AS/SC/MATH 4000 3.0 FW and 6.0 Y Individual Project

     2001/2002 Calendar copy: A project of a pure or applied nature in mathematics or statistics under the supervision of a faculty member. The project allows the student to apply mathematical or statistical knowledge to problems of current interest. A report is required at the conclusion of the project.
     The student works under supervision of a faculty member, who is selected by the Course Coordinator and the student. The project allows the student to apply mathematical or statistical knowledge to problems of current interest. A report is required at the conclusion of the project.
     Students in the Applied Mathematics Honours Programs are particularly encouraged to take this course. The procedure is as follows: Each year, faculty members who are interested in supervising projects will submit project descriptions to the Course Coordinator for Applied Mathematics. Students will meet with the CC for AM, and they will jointly decide on a faculty member to supervise the project, taking into account the background and interests of the student, as well as the availability and interests of faculty members.
     The amount of work expected of the student is approximately ten hours per week, that is, the equivalent of a standard full-year (for 4000 6.0) or half-year (for 4000 3.0) course. The supervisor is expected to spend about one or two hours per week with the student, averaged over the duration of the project. In addition to the final report, regular short progress reports will be expected at definite times during the course. The final grade will be based upon the final report as well as the interim progress reports.
Applied Mathematics Coordinator: Buks van Rensburg
Maths.for Commerce Coordinator: Morton Abramson
Pure Mathematics Coordinator: Morton Abramson
Statistics Coordinator:} Yuehua Wu

Prerequisites:  Open to all students in Honours programs in the Department of Mathematics and Statistics. Permission of the Program Director is required. Applied Mathematics students can enrol only after they have completed the core program in Applied Mathematics.

AS/SC/AK/MATH 4010 6.0 Real Analysis

     2001/2002 Calendar copy: Survey of the real and complex number systems, and inequalities. Metric space topology. The Riemann-Stieltjes integral. Some topics of advanced calculus, including more advanced theory of series and interchange of limit processes. Lebesgue measure and integration. Fourier series and Fourier integrals.
     This course provides a rigorous treatment of real analysis. All students should have completed the introductory analysis course MATH 3210 3.0.  Students contemplating graduate work in mathematics are strongly advised to take this course.
     The text will be W.Rudin, Principles of Mathematical Analysis (McGraw-Hill).  
     The final grade will be based on assignments and four term tests (60%) and a final exam (40%).

Prerequisite: AS/SC/AK/MATH 3210 3.0 or permission of the course coordinator.
Coordinator:  N.Purzitsky

AS/SC/AK/MATH 4020 6.0 Algebra II

     2001/2002 Calendar copy: Continuation of Algebra I, with applications: groups (finitely generated Abelian groups, solvable groups, simplicity of alternating groups, group actions, Sylow's theorems, generators and relations); fields (splitting fields, finite fields, Galois theory, solvability of equations); additional topics (lattices, Boolean algebras, modules).
     This course aims to broaden and deepen the student's knowledge and understanding of modern abstract algebra by building on the material of MATH 3020 6.0 (or a comparable course which students may have taken). In addition to the topics listed in the Calendar, the following will be expounded:
Group theory: Composition series.
Ring theory: General ring theory, Chinese Remainder
Theorem, factorization in domains, Noetherian rings.
Field theory: Ruler and compass constructions.
Linear algebra: The Jordan canonical form of a matrix.

Prerequisite:  AS/SC/AK/MATH 3020 6.0 or permission of the course coordinator.
Exclusion: AS/SC/MATH 4241 3.0.
Coordinator:  R.G.Burns

