Graduate Catalog
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Mathematical Sciences

JOHN HADDOCK, PhD
Interim Chair
Room 373, Winfield Dunn Building
(901) 678-2482

ANNA KAMINSKA, PhD
Mathematics Graduate Studies Coordinator

EBENEZER O. GEORGE, PhD
Statistics Graduate Studies Coordinator

E-mail: dfwilson@memphis.edu

www.msci.memphis.edu

I. The Department of Mathematical Sciences offers graduate programs leading to the Master of Science and Doctor of Philosophy degrees with a major in Mathematical Sciences.

The areas of concentration for the MS degree are Applied Mathematics, Mathematics, Teaching of Mathematics, and Statistics. Within the MS degree, students may complete up to twelve semester hours in a collateral area approved by their advisor.

The areas of concentration for the Doctor of Philosophy degree are Applied Statistics and Mathematics.

 All graduate students must comply with the general requirements of the Graduate School (see Admissions Regulations, Academic Regulations, and Minimum Degree Requirements) as well as the program requirements of the degree being pursued.

II. MS Degree Program, with concentrations in Applied Mathematics, Mathematics, Teaching of Mathematics, and Statistics.

Program objectives are (1) development of thorough background in mathematical sciences, including retention and integration of core knowledge; (2) development research skills in mathematics; and (3) development of interdisciplinary opportunities and good oral and written communication skills.

A. Program Prerequisites

  1. GRE scores are required and are an important factor for admission.
  2. Two letters of recommendation.
  3. A minimum score of 550 on the TOEFL (for students whose native language is not English).
  4. An undergraduate degree with a minimum GPA of 2.5 on a 4.0 scale.

B. Program Requirements

  1. At least 24 semester hours at the 7000 level
  2. A passing grade on a comprehensive examination
  3. Each of the concentration areas has additional program prerequisites and requirements, which are given below.
  4. Mathematics Concentration
    1. Prerequisite: An undergraduate degree with a major in mathematics or equivalent training.
    2. Requirements
      1. Satisfactory completion of 33 semester hours of graduate course work in a program approved by the department.
      2. Satisfactory completion of at least 21 semester hours of graduate course work in mathematics (A typical program will include at least two of the following two-course sequences: MATH 7350-7351, 7261-7262, 7411-7361.)
  5. Applied Mathematics Concentration
    1. Prerequisite: An undergraduate degree with a major in mathematics or equivalent training. Students should have some background in differential equations and linear algebra. Students whose major was in a related field but not mathematics will be considered on a case-by-case basis.
    2. Requirements
      1. Satisfactory completion of at least 33 semester hours of graduate course work in a program approved by the department.
      2. Satisfactory completion of at least 21 semester hours of graduate course work in mathematics. This course work should include training in real and complex analysis, ordinary and partial differential equations, mathematical modeling, numerical analysis, and calculus of variations. MATH 6350-6351 or equivalent is required. In addition this program must include at least one additional two-course sequence, typically MATH 6391-6392, 7375-7376, or 7393-7395.
      3. Sensible alternative programs, depending on the student’s motivation and goals, may be considered.
  6. Statistics Concentration
    1. Prerequisites: three semesters of calculus and one semester of linear algebra.
    2. Requirements
      1. Satisfactory completion of 30 semester hours of graduate course work with a thesis or 33 semester hours of graduate course work without a thesis in a program approved by the department.
      2. Satisfactory completion of the following courses: MATH 7642, 7643, 7647, 7654, 7685, 7762, and either MATH 7645 or MATH 7657, either MATH 7660 or MATH 7670.
      3. Graduate students in the Department of Mathematical Sciences may not receive credit for both MATH 6637 and MATH 7643.
  7. Teaching of Mathematics Concentration
    1. Prerequisite: In addition to the general prerequisites for the MS Degree program, students will be required to have an undergraduate degree in mathematics or the equivalent.
    2. Requirements
      1. Satisfactory completion of at least 33 semester hours of graduate course work in a program approved by the department.
        1. Core courses required for all students are: MATH 6151, MATH 7171, MATH 7174, MATH 7281; MATH 7282; MATH 7381; MATH 7282; MATH 7681; either ICL 7500 or ICL 7503.
        2. Elective courses must be approved by the department. Sample electives include: MATH 6242; MATH 6361; MATH 6411; MATH 7237; MATH 7996; ICL 7500; ICL 7503; ICL 7508.
      2. At least 27 hours must be at the 7000 or 8000 level and a minimum of 24 hours must be mathematics coursework (MATH 7996 does not count toward this requirement).
      3. Students may choose a thesis or non-thesis option.
        1. Thesis Option - Each student must submit a thesis acceptable to the student’s advisory committee. The thesis can be based on work done for Math 7996. A student may take 3-6 credit-hours in Math 7996; however, only 3 hours may be applied to the degree requirement. Students must complete a research project, submit a written thesis describing the research, orally present and defend the thesis before a faculty committee. Students are also required to earn a passing grade on a comprehensive written examination. The oral defense of the thesis will encompass material learned during course work and will count as the comprehensive examination. NOTE: Students should familiarize themselves with the Thesis/Dissertation Preparation Guide before starting to write.
        2. Non-thesis Option - Pass a final written and oral comprehensive examination which will be administered by the student’s Advisory Committee during the final semester of residence. The content for the comprehensive written examination will be based on the core curriculum of the program.

