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Graduate Calendar Archives: 1998 / 1999 |
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Ottawa-Carleton Institute of Mathematics and StatisticsHerzberg
Physics 4314 The InstituteDirector
of the Institute, David McDonald Students pursuing studies in pure mathematics, applied mathematics, probability and statistics at the graduate level leading to a M.Sc. or a Ph.D. degree do so in a joint program offered by the School of Mathematics and Statistics at Carleton University and the Department of Mathematics and Statistics at the University of Ottawa under the auspices of the Ottawa-Carleton Institute of Mathematics and Statistics. The Institute is responsible for supervising the programs, regulations, and student admissions, as well as providing a framework for interaction between the two departments at the research level. The list below of all members of the Institute along with their research interests can be used as a guide to possible supervisors. In addition to the programs administered by the Institute, the School of Mathematics and Statistics at Carleton University offers several other programs. In cooperation with the Department of Epidemiology and Community Medicine at the University of Ottawa, students may pursue a program leading to an M.Sc. with a Specialization in Biostatistics. For information, see p. 89. In cooperation with the Department of Systems and Computer Engineering and the School of Computer Science at Carleton University, students may pursue a program leading to an M.Sc. in Information and Systems Science. For information see p. 208. In cooperation with the School of Computer Science and the Department of Systems and Computer Engineering at Carleton University and the Department of Computer Science at the University of Ottawa, students may pursue a program leading to a Master of Computer Science (M.C.S.). For information see p. 140. The School of Mathematics and Statistics also offers a cooperative master’s program in statistics in collaboration with the federal government, emphasizing practical training through work experience, along with sound training in statistical inference and basic probability theory. Requests for information and completed applications should be sent to the Director or Associate Director of the Institute. Members of the InstituteThe home department of each member of the Institute is indicated by (C) for the School of Mathematics and Statistics, Carleton University and (UO) for the Department of Mathematics and Statistics, University of Ottawa
Master of ScienceAdmission RequirementsThe normal requirement for admission to the master’s program is an Honours bachelor’s degree in mathematics, or the equivalent, with at least high honours standing. Applicants holding a general (3 year) degree with at least high honours standing may be admitted to a qualifying-year program. Their subsequent admission to the regular master’s program depends on their performance during the qualifying-year program and will be decided no later than one year after admission to the qualifying-year program. Details are outlined in the general section of this calendar. Students with outstanding academic performance and research promise while in the M.Sc. program may be permitted to transfer to the Ph.D. program without completing the M.Sc. program. Program RequirementsThe two options for the M.Sc. program are:
The courses must be chosen from those at the graduate level except that a student may take up to 1.0 credit of undergraduate courses at the 400-level to satisfy these requirements. Not all these courses may be taken in the same field of mathematics; at least 1.0 credit must be in another field. All master’s students are required to participate actively in a seminar or project under the guidance of their adviser. A maximum of 1.0 credit taken outside of the School of Mathematics and Statistics at Carleton University or the Department of Mathematics and Statistics at the University of Ottawa may be allowed for credit. Students who plan to specialize in probability or statistics are strongly advised that during their master’s program they include, where possible, the courses 70.560, 70.551 in mathematical statistics; 70.452, 70.555 in applied statistics, and 70.451, 70.571 in probability, together with 1.0 credit further in the School of Mathematics and Statistics. In addition, a graduate course in another field, such as biology, biostatistics, economics, computer science, systems analysis, and stochastic modelling, is highly recommended. Doctor of PhilosophyAdmission RequirementsThe normal requirement for admission to the Ph.D. program is a master’s degree in mathematics, or the equivalent, with at least high honours standing. Details are outlined in the General Regulations section of this Calendar. Program RequirementsThe course requirements, which are determined at the time of admission, include a minimum of 3.0 credits and a suitable thesis. Not all of these courses may be taken in the same field of mathematics; at least 1.0 credit must be in another field. All candidates must take a comprehensive examination, and satisfy a language requirement. The language requirement is determined by the candidate’s advisory committee and normally requires the ability to read mathematical literature in a language considered useful for his/her research or career, and other than the candidate’s principal language of study. Students specializing in mathematics or probability undertake a comprehensive examination in the following areas:
