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NOTE: Faculty of Graduate Studies regulations regarding integrated courses (6000/5000 level course with a 4000 level course):  

Masters students who are enrolled in a thesis option must complete at least one full course (or equivalent) which is not integrated with an undergraduate course. Master’s students who are enrolled in a course work or research-review option must complete at least one and a half (or equivalent) courses, which are not integrated with an undergraduate course. Doctoral candidates shall not receive credit towards the Ph.D. degree for more than one full integrated course.

Math 5300.06


(FW 1999-2000, T 6-9, T107 Steacie)

The purpose of the course is to introduce students to the Maple language for doing symbolic as well as numerical mathematics. This computer system is able to do much of the tedious or routine calculations which need to be done in solving many mathematical problems, thereby, allowing the student to concentrate on the conceptual difficulties rather than on the mechanical calculations. Students will be introduced to the language by working on problems from high school algebra, calculus and other areas of elementary mathematics. The recommended text is "A Guide to Maple" by Kammerlich. The text is available in the campus book store. The course work will be consistent of submitting solutions to assigned projects. Students will be expected to provide high quality expositions of their solutions. Later in the course, students will learn how to use Maple to create web pages and will be expected to use this technology to provide solutions which can be used as resource material in high school courses.

Course director : J. Steprans, N536 Ross, 736-5250

Math 5410.06


(FW 1999-2000, R 6-9, 3005 VH)

Calculus (analysis) was invented by Newton and Leibniz in the 17th century, but it took another two centuries to provide it with rigorous foundations. At the same time, some calculus-type problems were solved by the Greeks ca. 250 B.C.

Calculus was geometric in the 17th century, algebraic in the 18th, and arithmetical in the 19th, at which time the limit concept took centre stage. The 19th century saw the "arithmetization of analysis" - its logical reduction to properties of the real numbers, and even further, to those of the positive integers. One of the objectives of the course will be to explain this process.

More generally, a major focus of the course will be on foundations; in particular, on showing that some of the fundamental concepts of calculus, such as the real numbers, functions, continuity, and integrability are considerably more subtle than might appear, and that intuition, though an important guide, can be misleading.

Calculus includes three major elements: a set of rules or algorithms (a "calculus"), a theory to explain why the rules work, and applications (of the theory and the rules) to fundamental problems in science. We will exhibit the power of the calculus by focusing on its solution of important classical problems such as the Catenary and Brachistochrone problems. The major device here is differential equations - both ordinary and partial.

Infinitesimals were a fundamental tool in calculus during the 17th, 18th, and part of the 19th centuries (what is an infinitesimal?), but were abandoned, for lack of logical justification, in favor of limits in the second half of the 19th century. About a century after they were banished from analysis "for good" (so we all thought until 1960), they were brought back to life (in 1960) as genuine and rigorously defined mathmatical objects in the "nonstandard analysis" of Abraham Robinson. We will describe the underlying ideas of Robinson's infinitesimals and how they "rehabilitated" the infinitesimals of Leibniz, Euler, Cauchy, et al.

Finally, we will examine several interesting results, some seemingly unrelated to calculus, which can be derived by using calculus. Among them are the irrationality and transcendentality of e and _, the infinitude of primes, Leibniz' formula _/4 = 1-1/3+1/5-1/7+. . ., and Euler's result _2/6 = 1+1/4+1/9+1/16+. . . .

Course director : I. Kleiner, N618 Ross

736-5250, ext. 66095


Math 6003K.03W


(W 2000, TR 11:30-1:00, S101A Ross)

Category theory has become an important tool not only in mathematics but also in theoretical computer science. In this course the basic elements of lattice and category theory are presented as needed in computer science. We then introduce several formal systems which may be viewed as very basic programming languages. For each of these type theories, a categorical semantis is derived from first principles, and soundness and completeness results are proved. Time permitting, specific examples of categorical models will be given.

A desirable corequisite is Math 6120.06 or equivalent, or permission by the instructor.

Text: Roy L. Crole, Categories for Types, Cambridge University Press, 1993.

