York University

Undergraduate Minicalendar

Fall/Winter 1997/98

**MATH4000.03FW/.06 Individual Project**

(pending approval)

*1995-95 York Calendar: *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 4000.06) or half-year (for 4000.03) 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:*
Open to 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/MATH4010.06 Real Analysis**

*1995-95 York Calendar: *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 an introductory analysis course, MATH3210.03. 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 final grade will probably be based on assignments and a project (30%), three class tests (40%), and a final examination (30%).

*Prerequisite:* AS/SC/AK/MATH3210.03 or permission of
the course coordinator.

**Coordinator: ** N. Madras

**AS/SC/AK/MATH4020.06 Algebra II**

*1995-95 York Calendar: *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 MATH3020.06 (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/MATH3020.06 or permission of the
course coordinator.

**Coordinator: ** A. I. Weiss

**AS/SC/MATH4110N.03W Ordinary Differential Equations**

(same as GS/MATH6340.03)

*1995-95 York Calendar: *MATH4110: Topics in Analysis:
One or two
topics which may be chosen from the following: special functions,
integral transforms, Fourier series, divergent series, asymptotic
expansions, theory of approximation, partial differential equations,
calculus of variations, calculus of manifolds, introduction to functional
analysis, difference equations.

This is an advanced introduction to a number of topics in ordinary differential equations. The topics will be chosen from the following: existence and uniqueness theorems, qualitative theory, osciallation and comparison theory, stability theory, bifurcation, dynamical systems, boundary value problems, asymptotic methods. The lectures will survey the above topics and students will be expected to make an in-depth study of some of them by doing assignments and projects.

Students should have a thorough knowledge of undergraduate analysis and linear algebra to the level of MATH2222 and MATH 3210. It would be helpful to have taken an undergraduate course in differential equations. Some exposure to real and complex analysis and topology would be desirable also.

The textbook and grading scheme have not yet been determined.

*Prerequisite:* Permission of the course coordinator.

**Coordinator: ** Y. Yang

**AS/SC/MATH4120A.03W Topics in Algebra: Advanced Linear Algebra**

*Note: As of September 1997, this course lacks official approval by the
Faculty of Arts and the FPAS, but it is hoped that it will be offered
in Winter term. *

One advantage of a 4000-level topics course is that the course content can be sufficiently flexible to accommodate the common interests of the professor and the students. Thus, a selection of advanced topics will be proposed for study, from which choices will be made at the beginning of the semester based on input from the students.

*Sample of possible topics:*
Detailed proofs of the theorems concerning determinants; the
orthogonal diagonalizability over $\Bbb{R}$
of real symmetric matrices; the unitary
diagonalizability over $\Bbb{C}$
of normal complex matrices; various proofs of the
Jordan canonical form theorem; the Gershgorin disk theorem; the
Perron--Frobenius theorem; Kronecker products of matrices; matrix norms;
generalised inverses; infinite-dimensional vector spaces; linear algebra when
the scalars form a ring rather than a field; singular-value decomposition of
matrices; incidence matrices of graphs.

We could also consider some of the applications of linear algebra that appear at the end of the textbook by Anton and Rorres that is used in the sequence MATH2221/2222 but that were not covered in that course for lack of time.

There is no textbook for the course. A collection of books and journal articles will be placed on reserve in Steacie Library and students will make their own personal photocopies of the relevant portions. Here is a partial list of books:

- A. Berman and R. Plemmons,
*Non-negative Matrices in the Mathematical Sciences*. - S. Friedberg, A. Insel, and L. Spence,
*Linear Algebra*. - R. Horn and C. Johnson,
*Matrix Analysis*. - B. Noble and J. Daniel,
*Applied Linear Algebra*. - J. Ortega,
*Matrix Theory, a Second Course*. - E. Seneta,
*Non-negative Matrices and Markov Chains*. - G. Strang,
*Linear Algebra and its Applications*.

The principal component of the grade will be based on class participation, with the balance based on homework. Students will be required to give 50-minute presentations in class on topics they will have prepared in writing and discussed with me in advance.

*Prerequisite: *AS/SC/MATH2022.03 or AS/SC/AK/MATH2222.03 or AK/MATH2220.06.

