Mathematics Commons^™

Differential Equations Models Of Pathogen-Induced Single- And Multi-Organ Tissue Damage, Fiona Lynch

Honors Theses

The rise of antibiotic resistance has created a significant burden on healthcare systems around the world. Antibiotic resistance arises from the increased use of antibiotic drugs and antimicrobial agents, which kill susceptible bacterial strains, but have little effect on strains that have a mutation allowing them to survive antibiotic treatment, defined as “resistant” strains. With no non-resistant bacteria to compete for resources, the resistant bacteria thrives in this environment, continuing to reproduce and infect the host with an infection that does not respond to traditional antibiotic treatment.

A number of strategies have been proposed to tackle the problem of antibiotic …

Differential Privacy For Growing Databases, Gi Heung (Robin) Kim

Honors Theses

Differential privacy [DMNS06] is a strong definition of database privacy that provides indi- viduals in a database with the guarantee that any particular person’s information has very little effect on the output of any analysis of the overall database. In order for this type of analysis to be practical, it must simultaneously preserve privacy and utility, where utility refers to how well the analysis describes the contents of the database.

An analyst may additionally wish to evaluate how a database’s composition changes over time. Consider a company, for example, that accumulates data from a growing base of customers. This company …

Toward A Scientific Investigation Of Convolutional Neural Networks, Anh Tran

Honors Theses

This thesis does not assume the reader is familiar with artificial neural networks. However, to keep the thesis concise, it assumes the reader is familiar with the standard Machine Learning concepts of training set, validation set, and test set [1]. Their usage is intended to help ensure that the Machine Learning system can generalize its training from input examples used during its training to “similar” kinds of examples never used during its training.

The concept of a Convolutional Neural Network (CNN) is one of the most successful computational concepts today for solving image classification problems. However, CNNs are difficult and …

Edge Colorings Of K(M,N) With M+N-1 Colors Which Forbid Rainbow Cycles, Peter Johnson, Claire Zhang

Theory & Applications of Graphs

For positive integers m, n, the greatest number of colors that can appear in an edge coloring of K(m,n) which avoids rainbow cycles is m + n - 1. Here these colorings are constructively characterized. It turns out that these colorings can be encoded by certain vertex labelings of full binary trees with m + n leafs.

An Eternal Domination Problem In Grids, William Klostermeyer, Margaret-Ellen Messinger, Alejandro Angeli Ayello

Theory & Applications of Graphs

A dynamic domination problem in graphs is considered in which an infinite sequence of attacks occur at vertices with mobile guards; the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard. Other guards are allowed to move at the same time, and before and after each attack, the vertices containing guards must form a dominating set of the graph. The minimum number of guards that can defend the graph against such an arbitrary sequence of attacks is called the m-eviction number of the graph. In this paper, the m-eviction …

Two Short Proofs Of The Perfect Forest Theorem, Yair Caro, Josef Lauri, Christina Zarb

Theory & Applications of Graphs

A perfect forest is a spanning forest of a connected graph G, all of whose components are induced subgraphs of G and such that all vertices have odd degree in the forest. A perfect forest can be thought of as a generalization of a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra.

We give here two very short proofs of the Perfect Forest Theorem which use only …

On Certain Semigroups Of Full Contraction Maps Of A Finite Chain, Goje Uba Garba, Muhammad Jamilu Ibrahim, Abdussamad Tanko Imam

Turkish Journal of Mathematics

Let $X_{n}=\{1,2,\ldots,n\}$ with its natural order and let ${\cal T}_{n}$ be the full transformation semigroup on $X_{n}$. A map $\alpha\in{\cal T}_{n}$ is said to be order-preserving if, for all $x,y\in X_{n}$, $x\leq y\Rightarrow x\alpha\leq y\alpha$. The map $\alpha\in{\cal T}_{n}$ is said to be a contraction if, for all $x,y\in X_{n}$, $ x\alpha-y\alpha \leq x-y $. Let ${\cal CT}_{n}$ and ${\cal OCT}_{n}$ denote, respectively, subsemigroups of all contraction maps and all order-preserving contraction maps in ${\cal T}_{n}$. In this paper we present characterisations of Green's relations on ${\cal CT}_{n}$ and starred Green's relations on both ${\cal CT}_{n}$ and ${\cal OCT}_{n}$.

