Mathematics

Chair

Hichem Ben-El-Mechaiekh

Professors Emeriti

Howard E. Bell, Charles F. Laywine, John P. Mayberry, Eric Muller

Professors

Stephen Anco, Hichem Ben-El-Mechaiekh, Mei Ling Huang, Ronald A. Kerman, Yuanlin Li, Jan Vrbik, Thomas Wolf

Associate Professors

Henryk Fuks, Omar Kihel, William J. Ralph, Wai Kong (John) Yuen

Assistant Professors

Chantal Buteau, Babak Farzad, Xiaojian Xu

Instructors

Dorothy Levay, Dorothy Miners

Mathematics Learning Resources Manager

Jodie Wallis

General Information

Administrative Assistants

Margaret Thomson, Pina McDonnell

905-688-5550, extension 3300

Mackenzie Chown J415

http://www.brocku.ca/mathematics/

The Department of Mathematics offers a unique program, Mathematics Integrated with Computers and Applications (MICA). This innovative program fully integrates computers and applications into a broad spectrum of courses that range over pure mathematics (the study of mathematics for its own sake), applied mathematics (mathematics for applications) and statistics. With its special focus on technology, the MICA program is especially suited for students desiring careers in applications of mathematics that involve computing. Within the MICA program, students can also form areas of concentration in applied and computational mathematics, mathematics education, pure mathematics or statistics, or they can choose to have no area of concentration.

Students in the MICA program get a solid grounding in mathematical theory and learn how to use computer and information technology to apply and present what they have learned. The core of the MICA program consists of MATH 1P40 and 2F40 in which students will confront problems from pure and applied mathematics that require experimental and heuristic approaches. In dealing with such problems, students will be expected to develop their own strategies and make their own choices about the best combination of mathematics and computing required to find solutions.

The Computer Science and Mathematics Co-op program combines academic and work terms over a period of four and one-half academic years. Students spend one and one-half years in an academic setting studying the fundamentals of Computer Science and Mathematics prior to their first work placement. Successful completion of courses in the core areas of Computer Science and Mathematics provides the necessary academic background for the work experience. In addition to the current fees for courses in academic study terms, Computer Science and Mathematics Co-op students are assessed an administrative fee for each work term (see the Schedule of Fees).

Eligibility to continue in the Computer Science and Mathematics Co-op program is based on the student's major and non-major averages. A student with a minimum 70 percent major average and a minimum 60 percent non-major average may continue. A student with a major average lower than 70 percent will not be permitted to continue in the Computer Science and Mathematics Co-op program, but may continue in the non-co-op Computer Science and Mathematics stream. If a student subsequently raises his/her major average to 70 percent, the student may be readmitted only if approved by the Co-op Admissions Committee. For further information, see the Co-op Programs section of the Calendar.

The Computer Science and Mathematics Co-op program designation will be awarded to those students who have honours standing and who have successfully completed a minimum of twelve months of Co-op work experience.

The Department has a special interest in Mathematics Education and offers several programs and courses specifically for prospective teachers. These include both Concurrent and Consecutive Education Programs as well as Minors for future teachers.

Certain courses are required for any degree in Mathematics (see below). Because a Mathematics major needs both facility in dealing with mathematical theories and experience in the application of mathematics to real-world problems, each student should discuss his or her particular interests with faculty before selecting elective courses.

Program Notes

Mathematics Integrated with Computers and Applications Honours Program (MICA)

Year 1

	MATH 1P01, 1P02, 1P12 and 1P40
	three elective credits (see program note 1)

Year 2

	MATH 2F40, 2P03, 2P08, 2P12, 2P81 and 2P82
	the Humanities context credit, Science context credit or Social Science context credit (not taken in Year 1)
	one-half elective credit

Year 3

	MATH 3F40 (see program note 4)
	two MATH credits numbered 3(alpha)00 or above
	two elective credits (see program note 3)

Year 4

	Four MATH credits (see program notes 2 and 3)
	one elective credit (see program note 3)

Mathematics Pass Program

Year 1

	MATH 1P01, 1P02, 1P12 and 1P40
	three elective credits (see program note 1)

Year 2

	MATH 2F40 and 2P03
	one of MATH 2P08 and 2P12, MATH 2P12 and 2P72, MATH 2P81 and 2P82
	the Humanities context credit, Science context credit or Social Science context credit (not taken in Year 1)
	one and one-half elective credits

Year 3

	Three MATH credits numbered 3(alpha)00 or above
	two elective credits

Combined Major Program

Combined major programs have been developed by the Department of Mathematics in co-operation with each of these departments: Biological Sciences, Chemistry, Computer Science, Economics and Physics. Program requirements are listed in the calendar sections of the co-major discipline. Students may take a combined major in Mathematics and a second discipline. For requirements in the other discipline, the student should consult the relevant department/centre. It should be noted that not all departments/centres provide a combined major option.

