166 courses in the 1972–73 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
D.G. Wertheim, B.A. (McMaster), Ph.D. (Toronto) [On Sabbatical Leave, 1972-73] C.F.A. Beaumont, B.A. (McMaster), M.A. (Toronto)
LJ. McGee, B.A.Sc (Toronto), M.Sc. (Waterloo), Ph.D. (Yale)
E.T. Davies, M.Sc. (Wales), Dottore in Matematica (Rome), Ph.D., D.Sc. (London) C. Froese Fischer (Mrs.), 4.4. (U.B.C.), Ph.D. (Cantab.)
B. Forte, Ph.D. (Pisa), Habil. D.Sc. (Rome) F.O. Goodman, Ph.D. (London, England) M.A. McKiernan, (Loyola), Ph.D. (LL.T.)
J. Cizek, R.N.Dr. (Charles University, Prague), C.Sc. (Czechoslovak Academy of Sciences, Prague)
J. Froese, B.A. (Manitoba), M.A. (Queen’s), Ph.D. (U.B.C.)
G.J. Lastman, M.A. (U.B.C.), Ph.D. (Texas)
D. Lovelock, Ph.D. (Natal, Durban)
J. Paldus, R.N.Dr. (Charles University, Prague), CSc. (Czechoslovak Academy of Sciences, Prague)
P.J. Ponzo, M.A. (Toronto), Ph.D. (Illinois)
R.A. Wentzell, B.Sc. (Acadia), Ph.D. (Western)
Z. Dvoracek, M.S, R.N.Dr. (Charles University, Prague), Ph.D. (Czechoslovak Academy of Sciences, Prague)
W.H. Hui, BSc. (Peking), Ph.D. (Southampton)
S.P. Lipshitz, B.Sc. Hons. (Natal), M.Sc. (South Africa), Ph.D. (Witwatersrand)
B.J. Marshman, (Mrs.), Ph.D. (Waterloo)
R.G. McLenaghan, M.Sc. (Queen’s), Ph.D. (Cambridge)
M.E. Snyder B.Sc. (Western), M.Sc. (Waterloo)
J. Wainwright, B.Sc. (Natal), Ph.D. (South Africa)
D.J. Henderson, B.A. (U.B.C.), Ph.D. (Utah), F. Inst. P. H. Rund, PA.D. (Cape Town), Habilitation (Freiburg)
Professor, Chairman of the Department
Professor, Associate Dean of the Faculty of Mathematics Professor, Associate Dean Mathematics Graduate Studies
Assistant Professor, (part-time), Associate Director af Computing
Centre ecturer, Administrative Assistants
Distinguished Professor
Professors
Associate Professors
Assistant Professors
Instructor
Adjunct Professors
Faculty of Mathematics 423
Department of Combinatorics and Optimization
G. Berman, M.A., Ph.D. (Toronto) K.D. Fryer, B.A. (Western), Ph.D. (Toronto)
R.C. Mullin, B.A. (Western), Ph.D. (Waterloo)
J.W. Dodd, B.A.Se. (Toronto), M.Sc. (Waterloo)
E. Anderson, B.A. (McMaster) R.G. Dunkley, B.A. (Western) W.I. Miller, B.A. (Queen’s)
W.T. Tutte, Ph.D. (Cantab.), F.R.S.C.
H.F. Davis, Ph.D. (M.I.T.)
J. Edmonds, B.A. (Geo. Washington), M.S. (Maryland)
P.L. Hammer, Ph.D. Math. (University of Bucharest, Romania) R.C. Read, M.A. (Cambridge), Ph.D. (London)
C.E. Haff, B.S. (Stanford), Ph.D. (Waterloo)
R.A. Honsberger, B.A. (Toronto), M.A. (Waterloo)
U.S.R. Murty, M4. (Osmania), Ph.D. (Indian Statistical Institute) H. Shank, M.Sc. (Chicago), Ph.D. (Cornell)
D.H. Younger, Ph.D. (Columbia).
M. Best, M@. Math. (Waterloo), Ph.D. (University of California at Berkeley) J.A. Bondy, D. Phil. (Oxon.)
R.N. Burns, B.Sc. (Toronto), Ph.D. (Waterloo)
A.R. Conn, B.Sc. (Imperial College), M.Sc. (Manitoba), Ph.D. (Waterloo) G.B. Faulkner, B.A.Sc. (Toronto), Ph.D. (Waterloo)
D.M. Jackson, Ph.D. (Cambridge)
P. Schellenberg, Ph.D. (Waterloo)
P. Zima, M.Sc. Physics (Charles University, Prague) ELL. Johnson, B.S. (Georgia Tech), Ph.D. (University of California at
Berkeley) C.St. J.A. Nash-Williams, Ph.D. (Cantab.), F.R.SE.
