Math 545 Advanced Calculus I 3 credits
Prerequisite: undergraduate calculus. Rigorous treatment of the calculus of
real-valued functions of one real variable: the real number system,
Epsilon-Delta theory of limit, continuity, derivative, and the Riemann
integral. The fundamental theory of calculus. Series and sequences including
Taylor series and uniform convergence. The inverse and implicit function
theorems.
Math 546 Advanced Calculus II 3 credits
Prerequisite: Math 545 or equivalent. Rigorous treatment of the calculus of
real-valued functions of several real variables: the geometry and algebra of
n-dimensional Euclidean space. Limit, continuity, derivative, and the Riemann
integral of functions of several variables. The inverse and implicit function
theorems. Series, including Taylor series. Optimization problems. Integration
on curves and surfaces, the divergence and related theorems.
Math 551 Engineering Mathematics 3 credits
Prerequisite: undergraduate differential equations. Mathematical methods useful
in the analysis of problems arising in applied mathematics and engineering.
Topics include Fourier series, general orthogonal systems, Laplace and Fourier
transforms, boundary-value problems, generalized functions, linear algebra and
systems of ordinary differential equations.
Math 560 Methods of Applied Mathematics I 3 credits
Prerequisite: Math 331, Math 337, Math 545 or departmental approval. This
course introduces relevant problems and techniques in applied mathematics. This
includes basic problems in linear algebra, and ordinary and partial
differential equations.
Math 561 Methods of Applied Mathematics II 3 credits
Prerequisite: Math 560. This course is a continuation of Math 560. Topics
include Green's functions, scattering, spectral theory, characteristics and
conservation laws. Applications to fluid and gas dynamics, traffic flow and
mathematical biology will be treated.
Math 573 Intermediate Differential Equations 3 credits
Prerequisites: undergraduate differential equations and linear algebra. Methods
and applications for systems of ordinary differential equations: existence and
uniqueness for solutions of ODEs, linear systems, and stability analysis. Phase
plane and geometrical methods. Sturm-Louville eigenvalue problems.
Math 590 BS/MS Co-op Work Experience I 3 additive credits
Prerequisites: enrollment and standing in the BS/MS program, and permission
from mathematics department and co-op office. Cooperative education/internship
providing on-the-job complement to academic programs in mathematics. Work
assignments and projects are developed by the co-op office in consultation with
the mathematics department.
Math 591 BS/MS Co-op Work Experience II 3 additive credits
Prerequisites: Math 590, and permission from mathematics department and co-op
office. Continuation of Math 590.
Math 592 Co-op Work Experience III 3 additive credits
Prerequisites: Math 591, graduate status and permission from mathematics
department and Office of Cooperative Education and Internships. Continuation of
Math 591.
Math 599 Teaching in Mathematics 3 credits
Prerequisite: full-time status in a graduate program. This course provides TA's
in mathematics with the skills and practice necessary for the effective
performance of teaching and related duties. Students are exposed to strategies
and methods for teaching undergraduate mathematics, and are required to
demonstrate their capability for teaching using techniques. Not counted toward
degree credit.
Math 611 Numerical Methods for Computation 3 credits
Prerequisites: undergraduate differential equations, linear algebra and
familiarity with a computer language (FORTRAN, C, or equivalent). A practical
introduction to the numerical methods of science and engineering. Numerical
solution of a linear system. Interpolation and quadrature. Iterative solution
of nonlinear systems. Computation of eigenvalues and eigenvectors. Numerical
solution of initial and boundary value problems for ODEs. Introduction to
numerical solution of PDEs. Includes examples requiring student use of a
computer with some use of software packages.
Math 613 Advanced Applied Mathematics I: Modeling 3 credits
Prerequisites: undergraduate differential equations and linear algebra.
Concepts and strategies of mathematical modeling are developed by investigation
of case studies in a selection of areas. Consistency of a model,
nondimensionalization and scaling, regular and singular effects are discussed.
Possible topics include continuum mechanics, vibrating strings, population
dynamics, traffic flow, and the Sommerfeld problem.
Math 614 Numerical Methods I 3 credits
Prerequisites: undergraduate differential equations, linear algebra and
familiarity with a computer language (FORTRAN, C, or equivalent). Theory and
techniques of scientific computation, with more emphasis on accuracy and rigor
than Math 611. Machine arithmetic. Numerical solution of a linear system and
pivoting. Interpolation and quadrature. Iterative solution of nonlinear
systems. Computation of eigenvalues and eigenvectors. Numerical solution of
initial and boundary value problems for systems of ODEs. Applications. The
class includes examples requiring student use of a computer.
