170 courses in the 1973–74 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
Notes
Faculty listed in 1973–74
Associate Professor, Chairman of A. Kerr-Lawson, B.A. (Toronto), M.A. (Chicago), PhD. (McMaster) the Department Assistant Professor, Associate E.M. Moskal, B.A. (Toronto), Ph.D. (Illinois) [On Sabbatical Leave, Chairman of the Department 1973-74] Associate Professor of Foundations P. de Witte, M.Sc. Theor. Phys. (Ghent), M.Sc. Math., Ph.D. (Brussels) . of Mathematics ‘ Professors H.H. Crapo, A.B. (Michigan), | Ph.D. (M.L.T.) [On Sabbatical Leave, 1973-74] G.E. Cross, M.A. (Dathousie), PAD. (U.B.C.) D.Z. Djokovic, Ph.D. (Belgrad) _* R.A. Staal, Ph.D. (Toronto) ° D.B. Summer, 44.Sc. (Cantab.), D.Phil. (Witwatersrand) J.W. Tucker,' B.Sc. (King’s College, London), Ph.D. (Birbeck College, University of London)
Associate Professors J.A. Baker,? M.A. (Saskatchewan), Ph.D. (Waterloo) S. Burris, Ph.D. (Oklahoma) Y. Chen, M.Ph. (Frankfurt), Ph.D. (Bochum) G. Dankert, Dip. Math (TU. Hanover), Ph.D. (Cologne) D.A. Higgs, ‘B.Sc. Hons. (Witwatersrand), M.A. (Cantab.), Ph.D. (McMaster) P. Hoffman, B.A. (Toronto), Ph.D. (Manchester) F.C.Y. Tang, B.Sc. (Hong Kong), M.S. (South Carolina), Ph.D. (Hlinois)
Professor, Chairman of the D.A. Sprott, Ph.D. (Toronto) Department Professor, Dean of the Faculty of W.F. Forbes, Ph.D., D.Sc. (London), DIC. Mathematics Professor, Associate Chairman of J.G. Kalbfleisch, B.Sc. (Toronto), Ph.D. (Waterloo) the Department Lecturer, Administrative Assistant C. Springer, M.Sc. (McGill)
Faculty members holding cross appointments as shown
' Statistics and Economics
? Applied Mathematics and Statistics and Applied Analysis and Computer Science
3Philosophy and Statistics
‘Chemical Engineering and Statistics
5 Psychology and Statistics
Applied Analysis and Computer Science and Statistics
Name Y.H. Au-Yeung
E. Berz N. Biggs
J.C. Du Plessis
W. Eichhorn
D. Fieldhouse
S.C. Johnson G.B. McMahon
P.J. O'Halloran H. Schneeweiss N. Starr
D.S. Tracy J.L.A. Urquhart P. Vamos
W.D. Wallis
Faculty of Mathematics _ 459
Visiting Professors with Faculty of Mathematics 1972-73
On Leave From Department University of Hong Kong, Pure Mathematics Hong Kong
Mathematisches Institut Applied Analysis and des Universitat Geissen, Computer Science Germany
Royal Holloway College, Combinatorics and University of London, Optimization Englefield Surrey, England
University af Witwatersrand, Applied Mathematics Johannesberg, S. Africa.
Institut fiir Wirtschaftstheorie Applied Analysis and und Operations Research der Computer Science
Universitat Karlsruhe, Karlsruhe, West Germany
University of Guelph, Pure Mathematics Guelph, Ontario Computer Science Research Applied Analysis and Centre, Bell Laboratories, Computer Science Murray Hill, New Jersey, U.S.A. School of Electrical Engineering, Combinatorics and University of South Wales, Optimization Kensington, N.S.W., Australia College of Advanced Education, Combinatorics and Canberra City, Canberra, Australia Optimization . University of Saarbriicken, Statistics Saarbriicken, West Germany , Amherst College, Combinatorics and Ambherst, Massachusetts, U.S.A. Optimization University af Windsor, Statistics Windsor, Ontario Hampton Institute, Combinatorics and Hampton, Virginia, U.S.A. Optimization University of Sheffield, _ Pure Mathematics
‘Sheffield, England
University of Newcastle, Combinatorics and New South Wales, Australia Optimization
Name J. Pfanzagl
A. Mostowski
Name G.A. Barnard
M. Gordon
A. Lichnerowitz R.E. Woolsey
K. Yano
M. Yoeli
Faculty of Mathematics
Division of Mathematical Research
The Division of Mathematical Research is a trans-departmental organization within the Faculty of Mathematics which is responsible for the general direction, encouragement and coordination of research and related activities. It arranges Colloquium lectures for the Mathematics Faculty (as distinct from research seminars), and organizes the _ Distinguished Lecturer Series in which individuals, outstanding both for their lecturing ability and research, are invited to present a series of lectures to undergraduates as well as graduates. It also brings mathematicians from other institutions to the University as Visiting Professors. . .
_ Faculty members working on related mathematical fields are encouraged to coordinate their activities through the formation of a Research Section. At present, there are seven such Research Sections, namely, Combinatorics, Differential Geometry and Relativity, Functional Equations, Numerical Analysis, Quanturn Theory, Statistics and Biometry, and Theory of Computing. Each of these holds regular research seminars. Graduate students should keep these seminars in mind when planning their graduate programme.
Distinguished Lecturers (1972-73)
University Area _ Math tische Insti der Universitat Statistics
zu Koln, Kéln, Germany
Wydzial Matematyki i Mechaniki
‘UW. Logic Istytuk Matematyki, Warszawa.