Course director : W. Tholen, N605 Ross


Math 6003L.03W


(Math 4130K.03W) (W 2000) T 3:30-5:00, 221 Stong; R 3:30-5:00, 122 CCB)

This course extends the ability of students in analyzing censored data which may not be encountered in other courses. As the interest in the topic mainly stems from medical statistics and industrial life-testing, we emphasize on the applications of this course in medical research, life-testing and related fields.

Topics include:

Accelerated life models, exponentially distributed failure times, nonparametric methods, dependence on explanatory variables and proportional hazard model, semi parametric models, time dependent covariates.

If time permits, we will also discuss bivariate survivor functions, competing risk, and Bayesian nonparametric methods.

Prerequisites: AS/SC/AK/Math 3131.3 or equivalent and one of the following:

1. AS/SC/Math 3033.3 or equivalent

2. AS/SC/AK/Math 3330.3 or equivalent

Text:  J.D. Kalbfleisch and R.L. Prentice, The Analysis of Failure Time Data, John Wiley and Sons, 1980.

Course director : M. Asgharian, N621B Ross


Math 6003M.03F


(F 1999, T 2:30-4:00, 122 CCB; R 2:30-4:00, 225 BC)

Hyperbolic n-space is constructed as one of the two sheets of the hyperbola x02-x12-x22...-xn2=1. The natural framework for this construction is the theory of quadratic forms. We shall start with a general introduction to quadratic forms. The geometry of the hyperbolic plane will be studied in detail leading to the relationship between hyperbolic geometry and discrete groups of isometries. If time allows, we shall prove the Poincaré's polygon theorem.

Prerequisites:  The general prerequisites are linear algebra, elementary point set topology and some familiarity with combinatorial group theory ("generators and relations").

Course director : A. Weiss, S618 Ross


Math 6004.00


This course provides students with a chance to work independently and to present the results of their work to other students. Each student gives two one-hour seminars on topics arranged with one or two faculty members. The topics may be related to other courses the student is taking, but should not actually be covered in those courses. They may be in the same field or two different fields. Students are expected to submit a written report prior to presenting each seminar. The seminars are graded separately and the course is graded on a pass/fail basis. Students in the course are expected to attend all seminars.

Math 6030.03F


(F 1999, T 10:30-12:00, S536 Ross; R 10:30-12:00, S170 Ross)

Logic is the study of reasoning. Mathematical Logic studies the kind of reasoning practiced by mathematicians. The modern use of the title suggests the inclusion of proof theory, model theory, recursion theory, and set theory.

Under the first subtitle the objects of study are (mathematical) proofs. In particular, questions such as ``what statements are `reachable' by proofs in a given axiomatic theory'', and ``is there an algorithm which separates the theorems from the non theorems of a particular theory'' are dealt with.

Model theory studies the connection between the syntax and semantics of mathematical theories. Gödel's completeness theorem as well as the Gödel-Mal'cev compactness theorem belong here. One of the most beautiful applications of model theory is the legalization of infinitesimals by Robinson.

Recursion theory, narrowly speaking, addresses the question of what is, and what is not, ``mechanically'', or ``effectively'', computable (once it gives a formal definition of what the terms in quotation marks ought to mean). Broadly speaking, it studies recursive (or inductive) definitions on all sorts of mathematical structures.

Set theory is the subtitle of logic that is most familiar to the working mathematician. An oversimplified view of the subject would be ``the study of the properties of collections of mathematical objects'' (for more details consult MATH 6040 3.0).

In this course we shall mostly deal with proof theory and recursion theory, and we will also do a fair bit of elementary model theory as outlined below. Our primary goal will be a thorough understanding of Gödel's incompleteness theorems as well as the related undecidability result of Church. We shall address these results at several levels of abstraction.

Recursion theory will be studied to the extent that it will serve the above goal. Aspects of model theory (compactness, completeness and Löwenheim-Skolem results, ultrafilters) will be covered, with a quick introduction to nonstandard analysis ā la Robinson.

Prerequisites: Nothing specific, except ``mathematical maturity''at such level that a 3rd--4th year undergraduate or 1st year graduate mathematics student ought to normally have.

Previous courses in recursion theory and/or (undergraduate) ``logic'' (this is, usually, mostly proof theory) would make life easier.

