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Statistical methodology of survey sampling. Data analysis and estimation methods for various experimental designs; fixed or random sample sizes, pre-and/or post-stratified samples, and multistage sampling. Estimates of totals, means, ratios and proportions with methods of estimating variances of such estimates. Prerequisite: A post-calculus probability or statistics course.
Methods requiring few assumptions about the populations sampled. Topics include quantile tests, tolerance limits, the sign test, contingency tables, rank-sum tests, and rank correlation. Prerequisite: Math 628 or permission of the instructor.
Counting problems, with an introduction to Polya's theory; Mobius functions; transversal theory, Ramsey's theorem; Sperner's theorem and related results.
Graphs; trees; connectivity; Menger's theorem; eulerian and hamiltonian graphs; planarity; coloring of graphs; factorization of graphs; matching theory; alternating chain methods; introduction to matroids with applications to graph theory.
A mathematical introduction to premeasure-theoretic probability. Topics include probability spaces, conditional probabilities and independent events, random variables and probability distributions, special discrete and continuous distributions with emphasis on parametric families used in applications, the distribution problem for functions of random variables, sequences of independent random variables, laws of large numbers, and the central limit theorem. Prerequisite: Math 123 and graduate standing or permission of the instructor.
An introduction to the mathematical methods of deterministic control theory is given by considering some specific examples and the general theory. The methods include dynamic programing the calculus of variations, and Pontryagin's maximum principle. Various problems of linear control systems, e.g., the linear regulator problem, are solved. Prerequisite Math 320 or equivalent.
Divisibility, the theory of congruences, primitive roots and indices, the quadratic reciprocity law, arithmetical functions and miscellaneous additional topics. Prerequisite: Math 123 or equivalent.
The identification and control of discrete and continuous time stochastic systems is investigated. Stochastic processes (Markov chains, Brownian motion), stochastic integrals, the Ito differential rule, stochastic differential equations, martingales, and estimation techniques are introduced, as well as some elements of optimal control theory. Some specific applications and simulation results of stochastic adaptive control theory are also presented. Prerequisite: Math 627.
Mathematics 765 and 766 are theoretical courses on the fundamental concepts of analysis and the methods of proof. These two courses include the concept of a real number; limits, continuity, and uniform convergence; derivatives and integrals of functions of one and of several real variables. Prerequisite: Multivariable calculus for Math765, and Math 765 for Math 766.
Finite and divided differences. Interpolation, numerical differentiation and integration. Gaussian quadrature. Numerical integration of ordinary differential equations. Curve fitting. (Sameas CS 781: ) Prerequisite: Math 320 and knowledge of a programming language.
Direct and iterative methods for solving systems of linear equations. Numerical solution of partial differential equations. Numerical determination of eigen-vectors and eigenvalues. Solution of nonlinear equations. (Same as CS 782: ) Prerequisite: Math 681 or 781.
Finite Difference methods applied to particular initial-value problems (both parabolic and hyperbolic), to illustrate the concepts of convergence and stability and to provide a background for treating more complicated problems arising in engineering and physics. Finite difference methods for elliptic boundary-value problems, with a discussion of convergence and methods for solving the resulting algebraic system. Variational methods for elliptic problems. Prerequisite: Math 647 or equivalent.
A theoretical course on the fundamental concepts and theorems oflinear algebra. Topics covered are: vector space, basis, dimension, subspace, norm, inner product, Banach space, Hilbert space, orthonormal basis, positive definite matrix, minimal polynomial, diagonalization and other canonical forms, Cayley-Hamilton, spectral radius, dual space, quotient space. Prerequisite: Math 590.
Mathematics 791 and 792 include the following topics: the number system; groups, rings and fields; matrices and linear transformations; lattices; Galois theory; linear algebras. Prerequisite: Multivariable calculus for Math 791 and Math 791 for Math 792.
Arranged as needed to present appropriate material for groups of students. May be repeated for credit. Prerequisite: variable.
Directed reading on a topic chosen by the student with the advice of an instructor. May be repeated for additional credit. Consent of the department required for enrollment.
Cauchy's theorem and contour integration; the argument principle; maximum modulus principle; Schwarz symmetry principle; analytic continuation; monodromy theorem; applications to the gamma functions and Riemann's zeta functions; entire and meromorphic functions; conformal mapping; Riemann mapping theorem; univalent functions. Prerequisite: Math 766 or concurrently with Math 766 for Math 800, and Math 800 for Math 801.
