Ordered alphabetically by student's last name
Clarke 
Johnson  Messina  Sabo 
Toupo 
Deaton  Hyatt 
Padron  Saponaro  
Devlin  Kress  Pippins  Tait 
An Experimental Investigation of the Theory of Electrostatic Deflections Robert Deaton, Eric Sabo, Nick Brubaker, and J. A. Pelesko Department of Mathematical Sciences The theory of electrostatic deflections is crucial to the design and implementation of electrostatically actuated micro and nanoelectromechanical systems (MEMS and NEMS). While the socalled “standard model” of electrostatic deflections has been investigated mathematically, it has yet to be thoroughly tested experimentally. We present a complete overview of the standard model and its implications and then discuss an experimental investigation of the theory. Our experimental setup allowed us to obtain results for both the static case, which has had little experimental verification, and the dynamic case, which has no known experimental verification. Finally, we show the need for a new, revised theory based on our experimental findings. Our study of the theory of electrostatic deflections reveals limits in the current standard model. Understanding these limits can improve the design of future MEMS and NEMS devices. This project was funded by the National Science Foundation #312154 and University of Delaware Science and Engineering Scholars. 
DiscreteTime
Simple Random Walks with Fluctuating Absorbing Barriers
Patrick R. Devlin and Wenbo V. Li Department of Mathematical Sciences A random walk is the path
traced by a particle whose velocity at each time increment is
determined randomly according to some probability function.
Random walks have vast applications in many fields including economics,
physics, and computer science, and they are used to model systems such
as the stock market, predatorpray situations, and the movement of
atoms. Here, we consider two problems involving a simple random
walk on the integers with two absorbing boundaries. The position
of the upper boundary is given as an independent random walk, and the
position of the lower boundary is either (1) stationary or (2) given as
a third random walk. We concern ourselves with questions
involving the distribution of the first exit time, the probabilities of
reaching a given boundary, and the distribution of the various possible
states as a function of time. / Attempting to solve the problems, we
used generating functions; Markov chains; the method of partial
difference equations; bounds derived from infinite sums; martingales;
alternate coordinate systems; and data from computer programs written
using OpenOffice.org, Maple, and MATLAB. Ultimately, by deriving
a type of reflection principle for two dimensions, we were able to
obtain an exact closed formula for the distribution of the various
possible states in the first problem. For the second problem, we
have obtained functions that appear to approximate exact solutions very
well, but we have not yet mathematically proved their validity.

Quantifying Interspecific Competition from BBS TimeSeries Data Zariel I. Johnson, Kelly A. Pippins, John A. Pelesko, and Chris Williams Department of Mathematical Sciences and Department of Entomology and Wildlife Ecology Understanding competition between species is a critical aspect of population ecology and plays an important role in predicting the effects of species interactions. Although competition can be quantified experimentally in artificially controlled environments, these experiments are often expensive and impractical for the majority of species. Therefore, we propose a method to quantify competition from timeseries data, which may be readily gathered or accessed from existing databases. We fit the thetalogistic twospecies population model to timeseries data from the North American Breeding Bird Survey (BBS) to obtain competition coefficients for two species of wren. A competitive dynamic between house wrens and Bewick’s wrens has been widely observed and well documented as coinciding with a recent increase the range of the house wren. We present approaches for establishing reliable parameter values for the model, using BBS data from physiographic stratum #14, the Highland Rim, and life history information. We estimated competition coefficients for Bewick’s wren (0.20) and house wren (0.75) populations. The model predicts stable coexistence of the two species, at an extremely low Bewick’s wren population suggesting possible remnant Bewick’s populations can remain in the face of house wren invasion. With this methodological foundation, future biologists will have a tool to better understand competitive effects on invasive species on existing native populations. Funding was provided by the Howard Hughes Medical Institute and by Delaware EPSCoR. 
Bioinformatics of the unknown: Computational exploration of environmental viral genes William Kress^{1}, Mara Hyatt^{2}, Jennifer Clarke^{3}, Rachel Marine, Michael Dumas, Megan Furman, Sanchita Jamindar, Sharath Srinivasiah, Jaysheel Bhavsar, Sandeep Kumar, Shawn Polson, and K. Eric Wommack Department of Civil and Environmental Engineering^{1}, University of Delaware, Delaware Technical and Community College^{2}, Department of Biological Sciences^{3}, Lincoln University Viruses are the single most
abundant biological entity on the planet. Ubiquitous in natural
environments, their global population is estimated to be on the order
of 1031 individuals. Bioinformatic analysis of fragmentary viral
metagenome data allows for detailed comparisons among unknown viral
genes. Here, open reading frames (ORFs) were predicted from
metagenomic sequences within two single stranded DNA viral libraries
using Metagene Annotator software. Predicted ORFs were clustered based
on sequence homology, by a hybrid application of CDHit (60% sequence
identity) and PSIBLAST (evalue: 1e7). A multiple sequence
alignment (MSA) was performed and primers were designed for the 14th,
19th, and 20th most abundant clusters. In addition, the genomic
association of these unknown genes with other genes was assessed
through examination of ORF calls on individual sequence
fragments. ORFs within the 19th most abundant cluster were
homologus to a ssDNAbinding protein of T7like podoviruses infecting
cyanobacteria, while the 14th and 20th most abundant clusters were
novel, having no homologs within the UniRef 100 database. The peptide
sequence of the representative ORF from cluster 19 was
significantly shorter than other known ssDNAbinding proteins from
T7like podoviruses. It remains to be determined if this
demonstrates a novel protein or an error in the Metagene
annotation. Multiple sequence alignments showed that while ORFs
within clusters 14, 19, and 20 were highly conserved, while their
neighboring ORFs were highly polymorphic. Thus, it is likely that ORFs
14, 19, and 20 occur within multiple virus strains. Research funded by
NSF EPSCoR.

