Individual Project

1998/99 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 Programmes 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:* Martin Muldoon

*Maths. for Commerce Coordinator:* Morton Abramson

*Pure Mathematics Coordinator:* Walter Whiteley

*Statistics Coordinator:* Augustine Wong

*Prerequisites: *May be taken for major credit by
students in Honours Programmes in Applied
Mathematics, Mathematics, Mathematics for Commerce,
and Statistics. Permission of the appropriate Coordinator
(above) is required. Applied Mathematics students can enrol only
after they have completed the core programme in Applied Mathematics.

**AS/SC/AK/MATH4010 6.0
Real Analysis**

1998/99 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 MATH3210 3.0. Students contemplating graduate work in mathematics are strongly advised to take this course.

The text is W. Rudin,
*Principles of Mathematical Analysis* (McGraw-Hill).

The course grade will be based on term tests and assignments (60%) and a final exam (40%).

*Prerequisite: *AS/SC/AK/MATH3210 3.0 or permission of the course
coordinator.

**Coordinator: ** S. Scull

**AS/SC/AK/MATH4020 6.0
Algebra II**

1998/99 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 abstract algebra by building on the material
of MATH3020 6.0 (or a comparable course which students may
have taken).Further possible topics:

*Group theory*: permutation
groups, simple groups, symmetry groups.

*Ring theory*: divisibility in integral domains with
applications
to diophantine equations, elements of algebraic number theory,
rings with chain conditions.

*Field theory*: field extensions with applications to
constructions with straightedge and compass.

*Boolean algebra* (time permitting): applications to circuitry
and logic, boolean rings, finite boolean algebras.

The text will be announced later.

The grade breakdown has not yet been decided.

*Prerequisite: * AS/SC/AK/MATH3020 6.0 or permission of the course
coordinator.

*Degree credit exclusion: *AS/SC/MATH4241 3.0.

**Coordinator: ** R. Burns

**AS/SC/AK/MATH4080 6.0
Topology**

1998/99 CALENDAR COPY:*Topological spaces, continuity, connectedness, compactness,
nets, filters, metrization theorems, complete metric spaces, function
spaces, fundamental group, covering spaces.*

Topology is a large and core branch of modern mathematics (along with geometry, algebra and analysis). Major parts of it can be viewed as a synthesis of geometry and analysis, strongly influenced by algebraic methods. Other components can best be described as point-set topology and will offer some insight into modern set theory. Its study clarifies the nature of concepts learned in analysis and geometry such as proximity, continuity and distance. It studies objects called topological spaces by studying the maps (functions) that they support, and their invariants.

More advanced topics in topology that this course leads to are point-set topology, algebraic topology (homology theory, homotopy theory) and differential topology (manifold theory). Topology has many applications within mathematics, but nowadays it also is used in physics and physical astronomy (e.g. cosmology) as well as in catastrophe theory and physiology, economics and sociology.

The text will be C. Wayne Patty, *Foundations of Topology*
(Waveland Press, ISBN 0-88133-955-5).

The final grade will be based on assignments (20%), class tests (40%), a project (10%) and a final examination (30%).

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: ** A. Dow

**AS/SC/MATH4110N 3.0 F
Topics in Analysis:
Ordinary Differential Equations**

(same as GS/MATH6340 3.0) 1998/99 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 AS/SC/AK/MATH2221 3.0 and AS/SC/AK/MATH3210 3.0, or seek permission to take this course from the course coordinator.

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.

*Prerequisite: *Permission of the course coordinator.

**Coordinator: ** J. Wu

**AS/SC/MATH4130B 3.0 F
Topics in Probability and
Statistics: Introduction to the Theory and Methods
of Time Series Analysis**

(same as GS/MATH6630 3.0) 1998/99 CALENDAR COPY:*A systematic presentation of many statistical techniques for the
analysis of time series data. The core topics include time
dependence and randomness, trend, seasonality and error,
stationary processes, ARMA and ARIMA processes, multivariate time
series models and state-space models.*

An additional topic is forecasting. The emphasis is on the theory and methodology of the time-domain analysis based on ARIMA and state-space models. An important component of the course is that the analysis of data sets is illustrated throughout.

The text is P.J. Brockwell and R.A. Davis,
*Introduction to Time Series
and Forecasting*, Springer-Verlag. Some materials from P.J. Brockwell
and
R.A. Davis, *Time Series: Theory and Methods*, Springer-Verlag, are
imported as necessary.

The course will be evaluated by assignments, one midterm, and a final exam.

*Prerequisite: *AS/SC/MATH3033 3.0 and AS/SC/AK/

MATH3131 3.0, or permission of the course coordinator.

*Degree credit exclusions: *SC/AS/COSC4242 3.0, SC/EATS

4020 3.0, AS/SC/MATH4830 3.0, AS/SC/MATH4930C 3.0,

SC/PHYS4060 3.0, SC/PHYS4250 3.0.

