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 Title
 ON WELLQUASIORDERINGS.
 Creator

Thurman, Forrest, Brennan, Joseph, University of Central Florida
 Abstract / Description

A quasiorder is a relation on a set which is both reflexive and transitive, while a wellquasiorder has the additional property that there exist no infinite strictly descending chains nor infinite antichains. Wellquasiorderings have many interesting applications to a variety of areas which includes the strength of certain logical systems, the termination of algorithms, and the classification of sets of graphs in terms of excluded minors. My thesis explores how wellquasiorderings are...
Show moreA quasiorder is a relation on a set which is both reflexive and transitive, while a wellquasiorder has the additional property that there exist no infinite strictly descending chains nor infinite antichains. Wellquasiorderings have many interesting applications to a variety of areas which includes the strength of certain logical systems, the termination of algorithms, and the classification of sets of graphs in terms of excluded minors. My thesis explores how wellquasiorderings are related to these topics through examples of four known wellquasiorderings which are given by Dickson's Lemma, Higmans's Lemma, Kruskal's Tree Theorem, and the RobertsonSeymour Theorem. The wellquasiordering conjecture for matroids is also discussed, and an original proof of Higman's Lemma is presented.
Show less  Date Issued
 2013
 Identifier
 CFH0004455, ucf:45082
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFH0004455
 Title
 ALGEBRAIC ASPECTS OF (BIO) NANOCHEMICAL REACTION NETWORKS AND BIFURCATIONS IN VARIOUS DYNAMICAL SYSTEMS.
 Creator

Chen, Teng, Brennan, Joseph, University of Central Florida
 Abstract / Description

The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question...
Show moreThe dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle con gurations. We also investigate Hopf bifurcation surfaces of various dynamical systems.
Show less  Date Issued
 2011
 Identifier
 CFE0003933, ucf:48689
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0003933
 Title
 Analysis and Simulation for Homogeneous and Heterogeneous SIR Models.
 Creator

Wilda, Joseph, Shuai, Zhisheng, Brennan, Joseph, Nevai, A, University of Central Florida
 Abstract / Description

In mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous SusceptibleInfectiousRemoved (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement...
Show moreIn mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous SusceptibleInfectiousRemoved (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement between populations is incorporated into a heterogeneous SIR model with nonlinear incidence. The analysis of the combined effects of the spatial heterogeneity and nonlinear incidence on the disease dynamics of our model is presented along with supporting simulations. New global stability results are established for the heterogeneous model utilizing a graphtheoretic approach and Lyapunov functions. Numerical simulations confirm nonlinear incidence gives raise to rich dynamics such as synchronization and phaselock oscillations.
Show less  Date Issued
 2015
 Identifier
 CFE0005906, ucf:50872
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005906
 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
 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
 On the security of NoSQL cloud database services.
 Creator

Ahmadian, Mohammad, Marinescu, Dan, Wocjan, Pawel, Heinrich, Mark, Brennan, Joseph, University of Central Florida
 Abstract / Description

Processing a vast volume of data generated by web, mobile and Internetenabled devices, necessitates a scalable and flexible data management system. DatabaseasaService (DBaaS) is a new cloud computing paradigm, promising a costeffective and scalable, fullymanaged database functionality meeting the requirements of online data processing. Although DBaaS offers many benefits it also introduces new threats and vulnerabilities. While many traditional data processing threats remain, DBaaS...
Show moreProcessing a vast volume of data generated by web, mobile and Internetenabled devices, necessitates a scalable and flexible data management system. DatabaseasaService (DBaaS) is a new cloud computing paradigm, promising a costeffective and scalable, fullymanaged database functionality meeting the requirements of online data processing. Although DBaaS offers many benefits it also introduces new threats and vulnerabilities. While many traditional data processing threats remain, DBaaS introduces new challenges such as confidentiality violation and information leakage in the presence of privileged malicious insiders and adds new dimension to the data security. We address the problem of building a secure DBaaS for a public cloud infrastructure where, the Cloud Service Provider (CSP) is not completely trusted by the data owner. We present a high level description of several architectures combining modern cryptographic primitives for achieving this goal. A novel searchable security scheme is proposed to leverage secure query processing in presence of a malicious cloud insider without disclosing sensitive information. A holistic database security scheme comprised of data confidentiality and information leakage prevention is proposed in this dissertation. The main contributions of our work are:(i) A searchable security scheme for nonrelational databases of the cloud DBaaS; (ii) Leakage minimization in the untrusted cloud.The analysis of experiments that employ a set of established cryptographic techniques to protect databases and minimize information leakage, proves that the performance of the proposed solution is bounded by communication cost rather than by the cryptographic computational effort.
Show less  Date Issued
 2017
 Identifier
 CFE0006848, ucf:51777
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006848
 Title
 Interval EdgeColorings of Graphs.
 Creator

