#### List of acceptable courses that satisfy Advanced Engineering Mathematics requirements for the PhD in Engineering program

This list may not be comprehensive. Not all of these courses are offered every semester. Students should plan in advance and check on CampusNet for the listings. Other courses may be added to this list, with approval by the Graduate Affairs Committee of the Washkewicz College of Engineering. Doctoral students can contact the Director of the Doctoral programs for any questions.

- ESC 702 Advanced Optimization (4 credit hours).

Methods of optimization for engineering systems; classical optimization, Taylor’s theorem, Lagrange Multipliers, and Kuhn-Tucker theorem; direct methods, Newton and quasi-Newton methods, penalty and Barrier methods, linear and nonlinear programming.

- ESC 704 Stochastic Systems (4 credit hours).

Prerequisite: Engineering Statistics. Optimization in engineering economics; application of renewal theory; inventory and Markov decision models; Bayesian decision analysis.

- ESC 706 Advanced Partial Differential Equations (4 credit hours).

Engineering applications and solution techniques for partial differential equations; variational derivation of differential equations and boundary conditions; Hamilton’s principle and Lagrange’s equation; numerical methods and computer solutions for differential equations.

- BME 570/770 Biomedical Signal Processing (3-0-3).

Prerequisite: Graduate standing in Engineering or permission of instructor. Signals and biomedical signal processing; the Fourier transform; image filtering, enhancement, and restoration; edge detection and image segmentation; wavelet transform; clustering and classification; processing of biomedical signals; processing of biomedical images.

- CVE 604/704 Elasticity (4-0-4).

Prerequisite: CVE 513. Elasticity topics include tensor algebra, fundamentals of stress analysis, fundamentals of deformation theory, thermoelastic constitutive relationships, uniqueness of solution, Airy’s stress function, and various solution techniques for two-dimensional problems.

- EEC 643/743 or MCE 693/793 Nonlinear Systems (4-0-4)

Prerequisite: EEC 510. State-space and frequency domain analysis and design of nonlinear feedback systems. Methods include Liapunov’s stability analysis, singular perturbations, describing functions, and absolute stability criteria. Feedback linearization and variable structure/sliding mode control.

- EEC 645/745 Intelligent Control Systems (4-0-4)

Prerequisite: EEC 510. Artificial intelligence techniques applied to control system design. Topics include fuzzy sets, artificial neural networks, methods for designing fuzzy-logic controllers and neural network controllers; application of computer-aided design techniques for designing fuzzy-logic and neural-network controllers.

- EEC 644/744 Optimal Control Systems (4-0-4)

Prerequisite: EEC 510. Introduction to the principles and methods of the optimal control approach: performance measures; dynamic programming; calculus of variations; Pontryagin’s principle; optimal linear regulators; minimum-time and minimum-fuel problems; steepest descent; and quasi-linearization methods for determining optimal trajectories.

- EEC 693/793 Population-Based Optimization (4-0-4)

This course discusses the theory, history, mathematics, and applications of population-based optimization algorithms, most of which are based on biological processes. Some of the algorithms that are covered include genetic algorithms, evolutionary computing, ant colony optimization, biogeography-based optimization, differential evolution, and artificial immune systems. Students will write computer-based simulations of optimization algorithms using MATLAB. After taking this course the student will be able to apply population-based algorithms using MATLAB (or some other high-level programming language) to realistic engineering problems. This course will make the student aware of the current state-of-the-art in the field, and will prepare the student to conduct independent research in the field.

- ESC 794 Selected Topics Mathematics of Control and Systems Theory (4-0-4)

Selected mathematical topics to prepare the student for independent, advanced study in systems and control theory and related fields, fundamental notions, real analysis methods, and geometric methods. Open to doctoral students only unless permission is obtained from the instructor. One course in linear algebra and at least one graduate course in control systems are required.

- CHE 504 ADVANCED REACTOR DESIGN [4 credit(s)]

Prerequisite: Graduate standing in chemical engineering or permission of instructor. Flow patterns in ideal and real reactors. Residence time distribution as a reactor design tool. Reactor design for multiple reactions, yield and selectivity concepts. Parametric sensitivity. Reactor dynamics and stability. Introduction to high-temperature non-catalytic reactions.