**Coordinator: ** P. Song

**AS/SC/MATH4130G 3.0 W
Topics in Applied Statistics:
Applied Categorical Data Analysis**

*Note:* This course will be offered, pending approval by the
appropriate Faculty curriculum committees after we go to press.

1998/99 CALENDAR COPY:*This course demonstrates the use of categorical data analysis
techniques within the context of epidemiology, bioassay and survival
analysis. The emphasis is on the analysis and interpretation of real
world data sets using the SAS statistical software package.*

This course is aimed at the student who wishes to gain a working knowledge of categorical data analysis methods as applied to Epidemiology, Bioassay, and Survival Analysis. The emphasis is on methods and the analysis of data sets using SAS software. The course will cover Logistic Regression, Sensitivity/Specificity Analysis, Contingency Table Analysis, Odd's Ratios, Attributable Fractions, Indirect and Direct Quantal and Quantitative Parallel Line and Slope Ratio Bioassay, Proportional Hazard and Failure Time Survival Analysis strategies.

The text will probably be A. Agresti, *Categorical Data Analysis*.

The grade will be based upon ten assignments (20%), two tests (40%) and a final exam (40%).

*Prerequisites: *AS/SC/AK/MATH3330 3.0; either

AS/SC/AK/MATH3230 3.0 or AS/SC/AK/MATH3430 3.0.

**Coordinator: ** D. Montgomery

**AS/SC/MATH4140A 3.0 F
Topics in Number Theory:
Algebraic Number Theory**

(*Note:* MATH4140 3.0 is offered on an irregular basis. Subject to
approval by the Faculty of Arts, it will be offered in the fall.)

1998/99 CALENDAR COPY:*The course covers
prime numbers, modular arithmetic, Diophantine problems,
cryptography and, possibly, some other contemporary applications.*

Number theory is an old branch of mathematics and its content is vast. We shall focus on topics in the direction of algebraic number theory. In short, this theory can be described as the study of the divisibility properties of integers. Since every integer can be factored into a product of primes, the prime numbers are of the utmost importance in the theory.

We shall begin by discussing prime numbers and modular arithmetic. We shall also study Diophantine problems, that is, integral solutions of polynomial equations with integer coefficients. This introduction to the classical theory will be followed by some contemporary applications. In particular, we shall study some elementary cryptography.

The text is Ramanujachary Kumandury and Cristina Romero, *Number
Theory with Computer Applications*, Prentice-Hall (1998).

The final grade will be based on two tests (60%) and a final exam (40%).

*Prerequisite: *AS/SC/AK/MATH3020 6.0 or permission of the course
coordinator.

**Coordinator: ** A. Ivic Weiss

[See comment after "Coordinators", MATH1310, for spelling of "Ivic".]

**AS/SC/MATH4141 3.0 F**

**Advanced
Numerical Methods**

(same as GS/MATH6651 3.0, GS/PHYS5070A 3.0) 1998/99 CALENDAR COPY:*Systems
of non-linear equations: 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, Padé
approximation; matrix eigenvalues: power method, Householder, QL and QR
algorithms.*

The mark will be based on a combination of computer-based assignments, tests and a final exam.

*Prerequisite: *AS/SC/MATH3242 3.0

or
SC/AS/COSC3122 3.0.

**Coordinator: ** A.D. Stauffer

**AS/SC/MATH4142 3.0 W
Numerical Solutions to Partial Differential Equations**

(same as GS/MATH6652 3.0) 1998/99 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/MATH2270 3.0;

AS/SC/MATH3242 3.0 or SC/AS/COSC3122 3.0.

**Coordinator: ** J. Laframboise

**AS/SC/MATH4170 6.0
Operations Research II**

(same as GS/MATH6900 3.0 F plus GS/MATH6901 3.0 W) 1998/99 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,
*Introduction to
Operations Research*; (b) H.A. Taha,
*Operations Research*.

The final grade will be based on three tests (60%) and a final examination (40%).

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.

(See next page.)

*Prerequisites: *AS/SC/MATH2010 3.0 or

AS/SC/MATH2015 3.0 or AS/SC/AK/MATH2310 3.0;

AS/SC/MATH1132 3.0 or AS/SC/AK/MATH2030 3.0;

AS/SC/AK/MATH3170 6.0; or permission

of the course coordinator.

*Degree credit exclusion: *AS/MATH4570 6.0.

**Coordinator: ** Silviu Guiasu

**AS/SC/AK/MATH4210 3.0 W
Complex Analysis**

(*Note:* This course is offered on an irregular basis.)

1998/99 CALENDAR COPY:*Development of the
principal results in complex variable theory, including Taylor
and Laurent series, the calculus of residues, the maximum modulus
theorem and some special functions. Introduction to some more
advanced topics.*

This course is the developmental side of MATH3410. In it we will develop the theory of complex differential and integral calculus. Time should permit the application of this theory to the study of certain integrals from real analysis as well as to the study of some classical functions such as the gamma function. Topics to be included are complex differentiation, the Cauchy-Riemann equations, conformal mappings, complex integration, Cauchy's theorems, Cauchy's integral formula, the theory of residues, power series, and applications.

