# Mathematics, Applied (MATH) and Statistics (STAT)

##### Department of Mathematics and Statistics

NEERCHAL, NAGARAJ, Department Chair

BISWAS, ANIMIKH, Graduate Program Director, Applied Mathematics

PARK, JUNYONG, Graduate Program Director, Statistics

**Professors**

ARMSTRONG, THOMAS E., Ph.D., Princeton University; Functional analysis and measure theory, probability, mathematical economics

BELL, JONATHAN, Ph.D., University of California, Los Angeles; Mathematical biology, mathematical finance, partial differential equations, applied mathematics

GOBBERT, MATTHIAS K., Ph.D., Arizona State University; Computational and industrial mathematics, scientific computing

GOWDA, MUDDAPPA S., Ph.D., University of Wisconsin; Applied analysis, optimization

GULER, OSMAN K., Ph.D., University of Chicago; Convex programming, computational complexity, interior point methods

LO, JAMES T., Ph.D., University of Southern California; Computational intelligence, intelligent systems, neural networks approach to systems control and signal processing, stochastic systems

LYNN, YEN-MOW, Ph.D., California Institute of Technology; Fluid dynamics, mathematical physics

MATHEW, THOMAS, Ph.D., Indian Statistical Institute, India; Inference in linear models and variance component models, design of experiments

NEERCHAL, NAGARAJ K., Ph.D., Iowa State University; Time series analysis, over-dispersion models, environmental statistics, data analysis

KOGAN, JACOB, Ph.D., Weizmann Institute, Israel; Calculus of variations, optimal control theory, optimization

POTRA, FLORIAN, Ph.D., University of Bucharest, Romania; Numerical optimization, mathematical modeling, parallel computing

ROSTAMIAN, ROUBEN, Ph.D., Brown University; Differential equations, mathematical modeling, mechanics

ROY, ANINDYA, Ph.D., Iowa State University; ; Bayesian Analysis, Bioinformatics, Brain Imaging, Sequential Design, Time Series and Signal Processing

SEIDMAN, THOMAS I., Ph.D., New York University; Control theory, non-linear partial differential equations, inverse problems

SINHA, BIMAL K., Ph.D., University of Calcutta, India; Multi-variate analysis, statistical inference, linear models, decision theory, robustness, asymptotic theory

SURI, MANIL, Ph.D., Carnegie Mellon University; Numerical analysis, partial differential equations

**Associate Professors**

HOFFMAN, KATHLEEN, Ph.D., University of Maryland, College Park; Calculus of variations, differential equations, mathematical biology, coupled oscillator theory

MINKOFF, SUSAN, Ph.D., Rice University; Large-scale scientific computing, seismic and electromagnetic inverse problems, fluid flow and geomechanical deformation modeling, multi-physics simulation, uncertainty quantification

RATHINAM, MURUHAN, Ph.D., California Institute of Technology; Stochastic dynamics, non-linear dynamics, geometric control theory

ZWECK, JOHN, Ph.D., Rice University; Modeling and simulation of optical systems, applications of differential geometry

**Assistant Professors**

DRAGANESCU, ANDREI, Ph.D., University of Chicago; Numerical analysis of partial differential equations, multi-level algorithms for inverse problems

KANG, WEINING, PH.D., University of California, San Diego; Probability theory, stochastic processes, and their applications.
PARK, DOHWAN, Ph.D. University of Missouri, Columbia; Longitudinal data analysis, survival analysis, and biostatistics.

PARK, JUNYONG, Ph.D., Purdue University; High-dimensional data analysis, classification, asymptotic theory

PEERCY, BRAD, Ph.D., University of Utah; Mathematical biology, partial differential equations, bifurcation theory, dynamical systems.

SHEN, JINGLAI, Ph.D., University of Michigan; Differential equations and dynamical systems; dynamic optimization; control theory; variational integrators with applications in operations research, mechanics and engineering

HUANG, YI, Ph.D., Johns Hopkins University; Average treatment/exposure effect evaluation, biostatistical methods and applications in biomedical, public health and policy research.

**Professors Emeritus**

AZIZ, A. KADIR, Ph.D., University of Maryland, College Park; Functional and numerical analysis, control theory for partial differential equations

GROSS, FRED, Ph.D., University of California, Los Angeles; Functional equations, complex function theory, meromorphic functions

###### Degrees offered

M.S. and Ph.D in Applied Mathematics, M.S. and Ph.D in Statistics

###### Program Description

The Department of Mathematics and Statistics offers graduate programs leading to the M.S. and Ph.D. degrees in both applied mathematics and statistics. The department has had an active graduate program in applied mathematics since 1970. It expanded to include a full graduate program in statistics in 1984. The strength of these programs lies in its graduate faculty, who are actively engaged in research in applications of mathematics and statistics in a wide variety of real-world problems, as well as in investigations of fundamental and theoretical questions. The faculty designs and implements courses and curricula with emphasis on innovative research directed toward practical applications, as mandated by the charter from the University System of Maryland Board of Regents.

Both the applied mathematics and statistics programs are intended for those students who are interested in pursuing an advanced degree and who have earned the equivalent of a bachelor’s or master’s degree in mathematics, statistics or in other mathematically oriented disciplines. Students who already hold a master’s degree may apply and enter the doctoral program directly. The doctoral programs provide training suitable for employment in academia, industrial research and development organizations, as well as research-oriented government agencies. The master’s degree programs provide training in applications of mathematics and statistics in areas suitable for employment in industry or government agencies. They also can serve as preparatory steps toward advancing to a doctoral program.

To serve students’ varying range of backgrounds and goals, the department has instituted several tracks within its master’s degree programs. Each track defines a set of well-focused graduation requirements. Students who intend to continue to the doctoral programs should consider the traditional tracks in applied mathematics or statistics. A student whose final goal is a master’s degree should consider the industrial track in applied mathematics or the applications-oriented track in statistics. Most graduate courses are offered in the late afternoon or in the evening to enable the participation of those who hold full-time employment.