Math 621 Applied Exterior Calculus 3 credits
Prerequisites: undergraduate calculus and linear algebra. Development of
exterior calculus: the method of characteristics, first order linear and
quasilinear PDEs and systems of PDEs. Lie subalgebras and reduction to Jacobi
normal form. Theorems of Froebenius, Darboux, and Cartan. Antiexact forms and
solution of exterior differential equations. Applications to nonlinear second
order PDEs, calculus of variations, and examples from physics.
Math 630 Linear Algebra and Applications 3 credits
Prerequisites: undergraduate calculus and differential equations. Development
of the concepts needed to study applications of linear algebra and matrix
theory to science and engineering. Topics include linear systems of equations,
matrix algebra, orthogonality, eigenvalues and eigenvectors, diagonalization,
and matrix decomposition.
Math 631 Linear Algebra 3 credits
Prerequisites: undergraduate calculus and differential equations. Similar in
aim and content to Math 630 but with more emphasis on mathematical rigor.
Linear systems of equations, matrix algebra, linear spaces, orthogonality,
eigenvalues and eigenvectors, diagonalization, and matrix decomposition.
Applications.
Math 645 Analysis I 3 credits
Prerequisite: a background in advanced calculus. Review and extension of the
fundamental concepts of advanced calculus: the real number system, limit,
continuity, differentiation, the Riemann integral, sequences and series. Point
set topology in metric spaces. Uniform convergence and its applications.
Math 651 Applied Mathematics I 3 credits
Prerequisite: undergraduate ordinary differential equations. A survey of
mathematical methods for the solution of problems in the applied sciences and
engineering. Topics include: ordinary differential equations, Fourier series,
Fourier and Laplace transforms, and eigenfunction expansion.
Math 652 Applied Mathematics II 3 credits
Prerequisite: Math 651 or equivalent. Continuation of Math 651. Topics include:
partial differential equations, functions of a complex variable, and the
calculus of variations.
Math 656 Complex Variables I 3 credits
Prerequisite: undergraduate calculus. The theory and applications of analytic
functions of one complex variable: elementary properties of complex numbers,
analytic functions, elementary complex functions, conformal mapping, Cauchy
integral formula, maximum modulus principle, Laurent series, classification of
isolated singularities, residue theorem and applications.
Math 660 Differential Geometry of Curves and Surfaces II 3 credits
Prerequisites: Math 460 or equivalent. Differential forms, the Euler
characteristic, the Gauss-Bonnet theorem, and the fundamental group. Outline of
the topological classification of compact surfaces, vector fields, geodesics,
and Jacobi fields. Calculus of variations. The global differential geometry of
surfaces and the elementary theory of Riemann surfaces.
Math 661 Applied Statistics 3 credits
Prerequisite: undergraduate calculus. Data collection, elementary sampling
methods and experimental design, descriptive statistics, summary measures for
quantitative and qualitative data, graphical data analysis. Computational
statistical inference, confidence intervals and tests on sample means,
variances, and proportions. Curve fitting, lines, curves, and surfaces using
least squares, data transformation, inference for curve fitting analysis,
one-way and two-way ANOVA, Shewhart control charts.
Math 662 Mathematical Statistics I 3 credits
Prerequisite: a background in undergraduate statistics. Probability,
conditional probability, random variables and distributions, independence,
expectation, moment generating functions, special distributions, sampling
distributions, the central limit theorem.
Math 668 Probability Theory 3 credits
Prerequisite: Math 662 or equivalent. Introduction to measure theory and
integration, axiomatic probability, random variables, distribution function,
expectation, independence, modes of convergence, characteristic functions, sums
of identically distributed random variables, conditional expectation.
Math 671 Asymptotic Methods I 3 credits
Prerequisite: Math 545 or Math 645, Math 656, or equivalent. Asymptotic
sequences and series. Use of asymptotic series. Regular and singular
perturbation methods. Asymptotic methods for the solution of ODEs, including:
boundary layer methods and asymptotic matching, multiple scales, the method of
averaging, and simple WKB theory. Asymptotic expansion of integrals, including:
Watson's lemma, stationary phase, Laplace's method, and the method of steepest
descent.
Math 672 Biomathematics I: Biological Waves and Oscillations 3 credits
Prerequisites: differential equations and linear algebra, or permission of the
instructor. Models of wave propagation and oscillatory phenomena in nerve,
muscle, and arteries: Hodgkin-Huxley theory of nerve conduction,
synchronization of the cardiac pacemaker, conduction and rhythm abnormalities
of the heart, excitation-contraction coupling, and calcium induced waves, wave
propagation in elastic arteries, models of periodic human locomotion.
