Faculty of Science

Mathematics Courses

MA103 Calculus I 0.5

Limits and continuity; differential and integral calculus of functions of a single variable; the Mean Value Theorem; determination of extrema; the Fundamental Theorem of Calculus and techniques of integration; introduction to partial derivatives.

Prerequisite: OAC or Grade 12 U Calculus.

Exclusion: MA100*, MA101, MA102, MA105, MA110*, MA130.

3 lecture hours, 2 lab hours

MA104 Calculus II 0.5

Applications of integration; polar coordinates and parametric equations; infinite sequences and series; applications of partial derivatives.

Prerequisite: MA103 (or MA110*) or MA105.

Exclusion: MA100*, MA102, MA200.

3 lecture hours, 2 lab hours

MA105 Calculus for the Sciences 0.5

Differential calculus of the algebraic, exponential, logarithmic, trigonometric and inverse trigonometric functions of a single variable; introduction to integral calculus; techniques of integration including integration by parts; introduction to partial derivatives. Applications to problems in science are emphasized.

Prerequisite: OAC or Grade 12 U Calculus.

Exclusion: MA100*, MA101, MA102, MA103, MA110*, MA130; Honours Mathematics programs; Mathematics major or minor.

3 lecture hours, 2 lab hours

MA110* Introduction to Differential and Integral Calculus 1.0

A thorough introduction to limits of functions. Continuity and its consequences. Rational, algebraic and transcendental functions and geometric relationships. Theory and applications of differential and integral calculus of functions of a single variable. The Fundamental Theorem of Calculus and techniques of integration. Introduction to multivariable calculus and applications.

Prerequisite: Registration in a Faculty of Science program and permission of the Department of Mathematics (based on the results of a diagnostic or skills assessment test).

Exclusion: MA101, MA102, MA103, MA105, MA130.

3 lecture hours, 1.5 lab hours

MA121 Introduction to Sets and Algebra 0.5

Introduction to sets; algebra of sets; Boolean algebra and truth tables; functions and binary relations; induction and other methods of proof; Binomial Theorem; properties of the integers including the division algorithm and the Euclidean algorithm; algebra of complex numbers including de Moivre's Theorem.

MA122 Introductory Linear Algebra 0.5

Systems of linear equations; matrix algebra; determinants and their applications; vector geometry in R2 and R3; spanning, linear independence and linear transformations in Rn; introduction to eigenvalues and eigenvectors; applications of linear algebra.

MA130 Calculus for Students of Business and Economics 0.5

Differential calculus of the algebraic, exponential, and logarithmic functions of a single variable; introduction to integral calculus; introduction to the differential calculus of functions of several variables. Applications to problems in business and economics are emphasized.

Prerequisite: OAC or Grade 12 U Calculus.

Exclusion: MA100*, MA101, MA102, MA103, MA105, MA110*; Honours Mathematics programs; Mathematics major or minor; BSc programs.

MA141 Introduction to Applied Statistical Analysis 0.5

Numerical and graphical methods of descriptive statistics; basic probability; introduction to discrete and continuous random variables; sampling distributions; estimation and hypothesis testing for data from one and two normal populations. Applications to problems in science are emphasized and statistical software packages are used for data analysis.

Exclusion: MA240, BU205/EC205, BU255/EC255, EC235, EC285, KP262; all Honours Business, Honours Economics and Honours Mathematics programs. [Note: Students holding credit in statistical quantitative methods courses other than those listed above are strongly advised to consult with their home departments before registering in MA141].

3 lecture hours, 2 lab hours

MA201 Multivariable Calculus 0.5

Vector functions; differential and integral calculus of functions of several variables, including vector fields; line and surface integrals including Green's Theorem, Stokes' Theorem and the Divergence Theorem.

Prerequisite: MA122, and one of MA102, MA103 (or MA110*) or MA105 (MA104 is recommended).

3 lecture hours, 2 lab hours every other week

MA205 Differential Equations I 0.5

First order differential equations; linear differential equations of second and higher order; methods of undetermined coefficients and variation of parameters; Laplace transforms; power series solutions.

Prerequisite: One of MA102, MA103 (or MA110*), or MA105.

3 lecture hours, 2 lab hours every other week

MA215 Set Theory 0.5

Equivalence relations and partitions; countable and uncountable sets; ordered sets; development of number systems.

Prerequisite: MA121.

MA222 Linear Algebra 0.5

Vector spaces; linear transformations; diagonalizability; applications.

