Mathematics Commons^™

Rigorous Analysis Of An Edge-Based Network Disease Model, Sabrina Mai

Honors Undergraduate Theses

Edge-based network disease models, in comparison to classic compartmental epidemiological models, better capture social factors affecting disease spread such as contact duration and social heterogeneity. We reason that there should exist infinitely many equilibria rather than only an endemic equilibrium and a disease-free equilibrium for the edge-based network disease model commonly used in the literature, as there do not exist any changes in demographic in the model. We modify the commonly used network model by relaxing some assumed conditions and factor in a dependency on initial conditions. We find that this modification still accounts for realistic dynamics of disease spread …

Comparison Of Green's Functions For A Family Of Boundary Value Problems For Fractional Difference Equations, Paul W. Eloe, Catherine Kublik, Jeffrey T. Neugebauer

Mathematics Faculty Publications

In this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.

On The Time Evolution Of Resonant Triads In Rotational Capillary-Gravity Water Waves, Rossen Ivanov, Calin Martin

Articles

We investigate an effect of the resonant interaction in the case of one-directional propagation of capillary-gravity surface waves arising as the free surface of a rotational water flow. Specifically, we assume constant vorticity in the body of the fluid which physically corresponds to an underlying current with a linear horizontal velocity profile. We consider the interaction of three distinct modes, and we obtain the dynamic equations for a resonant triad. Setting the constant vorticity equal to zero, we recover the well known integrable three-wave system.

Effects Of Advective-Diffusive Transport Of Multiple Chemoattractants On Motility Of Engineered Chemosensory Particles In Fluidic Environments, D. King, H. Başağaoğlu, Hoa Nguyen, Frank G. Healy, Melissa Whitman, Sauro Succi

Mathematics Faculty Research

Motility behavior of an engineered chemosensory particle (ECP) in fluidic environments is driven by its responses to chemical stimuli. One of the challenges to understanding such behaviors lies in tracking changes in chemical signal gradients of chemoattractants and ECP-fluid dynamics as the fluid is continuously disturbed by ECP motion. To address this challenge, we introduce a new multiscale numerical model to simulate chemotactic swimming of an ECP in confined fluidic environments by accounting for motility-induced disturbances in spatiotemporal chemoattractant distributions. The model accommodates advective-diffusive transport of unmixed chemoattractants, ECP-fluid hydrodynamics at the ECP-fluid interface, and spatiotemporal disturbances in the chemoattractant …

Use Of Symmetries In Economics: An Overview, Vladik Kreinovich, Olga Kosheleva, Nguyen N. Thach, Nguyen Duc Trung

Departmental Technical Reports (CS)

In this paper, we show that many semi-heuristic econometric formulas can be derived from the natural symmetry requirements. The list of such formulas includes many famous formulas provided by Nobel-prize winners, such as Hurwicz optimism-pessimism criterion for decision making under uncertainty, McFadden's formula for probabilistic decision making, Nash's formula for bargaining solution -- as well as Cobb-Douglas formula for production, gravity model for trade, etc.

Why Bilingualism Helps Autistic Children Function: A Symmetry-Based Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems of autistic children is that it is very difficult for them to switch to a different state, to a different activity -- and such switches are often needed. Researchers have recently shown that bilingualism helps autistic children function: namely, it is somewhat easier for bilingual children to switch to a new activity. In this paper, we provide a possible explanation for this empirical phenomenon. Namely, we show that, in general terms, autism means difficulty with breaking symmetries of a state, and we describe how this general reformulation indeed explains the above recently discovered phenomenon.

Applications Of Geometric And Spectral Methods In Graph Theory, Lauren Morey Nelsen

Electronic Theses and Dissertations

Networks, or graphs, are useful for studying many things in today’s world. Graphs can be used to represent connections on social media, transportation networks, or even the internet. Because of this, it’s helpful to study graphs and learn what we can say about the structure of a given graph or what properties it might have. This dissertation focuses on the use of the probabilistic method and spectral graph theory to understand the geometric structure of graphs and find structures in graphs. We will also discuss graph curvature and how curvature lower bounds can be used to give us information about …

Decidability For Residuated Lattices And Substructural Logics, Gavin St. John

Electronic Theses and Dissertations

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical …

Bottle Filling Task Reasoning: A Comparison Of Matching Versus Constructed Student Responses, Dominique Banner, Laurie Overman Cavey

Mathematics Faculty Publications and Presentations

In this paper, we compare the levels of reasoning elicited during the completion of three versions of a bottle filling task: high school level matching; middle school level matching and constructed response. The goal of the tasks was to make visible secondary students covariational reasoning methods. Video of students completing the task while explaining their reasoning during one-onone interviews were analyzed. Analysis demonstrated a wide range of reasoning when provided a matching version with a greater incidence of accuracy with students who exhibited lower levels of reasoning. Conversely, the constructed response task demonstrated higher levels of reasoning more consistently with …

Mathematics Of The Infinite God In The Works Of Nicholas Of Cusa, Lauren Henderson

Beets Paper Contest

The geometrical analyses of infinity exemplified in the work of Nicholas of Cusa can aid in understanding the complex nature of God’s attributes, specifically the perceived tension between his simplicity and his quality of encompassing and upholding all things.