Students specializing in statistics must write an examination in the following areas:
In all cases, the examination must be completed successfully within twenty months of initial registration in the Ph.D. program in the case of full-time students, and within thirty-eight months of initial registration in the case of part-time students. All Ph.D. candidates are also required to undertake a final oral examination on the subject of their thesis. Selection of CoursesThe following undergraduate courses may, with the approval of the Department of Mathematics and Statistics, be selected by master’s candidates in partial fulfilment of their degree requirements: Mathematics and Statistics
Graduate CoursesNot all of the following courses are offered in a given year. For an up-to-date statement of course offerings for 1998-99, please consult the Registration Instructions and Class Schedule booklet published in the summer. F,W,S indicates term of offering. Courses offered in the fall and winter are followed by T. The number following the letter indicates the credit weight of the course: 1 denotes 0.5 credit, 2 denotes 1.0 credit, etc. Mathematics
70.501W1 (MAT5120) Abstract measure
and integral, L-spaces, complex measures, product measures,
differentiation theory, Fourier transforms. Mathematics
70.503F1 (MAT5122) Commutative Banach algebras; the space of maximal ideals; representation of Banach algebras as function algebras and as operator algebras; the spectrum of an element. Special types of Banach algebras: for example, regular algebras with involution, applications. Mathematics
70.504F1 (MAT5129) A survey of the
main results in the theory of non-singular linear integral
equations; Volterra and Fredholm equations of first and second
kind in the L2 case, with special results for the
continuous case; Hermitian kernels; eigen-function expansions;
compact operators. Mathematics
70.505F1 (MAT5127) Complex differentiation and integration, harmonic functions, maximum modulus principle, Runge’s theorem, conformal mapping, entire and meromorphic functions, analytic continuation. Mathematics
70.506F1 (MAT5316) Construction of
new topological vector spaces out of given ones; local convexity
and the Hahn-Banach theorem; compactness and the Krein-Milman
theorem; conjugate spaces, polar sets. Mathematics
70.507F1 (MAT5125) General measure
and integral, Lebesgue measure and integration on R,
Fubini’s theorem, Lebesgue-Radon-Nikodym theorem, absolute
continuity and differentiation, LP-spaces. Selected topics such
as Daniell-Stone theory. Also offered at the undergraduate level,
with different requirements, as Mathematics 70.407«, for which additional credit is
precluded. Mathematics
70.508W1 (MAT5126) Banach and
Hilbert spaces, bounded linear operators, dual spaces. Topics
selected from: weak-topologies, Alaoglu’s theorem, compact
operators, differential calculus in Banach spaces, Riesz
representation theorems. Also offered at the undergraduate level,
with different requirements, as Mathematics 70.403«, for which additional credit is
precluded. Mathematics
70.509F1 (MAT5121) Geometry of
Hilbert Space, spectral theory of linear operators in Hilbert
Space. Mathematics
70.512F1 (MAT5148) An introduction to group representations and character theory, with selected applications. Mathematics
70.513F1 (MAT5146) Generalizations of the Wedderburn-Artin theorem and applications, homological algebra. Mathematics
70.514F1 (MAT5143) Basic concepts;
ideals, homomorphisms, nilpotent, solvable, semi-simple.
Representations, universal enveloping algebra. Semi-simple Lie
algebras: structure theory, classification, representation
theory. Mathematics
70.516W1 (MAT5145) Fundamental
principles as applied to abelian, nilpotent, solvable, free, and
finite groups; representations. Mathematics
70.517F1 (MAT5141) Groups, Sylow
subgroups, finitely generated abelian groups. Rings, field of
fractions, principal ideal domains, modules. Polynomial algebra,
Euclidean algorithm, unique factorization. Mathematics
70.518W1 (MAT5147) Axioms of set
theory, categories, functors, natural transformations; free,
projective, injective and flat modules; tensor products and
homology functors, derived functors; dimension theory. Also
offered at the undergraduate level, with different requirements,
as Mathematics 70.418«, for which additional credit is
precluded. Mathematics
70.519W1 (MAT5142) Field theory,
algebraic and transcendental extensions, finite fields, Galois
groups. Modules over principal ideal domains, decomposition of a
linear transformation, Jordan normal form. Mathematics
70.521W1 (MAT5150) Various axiom
systems of geometry. Detailed examinations of at least one modern
approach to foundations, with emphasis upon the connections with
group theory. Mathematics
70.522F1 (MAT5168) The
Eilenberg-Steenrod axioms and their consequences, singular
homology theory, applications to topology and algebra. Mathematics
70.525F1 (MAT5151) Topological
spaces, product and identification topologies, countability and
separation axioms, compactness, connectedness, metrization, net
