122 courses in the 1971–72 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 1971–72
Associate Professor
Associate Professor
Associate Professor
Associate Professor
Associate Chairman of the Department
Faculty listed in 1971–72
Associate Professor
Associate Professor of Foundations of Mathematics
Assistant Professor
Assistant Professor
Assistant Professor
Faculty of Mathematics 337
Faculty listed in 1971–72
K.D. Fryer, B.A. (Western), Ph.D. (Toronto)
R.C. Mullin, B.A. (Western), Ph.D. (Waterloo)
C.St.J.A. Nash-Williams, Ph.D. (Cantab.), F RSE.
Faculty listed in 1971–72
R.C. Read, M.A. (Cambridge), Ph.D. (London)
C.E. Haff, B.S. (Stanford), Ph.D. (Waterloo)
D.H. Younger, Ph.D. (Columbia)
J.A. Bondy, D. Phil. (Oxon.)
Faculty listed in 1971–72
R.N. Burns, B.Sc. (Toronto,) Ph.D. (Waterloo)
R. P. Gupta, M.Sc. (AGRA University), Ph.D. (Inoian Statistical Institute)
R.A. Honsberger, 8.4. (Toronto), M.A. (Waterloo)
Faculty listed in 1971–72
U.S.R. Murty, M.A. (Osmania), Ph.D. (Indian Statistical Institue) E. Anderson, B.A. (McMaster)
V. Taht, 44.4. (Toronto) J. Holt, M4. (Toronto), Ph.D. (Waterloo) J.E. Gentleman (Mrs.), B.S. Math. M.S. Stat. (Chicago)
Faculty listed in 1971–72
E. Haag, B.Sc. (Queen’s ), M.A. (Waterloo)
E. Paldus, M.U.Dr. (Charles University, Prague)
P. Robinson, Dipl Math. Stat. (Cantab.), Ph.D. (Cape Town)
Faculty listed in 1971–72
M. Zelen, B.S. (City College, New York), M.A. (North Carolina), Ph.D. (American University, Washington, D. C.)
Name P. Erdds
J.J. Florentin
D.R. Fulkerson
fF. Harary
E.L. Johnson A. 0. Kaires
S. Kurepa
W. Leissner
A. Lundberg
Hi. Schneeweiss
L. Vineze
Faculty of Mathematics
339
Visiting Professors with Faculty of Mathematics
(1970-1971)
On Leave From Mathematical Institute of the Hungarian Academy of Sciences.
and Division of
Budapest, Hungary
Imperial College, London, England
Rand Corporation, Santa Monica, California
University of Michigan, Ann Arbor, Michigan
IBM Research.
York Town Heights, New York Institute of Mathematics, Braunschweig W. Germany
University of Zagreb, Zagreb, Yugoslavia
Ruhr University, Bochum, West Germany
University of Stockholm, Stockholm, Sweden
University of Saarbriicken, Saarbriicken, West Germany
Mathematical Institute of the Hungarian Academy of Sciences.
Budapest, Hungary
Division of Mathematical Research
The Division of Mathematical Research is a trans-departmental
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Department Combinatorics and Optimization.
Mathematical Research Applied Analysis and Computer Science, and Division of Mathematical Research Combinatorics and Optimization, and Division of Mathematical Research Combinatics and Optimization and Division of Mathematical Research Combinatorics and Optimization Division of Mathematical Research Statistics, and Division of Mathematical Research
Pure Mathematics,and Division of Mathematical Research Division of Mathematical Research Statistics, and Division of Mathematical Research (Statistics and Division of Mathematical Research)
Note
12 Calculus 1
11 Algebra and Vector Geometry
22 Calculus 2
23 Introduction to Computer Programming
Faculty of Mathematics
related activities. It arranges a regular programme of Colloquium lectures and research seminars, and brings mathematicians from other academic institutions in many parts of the world to the University as Visiting Professors. The Division has an organizing Committee of senior Faculty members, and Research Sections concerned with certain particular mathematical fields. At present there are six such Research Sections, namely, Theory of Computing, Combinatorics, Functional Equations, Statistics and Biometry, Differential Geometry and Relativity, Quantum Theory. Each of these holds research seminars weekly. These Research Sections cover only a relatively small part of the range of the Division’s research interests, which extend widely over the field of Mathematics.
