## Mathematics

### College of Science

### Chair Jianzhong Su

Email math@uta.edu

Phone 817.272.3261

Fax 817.272.5802

#### Degrees / Certificates

**Master’s Degrees**

Mathematics (General Mathematics), M.S.

Mathematics (General Statistics), M.S.

Mathematics Teaching, M.A.

Mathematics, M.A.

**Doctoral Degrees**

Mathematical Sciences, Mathematics, Ph.D.

Mathematics (General Mathematics), B.S. to Ph.D.

Mathematics (General Mathematics), Ph.D.

Mathematics (General Statistics), B.S. to Ph.D.

Mathematics (General Statistics), Ph.D.

**Certificates**

Applied Statistics Certificate

#### Graduate Faculty

**Professor**

Tuncay Aktosun, Graduate Advisor:

*Mathematical Sciences, Mathematics, Ph.D.*

*Mathematics (General Mathematics), B.S. to Ph.D.*

*Mathematics (General Mathematics), M.S.*

*Mathematics (General Mathematics), Ph.D.*

*Mathematics (General Statistics), B.S. to Ph.D.*

*Mathematics (General Statistics), M.S.*

*Mathematics (General Statistics), Ph.D.*

**Associate Professor**

James Epperson, Graduate Advisor:

*Mathematics, M.A.*

**Assistant Professor**

#### Department Information

#### Courses

- Master of Science Program
- Master of Arts Program
- Certificate of Applied Statistics Program
- Doctoral Program
- Doctoral Program (B.S.-Ph.D. track)

Master of Science Degree Requirements

Master of Arts Degree Requirements

Certificate of Applied Statistics Requirements

Ph.D. Degree Requirements for the B.S.-Ph.D. track

### Objective

The objectives of the Mathematics Department's program at the master's level are (1) to develop the student's ability to do independent research and prepare for more advanced study in mathematics, and (2) to give advanced training to professional mathematicians, mathematics teachers, and those employed in engineering, scientific, and business areas.

Graduate work will be offered in algebra, complex and real variables, differential equations, functional analysis, geometry, mathematics education, numerical analysis, operations research, probability, statistics and topology.

### Admissions Requirements

#### Master of Science Program

For unconditional admission, a student must meet the following requirements:

- A B.A. or B.S. degree in mathematics or closely related field.
- An overall GPA in the final 60 hours of coursework of a 3.0 or better, as calculated by the Graduate School, on a 4.0 scale.
- Minimum of 350 on the verbal and 650 on the quantitative portions of the Graduate Record Examination (GRE) if taken prior to August 2011. Minimum of 143 on the verbal and 151 on the quantitative portions of the GRE it faken after August 2011.
- For applicants whose native language is not English, a minimum score of 550 on the Test of English as a Foreign Language (or a minimum score of 213 on a computer-based test, or a minimum score of 79 on an internet-based test) or a minimum score of 40 on the Test of Spoken English.
- Three favorable letters of recommendation from people familiar with the applicant's academic work.

Applicants who do not satisfy requirements 2 or 3 above may be considered for unconditional admission if further review of their undergraduate transcript, recommendation letters, correspondence or direct interactions with mathematics faculty, and statement of professional or research interests indicates that they are qualified to enter the Master's Program without deficiency.

If an applicant does not meet a majority of standards for unconditional admission outlined above, they may be considered for probationary admission after careful examination of their application materials. Probationary admission requires that the applicant receive a B or better in the first 12 hours of graduate coursework at UT Arlington.

Students who are unconditionally admitted or admitted on probation will be eligible for available scholarship and/or fellowship support. Award of scholarships or fellowships will be based on consideration of the same criteria utilized in admission decisions. To be eligible, candidates must be new students coming to UT Arlington in the fall semester, must have a GPA of 3.0 in the last 60 undergraduate credit hours plus any graduate credit hours as calculated by the Graduate School, and must be enrolled in a minimum of 6 hours of coursework in both long semesters to retain the fellowship.

Applicants may be denied admission if they have less than satisfactory performance on a majority of the admission criteria described above.

A deferred decision may be granted when a file is incomplete or when a denied decision is not appropriate. An applicant unable to supply all required documentation prior to the admission deadline, but who otherwise appears to meet admission requirements, may be granted provisional admission.

#### Master of Arts Program

For unconditional admission a student must meet items 1-3 or 3-5.

- A B.S. or B.A. degree with at least 24 hours of mathematics coursework with a GPA of at least 3.0, as calculated by the Graduate School on a 4.0 scale.
- Minimum of 400 on the verbal and 600 on the quantitative portions of the Graduate Record Examination (GRE) if taken prior to August 2011. Minimum of 146 on the verbal and 148 on the quantitative portions of the GRE if taken after August 2011.
- Three favorable letters of recommendation from people familiar with the applicant's academic work and/or professional work.
- A B.S. or B.A. degree with a GPA of at least 3.0, as calculated by the Graduate School on a 4.0 scale.
- Certified to teach mathematics at the Secondary Level (Secondary Mathematics Certification).

Applicants who do not satisfy requirements 1 or 2 above may be considered for unconditional admission if further review of their undergraduate transcript, recommendation letters, correspondence or direct interactions with mathematics faculty, and statement of professional or research interests indicates that they are qualified to enter the Master's Program without deficiency.

If an applicant does not meet a majority of standards for unconditional admission outlined above, they may be considered for probationary admission after careful examination of their application materials. Probationary admission requires that the applicant receive a B or better in the first 12 hours of graduate coursework at UT Arlington.

Applicants may be denied admission if they have less than satisfactory performance on a majority of the admission criteria described above.

A deferred decision may be granted when a file is incomplete or when a denied decision is not appropriate. An applicant unable to supply all required documentation prior to the admission deadline, but who otherwise appears to meet admission requirements, may be granted provisional admission.

#### Certificate of Applied Statistics Program

The admission standard is the same as that of Master of Science Program.

