# Mathematics and Statistics

## REU@NAU 2012 Project Descriptions

The projects you can work on as a student in the Research Experience for Undergraduates program in 2012 are described below.

#### Shafiu Jibrin—Operations Research (2 students)

Operations research is concerned with decision-making where one chooses decision variables that maximize or minimize an objective function, subject to the requirement that the decision variables satisfy certain constraints. For example, one may wish to find the amount of each product to produce in a factory in order to maximize the profit subject to restriction on labor, raw materials and demand on the products. An important area of operations research that has many applications is called semidefinite programming. In a semidefinite programming problem one optimizes a linear objective function subject to a system of constraints called linear matrix inequalities (LMIs). This project concerns the strict feasibility problem for a system of LMIs, where we seek to find a strictly feasible point for the system. The project would study and develop iterative methods to solve this problem. One part of the project would use orthogonal projections based on gradients or sub-gradients. The second part of the project would study a variant of the alternating projection method for the strict feasibility problem. One would have an option to do either part of the project. For more information on semidefinite programming, go to the link: Helmberg - Semidefinite Programming Prerequisites: A background in Linear Algebra is required. Some experience with a programming language or mathematics software (e.g., Matlab, C, C++, java, Maple, or Mathematica) is also necessary.

#### John Neuberger and James W. Swift - Partial Differential Equations (4 students)

Our recent focus has been on using the Gradient Newton Galerkin Algorithm (GNGA) to investigate symmetry and bifurcation of various nonlinear elliptic PDE. The first paper on the GNGA is here Our past work includes topics such as fractals, parallel algorithms, dynamical systems, numerical integration, graphical representation of solutions, and automated symmetry and bifurcation analysis. A list of John Neuberger's papers is here, and Jim Swift's page of REU project ideas is here.

We write computer programs to implement ideas from multivariate calculus, differential equations, algebra, and especially linear algebra in order to explore the underlying structure of our problem in function space. The REU students that we will ultimately mentor will learn a lot of background material in these subjects, write and test codes, use our codes, and after an initial period of general study, seek new results under our guidance for related problems.

Prerequisites: A background in Linear Algebra is required. Course work through Calculus III and Differential Equations is required, and a course on PDE is a plus. Some experience with a programming language and mathematics software (e.g., Matlab, C++, or Mathematica) is also necessary.