Concordia University

# Mathematics and Statistics Courses

## Actuarial Mathematics Courses

#### Prerequisite/Corequisite:

The following course must be completed previously or concurrently:. Permission of the Department is required.

#### Description:

This course is an introduction to interest theory and provides an understanding of the fundamental concepts of financial mathematics used in valuing cash flows, investment income and asset/liability management. In this course, students examine measurement of interest; annuities and perpetuities; amortization and sinking funds; rates of return; bonds and related securities. Further special topics may be explored.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: .

#### Description:

This course introduces students to the mathematical models for present values of claims contingent on some event, e.g. survival (Life) or sickness (Health). In this course, students examine measurement of mortality; pure endowments; life insurance; net single premiums; life annuities; and net annual premiums. Further special topics may be explored.

Lecture

#### Description:

This lab features problem‑solving sessions for the professional examination on financial mathematics of the Society of Actuaries and the Casualty Actuarial Society.

#### Prerequisite/Corequisite:

The following course must be completed previously: .

#### Description:

This course is a continuation of , and considers pricing of more complicated life insurance products. Students examine net level premium reserves; multiple life functions; multiple decrements, and the expense factor. Further special topics may be explored.

Lecture

#### Description:

In this lab, students use programming languages and software applications specific to the pension and insurance industry. Students learn how to calculate and apply actuarial concepts and how to communicate their results using the softwares (such as Excel, Access, and Axis) introduced in this course.

Laboratory

#### Prerequisite/Corequisite:

The following course must be completed previously: .

#### Description:

Valuation methods; gains and losses; dynamic control; special topics.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously:. The following course must be completed previously or concurrently:

#### Description:

This course offers an introduction to classical models and applies them to relevant problems in risk theory, which is a core component of Property-Casualty Insurance mathematics. Students learn about the applications of contingency theory in health insurance, individual and collective risk theory, and ruin theory. Further special topics may be explored.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: . The following course must be completed previously or concurrently: .

#### Description:

Credibility approach to inference for heterogeneous data; classical, regression and Bayesian models; illustrations with insurance data.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: ; and .

#### Description:

Probability model fitting to loss data; estimation and testing under variety of procedures and sampling designs.

Lecture

#### Description:

This lab is designed to prepare students for the Actuarial Models examination of the Society of Actuaries and the Casualty Actuarial Society.

Laboratory

#### Description:

Specific topics for this courses, and relevant prerequisites, are stated in the Undergraduate Class Schedule.

Lecture

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for this courses, and relevant prerequisites, are stated in the Undergraduate Class Schedule.

Lecture

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for this courses, and relevant prerequisites, are stated in the Undergraduate Class Schedule.

Research

## Mathematical and Computational Finance Courses

#### Prerequisite/Corequisite:

The following courses must be completed previously: or; or.

#### Description:

This course is an introduction to topics related to quantitative finance. Topics may include: financial derivatives, binomial option pricing models, Black-Scholes option pricing model, derivatives risk management, mean-variance portfolio theory, asset pricing models, investment risks, and behavioral finance.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: or ; or ; or .

#### Description:

This course is a rigorous introduction to the theory of mathematical and computational finance. Topics include multi-period binomial model; state prices; change of measure; stopping times; European and American derivative securities; interest-rate models; interest-rate derivatives; hedging; and convergence to the Black-Scholes model.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: .

#### Description:

This course is a continuation of and focuses on modelling and computational techniques beyond the binomial model. Topics include simulation; Monte- Carlo methods in finance; option valuation; hedging; heat equation; finite difference techniques; stability and convergence; exotic derivatives; risk management; and calibration and parameter estimation.

Lecture

#### Description:

Specific topics for this course, and relevant prerequisites, are stated in the Undergraduate Class Schedule.

Lecture

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for this course, and relevant prerequisites, are stated in the Undergraduate Class Schedule.

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

## Mathematics Courses

#### Description:

This course is designed to give students the background necessary for. Some previous exposure to algebra is assumed. Sets, algebraic techniques, inequalities, graphs of equations.

Lecture

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or exemption for a course at the level of or above may not take this course for credit.

#### Description:

This course focuses on basic functions (power functions, polynomials, rational and algebraic function, exponentials and logarithms, trigonometric functions), and operations on functions, including inverse functions, transformations and composition of functions. The course also includes a detailed study of trigonometry of angles, geometry of triangles and related trigonometric transformations.