Mathematics and Computer Science Co-op (Honours only)

Students admitted to the Mathematics and Computer Science Co-op program must follow an approved program pattern. The most common pattern is listed below. For other approved patterns, consult the Co-op Office.

Year 1

	MATH 1P01, 1P02, 1P12 and 1P40
	COSC 1P02, 1P03 and 1P50
	one Science context credit
	one-half elective credit

Year 2

Fall Term:

	MATH 2P03 and 2P81
	COSC 2P03, 2P12 and 2P90
	SCIE 0N90

Winter Term:

COSC 0N01

Spring/Summer Sessions:

	MATH 1P66 and 1P67
	COSC 2P32
	one-half COSC credit

Year 3

	COSC 3F00
	MATH 2F40 and 3F65
	one Humanities context credit
	one Social Science context credit

Spring/Summer Sessions:

COSC 0N02

Year 4

Fall Term:

COSC 0N03

Winter Term:

	MATH 2P82, 3P60 and 4P61
	COSC 2P13
	one COSC credit (see program note 4)

Year 5

Fall Term:

	One COSC credit (see program note 4)
	one MATH credit (see program note 4)
	one-half elective credit (see program note 4)

Programs and Courses for Future Teachers

The Department of Mathematics has identified courses that are particularly appropriate for students preparing to become teachers at either the elementary or secondary levels. Students should consult the Chair in the selection of courses.

Elementary

To help students meet Primary/Junior Pre-service Education admission requirements at Brock University - MATH 2P52.

Three credits for a teachable subject at the Junior/Intermediate level. May include MATH 1F92, 1P05, 1P06, 1P12, 1P66, 2P90, 2P93 and 3P91.

Secondary

For Mathematics as the first teachable subject (a minimum of five credits), an Honours degree in Mathematics is recommended.

For Mathematics as the second teachable subject (a minimum of three credits); for example: MATH 1P01, 1P02, 1P12, 2P90, 2P93 and one-half MATH credit.

Concurrent BSc/BEd

The Department of Mathematics and the Faculty of Education co-operate in offering two Concurrent BSc (Honours)/BEd programs. The Mathematics BSc (Honours)/BEd programs combines the BSc Honours program or BSc Integrated Studies Honours program with the teacher education program for students interested in teaching at the Intermediate/Senior level (grades 7-12) and at the Junior/Intermediate level (grades 4-10). Refer to the Education - Concurrent BSc (Honours)/BEd (Intermediate/Senior) or Education - Concurrent BSc Integrated Studies (Honours)/BEd (Junior/Intermediate) program listings for futher information.

Certificate in Statistics

The Mathematics Department offers a program leading to a Certificate in Statistics normally for those with a degree in another discipline.

The certificate in Statistics is awarded upon completion the following courses with a minimum 60 percent overall average:

	One university Calculus credit
	MATH 2P12, 2P81, 2P82, 3P81, 3P82, 3P85, 4P81 and 4P82

See "Certificate Requirements" under Academic Regulations.

Concentration Program

Concentration in Applied and Computational Mathematics

Students may earn a Concentration in Applied and Computational Mathematics by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics Integrated with Computers and Applications:

	MATH 2F40 and 3F40
	two and one-half credits from MATH 3P04, 3P08, 3P09, 3P12, 3P60, 3P72, 3P75
	two credits from MATH 4P05, 4P07, 4P09, 4P84, 4P93, 4P94

Concentration in Mathematics Education

Students may earn a Concentration in Mathematics Education by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics Integrated with Computers and Applications:

MATH 2F40, 2P03, 2P08, 2P12, 2P71, 2P90, 2P93, 3P12, 3P90, 3P91 and 3P93

Concentration in Pure Mathematics

Students may earn a Concentration in Pure Mathematics by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics Integrated with Computers and Applications (with the possible exception of MATH 2P72):

	MATH 2P04, 2P12, 2P13, 3P03, 3P04, 3P12 and 3P13
	MATH 2P71 (recommended) or 2P72
	two credits from MATH 4F90, 4P03, 4P04, 4P11, 4P12, 4P71, 4P92, 4P93

Concentration in Statistics

Students may earn a Concentration in Statistics by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics Integrated with Computers and Applications:

