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The "View now" link directs to the finding aid only. Please email wvrhcref@westvirginia.libanswers.com or call 304-293-3536 for more information about accessing collection A&M 0892 Preston County Miscellaneous Papers, Land Grant, 1835-1840. A land grant signed by Wyndham Robertson, Esq; Lt. Gov. of Virginia, granting land in Preston County to Abraham Listons, Mar 17, 1835 ; and a bound manuscript workbook containing mathematical exercises in reduction of vulgar (improper) fractions, ca. 1840

This paper explains the steps involved in the proof of the Eichler-Selberg Trace Formula for Hecke operators of level one. It is based on an appendix in Serge Lang's Introduction to Modular Forms written by Don Zagier, though I also draw heavily from sections of Toshitsune Miyake's Modular Forms and Xueli Wang's and Dingyi Pei's Modular Forms with Integral and Half-Integral Weights. Section 2 summarizes the necessary background in the theory of modular forms. The material covered here is standard, so I've left out most of the details and proofs. Most of the section is based on Chapter VII of Jean-Pierre Serre's A Course in Arithmetic. Section 3 covers the first major step in Zagier's terse proof of the Eichler-Selberg Formula. I've tried to explain where Zaiger's proof comes from and why such a formula should even exist, since the original proof lacks this sort of motivation for the reader. Section 4 completes the proof of the E-S Formula. This last part requires several results from the theory of binary quadratic forms. There are several well-known number theory books that cover these facts in detail, so I assume them without proof.

The "chaos game" is a well-known algorithm by which one may construct a pictorial representation of an iterative process. The resulting sets are known as fractals and can be mathematically characterized by measures of dimension as well as by their associated recurrence relations. Using the chaos game algorithm, is it possible to derive meaningful structure out of our own genetic encoding, and that of other organisms? In this paper, I will present one method of applying the chaos game to biological data and subsequently will discuss both the mathematical and biological implications of the results.

We explore methods that allow us to change an existing network's structure to achieve a desired PageRank. Being able to change the ranking of the nodes is desirable for any network for which the PageRank determines the importance of the nodes. We search for a perturbation to the original network that when added will give the desired ranking. We derive multiple algorithms to solving this Inverse PageRank problem using both constrained optimization techniques and root finding approaches. In these approaches, we ensure that the answer we find satisfies the conditions for an adjacency matrix of an undirected graph, and we explore techniques that promote sparsity of the final solution. We want sparse solutions so that we only have to change a fraction of the existing edges. We develop a general approach that can be applied to any network. We describe C++ implementations for a set of algorithms, the best of which is scalable to thousands of vertices and much faster than Matlab-based approaches we have tried. This same set of algorithms is subjected to a number of tests designed to study the relationship between the initial guess and the final solution. Our ultimate goal is to solve the inverse PageRank problem on a social network of over 6000 venture capitalists to try and reproduce the top 10 list from a previously published set of rankings. Using a root-finding algorithm where nonnegativity constraints are handled implicitly, we are able to find a highly sparse solution to this inverse problem. This thesis consists of a draft of a research paper, co-authored by my advisor and me, that we plan to submit to a peer-reviewed journal of applied/computational mathematics.