutations, combinations, partitions, etc. The use of generating functions, and methods of handling them on a computer. Enumeration problems: Polya’s theorem and variations. Applications. oo. Useful for students wishing to specialize in the more computationally-oriented aspects of graph theory or operations research, 2 hours lectures, one term Prerequisites: Practical knowledge of at least one computer programming language. Algebra (M319/329)
Faculty of Mathematics
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MATH 418 (b) Combinatorial Applications of Computers — B
Methods for searching and sorting. Tree-like data structures: branch and bound methods. Back-track methods. Boolean programming. Construction problems: construction of graphs , digraphs, trees. The cell-growth problem and its variations. Useful for students wishing to specialize in the more computationally-oriented aspects of graph theory or operations research. 2 hours lectures, one term Prerequisites: Practical knowledge of at least one computer programming language. Algebra (M329). M418 (a) is a prerequisite for M418 (b)
MATH 419 (a) Applications of Graph Theory — A
Not offered in 1974–75.
MATH 419 (b) Applications of Graph Theory — B
Not offered in 1974–75.
MATH 420 (a) Combinatorial Methods and Probability Models — A
Not offered in 1974–75.
MATH 420 (b) Combinatorial Methods and Probability Models — B
Not offered in 1974–75.
MATH 425 Theory of Numbers
Multiplicative-algebraic theory of numbers. Foundations of natural number theory. Elements of additive-combinatorial number theory. 2 hours lectures Corequisite: M329 or 341
MATH 426 Topology
Continuation of general topology; selected topics from other branches of topology. 2 hours lectures Prerequisite: M344
MATH 428 Lattice Theory
Ordered sets, lattices, and Galois connections. Applications in algebra, geometry and logic. 2 hours lectures “Consent of instructor.
MATH 429 Combinatorial Topology
Homology, theory of complexes. Theorems of invariance, covering, and duality. 2 hours lectures
MATH 430 Finite Geometries
Finite incidence structures. Designs. Finite projective and affine planes. 2 hours lectures Corequisite: M330 (a) Projective Geometry
MATH 432 Complex Analysis 2
Further properties of holomorphic and meromorphic functions. Riemann surfaces. 2 hours lectures Prerequisite: M332 (a) or M343
MATH 542 Faculty of Mathematics
MATH 433 Real Analysis 2
An introduction to integration and measure theory with emphasis on the real lines. 2 hours lectures . Prerequisite: M332 (a) or M342
MATH 434 Differential Equations
Green’s functions. Cauchy problems. Classification of partial differential equations. Orthogonal functions. Bessel functions. Legendre polynomials. Fourier Integral and Laplace transform. Variational properties of eigenvalues and eigenfunctions. 2 hours lectures Prerequisite: M363 or equivalent, or consent of instructor
MATH 435 Laboratory Note
a) Numerical problems arising in.actuarial science and statistics. 2 hours laboratory, one term Prerequisite — Corequisite: M335b, M461a, M437a, or consent of instructor
b) Numerical problems arising in actuarial science and statistics. 2 hours laboratory, one term Prerequisite: M336a and b or consent of instructor
Only advanced-actuarial students are advised to take this course.