MATH 2F40, 2P81, 2P82, 3P81, 3P82, 3P85, 3P86, 4P81, 4P82, 4P84 and 4P85

Minor Program

Minor in Mathematics

Students in other disciplines may obtain a Minor in Mathematics within their degree program by completing the following courses with a minimum 60 percent average:

	MATH 1P01, 1P02, 1P12, 1P40 and 2F40
	one MATH credit numbered 2(alpha)00 or above
	one MATH credit numbered 3(alpha)00 or above

Minor Programs for Teachers

Students intending to become elementary teachers, who are in another discipline, can obtain a Minor in Elementary Teaching Mathematics within their degree program by completing the following courses with a minimum 60 percent overall average:

	MATH 1P12, 1P66, 1P97, 1P98, 2P90, 2P93 and 3P91
	one-half MATH credit

Students intending to become secondary teachers, who are in another discipline, can obtain a Minor in Secondary Teaching Mathematics within their degree program by completing the following courses with a minimum 60 percent overall average:

	MATH 1P01, 1P02, 1P12, 1P40, 2P90 and 2P93
	two MATH credits numbered 2(alpha)00 or above

Course Descriptions

Note that not all courses are offered in every session. Refer to the applicable term timetable for details.

# Indicates a cross listed course

* Indicates primary offering of a cross listed course

Prerequisites and Restrictions

Students must check to ensure that prerequisites are met. Students may be deregistered, at the request of the instructor, from any course for which prerequisites and/or restrictions have not been met.

MATH 1F92

Introductory Statistics

Describing and comparing data sets, linear regression analysis, basic probability theory, discrete probability distributions, binomial and normal distributions, Central Limit Theorem, confidence intervals and hypothesis tests on means and proportions, properties of t-, F- and chi-squared distributions, analysis of variance, inference on regression. Emphasis on interpretation of numerical results for all topics. Use of Minitab.

Lectures, 3 hours per week.

Prerequisite: one grade 11 mathematics credit.

Note: designed for non-science majors. Not open to students with credit in any university mathematics or statistics course.

MATH 1P01

Calculus Concepts I

Differential calculus with an emphasis on concepts and the use of both theory and computers to solve problems. Precalculus topics, limits, continuity and the intermediate value theorem, derivatives and differentiability, implicit differentiation, linear approximation, mean value theorem with proof and applications, max and min, related rates, curve sketching, l'Hospital's rule, antiderivatives, Riemann sums, FTC with proof, integration by substitution. Use of Maple.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: two grade 12 mathematics credits including MCV4U or permission of the instructor.

Note: open to all, but primarily intended for mathematics majors and/or future teachers.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P05.

MATH 1P02

Calculus Concepts II

Integral calculus with emphasis on concepts, theory, and computers to solve problems. Further integration techniques. Applications to areas between curves, volumes, arc length and probabilities. Multivariable calculus: partial derivatives, optimization of functions of two variables. Sequences and series: convergence tests, Taylor and Maclaurin series and applications. Differential Equations: direction fields, separable equations, growth and decay, the logistic equation, linear equations. Use of Maple.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 1P01 or 1P05 and permission of instructor.

Note: open to all, but primarily intended for mathematics majors and/or future teachers.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P06.

MATH 1P05

Applied Calculus I

Differential calculus emphasizing problem solving, calculation and applications. Precalculus topics, limits, continuity, derivatives and differentiability, implicit differentiation, linear approximation, max and min, related rates, curve sketching, l'Hospital's rule, antiderivatives, integrals, FTC without proof, integration by substitution. Use of Maple.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: two grade 12 mathematics credits including MCV4U or permission of the instructor.

Note: designed for students in the sciences, computer science, and future teachers.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P01.

MATH 1P06

Applied Calculus II

Integral calculus emphasizing problem solving, calculations and applications. Further techniques of integration. Applications to areas between curves, volumes, arc length and probabilities. Multivariable calculus: partial derivatives, optimization of functions of two variables. Sequences and series: convergence tests, Taylor and Maclaurin series and applications. Differential Equations: direction fields, separable equations, growth and decay, the logistic equation, linear equations. Use of Maple.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 1P01 or 1P05.

Note: designed for students in the sciences, computer science, and future teachers.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P02.

MATH 1P12

Linear Algebra I

Introduction to finite dimensional real vector spaces; systems of linear equations: matrix operations and inverses, determinants. Vectors in R² and R³: Dot product and norm, cross product, the geometry of lines and planes in R³; Euclidean n-space, linear transformations for Rⁿ to R^m, complex numbers, selected applications and use of a computer algebra system.

Lectures, 4 hours per week.

Prerequisites: two grade 12 mathematics credits or permission of instructor.