and filter convergence. Also offered at the undergraduate level,
with different requirements, as Mathematics 70.425«, for which additional credit is
precluded. Mathematics
70.526W1 (MAT5152) Covering spaces,
homology via the Eilenberg-Steenrod Axioms, applications,
construction of a homology functor. Also offered at the
undergraduate level, with different requirements, as Mathematics
70.426«, for which additional
credit is precluded. Mathematics
70.527F1 (MAT5169) A study of at
least one modern axiom system of Euclidean and non-Euclidean
geometry, embedding of hyperbolic and Euclidean geometries in the
projective plane, groups of motions, models of non-Euclidean
geometry. Mathematics
70.528F1 (MAT5155) A study of
differentiable manifolds from the point of view of either
differential topology or differential geometry. Topics such as
smooth mappings, transversality, intersection theory, vector
fields on manifolds, Gaussian curvature, Riemannian manifolds,
differential forms, tensors, and connections are included. Mathematics
70.531F1 (MAT5161) A basic graduate
course in mathematical logic. Propositional and predicate logic,
proof theory, Gentzen’s Cut-Elimination, completeness,
compactness, Henkin models, model theory, arithmetic and
undecidability. Special topics (time permitting) depending on
interests of instructor and audience. Mathematics
70.535F1 (MAT5163) Dirichlet series,
characters, Zeta-functions, prime number theorem,
Dirichlet’s theorem on primes in arithmetic progressions,
binary quadratic forms. Mathematics
70.536W1 (MAT5164) Algebraic number
fields, bases, algebraic integers, integral bases, arithmetic in
algebraic number fields, ideal theory, class number. Mathematics
70.543 (MAT5187) Mathematics
70.545F1 (MAT5131) Existence and
uniqueness theorems, boundary value problems, qualitative theory. Mathematics
70.546F1 (MAT5133) First order
linear, quasi-linear, and nonlinear equations; second order
equations in two or more variables; systems of equations; the
wave equation; Laplace and Poisson equations; Dirichlet and
Neumann problems; Green’s functions. Also offered at the
undergraduate level, with different requirements, as Mathematics
70.470«, for which additional
credit is precluded. Mathematics
70.547W1 (MAT5134) Theory of
distributions, initial-value problems based on two-dimensional
wave equations, Laplace transform, Fourier integral transform,
diffusion problems, Helmholtz equation with application to
boundary and initial-value problems in cylindrical and spherical
coordinates. Also offered at the undergraduate level, with
different requirements, as Mathematics 70.471«, for which additional credit is
precluded. Mathematics
70.550F1 (MAT5177) Multivariate
normal distribution properties, characterization, estimation of
means, and covariance matrix. Regression approach to distribution
theory of statistics; multivariate tests; correlations;
classification of observations; Wilks’ criteria. Mathematics
70.551W1 (MAT5191) Confidence
intervals and pivotals; Bayesian intervals; optimal tests and
Neyman-Pearson theory; likelihood ratio and score tests;
significance tests; goodness-of-fit-tests; large sample theory
and applications to maximum likelihood and robust estimation.
Also offered at the undergraduate level, with different
requirements, as Mathematics 70.457«, for which additional credit is
precluded. Mathematics
70.552W1 (MAT5192) Unequal
probability sampling with and without replacement; unified theory
for standard errors; prediction approach; ratio and regression
estimation; stratification and optimal designs; multistage
cluster sampling; double sampling; domains of study;
post-stratification; nonresponse; measurement errors; related
topics. Mathematics
70.553F1 (MAT5193) Theory of non
full rank linear models; estimable functions, best linear
unbiased estimators, hypotheses testing, confidence regions;
multi-way classifications; analysis of covariance; variance
component models; maximum likelihood estimation, Minque, Anova
methods; miscellaneous topics. Mathematics
70.554F1 (MAT5194) Stationary
stochastic processes, inference for stochastic processes,
applications to time series and spatial series analysis. Mathematics
70.555W1 (MAT5195) Overview of
linear model theory; orthogonality; randomized block and split
plot designs; latin square designs; randomization theory;
incomplete block designs; factorial experiments: confounding and
fractional replication; response surface methodology.
Miscellaneous topics. Mathematics
70.556W1 (MAT5175) Nonparametric
tests for location, scale, and regression parameters; derivation
of rank tests; distribution theory of linear rank statistics and
their efficiency. Robust estimation of location, scale and
regression parameters; Huber’s M-estimators, Rank-methods,
L-estimators. Influence function. Adaptive procedures. Mathematics
70.557W1 (MAT5176) Pure significance
test; uniformly most powerful unbiased and invariant tests;
asymptotic comparison of tests; confidence intervals;
large-sample theory of likelihood ratio and chi-square tests;
likelihood inference; Bayesian inference and topics such as
empirical Bayes inference; fiducial and structural methods;
resampling methods. Mathematics
70.558F1 (MAT5172) Course contents
will vary, but will include topics drawn from Markov processes.