Work load/grading: Several homework assignments, equally weighted. No exams/tests in class.


_ H. Enderton, A Mathematical Introduction to Logic, Academic Press (this is our ``official text'').

_ E. Mendelson, Introduction to Mathematical Logic, 3rd Edition, Wadsworth Math. Series.

_ Chang and Keisler, Model Theory, North Holland.

_ Yu. I. Manin, A Course in Mathematical Logic, GTM, Springer-Verlag.

_ J.R. Shoenfield, Mathematical Logic, Addison Wesley.

_ R. Smullyan, First-Order Logic, Springer-Verlag.

_ R. Smullyan, Gödel's Incompleteness Theorems, Oxford Logic Guides, 19, Oxford University Press.

_ G. Tourlakis, Computability, Prentice-Hall.

_ J. Barwise, Handbook of Mathematical Logic, in particular A.1, C.1; possibly D.1.

Course director : G. Tourlakis, Rm. 354, CCB.

736-2100 ext. 66674.

E-mail: gt@cs.yorku.ca (NeXT or MIME mail OK)

Math 6120.06


(FW 1999-2000, MWF 11:30-12:30, S101A Ross)

This course presents the most important elements of modern algebra, as described by the following headings: group theory (Sylow theorems, finitely generated abelian groups); ring- and module theory (unique factorization domains, free modules, finitely generated modules over a PID); field theory (finite fields, Galois theory).

Final grades are based on homework assignments, class test and possibly an oral presentation.

Text:  D. Dummit & R. Foote, Abstract Algebra, Prentice Hall, 1991.

Course director : N. Bergeron, S626 Ross


Math 6280.03F


(F 1999, MWF 12:30-1:30, S101A Ross)

A measure is a function that assigns numbers to subsets of a given set. For example, the Lebesgue measure of a subset of the plane is the area of the subset for those sets that have a classically defined area, and it also extends the definition of area to a much wider class of subsets. Measures are also important in probability theory, where

events correspond to subsets and measures assign probabilities to events. More generally, measure theory is central to analysis, since measures are used to construct integrals and conversely integrals give rise to measures.

This course begins with the classical theory of Lebesgue measure and Lebesgue integration on the real line. We will then examine the general theory of measures on abstract spaces.

Topics include: Sigma-algebras, measure spaces, measurable functions, outer measure and measurability, extension theorems, integration, convergence theorems, signed measures, Hahn-Jordan decomposition, Radon-Nikodym theorem, product measures, Fubini theorem.

Text: H.L. Royden, Real Analysis, third edition, Macmillan 1988.

Course director : N. Madras, N623 Ross


Math 6340.03F


(Math 4110N.03F) (F 1999, MF 3:30-5:00, 122 CCB)

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, and dynamical systems.

Students should have a thorough knowledge of undergraduate analysis and linear algebra to the level of MATH 2220 and MATH 3210. It would be desirable but not essential that they have taken an undergraduate course in differential equations. Some exposure to real analysis, complex analysis and topology would be desirable also.

Text:  Lawrence Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 1991.


-J.K. Hale and H. Kocak, Dynamics and Bifurcations, Springer-Verlag, 1991.

-M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974.

Course director : J. Wu, N518 Ross


Math 6350.03W


(W 2000, T 10:00-11:30, 301 SSB; R 10:00-11:30, 326 BC)

This is primarily a basic course in the modern theory of partial differential equations using pseudo-differential operators. We shall begin the course with a self-contained treatment of Fourier analysis and the theory of a standard class of pseudo-differential operators, and then proceed to use the techniques so developed to study the existence and regularity of weak solutions of elliptic partial and pseudo-differential equations.

The main prerequisite for the course is familiarity with the theory of Lebesgue measure and integration.

The final grade will be based on several homework assignments (60%) and a final examination (40%).

Text: M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999.

Course director : M.W. Wong, S518 Ross


Math 6461.03W


(W 2000, MWF 1:30-2:30, 327 BC)

Functional Analysis is a subject of great importance, with connections to both pure and applied mathematics, as well as many branches of physics. It deals primarily with linear functionals and operators on infinite dimensional linear spaces. The subject was motivated by certain problems in classical analysis, which considered various spaces of functions, together with integral and differential operators on these spaces.