Professor, Chairman of the Department Associate Professor of Foundations of Mathematics Professors
(Philosophy and Pure Mathematics)
Associate Professors
Assistant Professors
(St. Jerome’s and Pure Mathematics)
Professor, Chairman of the Department
Faculty of Mathematics
Department of Pure Mathematics
A. Kerr-Lawson, B.A. (Toronto), M.A. (Chicago), Ph.D. (McMaster)
Paul de Witte, M.Sc. Theor. Phys. (Ghent), M.Sc. Math., Ph.D. (Brussels)
H.H. Crapo, 4.B. (Michigan), Ph.D. (M.I.T.)
G.E. Cross, M.A. (Dalhousie), Ph.D. (U.B.C.)
D.Z. Djokovic, Ph.D. (Belgrad)
R.A. Staal, Ph.D. (Toronto)
D.B. Summer, M.Sc. (Cantab.), D.Phil. (Witwatersrand)
J.W. Tucker, B.Sc. (King’s College, London}, Ph.D. (Birbeck College, University of London)
S. Burris, Ph.D. (Oklahoma)
Y. Chen, M.Ph. (Frankfurt), Ph.D. (Bochum)
G. Dankert, Dip. Math (T.U. Hanover), Ph.D. (Cologne)
D.A. Higgs, B.Sc. Hons. (Witwatersrand), M.A. (Cantab.), Ph.D. (McMaster) {On Sabbatical Leave, 1972-73}
P. Hoffman, B.A. (Toronto), Ph.D. (Manchester)
F.C.Y. Tang, B.Sc. (Hong Kong), M.S. (South Carolina), Ph.D. (Illinois)
J.G. Anderson, M.Sc. (Durham), Ph.D. (Newcastle upon Tyne) L.J. Cummings, Ph.D. (U.B.C)
L.J. Dickey, 4.4. (Arizona), Ph.D. (Wisconsin)
W.J. Gilbert, M.A. (Cantab.), D. Phil. (Oxon.)
J. Malzan, Ph.D. (Toronto)
E.M. Moskal, B.A. (Toronto), Ph.D. (Illinois)
D. Mowat, Ph.D. (Waterloo)
K.A. Rowe, B.A. (Toronto), M.S. (Wisconsin), Ph.D. (fllinois)
Department of Statistics
D.A. Sprott, Ph.D. (Toronto)
Professor, Dean of the Faculty of W.F. Forbes, Ph.D., D.Sc. (London), D.C.
Mathematics
Professor, Associate Chairman of J.G. Kalbfleisch, B.Sc. (Toronto), Ph.D. (Waterloo)
the Department Lecturer, Administrative Assistant
Professors (Economics and Statistics)
(Philosophy and Statistics) (Chemical Engineering and Statistics)
(Economics and Management Science and Statistics) (Psychology and Statistics)
Associate Professors
C. Springer, M.Sc. (McGill)
S.N. Afriat, M.A. (Cambridge), Ph.D. (Oxford)
V.P. Godambe, M.Sc. (Bombay), Ph.D. (London)
J.S. Minas, B.A. (Wayne), Ph.D. (Illinois)
P.M. Reilly, B.A.Sc. (Toronto), D.I.C., Ph.D. (London), FSS. G. Tintner, PAD. (Vienna)
M.D. Vogel-Sprott, B.A. (McMaster), Ph.D. (Toronto)
M.A. Bennett, B.A. (Nottingham), F.S.A. F.C.ILA.
W.H. Cherry, Ph.D. (Melbourne) K.R. Shah. MA fRomhav) Ph Ondian Ctasictinn] Tuntitata)
(Psychology and Statistics) (part-time)
Assistant Professors
(Psychology and Statistics)
(part-time) (part-time)
Lecturer Instructors (A.A. and C.S., and Statistics)
Postdoctoral Fellows
Adjunct Professors
Adjunct Associate Professor
Faculty of Mathematics 425
R.V. Thysell, B.S. (Montana), Ph.D. (Iowa) J.C. Young, B.A.Se. (Toronto), M.Sc. (Waterloo), Ph.D. (Edinburgh) G.W. Bennett, PA.D. (Adelaide)
L. Billard, Ph.D. (New South Wales)
J.F. Lawless, Ph.D. (Waterloo)
R.L. Prentice, B.Sc. (Waterloo), Ph.D. (Toronto)
W.S. Rickert, Ph.D. (Waterloo)
J.C. Robinson, M.A.Sc., P.Eng., Ph.D. (Waterloo)