Individuals wishing to benefit from the department’s course offerings without enrolling as degree-seeking students may do so by filing a non-degree seeking student application form. For students who do not already hold an undergraduate degree, a combined B.S./M.S. program leading to a master’s degree in either applied mathematics or statistics is also offered by the department.

###### Program Admission Requirements

Applications for the master’s and the doctoral programs must be made on prescribed forms, obtainable from the Graduate School’s Web site, www.umbc.edu/gradschool/admissions. Admission to the two programs, applied mathematics and statistics, are made separately and are not interchangeable. The completed application packet must include: (a) completed application forms; (b) transcripts of college and university academic records; (c) three recommendation letters; (d) Graduate Record Examination (GRE) scores for the General Test; and (e) TOEFL score for international applicants whose native language is not English. All original application documents must be sent directly to the Graduate School, not the department. Requests for exemption from submitting some of the required documents may be considered if reasonable justification is given. Criteria for admission include:

- A bachelor’s degree with a grade point average higher than 3.0 out of 4.0
- A sound background in mathematics and/or statistics
- A GRE score in the 85th percentile in the quantitative section
- A TOEFL score of at least 600 (paper-based test) or 250 (computer-based test) or 100 (iBT) for international students

Applicants whose backgrounds are considered deficient occasionally may be admitted on provisional basis. In such cases, the department specifies conditions in the form of courses to be completed in a specified time before a recommendation for change to regular status is made.

###### Degree Requirements

## Master’s Degree Programs in Applied Mathematics

The program offers two tracks for the master’s degrees in applied mathematics. The traditional track is designed for students wishing to continue toward a doctorate in mathematics either at UMBC or another academic institution. The industrial track is designed for students interested in master’s degrees that will prepare them for employment in industry. Each entering student will have an advisor who will help design a program meeting the degree requirements set forth below. If necessary, a student may be advised to take preparatory courses at the undergraduate or graduate level. These preparatory courses may not count toward the degree.

**Traditional Track**

In this track, there are two options.

The comprehensive examination option requires

- the completion of 30 graduate credits with an average grade of B or better, with at least 24 (of those) credits at the 600-level or above;
- passing a comprehensive examination based on topics at the level of, and contained, in Math 600 and 603.

The curriculum must include the following core courses:

- MATH 600: Real Analysis
- MATH 603: Matrix Analysis
- MATH 620 or 630: Numerical Analysis
- MATH 650 or 651: Optimization
- Math 612 or 614: Differential Equations.

In addition to the core courses above, a student must complete 15 graduate credits (or 5 elective courses) approved by the Graduate Program Director. Not more than six credit hours of STAT courses may be used to fulfill the 30 credit hours course requirement.

The thesis option requires:

- the completion of 24 graduate credits with an average grade of B or better, with at least 18 of (those) at the 600-level or above;
- completing 6 credits of Math 799 (Master's Research);
- Writing and defending a master's thesis.

The curriculum must include the following core courses:

- MATH 600: Real Analysis
- MATH 603: Matrix Analysis
- MATH 620 or 630: Numerical Analysis
- MATH 650 or 651: Optimization
- MATH 612 or 614: Differential Equations.

In addition to the core courses above, a student must complete 9 graduate credits (or 3 elective courses) approved by the Graduate Program Director. Not more than six credit hours of STAT courses may be used to fulfill the 24 credit hours course requirement. Regarding Master's Research and Thesis: After establishing an area of concentration, the student should seek a thesis advisor from the university graduate faculty. The student should submit a thesis proposal to the graduate program director with an endorsement from the prospective thesis advisor. It is expected that the completed thesis will be a significant exposition of the approved topic or will concentrate on developing better methods for solving practical problems. The final acceptance and earning of credit for the masters thesis require passing an oral defense of masters thesis examination. This examination is conducted in accordance with the general Graduate School regulations.

**Industrial Track**

An industrial-project-oriented track is offered under the master’s program in applied mathematics to meet the educational needs of students who intend to obtain employment in industry or government. This track also may be attractive to part-time students from the government and local industries. The track is designed to provide students with the basic tools of applied and computational mathematics, as well as statistics, fused with mathematical model building. The program is capped with a significant industrial-oriented project. Another aspect of the program is incorporating some experience with technical report writing and oral presentation, a valuable skill in all career options. The track requires the completion of 30 graduate credit hours.

The student should declare the intention to pursue this specific track soon after entering the master’s program. Approval of a plan of study must be obtained from the department before the student starts taking courses toward satisfying the track requirements. If the student wishes to transfer courses taken elsewhere to satisfy specific track requirements, approval must be obtained before taking more than six credit hours toward satisfying the track requirements.

The course requirements of the industrial track are satisfied by Math 617: Introduction to Industrial Mathematics, taken in the first two semesters in the program; five graduate-level courses, taken within the department, at least one of which is a statistics course at the 600 level or higher; two courses in a focused area taken outside the department with the approval of the graduate program director; and Math 717: Projects in Industrial Mathematics, taken in the second year in the program. It will be strongly advised that no more than two courses come from any one area of applied mathematics.

The program culminates in a capstone event by completing a project (MATH 699 for three credit hours) under the direction of a faculty member in the department or under the joint direction of a faculty member in the department and an expert in the area of the project from outside the department. The topic should be associated with an industry- or government-related problem in applied mathematics.

## Master’s Degree Programs in Statistics

The programs offer various tracks leading to the master’s degrees in statistics. The traditional track is designed for students wishing to continue toward a Ph.D. in Statistics, either at UMBC or elsewhere. The applications-oriented tracks are designed for students interested in master’s degrees that will prepare them for employment in industry. The student should declare the intention to pursue a specific track soon after entering the master’s program. Approval of a plan of study must be obtained from the department before the student starts taking courses toward satisfying the track requirements. Each entering student will have a program advisor who will help design a program meeting the degree requirements set forth below.