III. PhD Degree Program

A. Admission Requirements

  1. GRE scores are required and are an important factor for admission.
  2. Three letters of recommendation
  3. A score of at least 550 on the TOEFL, (for students whose native language is not English)
  4. An undergraduate degree in an appropriate discipline with a minimum GPA of 2.5 (on a 4.0 scale) or equivalent preparation

B. Program Requirements

  1. The doctoral degree program requires satisfactory completion of a minimum of 72 semester hours of graduate credit (a minimum of 36 hours for a student entering with an approved master’s degree). The 72 hours:
    1. may include a maximum of 12 hours of 6000 level coursework, but must include at least 18 hours of 8000 level coursework;
    2. may include between 9 and 15 hours of dissertation (9000); and
    3. must include the satisfactory completion of one of the concentration requirements listed below.
  2. Each student must:
    1. obtain a passing grade on a qualifying examination prior to the end of the first 13 months of study in the program;
    2. obtain a passing grade on a comprehensive examination;
    3. complete an acceptable dissertation (Students should familiarize themselves with the Thesis/Dissertation Preparation Guide before starting to write.); and
    4. pass a final examination given by a committee composed of departmental and university representatives.
      Detailed information can be obtained by contacting the graduate coordinator of the department.
  3. Mathematics Concentration
    1. The PhD concentration in mathematics is designed so that students may pursue a degree based on independent research or may choose a more broadly based program aimed toward a college teaching career. Students may contact the department for more detailed information.
  4. Applied Statistics Concentration
    1. Students must complete the following courses: MATH 7-8642, 7-8651, 7-8670, 7-8692, 7-8695, and two courses from MATH 7-8759, 7-8763, 7-8764, and 7-8765. In addition, students are required to give at least two formal presentations through taking MATH 7-8691
    2. Presentation of an acceptable dissertation proposal within six months after passing the comprehensive examination. Students should familiarize themselves with the Thesis/Dissertation Preparation Guide before starting to write.

MATHEMATICS (MATH)

In addition to the courses below, the department may offer the following Special Topics courses:
MATH 6010-19. Special Topics in Mathematics and Statistics. (1-3). Topics are varied and announced in online class listings. PREREQUISITE: Permission of instructor.

MATH 7020-40–8020-40. Special Topics in Mathematics (3).


MATH 7630-7639-8630-8639. Special Topics in Statistics. (1-3).
Topics are varied and announced in online class listings.


MATH 6012 - Differential Geometry (3)
Gaussian curvature; metric tensor, general relativity, hyperbolic geometry; Gauss-Bonnet Theorem; Poincare Conjecture. PREREQUISITE: MATH 2110, plus at least one upper division course in applied mathematics.

MATH 6020 - Actuarial Mathematics (3)
Preparation for SOA Exam P, CAS Exam 1; conditional probability, dependence, combinatorial principles, random variables, discrete and continuous probability distributions, expectations, marginal distributions, risk management concepts. PREREQUISITES: MATH 4635.

MATH 6022 - Fin Math I/Theory of Interest (3)
Preparation for SOA Exam FM, CAS Exam 2. Interest rates and time value of money, annuity valuation, loan repayment, bond valuation and amortization, internal rates of return, the term structure of interest rates, asset liability management, duration and immunization. PREREQUISITE: MATH 1920.

MATH 6025 - Fin Math II/Derivatives (3)
Preparation for SOA Exam FM, CAS Exam 2. Financial risk concepts; derivatives, forwards, futures, short and long positions, call and put options, spreads, collars, hedging, arbitrage, swaps. Definitions and evaluations of basic derivatives contracts and trading strategies. PREREQUISITE: MATH 1920

MATH 6028 - Models for Fin Econ/Options (3)
Various aspects of theory and practice of options pricing and related topics: put-call parity, binomial trees, arbitrage, risk-neutral pricing, random walk model, lognormality and the binomial model, estimating volatility, Black-Scholes formula, option Greeks, market making, delta hedging, Asian, barrier, compound, gap and exchange options. PREREQUISITE: MATH 6025.

MATH 6030 - Model Fin Econ/Adv Pre Thry (3)
Continuation of MATH 6028; lognormal model of stock prices, distribution of asset prices, risk neutral valuation, true valuation, simulated stock prices, Monte Carlo valuation, geometric Brownian motion, Sharpe ratio, Ito's lemma, Black-Scholes equation, all-or-nothing options, measurement and behavior of volatility, bond price models, Black-Derman-Toy model. PREREQUISITE: MATH 6028.

MATH 6051 - Methods of Proofs for Tchrs (3)
Enhance mathematical communication skills by learning methods to prove inductive statements, statements about size, and statements about relationships among objects using the language of functions and relations. This course will not be counted as credit for a graduate program in Mathematics except the Masters of Science in Mathematics with concentration in the Teaching of Mathematics. PREREQUISITE: MATH 1920 or permission of instructor.

MATH 6083 - Dynamical Systems/Chaos (3)
Examples of dynamic systems, one dimensional maps (periodic points, stability of fixed points, sensitivity dependence on initial conditions), two dimensional maps (sinks, sources and saddles, linear and nonlinear maps, Julia and Mandelbrot sets), chaos (Lyapunov exponents, chaotic orbits, basins of attractions), fractals (probabilistic and deterministic constructions, fractals dimension), differential equations (one and higher dimensional linear equations, periodic orbits and limit sets). COREQUISITE: MATH 3120 or MATH 3242.

MATH 6084 - Introduction to Graph Theory (3)
Applications, connectivity, trees, paths and cycles, factors, matching and coverings, vertex and edge colorings, planar graphs, directed graphs, max-flow min-cut theorem, basic algorithms. PREREQUISITE: MATH 2701, or MATH 2702 and 3221, or MATH 3581, or permission of instructor.

MATH 6085 - Combinatorial Geometry (3)
Convexity and fundamental theorems (Radon's Theorem, Helly's Theorem), geometric incidences, geometric graphs (planar graphs, proximity graphs), Pick's Theorem, distance problems in the plane, geometric transversals and covers. PREREQUISITE: MATH 2701 or MATH 2702, and MATH 3221 or MATH 3581.

MATH 6086 - Analytic Number Theory (3)
Partial summation, Euler-Maclaurin summation formula, basic arithmetic functions and their mean values; Dirichlet series, Euler products; Meilin function and prime number theorem; characters and primes in arithmetic progressions, basic sieve methods. PREREQUISITE: MATH 3221. COREQUISITE: MATH 6361

MATH 6151 - History Of Math (3)
The development of mathematics from the earliest times to the present; problem studies; parallel reading and class reports. PREREQUISITE: 21 hours in MATH courses including MATH 2110 and one of MATH 2701, 2702, or permission of instructor.