V. Taht, 4.4. (Toronto), A.S.A.
T.G. Waller, M.S. (Southern Mississippi), Ph.D. (Vanderbilt) J.B. Whitney, M.A. (Western), Ph.D. (Toronto)
R.C. Frecker, B.Sc. (Memorial), M.D. (Dalhousie)
M.E. Thompson (Mrs.), B.Sc. (Toronto), Ph.D. (Hlinois)
F. Reynolds, M.Sc. (Manitoba), F.S.A., F.CLA. R.L. Brown, B. Math. (Waterloo) J.F. Gentleman (Mrs.), B.S. Math., M.S.Stat. (Chicago)
N.T. Ison, M.A. (University of the Philipines), Ph.D. (Kansas State) J. Jackson, M.A. (Cambridge), M.B., B.Chir. (Cambridge)
H. Moraal, Nat Phil.Cand., Doctorandus (Leiden), Ph.D. (Waterloo) J.R.S. Wallis (Mrs.), B.Sc. (University of N.S.W.), Ph.D. (La Trobe University, Bundoora, Australia)
P. Robinson, Dip. Math. Stat. (Cantab.), Ph.D. (Cape Town) M. Zelen, B.S. (City College, New York), M.A. (North Carolina), Ph.D. (American University, Washington, D.C.)
J.G. Spiro, P.Eng. (Budapest), M.Sc. (Carnegie-Mellon University), Ph.D. (McGill)
Name A.P, Anisimov
M. Barner
P.L. Bhatnagar L. Billard
P. Erdés
JJ. Florentin P.W. Kasteleyn W. Laidlaw HT. Loh
R. Rado
BH. Worsley
Note
12 Calculus 1
21 Algebra and Vector Geometry
Faculty of Mathematics
Visiting Professors with Faculty of Mathematics 1971-72
On Leave From
U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. Mathematisches Forschungsinstitut, Oberwolfach.
West Germany
Himachel Pradesh University, Simla, India
University of New South Wales Kensington, N.S.W., Australia Math. Inst. of the
Hungarian Academy of Sciences, Budapest, Hungary Imperial College.
London, England Instituut-Lorenz.
Leiden, The Netherlands University of Calgary, Calgary, Alberta
University of Malaya.
Kuala Lampur, Malaysia University of Reading, Reading, Berkshire, England Queen’s University.
Kingston, Ontario
Department Statistics
Pure Mathematics
Applied Mathematics Statistics
Combinatorics and Optimization
Applied Analysis and Computer Science Combinatorics and Optimization Applied Mathematics
Combinatorics and Optimization Combinatorics and Optimization Applied Analysis and Computer Science
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.
2 hours lectures, 14 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. Row-space, column-space, null-space, rank. Determinants. Linear transformations. Eigenvalues and Eigenvectors. Functions of complex variables.
2 hours lectures, 144 hours problems.
22 Calculus 2 (for Chemical Engineers)
23 Introduction to Computer Programming
25 Calculus 2 (for Electrical Engineers)
31 Differential Equations
32 Numerical Analysis
33 Differential Calculus
34 Integral Calculus
35 Advanced Caiculus (for Electrical Engineers)
41 Applied Analysis
Faculty of Mathematics 427
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.
Concept and properties of an algorithm, language and notation for describing algorithms. An analysis of compututional problems and development of algorithms for their solution. An introduction to the Fortran 4 programming language for use in implementing algorithms on the computer.
2 hours lectures, 1 hour problems, one term.
No prerequisite.
Differential calculus of fuctions of several variables. Differential Equations. Multiple integrals.
2 hours lectures, 2 hours tutorial; one term.
Prerequisite: Math 12 or equivalent.
Standard methods of solution. Applications to physical engineering problems. Linear equations with constant coeficients. 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 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.
Prerequisite: Mathematics 23 or equivalent.
2 hours lectures, 2 hours problems, one term.
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.
Prerequisite: Math 22, 31.
3 lectures, one term.
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.
Prerequisites: Math. 21, 31.
3 lectures, one term.
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.
428 § 44 Complex Variable 51 Probability and Statistics Note 1 Note 2 81 An Introduction to Functions, Analytic Geometry, and Calculus 82 Introduction to Algebra
Faculty of Mathematics
Cauchy-Riemann equation, the Cauchy intergral 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.
Polynomial functions. Inverse of a function. Rational functions. Exponential functions with particular emphasis on logarithms. Trigonometric functions and identities. The straight line. Conic sections. problems solved by analytic means. Simple transformations. The limit concept and an introduction to Calculus including the derivative and its application in graph sketching. 2 hours lectures.
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.
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 on 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.
biology and economics; simpler parts of vector and matrix theory, ideas of Stochastic process and Markov chains with application. Prerequisite: Grade 13 Mathematics.
Faculty of Mathematics
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, I hour tutorial.
119 is not a course for Honours Mathematics students.
Funcations 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, I hour tutorial.
120 is not a course for Honours Mathematics students.
A conventional course for freshmen. Calculus but with more time spent on concepts of algebra and analytic geometry intimately connected with the Calculus. 4 hours lectures.
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 functions. 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 algorithms, stored-programme computers and programming languages. Concept and properties of an algorithm, language and notation for describing algorithms. Analysis of computational problems and development of algorithms for their solution. A procedure-oriented language (FORTRAN IV) and machine and assembly languages are used to implement algorithms on the computer. Introductory ideas from Boolean Algebra switching theory and hardware design. 3 hours lectures, 2 hour problems. Prerequisite; Grade 13 Mathematics recommended.
