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

Notes:

For listings of related courses, see Actuarial Science, Mathematics, Pure Mathematics and Statistics.

Effective Fall 2014, Mathematics 265, 267, 367, Mathematics 275, 277, 375 and 377 replaced respectively Mathematics 251, 253, 353, Applied Mathematics 217, 219, 307 and 309 and serves as prerequisites for appropriate courses. In some special cases, Mathematics 267 replaces Mathematics 349 or 353. For these and other deviations from the general rule, see individual course entries for details. Mathematics 267 supplemented by Mathematics 177 will be accepted as equivalent to Mathematics 277.

Senior Courses

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:3 units; H(3-1T) Prerequisite(s):Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219. Antirequisite(s):Credit for Applied Mathematics 311 and either Mathematics 375 or Applied Mathematics 307 will not be allowed.

Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, Sturm-Liouville problems, systems of equations, phase portraits, stability. Course Hours:3 units; H(3-1T) Prerequisite(s): One of Mathematics 375, Applied Mathematics 311 or 307; and one of Mathematics 331, 353, 367, 377, 381, Applied Mathematics 309. Notes: It is recommended that students complete Mathematics 335 or 355 or Pure Mathematics 435 or 455 before taking this course.

First order partial differential equations, Sturm-Liouville systems, Fourier series, Double Fourier series, Fourier integrals, Applications to boundary value problems in bounded and unbounded domains, Bessel function with applications. Course Hours:3 units; H(3-1T) Prerequisite(s):Mathematics 375 or 331; or one of: Mathematics 353 or 367 or 381 or Applied Mathematics 309 and one of: Applied Mathematics 307 or 311. Antirequisite(s):Credit for Applied Mathematics 413 and 415 will not be allowed in the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, Statistics programs, and the Natural Sciences program with a Concentration in General Mathematics.

Mathematical analysis of linear systems. Fourier and Laplace transforms, applications and numerical methods. Functions of a complex variable and applications. Course Hours:3 units; H(3-1T) Prerequisite(s):One of Applied Mathematics 311, 307, Mathematics 331, 353, 367, 375, 381, or Applied Mathematics 309. Antirequisite(s):Credit for Applied Mathematics 415 and 413 in the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, Statistics, and the Natural Sciences program with a Concentration in General Mathematics will be not be allowed. Notes:Credit in an introductory Computer Science course is highly recommended as a prerequisite.

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:3 units; H(3-1T) Prerequisite(s):Mathematics 311 or 313; and one of Mathematics 331 or 353 or 367 or 377 or 381 or Applied Mathematics 309.

Fourier analysis. Laplace Transforms. Partial differential equations. Complex analysis. Residue integrals. Extensive physical applications. Course Hours:3 units; H(3-1T) Prerequisite(s):Mathematics 211 or 213 and one of Applied Mathematics 307 or 311 or Mathematics 375; and one of Applied Mathematics 309 or Mathematics 353 or 367 or 377 or 381 or 331. Antirequisite(s):Credit for Applied Mathematics 433 in the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, Statistics programs, and the Natural Sciences program with a Concentration in General Mathematics will be not be allowed.

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:3 units; H(3-1T) Prerequisite(s):Mathematics 321 or Statistics 321.

Interpolation and approximation, numerical integration, numerical methods for the solution of non-linear equations, systems of linear equations and the eigenvalue problem. Course Hours:3 units; H(3-1T) Prerequisite(s):Computer Science 231 or 217; and Mathematics 311 or 313; and one of Mathematics 381 or 367 or 377 or 349 and 353 or Applied Mathematics 309. Antirequisite(s):Credit for Applied Mathematics 491 and Computer Science 491 will not be allowed.

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:3 units; H(3-0) Prerequisite(s): Consent of the Department. MAY BE REPEATED FOR CREDIT

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:3 units; H(3-0) Prerequisite(s):Applied Mathematics 491 or Computer Science 491.

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:3 units; H(3-0) Prerequisite(s): Pure Mathematics 445 or 545; and one of Mathematics 375 or Applied Mathematics 311 or 307.

Mathematical theories of space and time. Special Relativity. Electro-dynamics. General Relativity. Course Hours:3 units; H(3-0) Prerequisite(s):Applied Mathematics 505.

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:3 units; H(3-0) Prerequisite(s):Applied Mathematics 481. Antirequisite(s):Credit for Applied Mathematics 581 and 681 will not be allowed.

Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods. Course Hours:3 units; H(3-0) Prerequisite(s):Applied Mathematics 481 and 491. Antirequisite(s):Credit for Applied Mathematics 583 and 683 will not be allowed.