Prerequisite: MA121, MA122.

3 lecture hours, 2 lab hours every other week

MA233 Problem Solving in Mathematics 0.5

A comprehensive study of techniques in mathematical problem solving, including topics from classical and contemporary mathematics. Examples will be taken from such areas as calculus, algebra, logic, geometry, elementary number theory, combinatorics and graph theory. Problems from various mathematical contests and competitions will be discussed.

Prerequisite: MA103 (or MA110*), MA121.

MA234 Introduction to Combinatorics 0.5

Basic counting principles; permutations and combinations, with applications; Multinomial Theorem; combinatorial identities and combinatorial arguments; recurrence relations; principle of inclusion and exclusion; pigeonhole principle.†

Prerequisite: MA121, and one of MA102, MA103 (or MA110*), MA105 or MA130.

MA235 Introduction to Game Theory 0.5

An introduction to game-theoretic methods and their applications. Mathematical game models are used to study problems such as the free-rider problem of economics and political science, and the choices of firms in oligopolistic markets.

Prerequisite: One of MA102, MA103 (or MA110*), MA105 or MA130.

Exclusion: EC325.

MA238 Discrete Mathematics 0.5

Basic graph theory, Euler circuits and Hamilton cycles in graphs, planar graphs, graph colouring, trees, relations, partial orders, introduction to counting, recurrence relations, inclusion-exclusion.

Prerequisite: MA121 and an additional 0.5 MA credit.

MA240 Introduction to Probability and Statistics 0.5

Data collection and description including univariate and bivariate frequency tables, histograms and summary statistics; elementary probability theory; random variables and expectations; sampling theory and the Central Limit Theorem; estimation and hypothesis testing for data from one and two normal populations.

Prerequisite: MA102 or MA104 or (one of MA103 and MA105 as well as either MA121 or MA122).

Exclusion: MA141, BU205/EC205, BU255/EC255, EC235, EC285 or any other quantitative statistical methods course.

3 lecture hours, 2 lab hours

MA242 Applied Regression Analysis 0.5

Simple linear regression and bivariate correlation models; multiple regression analysis including polynomial regression and variable selection techniques; regression diagnostics. Data will be processed using available statistical software packages.

Prerequisite: MA122, and one of MA141 or MA240 (or permission of the department).

Exclusion: MA245, MA342, EC295.

1 lab hour every other week

MA250 Introduction to Analysis 0.5

A rigorous development of calculus. Topics include sequences, convergence, limits, continuity, differentiability and the Riemann integral.

Prerequisite: MA121, and one of MA102 or MA103 (or MA110*).

MA255 Continuous and Discrete Transforms W 0.5

Properties of continuous and discrete Fourier transforms; the Sampling Theorem; Inverse Fourier Transforms and convolution; introduction to wavelet analysis; Fast Fourier Transforms and other algorithms. Applications will be selected from areas such as sound, vision and optics.

Prerequisite: MA104, MA201, MA205.

3 lecture hours, 3 lab hours every other week

MA270 Financial Mathematics I 0.5

An introduction to mathematical methods used in financial analysis. Linear algebra, calculus and probability and statistics are used in the analysis of problems in areas such as input-output analysis, inventory problems, bonds, annuities, amortization, futures, options, profit/cost optimization, decisions under certainty/uncertainty, capital budgeting and risk and return.

Prerequisite: MA103 (or MA110*), MA122, MA240; CP102 or CP104.

3 lecture hours; 3 lab hours every other week

MA303 Real Analysis 0.5

The topology of the reals including properties of open, closed, connected and compact sets, continuity and uniform continuity; definition and properties of the Riemann integral.†

Prerequisite: MA215, MA250.

MA304 Introduction to Complex Analysis 0.5

Functions of a complex variable; transformations; integration; Taylor and Laurent expansions; theory of residues.†

Prerequisite: MA104 or MA200, and MA201.

MA305 Differential Equations II 0.5

Numerical solutions of differential equations and boundary value problems; linear systems of differential and difference equations including their solution by matrix methods and their stability; introduction to dynamical systems. Numerical methods will be illustrated by exercises requiring the use of a computer.†

Prerequisite: MA122, MA104 or MA200, MA205 and either a 0.5 credit in computer programming or permission of the department.

Exclusion: MA308.

