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MATHEMATICS AND STATISTICS: COURSES
Students are reminded that, as indicated in the course descriptions, certain Mathematics and Statistics courses may not be available for credit in some or all of the degree programs.
All courses listed will not necessarily be offered each year.
MATHEMATICS
62101. Access to Calculus
A variety of precalculus topics including coordinate geometry, trigonometric, exponential and logarithmic functions, and algebraic procedures. Introduction to differential calculus. (This course satisfies the prerequisite or admission requirement of Grade 12 "U" Advanced Functions. May not be taken for credit by by majors in the Faculty of Science or the Faculty of Engineering.) (Antirequisites: 62130, 62139, 62140, 62194, a grade of 70% or greater in Grade 12 Advanced Functions) (3 credit hours, one hour tutorial per week.)
62102. Access to Algebra
This course enables students to broaden their mathematical knowledge and skills related to the mathematical topics of vectors, intersections of lines and planes in three dimensional space counting techniques, and mathematical induction. Students will develop an understanding of proofs, using deductive, algebraic, vector and indirect methods. Students will use vector concepts to solve physical problems. (This course and 62101 satisfy the prerequisite or admission requirement of Grade 12 Advanced Functions and Grade 12 Calculus and Vectors or equivalent. May not be taken for credit by majors in the Faculty of Science or the Faculty of Engineering.) (3 credit hours, one hour tutorial per week.)
62120. Linear Algebra I
Linear systems, matrix algebra, determinants, vectors in Rn , dot product, orthogonalization, eigenvalues, and diagonalization. (Prerequisite: 62102 or Grade 12 Advanced Functions and Grade 12 Calculus and Vectors or equivalent.) (Antirequisite: 62125 or 62126) (3 lecture hours, 1 tutorial hour a week.)
62125 Vectors and Linear Algebra
Vectors, three dimensional geometry, linear systems, matrix algebra, determinants, vector spaces, dot products, cross products, eigenvalues and eigenvectors, and diagonalization, orthogonalization. (This is required for students who do not have credit for Ontario grade 12 Calculus and Vectors. The course is equivalent to 62120/126 for all prerequisite purposes.) (Prerequisite: Grade 12 Advanced Functions or equivalent.) (Antirequisites: 62120, 62126.) (4 lecture hours, 1 tutorial hour a week.)
62126. Linear Algebra (Engineering)
Linear systems, matrix algebra, determinants, vectors in Rn, dot product, orthogonalization, and eigenvalues. (Prerequisite: 62102 or Grade 12 Advanced Functions and Grade 12 Calculus and Vectors, or equivalent.) (Antirequisite: 62120, or 62125.) (3 lectures hours, 1 tutorial hour a week.)
62130. Elements of Calculus
Review of functions. Limits and continuity. Derivatives and applications. Indefinite integrals and methods of integration. Partial derivatives. A variety of applications. Prerequisite: Grade 12 Advanced Functions or 62101.) (May not be taken for credit concurrently with, or subsequent to having obtained credit in 62139 or 62140. This course is not a sufficient prerequisite to 62141, but may serve as preparation for 62140.) (3 lecture hours, 1 tutorial hour a week.)
62139. Functions and Differential Calculus
Trigonometric functions and identities, inverse trigonometric functions, limits and continuity, derivatives and applications, Mean value theorem, indeterminate forms and l’Hospital’s rule, antiderivatives, introduction to indefinite integrals. (This course is required for students who do not have credit for Ontario grade 12 Calculus and Vectors. The course is equivalent to 62140 for all prerequisite purposes.) (Prerequisite: Grade 12 Advanced Functions or equivalent.) (Antirequisite: 62140.) (4 lecture hours, 1 tutorial hour a week.) [Fall 2010Winter 2013: 4 lecture hours, 2 tutorial hour a week.]
62140. Differential Calculus
Trigonometric functions and identities. Inverse trigonometric functions. Limits and continuity. Derivatives and applications. Mean Value Theorem. Indeterminate forms and l'Hopital's Rule. Antiderivatives. Introduction to definite integrals. (Prerequisite: Grade 12 Advanced Functions and Grade 12 Calculus and Vectors or equivalent, or 62101.) (Antirequisite: 62139) (3 lecture hours, 1 tutorial hour a week.) [Fall 2010Winter 2013: 4 lecture hours, 2 tutorial hour a week.]
62141. Integral Calculus
Antiderivatives. The definite integral and Fundamental Theorem. Techniques of integration. Applications of the definite integral. Improper integrals. Separable differential equations. Polar and parametric coordinates. (Prerequisite: 62139 or 62140.) (3 lecture hours, 1 tutorial hour a week.)
