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 Title
 MACMAHON'S MASTER THEOREM AND INFINITE DIMENSIONAL MATRIX INVERSION.
 Creator

Wong, Vivian Lola, Martin, Heath, University of Central Florida
 Abstract / Description

MacMahon's Master Theorem is an important result in the theory of algebraic combinatorics. It gives a precise connection between coefficients of certain power series defined by linear relations. We give a complete proof of MacMahon's Master Theorem based on MacMahon's original 1960 proof. We also study a specific infinite dimensional matrix inverse due to C. Krattenthaler.
 Date Issued
 2004
 Identifier
 CFE0000032, ucf:46084
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0000032
 Title
 Weierstrass vertices and divisor theory of graphs.
 Creator

De Vas Gunasekara, Ajani Ruwandhika, Brennan, Joseph, Song, Zixia, Martin, Heath, University of Central Florida
 Abstract / Description

Chipfiring games and divisor theory on finite, connected, undirected and unweighted graphs have been studied as analogs of divisor theory on Riemann Surfaces. As part of this theory, a version of the onedimensional RiemannRoch theorem was introduced for graphs by Matt Baker in 2007. Properties of algebraic curves that have been studied can be applied to study graphs by means of the divisor theory of graphs.In this research, we investigate the property of a vertex of a graph having the...
Show moreChipfiring games and divisor theory on finite, connected, undirected and unweighted graphs have been studied as analogs of divisor theory on Riemann Surfaces. As part of this theory, a version of the onedimensional RiemannRoch theorem was introduced for graphs by Matt Baker in 2007. Properties of algebraic curves that have been studied can be applied to study graphs by means of the divisor theory of graphs.In this research, we investigate the property of a vertex of a graph having the Weierstrass property in analogy to the theory of Weierstrass points on algebraic curves. The weight of the Weierstrass vertices is then calculated in a manner analogous to the algebraic curve case. Although there are many graphs for which all vertices are Weierstrass vertices, there are bounds on the total weight of the Weierstrass vertices as a function of the arithmetic genus.For complete graphs, all of the vertices are Weierstrass when the number of vertices (n) is greater than or equals to $4$ and no vertex is Weierstrass for $n$ strictly less than 4. We study the complete graphs on 4, 5 and 6 vertices and reveal a pattern in the gap sequence for higher cases of n.Furthermore, we introduce a formula to calculate the Weierstrass weight of a vertex of the complete graph on n vertices. Additionally, we prove that Weierstrass semigroup of complete graphs is 2  generated. Moreover, we show that there are no graphs of genus 2 and 6 vertices with all the vertices being normal Weierstrass vertices and generalize this result to any graph with genus g.
Show less  Date Issued
 2018
 Identifier
 CFE0007397, ucf:52072
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007397
 Title
 QuasiGorenstein Modules.
 Creator

York, Alexander, Brennan, Joseph, Martin, Heath, Ismail, Mourad, Kuebler, Stephen, University of Central Florida
 Abstract / Description

This thesis will study the various roles that quasiGorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An $R$module $M$ of grade $g$ will be quasiGorenstein if $\Ext_R^i(M,R)=0$ for $i\neq g$ and there is an isomorphism $M\cong\Ext_R^g(M,R)$. Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a...
Show moreThis thesis will study the various roles that quasiGorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An $R$module $M$ of grade $g$ will be quasiGorenstein if $\Ext_R^i(M,R)=0$ for $i\neq g$ and there is an isomorphism $M\cong\Ext_R^g(M,R)$. Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a characterization of diagonalizable matrices over principal ideal domains to more general rings. We will use their properties to help lay a foundation for a study of homological dimensions, helping to generalize the concept of Gorenstein dimension to modules of larger grade and present a connection to these new dimensions with certain generalized Serre conditions.We then give a categorical construction to the concept of linkage. The main motivation of such a construction is to generalize ideal and module linkage into one unified theory. By using the defintion of linkage presented by Nagel \cite{NagelLiaison}, we can use categorical language to define linkage between categories. One of the focuses of this thesis is to show that the history of linkage has been wrought with a misunderstanding of which classes of objects to study. We give very compelling evidence to suggest that linkage is a tool to gain information about the even linkage classes of objects. Further, scattered among the literature is a wide array of results pertaining to module linkage, homological dimensions, duality, and adjoint functor pairs and for which we show that these fall under the umbrella of this unified theory. This leads to an intimate relationship between associated homological dimensions and the linkage of objects in a category. We will give many applications of the theory to modules allowing one to cover vast grounds from Gorenstein dimensions to Auslander and Bass classes to local cohomology and local homology. Each of these gives useful insight into certain classes of modules by applying this categorical approach to linkage.
Show less  Date Issued
 2018
 Identifier
 CFE0007268, ucf:52202
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007268
 Title
 Two Ramseyrelated Problems.
 Creator

