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Graduate Catalog 2014-15

MATHEMATICS (MATH)

400-3 DEVELOPMENT OF MODERN MATHEMATICS. The development of mathematics since the discovery of calculus. Prerequisites: MATH 152; 223.

416a-i 1 to 3 each MATHEMATICS TOPICS FOR TEACHERS. (a) Analysis; (b) Algebra; (c) Number theory; (d) Probability and statistics; (e) Mathematical concepts; (f) Geometry; (g) History of mathematics; (h) Applied mathematics; (i) Logic and foundations. Students may earn a maximum of 6 hours in each section provided no topic is repeated. Does not count toward a concentration or minor in mathematics. Prerequisite: consent of instructor.

420-3 ABSTRACT ALGEBRA. Rings, fields, integral domains homomorphisms, factor rings, rings of polynomials, prime ideals, maximal ideals, extension fields, and vector spaces. Prerequisite: MATH 320 with a grade of C or better or consent of instructor.

421-3 LINEAR ALGEBRA II. Advanced study of vector spaces: Cayley-Hamilton Theorem, minimal and characteristic polynomials, eigenspaces, canonical forms, Lagrange-Sylvester Theorem, applications. Prerequisite: MATH 321 or consent of instructor.

423-3 COMBINATORICS AND GRAPH THEORY. Solving discrete problems. Counting techniques, combinatorial reasoning and modeling, generating functions and recurrence relations. Graphs: definitions, examples, basic properties, applications, and algorithms. Prerequisites: MATH 223; some knowledge of programming recommended.

430-3 A GEOMETRIC INTRODUCTION TO TOPOLOGY. Topological spaces and equivalence through the study of knots, links, surfaces, 3-manifolds and other selected topics. Prerequisite: MATH 350.

435-3 FOUNDATIONS FOR EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY. Points, lines, planes, space, separations, congruence, parallelism and similarity, non-Euclidean geometries, independence of the parallel axiom. Riemannian and Bolyai-Lobachevskian geometries. Prerequisites: MATH 250; 321; MATH 320 or 350, consent of instructor.

437-3 DIFFERENTIAL GEOMETRY. Curves and surfaces in Euclidean 3- space from the perspective of classical differential geometry. Topics include: Frenet frames, fundamental surface forms, geodesics, and the Gauss-Bonnet theorem. Prerequisites: MATH 250 and 321.

450-3 REAL ANALYSIS I. Differentiation and Riemann integration of functions of one variable. Taylor series. Improper integrals. Lebesgue measure and integration. Prerequisite: MATH 350.

451-3 INTRODUCTION TO COMPLEX ANALYSIS. Analytic functions, Cauchy-Riemann equations, harmonic functions, elements of conformal mapping, line integrals, Cauchy-Goursat theorem, Cauchy integral formula, power series, the residue theorem and applications. Prerequisite: MATH 350 with a grade of C or better or consent of instructor.

462-3 ENGINEERING NUMERICAL ANALYSIS. Polynomial interpolation and approximations, numerical integration, differentiation, direct and iterative methods for linear systems. Numerical solutions for ODE's and PDE's. MATLAB programming required. Prerequisites: MATH 250; 305; CS 140 or 141, or consent of instructor. Not for MATH majors.

464-3 PARTIAL DIFFERENTIAL EQUATIONS. Partial differential equations; Fourier series and integrals; wave equation; heat equation; Laplace equation; and Sturm-Liouville theory. Prerequisites: MATH 250, 305, and 321.

465-3 NUMERICAL ANALYSIS. Error analysis, solution of nonlinear equations, interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, solution of linear systems of equations. Prerequisites: MATH 305; CS 140 or 141.

466-3 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS. Direct and iterative methods for linear systems, approximation of eigenvalues, solution of nonlinear systems, numerical solution of ODE and PDE boundary value problems, function approximation. Prerequisites: MATH 305; 321; CS 140 or 141.

495a-g 1 to 3 each INDEPENDENT STUDY. Research and reading in specified area of interest. (a) Algebra; (b) Geometry; (c) Analysis; (d) Mathematics education; (e) Logic and foundations; (f) Topology; (g) Numerical analysis. May be repeated to a maximum of 9 hours provided no topic is repeated and not more than 3 hours are accumulated in a single segment nor more than 6 hours in one semester. Prerequisites: written consent of adviser and instructor.

501-3 DIFFERENTIAL EQUATIONS AND THE FOURIER ANALYSIS. Brief review of ODE. Legendre and Bessel functions. Fourier series, integrals, and transforms. Wave equation, heat equation, Laplace equation. Not for MATH majors. Prerequisite: MATH 250, MATH 305, or consent of instructor.

502-3 ADVANCED CALCULUS FOR ENGINEERS. Review of vector calculus, Green's theorem, Gauss' theorem, and Stokes' theorem. Complex analysis up to contour integrals and residue theorem. Not for MATH majors. Prerequisite: MATH 250 or consent of instructor.