**Traditional Track**

There are two major options within the traditional track. The comprehensive examination option requires the completion of 30 graduate credit hours of courses and passing a comprehensive examination. The thesis option requires the completion of 24 graduate credit hours of courses and six credit hours for a master’s thesis. The choice of option must be established by the time the student has completed 18 credit hours of coursework toward the master’s degree. The curriculum must include the following core courses:

STAT 601: Applied Statistics I

STAT 602: Applied Statistics II

STAT 611: Mathematical Statistics I

STAT 612: Mathematical Statistics II

MATH 601 and MATH 630 are highly recommended.

The comprehensive examination option requires the completion of at least 30 credit hours of approved graduate-level course work, including the four core courses above, with an average grade of “B” or better, and the passing of two written comprehensive examinations, one in applied statistics (based on STAT 601 and STAT 602) and one in mathematical statistics (based on STAT 611 and STAT 612).

The thesis option requires the completion of 24 credit hours of approved courses with an average grade of “B” or better, including the four core courses above. In addition, six credit hours of STAT 799: Master’s Research must be completed, as follows. After establishing an area of concentration, the student should seek a thesis advisor from the graduate faculty. The student should submit a thesis proposal to the graduate program director with an endorsement from the prospective thesis advisor. It is expected that the completed thesis will be a significant exposition of the approved topic or will concentrate on developing better methods for solving practical problems. The final acceptance and earning of credit for the master’s thesis require passing an oral defense of master’s thesis examination. This examination is conducted in accordance with the general Graduate School regulations.

**Applications-Oriented Tracks**

Applications-oriented tracks are offered
under the master’s program in statistics to
meet the educational needs of students who
intend to obtain employment in industry
and government. The program offers a track
in environmental statistics and a track in
biostatistics. Within each track, the student
can either choose the comprehensive exam
option or the master’s thesis option.

**Environmental Statistics Track**

This track is recommended for students with an interest in environmental applications. An undergraduate major in a field related to environmental sciences with sufficient background in mathematics or statistics will be adequate to pursue this track. Students completing the track will gain working knowledge of statistical methodology and software packages used in environmental applications. This track also will be attractive to part-time students from the government and local industries, who are involved with environmental applications.

The course requirements are:

- STAT 651: Basic Probability
- STAT 653: Basic Mathematical Statistics
- STAT 601: Applied Statistics I
- STAT 602: Applied Statistics II
- STAT 614: Environmental Statistics

Additionally, one advanced courses focusing on a specific topic in environmental sciences and two additional graduate-level applied statistics courses are required. In exceptional circumstances, (e.g., in the case of a student who already has a strong background in mathematical statistics) substitutions may be allowed with approval of the graduate program director. Students can take advantage of such substitutions to strengthen their background in other relevant areas.

In the comprehensive exam option, the student must pass the written comprehensive examination based on STAT 651 and STAT 653. In addition, the student is required to complete three credits of capstone project (Stat699). The total number of course credits including 3 credits for Stat 699 should be at least 30.

STAT 699 in environmental statistics constitutes a project under the direction of a faculty member in the department or under the joint direction of a faculty member in the department and an expert in this area from outside the department. The work should include substantial data analysis. The project report should be submitted and approved prior to graduation. Approval may be sought for combinations of courses consisting of STAT 614: Environmental Statistics and two courses outside the department focusing on environmental applications. Several such courses are offered by the Department of Geography and Environmental Systems and the Toxicology.

The thesis option requirements are same as those in the traditional track (see the description in the section for traditional track) except the coursework must include the list of core courses given in the environmental track section.

If necessary, some students may be advised to take preparatory courses at the undergraduate or graduate level. These preparatory courses may not count toward the degree.

**Biostatistics Track**

This track is recommended for students interested in designing and analyzing biomedical studies, including pharmaceutical clinical trials. The program is a joint undertaking of the Department of Mathematics and Statistics at UMBC and the Department of Epidemiology and Preventive Medicine at the University of Maryland, Baltimore. Applicants to the program are expected to have had courses in multi-variable calculus, linear algebra and introductory courses in statistics.

The course requirements are:

- PREV 600: Principles of Epidemiology
- STAT 651: Basic Probability
- STAT 653: Basic Mathematical Statistics
- STAT 601: Applied Statistics I

In addition, students are required to take two of the following three courses:

- STAT 602: Applied Statistics II (Design of Experiments)
- STAT 603: Categorical Data Analysis
- STAT 619: Biostatistics: Principles and Design

In the comprehensive exam option the students are required to pass the written comprehensive examination based on STAT 651 and STAT 653. In addition the student must complete 30 credits of coursework including the list of courses given in this section and 3 credits of capstone project (STAT 699 or PREV 789: Project)

STAT 699 in biostatistics constitutes a project under the direction of a faculty member in the department or under the joint direction of a faculty member in the department and an expert in this area from outside the department. The work should include sophisticated data analysis, a simulation study, a review of literature, statistical software development or other activities related to biostatistics. The project must result in a report of 10-15 pages. The student will register for the project as STAT 699 or as PREV 789, depending on whether the faculty advisor is at UMBC or at UMB.

There are a number of course offerings at both schools that would be relevant to the degree and recommended through advising. This would include up to two 400-level statistics courses at the consultation and approval of the student’s advisor. Students are required to take at least one elective course at UMB. Students are required to attend a monthly biostatistics seminar.

The thesis option requirements are same as those in the traditional track (see the description in the section for traditional track) except the coursework must include the list of core courses given in the environmental track section.

If necessary, some students may be advised to take preparatory courses at the undergraduate or graduate level. These preparatory courses may not count toward the degree.

The master’s biostat track is recommended both for students seeking terminal master’s degree and students who wish to continue in the newly instated ‘Biostatistics Track’ within the statistics Ph.D. program. For more details about the Ph.D. biostatistics track see the section for Ph.D. program in statistics.