62188. Work Term I
Supervised experience in an approved careerrelated setting with a focus on the application of theory and the development of transferable skills. The coop work experience is designed to provide students with an enriched learning opportunity to integrate academic theory and concepts in an applied setting. (Prerequisite: Student must be enrolled in a cooperative education program. Offered on a Pass/nonPass basis. Supervised practicum requires the successful completion of a minimum of 420 hours. Students who do not pass the course can not continue in the coop program.)
62190. Mathematical Foundations
Logic, sets, relations, functions. Development of skills in theoretical mathematics. (Prerequisite: 60100 or one of 62120, 62125, or 62126.) (2 lecture, 2 tutorial hours a week.)
62194. Mathematics for Business
An introduction to concepts and techniques of mathematics useful in business situations. Topics include mathematical modeling of qualitative scenarios, linear simultaneous equations, inequalities, exponential and logarithmic functions, graphical linear programming, and probability. (Prerequisite: Any grade 12 “U” math course, or 62101).(3 lecture hours, 1 tutorial hour a week.) (This course is intended for students in Business Administration only. May not be taken for credit by BSc and BCS majors in the Faculty of Science and Mathematics and Statistics majors.)
62215. Vector Calculus
Quadric surfaces. Vector differential calculus. Multiple integration. Line and surface integrals. (Prerequisites: 62141, and one of 62120, 62126 or 62125.) (3 lecture hours, 1 tutorial hour a week.)
62216. Differential Equations
Differential equations and Laplace transforms. Series solutions of differential equations. Applications to science and engineering. (Prerequisites: 62141, and one of 62120 or 62126.) (3 lecture hours, 1 tutorial hour a week.)
62220. Linear Algebra II
Rigorous study of the following topics: linear systems, vector spaces, linear transformations, projections, pseudoinverses, determinants, inner product spaces and applications. (Prerequisites: 62190 and one of 62120, 62125 or 62126.) (3 lecture hours, 1 tutorial hour a week.)
62221. Linear Algebra III
A rigourous treatment of eigenvalues and eigenvectors, diagonalization, similarity problem and canonical form for real and complex matrices; positive definite matrices; computational methods for approximating solutions to systems of linear equations and eigenvalues. (Prerequisite: 62220.) (3 lecture hours, 1 tutorial hour a week.)
62288. Work Term II
Supervised experience in an approved careerrelated setting with a focus on the application of theory and the development of transferable skills. The coop work experience is designed to provide students with an enriched learning opportunity to integrate academic theory and concepts in an applied setting. (Prerequisite: Student must be enrolled in a cooperative education program. Offered on a Pass/nonPass basis. Supervised practicum requires the successful completion of a minimum of 420 hours. Students who do not pass the course can not continue in the coop program.)
62314. Introduction to Analysis I
Real numbers. Limits, sequences, and continuity. Differentiation. (Prerequisites: 62141, 62190 and one of 62120, 62125 or 62126.) (3 lecture hours, 1 tutorial hour a week.)
62315. Introduction to Analysis II
Sequences and series of functions. Uniform and absolute convergence. Power Series. Integration. (Prerequisite: 62314.) (3 lecture hours, 1 tutorial hour a week.)
62318. Complex Variables
Complex numbers. Analytic functions. Contour integration. Series, Laurent expansions, residues. Application to real integrals. (Prerequisite: 62215; corequisite: 62216.) (3 lecture hours, 1 tutorial hour per week.)
62321. Abstract Algebra
Introduction to groups, rings, and fields. (Prerequisite: 62220 or 62322.) (3 lecture hours a week.)
62322. Number Theory
Divisibility, congruences, numbertheoretic functions. Theorems of Euler, Fermat, and Wilson. Theory of primes and quadratic residues. (Prerequisites: one of 62120, 62125 or 62190.) (3 lecture hours a week.)
62342. Combinatorics
Finite combinatorics; counting problems involving set operations, relations and functions; principle of inclusion and exclusion; ordinary and exponential generating functions; recurrence relations. (Prerequisites: 62141 and 62190.) (3 lecture hours a week.)
62343 Introduction to graph theory
This is an introductory course in graph theory. Topics include: paths and cycles, bipartite graphs, graph isomorphism, connectivity, Eulerian graphs, Hamiltonian graphs, trees, properties of trees, planarity, Euler’s formula, dual graphs, coloring graphs, Brooks’ theorem, coloring maps, chromatic polynomials, digraphs, matchings, Menger’s theorem, Hall’s theorem, Tutte’s theorem. (Prerequisites: 62220 or 60231).
