# Graduate Mathematical Courses

##### MA 4133/6133. Discrete Mathematics. (3)

(Prerequisites: MA 3163 or consent of instructor). Three hours lecture. Sets, relations, functions, combinatorics, review of group and ring theory, Burnside’s theorem, Polya’s counting theory, group codes, finite fields, cyclic codes, and error-correcting codes.

##### MA 4143/6143. Graph Theory. (3)

(Prerequisites: MA 3113 or consent of instructor). Three hours lecture. Basic concepts, graphs, and matrices, algebraic graph theory, planarity and nonplanarity, Hamiltonian graphs, digraphs, network flows, and applications.

##### MA 4153/6153. Matrices and Linear Algebra. (3)

(Prerequisites: MA 3113 and MA 3253). Three hours lecture. Linear transformations and matrices; eigenvalues and similarity transformations; linear functionals, bilinear and quadratic forms; orthogonal and unitary transformations; normal matrices; applications of linear algebra.

##### MA 4163/6163. Group Theory. (3)

(Prerequisite: MA 3163 or consent of the instructor). Three hours lecture. Elementary properties: normal subgroups; factor groups; homomorphisms and isomorphisms; Abelian groups; Sylow theorems; composition series; solvable groups.

##### MA 4173/6173. Number Theory. (3)

(Prerequisite: MA 3113). Three hours lecture. Divisibility: congruences; quadratic reciprocity; Diophantine equations; continued fractions.

##### MA 4213. Senior Seminar in Mathematics. (3)

(Prerequisites: MA 3163 and MA 3253 and MA 4633). Three hours lecture. Students explore topics in current mathematical research, write expository articles, and give oral presentations. Refinement of specialized writing skills needed for effective mathematical communication.

##### MA 4243/6243 Data Analysis I. (3)

(Prerequisite: MA 2743. Co-requisite: MA 3113). Three hours lecture. Data description and descriptive statistics, probability and probability distributions, parametric one-sample and two-sample inference procedures, simple linear regressions, one-way ANOVA. Use of SAS. (Same as ST 4243/6243.)

##### MA 4253/6253 Data Analysis II. (3)

(Prerequisites: MA 4243/6243 and MA 3113). Three hours lecture. Multiple linear regression; fixed, mixed and random effect models; block designs; two-factor analysis of variance; three-factor analysis of variance; analysis of covariance. Use of SAS. (Same as ST 4253/6253.)

##### MA 4313/6313. Numerical Analysis I. (3)

(Prerequisites: CSE 1213, MA 3113, and MA 2743). Three hours lecture. Matrix operations; error analysis; norms of vectors and matrices; transformations; matrix functions; numerical solutions of systems of linear equations; stability; matrix inversion; eigen value problems; approximations.

##### MA 4323/6323. Numerical Analysis II. (3)

(Prerequisites: CSE 1213 or equivalent. MA 3113 and MA 3253). Three hours lecture. Numerical solution of equations; error analysis; finite difference methods; numerical differentiation and integration; series expansions; difference equations; numerical solution of differential equations.

##### MA 4373/6373. Introduction to Partial Differential Equations. (3)

(Prerequisite: MA 3253). Three hours lecture. Linear operators: linear first order equations; the wave equation; Green’s function and Sturm—Liouville problems; Fourier series; the heat equation; Laplace’s equation.

##### MA 4513/6513. Applied Probability and Statistics for Secondary Teachers. (3)

(Prerequisite: MA 1723). Three hours lecture. (Credit not available for students with credit in MA-ST 4543/6543). Graphical methods of presenting data; analysis of data; probability, binomial distribution, normal distribution; random sampling; linear regression and correlation.

##### MA 4523/6523. Introduction to Probability. (3)

(Prerequisite: MA 2733). Three hours lecture. Basic concepts of probability, conditional probability, independence, random variables, discrete and continuous probability distributions, moment generating function, moments, special distributions, central limit theorem. (Same as ST 4523/6523).

##### MA 4533/6533. Introductory Probability and Random Processes. (3)

(Prerequisites: MA 3113 and MA 2743). Three hours lecture. Probability, law of large numbers, central limit theorem, sampling distributions, confi dence intervals, hypothesis testing, linear regression, random processes, correlation functions, frequency and time domain analysis. (Credit can not be earned for this course and MA/ST 4523/6523.)

##### MA 4543/6543. Introduction to Mathematical Statistics I. (3)

(Prerequisite: MA 2743.) Three hours lecture. Combinatorics; probability, random variables, discrete and continuous distributions, generating functions, moments, special distributions, multivariate distributions, independence, distributions of functions of random variables. (Same as ST 4543/6543.)

