# Industrial Mathematics Course Descriptions

**56:645:556 Visualizing Mathematics by Computer.** (3)

Introduction to symbolic computational packages and scientific visualization through examples from calculus and geometry. Covers 2-D, 3-D, and animated computer graphics using Maple, Mathematica, and Geomview. No programming knowledge required.

**56:645:560 Industrial Mathematics.** (3)

This course covers various problems that can be found in industry. A problem based learning approach is used. Each problem studied motivates the need for learning the mathematical techniques necessary to solve the problem. The problems will be solved using MATLAB and C++ programs. Part of the course involves writing a report on a project and giving a presentation of the results. It is suggested that students learn to use LaTeX. Previous problems include Monte Carlo methods for a financial application, circadian rhythm analysis, atmospheric refraction correction, and the Fourier synthesis of ocean scenes. Prerequisites include a strong background in undergraduate mathematics and knowledge of C++ and MATLAB. This course is suitable for graduate students and advanced undergraduates.

**56:645:562 Mathematical Modeling.** (3)

This course introduces concepts of mathematical modeling through a hand-ons problem solving approach. Depending on course enrollment, the students are divided into groups for two projects. The first project is a competition where each group is solving the same problem. The groups will develop a design and submit a paper describing the design along with C++ code of their design. The second project is a class project where the entire class will solve a problem working in groups but in this project, each group is working on a different part of the bigger problem. The groups must coordinate their efforts and integrate the solutions to solve the main problem. Previous modeling problems include problems from various sports (football, basketball, and tennis), radar system modeling and tracking, ballistic missile modeling, and thermal expansion in a hot water heater. Prerequisites include a strong background in undergraduate mathematics and knowledge of C++ and MATLAB. This course is suitable for graduate students and advanced undergraduates.

**56:645:563 Statistical Reasoning. **(3)

Random variables, uniform, Gaussian, binomial, Poisson distributions, probability theory, stationary processes, central limit theorem, Markov chains, Taguchi quality control.

**56:645:571 Computational Mathematics I **(3)

This first semester of a one-year sequence covers the following numerical techniques for solving mathematical problems on a computer: the IEEE internal representation of floating point numbers, interpolation, root finding, numerical integration, numerical differentiation, and function minimization. The material is presented so that topics build on one another and applications are given to illustrate the use of the techniques. Prerequisites include calculus and knowledge of C++. The course is suitable for graduate students and advanced undergraduates who have the prerequisites. It is preferred that the first semester course 645:571 is taken before the second semester course 645:572.

**56:645:572 Computational Mathematics II **(3)

This second semester of a one-year sequence covers the following numerical techniques for solving mathematical problems on a computer: numerical linear algebra (including the numerical solution of linear systems of equations and the algebraic eigenvalue problem) and the numerical solution of differential equations. The material is presented so that topics build on one another and applications are given to illustrate the use of the techniques. Prerequisites include advanced calculus, linear algebra, differential equations and knowledge of C++. The course is suitable for graduate students and advanced undergraduates who have the prerequisites.**
56:645:535-536 Algebra for Computer Scientists I,II **(3 + 3 credits)

Linear and abstract algebra, including group theory, with applications to image processing, data compression, error correcting codes, and encryption.

**56:645:537 Computer Algorithms.** (3)

Algorithm design techniques: divide-and-conquer, greedy method, dynamic programming, backtracking, and branch-and-bound. Advanced data structures, graph algorithms, and algebraic algorithms. Complexity analysis, complexity classes, and NP-completeness. Introduction to approximation algorithms and parallel algorithms.

**56:645:540 Computational Number Theory and Cryptography.** (3)

Primes and prime number theorems and numerical applications; the Chinese remainder theorem and its applications to computers and Hashing functions; factoring numbers; cryptography; computation aspects of the topics emphasized. Students required to do some simple programming.

**56:645:541 Introduction to Computational Geometry. **(3)

Algorithms and data structures for geometric problems that arise in various applications, such as computer graphics, CAD/CAM, robotics, and geographical information systems (GIS). Topics include point location, range searching, intersection, decomposition of polygons, convex hulls, and Voronoi diagrams.

**56:645:557 Signal Processing.** (3)

Signal modeling: periodic, stationary, and Gaussian signals. System representation: Volterra representation, state space representation, simulation. Themes in system design: least square estimation, system identification, adaptive signal processing. Representation of discrete causal signals: role of Fourier analysis, convolutions, fast Fourier transforms. Realization of linear recurrent structures: controllability, observability and minimal realization, frequency domain analysis of signals, and the role Laplace transforms. Stability analysis: Lyapunov and linearization methods. Prediction, filtering, and identification: linear prediction, the LQR problem, Kalman filter.

**56:645:558 Theory and Computation in Probability and Queuing Theory.** (3)

Basic probability structures, probability distributions, random number generations and simulations, queuing models, analysis of single queue, queuing networks, applications of queuing theory.

**56:645:561 Optimization Theory.** (3)

Linear programming: optimization, simplex algorithm, nonlinear programming, game theory.

**56:645:574 Control Theory and Optimization.** (3)

Controllability, observability, and stabilization for linear and nonlinear systems. Kalman and Nyquist criteria. Frequency domain methods, Liapunov functions.

**56:645:575 Qualitative Theory of Ordinary Differential Equations. **(3)

Cauchy-Picard existence and uniqueness theorem. Stability of linear and nonlinear systems. Applications to equations arising in biology and engineering.

**56:645:580 Special Topics in Applied Mathematics.** (3)

Topics vary from semester to semester. Prerequisite: Permission of instructor. Course may be taken more than once.