Foster, Austin, Song, Zixia, Reid, Michael, Brennan, Joseph, University of Central Florida
 Abstract / Description

A proper edgecoloring of a graph G by positive integers is called an interval edgecoloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edgecolorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edgecolorings to schedule parentteacher conferences so...
Show moreA proper edgecoloring of a graph G by positive integers is called an interval edgecoloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edgecolorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edgecolorings to schedule parentteacher conferences so that every person's conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for parents and teachers, and edges for the required meetings, has an interval edgecoloring.A wellknown result of Vizing states that for any simple graph $G$, $\chi'(G) \leq \Delta(G) + 1$, where $\chi'(G)$ and $\Delta(G)$ denote the edgechromatic number and maximum degree of $G$, respectively. A graph $G$ is called class 1 if $\chi'(G) = \Delta(G)$, and class 2 if $\chi'(G) = \Delta(G) + 1$. One can see that any graph admitting an interval edgecoloring must be of class 1, and thus every graph of class 2 does not have such a coloring.Finding an interval edgecoloring of a given graph is hard. In fact, it has been shown that determining whether a bipartite graph has an interval edgecoloring is NPcomplete. In this thesis, we survey known results on interval edgecolorings of graphs, with a focus on the progress of $(a, b)$biregular bipartite graphs. Discussion of related topics and future work is included at the end. We also give a new proof of Theorem 3.15 on the existence of proper path factors of $(3, 4)$biregular graphs. Finally, we obtain a new result, Theorem 3.18, which states that if a proper path factor of any $(3, 4)$biregular graph has no path of length 8, then it contains paths of length 6 only. The new result we obtained and the method we developed in the proof of Theorem 3.15 might be helpful in attacking the open problems mentioned in the Future Work section of Chapter 5.
Show less  Date Issued
 2016
 Identifier
 CFE0006301, ucf:51609
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006301
 Title
 On Randic Energy of Graphs.
 Creator

Burns, Brittany, Mohapatra, Ram, Song, Zixia, Brennan, Joseph, University of Central Florida
 Abstract / Description

In this research, we explore the subject of graph energy. We first discuss the connections between linear algebra and graph theory and review some important definitions and facts of these two fields. We introduce graph energy and provide some historical perspectives on the subject. Known results of graph energy are also mentioned and some relevant results are proven. We discuss some applications of graph energy in the physical sciences. Then, Randic energy is defined and results are given and...
Show moreIn this research, we explore the subject of graph energy. We first discuss the connections between linear algebra and graph theory and review some important definitions and facts of these two fields. We introduce graph energy and provide some historical perspectives on the subject. Known results of graph energy are also mentioned and some relevant results are proven. We discuss some applications of graph energy in the physical sciences. Then, Randic energy is defined and results are given and proved for specific families of graphs. We focus on simple, connected graphs that are commonly studied in graph theory. Also, the Laplacian energy of a graph is defined. We then examine the connections between the different types of energies for graphs, beginning with graph energy and Randic energy, followed by Laplacian energy and Randic energy. In our results chapter, we introduce the Petersen graph and calculate the Randic energy of this graph. We also define StackedBook graphs and perform some calculations on these graphs. From these calculations, we form a conjecture and discuss some details on how to proceed with the proof of this conjecture. Finally, we summarize our work and details are provided on how this research can be continued.
Show less  Date Issued
 2016
 Identifier
 CFE0006273, ucf:51043
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006273
 Title
 On the Theory of Zetafunctions and Lfunctions.
 Creator

Awan, Almuatazbellah, Mohapatra, Ram, Li, Xin, Brennan, Joseph, University of Central Florida
 Abstract / Description