##### MA 4573/6573. Introduction to Mathematical Statistics II. (3)

(Prerequisite: MA 4543/6543.) Three hours lecture. Continuation of MA-ST 4543/6543. Transformations, sampling distributions, limiting distributions, point estimation, interval estimation, hypothesis testing, likelihood ratio tests, analysis of variance, regression, chi-square tests. (Same as ST 4573/6573.)

##### MA 4633/6633. Advanced Calculus I. (3)

(Prerequisite: MA 2743 and MA 3053). Three hours lecture. Theoretical investigation of functions; limits; differentiability and related topics in calculus.

##### MA 4643/6643. Advanced Calculus II. (3)

(Prerequisite: MA 4633/6633). Three hours lecture. Rigorous development of the defi nite integral; sequences and series of functions; convergence criteria; improper integrals.

##### MA 4733/6733. Linear Programming. (3)

(Prerequisites: MA 3113). Three hours lecture. Theory and application of linear programming; simplex algorithm, revised simplex algorithm, duality and sensitivity analysis, transportation and assignment problem algorithms, integer and goal programming. (Same as IE 4733/6733).

##### MA 4753/6753. Applied Complex Variables. (3)

(Prerequisite: MA 2743). Three hours lecture. Analytic functions: Taylor and Laurent expansions; Cauchy theorems and integrals; residues; contour integration; introduction to conformal mapping.

##### MA 4933/6933. Mathematical Analysis I. (3)

(Prerequisite: MA 4633/6633 or equivalent). Three hours lecture. Metric and topological spaces; functions of bounded variation and differentiability in normed spaces.

##### MA 4943/6943. Mathematical Analysis II. (3)

(Prerequisite: MA 4933/6933). Three hours lecture. Riemann-Stieltjes integration, sequences and series of functions; implicit function theorem; multiple integration.

##### MA 4953/6953. Elementary Topology. (3)

(Prerequisite: MA 4633/6633). Three hours lecture. Definition of a topological space, metric space, continuity in metric spaces and topological spaces; sequences; accumulation points.

##### MA 8113. Modern Higher Algebra I. (3)

(Prerequisite: MA 4163/6163). Three hours lecture. A study of the basic mathematical systems with emphasis on rings, fi elds, and vector spaces.

##### MA 8123. Modern Higher Algebra II. (3)

(Prerequisite: MA 8113). Three hours lecture. A continuation of the topics introduced in MA 8113.

##### MA 8203. Foundations of Applied Mathematics I. (3)

(Prerequisites: MA 3113, MA 3253 or consent of instructor.) Three hours lecture. Principles of applied mathematics including topics from perturbation theory, calculus of variations, and partial differential equations. Emphasis of applications from heat transfer, mechanics, fluids.

##### MA 8213. Foundations of Applied Mathematics II. (3)

(Prerequisite: MA 8203). Three hours lecture. A continuation of MA 8203 including topics from wave propagation, stability, and similarity methods.

##### MA 8253. Operational Mathematics. (3)

(Prerequisite: MA 4753/6753). Three hours lecture. Theory and applications of Laplace, Fourier, and other integral transformations: introduction to the theory of generalized functions.

##### Courses numbered MA 8273, 8283, 8293 and 8313 have as prerequisites at least one of the courses MA 4633/6633, MA 4153/6153, 4753/6753.

##### MA 8273. Special Functions. (3)

Three hours lecture. Infi nite products: asymptotic series; origin and properties of the special functions of mathematical physics.

##### MA 8283. Calculus of Variations. (3)

Three hours lecture. Functionals: weak and strong extrema; necessary conditions for extrema; sufficient conditions for extrema; constrained extrema; direct methods; applications.

##### MA 8293. Integral Equations. (3)

Three hours lecture. Equations of Fredholm type: symmetric kernels; Hilbert-Schmidt theory; singular integral equations; applications; selected topics.

##### MA 8313. Ordinary Differential Equations I. (3)

Three hours lecture. Linear systems of differential equations; existence and uniqueness; second order systems; systems with constant coefficients; periodic systems; matrix comparison theorems; applications and selected topics.

##### MA 8323. Ordinary Differential Equations II. (3)

(Prerequisite: MA 8313). Three hours lecture. Existence, uniqueness, continuation of solutions of nonlinear systems; properties of solutions of linear and nonlinear equations including boundedness, oscillation, asymptotic behavior, stability, and periodicity; application.