The following Research Seminars of the Division of Mathematical Research should be borne in mind by graduate students in shaping their schedules.
Research Seminar in Theory of Computing
Research Seminar in Functional Equations
Research Seminar in Combinatorics
Research Seminar in Statistics and Biometry
Research Seminar in Differential Geometry and Relativity
Research Seminar in Quantum Theory
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 equation with applications. Students will work selected problems under supervision.
2 hours lectures, I’ hours problems.
Mathematical induction, Determinants, vectors and solid geometry, matrices, complex numbers, polynomial equations, linear transformations, infinite series.
2 hours lectures, 1'fz 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.
Concept and properties of an algorithm, language and notation for describing algorithms. An analysis of computational problems amd 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.
31 Differential Equation
32 Numerical Analysis
33 Differential Calculus
34 Integral Calculus
35 Advanced Calculus (for
Electrical Engineers)
41 Applied Analysis
44 Complex Variable
51 Probability and Statistics
Note 1
Note 2
Faculty of Mathematics 341
Standard methods of solution. Applications to physical engineering problems. Linear equations with constant coeffcients. Systems of differential equations. Solution of differential equations by theLaplace 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.
Vector differential Calculus—gradient, divergence, curl. Vector integral Calculus—tine integrals and surface integrals. Green’s theorem. Partial differential equations. Orthogonal funtions.
2 hours lectures, 2 hours problems.
Partial differentiation. Laplace transforms, special functions. Fourier series, second order differential equations. 3 lectures, one term.
Cauchy-Riemann equations, 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, qualtiy 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.
81 An Introduction to Functions, Analytic Geometry, and Calculus
82 Introduction to Algebra
Faculty of Mathematics
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 mathematics course in the liberal arts tradition. A selection of mathematical topics in a context of history, philosophy and applications. The growth of the number system: rational, real, complex, hyper-complex and infinite numbers, vectors and matrices; paradoxes and applications. The limit concept, including an introduction to Calculus; Newton and the influence of Newtonian physics. Abstraction, symbolic logic and recent trends in mathematics education. Little is assumed in the way of previous knowledge, but the student should have some affinity for mathematics. The course is library-oriented rather than textbook-oriented and the student is encouraged to find and pursue some interest of his own, subject to approval. 3 lectures
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.
M 119 is not a course for Honours Mathematics students. 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.
MATH 125Algebra, Geometry, and Calculus for Optometry Students
A conventional course for freshman Calculus but with more time spent on concepts of algebra and analytic geometry intimately connected with the Calculus. 4 hours/week.
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.
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 1V) and machine and assembly languages are used to implement algorithms on the computer. Introductory ideas from Boolean Algebra switching theory and hardware design. 2 hours lectures, 2 hours problems Prerequisite: Grade 13 Mathematics recommended
MATH 201(a) 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, Prerequisite: M. 130, Phys. 131 or equivalents.
Math. 201, 301, 401, 402 may not be counted as Mathematics credits but may be counted as elective credits.
MATH 201(b) Applications of Mathematics to the Physical Sciences
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. 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.
An introduction to probability theory and statistics, with emphasis on their applications to practical problems, and on the logical principles involved. Several of the 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: M 130, M 132
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. Requisite: M237 (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. M 234 (a) or M 236 or M 31
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, 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 parial differential equations. 2 lectures.
M 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: M 130
Existence and non-existence. Enumeration. Combinations and Permutations. Inclusion-exclusion principle. Linear equations with unit coefficients. Recursion. Generating functions. 2 hours lectures.