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. Prerequisite: M. 130, Phys. 131 or equivalents.
201 may not be counted as Mathematics credit but may be counted as an elective credit.
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.
Differential calculus of functions of several variables. Multiple integrals. Line and Surface integrals; Green’s Theorem, Divergence Theorem, Stokes’ Theorem. Inifinte Series. 3 hours lectures.
is not a course for Honours Mathematics students. 217
Vector spaces, linear maps. Inner products. Bilinear and quadratic forms. Matrix theory. 3 hours lectures.
219 is not a course for Honours Mathematics students.
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.
Vector spaces, linear maps. Inner products. Bilinear and quadratic forms. Matrix theory. 3 hours lectures. Prerequisite: 129 or 131.
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; exact and approximate tests of significance, tests of fit, contingency tables. 3 hours lectures. Prerequisite: 130, 132 (may be taken concurrently).
Faculty of Mathematics
First order equations; second order equations with constant coefficients. Applications to problems in the physical sciences. Solution of differential equations by series. 2 hours lectures, one term. Prerequisite: 237 (may be taken concurrently).
Newtonian mechanics, central forces, accelerated reference frames. Constrained motion of a particle. Further methods of solution of the differential equations of particle mechanics. 2 hours lectures, one term. Prerequisites: First year Physics or consent of instructor. 31 or 234 (a) or 236.
The theory of rates of interest and discount, annuities and sinking funds; application to financial problems, including the determination of mortgage payments and the price and yield of bonds. 2 hours lectures, Fall Term.
Applications of probability to problems of life and death. The determination of single and annual premiums for assurances and annuities. Reserves. Company expenses and their incorporation into premium and cash value calculations. 2 hours lectures, Winter term.
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.
236 is not a course for Honours Mathematics Students.
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 130.
Existence and non-existence. Enumeration. Combinations and Permutations. Inclusion-exclusion principle. Linear equations with unit coefficients. Recursion. Generating functions. 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. Prerequisites: Ist year algebra (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: Ist year algebra (129); Note that 239a is Not a prerequisite but is highly recommended.
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. 2 hours lectures, one term. Prerequisites: 119 or 129, and 120 or 130, and 132. There is a special section for non-Mathematics students.
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; program as an object for manipulation; syntactic analysis, generation of object code - macros. 2 hours lectures, one term. Prerequisite: Mathematics 132 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. 2 hours lectures, 1 hour laboratory.
This course is not for Honours Mathematics students.
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.
Db Mathematical Discovery and Invention-B
Note
Faculty of Mathematics
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.
300a is Not a prerequisite to 300b.
See the preamble to Mathematics 201. Partial differential equations are applied to classical wave problems. 2 hours lectures. Fall term. or equivalent. Prerequisites: 201 and 237
301 may not be counted as a mathematics credit, but may be counted as elective credit.
Vector and tensor analysis are related to problems in continuum mechanics. 2 hours lectures. Winter term. Prerequisite: 201 and 237 or equivalent.
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 Biological 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. Two lectures per week for two terms.
A study of combinatorial properties of plane figures, with particular attention to convex 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. Prerequisites: First year calculus (130).
Metric properties of r", functions from R", to R; differentials; Riemann integrals; and one of a) and 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.
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.
Rings, integral domains, fields, groups. Quotient groups and quotient rings.
Emphasis will be on examples rather than proofs of theorems.
319 is not a course for Honours Mathematics students.
Relations and operations in sets. Rings, integral domains, fields, groups. Quotient structures. Linear maps of inner-product spaces. 2 hours lectures.
Projective spaces over fields, collineations and correlations, quadric curves and surfaces. References to non-euclidean geometries. 3 hours lectures, one term. Prerequisite: 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: 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", 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.
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
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. 2 hours lectures, 2 hours problems, one term. Prerequisite: 240 (a). Also recommended are 219 or 229 and 217 or 237.
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. Z hours lectures, 2 hours problems, one term. Prerequisite: 240 (A). Also recommended are 219 or 229 and 217 or 237.
A course in the calculus of finite differences, to include: summation, differences of zero, numerical integration, relation between integration and summation. 2 hours lectures.
Applications of finite differences to actuarial problems in graduation of Statistical tables. 2 hours lectures.
336 (a)
An advanced course on problems with single lives. 3 hours lectures, one term.
An advanced course on problems with multiple lives; population and multiple decrement theory. 3 hours lectures, one term.
Derivation and applications of statistical theory for normally distributed measurements, Part (a) may be taken separately for half credit.
Continuous random variables; moments and moment generating function; distribution of t, chi-squared, and F, and their applications. Simple analysis of variance and regression. 2 hours lectures. Prerequisites: 233, 237.
