The 5th Annual

NC State University

Undergraduate Summer Research Symposium

 

REU Mathematics:

Modeling and Industrial Applied Mathematics


Abstracts are listed in alphabetical order by the last name of the corresponding author.

 

 

 

 

 

 

Student Author(s): 

Appel, Ian R.

Ho, Nancy

Pickelsimer, BreAnne

Tran, Thu A.

Home Institution:

Duke University

Mills College

Elizabethtown College

Bryn Mawr College

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Alun Lloyd/Mathematics

Title of Presentation:

Evaluating Strategies for Limiting the Extent of an Influenza Pandemic

 

 

Fears of a global influenza pandemic have been heightened by the occurrence of over 200 confirmed human cases of H5N1 avian influenza. In the absence of a suitable vaccine, antiviral drugs, such as Tamiflu, can be used in an attempt to mitigate the  impact of a pandemic. In order to analyze mitigation strategies, we extended a mathematical model proposed by Stilianakis et al (1997).  Their model concluded that chemoprophylaxis of susceptible persons, as opposed to treatment of those who are symptomatic, is the optimal strategy for minimizing the incidence of influenza in a closed population. However, the authors did not account for the supply of antiviral drugs being finite and for the death of symptomatic persons. Our model, which includes both of these factors, predicts that chemoprophylaxis is not  necessarily the optimal strategy, especially when minimizing the number of deaths for a given amount of antiviral drugs.

 

 

 

 

Student Author(s): 

Attarian, Adam

Tarko, Laura

Zuckero, Jeremy

Home Institution:

North Carolina State University

Mount Holyoke College

Wilkes University

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Hien T. Tran/Mathematics

Title of Presentation:

Optimization of Traveling Wave Tubes Using Large Signal Codes and Optical Beam Analysis

 

 

Traveling wave tubes are common devices with widespread use in both satellite and terrestial comunications as well as areas such as electronic counter measures.  For instance in satellite communications due to constraints on power usage and weight, it is extremely desirable to optimize these devices so that they perform as well as possible.  For example, an improvement of 1% in efficiency will result in a savings of $15 million over the satellite lifetime.  We utilize the CHRISTINE suite of large signal codes to model the slow wave portion of the TWT and Beam Optics Analysis (BOA) to model a multi staged depressed collector, where the goal is to optimize the collector given the spent beam data from an optimal circuit. The optimization techniques, including cost functions and methodology, as well as the optimized results will be discussed in detail.

 

 

 

 

 

Student Author(s): 

Backes, Tracy M.

Latterman, Russell

Small, Stephen A.

Home Institution:

Harvey Mudd College

Arizona State University

Norfolk State University

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Sharon R. Lubkin/Mathematics

Title of Presentation:

Cell Based Model of Convergent

Extension in the Ascidian Notochord

 

We present a three-dimensional cellular Potts model (a discrete cell based Monte-Carlo model) of convergent extension of the ascidian notochord.  Our work derives from recent research that explores the coupling of invagination and convergent extension in ascidian notochord formation (Odell and Munro, 2002).  Modeling cells individually allows us to assign different physical properties and behaviors to each cell type and to study how these properties affect notochord elongation. We have tested the roles of cell-cell adhesion, cell-extracellular matrix adhesion, random motion, and extension of individual cells, as well as the presence or absence of various tissue types, and determined which factors are necessary and/or sufficient for convergent extension.

 

 

 

 

Student Author(s): 

Cahn, Patricia R.

Li, Juan

Schwartz, Jeremy R.

Home Institution:

Smith College

Smith College

Brandeis University

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Ruth Haas/Mathematics

Aloysius Helminck/Mathematics

Title of Presentation:

Efficiency and Notation in Weyl Groups

 

 

Packages for Weyl group computations typically do not represent group elements uniquely due to their use of  generators and relations.  A unique representation for Weyl group elements in An, Bn, and Dn as signed permutation vectors has already been developed.  We present an extension of this notation for Weyl groups of types E, F, and G.  We also present algorithms for computing properties of group elements such as length.  A future goal is to implement this notation and its associated algorithms in Mathematica.

 

 

 

 

Student Author(s): 

Elliott, Kadie J.

Lo, Ka-Ho

Hendrick, Ken

Home Institution:

Blackburn College

Stony Brook University

California State Polytechnic University, Pomona

Program:

REU Mathematics

Department(s):

Mathematics

Physics

Mathematics

Research Mentor(s): 

Mansoor Haider/Mathematics

Brandy Benedict/Mathematics

Title of Presentation:

A Mechano-chemical Model for Chondron Deformation in Articular Cartilage Subjected to Osmotic Loading

 

 

Articular cartilage, the load-bearing surface of bones at joints, is composed primarily of cartilage cells dispersed within a cross-linked network of collagen fibers and negatively-charged proteins, called proteoglycans. A thin region of fibers 

surrounding each cell has a higher concentration of proteoglycans than the rest of the fibrous network. Together cells and their surrounding regions are called chondrons, forming distinct structural units that make up less than 10% of articular cartilage tissue, by volume.