MATH 239(b) An Introduction to Optimization 0 (a) Numerical Applications in Computer Science 0 (b) Non-Numeric Applications in Computer Science
Convex sets. Linear Programming. Pre-PERT. Ranking. Applications of Boolean Algebra to Combinatorial Circuits. Probability; Markov chains; random walks. Theory of Games. 2 hours lectures.
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 intergration, matrix operations and solution of systems of linear equations, numerical solution of ordinary differential equations. 2 hours lectures, one term. Prerequisites: M 132, and M 120 or M 130, and M 119 or M 129 There is a special section for Non-Mathematics students. Prerequisites: Mathematics 130 Mathematics 132
Introduction to: Number Systems, symbol manipulation, Boolean Algebra, concept of a list and elementary list processing techniques, sorting and searching algorithms, basic techniques of compilation. 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.
The art technique of problem solving from the point of view of discovery and invention (heuristic); induction, analogy, pattern generalization, analysis, similarity and plausible reasoning. Problems will be selected from various areas of mathematics including Geometry (Euclid to Hilbert), Algebraic Structure (Groups in Algebra and Geometry), Set theory (point sets, Boolean Algebra) and Logic. 2 hours lectures.
MATH 301(a) Applications of Mathematics to the Physical Sciences
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.
Introductory problems and a brief survey; convex sets; covering problems; graphs; addition of figures; maxima and minima problems; curves of constant breadth 4- curves; equidecomposable figures. 2 hours lectures.
MATH 312(a) Elements of Real Variable Theory Note Note
Metric properties of IR", functions from IR, to IR; 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.
M 312 (a) is not a course for Honours Mathematics students.
MATH 312(b) An Introduction to Complex Variable Theory Note Note
Complex numbers; continuity, differentiability, analyticity of functions; the Cauchy-Riemann equations; solutions of Laplace’s equation; conformal mapping by elementary functions, and applications; the Cauchy and allied theorems; Taylor and Laurent expansions; uniform convergence and power series; the residue calculus, and applications. 3 hours lectures.
Emphasis will be on applications rather than theory.
M 312 (b) is not a course for Honours Mathematics students.
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 geometries. 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.
MATH 332(b) An Introduction to Complex Variable Theory
Complex numbers; continuity, differentiability, analyticity of functions; the Cauchy Riemann equations; solutions of Laplace’s equation; conformal mapping by elementary functions, and applications; the Cauchy and allied theorems; Taylor and Laurent expansions; uniform convergence and power series; the residue calculus, and applications. 3 hours lectures.
MATH 334(a) Numerical Algebra 34 (b) Numerical Approximation
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: M 240 (a). Also recommended are M 219 or 229 and M 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. 2 hours lectures, 2 hours problems, one term. Prerequisite: Math. 240 (a). Also recommended are M 219 or 229 and M or 237. 217
A course in the calculus of finite differences, to include: summation, differences of zero, numerical integration, relation between integration and summation. 2 hours lectures.
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; M 233, M 237.
The multivariate normal distribution, its properties and uses; quadratic forms and Cochran’s theorem; regression and auto-regression with k independent variables. 2 hours lectures. Denwaneiettac: AM 220 /-1 34 2990
A discussion of the hardware and software required in a computer system. Overlapped channels, interrupt facilities, memory protection, buffers, input-output control systems, macro-programming, monitor systems. Relocation schemes, multi-programming, multi-processing, dynamic memory allocation, time-sharing. Special purpose computer systems and simulation of computer systems. Peripheral equipment. Introduction to the theory of sequential machines and the logical design of computers. 2 hours lectures. Prerequisite: Math. 240 (b) or Consent of Instructor
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
MATH 345Topics in Pure Mathematics for Combinatorial Mathematicians
Elementary introduction to cardinal and ordinal numbers; convexity and its applications to optimization; introduction to the main concepts of projective geometry, metric and topological spaces. 2 hours lectures.