Brownian motion, stochastic differential equations, martingales,
Markov random fields, random measures, and infinite particle
systems, advanced topics in modelling, population models, etc. Mathematics
70.559F1 (MAT5196) Multivariate
methods of data analysis, including principal components, cluster
analysis, factor analysis, canonical correlation, MANOVA, profile
analysis, discriminant analysis, path analysis. Mathematics
70.560F1(MAT5190) Statistical
decision theory; likelihood functions; sufficiency; factorization
theorem; exponential families; UMVU estimators; Fisher’s
information; Cramer-Rao lower bound; maximum likelihood and
moment estimation; invariant and robust point estimation;
asymptotic properties; Bayesian point estimation. Also offered at
the undergraduate level, with different requirements, as
Mathematics 70.450«, for which additional
credit is precluded. Mathematics
70.561F1 (MAT5197) Topics chosen
from stochastic dynamic programing, Markov decision processes,
search theory, optimal stopping. Mathematics
70.562F1 (MAT5317) Analysis of
one-way and two-way tables of nominal data; multi-dimensional
contingency tables and log-linear models; tests of symmetry and
marginal homogeneity in square tables; incomplete tables; tables
with ordered categories; fixed margins and logistic models with
binary response; measures of association and agreement;
applications in biological, social and medical sciences. Mathematics
70.563W1 (MAT5318) Types of censored
data; nonparametric estimation of survival function; graphical
procedures for model identification; parametric models and
maximum likelihood estimation; exponential and Weibull regression
models; nonparametric hazard function models and associate
statistical inference; rank tests with censored data;
engineering, medical and biological sciences applications. Mathematics
70.564F1 (MAT5173) Brownian motion,
continuous martingales, and stochastic integration. Mathematics
70.565F1 (MAT5165) Algebraic
structure of sequential machines, de-composition of machines;
finite automata, formal languages; complexity. Also offered at
the undergraduate level, with different requirements, as
Mathematics 70.485«/Computer Science 95.485«, for which additional credit is
precluded. Mathematics
70.567F1 (MAT5324) Two-person
zero-sum games; infinite games; multi-stage games; differential
games; utility theory; two-person general-sum games; bargaining
problem; n-person games; games with a continuum of players. Also
offered at the undergraduate level, with different requirements,
as Mathematics 70.487«, for which additional credit is
precluded. Mathematics
70.569F1 (MAT5301) Prerequisite: Permission of the Department. Mathematics
70.571W1 (MAT5198) Markov systems,
stochastic networks, queuing networks, spatial processes,
approximation methods in stochastic processes and queuing theory.
Applications to the modelling and analysis of
computer-communications systems and other distributed networks. Mathematics
70.578F1 (MAT5170) Probability
spaces, random variables, expected values as integrals, joint
distributions, independence and product measures, cumulative
distribution functions and extensions of probability measures,
Borel-Cantelli lemmas, convergence concepts, independent
identically distributed sequences of random variables. Mathematics
70.579W1 (MAT5171) Laws of large
numbers, characteristic functions, central limit theorem,
conditional probabilities and expectations, basic properties and
convergence theorems for martingales, introduction to Brownian
motion. Mathematics
70.581F1 (MAT5303) Linear programing
problems; simplex method, upper bounded variables, free
variables; duality; postoptimality analysis; linear programs
having special structures; integer programing problems;
unimodularity; knapsack problem. Mathematics
70.582F1 (MAT5325) An introduction to the process of applying computers in problem-solving. Emphasis is placed on the design and analysis of efficient computer algorithms for large, complex problems. Applications in a number of areas are presented: data manipulation, databases, computer networks, queuing systems, optimization. (Also listed as Engineering 94.582, Computer Science 95.582 and Information and Systems Science 93.582) Mathematics
70.583W1 (MAT5304) Methods for
unconstrained and constrained optimization problems; Kuhn-Tucker
conditions; penalty functions; duality; quadratic programing;
geometric programing; separable programing; integer nonlinear
programing; pseudo-Boolean programing; dynamic programing. Mathematics
70.584F1, W1, S1 (MAT5307) Mathematics
70.585F1, W1, S1 (MAT5308) Mathematics
70.586F1 (MAT5180) Error analysis
for fixed and floating point arithmetic; systems of linear
equations; eigen-value problems; sparse matrices; interpolation
and approximation, including Fourier approximation; numerical
solution of ordinary and partial differential equations. Mathematics
70/95.587F1 (MAT5167) Computability,
unsolvable and NP-hard problems. Formal languages, classes of
language automata. Principles of compiler design, syntax
analysis, parsing (top-down, bottom-up), ambiguity, operator
precedence, automatic construction of efficient parsers, LR,
LR(O), LR(k), SLR, LL(k). Syntax directed translation. Mathematics
70.588W1 (MAT5305) Network flow
theory and related material. Topics will include shortest paths,
minimum spanning trees, maximum flows, minimum cost flows.
