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71 - 80 of 165 results for: MATH

MATH 220: Partial Differential Equations of Applied Mathematics (CME 303)

First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: foundation in multivariable calculus and ordinary differential equations.
Terms: Aut | Units: 3
Instructors: Ryzhik, L. (PI)

MATH 226: Numerical Solution of Partial Differential Equations (CME 306)

Hyperbolic partial differential equations: stability, convergence and qualitative properties; nonlinear hyperbolic equations and systems; combined solution methods from elliptic, parabolic, and hyperbolic problems. Examples include: Burger's equation, Euler equations for compressible flow, Navier-Stokes equations for incompressible flow. Prerequisites: MATH 220A or CME 302.
Terms: Spr | Units: 3
Instructors: Ying, L. (PI)

MATH 227: Partial Differential Equations and Diffusion Processes

Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites: MATH 131 and MATH 136/ STATS 219, or equivalents.
Terms: Win | Units: 3
Instructors: Menz, G. (PI)

MATH 228: Stochastic Methods in Engineering (CME 308)

Review of basic probability; Monte Carlo simulation; state space models and time series; parameter estimation, prediction, and filtering; Markov chains and processes; stochastic control; and stochastic differential equations. Examples from various engineering disciplines. Prerequisites: exposure to probability; background in real variables and analysis.
Terms: Spr | Units: 3

MATH 230A: Theory of Probability (STATS 310A)

Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: 116, MATH 171.
Terms: Aut | Units: 2-4
Instructors: Diaconis, P. (PI)

MATH 230B: Theory of Probability (STATS 310B)

Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,nn(v) ergodic theory. Prerequisite: 310A or MATH 230A.
Terms: Win | Units: 2-3
Instructors: Dembo, A. (PI)

MATH 230C: Theory of Probability (STATS 310C)

Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Invariance principle, random walks, LIL and functional CLT. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Prerequisite: 310B or MATH 230B.
Terms: Spr | Units: 2-4

MATH 231: Orthogonal Polynomials and the Moment Problem

Orthogonal polynomials in one variable (three term recurrence, Favard's theorem, distribution of zeros Verblunsky coefficients). Classical examples (Hermite, Chebychev, Jacobi, Meixner, Askey-Wilson). Applications in probability (Markov chains), Statistics (multivariate distributions with given margins), Numerical analysis (Gaussian Quadriture), Combinatorics (combinatorial interpretation of the classical orthogonal polynomials). The moment problem on R (when is a measure determined by its moments, what happens if not?). Multivariate orthogonal polynomials (with an introduction to symmetric function theory). Connections to group representations.
Terms: Spr | Units: 3
Instructors: Diaconis, P. (PI)

MATH 232: Topics in Probability: Percolation Theory

An introduction to some of the most important theorems and open problems in percolation theory. Topics include some of the difficult early breakthroughs of Kesten, Menshikov, Aizenman and others, and recent fields-medal winning works of Schramm, Lawler, Werner and Smirnov. Prerequisites: graduate-level probability. Offered every 1-2 years.
Last offered: Winter 2013 | Repeatable for credit

MATH 236: Introduction to Stochastic Differential Equations

Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: 136 or equivalent and differential equations.
Terms: Win | Units: 3
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