The Influence Of Language On The Teaching And Learning Of Mathematics, Oneil St.Orbine Smith

Walden Dissertations and Doctoral Studies

A majority of students at the local University College of Science and Education (UCSE, pseudonym) in Jamaica do not have the conceptual understanding of mathematical principles to function in a competitive and highly globalized marketplace. In 2013 and 2014, 88% and 92% of freshmen education students scored at the lowest 2 levels on the Mathematics Diagnostic Test (MDT). The instructional language at UCSE is Standard English (SE) whereas most students speak Jamaican dialect (JD). The purpose of this study was to determine the effect that the language of instruction has on student achievement in math as measured by the MDT. …

The Largest Bond In 3-Connected Graphs, Melissa Flynn

Honors Theses

A graph G is connected if given any two vertices, there is a path between them. A bond B is a minimal edge set in G such that G − B has more components than G. We say that a connected graph is dual Hamiltonian if its largest bond has size |E(G)|−|V (G)|+2. In this thesis we verify the conjecture that any simple 3-connected graph G has a largest bond with size at least Ω(nlog32) (Ding, Dziobiak, Wu, 2015 [3]) for a variety of graph classes including planar graphs, complete graphs, ladders, Mo ̈bius ladders and circular ladders, complete bipartite …

Teachers' Theories Of Teaching And Learning And The Use Of Math Interventions, Nicole P. Jones

Walden Dissertations and Doctoral Studies

Despite the academic gap between students with learning disabilities (LD) and their nondisabled peers, schools continue to educate students with LD in regular education classrooms. In secondary math classes, such as Algebra 1, students with LD have high percentages of failure. The purpose of this cross-sectional study was to examine the relationship between teachers' personal theories of teaching and learning and their use of math interventions. Fox's (1983) theoretical framework of teaching and learning was used as a conceptual lens. Surveys were administered to 20 high school math teachers in an urban Northeastern U.S. school district. An ordinal logistic regression …

Special Issue: Neutrosophic Theories Applied In Engineering, Florentin Smarandache, Jun Ye

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic sets and logic are generalizations of fuzzy and intuitionistic fuzzy sets and logic. Neutrosophic sets and logic are gaining significant attention in solving many real life decision making problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. They have been applied in computational intelligence, multiple criteria decision making, image processing, medical diagnoses, etc. This Special Issue presents original research papers that report on state-of-the-art and recent advancements in neutrosophic sets and logic in soft computing, artificial intelligence, big and small data mining, decision making problems, and practical achievements.

P-Union And P-Intersection Of Neutrosophic Cubic Sets, Florentin Smarandache, Young Bae Jun, Chang Su Kim

Branch Mathematics and Statistics Faculty and Staff Publications

Conditions for the P-intersection and P-intersection of falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic sets to be an falsity-external (resp. indeterminacy-external and truthexternal) neutrosophic cubic set are provided. Conditions for the Punion and the P-intersection of two truth-external (resp. indeterminacyexternal and falsity-external) neutrosophic cubic sets to be a truthinternal (resp. indeterminacy-internal and falsity-internal) neutrosophic cubic set are discussed.

Freedericksz Transition In Nematic Liquid Crystal Couette Flow, Arup Mukherjee

Department of Mathematics Faculty Scholarship and Creative Works

This article investigates the effects of an external magnetic field on the Freedericksz transition for an elastically anisotropic nematic liquid crystal sample occupying the annular region between two concentric cylinders in relative (slow) rotation. Assuming both azimuthal and radial magnetic fields and strong anchoring conditions for the liquid crystal director perpendicular to the surface of the cylinders, we investigate and characterize the differences in the director distortions and the critical field value for the onset of the transition.