MATH 6171 - Spec Prob In Math (1-3)
Directed individual study in a selected area of mathematics chosen in consultation with the instructor. Repeatable for a maximum of 3 credit hours by permission of the Chair of the Department. PREREQUISITE: Permission of the instructor.

MATH 6242 - Linear Algebra (3)
Linear transformations polynomials, determinants, direct-sum decompositions diagonalizable operators, rational and Jordan form, inner product spaces, the spectral theorem. PREREQUISITE: MATH 3242.

MATH 6261 - Abstract Algebra (3)
Groups, homomorphisms, rings, integral domains, fields, polynomials. PREREQUISITE: MATH 2702 and 3242, or equivalent.

MATH 6350 - Intro Real Analysis I (3)
The real number system, functions and sequences, limits, continuity, differentiation; Riemann-Stieltjes integration, series of functions. PREREQUISITE: MATH 2110, 2702 and 3242, or equivalent.

MATH 6351 - Intro Real Analysis II (3)
Integration theory; Riemann and Lebesgue integrals; partial differentiation; implicit function theorem. PREREQUISITE: MATH 6350 or permission of instructor.

MATH 6361 - Complex Variables (3)
Complex numbers, analytic functions, Cauchy-Riemann conditions, Taylor and Laurent series, integration. PREREQUISITE: MATH 2110.

MATH 6391 - Partial Diffrntl Equation I (3)
Laplace transforms; Fourier series; introduction to partial differential equations. PREREQUISITE: MATH 3120.

MATH 6392 - Partial Diffrntl Equation II (3)
Methods of characteristics; Greens functions; existence and regularity of solutions of boundary value and Cauchy problems. PREREQUISITE: MATH 6391.

MATH 6396 - Perturbation Methods (3)
Asymptotic approximations, boundary layers, matched asymptotic expansions, multiple scales, geometric optics approximation (WKB), homogenization, application to differential equations. PREREQUISITE: MATH 2110 and MATH 3120.

MATH 6411 - Topology (3)
Introductory set theory, metric spaces, topological spaces, continuous functions, separation axioms, separability and countability axioms, connectedness, and compactness. PREREQUISITE: MATH 2702 and either 3242 or 4350, or equivalent.

MATH 6607 - Intro SAS Programming (3)
SAS program statement syntax and flow control; selecting and summarizing observations; combining, dividing, and updating SAS dataset; input tailoring and output customization; SAS built-in functions; SAS Macro Language Programming; other SAS packages like SAS/GRAPH and SAS/IML. NOTE: Introductory statistical courses are recommended.

MATH 6611 - Intro Applied Statistics (3)
Binomial, hypergeometric, Poisson, multinomial and normal distributions; test of hypotheses, chi-square test, t-tests, F- test, etc.; nonparametric tests; correlation analysis. PREREQUISITE: 6 hours in Mathematics at level of MATH 1710 or above. NOTE: Students majoring in Mathematical Sciences may not apply credit for this course to their degree requirements. Students majoring in other areas such as Physics or Engineering and who have a calculus background should take MATH 6635.

MATH 6614 - Probability/Statistics (3)
Probability distribution; statistical methods of parameter estimation and hypothesis testing; comparisons of two population means, proportions, and variances; analysis of variance, linear models, and multiple regression. NOTE: Students may not receive credit for both MATH 6614 and MATH 6635. PREREQUISITES: MATH 1920 and MATH 2701.

MATH 6635 - Intro Probability Theory (3)
Basic probability theory, random variables, discrete and continuous probability distributions, functions of one or more random variables, multivariate distributions including multinomial and bivariate normal distributions. NOTE: Students may not receive credit for both MATH 6635 and MATH 6614. PREREQUISITE: MATH 1920.

MATH 6636 - Intro Statistical Theory (3)
Functions of two random variables; gamma, beta, multinomial, and bivariate normal distributions; Bayes estimators; maximum likelihood and method of moments estimators; sufficient statistics, unbiasedness, confidence intervals, and hypothesis testing. PREREQUISITE: MATH 6635.

MATH 6637 - Intro/Stat Models/Analysis (3)
Basic concepts of statistical modeling and analysis with extensive us of R; topics include hypothesis testing; means, proportions, and variances; analysis of variance; completely randomized designs, randomized block designs, Latin square designs; multiple comparisons; simple linear model and multiple regression; analysis of covariance. PREREQUISITE: MATH 6611 or MATH 6635.

MATH 6640 - Intro Probability Models (3)
Basic concepts of discrete Markov chains; branching processes; Poisson processes; applications to modeling of population growth; applications to modeling of the spread of infectious disease. PREREQUISITE: MATH 6635.

MATH 6643 - Intro Regression/Time Ser Anal (3)
Hypothesis testing and confidence intervals for linear regression models, examination of residuals, calculation of elasticities and partial correlations, heteroscedasticity, serial correlation, multicolinearity, non-linearity, deterministic and stochastic time series models, stationary time series and autocorrelation functions, diagnostic checks, forecasting using ARIMA models. PREREQUISITE: MATH 6636.

MATH 6721 - Numerical Analysis (3)
Derivation and application of computer-oriented numerical methods for functional approximation, differentiation, quadrature, and the solution of ordinary differential equations. PREREQUISITES: MATH 1920 and knowledge of some structured programming language.

MATH 7016 - Fourier Analysis (3)
Facilitates understanding of some important facts abut Fourier series, Fourier transforms, and finite Fourier analysis, including applications to other sciences (optics, acoustics, particle physics, uncertainty principle) as well as links within mathematics (infinitude of primes, isoperimetric inequality). May be repeated for a maximum of 6 credit hours when topics change. PREREQUISITE: MATH 6350 or equivalent, or permission of instructor.

MATH 7171 - Wksp Middle Sch Math (3)
This course is designed to provide in-service training, with emphasis on new course content.

MATH 7174 - Workshop Sr Hi Math (3)
This course is designed to provide in-service training, with emphasis on transformation geometry.

