Mathematics

The faculty has approval to offer the following courses in the academic years 1999-2000 and 2000-2001; however, not all courses are taught each semester or summer session. Students should consult the Course Schedule to determine which courses and topics will be offered during a particular semester or summer session. The Course Schedule may also reflect changes that have been made to the courses listed here since this catalog was published.

Unless otherwise stated below, each course meets for three lecture hours a week for one semester.

Mathematics: M

380C. Algebra.
A survey of algebraic structures, including groups, fields, rings, and modules. Mathematics 380C and 680CA may not both be counted. Mathematics 380C and 680CB may both be counted. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

380D. Algebra.
Continuation of Mathematics 380C. Mathematics 680CB and 380D may not both be counted. Prerequisite: Graduate standing, consent of instructor or the graduate adviser, and Mathematics 380C (or 680CA).

381C. Real Analysis.
Same as Computational and Applied Mathematics 381R. Measure and integration over abstract spaces; Lebesgue's theory of integration and differentiation on the real line. Only one of the following may be counted: Computational and Applied Mathematics 681EA, Mathematics 381C, 681CA. Mathematics 381C and 681CB may both be counted. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

381D. Complex Analysis.
Same as Computational and Applied Mathematics 381D. Introduction to complex analysis. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

381E. Functional Analysis.
Same as Computational and Applied Mathematics 381S. Introduction to functional analysis. Only one of the following may be counted: Computational and Applied Mathematics 681EB, Mathematics 381E, 681CB. Prerequisite: Graduate standing, consent of instructor or the graduate adviser, and Computational and Applied Mathematics 381R (or 681EA) or Mathematics 381C (or 681CA).

382C. Topology.
Mathematics 382C and 682CA may not both be counted. Mathematics 382C and 682CB may both be counted. Prerequisite: Graduate standing, an undergraduate course in topology, and consent of instructor or the graduate adviser.

382D. Differential Topology.
Continuation of Mathematics 382C. Mathematics 682CB and 382D may not both be counted. Mathematics 382D and 682DA may both be counted; Mathematics 382D and 682DB may both be counted. Prerequisite: Graduate standing, consent of instructor or the graduate adviser, and Mathematics 382C (or 682CA).

382E. Algebraic Topology.
Mathematics 682DA and 382E (Algebraic Topology) may not both be counted. Mathematics 382E (Algebraic Topology) and 382E (Differential Geometry) may both be counted. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

382F. Algebraic Topology.
Continuation of Mathematics 382E. Mathematics 682DB and 382F may not both be counted. Prerequisite: Graduate standing, consent of instructor, or the graduate adviser, and Mathematics 382E (Algebraic Topology) (or 682DA).

382G. Differential Geometry.
Mathematics 382E (Differential Geometry) and 382G may not both be counted. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

383C. Methods of Applied Mathematics.
Same as Computational and Applied Mathematics 385C. Topics include basic normed linear space theory; fixed-point theorems and applications to differential and integral equations; Hilbert spaces and the spectral theorem; applications to Sturm-Liouville problems; approximation and computational methods such as the Galerkin, Rayleigh-Ritz, and Newton procedures. Only one of the following may be counted: Computational and Applied Mathematics 385, 393C (Topic: Methods of Applied Mathematics I), Mathematics 383C (Methods of Applied Mathematics), 685CA. Mathematics 383C (Methods of Applied Mathematics) and 383C (Numerical Analysis: Linear Algebra) may both be counted.Prerequisite: Graduate standing.

383D. Methods of Applied Mathematics.
Same as Computational and Applied Mathematics 385D. Topics include distributions, fundamental solutions of partial differential equations, the Schwartz space and tempered distributions, Fourier transform, Plancherel theorem, Green's functions, Sobolev spaces, weak solutions, differential calculus in normed spaces, implicit function theorems, applications to nonlinear equations, smooth variational problems, applications to classical mechanics, constrained variational problems. Mathematics 383D (Methods of Applied Mathematics) and 685CB may not both be counted. Mathematics 383D (Methods of Applied Mathematics) and 383D (Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations) may both be counted. Prerequisite: Graduate standing and Mathematics 383C (Methods of Applied Mathematics) (or 685CA).

383E. Numerical Analysis: Linear Algebra.
Same as Computational and Applied Mathematics 383C and Computer Sciences 383C. Survey of numerical methods in linear algebra: floating-point computation, solution of linear equations, least squares problems, algebraic eigenvalue problems. Mathematics 383C (Numerical Analysis: Linear Algebra) and 383E may not both be counted. Prerequisite: Graduate standing, either consent of instructor or Mathematics 311 or 340L, and either Mathematics 368K or Computer Sciences 367.

