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 Áù¾ÅÉ«Ìà Calendar 2012-2013 COURSES OF INSTRUCTION Course Descriptions A Applied Mathematics AMAT
Applied Mathematics AMAT
AMAT 217 219 307 309 311 411 413 415 425 433 481 491 493 501 503 505 507 509 581 583 601 605 613 617 621 643 651 653 671 673 677 681 683
 

Instruction offered by members of the Department of Mathematics and Statistics in the Faculty of Science.

Department Head - M. Lamoureux

Note: For listings of related courses, see Actuarial Science, Mathematics, Pure Mathematics and Statistics.
 
Junior Courses
 
Applied Mathematics 217       Calculus for Engineers and Scientists
Functions, limits, continuity, derivatives, Mean Value Theorem, integrals, Fundamental Theorem of Calculus, applications to the physical sciences.
Course Hours: H(3-1T-1.5)
Prerequisite(s): A grade of 70% or higher in Pure Mathematics 30 and credit in Mathematics 31; or admission to the Faculty of Engineering including credit in Pure Mathematics 30 and Mathematics 31.
Antirequisite(s): Credit for more than one of Applied Mathematics 217 and Mathematics 249, 251, or 281 will not be allowed.
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Applied Mathematics 219       Multivariable Calculus for Engineers
Techniques of integration, double and triple integrals, partial derivatives, applications.
Course Hours: H(3-1T-1.5)
Prerequisite(s): Applied Mathematics 217; or Mathematics 249 or 251 or 281 plus Mathematics 117; or consent of Applied Mathematics Division.
Antirequisite(s): Credit for more than one of Mathematics 253, 283 or Applied Mathematics 219 will not be allowed.
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Senior Courses
 