**Coordinator: ** Donald H. Pelletier

**AS/SC/MATH4130E.03W Topics in Probability and Statistics: Bayesian Statistics**

*1995-95 York Calendar: *The course first presents the Bayesian
approach to single- and multi-parameter statistical problems and its
link to major concepts of non-Bayesian statistics. The course then
studies some hierarchical models and regression models using a
Bayesian approach with theory and examples.

Topics covered in the course include: Setting up a probability model, using the Bayesian approach; some standard univariate models including the normal model, noninformative prior distributions; multiparameter models, normal with unknown mean and variance, the multivariate normal distribution, multinomial models; computation and simulation from arbitrary posterior distribution in two parameters; inference from large samples and comparison to standard non-Bayesian methods; estimating population parameters from data; estimating normal means; model checking, comparison to data and prior knowledge sensitivity analysis. Each topic will be illustrated by at least one example.

*Prerequisite:* Permission of the course coordinator.

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

**AS/SC/MATH4141.03F Advanced Numerical Methods**

*1995-95 York Calendar: *Systems
of non-linear equations: Newton-Raphson iteration,
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/MATH3242.03 or SC/AS/COSC3122.03.

**Coordinator: ** A.D. Stauffer

**AS/SC/MATH4142.03W Numerical Solutions to Partial Differential Equations**

*1995-95 York Calendar: *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/MATH2270.03; AS/SC/MATH3242.03 or SC/AS/COSC3122.03;
AS/SC/MATH3272.03 is strongly recommended. (This last course
is moribund; many of its topics have found a home in MATH3271.03.)

**Coordinator: ** J. Laframboise

**AS/SC/MATH4160.03F Combinatorial Mathematics**

*1995-95 York Calendar: *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.

The above list of topics is quite ambitious for a one-term course; we may have to forego some of them.

The text will be Alan Tucker's *Applied Combinatorics* (3rd ed.), John
Wiley and Sons, 1995. An optional reference is Ivan Niven's
*Mathematics of Choice*, New Mathematical Library,
Math. Association of America.

The final grade will be based on two term tests (25% each), graded assignments (15%), and a final examination (35%).

*Prerequisites*:
AS/SC/MATH2022.03 or AS/SC/AK/MATH2220.03; six credits from 3000-level MATH courses (without second digit 5); or permission of the course coordinator.

**Coordinator: ** H. Botta

**AS/SC/MATH4170.06 Operations Research II**

*1995-95 York Calendar: *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 prerequisites below 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 permission of the course coordinator to enrol.

*Prerequisites*:
AS/SC/MATH2010.03 or AS/SC/MATH2015.03 or AS/SC/AK/MATH2310.03; AS/SC/MATH1132.03 or AS/SC/MATH2030.06 or AS/SC/AK/MATH2030.03; AS/SC/AK/MATH3170.06; or permission of the course coordinator.

*Degree credit exclusion:*
AS/MATH4570.06.

**Coordinator: ** Silviu Guiasu

**AS/SC/MATH4230.03F Nonparametric Methods in Statistics**

*1995-95 York Calendar: *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 will be announced later.

The final grade will be determined 20% by assignment, 30% by test, 50% by final exam.

*Prerequisite*
AS/SC/MATH3030.03 (taken before 1993/94) or AS/SC/AK/MATH3131.03; AS/SC/MATH3031.03 or AS/SC/AK/MATH3132.03 is recommended but not required.

**Coordinator: ** D.A.S. Fraser

**AS/SC/MATH4240.03 Independent Studies in Applied Mathematics**

*1995-95 York Calendar: *(Cross-listing to Arts and course description
pending University approval.) Independent study under direction of a
faculty supervisor. Areas: applied and numerical analysis, discrete
applied mathematics, operations research, mathematical physics,
mathematical biology, mathematical modeling. The area is restricted by
the availability of a supervisor.

This is essentially a directed reading course. To take it, the interested student should approach either a prospective faculty supervisor or the Course Coordinator (below).

*Prerequisite*(pending approval) Permission of the Applied Mathematics Programme Director, and restricted to students who have completed the Applied Mathematics core.

**Coordinator: ** Martin Muldoon

**AS/SC/MATH4241.03W Applied Group Theory**

(formerly AS/SC/MATH4120M.03)

*1995-95 York Calendar: *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.

Group theory is widely used in many fields outside mathematics. This is a consequence of the fact that the algebraic structure which defines 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), it is often possible to 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 MATH2015 and MATH 2270.

*Prerequisites:* AS/SC/AK/MATH2222.03 -- or equivalent.

