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# 3000-level Courses

This is a course that formally studies the convergence of sequences and series of functions. The course will start with discussions of problems motivating the development of this subject, conditional convergence of series of real numbers, absolute convergence of series of complex numbers, and uniform convergence of series of functions. Subsequently, we will transition into a formal treatment of Fourier series by defining the Fourier coefficients of an integrable function and develop the basic properties of Fourier series. It will be examined when a Fourier series converges pointwise and kernels and Cesaro summability will be explored. This will be followed by the proof of mean square convergence of Fourier series and Parseval’s formula. The final part of the course will examine several applications of these ideas: the isoperimetric inequality, Weyl’s equidistribution theorem, a proof of the Central Limit Theorem, and the existence of a continuous nowhere differentiable function. Familiarity with linear algebra would also be useful.

Prerequisites: SC/ MATH 2001 3.00; SC/MATH 1310 3.00 or SC/ISCI 1401 3.00 or SC/ISCI 1410 6.00.

Course credit exclusion: GL/MATH 3320 3.00, GL/MATH 4240 6.00.

Note: This course is a prerequisite for SC/MATH 4011 3.00, SC/MATH 4012 3.00 and SC/MATH 4081 3.00.

This course will be a rigorous study of functions of several variables. Linear algebra is used to extend the concepts of single variable differential and integral calculus to multivariate functions of one and several variables. The topics included are limits, continuity, differentiation, the inverse and implicit function theorems, integration, Fubini's theorem, vector fields, line and surface integrals, elementary differential forms and integration on manifolds.

Prerequisite: SC/MATH 2310 3.00; or SC/MATH 2015 3.00 and written permission of the mathematics undergraduate director (normally granted only to students proceeding in Honours programs in mathematics or in the Specialized Honours program in statistics). Prerequisite or corequisite: SC/MATH 2022 3.00 or SC/MATH 2222 3.00.

Note: This course is a prerequisite for SC/MATH 3410 3.00.

This is the first basic course on the structures of Algebra embracing Groups, Rings and Fields and other algebraic topics, The focus of this course is Group Theory, We begin with the general concepts of groups, homomorphisms, cosets, normal subgroups, and operations of a group on a set, i.e., group actions. Then we look at different types of groups in detail, e.g., cyclic groups, permutation groups, finite abelian groups and Sylow subgroups. If time permits, some matrix groups are studied.

Prerequisites: SC/MATH 1019 3.00 or SC/MATH 1190 3.00 or SC/MATH 1200 3.00; SC/MATH 2022 3.00 or SC/MATH 2222 3.00.

Course credit exclusions: SC/MATH 3020 6.00, GL/MATH 3650 6.00, GL/MODR 3650 6.00, GL/MATH 3510 3.00.

Note: This course is a prerequisite for SC/MATH 3022 3.00.

This is a continuation of Math 3021. Algebra is one of the core parts of mathematics. It is concerned with properties of mathematical objects that can be expressed symbolically and it is used in almost every branch of mathematics. Algebraic methods provide not only solutions to problems in other areas of mathematics, but they also provide proofs that certain problems do not have solution. This is the case with some familiar problems such as trisection of an angle and solution of quintic equation in radicals. This course also supports students in learning how to write clear and concise proofs and how to communicate mathematical ideas effectively.

Prerequisite: SC/MATH 3021 3.00 or permission of the course coordinator.

Course credit exclusions: SC/MATH 3020 6.00, GL/MATH 3650 6.00, GL/MODR 3650 6.00, GL/MATH 3515 3.00.

Note: This course is a prerequisite for SC/MATH 4021 3.00.