MATH 7221 - Stat Gene Expression (3)
Design of microarray experiements; normalization procedures for Oligonucleotide and cDNA microarrays; clustering procedures: hierarchical clustering, principal compenents and analysis, discriminant analysis, eigenvalue decomposition discriminant analysis and nonparametric clustering methods; controlling error rates in multiple testing through resampling methods, false discovery rates, Bayesian and empirical Bayes techniques, Support Vector Machines. PREREQUISITE: MATH 7643.

MATH 7235 - Combinatorics (3)
(MATH 7793). Principles and techniques of combinatorial mathematics with a view toward applications in computer science; methods of enumeration, matching theory, paths and cycles, planarity, coloring problems, extremal problems. PREREQUISITE: Permission of instructor.

MATH 7237 - Graph Theory (3)
Connectivity, Euler tours, and Hamilton cycles, matchings, coloring problems, planarity, and network flows; study of classical theorems due to Brooks, Menger, Kuratowski, Schur, Tutte, and Vizing. PREREQUISITE: MATH 6242 or permission of instructor.

MATH 7261 - Algebraic Theory I (3)
Studies in group theory and ring theory, including Sylow theory and factorization theory. PREREQUISITE: MATH 6261.

MATH 7262 - Algebraic Theory II (3)
A continuation of Math 7261. Studies in field theory and modules, including free algebras, Galois theory, tensor products. PREREQUISITE: MATH 7261.

MATH 7281 - Linear Alg For Tchrs (3)
Euclidean n-space; vector spaces; subspaces; linear independence and bases; linear transformations; matrices; systems of linear conditions; characteristic values and vectors of linear transformations. PREREQUISITE: MATH 1920.

MATH 7282 - Abstract Alg For Tchrs (3)
A basic abstract algebra course designed especially for teachers. Topics will include: groups, rings, integral domains, fields; an axiomatic approach to the development of algebra; concepts of proof. PREREQUISITE: MATH 7281 or equivalent.

MATH 7291 - Number Theory for Tchrs (3)
Divisibility properties of the integers and modular arithmetic. Greatest common divisors, Euclidean algorithm, and linear Diophantine equations. Tests for Divisibility. Systems of linear congruences and Chinese remainder theorem. Prime numbers, distribution of prime numbers, and Mersenne primes. Fermat?s little theorem, Euler?s Theorem and Wilson?s Theorem. Applications to RSA encryption. This course will not be counted as credit for a graduate program in Mathematics except the Masters of Science in Mathematics with concentration in the Teaching of Mathematics. PREREQUISITE: MATH 6051 or Permission of Instructor

MATH 7296 - Geometry for Tchrs (3)
Axiomatic development of Euclidean geometry. Comparisons of hyperbolic, spherical, and projective geometries. Focus is on constructing geometric proofs. This course will not be counted as credit for a graduate program in Mathematics except the Masters of Science in Mathematics with a concentration in the Teaching of Mathematics. PREREQUISITE: MATH 6051 or Permission of Instructor

MATH 7311 - Topics In Analysis (1-3)
Repeatable by permission. PREREQUISITE: MATH 7350.

MATH 7321 - Modeling & Computation (3)
Introduction to process of formulating, solving, and interpreting mathematical models of real phenomena; both formal analysis and numerical techniques for variety of models. PREREQUISITE: MATH 6391.

MATH 7350 - Real Variables I (3)
s-algebra, outer measure, Lebesque measure, measurable functions, differentiation, absolute continuity, Lp-spaces. PREREQUISITE: MATH 6351.

MATH 7351 - Real Variables II (3)
Metric spaces, Baire category theorem, Hahn Banach theorem, uniform boundedness principle, closed graph theorem, general measure, signed measures, Radon-Nikodym theorem, product measures, Fubini theorem. PREREQUISITE: MATH 7350.

MATH 7352 - Ergodic Theory (3)
Examples of measure preserving transformations, Von Neumann and Birkhoff ergodic theorem, isomorphism, factors, ergodic decomposition, weak mixing, strong mixing, invariant measures for continuous transformations, unique ergodicity, applications to combinatorics and number theory (uniform distribution, continued fractions, Furstenberg correspondence principle, Roth and Sarkozy's theorem), entropy, asymptotic equipartition property. PREREQUISITE: MATH 7350.

MATH 7355 - Functional Analysis I (3)
Vector spaces, Banach spaces, Hilbert spaces; linear functionals and operators in such spaces; spectral theory. PREREQUISITE: MATH 7350.

MATH 7356 - Functional Analysis (3)
A continuation of MATH 7355-8355. PREREQUISITE: MATH 7355-8355.

MATH 7361 - Complex Analysis (3)
Analytic functions, power series, mapping properties, complex integration, Cauchy's theorem and its consequences, sequences of analytic functions. PREREQUISITE: MATH 6351.

MATH 7371 - Calculus Of Variations (3)
Introduction to calculus of variations, Euler-Lagrange equations, and optimization in infinite dimensional spaces. Applications could include various topics in science, engineering, economics, or geometry, such as ground state density theories, Dirichlet's principle and differential equations, theory of least action, depending on interests of class. PREREQUISITE: Permission of instructor.

MATH 7375 - Methods Math Physics I (3)
(Same as ESCI 7375, PHYS 7375). Vector spaces, matrices, tensors, vector fields, function spaces, differential and integral operators, transform theory, partial differential equations. PREREQUISITE: MATH 3120, 4242, and 4350; or permission of instructor.

MATH 7376 - Mthds Math Physics II (3)
(Same as ESCI 7376, PHYS 7376). Complex variables, asymptotic expansions, special functions, calculus of variations, additional topics on matrices and operators, topics in non-linear analysis. PREREQUISITE: MATH 7375 or permission of the instructor.

MATH 7381 - Real Analy For Tchrs I (3)
Properties of real number system, elementary functions, plane analytic geometry, nature of the derivative, techniques of differentiation, periodic functions, differentiation of trigonometric functions, applications of the derivative, concepts of integration. PREREQUISITE: MATH 1920.

