## Mathematics Department

Undergraduate Research Symposium

Fall 2014

## Monday December 15 - 10:30am-12:30pm - Room LCB 323

10:30-10:50am,

**Alex Henabray**,

*Numerical Solutions for the Heat Equation*

*Mentors:*Yekaterina Epshteyn and Jason Albright

*Abstract:*Differential equations play an important role in science and engineering because they are employed to model many natural phenomena, including diffusion. Diffusion is defined as the movement of a substance (flow of heat energy, chemicals, etc.) from regions of high concentration to regions of low concentration. In this talk, I will consider the heat equation, which is one example of a model for diffusion processes. For instance, I will use the heat equation to model heat transfer along a metal rod and I will present two methods to approximate solutions of the heat equation numerically.

10:50-11:10am,

**Nathan Simonsen**,

*Simulation of Texas hold'em*

*Mentor:*Stewart Ethier

*Abstract:*TBA

11:10-11:30am,

**Michael Primrose**,

*Imaging with waves in random media*

*Mentor:*Fernando Guevara Vasquez

*Abstract:*We explored different imaging functionals to determine the placement of wave sources in a random medium. We were interested in generating images that had good resolution and statistical stability. The first method we explored was Kirchhoff migration. This method varied greatly from one realization of the medium to the next. Time reversal was implemented next with better results, however, the method assumes a perfect knowledge of the medium which is unrealistic. Then we used interferometric imaging which exploits the statistical stability of the cross-correlation to generate images. The images lacked range resolution. To get a better range resolution, we worked with coherent interferometry (CINT). CINT introduces additional constraints on the cross-correlations resulting in better range resolution. The statistical stability of CINT was then studied as we varied the constraints.

(Supported by NSF DMS-1411577)

11:30-11:50am,

**Wenyi Wang**,

*Imaging in a homogeneous aluminum plate by using ultrasonic waves*

*Mentor:*Fernando Guevara Vasquez

*Abstract:*This project is about detecting and imaging damage (such as cracks) in a plate by using ultrasonic waves. The waves are generated by a source (an ultrasonic transducer) that is part of a robot that can move on the plate. The waves traveling in the plate are recorded at a receiver (another ultrasonic transducer) that is also carried by the robot. The imaging method we use is Kirchhoff migration and we do a rigorous resolution study of this imaging method for two different configurations of the source/receiver pair, assuming the robot follows a straight path and that certain length scalings hold. Our results reveal that one of the setups gives images with much better resolution than the other one. Imaging on other paths is illustrated with numerical experiments. The application of this research is to aircraft structural health monitoring and is done in collaboration with Thomas Henderson (School of Computing, University of Utah).

11:50-12:30pm,

**Jin Zhao and Yu Tao**,

*Construction of Optimal Portfolio Strategies*

*Mentor:*Jingyi Zhu

*Abstract:*One of common practices in financial industry to calculate correlation between different stock returns is to use daily closed prices. In this project, we propose new methods to compute correlation matrix that uses real stock data based on transaction level. We use both Brownian bridge and linear interpolation to interpolate prices of all stocks at any given specific time.

## Tuesday December 16 - 1pm-3pm - Room LCB 225

01:00-01:20pm,

**Trevor Dick**,

*On the ideal dynamic climbing rope*

*Mentor:*Davit Harutyunyan

*Abstract:*We consider the rope climber fall problem, the simplest formulation of which is when the climber falls from a point being attached to one end of the rope and the other end of the rope is attached to the rock. The problem of our consideration is to minimize the maximal value of the force that the climber feels during the fall, which is the tension of the rope at the point attached to the climber. Given the initial height of the rock attached point, the altitude and the mass of the climber and the length of the rope, we find the so called best dynamic rope in the framework of nonlinear elasticity.

01:20-01:40pm,

**Michael Senter**,

*Numerical investigation of First Passage Time through a Fluid Layer*

*Mentor:*Christel Hohenegger

*Abstract:*Modeling the motion of passive particles in a viscous fluid is well studied and understood. Extensions to passive motion in a complex fluid which exhibits both viscous and elastic properties have been developed in recent years. However, questions remain on the characterization of mean-square displacement and mean first passage for different theoretical models. We will present a statistically exact covariance based algorithm implemented in parallel C++ to generate particle paths to answer these questions.

(Supported by NSF DMS-1413378)

01:40-02:00pm,

**Michael Zhao and Nathan Briggs**,

*An inverse problem: finding boundary fields which produce breakdown*

*Mentor:*Graeme Milton

*Abstract:*We study the inverse problem of finding the lower bound on the maximum value the electric field/stress of a two-phase body of unknown interior geometry, fields which exceed these bounds will cause breakdown. We apply the splitting method to obtain bounds without using variational principles. Knowing the materials' properties (density, threshold for breakdown) and weight of the object, we are able to determine necessary criteria for breakdown. We also find optimal $E_\Omega$ inclusions for which these bounds are sharp. The $E_\Omega$ inclusions are computed by finding geometries such that the field is uniform in the inclusion. The geometry of the inclusions can then be determined by enforcing boundary conditions and physical constraints.

