110 courses in the 1970–71 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 1970–71
Professor
(Applied Analysis and Computer Science, and Applied Mathematics) Professor
Faculty listed in 1970–71
Professor
Associate Professor Associate Dean of the Faculty of Mathematics Associate Professor
Associate Professor (Physics Department and Applied Mathematics) Associate Professor Associate Professor Associate Professor
Associate Professor Assistant Professor Assistant Professor Assistant Professor Assistant Professor
Chairman of the Department Distinguished Professor Professor
Faculty listed in 1970–71
Professor
Associate Dean of the Faculty of Mathematics Professor
Faculty listed in 1970–71
Professor
Faculty of Mathematics 317
Faculty listed in 1970–71
Department of Applied Mathematics P.J. Ponzo, M.A. (Toronto), Ph.D. (Illinois)
C. Froese Fischer (Mrs.), M.A. (British Columbia), Ph.D. (Cantab.)
M.A. McKiernan, M.A. (Loyola), Ph.D. (Illinois Institute of Technology)
D.G. Wertheim, B.A. (McMaster), Ph.D. (Toronto) H. Rund, PA.D. (University of Cape Town) C.F.A. Beaumont, B.A. (McMaster), M.A. (Toronto)
J. Cizek, R.N.Dr. (Charles University of Prague).
C.Sc. (Institute of Phys. Chem., Czechoslovakian Academy of Science, Prague)
S.G. Davison, Ph.D. (Manchester), A.Inst.P.
D. Lovelock, Ph.D. (University of Natal, Durban)
Faculty listed in 1970–71
IJ. McGee, B.A.Sce. (Toronto), M.Sc. (Waterloo), Ph.D. (Yale) J. Paldus, R.N.Dr. Fac. of Math. and Physics (Charles University of Prague), C.Sc. (Institute of Phys. Chem., Czechoslovakian Academy of Science, Prague)
R.A. Wentzell, B.Sc. (Acadia), Ph.D. (Western)
J. Froese, B.A. (Manitoba), M.A. (Queen’s),Ph.D. (U.B.C.) G.J. Lastman, M.A. (British Columbia), Ph.D. (Texas) M.E. Snyder (Mrs.), B.Sc. (Western), M.Sc. (Waterloo)
J. Wainwright, B.Sc. (University of Natal), Ph.D, (University of South Africa)
D.W. Trim, M.Sc. (Waterloo)
F. Burkowski, M.Sc. (Waterloo)
D.I. MacLeod, M.A. (Waterloo)
B.J. Marshman (Mrs.), M.Sc. (Waterloo)
V.M. Bobetic, M.A. (University of Zagreb, Yugoslavia)
Z. Dvoracek, M.S. (Charles University, Prague).
Ph.D. (Czechoslovak Academy of Sciences, Prague), R.N.Dr. (Charles University, Prague)
M. Steslicka, Ph.D. (Warsaw)
Faculty listed in 1970–71
D.J. Henderson, B.A. (U.B.C.), Ph.D. (Utah), F. Inst. P.
Department of Combinatorics and Optimization G. Berman, M.A., Ph.D. (Toronto) W.T. Tutte, PD. (Cantab.), F.R.S.C.
H.F. Davis, Ph.D. (Massachusetts Institute of Technology) K.D. Fryer, B.A. (Western), Ph.D. (Toronto)
P. Zima, M.Sc. Physics (Charles University, Prague) M.A. Economics (University of Economics, Prague, Czech.) J.A. Zimmer, M.A. (University of Nebraska) C.Cadogan, B.Sc. (Lond.-U.C.W.L.).
Ph.D. (University of West Indies)
P.N. Rathie, Ph.D. (Jodhpur University, India)
Department of Pure Mathematics
W. Benz, Ph.D. (Mainz)
H.H. Crapo, A.B. (Michigan), Ph.D. (Massachusetts Institute of Technology)
Faculty listed in 1970–71
GE. Cross, M.A. (Dalhousie), Ph.D. (British Columbia) (On Sabbatical Leave, 1969-70)
Y. Chen, M.Ph. (University of Frankfort), Ph.D. (U. of Bochum) G. Dankert, Dipl. Math. (T.U. Hanover).
Ph.D. (University of Cologne)
Faculty listed in 1970–71
P. Hoffman, B.A. (Toronto), Ph.D. (Manchester)
A. Kerr-Lawson, B.A. (Toronto), §.M. (Chicago).
Ph.D. (McMaster)
Faculty listed in 1970–71
F.C.Y. Tang, B.Sc. (Hong Kong), M.S. (South Carolina), Ph.D. (Illinois)
Paul de Witte, M.Sc. Theor. Phys. (Ghent).
