^{MATHEMATICS
AND COMPUTING
RESEARCH EXPERIENCES FOR UNDERGRADUATES AT IOWA STATE UNIVERSITY}^{supported by the National Science
Foundation
(DMS-0353880)} |

Elizabeth Blankenship Iowa State University

John Bowers Grinnell College

Denrol Carayol Jackson State University

Brian Davis Western Oregon University

Laura Dev Tufts University

Jacob Harper University of Denver

John Hegeman Stanford University

Lori Kraus The College of New Jersey

Jeff Langford Drake Unversity

Sara Lapan Univeristy of Chicago

Jessica Poole Texas Southern University

Monica Robinson Jackson State University

Karyn Snider Unversity of Richmond

Misha Teplitskiy Rice Unversity

Laura Walters Culver-Stockton College

- Polygonal Designs: Existence and Construction Prof.
Sung-Yell Song, Gargi Bhattacharyya, Joohyung Kim, John Hegeman,
Jeff Langford.
Polygonal
designs form a special class of partially balanced incomplete block
designs.
We resolve the existence problem for polygonal designs with various
parameter
sets and find several construction methods with blocks of small sizes. More
details. Paper will
appear in
*European Journal of Combinatorics.*John Hegeman presented these results at the Young Mathematicians Conference 2004 at Ohio State University. - Frames, Data Communication, and Security
Prof.
Eric Weber, Ghanshyam Bhatt, Lori Kraus, Laura Walters. A
system
using an oversampled Fourier transform for hiding data is given in [J.
R. Miotke and L. Rebollo-Neira, Oversampling of Fourier coefficients
for
hiding messages, Appl. Comput. Harmon. Anal. 16 (2004), no. 3,
203-207.]
When viewed as a cryptographic algorithm, we demonstrate that the
system
is susceptible to a known plaintext attack, therefore providing little
added security when used to hide messages. Paper
appeared in
*Journal of Mathematical Analysis and Applications.*Lori Kraus presented these results at the Young Mathematicians Conference 2004 at Ohio State University. Laura Walters presented this work at Argonne Lab's Undergraduate Research Symposium in November 2004. - Partial Semigroups and Primality Indicators in the Fractal Generation of Binomial Coefficients to a Prime Square Modulus Prof. Jonathan D. H. Smith, Benard Kivunge, Jessica Poole, Misha Teplitskiy. This project builds on the work of the summer 2003 group and examines the congruence classes of binomial coefficients to a prime square modulus as given by a fractal generation process for lattice path counts. The process depends on the isomorphism of partial semigroup structures associated with each iteration. We also consider integrality properties of certain critical coefficients that arise in the generation process. Generalizing the application of these coefficients to arbitrary arguments, instead of just to the prime arguments appearing in their original function, it transpires that integrality of the coefficients is indicative of the primality of the argument. Paper (joint with 2003 group) submitted. Misha Teplitskiy presented these results at the Young Mathematicians Conference 2004 at Ohio State University. Matlab codes IteratedMatrixModp2.m, LambdaPrimalityTest.m, TriangleGeneratorModN.m.
- The Fischer Matrix Completion Problem Prof.
Leslie Hogben , Amy Wangsness, John Bowers, Karyn Snider. A partial
matrix is a matrix in which some entries are specified and others are
not.
A completion of a partial matrix is a matrix obtained by choosing
values
for the unspecified entries. A matrix completion problem for the
class of matrices X asks: Does a partial X-matrix have a completion to
an X-matrix? Applications arise in situations where only
partial
data are known or available. Examples include seismic
reconstruction
problems and image enhancement, data transmission and coding
problem.
A Fischer matrix is a P-matrix that satisfies Fischer's inequality for
all principal submatrices. A pattern of positions in an n x n real
matrix
is said to have Fischer completion if every partial Fischer matrix
which
specifies that pattern can be completed to a Fischer matrix. All
patterns
of entries of size up through 4, along with symmetric patterns up
through
size 5, are classified as to whether or not they have Fischer
completion. Paper (joint with 2003 group)
appeared in
*Linear Algebra and Its Applications.* - Modeling Cancer Mathematically Prof. Howard Levine, Laura Dev, Sara Lapan. The purpose of this project was to create a mathematical model of tumor angiogenesis. This model analyzes the effects of cell density and the concentrations of fibronectin, a protease enzyme, growth factor, and various inhibitors on the movement of endothelial cells along the capillary wall. We used enzyme kinetics, random walks, and systems of differential equations to derive mathematical relationships among the above 5 components. From these equations and the help of MATLAB, we were able to simulate the onset of angiogenesis and show that certain systems are inherently unstable, while others can be controlled by the presence of certain inhibiting factors. Laura Dev and Sara Lapan presented thse results at the Young Mathematicians Conference 2004 at Ohio State University
- Dynamical Systems Prof. Justin R. Peters, Prof. Wolfgang Kliemann , Ajith Gunaratne, Rajeev Rajaram, Michael Anderson, Denrol Carayol. We look at various ways of mapping one linear differential system onto another. In R^d one can create a C^k conjugacy (with k >= 1) that preserves the entire Jordan structure except with eigenvalues that are multiples of each other, while a C^0 conjugacy only preserves the dimensions of the unstable and stable eigenspaces. In an attempt to find a result in between these two, we project dynamical systems form R^d onto a projective space P^d
- Evolutionary Robotics Prof. Dan Ashlock, Eun-Youn Kim, Elizabeth Blankenship, Brian Davis, Jacob Harper. Grid robots, virtual robots living on a rectilinear grid, are capableof performing a wide range of tasks. These tasks vary widely in difficulty and complexity in an unintuitive fashion. This study seeks to find generic improvements in the performance of grid robots by comparing two representations, GP-Automata and ISAc lists, on several different tasks. These include a multi-agent version of the Tartarus task, the builder task, a version of the game tag, and the follower task. In addition to testing these two representations on all four tasks we assess the worth of a generic technique called genetic hybridization. This technique is inspired by techniques used in stock breeding and should not be confused with algorithmic hybridization in which evolutionary algorithm techniques are blended with non-evolutionary techniques. Overall the ISAc list representation is found to be superior for the grid robot tasks studied. Genetic hybridization is found to improve performance for all the tasks studied.

**Thursday**, July 22, 3:30 PM, 408 Carver, refreshments in 404
at 3 PM

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