MATH 1P40

Mathematics Integrated with Computers and Applications I

Exploration of ideas and problems in algebra differential equations and dynamical systems using computers. Topics include number theory, integers mod p, roots of equations, fractals, predator-prey models and the discrete logistic equation for popular growth.

Lectures, 2 hours per week; lab, 2 hours per week.

Prerequisite: MATH 1P01 or 1P05.

MATH 1P66

Mathematics for Computer Science I

Development, analysis and applications of algorithms in computation; elementary logic, proofs; graphs and trees.

Lectures, 3 hours per week.

Prerequisite: one grade 12 mathematics credit.

Note: MCB4U recommended. Designed for students in Computer Science.

MATH 1P67

Mathematics for Computer Science II

Development, analysis and applications of algorithms in combinatorial analysis; discrete probability models; recursion; limiting procedures and summation; difference equations; introduction to automata theory.

Lectures, 3 hours per week.

Prerequisite: MATH 1P66.

Note: designed for students in Computer Science.

MATH 1P97

Differential and Integral Methods

Elementary functions, particularly the power function, the logarithm and the exponential; the derivative and its application; integration; approximation to the area under a curve; the definite integral; partial differentiation; simple differential equations; numerical methods; and the use of computer algebra systems.

Lectures, 4 hours per week.

Prerequisite: one grade 12 mathematics credit.

Note: MCB4U recommended. Designed for students in Biological Sciences, Biotechnology, Business, Earth Sciences, Environment, Economics, Geography and Health Sciences. Not open to students with credit in any university calculus course.

MATH 1P98

Basic Statistical Methods

Descriptive statistics; probability distributions, estimation; hypothesis testing; normal, chi-squared, t- and F-distributions; mean and variance tests; regression and correlation; and the use of statistical computer software.

Lectures, 3 hours per week.

Prerequisite: one grade 12 mathematics credit.

Note: designed for students in Biological Sciences, Biotechnology, Business, Earth Sciences, Economics, Environment, Geography and Health Sciences. Not open to students with credit in any university statistics course.

MATH 2F05

Applied Advanced Calculus

First and second order differential equations, vector functions, curves, surfaces; tangent lines and tangent planes, linear approximations, local extrema; cylindrical and spherical co-ordinates; gradient, divergence, curl; double and triple integrals, line and surface integrals; Green's theorem, Stokes' theorem, Gauss' theorem; elementary complex analysis. Emphasis on applications to physical sciences. Use of Maple.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 1P02 or 1P06.

Students will not receive earned credit in MATH 2F05 if MATH 2P03 has been successfully completed.

MATH 2F40

Mathematics Integrated with Computers and Applications II

Theory and application of mathematical models; discrete dynamical systems; time series and their application to the prediction of weather and sunspots; Markov chains; empirical models using interpolation and regression; continuous stochastic models; simulation of normal, exponential and chi-square random variables; queuing models and simulations, use of a computer algebra system.

Lectures, lab, 4 hours per week.

Prerequisites: MATH 1P02 and 1P40 or permission of the instructor.

MATH 2P03

Multivariate and Vector Calculus

Multivariable integration, polar, cylindrical and spherical coordinates, vector algebra, parameterized curves and surfaces, vector calculus, arc length, curvature and torsion, projectile and planetary motion, line integrals, vector fields, Green's theorem, Stokes' theorem, the use of computer algebra systems to manipulate vectors, plot surfaces and curves, determine line integrals and analyze vector fields.

Lectures, 3 hours per week, lab/tutorial, 1 hour per week.

Prerequisite: MATH 1P02, 1P06 or permission of the instructor.

MATH 2P04

Basic Concepts of Analysis

Sets; mappings, count ability; properties of the real number system; inner product, norm, and the Cauchy-Schwarz inequality; compactness and basic compactness theorems (Cantor's theorem, the Bolzano-Weierstrass theorem, the Heine-Borel theorem); connectedness; convergence of sequences; Cauchy sequences; continuous and uniformly continuous functions.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2P03.

MATH 2P08

Ordinary Differential Equations

Linear and nonlinear differential equations and autonomous systems; analytical and numerical solution methods, basic existence and uniqueness theory, qualitative analysis of solutions including periodic cycles and steady-states, attractors, chaos, asymptotic behaviour; modelling and applications of differential equations, use of a computer algebra system.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: MATH 1P02 and 1P05 or permission of the instructor.

MATH 2P12

Linear Algebra II

Finite dimensional real vector spaces and inner product spaces; matrix and linear transformation; eigenvalues and eigenvectors; the characteristic equation and roots of polynomials; diagonalization; complex vector spaces and inner product spaces; selected applications; use of a computer algebra system and selected applications.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 1P12.