AS/SC/AK/MATH 4080 6.0 Topology

     2001/2002 Calendar copy: Topological spaces, continuity, connectedness, compactness, nets, filters, metrization theorems, complete metric spaces, function spaces, fundamental group, covering spaces.
Note: This course will probably NOT
be offered in FW 2002.
     Unlike a geometer, who will consider only non-distorting transformations of geometric objects, such as reflections and rotations, a topologist studies properties that are invariant under bending, stretching, compressing, etc.  Hence, a circle is considered topologically equivalent not only to an ellipse, but even to a triangle or a square! While this seems mathematically rough and disturbing, a second look quickly reveals how fine and powerful a tool topological equivalences (called homeomorphisms) are, since they turn out to be able to distinguish such seemingly similar objects as open, closed and half-open intervals of the same length!
     The main purpose of this course is to study so-called topological spaces, and those properties which are invariant under homeomorphisms and isotopies. To get a feel for the latter concept, perform the following (rather theoretical) experiment: Using both of your hands, make LINKED rings with your thumbs and index fingers, and now assume that your whole body is made of VERY elastic material, so that its shape may be changed at will, by bending, stretching and compressing it as much as you like (but tearing and gluing is forbidden). Question: can you move your hands apart without separating the joined fingertips? Surprisingly, the answer turns out to be positive! 
     Topology is an integral part of modern mathematics, just like geometry, algebra and analysis. It has many applications to almost all mathematical fields and is increasingly used in other subjects, such as physics and economics. The course will present basic constructions and concepts of general topology, such as separation axioms, compactness, connectedness, metrizability, as well as the fundamentals of homotopy theory.
     A text has not been chosen yet.
     The final grade will be based on assignments (20%), class tests (40%) and a final examination (40%).
     The course can be used to fulfill the Pure Mathematics Honours requirement.

Prerequisite:AS/SC/AK/MATH3210 3.0 or permission of the course coordinator.
Coordinator:  P.Szeptycki

AS/SC/MATH 4110N 3.0 WTopics in Analysis: Ordinary Differential Equations
Same as: GS/MATH 6340 3.0

     2001/2002 Calendar copy: This course is an advanced introduction to a number of topics in ordinary differential equations. The topics are chosen from the following: existence and uniqueness theorems, qualitative theory, oscillation and comparison theory, stability theory, bifurcation, dynamical systems, boundary value problems, asymptotic methods.
     The last two topics above will be omitted. The lectures will survey the others, and students will be expected to make an in-depth study of some, by doing assignments and projects.
     Students should have passed MATH 2221 and MATH 3210, or seek permission from the course coordinator to take this course.
     The text will be Lawrence Perko, Differential Equations and Dynamical Systems (Springer--Verlag, 1991).
     Other references include J.K.Hale and H.Kocak, Dynamics and Bifurcations (Springer--Verlag, 1991), and M.W.Hirsch and S.Smale, Differential Equations, Dynamical Systems and Linear Algebra (Academic Press, 1974).
     The grade will be based upon term tests and assignments (60%) and a final exam (40%).

Prerequisite: Permission of the course coordinator.
Coordinator:  J.Wu

AS/SC/MATH 4130K 3.0 F Topics in Probability and Statistics: Survival Analysis

     2001/2002 Calendar copy: This course provides students with an introduction to the statistical methods for analyzing censored data which are common in medical research, industrial life-testing and related fields. Topics include accelerated life models, proportional hazards model, time dependent covariates.
It has not yet been determined whether this course will be offered in FW 2002.
     We start with some parametric models and show how censored data can be incorporated in the analysis. Then we proceed to nonparametric methods and discuss Kaplan-Meier and Actuarial estimators. Semiparametric models, proportional hazards model and time dependent covariates will also be discussed. The computer will be extensively used, and familiarity with elementary use of S+ and SAS will be assumed.
     Evaluation will be based on a combination of assignments, midterm exam, final exam and a project.

Prerequisites: AS/SC/AK/MATH 3131 3.0; either AS/SC/MATH 3033 3.0 or AS/SC/AK/MATH 3330 3.0.
Note: Computer/Internet use is essential for course work.
Coordinator:  Stephen Chamberlin

AS/SC/MATH 4141 3.0 F Advanced Numerical Methods

     2001/2002 Calendar copy: Newton-Raphson, quasi-Newton methods; optimization problems: steepest descents, conjugate gradient methods; approximation theory: least squares, singular value decomposition, orthogonal polynomials, Chebyshev and Fourier approximation, Pad\'{e} approximation; matrix eigenvalues: power method, Householder, QL and QR algorithms.
     Additional topics will include optimization problems: simplex, conjugate directions; fast Fourier transforms. 
     Some assignment questions will be done on computer. Students should be familiar with either the C or Fortran programming language.
     The final grade will be based on assignments, tests and a final examination.

