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Program Requirements
The Minor Concentration Mathematics is offered in two versions: an expandable version, for students who wish to leave open the option of expanding the program into a Major Concentration Mathematics, and a nonexpandable version for students who know on entry into the Minor that they do not wish to expand it into a major concentration. The Minor Concentration Mathematics may be taken in conjunction with a major concentration in some other discipline under option A of the Multitrack System. Students planning on taking the Major Concentration Mathematics and the Minor Concentration Mathematics as part of Multitrack option C should select the Supplementary Minor Concentration in Mathematics in place of this Minor concentration. Under option C, it is not possible to combine the Minor Concentration Mathematics and the Minor Concentration Statistics. Students wishing to do this should instead take the Major Concentration Mathematics under option B (two major concentrations) and select a large number of statistics complementaries. For more information about the Multitrack System options please refer to the Faculty of Arts regulations under "Faculty Degree Requirements", "About Program Requirements", and "Departmental Programs". No overlap is permitted with other programs.Program Prerequisites
Students who have not completed the program prerequisite courses listed below or their equivalents will be required to make up any deficiencies in these courses over and above the 18 credits required for the program.

MATH 133 Linear Algebra and Geometry (3 credits)
Overview
Mathematics & Statistics (Sci) : Systems of linear equations, matrices, inverses, determinants; geometric vectors in three dimensions, dot product, cross product, lines and planes; introduction to vector spaces, linear dependence and independence, bases; quadratic loci in two and three dimensions.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: Shabnam Akhtari, Djivede Kelome, William J Anderson, Adam Clay, Sidney Trudeau (Fall) Djivede Kelome, Wilbur Jonsson (Winter) Nabil Kahouadji (Summer)
3 hours lecture, 1 hour tutorial
Prerequisite: a course in functions
Restriction: Not open to students who have taken MATH 221 or CEGEP objective 00UQ or equivalent.
Restriction Note B: Not open to students who have taken or are taking MATH 123, MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics.

MATH 140 Calculus 1 (3 credits)
Overview
Mathematics & Statistics (Sci) : Review of functions and graphs. Limits, continuity, derivative. Differentiation of elementary functions. Antidifferentiation. Applications.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: Stephen W Drury, David Shirokoff, Mark Eric Fels (Fall) Wilbur Jonsson (Winter) Yaiza Canzani Garcia (Summer)
3 hours lecture, 1 hour tutorial
Prerequisite: High School Calculus
Restriction: Not open to students who have taken MATH 120, MATH 139 or CEGEP objective 00UN or equivalent
Restriction: Not open to students who have taken or are taking MATH 122 or MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics
Each Tutorial section is enrolment limited

MATH 141 Calculus 2 (4 credits)
Overview
Mathematics & Statistics (Sci) : The definite integral. Techniques of integration. Applications. Introduction to sequences and series.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: Sungmo Kang (Fall) Antonio Lei, Stephen W Drury, Layan El Hajj (Winter) Hadi Bigdely, Philip Grech (Summer)
Restriction: Not open to students who have taken MATH 121 or CEGEP objective 00UP or equivalent
Restriction Note B: Not open to students who have taken or are taking MATH 122 or MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics.
Each Tutorial section is enrolment limited
Expandable Version: Required Courses (12 credits)
* Note: Credit cannot be received for both MATH 236 and MATH 223 (listed as a required course in the nonexpandable version of this Minor concentration).

MATH 222 Calculus 3 (3 credits)
Overview
Mathematics & Statistics (Sci) : Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extreme of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: James G Loveys, Mathew Donald Rogers (Fall) James G Loveys (Winter) Nicolás Fraiman (Summer)

MATH 235 Algebra 1 (3 credits)
Overview
Mathematics & Statistics (Sci) : Sets, functions and relations. Methods of proof. Complex numbers. Divisibility theory for integers and modular arithmetic. Divisibility theory for polynomials. Rings, ideals and quotient rings. Fields and construction of fields from polynomial rings. Groups, subgroups and cosets; group actions on sets.
Terms: Fall 2011
Instructors: Heekyoung Hahn (Fall)
Fall
3 hours lecture; 1 hour tutorial
Prerequisite: MATH 133 or equivalent

MATH 236 Algebra 2 (3 credits) *
Overview
Mathematics & Statistics (Sci) : Linear equations over a field. Introduction to vector spaces. Linear mappings. Matrix representation of linear mappings. Determinants. Eigenvectors and eigenvalues. Diagonalizable operators. CayleyHamilton theorem. Bilinear and quadratic forms. Inner product spaces, orthogonal diagonalization of symmetric matrices. Canonical forms.
Terms: Winter 2012
Instructors: Heekyoung Hahn (Winter)
Winter
Prerequisite: MATH 235

MATH 315 Ordinary Differential Equations (3 credits)
Overview
Mathematics & Statistics (Sci) : First order ordinary differential equations including elementary numerical methods. Linear differential equations. Laplace transforms. Series solutions.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: JianJun Xu (Fall) JianJun Xu (Winter) Suresh Eswarathasan (Summer)
Expandable Version: Complementary Courses (6 credits)
Students selecting the expandable version of this program complete six complementary courses from the Complementary Course List.
It is strongly recommended that students take MATH 323 as a complementary course.
NonExpandable Version: Required Courses (9 credits)
* Note: Credit cannot be received for both MATH 223 and MATH 236 (listed as a required course in the expandable version of this Minor concentration).