Geometry is one of the oldest fields of mathematics that goes back to the times of Euclid, Pythagoras, and other famous ancient Greek mathematicians. Initially concerned with fundamental concepts of point, line, plane, distance, angle, surface, and curve, the scope of geometry has been expanded during the last two centuries, leading to the creation of several subfields that include Riemannian geometry, algebraic geometry, etc. Geometry is a good subject to explore the role of visual and spatial reasoning in the practice of mathematics: generating insights, problem-solving, communication, remembering, proofs, etc. In this course, students will explore a variety of topics in the exciting field of geometry, including axiomatic and analytic treatment of incident geometry, plane geometry, spherical geometry, hyperbolic geometry, and Poincaré disk model from Riemannian geometry. Important themes for each of the topics include axioms & common notions, geodesics & straightness, congruency & similarity, area & holonomy, isometries, projections, parallel postulates in different geometries, isometries with matrices & eigenvectors, etc. Coursework will explore geometric questions through investigation, hands-on materials and dynamic geometry software such as Geometer's Sketchpad, GeoGebra 3D, GeoGebra Geometry, Desmos, Maple and also emphasize the many applications of geometry in the areas of computer-aided design, computer graphics, robotics, architecture, virtual reality, video game programming, and engineering.

Prerequisite: SC/MATH 2022 3.00 or SC/MATH 2222 3.00 or permission of the Instructor.

Course credit exclusion: SC/MATH 3050 6.00.

This course is an introduction to modelling via computer-based methods. Topics include root-finding, curve fitting and interpolation, eigenvalues and matrix decomposition, matrix exponentiation, numerical methods for solving differential equations, and random number generation.

Prerequisites: SC/MATH 2022 3.00; SC/MATH 2030 3.00; LE/CSE 1560 3.00, or LE/CSE 2031 3.00 and SC/MATH 2041 3.00, or LE/CSE 1540 3.00 and SC/MATH 2041 3.00.

After a review of the basic concepts introduced in MATH 2131, we will cover the following topics: some standard multivariate distributions, some special distributions related to the normal distribution, con­ver­gence in probability and convergence in distribution, order statistics, maximum likelihood methods, sufficiency and the basis of hypothesis testing. If time permits, we will look at the distribution of special quadratic forms.

Prerequisite: SC/MATH 2131 3.00 or permission of the course coordinator.

Note: This course is a prerequisite for SC/MATH 3132 3.00, SC/MATH 4130B 3.00, SC/MATH 4230 3.00, SC/MATH 4630 3.00 and SC/MATH 4939 3.00.

This course is a continuation of Math 3131.03. The basic nature of statistical inference will be studied. Topics include sufficiency, exponential family, decision theory, most powerful tests, likelihood ratio tests, Bayesian statistics, linear models, etc. The final grade will be based on tests, a presentation, and the final exam.

Prerequisite: SC/MATH 3131 3.00.

Note: This course is a prerequisite for SC/MATH 4230 3.00 and SC/MATH 4939 3.00.

This course is an introduction to number theory, the branch of mathematics that deals with the properties of numbers in general and integers in particular. It is also one of the oldest branches of mathematics, and one of the largest. This course is a rigorous mathematical course, so there will be a certain amount of definitions, theorems, and proofs. However, there will also be a good deal of concrete, hands-on computations, and many interesting and fun applications. Topics include divisibility and congruences, quadratic residues and the law of quadratic reciprocity, arithmetic functions and the Mobius inversion formula, continued fractions, Diophantine equations, primitive roots, and the distribution of prime numbers.

Prerequisites: SC/MATH 1200 3.00 (or SC/MATH 2200 3.00), and one of SC/MATH 1021 3.00 or SC/MATH 1025 3.00.

This course introduces students to theories, algorithms and applications of linear programming (LP), including the classical simplex method, duality theory, sensitivity analysis, how to model many important business and industrial decision problems as LP, and how to use numerical solvers to solve LP.

Prerequisite: SC/MATH 1021 3.00 or SC/MATH 1025 3.00 or SC/MATH 2221 3.00.

Course credit exclusions: AP/ECON 3120 3.00, AP/ADMS 3331 3.00, AP/ADMS 3351 3.00, GL/MATH 3660 6.00, SC/MATH 3170 6.00.

Note: This course is a prerequisite for SC/MATH 3172 3.00.

