MATHEMATICS /123 comprehensive preliminary examination administered by the department to become a candidate for the degree. The doctoral student must pass reading knowledge examina- tions in one of the following foreign languages: French, German, or Russian. The dissertation is an important requirement for the doctoral degree in m matics. The topics for the disser- tation may be chosen from a number of areas of current research in pure and applied mathematics. Every graduate student is expected to attend the regular colloquium. Details concerning all requirements for graduate de- grees in mathematics can be obtained by writing the Mathematics Department Graduate Selection Commit- tee. MAA 5104-Advanced Calculus for Engineers and Physical Sci- entists I (3) MAA 5105-Advanced Calculus for Engineers and Physical Sci- entists 11 (3) Prereq: MAA 5104. MAA 5228-Modern Analysis I (3) Prereq: MAC 3313. Topology of metric spaces, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, the Stone-Weierstrass theorem, functions of several variables. Stokes' theorem, the Lebesgue theory. MAA 5229-Modern Analysis II (3) Prereq: MAA 5228. MAA 5404-Introduction to Complex Variables for Engineers and Physical Scientists (3) MAA 5506-Introduction to Functional Analysis I (3) MAA 5507-Introduction to Functional Analysis II (3) Prereq: MAA 5506. MAA 6236-Mathematical Analysis for Statisticians (4) Coreq: STA 6326. Numerical sequences and series, limits, continuity, differentiation, integration, series of functions. Applications to probability and statistics stressed. MAA 6406-Complex Analysis I (3) Prereq: MAA 5229. Rapid survey of properties of complex numbers, linear transformations, geometric forms and necessary concepts from topology. Com- plex integration. Cauchy's theorem and its corollaries. Taylor's series and the implicit function theorem in complex form. Conformality andthe Riemann-Caratheodory mapping theorem. Theorems of Bloch, Schottky, and the big and little theorems of Picard. Harmonicity and Dirichlet's problems. MAA 6407-Complex Analysis 11 (3) Prereq: MAA 6406. MAA 6616-General Theory of Measure and Integration I (3) Prereq: MAA 5229. Borel sets, measurable functions and the monotone class theorem. Measures and their extension theo- rems, Lebesgue integral, convergence theorems. Product meas- ures and Fubini's, theorem, the Radon-Nikodym theorem and differentiation. Elementary Hilbert and Banach space theory, LP- spaces. MAA 6617-General Theory of Measure and Integration II (3) Prereq: MAA 6616. MAA 7526-Advanced Topics in Functional Analysis I (3; max: 6) Prereq: MAA 6617, MAS 6332. Algebraic and topological ap- proach to current material and methods in analysis. MAA 7527-Advanced Topics in Functional Analysis 11 (3; max: 6) Prereq: MAA 7526. MAD 5405-Numerical Methods of Differential Equations (3) Prereq: MAS 3113, 3114, or 4105; and programming language. Numerical methods for the solution of ordinary differential equations. MAD 6206-Combinatorial Theory I (3) Matching theory, Ramsey's theorem, lattice theory, Mobius inversion, generating functions. Polya's theorem, matroids, applications, block de- signs, graph theory. MAD 6207-Combinatorial Theory II (3) Prereq: MAD 6206. MAD 6406-Numberical Linear Algebra (3) Prereq: MAS 3114, 4105, or 4124; and programming language. Topics most useful in applications with emphasis on numerical techniques: systems of linear equations, positive definite and toeplitz systems, least squares problems, singular value decomposistion, and eigenval- ues. Numerical stability and efficiency of algorithms as well as effect of perturbations on the problem. Companion to MAD 6407. MAD 6407-Numerical Analysis (3) Prereq:MAA 4212, 5105, or 5229; and. programming language. Numerical techniques to solve systems of nonlinear equations, to approximate functions, to compute derivatives, to evaluate integrals, and to integrate systems of differential equations. Introduction to numerical techniques for partial differential equations. Companion to MAD 6406. MAD 7396-Topics in Combinatorial Theory I (3; max 6) Prereq: MAS 5312. Topics