**AS/SC/AK/MATH3010 3.0 W
Vector Integral Calculus**

1998/99 CALENDAR COPY:*Integrability
of continuous functions over suitable domains, iterated
integrals and Fubini's theorem, counterexamples, change of variables,
Jacobian determinants, polar and spherical coordinates, volumes, vector
fields, divergence, curl, line and surface integrals, Green's and Stokes's
theorems, differential forms, general Stokes's theorem.*
The text and marking scheme have not yet been determined.

*Prerequisite: *AS/SC/MATH2010 3.0; or AS/SC/AK/MATH

2310 3.0; or
AS/SC/MATH2015 3.0 and written permission of the Mathematics
Undergraduate Director (normally granted only to students
proceeding in Honours programmes in Mathematics or in the
Specialized Honours Programme in Statistics).

*Corequisite: *(or prerequisite)
AS/SC/MATH2022 3.0

or AS/SC/AK/MATH2222 3.0.

**Coordinator: ** M-W. Wong

**AS/SC/AK/MATH3020 6.0
Algebra I**

1998/99 CALENDAR
COPY:*Introduction
to the basic concepts of abstract algebra, with applications:
groups (cyclic, symmetric, Lagrange's theorem, quotients, homomorphism
theorems); rings (congruences, quotients, polynomials, integral domains,
PID's and UFD's); fields (field extensions,
constructions with ruler and compass, coding theory). *

Algebra is the study of algebraic systems, that is, sets of elements endowed with certain operations. A familiar example is the set of integers with the operations of addition and multiplication.

Algebra is used in almost every branch of mathematics; indeed, it has simplified the study of mathematics by indicating connections between seemingly unrelated topics. In addition the success of the methods of algebra in unravelling the structure of complicated systems has led to its use in many fields outside of mathematics.

One aim of this course is to help students learn to write clear and concise proofs, read the mathematical literature, and communicate mathematical ideas effectively, both orally and in writing.

Any student who performed well in the prerequisite linear algebra course is welcome to enrol, but THIS COURSE IS INTENDED PRIMARILY FOR STUDENTS WHO HAVE TAKEN THE HONOURS VERSIONS OF FIRST AND SECOND YEAR COURSES.

The text will be Fraleigh, *A First Course in Abstract Algebra*, 5th
Edition (Addison-Wesley).

The final grade will be based on assignments, class participation, quizzes, class tests, and a final examination.

*Prerequisite: *AS/SC/MATH2022 3.0

or AS/SC/AK/MATH2222 3.0.

*Degree credit exclusion: *AK/MATH3420 6.0

**Coordinator:
** J.W. Pelletier

**AS/SC/MATH3033 3.0 F
Classical Regression Analysis**

1998/99 CALENDAR COPY:*General
linear model. Properties and geometry of least-squares estimation.
General linear hypothesis, confidence regions and intervals.
Multicollinearity. Relationship between ANOVA models and linear models.
Residual analysis, outliers, partial and added variable plots.*

This course is closely linked with MATH3034 3.0, Modern Regression Analysis, for which it is a prerequisite. The emphasis, in contrast to MATH3330 3.0 and MATH3230 3.0, will be a more mathematical development of linear models including modern regression techniques. To develop a solid knowledge of regression models, it is strongly advised that you take both MATH3033 3.0 and MATH3034 3.0.

Students will use the computer for some exercises. No previous courses in computing are required. The statistical software package SPLUS in a UNIX enrivonment will be used and instructions will be given in class.

The text is R. H. Myers (ed.), *Classical and Modern
Regression with Applications* (Duxbury).

The final grade may be based (in each term) on assignments, quizzes, a project, one midterm, and a final examination.

*Prerequisites: *AS/SC/MATH1132 3.0 , or an average of

B or higher
in AS/SC/AK/MATH2560 3.0 and

AS/SC/AK/MATH2570 3.0;
AS/SC/MATH2022 3.0

or AS/SC/AK/MATH2222 3.0.

*Corequisite: *AS/SC/AK/MATH3131 3.0 or permission of the course
coordinator.

*Degree credit exclusions: *AS/SC/AK/MATH3330 3.0, AS/SC/

GEOG3421 3.0,
AS/SC/PSYC3030 6.0, AK/PSYC3110 3.0.