Poland
Visiting Professors (1972-73)
University Research Section
University of Essex, Golchester, Statistics
Essex, England
University of Essex, Colchester, Combinatorics
Essex, England
College de France, Paris, France Differential Geometry & Relativity
Colorado School of Mines, Combinatorics Golden, Colorado, U.S.A. Institute of Technology, Differential . Tokyo, Japan Geometry & Relativity
Technion, Haifa, Israel Theory of Computing
” Note
12 Calculus 1
!
21 Algebra and Vector Geometry
22 Calculus 2 (For Chemical Engineers)
25 Calculus 2 (For Electrical Engineers)
31 Differential Equations
32 Numerical Analysis
Note
/ 4 x.
Féculty of Mathematics 461
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 elected problems under supervision.
Systems of linear equations, Gaussian Elimination. Matrices, augumented matrix, row-reduced matrices, elementary matrices, inverses. Vector spaces, spanning sets, bases, dimension. Row-space, column-space, null-space, rank. Determinants. Linear transformations. Eigenvalues and Eigenvectors. Functions of complex variables.
2 hours lectures, 14 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.
Differential calculus of functions of several variables. Differential Equations. Multiple integrals. Prerequisite: M 12 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).
Ta) Introduction to Computer Programming in Business and
Economics
‘
‘Faculty of Mathematics
*
‘Real numbers, sequences, limits, continuity. The. derivative. General
Theorem of Mean Value. Functions of several variables. Implicit functions, Jacobians, Power series with complex terms, the Taylor series for functions of several variables, constrained extrema. The elementary functions for a complex variable.
3 lectures, one term.
Prerequisites: M 22, 31.
Systematic treatment of formal integration. Differential equations. Calculus of variations. Fourier series. Riemann integration, improper integrals, multiple integrals with applications. Line integrals. Theorems of Green and Stokes.
‘3 lectures, one term.
Prerequisites: M 21, 31.
Fourier series. Partial Differential Equations: separation of variables, wave equation, heat equation and Laplace’s equation. Fourier integral. Properties of complex analytic functions. Complex integration.
Partial differentiation. Laplace transforms, special functions. Fourier series, second order differential equations. 3 lectures, one term.
Cauchy-Riemann equation, 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, and Poisson laws. Sampling theory. Distribution of Chi-squared, 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.
The following two courses, Mathematics 81 and 82, have been designed primarily for elementary school teachers taking a degree at the University of Waterloo. These two courses should be open only to persons who have not completed Grade 13 Mathematics and/or equivalent and are not to be counted as credits towards a B.Math. degree. / This course is an introduction to computing which requires no pre-requisites. The emphasis in the course will be on problems associated with business and economics and their solution using the COBOL language. There will be an emphasis on data management including file processing, information retrieval, report generation, sorting, etc.
Some applications in business and economics such as simulation, modelling, forecasting, and regression analysis require the use of existing packages such as GPSS, SPSS, CPM and PERT. In some cases these
. may also be solved using the FORTRAN language. Elements of the
FORTRAN language will be introduced to the extent necessary to solve
__ these problems.
2 hours lectures, 1 hour tutorial one term.
Offered in the Fall and Winter term.
Credit will only be granted for one of M72, M112 or M132. M72 cannot be counted as a credit towards a B.Math degree.
81 Introduction to Functions, Analytic Geometry, and Calculus
‘
82 Introduction to Algebra
83 Mathematics for Social Scientists
4 Introductory Mathematics for Social Scientists
85 Mathematics for Environmental Studies
wor 1
Faculty of Mathematics 0 . 463
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.
Properties of rea] 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.
Polynomial, rational, exponential and trigonometric functions. Elementary analytic geometry. Polar coordinates. 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. Winter term only.
MATH 100Fundamental Concepts of Modern Mathematics 1 Applications of Mathematics in the Sciences
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.
(a) First term: Formulation of problems from biology, information theory and economics in terms of difference equations; techniques of solution and interpretation of results. The emphasis will be on
’ formulation, interpretation and the similarities between problems in
apparently separate disciplines — techniques are relatively simple. Typical problems: predator prey; host parasitoid; economic dynamics, inventory analysis, national income: transmission of information; learning models.
(b} Second term: Simple probability theory with applications to biology and economics; simpler parts of vector and matrix theory, ideas of Stochastic process and Markov chains with application..
The purpose of Mathematics 112 is to. provide students in programs 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 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 or M132. M112 cannot be counted as a credit towards a B.Math Honours degree.
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. i M 119 is not a course for Honours Mathematics students.
MATH 120Calculus Note 122A; Introduction to Computer Programming
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.
M 120 is not a course for Honours Mathematics students.
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 procedures and graph plotting. ~ 2 hours lectures, 2 hours tutorial (alternate weeks) Offered in the Fall and Winter terms. Prerequisite: Grade 13 Mathematics is recommended. wana 1 s4%4 . 8 , e wee
MATH 125Algebra, Geometry, and alculus for Optometry Students
A conventional course for freshman Calculus but with miore time spent on concepts of algebra and analytic geometry intimately connected with the Calculus. 4 hours lectures.
MATH 129Algebra and Geometry ~ 130 Calculus \1 Algebra and Solid Geometry 2 (a) Introduction to Computer Science ~ Programming : (b) Introduction to Computer Science ~ Characteristics of Computers
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, I hour tutorial.
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, volumes and movements. Power Series. Partial differentiation. 3 hours lectures, 1 hour tutorial.