*Degree credit exclusion:*
AS/SC/MATH4120M.03.

**Coordinator: ** Kim Maltman

**AS/SC/MATH4270.03W Integral Transforms and Equations**

*1995-95 York Calendar: *This course
studies the Laplace, Fourier, Hankel and Mellin transforms; the
solution of integral equations; and the treatment of asymptotic expansions.
The applications are to problems in circuit theory, heat flow, elasticity,
transport theory and scattering theory.

*Prerequisites:* AS/SC/AK/MATH2270.03; AS/SC/AK/MATH3410.03.

*Corequisite:* AS/SC/MATH3271.03. May also be taken before 4270.

**Coordinator: ** J. Laframboise
**AS/SC/MATH4280.03F Risk Theory**

*1995-95 York Calendar: *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 MATH2030.03.

The text will be N.L. Bowers et al.,
*Actuarial Mathematics* (Society of
Actuaries). This is the same text as for MATH 3280.06. (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/MATH2030.03; AS/SC/MATH 3280.06
is recommended but not required.

**Coordinator: ** N. Madras

**AS/SC/AK/MATH4290.03F Mathematical Logic**

(same as GS/MATH6030.03)

*1995-95 York Calendar: *Predicate
logic, rules of inference, elimination of quantifiers, semantics
and model theory, the completeness and compactness theorems, ultrapowers and
non-standard analysis.

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*.

The objects studied in proof theory are (mathematical) proofs; it deals with such questions 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?".

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 GS/MATH 6040.03).

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 à la Robinson.

Prerequisite beyond the official one below: The degree of "mathematical maturity" normally found in a 3rd or 4th year undergraduate or 1st year graduate mathematics student. A previous course in recursion theory and/or (undergraduate) "logic" would make life easier.

The official text is
H. Enderton, *A Mathematical Introduction to Logic*, Academic Press.

*Work load/grading:* There will be
several homework assignments, equally weighted, and no exams/tests
in class. There will be a differential in workload between graduate and
undergraduate students taking the course (as required by Senate).
Graduate students will be guided to more esoteric aspects of the literature
for study (in recursion, proof and model theory) and also will do
approximately 25% more homework problems than the undergraduate students (on
Gödel's incompleteness theorems,
Robinson's nonstandard analysis, and some work on *reducibility*).

*References*:

- E. Mendelson,
*Introduction to Mathematical Logic*, 3rd ed. - Chang and Keisler,
*Model Theory*, North Holland - Yu. I. Manin,
*A Course in Mathematical Logic* - J.R. Shoenfield,
*Mathematical Logic*, Addison Wesley - R. Smullyan,
*First-Order Logic*, Springer-Verlag - R. Smullyan,
*Gödel's Incompleteness Theorems* - G. Tourlakis,
*Computability*, Prentice Hall. - J. Barwise,
*Handbook of Mathematical Logic*, in particular A.1, C.1; possibly D.1.

*Prerequisite:* AS/SC/MATH2090.03 or permission of the course
coordinator.

**Coordinator: ** G. Tourlakis

**AS/SC/MATH4400.06 The History of Mathematics**

*1995-95 York Calendar: *Selected
topics in the history of mathematics, discussed in full technical
detail but with stress on the underlying ideas, their evolution and their
context.

This course will focus on the historical and cultural context of mathematical ideas, while examining their shifting technical content. Examples drawn from geometry, algebra and analysis will be looked at from the point of view of various cultures and different historical periods with the aim of observing the physical, social and philosophical influences that shaped the resulting mathematics.

Students will be expected to write two papers on topics agreed to by the instructor, and to participate in class discussions of the material being presented.

The final mark will be based on these two papers (60%), and on the class discussions (40%).

It is recommended that you obtain a copy of Struik's A Concise History of Mathematics.

*Prerequisite*:
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: ** T. MacHenry

**AS/SC/AK/MATH4430.03W Stochastic Processes II**

*1995-95 York Calendar: *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.

The text will be Howard M. Taylor and Samuel Karlin,
*An Introduction to Stochastic Modelling*
(revised edition,
Academic Press).

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

*Prerequisite: AS/SC/AK/MATH2030.03 or AS/SC/AK/MATH3030.03. *

**Coordinator: ** N. Madras

**AS/MATH4570.06 Applied Optimization**

*1995-95 York Calendar: *Topics chosen
from decision theory, game theory, inventory control, Markov
chains, dynamic programming, queueing theory, reliability theory, simulation,
nonlinear programming. This course is designed primarily for students in
the General Stream of Honours Mathematics for Commerce.

