Mathematics Commons^™

Predicting Risk Of Adverse Outcomes In Knee Replacement Surgery With Reconstructability Analysis, Cecily Corrine Froemke, Martin Zwick

Complex Systems Faculty Publications and Presentations

Reconstructability Analysis (RA) is a data mining method that searches for relations in data, especially non-linear and higher order relations. This study shows that RA can provide useful predictions of complications in knee replacement surgery.

Evolution Equations With A Parameter And Application To Transport-Convection Differential Equations, Emile Franc Doungmo Goufo

Turkish Journal of Mathematics

We deeply investigate the well-posedness of models taking the form $_0^AD^{\beta }_tu(t) = Au(t),\;\; u(0)= \,f,\;\;\;00$ where $_0^AD^{\beta }_t$ is a derivative with the fractional parameter $\beta$ and $A$ is a closed densely defined operator in a Banach space. We show that, unlike other systems, solutions of our models are not governed by Mittag--Leffler functions and their variants. We extend and adapt Peano's idea to our models and establish conditions for existence and uniqueness of solutions. In particular, relations between the two-parameter solution operator, its resolvent, and its generator are provided; the issue of subordination and prolongation principles are addressed; …

General Pseudo-Polynomials In Fq((1/X)), Rattiya Meesa

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this thesis, we study the integer-valued functions and pseudo-polynomials over a finite field. We establish explicit shapes of them and prove that the set of all pseudo-polynomials is an integral domain that is not a unique factorization domain. Difference and higher order differences of integer-valued functions and their properties are also presented.

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 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 …

Identifying Clouds With Convolutional Neural Networks, Jeff Mullins, Sean Richardson, Peter Drake

Portland Institute for Computational Science Publications

The greatest source of uncertainty in model estimates of projected climate change involve clouds and aerosols. Photographic images of clouds in the sky are simple to acquire and archive, but climate scientists need an automated process for identifying clouds in these images. We bring machine learning to bear on this problem. Specifically, we use convolutional neural networks, which to our knowledge have not previously been applied to this task. We trained a network to identify clear sky, thin cloud, thick cloud, and non-sky pixels in photos taken by the Total Sky Imager. The trained network is capable of classifying 91.9% …

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.

Question 1: Time Texting; Question 2: Opening Doors, Larry Weinstein

Physics Faculty Publications

[Introduction] How much total time do Americans spend texting every year?

Can a typical human open a 20-ton door (swinging on hinges around a vertical axis like a normal room door) with no mechanical assistance?

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 …

Application Of An Extremal Result Of Erdős And Gallai To The (N,K,T) Problem, Matt Noble, Peter Johnson, Dean Hoffman, Jessica Mcdonald

Theory & Applications of Graphs

An extremal result about vertex covers, attributed by Hajnal to Erdős and Gallai, is applied to prove the following: If n, k, and t are integers satisfying n ≥ k ≥ t ≥ 3 and k ≤ 2t - 2, and G is a graph with the minimum number of edges among graphs on n vertices with the property that every induced subgraph on k vertices contains a complete subgraph on t vertices, then every component of G is complete.

Electric Ion Dispersion As A New Type Of Mass Spectrometer, Michael R. Lindstrom, Iain Moyles, Kevin Ryczko

School of Mathematical & Statistical Sciences Faculty Publications

At the 2014 Fields-MPrime Industrial Problem Solving Workshop, PerkinElmer presented a design problem for mass spectrometry. Traditionally, mass spectrometry is done via three methods: using magnetic fields to deflect charged particles whereby different masses bend differently; using a time-of-flight procedure where particles of different mass arrive at different times at a target; and using an electric quadrupole that filters out all masses except for one very narrow band. The challenge posed in the problem was to come up with a new design for mass spectrometry that did not involve magnetic fields and where mass fractions could be measured in an …

Orientable ℤ < Inf> N -Distance Magic Labeling Of The Cartesian Product Of Many Cycles, Bryan Freyberg, Melissa S. Keranen

Michigan Tech Publications, Part 1

The following generalization of distance magic graphs was introduced in [2]. A directed ℤn- distance magic labeling of an oriented graph G = (V,A) of order n is a bijection ℓ: V → ℤn with the property that there is a μ ∈ ℤn (called the magic constant) such that If for a graph G there exists an orientation G such that there is a directed ℤn-distance magic labeling ℓ for G, we say that G is orientable ℤn-distance magic and the directed ℤn-distance magic labeling ℓ we call an orientable ℤn-distance magic labeling. In this paper, we find orientable …

Unconditionally Energy Stable Numerical Schemes For Hydrodynamics Coupled Fluids Systems, Alexander Yuryevich Brylev

Theses and Dissertations

The thesis consists of two parts. In the first part we propose several second order in time, fully discrete, linear and nonlinear numerical schemes to solve the phase-field model of two-phase incompressible flows in the framework of finite element method. The schemes are based on the second order Crank-Nicolson method for time disretizations, projection method for Navier-Stokes equations, as well as several implicit-explicit treatments for phase-field equations. The energy stability, solvability, and uniqueness for numerical solutions of proposed schemes are further proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes thereafter.