383F. Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations.
Same as Computational and Applied Mathematics 383D and Computer Sciences 383D. Survey of numerical methods for interpolation, functional approximation, integration, and solution of differential equations. Mathematics 383D (Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations) and 383F may not both be counted. Prerequisite: Graduate standing; either consent of instructor or Mathematics 427K and 365C; and Computational and Applied Mathematics 383C, Computer Sciences 383C, or Mathematics 383E (or 383C [Topic: Numerical Analysis: Linear Algebra]).

383G. Numerical Treatment of Differential Equations.
Same as Computational and Applied Mathematics 386K and Computer Sciences 386K. The analysis of numerical methods for solving ordinary and partial differential equations. Mathematics 383G and 386K may not both be counted. Prerequisite: Graduate standing; and Computational and Applied Mathematics 383D, Computer Sciences 383D, Mathematics 368K, 383F (or 383D [Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations]), or consent of instructor.

384C. Mathematical Statistics.
Same as Computational and Applied Mathematics 384R. General theory of mathematical statistics. Hypothesis testing, estimation, decision theory. Only one of the following may be counted: Computational and Applied Mathematics 684DA, Mathematics 384C, 684DA. Mathematics 384C and 684CA may both be counted; Mathematics 384C and 684CB may both be counted. Prerequisite: Graduate standing and consent of instructor.

384D. Mathematical Statistics.
Same as Computational and Applied Mathematics 384S. Continuation of Mathematics 384C. Only one of the following may be counted: Computational and Applied Mathematics 684DB, Mathematics 384D, 684DB. Prerequisite: Graduate standing, consent of instructor, and Computational and Applied Mathematics 384R (or 684DA) or Mathematics 384C (or 684DA).

384E. Analysis of Variance and Design of Experiments.
Analysis of variance, including the one-way layout and multifactor experiments; mixed, fixed, and random models; crossed and nested classifications. Mathematics 384E and 684EA may not both be counted. Mathematics 384E and 684EB may both be counted. Prerequisite: Graduate standing and consent of instructor.

384F. Analysis of Variance and Design of Experiments.
Experiments with factors at two or three levels, confounding, fractional factorials, response surface methodology. Mathematics 684EB and 384F (Analysis of Variance and Design of Experiments) may not both be counted. Mathematics 384F (Analysis of Variance and Design of Experiments) and 384F (Regression Analysis) may both be counted. Prerequisite: Graduate standing, consent of instructor, and Mathematics 384E (or 684EA).

384G. Regression Analysis.
Fitting linear models to data by the method of least squares, choosing best subsets of predictors, and related materials. Mathematics 384F (Regression Analysis) and 384G (Regression Analysis) may not both be counted. Mathematics 384G (Multivariate Statistical Analysis) and 384G (Regression Analysis) may both be counted. Prerequisite: Graduate standing and consent of instructor.

384H. Multivariate Statistical Analysis.
Introduction to the general multivariate linear model; a selection of techniques, such as principle component, factor, and discriminant analysis. Mathematics 384G (Multivariate Statistical Analysis) and 384H may not both be counted. Prerequisite: Graduate standing and consent of instructor.

385C. Theory of Probability.
Same as Computational and Applied Mathematics 384K. Only one of the following may be counted: Computational and Applied Mathematics 684CA, Mathematics 684CA, 385C. Mathematics 385C and 685CA may both be counted; Mathematics 385C and 685CB may both be counted. Prerequisite: Graduate standing and consent of instructor.

385D. Theory of Probability.
Same as Computational and Applied Mathematics 384L. Continuation of Mathematics 385C. Only one of the following may be counted: Computational and Applied Mathematics 684CB, Mathematics 684CB, 385D. Prerequisite: Graduate standing, consent of instructor, and Computational and Applied Mathematics 384K (or 684CA) or Mathematics 385C (or 684CA).

389J. Probability Models with Actuarial Applications.
Probability models with actuarial applications, including Markov chains, Brownian motion, the Black-Scholes formula, frequency-of-loss and severity-of-loss random variables, compound distributions, and ruin theory. With Mathematics 389U and 389V, covers the syllabus for exam #3 of the Society of Actuaries and the Casualty Actuarial Society. Four lecture hours a week for one semester. Prerequisite: Graduate standing and Mathematics 362K.

389L. Actuarial Mathematics I.
Probabilistic theory of single-event losses, with applications to insurance. With Mathematics 389M, includes the syllabus for the Society of Actuaries Exam 150. Mathematics 469K and 389L may not both be counted. Prerequisite: Graduate standing, Actuarial Foundations 309, Mathematics 362K, and consent of the director of the concentration in actuarial studies.

389M. Actuarial Mathematics II.
Probabilistic theory of multiple-event losses, with applications to insurance. With Mathematics 389L, includes the syllabus for the Society of Actuaries Exam 150. Mathematics 469L and 389M may not both be counted. Prerequisite: Graduate standing, Mathematics 389L, and consent of the director of the concentration in actuarial studies.

