supported by the National Science Foundation (DMS-0353880)

Summer 2004



Michael Anderson                         Taylor University
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

Project Descriptions

  • 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.



The 2004 ISU Math REU had an extensive program of activities, both academic and social.  Students heard lectures on topics as varied as The mathematics of secrecy: an introduction to public key cyptography, to Graduate school: what is it REALLY like?  See Calendar. Weekly REU lunches included pasta, tacos, pizza, sandwiches, a Chinese buffet, and the 4th of July picnic at Big Creek Lake including boating.  Students attended a concert by a local fock singer, went bowling, skating, and to Worlds of Fun/Oceans of Fun in Kansas City.  See pictures.  A typical weekly schedule appears below.

July 19 - July 23

Mathematical Modeling of Phytoplankton, Prof. Khalid Boushaba, ISU
Mathematcs Summer Colloquium
Tuesday, July 20, 3:30 PM, 408 Carver, refreshments in 404 at 3 PM
Graduate school: What is it REALLY like?
REU Panel Discussion with graduate students

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

REU lunch
Friday July 23, 12 noon - 1 PM, pasta from DaVinci's, let Melanie know whether you will attend by noon Thursday.

Undergraduate Research Conferences and Journals Link

AMS Undergraduate Site, has links to many conferences and also a lot of other good information.

Mathematics Department Homepage
Web page maintained by Leslie Hogben
Last Update: 4-May-07