The real and complex number systems. Polynomial functjons. Theory of equations. Determinants. Vector and matrix notation. Elementary solid geometry, planes and lines, the standard quadric surfaces. Co-ordinate systems. 2 lectures, 1 hour problems.
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 FORTRAN IV). The preparation implementation and debugging of programs in these languages. Students are expected to spend considerable time running programs on the computer. 2 hours lectures, 2 hours tutorial (laboratory), one term. Normally offered in the Fall term. Prerequisite: Grade 13, mathematics.
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 programs: 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.
Note
Prerequisite: Grade 13 mathematics.
Credit will only be granted for one of M72, M112. or M132.
MATH 201(b) Applications of Mathematics to the Physical Sciences Note
Mathematics 201 (a and b) is the first of three courses (201, 301, 401), designed to show the interplay between mathematics and science; each course is a study of the application of mathematical ideas and techniques to the sciences and the influence of physical problems on the formulation of mathematical concepts. In Mathematics 201 (a) calculus is applied to an examination of the basic concepts of Newtonian mechanics, planetary motion, rocket flight and related topics; vectors, tensors and groups are given added meaning through a discussion of various physical laws and crystalline structure. 2 hours lectures. Fall term. Prerequisites: M 130, Phys.-131 or equivalents.
The concepts of probability and statistics are exemplified in discussions of some of the following topics: Kinetic theory of gases, Maxwell’s distribution of velocities, statistical meaning of entropy, random numbers and random walk, Brownian motion, Monte Carlo method, design and evaluation of experiments, Markov chains and notion of information. 2 hours lectures. Winter term. Prerequisite: Phys. 131 or equivalent.
M 201 may not be counted as Mathematics credit but may be counted as an elective credit.
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.
is not a course for Honours Mathematics students. M 217
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.
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 symetric matrices. . 3 hours lectures.
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. Pee a en 2? ae 2 Y, ae 2 oe 7 ee eee ee ees el
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. Requisite: M 237 (may be taken concurrently).
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. M 31 or M 234 (a) or M 236.
M 234 (a) and M 236 may not both be taken for credit.
MATH 235(a) Actuarial Mathematics Mathematics of Investment
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, 1 hour problems.
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, 1 hour problems.
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 hour lectures.
M 236 is not a course for Honours Mathematics Students.
MATH 237Advanced Calculus \ ‘ ) Introduction to Combinatorics and Optimization
Differential calculus of functions of several variables. Multiple integrals. Line and Surface integrals; Green’s Theorem, Divergence Theorem, Stokes’ Theorem, Infinite Series. Uniform Convergence. 3 hours lectures. Prerequisite: M 130.
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, one term. Prerequisite: First pear algebra (M 119/129). .
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 and optimization. 2 hours lectures, one term. Prerequisites: First year algebra (M 129); 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. 3 hours lectures, 1 hour tutorial, one term. . Offered in the Fail, Winter and Spring terms. Prerequisites: M132, or equivalent, and M120 or M130 and M119 or M129. There is a special section for non-Mathematics students.
MATH 240(b) Non-Numeric Applications _ in Computer Science
Primitive numerical data representation; character strings: representation, — manipulation, and applications; logical operations; introduction to data structures; processing linearly linked lists, multi-linked lists; searching and sorting routines; programme as an object for manipulation; syntactic analysis, generation of object code-macros. 3 hours lectures, one term. Offered in the Fall and Winter terms. Prerequisite: M132 or equivalent.
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.
Note
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
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.
MATH 300(b) Mathematical Discovery and Invention - B Nate
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.
MATH 301(a) Applications of Mathematics to the Physical Sciences Note
‘See the preamble to Mathematics 201. Partial differential equations are applied to classical wave problems. . 2 hours lectures. Fall term. Prerequisites: M 201 and M 237 or equivalent.
M 301 (a) and (b) may not be counted as a Mathematics credit, but may be counted as elective credit.
MATH 301(b) Applications of Mathematics to the Physical Sciences
Vector and tensor analysis are related to problems in continuum mechanics. 2 hours lectures. Winter term. Prerequisites: M 201 and M 237 or equivalent.
MATH 302Applications of Mathematics to the Biological Sciences * Note
The aim of Mathematics 302 is mainly to acquaint Mathematics students with the fundamentals of the Biological Sciences, so that the student can see how his discipline can interact with the Biologica) Sciences. Subject areas such as probability, design of experiments, distributions, data processing, differential equations, and their applications in the Biological Sciences, will form part of the course. 2 hours lectures. Two terms.
302 may not be counted as a Mathematics credit but may be counted as an elective credit..
MATH 307Combinatorial Geometry 12 (a) Elements of Real Variable Theory Note
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 is helpful, but is certainly not necessary. 2 hours lectures. Prerequisite: First year calculus (M 120/130).
Metric properties of R", functions from R" to R; differentials; Riemann integrals; and one of a) an introduction to the Lebesgue integral; b) an introduction to metric spaces; c) orthogonal systems. 3 hours lectures. Emphasis will be on applications rather than theory.
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.
Projective spaces over fields, collineations and correlations, quadric curves and surfaces. References to non-euclidean geometries. 3 hours lectures, one term. Prerequisite: M 229.
The plane of complex numbers. The group of circle preserving mappings and its subgroups. Connections with non-euclidean geometrics. Other number systems and their geometries. (Laguerre, Minkowski). 3 hours lectures, one term. Prerequisite: M 229.
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.
Metric properties of R", functions from R" to R; differentials; Riemann integrals; and one of: a) an introduction to the Lebesgue integral; b) an introduction to metric spaces; c) orthogonal systems. 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.