The multivariate norma! distribution, its properties and uses; quadratic forms and Cochran’s theorem; regression and auto-regression with K independent variables. 2 hours lectures. Prerequisites: 229, 338(a).
Probability distributions, joint variation, central limit theorem. Introduction to continuous time stochastic processes. Poisson process and application to simple congestion situations. 2 hours lectures, Fall or Spring term. Prerequisite: 233 or 243.
Discrete probability models. Random walks. Introduction to theory of Markov chains. Applications to biological and social processes. 2 hours lectures, Winter term. Prerequisite: 233 or 243.
Representation of numbers and strings. Addressing schemes. Compound data structures. Programme structure. 3 hours lectures, 2 hours problems, one term. Prerequisite: 240 (6).
Searching and sorting. Run time representation of a high level language. Dynamic storage management. Text Manipulation. 3 hours lectures, 2 hours problems, one term. Prerequisite: 240 (b).
Fundamentals of group, ring, and field theory. 3 hours lectures. Prerequisite: 229.
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: 237.
Analysis of complex numbers: fundamental theorems of holomorphic functions; meromorphic functions. 2 hours lectures. Prerequisite: 237.
Intuitive set theory, metric spaces, point set topology. 3 hours lectures, one term. Consent of instructor.
Elementary introduction to cardinal and ordinal numbers; convexity and its applications to optimization; introduction to the main concepts of projective geometry, metric and topological spaces. 2 hours lectures.
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. Prerequisites: No formal prerequisites, 239 recommended.
Faculty of Mathematics
Applications of graph theory to communication networks, timetabling, ranking, and other combinatorial problems. This course supplements the material covered in Math 352, but with greater emphasis on combinatorial aspects. 2 hours lectures, one term. Prerequisites: 35la.
Planar graphs, Euler’s formula, Kuratowski’s theorem, five colour theorem; orientable and non-orientable surfaces, embedding of graphs, genus, crossing number, Heawood’s conjecture. For students interested in theoretical aspects of combinatorics and graph theory. 2 hours lectures, one term. (not offered in 1972-73). Prerequisites: 351a.
Latin squares, finite fields, orthogonal Latin squares; balanced incompleted block designs, difference sets, finite geometries, steiner triple systems; Coding theory, linear codes, hamming codes. Recommended for students interested in applications of algebra and statistical theory of design and analysis of experiments, and for those wishing background for advanced courses in this area of combinatories. 2 hours lectures, one term. Prerequisites: No formal prerequisite, 239 recommended.
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: Linear algebra (Math 219/229) or consent of instructor.
Introductory study of optimization techniques complementary to those covered in 352a. 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: Differential calculus (Math 217/237), some knowledge of probality and statistics, or consent of instructor.
352a is Not a prerequisite for 352b.
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: 237, Consent of instructor.
Hamilton’s principle. Lagrangian and canonical Equations of motion. Relativistic mechanics. 2 hours lectures, 1 hour tutorial. Winter Term. Prerequisites: 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, Fall term. Prerequisites: 234, 237 or consent of instructor.
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, Winter term. Prerequisites: 234, 237 or consent of instructor.
Existence of solutions. Second-order linear equations. Green’s functions. Series solutions. Laplace transforms. Sturm-Liouville boundary-value problems. Fourier series. Partial differential equations. 2 hours lectures. Prerequisites: 234, 237 or equivalent 332 (real variables) or equivalent may be taken concurrently.
Two hours lectures.
)1 Applications of Mathematics to the Physical Sciences
Note
Faculty of Mathematics
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.
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.
Continuation of linear algebra. Main topics; representations of endomorphisms, structures of bilinear forms, multilinear products. 2 hours lectures. Prerequisite: 239 or 341.
An introduction to the theory of algebraic varieties. Special topics such as the Theorem of Riemann-Roch. 2 hours lectures. Prerequisite: 329 or 341.
First order languages and theories. A treatment of at least one of the following: set theory, model theory, undecidability. 2 hours lectures.
An axiomatic treatment of geometry. Geometrical structures, such as projective planes and inversive planes. 2 hours lectures.
Continuation of the theory of rings and modules. 2 hours lectures. Prerequisite: 329 or 341.
Continuation of group theory. 2 hours lectures. Corequisite: 329 or 341.
Field extensions and Galois theory. 2 hours lectures. Prerequisite: 329 or 341.
Riccati equation, conflict and pursuit problems, elliptic integrals and functions, phase plane, stability, period solutions, non-linear mechanics. Non-linear integral equations. 2 hours lectures. Prerequisites: 363 (may be taken concurrently), Consent of instructor.
Not offered in 1972–73.
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. 329-algebra.
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. 329-Algebra, 418a is a prerequisite for 418b.
Not offered in 1972–73.
Not offered in 1972–73.
Multiplicative-algebraic theory of numbers. Foundations of natural number theory. Elements of additive-combinatorial number theory. 2 hours lectures. Corequisite: 329 or 341.