MATH 222 Calculus 3 (3 credits)
Overview
Mathematics & Statistics (Sci) : Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extreme of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: James G Loveys, Mathew Donald Rogers (Fall) James G Loveys (Winter) Nicolás Fraiman (Summer)

MATH 223 Linear Algebra (3 credits) *
Overview
Mathematics & Statistics (Sci) : Review of matrix algebra, determinants and systems of linear equations. Vector spaces, linear operators and their matrix representations, orthogonality. Eigenvalues and eigenvectors, diagonalization of Hermitian matrices. Applications.
Terms: Fall 2011, Winter 2012
Instructors: Wilbur Jonsson (Fall) Wilbur Jonsson (Winter)

MATH 315 Ordinary Differential Equations (3 credits)
Overview
Mathematics & Statistics (Sci) : First order ordinary differential equations including elementary numerical methods. Linear differential equations. Laplace transforms. Series solutions.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: JianJun Xu (Fall) JianJun Xu (Winter) Suresh Eswarathasan (Summer)
NonExpandable Version: Complementary Courses (9 credits)
Students selecting the nonexpandable version of this program complete nine complementary courses from the Complementary Course List.
It is strongly recommended that students take MATH 323 as a complementary course.
Complementary Course List
* Note: Either MATH 249 or MATH 316 may be taken but not both.

MATH 249 Honours Complex Variables (3 credits) *
Overview
Mathematics & Statistics (Sci) : Functions of a complex variable; CauchyRiemann equations; Cauchy's theorem and consequences. Taylor and Laurent expansions. Residue calculus; evaluation of real integrals; integral representation of special functions; the complex inversion integral. Conformal mapping; SchwarzChristoffel transformation; Poisson's integral formulas; applications.
Terms: Winter 2012
Instructors: Robert Seiringer (Winter)

MATH 314 Advanced Calculus (3 credits)
Overview
Mathematics & Statistics (Sci) : Derivative as a matrix. Chain rule. Implicit functions. Constrained maxima and minima. Jacobians. Multiple integration. Line and surface integrals. Theorems of Green, Stokes and Gauss.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: Wilbur Jonsson (Fall) Ivo Klemes (Winter) Charles Roth (Summer)

MATH 316 Complex Variables (3 credits) *
Overview
Mathematics & Statistics (Sci) : Algebra of complex numbers, CauchyRiemann equations, complex integral, Cauchy's theorems. Taylor and Laurent series, residue theory and applications.
Terms: Fall 2011
Instructors: Heekyoung Hahn (Fall)

MATH 317 Numerical Analysis (3 credits)
Overview
Mathematics & Statistics (Sci) : Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations.
Terms: Fall 2011
Instructors: Peter Bartello (Fall)

MATH 318 Mathematical Logic (3 credits)
Overview
Mathematics & Statistics (Sci) : Propositional calculus, truthtables, switching circuits, natural deduction, first order predicate calculus, axiomatic theories, set theory.
Terms: Fall 2011
Instructors: James G Loveys (Fall)
Fall
Restriction: Not open to students who are taking or have taken PHIL 210

MATH 319 Introduction to Partial Differential Equations (3 credits)
Overview
Mathematics & Statistics (Sci) : First order equations, geometric theory; second order equations, classification; Laplace, wave and heat equations, SturmLiouville theory, Fourier series, boundary and initial value problems.
Terms: Winter 2012
Instructors: Gantumur Tsogtgerel (Winter)

MATH 320 Differential Geometry (3 credits)
Overview
Mathematics & Statistics (Sci) : Review of Euclidean geometry. Local theory of plane and space curves: the Frenet formulas. Local theory of surfaces: the first and second fundamental forms, the shape operator, the mean and Gaussian curvatures, surfaces of revolution with prescribed curvature, ruled and developable surfaces. Geodesic curves on surfaces of revolution. The GaussCodazzi equations, rigidity.
Terms: This course is not scheduled for the 20112012 academic year.
Instructors: There are no professors associated with this course for the 20112012 academic year.