Ua84 Sigma Chi, Wku Archives

WKU Archives Collection Inventories

Records created by and about the WKU chapter of Sigma Xi.

Student Help-Seeking Behaviors And Teacher Instructional Practices: Examining Their Relationship With U.S. Student Mathematics Achievement, Michael C. Osborne

Theses and Dissertations--Curriculum and Instruction

Even though the United States (U.S.) spends, on average, more money per student than most Organisation for Economic Co-operation and Development (OECD) countries, it continues to lag behind its international peers in mathematics achievement. This study, which responded to the call for educational reforms that improve the mathematics achievement of U.S. students, aimed to examine the issue of student help-seeking behaviors and teacher instructional practices as they interact to affect student mathematics achievement. The Programme for International Student Assessment (PISA) defines student help-seeking behaviors as the ways in which students have a propensity to depend on the knowledge and intellect …

L(2, 1)-Labeling Of Circulant Graphs, Sarbari Mitra, Soumya Bhoumik

Mathematics Faculty Publications

An L(2, 1)-labeling of a graph Γ is an assignment of non-negative integers to the vertices such that adjacent vertices receive labels that differ by at least 2, and those at a distance of two receive labels that differ by at least one. Let λ12(Γ) denote the least λ such that Γ admits an L(2, 1)-labeling using labels from {0, 1, . . ., λ}. A Cayley graph of group G is called a circulant graph of order n, if G = Zn. In this paper initially we investigate the upper bound for the span of the L(2, 1)-labeling for …

Using A Discursive Framework To Analyze Geometric Learning And Instruction, Peter Wiles, Rick Anderson

Faculty Research and Creative Activity

In this study we applied a discursive perspective of learning (Sfard, 2008) to a sequence of 21 geometry mini-lessons taught in a fourth grade classroom. From this perspective, learning is defined as changes in mathematical discourse. We first characterize and then compare discourse from the beginning and the end of the mini-lesson sequence. We identify shifts in the discourse that occurred during the sequence. We then discuss how the characteristics of the st geometric discourse informed task design and instruction. This perspective provided a useful means for linking instruction to student learning in an operationalized manner.

A Second Order Bdf Numerical Scheme With Variable Steps For The Cahn-Hilliard Equation, Wenbin Chen, Xiaoming Wang, Yue Yan, Zhuying Zhang

Mathematics and Statistics Faculty Research & Creative Works

We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn-Hilliard equation. the construction relies on a second order backward difference, convex-splitting technique and viscous regularizing at the discrete level. We show that the scheme is unconditionally stable and uniquely solvable. in addition, under mild restriction on the ratio of adjacent time-steps, an optimal second order in time convergence rate is established. the proof involves a novel generalized discrete Gronwall-type inequality. as far as we know, this is the first rigorous proof of second order convergence for a variable step BDF2 scheme, even in …

Automating The Calculation Of The Hilbert–Kunz Multiplicity And F-Signature, Gabriel Johnson, Sandra Spiroff

Faculty and Student Publications

© 2018 The Authors The Hilbert–Kunz multiplicity and F-signature are important invariants for researchers in commutative algebra and algebraic geometry. We provide software, and describe the automation, for the calculations of the two invariants in the case of intersection algebras over polynomial rings.

Ua66/10/2 Ogden College Of Science & Engineering Mathematics Publications, Wku Archives

WKU Archives Collection Inventories

Publications created by and about the Mathematics department.

Enhancing The Quandle Coloring Invariant For Knots And Links, Karina Elle Cho

HMC Senior Theses

Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.

Radial Solutions To Semipositone Dirichlet Problems, Ethan Sargent

HMC Senior Theses

We study a Dirichlet problem, investigating existence and uniqueness for semipositone and superlinear nonlinearities. We make use of Pohozaev identities, energy arguments, and bifurcation from a simple eigenvalue.