Normal Subgroups Of Wreath Product 3-Groups, Ryan Gopp

Williams Honors College, Honors Research Projects

Consider the regular wreath product group P of Z₉ with (Z₃ x Z₃). The problem of determining all normal subgroups of P that are contained in its base subgroup is equivalent to determining the subgroups of a certain matrix group M that are invariant under two particular endomorphisms of M. This thesis is a partial solution to the latter. We use concepts from linear algebra and group theory to find and count so-called doubly-invariant subgroups of M.

Orthogonal Polynomials On An Arc Of The Unit Circle With Respect To A Generalized Jacobi Weight: A Riemann-Hilbert Method Approach, Lynsey Cargile Naugle

Electronic Theses and Dissertations

We investigate the asymptotic behavior of polynomials orthogonal over a symmetric arc of the unit circle with respect to a generalized Jacobi-type weight. Full asymptotic expansions for the orthogonal polynomials are obtained at every point of the complex plane. Our method of proof is based on a characterization of the orthogonal polynomials as solutions of a 2X2 matrix Riemann-Hilbert problem, which extends to the unit circle the original Riemann-Hilbert characterization for orthogonal polynomials on the real line, first discovered by Fokas, Its, and Kitaev. In order to extricate the behavior of the polynomials from its Riemann-Hilbert matrix representation, we follow …

On A Time Domain Boundary Integral Equation Formulation For Acoustic Scattering By Rigid Bodies In Uniform Mean Flow, Fang Q. Hu, Michelle E. Pizzo, Douglas M. Nark

Mathematics & Statistics Faculty Publications

It has been well-known that under the assumption of a uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation. However, the constant mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the assumed uniform flow. A customary boundary condition for rigid surfaces is that the normal acoustic velocity be zero. In this paper, a careful study of the acoustic energy conservation equation is presented that shows such a boundary condition would in fact lead to source …

Construction Of Weavings In The Plane, Eden Delight Miro, Aliw-Iw Zambrano, Agnes Garciano

Mathematics Faculty Publications

This work develops, in graph-theoretic terms, a methodology for systematically constructing weavings of overlapping nets derived from 2-colorings of the plane. From a 2-coloring, two disjoint simple, connected graphs called nets are constructed. The union of these nets forms an overlapping net, and a weaving map is defined on the intersection points of the overlapping net to form a weaving. Furthermore, a procedure is given for the construction of mixed overlapping nets and for deriving weavings from them.

Discovering New Tessellations Using Dynamic Geometry Software, Ma. Louise Antonette N. De Las Peñas, Eduard C. Taganap

Mathematics Faculty Publications

In this paper we use dynamic geometry software to investigate a class of tilings called k-uniform tilings or tessellations. A tiling consisting of regular polygons whose vertices belong to k-transitivity classes under the action of its symmetry group (vertex-k-transitive) is said to be k-uniform. We also present constructions of tilings consisting of irregular polygons that are vertex-k-transitive.

On Homology Of Associative Shelves, Alissa Crans

Mathematics, Statistics and Data Science Faculty Works

Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for selfdistributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study associative self-distributive algebraic structures and their one-term and two-term (rack) homology groups.

Optimization And Control Of Agent-Based Models In Biology: A Perspective, G. An, B. G. Fitzpatrick, S. Christley, P. Federico, A. Kanarek, R. Miller Neilan, M. Oremland, R. Salinas, R. Laubeanbacher, S. Lenhart

Mathematics, Statistics and Data Science Faculty Works

Agent-based models (ABMs) have become an increasingly important mode of inquiry for the life sciences. They are particularly valuable for systems that are not understood well enough to build an equation-based model. These advantages, however, are counterbalanced by the difficulty of analyzing and using ABMs, due to the lack of the type of mathematical tools available for more traditional models, which leaves simulation as the primary approach. As models become large, simulation becomes challenging. This paper proposes a novel approach to two mathematical aspects of ABMs, optimization and control, and it presents a few first steps outlining how one might …