Faculty listed in 1970–71
M.Sc.Math., Ph.D. (Brussels)
J.G. Anderson, M.Sc. (Durham), Ph.D. (Newcastle upon Tyne) S. Burris, Ph.D. (Oklahoma University)
L.J. Cummings, Ph.D. (British Columbia)
W.J. Gilbert, 4.4. (Cantab.), D.Phil. (Oxon.)
Faculty listed in 1970–71
D. Higgs, B.Sc. Hons. (Witwatersrand), M.A. (Cantab.) J. Malzan, Ph.D. (Toronto)
E.M. Moskal, B.A. (Toronto), Ph.D, (Illinois)
Assistant Professor
(St. Jerome’s and Pure Mathematics) Assistant Professor
Faculty listed in 1970–71
Lecturer
Professor
Chairman of the Department Dean of the Faculty
of Mathematics
Faculty listed in 1970–71
Professor
Acting Dean of the Faculty of Mathematics
Faculty listed in 1970–71
Professor
Professor
(Philosophy and Statistics)
Faculty listed in 1970–71
Associate Professor
Associate Professor Acting Chairman
of the Department Associate Professor Associate Professor (Psychology and Statistics) Associate Professor (Psychology and Statistics) Assistant Professor Assistant Professor Assistant Professor Assistant Professor Assistant Professor Lecturer
The Division of Mathematical Research is a trans-departmental organization within the Faculty of Mathematics which is responsible for the general direction, encouragement and coordination of research and related activities. It arranges a regular programme of
Faculty of Mathematics
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 five such Research Sections, namely, Theory of Computing, Combinatorics, Functional Equations, Statistics and Biometry, 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 list of Visiting Professors for the academic year 1969-70 is as follows.
Name
V.D. Belousov Z. Daroczy
J. Edmonds
W. M. Fairbairn B. Forte
G. Haggard
D.W. Hall
U. Melchior R.S. Mishra F. Newman F. Rado
R.S. Varma
D.J.A. Welsh
On Leave From
Mathematical Institute, Moldav, Academy of Science, Kishinev, Moldav, U.S.S.R.
L. Kossuth University, Debrecen, Hungary
National Bureau of Standards, Washington, D.C., U.S.A. University of Lancaster, Bailaigg, Lancaster, England. University of Pavia.
Pavia, Italy.
University of California, Santa Cruz, California, U.S.A. Harpur College.
State University of New York, Binghamton, New York, U.S.A. Ruhr University.
Bochum, West Germany. Banaras Hindu University, Varanasi, India.
University of J.E. Purkyne, Brno, Czechoslovakia. Babes-Bolyai University.
Cluj, Romania.
University of Delhi.
Delhi, India.
Merton College.
Oxford, England.
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 Quantum Theory
Note
12 Calculus 1
21 Algebra and Vector Geometry
22 Calculus 2
23 Introduction to
Computer Programming
31 Differential Equations
32 Numerical Analysis
33 Differential Calculus
Faculty of Mathematics 321
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. Mean value theorem. Introduction to partial derivatives. Introduction to differential equations with applications. Students will work selected problems under supervision.
2 hours lectures, 1% hours problems.
Mathematical induction, Determinants, vectors and solid geometry, matrices, complex numbers, polynomial equations, linear transformations, infinite series.
2 hours lectures, 1% 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 and development of algorithms for their solution. An introduction to the Fortran 4 programming language for use in implementing algorithms on the computer.
1 hour lecture, 2 hours problems, one term.
Standard methods of solution. Applications to physical engineering problems. Linear equations with constant coefficients. Systems of differential equations. Solution of differential equations by the Laplace Transformation.
3 hours lectures.
A survey of numerical procedures with emphasis upon computer implementation using the Fortran 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.
322 § 34 Integral Calculus 35 Advanced Calculus (for Electrical Engineers) 41 Applied Analysis 44 Complex Variable 51 Probability and Statistics Note 1 Note 2 81 An Introduction to Funtions, Analytic Geometry, and Calculus 82 Introduction to Algebra
Faculty of Mathematics
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.
Partial differentiation. Laplace transforms, special functions. Fourier series, second order differential equations. 3 lectures, one term.
Cauchy-Riemann equations, the Cauchy integral theorems, conformal mapping. the taylor and Laurent series, contour integration. 2 lectures, 1 hour problems, one term.