#### Component(s):

Lecture; Tutorial

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or exemption for or equivalent, or for a course having or equivalent in its sequence of prerequisites, may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course introduces students to the fundamental notions of algebra. Topics may include progressions, combinations, permutations, mathematical induction, inequalities, polynomials, and the Cartesian and polar forms of complex numbers.

#### Component(s):

Lecture; Tutorial

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or an exemption for a course at the level of or above; or above; or above; or above; or for a course having any of these courses in its sequence of prerequisites, may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course is the first of two connected calculus courses (followed by. This course focuses on an overview of functions and limits; derivative as rate of change, differentiation of elementary functions (power, exponential, logarithmic, trigonometric); differentiation rules (product, quotient, and chain rules); implicit differentiation; higher derivatives; approximations using linearization and the differential. Applications include related rates; optimization, analysis of functions (determining maxima, minima and inflection points) and graphing.

#### Component(s):

Lecture; Tutorial

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or an exemption for a course at the level of or above; or above; or above; or above; or for a course having any of these courses in its sequence of prerequisites, may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This is an introductory course in elementary linear algebra. In this course, students learn algebra and geometry of vectors, dot and cross products, lines and planes. A key objective is learning methods to solve systems of linear equations, which are reformulated with matrix multiplication, matrix inversion, and determinants. Finally, these methods are clarified through linear transformations.

#### Component(s):

Lecture; Tutorial

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or an exemption for a course at the level of or above; or above; or above; or above; or for a course having any of these courses in its sequence of prerequisites, may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course is the second of two connected courses on differential and integral calculus (preceded by). In this course, students learn about definite integrals and antiderivatives, the fundamental theorem of calculus, indefinite integrals and techniques of integration (substitutions, integration by parts, partial fractions, trigonometric functions), improper integrals, applications of integration (area between curves, volumes of solids of revolurion); infinite sequences and series: tests for convergence; power series, Taylor and Maclaurin series.

#### Component(s):

Lecture; Tutorial

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or an exemption for a course at the level of or above; or above; or above; or above; or for a course having any of these courses in its sequence of prerequisites, may not take this course for credit.

#### Description:

Coordinate systems. Radicals and distance formula. Polynomials, factoring, and graphing. Relations and functions. Linear and quadratic functions, equations, and systems. Exponents, exponential and logarithmic functions and equations.

#### Component(s):

Lecture; Tutorial

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or exemption for a course at the level of or above may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

Topics covered in this course include matrices, Gaussian elimination, input‑output analysis, progressions, compound interest, annuities, permutations and combinations, probability, exponential and logarithmic functions, inequalities, linear programming.

Lecture

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• This is a prerequisite course for John Molson School of Business students. See
• Students who have received credit or an exemption for a course at the level of or above; or above; or above; or above; or for a course having any of these courses in its sequence of prerequisites, may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course is a prerequisite course for John Molson School of Business students*. Limits; differentiation of rational, exponential, and logarithmic functions; theory of maxima and minima; integration.

Lecture

#### Notes:

• Students in programs leading to the BSc degree or the BA programs in Mathematics and Statistics may not take this course for credit to be applied to their program of concentration.

• Students who have received credit or exemption for or equivalent may not take this course for credit.

#### Description:

This course deals with a blend of fascinating mathematical themes in various contexts: historical, cultural, and practical. It is intended for non‑mathematics students. One of the aims of the course is to demonstrate the presence of mathematics and mathematical ideas in many aspects of modern life. At a deeper level, it is also intended to explain what mathematics is all about and why some easily stated assertions, such as Fermat’s last theorem, are so difficult to prove. Students who complete the course successfully should have enough understanding and knowledge of fundamental ideas and techniques of mathematics to appreciate its power, its beauty, and its relevance in so many different fields such as architecture, art, commerce, engineering, music, and all of the sciences.

Lecture

#### Notes:

• Students enrolled in a Mathematics and Statistics program and students who have taken mathematics beyond the pre-calculus level may not take this course for credit.

• Students who have received credit for this topic under a MATH 298 number may not take this course for credit.

#### Description:

Mathematics is used to unravel the secrets of nature. This course introduces students to the world of mathematical ideas and mathematical thinking. Without being overly technical, that is, without requiring any formal background from the student other than high school mathematics, the course delves into some of the great ideas of mathematics. The topics discussed range from the geometric results of the Ancient Greeks to the notion of infinity to more modern developments.

Lecture

#### Notes:

• This course is designed as a suitable elective for students following an undergraduate program. It has no formal prerequisites and will not qualify students to enrol for any other Mathematics course, and cannot be used to satisfy a Mathematics requirement in any BSc or BA program.