Math 673 Biomathematics II: Pattern Formation in Biological Systems 3
credits
Prerequisites: differential equations and linear algebra, or permission of the
instructor. Emergence of spatial and temporal order in biological and
ecological systems: Hopf and Turing bifurcation in reaction-diffusion systems,
how do zebras get their stripes, patterns on snake skins and butterfly wings,
spatial organization in the visual cortex, symmetry breaking in hormonal
interactions, how do the ovaries count. Basic techniques of mathematics are
introduced and applied to significant biological phenomena that cannot be fully
understood without their use.
Math 675 Partial Differential Equations 3 credits
Prerequisite: Math 690 or equivalent. A survey of the mathematical theory of
partial differential equations: first order equations, classification of second
order equations, the Cauchy-Kovalevsky theorem, properties of harmonic
functions, the Dirichlet principle. Initial and boundary value problems for
hyperbolic, elliptic, and parabolic equations. Systems of equations.
Math 676 Advanced Ordinary Differential Equations 3 credits
Prerequisites: undergraduate differential equations and Math 545 or equivalent.
A rigorous treatment of the theory of systems of differential equations:
existence and uniqueness of solutions, dependence on initial conditions and
parameters. Linear systems, stability, and asymptotic behavior of solutions.
Nonlinear systems, perturbation of periodic solutions, and geometric theory of
systems of ODEs.
Math 677 Calculus of Variations 3 credits
Prerequisite: Math 676 or equivalent. Necessary conditions for existence of
extrema. Variation of a functional, Euler's equation, constrained extrema,
first integrals, Hamilton-Jacobi equation, quadratic functionals. Sufficient
conditions for the existence of extrema. Applications to mechanics.
Math 683 Functional Analysis 3 credits
Prerequisite: Math 645 or equivalent. Principles of linear analysis:
Hahn-Banach, uniform boundedness and closed graph theorems. Riesz
representation theorem; weak topologies; Riesz theory of compact operators.
Spectral theory of operators on Hilbert space. Applications to differential and
integral equations.
Math 685 Combinatorics 3 credits
Prerequisite: Math 545 or equivalent. Generating functions, principle of
inclusion-exclusion, pigeonhole principle, partitions. Polya's theory of
counting, graph theory and applications.
Math 687 Quantitative Analysis for Environmental Design Research 3
credits
Prerequisites: college level statistics course and permission of instructor.
Fundamental concepts in the theory of probability and statistics including
descriptive data analysis, inferential statistics, sampling theory, linear
regression and correlation, and analysis of variance. Also includes an
introduction to linear programming and nonlinear models concluding with some
discussion of optimization theory.
Math 689 Advanced Applied Mathematics II: ODEs 3 credits
Prerequisites: Math 545 and Math 631, or equivalent. A practical and
theoretical treatment of boundary value problems for ordinary differential
equations: generalized functions, Green's functions, spectral theory,
variational principles, and allied numerical procedures. Examples will be drawn
from applications in science and engineering.
Math 690 Advanced Applied Mathematics III: PDEs 3 credits
Prerequisite: Math 689 or equivalent. A practical and theoretical treatment of
initial and boundary value problems for partial differential equations: Green's
functions, spectral theory, variational principles, transform methods, and
allied numerical procedures. Examples will be drawn from applications in
science and engineering.
Math 691 Stochastic Processes with Applications 3 credits
Prerequisite: Math 662 or equivalent. Renewal theory, renewal reward processes
and applications. Homogeneous, non-homogeneous and compound Poisson processes
with illustrative applications. Introduction to Markov chains in discrete and
continuous time with selected applications.
Math 698 Sampling Theory 3 credits
Prerequisite: Math 662 or equivalent. Role of sample surveys. Sampling from
finite populations, the conceptual framework. Sampling designs, the
Horowitz-Thompson estimator of the population mean. Different sampling methods,
simple random sampling, stratified sampling, ratio and regression estimates,
cluster sampling, systematic sampling.
Math 699 Design and Analysis of Experiments 3 credits
Prerequisite: Math 662 or equivalent. Statistically designed experiments and
their importance in data analysis, industrial experiments. Role of
randomization. Fixed and random effect models and ANOVA, block design, latin
square design, factorial and fractional factorial designs and their analysis.
Math 700 Master's Project 3 credits
Prerequisites: matriculation for the master's degree and departmental approval.
Work must be initiated with the approval of a faculty member, who will be the
student's project advisor. Work of sufficient quality may qualify for extension
into a master's thesis, see Math 701.
Math 701 Master's Thesis 6 credits
Prerequisite: matriculation for the master's degree and departmental approval.
