Introductory Statistics
Types of scales; frequency distribution, mean, mode, median and measures of dispersion; elements of probability theory, probability distributions, nonparametric tests; normal, chi-squared, t- and F-distributions; means and variance tests; analysis of variance, correlation and regression, applications and use of a computer package.
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.
Calculus I
Applications of differential calculus, linearization and optimization; antiderivatives, definite integrals, the fundamental theorem of calculus, numerical integration; logarithms, exponentials, and inverse trigonometric functions, ordinary differential equations and their applications, improper integrals, the use of computer algebra systems.
Lectures, 4 hours per week; lab/tutorial, 1 hour per week.
Prerequisites: two OAC mathematics credits including OAC calculus or two grade 12 mathematics credits including MCB4U, or permission of instructor.
Calculus II
Applications of the definite integral: areas, volumes, and work; infinite series, Taylor's theorem, Taylor series; functions of several variables: and partial differentiation, limits and continuity, gradients, extrema with and without constraints, double integrals; the use of computer algebra systems to solve systems of equations, plot surfaces, compute partial derivatives and evaluate multiple integrals.
Lectures, 4 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 1P01.
Linear Algebra I
Introduction to finite dimensional real vector spaces; systems of linear equations: Gaussian elimination, 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 Rm, eigenvalues and eigenvectors; selected applications and use of a computer algebra system.
Lectures, 4 hours per week.
Prerequisites: two grade 12 or OAC mathematics credits or permission of instructor.
Note: MCB4U recommended.
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.
Prerequisites: MATH 1P01 and 1P12.
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 or OAC mathematics credit.
Note: MCB4U recommended. Designed for students in Computer Science.
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.
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.
Basic Statistical Methods
Descriptive statistics; probability distributions, estimation; hypothesis testing; nonparametric tests; 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.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 1P02.
Completion of this course will replace previous assigned grade in MATH 2P03.
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.
Lectures, lab, 4 hours per week.
Prerequisites: MATH 1P02 and 1P40.
Calculus III
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 .
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.
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 behavior; modeling and applications of differential equations.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisites: MATH 1P02 and 1P12.
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 application.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 1P12.
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.
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 (Advanced)/OAC or university mathematics course.
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 12(OAC) mathematics credits or permission of the instructor.
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.
Completion of this course will replace previous assigned grade in MATH 2P60.
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.
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. Discrete and continuous distributions. Transforming random variables, central limit theorem and its applications.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2F05 or 2P03.
Note: MATH 2F05 or 2P03 may be taken concurrently.
Mathematical Statistics
Random sample, sampling distributions of sample mean and variance. Law of large numbers and central limit theorem. Chi-square, t, F distributions. Order statistics. Unbiased, consistent, efficient and robust estimators. Moments, maximum likelihood and Bayesian estimation. Confidence intervals. Hypothesis testing,, power function, Neyman-Pearson Lemma, likelihood-ratio test. Linear regression and correlation analysis. Analysis of variance. Nonparametric methods.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2P81.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: one MATH credit.
Completion of this course will replace previous assigned grade in MATH 2P50.
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 in MATH 2P51.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 1F92 or 1P98.
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.
Lectures, lab, 4 hours per week.
Prerequisites: MATH 2P03, 2F40 and 2P82.
Co-requisite: MATH 2P72.
Note: projects demonstrating creative application of the course content.
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; tutorial, 1 hour per week.
Prerequisites: MATH 1P12, 1P66, 1P67 and 1P97.
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.
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.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2F05 or 2P08.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2F05 or 2P08.
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 1P12.
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.
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.
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).
Game Theory
(also offered as ECON 3P73)
Applications of modelling, review of elementary decision theory and subjective probability theory, game theory (Nash equilibrium, two player NZS games, Nash cooperative solution), Shapley value, voting power, selected cases from economics and other applications.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2P72 (2P60) or ECON 3P91.
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.
Experimental Design
Analysis of variance; single-factor experiments; randomized block designs; Latin squares designs; factorial designs; 2f and 3f 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 computer packages, related topics.
Lectures, 3 hours per week; lab, 1 hour per week.
Prerequisite: MATH 2P82.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2P82.
Euclidean and NonEuclidean 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 in MATH 3P50.
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.
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 in MATH 3P51.
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.
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.
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.
Advanced Real Analysis
Lebesgue integration on Rn; differentiation and absolute continuity; Fubini's theorem; Lp spaces, elementary theory of Banach and Hilbert spaces.
Lectures, 3 hours per week; tutorial, 1 hour per week.
Prerequisite: MATH 3P03.
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 in MATH 4P04.
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 in MATH 4F08.
Topics in Nonlinear Partial Differential Equations
Topics may include nonlinear partial differential equations, exact solutions and symmetry methods, global existence of solutions, finite element methods. Quasilinear elliptic equations and variational inequalities. Nonlinear wave equations, solitons and solitary wave solutions. Integrable systems and their properties. Field equations in mathematical physics.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 3P09.
Completion of this course will replace previous assigned grade in MATH 4F08.
Topics in Groups and Rings
Advanced topics from group theory and ring theory. Topics may include the Sylow theorems, free groups, nilpotent and solvable groups and Galois theory.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 3P13.
Completion of this course will replace previous assigned grade in MATH 4F10.
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 in MATH 4F10.
Theory of Computation
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; tutorial, 1 hour per week.
Restriction: open to COSC (single or combined) majors.
Prerequisite: MATH 1P67.
Note: MATH students may take this course with permission of Department.
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.
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 computer packages; related topics.
Lectures, 3 hours per week; lab, 1 hour per week.
Prerequisite: MATH 2P82.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2P82.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2P82.
Completion of this course will replace previous assigned grade in MATH 4F83.
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.
Lectures, 3 hours per week; lab/tutorial, 1 hour per week.
Prerequisite: MATH 2P82.
Completion of this course will replace previous assigned grade in MATH 4F83.
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 in MATH 4F91.
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 in MATH 4F91.