The 6th 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): 

Abatzoglou, Alex W.

Smith, Amanda J.

Thompson, Katie I.

WebsterLove, Jessica E.

Home Institution:

Centre College

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Irina A. Kogan/Mathematics

Title of Presentation:

Geometric Invariants in Computer Vision

 

 

This research is focused on the problem of classifying curves up to rotations and translations with application to automated solving of an a pictorial jigsaw puzzle. Two different methods are studied. The first, a more classical method based on differential invariants, produces a signature as a plot of curvature versus its derivative with respect to arc length. The second method involves using integral invariants to construct signatures. We studied integral invariants from both theoretical and computational perspectives. We proved that integral invariants classify curves up to rotation and translation. We also coded numerical approximations and experimented with integral signatures. The advantage of the integral invariants over the differential invariants is that the integration reduces the effect of noise whereas differentiation amplifies it. We have conducted numerous experiments verifying advantages and disadvantages of the differential and integral signatures.

 

 

 

 

Student Author(s): 

Abbey, Ralph W.

Diependbrock, Jeremy

Zhou, Dexin

Home Institution:

North Carolina State University

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Carl Meyer/Mathematics

Title of Presentation:

Data Clustering: A Comparison of Algorithms

 

 

As the ease of collecting large amounts of data increases, the ability to obtain useful information out of large datasets becomes significantly more important. The research investigated current clustering algorithms and experimented with modifications on those algorithms. As an initial basis for experimentation, a sample document collection was formed. Comparisons were made among the clusters formed by various algorithms. These algorithms were then tested on larger benchmark document sets to determine the ability of the algorithms to work in a less contrived environment. In many cases, the adaptations to preexisting algorithms performed better than the basic algorithm. Based on these comparisons between the algorithms, they were applied to unfamiliar datasets in order to draw out hidden information. Some correlation among the data was found and conclusive results could be seen.

 

 

 

 

 

Student Author(s): 

Anderson, Marlana N.

Brasfield, Chris A.

Maschmeyer, Katherine T.

McGoff, Kevin A.

Siloti, Julie A.

Home Institution:

Washington University in St. Louis

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Pierre Gremaud/Mathematics

Title of Presentation:

Edge Detection by Multi-Dimensional Wavelets

 

 

Wavelets are functions that can be used to decompose signals into various frequency components at an appropriate resolution for a range of spatial scales. Edges can be defined as sharp changes of the intensity in a signal. Applications of edge detection technology can be found in many fields, including medical imaging. The objective of this project was to explore of the latest generation of wavelets in order to create improved edge detectors. Toward this end, several known signal-processing methods were studied and applied. These included methods based on well-known Fourier transform and wavelet transforms in both one and two dimensions. Theoretical results concerning edge detection in one dimension were reviewed and the corresponding algorithms were implemented. Furthermore, tests were run on images by applying one-dimensional decompositions in both the horizontal and vertical directions independently. These results were compared to edge detection schemes based on gradient methods, which capture sharp changes in intensity. It is well known that one-dimensional wavelet techniques are suboptimal in the representation of images. Recently a new generation of intrinsically two-dimensional wavelets, e.g. shearlets, has been introduced to alleviate these deficiencies. In this project, new edge detection methods were developed based on the shearlet transform. As a refinement of these methods, subdomain decomposition was introduced to preserve less dominant edges. Furthermore, several basic post-processing schemes were used to provide more distinct edges. All of the above methods were applied to both artificially generated and natural images. In order to measure the accuracy of the various methods, the Hausdorff distance between the actual and approximate edges of artificial images was computed. Through this analysis, it was concluded that edge detection methods based on shearlets were at least as accurate as popular methods, such as Canny and Sobel, when applied to artificial images.

 

 

 

 

Student Author(s): 

Baltera, Constance G.

Brenneman, Kathryn A.

Hatfield, Ashley E.

Tramel, Rebecca L.