MATH 436 Introduction to Logic
An introduction to the logic of sentences and predicates with some emphasis on familiar algebraic structures. At first informal, then leading to a formal axiomatic treatment with proofs of consistency and conipleteness. 2 hours lectures Prerequisite: Consent of instructor
MATH 437 (a) Measurement of Mortality
Methods of analysis of data to produce raw rates for mortality and other tables. 2 hours lectures, one term Prerequisite: M235 or consent of instructor
MATH 437 (b) Risk Theory
Individual and collective risk theory. Ruin theory. Stop loss reinsurance premiums, Monte Carlo methods. 2 hours lectures, one term Prerequisite: M233 or consent of. instructor
MATH 438 (a) Topics in Estimation and Hypothesis Testing
Discussion of general inference problems under the headings of © point and interval estimation, hypothesis testing, and decision theory. Large sample normal likelihoods, maximum likelihood estimation, theory of UMV estimation, least squares, Neyman- Pearson theory of hypothesis testing. 2 hours lectures, one term Prerequisite: M338 (a)
MATH 438 (b) Statistical Inference with Small Samples
Logic of tests of significance; exact tests and confidence intervals; normal approximations accurate for small samples and relationship to the normality of the likelihood function. Introduction to problems involving more than one parameter. 2 hours lectures, one term Prerequisite: M438 (a) or permission of instructor
Faculty of Mathematics
MATH 439 (a) Theory of Experimental Design — A
The logical requirements of designed experiments. The design and analysis of basic complete designs with fixed effects or both. The analysis of covariance. The latin square. An introduction to sampling theory. Applications. 3 hours lectures, one term Prerequisite: M338 (b) or consent of instructor
MATH 439 (b) Theory of Experimental Design — B
‘Construction and analysis of incomplete designs: latin square, confounded, fractional factorial, incomplete block. Applications. 3 hours lectures, one term Prerequisite: M439 (a)
MATH 441 Quantum Mechanics
Schrodinger equation. Solution of one dimensional problems, commutation relations, uncertainty principle, W.K.B. approximation. Hydrogen atom, angular momentum. Scattering theory. Matrix mechanics, symmetry groups. Introduction to relativistic wave equations. 2 hours lectures
MATH 442 Theory of Relativity
Flat space-time and special relativity, curved space-time and the Finstein gravitational field equations. The Schwarzschild solutions Maxwell’s equations in and the experimental tests of relativity. space time and the Reissner-Nordstrom solution. Selected topics from cosmology, Petrov classification, spinors, gravitational waves, special solutions, mathematical aspects of the Einstein field equations. 2 hours lectures, Winter term Prerequisite: M360 (a). Consent of instructor
MATH 443 Electromagnetism
Applications of the Maxwell equations. Reficction and refraction. Introduction to wave guides and antennae. 2 hours lectures, Winter term ; Prerequisite: Consent of instructor
MATH 444 Elasticity
Basic equations of elasticity for homogeneous isotropic bodies; bending of beams; plane elastic waves; Rayleigh surface waves, Love waves. Solution of problems by potentials, variational methods and Saint Venant’s principle. 2 hours lectures + I hour, tutorial, Winter term , Prerequisite: M455 (a)
MATH 445 (a) Fluid Mechanics
Fundamental equations of invisicid fluids, compressibility, vorticity; two and three dimensional irrotational, incompressible Joukowski hypothesis. flow, Blasius’ theorem, 2 hours lectures, I hour tutorial, Fall term Prerequisite: M363 (b )
MATH 445 (b) Fluid Dynamics
Shock wave theory, supersonic flow around a corner, Prandtl- Meyer flow. Dynamics of real fluids, Navier-Stokes equations, exact solutions, Stokes and Oseen flow; introduction to boundary layer theory. 2 hours lectures, 1 hour tutorial, Winter term Prerequisite: M445 (a)
MATH 344 Faculty of Mathematics
MATH 446 (a) History of Mathematics — A
A study of selected topics from Greek geometry. Some related work of post-renaissance scholars is included. Topics include: famous construction problems, pythagorean arithmetic, regular solids, four discoveries of Archimedes, the problem of Apollonius; selected works of Archimedes, Euclid, Apollonius, Euler, Steiner, and many others. Emphasis is placed on the mathematics, with little attention to non-technical historical aspects. This course was designed as a final-year course for those planning to go into high school teaching. However, it may appeal to anyone interested in mathematics. 2 hours lectures, one term Prerequisites: No formal prerequisites are demanded. However, it is recommended that a student have a working knowledge “A of elementary geometry (e.g. the material contained in Modern Geometry for High Schools” by Lougheed and Workman)