The text will be H.A. Priestly, *Introduction to Complex
Analysis*.

*Prerequisite: *AS/SC/AK/MATH3410 3.0 or permission of the course
coordinator.

**Coordinator: ** N. Purzitsky

**AS/SC/MATH4230 3.0 F
Nonparametric Methods in
Statistics**

(same as GS/MATH6639A 3.0) 1998/99 CALENDAR COPY:*Order statistics; general rank statistics; one-sample, two-sample, and
k-sample problems; Kolmogorov-Smirnov statistics; tests of independence and
relative efficiencies.*

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 as we go to press.

*Prerequisite: *AS/SC/AK/MATH3131 3.0.

AS/SC/AK/MATH3132 3.0 is recommended but not required.

**Coordinator: ** T.B.A.

**AS/SC/MATH4280 3.0 F
Risk Theory**

1998/99 CALENDAR COPY:*A
study of the stochastic aspects of risk with emphasis on insurance
applications. Topics include an introduction to utility theory, individual
and collective risk theory, compound Poisson processes, ruin theory,
non-proportional reinsurance.*

This course is intended mainly for students contemplating a career in the actuarial profession. It will cover the complete course of reading for Examination 151 of the Society of Actuaries, which is chapters 1, 2, 11, 12, and 13 of the text mentioned below. It is essential for students to have a sound knowledge of probability theory, as would be taught in MATH2030 3.0.

The text will be N.L. Bowers et al.,
*Actuarial Mathematics* (Society of
Actuaries). This is the same text as for MATH

3280 6.0. (NOTE: It seems
likely that a second edition of the text will be available by September.
In the second edition, the relevant chapters are 1, 2, 12, 13, and 14.
Either edition could be used for this course.)

The final grade is likely to be based on assignments (20%), two tests (20% each), and a final exam (40%).

*Prerequisite: *AS/SC/AK/MATH2030 3.0;
AS/SC/MATH

3280 6.0 is recommended but not required.

**Coordinator: ** D. Salopek

**AS/SC/AK/MATH4430 3.0 W
Stochastic Processes II**

(same as GS/MATH6602 3.0) 1998/99 CALENDAR COPY:*Continuous
parameter stochastic processes: Markov jump processes, Poisson
processes, renewal theory. Topics from queuing theory, Brownian motion,
stationary processes.*

This course is an introduction to stochastic, or random, processes. Stochastic processes are models which represent phenomena that change in a random way over time. Simple examples are (a) the amount of money a gambler has after each play of a game and (b) the number of people waiting for service at a bank at various times. This course studies some of the most basic stochastic processes, including Markov chains and Poisson processes. A Markov chain is a stochastic process in which predictions for the future depend only on the present state of affairs, but not on knowledge of the past behaviour of the process. Markov chains are relatively easy to analyze, and they have been used as models in many areas of science, management, and social science. A Poisson process is a model for the occurrence of random events (such as oil spills in the Atlantic Ocean). This course will treat both the theory and applications of these stochastic processes.

*Prerequisite: * AS/SC/AK/MATH2030 3.0

or AS/SC/AK/MATH3030 3.0.

**Coordinator: ** T.B.A.

**AS/SC/MATH4630 3.0 W
Applied Multivariate Statistical Analysis**

(same as GS/MATH6625 3.0) 1998/99 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.*

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 students to enrol who have background ``equivalent to'' the formal prerequisites below.

*Prerequisites: *AS/SC/AK/MATH3131 3.0;

AS/SC/MATH3034 3.0 or AS/SC/MATH3230 3.0;

AS/SC/MATH2022 3.0 or AS/SC/AK/MATH2222 3.0.

**Coordinator: ** G. Denzel

**AS/SC/MATH4730 3.0 F
Experimental Design**

(same as GS/MATH6626 3.0) 1998/99 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.*

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.

For the text and references, please see the web site:

http://www.math.yorku.ca/Who/Faculty/Ng/menu.html

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

*Prerequisites: * A second 6 credits in statistics,

including either AS/SC/MATH3033 3.0, or both

AS/SC/AK/MATH

3230 3.0 and AS/SC/AK/MATH3330 3.0,

or permission of the course coordinator.

**Coordinator: ** P. Ng

**AS/SC/MATH4930A 3.0 F**

**Topics in Applied Statistics:
Statistical Quality Control**

1998/99 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.*

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*, 2nd ed. (Wiley)

The final grade may be based on assignments (15%), a class test (35%), and a final examination (50%).

*Prerequisites: * AS/SC/AK/MATH3330 3.0;

AS/SC/AK/MATH3230 3.0 or AS/SC/AK/MATH3430 3.0.

*Corequisite: *AS/SC/MATH4730 3.0.

**Coordinator: ** P. Peskun

Contents Page, 1998/99 Minicalendar

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