MATH 2P13

Abstract Linear Algebra

Vector spaces over fields; linear transformations; diagonalization and the Cayley-Hamilton theorem; Jordan canonical form; linear operators on inner product spaces; the spectral theorem; bilinear and quadratic forms.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2P12.

MATH 2P31

Mathematics and Music

Scales and temperaments, history of the connections between mathematics and music, set theory in atonal music, group theory applied to composition and analysis, enumeration of rhythmic canons, measurement of melodic similarity using metrics, topics in mathematical music theory, applications of statistics to composition and analysis.

Lectures, 3 hours per week; lab/tutorial 1 hour per week.

Prerequisites: one of MATH 1P01, 1P02, 1P05, 1P06, 1P97; MATH 1P12 or permission of the instructor.

MATH 2P52

Principles of Mathematics for Primary and Junior Teachers

Mathematical concepts and ideas in number systems; geometry and probability arising in the Primary and Junior school curriculum.

Lectures, seminar, 4 hours per week.

Restriction: students must have a minimum of 5.0 overall credits.

Note: designed to meet the mathematics admission requirement for the Primary/Junior Pre-service program of the Faculty of Education at Brock University. Not open to students holding credit in any grade 12 or university mathematics course.

MATH 2P71

Introduction to Combinatorics

Permutations, combinations, binomial and multinomial expansions; the inclusion-exclusion principle; recurrence relations; ordinary and exponential generating functions. Introduction to graph theory including isomorphism, trees, Euler and Hamilton path problems, planarity and map colouring. Pigeonhole principle and an introduction to classical Ramsey theory.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Prerequisites: two grade 12U mathematics credits or permission of the instructor.

MATH 2P72

Discrete Optimization

Problems and methods in discrete optimization. Linear programming: problem formulation, the simplex method, software, and applications. Network models: assignment problems, max-flow problem. Directed graphs: topological sorting, dynamic programming and path problems, and the travelling salesman's problem. General graphs: Eulerian and Hamiltonian paths and circuits, matchings.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisite: MATH 1P12.

MATH 2P75

Introductory Financial Mathematics

Applications of mathematics to financial markets. Models for option pricing, rates of interest, price/yield, pricing contracts and futures, arbitrage-free conditions, market risk, hedging and sensitivities, volatility; stock process as random walks and Brownian motions; Black-Scholes formula; finite difference methods and VaR.

Lectures, lab, 4 hours per week.

Prerequisites: MATH 1P97 and 1P98.

MATH 2P81

Probability

Probability, events, algebra of sets, independence, conditional probability, Bayes' theorem; random variables and their univariate, multivariate, marginal and conditional distributions. Expected value of a random variable, the mean, variance and higher moments, moment generating function, Chebyshev's theorem. Some common discrete and continuous distributions: Binomial, Poisson, hypergeometric, normal, uniform and exponential. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH2P03 or permission of the instructor.

Note: may be taken concurrently with MATH 2P03.

MATH 2P82

Mathematical Statistics I

Transforming random variables, central limit theorem, law of large numbers. Random sample; sample mean and variance. Sampling from normal population: chi-square, t and F distributions, sample median and order statistics. Point and interval estimation of population parameters: method of moments, maximum-likelihood technique, consistent, unbiased and efficient estimators, confidence intervals. Hypotheses testing: type I and II errors, most powerful tests. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2P81.

MATH 2P90

Euclidean and Non-Euclidean Geometry I

The development of Euclidean and non-Euclidean geometry from Euclid to the 19th century. The deductive nature of plane Euclidean geometry as an axiomatic system, the central role of the parallel postulate and the general consideration of axiomatic systems for geometry in general and non-Euclidean geometry in particular. Introduction to transformation geometry. Use of Geometer's Sketchpad.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: one MATH credit.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2P50.

MATH 2P93

Great Moments in Mathematics I

Triumphs in mathematical thinking emphasizing many cultures up to 1000 AD. Special attention is given to analytical understanding of mathematical problems from the past, with reference to the stories and times behind the people who solved them. Students will be encouraged to match wits with great mathematicians by solving problems and developing activities related to their discoveries.

Lectures, 4 hours per week.

Prerequisite: one MATH credit.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2P51.

MATH 2P98

Applied Statistics

Single-factor and factorial experimental design methods; nested-factorial experiments. Simple and multiple linear regression methods, correlation analysis, indicator regression; regression model building and transformations. Contingency tables, binomial tests, nonparametric rank tests. Simple random and stratified sampling techniques, estimation of sample size and related topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 1F92 or 1P98.