**Coordinator: ** P. Song

**AS/SC/MATH3034 3.0 W
Modern Regression Analysis**

1998/99 CALENDAR COPY:*Selecting
best model, cross-validation. Influence diagnostics. Weighted
least squares, correlated errors, transformations, Box-Cox transformations.
Logistic and Poisson regression. Generalized linear models.
Multicollinearity, ridge regression. Topics selected from non-linear
regression, scatterplot smoothing, non-parametric regression, additive
non-linear regression, projection pursuit, robust regression.*

For course description see AS/SC/MATH3033 3.0 F.

*Prerequisite: *AS/SC/MATH3033 3.0.

*Degree credit exclusions: *AS/SC/AK/MATH3230 3.0,

AS/SC/GEOG3421 3.0, AS/SC/PSYC3030 6.0,

AK/GEOG4200 6.0, AK/PSYC3110 3.0.

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

**AS/SC/AK/MATH3050 6.0
Introduction to Geometries**

(*Note:* This course will probably not be offered in 1999/2000.)
1998/99 CALENDAR COPY:*Analytic geometry over a field with vector and barycentric
coordinate methods, affine and projective transformations, inversive
geometry, foundations of Euclidean and non-Euclidean geometry,
applications throughout to Euclidean geometry.*

Geometry has an important classical side, moving from the Greeks to non-Euclidean geometries (which differ in their assumptions about ``parallel lines''), including spherical, hyperbolic and projective geometries. This shift from Euclidean geometry to multiple geometries is one of the critical ``paradigm shifts'' in the history of mathematics. In modern geometry, the fertile interplay of synthetic (constructive and visual) methods, groups of transformations, analytic methods and axiomatics presents a rich mix of problems and methods to be explored. Through multiple mathematical and pedagogical approaches we will introduce these geometries in their classical and modern forms.

Modern geometry has important applications to areas involving shape and computing: Computer Aided Design; Robotics; computer graphics; physics and engineering. Geometry also has a critical role in developing our skills and vocabulary for reasoning with visual representations. Both why we practice geometry and how we practice (and teach) geometry are changed by computers.

The course is designed to prepare the student for further studies in: (i) pure mathematics, (ii) applications of geometry, or (iii) teaching geometry. The formal prerequisites are minimal: familiarity with linear algebra and some mathematical maturity. Other background will be developed as needed. We will expect students to join in group work, to work with and build physical models in class (using spheres for spherical geometry, plastic Polydron for polyhedra, mirrors for symmetry, etc.), to use a dynamic geometry program, The Geometer's Sketchpad (available in the classroom and at Steacie Labs), and to develop their own geometric questions and projects.

The text for the course is
David Henderson, *Experiencing Geometry on Plane and Sphere*
(Prentice-Hall), 1996.

Graded work will include regular assignments, including proofs, conjectures and open-ended explorations, oral presentations, written (and drawn) projects and possibly quizzes.

*Prerequisite: *AS/SC/MATH2022 3.0 or AS/SC/AK/MATH2222 3.0 or
permission of the Course Coordinator.

*Degree credit exclusion: *AK/MATH/3550 6.0

**Coordinator: ** Walter Whiteley

**AS/SC/MATH3100 3.0 W
Famous Problems in Mathematics**

(*Note:* This course will probably not be offered in 1999/2000.)

1998/99 CALENDAR
COPY:*An attempt to
foster an appreciation of the history, the personalities and some
of the content of different areas of mathematics, by means of a
study of some specific problems which have exercised the minds of
mathematicians.*

The problems will range from ancient to recent times, and will be selected from the fields of algebra, analysis, geometry, number theory, set theory, and foundations of mathematics.

The course will deal wih mathematical ideas in the context of mathematical techniques. Philosophical issues in the develoment of mathematics will also be discussed.

The final grade will be based on term work (assignments, class test(s), and possibly presentations) worth 70%, and a final examination worth 30%.

There will be no text for the course, but many references will be provided. Students may get a flavor of the course by consulting the following books:

P. Davis & R. Hersh, *The Mathematical Experience*.

W. Dunham, *Journey Through Genius: The Great Theorems
of Mathematics*. H. Eves,

*Prerequisites: *At least 12 credits from 2000-level MATH courses without
second digit 5, or permission of the Course Coordinator.

