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MATHEMATICS
AND COMPUTING
RESEARCH EXPERIENCES FOR UNDERGRADUATES AT IOWA STATE UNIVERSITY
supported by the
National Science
Foundation
(DMS-0353880)
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Publications:
2004 REU
John Bowers, Job Evers, Leslie Hogben, Steve Shaner, Karyn Snider, Amy
Wangsness, On completion problems for various classes of P-matrices, Linear Algebra and Its Applications,
413 (2006) 342-354. [PDF preprint]
Abstract: A pattern of positions in an n x n real matrix is
said to have -completion if every partial -matrix which
specifies that pattern can be completed to a -matrix. A Fischer matrix
is a P -matrix that satisfies Fischer’s inequality for all principal
submatrices. In this paper, all patterns of positions of size up
through 4 are classified as to whether or not every partial -matrix
can be completed to a -matrix for any of the classes positive P -,
nonnegative P -, or Fischer matrices. Also, all symmetric patterns up
through size 5 are classified as to whether or not they have Fischer
completion, and all but 2 such patterns are classified as to whether or
not they have positive P - or nonnegative P -completion. We also show
that any pattern whose digraph contains a minimally chordal
symmetric-Hamiltonian induced subdigraph does not have -completion for
any of the classes positive P -, nonnegative P -, Fischer matrices.
Ghanshyam Bhatt, Lorraine Kraus, Laura Walters, Eric Weber, On Hiding
Messages in the Oversampled Fourier Coefficients, Journal of Mathematical Analysis and
Applications, 320 (2006) 492-498. [PDF
preprint]
Abstract: A system using an oversampled Fourier transform for
hiding data is given in [J.R. Miotke and L. Rebollo-Neira, Applied
Comp. Harmonic Anal., 16 (2004) no. 3, 203– 207]. When viewed as a
cryptographic algorithm, we demonstrate here that the system is
susceptible to a known plaintext attack
G. Bhattacharyya, J. Hegeman, J. Kim, J. Langford, S. Y. Song, Some
Existence and Construction Results of Polygonal Designs, European Journal of Combinatorics,
29 (2008) 1396--1407. [PDF preprint]
Abstract: This paper revisits the existence and construction
problems for polygonal designs (a special class of partially
balanced incomplete block designs associated with regular polygons). We
present new polygonal designs with various parameter sets by explicit
construction. In doing so we employ several construction methods – some
conventional and some new. We also establish a link between a
class of polygonal designs of block size 3 and the cyclically
generated ‘λ-fold triple systems’. Finally, we show that the
existence question for a certain class of polygonal designs is
equivalent to the existence question for ‘perfect grouping systems’
which we introduce.
D.Doan, B.Kivunge, J.J.Poole, J.D.H. Smith, T. Sykes, M.Teplitskiy,
Partial semigroups and and primality indicators in the fractal
generation of binomial coefficients to a prime square modulus, to
appear in Demonstratio Mathematica.
2005 REU
Justin P. Peters, Khalid Boushaba, Marit Nilsen-Hamilton, A
Mathematical Model for Fibroblast Growth Factor Competition Based on
Enzyme Kinetics, Mathematical
Biosciences and Engineering, 2 no. 4 (2005) pp. 789-810. Cover.
Abstract: In this paper, we develop a mathematical model for
the competition of two species of fibroblast growth factor, FGF-1 and
FGF-2, for the same cell surface receptor. We provide pathways for this
interaction using experimental data obtained by Neufeld and
Gospodarowicz reported in 1986 [9]. These pathways demonstrate how the
interaction of two fibroblast growth factors affects cell
proliferation. Upon development of these pathways, we use simulations
in MATLAB and optimization to extrapolate the values of a variety of
biochemical parameters imbedded within the model. Furthermore, it
should be possible to use the model as the basis for a testable
hypothesis. We explore this predictive ability with further simulations
in MATLAB.
Atoshi Chowdhury, Leslie Hogben, Rana Mikkelson, Jude Melancon,
Rational Realization of Maximum Eigenvalue Multiplicity of Symmetric
Tree Sign Patterns, Linear Algebra
and Its Applications, 418 (2006) 380-393. [PDF
preprint]
Abstract: A sign pattern is a matrix whose entries are elements
of {+, −, 0}; it describes the set of real matrices whose entries have
the signs in the pattern. A graph (that allows loops but not multiple
edges) describes the set of symmetric matrices having a zero-nonzero
pattern of entries determined by the absence or presence of edges in
the graph. DeAlba et al. [3] gave algorithms for the computation of
maximum multiplicity and minimum rank of matrices associated with a
tree sign pattern or tree, and an algorithm to obtain an integer matrix
realizing minimum rank. We extend these results by giving algorithms to
obtain a symmetric rational matrix realizing the maximum multiplicity
of a rational eigenvalue among symmetric matrices associated with a
symmetric tree sign pattern or tree.
Jose Ayala-Hoffman, Patrick Corbin, Fritz Colonius, Wolfgang Kliemann,
Justin R. Peters, Morse Decompositions, Attractors and Chain
Recurrence, Proyecciones 25
(2006), no. 1, 79--110. [PDF
preprint]
Abstract: The global behavior of a dynamical system can be
described by its Morse decompositions or its attractor and repeller
conÖgurations. There
is a close relation between these two approaches and also with
(maximal) chain recurrent sets that describe the system behavior on
Önest Morse sets. These sets depend upper semicontinuously on
parameters. The connection with ergodic theory is provided through the
construction of invariant measures based on chains.
Clifford Bergman, Jennifer Davidson, Eric Bartlett, Tracy McKay, Aleka
McAJennJenndams. ANNTS: Artificial Neural Network Technology for
Stegnanalysis, submitted to IEEE Transactions on Information Forensics
and Security.
One paper from 2005 is still under revision.
2006 REU
Nathan L. Chenette, Sean V. Droms, Leslie Hogben, Rana Mikkelson,
Olga Pryporova. Minimum Rank Of A Tree Over An Arbitrary
Field,
Electronic
Journal of
Linear
Algebra 16 (2007): 183-186.
Abstract: For a field F and graph G of order n, the minimum
rank of G over F is defined to be the smallest possible rank over all
symmetric n by n matrices A over F whose (i, j )th entry
(for i not j ) is nonzero whenever {i, j } is an edge in G and is zero
otherwise. We show that the minimum rank
of a tree is independent of the field.
Stephen
Dupal, Michael Yoshizawa. Design
and Optimization of Explicit Runge-Kutta Formulas, Rose-Hulman Undergraduate Mathematics
Journal, Vol. 8, Issue 1,
2007.
Abstract: Explicit Runge-Kutta methods have been studied for
over a century and have applications in the sciences as well as
mathematical software such as Matlab’s ode45 solver. We have taken a
new look at fourth- and fifth-order Runge-Kutta methods by utilizing
techniques based on Gr ̈obner bases to design explicit fourth-order
Runge-Kutta formulas with step doubling and a family of (4,5) formula
pairs that minimize the higher-order truncation error. Gr ̈obner bases,
useful tools for eliminating variables, also helped to reveal patterns
among the error terms. A Matlab program based on step doubling was then
developed to compare the accuracy and efficiency of fourth-order
Runge-Kutta formulas with that of ode45.
Shiliang Chen, Howard A. Levine, Yeon Jung Seo, Marit
Nilsen-Hamilton. A mathematical analysis of multiple-target SELEX, in
preparation.
Some papers from 2006 are still in preparation/under review.