MATH 351Introduction to Graph Theory and Combinatorial Analysis (a)(b)
Graphs, connectivity, bipartite graphs, factors, Menger’s Theorem, directed graphs, planar graphs and duality, with applications.
Theory of enumeration, generating functions, Ramsey’s Theorem and applications, systems of distinct representatives, Latin Squares, and Balanced Incomplete Block Designs, with applications. 2 hours lectures.
Vector analysis im curvilinear coordinates; gradient, divergence, curl. Transformation laws of covariant, contravariant, relative tensors. Covariant and intrinsic differentiation. The Riemann-Christoffel tensors, Bianchi identities, Ricci and Einstein tensors. Integrability theorems. Applications to classical mechanics and elasticity. Introduction to general relativity. 9 houre lectures Fall term.
The metric tensor, and elementary differential geometry of surfaces, introduction to Riemannian geometry; Riemannian curvature, Gaussian curvature and Einstein curvature tensors. Spaces of affine connection and Finsler spaces with application to electromagnetic theory. 2 hours lectures, Winter term. Prerequisites: M 237, 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, Fall term. Prerequisites: M 237, 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: M 237, M 234 or equivalent M 332 (real variables) or equivalent (may be taken concurrently)
MATH 399Reading in Mathematics 01 Applications of Mathematics to the Physical Sciences
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, 2 terms. Prerequisite: Consent of Instructors.
MATH 402Topics in Mathematical Aspects of Chemistry, Biology, and the Medical Sciences
Subjects will be selected from areas such as introductory wave mechanics, spectroscopy, molecular biology, and mathematical models of disease processes. 2 hours lectures.
Continuation of linear algebra. Main topics: representations of endomorphisms, structures of bilinear forms, multilinear products. 2 hours lectures. Prerequisite: M 341 or M 239
An introduction to the theory of algebraic varieties. Special topics such as the Theorem of Riemann-Roch. 2 hours lectures. Prerequisite: M 341 or M 329
Riccati equation, conflict and pursuit problems, elliptic integrals and functions, phase plane, non-linear mechanics. Non-linear integral equations. 2 hours lectures. Prerequisites: M 363 (may be taken concurrently) Consent of Instructor
Algorithms and theory in the intersection of combinatorics and optimization. The Hitchcock problems. Optimum paths and flows in networks. Efficiency of algorithms. Convex polyhedra. Linear programming and max-min relations. Totally unimodular matrices. Elementary matroid theory, illustrated by algebraic dependence, forests, transversals, and geometry. Sub-modular functions, polymatroids, and the greedy algorithm. Boolean and Integer programming. The Knapsack problem. Matchings and constrained subgraphs. Branchings in networks. Matroid partitions and intersections. Bottleneck Extrema. 2 hours lectures.
A general survey of those problems in graph theory that are amenable to the use of computing techniques. Algorithms for testing graphs for given properties, e.g. planarity, isomorphism. Algorithms for constructing graphs with given properties, or finding all subgraphs of a given kind, e.g. spanning trees, circuits. Combinatorial algorithms, etc. The emphasis of the course will be on the algorithms, rather than the details of computation; but students will be expected to implement these algorithms on the computer. 2.hours lectures.
A course on Graph Theory with applications to psychology, electrical engineering, operational research, chemistry, and perhaps other topics. 4 hours/week
MATH 420Combinatorial Methods and Probability Models
Combinatorial techniques applied to a variety of optimization problems including inventory control, replacement, discrete dynamic programming, and scheduling. 2 hours lectures.