• Students who have received credit for INTE 215 may not take this course for credit.

#### Description:

This course is designed as an elective course for students who are not registered in a Mathematics and Statistics program. The particular topic varies from one term to the next and the material is dealt with in a manner appropriate for students who have no background in university‑level mathematics.

Lecture

#### Notes:

• Students registered in a Mathematics and Statistics program may not take this course for credit.

#### Description:

This course is designed as an elective course for students who are not registered in a Mathematics and Statistics program. The particular topic varies from one term to the next and the material is dealt with in a manner appropriate for students who have no background in university‑level mathematics.

#### Notes:

• Students registered in a Mathematics and Statistics program may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: Cegep Mathematics 203 or 201-NYB or .

#### Description:

First‑order differential equations (first‑ and second‑order chemical reactions). Hermite, Laguerre, and Legendre equations. Solutions by power series. Eigenfunctions and eigenvalues, Sturm‑Liouville theory.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: , or equivalent.

#### Description:

This course provides an introduction to formal logic and standard techniques for constructing proofs from a mathematical perspective. This is done in tandem with an introduction to a variety of fundamental mathematical structures so that students learn to identify, understand, and create rigorous mathematical proofs. Topics may include logic and proofs, basic structures, algorithms, number theory and cryptography, induction and recursion, counting techniques, relations and their properties, graphs, trees, Boolean algebra, and modelling computation.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: and or equivalent.

#### Description:

This course is an introduction to vector spaces and linear transformations. The following topics are treated with mathematical rigour: matrices and linear equations; vector spaces; bases, dimension and rank; linear mappings and algebra of linear operators; matrix representation of linear operators; determinants; eigenvalues and eigenvectors; diagonalization.

Lecture

#### Notes:

• Students who have received credit for or may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course continues the study of vector spaces and linear transformations, with mathematical rigour. Topics may include characteristic and minimum polynomials; invariant subspaces, invariant direct sums; cyclic subspaces; rational canonical form; bilinear and quadratic forms; inner product; orthogonality; adjoint operators and orthogonal operators; Jordan canonical form.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: and or equivalent.

#### Description:

Introduction to limits and continuity in Rn. Multivariate calculus: the derivative as a linear approximation; matrix representation of derivatives; tangent spaces; gradients, extrema, including Lagrange multipliers, Taylor’s formula and the classification of critical points.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

Implicit functions and the implicit function theorem. Multiple integrals and change of variables. Curves, surfaces and vector calculus.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

18 credits in post-Cegep Mathematics.

#### Description:

General principles of counting, permutations, combinations, identities, partitions, generating functions, Fibonacci numbers, Stirling numbers, Catalan numbers, principle of inclusion‑exclusion. Graphs, subgraphs, isomorphism, Euler graphs, Hamilton paths and cycles, planar graphs, Kuratowski’s Theorem, trees, colouring, 5‑colour theorem, matching, Hall’s theorem.

#### Component(s):

Lecture; Tutorial

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: .

#### Description:

Matrices, linear transformations, determinants, metric concepts, inner‑product spaces, dual spaces, spectral theorem, bilinear and quadratic forms, canonical forms for linear transformation, matrix functions, selected topics.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent; The following course must be completed previously or concurrently: or equivalent.

#### Description:

Error analysis in numerical algorithms; solution of non‑linear equations; fixed point iterations, rate of convergence. Interpolations and approximations, Legendre polynomials. Numerical integration and quadrature.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course covers the main topics of linear optimization and an introduction to nonlinear optimization. A thorough treatment of the simplex algorithm is given, along with applications such as duality, the transportation problem, and an introduction to game theory. The course also introduces unconstrained nonlinear optimization and the rudiments of constrained nonlinear optimization via basic facts about convex sets and functions. Further special topics may be explored.

Lecture

#### Notes:

• Students who have received credit for MAST 224 or MATH 324 may not take this course for credit.

#### Prerequisite/Corequisite:

Students must complete 12 credits in university-level mathematics (MATH 251 or higher) prior to enrolling. If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This course provides an introduction to mathematical analysis in one variable. Topics may include mathematical rigour: proofs and counter-examples, quantifiers; number systems, cardinality; decimal representation, density of the rationals, least upper bound, completeness; sequences of real numbers, limits; continuous functions, differentiability, intermediate value theorems, mean value theorem and Taylor’s formula.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course continues the study of mathematical analysis with more advanced topics and greater precision. The Riemann integral is defined rigorously. Examples and counter-examples are given. Families of Riemann integrable functions are studied as are improper integrals. Series of real numbers are studied, under the umbrella of absolute and conditional convergence. Important and useful tests for convergence and divergence are studied, including results by Abel and Dirichlet. The regrouping and rearrangement of series are also covered. The work on series of reals is followed by the topic of sequences and series of functions. The fundamental idea of uniform convergence of sequences and series of functions is studied in detail. This leads to the three preservation theorems and their applications to the M-test and to Taylor series. Further special topics may be explored.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

Algebra and geometry of complex numbers, linear transformations, analytic functions, Laurent’s series, calculus of residues, special functions.