Home Institution:

Smith College

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Ruth Haas/Mathematics

Aloysius G. Helminck/Mathematics

Title of Presentation:

Combinatorics and Characterizations for Involutions and Twisted Involutions

 

 Weyl groups generalize permutation groups and have many interesting properties. For a Weyl group W, the set of involutions is I = {w ∈ W | w = w-1}, and the set of twisted involutions is Iθ = {w ∈ W | θ (w) = w-1} for a group automorphism θ. Although the partially ordered set, or poset, of involutions has undergone some previous study, the twisted case is not as well understood. In this work we examine posets of involutions and twisted involutions in several families of Weyl groups. We present rank formulae for the posets of twisted involutions in An D2n+1 and D2n. We also offer more results about I and Iθ in D2n, and formulae for the number of elements in each of these sets.

 

 

 

 

 

Student Author(s): 

Behrend, Sam J.

Berman, Ben P.

Smith, Jason R.

Wright, Justin P.

Home Institution:

Denison University

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Alun Lloyd/Mathematics

Alex Capaldi/Mathematics

Title of Presentation:

Designing Control Strategies for Infectious Diseases in Farm Crops

 

 

Infectious diseases among plants cause billions of dollars of damage each year in the United States, much of which can be attributed to loss of farm crops. It is important to control these diseases in the most efficient manner possible to maintain maximum crop yields. We developed probabilistic cellular automaton models of epidemics on a two-dimensional landscape, much like a common plot of farmland. Plants were arranged on a grid, and we considered an infection for which disease is only spread to a plant’s immediate neighbors. Individuals can be susceptible, infectious or recovered and there is a simple set of rules governing transitions between states. Provided that the disease is sufficiently infectious, introducing a single diseased individual leads to wave-like spread of infection; allowing a small degree of non-local transmission, similar to a “small world” network, facilitates this spread. One possible disease control method involves removing plants in a predetermined or random pattern. We investigated the impact of various removal strategies, both in terms of minimizing the size of the outbreak and maximizing the crop yield. Over a wide range of disease parameter values, we found that randomly removing 65% of a crop will, in most cases, ensure that a severe epidemic cannot occur. Significantly fewer plants have to be sacrificed if a lattice-like pattern is used. We hope that further development of our modeling framework will lead to the creation of even more effective disease control measures.

 

 

 

 

 

Student Author(s): 

Draper, Bailey K.

Margolskee, Alison

Murden, Raphiel

Marcin, Daniel

Attarian, Adam

Home Institution:

Colorado State University

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Marina Evans/Biomedical Engineering, EPA

Karen Yokley/Applied Mathematics, UNC

Title of Presentation:

Feasibility of Metabolic Parameter Estimation in Pharmocokinetic Models of Carbon Tetrachloride Exposure in Rats

 

Carbon tetrachloride (CCl4) is a dangerous chemical that was once used in degreasers and detergents, but since 1970 has been banned from consumer products. Scientists are still concerned that some remnants of the chemical may get into the water supply. CCl4 itself is nontoxic, but is metabolized by an enzyme in the liver into harmful chemicals such as chloroform and hexachloroethane. The physiological parameters describing the metabolism of CCl4 are not well known and studies have been conducted with rats to gather information that can then be extrapolated to humans. The goal of this research was to more accurately estimate these unknown values in rats. Four different physiologically based pharmocokinetic (PBPK) models were constructed to describe CCl4_ exposure in rats via inhalation, oral ingestion, and venous injection. Each of these models was compared and verified with experimental data extracted from previous studies. Sensitivity analysis was performed for each model to determine whether the available data could be used to accurately determine the metabolic parameters of interest. These parameter sensitivities for the experimental data sets were so low optimization yielded physiologically unrealistic results. Model sensitivities were analyzed for different exposure doses in order to find experimental conditions that would allow for greater identifiability of the metabolic parameters. Data were simulated from these models at optimal conditions with varying levels of noise from a normal distribution. Optimizations were then performed to confirm that the original values could be obtained. The experiments developed are left as suggestions for scientists that wish to further pursue finding these metabolic parameters.

