University of Delaware
Graduate Catalog 1995-1996
College of Arts and Science
Department of Mathematical Sciences
Mathematics Program
Course Descriptions
Course offerings for each semester are chosen from the list below. In
addition, the department offers advanced topics courses in Algebra and
Combinatorics, Integral and Differential Equations, Numerical Analysis and
Scientific Computing, and Topology and Dynamical Systems.
MATH 503 Advanced Calculus for Applications 3
Multivariable calculus, vector calculus, infinite series, uniform
convergence and Fourier analysis.
PREREQ: MATH302.
MATH 508 Introduction to Complex Variables and Applications 3
Introduction to analytic functions, contour integration, power series,
residues and conformal mapping.
PREREQ: MATH243.
MATH 514 Topics in Advanced Mathematics for Engineers 3
Basic methods of analysis: introduction to complex variables; special
functions including Bessel functions and Legendre polynomials; Fourier
series and integrals; partial differential equations; and emphasis on
engineering applications.
PREREQ: MATH302.
RESTRICTIONS: For engineering students.
MATH 518 Mathematical Models and Applications 3
Illustration and analysis of mathematical models for problems in the
biological, physical and social sciences.
PREREQ: Either MATH230, or MATH349 and STAT370.
MATH 529 Linear Programming: Methods and Applications 3
Theory of linear programming (linear inequalities, convex polyhedra,
duality), related topics (games, integer programming), main algorithms
(simplex, dual) and representative applications in agriculture,
economics, engineering, operations research and mathematics. Familiarity
with computer implementation of LP methods acquired by individual (or
small group) projects of applying LP to the students' chosen areas.
PREREQ: MATH349.
MATH 540 Geometry 3
Axiomatic systems; transformations; Euclidean, projective and hyperbolic
geometry.
PREREQ: MATH349.
RESTRICTIONS: Graduate credit only for teachers.
MATH 555 Applied Calculus for Business and Economics 3
Accelerated version of the usual two-semester undergraduate preparation
in calculus.
RESTRICTIONS: Requires high school algebra. Designed for students
enrolling in M.B.A. program.
MATH 600 Fundamentals of Real Analysis 3
Rigorous introduction to classical real analysis. Brief review of real
numbers and a thorough discussion of the basic topology of metric
spaces. Covers in detail the following topics: the analysis of sequences
and series, continuity, differentiation and Taylors theorem, and the
analysis of sequences and series of functions.
RESTRICTIONS: Credit not given for both MATH600 and MATH601.
MATH 601 Advanced Calculus - Introduction to Analysis I 3
Limits, continuity, sequences and series, theory of differentiation and
integration, and several variable calculus.
PREREQ: MATH260.
RESTRICTIONS: Credit not given for both MATH600 and MATH601.
MATH 602 Advanced Calculus - Introduction to Analysis II 3
Continuation of MATH600 OR MATH601.
PREREQ: MATH600 or MATH601.
MATH 605 Applied Functional Analysis 3
Introduction to formulation and solution of problems of engineering and
science by means of functional analytic methods in Hilbert and Banach
spaces. Includes boundary and initial value problems in ordinary and
partial differential equations as well as integral equations. Emphasis
on constructive techniques: variational methods, approximate solutions,
bounds for eigenvalues, etc.
PREREQ: MATH514, PHYS608, MEEG864 or advanced calculus.
MATH 609 Intermediate Ordinary Differential Equations with Applications 3
Theory and applications of ordinary differential equations; existence
theorems of linear and nonlinear systems, oscillation theorems,
stability theory, and Sturm-Liouville theory.
PREREQ: MATH302, MATH349 and one semester of advanced calculus.
MATH 610 Introduction to Partial Differential Equations with Applications
3
Introduction to partial differential equations: equations of
mathematical physics and their classical theories emphasizing boundary
and initial value problems and their interpretations.
PREREQ: Two semesters of advanced calculus.
MATH 611 Introduction to Numerical Analysis and Scientific Computing I 3
Introduction to numerical computing, analysis and solution of systems of
linear equations, linear least-squares, eigenvalue problems, methods for
unconstrained optimization, solution of systems of nonlinear equations.
Experience with standard computer packages, code development and
simulations of applied problems.
PREREQ: MATH503 or MEEG863 or PHYS207.
MATH 612 Introduction to Numerical Analysis and Scientific Computing II 3
Approximation, interpolation, data fitting and smoothing, numerical
methods for ordinary differential equations. Additional topic selected
at discretion of instructor. Experience with standard computer packages,
code development and simulations of applied problems.
PREREQ: MATH503 or MEEG863 or PHYS207.
MATH 613 Applied Symbolic Computation 3
See CISC623 for course description.
MATH 616 Introduction to Applied Mathematics I 3
Introduction to formulation of mathematical problems for systems of
interest outside mathematics, particularly those from engineering and
physics; systems studied vary; emphasis on interplay between system and
mathematical model.