MA307 Numerical Analysis 0.5

Numerical solution of equations and systems of equations; numerical integration; methods of interpolation, extrapolation and curve-fitting; error analysis. Methods will be illustrated by exercises requiring the use of a computer.†

Prerequisite: MA122, MA104 or MA200, MA201, and either a 0.5 credit in computer programming or permission of the department.

MA315 Metric Spaces 0.5

Examples of metric spaces; basic properties of open and closed sets; convergent sequences and continuous and uniformly continuous functions; connected sets; compactness, sequential compactness and the Bolzano-Weierstrass property; completion of a metric space.††

Prerequisite: MA104 or MA200; MA215, MA250.

MA317 Number Theory 0.5

Elementary theory of numbers; arithmetic functions; congruences; quadratic reciprocity; solutions to Diophantine equations.†

Prerequisite: MA121 and 1.0 credit of 200 level Mathematics.

MA318 Geometry 0.5

General survey of geometries and their transformations emphasizing the axiomatic approach; finite geometries; Euclidean geometry; congruence and similarity; non-Euclidean geometries; affine and projective geometry; Desargues' Theorem and Pappus' Theorem.†

Prerequisite: MA215 or MA222.

MA323 Introduction to Groups and Rings 0.5

Examples and basic properties of groups and rings including their substructures, quotient structures and homomorphisms.

Prerequisite: MA215 or MA222.

Exclusion: MA225.

MA331 Algorithm Analysis 0.5

Introduction to the construction and analysis of algorithms; describing algorithms in pseudocode; recursive procedures; complexity of functions and algorithms; study of several specific non-numerical and numerical algorithms.††

Prerequisite: MA121, MA122, a 0.5 credit in computer programming and one of MA102, MA103 (or MA110*), MA105 (or permission of the department).

MA338 Graph Theory W 0.5

Selected topics may include graph colouring, extremal graph theory, planar graphs, random graphs, network flows, algebraic methods in graph theory, Ramsay theory for graphs, matching theory, graph algorithms; application of graph theory, such as applications to scheduling, VLSI circuits, compiler design, computer vision and the design of internet search engines.

Prerequisite: MA238.

MA339 Directed Studies in Mathematics 0.5

Prerequisite: Permission of the department.††

Irregular course

MA340 Introduction to Probability Theory 0.5

Discrete and continuous probability models; random variables; univariate and multivariate distributions; conditional distributions; expectation operator, moments and moment generating functions; functions of random variables; limit theorems.†

Prerequisite: MA201, and one of MA141 or MA240 (or permission of the department).

MA341 Introduction to the Theory of Statistics 0.5

Review of problems and objectives of parametric statistics; likelihood function; methods of finding estimators; properties of estimators; interval estimation; hypothesis testing procedures; Neyman-Pearson Lemma and likelihood ratio testing.†

Prerequisite: MA201, and one of MA141, or MA240 (or permission of the department).

MA343 Introduction to Multivariate Analysis 0.5

Multivariate normal and Wishart distributions; estimation and hypothesis testing for data from one and two independent normal populations; classification procedures including discriminant and cluster analyses; principal component and factor analyses. Data may be processed using available statistical software packages.†

Prerequisite: MA122, and one of MA141 or MA240 (or permission of the department).

MA344 Introduction to Experimental Design and Analysis 0.5

Randomization and replication; one and two way classifications, including completely randomized and randomized complete block experiments; designs and analysis of variance (ANOVA) tables for fixed, random and mixed effects models; efficiency of designs; a priori treatment comparisons based on orthogonal contrasts and orthogonal polynomials; multiple comparison procedures, including multiple range tests. Data will be processed using available computer software packages.†

Prerequisite: MA242 (or permission of the department).

Exclusion: MA245.

MA349 Sampling Theory 0.5

Simple random samples, survey design, ratio and regression estimators, proportional allocation, stratified sampling, cluster sampling, complex surveys, nonresponse.H.††

Prerequisite: MA141 or MA240 (or permission of the department).

MA351 Introduction to Stochastic Calculus 0.5

Stochastic integration, stochastic differential equations, Martingales and semimartingales and applications.

Prerequisite: MA340.

MA360 Topics in Applied Mathematics 0.5

Selected topics may include Fourier analysis, fast Fourier transforms, filtering and wavelet theory, with emphasis on the application of mathematical models to problems in the sciences (e.g., biomechanical and physiological feedback).†

Prerequisite: MA205, MA222, and one of MA200, MA201, MA250.