This course provides an introduction to the subject. We consider general topological vector spaces, but with an emphasis on the more concrete examples of Banach spaces and Hilbert spaces. Topics include: the Hahn-Banach Theorem, duality theory, uniform boundedness principle, open mapping and closed graph theorems, spectral theory of compact operators.

The main prerequisite for the course is a good undergraduate course in real analysis (such as the York course Math 4010.06). Students should also be familiar with basic concepts of linear algebra. A knowledge of measure theory will be helpful, but it is not essential.

The evaluation in the course will depend on written assignments (65%) and a final examination (35%).

There is no official textbook. Following is alist of references which cover most of the material. They have been placed on reserve in the Steacie Library.

_ Bollobas B., Linear Analysis, Cambridge University Press, Cambridge, 1990. Chapters 2-6, 8-13.

_ Kadison, R.V, & Ringrose, J.R., Fundamentals of the Theory of Operator Algebras, Vol 1, Academic Press, N.Y., 1983. Chapter 1, Chapter 2 (sections 1-4).

_ Pedersen, G.K., Analysis Now, Springer-Verlag, N.Y., 1989. Chapter 2, Chapter 3 (sections 1,2).

_ Royden, H.L., Real Analysis, 3rd. ed., Macmillan, N.Y., 1988. Chapter 10.

_ Rudin, Real and Complex Analysis, 2nd. ed., McGraw Hill, N.Y., 1974. Chapters 3-5.

Course director : S.D. Promislow, N518 Ross


Math 6540.03F


(F 1999, TR 4:00-5:30, 120 CCB)

Topological ideas form the foundations of many fields in mathematics. The aim of the course is to provide and introduction to these ideas which is intended to be an enjoyable excursion rather than an encyclopedic treatment as one might find in a reference manual. Most

topics will be developed in a quasi historic manner and so that the student can see how topologists have gradually refined and extended work of their predecessors. Throughout the course there will be a definite geometric flavor.

The course may considered to consist of three parts: Part 1 will deal with the toplogy of complete metric spaces including their hyperspaces of sequentially compact subspaces which in turn will lead to a discussion of fractional Hausdorff dimension and a brief introduction to fractals. Part 2 will develop the ususal topics in topics in general topology, including nets, filters, adjunction spaces, topological bases, connectedness, separation axioms, various forms of compactness, normal and regular spaces, metrizations theorems, the Urysohn Lemma, and the Tychonoff Theorem. Part 3 will consist of a very brief introduction to ideas in homotopy theory and the classification of surfaces.

Weekly assignments will be given and a good part of the course will focus on the solutions of various exercises. There will be a mid-term test in late October and a final take-home examination in mid December.

Grading will be done according to the following formula:

Final exam 50%

Mid-term test 30%

Homework 20%

Text: Sierdaski, A., An Introduction to Topology and Homotopy, PWS Kent (Boston), 1992.

Course director : M. Walker, 528 Atkinson College


Math 6550.03W


(W 2000, MWF 10:30-11:30, 327 BC)

Algebraic topology solves problems in topology by studying algebraic invariants associated to topological spaces. This point of view has been useful for studying certain problems in analysis and algebra as well as in topology.

We will begin by studying the fundamental group using covering spaces. Then we will define singular homology and determine its basic properties. Cell complexes will be introduced to make computations. Applications will be made to the study of surfaces, the generalized Jordan curve theorem and invariance of domain. If time permits we will study singular cohomology and prove various duality theorems.

Prerequisite: Mathematics 6540.03 or a knowledge of basic general topology; undergraduate algebra including the fundamental theorem of abelian groups.

Text:  Greenberg and Harper, Algebraic Topology, A First Course, Addison-Wesley Publishing Co., 1981.

Course director : S.O. Kochman, N510 Ross


Math 6602.03W


(Math 4430.03MW) (W 2000, MWF 9:30-10:30, 122 CCB)

This course is an introduction to stochastic processes. Informally, a stochastic process is a random quantity that evolves over time. Examples of this phenomenon include the total number of emissions of a radioactive substance measured from a given time, a gambler’s net fortune, and the price fluctuations of a stock on the New York Stock Exchange. This course will cover discrete- and continuous-time Markov chains, Poisson process, and Brownian motion. The course will emphasize both theory and applications of these stochastic processes.