MATH 7382 - Real Analy For Tchrs II (3)
Continuation of MATH 7381; definite integral with applications, integration of elementary transcendental functions, techniques of integration, indeterminate forms and improper integrals, infinite sequences and infinite series with tests for convergence. PREREQUISITE: MATH 7381 or equivalent.

MATH 7393 - Differl Equatns/App (3)
Basic concepts in ordinary and partial differential equations (possibly functional or stochastic differential equations); existence, uniqueness, continuous dependence theorems. Application areas could include diffusion, wave propagation, population dynamics, neural networks, mathematical biology and ecology, quantum theory, kinetic theory, depending on interests of class. PREREQUISITE: MATH 3120 or consent of instructor.

MATH 7395 - Theory Diff Equatns (3)
Qualitative aspects of linear and nonlinear differential equations including asymptotic behavior and regularity; geometric, functional analytic, and harmonic analytic methods. The asymptotic could include ergodic limits and chaos. The regularity might range from analyticity to discontinuous solutions (shocks, liquid crystals, etc.). PREREQUISITES: MATH 6350 and 6242.

MATH 7411 - Point Set Topology (3)
An axiomatic approach to compactness, separability, connnectedness, metrizability and other topological properties. PREREQUISITE: MATH 6411.

MATH 7501 - Nonlinear Wave Phenomena (3)
KdV-equation, regularized long wave BBM-equation, explicit solitary and cnoidal waves, orbital stability of solitary and cnoidal waves, Boussinesq equation, B oussinesq systems of equations, pseudo differential equations as internal wave models, Krasnosell'skii's topological degree theory, P.L. Lions' concentration-compactness principle, existence and stability of traveling waves. PREREQUISITE: MATH 4392, 7350, or permission of instructor.

MATH 7502 - Semigroups of Linear Operators (3)
Generation of linear semigroups, perturbation and approximation, applications to partial differential equations, probability theory, quantum theory and Feynman integrals. PREREQUISITE: Permission of instructor.

MATH 7503 - Semigroups Nonlinear Operators (3)
Generation of nonlinear semigroups, mild solutions and limit solutions, approximation and perturbation theory, convex analysis, applications to partial differential equations, nonlinear parabolic problems, conservation laws, Hamilton-Jacobi equation, vixcosity solutions, variational calculuc and elliptic problems. PREREQUISITE: Permission of instructor.

MATH 7504 - Partial Differential Equations (3)
Explicit and semi-explicit formulas for some classical partial differential equations, Maximum Principle, Sobolev spaces, harmonic analysis methods, parabolic, hyperbolic and elliptic equations, introduction to nonlinear partial differential equations. PREREQUISITE: Permission of instructor.

MATH 7521 - ADP Stoch Optim & Control (3)
Basic concepts and mathematical foundations of neural networks, learning, nonlinear optimization and control. Exact and approximate optimization of the utility function. Bellman equation, approximate Bellman equation for solving multivariate optimization problems in real time. Partially observable variables, with random noise and tactical objectives varying in time. PREREQUISITES: Background in calculus and functional analysis, linear algebra MATH 4/6242, or permission of instructor.

MATH 7601 - Statistics for Tchrs (3)
Binomial and geometric random variables; sampling distributions; basic concepts of hypothesis testing; inference for two population means, proportions, and variances; simple linear regression; inference for regression coefficients. This course will not be counted as credit for a graduate program in Mathematics except the Masters of Science in Mathematics with concentration in the Teaching of Mathematics. PREREQUISITE: MATH 1530 or MATH 4611 or MATH 4614 or MATH 4635 or Permission of Instructor

MATH 7607 - Adv Prog In Sas (3)
Covers SAS macro language and SAS SOL; topics include macro variables, macro processing, Marco expressions, Marco quoting; Proc SQL, retrieving data from tables, creating and updating tables and views; applications in statistics. PREREQUISITE: MATH 6607.

MATH 7608 - Statistical Programming with R (3)
Covers R programming language for statistical computation; Topics include: Input/output, R objects, functions, graphics, numerical techniques, optimization, simulation, Monte Carlo techniques. PREREQUISITE: Permission of the Instructor.

MATH 7613 - Probability Theory (3)
Probability measures; distribution functions; independence; mathematical expectation, modes of convergence; Borel-Cantelli Lemma, weak and strong laws of large numbers; Glinvenko-Cantelli lemma; characteristic functions inversion theorems; Slustky's theorem, central limit theorem, Liapounov and Lindberg-Levy and Lindberg-Feller theorems; multivariate extensions; Berry-Esseen theorem. PREREQUISITES: MATH 6350. Knowledge of MATH 6635 is recommended.

MATH 7641 - Analysis Of Variance (3)
Basic concepts of ANOVA, partitioning of the sums of squares, fixed effects models, t- and F-tests, multiple comparison procedures, random effect models, variance component models, analysis of covariance and introduction to MANOVA (SAS or comparable statistical packages used extensively to analyze different types of designs). PREREQUISITE: MATH 7643 or MATH 6636.

MATH 7642 - Experimental Design (3)
Fundamental concepts in designing experiments, justification of linear models, randomization, principle of blocking, use of concomitant observations, principle of confounding, fractional replication, composite designs, incomplete block designs. PREREQUISITE: MATH 7641 or 7643.

MATH 7643 - Least Sq/Regr Analysis (3)
Basic concepts of hypothesis testing and confidence intervals; simple and multiple regression analyses, model selection, Mallow's Cp, examination of residuals, Box-Cox transformation, influence diagnostics, multicolinearity, ridge-regression, probit, logit, and log-linear analyses; intensive use of SAS or other statistical packages. PREREQUISITE: MATH 6635.

MATH 7645 - Sampling Techniques (3)
Planning, execution, and analysis of sampling from finite populations; simple, stratified, multistage cluster and systematic sampling; ratio and regression estimates, estimation of variance. PREREQUISITE: MATH 6635; COREQUISITE: MATH 6636.