chosen from among graph theory, coding theory, matroid theory, finite geometries, projective geometry, difference methods, and Latin squares. MAD 7397-Topics in Combinatorial Theory II (3; max: 6) Prereq: MAD 7396 MAE 6940-Supervised Teaching (1-5; max: 5) S/U. MAE 6943-Internship in College Teaching (3; max: 6) Prereq: consent of graduate adviser. MAP 5304-Intermediate Differential Equations for Engineers and Physical Scientists (3) MAP 5345-Introduction to Partial Differential Equations (3) MAP 6208- Numerical Optimization (3) Prereq:MAD 6406 and 6407 or permission of instructor. Unconstrained and con- strained optimization, linear and nonlinear programming, gradi- ent, multplier, and quasi-Newton methods. Penalty, multiplier, and projection methods for constrained problems. MAP 6217-Introduction to Calculus of Variations for Engineers and Physical Scientists (3) Prereq: MAP 5304, MAS 5157 or equivalent. Extremum problems, first variation. Euler equation problems with fixed and movable boundaries. Lagrange multi- plier methods for problems with constraints, canonical form, second variation, applications to physics and engineering. MAP 6327-Applied Differential Equations I (3) Prereq: MAA 5229. Theory and methods for solving linear and nonlinear systems of differential equations and partial differential equa- tions. Applications and computer techniques included. MAP 6328-Applied Differential Equations II (3) Prereq: MAP 6327. MAP 6356-Partial Differential Equations I (3) Prereq: MAA 5229, MAP 5345 or 6506. Cauchy-Kowalewski theorem, first order equations, classification of equations, hyperbolic equa- tions, elliptic equations, parabolic equations, hyperbolic sys- tems, nonlinear hyperbolic systems, existence theory based on functional analysis. Applications to physical sciences. MAP 6357-Partial Differential Equations II (3) Prereq: MAP 6356. MAP 6375-Numerical Partial Differential Equations (3) Prereq: MAD 6406 and 6407 or permission of instructor. Introduction to partial differential equations and fundamental concepts. Para- bolic equations: finite differences, consistency, convergence and stability, two- and three-dimensional problems. Elliptic equations, boundary integral equation methods. Hyperbolic equations: finite differences and method of characteristics. Intro- duction to finite elements. Method of lines. MAP 6376-Finite Element Method (3) Prereq: MAD 6406 and 6407 or permission of instructor. Variational formulations of partial differential equations, finite element approximations, both theoretical framework and numerical issues addressed. Finite element spaces in one, two, and three dimension, error analysis, nonconforming finite element spaces, isoparamatic approximations to boundary conditions. MAP 6417-Fourier Series I (3) Prereq:MAP 6505. Fundamental theorems on convergence, differentiation, and integration. Applications to boundary value problems. MAP 6418-Fourier Series II (3) Prereq: MAP 6417. MAP 6467-Stochastic Differential Equations and Filtering Theory I (3) Introduction to random functions; the Brownian motion process. Ito's stochastic integral; Ito's stochastic calculus; sto- chastic differential equations. Linear filtering; Kalman filtering; nonlinear filtering theory. MAP6468-Stochastic Differential Equationsand FilteringTheory II (3) Prereq: MAP 6467. MAP 6472-Probability and Potential Theory 1 (3) Prereq: MAA 5229 orSTA 6326. Random variables, independence and condi- tioning. Laws of large numbers and the Central Limit Theorem. Stochastic processes, martingales, Gaussian processes, Markov processes, potentials and excessive functions. MAP 6473-Probability and Potential Theory II (3) Prereq: MAP 6472. MAP 6487-Biomathematics Seminar I (3) Prereq: MAC 3312, MAP 3302, STA 6326 or MAP 4102. Stochastic processes, differential equations, and reaction-diffusion equations used to model various biological processes. Among the applications covered are the following: population dynamics, epidemiology, genetics, enzyme kinetics, cell differentiation and morphogene- sis, nerve impulse generation, and aggregation of slime mold.