 

 

 

 

 

Student Author(s): 

Kinnaird, Katherine

Carlin, Daniel

Flagg, Garret

Home Institution:

Wellesley College

Duke University

Virginia Polytechnic Institute and State University

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Mansoor Haider/Mathematics

Eunjung Kim/Applied Mathematics

Title of Presentation:

Monte Carlo Simulation of Diffusion in Hyaluronan-based Scaffolds with Applications to Tissue Engineering of Articular Cartilage

 

 

The development of biocompatible scaffold materials for engineered articular cartilage must balance the need for sufficient diffusion of nutrients with careful integration with the surrounding native tissue. A microstructural model of a hyaluronan scaffold was developed for use in simulating random walk diffusion through such a gel medium. The scaffold was modeled via a cubic lattice with cylinders along the diagonals representing individual biopolymer chains. The length and radius of the cylinders were determined via Monte Carlo simulation of polymer chains in a background medium of fixed porosity via the chemical formula of hyaluronic acid and a principal components analysis. An alternate scaffold representation was also considered in which the cylinders were constrained to lie parallel to the three principal directions of the cubic lattice. Using these models, random walk simulations were performed to quantify the effects of polymer chain length, polymer chain diameter, polymer arrangement and solute size on the diffusion coefficient of the scaffold.

 

 

 

 

 

Student Author(s): 

Pauley, Gwyn C.

Mattis, Steve A.

Reilly, Emily M.

Home Institution:

University of South Carolina

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Sharon R. Lubkin/Mathematics

Title of Presentation:

Cell Rearrangement: Mathematical Modeling of Branching Morphogenesis and other Biological Phenomena

 

Mathematical modeling of cellular rearrangement is of interest to developmental biologists, particularly in the area of branching morphogenesis. The goal of this project is to help identify morphogenetic mechanisms, which may be capable of producing certain observed shape changes in developing tissues. In both two and three dimensions, we use a grid system to model cells as well as different cell types. Cells begin as rectangles of given dimensions surrounded by medium, a neutral non-cellular substance. The discrete stochastic model incorporates interface energies as well as random movements. In order for cells to maintain biologically realistic shapes, movements occur with greater probability if the cells will remain convex, reach a given aspect ratio, and stay with cells of their own type. The algorithm developed aids in the modeling of cell growth, mitosis, and convergent extension. This model is important because cell rearrangements are imperative in determining the shapes of tissues, which helps to explain how complex organisms grow from a single cell.

 

 

 

 

Student Author(s): 

Rodriguez, Roberto

King, Shawn

Rutter, Erica

Vansiclen, Emily

Home Institution:

North Carolina State University

Program:

REU Mathematics:  Modeling and Industrial Applied Mathematics

College:

PAMS

Department(s):

Mathematics

Research Mentor(s): 

Mette Olfusen/Mathematics

Hien Tran/Mathematics

Title of Presentation:

Sensitivity Analysis of Mathematical Models of Blood Flow and Pressure Regulation

 

 

Cardiovascular circulation in the human body represents a complex system concerning factors such as blood flow, pressure, velocity, and resistance.  While this system is involved in the development and progression of human diseases, its many dynamics are still poorly understood.  A seven-compartment model of sympathetic and cerebral blood circulation exists to simulate the cardiovascular system, but the sensitivities of its many parameters are unfortunately unknown.  The goal of our research is to apply the process of sensitivity analysis on two simplified Windkessel models of blood flow, interpret the results of this analysis, and then use this information to optimize all parameters and improve the solutions of both models.  Through these simplified applications, we develop a procedure to analyze the sensitivity and improve the results of the entire seven-compartment model. Our first application investigates the three-parameter, single equation Windkessel model of blood circulation.  After computing the sensitivities of all parameters, we non-dimensionalize the system to one of two parameters and recalculate these values in order to compare and confirm them against our analytic results.  Using this sensitivity information, all parameters are optimized using a Nelder-Mead algorithm.  This same process is then applied to a five parameter, two equation Windkessel model of blood circulation.  Due to the complexity of this system, certain sensitivity information must be ranked through an eigenvalue decomposition using QR factorization.  Based on the procedures developed through the analysis of the Windkessel models, the sensitivity information of the seven-compartment model is also ranked and calculated. Researchers will now be able to use this sensitivity information to improve the accuracy of the seven-compartment model and achieve a better understanding of the dynamics of cardiovascular circulation in the human body.

 

 

 

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Last modified June 2007 by Sharon E. Hunt