###### The Doctor of Philosophy (Ph.D.) Programs

The department offers doctoral study in a broad spectrum of both classical and modern applied mathematics and statistics. Admission to this program presupposes a strong background in mathematics and/or statistics. Doctoral students continue with advanced study and dissertation research, with specialization in any of the departmental fields or in an interdisciplinary area. Particular emphasis is given to the following areas in applied mathematics: differential equations and applications, numerical analysis and scientific computation, dynamical systems, stochastic processes, mathematical biology, optimization theory and algorithms. In statistics, the areas of emphasis are : Bayesian analysis, biostatistics, data mining, design of experiments, environmental statistics, nonparametric statistics, reliability, spatial statistics and image analysis, statistical decision theory and inference, survival analysis, time series analysis.

Requirements for the Ph.D. Degree Programs in Applied Mathematics:

- (1) Completing the course work.
- (2) Passing the written Master's comprehensive examination.
- (3) Passing the written Ph.D. qualifying examination.
- (4) Passing the oral Ph.D. qualifying examination.
- (5) Admission to candidacy.
- (6) Completing residency requirements of the university.
- (7) Completing and successfully defending a doctoral dissertation.

The *course work* consists of passing Math 600 and 603 (exempted if able to pass the master's comprehensive exam), Math 611, 4 courses listed below with at least one from each area/category:

Differential Equations: Math 612, Math 614

Optimization: Math 650, Math 651

Numerical Analysis: Math 620, Math 630.

Additionally, each student is required to complete a 3 credit Math 699 (Independent Study) course in the first two years and make a presentation.

The written Master's comprehensive examination is based on the material from Math 600 and Math 603; The written Ph.D. qualifying examination is based on two courses in one of the areas of Differential Equations, Optimization, or Numerical Analysis (see the course work description above).

*The oral Ph.D. qualifying examination:*

Upon successful completion of written master's comprehensive and PhD qualifying examinations, a student will commence advanced study under close supervision of a faculty advisor. An oral Ph.D. qualifying examination
committee of at least four members will be appointed by the graduate program director on nomination by the students advisor. Two members, besides the student’s advisor, must be from the graduate faculty, but additional members who are knowledgeable in the area of intended research may be included.
The student, the advisor and the graduate program director will agree on the scope of the oral Ph.D. qualifying examination to be administered by the committee. The scope of the examination may include topics from the courses taken, research papers read, and preliminary results obtained by the student.
The committee must be notified, at least two weeks in advance, of the date, time and place of the oral Ph.D. qualifying examination.

## Requirements for the Ph.D. Degree Programs in Statistics

Every entering student will have a doctoral program advisor. The program advisor will determine if the student needs to complete any specific courses before he or she proceeds along the following steps:

- A doctoral-level pass on the department’s written comprehensive examinations. For students in the statistics program, the examinations are based on the subjects of STAT 601, STAT 602, STAT 611 and STAT 612 for the traditional track and they are based on the subjects of STAT 601, STAT 602, STAT 651 and STAT 653 for the biostatistics track. The examinations must be passed within three semesters of entering the doctoral program.
- Passing the doctoral qualifying oral examination
- Admission to candidacy
- Completing residency requirements of the university
- Completing and successfully defending a doctoral dissertation

**The Ph.D. qualifying oral examination **

Upon successful completion of requirement 1, a student will commence advanced study under close supervision of a faculty advisor. A doctoral qualifying oral examination committee of at least four members will be appointed by the graduate program director on nomination by the student’s advisor. Two members, besides the student’s advisor, must be from the graduate faculty, but additional members who are knowledgeable in the area of intended research may be included. The student, the advisor and the graduate program director will agree on the scope of the doctoral qualifying oral examination to be administered by the committee. The committee must be notified, at least two weeks in advance, of the date, time and place of the doctoral qualifying oral examination and must be provided with a study list of the candidate.

**Admission to Candidacy**

On receiving recommendation of the doctoral qualifying oral examination committee for admission to candidacy, the student must obtain the written consent of a graduate faculty member to act as the doctoral dissertation advisor and apply to the Graduate School for admission to doctoral candidacy. The application for admission to candidacy is made through the graduate program director, who certifies that all requirements for the doctorate except the doctoral dissertation, have been met and that the student has a doctoral dissertation advisor.

**Residency Requirements**

The residency requirements shall be those imposed by the university at the time of the student’s entrance to the doctoral program.

**Doctoral Dissertation**

The doctoral dissertation is the heart of the Ph.D. program. An acceptable dissertation will contain a significant advance in the current knowledge of applied mathematics or statistics. On completion of the dissertation in its final form, the dissertation advisor will recommend to the Graduate School, through the graduate program director, to appoint a final doctoral examining committee. The final oral defense of the dissertation will be conducted in accordance with the general Graduate School requirements.

**Scientific Minor**

In addition, the biostatistics track with
the statistics Ph.D. program requires that
the student complete a scientific minor
in epidemiology and/or cancer research.
The students will fulfll the scientific minor
requirement by carrying on project research
and taking at least three courses in one of
the following areas of active biostatistical
research:

- Clinical Trials
- Population Epidemiology
- Bioinformatics
- Aging Studies

(The choices for the area for the scientific Minor are not limited to the list given above. Students can do scientific minor in other research areas related to the existing faculty expertise with prior approval from the graduate program director. )

###### Facilities and Special Resources

The department’s focus on applied aspects of mathematics and statistics calls for state-of-the-art computing facilities. The students and faculty have easy access both to the university’s central computing facilities and the extensive departmental network of computers and workstations. Essentially all of the department’s computing is done on Linux workstations. Several Linux Beowulf clusters enable parallel processing of computationally intensive tasks. The department has its own full-time systems administrator, who oversees the installation and maintenance of hardware and software and manages the operation of department’s graduate student computer laboratory. The department and the university maintain site licenses for Maple, Mathematica, MATLAB, Femlab, SAS, S-Plus and other software of interest to mathematicians and statisticians.