MATH 3F40

Mathematics Integrated with Computers and Applications III

Advanced applications of mathematics involving computers. Topics may include deterministic models; equilibrium; optimal control; probabilistic models; models from physics such as the n-body problem, the heat equation and finite element methods, and the driven pendulum; image compressing; genetic algorithms; neural nets; optimization and stochastic processes. Use of Maple.

Lectures, lab, 4 hours per week.

Prerequisites: MATH 2F40, 2P03 and 2P82.

Co-requisite: MATH 2P72.

Note: projects demonstrating creative application of the course content.

MATH 3F65

Mathematical Methods for Computer Science

Applied probability, Markov chains, Poisson and exponential processes, renewal theory, queuing theory, applied differential equations. Networks, graph theory, reliability theory, NP-completeness.

Lectures, 3 hours per week.

Prerequisites: MATH 1P01 or 1P97; MATH 1P12, 1P66 and 1P67.

MATH 3P03

Real Analysis

Approximation of functions by algebraic and trigonometric polynomials (Taylor and Fourier series); Weierstrass approximation theorem; Riemann integral of functions on Rⁿ, the Riemann-Stieltjes integral on R; improper integrals; Fourier transforms.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Prerequisite: MATH 2P04.

MATH 3P04

Complex Analysis

Algebra and geometry of complex numbers, complex functions and their derivatives; analytic functions; harmonic functions; complex exponential and trigonometric functions and their inverses; contour integration; Cauchy's theorem and its consequences; Taylor and Laurent series; residues.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Prerequisite: MATH 2F05 or 2P03.

MATH 3P08

Advanced Differential Equations

Linear second-order differential equations. Integral transform methods, series solutions, special functions (Bessel, Legendre, Laguerre, Hermite). Boundary value problems and general Sturm-Liouville theory, orthogonal functions, series expansions. Linear autonomous systems and phase plane analysis. Emphasis on applications to physical sciences. Use of Maple.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2F05 or 2P08.

MATH 3P09

Partial Differential Equations

First-order equations and method of characteristics. Second-order linear equations, initial and boundary value problems for the heat equation, wave equation, and Laplace equation. Fourier series, cylindrical (Bessel) and spherical (Legendre) harmonic series. Eigenfunction problems and normal modes. Nonlinear wave equations. Emphasis on applications to physical sciences. Use of Maple.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2F05 or 2P08.

MATH 3P12

Applied Algebra

Group theory with applications. Topics include modular arithmetic, symmetry groups and the dihedral groups, subgroups, cyclic groups, permutation groups, group isomorphism, frieze and crystallographic groups, Burnside's theorem, cosets and Lagrange's theorem, direct products and cryptography.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2P12 or permission of the instructor.

MATH 3P13

Abstract Algebra

Further topics in group theory: normal subgroups and factor groups, homomorphisms and isomorphism theorems, structure of finite abelian groups. Rings and ideals; polynomial rings; quotient rings. Division rings and fields; field extensions; finite fields; constructability.

Lectures, 3 hours per week; lab/tutorial 1 hour per week.

Prerequisite: MATH 3P12.

MATH 3P60

Numerical Methods

Survey of computational methods and algorithms; basic concepts (algorithm, computational cost, convergence, stability); roots of functions; linear systems; numerical integration and differentiation; Runge-Kutta method for ordinary differential equations; finite-difference method for partial differential equations; fast Fourier transform; Monte Carlo methods. Implementation of numerical algorithms in a scientific programming language.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisites: MATH 1P02 and 1P12 or permission of the instructor.

MATH 3P72

Continuous Optimization

Problems and methods in non-linear optimization. Classical optimization in Rⁿ: inequality constraints, Lagrangian, duality, convexity. Non-linear programming. Search methods for unconstrained optimization. Gradient methods for unconstrained optimization. Constrained optimization. Dynamic programming.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: MATH 2F05 or 2P03; MATH 2P72 (2P60).

*MATH 3P73

Game Theory

(also offered as ECON 3P73)

Representation of Games. Strategies and payoff functions. Static and dynamic games of complete or incomplete information. Equilibria concepts: Nash, Bayesian Nash, and Perfect Bayesian Nash equilibria. Convexity concepts, fixed points for correspondences and minimax. Core and Shapley value of a game. Refinements and Applications.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2P72 or ECON 3P91.

MATH 3P75

Theory of Financial Mathematics

Probability, Brownian motion, martin-gales, Markov processes, differential equations, finite difference and tree models used in financial mathematics of options; stocks; one-dimensional Ito processes, Black-Scholes for both constant and non-constant inputs, continuous time hedging, valuing American and exotic options.