Exact Linear Algebra on Packed Dense ZeroOne Matrices Nicholas Messina and David Saunders Project LinBox, Department of Computer Science Exact
linear algebra typically involves large integer matrices or vectors and
performs various operations among such objects. A matrix is
described as zeroone if it only contains zeros or ones. Dense
matrices do not consist of primarily one number (typically zero), so
each element of a dense matrix is stored in memory. Packed
matrices use less memory by taking advantage of the number of unique
elements. For instance, since zeroone matrices only consist of
two different elements, zero and one, they can undergo a packing
scheme. To store a zero or a one in memory, all that is needed is
one bit, which is either off (zero) or on (one). As 64bit
integers use 64 bits of memory, we were able to pack 64 elements of
dense zeroone matrices into one 64bit integer. When compared to
storing each element in its own 64bit integer, packing matrices take
drastically less space in memory. The language C++ was used to
write the operations needed and all files used Project LinBox
conventions. Dense zeroone matrices arise as representations of
incidence relations. This packing scheme expands computational
capability when blackbox algorithms are used and when elimination is
used in arithmetic modulo two, which is important in
cryptography. We have seen large increases in efficiency by using
operations specifically designed for dense zeroone matrices.
Funded by NSF grant CCF0830130.

Improving Reproducibility and Stability of Numerically Intensive Applications on Graphics Processing Units. Omar Padron^{1}, Philip Saponaro, and Michela Taufer ^{1}New Jersey Center for Science, Technology, and Mathematics Education, Kean University, Union, NJ The advent of general purpose
graphics processing units (GPGPU's) brings about a whole new platform
for running numerically intensive applications at high speeds.
Their multicore architectures enable large degrees of parallelism via
a massively multithreaded environment. Molecular dynamics (MD)
simulations are particularly wellsuited for GPU's because their
computations are easily parallelizable. Significant performance
improvements are observed when single precision floating point
arithmetic is used. However, this performance comes at a cost of
accuracy that is not characterized simply by a lack of significant
bits. GPU single precision floating point operations truncate the
output in a fashion that is not IEEE compliant. As a result,
largescale simulations suffer from large errors accumulated from
intermediate calculations. This produces a drifting behavior in
the otherwise constant energy MD simulations performed by the Global
Computing Lab. Double precision arithmetic corrects this
drifting, but is eight times slower than single precision and
comparable to CPU performance. To solve this problem, we extend the
approaches of previous literature to guarantee numerical
reproducibility and stability in MD simulations, while assuring
efficiency and performance comparable to that when using the GPU
hardware implementation of single precision arithmetic. We
present development of a library of mathematical functions that use
fast and efficient algorithms to fix the rounding error produced by the
equivalent operations performed by GPU.

Line arrangements in the projective plane Michael Tait and Felix Lazebnik Department opf Mathematical Sciences A line arrangement is a
set of lines in a plane. An arrangement is called {\it simple} if no
three lines go through the same point and no two lines are
parallel. The question that our summer work was mostly focused on
was the following: When considering N lines, how many nonisomorphic,
i.e., ``essentially distinct" arrangements are possible? Early
work in the field had solved this problem through 7 lines, with a
conjecture about the number for 8. More recently the numbers have
been published through 11 lines, however we are still attempting to
verify the method used to find these and the accuracy of the
publications. To solve the latter problem, we wrote a
computer program that creates, we believe, all nonisomorphic
arrangements. Using it, we found a result on 8 lines
that was one short of the conjectured 135 arrangements.
Currently, we are attempting to figure out which of these numbers is
correct. Having the programs working properly will allow us to address
many other unsolved questions of combinatorial geometry related to line
arrangements.

In science and medicine, it is useful to be able to encapsulate microscopic objects. Unfortunately, this is not always easy to do. In Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet (2007), Py et. al introduce a technique called “Capillary Origami”. This is a method to fold planar surfaces into threedimensional objects. In our research, we have attempted to verify and extend their work with PolyDimethylSiloxane (PDMS), the most commonly used siliconbased organic polymer. We deposit a drop of water on the PDMS sheet, and as the liquid evaporates at room temperature, its surface tension pulls the membrane’s edges closer together. As a result, the membrane folds into the shape of a tetrahedron, tube, or sphere. By balancing the surface energy of the droplet with the bending energy of the PDMS, a “critical length” was found where the PDMS membrane will not fold completely. This critical length is a function of the membrane thickness. This technology has many applications including drug delivery, whereby we could manufacture very small containers to deliver microscopic amounts of drugs. Funding was provided by the NUCLEUS program, the Howard Hughes Medical Institute, and NSF #312154 