###### Financial Assistance

Teaching and research assistantships are available on a competitive basis. Students also have the opportunity to obtain summer teaching at the university and summer internships in industry and government agencies.

##### COURSE LISTING

###### Mathematics Courses

**MATH 600
Real Analysis [3] **

Review of properties of (usual) norm and inner product in Rn, countable and uncountable sets. Basic ideas of metric spaces. Compact, connected, and complete metric spaces, continuous and uniformly continuous functions on metric spaces. Fixed points and Banach contraction principle. Uniform convergence of sequences and series of functions, term-by-term differentiation and integration. Power series and radius/interval of convergence. Arzela-Ascoli theorem. The weierstrass approximation theorem on a compact interval. Differentiation in Rn. Inverse and implicit function theorems. Prerequisite: MATH 301.

**MATH 601
Measure Theory [3]**

Measure and integration: sigma-algebras, measures, outer measures, Borel measures on R, measurable functions, integration of non-negative functions, mode of convergence, product measures, Lebesgue integral on R, decomposition and differentiation of measures, signed measures, the Lebesgue- Radon-Nikodym theorem, differentiation on R, functions of bounded variation spaces, basic theory of Lp spaces and the dual of Lp. Prerequisite: MATH 302, MATH 401 or consent of instructor.

**MATH 602
Complex Analysis [3]**

Complex numbers and their algebraic and geometric properties, moduli, polar form, powers and roots, analytic functions, derivatives, Cauchy-Riemann equations, harmonic functions, elementary functions, definite integrals, contours, line integrals, Cauchy-Goursat theorem, Cauchy integral formula, derivatives, Morera’s theorem, maximum modulus theorem, Liouville’s theorem and fundamental theorem of algebra, Schwarz lemma and applications, Taylor and Laurent series, integration and differentiation of series, residues and poles, mapping by elementary functions and conformal mapping. Prerequisite: MATH 600 or consent of instructor.

**MATH 603 Matrix Analysis**

Topics in this course will include a review of basic matrix operations, determinants,rank, matrix inverse and solving linear equations. The course then will study partitioned matrices, eigenvalues and eigenvectors, spectral decomposition, singular-value decomposition, Jordan canonical form, orthogonal projections, idempotent matrices, quadratic forms, extrema of quadratic forms, non-negative definite and positive definite matrices, and matrix derivatives. Prerequisite: MATH 221, 251 and 301, or permission of the instructor.

**MATH 611
Applied Analysis [3]**

Hilbert and Banach spaces, linear operators and quadratic functional, orthogonal bases and the generalized Fourier series, variational problems and the methods of Ritz-Galerkin, least squares and steepest descent, applications to boundary value problems for ordinary and partial differential equations. Prerequisite: MATH 600 or consent of instructor.

**MATH 612
Ordinary Differential Equations [3]**

Matrix exponentials, linear systems of equilibria, phase diagrams, non-linear systems, existence, uniqueness and dependence on initial data, stability by linearization and by Liapunov’s direct method, limit sets and LaSalle’s invariance principle, periodic orbits and self-sustained oscillations, Poincare Bendixon theory, Floquet theory, gradient systems, applications to mechanical systems and predator-prey problems. Prerequisite: MATH 225, MATH 301, MATH 302 or consent of instructor.

**MATH 614
Partial Differential Equations [3]**

Quasi-linear, first-order PDEs; conservation laws; the method of characteristics; discontinuous solutions and shock waves; linear second-order PDEs and their classification; maximum principles; elliptic PDEs; Sobolev spaces and existence of weak solutions; and regularity. Prerequisite: MATH 600 or consent of instructor.

**MATH 617
Introduction to Industrial Mathematics [3]**

The objective of this course is the survey of mathematical and statistical techniques traditionally important to quantitative modeling in industry and government. Topics might include, but are not restricted to, data acquisition and manipulation, discrete Fourier transform and image processing, linear programming, regression, random variables and distributions, Monte Carlo method, ordinary and partial differential equations, with applications of these concepts to industrial problems. The course is not designed to substitute for any course covering these topics in detail, but rather to link a survey of techniques to applications. Prerequisite: MATH 251, MATH 225, MATH 221, STAT 355 or consent of the instructor. Some programming experience is recommended.

**MATH 620
Numerical Analysis [ 3]**

Interpolation, numerical differentiation and integration, solution of non-linear equations, acceleration of convergence, numerical treatment of differential equations. Topics will be supplemented with programming assignments. Prerequisites: MATH 221, MATH 301 and familiarity with a high-level programming language or consent of instructor.

**MATH 621
Numerical Methods For Partial Differential Equations [3] **

Finite difference methods for elliptic, parabolic and hyperbolic equations; first-order systems; the eigenvalue problems; variational formulation of elliptic problems, and the finite element method and its relation to finite difference methods. Prerequisites: Math 620 , Math 630 and familiarity with a high-level programming language or consent of instructor.

**MATH 625
Computational Mathematics and C Programming [3]**

Introduction to theory and computational algorithms in selected topics of interest to mathematicians, engineers and scientists. Includes design and implementation of algorithms as C programs. Representative topics: solution of linear systems, the wavelet transform and applications to image processing and image compression, minimization routines for multi-variable functions and application to determining the equilibria of statically indeterminate trusses, Legendre polynomials and their derivatives, Gaussian quadrature of arbitrary order, non-linear system solvers, mesh generation routines and other ancillary components leading to an implementation of a simple but general-purpose finite element solver for elliptic partial differential equations in the plane. The course includes a concurrent and intensive introduction to programming in C. Prerequisite: MATH 221, MATH 225, MATH 251 or consent of instructor.