PREREQ: Two semesters of advanced calculus and PHYS208.
MATH 617 Introduction to Applied Mathematics II 3
Methods of attack on mathematical problems for systems of interest
outside mathematics; calculus of variations techniques; and
interpretation of solutions to problems in terms of systems.
PREREQ: MATH616.
RESTRICTIONS: Familiarity with systems treated acceptable in lieu of
Prereq.
MATH 630 Probability Theory and Applications 3
Introduction to probability theory as background for further work in
statistics or stochastic processes. Sample spaces and axioms of
probability; discrete sample spaces having equally likely events;
conditional probability and independence; random variables, classical
discrete and continuous random variables; mathematical expectation and
moments of a distribution; Chebyshev's inequality; and law of large
numbers and central limit theorem. May be cross-listed with STAT601.
MATH 631 Introduction to Stochastic Processes 3
Classical stochastic processes with emphasis on their properties, which
do not involve measure theory. Course contents: Markov chains in
discrete and in continuous time with examples from random walk, birth
and death processes, branching processes and queueing theory. Renewal
and Markov renewal processes. Basic notions of Brownian motion and
second-order processes.
PREREQ: MATH630.
MATH 632 Topics in Applied Probability 3
The application of probability theory or stochastic processes in a
specific area of science. May include treatment of probabilistic methods
not ordinarily covered in other courses. Possible topics are the theory
of queues, dams and inventories, replacement and reliability,
probability models in population growth and biomathematics, Monte Carlo
simulation, algorithmic methods in probability and operational methods.
MATH 650 Abstract Algebra 3
Modular arithmetic, Chinese remainder theorem, rings (including
polynomial rings), ideals and quotient rings, groups and homomorphism
theorems, unique factorization and principal ideal domains, field
extensions.
PREREQ: MATH349.
MATH 672 Vector Spaces 3
Vector spaces, linear transformations, decomposition theorems and
bilinear forms.
PREREQ: MATH349.
MATH 688 Combinatorics and Graph Theory with Applications I 3
Permutations and combinations, generating functions and other
enumeration techniques, recurrence relations, basic graph theory,
partially ordered sets, combinatorial optimization and time complexity.
PREREQ: An undergraduate course in linear algebra.
MATH 689 Combinatorics and Graph Theory with Applications II 3
Selected topics from graph theory, combinatorial designs, finite
geometries, extremal and probabilistic combinatorics. Applications to
combinatorial optimization, experimental design and analysis of
algorithms.
PREREQ: MATH688.
COREQ: MATH650.
MATH 694 Methods of Optimization 3
Review of linear programming, unconstrained and constrained non-linear
programs, numerical methods, Kuhn-Tucker theory, duality and Lagrange
multipliers.
MATH 698 Thematic Seminar 2 PF
Problems oriented class, topics vary from year to year. Aim is to give
students research experience in mathematics and to show how mathematics
is used to solve problems.
MATH 801 Calculus of Variations 3
Comprehensive introduction to variational principles and methods in
science and engineering; classical calculus of variations with
applications to mechanics; problems of optimal control; direct methods
including the method of Faedo-Gelerkin. Emphasis on applications.
RESTRICTIONS: Requires familiarity with concepts of advanced calculus.
MATH 804 Topics in Optimization 3
Selected topics from the following: variational inequalities theory of
optimal control, complex analysis and nonsmooth optimization, game
theory, and optimization algorithms.
MATH 805 Analysis I 3
Topics include Lebesgue measure and integration, absolute continuity and
functions of bounded variations, Lp spaces and Fubini's theorem.
PREREQ: MATH602.
MATH 806 Analysis II 3
Fundamental structures of modern analysis with special emphasis on the
theory of Hilbert space, spectral theorems and application to integral
and differential equations.
PREREQ: MATH805.
MATH 807 Complex Analysis 3
Complex numbers; analytic functions; geometry of elementary functions;
integrals; power series; residues and poles.
PREREQ: MATH602.
MATH 808 Complex Analysis 3
Conformal mapping with applications; analytic continuation; Riemann
surfaces; elliptic functions; and infinite products.
PREREQ: MATH807.
MATH 811 Topics in Classical Analysis 3
Investigation of topics chosen from function theory such as geometric
function theory, Riemann surfaces, meromorphic and entire functions,
etc.
RESTRICTIONS: Requires permission of instructor.
MATH 815 Functional Analysis 3
Topological vector spaces with short introductory review of Banach
spaces.
PREREQ: MATH806.
MATH 818 Theory of Ordinary Differential Equations 3
Linear systems with isolated singularities and systems with periodic
coefficients; boundary value problems; Poincare-Bendixson theory.
PREREQ: MATH609 and MATH805.
MATH 819 Theory of Ordinary Differential Equations 3
Singular Sturm-Liouville theory; asymptotic behavior of linear and
nonlinear systems; and topics of current research.
