106 / FIELDS OF INSTRUCTION


Discrete and variational methods (numerical) for the solution
of ordinary and partial differential equations.
MAD 6206-Combinational Theory I (3) Matching theory,
Ramsey's theorem,. lattice theory, Mobius inversion, generating
functions. Polya's theorem, combinatorial geometries (for
matroids) applications.
MAD 6207-Combinatotial Theory II (3) Prereq: MAD 6206.
MAD 6407-Numerical Analysis (3) Prereq: MAA 4212, 5227
or'MAD 4401. Designed to acquaint research student with
numerical analysis, error analysis of differential equations,
integral equations, eigenvalue, and matrix problems with study
of errors.
MAD'7396-Topics in Combinatorial Theory I (3; max: 6)
Prereq:MAS 5332. 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) S/U.
MAP 5304-Intermediate Differential Equations for Engineers
and Physical Scientists (3)
MAP 5345-Introduction to Partial Differential Equations (3)
MAP 5484-Mathematics for Biological Sciences I (3) Prereq:
MAC 3311, Difference equations, differential equations, prob-
ability theory, matrix theory, and Markov chains with applica-
tions in the biological sciences. Among the applications covered
are population dynamics, epidemiology, and genetics.
MAP 5485-Mathematics for Biological Sciences II (3) Prereq:
MAP 5484. Continuation of MAP 5484.
MAP 6216-Introduction to Calculus of Variations for
Engineers and Physical Scientists (3) Prereq: MAP 5304, MAS
5156 or equivalent. Extremum problems, first variation. Euler
equation problems with fixed and movable boundaries.
Lagrange multiplier methods for problems with constraints,
canonical form, second variation, applications to physics and
engineering.
MAP 6327-Applied Differential Equations 1 (3) Prereq: MAA
5227. 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
5227, MAP 5341 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 6417-Fourier Series 1 (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;
stochastic differential equations. Linear filtering; Kalman filter-
ing; nonlinear filtering theory.
MAP 6468-Stochastic Differential Equations and Filtering
Theory II (3) Prereq: MAP 6467.
MAP 6472-Probability and Potential Theory I (3) Prereq: MAA
5226, 5227, STA 6326. Probability Laws. Uniformly integrable
random variables. Independence and conditioning. General
properties of stochastic processes. First and second canonical
process. Separability of stochastic processes. Decompositions.
Potentials. Excessive functions. Connections with Martingale
theory.
MAP 6473-Probability and Potential Theory II (3) Prereq: MAP
6472.
MAP 6487-Biomathematics Seminar I (3) Prereq: MAC 3312,
MAP3302, STA 6327. Stochastic processes, differential equa-
tions, and catastrophe theory 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 genera-
tion, 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: MAP6487.
Continuation of MAP 6487.
MAP 6505-Mathematical Methods of Physics and Engineer-
ing I (3) Prereq: MAA 5402, MAP 5304, 5345, MAS 5156 or
equivalent. Orthogonal functions; theory of distributions; integral
equations; eigenfunctions and Green's functions; special func-
tions; boundary and initial value problems, with emphasis on
potential theory (Laplace and Poisson equations); the wave equa-
tion; and the diffusion equation.
MAP 6506-Mathematical Methods of Physics and Engineer-
ing 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)
Prereq: admission to doctoral study. 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. May
be repeated.
MAP 7477-Introduction to Mathematical System Theory I (3)
Required for doctoral woik in system theory. Prereq: consent
of instructor: 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.
H.
MAP 7478-Introduction to Mathematical System Theory II
(3) May be repeated.
MAS 5156-Vector Analysis (3)
MAS 5331-Introductory Algebra I (3) Prereq: MAS 4103 or
4312. The basic algebraic systems: groups, rings, vector spaces,
and modules. Linear transformations, matrices, and deter-
minants, the Galois theory.
MAS 5332-Introductory Algebra II (3) Prereq: MAS 5331.
MAS 5334-Rings, Modules, and Linear Algebra (3) Prereq:
undergraduate linear algebra. Structure of principal ideal
domains and their modules, with application to abelian groups
and linear transformations.
MAS 6158-Tensor Analysis (3) Prereq: MAS 5156 or equivalent.
Tensor algebra, tensor calculus, covariant differentiation.
Riemannian spaces, curvature tensor, the Ricci and Einstein
tensors, geodesics, parallel displacement; applications to dif-
ferential geometry, physics, and engineering.
MAS 6346-Algebra I (3) Prereq: MAS 5332. Sylow theorems,
solvable and nilpotent groups, Jordan-Holder theorem, abelian
groups, Jacobson radical, Jacobson density theorem,
Wedderburn-Artin theorem.
MAS 6347-Algebra II (3) Prereq: MAS 6346.
MAS 7215-Theory of Numbers I (3) Prereq: 2 of MAA 6407,
6618, MAS 6347. Introduction to the theory of numbers;
theorems on divisibility; congruences, number-theoretic func-
tions; primitive 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 6347 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) S/U.
MAT 6932-Special Topics in Mathematics (3) 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 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,