Prerequisite: AS/SC/MATH 3242 3.0 or AK/AS/SC/COSC 3122 3.0.
Coordinator:  D.Liang

AS/SC/MATH 4142 3.0 W Numerical Solutions to Partial Differential Equations

     2001/2002 Calendar copy: Review of partial differential equations, elements of variational calculus; finite difference methods for elliptic problems, error analysis, boundary conditions, non-Cartesian variables, PDE-eigenvalue problems; hyperbolic and parabolic problems, explicit and implicit methods, stability analysis; Rayleigh-Ritz and Galerkin method for ODEs, finite element methods.

Prerequisites: AS/SC/AK/MATH 2270 3.0; AS/SC/MATH 3242 3.0 or AK/AS/SC/COSC 3122 3.0.
Coordinator:  H.Huang

AS/SC/MATH 4160 3.0 Combinatorial Mathematics

     2001/2002 Calendar copy: Topics from algebra of sets, permutations, combinations, occupancy problems, partitions of integers, generating functions, combinatorial identities, recurrence relations, inclusion-exclusion principle, Polya's theory of counting, permanents, systems of distinct representatives, Latin rectangles, block designs, finite projective planes, Steiner triple systems.
Note: This course will NOT be offered in FW 2001. It will probably be offered in FW 2002.

Prerequisites: AS/SC/AK/MATH 2022 3.0 or AS/SC/AK/MATH 2222 3.0; six credits from 3000-level MATH courses (without second digit 5); or permission of the course coordinator.

AS/SC/MATH 4170 6.0 Operations Research II 
Same as: GS/MATH 6900 3.0 F plus GS/MATH 6901 3.0 W

     2001/2002 Calendar copy: Selected topics from game theory, decision theory, simulation, reliability theory, queuing theory, nonlinear programming, classification, pattern-recognition and prediction. Each chapter contains an optimization problem and methods and algorithms for solving it. The course is rich in examples.
     This course deals mainly with probabilistic models based on optimization. The following topics will be discussed: (a) Game Theory: how to find the best strategies in a confrontation between two players with opposite interests. (b) Decision Theory: how to act in order to minimize the loss subject to the available data. (c) Simulation: how to get representative samples from probability distributions and accurately approximate multiple integrals using random numbers. (d) Reliability Theory: how to evaluate the lifetime of a system consisting of many interacting subsystems. (e) Queueing Theory: how to assess what may happen in a system where the customers arrive randomly, wait in line, and then get served. (f) Uncertainty: how to measure uncertainty in probabilistic modelling with applications to pattern-recognition and classification.
     There is no textbook, and the lecture notes are essential. Useful books are: (a) F.S.Hillier and G.J.Liberman, \ita{Introduction to Operations Research (b) H.A.Taha, Operations Research.
     The final grade will be based on two tests (25% each) and a final examination (50%).
     The following prerequisites indicate the sort of background in probability and statistics, in calculus of several variables, and in linear programming, needed for MATH4170. Students missing a prerequisite need the course coordinator's permission to enrol.

Prerequisites: AS/SC/MATH 2010 3.0 or
AS/SC/MATH 2015 3.0 or AS/SC/AK/MATH 2310 3.0; AS/SC/AK/MATH 2030 3.0; AS/SC/AK/MATH 3170 6.0; or permission of the course coordinator.
Exclusion: AS/MATH 4570 6.0.
Coordinator:  Silviu Guiasu

AS/SC/MATH 4230 3.0 W Nonparametric Methods in Statistics
Same as: GS/MATH6639A 3.0

     2001/2002 Calendar copy: Order statistics; general rank statistics; one-sample, two-sample, and k-sample problems; Kolmogorov-Smirnov statistics; tests of independence and relative efficiencies.
It has not yet been determined whether this course will be offered in FW 2002. 
     Survey of basic nonparametric test procedures together with the related theory for permutation, rank, and related techniques. 

     The text and grading scheme have not been determined.