Multiplicative-algebraic theory of numbers. Foundations of natural number theory. Elements of additive-combinatorial number theory. 2 hours lectures. Corequisite: M341 or M 329
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
MATH 436Introduction to Logic 7 (a) Measurement of Mortality(b) Risk Theory
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, I 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. 2 hours lectures Prerequisite: M 338 (a)
The Schroedinger wave equation, interpretation of wave functions, harmonic oscillator in wave mechanics. The hydrogen atom. Perturbation theory. The variation methods. The W.K.B. approximation. 2 lectures. 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 lectures, Fall term. Consent of Instructor
Kinematics and dynamics of flow. Equations of motion. Inviscid two-dimensional motion. Applications of conformal transformation to fluid flow. Viscous motion and similarity. Oseen’s approximation. 2 hours lectures. Consent of Instructor
The development of Mathematics from ancient to modern times, including study of prominent mathematicians and their works. Problems will be worked using both modern and historical approaches. 2 lectures.
Similar to Mathematics 439 but with more accent on the logic and methods than on the mathematics. Primarily for students in the Sciences. 2 hours lectures.
Applied Mathematics for Science majors. Ordinary and partial differential equations. Laplace and Fourier transform. Special functions. Fourier series. Vector and matric methods. Calculus of variations. Other tonics in advanced cralenInc 2? haure lorturoc
MATH 451Non-Linear Optimization(a) Classical Methods(b) Sequential Optimization 2 Linear Programming: (Theory, Applications, and Related Topics)(a)(b)
Discrete functions; continuous functions of n variables. Quadratic forms; convex functions. Equality constraints; Langrange multipliers and generalizations. Inequality constraints: Kuhn-Tucker theorem. 2 hours lectures.
Gradient methods; recent sequential optimization methods; direct search methods; application to the solution of equation and non-linear regression analysis. 2 hours lectures.
Preliminaries, convex sets, geometric motivation for the Simplex Algorithm. Simplex Algorithm. Revised Simplex Algorithm. Degeneracy. Duality Theory. 2 hours lectures.
Input and Service-time distributions; Poisson process, Erlangian distribution. Single channel queue; differential-difference equations, integral equations. Complex queueing models. Monte Carlo techniques. 2 hours lectures.
MATH 456Combinatorial Methods in Operations Research(a) Combinatorial Systems(b) Boolean Methods 457Applied Combinatorial Mathematics
Representations by networks; simulation; recurrence relations; Markov processes, application to queueing theory, random walks, renewal theory, activity scheduling, etc. 2 hours lectures.
Boolean algebras; pseudo Boolean programming; applications to switching circuits; selection problems; graph theory; integer programming, etc. 2 hours lectures.
This is a relatively nonspecialized course on modern applied algebra. Emphasis is on the combinatorial foundations of applied mathematics. 2 hours lectures.
A general survey of a number of topics in graph theory. Likely topics include: Eulerian and Hamiltonian lines in finite and infinite graphs, Menger’s Theorem, trees, directed graphs, enumerative results in graph theory, reconstruction of graphs from subgraphs, colouring of graphs, graphs and the Four Colour Problem, planarity and thickness of graphs, graphs and matroids, applications of matrices to graph theory. 2 hours lectures.
The existence and construction of projective geometries, orthogonal Latin squares, room designs, balanced incomplete block designs and other combinatorial configurations. Results on the football pool problem. 2 hours lectures.
Mathematical applications in demography. Accidental death and disability benefits. Pension fund mathematics. 2 hours lectures, one term. Prerequisites: M 332 (complex variables) M 332 (real variables) or equivalents.
Dynamical systems, controlability 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
An introduction to the basic theory of stochastic process, with applications to biology and operations research. Definition of stochastic process, waiting times. Random walk, Poisson process, Markov chains, Brownian motion. Applications to queueing theory, storage problems, population growth models; Monte Carlo methods; stochastic optimization problems. 2 hours lectures, Fall and Winter terms. Prerequisite: Math. 233, 243, or 51.