Lecture

#### Prerequisite/Corequisite:

12 credits in university-level mathematics (MATH 251 or higher) must be completed prior to enrolling. If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This course is the first of four connected courses in abstract algebra (followed by , and ). In this course, students learn about the fundamental notion of groups and several of their properties such as subgroups, quotients, homomorphisms and group actions. Students also learn about Lagrange's theorem, Cayley's theorem, Sylow’s theorem, and their applications. Further special topics may be explored.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: , or equivalent.

#### Description:

In this course, students are taught to recognize, interpret, and solve various differential equations and boundary value problems. Topics may include separable equations, exact equations, integrating factors, first order linear equations, second order equations, series solutions, reduction of order, variation of parameters, the Laplace transform, and higher-order linear equations with constant coefficients. Time permitting, the systems of linear differential equations are covered. Applications of these methods to population models, mechanical systems, and many other realistic situations are emphasized throughout.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Description:

This lab will demonstrate the use of MAPLE software for Calculus, Linear Algebra, and Statistics.

Laboratory

#### Notes:

• Students who have received credit for MATH 232 may not take this course for credit.

#### Prerequisite/Corequisite:

Students must complete 18 credits in university-level mathematics (MATH 251 or higher) prior to enrolling. If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This course introduces students to the basic arithmetic concepts and classical results in number theory. Students learn about divisibility, greatest common divisor, and unique factorization; congruences, Euler's and Fermat's theorems; applications to primality testing and public key cryptography; primitive roots, quadratic residues and quadratic reciprocity; Diophantine equations. Further special topics may be explored.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: ; . If prerequisites are not satisfied, permission of the Department is required.

#### Description:

Early mathematics, Greek mathematics, European mathematics in the Middle Ages, the origin and development of analytic geometry and calculus, mathematics as free creation, the generality of mathematics in the 20th century.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: , , or equivalent.

#### Description:

Nature of problems, weak variations, the first variation, Euler’s equation. The second variation, Jacobi’s equation, Legendre’s test, conjugate points. Relative maxima and minima, iso‑perimetrical problems. Integrals with variable end points. Applications to problems in pure and applied mathematics; the principle of least action. Strong variations, the Weierstrass E‑function.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: , or equivalent.

#### Description:

Metric spaces; function spaces; compactness, completeness, fixed‑point theorems, Ascoli‑Arzela theorem, Weierstrass approximation theorem.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: , , or equivalent.

#### Description:

Cauchy’s theorem, singularities, maximum modulus principle, uniqueness theorem, normal families, Riemann mapping theorem.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously:, The following course must be completed previously or concurrently: or equivalent.

#### Description:

In this course, students learn about the Lebesgue measure and integration on the real line, convergence theorems, Lp spaces, completeness of Lp[0,1], absolute continuity. Further special topics may be explored.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This is an advanced course in algebra, introducing the notion of rings and studying the properties of several classes of rings. Topics may include an introduction to rings, ideals, euclidean domains, principal ideal domains and unique factorization domains; polynomial rings; introduction to modules.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course continues the study of rings and modules from MATH 470. Topics may include the structure theorem of modules over principal ideal domains; Noetherian rings and modules (including Hilbert basis theorem for rings and modules), and Hilbert’s Nullstellensatz.

Lecture

#### Notes:

• Students who have received credit for MATH 491 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course studies fields and their extensions. Topics may include elements of field and Galois theory, including straight-edge-and-compass construction and unsolvability of equations of fifth degree by radicals.

Lecture

#### Notes:

• Students who have received credit for MATH 492 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course is an introduction to partial differential equations and focuses on recognizing, interpreting, and solving such equations. Particular attention is paid to the Laplace, Poisson, heat, and wave equations with various boundary conditions. Solution methods include separation of variables, Fourier expansions, variational methods, and energy methods. Theory developed in this course includes Green’s formula, the maximum principle, and properties of harmonic functions, such as the mean value theorem. Further special topics may be explored.