Lectures, lab, 4 hours per week.

Prerequisites: MATH 1P12 and 2P82; MATH 2F05 or MATH 2P03 and 2P08.

MATH 3P81

Experimental Design

Analysis of variance; single-factor experiments; randomized block designs; Latin squares designs; factorial designs; 2^f and 3^f factorial experiments; fixed, random and mixed models; nested and nested-factorial experiments; Taguchi experiments; split-plot and confounded in blocks factorial designs; factorial replication; regression models; computational techniques and use of SAS, Maple or other statistical packages; related topics.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisite: MATH 2P82.

MATH 3P82

Regression Analysis

Simple and multiple linear regression and correlation, measures of model adequacy, residual analysis, weighted least squares, polynomial regression, indicator variables, variable selection and model building, multicollinearity, time series, selected topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: MATH 2P12 and 2P82 or permission of the instructor.

MATH 3P85

Mathematical Statistics II

Review of distributional theory. Convergence types. Some special and limiting distributions. Review of point and interval estimations. Efficiency, sufficiency, robustness and completeness. Bayesian estimations, credible intervals, prediction intervals. Basic theory of hypotheses testing: Neyman-Pearson lemma, likelihood ratio test, chi-square test, Test of stochastic independence. Normal models: quadratic forms, noncentral chi-square and noncentral Fdistributions. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 2P82

MATH 3P86

Applied Multivariate Statistics

Matrix algebra and random vector, sample geometry and random sampling, multivariate normal distribution, inference about mean, comparison of several multivariate means, multivariate linear regression model, principle components, factor analysis, covariance analysis, canonical correlation analysis, discrimination and classification, cluster analysis, computational techniques and use of SAS, Maple or other statistical packages and related topics.

Lectures, 3 hours per week; lab 1 hour per week.

Prerequisites: MATH 2P12 and 3P85 or permission of the instructor.

MATH 3P90

Euclidean and Non Euclidean Geometry II

Topics in Euclidean and non-Euclidean geometry chosen from the classification of isometries in selected geometries, projective geometry, finite geometries and axiometic systems for plane Euclidean geometry.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: MATH 1P12 and 2P90 (2P50).

Completion of this course will replace previous assigned grade and credit obtained in MATH 3P50.

MATH 3P91

Mathematics at the Junior/Intermediate/Senior Level

A treatment of mathematics and its teaching and learning at the junior, intermediate and senior levels. A major portion of the course will involve directed projects.

Lectures, seminar, 4 hours per week.

Restriction: open to BSc/BEd MATH majors with a minimum of 9.0 overall credits.

Prerequisite: three MATH credits.

Note: students in the minor programs for Teachers may register. Contact the Mathematics Department.

MATH 3P93

Great Moments in Mathematics II

The development of modern mathematics from medieval times to the present. The course includes Fibonacci's calculation revolution, the disputes over cubic equations, Pascal and probability, Fermat's last theorem, Newton and Calculus, Euler and infinite series, set theory and the possibilities of inconsistencies in mathematics.

Lectures, 4 hours per week.

Prerequisites: MATH 1P02, 1P12 and 2P93.

Completion of this course will replace previous assigned grade and credit obtained in MATH 3P51.

MATH 3P97

Introductory Topology

Introduction to metric and topological spaces; connectedness, completeness, countability axioms, separation pro-perties, covering properties, metrization of topological spaces.

Lectures, 4 hours per week.

Prerequisites: MATH 2P04; MATH 2P12 and 2P13 or MATH 3P12 and 3P13.

MATH 3P98

Functional Analysis

Introduction to the theory of normed linear spaces, fixed-point theorems, Stone-Weierstrass approximation on metric spaces and preliminary applications on sequence spaces.

Lectures, 4 hours per week.

Prerequisite: MATH 3P97.

MATH 4F90

Honours Project

Independent project in an area of pure or applied mathematics, or mathematics education.

Restriction: open to MATH (single or combined) majors with either a minimum of 14.0 credits, a minimum 70 percent major average and a minimum 60 percent non-major average or approval to year 4 (honours) and permission of the instructor.

Note: carried out under the supervision of a faculty member. The supervisor must approve the topic in advance. Presentation of the project is required.

MATH 4P03

Advanced Real Analysis

Lebesgue integration on Rⁿ; differentiation and absolute continuity; Fubini's theorem; L^p spaces, elementary theory of Banach and Hilbert spaces.

Lectures, 3 hours per week.

Prerequisite: MATH 3P03.

MATH 4P05

Introduction to Wavelets

Wavelets as an orthonormal basis for Rⁿ, localized in space and frequency; wavelets on the real line; image compression (fingerprint files); wavelet-Galerkin numerical solution of differential equations with variable coefficients.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: MATH 2P08 and 2P12.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4P04.