#### Doctoral Program

For unconditional admission a student must meet the following requirements:

- A master's degree or at least 30 hours of graduate coursework in mathematics or closely related fields.
- A minimum GPA of 3.0, as calculated by the Graduate School, on a 4.0 scale in graduate coursework.
- Minimum of 350 on the verbal and 700 on the quantitative portions of the Graduate Record Examination (GRE) if taken prior to August 2011. Minimum of 143 on the verbal and 155 on the quantitative portions of the GRE if taken after August 2011.
- For applicants whose native language is not English, a minimum score of 550 on the Test of English as a Foreign Language (or a minimum score of 213 on a computer-based test, or a minimum score of 79 on an internet-based test) or a minimum score of 40 on the Test of Spoken English.
- Three favorable letters of recommendation from people familiar with the applicant's academic work and/or professional work.

Applicants who do not satisfy requirements 2 or 3 above may be considered for unconditional admission if further review of their undergraduate transcript, recommendation letters, correspondence or direct interactions with mathematics faculty, and statement of professional or research interests indicates that they are qualified to enter the Doctoral Program without deficiency.

If an applicant does not meet a majority of standards for unconditional admission outlined above, they may be considered for probationary admission after careful examination of their application materials. Probationary admission requires that the applicant receive a B or better in the first 12 hours of graduate coursework at UT Arlington.

Applicants may be denied admission if they have less than satisfactory performance on a majority of the admission criteria described above.

A deferred decision may be granted when a file is incomplete or when a denied decision is not appropriate. An applicant unable to supply all required documentation prior to the admission deadline, but who otherwise appears to meet admission requirements, may be granted provisional admission.

#### Doctoral Program (B.S.-Ph.D. track)

For unconditional admission a student must meet the following requirements:

- A bachelor's degree in mathematics or in a closely related field.
- A minimum GPA of 3.00 on the 4.00 scale in undergraduate course work, as calculated by the UT Arlington Graduate School.
- A minimum of 350 on the verbal part and 700 on the quantitative part of the Graduate Record Examination (GRE) if taken prior to August 2011. Minimum of 143 on the verbal and 155 on the quantitative portions of the GRE if taken after August 2011.
- For an applicant whose native language is not English, a minimum score of 550 on the Test of English as a Foreign Language (or a minimum score of 213 on a computer-based test, or a minimum score of 79 on an internet-based test) or a minimum score of 40 on the Test of Spoken English.
- At least three letters of recommendation from people familiar with the applicant's academic work and/or professional work.

Applicants who do not satisfy requirement 2 or/and 3 above may be considered for an unconditional admission if a further review of their undergraduate transcript(s), recommendation letters, correspondence or direct interactions with mathematics faculty, and statement of professional or research interests indicates that they are qualified to enter the B.S.-Ph.D. track program without deficiency.

If an applicant does not meet a majority of standards for an unconditional admission outlined above, he/she may be considered for a probationary admission after a careful examination of his/her application materials. A probationary admission requires that the applicant receive grades of B or better in the first 12 hours of graduate course work at UT Arlington.

An applicant may be denied admission if he/she has less than satisfactory performance on a majority of the admission criteria described above.

A deferred decision may be granted when the applicant's file is incomplete or when a denial on his/her admission is not appropriate. An applicant who is unable to supply all required documentation prior to the admission deadline but who otherwise appears to have met admission requirements may be granted provisional admission.

Students who are unconditionally admitted or admitted on probation will be eligible for available scholarship and/or fellowship support. Award of scholarships or fellowships will be based on consideration of the same criteria utilized in admission decisions. To be eligible, candidates must be new students coming to UT Arlington in the fall semester, must have a GPA of 3.0 in the last 60 undergraduate credit hours plus any graduate credit hours as calculated by the Graduate School, and must be enrolled in a minimum of 6 hours of coursework in both long semesters to retain the fellowship.

### Master of Science Degree Requirements

The Department of Mathematics offers master's degree programs in mathematics with additional emphasis in applied mathematics, computer science, mathematics education, pure mathematics, and statistics. All students are to use either the thesis or thesis-substitute plan.

All students in Master of Science program must complete one of the following:

General Mathematics core requirements:

MATH 5300: Computer Programming and Applications

MATH 5307: Mathematical Analysis I

MATH 5308: Mathematical Analysis II

MATH 5333: Linear Algebra and Matrices

One of the following tracks:

Applied Mathematics: MATH 5350, 5351, and either 5320 or 5321

Computer Science: MATH 5338 and 5339, and either 5371 or 5373, and six approved hours in computer science engineering

Mathematics Education: Nine hours from MATH 5336, 5337, 5340-5348, 5352

Pure Mathematics: MATH 5331 (replaces MATH 5300), either 5317 or 5322, either 5332 or 5334, and either 5304 or 5326

Core requirements can also be fulfilled by completing core requirements in the BS-Ph.D. track in the Doctoral program.

General Statistics core requirements:

MATH 5300: Computer Programming and Applications

MATH 5307: Mathematical Analysis I

MATH 5333: Linear Algebra and Matrices

One of the following three courses:

MATH 5356: Applied Multivariate Statistical Analysis

MATH 5357: Sample Surveys

MATH 5392: Regression Analysis

MATH 5305: Statistical Methods

MATH 5312: Mathematical Statistics I

MATH 5313: Mathematical Statistics II

Core requirements can also be fulfilled by completing core requirements in the BS-Ph.D. track in the Doctoral program.

In addition:

- Those students enrolled in the thesis substitute plan must take MATH 5395, and all except those in the computer science track must take at least nine other hours of electives.
^{[1]} - Those students enrolled in the thesis plan must take at least six hours of MATH 5398-5698, and all except those in the computer science track must take at least three other hours of electives.
^{[1]}

- Those students enrolled in the thesis substitute plan must take MATH 5395, and all except those in the computer science track must take at least nine other hours of electives.

[1] Electives may not be chosen from MATH 5336, 5337, 5340-5348, 5352, 5370, 5375-5379.

Students in every degree plan must pass a final exam.

### Master of Arts Degree Requirements

The master of arts program in the Department of Mathematics is designed for those who are interested in strengthening their understanding of mathematics and enriching their mathematics teaching. The program focuses on enhancing mathematics teaching through preparation in topics grounded in secondary school mathematics from an advanced standpoint. The program embraces a philosophy of teaching and learning mathematics that is consistent with the landmark *Standards* documents produced by the National Council of Teachers of Mathematics.