This course introduces students to a variety of combinatorial optimization problems, such as the scheduling problem, travelling salesman problem, facility location problem, max-flow min-cut problem etc. We study a variety of algorithms for solving combinatorial problems: branch-and-bound method, and various searching methods, including the depth first search, the breadth first search, the greedy search, the iterative deepening search, and A-star search.

Prerequisites: SC/MATH 3171 3.00, SC/MATH 1021 3.00 or SC/MATH 1025 3.00 or SC/MATH 2221 3.00.

Course credit exclusions: AP/ECON 3120 3.00, AP/ADMS 3331 3.00, AP/ADMS 3351 3.00, GL/MATH 3660 6.00, SC/MATH 3170 6.00.

The course begins with a discussion of computer arithmetic and computational errors. Examples of ill-conditioned problems and unstable algorithms will be given. The first class of numerical methods introduced are those for nonlinear equations, i.e., the solution of a single equation in one variable. We then discuss the most basic problem of numerical linear algebra: the solution of a linear system of n equations in n unknowns. We discuss the Gauss algorithm and the concepts of error analysis, condition number and iterative refinement. We then use least squares to solve over determined systems of linear equations. The course emphasizes the development of numerical algorithms, the use of mathematical software, and interpretation of results obtained on some assigned problems.

Prerequisites: One of SC/MATH 1010 3.00, SC/MATH 1014 3.00, SC/MATH 1310 3.00; one of SC/MATH 1021 3.00, SC/MATH 1025 3.00, SC/MATH 2221 3.00; one of LE/EECS 1540 3.00, LE/EECS 2031 3.00, or LE/EECS 2501 1.00.

Course credit exclusions: LE/EECS 3121 3.00, LE/CSE 3121 3.00, SC/CSE 3121 3.00.

Note: This course is a prerequisite for SC/MATH 3242 3.00.

The course is a continuation of SC/MATH 3241 3.00 or LE/EECS 3121 3.00. The main topics include numerical differentiation, Richardson’s extrapolation, elements of numerical integration, composite numerical integration, Romberg integration, adaptive quadrature methods, Gaussian quadrature, numerical improper integrals; fixed points for functions of several variables, Newton’s method, quasi-Newton methods, steepest descent tech­niques, and homotopy methods; power method, Householder method and QR algorithms. The final grade will be based on assignments, tests and a final examination.

Prerequisite: SC/MATH 3241 3.00 or LE/EECS 3121 3.00.

Course credit exclusions: LE/EECS 3122 3.00, LE/CSE 3122 3.00, SC/CSE 3.00.

Note: This course is a prerequisite for MATH 4141 3.00.

This course introduces the student to mathematical modelling with applications in biology in related fields such as chemistry, ecology and health. There is an emphasis on case studies and problem solving skills. Topics include discrete and continuous models describing population dynamics, population health, chemical reactions and biological structures.

Prerequisites: Registration in an Honours Program in Mathematics and Statistics, completion of the SC/MATH core and SC/CSE 1560 3.00 or LE/EECS 1560 3.00 or permission of the Instructor

Introductory graph theory with applications. Topics include graphs, digraphs, Eulerian and Hamiltonian graphs; the traveling salesman; path algorithms; connectivity; trees; planarity; colourings; scheduling; minimal cost networks; tree searches and sorting; minimal connectors and ap­plications; matchings and Hall's marriage theorem; Menger's theorem; and matroids.

Prerequisite: At least six credits from 2000-level SC/MATH courses without second digit "5".

This course is about partial differential equations (PDEs) and their solutions. You will be introduced to the diffusion / heat equation in one dimension, the wave equation in one dimension, higher-dimensional partial differential equations . We will explore i) the meaning of a partial differential equations (PDEs) and  their  solutions; the concepts of initial and boundary value problems  ii) how physical laws such as the Fourier’s law of heat conduction, Fick’s law of diffusion, Newton’s law on a vibrating string, and the conservation of thermal energy are used to  derive the heat/diffusion, wave, and Laplace equations. The analytical methods we will learn include  Fourier series, separation of variables, method of characteristics,  Fourier transform,  Green functions, Laplace transform and  Sturm-Liouville theory.