MATH 7647 - Non-Param Stat Meth (3)
Use of distribution-free statistics for estimation, hypothesis testing, and correlation measures in designing and analyzing experiments. PREREQUISITE: MATH 6635; COREQUISITE: MATH 6636.

MATH 7651 - Linear Models (3)
Multivariate normal distributions, distribution of quadratic forms, general linear hypothesis of full rank, optimal point and interval estimations, applications to regression models; elements of generalized linear models, applications to logistic regression and log-linear models; use of SAS procedures. PREREQUISITE: MATH 7643.

MATH 7654 - Inference Theory (3)
Bayes and maximum likelihood estimators, sufficient statistics; Rao-Blackwell Theorem, sampling distributions; unbiasedness, completeness and UMVU estimators; efficient estimators, Cramer-Rao inequality; simple robust estimators; UMP-tests; likelihood ratio tests, t-tests and F-tests. PREREQUISITE: MATH 6636.

MATH 7656 - Adv Tchn Statistcl Infr (3)
Limit theorems; uniformly minimum variance unbiased and maximum likelihood estimators; information inequalities; large sample theory; robust estimators; uniformly most powerful unbiased and invariant tests; sequential and robust tests. PREREQUISITE: MATH 7654.

MATH 7657 - Multivar Stat Meth (3)
Basic contents: multivariate normal distributions; Wishart distribution, Hotelling-T2, Matric-t and Beta distributions; generalized regression models and growth curve models; multivariate analysis of variance; principal component analysis; discriminant analysis; factor analysis; curve fitting procedures in multivariate cases. All topics will be illustrated by practical examples. PREREQUISITE: MATH 6636 or permission of the instructor.

MATH 7660 - App Time Series Analy (3)
Basic concepts and examples of stationary and nonstationary time series; random harmonic analysis; spectral density functions, model building procedures for time series models; model identification; diagnostic checking, smooth, forecasting and control; Box-Jenkin approach of time series analysis; some seasonal models. PREREQUISITE: MATH 6636.

MATH 7670 - App Stochastic Models (3)
Markov chains with discrete time; classification of states, stationary distributions, absorption probabilities and absorption time; Markov chains with continuous time; birth-death processes, waiting time distributions, queuing models, population growth models, Kolmogorov forward and backward equations, diffusion processes, Fokker-Planck equation; applications to genetic problems, etc. PREREQUISITES: MATH 6636 and 6640.

MATH 7671 - Indiv Study Statistics (1-3)
Directed individual study of recent developments in statistics. Repeatable by permission. PREREQUISITE: Permission of the instructor. Grades of A-F, or IP will be given.

MATH 7672 - Spec Prob Statistics (1-3)
(6671). Recent developments in statistical methods and applications. PREREQUISITE: Permission of the instructor.

MATH 7680 - Bayesian Inference (3)
Nature of Bayesian inference; formulation and choice of prior distributions; advantages and disadvantages of Bayesian approach; applications of Bayesian approach to Behren-Fisher problems, to regression analysis, and to the analysis of random effect models; applications of Bayesian approach to the assessment of statistical assumptions; Bayesian prediction procedures. PREREQUISITE: MATH 6636.

MATH 7681 - Probability For Tchrs (3)
Probability spaces, theory of statistical inference, physical interpretations of probability. PREREQUISITE MATH 1920.

MATH 7685 - Simulation & Computing (3)
Uniform random number generation and testing, generation methods for non-uniform random variables, simulating random numbers from specific distributions, Metropolis-Hastings algorithm, Markov Chain Monet-Carlo (MCMC), Gibbs sampling. PREREQUISITE: MATH 6636 and some computer programming experience.

MATH 7691 - Sem Statistical Resch (1-3)
Recent developments in statistical methods and their applications. Basic topics cover "multivariate method," growth curve models, robustness and effects of departure from basic statistical assumptions on common inference procedures, multivariate contingency tables, bioassay, etc. PREREQUISITE: MATH 6636.

MATH 7692 - Statistical Consulting (3)
Methods and techniques of statistical consulting; students will participate in consulting practice supervised by graduate faculty in statistics. May be repeated for a total of 6 credit hours. PREREQUISITES: MATH 6611 and MATH 6637. Grades of A-F, or IP will be given.

MATH 7695 - Bootstrap/Other Methods (3)
Empirical distribution and plug-in principle; bias reduction; bootstrapping regression models; the jackknife; balanced repeated replication; bootstrap confidence intervals; parametric bootstrap; permutation tests. PREREQUISITE: MATH 7645 and MATH 7647.

MATH 7721 - Adv Numerical Analysis (3)
A continuation of Mathematics 6721; specialized methods and techniques in field of numerical analysis. PREREQUISITE: MATH 6721.

MATH 7759 - Categorical Analysis (3)
Exponential family of distributions and generalized linear models; binary variables and logistic regression; contingency tables and log-linear models; quasi-likelihood functions; estimating functions. PREREQUISITES: MATH 7643 and MATH 7654.

MATH 7762 - Survival Analysis (3)
Nonparametric estimation and comparison of survival functions: Kaplan-Meier Estimator and other estimators of hazard functions; parametric survival models; Gehan test, Mantel-Haenszel test and their extensions; Cox proportional hazard model: conditional likelihood, partial likelihood analysis, identification of prognostic and risk factors; applications to life-testing and analysis of survival data using statistical packages such as SAS. PREREQUISITES: MATH 7643 and MATH 7654.

MATH 7764 - Stat Methods Biom/Envir (3)
Penalized likelihood method, spline and nonparametric regression, use of E-M algorithm, Fourier transform method, error-in-variables, longitudinal models and repeated measures; generalized estimating equations; analysis and modeling of AIDS data; statistical risks assessment. PREREQUISITES: MATH 7643 and MATH 7654.

MATH 7765 - Adv Stochstic Mod Biom (3)
Stochastic models of the AIDS epidemic; chain multinomial models, Markov models, Non-Markov marker processes, diffusion processes for AIDS, stochastic models of carcinogenesis; two-stage, multi-event and multiple path models. PREREQUISITES: MATH 7654 and MATH 7-8670.

