W&L Dept. of Mathematicshttps://dspace.wlu.edu/handle/11021/357732024-03-28T18:43:05Z2024-03-28T18:43:05ZOn the Realizability of Projective Configurations (thesis)https://dspace.wlu.edu/handle/11021/358542022-07-28T11:07:22ZOn the Realizability of Projective Configurations (thesis)
Many natural questions emerge from the work of this thesis. The first chapter may inspire a budding mathematician to work (in the fashion of my thesis advisor) on area relations and equidissections. From the second chapter, one may wonder if a group of projective transformations exists over skew fields (or rings!) and if so, how to describe such an object. The third chapter bears the question of whether or not all combinatorially degenerate realizations of projective configurations hold over every projective plane. One may expand upon the fourth chapter and outline a technique for encoding any polynomial equation over F into a projective configuration. Finally, I hope that the fifth chapter compels future researches to infuse machinery from other disciplines into their studies. In addition to these listed above, Garst's thesis poses many open questions at its completion [7]. [From concluding section]
Thesis; [FULL-TEXT FREELY AVAILABLE ONLINE]; Troy James Larsen is a member of the Class of 2022 of Washington and Lee University.
Linear Algebraic Methods in Data Science and Neural Networks (thesis)https://dspace.wlu.edu/handle/11021/358532022-07-28T11:07:22ZLinear Algebraic Methods in Data Science and Neural Networks (thesis)
This thesis is about some of the methods and concepts of linear algebra that are particularly helpful for data analysis. After a brief review of some linear algebra concepts in chapter 1, the second chapter of the thesis centers around the singular value decomposition (SVD) which expresses any matrix A as a product of an orthogonal matrix, a diagonal matrix, and another orthogonal matrix. Understanding the SVD requires understanding the properties of symmetric matrices, which are explained first. Chapter 3 focuses on the applications of the SVD. It begins with using the SVD for low-rank approximation, and then explores how the SVD is applied in principle component analysis. Chapter 4 introduces neural networks, a machine learning architecture useful for image recognition among other applications. It introduces the structure of a neural network in linear algebraic notation; one of the main goals of chapter 4 is to reinforce the idea that neural networks can be seen as compositions of matrix transformations with non-linear activation functions. We then introduce how the parameters in a neural network are optimized. Chapter 5 deals with convolutional neural networks. It also focuses heavily on circulant matrices, and the relationship between convolution and circulant matrices. Understanding the properties of circulant matrices will be instrumental in understanding the benefits of convolution. We finish the chapter by
showing how PCA can be used as a data pre-processing tool before running the data through a convolutional neural network.
Thesis; [FULL-TEXT FREELY AVAILABLE ONLINE]; Jackson Mark Gazin is a member of the Class of 2022 of Washington and Lee University.
The Lambda Property and Isometries for Higher Order Schreier Spaces (thesis)https://dspace.wlu.edu/handle/11021/343692022-07-28T11:07:22ZThe Lambda Property and Isometries for Higher Order Schreier Spaces (thesis)
For each n in N, let Sn be the Schreier set of order n and XSn be the corresponding Schreier space of order n. In their 1989 paper "The lambda-property in Schreier space S and the Lorentz space d(a, 1)," Th. Shura and D. Trautman proved that the Schreier space of order 1 has the lambda-property. This thesis extends the theorem by proving the lambda-property for the Schreier spaces of any order and the uniform lambda-property (stronger than the lambda-property) for the p-convexification of these spaces. Furthermore, using what we know about extreme points of the unit balls, we are able to characterize all surjective linear isometries of these spaces.
Hung Viet Chu is a member of the Class of 2019 of Washington and Lee University.; Thesis; [FULL-TEXT FREELY AVAILABLE ONLINE]
Realizability of n-Vertex Graphs with Prescribed Vertex Connectivity, Edge Connectivity, Minimum Degree, and Maximum Degree (thesis)https://dspace.wlu.edu/handle/11021/335662022-07-28T11:07:22ZRealizability of n-Vertex Graphs with Prescribed Vertex Connectivity, Edge Connectivity, Minimum Degree, and Maximum Degree (thesis)
This is the fourth and nal
thesis that concludes ProfessorWayne M. Dymacek's research project Realizability
of n-Vertex Graphs with Prescribed Vertex Connectivity, Edge Connectivity,
Minimum Degree, and Maximum Degree. With the completion of this project,
working through hundreds of cases, Professor Dymacek's students have successfully
completed an exhaustive system to determine the realizability of any given
parameters and produce these simple and undirected graphs for any possible
order that is desired. [From Conclusion]
Thesis; [FULL-TEXT FREELY AVAILABLE ONLINE]; Lewis N. Sears is a member of the Class of 2016 of Washington and Lee University.