Applied Mathematics 307       Differential Equations for Engineers
Definition, existence and uniqueness of solutions, first and second order equations with applications, series solutions about regular points and singular points, special functions. Laplace transform, systems of equations.
Course Hours: H(3-1.5T)
Prerequisite(s): Mathematics 211 or 213; and Applied Mathematics 219 or Mathematics 253 or 283.
Antirequisite(s): Credit for both Applied Mathematics 307 and 311 will not be allowed.
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Applied Mathematics 309       Vector Calculus for Engineers
Functions of several variables, chain rule and differentials. Vector calculus, line, surface and volume integrals, Green's, Gauss' and Stokes' theorems. Students will complete a project using a computer algebra system.
Course Hours: H(3-1.5T)
Prerequisite(s): Applied Mathematics 219 or Mathematics 114; and Mathematics 253 or 283.
Antirequisite(s): Credit for more than one of Applied Mathematics 309 and Mathematics 353 or 381 will not be allowed.
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Applied Mathematics 311       Differential Equations I
Classification of ordinary differential equations, first and second order equations with applications, series solutions about regular points and singular points, special functions, Laplace transform.
Course Hours: H(3-1T)
Prerequisite(s): Mathematics 253 or 283 or Applied Mathematics 219.
Antirequisite(s): Credit for both Applied Mathematics 311 and 307 will not be allowed.
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Applied Mathematics 411       Differential Equations II
Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, Sturm-Liouville problems, systems of equations, phase portraits, stability.
Course Hours: H(3-1T)
Prerequisite(s): One of Applied Mathematics 311 or 307, and one of Mathematics 331, 353, 381, Applied Mathematics 309, or consent of the Division.
Notes: It is recommended that students complete Pure Mathematics 435 or 455 before taking this course.
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Applied Mathematics 413       Introduction to Partial Differential Equations
Orthogonal sets of functions, Fourier series, solution of potential equation, heat equation and wave equation. Numerical solution of partial differential equations.
Course Hours: H(3-1T)
Prerequisite(s): One of Mathematics 353, 381, Applied Mathematics 307 or 309 or 311, Mathematics 331; or consent of the Division.
Antirequisite(s): Credit for both Applied Mathematics 413 and 415 will not be allowed for the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, and Statistics programs.
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Applied Mathematics 415       Mathematical Methods
Mathematical analysis of linear systems. Fourier and Laplace transforms, applications and numerical methods. Functions of a complex variable and applications.
Course Hours: H(3-1T)
Prerequisite(s): One of Applied Mathematics 311, 307, Mathematics 331, 353, 381, or Applied Mathematics 309.
Antirequisite(s): Credit will be not be allowed for more than one of Applied Mathematics 415 and 413 for the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, and Statistics programs.
Notes: Credit in an introductory Computer Science course prior to taking Applied Mathematics 415 is highly recommended.
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Applied Mathematics 425       Introduction to Optimization
Examples of optimization problems. Quadratic forms, minimum energy and distance. Least squares, generalized inverse. Location and classification of critical points. Variational treatment of eigenvalues. Lagrange multipliers. Linear programming.
Course Hours: H(3-1T)
Prerequisite(s): Mathematics 311; and Mathematics 353 or 381 or Applied Mathematics 309 or Mathematics 331.
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Applied Mathematics 433       Mathematical Methods in Physics
Fourier analysis. Laplace Transforms. Partial differential equations. Complex analysis. Residue integrals.  Extensive physical applications.
Course Hours: H(3-1T)
Prerequisite(s): Applied Mathematics 307 or 311; one of Applied Mathematics 309 or Mathematics 353 or 381 or 331; and Mathematics 211 or 213.
Antirequisite(s): Credit will be not be allowed for Applied Mathematics 433 for the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, and Statistics programs.
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Applied Mathematics 481       Introduction to Mathematical Finance
This course is an introduction to the fundamental concepts of mathematical finance in an elementary setting. Topics include: risk, return, no arbitrage principle; basic financial derivatives: options, forwards and future contracts; risk free assets, time value of money, zero coupon bonds; risky assets, binomial tree model, fundamental theorem of asset pricing; portfolio management and capital asset pricing model; no arbitrage pricing of financial derivatives; hedging.
Course Hours: H(3-1T)
Prerequisite(s): Mathematics 321 or consent of the division.
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Applied Mathematics 491       Numerical Analysis I
Interpolation and approximation, numerical integration, numerical methods for the solution of nonlinear equations, systems of linear equations and the eigenvalue problem.
Course Hours: H(3-1T)
Prerequisite(s): Mathematics 311 and 381 or 349 and 353 or Applied Mathematics 309 and Computer Science 231 or 217; or consent of the Division.
Antirequisite(s): Not open to students with credit in Computer Science 491.
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Applied Mathematics 493       Numerical Analysis II
Numerical differentiation, numerical solution of ordinary and partial differential equations.
Course Hours: H(3-1T)
Prerequisite(s): Applied Mathematics 311, 413, and 491 or Computer Science 491.
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Applied Mathematics 501       Seminar in Applied Mathematics
Topics will be chosen according to the interests of instructors and students and could include analysis of optimization algorithms, approximation theory, control theory, differential equations, mathematical physics.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Division.
MAY BE REPEATED FOR CREDIT
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Applied Mathematics 503       The Mathematics of Wavelets, Signal and Image Processing
Continuous and discrete Fourier transforms, the Fast Fourier Transform, wavelet transforms, multiresolution analysis and orthogonal wavelet bases, and applications.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 491 or Computer Science 491.
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Applied Mathematics 505       Calculus on Manifolds
Integral and differential calculus on manifolds including tensor fields, covariant differentiation, Lie differentiation, differential forms, Frobenius' theorem, Stokes' theorem, flows of vector fields.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 445 or 545; and one of Applied Mathematics 311 or 307; or consent of the Division.
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Applied Mathematics 507       Introduction to Relativity Theory
Mathematical theories of space and time. Special Relativity. Electro-dynamics. General Relativity.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 505 or consent of the Division.
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Applied Mathematics 509       Analytical Dynamics
Symplectic geometry, Hamilton's equation, Hamilton-Jacobi theory, constraints and reduction.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 505 or consent of the Division.
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Applied Mathematics 581       Stochastic Calculus for Finance
Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanov’s theorem, Black-Scholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, value-at-risk and risk management.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 481.
Antirequisite(s): Credit for both Applied Mathematics 581 and 681 will not be allowed.
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Applied Mathematics 583       Computational Finance
Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 481 and 491.
Antirequisite(s): Credit for both Applied Mathematics 583 and 683 will not be allowed.
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Graduate Courses

In addition to the prerequisites listed below, consent of the Applied Mathematics Division is a prerequisite for all graduate courses in Applied Mathematics.
 
