Shenandoah Undergraduate Mathematics and Statistics Conference
at James Madison University, October 18, 2008
Abstracts of Contributed Talks and Posters
| Abstracts of contributed
talks and posters: Michael Atkins, George Mason University A numerical and analytical study of modeling techniques for solidification The nonlinear and metastable nature of the mesoscale phenomenon of solidification has given rise to various numerical models that attempt to describe it. We provide a comparison of several computational models that describe microstructure evolution based on their ability to predict statistical properties of different materials. Margaret Beckom and Matthew Spencer, James Madison University Nearest neighbor distance measures for mixed variables in an ecological setting We discuss a new way to more effectively use distance measures in an ecological setting where there are both continuous and categorical variables used. We present three approaches that use local information to adjust for natural conditions and methods for statistically evaluating conditions at a test site using biological metrics. Brian Beruete and Teri Swinson, James Madison University Eigenvalue decomposition of a symmetric Matrix in parallel Using the Jacobi algorithm, we implement an algorithm to compute an eigenvalue decomposition of a symmetric matrix. Although the Jacobi algorithm is generally slow, it is very powerful when implemented in parallel. We will not actually implement it in parallel, but we can still develop it in a standard way. Rujira Chaysiri, University of Virginia Enhancement of stability of LQR problem in a strongly damped wave equation A model of strong damped wave equation with both punctual and distributed control are studied in order to enhance the stability of a vibrating structure. Using LQR and associated Riccati equation, we compare enhancement of stability due to different models of controls and location of actuators. Xuanyi Chen, Southwest Virginia Governor's School Strategy in a random walking game In studies with penalties, the best strategy in a random walking game on a one-dimensional basis was the "closer, shorter side" strategy when there were few points. When there were more points on the line in a game with switching penalty, the "closer, same token" strategy was best. Anthony Clifford, Governor's School of Southside Virginia Evaluation of simple functions to produce the best strategic advantages in Go This study evaluates which simple functions produce the best opportunity to achieve a strategic advantage in the ancient board game Go. Some criteria include position, defensibility, and simplicity. Two constant functions, an absolute value function, and a radical function were judged to have superior strategic advantages. Julianne Coxe, James Madison University A nano-sculptor's knife: Cutting into infinite space Pair distribution functions are often used in the analysis of samples with neutron scattering. For most samples, infinite length and width are assumed. When samples include nano-particles the corresponding correction function is needed. A computer program was developed to implement this equation. Michael Dankwa and Juan Ortega, James Madison University Jan Herburt-Hewell and Lianne Loizou, James Madison University Sandwheel Parts I and II NREUP participants at JMU analyzed differences between a chaotic waterwheel and sandwheel. We ran experiments in order to see the differences between the rate of granular material and liquids. Linear stability theory was used to explore stability and numerical experiments indicated that the center of mass could be used to classify the system's behavior, including constant, rolling, periodic or chaotic states. Matthew DiGiosaffatte, James Madison University A function from N to N that grows so fast it's silly We define a recursive function fancy D:N->N, the Ackermann function, and hyperoperations, and show that fancy D grows faster than Ack(n,n) which grows faster than n hyper-n n. Thomas Dowd, James Madison University Circulant matrices Circulant Matrices are matrices where each row vector is rotated to the right of the preceding row vector. Circulant matrices are useful in numerical analysis because they are diagonalized by the discrete Fourier transform. I will be exploring circulant matrices using MATLAB and applications to problems in physics. Douglas Fordham Programming the modified Picard method The Parker-Sochacki method has successfully shown that by using integration techniques a Taylor series can be produced for any given ordinary differenial equation. This presentation will be focused on programming the Parker-Sochacki method using Matlab, and the discovery of patterns using the algorithm to solve any order ODE. Douglas Fordham and Glenn Young, James Madison University Color image compression using singular value decomposition We are using singular value decomposition to compress color images. Color images are stored as three dimensional matrices, so standard SVD techniques are difficult to apply. Our algorithm uses and compares techniques for applying SVD to color image compression. Lydia Garcia, St. Mary's College of Maryland The search for a Jones knot This talk discusses algorithms for quickly calculating the Jones Polynomial for 3-braids. It shows how to use this technique, along with a general mutation, to attempt to discover a nontrivial knot with Jones Polynomial of 1. Deanna Hannoun, James Madison University An optimal energy allocation strategy for multiple constrained resources The goal of this project is to use difference equations to discover how a subpopulation can divide its energy most effectively between searching for food and searching for shelter such that it can have more reproductive success than its competitor and can consequently dominate its niche. Deanna Hannoun and Abigayle Wood, James Madison University e^(xtra) special matrix operations It's easy to compute e^a for any value of a, but for this project we will be exploring what happens when the exponent is a matrix. Kevin Kelbaugh and S. Minerva Venuti, George Mason University Modeling, analysis and computation of fluid structure interaction models for biological systems This undergraduate research presents