Continuation of general topology; selected topics from other branches of topology. 2 hours lectures. Prerequisite: 344.
Ordered sets, lattices, and Galois connections. Applications in algebra, geometry and logic. 2 hours lectures. Consent of instructor.
Homology theory of complexes. Theorems of invariance, covering, and duality. 2 hours lectures.
Finite incidence structures. Designs. Finite projective and affine planes. 2 hours lectures. Corequisite: 330 (a) (Projective Geometry).
SY Faculty of Mathematics
Further properties of holomorphic and meromorphic functions. Riemann surfaces. 2 hours lectures. Prerequisite: 332 (b) or 343.
An introduction to integration and measure theory with emphasis on the real line. 2 hours lectures. Prerequisite: 332 (a) or 342.
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: 363 or equivalent, or consent of instructor.
Numerical problems arising in actuarial science and statistics. 2 hours laboratory.
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.
Individual and collective risk theory. Ruin theory. Stop loss reinsurance premiums, Monte Carlo methods. 2 hours lectures, one term.
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 sample theory. 3 hours lectures. Prerequisite: 338 (a).
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: 338A or consent of instructor.
Schrodinger equation. Solution of one dimensional problems, computation 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.
Flat space-time and special relativity, curved spacetime 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: 360(a) Consent of instructor.
Applications of the Maxwell equations. Reflection and refraction. Introduction to wave guides and antennae. 2 hours lectures, Fall term. 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: 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.
Viscosity, compressibility, vorticity, circulation; Stokes and Kelvin’s theorems, boundary conditions; two and three dimensional irrotational, incompressible flow, elements of compressible flow, Navier-Stokes equations. 2 hours lectures, 1 hour tutorial, Winter term. Prerequisite: 445 (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; taken from the work of such men as 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 (e.g. the material contained in “‘A Modern Geometry For High Schools” by Lougheed and Workman).
Faculty of Mathematics
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.
Applications of probability theory to theoretical Physics. 2 lectures, Fall term.
Differentiable manifolds, Tensors and Forms, Connexions, Riemannian manifolds. 2 hours lectures. Prerequisite: 229 and 237.
Similar to Mathematics 439 but with more accent on the logic and methods than on the mathematics. 2 hours lectures.
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)
A study of recent algorithms for minimizing nonlinear differentiable functions subject to nonlinear constraints. Extentions of classical calculus methods to semidefinite quadratic forms. Unconstrained univariate optimization with examples of the four basic approaches. Unconstrained multivariate optimization including first and second order Newton methods, quasi-Newton methods, and conjugate gradient directions. Constrained optimization including first and second order Newton methods, quasi-Newton methods, and conjugate gradient directions. Constrained optimization including nonlinear transformations, feasible direction methods, projection algorithms and penalty functions. Recommended for graduate level nonlinear programming. 2 hours lectures, one term. Prerequisites: Differential Calculus (237); Elementary Fortran.
Faculty of Mathematics
A study of recent mathematical programming algorithms that do not require differentiability. Unicyclic methods such as Simplex, and Complex; Simple Cyclic algorithms; conjugate directions of Powell; pattern search; lattice minimization with constraints, difference approximations to differentiable methods. Applications of optimization to linear and nonlinear least squares problems. Special topics and extensions. Recommended for graduate level nonlinear programming. 2 hours lectures. Prerequisites: Differential Calculus (237); Elementary Fortran.
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. Trans-shipment 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. Prerequisites: 229 (or equivalent). Students will be required to use the computer on occasion and 132 (or its equivalent) would be a definite asset.
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 453b. Recommended for operations research and computer sicence students. 2 hours lectures, one term. Prerequisites: Introductory Probability.
Faculty of Mathematics
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. Prerequisites: Math 453a.
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 (Math 229); fundamentals of linear programming and probability theory.
454b Selected Topics in Graph Theory
(Not offered in 1972-73)
155 Mathematical Programming
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: Math 352a.
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. Prerequisites: Math 455a.
456a Combinatorial Systems
(Not offered in 1972-73)
456b Boolean Methods
(Not offered in 1972-73)
A selection of combinatorial problems from the applied sciences. Topics 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: Math 35la and Math 233.
Not offered in 1972–73.
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. Prerequisites: Math 35la or consent of instructor.
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. Prerequisites: Math 352a or consent of instructor.
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. Prerequisites: Math 459a 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 Math 785 ‘Planar graphs”. 2 hours lectures, one term. Prerequisites: A knowledge of elementary group theory (Math 329).
Faculty of Mathematics
The existence and construction of error correcting codes, projective geometrics, orthogonal Latin squares, balanced incomplete block designs and other combinatorial configurations. Recommended for Math 780, Combinatorial Analysis. Prerequisites: Theory of finite groups rings and fields (Math 329).
Mathematical applications in demography. Accidental death and disability benefits. Pension fund mathematics. 2 hours lectures, one term. Prerequisite or Corequisite: M336.