**Coordinator: ** I. Kleiner

**AS/SC/AK/MATH3110 3.0 F
Introduction to Mathematical Analysis**

1998/99 CALENDAR COPY:*Proofs
in calculus and analysis. Topics include sets, functions, axioms for
R,
applications of the completeness axiom, countability, sequences and their
limits, monotone sequences, limits of functions, continuity.*

This course provides a path towards an honours degree for those students who have not taken the honours first year calculus course MATH1010 3.0. The course MATH3210 3.0, which is required for several honours programmes, has this course as an alternative to MATH1010 3.0 as a prerequisite.

The course will emphasize the theoretical aspects of the subject. A principal goal of the course is learning to understand the various definitions and to use them to prove basic properties of the objects being defined. The structure of proofs and the basic logic underlying them will be carefully considered. Relatively little effort will be devoted to problems involving calculations, except when they are useful for explaining the concepts.

The text has not yet been selected.

The final grade will be based 60% on term work, 40% on the final exam.

*Prerequisite: *AS/SC/AK/MATH1310 3.0

or AS/SC/MATH1014 3.0.

*Corequisites: *AS/SC/AK/MATH2310 3.0 or AS/SC/MATH

2010 3.0 or AS/SC/MATH2015 3.0; AS/SC/MATH

2021 3.0

or AS/SC/AK/MATH2221 3.0 or AS/SC/MATH1025 3.0.

*Degree credit exclusions: *AS/SC/MATH1010 3.0,

AK/MATH2400 6.0.

**Coordinator: ** G. O'Brien

**AS/SC/AK/MATH3131 3.0 F
Mathematical Statistics I**

1998/99 CALENDAR COPY:*Topics
include common density functions, probability functions, principle of
likelihood, the likelihood function, the method of maximum likelihood,
likelihood regions, tests of hypotheses, likelihood ratio tests, goodness of
fit tests, conditional tests, and confidence sets with a view towards
applications.*

This course is intended for students who need a theoretical foundation in mathematical statistics. Students who have taken it normally take MATH3132 in the second term. It continues where MATH2030 left off, while providing a theoretical foundation for many of the statistical procedures learned in MATH1131 and MATH1132.

*Prerequisite: *AS/SC/AK/MATH2030 3.0 or permission of the course
coordinator.

*Degree credit exclusions: *AK/MATH3030 3.0 (before SU95),

AS/SC/MATH3030 3.0 (before 1993/94), AK/MATH3530 6.0.

**Coordinator: ** Y. Wu

**AS/SC/AK/MATH3132 3.0 W
Mathematical Statistics II**

(formerly MATH3031) 1998/99 CALENDAR COPY:*Important
examples and methods of statistical estimation and hypothesis
testing are discussed in terms of their mathematical and statistical
properties. Topics include sufficiency, Bayesian statistics, decision
theory, most powerful tests, likelihood ratio tests.*

This course is a continuation of MATH3131 3.0. The basic nature of statistical inference will be studied. Joining the above list of topics is that of asymptotics.

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

*Degree credit exclusions: *AS/SC/MATH3031 3.0,

AK/MATH3530 6.0.

**Coordinator: ** Y. Wu

**AS/SC/AK/MATH3140 6.0
Number Theory and Theory of Equations**

This course will not be offered in 1998/99. It is expected to be
offered in 1999/2000.

**AS/SC/AK/MATH3170 6.0
Operations Research I**

1998/99 CALENDAR COPY:*A study
of linear programming; transportation problems, including network
flows, assignment problems and critical path analysis; integer programming;
dynamic programming and an introduction to stochastic models. Application to
a set of problems representative of the field of operations research.*

This course deals with standard optimization techniques used in
Operations Research. The main topics include:

(a) *Linear Programming:*
the theory and applications of linear programming including the simplex
algorithm, duality theorem, postoptimality analysis, and a discussion
of the types of problems that lead to linear programming problems.
(b) *Transportation Problems:*
the transportation algorithm with applications
to network flows, assignment problems, shortest-route problems, and
critical path scheduling. (c) *Integer Programming:* a study of the
situations leading to integer-programming problems, branch-and-bound
algorithm for solving such problems. (d) *Dynamic Programming:* an
introduction to the concepts of dynamic programming with a discussion
of typical problems and their solutions.

The text will be W.L. Winston, *Operations Research. Applications and
Algorithms*, 3rd ed.

(Wadsworth Publishing Co., Duxbury Press), 1994.

The final grade will be based on two computer assignments (5%), three tests (55%), and a final examination (40%).

*Prerequisites: *AS/SC/MATH2021 3.0 or AS/SC/AK/

MATH2221 3.0 or AS/SC/MATH1025 3.0,

plus SC/AS/COSC1530 3.0 or SC/AS/

COSC1540 3.0 or equivalent. Students who have not taken

these courses need the permission of the course coordinator.

