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MATHEMATICS AND STATISTICS 251
540. Stochastic Processes with Applications. (3)
(Also offered as STAT 565.) Markov chains and processes
with applications. Classification of states. Decompositions.
Stationary distributions. Probability of absorption, the gambler’s
ruin and mean time problems. Queuing and branching
processes. Introduction to continuous time Markov processes.
Jump processes and Brownian motion.
Prerequisite: 527. {Offered on demand}
541. Advanced Probability. (3)
(Also offered as STAT 567.) A measure theoretic introduction
to probability theory. Construction of probability measures.
Distribution and characteristic functions, independence and
zero-one laws. Sequences of independent random variables,
strong law of large numbers and central limit theorem.
Conditional expectation. Martingales.
Prerequisite: 563. {Alternate Springs}
549. Selected Topics in Probability Theory. (3, no
limit) ∆
(Also offered as STAT 569.)
551. Problems. (1-3, no limit) †
557. Selected Topics in Numerical Analysis. (3, no
limit) ∆
(Also offered as CS 557.) Possible topics include approximation
theory, two point boundary value problems, quadrature,
integral equations and roots of nonlinear equations.
561. Functions of a Complex Variable I. (3)
Analyticity, Cauchy theorem and formulas, Taylor and Laurent
series, singularities and residues, conformal mapping,
selected topics.
Prerequisite: 311 or 402. {Fall}
562. Functions of a Complex Variable II. (3)
The Mittag-Leffler theorem, series and product expansions,
introduction to asymptotics and the properties of the gamma
and zeta functions. The Riemann mapping theorem, harmonic
functions and Dirichlet’s problem. Introduction to
elliptic functions. Selected topics.
Prerequisite: 561. {Fall}
563. Measure Theory. (3)
Functions of one and several real variables, measure theory,
starting with Lebesque measure and integration. Product
measures. Measure on spaces of functions.
Prerequisite: 401 or 510. {Fall}
565. Harmonic Analysis. (3)
Fourier analysis on the circle, real line and on compact and
locally compact groups.
Prerequisite: 563. {Offered upon demand}
568. Stochastic Differential Equations. (3)
Basic theory of stochastic differential equations with applications.
The presentation will be at a level accessible to scientists,
engineers and applied mathematicians.
Prerequisites: 316, 441 and some familiarity with elementary
PDEs. {Offered upon demand}
569. Selected Topics in Analysis. (3, no limit) ∆
572./472. Fourier Analysis and Wavelets. (3)
Discrete Fourier and Wavelet Transform. Fourier series and
integrals. Expansions in series of orthogond wavelets and
other functions. Multiresolution and time/frequency analysis.
Applications to signal processing and statistics.
Prerequisite: 314, 321 or 401 or permission of the instructor.
{Offered upon demand}
573. Partial Differential Equations. (3)
Equations of first order, classification of equations and systems,
elliptic equations and introduction to potential theory,
hyperbolic equations and systems, parabolic equations.
Prerequisite: 463. {Alternate Falls}
576. Numerical Linear Algebra. (3)
Selected advanced topics in numerical linear algebra.
Prerequisite: 504. {Alternate Springs}
577. Numerical Ordinary Differential Equations. (3)
Numerical methods for initial value and/or boundary value
problems.
Prerequisites: 462, 504, 505. {Offered upon demand}
578. Numerical Partial Differential Equations. (3)
Introduction to the numerical analysis of partial differential
equations.
Prerequisites: 463, 504, 505. {Alternate Falls}
579. Selected Topics in Applied Mathematics. (3, no
limit) ∆
581. Functional Analysis I. (3)
Normed vector spaces, including Hilbert and Banach spaces.
Linear operators on these spaces, with an emphasis on
applications.
Prerequisite: 510. {Offered upon demand}
582. Functional Analysis II. (3)
Advanced topics in function spaces and linear operators.
Prerequisite: 581.
583. Methods of Applied Mathematics I. (3)
Approximation in Hilbert spaces, basic operator theory,
integral equations, distribution theory, Green’s functions, differential
operators, boundary value problems and nonlinear
problems.
Prerequisites: 312, 314, 316, 401. {Alternate Falls}
584. Methods of Applied Mathematics II. (3)
Eigenfunction expansions for ordinary and partial differential
operators, Euler-Lagrange equations, Hamilton’s principle, calculus
of variations, brief complex variable theory, special functions,
transform and spectral theory, asymptotic expansions.
Prerequisites: 312 and 314 and 316 and 401. {Alternate
Springs}
598. Practicum. (1-6 to a maximum of 6) †
Practicum involves a project of an applied nature which may
be done in conjunction with an industrial laboratory, a research
institution or another department of the University. It is expected
the student will become acquainted with a field of application in
science or engineering and complete a project of use and interest
to workers in that field. A final written report is required.
ARTS AND
SCIENCES
570. Singular Perturbations. (3)
Singularly perturbed boundary value problems, layer type
expansions and matching. Initial value problems and multiscaling
methods for ordinary and partial differential equations.
Phase plane and qualitative ideas. Applications. Perturbations
of Hamiltonian systems.
Prerequisites: 462, 463. {Alternate Springs}
571. Ordinary Differential Equations. (3)
Existence and uniqueness of solutions, linear systems,
asymptotic behavior of solutions to nonlinear systems, integral
manifolds and linearizations, perturbation theory, bifurcation
theory, dichotomies for solutions of linear systems.
Prerequisite: 462. {Alternate Springs}
599. Master’s Thesis. (1-6)
Offered on a CR/NC basis only.
605. Graduate Colloquium. (1-1 to a maximum of 4)
Students present their current research.
639. Seminar in Geometry and Topology. (1-3) ∆
May be repeated for credit, no limit.
649. Seminar in Probability and Statistics. (1-3, no
limit) ∆
(Also offered as STAT 649.)
650. Reading and Research. (1-6 to a maximum of 12) †
669. Seminar in Analysis. (1-3, no limit) ∆
UNM CATALOG 2006–2007 Symbols, page 611.