## Reconstruction Problems in Graph Theory

## Project Mentor

Professor Stephen Hartke (Department of Mathematics) and Student Mentor, Derrick Stolee (Department of Mathematics)

## Prerequisites

There aren't many prerequisites in terms of material studied, but students must have had a proof-based course. Background in discrete mathematics (such as graph theory and combinatorics) will be beneficial.

## Project Description

Graph theory is a very broad area focusing on the mathematical study of networks. We will work on studying the question of graph reconstruction: Suppose that G is a fixed but unknown graph. What information about pieces of G would allow you to "reconstruct" G exactly? One famous conjecture is that G can be reconstructed from the collection of subgraphs formed by deleting one vertex from G. We will study variants of the question when extra information is also provided.

## Differential/Difference Equations

## Project Mentors

Professor Allan Peterson (Department of Mathematics) and Student Mentor, Chris Ahrendt (Department of Mathematics)

## Prerequisites

A calculus sequence and a one-semester course in Differential Equations.

## Project Description

We will be concerned with problems that come up in differential equations and difference equations. One of our interests will be to see how these two theories can be unified and extended. For example, calculus is very useful in differential equations and when one studies difference equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general time scales calculus (which for example also includes quantum calculus). Consequently we get a generalization of difference equations and differential equations to so-called dynamic equations on time scales. A simple application of dynamic equations on time scales is a population model which is discrete in season, dies out in winter, while their eggs are incubating or dormant, and, in season again, when hatching gives rise to a overlapping population. Other potential areas of applications include engineering, biology, economics and finance, and mathematics education. Currently there are about 300 researchers worldwide who have published about 500 research articles in the area of dynamic equations on time scales.

## Research Activities

Previous REU teams have studied the so-called nabla exponential function (which is a generalization of the usual exponential function) oscillation of a Euler--Cauchy dynamic equation, oscillation of factored dynamic equations, first order dynamic equations, the Henstock--Kurzweil delta integral, stability theory, and existence and nonexistence of periodic points, and the asymptotic behavior and stability of the exponential function on certain periodic time scales. We will work on similar projects this summer. Some of our time will be devoted to studying difference equations -- no prior knowledge of difference equations is required. As part of our REU experience this year we will spend eleven days (travel, food, and lodging provided) at the University of Wyoming Rocky Mountain Mathematics Consortium Summer Conference, attending introductory lectures that will be useful for your research project.

## Stabilization of Underactuated Mechanical Systems

## Project Mentors

Professor Mikil Foss (Department of Mathematics) and Student Mentor, Joseph Geisbauer (Department of Mathematics)

## Prerequisites

The equivalent of a three-semester sequence in college Calculus and a one-semester course in Differential Equations. Familiarity with a symbolic programming language such as MAPLE would be useful but is not a requirement.

## Project Description

First, a bit of jargon, which will be illustrated by an example: A mechanical system is called underactuated if it has fewer actuators than degrees of freedom. A degree of freedom is called actuated if there is some device that can directly influence the system with respect to that degree of freedom. An example of an underactuated system is a person balancing a broom stick in the horizontal palm of their hand. This system has five degrees of freedom: three for the position of the hand in space and two that describe the angle of the broom relative to the horizontal palm. The person has direct control of the hand's position coordinates in space, so these coordinates are the actuated degrees of freedom. The angles for the broom stick, however, can only be indirectly influenced by the motion of the hand, and these angles are the unactuated degrees of freedom. Underactuated systems are quite prevalant; some other examples include spacecraft, aerial rockets, underwater vehicles, vertical takeoff aircraft,satellites, hovercraft, and ship-to-shore cargo transport cranes. A basic problem for these systems is to design a controller that can take the system from an initial state and steer the system to a state of equilibrium. Such a controller is called a stabilizing control law. In general, the system of differential equations governing the dynamics of these mechanical systems are nonlinear, and methods from nonlinear control theory are required for the controller design. The objective of this project is to introduce nonlinear systems and control theory and to develop and implement a general strategy for producing stabilizing control laws for underactuated mechanical systems. Some of the components of the project will involve deriving mathematical models and using numerical methods to design stabilizing control laws and simulate system responses.