MATH 7821 - Special Prob In Math (1-3)
Directed individual study in a selected area of mathematics chosen in consultation with the instructor and the student's advisor. Repeatable by permission. PREREQUISITE: Permission of the instructor. Grades of A-F, or IP will be given.

MATH 7921 - Spec Prob Diff Equation (1-3)
Repeatable by permission. PREREQUISITE: MATH 7393. Grades of A-F, or IP will be given.

MATH 7922 - Spec Prob Applied Math (1-3)
Repeatable by permission. PREREQUISITE: Permission of the instructor.

MATH 7960 - Sem Teachng/Res/Consult (3)
Non-traditional setting in which master's students develop skills in areas of teaching, research, and consulting. Required of all graduate assistants in the department. Grades of S, U, or IP will be given.

MATH 7995 - Project Applied Math (1-3)
Mathematical modeling problem related to science or industry, selected in consultation with a faculty advisor, and leading to final report. Repeatable by permission. PREREQUISITE: MATH 7321. Grades of A-F, or IP will be given.

MATH 7996 - Thesis (1-6)
Grades of S, U, or IP will be given.

MATH 8221 - Stat Gene Expression (3)
Design of microarray experiements; normalization procedures for Oligonucleotide and cDNA microarrays; clustering procedures: hierarchical clustering, principal compenents and analysis, discriminant analysis, eigenvalue decomposition discriminant analysis and nonparametric clustering methods; controlling error rates in multiple testing through resampling methods, false discovery rates, Bayesian and empirical Bayes techniques, Support Vector Machines. PREREQUISITE: MATH 7643.

MATH 8235 - Combinatorics (3)
(MATH 7793). Principles and techniques of combinatorial mathematics with a view toward applications in computer science; methods of enumeration, matching theory, paths and cycles, planarity, coloring problems, extremal problems. PREREQUISITE: Permission of instructor.

MATH 8237 - Graph Theory (3)
Connectivity, Euler tours, and Hamilton cycles, matchings, coloring problems, planarity, and network flows; study of classical theorems due to Brooks, Menger, Kuratowski, Schur, Tutte, and Vizing. PREREQUISITE: MATH 6242 or permission of instructor.

MATH 8311 - Topics In Analysis (1-3)
Repeatable by permission. PREREQUISITE: MATH 7350.

MATH 8355 - Functional Analysis I (3)
Vector spaces, Banach spaces, Hilbert spaces; linear functionals and operators in such spaces; spectral theory. PREREQUISITE: MATH 7350.

MATH 8356 - Functional Analysis (3)
A continuation of MATH 7355-8355. PREREQUISITE: MATH 7355-8355.

MATH 8393 - Differl Equatns/App (3)
Basic concepts in ordinary and partial differential equations (possibly functional or stochastic differential equations); existence, uniqueness, continuous dependence theorems. Application areas could include diffusion, wave propagation, population dynamics, neural networks, mathematical biology and ecology, quantum theory, kinetic theory, depending on interests of class. PREREQUISITE: MATH 3120 or consent of instructor.

MATH 8395 - Theory Diff Equatns (3)
Qualitative aspects of linear and nonlinear differential equations including asymptotic behavior and regularity; geometric, functional analytic, and harmonic analytic methods. The asymptotic could include ergodic limits and chaos. The regularity might range from analyticity to discontinuous solutions (shocks, liquid crystals, etc.). PREREQUISITES: MATH 6350 and 6242.

MATH 8501 - Nonlinear Wave Phenomena (3)
KdV-equation, regularized long wave BBM-equation, explicit solitary and cnoidal waves, orbital stability of solitary and cnoidal waves, Boussinesq equation, B oussinesq systems of equations, pseudo differential equations as internal wave models, Krasnosell'skii's topological degree theory, P.L. Lions' concentration-compactness principle, existence and stability of traveling waves. PREREQUISITE: MATH 4392, 7350, or permission of instructor.

MATH 8502 - Semigroups of Linear Operators (3)
Generation of linear semigroups, perturbation and approximation, applications to partial differential equations, probability theory, quantum theory and Feynman integrals. PREREQUISITE: Permission of instructor.

MATH 8503 - Semigroups Nonlinear Operators (3)
Generation of nonlinear semigroups, mild solutions and limit solutions, approximation and perturbation theory, convex analysis, applications to partial differential equations, nonlinear parabolic problems, conservation laws, Hamilton-Jacobi equation, vixcosity solutions, variational calculuc and elliptic problems. PREREQUISITE: Permission of instructor.

MATH 8504 - Partial Differential Equations (3)
Explicit and semi-explicit formulas for some classical partial differential equations, Maximum Principle, Sobolev spaces, harmonic analysis methods, parabolic, hyperbolic and elliptic equations, introduction to nonlinear partial differential equations. PREREQUISITE: Permission of instructor.

MATH 8521 - ADP Stoch Optim & Control (3)
Basic concepts and mathematical foundations of neural networks, learning, nonlinear optimization and control. Exact and approximate optimization of the utility function. Bellman equation, approximate Bellman equation for solving multivariate optimization problems in real time. Partially observable variables, with random noise and tactical objectives varying in time. PREREQUISITES: Background in calculus and functional analysis, linear algebra MATH 4/6242, or permission of instructor.

MATH 8642 - Experimental Design (3)
Fundamental concepts in designing experiments, justification of linear models, randomization, principle of blocking, use of concomitant observations, principle of confounding, fractional replication, composite designs, incomplete block designs. PREREQUISITE: MATH 7641 or 7643.

MATH 8656 - Adv Tchn Statistcl Infr (3)
Limit theorems; uniformly minimum variance unbiased and maximum likelihood estimators; information inequalities; large sample theory; robust estimators; uniformly most powerful unbiased and invariant tests; sequential and robust tests. PREREQUISITE: MATH 7654.

MATH 8657 - Multivar Stat Meth (3)
Basic contents: multivariate normal distributions; Wishart distribution, Hotelling-T2, Matric-t and Beta distributions; generalized regression models and growth curve models; multivariate analysis of variance; principal component analysis; discriminant analysis; factor analysis; curve fitting procedures in multivariate cases. All topics will be illustrated by practical examples. PREREQUISITE: MATH 6636 or permission of the instructor.