MATH 446 (b) History of Mathematics — B
A study of selected topics from post-renaissance mathematics. Topics include material on prime numbers, Fermat’s Last Theorem, the Gaussian Integers, the Fibonacci Sequence, other topics for elementary number theory, a collection of outstanding problems in geometry (Fagnano, Steiner-Lehmus, Morley, and several miscellaneous items such as the Cantor ternary set and Ramanujan’s Highly Composite Numbers. Emphasis is placed on the mathematics, with little attention to non-technical historical aspects. This course was designed as a final-year course for those planning to go into high school teaching. However, it may appeal to anyone interested in mathematics. 2 hours lectures, one term Prerequisites: No formal prerequisites are demanded. However, it is recommended that the student possess a reasonable mathematical maturity
MATH 447 Statistical Mechanics
Applications of probability theory to theoretical Physics. 2 hours lectures, Fall term
MATH 448 Differential Geometry
Differentiable manifolds, Tensors and Forms, Connexions, Riemannian manifolds. 2 hours lectures Prerequisites: M229 and M237
MATH 449 (a) Experimental Design - A
The requirements for a good experimental design prior to the accumulation of data. The concepts of randomization, replication and experimental error is applied to basic designs; the completely randomized, randomized block and latin square designs. Analysis of variance. 2 hours lectures, one term Prerequisite: One of M223 or M233 or M243 or consent of instructor
MATH 449 (b) Experimental Design — B
Factorial experiments, confounding, fractional replication. Application of designs. Incomplete block designs. Analysis of covariance. 2 hours lectures, one term Prerequisite: M449 (a)
Faculty of Mathematics
MATH 450 Applied Analysis Note
Applied Mathematics for Science majors. Ordinary and partial differential equations. Laplace and Fourier transforms. Special functions. , Fourier Series. Vector and matric methods. Calculus of variations. Other topics in advanced calculus. 2 hours lectures a This course is not for Honours Mathematics students.
MATH 451 (a) Nonlinear Optimization — A
Recent algorithms for constrained and unconstrained differentiable optimization problems will be presented with stress on practical implementation. Topics will include first and second order Newton methods, quasi-Newton methods, conjugate gradient directions, feasible direction methods, and penalty algorithms. Computational concepts wil] include aspects of scaling variables, nonlinear transformations, nonlinear equality constraints, roundoff and computer size. Applications will be given to problems such as production planning and engineering design. : Recommended for students interested in operations research, industrial engineering, computer science and for those wishing to pursue graduate courses in mathematical programming. 2 hours lectures, one term
MATH 451 (b) Nonlinear Optimization — B
Recent mathematical programming algorithms not requiring differentiability or continuity will be presented with the stress on practical implementation. Topics will include the Simplex and Complex algorithms, Powell’s conjugate direction method, difference approximations to differentiable methods. Computational aspects will include evaluating gradients by difference methods, reliability of answers, stopping criteria, and problem formulation. Applications will be given to problems such as warehouse utilization and automated machine controls industry. Recommended for students interested in operations research, industrial engineering, computer science and for those wishing to pursue graduate courses in mathematical programming. 2 hours lectures, one term Prerequisite: M45la
MATH 546 Faculty of Mathematics
MATH 452 Linear Programming (Theory, Applications, and Related Topics)
Preliminaries, convex sets, convex functions, geometric motivation for the Simplex Algorithm. Simplex Algorithm. Revised Simplex Algorithm. Degeneracy. Duality Theory. Transportation Problem, Shortest Path Problem. Primal Dual Algorithms. Maximum flow problems, Max-flow Min-Cut Theorem, Ford- Fulkerson Labelling Algorithm. Assignment Problems. Transshipment Problem. Capacitated Transportation Problem. Bounded Variables. Dual Simplex Algorithm. Post-optimal analysis (parametric programming, sensitivity analysis, etc.). Discrete Linear Programming. Branch and Bound Algorithms.
* The Travelling Salesman Problem. Integer Programming.