Applied Mathematics 601       Topics in Applied Mathematics
Topics will be chosen according to the interests of instructors and students.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Division.
MAY BE REPEATED FOR CREDIT
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Applied Mathematics 605       Differential Equations III
Linear systems, classification. Nonlinear systems: Existence and uniqueness. Flow and one parameter groups of transformations. Stability theory. Hyperbolicity, Unstable/Stable/Center manifold theorems. Poincare-Bendixson.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 411 and Pure Mathematics 445 or 545 or equivalents.
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Applied Mathematics 613       Partial Differential Equations II
Fundamental solutions, integral equations, eigenvalue problems, non-linear problems.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Division.
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Applied Mathematics 617       Analysis IV
Analysis in abstract spaces. Function spaces.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 545.
Antirequisite(s): Credit for Applied Mathematics 617 and Pure Mathematics 617 will not be allowed.
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Applied Mathematics 621       Research Seminar
Reports on studies of the literature or of current research.
Course Hours: Q(2S-0)
Notes: All graduate students in Mathematics and Statistics are required to participate in one of Applied Mathematics 621, Pure Mathematics 621, Statistics 621 each semester.
MAY BE REPEATED FOR CREDIT NOT INCLUDED IN GPA
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Applied Mathematics 643       Perturbation Theory
Perturbation problems for ordinary differential equations, matrices and more general operators. Applications. Methods will be motivated by discussion of physical problems.
Course Hours: H(3-0)
Prerequisite(s): Familiarity with complex variables, linear algebra and differential equations.
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Applied Mathematics 651       Monte Carlo Methods for Quantitative Finance
Fundamental concepts of Monte Carlo methods; review of quantitative finance; random number generation; simulating stochastic differential equations; variance reduction; quasi-Monte Carlo methods; computing sensitivities; early exercise options; Levy processes and other price models; applications to risk management.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Department.
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Applied Mathematics 653       Introduction to Levy Processes with Applications
Infinite divisibility, Levy processes (LP), the Levy-Khintchine formula; examples of LP; Poisson integration, the Levy-Ito decomposition, subordinators; Markov processes, semi-groups and generators of LP; Ito-formula for LP, quadratic variation; LP as time-changed Brownian motion, change of measure (Girsanov theorem); stochastic differential equations driven by LP; Feynman-Kac formula  and martingale problem for LP; applications of LP; simulation of LPs.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Department.
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Applied Mathematics 671       Numerical Linear Algebra
Iterative and elimination methods for linear systems of equations, determination of eigenvalues, linear and convex programming.
Course Hours: H(3-0)
Prerequisite(s): Mathematics 411 and Applied Mathematics 491.
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Applied Mathematics 673       Approximation Theory
Existence, uniqueness of minimal solutions, Haar systems, characterization by alternation, Remez algorithm, monotone operators, spline approximation.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 491; and Pure Mathematics 435 or 455.
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Applied Mathematics 677       Numerical Solution of Partial Differential Equations
Explicit and implicit methods for PDE, difference equations.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 311 and 491.
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Applied Mathematics 681       Stochastic Calculus for Finance
Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanov’s theorem, Black-Scholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, value-at-risk and risk management.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 481.
Antirequisite(s): Credit for both Applied Mathematics 681 and 581 will not be allowed.
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Applied Mathematics 683       Computational Finance
Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 481 and 491.
Antirequisite(s): Credit for both Applied Mathematics 683 and 583 will not be allowed.
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In addition to the numbered and titled courses shown above, the department offers a selection of advanced level graduate courses specifically designed to meet the needs of individuals or small groups of students at the advanced doctoral level. These courses are numbered in the series 800.01 to 899.99. Such offerings are, of course, conditional upon the availability of staff resources.