*Degree credit exclusions: *AK/MATH2751 3.0,

AK/MATH3490 6.0, AK/ADMS3351 3.0,

AK/COSC3450 6.0, AK/ECON3120 3.0.

**Coordinator: ** Silviu Guiasu

**AK/AS/SC/MATH3190 3.0 W
Set Theory and Foundations of Mathematics**

(*Note:* This course is offered on an irregular basis.)
1998/99 CALENDAR COPY:*The following topics are covered: paradoxes in naive set theory;
functions and relations, transfinite numbers, their ordering and their
arithmetic; well-ordered sets and ordinal numbers; Zorn's lemma; an
introduction to axiomatic set theory.*

The relevance of set theory to a mathematician (student or otherwise) is equivalent to the relevance to an intelligent human being of the ability to speak, read, and write. Practically the entire ``modern'' literature in mathematics (Topology, Analysis, Algebra, etc.) relies heavily on the ``language'' of set theory, but also on the deeper results involving cardinal and ordinal numbers.

Additionally to the above considerations of ``relevance'', one will want to study set theory for its own sake.

We shall first look into the basic (informal) definitions and
notations, eventually leading to the notions of relations and
functions,
equivalence relations and partial orders. *En route* we will
get a flavour of the *foundational* difficulties that a purely
``informal'' approach entails.
We will see--in a ``naive'' manner--how the introduction of axioms
helps to get around these paradoxes. We will be ``fixing'' the theory
as we go
by introducing appropriate ``assumptions'' (axioms)--as
needed--regarding the
behaviour of sets.

The Axiom of Choice and a number of its equivalent variants (including ``Zorn's Lemma'') will be discussed, and some of its elementary consequences will be considered.

The final grade will be determined by assignments (70%) and a final exam (30%).

The text will consist of typeset notes from a preprint of
*Set Theory*, by G. Tourlakis.

*Prerequisite: *Six credits from 2000-level MATH courses without second
digit 5.

**Coordinator: ** G. Tourlakis

**AS/SC/AK/MATH3210 3.0 W
Principles of Mathematical Analysis**

1998/99 CALENDAR COPY:*Rigorous
presentation, with proofs, of fundamental concepts of analysis:
limits, continuity, differentiation, integration, fundamental theorem,
power series, uniform convergence.*

The origins of some ideas of mathematical analysis are lost in antiquity. About 300 years ago, Newton and Leibniz independently created the calculus. This was used with great success, but for the most part uncritically, for about 200 years. In the last century, mathematicians began to examine the foundations of analysis, giving the concepts of function, continuity, convergence, derivative and integral the firm basis they required. These developments continue today, for example with the study of calculus on infinite dimensional spaces.

This course is a continuation of first and second year calculus. The material will be presented in a rigorous manner, putting emphasis on careful mathematical arguments, proofs and illustrative examples. The objectives of the course include improved skill at understanding and writing mathematical arguments as well as understanding of the concepts of the course. The course will provide a useful theoretical background for a variety of higher level and graduate courses including those in analysis, probability, topology, mathematical statistics, and numerical analysis.

The text for the course has not yet been chosen.

The final grade will be based on assignments, tests and a final examination.

*Prerequisite: *

AS/SC/MATH2010 3.0 or AS/SC/AK/MATH3110 3.0.

*Note:* Subject to approval by the Faculty of Arts after we go
to press, these prerequisites will be augmented to include
MATH1010 + MATH2310, or MATH1010 + MATH2015, as alternatives.
Please inquire at the Undergraduate Office (N502 Ross).

**Coordinator: ** G. O'Brien

**AS/SC/AK/MATH3230 3.0 W
Analysis of Variance**

(*Note:* This course will not be offered after 1999/2000. At
that time, a slightly modified version of MATH3034 3.0
will replace it as prerequisite for other courses.)

1998/99 CALENDAR COPY:*Categorical
variables; one factor and two factor analysis; fixed, random and
mixed models; nested designs; an introduction to randomized block and Latin
square designs. Second term.*

A major focus will be on models with categorical variables as predictors (classical ANOVA, or Analysis of Variance). The computer will be used heavily, but no previous computing courses are required. See also the course description for MATH3330, with which this course is closely linked.

*Prerequisite: *AS/SC/AK/MATH3330 3.0.

