W&L Dept. of Mathematicshttp://hdl.handle.net/11021/270742019-05-25T06:59:44Z2019-05-25T06:59:44ZThe Lambda Property and Isometries for Higher Order Schreier Spaceshttp://hdl.handle.net/11021/343692019-04-29T15:13:25ZThe Lambda Property and Isometries for Higher Order Schreier Spaces
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)http://hdl.handle.net/11021/335662018-05-12T03:54:18ZRealizability 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.
Up-growing On-line Linear Discrepancy of Triple-optimal Partially Ordered Sets (thesis)http://hdl.handle.net/11021/335652018-05-12T03:52:25ZUp-growing On-line Linear Discrepancy of Triple-optimal Partially Ordered Sets (thesis)
Whether we acknowledge it as a poset or not, posets arise in many natural contexts, and many also seem to warrant linear extensions (or rankings) of the poset. In some sense, the linear discrepancy of a linear extension L of a poset P indicates the unfairness of L. We describe triple-optimal posets, a class of posets where there exists at least one linear extension which has linear discrepancy three times the minimum linear discrepancy l. This is the worst case scenario; there is no way to have a worse linear discrepancy than triple the optimal linear discrepancy. Two players, a Builder and an Assigner, play an on-line game to construct a linear extension. The Builder gives the Assigner points from P that the Assigner subsequently irrevocably places in a linear extension LA using an algorithm. The Builder's goal is to maximize the linear discrepancy of LA while the Assigner battles to minimize the linear discrepancy of LA. Restrictions can be placed on the Builder, such as up-growing where the Builder cannot give points less than those points already given. In the context of up-growing, we play this on-line game using triple-optimal posets and develop an algorithm that caps the linear discrepancy of LA at 2l on triple-optimal posets with linear discrepancy l.
Thesis; [FULL-TEXT FREELY AVAILABLE ONLINE]; Matthew R. (Matt) Kiser is a member of the Class of 2016 of Washington and Lee University.
Exploring Extreme Points and Related Properties of Tsirelson Space (thesis)http://hdl.handle.net/11021/335642018-05-12T03:52:29ZExploring Extreme Points and Related Properties of Tsirelson Space (thesis)
Tsirelson space was constructed in 1974 as the first example of a Banach space without an embedded c0 or lp space. In 1989, Casazza and Shura wrote a book Tsirelson's Space devoted to Tsirelson space and its many properties. In this thesis, we give two representations of Tsirelson space and give an exposition of many results found in the Casazza-Shura book. In the final chapters, we make two mathematical contributions. First, we give new examples of extreme points of the unit ball of Tsirelson space, which expands the list of known ones from the Casazza-Shura book. Secondly, we improve the bounds on j(n), which roughly measures the complexity of norming a vector of length n. We give an O(log2(n)) lower bound and an O( p n) upper bound. Both of these results answer questions from Tsirelson's Space, the second of which improves upon the O(n) upper bound given by Casazza and Shura. This thesis is written to be accessible to mathematicians outside of functional analysis.
Thesis; [FULL-TEXT FREELY AVAILABLE ONLINE]; Michael W. Holt is a member of the Class of 2016 of Washington and Lee University.