Faculty of Mathematics an
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: M 240 (a). Also recommended are 219 or 229 and 21 7 or 237.
-Interpolatory, least squares, and minimax approximation of functions by t 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: M 240(a). Also recommended are M 219 or 229 and M 27 or 237.
A course in the calculus of finite differences, to include: summation, numerical integration and differentiation, relation between integration and summation; and error theory. 2 hours lectures.
An advanced course on problems with multiple lives; multiple decrement theory; accidental death and disability benefits. 3 hours lectures, one term. Prerequisite: M235b.
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: M 233, M 237.
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: M 229, M 338 (a).
Probability distributions, joint variation, central limit theorem. Introduction to continuous time stochastic processes. Poisson process and application to simple congestion situations. 3 hours lectures, Fall or Spring term. Prerequisite: M 243.
Discrete probability models. Random walks. Introduction to theory of Markov chains. Applications to biological and social processes. 3 hours lectures, Winter term. Prerequisite: M 233 or. 243.
MATH 340(a) Machine, Data and Program Structures 40 (b) Implementation Schemes for Programming 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). 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)
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: M 237.
Analysis of complex numbers; fundamental theorems of holomorphic functions; meromorphit functions. 2 hours lectures. Prerequisite: M 237.
Review of Normal, t, Chi-squared and F distributions and their applications. Introduction to the design of experiments and surveys. Analysis of variance. Multiple regression. 3 hours lectures, Winter term. Prerequisite: 233 or 243
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. M 239 recommended. Prerequisites: No formal prerequisites.
Applications of graph theory to communication networks, timetabling, _ ranking, and other combinatorial problems. 2 hours lectures, one term. Prerequisites: M 351 (a).
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. Prerequisites: Completion of M229, 237, 239 or consent of instructor.
Introductory study of optimization techniques complementary to those covered in 352 (a). Extrema of functions of n-variables, implicit functions and Lagrange multipliers, inventory problems; difference methods; queuing, recurrent events and replacement policies; games; job-sequencing 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(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; minimization techniques; local and global methods; retrieval performance; measures of effectiveness; problems of retrieval relevance judgments. 2 hours lectures, one term.
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: 237, Consent of instructor.
Intermediate level course on Riemannian spaces (continuation of 360 (a). Conformal spaces, spaces with corresponding geodesics, theory of subspaces. 2 hours lectures + I hour tutorial, Winter term. Prerequisites: 237, 360 (a) or Consent of instructor.
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, 1 hour tutorial. Fall Term. Prerequisites: M 237, Consent of instructor.
MATH 361(b) Mechanics , (a) Introduction to Continuum Mechanics
Hamilton’s principle. Lagrangian and canonical equations of motion. : Relativistic mechanics. 2 hours lectures, 1 hour tutorial. Winter Term. Prerequisites: M 237, Consent of instructor.
Ordinary differential equations applied to the motion of a system of particles stability theory applied to the motion of a system of particles, applications to celestial mechanics; the stress tensor; partial differential equations in continuum mechanics; collisions between particles; from ordinary mechanics to impulsive mechanics by distribution theory; introduction to statistical mechanics. 2 hours lectures, + 1 hour tutorial. Fail.term. Prerequisites: 234, 237 or consent of instructor.
MATH 362(b) Partial Differential ations of Applied Mathematics
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, I hour tutorial. Winter term. Prerequisites: M 234, M 237 or consent of instructor.
2 hours lectures, + 1 hour tutorial. Prerequisites: M 234, M 237 or equivalent, M 332 (Real Variables) may be taken concurrently..
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. 3 hours lectures, one term. Offered in the Fall, Winter and Spring terms. Prerequisite: M132. Recommended Corequisite: M319 or M329.
This course will be offered as M471 (a) during the Fall term, 1973 for Co-op students registered in fourth year only. Credit will not be given for both M371 (a) and M47] (a).
MATH 371(b) Computer System * Organization and Logic Design Note
Part 1) 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.
Part 2) System Organization: Definition of total system requirements for batch processing, time-sharing, real-time processing and other applications; hardware-software tradeoffs and-the impact of large-scale integration; unorthodox. hardware organizations; evaluation of solutions. Some attention will be paid to design for reliability and fault diagnosis as additional system requirements. 3 hours lectures, one term. Offered in the Fall and Winter terms. Prerequisite: M371 (a) Recommended Corequisite: M319 or M329.
This course will be offered as M471 (b) during the Winter term, 1974 for Co-op students registered in fourth year only. Credit will not be given for both M371 (b) and M471 (b).
Sequential machine models. Reduction of completely and incompletely specified.machines. Decomposition of machines. Non-deterministic machines. Finite automata and regular events. Multitape finite-state nonwriting automata. 3 hours lectures. Winter term. Prerequisite: M371 (a) Recommended Corequisite: M319 or M329.
This course will be offered as M472 (a) during the Winter term, 1974 for Co-op students registered in fourth year only. Credit will not be given for both M372 (b) and M472 (a).
MATH 380Introduction 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.
MATH 401Applications of Mathematics to the Physical Sciences Note
See the preamble to Mathematics 201. The concepts and techniques of group theory, special functions and theory of distributions are illustrated by the problems of quantum physics. 2 hours lectures. Prerequisite: Consent of instructor.
Math. 401 may not be counted as a Mathematics credit, but may be counted as an elective credit.
MATH 402Topics in Mathematical Aspects of Chemistry, Biology, and the Medical Sciences Note ,
Subjects will be selected from areas such as introductory wave mechanics, spectroscopy, molecular biology, and mathematical models of disease processes. 2 hours lectures.
