Advanced Engineering Mathematics

[Yazmín Bohon]

ENGR3300 Advanced Engineering Mathematics (3 semester credit hours) Survey of advanced mathematics topics needed in the study of engineering. Topics include use of complex numbers, properties of complex-valued functions, scalar and vector fields, introduction to partial differential equations, and Fourier series. Examples are provided from electromagnetics, fluid mechanics, thermodynamics, and engineered systems. This course includes a required laboratory. Prerequisites: (MATH 2415 or MATH 2419) and ENGR 2300. Prerequisite or Corequisite: MATH 2420. (3-1) S

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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