MA370 Financial Mathematics II 0.5

Discrete-time models, riskless asset pricing, asset pricing under risk, arbitrage, utility theory. Complete and incomplete markets. Introduction to options and risk-neutral pricing.

Prerequisite: MA270, MA340.

3 lecture hours; 3 lab hours every other week

MA371 Computational Methods in Mathematics and Statistics 0.5

Solution techniques using computational methods in matrix, differential and integral equations. Statistical simulation, Fourier and other transform methods. The use of languages such as Fortran, C++, Java, Maple, Matlab in numeric and symbolic solution methods.

Prerequisite: MA201, MA222; CP104.

3 lecture hours; 3 lab hours every other week

MA372 Optimization 0.5

Optimality conditions, Lagrangian relaxation and duality, linear and integer programming, non-linear programming. Deterministic and stochastic dynamic programming.

Prerequisite: MA201, MA222, MA240.

MA395 Special Topics 0.5

Prerequisite: Permission of the department.

Irregular course

MA422 Advanced Linear Algebra 0.5

Cayley-Hamilton Theorem; real and complex inner product spaces; Spectral Theorem; bilinear and quadratic forms; canonical forms.

Prerequisite: MA215, MA222.

Exclusion: MA322.

MA425 Group Theory 0.5

Monoids and groups, subgroups, quotient groups and group homomorphisms: groups acting on sets, conjugacy and the class equation; the Sylow theorems; free groups; finitely generated Abelian groups.††

Prerequisite: MA222, MA323.

Exclusion: MA325.

MA450 Measure and Integration 0.5

Sigma-algebras of sets; set functions; outer measures, measurable sets and Lebesgue measure; Riemann and Lebesgue integrals; convergence in measure.†

Prerequisite: MA104 or MA200, MA215, MA250 and a 0.5 MA credit at the 300 level.

MA455 Partial Differential Equations 0.5

Hyperbolic, parabolic and elliptic differential equations; boundary value problems of applied mathematics including such partial differential equations as the heat equation, the wave equation and Laplace's equation. Techniques will include separation of variables, canonical transformations and integral transform methods.†

Prerequisite: MA104 or MA200, MA201 and MA205, and a 0.5 MA credit at the 300 level.

MA465 General Topology 0.5

Topological spaces: bases and subbases of topologies; closures and interiors; subspaces, continuity; the quotient topology; product spaces; the separation axioms; nets, filters and convergence; countability properties; compactness and Tychonoff's Theorem; connectedness.††

Prerequisite: MA315.

MA470 Financial Mathematics III 0.5

European and American options, Black-Scholes model, path dependent options, bonds, interest rate models.

Prerequisite: MA351, MA370, MA455.

3 lecture hours; 3 lab hours every other week

MA475 Ring and Field Theory 0.5

Rings; subrings, quotient rings and ring homomorphisms; ideal theory; polynomial rings; integral domains and divisor theory; fields and field extensions; the Fundamental Theorem of Galois Theory.††

Prerequisite: MA222, MA323.

MA480* Directed Research 1.0

A detailed study of a topic under faculty supervision including the submission of a formal report.††

Prerequisite: Permission of the department.

Irregular course

MA485 Directed Studies 0.5

Prerequisite: Permission of the department.†

Irregular course

MA488 Computability and Logic 0.5

Turing machines and concept of computability; automata; recursive functions; recursive and recursively enumerable sets; Church's Thesis; the halting problem and other unsolvable problems; NP-completeness; undecidability and Gφdel's Incompleteness Theorem.††

Prerequisite: MA215, MA238 and a 0.5 MA credit at the 300 level.

MA489 Honours Seminar 0.5

Completion of an appropriate individual project under faculty supervision, including submission of a final report and presentation in a department seminar. (See department for details.)

Prerequisite: Permission of the department.

MA490 Stochastic Processes 0.5

Markov Chains in discrete and continuous time; birth-death processes; renewal theory; renewal-reward theory; Markov processes; stationary processes; introduction to queuing theory.††

Prerequisite: MA340.

MA492 Time Series Analysis 0.5

(Cross-listed as EC455.)

Prerequisite: MA242 or permission of the department.

1 lab hour every other week.

MA495 Special Topics 0.5

Prerequisite: Permission of the department.††

Irregular course

Faculty of Science Faculty of Science – Departments, Programs and Courses Mathematics

Official electronic version updated at 10:33 a.m. March 31, 2005