The requirements for the master of arts degree are 30 hours of graduate courses from the Department of Mathematics and a 3 hour project.

All students must complete the following:

Required Courses (6) and Project:

MATH 5341: Concepts and Techniques in Geometry

MATH 5342: Concepts and Techniques in Algebra

MATH 5343: Concepts and Techniques in Probability and Statistics

MATH 5344: Mathematics-Specific Technologies

MATH 5345: Concepts and Techniques in Analysis

MATH 5346: Concepts and Technique in Problem Solving

MATH 5395: Project - Individual, Director-Approved Research

Elective Courses (4):

MATH 5300: Computer Programming and Applications

MATH 5305: Statistical Methods

MATH 5307: Mathematical Analysis I

MATH 5308: Mathematical Analysis II

MATH 5333: Linear Algebra and Matrices

MATH 5336: Concepts and Techniques in Number Theory

MATH 5337: Concepts and Techniques in Calculus

MATH 5340: Concepts and Techniques in Discrete Mathematics

MATH 5347: Concepts and Techniques in Modeling and Applications

MATH 5348: Advanced Algebra in Secondary School Mathematics

MATH 5352: Concepts and Techniques in Precalculus

MATH 5380: Seminar - Study of Current Mathematics Topics

MATH 5392: Selected Topics in Mathematics

### Certificate of Applied Statistics Requirements

The Certificate in Applied Statistics offers individuals with an undergraduate degree an opportunity to receive graduate instruction in applied statistics as a means of maintaining and enhancing their professional development. The certificate program will provide coursework in statistics to an individual whose undergraduate major was outside the area of statistics. Since the requirements for the certificate are substantially less than those for the Master’s Degree in Mathematics with a concentration in Statistics, the certificate can be earned in a much shorter time span.

The Certificate in Applied Statistics requires that the students take and successfully complete the following courses.

1: Required Courses (2):

STATS 5312: Mathematical Statistics I

STATS 5313: Mathematical Statistics II

2: Electives (3 courses chosen from the following list of courses)

STATS 5305: Statistical Methods

STATS 5314: Experimental Design

STATS 5353: Applied Linear Model

STATS 5356: Applied Multivariate Statistical Analysis

STATS 5357: Sample Surveys

STATS 5358: Regression Analysis

MATH 5392: Selected Topics in Statistical Quality Control

MATH 5392: Selected Topics in Statistical Methods in Clinical Research

Upon completion of the 15 hours of graduate courses from lists 1 and 2 with a minimum GPA of 3.0, the student is awarded the Certificate in Applied Statistics. The expected time to completion is 1 to 2 years. The time limit for completion of the certificate program is 6 years.

### Ph.D. Degree Requirements

A dynamic program leading to the Doctor of Philosophy degree in the mathematics will aim at both real and demonstrated competency on the part of the student over material from various branches of mathematics. The Doctor of Philosophy degree in Mathematics provides a program of study that may be tailored to meet the needs of those interested in applied or academic careers. This program allows students to pursue topics ranging from traditional mathematics studies to applied mathematical problems in engineering and sciences. The nature of the dissertation will range from research in mathematics to the discovery and testing of mathematical models for analyzing given problems in engineering and sciences and in locating and developing mathematical and computational techniques for deducing the properties of these models as to solve these problems effectively and efficiently. Such dissertations will be concerned with research problems from pure mathematics, applied mathematics, mathematics education and statistics.

The Department of Mathematics offers doctoral degree programs in Mathematics (algebra, applied mathematics, geometry, mathematics education, numerical analysis and statistics).

All doctoral students must complete one of the following:

- General MATHEMATICS core requirements:
- MATH 5308: Mathematical Analysis II
- MATH 5317: Real Analysis
- MATH 5320: Ordinary Differential Equations
- MATH 5322: Complex Variables
- MATH 5327: Functional Analysis I
- MATH 5331: Abstract Algebra I
- One of the following four courses:
- MATH 5319: Probability Theory
- MATH 5321: Partial Differential Equations
- MATH 5334: Differential Geometry
- MATH 5339: Numerical Analysis II

- General STATISTICS core requirements:
- MATH 5308: Mathematical Analysis II
- MATH 5312: Mathematical Statistics I
- MATH 5313: Mathematical Statistics II
- MATH 5314: Experimental Design
- MATH 5317: Real Analysis
- MATH 5319: Probability Theory
- MATH 5322: Complex Variables or MATH 5327: Functional Analysis I
- MATH 5356: Applied Multivariate Statistical Analysis

Students in every degree plan must pass the preliminary and comprehensive examinations.

### Ph.D. Degree Requirements for the B.S.-Ph.D. track

The student must complete either the mathematics or statistics core requirements.

- General MATHEMATICS core requirements:
- MATH 5307: Mathematical Analysis I
- MATH 5308: Mathematical Analysis II
- MATH 5317: Real Analysis
- MATH 5320: Ordinary Differential Equations
- MATH 5322: Complex Variables
- MATH 5327: Functional Analysis I
- MATH 5331: Abstract Algebra I
- MATH 5333: Linear Algebra
- One of the following four courses:
- MATH 5319: Probability Theory
- MATH 5321: Partial Differential Equations
- MATH 5334: Differential Geometry
- MATH 5339: Numerical Analysis II

- General STATISTICS core requirements:
- MATH 5307: Mathematical Analysis I
- MATH 5308: Mathematical Analysis II
- MATH 5312: Mathematical Statistics I
- MATH 5313: Mathematical Statistics II
- MATH 5314: Experimental Design
- MATH 5317: Real Analysis
- MATH 5319: Probability Theory
- MATH 5322: Complex Variables or MATH 5327: Functional Analysis I
- MATH 5333: Linear Algebra
- MATH 5356: Applied Multivariate Statistical Analysis

The requirements for the preliminary and comprehensive examinations are the same as the other tracks in the Ph.D. program.

For additional information on the mathematics program, see the program entry in the Interdepartmental and Intercampus Programs section of this catalog.