Lecture

#### Notes:

• Students who have received credit for MATH 371 may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: , or equivalent. If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This course is an introduction to dynamical systems. Topics may include systems of linear differential equations; fundamental matrices; non-homogeneous linear systems; non-linear systems; solutions and trajectories; the phase plane; stability concepts; Liapounov’s second method; periodic solutions and limit cycles.

Lecture

#### Notes:

• Students who have received credit for MATH 373 may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: , or equivalent. If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This course provides a comprehensive investigation of complex dynamics and chaos in the context of discrete dynamical systems. Topics may include modelling with discrete equations, bifurcations, the period three theorem, symbolic dynamics, transitivity, conjugacies, Julia and Fatou sets, and fractals. Complex behaviour in discrete models is analyzed through computer simulations and in the context of iterated function systems.

Lecture

#### Notes:

• Students who have received credit for MATH 379 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: . If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This course provides an overview of nonlinear optimization and related topics. Students review unconstrained nonlinear optimization using methods from advanced calculus. The rudiments of convex analysis are covered and applied to deriving optimality conditions in constrained convex optimization via the classical Lagrangian approach. Nonlinear duality is covered as well. Further special topics (such as an introduction to nonsmooth analysis) may be explored.

Lecture

#### Notes:

• Students who have received credit for MATH 436 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: . If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This is an introduction to convex analysis. Topics may include support and separation of convex sets, extreme point characterizations, convex and dual cones, Farkas’ theorem; convex functions, criteria for convexity, subgradient, Legendre-Fenchel conjugate, Young's inequality; Lagrangians, necessary and sufficient conditions for optimality in constrained minimization; the dual problem; functional inequalities and other applications.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: , , .

#### Description:

In this course, students learn about differential geometry and the geometric topology of surfaces. The topics covered are selected from orientation, geodesics, curvature, Theorema Egregium, Gauss‑Bonnet theorem, Euler characteristic, classification of surfaces, cohomology, homotopy groups, applications of ideas and techniques from geometry and topology in knot or graph theory and map colourings.

Lecture

#### Notes:

• Students who have received credit for MATH 380 may not take this course for credit.

#### Description:

Specific topics for this course, and prerequisites relevant in each case, are stated in the Undergraduate Class Schedule.

Lecture

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for this course, and prerequisites relevant in each case, are stated in the Undergraduate Class Schedule.

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for this course, and prerequisites relevant in each case, are stated in the Undergraduate Class Schedule.

Research

## Mathematics and Statistics Courses

#### Prerequisite/Corequisite:

The following courses must be completed previously: Cegep Mathematics 105 or 201-NYC, 203 or 201- NYB.

#### Description:

Functions; maxima and minima. Velocity and acceleration. Iterative solution of equations, parametric equation of curves. Integrals; change of variables, integration by parts, double integrals, numerical integration. Conic sections. Matrices, determinants, eigen‑values, eigenvectors, system of equations. Series and their convergence. Introduction to vector space and complex numbers. Word problems.

Lecture

#### Notes:

• This course can be counted as an elective towards a 90-credit degree program, but must be taken before any other post-Cegep Mathematics course except for , which may be taken concurrently. It must be taken, upon entry, by newly admitted students in the MATH/STAT Major who have less than 70% average in Cegep Mathematics courses.

#### Prerequisite/Corequisite:

The following courses must be completed previously: and , or equivalent.

#### Description:

This course aims to foster analytical thinking through a problem‑solving approach. Topics include construction of proofs, number systems, ordinality and cardinality, role of examples and counter examples, role of generalizations and specializations; role of symbols, notations and definitions; styles of mathematical discourse.

Lecture

#### Notes:

• Students with more than 12 credits in post-Cegep Mathematics (excluding ) may not take this course for credit.

• Students who have received credit for or COMP 238 may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: and , or equivalent.

#### Description:

This is an introductory course in the theory of functions of several variables addressing differential and vector calculus. In this course, students focus on vector geometry; lines and planes; curves in Rn; vector functions; vector differential calculus; functions of several variables, chain rule and implicit differentiation, extrema, classification of extremal values and Lagrange multipliers.

Lecture

#### Notes:

• Students who have received credit for MATH 264 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course is a continuation of the theory of functions of several variables from MAST 218. In this course, students learn about the theory of integration of such functions and the applications of multiple integrals to physics. Topics include vector integral calculus; line and surface integrals; fundamental theorem of line integrals; Green’s, Stokes’ and Gauss’ theorems and Gauss' law; change of integration variables in multiple integrals and Jacobians; applications to multivariate random variables.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: and , or equivalent. The following courses must be completed previously or concurrently: or equivalent.