Prerequisite: AS/SC/AK/MATH3131 3.0. AS/SC/AK/MATH3132 3.0 is recommended but not required.
Coordinator:  P.Ng

AS/SC/MATH 4241 3.0 F Applied Group Theory

     2001/2002 Calendar copy: Introduction to group theory and its applications in the physical sciences. Finite groups. Compact Lie groups. Representation theory, tensor representations of classical Lie groups, classification of semi-simple Lie groups.
Note: This course will probably NOT be offered in FW 2002. 
     Group theory is widely used in many fields outside mathematics. This is a consequence of the fact that the algebraic structure which definesF a group is naturally the property of the set of symmetries of a physical system. Paying attention to this fact, and using results from group representation theory (some of which we will study in this course), one can often radically simplify practical calculations in these fields.
     The course will provide an introduction to group theory, both for finite groups and continuous groups, as well as an introduction to group representation theory. No previous knowledge of group theory will be assumed, but a background in linear algebra is essential. The course will begin with a review of the formal aspects of linear algebra (vector spaces, linear transformations, dimensions, bases, inner products for complex vector spaces, etc.) necessary for the remainder of the course.
     There is no official text. A copy of the class notes will be available at the library, and a series of reference texts will be on reserve at the library. In addition to the official prerequisites, it is highly recommended that students also have passed MATH 2015 and MATH 2270. 

Prerequisites: AS/SC/MATH 2015 3.0; AS/SC/AK/MATH 2022 3.0 or
AS/SC/AK/MATH 2222 3.0; AS/SC/AK/MATH 2270 3.0.
Exclusion: AS/SC/AK/MATH 4020 6.0.
Coordinator: Kim Maltman

AS/SC/AK/MATH 4271 3.0 Dynamical Systems

     2001/2002 Calendar copy: Iterations of maps and differential equations; phase portraits, flows; fixed points, periodic solutions and homoclinic orbits; stability, attraction, repulsion; Poincar\'{e} maps, transition to chaos. Applications: logistic maps, interacting populations, reaction kinetics, forced Van der Pol, damped Duffing, and Lorenz equations.
This course will not be offered in FW 2001. In recent years it has run fairly frequently. It has not been determined whether it will be offered in FW 2002.

Prerequisites: AS/SC/AK/MATH 3010 3.0; AS/SC/AK/MATH 2022 3.0 or AS/SC/AK/MATH 2222 3.0; or permission of the course coordinator.
Exclusion: AS/SC/MATH 3450 3.0.

AS/SC/MATH 4280 3.0 F Risk Theory

     2001/2002 Calendar copy: Frequency and severity models in insurance, compound distributions, compound Poisson processes, ruin theory, non-proportional reinsurance, related topics in loss models and stochastic processes.
     This course is intended mainly for students contemplating a career in the actuarial profession. It will cover a portion of the material on the new Exam 3 in Actuarial Models, including the topics of Frequency and Severity Models, Compound Distribution Models, and Ruin Models.
     The text has not yet been determined.
     There will be a final exam, two class tests, and written assignments. The exact grading scheme will be announced during the first week of classes.

Prerequisite: AS/SC/AK/MATH 2131 3.0.
Coordinator:  S.D.Promislow

AS/SC/MATH 4400 6.0 The History of Mathematics

     2001/2002 Calendar copy:  Selected topics in the history of mathematics, discussed in full technical detail but with stress on the underlying ideas, their evolution and their context.
Note: This course will probably NOT be offered in FW 2002. 
     The aim of the course is to give students an overview of some of the main currents of mathematical thought from ancient to modern times so that they gain a better understanding of, and acquire a broader perspective on, mathematics.
     The course will trace the evolution of various areas of mathematics, such as algebra, analysis, geometry, and set theory. While it will involve a great deal of technical mathematics, the course will also explore issues closely bound up with its progress, such as the changing standards of rigor in mathematics, the cultural context of mathematics, the roles of problems and crises in the evolution of mathematics, and the roles of intuition and logic in its development.
     Students will be expected to write a major paper depicting the evolution of a given problem, general principle, concept, theory, or subfield of mathematics; for example, function, the real numbers, the distribution of primes among the integers, the arithmetization of analysis, fashions in mathematics, the rise and fall of rigorous proof, or Fermat's Last Theorem.
     The course will not follow a prescribed text, but the following book is recommended: V.J.Katz, {\it A History of Mathematics}, 2nd Ed. (Addison-Wesley, 1998). In addition, many references on various topics will be given. Some books will be put on reserve.
     The final grade will be based on a major paper (20%), other assignments (30%), two tests (20%), and a final exam (30%).