MATH 470(a) Ordinary Differential Equations and Integral Equations Initial value problems: Boundary value problems: Integral equations:
existence and uniqueness of solutions, one step methods, multistep methods, stability, error analysis. shooting and discretization 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: M 334 (b) 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, Fail term. Prerequisites: M 132 and 329, or Consent of Instructor
MATH 471(b) Computer System Organization and Logic Design
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. 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: M 132 and 329, or Consent of Instructor
MATH 472BIntroduction to Turing Machines and Computability Theory
Formalisms of 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. 2 hours lectures, Winter term. Prerequisite: M 472 (a) or 436, or Consent of Instructor
MATH 473(a) Programming Languages and Data Structures 1
Storage systems and Structure. Allocation and Collection of Storage. Multilinked Structures. List Processing Languages. Formal Specification of Data Structures. Data Structures in Programming Languages. 2 hours lectures, one term. Prerequisite: Math. 240 (b) or Consent of Instructor.
MATH 473(b) Programming Language and Data Structures 2,
Formal Definition of Programming Languages. Syntax and Semantics. Parsing Techniques. Object Code Generation. Organization of a Compiler. Data Structures for Symbol Tables and Language Definition. 2 hours lectures, one term. Prerequisite: Math 473 (a) or Consent of Instructor.
Cauchy's, Pexider’s and similar equations. Equations for polynomials and for trigonometric functions. Reduction to differential equations. General methods and theorems. Iteration. Applications. Further topics, as Equations for functions of several variables, or Equations for analytic functions, or Equations on algebraic structures. 2 hours lectures.
The Faculty offers graduate work over a broad spectrum of the various branches of Mathematics and fields of special interest include: Abstract Algebra, Biometry, Calculus of Variations, Combinatorial Analysis, Computer Science, Control Theory, Differential Equations, Differential Geometry, Foundations, Functional Analysis, Functional Equations, Graph Theory, Harmonic Analysis, Information Theory and Nets, Mathematical Logic, Number Theory, Numerical Analysis, Optimization, Probability, Quantum Theory, Relativity Theory, Statistical Inference, Switching and Automata Theory. The University of Waterloo has at present five computer systems available to Mathematics faculty and graduate students. These are: an IBM System/360 Model 75, and IBM System/360 Model 44, an IBM/360 Model 50, an IBM 1710, an IBM 1620 and peripheral equipment which includes an IBM 2250 Graphic Display and a CALCOMP plotter. The IBM 1620 computer is available to undergraduate students at all times. The IBM 1710 is used by undergraduates, graduates, faculty and staff.. The System/360 Model 75 is run on a closed shop basis but all members of the University can get programmes processed within a reasonably short period of time. The System/360 Model 44 is used in process control applications and provides the University with a centralized process control facility. The System 360 Model 50 is used by undergraduates, graduates and faculty for development of software in Computer Science as well as a facility for handling the conversational language APL. Extensive financial assistance is available through National Research Council Scholarships and Bursaries and Province of Ontario Graduate Fellowships. In addition, the Faculty of Mathematics offers a number of Teaching Fellowships and some Research Assistantships. For details, correspondence should be addressed to the Graduate Office of the Faculty or Department Chairmen. Candidates for the degrees of M.Math., M.Phil. and Ph.D. are accepted under the general regulations set forth in the section of the
Applied Analysis and Computer Science
Applied Mathematics
Faculty of Mathematics 357
M.Math. degree are encouraged to present theses. Those proceeding by course work must submit a substantial essay in addition to their programme of courses.
Faculty members of the Department of Applied Analysis and Computer Science are currently engaged in a varied and extensive research programme including: i
Programming: Software developed at the University of Waterloo is currently being used by Computing Centres throughout the world. In particular, its well-known WATFOR and WATFIV compilers are currently being used by over 350 System/360 installations. Several installations are using a similar compiler developed in 1965 for 7040-7044 Series machines. Many other topics are being studied including coroutine systems, symbol manipulation, and design of programming languages, both general purpose and also with particular emphasis on software implementation.