##### MA 8333. Partial Differential Equations I. (3)

(Prerequisite: MA 4373/6373 or consent of instructor). Three hours lecture. Solution techniques; existence and uniqueness of solutions to elliptic, parabolic, and hyperbolic equations; Green’s functions.

##### MA 8343. Partial Differential Equations II. (3)

(Prerequisite: MA 8333). Three hours lecture. A continuation of the topics introduced in MA 8333.

##### MA 8363. Numerical Solution of Systems of Nonlinear Equations. (3)

(Prerequisites: MA 4313/6313 and MA 4323/6323). Three hours lecture. Basic concepts in the numerical solution of systems of nonlinear equations with applications to unconstrained optimization.

##### MA 8383. Numerical Solution of Ordinary Differential Equations I. (3)

(Prerequisites: MA 4313/6313 and MA 4323/6323). Three hours lecture. General single-step, multistep, multivalue, and extrapolation methods for systems of nonlinear equations; convergence; error bounds; error estimates; stability; methods for stiff systems; current literature.

##### MA 8443. Numerical Solution of Partial Differential Equations I. (3)

(Prerequisites: MA 4313/6313, MA 4323/6323, and MA 4373/6373 or consent of instructor). Three hours lecture. Basic concepts in the finite difference and finite element methods; methods for parabolic, hyperbolic and elliptic equations; analysis of stability and convergence.

##### MA 8453. Numerical Solution of Partial Differential Equations II. (3)

(Prerequisite: MA 8443). Three hours lecture. Methods for elliptic equations; iterative procedures; integral equation methods; methods for hyperbolic equations; stability; dissipation and dispersion.

##### MA 8463. Numerical Linear Algebra. (3)

(Prerequisite: MA 4323/6323). Three hours lecture. Basic concepts of numerical linear algebra.

##### MA 8633. Real Analysis I. (3)

(Prerequisite: MA 4943/6943). Three hours lecture. Lebesgue measure and Lebesgue integrals; convergence theorems, differentiation and L spaces.

##### MA 8643. Real Analysis II. (3)

(Prerequisite: MA 8633). Three hours lecture. General measures; the Radon-Nikodym theorem and other topics.

##### MA 8663. Functional Analysis I. (3)

(Prerequisite: MA 8643). Three hours lecture. Hilbert spaces; Banach spaces; locally convex spaces; Hahn-Banach and closed graph theorems; principle of uniform boundedness; weak topologies.

##### MA 8673. Functional Analysis II. (3)

(Prerequisite: MA 8663). Three hours lecture. Continuation of topics introduced in MA 8663.

##### MA 8713. Complex Analysis I. (3)

(Prerequisite MA 4943/6943 or consent of instructor). Three hours lecture. Complex numbers: functions of a complex variable; continuity; differentiation and integration of complex functions; transformations in the complex plane.

##### MA 8723. Complex Analysis II. (3)

(Prerequisite: MA 8713). Three hours lecture. Series; analytic continuation; Riemann surfaces; theory of residues.

##### MA 8913. Introduction to Topology I. (3)

(Prerequisite: MA 4643/6643 or MA 4953/6953). Three hours lecture. Basic general topology; introduction of homotopy and homology groups.

##### MA 8923. Introduction to Topology II. (3)

(Prerequisite: MA 8913). Three hours lecture. Continuation of topics introduced in MA 8913.

##### MA 8981. Teaching Seminar. (1)

One hour lecture. Preparation for service as instructors in mathematics and statistics courses; includes practice lectures and exam preparation. (May be taken for credit more than once.)

##### MA 9313. Selected Topics in Ordinary Differential Equations. (3)

(Prerequisite: MA 8313 and consent of instructor). (May be taken for credit more than once). Three hours lecture. Topics to be chosen from such areas as Bifurcation Theory, Biological Modeling, Control Theory, Dynamical Systems, Functional Differential Equations, Nonlinear Oscillations, and Quantitative Behavior.

##### MA 9333. Selected Topics in Partial Differential Equations. (3)

(Prerequisite: MA 8333 and consent of instructor). (May be taken for credit more than once). Three hours lecture. Topics to be chosen from such areas as Bifurcation Theory, Boundary Integral Methods, Evolution Equations, Maximum and Variational Principles, and Spectral Methods.

##### MA 9413. Selected Topics in Numerical Analysis. (3)

(Prerequisite: Consent of instructor). (May be taken for credit more than once). Three hours lecture. Current topics in Numerical Analysis. The subject matter may vary from year to year.

##### MA 9633. Selected Topics in Analysis. (3)

(Prerequisite: MA 8643 and consent of instructor). (May be taken for credit more than once). Three hours lecture. Topics will be chosen from areas of analysis of current interest.