41 - 50 of 71 results for: cme

Computational symbolic mathematics is a one-unit hands-on seminar course on the use of sophisticated computer algebra systems for addressing mathematical problems that are primarily or entirely symbolic (rather than numerical). Examples will come from the undergraduate curriculum including calculus, differential equations, linear algebra, probability and statistics, and symbolic logic. Students will program in Mathematica but the principles presented apply as well to Mathics, SymPy, Matlab's Symbolic math toolbox, and other systems. (No prior experience with symbol-manipulation programs is assumed.) Students will learn how to exploit special functionality of symbol-manipulating programs, such as Gr¿bner bases. Class lectures will be highly interactive, where both the instructor and students program. There will be weekly homework assignments and a final programming exam.

Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with some applications in science and/or engineering. Prerequisites: Math 53 and introductory probability (such as Stats 116 or Math 151).

Required for first-year ICME Ph.D. students; recommended for first-year ICME M.S. students interested in research. Presentations about research by Stanford faculty and researchers. May be repeated for credit.

Prepares ICME students for the qualifying exams by reviewing relevant course topics and problem solving strategies. Senior ICME students share experiences and lead discussions revolving around ICME core courses.

Solution of linear systems, accuracy, stability, LU, Cholesky, QR, least squares problems, singular value decomposition, eigenvalue computation, iterative methods, Krylov subspace, Lanczos and Arnoldi processes, conjugate gradient, GMRES, direct methods for sparse matrices. Prerequisites: CME 108/ Math 114 and one of Math 104 or Math 113.

Introduction to partial differential equations: basic properties of elliptic, parabolic, and hyperbolic equations; Hamilton-Jacobi equations and applications to optimal control; stochastic modeling, forward and backward Kolmogorov equations; Fourier transform and Fourier series. Prerequisite: multivariable calculus, rigorous courses on basic real analysis and ordinary differential equations. Note: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Introduction to theoretical foundations of discrete mathematics and algorithms. Emphasis on providing mathematical tools for combinatorial optimization, i.e. how to efficiently optimize over large finite sets and reason about the complexity of such problems. Topics include: graph theory, minimum cut, minimum spanning trees, matroids, maximum flow, non-bipartite matching, NP-hardness, approximation algorithms, spectral graph theory, and Laplacian systems. Prerequisites: CS 161 is highly recommended, although not required.

Numerical methods for solving elliptic, parabolic, and hyperbolic partial differential equations. Algorithms for gradient and Hamiltonian systems. Algorithms for stochastic differential equations and Monte Carlo methods. Algorithms for computational harmonic analysis. Prerequisites: advanced undergraduate level PDE and advanced undergraduate level numerical analysis. Note: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Applications, theories, and algorithms for finite-dimensional linear and nonlinear optimization problems with continuous variables. Elements of convex analysis, first- and second-order optimality conditions, sensitivity and duality. Algorithms for unconstrained optimization, and linearly and nonlinearly constrained problems. Modern applications in communication, game theory, auction, and economics. Prerequisites: MATH 113, 115, or equivalent.