Prerequisites:   36 credits from MATH courses without second digit 5, including at least 12 credits at or above the 3000 level. (12 of the 36 credits may be taken as corequisites.)
Coordinator:  Israel Kleiner 

AS/SC/AK/MATH 4431 3.0 W Probability Models

     2001/2002 Calendar copy: This course introduces the theory and applications of several kinds of probabilistic models, including renewal theory, branching processes, and martingales. Additional topics may include stationary processes, large deviations, or models from the sciences.
Note: This course will probably NOT be offered in FW 2002. MATH 4430 3.0 is expected to be offered that year instead. These two courses are normally offered in alternate years. 
     Renewal processes are used to model an event that occurs repeatedly at random times, such as the failure of a machine component. The focus of study is on the long-run average behaviour of such processes. 
     Branching processes are a class of simple population growth models.
One important question is how the distribution of the number of offspring of one parent can be used to predict the probability that the population eventually dies out. Generating functions will be introduced and used to derive results. 
     Martingales are models of "fair games". They have been used to study stock market behaviour and are an important theoretical tool for a wide variety of probability problems. Important results include descriptions of how expected value is affected if your decision of when to stop (e.g.when to sell the stock) is determined by the behaviour of the process itself.
     In addition, we will study some probability models that arise in other areas of science.
     The text has not been chosen yet.
     The final grade will be based on assignments (20%), two class tests (40%), and a final exam (40%).

Prerequisite: AS/SC/AK/MATH 2030 3.0.
Corequisite: A MATH course at the 3000 level or higher, without
second digit 5 (Atkinson: second digit 7).
Coordinator:  N.Madras

AS/MATH 4570 6.0 Applied Optimization

     2001/2002 Calendar copy: Topics chosen from decision theory, game theory, inventory control, Markov chains, dynamic programming, queuing theory, reliability theory, simulation, non-linear programming.
     The course will cover several topics in optimization with an emphasis on implementing algorithms on the computer to solve practical problems. Of the official prerequisites (see below) the most important is MATH 3170. The text is the one used in that course: W.L.Winston, {\it Operations Research, Applications and Algorithms}, 3rd Ed. (Wadsworth Publishing, Duxbury Press, 1994). 
     The final grade will be based on assignments (30%), four class tests (10% each), and a final exam (30%).

Prerequisites: AS/SC/AK/MATH 3170 6.0; AS/SC/AK/MATH 3330 3.0; AS/SC/MATH3230 3.0 or AS/SC/AK/MATH 3430 3.0.
Exclusion: AS/SC/MATH 4170 6.0.
Coordinator:  R.L.W.Brown

AS/SC/MATH 4630 3.0 W Applied Multivariate Statistical Analysis
Same as: GS/MATH 6625 3.0

     2001/2002 Calendar copy: The course covers the basic theory of the multivariate normal distribution and its application to multivariate inference about a single mean, comparison of several means and multivariate linear regression. As time and interest permit, further related topics may also be covered. 
It has not yet been determined whether this course will be offered in FW 2002. 
     We will study methods of analysis for data which consist of observations on a number of variables. The primary aim will be interpretation of the data, starting with the multivariate normal distribution and proceeding to the standing multivariate inference theory. Sufficient theory will be developed to facilitate an understanding of the main ideas. This will necessitate a good background in matrix algebra, and some knowledge of vector spaces as well. Computers will be used extensively, and familiarity with elementary use of SAS will be assumed. Topics covered will include multivariate normal population, inference about means and linear models, principal component analysis, canonical correlation analysis, and some discussion of discriminant analysis, and factor analysis and cluster analysis, if time permits.
     Grades will be based on a combination of class tests and final examination, plus routine homework.  The text has not yet been determined.