### Please Note:

The grade of **R** (research in progress) is a permanent grade; completing course requirements in a later semester cannot change it. To receive credit for an **R**-graded course, the student must continue to enroll in the course until a passing grade is received.

An incomplete grade (the grade of **I**) cannot be given in a course that is graded **R**, nor can the grade of **R** be given in a course that is graded **I**. To receive credit for a course in which the student earned an **I**, the student must complete the course requirements. Enrolling again in the course in which an **I** was earned cannot change a grade of **I**. At the discretion of the instructor, a final grade can be assigned through a change of grade form.

Three-hour thesis courses and three- and six-hour dissertation courses are graded **R/F/W** only (except social work thesis courses.) The grade of **P** (required for degree completion for students enrolled in thesis or dissertation programs) can be earned only in six- or nine-hour dissertation courses and nine-hour thesis courses. In the course listings below, **R**-graded courses are designated either “Graded **P/F/R**” or “Graded **R**.” Occasionally, the valid grades for a course change. Students should consult the appropriate graduate advisor or instructor for valid grade information for particular courses. (See also the sections titled “**R**” Grade, Credit for Research, Internship, Thesis or Dissertation Courses and Incomplete Grade in this catalog.)

### Courses (MATH)

#### MATH5191 – SEMINAR FOR TEACHING ASSISTANTS

**0** Lecture Hours · **1** Lab Hour

This course is mandatory for all mathematics graduate teaching assistants. Students will be instructed on classroom procedures and strategies and will be required to deliver lectures under the supervision of math faculty. The purpose is to develop students to be effective lecturers. Admittance to this course is restricted to Math TAs.

#### MATH5300 – INTRODUCTION TO SCIENTIFIC COMPUTING

**3** Lecture Hours · **0** Lab Hours

Introduction to scientific computing utilizing algorithmic languages and operating environment such as Fortran, MATLAB, C, and C and UNIX (LINUX) operating system. Prerequisite: consent of the instructor.

#### MATH5302 – FUNDAMENTALS OF MATHEMATICAL SCIENCES I

**3** Lecture Hours · **0** Lab Hours

Matrices and operators, linear spaces, multivariable calculus, dynamical systems, applications. Prerequisites: MATH 3318 and 3330 or consent of the instructor.

#### MATH5303 – FUNDAMENTALS OF MATHEMATICAL SCIENCES II

**3** Lecture Hours · **0** Lab Hours

Wave propagation, potential theory, complex variables, transform techniques, perturbation techniques, diffusion, applications. Prerequisite: MATH 5302 or consent of the instructor.

#### MATH5304 – GENERAL TOPOLOGY

**3** Lecture Hours · **0** Lab Hours

Introduction to fundamentals of general topology. Topics include product spaces, the Tychonoff theorem, Tietzes Extension theorem, and metrization theorems. Prerequisite: MATH 4304 or 4335.

#### MATH5305 – STATISTICAL METHODS

**3** Lecture Hours · **0** Lab Hours

Topics include descriptive statistics, numeracy, and report writing; basic principles of experimental design and analysis; regression analysis; data analysis using the SAS package. Prerequisite: consent of the instructor.

#### MATH5307 – MATHEMATICAL ANALYSIS I

**3** Lecture Hours · **0** Lab Hours

Elements of topology, real and complex numbers, limits, continuity, and differentiation, functions of bounded variation, Riemann-Stieltjes integrals. Prerequisite: MATH 4335 or consent of Graduate Advisor.

#### MATH5308 – MATHEMATICAL ANALYSIS II

**3** Lecture Hours · **0** Lab Hours

Analysis in Rn, limits, continuity, Jacobian, extremum problems, multiple integrals, sequences and series of functions, Lebesgue integral. Prerequisite: MATH 5307 or consent of Graduate Advisor.

#### MATH5310 – MATHEMATICAL GAME THEORY

**3** Lecture Hours · **0** Lab Hours

Two person null sum games. Bimatrix games and Nash equilibrium points. Noncooperative games, existence theorem. Cooperative games, core, Shapley value, the nucleolus. Cost allocation. Market games. Simple games and voting. Prerequisite: MATH 5330.

#### MATH5311 – APPLIED PROBABILITY AND STOCHASTIC PROCESSES

**3** Lecture Hours · **0** Lab Hours

Topics include conditional expectations, law of large numbers and central limit theorem, stochastic processes, including Poisson, renewal, birth-death, and Brownian motion. Prerequisite: MATH 3313 or equivalent.

#### MATH5312 – MATHEMATICAL STATISTICS I

**3** Lecture Hours · **0** Lab Hours

Basic probability theory, random variables, expectation, probability models, generating functions, transformations of random variables, limit theory. Prerequisite: MATH 5307 or concurrent registration or consent of instructor.

#### MATH5313 – MATHEMATICAL STATISTICS II

**3** Lecture Hours · **0** Lab Hours

Theories of point estimation (minimum variance unbiased and maximum likelihood), interval estimation and hypothesis testing (Neyman-Pearson and likelihood ratio tests), regression analysis and Bayesian inference. Prerequisite: MATH/STATS 5312.

#### MATH5314 – EXPERIMENTAL DESIGN

**3** Lecture Hours · **0** Lab Hours

This course covers the classical theory and methods of experimental design, including randomization, blocking, one-way and factorial treatment structures, confounding, statistical models, analysis of variance tables and multiple comparisons procedures. Prerequisite: MATH/STATS 5305 or equivalent.

#### MATH5315 – GRAPH THEORY

**3** Lecture Hours · **0** Lab Hours

Algorithms for problems on graphs. Trees, spanning trees, connectedness, fundamental circuits. Eulerian graphs and Hamiltonian graphs. Graphs and vector spaces, matrices of a graph. Covering and coloring. Flows. Prerequisite: MATH 3314.

#### MATH5316 – COMBINATORIAL OPTIMIZATION

**3** Lecture Hours · **0** Lab Hours

Shortest paths. Minimum weight spanning trees and matroids. Matchings and optimal assignment. Connectivity. Flows in networks, applications. Prerequisite: MATH 3314.