62360. Special Functions
Uniform convergence, Fourier Series, Orthonormal bases, SturmLiouville eigenvalue problems, eigenfunction expansions, Gamma function, Bessel functions, Legendre polynomials and functions, and the hypergeometric functions. (Prerequisite: 62215 and 62216.) (3 lecture hours a week.)
62369. Numerical Analysis for Computer Scientists
Introductory course in the application of numerical methods using computer oriented algorithms such as finding roots, solving systems of equations, differentiation, integration and optimization. (Restricted to students in Computer Science.) (Prerequisites: 60141, 62141 and one of 62120, 62125 or 62126.)
62374. Linear Programming
Topics covered are: geometric linear programming, the Simplex method, the revised Simplex method, duality theory, sensitivity analysis, project planning and integer programming. Optional topics include: the transportation problem, the upper bounding technique, the dual Simplex method, parametric linear programming, game theory, and goal planning. Completion of some assignments will require the use of computer software packages. This course is intended to help students prepare for some parts of the Society of Actuaries examination on Operations Research (Course 130). Interested students should also take 65376. (Prerequisite: 62220 or consent of instructor.) (Antirequisite: 91312.) (3 lecture hours a week.)
62380. Numerical Methods
Topics covered are: nonlinear equations in one variable, interpolation, numerical integration (quadrature), and linear systems (direct methods). Optional topics are: numerical differentiation, iterative methods for boundary value problems. Completion of some assignments will require the use of computer software packages. This course is intended to help students prepare for some parts of the Society of Actuaries examination on Numerical Methods (Course 135). (Prerequisites: 62215, 62216, and one of 62120, 62125 or 62126.) (3 lecture hours a week.)
62388. Work Term III
Supervised experience in an approved careerrelated setting with a focus on the application of theory and the development of transferable skills. The coop work experience is designed to provide students with an enriched learning opportunity to integrate academic theory and concepts in an applied setting. (Prerequisite: Student must be enrolled in a cooperative education program. Offered on a Pass/nonPass basis. Supervised practicum requires the successful completion of a minimum of 420 hours. Students who do not pass the course can not continue in the coop program.)
62392. Theory of Interest
Measurement of interest, elementary and general annuities, amortization schedules and sinking funds, bonds, depreciation, depletion, and capitalized cost. This course helps prepare students for the Society of Actuaries examinations. (Prerequisite: 62141 or consent of instructor.) (3 lecture hours a week.)
62410. Real Analysis I
Lebesgue measure and Lebesgue integral. Differentiation and integration. RadonNikodym theorem. (Prerequisite: 62315.) (3 lecture hours a week.)
62411. Real Analysis II
Metric spaces. Topological spaces. StoneWeierstrass and Ascoli theorems. Classical Banach spaces. (Prerequisite: 62410.) (3 lecture hours a week.)
62413. Functional Analysis
Normed spaces, bounded linear operators, and the Banach dual spaces. The HahnBanach Theorem, the Uniform Boundedness Principle, and the Open Mapping Theorem. Weak and weak* topologies. Hilbert spaces and operators on Hilbert space. (Prerequisite: 62410.) (3 lecture hours per week.)
62420. Introduction to Group Theory
Abstract groups, subgroups, isomorphism theorems, orbits, class equation, quotient groups, Sylow's theorems, metric vector spaces, quadratic forms, basic concepts of orthogonal geometry, the classical groups. (Prerequisites: 62221 and 62321.) (3 lecture hours a week.)
62422. Introduction to Field Theory
Polynomial rings, splitting fields, The Fundamental Theorem of Galois Theory, Galois' criterion for solvability by radicals, algebraically closed fields, finite fields. (Prerequisites: 62221 and 62321.) (3 lecture hours a week.)
62482. Mathematical Programming
Topics include: unconstrained optimization, convexity, least squares problems, optimality conditions, penalty methods. Completion of some assignments will require the use of computer software packages. (Prerequisites: 62215, 62314, 62221, and one of 62374 or 65376.) (3 lecture hours a week.)
62488. Work Term IV
Supervised experience in an approved careerrelated setting with a focus on the application of theory and the development of transferable skills. The coop work experience is designed to provide students with an enriched learning opportunity to integrate academic theory and concepts in an applied setting. (Prerequisite: Student must be enrolled in a cooperative education program. Offered on a Pass/nonPass basis. Supervised practicum requires the successful completion of a minimum of 420 hours. Students who do not pass the course can not continue in the coop program.)