A student must register for a minimum of 3 credits per semester until
completion. The work will be carried out under the supervision of a designated
member of the faculty.
Math 707 Advanced Applied Mathematics IV: Special Topics 3 credits
Prerequisite: permission of the instructor. A current research topic of
interest to departmental faculty. Typical topics include: computational fluid
dynamics, theoretical fluid dynamics, acoustics, wave propagation, dynamical
systems, numerical analysis and scientific computation, theoretical and
numerical aspects of combustion, and various topics in statistics.
Math 711 Logic and Set Theory 3 credits
Prerequisite: permission of the instructor. Propositional calculus, predicate
calculus, first-order theories and concepts of consistency, completeness and
decidability. Theorems of Church, Kleene, Godel, Mostowski and Turing.
Axiomatic set theory according to von Neumann, Bernays, Godel, and others.
Recursive functions, effective computability and Turing machines.
Math 712 Numerical Methods II 3 credits
Prerequisites: Math 614 and introductory partial differential equations, or
equivalent. Numerical methods for the solution of initial and boundary value
problems for partial differential equations, with emphasis on finite difference
methods. Consistency, stability, convergence, and implementation are considered.
Math 720 Tensor Analysis 3 credits
Prerequisite: permission of the instructor. Review of vector analysis in
general curvilinear coordinates. Algebra and differential calculus of tensors.
Applications to differential geometry, analytical mechanics and mechanics of
continuous media. The choice of applications will be determined by the
interests of the class.
Math 730 Applied Algebra 3 credits
Prerequisite: Math 631 or equivalent. An introduction to groups, rings, fields
and their applications in science and engineering. Topics that are usually
emphasized include permutation groups, cyclic groups, polynomial algebras and
finite fields.
Math 745 Analysis II 3 credits
Prerequisite: Math 645 or equivalent. Lebesgue measure and integration,
including the Lebesgue dominated convergence theorem and Riesz-Fischer theorem.
Elements of Hilbert spaces and Lp-spaces. Fourier series and harmonic analysis.
Multivariate calculus.
Math 756 Complex Variables II 3 credits
Prerequisite: Math 656 or equivalent. Selected topics from: conformal mapping
and applications of the Schwarz-Christoffel transformation; applications of
calculus of residues; singularities, principle of the argument, Rouché's theorem, Mittag-Lefler's theorem, Casorati-Weierstrass theorem;
analytic continuation and applications, Schwarz reflection principle, monodromy
theorem, Wiener-Hopf technique; asymptotic expansion of integrals; integral
transform techniques; special functions.
Math 761 Statistical Reliability Theory and Applications 3 credits
Prerequisite: Math 662 or equivalent. Survival distributions, failure rate and
hazard functions, residual life. Common parametric families used in modeling
life data. Introduction to nonparametric ageing classes. Coherent structures,
fault tree analysis, redundancy and standby systems, system availability,
repairable systems, selected applications such as software reliability.
Math 762 Mathematical Statistics II 3 credits
Prerequisite: Math 662 or equivalent. Estimation and hypothesis testing.
Sufficiency, completeness, Rao-Cramer inequality, Neyman-Pearson lemma;
uniformly most powerful tests, nonparametric tests and likelihood ratio tests.
Regression and correlation, analysis of variance.
Math 771 Asymptotic Methods II 3 credits
Prerequisite: Math 671 or equivalent. Continuation of Math 671. Asymptotic
methods for the solution of PDEs, including: matched asymptotic expansions,
multiple scales, the WKB method or geometrical optics, and near-field far-field
expansions. Applications to elliptic, parabolic, and hyperbolic problems.
Further topics in the asymptotic expansion of integrals and the WKB method.
Emphasis on examples drawn from applications in science and engineering.
Math 790 Doctoral Dissertation and Research Credits as designated
Prerequisite: completion of the doctoral qualifying examination. A minimum of
36 credits is required of all candidates for the Ph.D. degree. Registration
between a minimum of 6 credits per semester and a maximum of 12 credits per
semester is determined by a designated thesis advisor. When 36 credits are
reached, students must continue to register for 3 credits each semester until
degree completion.
Math 791 Graduate Seminar Non-credit
Required each semester of all doctoral students and master's students receiving
departmental or research-based awards.
Math 792 Pre-Doctoral Research 3 credits
Prerequisite: departmental approval. For students admitted to the program
leading to the degree of Ph.D. in the mathematical sciences. Research is
performed under the supervision of a designated faculty member. If the work
culminates in doctoral research in the same area, up to 6 credits may be
counted toward Math 790. See Math 790.