In this thesis we provide a body of knowledge that concerns Riemann zetafunction and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet's Lfunction. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we also discuss different formulations and approaches to the proof of the Prime Number Theorem and the...
Show moreIn this thesis we provide a body of knowledge that concerns Riemann zetafunction and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet's Lfunction. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we also discuss different formulations and approaches to the proof of the Prime Number Theorem and the Riemann Hypothesis. These two topics constitute the main theme of this thesis. For the Prime Number Theorem, we provide a thorough discussion that compares and contrasts Norbert Wiener's proof with that of Newman's short proof. We have also related them to Hadamard's and de la Vallee Poussin's original proofs written in 1896. As far as the Riemann Hypothesis is concerned, we discuss some recent results related to equivalent formulations of the Riemann Hypothesis as well as the Generalized Riemann Hypothesis.
Show less  Date Issued
 2015
 Identifier
 CFE0005576, ucf:50268
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005576
 Title
 Dynamical Invariants and the Fluid Impulse in Plasma Models.
 Creator

Michalak, Martin, Shivamoggi, Bhimsen, Mohapatra, Ram, Brennan, Joseph, Eastes, Richard, University of Central Florida
 Abstract / Description

Much progress has been made in understanding of plasmas through the use of the MHD equations and newer models such as Hall MHD and electron MHD. As with most equations of fluid behavior, these equations are nonlinear, and no general solutions can be found. The use of invariant structures allows limited predictions of fluid behavior without requiring a full solution of the underlying equations. The use of gauge transformation can allow the creation of new invariants, while differential...
Show moreMuch progress has been made in understanding of plasmas through the use of the MHD equations and newer models such as Hall MHD and electron MHD. As with most equations of fluid behavior, these equations are nonlinear, and no general solutions can be found. The use of invariant structures allows limited predictions of fluid behavior without requiring a full solution of the underlying equations. The use of gauge transformation can allow the creation of new invariants, while differential geometry offers useful tools for constructing additional invariants from those that are already known. Using these techniques, new geometric, integral and topological invariants are constructed for Hall and electron MHD models. Both compressible and incompressible models are considered, where applicable. An application of topological invariants to magnetic reconnection is provided. Finally, a particular geometric invariant, which can be interpreted as the fluid impulse density, is studied in greater detail, its nature and invariance in plasma models is demonstrated, and its behavior is predicted in particular geometries under different models.
Show less  Date Issued
 2013
 Identifier
 CFE0005382, ucf:50442
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005382
 Title
 Numerical Simulations for the Flow of Rocket Exhaust Through a Granular Medium.
 Creator

Kraakmo, Kristina, Moore, Brian, Brennan, Joseph, Rollins, David, University of Central Florida
 Abstract / Description

Physical lab experiments have shown that the pressure caused by an impinging jet on a granular bed has the potential to form craters. This poses a danger to landing success and nearby spacecraft for future rocket missions. Current numerical simulations for this process do not accurately reproduce experimental results. Our goal is to produce improved simulations to more accurately and efficiently model the changes in pressure as gas flows through a porous medium. A twodimensional model in...
Show morePhysical lab experiments have shown that the pressure caused by an impinging jet on a granular bed has the potential to form craters. This poses a danger to landing success and nearby spacecraft for future rocket missions. Current numerical simulations for this process do not accurately reproduce experimental results. Our goal is to produce improved simulations to more accurately and efficiently model the changes in pressure as gas flows through a porous medium. A twodimensional model in space known as the nonlinear Porous Medium Equation as it is derived from Darcy's law is used. An AlternatingDirection Implicit (ADI) temporal scheme is presented and implemented which reduces our multidimensional problem into a series of onedimensional problems. We take advantage of explicit approximations for the nonlinear terms using extrapolation formulas derived from Taylorseries, which increases efficiency when compared to other common methods. We couple our ADI temporal scheme with different spatial discretizations including a secondorder Finite Difference (FD) method, a fourthorder Orthogonal Spline Collocation (OSC) method, and an Nthorder Chebyshev Spectral method. Accuracy and runtime are compared among the three methods for comparison in a linear analogue of our problem. We see the best results for accuracy when using an ADISpectral method in the linear case, but discuss possibilities for increased efficiency using an ADIOSC scheme. Nonlinear results are presented using the ADISpectral method and the ADIFD method.
Show less  Date Issued
 2013
 Identifier
 CFE0005017, ucf:49998
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005017
 Title
 Valuation of OverTheCounter (OTC) Derivatives with Collateralization.
 Creator