Zhang, Jingmei, Song, Zixia, Zhao, Yue, Martin, Heath, Turgut, Damla, University of Central Florida
 Abstract / Description

Extremal combinatorics is one of the central branches of discrete mathematics and has experienced an impressive growth during the last few decades. It deals with the problem of determining or estimating the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements. In this dissertation, we focus on studying the minimum number of edges of certain cocritical graphs. Given an integer r \geq 1 and graphs G, H_1, . . . , H_r, we write G \rightarrow (H_1, ....
Show moreExtremal combinatorics is one of the central branches of discrete mathematics and has experienced an impressive growth during the last few decades. It deals with the problem of determining or estimating the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements. In this dissertation, we focus on studying the minimum number of edges of certain cocritical graphs. Given an integer r \geq 1 and graphs G, H_1, . . . , H_r, we write G \rightarrow (H_1, . . . , H_r) if every rcoloring of the edges of G contains a monochromatic copy of H_i in color i for some i \in {1, . . . , r}. A graph G is (H_1, . . . , H_r)cocritical if G \nrightarrow (H_1, . . . , H_r), but G+uv \rightarrow (H_1, . . . , H_r) for every pair of nonadjacent vertices u, v in G. Motivated in part by Hanson and Toft's conjecture from 1987, we study the minimum number of edges over all (K_t,\mathcal{T}_k)cocritical graphs on n vertices, where \mathcal{T}_k denotes the family of all trees on k vertices. We apply graph bootstrap percolation on a not necessarily K_tsaturated graph to prove that for all t \geq 4 and k \geq max{6, t}, there exists a constant c(t,k) such that, for all n \geq (t1)(k1)+1, if G is a (K_t,\mathcal{T}_k)cocritical graph on n vertices, then e(G) \geq ((4t9)/2+\lceil k/2 \rceil /2)nc(t,k). We then show that this is asymptotically best possible for all sufficiently large n when t \in {4, 5} and k \geq 6. The method we developed may shed some light on solving Hanson and Toft's conjecture, which is wide open.We also study Ramsey numbers of even cycles and paths under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai kcoloring is a Gallai coloring that uses at most k colors. Given an integer k \geq 1 and graphs H_1, . . . , H_k, the GallaiRamsey number GR(H_1, . . . , H_k) is the least integer n such that every Gallai kcoloring of the complete graph K_n contains a monochromatic copy of H_i in color i for some i \in {1, . . . , k}. We completely determine the exact values of GR(H_1, . . . , H_k) for all k \geq 2 when each H_i is a path or an even cycle on at most 13 vertices.
Show less  Date Issued
 2019
 Identifier
 CFE0007745, ucf:52404
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007745
 Title
 Improving Student Learning in Undergraduate Mathematics.
 Creator

Rejniak, Gabrielle, Young, Cynthia, Brennan, Joseph, Martin, Heath, University of Central Florida
 Abstract / Description

