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
62-101. Access to Calculus
A variety of pre-calculus topics including coordinate geometry, trigonometric, exponential and logarithmic functions, and algebraic procedures. Introduction to differential calculus. (This course and 62-102 satisfy the prerequisite or admission requirement of Grade 12 “U” Advanced Functions and Grade 12 “U” Calculus and Vectors (or equivalent), respectively. May not be taken for credit by (a) majors in the Faculty of Science or the Faculty of Engineering;(b) students who successfully completed 62-130, 62-139, 62-140; (c) students who achieved a grade of 70% or greater in MHF4U Advanced Functions or equivalent.) (3 lecture hours, 1 tutorial hour per week.)
62-102. 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 62-101 satisfy the prerequisite or admission requirement of Grade 12 “U” Calculus and Vectors (or equivalent) and Grade 12 “U” Advanced Functions, respectively. May not be taken for credit in any program within the Faculty of Science or the Faculty of Engineering.) (3 lecture hours, 1 tutorial hour per week.)
62-120. Linear Algebra I
Linear systems, matrix algebra, determinants, vectors in Rn , dot product, orthogonalization, eigenvalues, and diagonalization. (Prerequisite: 62-102 or Grade 12 Advanced Functions and Grade 12 Calculus and Vectors or equivalent.) (Antirequisite: 62-125 or 62-126) (3 lecture hours, 1 tutorial hour a week.)
62-125 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 62-120/126 for all prerequisite purposes.) (Prerequisite: Grade 12 Advanced Functions or equivalent.) (Antirequisites: 62-120, 62-126.) (4 lecture hours, 1 tutorial hour a week.)
62-126. Linear Algebra (Engineering)
Linear systems, matrix algebra, determinants, vectors in Rn, dot product, orthogonalization, and eigenvalues. (Prerequisite: 62-102 or Grade 12 Advanced Functions and Grade 12 Calculus and Vectors, or equivalent.) (Antirequisite: 62-120, or 62-125.) (3 lectures hours, 1 tutorial hour a week.)
62-130. 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 62-101.) (May not be taken for credit concurrently with, or subsequent to having obtained credit in 62-139 or 62-140. This course is not a sufficient prerequisite to 62-141, but may serve as preparation for 62-140.) (3 lecture hours, 1 tutorial hour a week.)
62-139. 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 62-140 for all prerequisite purposes.) (Prerequisite: Grade 12 Advanced Functions or equivalent.) (Antirequisite: 62-140.) (4 lecture hours, 1 tutorial hour a week; or 4 lecture hours, 2 tutorial hour a week.)
62-140. 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 62-101.) (Antirequisite: 62-139) (3 lecture hours, 1 tutorial hour a week; or 4 lecture hours, 2 tutorial hour a week.)
62-141. 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: 62-139 or 62-140.) (3 lecture hours, 1 tutorial hour a week.)
62-188. Work Term I
Supervised experience in an approved career-related setting with a focus on the application of theory and the development of transferable skills. The co-op 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 co-operative education program. Offered on a Pass/non-Pass 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 co-op program.)
62-190. Mathematical Foundations
Logic, sets, relations, functions. Development of skills in theoretical mathematics. (Prerequisite: 60-100 or one of 62-120, 62-125, or 62-126.) (2 lecture, 2 tutorial hours a week.)
62-194. 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 62-101.) (This course is intended for students in Business Administration only. May not be taken for credit in any program within the Faculty of Science.) (3 lecture hours, 1 tutorial hour per week.)
62-215. Vector Calculus
Quadric surfaces. Vector differential calculus. Multiple integration. Line and surface integrals. (Prerequisites: 62-141, and one of 62-120, 62-126 or 62-125.) (3 lecture hours, 1 tutorial hour a week.)
62-216. Differential Equations
Differential equations and Laplace transforms. Series solutions of differential equations. Applications to science and engineering. (Prerequisites: 62-141, and one of 62-120 or 62-126.) (3 lecture hours, 1 tutorial hour a week.)
62-220. Linear Algebra II
Rigorous study of the following topics: linear systems, vector spaces, linear transformations, projections, pseudo-inverses, determinants, inner product spaces and applications. (Prerequisites: 62-190 and one of 62-120, 62-125 or 62-126.) (3 lecture hours, 1 tutorial hour a week.)
62-221. 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: 62-220.) (3 lecture hours, 1 tutorial hour a week.)
62-288. Work Term II
Supervised experience in an approved career-related setting with a focus on the application of theory and the development of transferable skills. The co-op 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 co-operative education program. Offered on a Pass/non-Pass 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 co-op program.)
62-314. Introduction to Analysis I
Real numbers. Limits, sequences, and continuity. Differentiation. (Prerequisites: 62-141, 62-190 and one of 62-120, 62-125 or 62-126.) (3 lecture hours, 1 tutorial hour a week.)
62-315. Introduction to Analysis II
Sequences and series of functions. Uniform and absolute convergence. Power Series. Integration. (Prerequisite: 62-314.) (3 lecture hours, 1 tutorial hour a week.)
62-318. Complex Variables
Complex numbers. Analytic functions. Contour integration. Series, Laurent expansions, residues. Application to real integrals. (Prerequisite: 62-215; corequisite: 62-216.) (3 lecture hours, 1 tutorial hour per week.)