mathematical models for the interaction of blood flow through arterial walls which are surrounded by cerebral spinal fluid. The arterial wall and the cerebral spinal fluid will be coupled using appropriate partial differential equations. Applications of the model studied to intracranial saccular aneurysms will be presented. Alex Kuberry, Indiana University of Pennsylvania A correlation study of mathematics students and multiple extra curricular studies No abstract submitted. Kristin McNamara, James Madison University Intrinsically linked and intrinsically triple-linked graphs in real projective space Many things are known about many graphs in 3-space. Fewer things are known about graphs in projective space. And even fewer things are known about the Petersen family graphs, intrinsically linked graphs, and intrinsically triple-linked graphs in projective space. I will tell you about them. Sia Minja, Indiana University of Pennsylvania Application of Calculus In Economics The basic application of modern mathematics in the world of economics.The fundaments and the uses of calculus are under discussion. A look into how maths is important in Economics. Scott Landenheim, Syracuse University Emily Miller, The College of New Jersey Higher order tensor operations and their applications A tensor, or n-dimensional array, can be manipulated in a similar manner to matrices. We define the operations for tensors in such a way that the set of nxnxn invertible tensors form a group, develop the definition of the tensor Singular Value Decomposition (SVD), and utilize it in two applications. Philip Lenzini, Dominican University Nonconstant harvesting on ratio-dependent predator-prey models Population dynamics have received great attention due to the dynamics of the models. The dynamics of a predator-prey model with non-constant harvesting is studied. Equilibria and periodic orbits are computed and their stability properties are analyzed. Bifurcations are detected as well as connecting orbits. Related topics are also addressed. Jose Manuel Lopez, University of Houston Numerical methods for stochastic population models Stochastic population play a central role in theoretical biology. Such models are frequently intractable, necessitating the use of numerical simulations. Conventional algorithms are not applicable in cases of interacting subpopulations of different sizes. A new, hybrid algorithm which reduces simulation time significantly with modest sacrifices in accuracy will be presented. Daniel Mertz, Indiana University of Pennsylvania Calculus applications: Aerospace engineering No abstract submitted. Audrey Poe, Shenandoah Valley Governor's School Row and column summations of the Narayana triangle Combinatorics often relates to the Catalan numbers. The Narayana triangle, which displays the amount of expressions containing sets, correctly matched with distinct nestings, is derived from these numbers. A comparison between the row summations and column summations can be shown, in relation to each O and the Catalan numbers. John Ross, St. Mary's College of Maryland Untangle: Game theory on the unknot In this talk, we introduce a combinatorial game called Untangle. The game is played by performing Reidemeister moves on a projection of the unknot. Using game theory and basic knot theory, game positions are identified and analyzed to determine winning strategies. Brian Salyer, Morehead State University Equivalence numbers of graphs The equivalence number eq(G) of a graph G is the minimum number of equivalence relations needed to cover the edges of that graph. We determine eq(G) for certain graphs based on chess piece moves and describe our attempts to find an upper bound for eq(G). Megan Skebeck, Indiana University of Pennsylvania Calculus applications: Suspension bridges and cantenary arches This paper considers the integration of engineering and calculus as applied to real-world situations. It shows how applications such as polar coordinates and vectors are used in engineering feats. It also examines how calculus is applied to catenary structures, especially the Gateway Arch. Tom Stephens, George Mason University Nonlocal extensions of the classical phase field model The classical phase field model represents a coupling of an Allen-Cahn type nonlinear equation with a standard diffusion equation. In this talk, I will discuss an extension of that model which takes into account nonlocal effects by introducing a convolution term involving the phase feld variable. Jamey Szalay, James Madison University Deblurring an image using linear algebra Approximate an original image given a blurred, noisy image. Using linear algebra, it is possible to reconstruct an image with a reasonable degree of accuracy given the conditions under which it was taken. Lok-kun Tsui Latent semantic indexing LSI is a method used to search a database of documents and is often applied to the help function in computer programs. We investigated LSI by using Singular Value Decomposition algorithms in Matlab on a sample database of Matlab's help files. Olivia Walch, College of William and Mary The commutant of the full tridiagonal pattern A number of barriers to commutativity with the tridiagonal pattern are discussed, including combinatorial asymmetry and multi-order local barriers. Additionally, the concept of commutative ratio equations is introduced and extended to apply to any polynomial of a matrix whose graph is a tree. Daniel Wilberger, James Madison University The world's largest linear algebra problem Google's PageRank algorithm assigns an importance ranking to each page on the web, allowing Google to rank the pages and present the user more important pages first. This project will involve computing the PageRank which involves computing eigenvalues. Vincent Zimmern, University of Virginia Boundary value problems on polygonal domains Some recent results regarding the employment of the method of successive approximations to the treatment of boundary value problems associated with higher-order elliptic operators in domains with polygonal boundaries. These problems naturally arise in mathematical physics and engineering in the context of modeling shallow shells, beam bending, and clamped plates. |