Optimal matching in bipartite graphs. Mathematics
70.589W1 (MAT5306) Topics include
optimal matching in non-bipartite graphs, Euler tours and the
Chinese Postman problem. Other extensions of network flows:
dynamic flows, multicommodity flows, and flows with gains,
Bottleneck problems. Matroid optimization. Enumerative and
heuristic algorithms for the Travelling Salesman and other
“hard” problems. Mathematics
70.590F1, W1, S1 (MAT5990) Mathematics
70.591F1, W1, S1 (MAT5991) Mathematics
70.593F1, W1, S1 This course is intended for students registered in the M.Sc. degree program in Information and Systems Science and the M.C.S. program. Students pursuing the non-thesis option will conduct a study, analysis, and/or design project under the supervision of a faculty member. Results will be given in the form of a typewritten report and presented at a departmental seminar. Mathematics
70.594F1, W1, S1 This course is project-oriented and affords students the opportunity to undertake statistical research and data analysis projects either within the Statistical Consulting Centre or as a cooperative project with governmental or industrial sponsors. In addition to project work, seminars on related topics will be conducted. Practical data analysis and consulting skills will be emphasized. The grade assigned in this course will be based upon oral and written presentation of analysis results and will be determined in consultation with the faculty adviser and the sponsor. Permission of the Institute is required for registration in this course. Mathematics
70/94/95.595F4, W4, S4 Mathematics
70/93/94/95.598 F3, W3, S3 Mathematics
70.599F3, W3, S3 Mathematics
70.602W1 (MAT5309) Transformation groups; Haar measure; unitary representations of locally compact groups; completeness and compact groups; character theory; decomposition. Mathematics
70.608F1, W1, S1 (MAT5326) Mathematics
70.609F1, W1, S1 (MAT5329) Mathematics
70.611F1, W1, S1 (MAT5327) Mathematics
70.612F1, W1, S1 (MAT5330) Mathematics
70.613F1, W1, S1 (MAT5331) Mathematics
70.614W1 (MAT5158) Matrix groups:
one-parameter groups, exponential map, Campbell-Hausdorff
formula, Lie algebra of a matrix group, integration on matrix
groups. Abstract Lie groups. Mathematics
70.621F1, W1, S1 (MAT5312) Mathematics
70.657F1, W1, S1 (MAT5313) Mathematics
70.658F1, W1, S1 (MAT5314) Mathematics
70.686F1, W1, S1 (MAT5361) Mathematics
70.687F1 (MAT5162) Foundations of
functional languages, lambda calculi (typed, polymorphically
typed, untyped), Curry-Howard Isomorphism, proofs-as-programs,
normalization and rewriting theory, operational semantics, type
assignment, introduction to denotational semantics of programs,
fixed-point programing. Topics chosen from: denotational
semantics for lambda calculi, models of programing languages,
complexity theory and logic of computation, models of concurrent
and distrubted systems, etc. Mathematics
70.690F1, W1, S1 (MAT6990) Mathematics
70.691F1, W1, S1 (MAT6991) Mathematics
70.699F, W, S Mathematics
70.621F1, W1, S1 (MAT5312) Mathematics
70.657F1, W1, S1 (MAT5313) Mathematics
70.658F1, W1, S1 (MAT5314) Mathematics
70.686F1, W1, S1 (MAT5361) Mathematics
70.687F1 (MAT5162) Foundations of
functional languages, lambda calculi (typed, polymorphically
typed, untyped), Curry-Howard Isomorphism, proofs-as-programs,
normalization and rewriting theory, operational semantics, type
assignment, introduction to denotational semantics of programs,
fixed-point programing. Topics chosen from: denotational
semantics for lambda calculi, models of programing languages,
complexity theory and logic of computation, models of concurrent
and distrubted systems, etc. Mathematics
70.690F1, W1, S1 (MAT6990) Mathematics
70.691F1, W1, S1 (MAT6991) Mathematics
70.699F, W, S |
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