*Degree credit exclusions: *AS/SC/MATH3034 3.0,

AS/SC/GEOG3421 3.0, AS/SC/PSYC3030 6.0,

AK/PSYC3110 3.0.

**Coordinator: ** D. Montgomery

**AS/SC/MATH3241 3.0 F
Numerical Methods I**

1998/99 CALENDAR COPY:*An
introductory course in computational linear algebra. Topics include
simple error analysis, linear systems of equations, nonlinear equations,
linear least squares and interpolation. (Same as SC/AS/COSC3121 3.0.)*

The course begins with a general discussion of computer
arithmetic and computational errors. Examples of ill-conditioned
problems and unstable algorithms will be given. The first class of
numerical methods we introduce are those for nonlinear equations,
i.e., the solution of a single equation in one variable. We then turn
to a discussion of the most basic problem of numerical linear algebra:
the solution of a linear system of *n* equations in *n* unknowns. The
Gaussian elimination algorithm will be discussed as well as the
concepts of error analysis, condition number and iterative refinement.
We then turn to the least squares methods for solving overdetermined
systems of linear equations. Finally we discuss polynomial
interpolations. The emphasis in the course is on the development of
numerical algorithms, the use of mathematical software, and the
interpretation of the results obtained on some assigned problems.

A possible textbook is
R.L. Burden and J.D. Faires, *Numerical Analysis* (6th ed.),
PWS, 1997. (The description of this course continues on the next page.)

The final grade will be based on assignments (including computer assignments), tests and a final examination. Details will be announced.

*Prerequisites: *One of AS/SC/MATH1010 3.0,

AC/SC/MATH1014 3.0, AS/SC/AK/MATH1310 3.0; one

of AS/SC/MATH1025 3.0, AS/SC/AK/MATH2221 3.0,

AS/SC/MATH2021 3.0; one of SC/AS/COSC1540 3.0,

SC/AS/COSC2011 3.0, SC/AS/COSC2031 3.0.

*Degree credit exclusions: * SC/AS/COSC3121 3.0,

AK/COSC3511 3.0.

**Coordinator: ** Martin Muldoon

**AS/SC/MATH3242 3.0 W
Numerical Methods II**

1998/99 CALENDAR COPY:*Algorithms
and computer methods for solving problems of differentiation,
integration, differential equations, and an introduction
to systems of non-linear
equations. (Same as SC/AS/COSC3122 3.0.)*

The textbook will be the same as for MATH3241.

The final grade will be based on assignments (including computer assignments), tests and a final examination. Details will be announced.

*Prerequisites: *AS/SC/AK/MATH2270 3.0;

AS/SC/MATH3241 3.0 or SC/AS/COSC3121 3.0.

*Degree credit exclusions: *SC/AS/COSC3122 3.0,

AK/COSC3512 3.0.

**Coordinator: ** A.D. Stauffer

**AS/SC/AK/MATH3260 3.0 W
Introduction to Graph Theory**

1998/99 CALENDAR COPY:
*Introductory
graph theory with applications. Graphs, digraphs. Eulerian and
Hamiltonian graphs. The travelling salesman. Path algorithms; connectivity;
trees; planarity; colourings; scheduling; minimal cost networks. Tree
searches and sortings, minimal connectors and applications from physical and
biological sciences.*

This is a first course in graph theory. After an introduction to graphs, we consider trees, circuits, cycles and connectedness. We may also consider extremal problems, and counting and labelings of graphs.

The text and grading scheme have not been determined as we go to press.

*Prerequisite: *At least six credits from 2000-level (or higher) MATH courses
(without second digit 5), or permission of the instructor.

**Coordinator: ** Richard Ganong

**AS/SC/MATH3271 3.0 F
Partial Differential Equations**

1998/99 CALENDAR COPY:*Partial differential
equations of mathematical physics and their solutions
in various coordinates, separation of variables in Cartesian coordinates,
application of boundary conditions; Fourier series and eigenfunction
expansions; generalized curvilinear coordinates; separation of variables in
spherical and polar coordinates.*

Further topics include Sturm-Liouville theory, the gamma function, and Bessel, Legendre, Laguerre, Hermite, Chebyshev, hypergeometric, and confluent hypergeometric equations and functions and their properties.

The principal reference text will be G. Arfken,

*Mathematical
Methods for Physicists*.

The final grade may be based on assignments, two tests, and a final exam.