Axiomatic set theory; transfinite induction; regularity and choice; ordinal and cardinal arithmetic; miscellaneous additional topics (e.g., extra axioms such as GCH or MA; infinite combinatorics; large cardinals). Prerequisite: MATH 765 or MATH 791, or concurrent enrollment in Math 765 or Math 791, or equivalent evidence of mathematical maturity.
Measurable spaces and functions. Measure spaces and integration. Extensions of set functions, outer measures, Lebesque measure. Signed and complex measures. Differentiation of set functions. Miscellaneous additional topics and applications. Prereq: Math 766 / Math 810, Math 810 / Math 811.
General topology. Set theory; topological spaces; connected sets; continuous functions; generalized convergence; product and quotient spaces; embedding in cubes; metric spaces and metrization; compact spaces; function spaces. Prerequisite: Math 765.
Fundamental groups; covering spaces; simplicial complexes and polyhedra; simplicial homology theory; mapping into spheres; simplicial approximation; relative homology groups; and exact sequences. Prerequisite: Math 791 and 820.
A study of some structures, theorems, and techniques in algebra whose use has become common in many branches of mathematics. Prerequisite:Math 792 for Math 830 and Math 830 for Math 831.
Multilinear algebra of finite dimensional vector spaces over fields; differentiable structures and tangent and tensor bundles; differentiable mappings and differentials; exterior differential forms; curves and surfaces as differentiable manifolds; affine connections and covariant differentiation; Riemannian manifolds. Prerequisite: Math 792 and 820 for Math 840.
Markov chains; Markov processes; diffusion processes; stationary processes. Emphasis is placed on applications: random walks; branching theory; Brownian motion; Poisson process; birth and death processes. Prerequisites: Math 627 and Math 765.
The general linear hypothesis with fixed effects; the Gauss-Markovtheorem, confidence ellipsoids, and tests under normal theory; multiple comparisons and the effect of departures from the underlying assumptions; analysis of variance for various experimental designs and analysis of covariance. Prerequisites: Math 628 and either linear algebra or Math 792.
The multivariate normal distribution; tests of hypotheses on means and covariance matrices; estimation; correlation; multivariate analysis of variance; principal components; canonical correlation. Prerequisite: Math 628 and either Math 590 or Math 792.
Algebraic sets, varieties, plane curves, morphisms and rational maps, resolution of singularities, Reimann-Roch theorem. Prerequisite: MATH 791 and MATH 792.
Injective and projective resolutions, homological dimension, chain complexes and derived functors (including Tor and Ext). Prerequisite: MATH 830 and MATH 831, or consent of instructor.
General properties of Lie groups, closed subgroups, one-parameter subgroups, homogeneous spaces, Lie bracket, Lie algebras, exponential map, structure of semisimple Lie algebras, invariant forms, Maurer-Cartan equation, covering groups, spinor groups. Prerequisite: MATH 791 and MATH 820.
Various topics will be discussed. These topics will include some of the following: Cardinal functions of topological spaces, trees and linear orders, normality of products, subspaces of Stone-Cech remainders, Infinitary Ramsey theory, generalizations of metrizability and paracompactness, and pathological subsets of the real line. Prerequisite: Math 820.
Probability measures, random variables, distribution functions, characteristic functions, types of convergence, central limit theorem. Laws of large numbers and other limit theorems. Conditional probability, Markov processes, and other topics in the theory of stochastic processes. Prerequisite: Math 811.
The course will cover topics in classical theory of PDE using the method of energy type inequalities. As time permits some topics of current interest will be outlined (Pseudodifferential operators, nonlinear problems, etc.).
Topological vector spaces, Banach spaces, basic principles of functional analysis. Weak and weak* topologies, operators and adjoints. Hilbert spaces, elements of spectral theory. Locally convex spaces. Duality and related topics. Applications. Prerequisite: Math 810 and Math 820 or concurrent with Math 820 for Math 960, and Math 960 for Math 961.
Courses offered in the recent past include: Several Complex Variables, Commutative Algebra, C*-Algebras, Stochastic Control, Fourier Analysis, Topological Groups, Harmonic Analysis.
Department of Mathematics
University of Kansas
405 Snow Hall
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Lawrence, Kansas 66045-7523
Phone: (785) 864-3651
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E-mail: kumath@math.ku.edu