Comments: Throughout this course an attempt will be made to indicate the various areas in which the techniques mentioned above may be applied. These areas include: inventory control, production scheduling, optimal product mix problems, blending problems, game theory, economic input-output models, project scheduling (Critical Path, Decision C.P.M., etc.), bottleneck problems, cost-benefit analysis, cutting stock problem, etc. Students will be required to use the computer to solve “semi-practical” problems, and some knowledge of computer programming is a definite asset. 2 hours lectures, one year Prerequisites: M229 (or equivalent). Students will be required to use the computer on occasion and M132 (or its equivalent) would be a definite asset
MATH 453 (a) Queueing Theory — A
Introductory study of queueing models from simple queues with Poisson arrival and Exponential service distributions to multiserver queueing models including: homogeneous and heterogeneous parallel servers; servers in series; and phase type service.- Methods of analysis introduced include differential-difference equations, probability generating functions and simulation. Recommended as a prerequisile for Math 453 (b). Recommended for operations research and computer science students. 2 hours lectures, one term Prerequisite: Introductory Probability
MATH 453 (b) Queueing Theory — B
Study of advanced queueing models including single server modeis with balking and reneging; jockeying between queues; pre-emptive and non-pre-emptive priority queues; queueing _models with batch arrival or service processes. Also included: advanced. simulation models_and the algebraic analysis of complex queueing systems. Recommended as a prerequisite course for Math 885. Recommended for students interested in modelling techniques. 2 hours lectures, one term Prerequisite: M453 (a)
Faculty of Mathematics
MATH 454 (a) Game Theory
A mathematically-oriented course on the basics of game theory, with applications to economics, bargaining, and strategy. Classification of games; sotution of matrix games and their relationship to linear programming; infinite zero sum games; utility theory; bimatrix games and the bargaining problem; n-person games; Shapley value; metagames, 2 hours lectures, one term Prerequisites: Formal Second year linear algebra (M229); fundamentals of linear programming and probability theory.
MATH 454 (b) Selected Topics in Game Theory
Not offered in 1974–75.
MATH 455 (a) Mathematical Programming — A
Dynamic Programming; deterministic decision process problems, monotonic path problems, equipment replacement, resource allocation, reduction of dimension by use of Lagrange multipliers, shortest path problems, cargo loading problem, the travelling salesman problem; introduction to optimal control. Convex functions; first and second order characterization of convex functions. 2 hours lectures, one term Prerequisites: M352 or consent of instructor
MATH 455 (b) Mathematical Programming — B
Non-linear Programming; convergence conditions and convergence rates for some unconstrained optimization methods, introduction to the Kuhn-Tucker conditions and duality theory for non-linear programmes, optimality conditions for a quadratic programme - . Quadratic Programming; Lemke’s complementary pivot scheme to solve a quadratic programme. 2 hours lectures, one term Prerequisites: M455 (a) or consent of the instructor
MATH 456 Combinatorial Methods in Operating Research
MATH 456 (a) Combinatorial Systems
Production scheduling including finite sequencing for a single machine, parallel machines, flow-shop scheduling, the general - n/m job-shop processes. Capital budgeting with time-shared experierice. Non-differentiable optimization techniques used in warehouse design and automated machine control. Primarily for students interested in applying mathematics to problems of business, industry and operations research. 2 hours lectures, one term Prerequisites: M352 is recommended but not required
MATH 456 (b) Boolean Methods
MATH 457 Applied Combinatorial ; Mathematics
Not offered in 1974–75.
MATH 548 Faculty of Mathematics
MATH 457 (a) Applied Combinatorial Mathematics — A
A selection of combinatorial problems from the applied sciences. Topics which might be included are: Markov chains, random walk, the Poly problem, the dimer (and monomer-dimer) problem, the Ising problem, percolation processes, probabilistic graphs, electrical network theory. 2 hours lectures, one term Prerequisites: M351 (a) and M233
MATH 457 (b) Selected Topics in Applied Combinatorics
Brief recapitulation of information retrieval and the use of classificatory methods; formal models of classification methods; necessary conditions; equivalence of structure; quantification of equivalence; order independence; stability; training methods and their implication in information retrieval; optimization problems; perturbation functions and their algebraic structure; heuristics. 2 hours lectures, one term Prerequisite: M353 (a) (not offered in 1974-75)
MATH 458 (a) Graph Theory — A ‘ 458 (b) Graph Theory — B
Topics in graph theory. These may include symmetry in graphs, planarity, bipartite graphs, minimax theorems, directed graphs, enumeration, algorithms, colouring problems, matrices and graphs. Recommended for students who have taken a course in graph theory and wish to study additional topics not normally covered in an introductory course. 2 hours lectures, one term Prerequisite: M351 (a) or consent of instructor
Continuation of topics covered in 458 (a). 2 hours lectures, one term Prerequisite: M458 (a)
MATH 459 (a) Algebraic Graph Theory —-A
(not offered in 1974-75)
459 (b) Algebraic Graph Theory - B
(not offered in 1974-75)
MATH 460 (a) Combinatorics — A
Enumerative Mathematics Combinatorial identities, generating functions, counting of labelled and unlabelled objects, theorems © of Polya, Redfield-Fead, and de Bruijn. Recommended as a useful preliminary for M785 “Planar graphs”. 2 hours lectures, one term Prerequisite: A knowledge of elementary group theory (M329)
MATH 460 (b) Combinatorics — B
The existence and construction of error correcting codes, projective geometries, orthogonal Latin squares, balanced incomplete block designs and other combinatorial configurations. Recommended for M780, Combinatorial Analysis. 2 hours lectures, one term Prerequisite: Theory of finite groups, rings and fields (M329)
MATH 461 (a) Demography
Mathematical applications in demography. 2 hours lectures, one term Prerequisite or Corequisite: M335
Faculty of Mathematics
MATH 461 (b) Topics in Actuarial Mathematics
A selection of topics for the advanced actuarial student. 3 hours lecture, one term .