#### Description:

Counting rules, discrete probability distributions; random sampling; conditional probability; means and variances, normal and other continuous sampling distributions. Applications. Use of statistical software, e.g. MINITAB.

Lecture

#### Notes:

• Students enrolled in a Mathematics and Statistics program who take probability/statistics courses in other departments may not receive credit for this course. Please consult the Mathematics and Statistics undergraduate program advisor.

• Students who have received credit for , or may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course is an introduction to stochastic methods of operations research. Topics may include Markov chains; queuing theory; inventory theory; Markov decision processes; applications to reliability.

Lecture

#### Notes:

• Students enrolled in a Mathematics and Statistics program who take probability/statistics courses in other departments may not receive credit for this course. Please consult the Mathematics and Statistics undergraduate program advisor.

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: and , or equivalent.

#### Description:

An introduction to the use of a high‑level mathematical programming language (MAPLE or MATHEMATICA) as a practical aid in doing mathematics. Most classes are given in an interactive way in the computer laboratory. The emphasis is on applications, not on general programming techniques or abstract structures. The aim is to arrive at a sufficient working familiarity with the computer algebra language to permit its regular use in subsequent studies and applications. The commands and online resources are introduced through a review of arithmetic, complex numbers, algebra, Euclidean geometry, trigonometry, coordinate systems and graphing, elementary functions and transformations, series, derivatives, integrals, vectors and matrices. There may be additional topics from domains such as number theory, differential equations, integral transforms, probability and statistics.

Lecture

#### Notes:

• Students who have received credit for or or COMP 467 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

In this course, students focus on two major concepts: vector spaces and linear transformations. Topics include system of linear equations, matrix operations, row echelon form; linear dependence; Rn and general vector spaces, subspaces; bases and coordinates; linear operators and their matrix representations; similar matrices; applications of determinants; eigenvalues and eigenvectors; diagonalization of matrices; and the applications of diagonalization (e.g. Markov chains). Computing environments/software may be used in the course as tools and/or assignment platforms, but not as objects of study.

Lecture

#### Notes:

• Students who have received credit for or may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

In this course, the theory studied in is applied and further developed. Topics may include inner product spaces; orthogonal projection and Gram-Schmidt algorithm; orthogonal and unitary matrices; self-adjoint operators and Hermitian matrices; spectral theorem. Applications to economic models, networks, dynamical systems, normal equations and least square solutions to inconsistent systems, principal axes theorem and quadratic forms are examined. Further special topics (e.g. Cayley-Hamilton Theorem, singular values and SVD factorization) may be explored. Computing environments/software may be used in the course as tools and/or assignment platforms, but not as objects of study.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course is an introduction to optimization. Topics may include linear models, linear programming (simplex algorithm, sensitivity analysis, duality and dual simplex algorithm), and the transportation problem (transportation simplex algorithm and sensitivity analysis).

Lecture

#### Notes:

• Students who have received credit for MAST 224 or may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: , and MAST 234 or equivalent.

#### Description:

This course introduces students to important applications of the techniques developed in calculus to everyday problems. These include methods for solving first- and second-order differential equations and their application to the solution of practical problems. Topics may include mathematical models of natural phenomena; method of linearization; and systems of first-order equations.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: , (or equivalent). The following course must be completed previously or concurrently: .

#### Description:

This course presents a dynamical systems approach to solving discrete-time equations that have practical applications. The course focuses on studying the long-term behaviour of linear and nonlinear systems. Students learn about fixed points, periodic orbits, stability of two-and-higher-dimensional models and the important notion of bifurcations for families of maps. Students are introduced to the idea of chaotic systems that play an important role in understanding complex nonlinear systems. Topics may include mathematical models as discrete-time dynamical systems; family of functions and bifurcations; Cantor sets; one- and two-dimensional chaos; fractals; Iterated Function Systems.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: or or equivalent; or equivalent.

#### Description:

This course is an application-oriented introduction to algebraic methods involved in symbolic computation, as applied to number theory and modular algebra. Topics may include numbers, primes, modular arithmetic, Diophantine equations; congruence classes and applications, finite fields and rings; Fermat’s and Euler’s theorems; Chinese Remainder Theorem and applications; polynomial congruences and rings. Applications to error-correcting codes (Humming codes), Hill Cryptosystem, public key encryption schemes, polynomial factorization and polynomial interpolation are also covered. Computing environments/software may be used in the course as tools and/or assignment platforms, but not as objects of study.