402 may not be counted as a Mathematics credit, but may be counted as an elective credit.
2 Continuation of linear algebra. Main topics: representations of endomorphisms, structures of bilinear forms, multilinear products. 2 hours lectures. Prerequisite: 329 or 341.
An introduction to the theory of algebraic varieties. Special topics such as the Theorem of Riemann-Roch. 2 hours lectures. Prerequisite: M 329 or M 341
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.
Riccati equation, conflict and pursuit problems, elliptic integrals and functions, phase plane, stability, periodic solutions, non-linear mechanics. Non-linear integral equations. 2 hours lectures. Prerequisites: M 363 (may be taken concurrently), Consent of instructor.
Algorithms and theory relating to both conbinatorics 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: M 352 (a) or consent of instructor.
Branchings in networks. Elementary matroid theory illustrated by forests, transversals, algebraic dependance, 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: M 417 (a) or consent of instructor.
MATH 118Combinatorial Applications of Computers ‘418 (a) Combinatorial Applications of Computers-A
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. 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 (M 319/329).
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. Gonstruction 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 (M 329). M 418 (a) is a prerequisite for M 418 (b)
Multiplicative-algebraic theory of numbers. Foundations of natural number theory. Elements of additive-combinatorial number theory. 2 hours lectures. Corequisite: M 329 or M 341
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: M 363 or equivalent, or consent of instructor.
a) Numerical problems arising in actuarial science and statistics. 2 hours laboratory, one term. Prerequisite — Corequisite: M335b, M46la, M437a, or consent of instructor.
b) Numerical problems arising in actuarial science and statistics. 2 hours laboratory, one term. Prerequisite: M336a and 6 or consent of instructor.
Only advanced students are advised to take this course.
MATH 436Introduction to Logic 37 (a) Measurement of Mortality
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 completeness. 2 hours lectures. Prerequisite: consent of instructor.
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.
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.
Several approaches to the controversial problems of statistical inference are compared. Likelihood, fiducial probability, Bayes’ theorem, confidence intervals. Sufficiency and ancillarity. Consistency and efficiency of estimates. Information. Maximum likelihood and large
The logic of experimental design. The design and analysis of sample surveys. The course covers the theory of the construction and analysis of various kinds of designs with emphasis on the important basic principles (e.g. randomization, replication) and the application of the designs to practical problems. ‘ 3 hours lectures. Prerequisite: M 338 (a) or consent of instructor.
Flat space-time and special relativity, curved space-time and the Einstein gravitational field equations. The Schwarzchild solutions and the experimental tests of relativity. Maxwell’s equations in 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: M 360 (a), Consent of instructor.
Applications of the Maxwell equations. Reflection and refraction. Introduction to wave guides and antennae. 2 hours lectures, Fall term. Prerequisite: Consent of instructor.
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 : Venants’ principle. 2 hours lectures + 1 hour tutorial, Winter term. Prerequisite: M 445 (a).
Stress and strain tensors, rate of strain tensor, generalized Hookes’ law, conservation laws, equations of motion; thermodynamic and energy considerations. 2 hours lectures + 1 hour tutorial, Fall term.
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 of elementary geometry (¢.g. the material contained in “A Modern Geometry For High Schools” by Lougheed and Workman).
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 from elementary
number theory, a collection of outstanding problems in geometry (Fagnano, Steiner-Lehrus, 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. ‘
Similar to Mathematics 439 but with more accent on the logic and methods than on the mathematics. 2 hours lectures. Prerequsite: M233 or M243 or consent of instructor.
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.
(This course is not for Honours Mathematics students)
Recent algorithms for constrained and unconstrained differentiable optimization problems will be presented with the stress on practical implementation. Topics will include first and second arder Newton methods, quasi-Newton methods, conjugate gradient directions, feasible direction methods, and penalty algorithms. Computational concepts will include aspects of scaling of 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. Prerequisite: Differential Calculus (M237), M352 (b). Texts and References: announced in class.
12. _ Faculty of Mathematics .
51 (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. : Texts and references to be announced in Class.
i2 Linear Programming (Theory, Preliminaries, convex sets, convex functions, geometric motivation for ipplications, and Related Topics) the Simplex Algorithm. Simplex Algorithm. Revised Simplex Algorithm. . Degeneracy. Duality Theory. Transportation Problem. Shortest Path
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: M 229 (or equivalent). Students will be required to use the computer on occasion and M 132 (or its equivalent) would be a definite asset.
MATH 453AQueuing Theory-A Introductory study of queuing models from simple queues with Poisson
. arrival and Exponential service distributions to multi-server queuing 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 pre-requisite for Math 453 (b). Recommended for operations research and computer science students. 2 hours lectures, one term. Prerequisites: Introductory Probability.
Study of advanced queuing models including single server models with balking and reneging; jockeying between. queues; pre-emptive and non-preemptive priority queues; queuing models with batch arrival or service processes. Also included: advanced simulation models and the algebraic analysis of complex queuing systems. Recommended as a pre-requisite course for Math 885. Recommended for students interested in modelling techniques. 2 hours lectures, one term. Prerequisite: M 453 (a)
A mathematically-oriented course on the basics of game theory, with applications to economics, bargaining, and strategy. Classification of games; solution 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 (M 229); fundamentals of linear progr ing and probability theory.