#### MATH5317 – REAL ANALYSIS FOR THE MATHEMATICAL SCIENCES

**3** Lecture Hours · **0** Lab Hours

Sigma-fields, measures, measurable functions, convergence in measure and almost everywhere, integration, Fatou's Lemma, Lebesgue-dominated convergence, signed measures, Radon-Nikodym Theorem, product measures, Fubini's Theorem. Prerequisite: MATH 5308.

#### MATH5318 – FUNDAMENTALS OF STOCHASTIC ANALYSIS

**3** Lecture Hours · **0** Lab Hours

General properties of stochastic processes, processes with independent increments, martingales, limit theorems including invariance principle, Markov processes, stochastic integral, stochastic differential. Prerequisite: MATH 5308.

#### MATH5319 – PROBABILITY THEORY

**3** Lecture Hours · **0** Lab Hours

Probability spaces, random variables, filtrations, conditional expectations, martingales, strong law of large numbers, ergodic theorem, central limit theorem, Brownian motion and its properties. Prerequisite: MATH 5308.

#### MATH5320 – APPLIED DIFFERENTIAL EQUATIONS

**3** Lecture Hours · **0** Lab Hours

Fundamentals of the theory of systems of ordinary differential equations: existence, uniqueness, and continuous dependence of solutions on data; linear equations, stability theory and its applications, periodic and oscillatory solutions. Prerequisite: MATH 5307 and 5333.

#### MATH5321 – APPLIED PARTIAL DIFFERENTIAL EQUATIONS

**3** Lecture Hours · **0** Lab Hours

General first order equations. Basic linear theory for elliptic, hyperbolic, and parabolic second order equations, including existence and uniqueness for initial and boundary value problems. Prerequisites: MATH 5307 and 5333.

#### MATH5322 – COMPLEX VARIABLES I

**3** Lecture Hours · **0** Lab Hours

Fundamental theory of analytic functions, residues, conformal mapping and applications. Prerequisite: MATH 5307.

#### MATH5325 – ALGEBRAIC NUMBER THEORY

**3** Lecture Hours · **0** Lab Hours

Field extensions, number fields and number rings, ramification theory, class groups, elliptic curves and their group structure, applications to Fermat's last theorem. Prerequisite: MATH 3321.

#### MATH5326 – ALGEBRAIC TOPOLOGY

**3** Lecture Hours · **0** Lab Hours

Fundamental groups, covering space, singular homology, relative homology, Mayer-Vietoris sequence, Betti numbers, Euler characteristic. Prerequisites: MATH 3321, MATH 3335.

#### MATH5327 – FUNCTIONAL ANALYSIS I

**3** Lecture Hours · **0** Lab Hours

Introduction to Hilbert and Banach spaces: Hahn-Banach, Banach-Steinhaus, and closed graph theorems. Riesz representation theorem and bounded linear operators in Hilbert space. Prerequisite: MATH 5308.

#### MATH5328 – FUNCTIONAL ANALYSIS II

**3** Lecture Hours · **0** Lab Hours

The theory of distributions and Sobolev spaces, with applications to differential equations. Compact operators and Fredholm theory. Spectral theory for unbounded operators. Prerequisite: MATH 5327.

#### MATH5330 – ALGEBRAIC GEOMETRY

**3** Lecture Hours · **0** Lab Hours

Theory of ideals in polynomial rings, Nullstellensatz, Hilbert's basis theorem, computation in polynomial rings, affine and projective varieties, singular and smooth points on varieties. Prerequisite: MATH 4321.

#### MATH5331 – ABSTRACT ALGEBRA I

**3** Lecture Hours · **0** Lab Hours

Zorn's Lemma, groups, including free groups and dihedral groups. Rings including factorization, localization, rings of polynomials, and formal power series. An introduction to modules. Prerequisite: MATH 3321.

#### MATH5332 – ABSTRACT ALGEBRA II

**3** Lecture Hours · **0** Lab Hours

Modules, including free, projective, and injective. Exact sequences and tensor products of modules. Chain conditions, primary decomposition, Noetherian rings and modules. Prerequisite: MATH 5331.

#### MATH5333 – LINEAR ALGEBRA AND MATRICES

**3** Lecture Hours · **0** Lab Hours

Vector spaces, their sums, linear (in)dependence, bases, linear maps and their matrices, change of basis, inner-products, adjoints, diagonalization, eigenvectors and generalized eigenvectors, eigenvalues, Jordan form, characteristic and minimal polynomials, dual vector spaces, bilinear and quadratic forms. Prerequisite: MATH 3330 or consent of instructor.

#### MATH5334 – DIFFERENTIAL GEOMETRY

**3** Lecture Hours · **0** Lab Hours

Introduction to the theory of curves and surfaces in three dimensional Euclidean space. Prerequisite: MATH 4334 or 4335.

#### MATH5336 – CONCEPTS AND TECHNIQUES IN NUMBER THEORY

**3** Lecture Hours · **0** Lab Hours

Topics include mathematical induction, fundamental theorem or arithmetic, inequalities, special sequences and sums, divisibility properties, greatest common divisor, division and Euclidean algorithm, properties of congruence and Diophantine equations.

#### MATH5337 – CONCEPTS AND TECHNIQUES IN CALCULUS

**3** Lecture Hours · **0** Lab Hours

Topics studied include limits, continuity, differentiation, integration, numerical approximations, applications and Taylor series.

#### MATH5338 – NUMERICAL ANALYSIS I

**3** Lecture Hours · **0** Lab Hours

Solution of equations including linear and nonlinear systems, interpolation and approximation, spline, numerical differentiation and quadrature. Prerequisite: MATH 2425 or consent of the instructor.

#### MATH5339 – NUMERICAL ANALYSIS II

**3** Lecture Hours · **0** Lab Hours

Rigorous treatment of numerical aspects of linear algebra and numerical solution of ordinary differential equations, boundary value problems, introduction to numerical solution of partial differential equations. Prerequisite: MATH 5338 or consent of the instructor.

#### MATH5340 – CONCEPTS AND TECHNIQUES IN DISCRETE MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

Topics include functions, mathematical induction, principles of counting, combinatorics, sequences and recurrence relations, and finite graph theory.

#### MATH5341 – CONCEPTS AND TECHNIQUES IN GEOMETRY

**3** Lecture Hours · **0** Lab Hours

Selected materials from geometry.