30% Two tests (15% each)

10% Project

20% Assignments (6 assignments). You can work in GROUPS OF UP TO 3 people. Hand in one copy of the solutions with ALL the names on it. Assignments are due in class.

40% Final exam

I will take the best 5 grades on the assignments. All assignments, tests, and exam marks should be interpreted as raw scores and not as percentage. That is, they can not necessarily be converted into letter grades according to York’s default scheme for such conversions. Cutoffs will be announced for converting midterm scores into letter grades.

Text: Taylor and Karlin, An Introduction to Stochastic Modelling (3rd edition), Academic Press.

Office Hours: Mondays & Wednesdays 2:00-3:00 or by appointment.

Course director : D. Salopek, 227 Petrie

736-5250 ext. 20882

Math 6620A.03F


(F 1999, TR 1:00-2:30, 209 FS)

The topics of the course include: Review undergraduate level mathematical statistics, exponential family and group family, minimal sufficiency, completeness and Basu's Theorem, decision theory, UMVU and equivariant estimators, Baysian and minimax estimators, admissibility and Stein paradox. As time and interest permit, further related topics may also be covered.

Prerequisite:  Math 3132 or equivalent.

Course director :  S. Chamberlin, N628 Ross


Math 6620B.03W


(W 2000, MWF 12:30-1:30, 107 Stedman)

The topics of the course include: Methods of large sample theory including laws of large numbers and central limit theorems, optimality theory, theory of hypothesis testing, confidence regions, etc. As time and interest permit, further related topics may also be covered.

Prerequisites:  Mathematical Statistics I

Course director :  S. Chamberlin, N628 Ross


Math 6621.03F


(F 1999, MF 1:30-2:30, N143 Ross; W 1:30-2:30, 115 CCB)

This course will assume a familiarity with multiple regression, as covered, for instance, in MATH 3033 or MATH 3330, and with basic mathematical statistical theory, at the level of Hogg and Craig or higher and basic matrix algebra.

In this course, a theoretical basis in linear models is provided, including standard topics as distribution theory, least square point estimation and hypothesis testing in the full or non-full rank cases, multiple comparison procedures and analysis of variance. Theory will be applied to a variety of data sets with intensive use of statistical computing packages. Regression diagnostics including the identification of outliers and model selection procedures will also be developed.

The final grade will be determined by a combination of exams, homework assignments and projects.

All students will need access to the departmental computer lab in N604 Ross. The lab has a variety of terminals and manuals for faculty/student use.


_ R. Christensen, Plane Answers to Complex Questions: The Theory of Linear Models, 2nd. ed., Springer-Verlag, 1996.

_ N.R. Draper and H. Smith, Applied Regression Analysis, 3rd. ed., Wiley, 1998.

Course director : Y. Wu, N609 Ross


Math 6622.03W


(W 2000, M 1:30-2:30, 101A CS; W 1:30-2:30, 204 BS; F 1:30-2:30, 1158 VH)

The following topics in analyzing catagorical data will be covered: Analysis of contingency tables: log linear models, linear logit models; Generalized linear models: model specification, link functions, measures of discrepancy, fitting algorithms, examples of applications to continuous, binary and polytomous data.

The final grade will be determined by a combination of exams, homework assignments and projects.

All students will need access to the departmental computer lab in N604 Ross. The lab has a variety of terminals and manuals for faculty/student use.

The text will be announced later.

Course director : Y. Wu, N609 Ross


Math 6623.03F


(F 1999, T 10:00-11:30, S501Ross; R 10:00-11:30, S169 Ross )

A survey of statistical inference for location-scale models and exponential family models. Both exact and asymptotic methods will be examined.

Students should have a reasonable knowledge of basic undergraduate mathematical statistics before taking this course.

Grading will be based on assignments, presentations and tests.