Nonlinear programming convex sets and convex functions; convergence conditions and convergence rates for some unconstrained optimization methods; introduction to Kuhn-Tucker theory and duality theory for nonlinear programmes; economic interpretation of dual variables. 2 hours lectures, one term. Prerequisites: M 352 (a)
MATH 455(b) Dynamic Programming and Quadratic Programming
Quadratic Programming: Lemke’s complementary pivot scheme to solve a quadratic programme. : Dynamic Programming: deterministic decision process problems; monotomic path problems; equipment replacement; resource allocation; reduction of dimension by use of Lagrange multipliers; shortest path problems; the travelling salesman problem; introduction to optimal control theory. 2 hours lectures, one term. Prerequisite: M 455 (a)
A selection of combinatorial problems from the applied sciences. Topics
Mathematics-A
which might be included are: Markov chains, random walk, the Polya problem, the dimer (and monomer-dimer) problem, the Ising problem, percolation processes, probabilistic graphs, electrical network theory. 2 hours lectures, one term. Prerequisites: M 351 (a) and M 233.
57 (b) Selected Topics in Applied
Brief recapitulation of information retrieval and the use of classificatory
Combinatorics
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 1973-74)
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: M 351 (a) or consent of instructor.
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 M 785 “Planar graphs”. 2 hours lectures, one term. : Prerequisite: A knowledge of elementary group theory (M 329).
The existence and construction of error correcting codes, projective geometrics, orthogonal Latin squares, balanced incomplete block designs and other combinatorial configurations. Recommended for M 780, Combinatorial Analysis. 2 hours lectures, one term. Prerequisites: Theory of finite groups, rings and fields (M 329).
MATH 464Topics in Mathematical A selection of topics given by members of the Applied Mathematics
Physics 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 What is exploratory data analysis (according to Tukey)?*‘Summary
statistics. Transformation of data. Plotting techniques: histogram, stem and leaf, empirical cumulative distribution function; generalized P-P and Q-Q plots including computerized normal and half-normal probability plots; high dimensional data plotting technique. Comparisags of specific distributions via probability plots, Analysis of residuals, including plotting techniques and half-normal, other - Q-Q plots, rootagrams, etc. — why they work. 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. Prerequisite: 1) A course in FORTRAN Programming or equivalent. 2) M 338 or M 349. -
MATH 467(a) Sampling Theory and Introduction to sample theory and practice. Elementary sampling designs
Practice and estimation procedures. Statistical inference in survey sampling. Interrelationships between survey sampling and the design of experiments. 2 hours lectures. Winter t term. Prerequisite: M 233 or 243."
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. i Prerequisite: Consent of instructor.
Discretization methods for partial differential equations, convergence and stability, grid and characteristic methods 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.
See Mathematics 371 (a). 3 hours lectures. Fall term. Prerequisite: M132. Recommended Prerequisite: M319 or M329.
M471 (a) will be open only to Co-op students registered in fourth year. Credit will not be given for both M371 (a) and M471 (a). M471 (a) will not be offered after the Fall term, 1973.
MATH 471(b) Computer System Organization and Logic Design Note
See Mathematics 371 (b). 3 hours lectures. Winter term. Prerequisite: M 47] (a) ‘ Recommended Prerequisite: M319 or M329.
M471 (6) will be open only to Co-op students registered in fourth year. , Credit will not be given for both M371 (b) and M471 (b). M471 (b) will not be offered after the Winter term. 1973-74.
MATH 472(a) Introduction to Automata Theory Note ' \
See Mathematics 372 (b). 3 hours lectures. Winter term. Prerequisite: M471 (a). Recommended Prerequisite: M319 or M329.
M472 (a) will be open only to Co-op students registered in fourth year. Credit will not be given for both M372 (b) and M472 (a). It is anticipated that M472 (a) will not be offered after the Winter term, 1973-74.
MATH 472(b) Introduction to Turing Machines and Computability Theoryro
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. Offered in the Spring and Fall terms.
. Prerequisite: M371 (a) or consent of the instructor.
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 ih 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. Winter term. Prerequisite: M371 (a) or consent of the instructor.
The practical computing problems which are encountered in scientific applications: An overview of batch operating systems; the importance of loaders and overlays; Check points, post mortems and operating system services; FORTRAN as a programming language; Programming techniques including modularity, performance evaluation, no GO TO’s, documentation, portability; Design and use of program libraries and human engineering of input and output; Magnetic tapes and bulk storage; Sclected topics such as finite precision arithmetic, graphical output, analog to digital conversion, language interfacing. 3 hours lectures, one term. Prerequisite: M340 (a) and M340 (b)
MATH 474(b) Primciples of Operating Systems 4 (c) Business Systems Analysis
Topics in operating systems will include: basic concepts of computer hardware; program translation; program 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 system, 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). . . 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 communications 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 design, 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, some developments in information retrieval systems. 3 hours lectures, one term. Prerequisite: M340 (a) and M340 (b)
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 M340 (b).
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.
Details of the undergraduate programme in Mechanical Engineering are to be found on page 142. All courses extend over one term only, and consist of 3 hours of lectures per week unless otherwise specified. The hours of the core courses are listed on page 143. In general, the only prerequisites are the core courses, unless otherwise specified.
Undergraduate Course Descriptions
Infinite series: Tests for absolute, conditional, uniform convergence; power series; series expansions; differentiation and integration. Partial differentiation: total derivatives; estimation of errors; chain rule; geometry; maxima and minima; taylor series; jacobians. Multiple integration: areas, centroids, moments of inertia, centres of gravity. Vector analysis: gradient, divergence, curl, Laplacian; integral theorems.
