Abstracts Submitted from
Mathematics and Quantitative Biology
Undergraduate Summer Research Symposium August 12, 2009

Ordered alphabetically by student's last name

Johnson Messina Sabo
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 so-called “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.

Discrete-Time 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, predator-pray 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 Time-Series 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 time-series data, which may be readily gathered or accessed from existing databases. We fit the theta-logistic two-species population model to time-series 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 Kress1, Mara Hyatt2, Jennifer Clarke3, 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 Engineering1, University of Delaware, Delaware Technical and Community College2,
Department of Biological Sciences3, 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 CD-Hit (60% sequence identity) and PSI-BLAST (e-value: 1e-7).  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 ssDNA-binding protein of T7-like 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 ssDNA-binding proteins from T7-like 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 Zero-One 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 zero-one 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 zero-one 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 64-bit integers use 64 bits of memory, we were able to pack 64 elements of dense zero-one matrices into one 64-bit integer.  When compared to storing each element in its own 64-bit 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 zero-one 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 zero-one matrices.  Funded by NSF grant CCF-0830130.

Improving Reproducibility and Stability of Numerically Intensive Applications on Graphics Processing Units.

Omar Padron1, Philip Saponaro, and Michela Taufer
1New 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 multi-core architectures enable large degrees of parallelism via a massively multi-threaded environment.  Molecular dynamics (MD) simulations are particularly well-suited 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, large-scale 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 non-isomorphic, 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 non-isomorphic 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.

Folding an Elastic Membrane

Danielle Toupo, Elise Braun and J. A. Pelesko
Department of Mathematical Sciences

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 three-dimensional objects. In our research, we have attempted to verify and extend their work with PolyDimethylSiloxane (PDMS), the most commonly used silicon-based 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

Links: Summer 2009 Undergraduate Research Symposium, Symposium Abstracts from other Colleges and Departments,
2009 Undergraduate Research Summer Enrichment ProgramUnversity of Delaware Undergraduate Research Program, Howard Hughes Undergraduate Program.
Created  8 August 2009. Last up dated 15 August 2009 by Hal White
Copyright 2009, University of Delaware