520-3 TOPICS IN ALGEBRA. Advanced topics in algebra. Groups: Sylow theorems; simple groups. Fields: automorphisms, elementary Galois theory. Rings: noncommutative rings, Dedekind domains. Content may vary from year to year. May be repeated to a maximum of 9 hours provided no topic is repeated. Prerequisite: MATH 420.

531-3 ALEBRAIC CONTENT, PEDAGOGY, AND CONNECTIONS. focused look at algebraic content, best practices in pedagogy, and connections to other areas. Within the Department of Mathematics and Statistics credit can only be earned for the Post Secondary Mathematics option. Prerequisite: MATH 250 with a C or better or consent of instructor.

532-3 GEOMETRIC CONTENT, PEDAGOGY, AND CONNECTIONS. A focused look at geometric content, best practices in pedagogy, and connections to other areas. Within the Department of Mathematics and Statistics credit can only be earned for the Post Secondary Mathematics option. Prerequisite: MATH 250 with a C or better or consent of instructor.

533-3 DISCRETE MATHEMATICS CONTENT, PEDAGOGY, AND CONNECTIONS. A focused look at discrete mathematics content, best practices in pedagogy, and connections to other areas. Within the Department of Mathematics and Statistics credit can only be earned for the Post Secondary Mathematics option. Prerequisite: MATH 250 with a C or better or consent of instructor.

534-3 CALCULUS CONTENT, PEDAGOGY, AND CONNECTIONS. A focused look at calculus content including limits, differentiation, integration, and series; best practices in pedagogy, and connections to other areas. Within the Department of Mathematics and Statistics credit can only be earned for the Postsecondary Mathematics Education specialization. Prerequisites: MATH 350 with a C or better or consent of instructor.

545-3 REAL ANALYSIS II. Riemann, Riemann-Stieltjes, and Lebesgue integrals. Differentiation of functions of n variables. Multiple integrals. Measure and probability. Differential forms, Stokes' Theorem. Prerequisites: MATH 321 and 450.

550-3 TOPICS IN ANALYSIS. Advanced topics in analysis. Metric and topological spaces; completeness; compactness; connectedness; Hilbert and Banach spaces; measure theory and integration; probability theory. May be repeated to a maximum of 9 hours provided no topic is repeated. Prerequisite: MATH 545.

551-3 TOPICS IN COMPLEX ANALYSIS. Riemann mapping theorem, analytic continuation, theorems of Weierstrass and Mittag-Leffler. Content may vary from year to year. May be repeated to a maximum of 6 hours provided no topic is repeated. Prerequisites: MATH 451; 545.

552-3 THEORY OF ORDINARY DIFFERENTIAL EQUATIONS. Existence and uniqueness theorem, dynamical systems, stability, bifurcation theory, boundary value problems. Prerequisites: MATH 350; 421.

555-3 FUNCTIONAL ANALYSIS WITH APPLICATIONS. Normed and Banach spaces, inner product and Hilbert spaces, Open Mapping and Closed Graph Theorem, Hahn-Banach Theorem, dual spaces and weak topology. Prerequisite: MATH 421, 450.

563-3 OPTIMAL CONTROL THEORY. (Same as ECE 563 and ME 563) Description of system and evaluation of its performance; dynamic programming, calculus of variations and Pontryagin's minimum principle; iterative numerical techniques. Prerequisite: MATH 305 or ECE 365 or ME 450.

565-3 ADVANCED NUMERICAL ANALYSIS. Rigorous treatment of topics in numerical analysis including function approximation, numerical solutions to ordinary and partial differential equations. Convergence and stability of finite difference methods. Prerequisites: MATH 321; 350; 465; 466.

567-3 TOPICS IN APPLIED MATHEMATICAL ANALYSIS. Topics from the following areas: Fourier theory and applications, applied functional analysis, asymptotic analysis, perturbation theory, control theory, theory of equilibrium, partial differential equations. May be repeated to a maximum of 12 hours provided no topic is repeated. Prerequisites: MATH 421; 451; 545.

590a-g 1 to 3 SEMINAR. Intensive study of selected mathematical topics. (a) Algebra; (b) Geometry; (c) Analysis; (d) Mathematics education; (e) Logic and foundations; (f) Topology; (g) Numerical analysis. Each segment may be repeated to a maximum of 6 hours provided no topic is repeated. Prerequisites: written consent of adviser and instructor.

595a-g 1 to 3 SPECIAL PROJECT. Intensive study that may be used to satisfy research paper requirements for MS degree in mathematics. (a) Algebra; (b) Geometry; (c) Analysis; (d) Mathematics education; (e) Logic and foundations; (f) Topology; (g) Numerical analysis. May be repeated to a maximum of 7 hours. Prerequisite: written consent of research adviser.

599-1 to 6 THESIS. Directed research to satisfy thesis requirement. May be repeated to a maximum of 6 hours. Prerequisite: written consent of thesis adviser.

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