Reference Textbooks:

• Partial Differential Equations: Theory And Completely Solved Problems (2nd Ed.), by T. Hillen, E.I. Leonard, H. van Roessel, Friesen Press 2019;
• Applied partial differential equations with Fourier series and boundary value problems (5th ed.), by R. Haberman, Pearson Education 2013.

Pre-requisites: SC/MATH 2270 3.00 or SC/MATH 2271 3.00; SC/MATH 2015 3.00 or SC/MATH 2310 3.00; SC/MATH 3010 3.00 is also desirable, though not essential, as a prerequisite for students presenting SC/MATH 2310 3.00.

Note: This course is a prerequisite for SC/MATH 4090 3.00 and SC/MATH 4120.

Probabilistic introduction to the mathematics of life contingencies. The course develops a theoretical basis for modeling the future lifetime of certain financial objects with an emphasis on insurance. Topics include international actuarial notation, life tables, life statuses, (multivariate) survival distributions, dependence, multiple decrement theory, multiple state models. The course ensures an adequate preparation for the MLC exam of the Society of Actuaries. Three lecture hours per week plus one hour of faculty led tutorials per week.

Prerequisites: SC/ MATH 2131 3.00 and SC/MATH 2280 3.00.

Course credit exclusion: SC/MATH 3280 6.00.

Note: This course is a prerequisite for MATH 3281 3.00.

Intermediate level course on the mathematics of life contingencies. The course builds on SC/MATH 3280 3.00 and develops theoretical basis for pricing and supporting life‑contingent products. Topics include economics of insurance, general insurances and annuities, (benefit) premiums and reserves, analysis of reserves, Hattendorf’s theorem. The course, in conjunction with MATH 4430 and MATH 3281, ensures an adequate preparation for the Long Term Actuarial Models (LTAM) exam of the Society of Actuaries. Three lecture hours per week plus one hour of faculty led tutorials per week.

Prerequisite: SC/MATH 3280 3.00.

Course credit exclusion: SC/MATH 3280 6.00.

A comprehensive introduction to continuous-time Mathematical Finance. This course introduces Brownian motion and Ito calculus and covers interest rate models and derivatives, the Black-Scholes model and the Black-Scholes partial differential equation, implied volatility and Merton's optimal portfolio problem.

Prerequisites: SC/MATH 2131 3.00 and SC/MATH 2281 3.00.

The course explores linear regression models for the analysis of data involving a single quantitative response variable and one or more explanatory variables. The focus is on understanding the different models, statistical concepts, and their application. The approach will require the use of matrix representations of data and the geometry of vector spaces, which will be reviewed in the course. Topics include simple linear regression, inference assumptions, matrix algebra, multiple linear regression, multicollinearity, diagnostic statistics, model building, indicator variables, and variable selections.

Prerequisites: One of SC/MATH 1131 3.00, SC/MATH 2570 3.00, HH/PSYC 2020 6.00, or equivalent; some acquaintance with matrix algebra (such as is provided in SC/MATH 1021 3.00, SC/MATH 1025 3.00, SC/MATH 1505 6.00, SC/MATH 1550 6.00, or SC/MATH 2221 3.00).

Course credit exclusions: SC/MATH 3033 3.00, AP/ECON 4210 3.00, HH/PSYC 3030 6.00.

Note: This course is a prerequisite for SC/MATH 3430 3.00, SC/MATH 4130B 3.00, SC/MATH 4330 3.00, SC/MATH 4630 3.00, SC/MATH 4730 3.00, SC/MATH 4930A 3.00, SC/MATH 4931 3.00 and SC/MATH 4939 3.00.

The lectures will spend 60% of time on various methodologies and algorithms; whereas 40% of the lectures will teach students to use data analytics related software R to solve real life problems. The lecturers will provide students with business case studies to practice their analytical skills. The course will cover dimension reduction methods, logistic regression, Naïve Bayes Estimation, K-nearest neighbor algorithm, hierarchical and k-means clustering, classification and decision Trees, ensemble methods, text mining, network analysis, market association analysis, neural networks and application of neural network modeling.