MATH 4P07

Topics in Differential Equations

Topics may include ordinary differential equations: existence and uniqueness theory, strange attractors, chaos, singularities. Partial differential equations: Cauchy-Kovalevski theorem, well-posedness of classical linear heat equation and wave equation, weak solutions, global existence, uniqueness and asymptotic behaviour.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P08.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F08.

*MATH 4P09

Solitons and Nonlinear Wave Equations

(also offered as PHYS 4P09)

Introduction to solitons. Travelling waves, nonlinear wave and evolution equations (Korteweg de Vries, Bousinesq, nonlinear Schrodinger, sine-Gordon), soliton solutions and their interaction properties, Lax pairs and construction of single and multi soliton solutions.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P09 or 3F40.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F08.

MATH 4P11

Topics in Groups

Advanced topics from group theory. Topics may include the Sylow theorems, free groups, nilpotent and solvable groups and some simple Lie groups.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P13.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F10.

MATH 4P13

Topics in Rings and Modules

Advanced topics from ring theory. Topics may include radicals, Wedderburn-Artin theorems, modules over rings and some special rings.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P13.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F10.

MATH 4P14

Advanced Mathematical Structures

Topics may include modules, homological algebra, group algebra, algebraic geometry, lattice theory, graph theory and logic.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P13 or permission of the Department.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F10 or 4P12.

*MATH 4P61

Theory of Computation

(also offered as COSC 4P61)

Regular languages and finite state machines: deterministic and non-deterministic machines, Kleene's theorem, the pumping lemma, Myhill-Nerode Theorem and decidable questions. Context-free languages: generation by context-free grammars and acceptance by pushdown automata, pumping lemma, closure properties, decidability. Turing machines: recursively enumerable languages, universal Turing machines, halting problem and other undecidable questions.

Lectures, 3 hours per week.

Restriction: open to COSC (single or combined) majors.

Prerequisite: MATH 1P67.

Note: MATH students may take this course with permission of Department.

MATH 4P71

Combinatorics

Review of basic enumeration including distribution problems, inclusion-exclusion and generating functions. Polya theory. Finite fields. Orthogonal Latin squares, affine and projective planes. Coding theory and cryptography.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Restriction: permission of the Department.

Note: while no specific course is an essential prerequisite, students should have competence in abstraction equivalent to that obtained by successful completion of MATH 3P12.

MATH 4P81

Sampling Theory

Theory of finite population sampling; simple random sampling; sampling proportion; estimation of sample size; stratified random sampling; optimal allocation of sample sizes; ratio estimators; regression estimators; systematic and cluster sampling; multi-stage sampling; errors in surveys; computational techniques and use of SAS, Maple or other statistical packages and related topics.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisite: MATH 3P85 or permission of the instructor.

MATH 4P82

Nonparametric Statistics

Order statistics, rank statistics, methods based on the binomial distribution, contingency tables, Kolmogorov Smirnov statistics, nonparametric analysis of variance, nonparametric regression, comparisons with parametric methods. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P85 or permission of the instructor.

MATH 4P84

Topics in Stochastic Processes and Models

Topics may include general stochastic processes, Markov chains and processes, renewal process, branching theory, stationary processes, stochastic models, Monte Carlo simulations and related topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P85 or permission of the instructor.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F83.

MATH 4P85

Topics in Advanced Statistics

Topics may include advanced topics in stochastic processes and models, queueing theory, time series analysis, multivariate analysis, Bayesian statistics, advanced methods and theory in statistical inference, and related topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P85 or permission of the instructor.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F83.

MATH 4P92

Topics in Number Theory and Cryptography

Topics may include algebraic number theory, analytic number theory and cryptography.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Restriction: permission of the Department.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F91.

MATH 4P93

Topics in Topology and Dynamical Systems

Topics may include point set topology, differential geometry, algebraic topology and dynamical systems.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite: MATH 3P97 or permission of the Department.

Completion of this course will replace previous assigned grade and credit obtained in MATH 4F91.

*MATH 4P94

General Relativity and Black Holes

(also offered as PHYS 4P94)

Review of Special Relativity and Minkowski space-time. Introduction to General Relativity theory including gravitation and the space-time metric, light cones, horizons, asymptotic flatness; energy-momentum of particles and light rays (geodesics). Static black holes (Schwarzschild metric), properties of light rays and particle orbits. Rotating black holes (Kerr metric).

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisites: MATH 2F05, PHYS 2P20 and 2P50 or permission of the instructor.