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Undergraduate Calendar
Winter 2018

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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

62-101. Access to Differential Calculus
The course will cover straight lines, relations and functions, trigonometric functions, limits, derivatives, curve sketching, equations and inequalities, transformations, symmetry, exponential and logarithmic functions. This course serves as the prerequisite for 62-130 and 62-140. Majors in Science, majors in Engineering and students with at least 70% in Ontario Grade 12 Advanced Functions (MHF4U) will not be given credit for this course. (Antirequisites: 62-130, 62-139, or 62-140) (3 lecture hours, 1 tutorial hour per week.)

62-102. Access to Linear Algebra
This course will cover matrix algebra, linear systems, vectors, lines and planes in three- dimensional space, equations and inequalities in one variable and linear relations. This course serves as the prerequisite for 62-120 and 62-126. Majors in Science and majors in Engineering will not be given credit for this course. (3 lecture hours, 1 tutorial hour per week.)

62-120. Linear Algebra I
This course will cover linear systems, matrix algebra, determinants, n-dimensional vectors, dot product, cross product, orthogonalization, eigenvalues, eigenvectors, diagonalization and vector spaces. (Prerequisites: Both Ontario Grade 12 Advanced Functions (MHF4U) and Calculus and Vectors (MCV4U) or 62-102.) (Antirequisites: 62-125, 62-126.) (3 lecture hours, 2 tutorial hours per week.)

62-125. Vectors and Linear Algebra
This course is for students without Ontario Grade 12 Calculus and Vectors (MCV4U). The course 62-120 is for students with MCV4U. This course will cover vectors, three-dimensional geometry, linear systems, matrix algebra, determinants, n- dimensional vectors, dot product, cross product, orthogonalization, eigenvalues, eigenvectors, diagonalization and vector spaces. (Prerequisite: Ontario Grade 12 Advanced Functions (MHF4U).) (Antirequisites: 62-120, 62-126.) (4 lecture hours, 2 tutorial hours per 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
This course will cover a review of functions, limits and continuity, derivatives and applications, indefinite integrals, methods of integration, partial derivatives and applications to the biological sciences. (Prerequisite: Ontario Grade 12 Advanced Functions (MHF4U) or 62-101.) (Antirequisites: 62-139 or 62-140.) (3 lecture hours, 1 tutorial hour per week.)

62-139. Functions and Differential Calculus
This course will cover a review of functions, trigonometric functions and identities, transcendental functions, inverse trigonometric functions, introduction to limits, continuity, derivatives and applications, mean value theorem, indeterminate forms and l’H˘pital’s rule, antiderivatives and an introduction to definite integrals. This course is for students who have taken Ontario Grade 12 Advanced Functions (MHF4U), but have not taken Ontario Grade 12 Calculus and Vectors (MCV4U). Students who have credit for MCV4U should take 62-140. The course is equivalent to 62-140 for all prerequisite purposes. (Prerequisite: Ontario Grade 12Advanced Functions (MHF4U).) (Antirequisite: 62-140.) (4 lecture hours, 2 tutorial hours per week.)

62-140. Differential Calculus
This course will cover trigonometric functions and identities, inverse trigonometric functions, limits and ontinuity, derivatives and applications, mean value theorem, indeterminate forms and l’H˘pital’s rule, antiderivatives and an introduction to definite integrals. This course is for students who have taken both Ontario Grade 12 Advanced Functions (MHF4U) and Ontario Grade 12 Calculus and Vectors (MCV4U). Students who do not have credit for MCV4U should take 62-139. (Prerequisites: Ontario Grade 12 Advanced Functions (MHF4U) and Ontario Grade 12 Calculus and Vectors (MCV4U) or 62-101.) (Antirequisite: 62-139.) (3 lecture hours, 2 tutorial hours per week.)

62-141. Integral Calculus
This course will cover antiderivatives, the definite integral and the fundamental theorem of calculus, techniques of integration, applications, improper integrals, sequences and series, convergence tests, power series, Taylor and Maclaurin series, and polar and parametric coordinates. (Prerequisite: 62-139 or 62-140.) (3 lecture hours, 1 tutorial hour per week.)

62-190. Mathematical Foundations
This course will cover mathematical logic, proof methods and development of proof techniques, mathematical induction, sets, equivalence relations, partial ordering relations and functions. (Prerequisite: One of 60-100, 62-120, 62-125 or 62- 126.) (2 lecture hours, 2 tutorial hours per 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
This course will cover quadric surfaces, vector differential calculus, functions of several variables, maximum and minimum problems, multiple integrals, vector differential operators, line and surface integrals, Green’s theorem, Stokes’ theorem and Gauss’ theorem. (Prerequisites: 62-141, and one of 62-120, 62-125 or 62-126.) (3 lecture hours, 1 tutorial hour per week.)