62-321. Abstract Algebra
Introduction to groups, rings, and fields. (Prerequisite: 62-220 or 62-322.) (3 lecture hours a week.)
62-322. Number Theory
Divisibility, congruences, number-theoretic functions. Theorems of Euler, Fermat, and Wilson. Theory of primes and quadratic residues. (Prerequisites: one of 62-120, 62-125 or 62-126 and 62-190.) (3 lecture hours a week.)
62-342. Combinatorics
Finite combinatorics; counting problems involving set operations, relations and functions; principle of inclusion and exclusion; ordinary and exponential generating functions; recurrence relations. (Prerequisites: 62-141 and 62-190.) (3 lecture hours a week.)
62-343 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: 62-220 or 60-231).
62-360. Special Functions
Uniform convergence, Fourier Series, Orthonormal bases, Sturm-Liouville eigenvalue problems, eigenfunction expansions, Gamma function, Bessel functions, Legendre polynomials and functions, and the hypergeometric functions. (Prerequisite: 62-215 and 62-216.) (3 lecture hours a week.)
62-369. 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: 60-141, 62-141 and one of 62-120, 62-125 or 62-126.)
62-374. 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 65-376. (Prerequisite: 62-220 or consent of instructor.) (Antirequisite: 91-312.) (3 lecture hours a week.)
62-380. 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: 62-215, 62-216, and one of 62-120, 62-125 or 62-126.) (3 lecture hours a week.)
62-388. Work Term III
Supervised experience in an approved career-related setting with a focus on the application of theory and the development of transferable skills. The co-op 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 co-operative education program. Offered on a Pass/non-Pass 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 co-op program.)
62-392. 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: 62-141 or consent of instructor.) (3 lecture hours a week.)
62-410. Real Analysis I
Lebesgue measure and Lebesgue integral. Differentiation and integration. Radon-Nikodym theorem. (Prerequisite: 62-315.) (3 lecture hours a week.)
62-411. Real Analysis II
Metric spaces. Topological spaces. Stone-Weierstrass and Ascoli theorems. Classical Banach spaces. (Prerequisite: 62-410.) (3 lecture hours a week.)
62-413. Functional Analysis
Normed spaces, bounded linear operators, and the Banach dual spaces. The Hahn-Banach Theorem, the Uniform Boundedness Principle, and the Open Mapping Theorem. Weak and weak* topologies. Hilbert spaces and operators on Hilbert space. (Prerequisite: 62-410.) (3 lecture hours per week.)
62-420. 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: 62-221 and 62-321.) (3 lecture hours a week.)
62-422. 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: 62-221 and 62-321.) (3 lecture hours a week.)
62-482. 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: 62-215, 62-314, 62-221, and one of 62-374 or 65-376.) (3 lecture hours a week.)
62-488. Work Term IV
Supervised experience in an approved career-related setting with a focus on the application of theory and the development of transferable skills. The co-op 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 co-operative education program. Offered on a Pass/non-Pass 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 co-op program.)
62-490. 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: 62-215, 62-216, 62-392, and 65-251, or consent of instructor.) (3 lecture hours a week.)
62-492. 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: 62-490 or consent of instructor.) (3 lecture hours a week.)
62-498. 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.
65-205. Statistics for the Sciences
Descriptive statistics. Probability, discrete and continuous distributions. Point and interval estimation. Hypothesis testing. Goodness-of-fit. Contingency tables. (Prerequisite: Grade 12“U” Advanced Level Mathematics or equivalent, or Grade 11 Functions and Relations, or Grade 11 Functions.) (Antirequisites: 02-250, 73-101, 73-102, 73-105, 73-205, and 85-222.) (May not be taken for credit after taking 65-250 or 65-251.) (3 lecture hours, 1 tutorial hour a week.)
65-250. Introduction to Probability
Descriptive measures, combinatorics, probability, random variables, special discrete and continuous distributions, sampling distribution, point and interval estimation. (Prerequisite: 62-141.) (3 lecture hours, 1 tutorial hour a week.)
65-251. Introduction to Statistics
Distributions, point and interval estimation, hypothesis testing, contingency tables, analysis of variance, bivariate distributions, regression and correlation, non-parametric methods. (Prerequisite: 65-250.) (3 lecture hours, 1 tutorial hour a week.)
65-350. Probability
Axioms of theory of probability. Discrete and continuous distributions including binomial, Poisson, exponential, normal chi-square, 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: 65-251.) (3 lecture hours a week.)
65-351. Statistics
Point and interval estimations, properties of estimators, methods of estimation, least squares estimation and linear models, Bayesian estimation, Rao-Blackwell theorem, tests of hypotheses, Neyman-Pearson Lemma, analysis of variance. (Prerequisite: 65-350.) (3 lecture hours a week.)
65-376. 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 62-374. (Prerequisite: 65-205 or 65-250.) (Antirequisite: 91-412.) (3 lecture hours a week.)
65-452. 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: 65-251 or 65-350.) (3 lecture hours a week.)
65-454. Sampling Theory
Basic concepts. Simple random and stratified sampling. Ratio and regression methods. Systematic and cluster sampling. Multi-stage sampling, PPS sampling. Errors in surveys. Sampling methods in social investigation. (Prerequisite: 65-251 or 65-350.) (3 lecture hours a week.)
65-455. 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.)