Guerrero, Leon, Yong, Jiongmin, Li, Xin, Brennan, Joseph, University of Central Florida
 Abstract / Description

Collateralization in overthecounter (OTC) derivatives markets has grown rapidly overthe past decade, and even faster in the past few years, due to the impact of the recentfinancial crisis and the particularly important attention to the counterparty credit risk in derivatives contracts. The addition of collateralization to such contracts significantly reduces the counterparty credit risk and allows to offset liabilities in case of default.We study the problem of valuation of OTC derivatives...
Show moreCollateralization in overthecounter (OTC) derivatives markets has grown rapidly overthe past decade, and even faster in the past few years, due to the impact of the recentfinancial crisis and the particularly important attention to the counterparty credit risk in derivatives contracts. The addition of collateralization to such contracts significantly reduces the counterparty credit risk and allows to offset liabilities in case of default.We study the problem of valuation of OTC derivatives with payoff in a single currencyand with single underlying asset for the cases of zero, partial, and perfect collateralization. We assume the derivative is traded between two defaultfree counterparties and analyze the impact of collateralization on the fair present value of the derivative. We establish a uniform generalized derivative pricing framework for the three cases of collateralization and show how different approaches to pricing turn out to be consistent. We then generalize the results to include multiasset and crosscurrency arguments, where the underlyingand the derivative are in some domestic currency, but the collateral is posted in a foreign currency. We show that the results for the single currency, multiasset case are consistent with those obtained for the single currency, single asset case.
Show less  Date Issued
 2013
 Identifier
 CFE0004855, ucf:49688
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004855
 Title
 Advancing Practical Specification Techniques for Modern Software Systems.
 Creator

Singleton, John, Leavens, Gary, Jha, Sumit Kumar, Zou, Changchun, Hughes, Charles, Brennan, Joseph, University of Central Florida
 Abstract / Description

The pervasive nature of software (and the tendency for it to contain errors) has long been a concern of theoretical computer scientists. Many investigators have endeavored to produce theories, tools, and techniques for verifying the behavior of software systems. One of the most promising lines of research is that of formal specification, which is a subset of the larger field of formal methods. In formal specification, one composes a precise mathematical description of a software system and...
Show moreThe pervasive nature of software (and the tendency for it to contain errors) has long been a concern of theoretical computer scientists. Many investigators have endeavored to produce theories, tools, and techniques for verifying the behavior of software systems. One of the most promising lines of research is that of formal specification, which is a subset of the larger field of formal methods. In formal specification, one composes a precise mathematical description of a software system and uses tools and techniques to ensure that the software that has been written conforms to this specification. Examples of such systems are Z notation, the Java Modeling Language, and many others. However, a fundamental problem that plagues this line of research is that the specifications themselves are often costly to produce and difficult to reuse. If the field of formal specification is to advance, we must develop sound techniques for reducing the cost of producing and reusing software specifications. The work presented in this dissertation lays out a path to producing sophisticated, automated tools for inferring large, complex code bases, tools for allowing engineers to share and reuse specifications, and specification languages for specifying information flow policies that can be written separately from program code. This dissertation introduces three main lines of research. First, I discuss a system that facilitates the authoring, sharing, and reuse of software specifications. Next, I discuss a technique which aims to reduce the cost of producing specifications by automatically inferring them. Finally, I discuss a specification language called Evidently which aims to make information flow security policies easier to write, maintain, and enforce by untangling them from the code to which they are applied.
Show less  Date Issued
 2018
 Identifier
 CFE0007099, ucf:51953
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007099
 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
 Title
 Hadwiger Numbers and GallaiRamsey Numbers of Special Graphs.
 Creator

Bosse, Christian, Song, Zixia, Brennan, Joseph, Zhao, Yue, DeMara, Ronald, University of Central Florida
 Abstract / Description