**MATH 627
Introduction to Parallel Computing [3]**

This course introduces students to scientific computing on modern parallel computers. Examples of numerical algorithms will be taken from several areas of mathematics, including numerical analysis and numerical linear algebra. Students will discuss the implications of the parallel architecture on the design of numerical algorithms. Standard packages available for low-level parallel computing will be introduced to implement high-level algorithms. Parallel computing equipment will be made available to students enrolled in the course. The course includes a significant portion of instruction dedicated to learning the parallel programming language on that machine. Prerequisite: MATH 630, fluency in C or Fortran programming or consent of instructor.

**MATH 630
Numerical Linear Algebra [3] **

Review of basic matrix theory, algebraic properties of eigenvalues and eigenvectors, diagonalization of square matrices, vector and matrix norms, singular value decomposition, orthogonal projections and generalized inverses, Matrix decompositions, LU, QR and Cholesky decompositions, numerical algorithms and applications. Topics will be supplemented with programming assignments. Prerequisites: MATH 221, MATH 301 and familiarity with a high-level programming language or consent of instructor.

**MATH 635
Foundations of Finite Element Methods [3]**

Variational formulations of physical problems; Ritz-Galerkin method; h, p and hp finite element spaces; approximation theory. A selection of topics from: plate and shell elements, mixed methods, superconvergence, a posterior error estimation, reliability and adaptivity. Computational experience with finite element codes will be provided. Prerequisites: MATH 600, MATH 620 and familiarity with a high-level programming language or consent of instructor.

**MATH 640
Linear Systems [3]**

Impulsive functions, signal representations and input-output relations, dynamic equations in the time and frequency domains, observability and controllability for continuous and discrete time, real stability and stabilization of linear systems, quadratic regulators for continuous and discrete time systems, asymptotic observers and compressor design. Prerequisite: STAT 451 or consent of instructor.

**MATH 650
Foundations of Optimization [3]**

Study of the fundamental theory underlying linear and nonlinear optimization; unconstrained and constrained optimization; first- and second-order optimality conditions for unconstrained optimization; Fritz John and Kuhn-Tucker conditions; fundamental notions of convex sets and convex functions; theory of convex polyhedral; duality; linear programming, quadratic programming. Prerequisites: MATH 251, MATH 430, MATH 302 or consent of instructor.

**MATH 651
Optimization Algorithms [3]**

Design and analysis of algorithms for linear and non-linear optimization; first-order numerical methods for unconstrained optimization (line-search methods, steepest-descent method, trust-region method, conjugate-gradient method, quasi-Newton methods, methods for large scale problems); Newton’s method; numerical methods for linear programming (simplex methods, interior-point methods); numerical methods for constrained optimization (penalty, barrier, and augmented-Lagrangian methods, sequential quadratic programming method). Prerequisite: MATH 221, MATH 251, or consent of instructor. MATH 650 recommended.

**MATH 652
Stochastic Models in Operations Research [3]**

Stochastic programming, Markov decision processes, stochastic inventory models and sequential stochastic games. Prerequisites: MATH 650, MATH 681, STAT 611 or consent of instructor.

**MATH 655
Calculus of Variations [3]**

Classical calculus of variations, the simplest variational problem, examples, variation of a functional, the Euler-Lagrange equations, the isoperimetric problem, constrained variational problems, Legendre’s condition, Jacobi necessary condition, connection between Jacobi condition and the theory of quadratic forms, the field of a functional, Hilbert invariant integral, Weierstrass E-function, Riccati differential equation, Noether’s theorem, the principle of least action, conservation laws and the Hamilton-Jacobi equation.

**MATH 671
Topology [3]**

Metric spaces, topological spaces, derived topological spaces, separation axioms, generalized convergence, covering properties and compactness, connectedness, metrizability, complete metric spaces and introduction to homotopy theory. Prerequisite: MATH 301.

**MATH 673
Differential Geometry [3]**

The differential geometry of curves and surfaces, curvature and torsion, moving frames, fundamental differential forms and intrinsic geometry of a surface. Prerequisites: MATH 221 and MATH 251.

**MATH 681
Linear Algebra [3]**

Finite-dimensional vector spaces, subspaces, linear transformations and matrices. Further topics to be chosen from: convex sets and convex functional; dual space; direct sum and quotient space; minimal polynomials; Jordan canonical form; inner product; normal, symmetric and orthogonal transformations and applications. Prerequisite: MATH 221 or consent of instructor.

**MATH 683
Number Theory [3]**

Divisibility, prime numbers, modules and linear forms, unique factorization theorem, Euler’s function, Mobius function, cyclothymic polynomials, congruences and quadratic residues, Legendre’s and Jacobi’s symbol, reciprocity law of quadratic residues and introductory explanation of the method of algebraic number theory. Prerequisite: MATH 301.

**MATH 685
Combinatorics and Graph Theory [3]**

General enumeration methods, difference equations, generating functions. Elements of graph theory, including transport networks, matching theory and graph algorithms. Prerequisite: MAT 301 or consent of instructor.

**MATH 688
Abstract Algebra [3]**

Sets and mapping, groups, sub-groups, homomorphisms, Sylow and Cayley theorems, rings and ideals, Euclidean rings, extension fields and Galois theory. Prerequisites: MATH 301 and MATH 302 or consent of instructor.

**MATH 690
Mathematics Seminars [0]**

**MATH 699
Independent Study in Mathematics [1-6]**

**MATH 700
Special Topics in Numerical Analysis [1-3]**

**MATH 710
Special Topics in Applied Mathematics [1-3]**

**MATH 717
Projects in Industrial Mathematics [3]**

The primary objective of the course is to apply knowledge gained in students’ graduate program to a number of micro-projects in a way that reflects procedures done in non-academic settings. This includes working in a team on the projects; helping to prepare a written report on the problem formulation, analysis and results; and giving an oral presentation on the work. The industrial-oriented problems might need modeling involving methodologies from probability, statistics, ordinary or partial differential equations or discrete mathematics. The course is considered a core requirement for students taking the industrial mathematics master’s degree option, but other students are encouraged to participate. Prerequisite: MATH 617 or consent of instructor.