#### MATH5342 – CONCEPTS AND TECHNIQUES IN ALGEBRA

**3** Lecture Hours · **0** Lab Hours

Selected materials from algebra.

#### MATH5343 – CONCEPTS AND TECHNIQUES IN PROBABILITY AND STATISTICS

**3** Lecture Hours · **0** Lab Hours

Consideration of (1) exploring data: descriptive statistics of situations involving one and two variables; (2) anticipating patterns: probability and simulation; (3) design of experiments and planning a study; (4) statistical inference: confirming models. Use of a graphing calculator and other appropriate technology.

#### MATH5344 – MATHEMATICS-SPECIFIC TECHNOLOGIES

**3** Lecture Hours · **0** Lab Hours

Focus on use of current mathematics-specific technologies for enhancing mathematical understanding and mathematics teaching. May include use of Geometer's Sketchpad, Fathom, graphing calculators and computer algebra systems.

#### MATH5345 – CONCEPTS AND TECHNIQUES IN ANALYSIS

**3** Lecture Hours · **0** Lab Hours

Selected materials from analysis including concepts and topics consistent with precalculus and elementary calculus. Prerequisite: MATH 5337 or consent of the instructor.

#### MATH5346 – CONCEPTS AND TECHNIQUES IN PROBLEM SOLVING

**3** Lecture Hours · **0** Lab Hours

A study of the application of various heuristics and general problem strategies in mathematics, with application to the teaching and learning of secondary school and college-level mathematics. Topics include analyzing, classifying, and modifying tasks, assessment of problem solving, and implementing problem solving in the classroom. Assignments require interaction in secondary school or college field settings.

#### MATH5347 – CONCEPTS AND TECHNIQUES IN MATHEMATICAL MODELING WITH APPLICATIONS

**3** Lecture Hours · **0** Lab Hours

Topics studied include algebraic, graphical, geometrical and numerical techniques to model and solve applied problems.

#### MATH5348 – ADVANCED ALGEBRA IN SECONDARY SCHOOL MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

Major concepts of second-year algebra applied to the teaching and learning of secondary school mathematics. Topics include relations, algebraic, tabular, verbal and geometric representations of functions, transformations, including applications involving systems of equations and inequalities.

#### MATH5350 – APPLIED MATHEMATICS I

**3** Lecture Hours · **0** Lab Hours

Development of models arising in the natural sciences and in engineering. Emphasis will be on the mathematical techniques and theory needed to analyze such models; these include aspects of the theory of differential and integral equations, boundary value problems, theory of distributions and transforms. Prerequisites: MATH 5307 and 5333.

#### MATH5351 – APPLIED MATHEMATICS II

**3** Lecture Hours · **0** Lab Hours

Continuation of MATH 5350; models arising in the physical sciences whose analysis includes such topics as the theory of operators in a Hilbert space, variational principles, branching theory, perturbation and stability analysis. Prerequisite: MATH 5350.

#### MATH5352 – CONCEPTS AND TECHNIQUES IN PRECALCULUS

**3** Lecture Hours · **0** Lab Hours

Topics include functions (transcendental, inverse, parametric, polar, transformations), asymptotic behavior, conics, sequences, complex numbers.

#### MATH5353 – APPLIED LINEAR MODELS

**3** Lecture Hours · **0** Lab Hours

The course covers, at an operational level, three topics: 1) the univariate linear model, including a self-contained review of the relevant distribution theory, basic inference methods, several parameterizations for experimental design and covariate-adjustment models and applications, and power calculation; 2) the multivariate linear model, including basic inference (e.g. the four forms of test criteria and simultaneous methods), applications to repeated measures experiments and power calculation; and 3) the univariate mixed model, including a discussion of the likelihood function and its maximization, approximate likelihood inference, and applications to complex experimental designs, missing data, unbalanced data, time series observations, variance component estimation, random effects estimation, power calculation and a comparison of the mixed model's capabilities relative to those of the classical multivariate model. Knowledge of the SAS package is required. Prerequisite: MATH/STATS 5358 (Regression Analysis) or equivalent.

#### MATH5354 – CATEGORICAL DATA ANALYSIS

**3** Lecture Hours · **0** Lab Hours

This course covers classical methods for analyzing categorical data from a variety of response/factor structures (univariate or multivariate responses, with or without multivariate factors), based on several different statistical rationales (weighted least squares, maximum likelihood and randomization-based). Included are logistic regression, multiple logit analysis, mean scores analysis, observer agreement analysis, association measures, methods for complex experimental designs with categorical responses and Poisson regression. The classical log-linear model for the association structure of multivariate responses is briefly reviewed. Randomization-based inference (e.g. Mantel-Haenzel) is discussed as well. The necessary distribution theory (multinomial, asymptotics of weighted least squares and maximum likelihood) are discussed at an operational level. Knowledge of the SAS package is required. Prerequisite: MATH/STATS 5358 (Regression Analysis).

#### MATH5355 – STATISTICAL THEORY FOR RESEARCH WORKERS

**3** Lecture Hours · **0** Lab Hours

Designed for graduate students not majoring in mathematics. Topics include basic probability theory, distributions of random variables, point estimation, interval estimation, testing hypotheses, regression, and an introduction to analysis of variance. Graduate credit not given to math majors. Prerequisite: graduate standing.

#### MATH5356 – APPLIED MULTIVARIATE STATISTICAL ANALYSIS

**3** Lecture Hours · **0** Lab Hours

Statistical analysis for data collected in several variables, topics including sampling from multivariate normal distribution, Hotelling's T'2, multivariate analysis of variance, discriminant analysis, principal components, and factor analysis. Prerequisite: MATH/STATS 5312 or consent of instructor.

#### MATH5357 – SAMPLE SURVEYS

**3** Lecture Hours · **0** Lab Hours

A comprehensive account of sampling theory and methods, illustrations to show methodology and practice, simple random sampling, stratified random sample, ratio estimates, regression estimates, systematic sampling, cluster sampling, and nonsampling errors. Prerequisite: MATH/STATS 5312 or consent of instructor.