Lecture

#### Notes:

• Students who have received credit for or COMP 467 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

Graphical and numerical descriptive methods; Estimation and hypothesis testing; linear regression and correlation; one way ANOVA; contingency and goodness of fit tests. Use of statistical software, e.g. MINITAB.

Lecture

#### Notes:

• Students enrolled in a Mathematics and Statistics program who take probability/statistics courses in other departments may not receive credit for this course. Please consult the Mathematics and Statistics undergraduate program advisor.

• Students who have received credit for , , or may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: or equivalent; or equivalent.

#### Description:

In this course, students learn about numerical analysis, an algorithmic approach to finding approximate solutions when exact solutions are impossible or unreasonably complicated. Lying at the intersection of mathematics and computer science, numerical analysis is a key component of computational mathematics. Topics may include analysis of errors involved in computations; floating-point arithmetic; root-finding methods; interpolation theory and function approximation; orthogonal polynomials; numerical integration and quadrature formulas; error analysis of numerical algorithms.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course introduces the notion of interest and the basic principles of financial mathematics. Topics may include simple and compound interest; annuities; amortization and sinking funds; mortgage schemes; bonds and related securities; capital cost and depletion; spread‑sheet implementation.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent; or equivalent.

#### Description:

This class provides an overview of techniques used by life insurers, pension plans and Property and Casualty insurers to quantify and measure their liabilities. The course is subdivided into two main parts. The first aims at studying life-contingent liabilities such as life insurance and annuities. The second part provides an overview of methods utilized by Property and Casualty insurers to represent their liabilities.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: ; and or .

#### Description:

This lab course offers hands-on exposure to a broad array of problems and tasks frequently encountered in the data science practice. Examples of topics that are covered may include dataset and table construction, data curation and preparation, data exploration, non-traditional data types and large data sets (big data). Extensive programming duties and data analysis projects are assigned to students.

Laboratory

#### Description:

Specific topics for this courses, and relevant prerequisites, are stated in the Undergraduate Class Schedule.

Lecture

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for this courses, and relevant prerequisites, are stated in the Undergraduate Class Schedule.

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

## Statistics Courses

#### Prerequisite/Corequisite:

The following course must be completed previously or concurrently: or equivalent.

#### Description:

This course is an introduction to rigorous basis of probability theory. Topics include axiomatic approach to probability; combinatorial probability; random sampling and sampling distributions; discrete and continuous distributions; expectation; variances; joint variables and their distributions, conditional distributions, conditional expectation.

Lecture

#### Notes:

• Students who have received credit for may take for credit only with prior permission of the Department. Students enrolled in a Mathematics and Statistics program who take probability/statistics courses in other departments may not receive credit for this course. Please consult a Mathematics and Statistics undergraduate program advisor.

• Students who have received credit for or may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent. The following course must be completed previously or concurrently: MATH 265 or equivalent. If prerequisites are not satisfied, permission of the Department is required.

#### Description:

This course is an introduction to the theoretical basis of core statistics topics. Topics include point and interval estimation; hypothesis testing; likelihood function; Neyman Pearson Lemma and likelihood ratio tests.

Lecture

#### Notes:

• Students enrolled in a Mathematics and Statistics program who take probability/statistics courses in other departments may not receive credit for this course. Please consult a Mathematics and Statistics undergraduate program advisor.

#### Prerequisite/Corequisite:

The following courses must be completed previously: , or equivalent.

#### Description:

This course is an introduction to statistical programming and computational statistics using the R programming language. Basic programming concepts such as data types, control structures, and algorithms are introduced. The course illustrates data manipulation methods, descriptive analyses, and data visualization tools. The use of linear algebra, statistical simulation, and optimization functions is also illustrated. Applications and examples use real data sets.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Description:

This lab is associated with and and features problem‑solving sessions for the probability examination of the Society of Actuaries and the Casualty Actuarial Society.

Laboratory

#### Prerequisite/Corequisite:

The following course must be completed previously: or .

#### Description:

This course is an introduction to statistical tools for quality management and productivity. Topics may include control charts for variables and attributes, acceptance sampling, sampling inspection, process capability measures and process improvement methods.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: or .

#### Description:

This course examines the theory and applications of survey sampling. Topics may include basic sampling designs and estimators; simple random sampling, stratified, cluster and systematic sampling; sampling with unequal probabilities; ratio and regression methods of estimation.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: or .