Bounding The Number Of Compatible Simplices In Higher Dimensional Tournaments, Karthik Chandrasekhar

Theses and Dissertations--Mathematics

A tournament graph G is a vertex set V of size n, together with a directed edge set E ⊂ V × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, j ∈ V and (i, i) ∉ E for all i ∈ V. We explore the following generalization: For a fixed k we orient every k-subset of V by assigning it an orientation. That is, every facet of the (k − 1)-skeleton of the ( …

Squared Distance Matrix Of A Weighted Tree, Ravindra B. Bapat

Journal Articles

Let T be a tree with vertex set f1;: :: ; ng such that each edge is assigned a nonzero weight. The squared distance matrix of T; denoted by is the n n matrix with (i; j)-element d(i; j)2; where d(i; j) is the sum of the weights of the edges on the (ij)-path. We obtain a formula for the determinant of A formula for 1 is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.

Spacetime Groups, Ian M. Anderson, Charles G. Torre

Publications

A spacetime group is a connected 4-dimensional Lie group G endowed with a left invariant Lorentz metric h and such that the connected component of the isometry group of h is G itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs (g,η), with g being a 4-dimensional Lie algebra and η being a Lorentzian inner product on g. A full analysis of the equivalence problem for spacetime Lie algebras is given which leads to a completely algorithmic solution to the …

Ua66/10/1 Ogden College Of Science & Engineering Mathematics Administration, Wku Archives

WKU Archives Collection Inventories

Administrative records created by and about the Mathematics department.

Ua66/15/1 Ogden College Of Science & Engineering Mathematics & Computer Science Events, Wku Archives

WKU Archives Collection Inventories

Records related to events hosted by Mathematics & Computer Science.

Quantum Isometry Group Of Deformation: A Counterexample, Debashish Goswami, Arnab Mandal

Journal Articles

We give a counterexample to show that the quantum isometry group of a deformed finite dimensional spectral triple may not be isomorphic with a deformation of the quantum isometry group of the undeformed spectral triple.

Well-Posedness And Symmetry Properties Of Free Boundary Problems For Some Non-Linear Degenerate Elliptic Second Order Partial Differential Equations, Alaa Haj Ali

Wayne State University Dissertations

In the first part of this thesis, a bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the $p$-Laplacian, subject to given boundary condition is proved in the first chapter. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second chapter and third chapter, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.

In the second part of this thesis, …

Neandertal Mandibular Molars From Hortus Cave, France: A Comparison Of Crown Shapes Using Elliptical Fourier Analysis, Frank L. Williams, Juliet K. Brophy, Gregory J. Matthews, Emilee Hart, Marie-Antoinette De Lumley, Gaël Becam

Mathematics and Statistics: Faculty Publications and Other Works

Neandertal permanent mandibular molars are well represented at Hortus Cave, France, Sub-Phase IVb and Vb, providing an opportunity to assess intrapopulation and intersite shape variation. We expected the mandibular molars from Hortus cave, comprising Hortus II, Hortus IV, Hortus V and Hortus VI to cluster together, compared to Neandertals close in proximity, such as Malarnaud and La Quina 5 as well as those more distantly located such as Arcysur- Cure 1, Engis 2 and Scladina 4A-1. A single Neolithic cave burial from Maurenne Caverne de la Cave, Belgium (n = 11) is used to contextualize the degree of variation in …

10. Corrections To Design And Analysis Of Experiments, Angela Dean, Dan Voss, Danel Draguljic

Design and Analysis of Experiments

Corrections for the provided by the authors for Design and Analysis of Experiments.

On Monochromatic Sets With Nondecreasing Diameter, Adam O’Neal

Theses, Dissertations and Capstones

Our problem comes from the field of combinatorics known as Ramsey theory. Ramsey theory, in a general sense, is about identifying the threshold for which a family of objects, associated with a particular parameter, goes from never or sometimes satisfying a certain property to always satisfying that property. Research in Ramsey theory has applications in design theory and coding theory. For integers m, r, and t, we say that a set of n integers colored with r colors is (m, r, t)-permissible if there exist t monochromatic subsets B₁, B₂, . . . , B_{t …}

An Inference-Driven Branch And Bound Optimization Strategy For Planning Ambulance Services, Kevin Mcdaniel

Theses, Dissertations and Capstones

Strategic placement of ambulances is important to the efficient functioning of emergency services. As part of an ongoing collaboration with Wayne County 911, we developed a strategy to optimize the placement of ambulances throughout Wayne County based on de-identified call and response data from 2016 and 2017. The primary goals of the optimization were minimizing annual operating cost and mean response time, as well as providing a constructive solution that could naturally evolve from the existing plan. This thesis details the derivation and implementation of one of the optimization strategies used in this project. It is based on parametric statistical …