*Prerequisites:* AS/SC/AK/MATH3170.06; AS/SC/AK/MATH3330.03;
either AS/SC/AK/MATH3230.03 or AS/SC/AK/MATH3430.03.

*Degree credit exclusion:*
AS/SC/MATH4170.06.

**Coordinator: ** M. Mandelbaum

**AS/SC/MATH4630.03W Applied Multivariate Statistical Analysis**

*1995-95 York Calendar: *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. 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 covariance, multivariate linear models, principal component analysis, canonical correlation analysis, and possibly some discussion of discriminant and classification, factor analysis and cluster analysis, if time permits.

Grades will be based on a combination of class test and final examination, plus routine homework.

*Prerequisites:* AS/SC/MATH3030.03 (taken before 1993/94)
or AS/SC/AK/MATH3131.03; AS/SC/MATH3034.03 or
AS/SC/MATH3230.03; AS/SC/MATH2022.03 or AS/SC/AK/MATH2222.03.

**Coordinator: ** Y. Wu

**AS/SC/MATH4730.03F Experimental Design**

(same as GS6626.03)

*1995-95 York Calendar: *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 founding block of 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.

The text will be announced later.

The final grade may be based on assignments, test(s), and a final examination. Details will be announced.

*Prerequisites:* A second 6 credits in statistics, including
either AS/SC/MATH3033.03, or both AS/SC/AK/MATH3230.03 and
AS/SC/AK/MATH3330.03,
or permission of the course coordinator.

**Coordinator: ** P. Ng

**AS/SC/MATH4830.03F Time Series and Spectral Analysis**

(same as SC/EATS4020.03 and SC/PHYS4060.03)

*1995-97 York Calendar: *
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.

Further topics:

*Discrete, Equispaced Time Series:*Power and energy signals, wavelets convolution and the z-transform, expected value, autocorrelation and cross-correlation impulse, white noise and Wold decomposition, time reverse, properties of wavelets, linear, optimum filtering deconvolution, shaping and spiking filters.*Fourier Methods:*Fourier transform effects of sampling and record length, digital frequency filtering, the power spectrum.

*References:*

- E.R. Kanasewich,
*Time Sequence Analysis in Geophysics*(3rd ed.), 1981. - Enders and Treitel,
*Geophysical Signal Analysis*, 1980. - A.R. Enders,
*Multichannel Time Series Analysis with Digital Computer Programs*(revised ed.), 1978.

The grading scheme has not yet been determined.

*Prerequisites:*
SC/AS/COSC1540.03 or similar FORTRAN programming experience;
AS/SC/AK/MATH2270.03; AS/SC/MATH2015.03, AS/SC/AK/MATH3010.03.

*Degree credit exclusion:*
SC/AS/COSC4010B.03, SC/AS/COSC4242.03,
SC/EATS4020.03, SC/PHYS4060.03, AS/SC/MATH4930C.03.

**Coordinator: ** M.A. Jenkins

**AS/SC/MATH4930C.03W Topics in Applied Statistics: Forecasting and Time Series**

* 1995-97 York Calendar: * Introduction to the needs and uses
of forecasting. The components of a time series. Forecasting time
series using smoothing methods; using trend projection; using
classical decomposition; using regression models. Autoregression
moving average (ARMA) time series methods. Box-Jenkins methodology.

This course is aimed at the student who wishes to gain a working knowledge of time series and forecasting methods as applied in economics, engineering, and the natural and social sciences. The emphasis is on methods and the analysis of data sets. The logic and tools of model-building for stationary and non-stationary time series are developed in detail and numerous exercises considered. The course covers stationary processes, ARMA and ARIMA processes, and state-space models, with an optional chapter on spectral analysis.

The text will probably be Peter J. Brockwell and Richard A. Davis,
*An Introduction to Time Series and Forecasting*.

The grade will be based on assignments and a project (25%), a term test (25%), and a final exam (50%).

*Prerequisites:* AS/SC/AK/MATH3330.03; either
AS/SC/AK/MATH3230.03 or AS/SC/MATH3430.03.

* Degree credit exclusions: *
SC/EATS4020.03, AS/SC/MATH4830.03, SC/PHYS4060.03.

**Coordinator: ** G. Monette