**MATH 740
Special Topics in Systems Theory and Operations Research [1-3]**

**MATH 750
Introduction to Interdisciplinary Consulting [3]**

This course provides an introduction to professional consulting. Students will gain experience in approaching application problems, translating them into the language of mathematics and statistics, applying mathematical and statistical tools including software, and delivering the product to the client. People from across campus and from local companies and government agencies will serve as clients to provide real-life consulting experience. If the course is integral to a student's curriculum, the course can be repeated once for credit by petitioning the graduate program director.

**MATH 799
Master’s Thesis Research [1-6]**

Master’s thesis research under the direction of a UMBC MEES faculty member. Note: Six credit hours are required for the master’s degree.

**MATH 898
Pre-Candidacy Doctoral Research [1-6]**

Research on doctoral dissertation conducted under the direction of a faculty advisor before candidacy.

**MATH 899
Doctoral Dissertation Research [6]**

Doctoral dissertation research under the direction of a UMBC MEES faculty member. Note: A minimum of 12 credit hours are required for the doctorate.

###### Statistics Courses

**STAT 601
Applied Statistics I [3]**

Theory and applications of the linear regression model, least squares estimation, model building, influence diagnostics, multi-collinearity and graphical analysis of residuals, nonlinear regression, logistic regression. Data analysis using statistical packages and other topics as time permits. Prerequisite: STAT 453 or consent of instructor.

**STAT 602
Applied Statistics II [3]**

Principles of experimental design, the analysis of variance and covariance, randomized designs, Latin square designs, incomplete block designs, factorial designs, confounding and fractional replication, split-plot designs and use of statistical packages. Prerequisite: STAT 453 or consent of instructor.

**STAT 603
Categorical Data Analysis [3]**

Theory and applications related to the statistical analysis of categorical nominal and ordinal data, scales of measurement, epidemiologic measures of risk assessment, discrete distribution theory, exact confidence intervals, two-by-two tables, conditioning on margins, chi-square and Fisher’s exact tests, measures of agreement, inter-rater reliability, matched pairs, McNemar’s test, theory and applications of log-linear models, multi-nomial response models, methods for ordinal data analysis, missing data and data analysis using statistical software packages. Co- or prerequisite: STAT 653, STAT 601 or consent of instructor.

**STAT 605
Survey Sampling [3]**

Sampling versus total enumeration, planning of a survey sampling, statistical sampling methods and their analysis, simple random sampling, stratified sampling, systematic sampling, cluster sampling, and double and multi-stage sampling, problem of non-response and variance estimation, and practical case study. Prerequisite: STAT 453 or consent of instructor.

**STAT 607
Bayesian Inference [3]**

The course is designed to provide statistics graduate students with an introduction to Bayesian methodology. The first part of the course will deal with theoretical and methodological development in Bayesian statistics. The second half of the course also will introduce students to Bayesian computing and related software such as BUGS. Prerequisites: STAT 651 and 653, STAT 611 and 612, or consent of instructor.

**STAT 611
Mathematical Statistics I [3]**

Random variables and their distribution functions, probability density and frequency functions, mathematical expectations and moments, moment-generating functions, characteristic functions, probability inequalities, special discrete and continuous distributions, multi-variate random vectors and their distributions, conditional and marginal distributions, conditional expectation, transformation and functions of random variables, modes of convergence and central limit theorem. Prerequisite: STAT 451 or consent of instructor. Credit will not be given for both STAT 611 and STAT 651.

**STAT 612
Mathematical Statistics II [3]**

Sampling distribution, estimation and testing of statistical hypotheses, principles of estimation, unbiasedness, minimum variance, consistency, Rao-Cramer inequity, Rao-Blackwell theorem, sufficiency, multi-parameter estimation, maximum likelihood and moment estimates, the Neyman-Pearson approach, UMP test, unbiased and consistent tests and multi-parameter hypotheses. Prerequisite: STAT 611 or consent of instructor. Credit will not be given to both STAT 612 and STAT 653.

**STAT 613
Linear Models [3]**

Matrix operations and generalized inverses, linear models, least squares theory, best linear unbiased estimation and Gauss-Markov theorem, general linear hypotheses, distribution of quadratic forms and mixed and random effects models. Prerequisite: STAT 453 or consent of instructor.

**STAT 614
Environmental Statistics**

This is a graduate-level introduction to statistical methods used in environmental applications. The following will be emphasized throughout the course: non-parametric methods using environmental data; methods of analyzing data that are below the limit of detection; sampling designs, including stratified sampling, composite sampling and ranked set sampling; sampling to determine hot spots; trend estimation methods for uncorrelated, correlated and seasonal data; discussion of some basic ideas from spatial statistics; and environmental data analysis using statistical software. Prerequisite: STAT 453/653 or consent of instructor.

**STAT 615
Multi-Variate Statistical Analysis [3]**

Multi-variate normal distribution, its properties and inference; multiple and partial correlation; Wishart distribution; Hotelling’s T2 test; tests for co-variance structure; discriminant analysis; principal components; multi-variate analysis of variance; factor analysis; and canonical correlation. Prerequisite: STAT 453 or consent of instructor.

**STAT 616
Non-Parametric Statistics [3]**

Distribution-free statistics and asymptotically distribution-free statistics for the one-, two- and multi-sample problems and linear rank statistics; U-statistics and related statistics; treatment of ties; estimation of parameters based on rank statistics and other efficiency concepts and goodness-of-fit tests. Prerequisite: STAT 453 or consent of instructor.

**STAT 617
Time Series Analysis [3]**

Theory and applications of time series models, auto-regressive integrated models, Box-Jenkins methodology, forecasting, seasonal models and their applications, spectral theory and estimation of time series models and time series data analysis using statistical packages. Prerequisite: STAT 451 or consent of instructor.