Course director : A. Wong, N633 Ross


Math 6625.03W


(Math 4630.03MW) (W 2000, TR 1:30-3:00, 327 Bethune)

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 based on linear models. 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 or S+ will be assumed. Topics covered will include the multivariate normal population, inference about means and covariance, multivariate linear models, principal component analysis, and some discussion of canonical correlation analysis, discriminant and classification, factor analysis and cluster analysis, as time permits.

Grades will be based on a combination of class quizzes and a final examination, plus homework including a group project. The coordinator may permit sudents to enrol who have background "equivalent to" the formal preerequisites below.

Prerequisites: AS/SC/AK/MATH3131 3.0;

AS/SC/ MATH 3034 3.0 or AS/SC/ MATH 3230 3.0; AS/SC/ MATH 2022 3.0 or AS/SC/AK/MATH 2222 3.0

Course director : M. Asgharian, N621B


Math 6626.03F


(Math 4730.03AF) (F 1999, T 12:30-1:30, R 12:30-2:30; S201 Ross)

Statistical issues involved in ensuring an experiment to yield relevant information are examined. Topics include: completely randomized, randomized block, factorial, fractional factorial, nested, and Latin square designs. Further topics as time permits.

Good experimental design is the cornerstone for the generation of good data. It can be viewed as selecting the best feasible experiment to achieve some specific objective. This includes the choice of treatments, and the random allocation of experimental units to them. Method of estimation is an important component of the determination of a design. One must consider how the data will be analysed after the experiment is carried out and data are collected. With the analytical procedure in mind, a proper choice of experiment is then determined to achieve that goal.

Various designs will be discussed in this course through definition of objectives, analytical procedures, and feasibility of experimental constraints. Statistical programs for sample size and power analysis will also be introduced.

Text and References:

Available on web site:


The final grade may be based on assignments (15%), 1 project (10%), midterm test (30%), and a final examination (45%).

Prerequisite:   AS/SC/MATH3033.03, or AS/SC/ MATH 3330.03, and AS/SC/ MATH 3230.03, or permission of the course director.

Course director : P. Ng, N601B Ross


Math 6630.03W


(Math 4130B.03W) (W 2000, MWF 10:30-11:30, 228 BC)

This course is intended to provide a systematic presentation of many statistical concepts and techniques for time series analysis. The core topics include time dependence and randomness, trend, seasonality and error, stationary process, ARMA and ARIMA processes, multivariate time seires and state-space models.

Text: P.J. Brockwell and Davis, Time Series: Theory and Methods, Springer-Verlag.

Some materials from P.J. Brockwell and Davis, Introduction to Time Series and Forecasting, Springer-Verlag, may be imported as necessary.

The evaluation of the course will be based on assignments, one midterm, one project and a final exam.

Course director : P. Song, N636 Ross


Math 6638.03W


Many graduates from programmes in statistics enter jobs in which initial success depends on their effectiveness as statistical consultants - and they find themselves needing skills very different from those they developed while doing traditional course work.

Students have learned to expect that problems are well posed and contain all the information necessary for a correct statistical solution. In the real world of consulting clarifying and identifying the actual problem is often half the problem (and half the solution), the other half being the study design and statistical analysis of the data and its interpretation.

The purpose of the practicum course is to prepare students for the transition to the "real world" of consulting:

_ by developing the skills for independent study and research that will be needed to continually broaden and update their knowledge of statistics,

_ by gaining experience in the clarification and analysis of real problems as they are presented by clients,

_ by becoming aware of the human and ethical aspects of consulting that play an important role in the effectiveness of the statistician's contribution.

_ by gaining experience in the application of statistical methods to real problems in a context that simulates as closely as possible a real situation.

Course content: The student will be asked to attend four (likely) consulting sessions between clients and faculty consultants and a project in which s/he will act as a statistical consultant for a client with problems that can be handled most of the time with standard statistical techniques (multiple regression, analysis of variance, basic contingency tables, generalized linear models). Students are expected to learn more advanced techniques as required pertaining to the study objectives.

Sometimes, the student will also explore the literature on non standard issues arising in the course of their work on the project. The student will prepare an effective report using terms and concepts that would be understood by a client. Graphs and charts are strongly encouraged. The student may also be called upon to participate in and/or lead discussions in the statistical consulting service's regular meetings on various topics including: the client-consultant interaction, model finding, report preparation, the role of the statistician in scientific inference, principles and ethics, applying a deductive descipline to inductive problems.