MATH 323 Probability (3 credits)
Overview
Mathematics & Statistics (Sci) : Sample space, events, conditional probability, independence of events, Bayes' Theorem. Basic combinatorial probability, random variables, discrete and continuous univariate and multivariate distributions. Independence of random variables. Inequalities, weak law of large numbers, central limit theorem.
Terms: Fall 2011, Winter 2012, Summer 2012
Instructors: William J Anderson (Fall) David B Wolfson (Winter) Djivede Kelome (Summer)

MATH 324 Statistics (3 credits)
Overview
Mathematics & Statistics (Sci) : Sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, contingency tables, nonparametric inference, regression, Bayesian inference.
Terms: Fall 2011, Winter 2012
Instructors: Russell Steele (Fall) William J Anderson (Winter)
Fall and Winter
Prerequisite: MATH 323 or equivalent
Restriction: Not open to students who have taken or are taking MATH 357
You may not be able to receive credit for this course and other statistic courses. Be sure to check the Course Overlap section under Faculty Degree Requirements in the Arts or Science section of the Calendar.

MATH 326 Nonlinear Dynamics and Chaos (3 credits)
Overview
Mathematics & Statistics (Sci) : Linear systems of differential equations, linear stability theory. Nonlinear systems: existence and uniqueness, numerical methods, one and two dimensional flows, phase space, limit cycles, PoincareBendixson theorem, bifurcations, Hopf bifurcation, the Lorenz equations and chaos.
Terms: Fall 2011
Instructors: Renato Calleja Castillo (Fall)

MATH 327 Matrix Numerical Analysis (3 credits)
Overview
Mathematics & Statistics (Sci) : An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems.
Terms: This course is not scheduled for the 20112012 academic year.
Instructors: There are no professors associated with this course for the 20112012 academic year.

MATH 328 Computability and Mathematical Linguistics (3 credits)
Overview
Mathematics & Statistics (Sci) : Calculability on an infinite abacus is compared with recursive functions and Turing machines. Categorial, contextfree, generative and transformational grammars are studied for formal and natural languages, with some emphasis on English and French morphology. Machines for generating and recognizing sentences are discussed.
Terms: This course is not scheduled for the 20112012 academic year.
Instructors: There are no professors associated with this course for the 20112012 academic year.
 Winter

MATH 339 Foundations of Mathematics (3 credits)
Overview
Mathematics & Statistics (Sci) : A continuation of MATH 338. Topics are chosen mainly from 19th and 20th century mathematics, with some emphasis on philosophical and foundational problems. Sample topics are: progress in number theory, construction of the number system, infinity according to Cantor, logic and foundations from Aristotle to Cohen, Gödel's incompleteness theorem, calculability and programs, formalism versus intuitionism, abstract mathematics and categories.
Terms: This course is not scheduled for the 20112012 academic year.
Instructors: There are no professors associated with this course for the 20112012 academic year.

MATH 340 Discrete Structures 2 (3 credits)
Overview
Mathematics & Statistics (Sci) : Review of mathematical writing, proof techniques, graph theory and counting. Mathematical logic. Graph connectivity, planar graphs and colouring. Probability and graphs. Introductory group theory, isomorphisms and automorphisms of graphs. Enumeration and listing.
Terms: Winter 2012
Instructors: Adrian Roshan Vetta (Winter)

MATH 346 Number Theory (3 credits)
Overview
Mathematics & Statistics (Sci) : Divisibility. Congruences. Quadratic reciprocity. Diophantine equations. Arithmetical functions.
Terms: This course is not scheduled for the 20112012 academic year.
Instructors: There are no professors associated with this course for the 20112012 academic year.

MATH 348 Topics in Geometry (3 credits)
Overview
Mathematics & Statistics (Sci) : Selected topics  the particular selection may vary from year to year. Topics include: isometries in the plane, symmetry groups of frieze and ornamental patterns, equidecomposibility, nonEuclidean geometry and problems in discrete geometry.
Terms: Summer 2012
Instructors: Benjamin Smith (Summer)
Prerequisite: MATH 133 or equivalent or permission of instructor.

MATH 407 Dynamic Programming (3 credits)
Overview
Mathematics & Statistics (Sci) : Sequential decision problems, resource allocation, transportation problems, equipment replacement, integer programming, network analysis, inventory systems, project scheduling, queuing theory calculus of variations, markovian decision processes, stochastic path problems, reliability, discrete and continuous control processes.
Terms: Winter 2012
Instructors: Neville G F Sancho (Winter)

MATH 417 Mathematical Programming (3 credits)
Overview
Mathematics & Statistics (Sci) : An introductory course in optimization by linear algebra, and calculus methods. Linear programming (convex polyhedra, simplex method, duality, multicriteria problems), integer programming, and some topics in nonlinear programming (convex functions, optimality conditions, numerical methods). Representative applications to various disciplines.
Terms: Fall 2011
Instructors: Yori Zwols (Fall)