MATH 462 (a) Measure and Integration
The theory of measure and the Lebesgue integral. 2 hours lectures, Fall term
MATH 462 (b) Linear Operatorst
Linear operators in Hilbert spaces. Compact operators. Introduction to functional analysis. 2 hours lectures, Winter term
MATH 463 Control Theory
Dynamical systems, controllability and observability. Minimization of functions and functionals. Optimal control. Pontryagin’s Maximum Principle. 2 hours lectures. Fall term Prerequisite: Consent of instructor
MATH 464 Topics in Mathematical Physics
A selection of topics given by members of the Applied Mathematics Department. Topics covered in previous years have included continuum mechanics (elasticity, fluid mechanics), electromagnetic theory, statistical mechanics. 2 hours lectures, two terms Prerequisite: Consent of instructor
MATH 466 (a) Statistical Data Analysis
Summary statistics. Transformation of data. Plotting techniques. Comparisons of specific distributions via probability plots. Analysis of residuals. Random number generation techniques. Outliers in regression and analysis of variance data. W-statistic and tests for normality. Computational techniques in multiple unweighted and weighted regression. 2 hours lectures, Winter term Prerequisites: (1) A course in FORTRAN Programming or equivalent. (2) M338 (b) or M349 (a)
MATH 466 (b) Topics in Statistics
An introduction to several areas of Statistics, such as multivariate analysis, time series, data analysis, non-parametric theory, decision theory. 2 hours lectures, Winter term Prerequisite: M338 (b) or consent of instructor
MATH 467 (a) Sampling Theory and Practice
Introduction to sample theory and practice. Elementary sampling designs and estimation procedures. Statistical inference in survey sampling. Interrelationships between survey sampling and the design of experiments. 2 hours per week, Fall term Prerequisite: One of M223, M233 or M243
MATH 467 (b) Sample Survey Design Note
Procedures for construction of sampling designs commonly used in agricultural, economic and scientific surveys, and corresponding estimation techniques. Validation of survey results. 3 hours lectures, Winter term Prerequisite: M338 (a) or M349 (a)
M467 (a) is not a prerequisite
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MATH 468 (a) Introduction to Convexity and Extrema, and Economics Applications
Linear equations and inequalities; convex sets, cones and functions; goods and transformation possibilities; linear activities, axiomatic characterization; production functions and correspondences; cost and profit functions. Linear and convex programming; marginal values and shadow prices; centralization and decentralization. Von Neumann production model. Demand analysis; index construction.
MATH 468 (b) Introduction to Convexity and Extrema, and Economic Applications
Sperner’s Lemma; Brouwer’s fixed point theorem, von Neumann’s intersection theorem; Gale’s theorem on supply and demand equilibrium. Games, equilibria. Theory of exchange and general economic equilibrium; Scarf’s algorithm. Stability theory. Prerequisite: M352 (a) or equivalent.
MATH 469 (a) Statistical Methods with Socio-Economic Applications
Review of single equation linear models, Gauss-Markov theorem of least squares; testing independence of regression disturbances, Durbin-Watson and Theil-Nagar tests; serial correlation and heteroscedastic disturbances; serial correlation and heteroscedastic disturbances, Aitken’s generalized least squares method. Zellner’s multi-regression estimator. Introduction to simultaneous equation models, identification problem, two stage and three stage least squares estimation, asymptotic properties.