MATH 8660 - App Time Series Analy (3)
Basic concepts and examples of stationary and nonstationary time series; random harmonic analysis; spectral density functions, model building procedures for time series models; model identification; diagnostic checking, smooth, forecasting and control; Box-Jenkin approach of time series analysis; some seasonal models. PREREQUISITE: MATH 6636.

MATH 8670 - App Stochastic Models (3)
Markov chains with discrete time; classification of states, stationary distributions, absorption probabilities and absorption time; Markov chains with continuous time; birth-death processes, waiting time distributions, queuing models, population growth models, Kolmogorov forward and backward equations, diffusion processes, Fokker-Planck equation; applications to genetic problems, etc. PREREQUISITES: MATH 6636 and 6640.

MATH 8671 - Indiv Study Statistics (1-3)
Directed individual study of recent developments in statistics. Repeatable by permission. PREREQUISITE: Permission of the instructor. Grades of A-F, or IP will be given.

MATH 8672 - Spec Prob Statistics (1-3)
(6671). Recent developments in statistical methods and applications. PREREQUISITE: Permission of the instructor.

MATH 8680 - Bayesian Inference (3)
Nature of Bayesian inference; formulation and choice of prior distributions; advantages and disadvantages of Bayesian approach; applications of Bayesian approach to Behren-Fisher problems, to regression analysis, and to the analysis of random effect models; applications of Bayesian approach to the assessment of statistical assumptions; Bayesian prediction procedures. PREREQUISITE: MATH 6636.

MATH 8685 - Simulation And Computing (3)
Uniform random number generation and testing, generation methods for non-uniform random variables, simulating random numbers from specific distributions, Metropolis-Hastings algorithm, Markov Chain Monet-Carlo (MCMC), Gibbs sampling. PREREQUISITE: MATH 6636 and some computer programming experience.

MATH 8691 - Sem Statistical Rsrch (1-3)
Recent developments in statistical methods and their applications. Basic topics cover "multivariate method," growth curve models, robustness and effects of departure from basic statistical assumptions on common inference procedures, multivariate contingency tables, bioassay, etc. PREREQUISITE: MATH 6636.

MATH 8692 - Statistical Consulting (3)
Methods and techniques of statistical consulting; students will participate in consulting practice supervised by graduate faculty in statistics. May be repeated for a total of 6 credit hours. PREREQUISITES: MATH 6611 and MATH 6637. Grades of A-F, or IP will be given.

MATH 8695 - Bootstrap/Other Methods (3)
Empirical distribution and plug-in principle; bias reduction; bootstrapping regression models; the jackknife; balanced repeated replication; bootstrap confidence intervals; parametric bootstrap; permutation tests. PREREQUISITE: MATH 7645 and MATH 7647.

MATH 8759 - Categorical Analysis (3)
Exponential family of distributions and generalized linear models; binary variables and logistic regression; contingency tables and log-linear models; quasi-likelihood functions; estimating functions. PREREQUISITES: MATH 7643 and MATH 7654.

MATH 8762 - Survival Analysis (3)
Nonparametric estimation and comparison of survival functions: Kaplan-Meier Estimator and other estimators of hazard functions; parametric survival models; Gehan test, Mantel-Haenszel test and their extensions; Cox proportional hazard model: conditional likelihood, partial likelihood analysis, identification of prognostic and risk factors; applications to life-testing and analysis of survival data using statistical packages such as SAS. PREREQUISITES: MATH 7643 and MATH 7654.

MATH 8764 - Stat Methods Biom/Envir (3)
Penalized likelihood method, spline and nonparametric regression, use of E-M algorithm, Fourier transform method, error-in-variables, longitudinal models and repeated measures; generalized estimating equations; analysis and modeling of AIDS data; statistical risks assessment. PREREQUISITES: MATH 7643 and MATH 7654.

MATH 8765 - Adv Stochstic Mod Biom (3)
Stochastic models of the AIDS epidemic; chain multinomial models, Markov models, Non-Markov marker processes, diffusion processes for AIDS, stochastic models of carcinogenesis; two-stage, multi-event and multiple path models. PREREQUISITES: MATH 7654 and MATH 7-8670.

MATH 8811 - Advan Sem In Math (1-3)
PREREQUISITE: permission of instructor.

MATH 8812 - Ind Stdy Math/Stat (1-12)
Directed independent studies in an area selected by the student and approved by the student's advisory committee. Proposed plan of study must be approved prior to enrollment. Repeatable by permission. A maximum of 12 credit hours will count toward graduation. PREREQUISITE: The student must have passed the qualifying examination. Grades of A-F, or IP will be given.

MATH 8813 - Dir Rsrch Math/Stat (1-12)
Directed research in an area selected by the student and approved by the student's advisory committee. Proposed plan of study must be approved prior to enrollment. Repeatable by permission. A maximum of 12 credit hours will count toward graduation. PREREQUISITE: The student must have completed at least 6 credit hours in MATH 8812. Grades of A-F, or IP will be given.

MATH 8821 - Spec Prob In Math (1-3)
Directed individual study in a selected area of mathematics chosen in consultation with the instructor and the student's advisor. Repeatable by permission. PREREQUISITE: Permission of the instructor. Grades of A-F, or IP will be given.

MATH 8921 - Spec Prob Diff Equation (1-3)
Repeatable by permission. PREREQUISITE: MATH 7393. Grades of A-F, or IP will be given.

MATH 8922 - Spec Prob Applied Math (1-3)
Repeatable by permission. PREREQUISITE: Permission of the instructor.

MATH 8960 - Sem Teachng/Res/Consult (3)
Non-traditional setting in which master's students develop skills in areas of teaching, research, and consulting. Required of all graduate assistants in the department. Grades of S, U, or IP will be given.

MATH 9000 - Dissertation (1-12)
Independent research for the PhD degree. Grades of S, U, or IP will be given.

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