Prerequisites: AS/SC/AK/MATH 3131 3.0; AS/SC/AK/MATH 3034 3.0 or AS/SC/AK/MATH 3230 3.0; AS/SC/AK/MATH 2022 3.0 or AS/SC/AK/MATH 2222 3.0. 
Coordinator:  Y.Wu

AS/SC/MATH 4730 3.0 F Experimental Design
Same as: GS/MATH 6626 3.0

     2001/2002 Calendar copy: An examination of the statistical issues involved in ensuring that an experiment yields relevant information. Topics include randomized block, factorial, fractional factorial, nested, Latin square and related designs. Further topics as time permits. The emphasis is on applications.
     Experimental design is the process of planning an experiment so that appropriate data will be collected which may be analysed by statistical methods, resulting in valid and meaningful conclusions. This includes the choice of treatments, the required sample size, the random allocation of experimental units to treatments, the method of estimation, and a consideration of how the data will be analysed once collected.
     We will study various experimental situations in this course, considering how the principles of design can be applied to each to create a design that is appropriate to the objectives of the experiment. We will examine appropriate procedures for the analysis of the resulting data, including the underlying assumptions and limitations of the procedures.
     Students will use the statistical software SAS for data analysis. The final grade will be based on assignments, a midterm test, and a final exam. The text will be Dean and Voss, Design and Analysis of Experiments (Springer, 1999).

Prerequisites: AS/SC/AK/MATH 3034 3.0 or permission of the course coordinator.
Coordinator:  A.Gibbs

AS/SC/MATH 4830 3.0 F Time Series and Spectral Analysis

     2001/2002 Calendar copy: Treatment of discrete sampled data by linear optimum Wiener filtering, minimum error energy deconvolution, autocorrelation and spectral density estimation, discrete Fourier transforms and frequency domain filtering and the Fast Fourier Transform algorithm. (Same as SC/EATS 4020 3.0 and SC/PHYS 4060 3.0.)
Ed. note: In the absence of any input from Smylie, we provide no supplementary course description here.

Prerequisites:   AK/AS/SC/COSC 1540 3.0 or equivalent FOR
TRAN programming experience; AS/SC/AK/MATH 2270 3.0; AS/SC/MATH 2015 3.0 or AS/SC/AK/MATH 3010 3.0.
Exclusions: AK/AS/SC/COSC 4242 3.0,
AK/AS/SC/COSC 4451 3.0, SC/EATS 4020 3.0, AS/SC/MATH\4130B 3.0, AS/SC/MATH 4930C 3.0, SC/PHYS 4060 3.0.
Coordinator:  D.Smylie

AS/SC/MATH 4930A 3.0 W Topics in Applied Statistics: Statistical Quality Control

     2001/2002 Calendar copy: This course provides a comprehensive coverage of the modern practice of statistical quality control from basic principles to state-of-the-art concepts and applications.
Note: This course will probably NOT be offered in FW 2002. 
     This course presents the modern approach to quality through the use of statistical methods. The primary focus will be on the control chart whose use in modern-day business and industry is of tremendous value. Various control charts will be discussed, including EWMA and CUSUM charts. Time permitting, the important interrelationship between statistical process control and experimental design for process improvement will be discussed.
     The text will be D.C.Montgomery, Introduction to Statistical Quality Control, 4th Ed. (Wiley).
     The final grade may be based on assignments (15%), a class test (35%), and a final examination (50%).

Prerequisites:   AS/SC/AK/MATH 3330 3.0; AS/SC/AK/MATH 3034 3.0 or AS/SC/AK/MATH 3230 3.0 or AS/SC/AK/MATH 3430 3.0.
Corequisite: AS/SC/MATH 4730 3.0.
Coordinator:  P.Peskun

AS/SC/MATH 4930B 3.0 Topics in Applied Statistics: Simulation and the Monte Carlo Method
Same as: GS/MATH 6003V 3.0

     2001/2002 Calendar copy: Introduction to systems, models, simulation, and Monte Carlo methods. Random number generation. Random variate generation. Monte Carlo integration and variance reduction techniques. Applications to queuing systems and networks.
Note: This course will NOT be offered in FW 2001. It will probably be offered in FW 2002.
     Prerequisite: In addition to the official prerequisites below, a 3000-level course in statistics, preferably mathematical statistics, such as AS/SC/AK/MATH/3131 3.0, is recommended.

Prerequisites:   AS/SC/AK/MATH 3330 3.0;AS/SC/AK/MATH 3034 3.0 or AS/SC/AK/MATH 3230 3.0or AS/SC/AK/MATH 3430 3.0.
Exclusion: AK/AS/SC/COSC 3408 3.0.