Pre and Corequisites:  Before taking this practicum students should have had a good undergraduate background in statistics, including a full course in mathematical statistics, a full course in linear models (regression and analysis of variance) and the equivalent of two half-courses in more advanced topics such as time series, experimental design, sample survey methods, non-parametric methods or multivariate analysis. Students must have taken or must take concurrently the courses MATH6621.03F: Linear Models and Regression and MATH6622.03: Statistical Techniques.

Ph.D. students in statistics should enrol in this course to fulfil part of their degree requirements (see page 7 under "Breadth Requirement and Comprehensive Examinations"). The Regular M.A. Programme also includes this course in one of the sets of core course requirements.

Meetings: The class will meet every second Wednesday, starting September 15, 1999 (3:30-5:30 p.m., 120 CCB (fall); 122 CCB (winter)) for two terms.

Course director : P. Ng, N601B Ross


Math 6651.03F


(4141.03AF) (F 1999, MWF 2:30-3:30, S101 Ross)

Systems of nonlinear equations: Newton-Raphson iteration, quasi- Newton methods; optimization problems: steepest descents, conjugate gradient methods; linear and nonlinear approximation theory. Least squares, singular value decomposition, orthogonal polynomials, Chebyshev and Fourier approximation, Pade approximation; matrix eigenvalues: power method, Householder, QL and QR algorithms.

The text will be announced later.

The mark will be based on a combination of computer-based assignments, tests, a project and a final exam. The project will consist of solving a non-trivial numerical problem and preparing both a written and oral report.

Prerequisites: A previous course in Numerical Methods plus programming experience in a higher level language (e.g. C, FORTRAN).

Course director : A. Stauffer, 223 Petrie

736-5248, Ext. 77742

Math 6652.03W


(4142.03MW) (W 2000, MWF 2:30-3:30, S175 Ross)

Review of partial differential equations; well-posed boundary-value problems; finite-difference approximations of derivatives. Parabolic equations: reduction to dimensionless form; solution by explicit method, Crank-Nicholson method, and hopscotch method. Elliptic equations: review of Jacobi and Gauss-Seidel methods; successive overrelaxation method; non-Cartesian grids; finite-element grids; Bubnov-Galerkin method; direct solution of elliptic boundary-value problems. Hyperbolic equations: linear wave equation; method of characteristics. Alternating-Direction Implicit method; two-dimensional hopscotch method; Incomplete Cholesky-Conjugate Gradient method; finite-volume method. Convergence and stability of solution methods.

Although less elegant than the analytic methods studied in MATH 4270 3.0, Integral Transforms and Equations / MATH 6269 3.0, Advanced Topics in Analysis, the numerical methods studied here are applicable to a much wider variety of spatial domains, and are therefore of wide use in industrial calculations of heat flows, diffusion, fluid motions, stresses in solids, andelectromagnetic fields and waves.

There is no textbook, and the lecture notes are essential.

Useful references are:

_ W.H. Press et al, Numerical Recipies, Cambridge

_ R.L. Burden and J.D. Faires, Numerical Analysis, Brooks/Cole.

Course Director : J.G. Laframboise, 228 Petrie

736-5248, Ext. 55621 or 736-5621

Math 6900.03F


(Math 4170.06AY) (F 1999, TR 8:30-10:00, S201 Ross)

This course deals with deterministic and probabilistic models based on optimization. The following topics will be discussed: 1) game theory (how to find the best strategies in a confrontation between two players with opposite interests); 2) decision theory (how to act in order to minimize the loss subject to the available data); 3) simulation (how to sample from a probability distribution and accurately approximate multiple integrals using random numbers); 4) reliability theory (how to evaluate the lifetime of a system consisting of many interacting subsystems). Each chapter contains a specific optimization problem and methods and algorithms for solving it. The course is rich in examples.

