16:642:516Applied Partial Differential Equations (3) Theory and applications of partial differential equations. First order equations: linear and quasi-linear. Hyperbolic systems: shocks. Classification of second-order linear equations. Hyperbolic: characteristics, wave equation. Elliptic: maximum principles, Laplace's and reduced-wave equations. Parabolic: heat equation. Fourier and Laplace transforms. Prerequisites: Advanced calculus, differential equations.

16:642:527,528Methods of Applied Mathematics (3,3) Appropriate topics from linear algebra, linear operators in Hilbert space, linear integral equations, boundary-value problems, calculus of variations, numerical solution of ordinary and partial differential equations. Prerequisite: Permission of instructor. Credit not given for these courses and 16:650:567, 568.

16:642:550Linear Algebra and Applications (3) Vector spaces, bases, and dimension. Linear operators, quadratic forms, and their matrix representations. Eigenvalues, eigenvectors, diagonalizability, Jordan, and other canonical forms. Applications to systems of linear differential equations.

16:642:551Applied Algebra (3) Basic algebraic structures, including groups and their representations, finite fields, and Boolean algebra. Applications to physics, counting arguments, switching circuits, and coding theory. Automata theory. Prerequisite: 16:642:550.

16:642:561-562Introduction to Mathematical Physics (3,3) Study of models of classical and/or quantum mechanical physical systems, with emphasis on the use of rigorous mathematical techniques. Prerequisites: Linear algebra, advanced calculus.

16:642:563(F) Rigorous Results in Statistical Mechanics I: Equilibrium (3) Treats the subject ab initio. Deals with general questions such as the existence of the thermodynamic limit, covergence of low-density expansions, correlation inequalities, coexistence of phases. Prerequisite: Permission of instructor. Students should have either a general mathematical background equivalent to that of a second-year graduate student in mathematics or knowledge of statistical mechanics obtained from physics, chemistry, or engineering courses in the subject.

16:642:564(S) Rigorous Results in Statistical Mechanics II: Nonequilibrium (3) Ergodic theory, time evolution of infinite systems, heat flow in random systems, stationary nonequilibrium systems, exactly soluble models systems, stochastic processes. Prerequisite: Permission of instructor. Students should have either a general mathematical background equivalent to that of a second-year graduate student in mathematics or knowledge of statistical mechanics obtained from physics, chemistry, or engineering courses in the subject.

16:642:573,574Numerical Analysis (3,3) Ideas and techniques of numerical analysis illustrated by problems in the approximation of functions, numerical solution of linear and nonlinear systems of equations, approximation of matrix eigenvalues and eigenvectors, numerical quadrature, and numerical solution of ordinary differential equations. Prerequisites: Advanced calculus, linear algebra, and differential equations.

16:642:575Numerical Solutions of Partial Differential Equations (3) Finite-difference schemes, investigating stability and convergence, other methods such as those of Ritz-Galerkin type and collocation. Prerequisite: Partial differential equations.

16:642:577,578Selected Mathematical Topics in System Theory (3,3) Selection of topics from mathematical system theory (e.g., realization, control, stability, optimization, and feedback), with emphasis on qualitative aspects. Algebraic techniques in linear system theory, geometrical and functional analytic techniques in the study of nonlinear control systems. Prerequisites: Linear algebra, differential equations.

16:642:581(S) Graph Theory (3) Advanced introduction to graph theory. Topics include matching theory, connectivity, graph coloring, planarity, extremal graph theory, and the main techniques (elementary, probabilistic, algebraic, and polyhedral) for analyzing the structure and properties of graphs. Prerequisites: 01:640:350 and 411, or permission of instructor. 01:640:477 is recommended.

16:642:582,583Combinatorics (3,3) Advanced introduction to combinatorial theory and applications. Typical topics include hypergraphs, probabilistic methods, algebraic methods, matching theory, packing and covering, Ramsey theory, partially ordered sets and lattices, block designs, error-correcting codes, and matroids. Topics and emphasis vary depending on instructor. Prerequisites: 01:640:350 and 411, or permission of instructor. 01:640:477 and 16:640:551 are recommended.

16:642:585Mathematical Models of Social and Policy Problems (3) Mathematical models of problems in social sciences and the public and private policy areas, emphasizing discrete models. Transportation and communication networks. Energy modeling. Pollution models. Models from economics, psychology, sociology, and political science, dealing with such issues as currency movement, land development, learning, small group behavior, and power in legislatures. Development of requisite mathematical tools about graphs, signed graphs, Markov chains, and n-person games. Prerequisites: Linear algebra, elementary probability.

16:642:586(S) Theory of Measurement (3) Foundations of measurement from a mathematical point of view. Homomorphisms or relational systems; scale type; uniqueness theory; ordinal, extensive, difference, and conjoint measurements; utility and expected utility; subjective probability; applications to social and physical sciences. Prerequisite: Undergraduate modern algebra or permission of instructor.

16:642:587Selected Topics in Discrete Mathematics (3) Choice of topics depends on year and instructor. Prerequisite: Permission of instructor.

16:642:588(F) Introduction to Mathematical Techniques in Operations Research (3) Deterministic methods in operations research, emphasizing linear programming. Hyperplanes, duality, complementary slackness, simplex method, dual simplex method. Integer programs. Assignment, network, and transportation problems. Emphasis on theoretical underpinnings. Prerequisite: Linear algebra.

16:642:589(S) Topics in Mathematical Techniques in Operations Research (3) Special mathematical topics such as matching, matroids, dynamic programming, recent work in combinatorial optimization. Prerequisites: 16:642:588 or equivalent, permission of instructor.

16:642:591,592Topics in Probability and Ergodic Theory (3,3) Basic probability theory and its applications. Topics include stochastic independence, distributions and densities, Markov processes, stationary processes, the law of large numbers, and the central limit theorem. A broad range of applications to communications engineering, economics, biology, and physics. Corequisites: 16:640:501, 502.

16:642:593(F) Mathematical Foundations for Industrial and Systems Engineering (3) Underlying mathematical principles of system modeling. Foundations of the real number system and calculus of functions of one variable, with emphasis on logical principles and methods of proof. Prerequisite: Permission of instructor.

16:642:611,612Selected Topics in Applied Mathematics (3,3) Topics of current interest. Prerequisite: Permission of instructor.

16:642:613Selected Mathematical Topics from Physiology and Medicine (3) Problems in the qualitative theory of nonlinear ordinary and functional differential equations that arise in such subjects as the Hodgkin-Huxley theory, hormonal control systems, and rhythms in physiology. Prerequisite: Permission of instructor.

16:642:621,622Financial Mathematics I, II (3,3) Introduction to stochastic partial differential equations and extreme value theory. Applications to risk analysis and pricing financial securities, such as options and derivatives. Prerequisite: Permission of instructor.

16:642:661,662Selected Topics in Mathematical Physics (3,3) Topics of current interest in areas such as classical mechanics, statistical mechanics, ergodic theory, nonrelativistic quantum mechanics, and quantum field theory. Prerequisite: Permission of instructor.