(Math 4800: An inverse problem: finding boundary fields which produce breakdown)

02:00-02:20pm,

**Curtis Houston**,

*Algebraic varieties associated to simple statistical models*

*Mentor:*Sofia Tirabassi

*Abstract:*TBA

02:20-02:40pm,

**Braden Schaer and Anand Singh**,

*A Model for Restricted Diffusion of Evoked Dopamine*

*Mentors:*Sean Lawley and Heather Brooks

*Abstract:*A simple gap diffusion model is commonly used for short, unaided intercellular transport. One prominent example is dopamine diffusion in neural synapses, where intercellular space is often a crowded environment consisting of extracellular networks, biological waste, macromolecules, and other obstructions. The traditional model fails to represent many unique characteristics of restricted diffusion in complex cellular environments. We compare existing mathematical models of gap and restricted diffusion against data acquired by fast-scan voltammetry of evoked dopamine in the striatum of rat brains. Finally, we propose a more physically realistic model that accounts for the spatial structure of this system. (Presented by Braden Schaer.)

(Supported by Mathematical Biology RTG grant)

## Wednesday December 17 - 10:50am-12:30pm - Room LCB 323

10:50-11:10am,

**Marie Tuft and Oliver Richardson**,

*Mathematical model insights into neurobiology*

*Mentors:*Sean Lawley and Heather Brooks

*Abstract:*TBA

(Supported by Mathematical Biology RTG grant)

11:10-11:30am,

**Alexander Beams**,

*Epidemiology of dual-strain bacterial infections*

*Mentors:*Fred Adler and Damon Toth

*Abstract:*Antibiotic resistance in bacteria has big consequences for medicine and public health. A compartmental deterministic ODE model (a not-too-distant relative of the familiar SIS) shows how two different bacterial strains interact, and what makes this model stand out is the assumption that people can be infected with both strains at once. In some scenarios this class of infectives can help antibiotic resistance persist when it otherwise would not. Other situations lead to lower overall levels of resistance than would otherwise be expected. The idea of people being infected with two strains at once has empirical evidence supporting it, but in the model it creates an inconsistency when analyzed in the context of evolutionary epidemiology. When strains are highly differentiated the inconsistency may not be a critical issue, but future work will nevertheless deal with finding a satisfactory fix for the model's apparent shortcomings. An area of future study includes describing the dynamics when a hospital is embedded in a community.

11:30-11:50am,

**Hitesh Tolani**,

*Modeling the dynamics of influenza transmission in school-age population of Utah*

*Mentors:*Fred Adler and Damon Toth

*Abstract:*Dynamics of infectious disease transmission vary radically between smaller vs. larger sized populations. Phenomenological models that well predict the average outbreak sizes in cities generally do a poor job of simulating outbreaks in schools, hospitals etc. Infectious diseases transmit primarily through direct contact between individuals. Phenomenological models are usually based on simplifying assumptions which overlook the population native contact-network dynamics. The Center for Disease Control and Prevention, (CDC), quantified a contact-network in their project entitled "Contacts among Utah's School-age Population", (CUSP). Mechanistic simulations on CUSP network reveal that network dynamics dominate outbreaks in smaller populations. In this project we're using the CUSP network to identify essential features of a contact-network. These are necessary in order to construct simpler random-graph based epidemic models which accurately simulate influenza outbreaks in school-age population of Utah.

11:50-12:10pm,

**Yang Lou**,

*The PAM method for interface tracking*

*Mentor:*Qinghai Zhang

*Abstract:*Interface tracking is an essential problem in numerically simulating multiphase flows. At present, three main methods have been developed, namely front tracking methods, level set methods and VOF methods. Despite the success of these interface tracking methods, these methods have some shortcomings. A relatively new interface tracking method which evolved from VOF methods is called the polygonal area mapping (PAM) method, which represents material areas explicitly as piecewise polygons, traces characteristic points on polygon boundaries along pathlines, and calculated new material regions inside interface cells via polygon-clipping algorithms in a discrete manner. Translation test and vortex test are performed to test the accuracy and efficiency of the PAM method on a structured rectangular grid with a uniform grid size $h$.

12:10-12:30pm,

**Kejia Zhu and Logan Calder**,

*Damage in Lattice Model of Materials*

*Mentor:*Andrej Cherkaev

*Abstract:*Lattice structures are often used to create discrete models of materials. For engineering purposes, a lot of mathematics has been developed to model the behavior of different types of materials as they experience different forms of stress: heat, pressure, impacts, etc. Experiments have helped establish standards for when a material may be critically damaged. What may be lacking though, is a model that describes the changes in a material as it receives damage and a measure for how close a partially damaged material is to critical damage. This paper will report our attempt to construct a model for the distribution of forces on a stressed periodic lattice and for the progression of damage as the lattice is over stressed. We do not attempt to describe the progress of damage with in each edge, only the progress of damage within the lattice. That is, we lay the ground work to investigate pattern of broken edges while the lattice is over stressed. We will expound on our question and only discuss some of the ground work necessary to begin simulations.