               With age or disease, such as osteoarthritis, cartilage tissue becomes less stiff, yielding greater deformations of chondrons under applied loads.  In its early stages, osteoarthritis is associated with a loss of proteoglycans, which contribute to the apparent stiffness of the tissue due to repulsion of their fixed charges.  By formulating a triphasic  model of chondron deformation that incorporates the solid, fluid and ion composition of articular cartilage, we can separately quantify the effects of collagen stiffness and fixed charge density of the proteoglycans on the overall loss of apparent tissue stiffness with osteoarthritis.

               We couple previous macroscopic solutions, which model confined swelling of human cartilage tissue, with microscopic models that describe the chemical osmotic swelling and shrinking effects on chondrons isolated in sodium chloride solution.  This model reduces to a set of effective elastic governing equations which are solved numerically.

 

 

 

 

 

Student Author(s): 

Chen, Shi Y.

Koehler,  Erin L.

Li, Aolin

Mincheva, Martina Z.

Home Institution:

Amherst College

Clemson University

UC Berkeley

Franklin and Marshall

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Tao Pang/Mathematics

Title of Presentation:

Applications of the Monte Carlo Simulation in Financial Mathematics

 

In the area of financial mathematics, derivative  pricing has been a big challenge for a long time.  Among all methods, Monte Carlo method is widely used  by researchers and practitioners. In this project, we  consider some applications of Monte Carlo method in  derivative pricing issues that arise from the equity  market and the fixed-income market. First we use Monte  Carlo simulation to price European and Asian options  in the equity market. Then we consider the  applications of Monte Carlo simulation in the  fixed-income market and the numerical results of some  derivatives of bond, such as bond options, interest  rate caps and floors, are given. We also focus on  various variance-reducing techniques to make our  algorithms more efficient. In addition, we also test  different models for the interest rates by calibrating  the models to the real market data.

 

 

 

 

 

Student Author(s): 

Koser, Kelly R.

Moore, Julia L.

Zapata, Cheryl L.

Home Institution:

Carnegie Mellon University

Rensselaer Polytechnic Institute

North Carolina State University

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Mette S. Olufsen/Mathematics

Title of Presentation:

Blood Pressure and Blood Flow Variation during Postural Change from Sitting to Standing: Model Reduction and Improvement

 

 

 

Short term cardiovascular responses to postural change from sitting to standing involve complex interactions between the autonomic nervous system, which regulates blood pressure, and cerebral autoregulation, which maintains cerebral perfusion.  We have developed a model which can predict dynamic changes in beat-to-beat arterial blood pressure and middle cerebral artery blood flow velocity during postural change from sitting to standing.  This model uses an electrical circuit analogy, predicting changes in blood pressure (voltage) and blood flow (current) as functions of resistance and compliance (capacitance). The base model has more than 100 parameters that must be identified to predict regulatory response for individual subjects.  In preliminary work, an inverse least-squares problem was formulated to estimate all 100 parameters to minimize the difference between observed data and computed values. This optimization process is time-consuming and not feasible if the model is to be validated against multiple datasets. We have used sensitivity analysis to identify a small number (approximately 20) of sensitive parameters. Furthermore, we have shown, using sensitivity analysis, that it is possible to reduce the structure of  that model and that with additional data it is possible to identify more parameters. Finally, we have developed a physiological model for autoregulation that predicts cerebrovascular resistance as a function of blood flow and blood pressure. It is not well understood how autoregulation is mediated; it is known that several components play a role including myogenic, metabolic, and neural contributions. We plan to use our mathematical model to understand dynamics of each of these components.

 

 

 

 

Student Author(s): 

Roubos, Spyro

Oliver, Erin M.

Home Institution:

University of Rochester

Smith College

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Ruth Haas/Mathematics

Aloysius G. Helminck/Mathematics

Title of Presentation:

Conjugacy Classes of  Involution Posets and Algorithm Development

 

 

Given a group G, two elements x, y in G are conjugate if there exists another element g in G such that y =gxg^{-1}.  An involution, w in G is an element such that: w =w^{-1}.  We consider the conjugacy classes in the set of involutions of the group of permutations.  We will give some combinatorial and algebraic results about these classes.  Additionally, we discuss algorithmic and computational issues.

 

 

 

 

 

Student Author(s): 

Williams, Nathan F.

Cooley, Chelsey A.

Home Institution:

Carleton College

North Carolina State University

Program:

REU Mathematics

Department(s):

Mathematics

Research Mentor(s): 

Ruth Haas/Mathematics

Aloysius Helminck/Mathematics

Title of Presentation:

Combinatorial Formulas for an Involution Poset

 

 

Weyl groups generalize permutation groups and are useful in many branches of mathematics.  For a Weyl Group $W$, the set of involutions is $I = \{w \in W | w = w^{-1}\}$.  We studied a poset of involutions of the Weyl groups $A_{n}$ and $B_{n}$, generated in the manner described by Haas and Helminck.  We first give a formula for the rank of any element and use it to define a generating polynomial for the number of elements of each rank in both posets.  We then propose a formula for the number of paths to the top-most element using Young Tableaux, based on the work of Stanley, Edelman, and Greene.  In the last part of our talk, we discuss several interesting related results about the posets.

 

 

 

 

 

 

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