62-216. Differential Equations
This course will cover first-order ordinary differential equations (ODEs), higher-order ODEs with constant coefficients, Cauchy-Euler equations, systems of linear ODEs, Laplace transforms, and applications to science and engineering. (Prerequisites: 62-141, and one of 62-120, 62-125 or 62-126.) (3 lecture hours, 1 tutorial hour per week.)

62-220. Linear Algebra II
This course is a rigorous and proof-based study of 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 per week.)

62-221. Linear Algebra III
This course is a rigorous and proof-based study of
eigenvalues and eigenvectors, diagonalization, similarity problem, 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 per week.)

62-314. Introduction to Analysis I
This course is a rigorous and proof-based study of supremum and infimum, the real number system, countable and uncountable sets, metric spaces, compact sets, connected sets, Cauchy sequences, completeness, limits and continuity, maximum and minimum on compact sets, intermediate value theorem, differentiation and the mean value theorem. (Prerequisites: 62-141, 62-190 and one of 62-120, 62-125 or 62-126.) (3 lecture hours, 1 tutorial hour per week.)

62-315. Introduction to Analysis II
This course is a rigorous and proof-based study of Riemann-Stieltjes integral, sequences and series of functions, uniform and absolute convergence, equicontinuity, Arzela-Ascoli theorem, Stone- Weierstrass theorem, power series, and functions of several variables. (Prerequisite: 62-314.) (3 lecture hours, 1 tutorial hour per week.)

62-318. Complex Variables
This course will cover complex numbers, analytic functions, exponential and logarithm functions, contour integration, Cauchy’s integral formula, series, Taylor and Laurent expansions, residue theory, applications to real integrals. (Prerequisite:62-215; Corequisite: 62-216) (3 lecture hours, 1 tutorial hour per week.)

62-321. Abstract Algebra
This course will cover an introduction to groups, rings and fields. (Prerequisite: 62-220 or 62-322.) (3 lecture hours per week.)

62-322. Number Theory
This course will cover divisibility, primes, fundamental theorem of arithmetic, greatest common divisor, Euclidean algorithm, least common multiple, linear Diophantine equations, congruency, residue classes, Chinese remainder theorem, number theoretic functions, theorems of Euler, Fermat, Wilson, theory of primes, and quadratic residues. (Prerequisites: one of 62-120, 62-125 or 62-126, and 62-190.) (3 lecture hours per week.)

62-342. Combinatorics
This course will cover finite combinatorics, in
particular, the pigeonhole principle, permutations and combinations, binomial coefficients, the inclusion-exclusion principle, recurrence relations and generating functions, special counting sequences, Polya counting. (Prerequisites: 62-141 and 62-190.) (3 lecture hours per week.)

62-343 Introduction to Graph Theory
This course will cover paths and cycles, bipartite graphs, graph isomorphisms, 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, and Tutte’s theorem. (Prerequisites: 62-220 or 60-231.) (3 lecture hours per week.)

62-360. Introduction to Fourier Series and Special Functions
This course will cover Fourier series, Sturm- Liouville problems, heat and wave equations, Laplace equation, weighted L2 -spaces and orthogonal bases, Gamma function, Bessel functions, Legendre polynomials and hypergeometric functions. (Prerequisite: 62-215and 62-216.) (3 lecture hours per week.)

62-369. Numerical Analysis for Computer Scientists
This course is an introduction to the applications 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.) (3 lecture hours per week)

62-374. Linear Programming
This course will cover the graphical solution of two variable linear programs, the tableau simplex algorithm, the revised simplex algorithm, linear programming theory, sensitivity analysis, the transportation problem, the assignment problem and integer programming. (Prerequisite: 62-220 or consent of instructor.) (Antirequisite: 91-312.) (3 lecture hours per week.)

62-380. Numerical Methods
This course will cover iterative solution methods for nonlinear equations in one variable, Lagrange interpolation, cubic splines, Bezier curves, numerical differentiation and integration (quadrature), initial value problems, linear algebraic systems (direct methods) and Newton’s method for nonlinear systems. (Prerequisites: 62-215, 62-216, and one of 62-120, 62-125 or 62-126.) (3 lecture hours per week.)

62-392. Theory of Interest
This course will cover measurement of interest, elementary and general annuities, amortization schedules and sinking funds, bonds, depreciation, depletion and capitalized cost. (Prerequisite: 62-141 or consent of instructor.) (3 lecture hours per week.)