Prerequisites: SC/MATH 1131 or equivalent; LE/EECS 1560 or LE/EESC 1541 or equivalent.

Some polynomials, such as 𝑥²+1= 0, have no roots if we confine ourselves to the real number system, but do have roots if we extend the number system to complex numbers, which can be defined as the set of all numbers of the form 𝑥+𝑖𝑦 ,where 𝑥 and 𝑦 are real and 𝑖²=-1, with basic arithmetic operations having the same structure as those of the real number system. The complex numbers defined so, include the reals (as a case 𝑦=0), and the extended system has the desirable property that not only 𝑥²+1=0 but every polynomial equation has a root. In the system of complex numbers certain connections are seen between otherwise apparently unconnected real numbers. A striking example is one of the most beautiful identities in mathematics; namely Euler’s formula exp(2𝜋𝑖)=1 which is a simple consequence of the extension to complex variables of familiar exponential and trigonometric functions. The concepts and operations of calculus (differentiation, integration, power series, etc.) find their most natural setting in complex (rather than real) variables. The present course is intended to give the student a basic knowledge of complex numbers and functions and a basic facility in their use.

Prerequisite: SC/MATH 2010 3.00 or SC/MATH 2015 3.00 or SC/MATH 2310 3.00. (SC/MATH 3010 3.00 is also recommended as a prerequisite for students who have taken SC/MATH 2010 3.00.)

Course credit exclusion: GL/MATH 4230 3.00.

Note: This course is a prerequisite for SC/MATH 4650 3.00.

This course deals with the peculiarities of sampling and inference commonly encountered in sample surveys in medicine, business, the social sciences, political science, natural resource management and market research. Attention will be focused on the economics of purchasing a specific quantity of information. That is, methods for designing surveys that capitalize on characteristics of the population under study will be presented, along with associated estimators to reduce the cost of acquiring an estimate of specified accuracy. (The emphasis will be on the practical applications of theoretical results.)

Prerequisite: SC/MATH 2131 3.00 or SC/MATH 3330 3.00.

Note: This course is a prerequisite for SC/MATH 4034 3.00, SC/MATH 4731 3.00 and SC/MATH 4939 3.00.

Textbook: Elementary Survey Sampling by R. L. Schaeffer, W. Mendenhall, L. Ott, and K.G. Gerow.

Qualified Honours or Specialized Honours students gain relevant work experience as an integrated complement to their academic studies, reflected in the requirements of a learning agreement and work term report. Students are required to register in this course for each four month work term, with the maximum number of work term courses being four (16 months). Students in this course are assigned a Faculty Supervisor/Committee. During the course, students are expected to work at least 480 hours.

Prerequisites: Enrollment is by permission only.

Criteria for permission include:

1. that students have a cumulative GPA and an average of math GPA of at least 7.5;
2. that Applied Mathematics students have successfully completed SC/MATH 3241 3.00, SC/MATH 3271 3.00 and at least one of one SC/MATH 3242 3.00, SC/MATH 3260 3.00, SC/MATH 3171 3.00 and SC/MATH 3172 3.00; that Pure Mathematics students have successfully completed at least two of SC/MATH 3001 3.00, SC/MATH 3010 3.00 and SC/MATH 3021 3.00; that Statistics students have successfully completed SC/MATH 3131 3.00, SC/MATH 3330 3.00, SC/MATH 3132 3.00, SC/MATH 3430 3.00; that Actuarial Science students have successfully completed SC/MATH 2280 3.00, SC/MATH 2281 3.00, SC/MATH 2131 3.00 and passed at least one professional exam; that Mathematics for Education students have successfully completed SC/MATH 3052 3.00;
3. that students are enrolled full-time in the Honours or Specialized Honours program;
4. that students have not been absent for more than two consecutive years as a full-time student from their Honours or Specialized Honours degree studies;
5. that upon enrolling in their course students have a minimum of nine credits remaining toward their degree and need to return as a full-time student for at least one academic term to complete their degree after completion of their final work term.

Note: This is a pass/fail course.