A tutorial course for the advanced actuarial student. 2 hours lectures, one term.
The theory of measure and the Lebesgue integral. 2 hours lectures, Fall term.
Linear operators in Hilbert spaces. Compact operators. Introduction to functional analysis. 2 hours lectures, Winter term.
Dynamical systems, controllability and observability. Minimization of functions and functionals. Optimal control. Pontryagin’s Maximum Principle. 2 hours lectures, Fall term. Consent of instructor.
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. Consent of instructor.
Summary Statistics (mean, median, midmean, hinge, trimean, etc.). Transformation of data (to increase symmetry, remove strays, equalize spread). Residuals (from various measures of centrality, plottings, etc.). Outliers (effect on summary statistics and residuals, identification and removal). Plotting techniques (dot plots, histograms, stem-and-leaf box plots, schematic plots, probability plots —- normal, half-normal, other — Q-Q plots, rootagrams, etc.). Prerequisite: (1) A course in Fortran Programming or equivalent (2) Mathematics 338.
Existence and uniqueness of solutions, one step methods, multistep methods, stability, error analysis. Shooting and discretizations methods, implementation problems especially for non-linear equations. Correspondence to ODE initial value and boundary value problems, solution techniques. 2 hours lectures, one term. Prerequisite: M334 (6) or consent of instructor.
Discretization methods for PDE’s, convergence and stability, grid and characteristic methods for hyperbolic equations; iterative methods of solution for parabolic and elliptic equations, rate of convergence estimates. 2 hours lectures, one term. Prerequisite: Math. 334 (a) or consent of instructor.
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. 2 hours lectures, Fall term.
Part J. 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. 2 hours lectures, Winter term. Prerequisite: M 471 (a) or consent of instructor.
Sequential machine models. Reduction of completely and incompletely specified machines. Decomposition of machines. Nondeterministic machines. Finite automata and regular events. Multitape-finite-state non-writing automata. 2 hours lectures, Fall term. Prerequisite: 132 and 329, or consent of instructor.
Formalisms 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. 2 hours lectures, Winter term. Prerequisite: 436 or 472 (a) or consent of instructor.
Mathematics 340(a) and Mathematics 340(b) will be available under the numbers Mathematics 473(a) and Mathematics 473(b) for students who started Mathematics 340 before September, 1971. (i.e. had the earlier version of this course).
Faculty of Mathematics
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.
This course will be offered on an experimental basis during the Fall/1972 and Winter/1973 academic terms. Enrolment will normally be restricted to students in fourth year. Topics available will include: 1) Advanced Applications Programming 2) Advanced Systems Programming 3) Systems Analysis 4) Simulation
Department of Mechanical Engineering
Professor, Chairman of the E. Brundrett, B.S.A. (O.4.C), B.A.Sc., M.A.Sc., Ph.D. (Toronto) Department Professor, Associate Dean T.A. Brzustowski, B.A.Sc. (Toronto), A.M., Ph.D. (Toronto) Engineering Graduate Studies Professor, Associate Chairman D.J. Burns, B.Sc., Ph.D. (Bristol) Graduate Studies Professor, Associate Dean E.L. Holmes, B.Sc. (Bristol), M.A.Sc., Ph.D. (London) Engineering Undergraduate Studies Professor, Associate Chairman G.F. Pearce, B.A.Sc. (British Columbia), M.A.Sc. (Toronto) Undergraduate Studies
Professors S.A. Alpay, Dipl. Ing., Dr. Ing. (Berlin) G.T. Csanady, Dipl Ing. (Munich), Ph.D. (New South Wales) M.B. Danard, B.A.Sc. (British Columbia), M.A. (Toronto), Ph.D. (Chicago) H.L. Evans, B.Sc., M.Sc. (Wales), D.L.C.. Ph.D. (London) J.H.G. Howard, B.Sc. (Queen’s), M.Sc., Ph.D. (Birmingham) W.B. Nicoll, SM. (M.L.T.), Engineer (Stanford), Ph.D. (London) P. Niessen, B.Sc. (McMaster), M.A.Sc., Ph.D. (Toronto) M.M. Yovanovich, B.Sc. (Queens), M.S. (Buf), ME, Sc.D. (M.IT.)