**STAT 618
Applied Multi-Variate Methods [3]**

Multiple regression, partial and multiple correlation, multi-variate normal distribution, statistical inference for the mean vector and co-variance matrix, uni-variate and multi-variate analysis of variance, principal components, canonical correlation, discriminant analysis, factor analysis and cluster analysis. The methods will be illustrated using real data with the help of statistical packages. Prerequisite: STAT 453 or consent of instructor.

**STAT 619
Biostatistics: Principles and Design [3]**

Philosophy and statistical theory involved in designing sequential, randomized medical studies, particularly, clinical trials; Scientific method; randomization principle; randomization techniques and their associated distribution theory and permutation tests; adaptive designs and ethics behind randomized studies involving human subjects; statistical methods, including survival analysis, power and sample size; dealing with multiplicities; interim monitoring techniques, including conditional power, spending functions and Brownian motion; randomized does-response studies; likelihoods and introductory Martingale theory. Both theory and applications will be stressed. Prerequisite: STAT 601, STAT 453 or consent of instructor.

**STAT 620
Biostatistics: Advanced Analysis [3]**

Theory and applications related to the statistical analysis of certain incomplete data, including survival data and longitudinal data; survival time model and censoring; generalized linear and nonlinear mixed-effect models; Kaplan-Meier estimator; Nelson-Aalen estimator; log-rank and Wilcoxon rank tests; Weibull; log-normal gamma; log-logistic distribution; parametric regression; partial likelihood; relative risk regression models; Martingale; EM-algorithm; Laplaces approximation; adaptive Gausian quadratures point; MCMC; Metropolis-Hasting algorithm; and Gibbs sampling. Prerequisite: STAT 619 or consent of instructor.

**STAT 621
Probability Theory and Stochastic Processes I [3]**

Measure-theoretic approach to probability, conditional probability and random variables; distribution functions; modes of convergence; the zero-one law; probability inequalities; dependence and the Martingale theory; and Markov chains and their stationary distributions. Prerequisite: MATH 600 or MATH 601, STAT 611 or consent of instructor.

**STAT 622
Probability Theory and Stochastic Processes II [3]**

Advanced development of estimation techniques, including sufficiency, minimum variance and invariance; asymptotic properties of maximum likelihood estimators; Neyman-Pearson lemma and least favorable distributions; Bayesian inference and decision theory; and Stein estimator. Prerequisite: STAT 612 or consent of instructor.

**STAT 623
Sequential Analysis [3]**

Sequential estimation, sequential probability ratio tests, operating characteristic and average sample size, efficiency of sequential testing, extensions of the probability ratio tests and sequential test. Prerequisite: STAT 612 or consent of instructor.

**STAT 625
Spatial Statistics and Image Analysis [3]**

Analysis of geostatistical data, variogram, co-variogram, spatial prediction, kriging, Lattice models, Markov random fields and their use in Bayesian inference for spatial models, statistical image analysis, frequentist and Bayesian approaches and their applications. Additional topics in image analysis, such as mixture analysis, segmentation and in spatial point process theory may be covered.

**STAT 633
Methods of Statistical Computing [3]**

Pseudo-random number generation; sampling methods inversion; rejection sampling, ratio of uniforms; squeezing; variance reduction; importance sampling; antithetic variates; stochastic simulation; Gibbs sampling and Markov Chain Monte Carlo; resampling methods, including jackknife, bootstrap and randomization tests; use of the EM algorithm.

**STAT 651
Basic Probability [3]**

This is a graduate-level introduction to basic probability theory and its applications that is suitable for the applications-oriented master’s tracks and will be cross-listed with STAT 451, the undergraduate basic probability course. Emphasis will be given on problem-solving using tools of probability. The course will cover all basic notions of probability, random variables, standard discrete and continuous distributions, mean and variance, functions of random variables and central limit theorem.

**STAT 652
Stochastic Models in Operations Research [3]**

Stochastic programming, Markov decision processes and stochastic inventory models and sequential stochastic games. Prerequisite: MATH 650, MATH 681, STAT 611 or consent of instructor.

**STAT 653
Basic Mathematical Statistics [3]**

This is a graduate-level introduction to basic mathematical statistics and its applications that is suitable for the applications-oriented master’s tracks. It will be cross-listed with STAT 453, the undergraduate basic mathematical statistics course. Emphasis will be given on problem-solving using tools of mathematical statistics. The course will cover all basic notions of estimation, tests, confidence intervals, hypothesis testing, linear models, analysis of variance, control charts and non-parametric procedures. Prerequisite: STAT 651 or consent of instructor.

**STAT 690
Statistics Seminars [0-1]**

**STAT 699
Independent Study in Statistics [1-6]**

One-on-one instruction of a graduate student by a member of the faculty. The topic and format of the course are by mutual agreement between the professor and the student, subject to approval by the graduate program director.

**STAT 700
Special Topics in Statistical Methods and Data Analysis [1-3]**

**STAT 710
Special Topics in Mathematical Statistics and Statistical Inference [1-3]**

**STAT 720
Special Topics in Probability and Stochastic Processes [1-3]**

**STAT 750
Introduction to Interdisciplinary Consulting [3]**

A course designed to train students in the art of interdisciplinary consulting and prepare them for careers as professional statisticians. Prerequisites: STAT 601, STAT 602, STAT 651, STAT 653, familiarity with SAS and S-Plus or consent of instructor.

**STAT 799
Master’s Thesis Research [1-6]**

Master’s thesis research under the direction of a UMBC MEES faculty member. Note: Six credit hours are required for the master’s degree.

**STAT 898
Pre-Candidacy Doctoral Research [1-6]**

Research on doctoral dissertation conducted under the direction of a faculty advisor before candidacy.

**STAT 899
Doctoral Dissertation Research [6]**

Doctoral dissertation research under the direction of a UMBC MEES faculty member. Note: A minimum of 12 credit hours are required for the doctorate.