*Prerequisites: *AS/SC/AK/MATH2270 3.0; one of AS/

SC/MATH2010 3.0,
AS/SC/MATH2015 3.0, AS/SC/AK/

MATH2310 3.0;
AS/SC/AK/MATH3010 3.0 is also
desirable, though not essential, as prerequisite for students
presenting AS/SC/MATH2010 3.0 or AS/SC/AK/MATH2310 3.0.

*Degree credit exclusion: *AS/MATH4200A 6.0.

**Coordinator: ** H. Freedhoff

**AS/SC/MATH3280 6.0
Actuarial Mathematics**

1998/99 CALENDAR COPY:*Actuarial
mathematics at a level appropriate for examination 150 of the
Society of Actuaries. Topics include survival distributions and life tables,
premiums and reserves for life insurance and annuities, multiple life
functions, multiple decrement models, valuation theory of pension plans.*

This course is intended for students contemplating careers
in the actuarial profession. It will help prepare them
for Examination 150 of the Society of Actuaries. We will cover
most of Chapters 3-10 of the official text, N.L. Bowers
et al., *Actuarial Mathematics*, 2nd ed. (Society of
Actuaries). There is inadequate time in a one-year course to cover
Chapters 11, 15 and 16, the remaining material needed for Exam
150. However, students who complete this course should acquire
enough background to enable them to study the omitted
chapters on their own.

The prerequisites are a sound knowledge of both interest

theory and probability theory. For the probability prere-

quisite, students should have completed MATH2030 3.0. For interest
theory the preferred prerequisite is MATH2280 3.0. Those who have
completed MATH2580 6.0 with a grade of B+ or better may be
allowed to enrol, but such students should note that
MATH3280 6.0 is considerably more advanced, and requires much more
mathematical ability, than MATH2580 6.0.

The final grade will be based on a combination of assignments, midterm tests and a final examination.

*Prerequisites: *AS/SC/MATH2280 3.0;

AS/SC/AK/MATH2030 3.0.

**Coordinator: ** S.D. Promislow

**AS/SC/AK/MATH3330 3.0 FW
Regression Analysis**

1998/99 CALENDAR COPY:*Simple regression
analysis, multiple regression analysis, matrix form of the
multiple regression model, estimation, tests (t- and F-tests),
multicollinearity and other problems encountered in regression, diagnostics,
model building and variable selection, remedies for violations of regression
assumptions. First term.*

This course is closely linked with MATH3230 3.0, Analysis of Variance, for which it is a prerequisite. Students will use the computer heavily in these courses, but no previous courses in computing are required.

MATH3330 3.0 will focus on linear models for the analysis of data on several predictor variables and a single response. The emphasis will be on understanding the different models and statistical concepts used for these models and on practical applications, rather than on the formal derivations of the models. The approach will require the use of matrix representations of the data, and the geometry of vector spaces, which will be reviewed in the course.

The first term (MATH3330 3.0) will cover the basic ideas of multiple regression, having reviewed in depth the elements of simple linear regression. The second term (MATH3230 3.0) will have a major focus on models with categorical variables as predictors (classical ANOVA, or Analysis Of Variance).

The nature of the course requires that students be involved on a constant basis with the material, and not fall behind.

The text and grading scheme have not been determined as we go to press.

*Prerequisites: *One of AS/SC/MATH1132 3.0,

AS/SC/AK/MATH2570 3.0, AS/SC/PSYC2020 6.0,

or equivalent; some acquaintance with matrix algebra

(such as is provided in AS/SC/MATH1025 3.0,

AS/SC/MATH1505 6.0, AS/AK/MATH1550 6.0,

AS/SC/MATH2021 3.0, or AS/SC/AK/MATH2221 3.0).

*Degree credit exclusions: *AS/SC/MATH3033 3.0,

AS/SC/GEOG3421 3.0, AS/SC/PSYC3030 6.0,

AS/ECON4210 3.0, AK/PSYC3110 3.0.

