home - activity - teaching
As an undergraduate at UMW, I participated in three research projects, including an REU at Iowa State University. Details on those projects are below, as well as associated papers and other documents.
You can see a list of my publications on MathSciNet if you have access.
Semigroups of Partial Linear Transformations (UMW 06-07)
In my senior year, I worked with Dr. Janusz Konieczny and Roberto Palomba (UMW) on two independent research projects dealing with semigroup theory. In the first, we characterized the structure of semigroups of partial linear transformations on an arbitrary vector space. In the second, we classified all Sn-normal subsemigroups of the semigroup of partial transformations. This project serves as my honors thesis at UMW. A paper containing the results of our Sn-normal subsemigroup classification theorem has been submitted to Algebra Colloquium.
Partial Linear Transformations of a Vector Space (with R. Polomba), 2007, Honors Thesis. (view)
- Abstract: We investigate the structure of two semigroups of transformations: the semigroup of partial linear transformations and the semigroup of partial linear 1-1 transformations of an arbitrary vector space. For each of these semigroups, we determine Green's relations, show that the D-classes and J-classes are equal, and that the J-classes form a chain. We present formulas that, for every Green's relation K, count the number of elements in each K-class. We conclude with examples of the eggbox structure for semigroups of each type.
Minimum Ranks of Graphs (ISU 06)
In the summer of 2006, I worked with Dr. Leslie Hogben, Rana Mikkelson, Olga Pryporova and Nathan Chenette (Harvey Mudd College) at Iowa State University in an NSF-sponsored REU. We worked on determining the minimum ranks of graphs, where the entries of the matrices are over fields other than the real numbers. We were able to extend many results previously known for the real numbers to arbitrary fields, including minimum ranks of paths, cycles, and trees. This research was in conjunction with the Combinatorial Matrix Theory Research Group.
I was sponsored by UMW to attend the National Joint Mathematics Meeting in New Orleans, LA, where I presented our poster and won a place in the unranked top 36 posters, of about 170 at the conference.
Minimum rank of a tree over an arbitrary field (with N. Chenette, L. Hogben, R. Mikkelson, and O. Pryporova)
Appeared in the Electronic Journal of Linear Algebra, Volume 16 (2007).
View: Listing in ELA Remote PDF Local PDF
- Abstract: For a field F and graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all symmetric matrices A in Fn x n whose (i,j)th entry is nonzero whenever i ≠ j and {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.
Minimum Rank of a Graph over an Arbitrary Field (with N. Chenette), 2006, Research report. (view)
- Abstract: For a field F and graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all symmetric matrices A in Fn x n whose (i, j)th entry is nonzero whenever i ≠ j and {i, j} is an edge in G. We extend results that give an explicit computational method for the minimum rank of a tree from the real numbers to an arbitrary field and extend the characterization that if G has minimum rank equal to |G| - 1 then G is a path from the real numbers to any field that is not Z2 .
Codes Arising from Projective Geometry (UMW 05)
In the summer of 2005, I participated in a ten-week REU-type undergraduate research program at the University of Mary Washington, sponsored by the Jepson Summer Science Institute. Chris Meyer (UMW) and I worked under Dr. Keith Mellinger on the problem of creating classes of binary linear codes from incidence matrices generated by ovals and related structures in the classical projective plane. We were able to develop numerous new classes of codes and simulate them all using software we obtained from the internet.
At the conclusion of the program, we presented a poster and oral presentation of our results in the Jepson Science Center for all the participants of the Summer Institute. I then presented our poster at the Shenandoah Undergraduate Mathematics & Statistics Conference (SUMS) at James Madison University, where I tied for first place in the poster contest. Dr. Mellinger, Chris and I were sponsored by UMW to attend the National Joint Mathematics Meeting in San Antonio, Texas, where we presented our poster and won a place in the unranked top thirty posters, of about 140 at the conference.
LDPC codes generated by conics in the classical projective plane (with K. Mellinger and C. Meyer)
Appeared in Designs, Codes and Cryptography, September 2006, Volume 40, Number 3.
View: Online Abstract Local PDF
- Abstract: We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2,q). Each incidence structure is based on the various classes of points and lines created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distance. Some arguments involve the geometry of two conics in the plane. As a result, we prove, under mild conditions, the existence of two conics, one entirely internal or external to the other. We conclude with some simulation data to exhibit the effectiveness of our codes.
LDPC codes generated by conics in the classical projective plane (with C. Meyer), 2005, Research report. (view)
- A more detailed version of the published paper above.