62490. Actuarial Mathematics I
Life contingencies. Survival distributions and life tables, life insurance, life annuities, net premiums, net premium reserves. This course helps prepare students for the Society of Actuaries examinations. (Prerequisites: 62215, 62216, 62392, and 65251, or consent of instructor.) (3 lecture hours a week.)
62492. Actuarial Mathematics II
Selection of topics from: advanced life contingencies, risk theory, survival models, construction and graduation of mortality tables. This course helps prepare students for the Society of Actuaries examinations. (Prerequisite: 62490 or consent of instructor.) (3 lecture hours a week.)
62498. Topics in Mathematics
Advanced topics not covered in other courses. (May be repeated for credit when the topic is different.) (Prerequisite: consent of the instructor.) (3 lecture hours a week.)
STATISTICS
Undergraduate Statistics courses taught outside Mathematics and Statistics may not be taken for credit in any mathematics program.
65205. Statistics for the Sciences
Descriptive statistics. Probability, discrete and continuous distributions. Point and interval estimation. Hypothesis testing. Goodnessoffit. Contingency tables. (Prerequisite: Grade 12“U” Advanced Level Mathematics or equivalent, or Grade 11 Functions and Relations, or Grade 11 Functions.) (Antirequisites: 02250, 73101, 73102, 73105, 73205, and 85222.) (May not be taken for credit after taking 65250 or 65251.) (3 lecture hours, 1 tutorial hour a week.)
65250. Introduction to Probability
Descriptive measures, combinatorics, probability, random variables, special discrete and continuous distributions, sampling distribution, point and interval estimation. (Prerequisite: 62141.) (3 lecture hours, 1 tutorial hour a week.)
65251. Introduction to Statistics
Distributions, point and interval estimation, hypothesis testing, contingency tables, analysis of variance, bivariate distributions, regression and correlation, nonparametric methods. (Prerequisite: 65250.) (3 lecture hours, 1 tutorial hour a week.)
65340. Applied Probability
Conditional probabilities and expectations. Markov chains. Poisson processes, renewal theory, reliability, queueing theory. (Prerequisites: 65251, 62215 and 62216.) (3 lecture hours a week.)
65350. Probability
Axioms of theory of probability. Discrete and continuous distributions including binomial, Poisson, exponential, normal chisquare, gamma, t, and F distributions. Multivariate distributions, conditional distributions, independence, expectation, moment generating functions, characteristic functions, transformation of random variables, order statistics, law of large numbers, central limit theorem. (Prerequisite: 65251.) (3 lecture hours a week.)
65351. Statistics
Point and interval estimations, properties of estimators, methods of estimation, least squares estimation and linear models, Bayesian estimation, RaoBlackwell theorem, tests of hypotheses, NeymanPearson Lemma, analysis of variance. (Prerequisite: 65350.) (3 lecture hours a week.)
65376. Stochastic Operations Research
Topics covered are: deterministic and stochastic dynamic programming, queuing theory, decision analysis, and simulation. Optional topics include: inventory theory, forecasting, and Markov processes. Completion of some assignments will require the use of computer software packages. This course is intended to help students prepare for some parts of the Society of Actuaries examination on Operations Research (Course 130). Interested students should also take 62374. (Prerequisite: 65205 or 65250.) (Antirequisite: 91412.) (3 lecture hours a week.)
65452. Experimental Designs
ANOVA models without and with interactions; randomized block, Latin square, factorial, confounded factorial, balanced incomplete block, and other designs; response surface methodology. (Prerequisite: 65251 or 65350.) (3 lecture hours a week.)
65454. Sampling Theory
Basic concepts. Simple random and stratified sampling. Ratio and regression methods. Systematic and cluster sampling. Multistage sampling, PPS sampling. Errors in surveys. Sampling methods in social investigation. (Prerequisite: 65251 or 65350.) (3 lecture hours a week.)
65455. Topics in Statistics
Advanced topics in probability or statistics not covered in other courses. (Prerequisite: consent of the instructor.) (3 lecture hours a week.) (May be repeated for credit when the topic is different.)
65456. Regression
An applied course covering multiple linear regression, model assumptions, inference about regression parameters, residual analysis, polynomial regression, multicollinearity, transformations. Topics to be selected from stepwise regression, weighted least squares, indicator variables, nonlinear regression. (Prerequisites: 62120 and 65251.) (3 lecture hours a week.) 