MATH 469 (b) Statistical Methods with Socio-Economic Applications
Simultaneous equation estimation continued, full information and limited information maximum likelihood, simultaneous least squares, asymptotic properties; forecasting with simultaneous equation models. Exact sampling distribution of the two stage least-squares estimator and its exact moments; approximations to exact moments. Prerequisite: M338 or equivalent
MATH 470 (a) Ordinary Differential Equations and Integral Equations ;
Initial value problems: existence and uniqueness of solution, one step methods, multistep methods, stability, error analysis. Boundary value problems: shooting and discretization methods, implementation problems especially for non-linear equations. Integral equations: correspondence to ordinary differential equation initial value and boundary value problems, solution techniques. 3 hours lectures, Fall term Prerequisite: Consent of the instructor
MATH 470 (b) Partial Differential Equations
Discretization methods for partial differential equations, convergence and stability, grid and characteristic method for hyperbolic equations; iterative methods of solution for parabolic and elliptic equations, rate of convergence estimates. 3 hours lectures, Winter term Prerequisite: Consent of the instructor
Faculty of Mathematics
MATH 472 (b) Introduction to Turing Machines and Computability Theory
Formalism for 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.
3 hours lectures, one term Prerequisite: M371 (a) or consent of the instructor
MATH 472 () Formal Languages
Languages and their representations. Grammars — Chomsky hierarchy. Regular sets and sequential machines. Context-free grammars — normal forms, basic properties. Pushdown automata and transducers. Operations on languages. Undecidable problems in language theory. Applications to the design of programming languages and compiler constructions, including lexical analysis, syntactical parsing, syntax directed translations and formalization of semantics. 3 hours lectures, one term Prerequisite: M371 (a) or consent of the instructor
MATH 474 (a) Scientific Applications Software
The practical computing problems which are encountered in scientific applications. An overview of batch operating systems; the importance of Joaders and overlays; Check points, post mortems and operating system services; FORTRAN as a programming language; Programming techniques including modudocumentation, larity, performance evaluations, no GO TO’s, portability; Design and use of programme libraries and human engineering of input and output; Magnetic tapes and bulk storage; Selected topics such as finite precision arithmetic, graphical output, analog to digital conversion, language interfacing. 3 hours lectures, one term Prerequisites: Two of M334 (a), M334 (b), M340 (a) and M340 (b) \
MATH 474 (b) Principles of Operating Systems
Topics in operating systems will include: basic concepts of computer hardware programme translation; programme loading and linking; cooperating sequential processes — computational and data structures, critical section problem, process synchronization primitives (semaphores, etc.), parallel programming; Introduction to multiprogramming; Operating system nucleus — process control and communication, system resource managements, input/output control systems, traps and trap handling routines; File system; Reliability; Protection; System performance, measurement and evaluation. Normal course requirements include a project. 3 hours lectures, one term Prerequisite: M340 (a) and M340 (b) ~
MATH 552 Faculty of Mathematics
MATH 474 (c) Business Systems Analysis
Survey of organization and management theory, including the following topics: Organizations, management, information systems in organization; Systems theory and the systems approach including the following topics: Definition of system, role of the systems analyst within an organization, the problems of commun-
_ ications within an organizational framework, the analysis phase
of systems development, the role of management within the systems approach; systems design: the nature of design, the desigti phase of systems development, design aids e.g. software packages and “packaged” documentation systems, the data base concept; implementation of computer based information systems, including the following topics: File structures, input and output design, forms design, implementation planning and control; som edevelopments in information retrieval systems. 3 hours lectures, one term Prerequisites: M340 (a) and fourth year standing
MATH 474 (d) Simulation
Simulation techniques are used to study systems which do not lend themselves to analysis. This course is intended to introduce the basic notions of simulation as well as introduce the student to programming languages for simulation. The course contents will be: An introduction to simulation; Random number generators; Stochastic Processes; Modelling; Simulation programming languages; the GPSS language. 3 hours lectures, one term Prerequisites: M340 (a) and fourth year standing
MATH 479 Advanced Topics in Computer Science
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, such as equations for functions of several variables, or equations for analytic functions, or equations on algebraic structures.
MATH 499 Readings in Mathematics
2 hours lectures, one term
Professor and Chairman of the Department
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