Frequency distributions; measures of central tendency; standard deviation and other measures of dispersion; moments, skewness, kurtosis. Probability. Binomial, Poisson, normal, multinomial distributions. Theory of sampling and statistical estimation. Tests of hypotheses; significance. The t-test and chi-squared test. Curvefitting by least squares. Correlation and regression. Analysis of variance. Experimental design.
First and simple higher order equations. Solution of linear equations. Solution of linear equations by various methods; equations transformable to linear form. Simultaneous equations. Laplace transforms. Series solutions. Fourier series.
A survey of nifinerical procedures with emphasis upon computer implementation using the Fortran 4 programming language. In particular, the following topics are covered: 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.
5 Partial Differential Equations.
8 Review of Engineering Mathematics.
12 Dynamics
15 Structure and Properties of Materials 1
19 Mechanics of Deformable Solids 1
20 Mechanics of Deformable Solids
21 Kinematics and Dynamics of Machines
22 Mechanical Design 1
30 Structure and Properties of Materials 2
i
32 Physical Metallurgy 2
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Department of Mechanical Engineering 491
Modelling physical systems with distributed parameters. Boundary and initial conditions. Division into hyperbolic, parabolic and elliptic equations by means of discriminarit.
Characteristics. Separation of variables; eigenvalues and eigenfunctions. Orthogonality. Fourier series, gamma functions. Bessel and Legendre equations. Numerical solution of Laplace and diffusion equations. Analogue simulation techniques.
A revision course in engineering mathematics. Topics include: a review of elementary differential and integral galculus; series; partial derivatives; selected ordinary differential equations; Laplace transforms; Fourier series.
An introduction to the kinematics of particle and rigid body motion. Impulse-momentum equations. Work-energy methods and Euler’s equations. Simple gyroscopes. Vibrations.
Interatomic bonding, crystal and amorphous structures including typical polymers and ceramics, structural defects, phase diagrams, diffusion, transformations in metals, corrosion: Some aspects of electrical, magnetic and optical properties.
An analytical treatment of statics and resistance of materials. Equilibrium of particles, rigid bodies and deformable bodies. Vector fields. Introduction to stress and strain. Stress-strain relationships. Elastic and inelastic behaviour of prismatic members subjected to axial, shearing, torsional and flexural deformations.
A general treatment of the behaviour of structural components from the study of stress and strain in solids. Topics include super-position, energy theorems, theories of failure, elastic and inelastic analysis of unsymmetrical bending, torsion of circular members, columns and stability, and virtual work.
Principles of the geometry of motion. Uniform and non-uniform motion, linkage, gears, cams. Synthesis and analysis of mechanisms. Consideration of the dynamic forces in machines. Gyroscopic torque. Vibration analysis, response to shock, motion and force transmissibility, vibration isolation.
Analysis and synthesis of machine elements. Factors affecting working stresses, fatigue, creep and impact considerations. Design of shafting, springs, screws, clutches, brakes and gear.
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The general principles of stress-strain relationships in all types of materials, including the effects of temperature, strain rate, alloying and microstructure. Different fracture mechanisms and the factors which influence them.
Elasticity and plasticity of metals. Recovery, recrystallization and growth. Hardening mechanjsms. Heat treatment of metals and alloys. Shear transformations. Failure in metals.
33 Experimental Materials Science
35 Industrial Metallurgy
40 Manufacturing Processes
41 Manufacturing Science 1
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42 Manufacturing Science 3
43 Manufacturing Science 4
44 Production Engineering
45 Manufacturing Science 6
Department of Mechanical Engineering
This course is designed to acquaint students with materials phenomena using an experimental approach. Microstructural changes and their effects on the mechanical properties in various materials will be studied using such techniques as optical and scanning electron microscopy together with electron probe microanalysis and X-ray analysis. The student may work qn a project of his own choice provided it meets the ‘objectives of this course.
This course is intended for those students interested in acquiring a
«working knowledge of metallurgy. It will cover: Metals and alloy systems, iron-carbon alloys, heat treatment and the function of alloying elements in steel, corrosion and scale resistant alloys, copper and nickel base alloys, light metals and their alloys; casting, hot and cold working of metals; soldering, brazing and welding; corrosion and oxidation; metal failure analysis. ‘
The generation and forming of surfaces. Concepts of design of tooling for manufacturing processes. The influence of materials on processes and choice of processes on design. The behaviour of materials in processes. Process limitations, tolerances, accuracy, surface finish and economics. The elements of cost in manufacturing. Determination of manufacturing times and cost of production.
Models of a plastically deformed metal; yield and flow. Analysis and mechanics of mechanical working operations. Patterns of flow, limiting factors; force, instability and temperature. Influence of processes on work-piece properties. Applications to forging, drawing, extrusion, rolling, sheet-drawing and deep drawing. Modern developments: high velocity forming; chemical, electromagnetic and spark discharge methods.
Mechanics of metal cutting operations. Geometrical force and energy analysis of single and multipoint tools, rotary cutters and grinding. Modern developments: Electromechanical, spark discharge and ultrasonic methods. Machining economics.
Moulding from the liquid state. Review of applicable heat transfer and fluid flow theory. Influence of time in casting. Solidification time, freezing waves and riser design. Pouring time, fluidity, gating and running. Hot tears, distortion, shrinkage and residual stresses. Die casting; thefmal stability, pressure pulse effects. Other casting methods.
The use of group technology for product design, production planning and control. Resource allocation. The concepts of statistical quality control and reliability engineering. Waiting lines. Economic, batch size. Economic decision making recognizing risk. Maintenance replacement policy. Introduction to the use of computers in production planning and control..