In the …

Covering Subsets Of The Integers And A Result On Digits Of Fibonacci Numbers, Wilson Andrew Harvey

Theses and Dissertations

A covering system of the integers is a finite system of congruences where each integer satisfies at least one of the congruences. Two questions in covering systems have been of particular interest in the mathematical literature. First is the minimum modulus problem, whether the minimum modulus of a covering system of the integers with distinct moduli can be arbitrarily large, and the second is the odd covering problem, whether a covering system of the integers with distinct moduli can be constructed with all moduli odd. We consider these and similar questions for subsets of the integers, such as the set …

Series Solutions Of Polarized Gowdy Universes, Doniray Brusaferro

Theses and Dissertations

Einstein's field equations are a system of ten partial differential equations. For a special class of spacetimes known as Gowdy spacetimes, the number of equations is reduced due to additional structure of two dimensional isometry groups with mutually orthogonal Killing vectors. In this thesis, we focus on a particular model of Gowdy spacetimes known as the polarized T³ model, and provide an explicit solution to Einstein's equations.

Convergence And Rate Of Convergence Of Approximate Greedy-Type Algorithms, Anton Dereventsov

Theses and Dissertations

In this dissertation we study the questions of convergence and rate of convergence of greedy-type algorithms under imprecise step evaluations. Such algorithms are in demand as the issue of calculation errors appears naturally in applications.

We address the question of strong convergence of the Chebyshev Greedy Algorithm (CGA), which is a generalization of the Orthogonal Greedy Algorithm (also known as the Orthogonal Matching Pursuit), and show that the class of Banach spaces for which the CGA converges for all dictionaries and objective elements is strictly between smooth and uniformly smooth Banach spaces.

We analyze an application-oriented modification of the CGA, …

Deep Learning: An Exposition, Ryan Kingery

Theses and Dissertations

In this paper we describe and survey the field of deep learning, a type of machine learning that has seen tremendous growth and popularity over the past decade for its ability to substantially outperform other learning methods at important tasks. We focus on the problem of supervised learning with feedforward neural networks. After describing what these are we give an overview of the essential algorithms of deep learning, backpropagation and stochastic gradient descent. We then survey some of the issues that occur when applying deep learning in practice. Last, we conclude with an important application of deep learning to the …

Nonequispaced Fast Fourier Transform, David Hughey

Theses and Dissertations

Two algorithms for fast and accurate evaluation of high degree trigonometric polynomials at many scattered points are presented. Both methods rely on highly localized kernels and the Fast Fourier Transform. The first algorithm uses the function values at uniformly distributed grid points and kernels that reproduce trigonometric polynomials, while the second method uses kernels that approximate well the function on the frequency side. Both algorithm are termed Nonequispaced Fast Fourier Transform. The first algorithm is coded in MATLAB and shown to approximate well the function to be evaluated.

Polynomials Of Small Mahler Measure With No Newman Multiples, Spencer Victoria Saunders

Theses and Dissertations

A Newman polynomial is a polynomial with coefficients in f0;1g and with constant term 1. It is known that the roots of a Newman polynomial must lie in the slit annulus fz 2C: f��1 1 such that if a polynomial f (z) 2 Z[z] has Mahler measure less than s and has no nonnegative real roots, then it must divide a Newman polynomial. In this thesis, we present a new upper bound on such a s if it exists. We also show that there are infinitely many monic polynomials that have distinct Mahler measures which all lie below f, have …

Subdivision Of Measures Of Squares, Dylan Bates

Theses and Dissertations

The primary goal of our work is to establish a method to relate simple measures to a given set of moments. We calculate the moments of squares via linear polynomial weight measures and straight line cuts and use this to calculate the centre of mass of the square. The one-to-one correspondence that is found is needed to represent surfaces with gaps, which can estimate arbitrary measures on squares. From this, a subdivision scheme is developed, which successively quadrisects squares and uses the relation to estimate the new measures in order to provide a good representation of the original surface. One …

Graph Invariants Of Trees With Given Degree Sequence, Rachel Bass

College of Graduate Studies: Theses & Dissertations

Graph invariants are functions defined on the graph structures that stay the same under taking graph isomorphisms. Many such graph invariants, including some commonly used graph indices in Chemical Graph Theory, are defined on vertex degrees and distances between vertices. We explore generalizations of such graph indices and the corresponding extremal problems in trees. We will also briefly mention the applications of our results.