- MCE 501 MECHANICAL ENGINEERING ANALYSIS [4 credit(s)]

Mathematical modeling/analysis of physical systems; boundary value problems. Fourier series and integrals; diffusion equation, Sturm-Liouville theory; Wave equation, d’Alembert’s solution; Bessel and Legendre functions.

- MCE 505 NUMERICAL METHODS IN MECHANICAL ENGINEERING [4 credit(s)]

Numerical methods for linear algebra; interpolation; integration; solving nonlinear algebraic equations and ordinary differential equations; spectrum analysis; optimization; and modeling of data.

- MTH 514 LINEAR ALGEBRA AND FUNCTIONS OF SEVERAL VARIABLES [4 credit(s)]

Vector spaces, linear transformations, eigenvalues, eigenvectors, canonical forms of matrices, matrix decompositions, applications of linear algebra, calculus of functions of several variables, Jacobians, Taylor’s formula, multiple integrals, surface integrals, and change of variables formula.

- MTH 525 MATHEMATICAL METHODS IN ENGINEERING AND SCIENCE I [4 credit(s)]

Part one of a two-part sequence devoted to methods of applied mathematics, including various topics in ordinary and partial differential equations, integral equations, and calculus of variations, as well as specific applications to engineering and the sciences.

- MTH 532 PROBABILISTIC MODELS [4 credit(s)]

Modeling of real-world problems using methods of probability theory such as Markov chains, queuing theory, decision analysis, and simulation.

- MTH 587 DYNAMICAL SYSTEMS [4 credit(s)]

Systems of differential equations, local and global behavior of a vector field in the plane, discrete dynamical systems, structural stability, the Poincare-Bendixon theorem, bifurcations, chaos, and strange attractors.

- MTH 577 NUMERICAL METHODS I [4 credit(s)]

Introduction to the numerical methods of financial derivatives. Topics include an overview of the basic concepts of mathematical finance, computational tools such as binomial methods, finite-difference methods, and methods for evaluating American options and Monte Carlo simulation. Numerical experiments are conducted using software such as MATLAB, Microsoft Excel, and Maple, but no previous familiarity with these packages is assumed. Part one of a two-part sequence.

- MTH 678 NUMERICAL METHODS II [4 credit(s)]

Prerequisite: MTH 577 or departmental approval. Applications of numerical methods to real-life problems in science and engineering. Topics may include the following: initial value problems, the radar problem, the calibration problem, building exploratory environments, refined graphics, numerical approximation of orbits in the planar three-body problem, effect of spin on trajectories, least squares problems, and boundary value problems. Numerical experiments are conducted using software such as MATLAB and Maple, but no previous familiarity with these packages is assumed. Part two of a two-part sequence.

- MTH 626 MATHEMATICAL METHODS IN ENGINEERING AND SCIENCE II [4 credit(s)]

Prerequisite: MTH 525 or departmental approval. Part two of a two-part sequence devoted to methods of applied mathematics, including various topics in ordinary and partial differential equations, integral equations, and calculus of variations, as well as specific applications to engineering and the sciences.

- STA 524 PROBABILITY AND MATHEMATICAL STATISTICS [4 credit(s)]

STA 524 is an introduction to the mathematical theory of probability and statistics using calculus. It is the study of statistics from a mathematical standpoint and prepares students for further study of statistical inference. It provides a strong foundation in mathematical statistics for understanding the concepts and development of statistical methodology.

- STA 531 CATEGORICAL DATA ANALYSIS [4 credit(s)]

Prerequisite: STA 536 or STA 567 or departmental approval. The course will cover techniques of modeling data for data that are categorical rather than continuous in nature. Topics to be covered include joint, marginal, and conditional probabilities, relative risk, odds ratios, generalized linear models, logistic regression, multi-category logit models, and loglinear models. The course will utilize data examples from the fields of biology, medicine, health, epidemiology, environmental science, and psychology. The course will use a statistical programming language. The course will also require the completion of a categorical data analysis project.

- STA 536 DESIGN AND ANALYSIS OF EXPERIMENTS [4 credit(s)]

This course provides a review of basic statistical concepts and a comprehensive introduction to statistical methods of designing experiments and analyzing data. A variety of experimental designs are covered, and regression analysis is presented as the primary technique for analyzing data from designed experiments, and in discriminating between various possible statistical models. This course is designed for students who have completed the first course in statistical methods. This background course should include at least some techniques of descriptive statistics, the normal distribution and an introduction to basic concepts of confidence intervals and hypothesis testing. Students will learn how to use Statistical Software for data manipulation and data analysis.