62-410. Measure Theory and Integration
This course will cover measures, Lebesgue measure, Lebesgue integral, monotone and dominated convergence theorems, Fubini’s theorem, Lp-spaces, modes of convergence and Radon-Nikodym theorem. (Prerequisite: 62-315.) (3 lecture hours per week.)

62-411. Real Analysis II
This course will cover metric spaces, topological spaces, Stone-Weierstrass theorem, Ascoli’s theorem and classical Banach spaces. (Prerequisite: 62-410.) (3 lecture hours per week.)

62-413. Functional Analysis
This course will cover normed and Banach spaces, bounded linear operators, dual spaces, Hahn- Banach theorem, uniform boundedness principle, open mapping theorem, Hilbert spaces, operators on Hilbert spaces, and weak and weak* topologies. (Prerequisite: 62-410.) (3 lecture hours per week.)

62-420. Introduction to Group Theory
This course will cover abstract groups, subgroups,quotient groups, products, isomorphism theorems, group actions, orbits, class equation, Sylow theorems, finitely generated abelian groups. (Prerequisites: 62-221 and 62-321.) (3 lecture hours per week.)

62-422. Introduction to Field Theory
This course will cover polynomial rings, splitting fields, the fundamental theorem of Galois theory, Galois' criterion for solvability by radicals, algebraically closed fields and finite fields. (Prerequisites: 62-221 and 62-321.) (3 lecture hours per week.)

62-482. Portfolio Optimization
This is a first course on Markowitz mean-variance portfolio optimization. The course will cover quadratic programming, parametric quadratic programming, the efficient frontier, the capital asset pricing model, Sharpe ratios and implied risk free returns, portfolio optimization with constraints, and quadratic programming solution algorithms; also covered are professional writing and presentation skills and the use of optimization software. (Prerequisite: 62-221.) (3 lecture hours per 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
This course will cover life contingencies, survival distributions and life tables, life insurance, life annuities, net premiums and net premium reserves. (Prerequisites: 62-215, 62-216, 62-392, and 65-251, or consent of instructor.) (3 lecture hours per week.)

62-492. Actuarial Mathematics II
This course will cover advanced life contingencies, risk theory, survival models, and construction and graduation of mortality tables. (Prerequisite: 62-490 or consent of instructor.) (3 lecture hours per week.)

62-498. Topics in Mathematics
This course will cover advanced topics not covered in other courses. (May be repeated for credit when the topic is different.) (Prerequisite: consent of instructor.) (3 lecture hours per 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
This course will cover descriptive statistics,probability, discrete and continuous distributions, point and interval estimation, hypothesis testing, goodness-of-fit and contingency tables. (Prerequisite: Grade 12 “U” Advanced Level Mathematics (MHF4U, MCV4U, MDM4U) or Grade 11 Functions and Applications (MCF3M) or Grade 11 Functions (MCR3U).) (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 per week.)

65-250. Introduction to Probability
This course will cover descriptive measures, combinatorics, probability, random variables, special discrete and continuous distributions, sampling distribution, and point and interval estimation. (Prerequisite: 62-141.) (3 lecture hours, 1 tutorial hour per week.)

65-251. Introduction to Statistics
This course will cover distributions, point and interval estimation, hypothesis testing, contingency tables, analysis of variance, bivariate distributions, regression, correlation and non-parametric methods. (Prerequisite: 65-250.) (3 lecture hours, 1 tutorial hour per week.)

65-350. Probability
The course will cover the axioms of the 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 and central limit theorem. (Prerequisite: 65-251.) (3 lecture hours per week.)

65-351. Statistics
This course will cover 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 and analysis of variance. (Prerequisite: 65-350.) (3 lecture hours per week.)

65-376. Stochastic Operations Research
This course will cover deterministic dynamic programming, stochastic dynamic programming, queuing theory, Brownian motion, decision analysis and simulation. Optional topics are inventory theory and Markov processes. (Prerequisites: 65-250, 62-120 or 62-125, 62-141.) (Antirequisite: 91-412.) (3 lecture hours per week.)

65-452. Experimental Designs
This course will cover ANOVA models without and with interactions, randomized block, Latin square, factorial, confounded factorial, balanced incomplete block, other designs and response surface methodology. (Prerequisite: 65-251 or 65-350.) (3 lecture hours per week.)

65-454. Sampling Theory
This course will cover basic concepts, simple random and stratified sampling, ratio and regression methods, systematic and cluster sampling, multi-stage sampling, PPS sampling, and errors in surveys and sampling methods in social investigation. (Prerequisite: 65-251 or 65-350.) (3 lecture hours per week.)

65-455. Topics in Statistics
This course will cover advanced topics in probability or statistics not covered in other courses. (Prerequisite: consent of instructor.) (3 lecture hours per week.) (May be repeated for credit when the topic is different.)