Frequency distributions; probability; Binomial, Normal, 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 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 sketchng.
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.
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 librarv-oriented rather than textbook-oriented and the student is
Faculty of Mathematics
encouraged to find and pursue some interest of his own, subject to approval. 3 lectures.
Logic, sets, and relations. Number systems. Matrices and linear equations. Polynomial ring, Euclidean algorithm. Introduction to vector spaces, examples from two and three dimensional geometry. 3 hours lectures.
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 moments. Power series. Partial differentiation. 3 lectures.
The real and complex number systems. Polynomial functions. Theory of equations. Determinants. Vector and matrix notation. Elementary solid geometery, planes and lines, the standard quadratic surfaces. Co-ordinate systems. 2 lectures, one 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. 2 hours lectures, 2 hours problems.
Real numbers, sequences, Limits. Continuity. Derivatives. Functions of several variables. Implicit functions. Jacobians. Taylor’s series with geometric applications. Power series with complex terms. The elementary functons of a complex variable. 2 lectures.
Methods of formal integration. Differential equations. Fourier series. Riemann integration and applications. Calculus of variations. Line integrals. Stokes’ and Green’s theorems. 3 lectures.
Differential equations, including first order and constant coefficients. The Laplace transform. Applications of differential equations to problems in the Physical Sciences. Vector analysis. Newtonian mechanics. Central forces. Transformation of co-ordinate systems. 2 hours lectures. Prerequisite: First Year Physics or consent of Instructor.
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.
Mathematics 236 is not a course for Honours Mathematics students.
MATH 239Introduction to Combinatorics and Optimization
A selection of topics from discrete mathematics including an introduction to graph theory, difference equations, generating functions, convex sets, finite geometries and partially ordered sets. The ideas are illustrated with examples from modern applications to such topics as network theory, queueing theory, random walks, linear programming, design of experiments, ranking problems and PERT. 2 hours lectures.
MATH 240(a) Numerical Applications in Computer Science
An introduction to Numerical Procedures with emphasis upon computer implementation using the FORTRAN IV programming language. In particular, the following topics are introduced: concept of numerical errors, interpolation, curve fitting, solution of non-linear equations, numerical integration, matrix operations and solution of systems of linear equations, numerical solution of ordinary differential equations. 2 hours lectures, one term. Prerequisite: Mathematics 132 or equivalent.
MATH 240(b) Non-Numeric Applications in Computer Science
In Introduction to: Number Systems, Boolean algebra, concept of a list and elementary list processing techniques, sorting and searching algorithms, symbol manipulation. 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.
The real number system. Uniform convergence. Theory of Fourier series. The complex number system. Analytic functions. The Cauchy theorems. Contour integration. Singularities. Conformal gral equations. 3 lectures.
First order differential equations; existence theorems; singular solutions; solution by Laplace Transforms; numerical methods of solutions; solution in series; hypergeometric, Bessel and Legendre functions; introduction to partial differential equations and integgral equatons. 2 hours lectures.
A survey of various numerical procedures with emphasis upon their derivations and error analysis. In particular, the following topics are discussed: interpolation, curve fitting, computation with series, solution of non-linear equations, numerical integration, numerical solution of Ordinary Differential Equations, matrix operations and solution of systems of linear equations, problems in the solution of A knowledge of the FORTRAN IV partial differential equations. programming language is essential for the problem assignments. 2 hours lectures, 2 hours problems.
A course in the calculus of finite differences, to include: summation, differences of zero, numerical integration, relation between integration and summation. 2 hours lectures.
(a) An advanced course on problems with single lives. 3 lectures, one term. (b) An advanced course on problems with multiple lives; population and multiple decrement theory. 3 lectures, one term.
An advanced mathematical course including the multivariate normal distribution and related distributions. Normal Regression Theory. Application to hypothesis testing and analysis of variance. Maximum likelihood and large sample theory. Prerequisite: Mathematics 233 or equivalent. 2 hours lectures.
A discussion of the hardware and sofware required in a computer system. Overlapped channels, interrupt facilities, memory protection, buffers, input-output control systems, macro-programming, monitor systems. Relocation schemes, multi-programming, multiprocessing, 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.
Real number system; metric space setting for ideas of continuity, uniform continuity, compactness, convergence, open sets, closed sets, connnected sets. Cantor ternary set. Stone-Weierstrass theorem, uniform convergence, equicontinuity. Ascoli’s theorem, fixed point theorems and applications, tangent spaces, differential forms, exterior differentiation, Stokes’ Theorem. Introduction to Lebesgue integration. 2 hours lectures.