- STA 564 STATISTICS AND DATA ANALYSIS [4 credit(s)]

Introduction to statistics, including descriptive statistics, sampling, expected value, estimation, hypothesis testing, and statistical computing software.

- STA 567 APPLIED REGRESSION MODELS [4 credit(s)]

Students will learn techniques, ideas, and concepts associated with linear regression. In the context of linear regression, they will learn how to use specific statistical methods and general modes of statistical thinking to make inferences from data. The emphasis is on being able to build an appropriate regression model, on being able to assess the adequacy of a proposed model, and on drawing and formulating conclusions about the fitted model. They will also learn how to assess the relative merits and applicability of competing statistical techniques. Students will learn how to perform the techniques covered in this course by using a statistical software package. Topics may also include transformations, matrix representation, non-linear regression and other topics as time allows.

- STA 571 STATISTICAL METHODS FOR GENETIC DATA [4 credit(s)]

This is an introduction to quantitative methods associated with the analysis of human genetic data, with an emphasis on applied projects aimed at prediction of disease status of a new sample on the basis of observed samples and identification of biomarkers leading to human disease. Topics will include overview of microarray, proteomics, and metabolomics data, overview of supervised learning, linear methods for classification, kernel methods, boosting and additive trees, neural networks, support vector machines and flexible discriminants, and unsupervised learning. Students must be familiar with matrix notation and the statistical programming language R will be used in this course.

- STA 635 STATISTICAL CONSULTING AND PROGRAMMING [3 credit(s)]

This course assigns students to work in consulting teams with, when possible, university or community partners on real-world case studies of statistical methods learned in previous courses. Students prepare written reports and oral presentations that discuss the findings of the analysis. Topics specific to this course may include sample size determination, reliability, and validity, missing data imputation, random sampling, randomization schemas, data management techniques, ethics, IRB, or other topics chosen by the instructor. In addition, students learn data manipulation and graphics using a variety of software to include SAS, SPSS, Minitab, and R.

- STA 675 APPLIED MULTIVARIATE STATISTICS [4 credit(s)]

Applications of multivariate statistical methods to applications in medicine, biology, and the social sciences. The main topics of this course will address the issue of multiple measures of a response variable of interest. Topics will include multivariate analysis of variance (MANOVA), principal components, factor analysis, canonical correlation analysis, and discriminant analysis, among others. Students must be familiar with matrix notation, and statistical software will be used in the course.

- STA 685 ADVANCED DATA MODELS [4 credit(s)]

This course introduces various methods of modern, computationally-based methods for exploring and drawing inferences from data. After a brief review of probability and inference, the course covers resampling methods, non-parametric regression, prediction, and dimension reduction and clustering. Specifically, topics include: tree-based methods, boosting, ensemble learning, forests, neural networks, support vector machines, bootstrap, cross-validation, smoothing methods such as kernels, local regression, splines, smoothing in likelihood models, density estimation, shrinkage methods (ridge regression, lasso), longitudinal data analysis and high dimensional problems.

- STA 767 DATA ANALYSIS FOR DOCTORAL STUDENTS I [4 credit(s)]

An applied data analysis course that begins with a quick review of techniques for analyzing two independent samples with a quantitative response. Other covered methodologies include One- and Two-Way Analysis of Variance, nonparametric statistics, and regression. The statistical methods taught will explore the concepts of estimation, hypothesis testing, statistical significance and p-value. The course emphasizes the link between statistical graphics and formal statistical tests and involve the use of a statistical programming language. Part one of a two-part sequence.

- STA 768 DATA ANALYSIS FOR DOCTORAL STUDENTS II [4 credit(s)]

We will continue using the analysis of variance model developed in STA 767 to study model repeated measures. We will continue the multiple regression model to study serial correlation, multivariate response, as well as collinearity and leverage. We will also study categorical data techniques such as risk and odds as well as logistic and Poisson regression. Part 2 or a two-part sequence.

- BIO 740 BIOSTATISTICS [3 credit(s)]

Introductory course in biostatistics, including probability, statistical inference, hypothesis testing, regression, and other analytical statistical methods applicable to biology.