Associate Professors G.M. Bragg, B.S.Sc. (Toronto), Ph.D. (Cambridge) R.N. Dubey, B.Sc. (Hons) (Patna), B.Sc. (Eng.) (Bihar), Ph.D. (Waterloo) D. French, B.Sc, C. Eng. A.M. Hale, B.Sc., M.A. (New Brunswick), B.A.Sc. (Toronto), M.A.Sc., Ph.D. (Waterloo) K.G.T. Hollands, B.A.Sc. (Toronto), Ph.D. (McGill) C.E. Hermance, BE. (Yale), M.A.A., M.S.E., Ph.D. (Princeton) H.W. Kerr, B.A.Sc., Ph.D. (Toronto) W.M. Mansour, B.Sc. (Cairo), M.A.Sc., Ph.D. (Toronto) H.R. Martin, B.Sc., M.Sc. (Queen’s Belfast), Ph.D. (Nottingham) K.R. Piekarski, Dipl. Ing. (London), Ph.D. (Cambridge) A. Plumtree, B.Sc, Ph.D. (Nottingham) G.D. Raithby, B.E.Sc., ME.Sc. (Western), Ph.D. (Minnesota) RF. Scrutton, B.Sc, M.Sc. (Melbourne) P.R. Slawson, B.A.Sc., M.A.Sc., Ph.D. (Waterloo) A.B. Strong, B.A.Sce. (Waterloo), M.Sc. (Imperial College, London), Ph.D. (Waterloo) T.M.L. Wigley, B.Sc. B.Sc. (Hons. Math. & Phys.), Ph.D. (Adelaide)
Assistant Professors K.G. Adams, B.Sc. (Queen's), M.A.Sc., Ph.D. (Waterloo)
G.C. Andrews, B.A.Sc., M.A.Sc. (British Columbia), Ph.D. (Waterloo) T.A. Ledwell, B.Eng., M.Eng. (Nova Scotia), Ph.D. (Waterloo) W.K. Luk, DIC. (London), (Dipl. of Hong Kong Tech. College), Ph.D. (Waterloo) J. Moss, B.Sc. (Western), Ph.D. (McMaster) R.J. Pick, B.A.Sc. (British Columbia), M.S. (Imperial College, London), Ph.D. (Waterloo) HF. Sullivan, B.A.Sc. (Waterloo), A.M., Ph.D. (Princeton)
Lecturers J.D. Malcolm, B.A.Sc., M.A.Sc. (Waterloo)
Special Lecturers C.J. Beingessner, B.Sc., M.A.Sc., Ph.D. (Toronto) R.G.R. Lawrence, 2.C. U.H. Mohaupt, B.A.Sc. (Waterloo)
Adjunct Professors
Adjunct Research Professor
Laboratory Director
1 Advanced Calculus
2 Statistics for Engineers
Ordinary Differential Equations
4 Numerical Analysis
10 System Dynamics
Department of Mechanical Engineering 451
D.R. Hopkins, M.C. (Toronto), F.R.C.P. (C), F.A.C.C.P. D.R. McTavish, M.D., FR.CS. (C), FRSC. (C)
A.M. Wiley, B.A., M.B., B.Sch.B.A.O., FRCS (England), M. Ch. FRCS (Canada)
J.R. Cook, B.Sc. (Mt. Allison)
Undergraduate Programmes
Details of the undergraduate programme in Mechanical Engineering are to be found on page 140. 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 142. 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 numerical 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.
Generalized and unified dynamic concepts, physical laws governing dynamic systems. Laplace transform review and the transfer functions. Mathematical models and block diagrams for physical systems. Dynamics of first and second order systems due to deterministic inputs. Analogies among mechanical, electrical, fluid and thermal systems. Linear flow graph and Mason’s rule. Analog computer simulation. Stability from the pole zero configuration.
12 Dynamics
15 Structure and Properties of
Materials 1
16 Modern Physics
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
32 Physical Metallurgy 2
33 Experimental Materials Science
35 Industrial Metallurgy
Department of Mechanical Engineering
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, structural defects, phase equilibria, diffusion, non-equilibria, transformations, corrosion.
Introduction to relativistic mechanics. Fundamentals of quantum mechanics based on Schroedinger equation. Applications to the hydrogen atom and atomic systems. Concepts of nuclear physics and their applications.
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. Principles of lubrication.
Elasticity, inelasticity, plasticity, fracture, creep, fatigue, strengthening mechanisms, electrical, magnetic and optical properties.
Elasticity and plasticity of metals. Recovery, recrystallization and growth. Hardening mechanisms. Heat treatment of metals and alloys. Shear transformations. Failure in metals.
This course is designed to acquaint students with experimental techniques in Materials Science. For the initial part, the students will be required to perform assigned experiments which demonstrate common physical phenomena. For the remainder of the course, each student will work on a small experimental project.
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.
41 Manufacturing Science 1
42 Manufacturing Science 3
43 Manufacturing Science 4
45 Manufacturing Science 6
46 Manufacturing Science 7
47 Analysis and Design of
Manufacturing Systems I
48 Analysis and Design of Manufacturing Systems 2
49 Metrology
50 Thermodynamics
Department of Mechanical Engineering 453
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; thermal stability, pressure pulse effects. Other casting methods.
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.
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 the 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 effects 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.
51 Fluid Mechanics 1
52 Air Conditioning
53 Heat Transfer 1
54 Thermodynamics 2
55 Refrigeration Engineering
56 Heat Transfer 2
58 Internal Combustion Engines
59 Energy Conversion
60 Control Theory 1
62 Fluid Mechanics 2