This dissertation explores two separate topics on graphs.We first study a farreaching generalization of the Four Color Theorem. Given a graph G, we use chi(G) to denote the chromatic number; alpha(G) the independence number; and h(G) the Hadwiger number, which is the largest integer t such that the complete graph K_t can be obtained from a subgraph of G by contracting edges. Hadwiger's conjecture from 1943 states that for every graph G, h(G) is greater than or equal to chi(G). This is...
Show moreThis dissertation explores two separate topics on graphs.We first study a farreaching generalization of the Four Color Theorem. Given a graph G, we use chi(G) to denote the chromatic number; alpha(G) the independence number; and h(G) the Hadwiger number, which is the largest integer t such that the complete graph K_t can be obtained from a subgraph of G by contracting edges. Hadwiger's conjecture from 1943 states that for every graph G, h(G) is greater than or equal to chi(G). This is perhaps the most famous conjecture in Graph Theory and remains open even for graphs G with alpha(G) less than or equal to 2. Let W_5 denote the wheel on six vertices. We establish more evidence for Hadwiger's conjecture by proving that h(G) is greater than or equal to chi(G) for all graphs G such that alpha(G) is less than or equal to 2 and G does not contain W_5 as an induced subgraph.Our second topic is related to Ramsey theory, a field that has intrigued those who study combinatorics for many decades. Computing the classical Ramsey numbers is a notoriously difficult problem, leaving many basic questions unanswered even after more than 80 years. We study Ramsey numbers under Gallaicolorings. A Gallaicoloring of a complete graph is an edgecoloring such that no triangle is colored with three distinct colors. Given a graph H and an integer k at least 1, the GallaiRamsey number, denoted GR_k(H), is the least positive integer n such that every Gallaicoloring of K_n with at most k colors contains a monochromatic copy of H. It turns out that GR_k(H) is more wellbehaved than the classical Ramsey number R_k(H), though finding exact values of GR_k(H) is far from trivial. We show that for all k at least 3, GR_k(C_{2n+1}) = n2^k+1 where n is 4, 5, 6 or 7, and GR_k(C_{2n+1}) is at most (n ln n)2^k(k+1)n+1 for all n at least 8, where C_{2n+1} denotes a cycle on 2n+1 vertices.
Show less  Date Issued
 2019
 Identifier
 CFE0007603, ucf:52532
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007603
 Title
 Quantum Algorithms for: Quantum Phase Estimation, Approximation of the Tutte Polynomial and Blackbox Structures.
 Creator

Ahmadi Abhari, Seyed Hamed, Brennan, Joseph, Mucciolo, Eduardo, Li, Xin, Marinescu, Dan, University of Central Florida
 Abstract / Description

In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of blackbox modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date...
Show moreIn this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of blackbox modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT)) is highly constrained by the requirement of highprecision controlled phase shift operators, which remain difficult to realize. In the second part of this dissertation, we introduce an alternative approach to approximately implement QPE with arbitrary constantprecision controlled phase shift operators.The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev's original approach. For approximating the eigenphase precise to the nth bit, Kitaev's original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constantprecision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev's approach.The other problem we investigate relates to approximating the Tutte polynomial. We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q,1/q) of the Tutte plane is BQPcomplete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at a point is DQC1complete and at some points is in BQP.To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids.To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible JonesWenzl representations of the Hecke algebra have finite order. We show that for each braid we can efficiently construct a braid such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of the resulting braid are alternating links.The final part of the dissertation focuses on finding the structure of a blackbox module or algebra. Suppose we are given blackbox access to a finite module M or algebra over a finite ring R and a list of generators for M and R. We show how to find a linear basis and structure constants for M in quantum poly (logM) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for rings. We then show that our algorithm is a useful primitive allowing quantum computer to determine the structure of a finite associative algebra as a direct sum of simple algebras. Moreover, it solves a wide variety of problems regarding finite modules and rings. Although our quantum algorithm is based on Abelian Fourier transforms, it solves problems regarding the multiplicative structure of modules and algebras, which need not be commutative. Examples include finding the intersection and quotient of two modules, finding the additive and multiplicative identities in a module, computing the order of an module, solving linear equations over modules, deciding whether an ideal is maximal, finding annihilators, and testing the injectivity and surjectivity of ring homomorphisms. These problems appear to be exponentially hard classically.
Show less  Date Issued
 2012
 Identifier
 CFE0004239, ucf:49526
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004239
 Title
 LatticeValued TFilters and Induced Structures.
 Creator

Reid, Frederick, Richardson, Gary, Brennan, Joseph, Han, Deguang, Lang, SheauDong, University of Central Florida
 Abstract / Description