#### MATH5358 – REGRESSION ANALYSIS

**3** Lecture Hours · **0** Lab Hours

A comprehensive course including multiple linear regression, non-linear regression and logistic regression. Emphasis is on modeling, inference, diagnostics and application to real data sets. The course begins by developing a toolbox of methods via a sequence of guided homework assignments. It culminates with projects based on consulting-level data analysis problems involving stratification, covariate adjustment and messy data sets. Some knowledge of the SAS package is required. Prerequisites: MATH/STATS 5305, basic knowledge of matrices.

#### MATH5359 – SURVIVAL ANALYSIS

**3** Lecture Hours · **0** Lab Hours

This course covers analysis of lifetime data, which has applications to actuarial science and health fields. Topics include the survivor function, hazard function, censoring, parametric regression models (e.g. the weibull), nonparametric regression models (e.g. the Cox proportional hazards model), categorical survival data methods, competing risks and methods for multivariate survival data. Knowledge of the SAS package is required. Prerequisites: MATH/STATS 5358 (Regression Analysis) and preferably MATH/STATS 5313. (Students without 5313 can still succeed if they have some basic calculus-based probability, such as MATH 3313).

#### MATH5361 – APPLIED CALCULUS OF VARIATION

**3** Lecture Hours · **0** Lab Hours

Functionals, variation, extremization, Euler's equation, direct and indirect approximation methods; applications to mechanics and control theory. Prerequisite: MATH 5302.

#### MATH5362 – MATHEMATICS OF LINEAR PROGRAMMING

**3** Lecture Hours · **0** Lab Hours

The simplex method and the revised simplex method. Linear algebra for polyhedra and polytopes. Duality theory. Sensitivity analysis. Applications to transportation problems, network flow problems, matrix-games and scheduling problems. Integer programming. Quadratic programming. Prerequisite: MATH 3330.

#### MATH5363 – OSCILLATIONS AND WAVES

**3** Lecture Hours · **0** Lab Hours

Development of methods and results related to phenomena in nature that exhibit oscillatory motion; mathematical techniques include Fourier series, ordinary and partial differential equations, and the theory of almost periodic functions. Prerequisite: MATH 3318.

#### MATH5364 – INTRODUCTION TO MATHEMATICAL CONTROL THEORY

**3** Lecture Hours · **0** Lab Hours

Systems in science, engineering, and economics and their mathematical description by means of functional equations (ordinary, partial, integral, delay-type). Basic properties of various classes of systems: observability, controllability, stability, and oscillating systems; optimal control problems and applications. Prerequisite: MATH 3318 or 4320.

#### MATH5365 – BIOMATHEMATICS

**3** Lecture Hours · **0** Lab Hours

Mathematical techniques used in modeling such as perturbation theory, dimensional analysis, Fourier analysis, and differential equations. Applications to morphogenetics, population dynamics, compartmental systems, and chemical kinetics.

#### MATH5366 – INTRODUCTION TO NEURAL AND COGNITIVE MODELING

**3** Lecture Hours · **0** Lab Hours

Principles of neural network modeling; application of these principles to the simulation of cognitive processes in both brains and machines; models of associative learning, pattern recognition, and classification. Prerequisite: consent of instructor.

#### MATH5370 – PROBLEM SOLVING IN K-8 MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

A study of the nature and aspects of problem solving in mathematics, with application to the teaching and learning of K-8 mathematics. Topics include deconstructing and modifying tasks, assessment of problem solving, and the roles of representation, conjecture & proof, and technology in problem solving. Assignments require interaction in K-8 field settings. Prerequisite: graduate standing.

#### MATH5371 – APPLIED NUMERICAL LINEAR ALGEBRA

**3** Lecture Hours · **0** Lab Hours

Numerical solutions of linear algebraic systems, least squares problems, and eigenvalue problems; LU and QR decompositions, Schur and Singular Value decompositions, Gaussian elimination, QR algorithm, and Krylov subspace iterations for large and sparse linear algebra problems. Prerequisites: MATH 3330 or consent of the instructor.

#### MATH5372 – OPTIMIZATION METHODS & NUMERICAL SOLUTIONS OF NONLINEAR EQUATIONS

**3** Lecture Hours · **0** Lab Hours

Unconstrained and constrained optimization, solutions of nonlinear system of equations; Newton and quasi-Newton methods, secant methods and variations, nonlinear least squares problems. Prerequisite: MATH 5308 or consent of the instructor.

#### MATH5373 – NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

**3** Lecture Hours · **0** Lab Hours

Numerical methods for approximating solutions of initial value problems, boundary value problems, including linear multistep methods, Runge-Kutta methods, shooting methods. Prerequisite: MATH 5300, 3319 or consent of instructor.

#### MATH5374 – NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

**3** Lecture Hours · **0** Lab Hours

Numerical methods for elliptic, parabolic, hyperbolic, mixed, and systems of partial differential equations; finite difference methods, finite element methods, spectral methods. Prerequisite: MATH 5373 or consent of instructor.

#### MATH5375 – CONSTRUCTING WHOLE NUMBER ANDOPERATIONS IN K-8 MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

A study of the interaction between the structure of place-value numeration systems and the nature of the four arithmetic operations. The development of traditional and alternative computational arithmetic algorithms from conceptual and concrete models for operations, viewed through the lens of alternative numeration systems and research on children's mathematical thinking. Assignments require interaction in K-8 field settings. Prerequisite: graduate standing.

#### MATH5376 – CONSTRUCTING RATIONAL NUMBERAND OPERATIONS IN K-8 MATH

**3** Lecture Hours · **0** Lab Hours

The meanings and representations of rational numbers, and the development of computations on rational numbers from algorithms for whole numbers, including concrete models for operations on fractions and decimals. Discussion of research on the learning and teaching of operations on rational numbers. Also, divisibility tests and factoring. Assignments require interaction in K-8 field settings. Prerequisite: MATH 5375.

#### MATH5377 – ALGEBRAIC THINKING IN K-8 MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

A study of the practice of making and justifying generalizations in K-8 mathematics, including field properties of operations, modular arithmetic (with applications to odd/even), relations and equivalence relations, the introduction and use of variables and unknowns, and the influence of representations on the form of mathematical arguments. Assignments require interaction in K-8 field settings. Prerequisite: MATH 5375.