**Coordinator: ** Fall: D. Montgomery. Winter: P. Song

**AS/SC/AK/MATH3410 3.0 W
Complex Variables**

1998/99 CALENDAR COPY:*Analytic functions, the
Cauchy-Riemann equations, complex integrals, the Cauchy integral
theorem, maximum modulus theorem. Calculations of residues and
applications to definite integrals, two-dimensional potential
problems and conformal mappings.*

Some polynomials, such as *x^2 + 1*,
have no roots if we confine ourselves to
the real number system, but do have roots if we extend the number system
to the complex numbers, which can be defined as the set of all numbers of the
form *a* + *ib* , where *a* and *b* are real and
*i* is a new kind of number satisfying *i^2 + 1 = 0*,
where the basic arithmetic operations have the same structure as
those of the real numbers. The complex numbers include the reals
(case *b*
= 0), and the extended system has the desirable property that not
only * x^2 + 1* but every [nonconstant -- Ed.]
polynomial now has a root. In the system of complex numbers
certain connections are seen between otherwise apparently unconnected real
numbers. A striking example is Euler's formula *e^(pi i) = -1* ,
which is a
simple consequence of the extension to complex variables of the familiar
exponential and trigonometric functions. The concepts and operations of
calculus (differentiation, integration, power series, etc.) find their most
natural setting in complex (rather than real) variables. The present course
is intended to give the student a basic knowledge of complex numbers and
functions and a basic facility in their use.

Further topics include: Complex numbers and their representations; functions of a complex variable; extensions of elementary functions from real to complex variables; mapping of elementary functions; complex differentiation; Cauchy's theorem; Cauchy's integral formula and its applications; complex power series; the residue theorem and its applications.

*Prerequisite: *AS/SC/MATH2010 3.0 or AS/SC/MATH2015 3.0 or
AS/SC/AK/MATH2310 3.0. (AS/SC/AK/MATH3010 3.0
is also recommended
as a prerequisite for students who have taken AS/SC/MATH2010
3.0.)

**Coordinator: ** Martin Muldoon

**AS/SC/AK/MATH3430 3.0 W
Sample Survey Design**

1998/99 CALENDAR COPY:*Principal
steps in planning and conducting a sample survey. Sampling
techniques including simple random sampling, stratified random sampling,
cluster sampling, and sampling with probabilities proportional to size.
Estimation techniques including difference, ratio, and regression
estimation.*

This course deals with the peculiarities of sampling and inference commonly encountered in sample surveys in medicine, business, the social sciences, political science, natural resource management, and market research. Attention will be focused on the economics of purchasing a specific quantity of information. (See next page.)

That is, methods for designing surveys that capitalize on characteristics of the population under study will be presented, along with associated estimators to reduce the cost of acquiring an estimate of specified accuracy. (The emphasis will be on the practical applications of theoretical results.)

The text will be R. L. Scheaffer, W. Mendenhall, and L. Ott,
*Elementary Survey Sampling* (5th ed.), PWS-Kent.

The final grade may be based on assignments (5%), class tests (40%) and a final examination (55%).

*Prerequisite: *AS/SC/MATH2030 3.0

or AS/SC/MATH3030 3.0 (taken before 1993/94)

or AS/SC/AK/MATH3330 3.0 or AS/SC/PSYC3030 6.0.

*Degree credit exclusions: *These courses may not be taken for credit *after*
taking MATH3430: AK/MATH2752 3.0,

AK/ADMS3352 3.0,
AK/ECON3130 3.0.

**Coordinator: ** P. Peskun

**AS/SC/MATH3440 3.0 F
The Mathematics of Physics**

(*Note:* This course is offered on an irregular basis.)
1998/99 CALENDAR COPY:*Various topics in physics which require mathematical analysis
are discussed. The emphasis is on showing how such mathematical
techniques as multivariable calculus, ordinary and partial
differential equations, probability and calculus of variations arise
in the study of these topics. Normally offered in alternate years.*

Particular emphasis will be placed on the symbiotic evolution of the techniques of calculus and physics which, ultimately, resulted in Maxwell's formulation of the equations of electromagnetism and special relativity.

The text for the course will be David M.Bressoud,
*Second year Calculus*,
but this will be supplemented by various materials made
available in class. Students must be prepared to read this text on
their own since much of the class time will be devoted to student
presentations of assigned problems and extra readings.

These student presentations will form 20% of the final course mark. Of the remaining 80%, 60% will be based on written solutions to assigned exercises and 20% on a final examination.

For more information, a course web page will soon be available.

*Prerequisite: *AS/SC/AK/MATH2270 3.0.

*Prerequisite or corequisite: *

AS/SC/MATH2015 3.0 or AS/SC/AK/MATH3010 3.0.

*Degree credit exclusion: *Not open to Physics majors.

**Coordinator: ** J. Steprans

[See note to "Coordinator"
of MATH2041 for spelling of this name.]

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