A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a Tconvergence structure which is defined in terms of Tfilters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of...
Show moreA complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a Tconvergence structure which is defined in terms of Tfilters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional.Subcategories of compact and of complete spaces are investigated. It is shown that each Tconvergence space has a compactification with the extension property provided the frame is a Boolean algebra. TCauchy spaces are defined and sufficient conditions for the existence of a completion are given. Tuniform limit spaces are also defined and their completions are given in terms of the TCauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed Tconvergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded TCauchy spaces which induce the fixed space.
Show less  Date Issued
 2019
 Identifier
 CFE0007520, ucf:52586
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007520
 Title
 Coloring graphs with forbidden minors.
 Creator

Rolek, Martin, Song, Zixia, Brennan, Joseph, Reid, Michael, Zhao, Yue, DeMara, Ronald, University of Central Florida
 Abstract / Description

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger's Conjecture from 1943 which states that every graph with no Ktminor is (t  1)colorable. This conjecture has been proved true for t ? 6, but remains open for all t ? 7. For t = 7, it is not even yet known if a graph with no K7minor is 7colorable. We begin by showing that every graph with no K_tminor is (2t  6)colorable for t = 7, 8, 9, in...
Show moreA graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger's Conjecture from 1943 which states that every graph with no Ktminor is (t  1)colorable. This conjecture has been proved true for t ? 6, but remains open for all t ? 7. For t = 7, it is not even yet known if a graph with no K7minor is 7colorable. We begin by showing that every graph with no K_tminor is (2t  6)colorable for t = 7, 8, 9, in the process giving a shorter and computerfree proof of the known results for t = 7, 8. We also show that this result extends to larger values of t if Mader's bound for the extremal function for Ktminors is true. Additionally, we show that any graph with no K8?minor is 9colorable, and any graph with no K8?minor is 8colorable. The Kempechain method developed for our proofs of the above results may be of independent interest. We also use Mader's HWege theorem to establish some sufficient conditions for a graph to contain a K8minor.Another motivation for my research is a wellknown conjecture of Erd?s and Lov(&)#225;sz from 1968, the DoubleCritical Graph Conjecture. A connected graph G is doublecritical if for all edges xy ? E(G), ?(G  x  y) = ?(G)  2. Erd?s and Lov(&)#225;sz conjectured that the only doublecritical tchromatic graph is the complete graph Kt. This conjecture has been show to be true for t ? 5 and remains open for t ? 6. It has further been shown that any noncomplete, doublecritical, tchromatic graph contains Kt as a minor for t ? 8. We give a shorter proof of this result for t = 7, a computerfree proof for t = 8, and extend the result to show that G contains a K9minor for all t ? 9. Finally, we show that the DoubleCritical Graph Conjecture is true for doublecritical graphs with chromatic number t ? 8 if such graphs are clawfree.
Show less  Date Issued
 2017
 Identifier
 CFE0006649, ucf:51227
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006649
 Title
 Extensions of Sspaces.
 Creator

Losert, Bernd, Richardson, Gary, Mikusinski, Piotr, Dutkay, Dorin, Brennan, Joseph, Marinescu, Dan, University of Central Florida
 Abstract / Description

Given a convergence space X, a continuous action of a convergence semigroup S on X and a compactification Y of X, under what conditions on X and the action on X is it possible to extend the action to a continuous action on Y. Similarly, given a Cauchy space X, a Cauchy continuous action of a Cauchy semigroup S on X and a completion Y of X, under what conditions on X and the action on X is it possible to extend the action to a Cauchy continuous action on Y. We answer the first question for...
Show moreGiven a convergence space X, a continuous action of a convergence semigroup S on X and a compactification Y of X, under what conditions on X and the action on X is it possible to extend the action to a continuous action on Y. Similarly, given a Cauchy space X, a Cauchy continuous action of a Cauchy semigroup S on X and a completion Y of X, under what conditions on X and the action on X is it possible to extend the action to a Cauchy continuous action on Y. We answer the first question for some particular compactifications like the onepoint compactification and the star compactification as well as for the class of regular compactifications. We answer the second question for the class of regular strict completions. Using these results, we give sufficient conditions under which the pseudoquotient of a compactification/completion of a space is the compactification/completion of the pseudoquotient of the given space.
Show less  Date Issued
 2013
 Identifier
 CFE0004881, ucf:49661
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004881