389P. Actuarial Statistical Estimates.
Statistical estimates for frequency-of-loss and severity-of-loss random variables; credibility theory; statistics of simulation. Covers 40 percent of the syllabus for exam #4 of the Society of Actuaries and the Casualty Actuarial Society. Prerequisite: Graduate standing, Mathematics 389J or 439J, and Mathematics 358K or 378K.

389U. Actuarial Contingent Payments I.
Simulation of random samples, single-status survival models, present-value random variables for life insurance and annuities. With Mathematics 389J and 389V, covers the syllabus for exam #3 of the Society of Actuaries and the Casualty Actuarial Society. Prerequisite: Graduate standing and Mathematics 362K. Actuarial Foundations 309 is recommended.

389V. Actuarial Contingent Payments II.
Mathematical analysis of insurance premiums, reserves, multiple-status survival models, multiple-decrement survival models; applications to such areas as life insurance and property/casualty insurance. With Mathematics 389J and 389U, covers the syllabus for exam #3 of the Society of Actuaries and the Casualty Actuarial Society. Four lecture hours a week for one semester. Prerequisite: Graduate standing, and Mathematics 389U or 339U.

390C. Topics in Algebra.
Recent topics have included algebraic geometry, number theory, algebraic curves, algebraic number theory, algebraic functions, rational curves on varieties, homological algebra. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

391C. Topics in Analysis.
Same as Computational and Applied Mathematics 391C. Recent topics have included measure and integration, real variables, complex analysis, functional analysis, ordinary differential equations, partial differential equations, integral transforms, operator theory, approximation theory, abstract harmonic analysis. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

392C. Topics in Topology.
Recent topics have included algebraic topology, differential topology, geometric topology, Lie groups. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

393C. Topics in Applied Mathematics.
Same as Computational and Applied Mathematics 393C. Recent topics have included quantum mechanics, statistical physics, ergodic theory, group representations, statistical mechanics, quantum field theory, introductory partial differential equations, monotone operators and partial differential equations, Hilbert space methods for partial differential equations, Hamiltonian dynamics, nonlinear functional analysis, Euler and Navier-Stokes equations, microlocal calculus and spectral asymptotics, calculus of variations. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

393D. Topics in Numerical Analysis.
Same as Computational and Applied Mathematics 393D and Computer Sciences 393D. Recent topics have included numerical methods in ordinary differential equations, numerical methods in partial differential equations, computational problems in linear algebra, numerical solution of systems of equations, numerical methods in functional approximation, numerical integration. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

393N. Numerical Solution of Elliptic Partial Differential Equations.
Same as Computational and Applied Mathematics 393M and Computer Sciences 393N. The numerical solution of large systems of linear algebraic equations arising in the solution of elliptic partial differential equations by discretization methods. Prerequisite: Graduate standing; and Computational and Applied Mathematics 386K, Computer Sciences 386K, Mathematics 383G (or 386K), or consent of instructor.

394C. Topics in Probability and Statistics.
Same as Computational and Applied Mathematics 394C. Recent topics have included nonparametric statistics, advanced probability. May be repeated for credit when the topics vary. Some topics are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

395C. Topics in Logic and Foundations.
Recent topics have included set theory, model theory, proof theory, axiomatic theorem proving, automatic theorem proving, foundations of mathematics, recursion theory. May be repeated for credit when the topics vary. Some topics are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

396C. Topics in Mathematics.
Recent topics have included set theory, history of mathematics. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

197S, 397S. Seminar in Mathematics.
One or three lecture hours a week for one semester. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Prerequisite: Graduate standing and consent of instructor.

698. Thesis.
The equivalent of three lecture hours a week for two semesters. Offered on the letter-grade basis only. Prerequisite: For 698A, graduate standing in mathematics and consent of the graduate adviser; for 698B, Mathematics 698A.

398R. Master's Report.
Preparation of a report to fulfill the requirement for the master's degree under the report option. The equivalent of three lecture hours a week for one semester. Offered on the letter-grade basis only. Prerequisite: Graduate standing in mathematics and consent of the supervising professor and the graduate adviser.

398T. Supervised Teaching in Mathematics.
Prerequisite: Graduate standing and appointment as a teaching assistant.

399R, 699R, 999R. Dissertation.
Offered on the letter-grade basis only. Prerequisite: Admission to candidacy for the doctoral degree.

399W, 699W, 999W. Dissertation.
Offered on the letter-grade basis only. Prerequisite: Mathematics 399R, 699R, or 999R.

About the Program: Mathematics

Contents |  Chapter 1 |  Chapter 2 |  Chapter 3
Chapter 4 |  Chapter 5 |  Appendix

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