Complex numbers; continuity and differentiability, Cauchy-Riemann equations; solution of Laplace equation; bilinear transformations and conformal mapping; power Series; integral theorems; Taylor and Laurent expansions; calculus of residues; series solution of differential equations by integrals. 2 hours lectures.
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.
MATH 351Introduction to Graph Theory and Combinatorial Analysis
Graphs, paths, arcs, circuits, connected graphs, trees, spanning trees, digraphs, Euler paths, tournaments, complete paths in tournaments, factors of bipartite graphs, Menger’s Theorem and flows in networks, theory of enumeration, generating functions, Ramsey’s Theorem and applications, systems of distinct representatives, Latin squares, balanced incomplete block designs and (v.k.) configurations. Applications to practical problems and other areas of mathematics.
Review of mathematical techniques frequently used in Operations Research: including difference-differential equations, Lagrange
Faculty of Mathematics
Multipliers, generating functions, quadratic forms, minimax algorithms in directed graphs. Introduction to Linear programming, Dynamic programming, transport network theory. Applications to production planning, inventory problems, growth and survival models, scheduling problems, replacement policies and reliability optimization of complex systems. 2 hours lectures.
Vector analysis in curvilinear coordinates; gradient, divergence, curl. Transformation laws of covariant, contravariant, relative tensors. Covariant and intrinsic differentiation. The Riemann-Christoffel tensors, Bianchi identities, Ricci and Einstein tensors. Integrability theorems. Applications to classical mechanics and elasticity. Introduction to general relativity. 2 hours lectures, Fall term.
The metric tensor, and elementary differential geometry of surfaces, Introduction to Riemannian geometry; Riemannian curvature, Gaussian curvature and Einstein curvature tensors. Spaces of affine connection and Finsler spaces with application to electromagnetic theory. 2 hours lectures, Winter term.
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.
Elementary properties of Abelian groups, solvable groups, nilpotent groups. Direct products and semi-direct products, Free groups and free products. Elements of group representations. 2 hours lectures.
Continuation of general topology. Topological manifolds, a weak embedding theorem, classification of 1-manifolds. Triangulation, simplicial complexes, simplicial homology. Classification and homology groups of closed (triangulated) 2-manifolds. The fundamenttal group covering spaces, homotopy groups, the Hurewicz isomorphism theorem. 2 hours lectures.
Measure in Euclidean n-space, the Lebsegue integral, convergence theorems, the Fubini theorem, differentiability, absolute continuity. A study of Banach spaces. 2 lectures.
An informal introduction to the logic of sentences and predicates, with emphasis on analogies with familiar mathematical structures. Syllogisms and algebra of sets related to predicate logic. Simplification, logical deduction, duality, consistency and completeness. The concepts of constant, variable, function, and set. Axiomatics. 4 lectures.
The mathematics and logic of estimation and hypothesis testing. Consistency, efficiency and sufficiency of estimates. Fiducial and confidence intervals, likelihood, Bayes’ theorem, tests of significance. 2 hours lectures.
The logic of experimental design with reference to randomization, experimental and sampling errors. Construction and analysis of various kinds of designs. 2 hours lectures.
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.
Covariant and contravariant tensors. Riemannian space. Gaussian curvature. The Lorentz transformation and Einstein’s special theory. Introduction to the general theory of relativity. 2 lectures, Winter term.
Kinematics and dynamics of flow. Equations of motion. Inviscid two-dimensional motion. Applications of conformal transformation to fluid flow. Viscous mo tion and similarity. Oseen’s approximation. 2 hours lectures.
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 topics in advanced calculus. 2 lectures.
Discrete functions; continuous functions of n variables. Quadratic forms; convex functions. Equality constraints; Lagrange multipliers and generalizations. Inequality constraints; Kuhn-Tucker theorem. Steepest descent and gradient methods. Fibonnaci search, successive approximations, simplex method and other techniques. Applications to the solution of equations and non-linear regression. 2 hours lectures.
Input and Service-time distributions; Poisson process, Erlangian distribution. Single channel queue; differential-difference equations, integral equations. Complex queueing models. Monte Carlo tech- 2 hours lectures. niques.
May include any subset of the following topics: applications of combinatorial mathematics to optimization, selected topics from the Beckenbach book, structural models, graphs and electrical networks, graph theory and statistical mechanics, the Ising problem, percolation processes, the dimer problem, random walk, applications to chemistry, automata. 2 hours lectures.