Machine Organization and Fault Diagnosis: Current research includes the development of software aids to fault diagnosis, the investigation of unorthodox machine organization from the viewpoints of fault diagnosis and system availability, and the self-diagnosis problem for digital processors. Also under study are system problems arising from time sharing such as the problem of testing software, and computer networks. Computer-Aided Design: The improvement of design techniques through the use of digital computers and associated devices such as visual display units is presently under study.
Formal Languages, Automata and Switching Theory: Among the problems being investigated are the algebraic theory of languages and structure theory of sequential machines.
Computability Theory: Topics currently being studied are Turing machines, complexity classifications of computable functions, abstract models of digital computers and hierarchies of classes of recursive functions.
Artificial Intelligence: One of the problems presently being studied is computer-assisted theorem proving.
Numerical Analysis: Algorithms for two-point boundary problems, stiff systems of differential equations, atomic structure equations, unilateral matrix equations, optimization problems, and linear algebra techniques are presently under investigation.
Functional Equations: Besides basic research, the fields investigated include fundamental applications in nomography, information theory, theory of complex variables, algebraic structures, geometry and generalized functions in general spaces.
Combinatorics and Optimization, Mathematical Operations Research, Graph Theory, Statistics and Decision Theory: Research in these areas is being conducted by members of other departments in the Faculty of Mathematics.
Research in the department is being carried out in a number of areas related either directly or indirectly to problems in Mathematical Physics. These areas include Linear and Nonlinear Differential Equations, Special Theory of Differential Operators, Optimal Control Theory, Boundary Layer Problems in Fluid Dynamics and Particle Physics. A Relativity group is being formed within the department with interests in General Relativity, Differential Geometry and the Calculus of Variations. The department also has a very active Quantum Theory Group whose research involves the Quantum Theory of Crystals and Surfaces.
Combinatorics and Optimization
Pure Mathematics
Statistics
Faculty of Mathematics
The department sponsors a number of seminars, invited talks by outstanding people in the various fields and visiting Professors. Graduate students are encouraged to become involved in these programmes.
Research currently being pursued in the Department of Combinatorics and Optimization includes work in graph theory, chromatic polynomials, colouring problems and enumeration, design theory, matroid theory, well-quasi-ordered sets, directed graphs and relations, Boolean algebra and its applications to Optimization, applications of linear programming and convex polyhedra to Combinatorics and Optimization, extremal problems in graphs, minimax theorems, algorithms and computer analysis in Combinatorics and Optimization, theory of formal languages, and mathematics of elementary particle physics. Research and teaching in Optimization emphasize the underlying mathematical theory, and especially applications of Combinatorics. The department is developing a computerized reference retrieval system for graph theory, which may be extended to cover other areas of Combinatorics and Optimization.
The following graduate courses are offered by the Department of Combinatorics and Optimization: 780, 782, 785, 788, 789, 791, 792, 793, 798, 969, 971, 973, 981, 982.
There are several research activities in the department. These include work in Geometry (Foundations of Geometry, Combinatorial Geometry, Special Geometries including Geometries over Algebras and Finite Geometries, Topology), work in Algebraic Structures (General Algebraic Systems, Linear Algebra, Lattice Theory, Group Theory) and work in Logic and Foundations, moreover some aspects of Analysis and Number Theory.
The departments of Philosophy and Pure Mathematics have inaugurated a joint graduate programme for students interested in the foundations of mathematics and related problems.
There is a weekly meeting of a Research Seminar on “Geometrical and Algebraical Structures”, where original mathematical investigations by students and professors are presented.
Research in the Department of Statistics centres on the controversial problem of formulating exact statistical inferences in many diverse situations using generalizations of the approach introduced by the late Sir Ronald Fisher in his foundation of the theory of fiducial inference, the theory of the likelihood function and the theory of estimation in general. The extensive computing facilities available at the University of Waterloo have given rise to investigations into the uses of computers to obtain exact results on the basis of these theories in situations previously treated by asymptotic methods, and also research into exact tests in multivariate analysis.
