Intrinsic properties of spaces such as tori, Möbius strips, and Klein bottles and methods for classifying them. A: Basic point set topology, covering spaces, homotopy, simplicial complexes, genus, Fundamental Theorem of Surfaces. B: Manifolds, simplicial homology, differential forms, and DeRham cohomology.
Intrinsic properties of spaces such as tori, Möbius strips, and Klein bottles and methods for classifying them. A: Basic point set topology, covering spaces, homotopy, simplicial complexes, genus, Fundamental Theorem of Surfaces. B: Manifolds, simplicial homology, differential forms, and DeRham cohomology.
History and development of the calculus of variations; sufficient and necessary conditions for a minimum; Euler-Lagrange equations; Hamilton Jacobi theory; the theory of geodesics; parametric variational problems.
Classical probability theory. Random variables, expectation and moments, independence. The major families of distributions and their properties; characteristic functions, infinite divisibility. Limit theorems on sums of random variables, random walks, Markov processes, Martingales, ruin.
Topics include bivariate and multiple regression with residual, multicolinearity, and sequential selection analysis. ANOVA with multiple comparisons, unbalanced and missing data analysis and repeated measure models. Experimental designs (completely randomized block and splitplot). Statistical software packages.
Statistical decision theory as used for modeling and resolving problems in a variety of fields, from agriculture to artificial intelligence. Basic statistical models, decision trees, utilities and rewards, risk and loss functions, principles of assessing probabilities, Bayes decisions and Bayes estimation. Also group utilities, conflicts of interest, two person zero-sum games, optimal strategies, the minimax theorem.
The theory and application of stochastic processes as models for empirical phenomena (time-series), with special emphasis on the following processes: Wiener, Poisson, stationary, normal, counting, renewal, Markov, and birth-death.
Laboratory sequence illustrating topics covered in course 141. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 141 is required.
Symbolic integration and differentiation; symbolic manipulation of algebraic equations; linear algebra; graphs of functions with one or two variables; parametric surfaces in three-dimensional space; solutions of ordinary differential equations; error estimates; elementary programming.
Laboratory sequence illustrating topics covered in course 142. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 142 is required.
Solving mathematical problems in a Unix environment using the C++ programming language. In addition, software packages such as LAPACK, MACSYMA, MATLAB, and Numerical Recipes are used. Computational problems in both pure and applied mathematics are discussed. Use of other machines on the Internet, such as the Cray supercomputer in San Diego is explained. Students are expected to know how to program in some language, not necessarily C++. Concurrent enrollment in course 143L is required.
Laboratory sequence illustrating topics covered in course 143.One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 143 is required.
Solving mathematical problems in a C++ UNIX environment. The C++ programming language along with mathematical software tools such as MACSYMA and MATLAB are used to attack problems in pure and applied mathematics. A thorough knowledge of C is assumed. Concurrent enrollment in course 144L is required.
Laboratory sequence illustrating topics covered in course 144. One three-hour session per week in microcomputer laboratory. Concurrently enrollment in course 144 is required.
A survey of strategies for building dynamical models for complex natural systems, with exemplary models taken from the literature of the biological and social sciences. Concurrent enrollment in course 147L is required.
Laboratory sequence illustrating topics covered in course 147. One three-hour sessionper week in microcomputer laboratory. Concurrent enrollment in course 147 is required.
Study of random fractals and holomorphic dynamical systems. Percolation theory, diffusion limited aggregation models, Brownian motion, Julia sets for rational functions, bifurcation sets and the Mandelbrot set.
Laboratory sequence illustrating topics covered in course 151. One three-hour session per week in microcomputer laboratory.