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MACMAHON'S MASTER THEOREM AND INFINITE DIMENSIONAL MATRIX INVERSION
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
Weierstrass vertices and divisor theory of graphs
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
Chip-firing 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 one-dimensional Riemann-Roch 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 moreChip-firing 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 one-dimensional Riemann-Roch 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.
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Date Issued
2018
Identifier
CFE0007397, ucf:52072
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007397
Quasi-Gorenstein Modules
Title
Quasi-Gorenstein 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 quasi-Gorenstein 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 quasi-Gorenstein 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 quasi-Gorenstein 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 quasi-Gorenstein 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.
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Date Issued
2018
Identifier
CFE0007268, ucf:52202
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007268
Two Ramsey-related Problems
Title
Two Ramsey-related 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 co-critical 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 co-critical graphs. Given an integer r \geq 1 and graphs G, H_1, . . . , H_r, we write G \rightarrow (H_1, . . . , H_r) if every r-coloring 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)-co-critical if G \nrightarrow (H_1, . . . , H_r), but G+uv \rightarrow (H_1, . . . , H_r) for every pair of non-adjacent 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)-co-critical 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_t-saturated 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 (t-1)(k-1)+1, if G is a (K_t,\mathcal{T}_k)-co-critical graph on n vertices, then e(G) \geq ((4t-9)/2+\lceil k/2 \rceil /2)n-c(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 k-coloring is a Gallai coloring that uses at most k colors. Given an integer k \geq 1 and graphs H_1, . . . , H_k, the Gallai-Ramsey number GR(H_1, . . . , H_k) is the least integer n such that every Gallai k-coloring 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.
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Date Issued
2019
Identifier
CFE0007745, ucf:52404
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007745
Improving Student Learning in Undergraduate Mathematics
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, par-ticularly 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 teamproject-based learning, individual inquiry-based learning and the modied empo-rium 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, par-ticularly 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 teamproject-based learning, individual inquiry-based learning and the modied empo-rium 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 Empo-rium2. 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 pre-tests and post-tests 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, project-based learning - both in-class individual projects and out-of-class 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 pre-test 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 teacher-led instruction, the students tookthe same post-test 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 pre-test and the post-animation test.
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Date Issued
2012
Identifier
CFE0004320, ucf:49481
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004320
Reasoning Tradeoffs in Implicit Invocation and Aspect Oriented Languages
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. non-modular reasoning, case-by-case 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 well-known 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.
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Date Issued
2015
Identifier
CFE0005706, ucf:50133
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005706
Hilbert Series of Graphs, Hypergraphs, and Monomial Ideals
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