#### Description:

This course is an introduction to non-parametric statistics. Topics may include the theory of rank tests, sign test, Mann-Whitney and Wilcoxon one-sample and two-sample tests, Kruskal-Wallis test, goodness of fit tests, Kolmogorov-Smirnov test, Pearson chi-square test, rank correlation and Kendall’s tau.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

Markov decision process and applications. Poisson process, queuing theory, inventory theory; applications.

Lecture

#### Notes:

• Students who have received credit for may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or equivalent.

#### Description:

This course examines linear models. Topics may include least squares estimators and their properties; general linear model with full rank; analysis of residuals; adequacy of model, lack of fit test, weighted least squares; stepwise regression.

Lecture

#### Notes:

• Students who have received credit for or may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: or or equivalent. The following course must be completed previously or concurrently: or or equivalent.

#### Description:

This course is an introduction to statistical learning techniques, including supervised and unsupervised learning methods. Supervised learning methods for regression and classification include linear models, variable selection methods, shrinkage, linear and quadratic discriminant, classification and regression trees, K‑nearest neighbours, support vector machines and neural networks. Unsupervised learning methods include clustering approaches and principal component analysis. Some applications to data science are illustrated. Further special topics may be explored.

Lecture

#### Notes:

• Students who have received credit for this topic under a number may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: or ; or ; or .

#### Description:

This course offers an introduction to the theory of prediction with neural networks, demonstrating their construction, estimation, and use in predictive analysis. Various neural network architectures (feedforward, recurrent, convolutional) are presented. Advanced estimation techniques such as regularization and adaptive learning rates are also considered. Several applications of neural networks to common problems faced in practice are finally explored. Students also learn to apply methods seen in class; programming assignments using programming languages such as Python are included.

Lecture

#### Description:

This lab will use various softwares such as SYSTAT, SAS, SPLUS, MINITAB for data analysis.

Laboratory

#### Notes:

• Students who have received credit for MATH 232 may not take this course for credit.

#### Prerequisite/Corequisite:

The following courses must be completed previously: , .

#### Description:

Central limit theorems and law of large numbers, convergence of random variables, characteristic function, moment generating function, probability generating functions, random walk and reflection principle.

Lecture

#### Notes:

• Students who have received credit for MATH 451 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously: . The following course must be completed previously or concurrently: If prerequisites are not satisfied permission of the Department is required.

#### Description:

Derivation of standard sampling distributions; distribution of order-statistics; estimation, properties of estimators; Rao- Cramer inequality, Rao-Blackwell theorem, maximum likelihood and method of moments estimation, Neyman-Pearson theory, likelihood ratio tests and their properties.

Lecture

#### Notes:

• Students who have received credit for MATH 454 may not take this course for credit.

#### Prerequisite/Corequisite:

The following course must be completed previously:.

#### Description:

This course is an introduction to stochastic processes. Topics may include Martingale, Brownian motion, stochastic integral, Ito's formula, stochastic differential equation.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: .

#### Description:

This course examines the properties and processes of time series models. Topics may include time series, forecasting by trend and irregular components (using multiple regression analysis and exponential smoothing); forecasting seasonal time series, additive and multiplicative decomposition methods, Box‑Jenkins methodology, moving average, autoregressive and mixed models.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: .

#### Description:

This course is an introduction to the methods of simulation and the Monte Carlo techniques. Additional topics covered may include simulation of Poisson process, simulation of discrete event systems (such as queues, insurance risk model, machine-repair problem), variance reduction methods.

Lecture

#### Prerequisite/Corequisite:

The following courses must be completed previously: ; or equivalent.

#### Description:

Multivariate normal distribution; estimation and testing of hypothesis about mean vector; multiple and partial correlation; MANOVA; principal components analysis.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: .

#### Description:

Construction and analysis of standard designs, including balanced designs; block designs; orthogonal designs; response surface designs.

Lecture

#### Prerequisite/Corequisite:

The following course must be completed previously: . If prerequisites are not satisfied, permission of the Department is required.

#### Description:

Statistical software packages in SAS or R are used for the analysis of real‑life data sets. Topics may include techniques from generalized linear models, model selection, log‑linear models for categorical data, logistic regression, survival models.

Lecture

#### Description:

Specific topics for this course, and prerequisites relevant in each case, are stated in the Undergraduate Class Schedule.

Lecture

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for this course, and prerequisites relevant in each case, are stated in the Undergraduate Class Schedule.

#### Notes:

• The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated by the title of the course.

#### Description:

Specific topics for these courses, and prerequisites relevant in each case, are stated in the Undergraduate Class Schedule.

Research