PREREQ: MATH818.
MATH 822 Integral Equations 3
Fredholm and Hilbert-Schmidt theories of Fredholm integral equations of
the second kind. Equations of the first kind. Volterra equations.
Nonlinear eigenvalue problems. Applications to physics and engineering.
RESTRICTIONS: Requires permission of instructor.
MATH 823 Integral Equations 3
Singular integral equations (Carleman and Wiener-Hopf equations).
Nonlinear integral equations (Volterra and Hammerstein equations).
Nonlinear singular integral equations. Applications to physics and
engineering (the nonlinear oscillator, the airfoil equation, the Tricomi
Problem in partial differential equations, etc.).
PREREQ: MATH822.
MATH 824 Topics in Applied Mathematics 3
Topics chosen from asymptotic analysis, elasticity, electromagnetic
theory, fluid dynamics, optimal control theory and other areas.
RESTRICTIONS: Requires permission of instructor.
MATH 825 Topics in Applied Mathematics 3
Topics chosen from asymptotic analysis, elasticity, electromagnetic
theory, fluid dynamics, optimal control theory and other areas.
RESTRICTIONS: Requires permission of instructor.
MATH 827 General Topology I 3
Generation and properties of topological spaces. Continuity, separation
and countability properties; and convergence of nets and filters.
MATH 828 General Topology II 3
Compactness and connectedness, metrization, uniform spaces and basic
homotopy theory.
PREREQ: MATH827.
MATH 835 Partial Differential Equations I 3
First order differential equations and systems. Existence and uniqueness
for elliptic, parabolic and hyperbolic equations. Boundary and initial
value problems for equations of hyperbolic and parabolic type in two
independent variables. Classical approaches employed.
PREREQ: MATH610.
MATH 836 Partial Differential Equations II 3
Cauchy's problem and initial boundary value problems for hyperbolic and
parabolic equations and systems. Boundary value problems for elliptic
equations and systems. Equations of mixed type. Emphasis on modern
approaches.
PREREQ: MATH835 and MATH805.
MATH 838 Numerical Methods for Partial Differential Equations 3
Introduces concepts of consistency, stability and convergence of
numerical schemes. Emphasis on various finite difference schemes and
their applications to fundamental partial differential equations.
PREREQ: MATH610.
MATH 839 Numerical Methods for Partial Differential Equations 3
Emphasis on finite element method and its applications to physical
problems.
PREREQ: MATH838.
MATH 845 Group Theory with Applications 3
Groups acting on sets, the class equation, Sylow's theorems, free
groups, classical groups, Polya enumeration theory, groups and graphs,
Frieze groups and crystallographic groups, and the group Knapsack
problem.
PREREQ: MATH650.
MATH 846 Field Theory with Applications 3
Field extensions, structure of finite fields, and basics of Galois
theory. Applications of finite fields to block designs and finite
geometries. Additional applications may include impulse response
sequences, pseudorandom sequences, algebraic coding theory (BCH and
Goppa codes) and cryptosystems.
PREREQ: MATH650, MATH672, and MATH845.
MATH 850 Foundation of Probability Theory 3
Mathematically rigorous treatment of the foundations of probability
theory. Families of sets, semi-ring and sigma algebras, axioms of
probability, and extension theorem. Random variables, probability
distributions, and modes of convergence for sequences of random
variables. Product measure and independence and conditional expectation.
The weak and strong laws of large numbers, the central limit theorem and
the law of the iterated logarithm.
PREREQ: MATH630 and MATH805.
MATH 851 Stochastic Processes 3
Mathematically rigorous treatment of stochastic processes. Course
content: general definitions, separability, Kolmogorov consistency
condition. Markov processes and Brownian motion. In-depth discussion of
topics such as weak convergence of processes, martingale theory,
diffusion processes or second order processes, as announced by
instructor.
PREREQ: MATH850.
MATH 868 Research 1-6
MATH 869 Master's Thesis 1-6
MATH 870 Reading in Mathematics 1-6
MATH 887 Mathematical Methods of Physics and Engineering 3
Green's function and eigenfunction expansions for boundary value
problems, theory of distributions, weak solution, metric spaces,
contractions, integral equations and Hilbert spaces.
PREREQ: MATH503, MATH508 or MEEG863, MEEG864 or PHYS607, and PHYS608.
MATH 964 Pre-Candidacy Study 3-12 PF
Research and readings in preparation of dissertation topic and/or
qualifying examinations for doctoral students before admission to
candidacy but after completion of all required course work.
RESTRICTIONS: Not open to students who have been admitted to candidacy.
MATH 969 Doctoral Dissertation 1-12 PF
UNIV 895 Master's Sustaining: Non-Thesis 0 PF
UNIV 899 Master's Sustaining: Thesis 0 PF
UNIV 999 Doctoral Sustaining 0 PF