Related to this research, there is active interest in problems of experimental design, with particular reference to incomplete block designs, multivariate analysis, estimation theory, mathematical genetics and stochastic processes as applied, for example, to biometrical problems, such as the analysis of environmental factors contributing to diseases processes and aging.
Members of the department are also interested in problems in combinatorial mathematics, such as the construction of balanced incomplete block designs, which arise in the design of experiments. Other combinatorial questions under consideration include edge-chromate graphs and the Ramsey numbers.
Increasing contact with the Department of Psychology and with work in the Physical Sciences, undertaken as part of the above-mentioned biometrical studies, provides a variety of new problems and also gives advanced students a valuable opportunity to apply lecture material to useful practical situations. An arrangement with the Statistics Unit of the Department of Agriculture in Ottawa provides potential summer employment for advanced undergraduates and at the same time, a further opportunity for practical experience.
Courses
An asterisk (*) indicates courses given in 1970-1971 academic year. Other graduate courses may be given from time to time as demand warrants.
Survey of Algebra and its Applications.* Survey of Numerical Analysis. Computer Programming.* Mathematical Logic.*
Transfinite Arithmetic.
Axiomatic Set Theory.
Recursive Function Theory.* Algebraic Theory of Numbers. Seminar in Logic and Foundations.* Seminar in Number Theory.* Topics in Logic and Foundations.* Topics in Number Theory. Universal Algebra.*
Groupoids and Quasigroups. Infinite Groups.
Representation Theory. Homological Algebra.
Seminar in Group Theory.
Topics on General Algebraic Systems.* Topics in Group Theory.
Rings and Ideals*.
Multilinear Algebra.*
Galois Theory.
Topics on Rings and Fields. Differential Geometry.
Differential Manifolds.
Topological and Lie Groups. Algebraic Topology.*
Seminar in Homotopy Theory.* Special Topics in Geometry.* Topics in Topological Structures.* Combinatorial Analysis* Combinatorial Graph Theory.* Analytic Graph Theory.
Linear Graphs.
Planar Graphs.
Extrema in Graphs.
Directed Graphs.* Combinatorics.*
Matroid Theory.*
Boolean Methods in Combinatorics and Optimization.*
Functional Equations for Functions of One Variable.* Functional Equations in Several Variables. Functional Equations for Analytic Functions. Integral Equations.
Seminar on Functional Equations.*
Topics on Functional Equations.
Calculus of Variations.
Analytic Theory of Geometric Objects. Vector and Tensor Analysis.
Integral Transforms.*
Calculus of Operators.
Theory of Approximation.
Survey of Numerical Analysis.
Numerical Solution of Ordinary Differential Equations.* Numerical Methods in Linear Algebra.* Numerical Solutions of Partial Differential Equations.* Problems in Numerical Integration.
Seminar in Numerical Analysis.*
Topics in Applied Analysis.
Topics in Numerical Analysis.
Introduction to Switching and Automata Theory.* Time-Sharing Systems.*
Computer Systems.*
Computer Programming.*
The Theory and Construction of Compilers.* Data Structures | and 2.*
Data Structures 3.*
Finite Automata.*
Design of Sequential Machines.
Algebraic Structure of Sequential Machines. Formal Languages.*
Mathematical Theory of Relativity.* Statistical Mechanics.
Quantum Surface Physics.*
Control Theory.*
Quantum Crystal Physics.
Topics in Applied Mathematics. Mathematical Optimization.
Directed Graphs and Applications.* Mathematical Programming.* Network Programming.
Mathematical Operations Research. Seminar in Mathematical Operations Research Seminar in Combinatorics and Optimization.* Topics in Optimization.
Game Theory.*
Data Processing for Behavioural Scientists. Mathematical Methods for Science and Engineering.* Numerical Methods and Programming. Seminar in Mathematics Education. Topics in Mathematical Education. Topics in the History of Mathematics. Literature and Research Studies. M.Phil. Comprehensive Examination Ph.D. Comprehensive Examination Language Requirement