#### MATH5378 – GEOMETRY CONCEPTS IN K-8 MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

Major concepts of geometry applied to the teaching and learning of K-8 mathematics. Topics include dimension, development of definitions, meanings of angle, geometric comparison relations, notions of center, and non-Euclidean geometries. Assignments require interaction in K-8 field settings. Prerequisite: graduate standing.

#### MATH5379 – MEASUREMENT CONCEPTS IN K-8 MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

The development of measurement concepts as applied to the teaching and learning of K-8 mathematics. Topics include the development and properties of standard and nonstandard units, notions of size, decomposing space, relationships between boundaries and interiors, the algebra of units, measuring time, and notions of heaviness. Assignments require interaction in K-8 field settings. Prerequisite: graduate standing.

#### MATH5380 – SEMINAR

**3** Lecture Hours · **0** Lab Hours

Current topics in mathematics, may be repeated for credit twice. Prerequisite: consent of instructor.

#### MATH5391 – SPECIAL TOPICS IN MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

Topics in mathematics assigned individual students or small groups. Faculty members closely supervise the students in their research and study. In areas where there are only three hours offered, the special topics may be used by students to continue their study in the same area. Graded P/F/R. Prerequisite: permission of instructor.

#### MATH5392 – SELECTED TOPICS IN MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

May vary from semester to semester depending upon need and interest of the students. May be repeated for credit. Prerequisite: permission of Graduate Advisor.

#### MATH5395 – SPECIAL PROJECT

**3** Lecture Hours · **0** Lab Hours

Graded P/F/R. Prerequisite: permission of Graduate Advisor.

#### MATH5398 – THESIS

**3** Lecture Hours · **0** Lab Hours

5398 Graded R/F only; 5698 graded P/F/R. Prerequisite: permission of Graduate Advisor.

#### MATH5698 – THESIS

**6** Lecture Hours · **0** Lab Hours

Graded P/F/R. Prerequisite: permission of Graduate Advisor.

#### MATH6180 – SEMINAR FOR PROFESSIONAL DEVELOPMENT OF PHD STUDENTS IN SPECIAL PROJECTS

**1** Lecture Hour · **0** Lab Hours

This seminar class is for Ph.D. students enrolled in special University projects. Topics include a survey of new Math, Science, Technology and Engineering advancements, Ph.D. students professional development and mentoring. Prerequisite: Prior approval of Project Director.

#### MATH6313 – TOPICS IN PROBABILITY AND STATISTICS

**3** Lecture Hours · **0** Lab Hours

May be repeated for credit when the content changes.

#### MATH6353 – GENERALIZED LINEAR MODELS

**3** Lecture Hours · **0** Lab Hours

This course covers modern methods for analyzing Bernoulli, multinomial and count data. It begins with a development of generalized linear model theory, including the exponential family, link function and maximum likelihood. Second is a discussion of the case of models for independent observations. Next is a discussion of models for repeated measures, based on quasi-likelihood methods. These include models (such as Markov chains) for categorical time series. Next is a treatment of models with random effects. Finally is a discussion of methods for handling missing data. Knowledge of the SAS package is required. Prerequisites: MATH/STATS 5358 (Regression Analysis) and preferably MATH/STATS 5313. (Students without 5313 can still succeed but must deal with the slightly higher mathematical level of this course.)

#### MATH6356 – TIME SERIES ANALYSIS

**3** Lecture Hours · **0** Lab Hours

This course covers classical methods of time series analysis, for both the time and frequency domains. For covariance stationary series, these include ARIMA modeling and spectral analysis. For nonstationary series, they include methods for detrending and filtering. Also included is a treatment of multivariate series, as well as a discussion of the Kalman filter state-space model. Knowledge of the SAS package is required. Prerequisites: MATH/STATS 5358 (Regression Analysis) and MATH/STATS 5313.

#### MATH6357 – NONPARAMETRIC STATISTICS

**3** Lecture Hours · **0** Lab Hours

This is a survey of classical nonparametric methods for inference in standard observational settings (one-sample, two-sample, k-samples and the univariate linear model), and includes a development of U-statistics, rank statistics and their asymptotic distribution theory. The mathematical level is fairly high. Prerequisite: MATH/STATS 5313.

#### MATH6391 – SPECIAL TOPICS IN MATHEMATICS

**3** Lecture Hours · **0** Lab Hours

Faculty directed individual study and research. May be repeated for credit when the content changes.

#### MATH6399 – DISSERTATION

**3** Lecture Hours · **0** Lab Hours

Prerequisite: admission to candidacy for the Doctor of Philosophy degree in mathematics.

#### MATH6699 – DISSERTATION

**6** Lecture Hours · **0** Lab Hours

Prerequisite: admission to candidacy for the Doctor of Philosophy degree in mathematics.

#### MATH6999 – DISSERTATION

**9** Lecture Hours · **0** Lab Hours

Prerequisite: admission to candidacy for the Doctor of Philosophy degree in mathematics.

#### MATH7399 – DOCTORAL DEGREE COMPLETION

**3** Lecture Hours · **0** Lab Hours

This course may be taken during the semester in which a student expects to complete all requirements for the doctoral degree and graduate. Enrolling in this course meets minimum enrollment requirements for graduation, for holding fellowships awarded by The Office of Graduate Studies and for full-time GTA or GRA positions. Students should verify that enrollment in this course meets other applicable enrollment requirements. To remain eligible in their final semester of study for grants, loans or other forms of financial aid administered by the Financial Aid Office must enroll in a minimum of 5 hours as required by the Office of Financial Aid. Other funding sources may also require more than 3-hours of enrollment. Additional hours may also be required to meet to requirements set by immigration law or by the policies of the student's degree program. Students should contact the Financial Aid Office, other sources of funding, Office of International Education and/or their graduate advisor to verify enrollment requirements before registering for this course. This course may only be taken once and may not be repeated. Students who do not complete all graduation requirements while enrolled in this course must enroll in a minimum of 6 dissertation hours (6699 or 6999) in their graduation term. Graded P/F/R.