136 / FIELDS OF INSTRUCTION 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 orpermission of instructor. Unconstrained and constrained optimization, linear and nonlinear programming, gradient, mul- tiplier, and quasi-Newton methods. Penalty, multiplier, and pro- jection 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 equations. 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 systems, nonlinear hyperbolic systems, existence theory based on func- tional 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 equa- tions: finite difi rences, solution to linear equations, boundary integral equation methods. Hyperbolic equations: finite differ- ences and method of characteristics. Introduction to finite ele- ments. Methods 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 dimensions, error analysis, nonconforming finite element spaces, isoparametric approxima- tions to boundary conditions. MAP 6417-Fourier Series I (3) Prereq:MAP6505. Fundamental theorems on convergence, differentiation, and integration. Appli- cations to boundary value problems. MAP 6418-Fourier Series II (3) Prereq: MAP 6417. MAP 6445-Approximation Theory (3) Prereq: MAD 6406 and 6407 or permission of instructor. Polynomial approximation, splineapproximation, piecewisepolynomials, interpolation theory, quadrature theory, inequalities in polynomials. MAP6467-Stochastic Differential Equationsand FilteringTheory I (3) Introduction to random functions; the Brownian motion process. Ito's stochastic integral; Ito's stochastic calculus; stochas- tic differential equations. Linear filtering; Kalman filtering; nonlin- ear filtering theory. MAP6468-Stochastic Differential Equationsand FilteringTheory II (3) Prereq: MAP 6467. MAP 6472--Probability and Potential Theory I (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 1 (3) Prereq: MAC 3312, MAP3302, STA 6326 orMAP4102. Stochastic processes, differ- ential 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 morphogenesis, nerve impulse generation, and aggregation of slime mold. The course is designed to benefit graduate students in biological sciences, as well as mathematics. MAP 6488-Biomathematics Seminar II (3) Prereq: MAP 6487. Continuation of MAP 6487. MAP 6505-Mathematical Methods of Physics and Engineering I (3) Prereq: MAA 5404, MAP 5304, 5345, MAS 5157 or equiva- lent. Orthogonal functions; theory of distributions; integral equa- tions; eigenfunctions and Green's functions; special functions; boundary and initial value problems, with emphasis on potential theory (Laplace and Poisson equations); the wave equation; and the diffusion equation. MAP 6506-Mathematical Methods of Physics and Engineering II (3) Prereq: MAP 6505 MAP 7436-Seminar in Applied Mathematics I (3; max: 6) Various topics in applications of mathematics both classical and in areas of current research. MAP 7437-Seminar in Applied Mathematics II (3; max: 6) MAP 7475-Seminar in Mathematical System Theory (3; max: 18) Prereq: admission to doctoralstudy. Critical review of current developments in system theory, with strong emphasis on (but not limited to) questions of mathematical interest. Presentations by invited speakers as well as by students and faculty affiliated with the Center for Mathematical System Theory. Intensive discussions by participants rather than ex cathedra lectures. MAP 7477-Introduction to Mathematical System Theory I (3; max: 6) Prereq: consent of instructor. Required for doctoral work in system theory. Fundamental mathematical structures in the description of dynamical systems, especially linear system and finite automata. Problems of controllability, observability, struc- ture, and identification. Topics to reflect current developments. MAP 7478-Introduction to Mathematical System Theory II (3; max: 6) MAS 5157-Vector Analysis (3) MAS 5311-Introductory Algebra I (3) Prereq: MAS 4105 or 4302. The basic algebraic systems: groups, rings, vector spaces, and modules. Lineartransformations, matrices, and determinants. MAS 5312-Introductory Algebra II (3) Prereq: MAS 5311. MAS 6331-Algebra I (3) Prereq: MAS 5312. Solvable and nilpotent groups, Jordan-Holder theorem, abelian groups, Galois theory, Noetherian rings, Dedekind domains, Jacobson radical, Jacobson density theorem, Wedderburn-Artin theorem. MAS 6332-Algebra II (3) Prereq: MAS 6331. MAS 7215-Theory of Numbers 1 (3) Prereq: 2 of MAA 6407, 6617, MAS 6332. Introduction to the theory of numbers; theorems on divisibility; congruences, number-theoretic functions; primi- tive roots and indices; the quadratic reciprocity law; Diophantine equations and continued functions. MAS 7216-Theory of Numbers II (3) Prereq: MAS 7215. MAS 7396-Advanced Topics in Algebra I (3; max: 6) Prereq: MAA 6407, 6617, MAS 6332 or MTG 6347. Current topics in algebra. MAS 7397-Advanced Topics in Algebra II (3; max: 6) Prereq: MAS 7396. MAT 6905-Individual Work (3; max: 9) MAT 6910-Supervised Research (1-5; max: 5) S/U. MAT 6932-Special Topics in Mathematics (3; max: 9) Prereq: consent of graduate adviser, who should be consulted well in advance of registration. MAT 6971-Research for Master's Thesis (1-15) S/U. MAT 7979-Advanced Research (1-9) Research for doctoral students before admission to candidacy. Designed for students with a master's degree in the field of study or for students who have been accepted for a doctoral program. Not open to students who have been admitted to candidacy. S/U. MAT 7980-Research for Doctoral Dissertation (1-15) S/U. MHF 5107-Introduction to Set Theory (3) Basic axioms and concepts of set theory, axiom of choice, Zorn's lemma, Schroder- Bernstein theorem, cardinal numbers, ordinal numbers, and the continuum hypothesis. MHF 5207-Foundations of Mathematics (3) Models and proofs. Foundations of the real and natural number systems. Algorithms. Turing Machines, undecidability and independence. Examples and applications in algebra, analysis, geometry, and topology. MHF 6306-Mathematical Logic I (3) Languages, models, and