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 subgaphs, colouring of graphs, graphs and the Four Colour Problem, planarity and thickness of graphs, graphs and matroids, applications of matrices to graph theory. BS baneeen Tanteswnen
MATH 470Numerical Solution of Ordinary and Partial Differential Equations
Existence and uniqueness of solutions of ordinary differential equations, one-step methods for calar and systems of differential equations, multi-step methods, linear boundary value problems. Discrete approximations to partial differential equations, convergence, stability and convergence rate estimates, generalization to non-linear equations. 2 hours lectures.
Switching devices. Binary codes and arithmetic. Boolean algebra and its application to switching circuits. Canonical forms for Boolean functons. Simplification of switching functions. Analysis of sequential circuits. Synicronous and asynchronous operation. Design of sequential circuits. 2 hours lectures, Fall term.
MATH 471BComputer 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 reliabilty and fault diagnosis as additional system requrements. Prerequisite: Switching Circuits. 2 hours lectures, Winter term.
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. Prerequisite: Mathematics 427a or Consent of Instructor. 2 hours lectures, Winter term.
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, Combinatorial Analysis, Computer Science, Differential Equations, Differential Geometry, Foundations, Functional Analysis, Functional Equations, Graph Theory, Harmonic Analysis, Mathematical Logic, Number Theory, Numerical Analysis, Optimization, Probability, Statistical Inference, Switching and Automata Theory, Information Theory and Nets. The University of Waterloo has at present five computer systems, namely an IBM System/360 Model 75, and IBM System/360 Model 44, an IBM/360 Model 40, 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 conrtol applications and provides the University with a centralized process control facility. The System 360 Model 40 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 Ascintamtchine Far dataile sarracnandancrs chauld ha oddraccad tan tha
Applied Analysis and Computer Science
Applied Mathematics
Faculty of Mathematics 333
Graduate Office of the Faculty or Department Chairmen.
Candidates for the degrees of M.Math., M.Phil. and Ph.D. are accepted under the general regulations set forth in the section of the Calendar pertaining to the Graduate Studies. Students proceeding to the M.Math. degree are encouraged to present theses. Those proceeding by course work must submit a substantial essay in connection with a least one of the courses.
Faculty members of the Department of Applied Analysis and Computer Science are currently engaged in a varied and extensive research programme including.
Programming: Waterloo’s WATFOR fast FORTRAN IV compiler is currently being used by over 200 System/360 Computing Centres throughout the world. Several installations are using a similar compiler developed in 1965 for 7040-7044 Series machines. (Comm. of the ACM, January 1967). Also under study is the 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 machines organization from the viewpoints of fault diagnosis and system availability, and the selfdiagnosis problem for digital processors. Also under study are system problems arising from time sharing, and the problem of testing software.
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, structure theory of sequential machines, and design of arithmetic units for residue number systems.
Computability Theory: Topics currently being studied are Turning machines, complexity classifications of computable functions, abstract models of digital computers and hierarchies of classes of recursive functions.
Artificial Intelligence: Among 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.
Applied Analysis: In Applied Analysis, among others, functional equations are studied, i.e. the theory of determining functions from elementary equations containing them. It is a field of mathematics with a two century history although the somewhat more general theory has developed only in the last two decades. Functional equations have applications in many modern disciplines including information theory, mathematical psychology and universal algebra.
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.
Research in the department is being carried out in a number of areas related either directly or indirectly to problems in Mathema- .
Combinatorics and Optimization
Pure Mathematics
Statistics
Faculty of Mathematics
tical 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.
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.
Research activities in the department are representative of the main areas of Pure Mathematics. Current research includes studies in Number Theory, Logic and Foundations, Category Theory, General Algebraic Systems, Multi-linear Algebra, Group Theory, Lattice Theory, Foundations of Geometry, Combinatorial Geometry, Differential Geometry, Algebraic Topology and Functional Analysis.
The Departments of Philosophy and Pure Mathematics have inaugurated a joint graduate programme for students interested in the foundations of mathematics and related problems.
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 desiens. multivariate analvsis. 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 edgechromatc graphs and the Ramsey numbers.
Increasing contact with the Department of Psychology and with work in the Physical Sciences, undertaken as part of the abovementioned 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 1969-1970 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.*
Special Topics in Geometry.* Topics in Topological Structures.* Combinatorical Analysis.* Combinatorial Graph Theory.*
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 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. Math. Thesis.
M. Phil. Thesis.
Ph.D. Thesis.
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