Manufacturing for design engineers. Classification of processes. Influence of choice of materials on processes and of choice of processes on design. Process limitations; tolerances; influence of scale of manufacture. Economic considerations. Interactions of design and manufacturing functions. Information flow, data specification.
46 Manufacturing Science 7
47 Analysis and Design of Manufacturing Systems I
48 Analysis and Design of Manufacturing Systems 2
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49 Metrology
50 Thermodynamics
51 Fluid Mechanics 1
52: Air Conditioning
53 Heat Transfer 1
54, Thermodynamics 2
Department of Mechanical Engineering : : 493
Introduction to polymer processing for mechanical engineers. Elements of polymer science; plastics as a design material. Screw extrusion; elements of non-Newtonian flow and viscometry. Injection moulding. Parameters of polymer conversion operations and their estimation.
Organization for manufacture. Product design and development. Principles of processing for various production processes: unit, batch, and mass production. Line balancing. Economic decision models: concepts, breakeven and minimum cost analysis, allocation of resources, scheduling of resources, concepts of risk. Inventory control. Prerequisite: ME 45.
Analysis and design of thé plant layout and materials handling systems as basic components of a manufacturing facility and system. Product range and mix and their effect on these systems. Proven systematic analysis and synthesis and evaluation techniques for efficient and effective plant design.
Prerequisite: ME 45.
Theory and practice of high precision mechanic measurements under strict control conditions—super micrometry; measurements by comparators; profilometry; surface profilography; environmental cffccts on measurements accuracy; theodolite techniques in the measurements of large structures; collimator applications in machine installation. ° Tolerances and quality control.
The scope and limitations of thermodynamics. Macroscopic approach to energy work and the First Law. The fundamental energy equation. The Second Law of Thermodynamics, principle of increase of entropy, limiting cycle efficiencies, criteria for equilibrium. Introduction to statistical thermodynamics and the microscopic description of energy, work: and the First Law. Thermodynamic potentials.
Physical properties of fiuids and fundamental concepts in fluid mechanics. Hydrostatics. Conservation laws for mass, momentum and energy. Flow similarity and dimensional analysis as applied to engineering problems in fiuid mechanics. Laminar and turbulent flow. Engineering applications such as flow measurement, flow in pipes and fluid forces on moving bodies. Introduction to compressible flow.
Thermodynamic properties of moist air; psychrometric charts; humidity measurements; direct water contact processes; heating and cooling of moist air by extended surface coils; solar radiation; heating and cooling of loads on buildings; effects of the thermal environment; air conditioning calculations; air flow in and around buildings, diffusers. Prerequisite: ME 54.
Introduction to heat transfer mechanisms. The formulation and solution of steady and transient heat conduction. Radiant heat transfer including exchange laws and view factors. Introductory convective heat transfer.
Emphasis on applications of thermodynamics to flow processes, real fluids, evaluation of state functions of real fluids. Thermodynamic analysis of selected devices.
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55 Refrigeration Esigineering
56 Heat Transfer 2
58 Internal Combustion Engines
59 Energy Conversion
60 Introduction to Control Systems
62 Fluid Mechanics 2
63 Lubrication Mechanics
69 Introduction to the Environmental Sciences
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Methods of refrigeration, refrigerants and their thermodynamic properties, vapour compression systems; actual cycles, simple and complex; survey of refrigeration applications such as preservation of food by quick freeze and long term deep freeze techniques; cooling load calculations; calculation of thermal capacities of components such as evaporators, compressors and condensors; refrigerant controls; piping and accessories; defrost methods; miscellaneous processés, cryogenics. Prerequisite: ME 54.
Forced and free laminar and turbulent convection heat transfer in internal and external flows. Special topics selected from current applications.
Reciprocating SI and CI engines, gas turbines, jets, rockets. Prerequisite: ME 54.
Applications of fundamental principles of thermodynamics, fluid mechanics, combustion and heat transfer to the design of power plants using fossil and nuclear fuel heat sources. Economic and pollution problems associated with power equipment. Other energy conversion devices such as batteries, fuel cells, solar cells, thermionic and thermoelectric devices and MHD generators.
Prerequisite: ME 54.
Open toop and feedback control. Laws governing mechanical, electrical, fluid and thermal control components. Analogies. Analysis of some engineering control systems using block diagram algebra, transient and steady-state operations. Different modes of control. Review of Laplace Transform methods. Transfer function. First and second order systems. Concepts of stability. Principles of analog computer simulation. Brief treatment of linear flow graph and system graph representations.
Basic equations of two-dimensional flow, exact viscous solutions, introduction to lubrication, boundary layers, and introduction to turbulence. Turbomachinery. fundamentals and applications. Selected advanced topics. A) Lubrication Principles: dry friction, boundary lubrication, hydrodynamic lubrication, rolling elements, squeesefilms, metal working lubrication, wear, failure modes.
B) Lubrication applications: sources and types of lubricants, their composition and selection; sliding bearings, rolling bearings, gears, wire rope, hydraulic fluids, metalworking..
C) Project: related to a specific topic in Parts A and B. Lab or theoretical evaluations will be involved in the design of a suitable lubrication system for a specified industrial application.
To be offered in Fall and Spring Terms.
Composition and structure of the atmosphere and oceans. Thermodynamics of moist air. Hydrostatic equation. Stability. Winds and Currents on a Rotating Earth. Gradient, geostrophic and inertial flow. Vertical variation of wind, convection, thermals, plumes, basic diffusion theory-coastal currents, gulf stream, tides, internal waves.