The goal of this study was to investigate ways of improving student learning, particularly conceptual understanding, in undergraduate mathematics courses. This studyfocused on two areas: course design and animation. The methods of study were thefollowing: Assessing the improvement of student conceptual understanding as a result of teamprojectbased learning, individual inquirybased learning and the modied emporium model; and Assessing the impact of animated videos on student learning with...
Show moreThe goal of this study was to investigate ways of improving student learning, particularly conceptual understanding, in undergraduate mathematics courses. This studyfocused on two areas: course design and animation. The methods of study were thefollowing: Assessing the improvement of student conceptual understanding as a result of teamprojectbased learning, individual inquirybased learning and the modied emporium model; and Assessing the impact of animated videos on student learning with the emphasis onconcepts.For the first part of our study (impact of course design on student conceptual understanding) we began by comparing the following three groups in Fall 2010 and Fall2011:1. Fall 2010: MAC 1140 Traditional Lecture (&) Fall 2011: MAC 1140 Modied Emporium2. Fall 2010: MAC 1140H with Project (&) Fall 2011: MAC 1140H no Project3. Fall 2010: MAC 2147 with Projects (&) Fall 2011: MAC 2147 no ProjectsAnalysis of pretests and posttests show that all three courses showed statistically significant increases, according to their respective sample sizes, during Fall 2010. However, in Fall 2011 only MAC 2147 continued to show a statistically significant increase. Therefore in Fall 2010, projectbased learning  both inclass individual projects and outofclass team projects  conclusively impacted the students' conceptual understanding. Whereas, in Fall 2011, the data for the Modified Emporium model had no statistical significance and is therefore inconclusive as to its effectiveness. In addition the difference in percent ofincrease for MAC 1140 between Fall 2010  traditional lecture model  and Fall 2011 modified emporium model  is not statistically significant and we cannot say that either model is a better delivery mode for conceptual learning. For the second part of our study, the students enrolled in MAC 1140H Fall 2011 and MAC 2147 Fall 2011 were given a pretest on sequences and series before showing them an animated video related to the topic. After watching the video, students were then given the same 7 question post test to determine any improvement in the students' understanding of the topic. After two weeks of teacherled instruction, the students tookthe same posttest again. The results of this preliminary study indicate that animated videos do impact the conceptual understanding of students when used as an introduction into a new concept. Both courses that were shown the video had statistically significant increases in the conceptual understanding of the students between the pretest and the postanimation test.
Show less  Date Issued
 2012
 Identifier
 CFE0004320, ucf:49481
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004320
 Title
 Reasoning Tradeoffs in Implicit Invocation and Aspect Oriented Languages.
 Creator

Sanchez Salazar, Jose, Leavens, Gary, Turgut, Damla, Jha, Sumit, Martin, Heath, University of Central Florida
 Abstract / Description

To reason about a program means to state or conclude, by logical means, some properties the program exhibits; like its correctness according to certain expected behavior. The continuous need for more ambitious, more complex, and more dependable software systems demands for better mechanisms to modularize them and reason about their correctness. The reasoning process is affected by the design decisions made by the developer of the program and by the features supported by the programming...
Show moreTo reason about a program means to state or conclude, by logical means, some properties the program exhibits; like its correctness according to certain expected behavior. The continuous need for more ambitious, more complex, and more dependable software systems demands for better mechanisms to modularize them and reason about their correctness. The reasoning process is affected by the design decisions made by the developer of the program and by the features supported by the programming language used. Beyond Object Orientation, Implicit Invocation and Aspect Oriented languages pose very hard reasoning challenges. Important tradeoffs must be considered while reasoning about a program: modular vs. nonmodular reasoning, casebycase analysis vs. abstraction, explicitness vs. implicitness; are some of them. By deciding a series of tradeoffs one can configure a reasoning scenario. For example if one decides for modular reasoning and explicit invocation a wellknown object oriented reasoning scenario can be used.This dissertation identifies various important tradeoffs faced when reasoning about implicit invocation and aspect oriented programs, characterize scenarios derived from making choices regarding these tradeoffs, and provides sound proof rules for verification of programs covered by all these scenarios. Guidance for program developers and language designers is also given, so that reasoning about these types of programs becomes more tractable.
Show less  Date Issued
 2015
 Identifier
 CFE0005706, ucf:50133
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005706
 Title
 Hilbert Series of Graphs, Hypergraphs, and Monomial Ideals.
 Creator

Trainor, Kyle, Brennan, Joseph, Song, Zixia, Martin, Heath, Morey, Susan, Wocjan, Pawel, University of Central Florida
 Abstract / Description

In this dissertation, identities for Hilbert series of quotients of polynomial rings by monomial ideals are explored, beginning in the contexts of graph and hypergraph rings and later generalizing to general monomial ideals. These identities are modeled after constructive identities from graph theory, and can thus be used to construct Hilbert series iteratively from those of smaller algebraic structures.
 Date Issued
 2018
 Identifier
 CFE0007258, ucf:52176
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007258