There is no textbook and the lecture notes are essential.  Useful  books  are: (a) F.S. Hillier and G.J. Liberman, Introduction to Operations Research, 4th ed. (Holden-Day); (b) H.A. Taha, Operations Research, An Introduction, 4th ed. (MacMillan).

Course director : S. Guiasu, N530 Ross


Math 6901.03W


(Math 4170.06AY) (W 2000, TR 8:30-10:00, S201 Ross)

This course deals with deterministic and probabilistic models based on optimization. The following topics will be discussed:

1) queuing theory (how to assess what may happen in a system where the customers arrive randomly, wait in line, and then get served); 2) uncertainty (how to measure the amount of uncertainty contained by a probabilistic experiment, with applications in classification, pattern-recognition, and forecasting.) Each chapter contains a specific optimization problem and methods and algorithms for solving it. The course is rich in examples.

There is no textbook and the lecture notes are essential. Useful books are: (a) F.S. Hillier and G.J. Liberman, Introduction to Operations Research, 4th ed.

(Holden-Day); (b) H.A. Taha, Operations Research, An Introduction, 4th ed. (MacMillan).

Course director : S. Guiasu, N530 Ross


Math 6910.03F


(F 1999, WF 9:30-11:30, S173 Ross)

Stochastic Calculus is the mathematical foundation of the theory of no-arbitrage asset pricing, in the continuous-time setting. This course carries the developemnt of the subject to the point that students can understand the mathematics of risk-neutral asset valuation and the Black-Scholes-Merton option pricing formula. No prior exposure to measure theory or stochastic processes is assumed, so probabilistic and analytic background material will be introduced as needed. The prerequisites are calculus and some elementary probability or statistics.

The emphasis of the couse is not on a rigorous derivation of the mathematical theory, but rather on the basic ideas, the pitfalls, and the computational aspects of the subject.

Topics will include: Probability models, sigma-fields and filtrations, Markov chains, stochastic processes, Brownian motion, martingales, stochastic integrals, Ito's formula, stochastic differential equations, hitting probabilities and their relation to partial differential equations, the Black-Scholes-Merton formula, diffusions, Girsanov transformations, risk-neutral valuation, no-arbitrage option pricing, interest rate models.

Text: Oksendal, Stochastic Differential Equations (5th edition), Springer-Verlag.

Course director : T. Salisbury , N625 Ross

Course Page




Math 6911.03W


(W 2000, MF 3:30-5:00, 3006 VH)

Background materials in partial differential equations: classiciations of elliptic, parabolic and hyperbolic equations; examples of exact solutions including the Black-Schole equations and perpetual puts; information flow and propogation for problems in finance. Finite difference methods for parabolic equations; explicit methods; implicit methods, including Backward Euler method and Crank-Nicolson method; issues of stability and convergence; applications to finance, including the effects of boundary conditions, dividends and transaction costs; degeneracy and Asian option, higher order discretization techniques.

Course director : H. Huang, N622 Ross


Math 6940.03F


(F 1999, MWF 8:30-9:30, 1022VH)

Coding theory is the application of techniques in Algebra and Number Theory to the study of information storage, transmission and retrieval. It is of interest both to pure mathematicians and applied mathematicians because it provides a large variety of examples of applications of abstract algebra, and, conversely, many of the ideas in modern algebra and geometry have come from coding theory. The course will begin with a brief introduction to the principal concepts of coding theory, error correcting codes, parity check matrices, etc. Certain algebraic concepts, such as finite fields, etc. will be developed in class. The only background necessary will be a thorough understanding of linear algebra.

We will examine applications of coding theory to other branches of mathematics, both pure and applied, including:

_ Game Theory

_ Information Storage and Retrieval, Applications to Computers

_ Group Theory, especially the Sporadic Simple Groups

_ Sphere Packings and Lattices

_ Techniques for Coding Compact Disks

_ Coding Theory and Geometry

_ Cryptography

No textbook is required. We shall be referring to various books and papers, such as:

_ MacWilliams and Sloane, The Theory of Error Correcting Codes, North Holland.

_ Conway and Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag.

_ Conway and Sloane, Lexicographical Codes: error-correcting codes from game theory, PGIT 32(1986), 337-348.

Course director : A.P. Trojan, 532 Atkinson


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