Mathematics Commons^™

Professional Formation Of Engineers: Enhancing The First Year Student Experience, Eileen Goold

Conference Papers

This paper reports on a study investigating engineering students’ perceptions of engineering practice and whether engineering students’ cognitive engagement benefits from bridging the gap between the technical issues in their education and the practical realities of modern engineering practice.

The Least Prime Number That Splits Completely In S3-Sextic Number Fields, Zhenchao Ge

Electronic Theses and Dissertations

In number theory, an integer n is quadratic residue modulo an odd prime p if n is congruent to a perfect square modulo p. Otherwise, n is is called a quadratic nonresidue. Bounding the least prime quadratic residue and the least quadratic nonresidue are two very classical problems in number theory. These classical problems can be generalized to any number field K by asking for bounds the least for prime that splits completely or does not split completely, respectively, in the ring of integers of K. The goal of this thesis is to bound the least prime that splits completely …

Bias-Corrected Maximum Likelihood Estimation Of The Parameters Of The Weighted Lindley Distribution, Wentao Wang

Dissertations, Master's Theses and Master's Reports - Open

This report discusses the calculation of analytic second-order bias techniques for the maximum likelihood estimates (for short, MLEs) of the unknown parameters of the distribution in quality and reliability analysis. It is well-known that the MLEs are widely used to estimate the unknown parameters of the probability distributions due to their various desirable properties; for example, the MLEs are asymptotically unbiased, consistent, and asymptotically normal. However, many of these properties depend on an extremely large sample sizes. Those properties, such as unbiasedness, may not be valid for small or even moderate sample sizes, which are more practical in real data …

Neutrosophic Graphs: A New Dimension To Graph Theory, Florentin Smarandache, Wb. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time have made a through study of neutrosophic graphs. This study reveals that these neutrosophic graphs give a new dimension to graph theory. The important feature of this book is it contains over 200 neutrosophic graphs to provide better understanding of this concepts. Further these graphs happen to behave in a unique way inmost cases, for even the edge colouring problem is different from the classical one. Several directions and dimensions in graph theory are obtained from this study. Finally certainly these new notions of neutrosophic graphs in general and in particular the …

On Closed Subsets Of ℝ And Of ℝ2 Admitting Peano Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

In this note we describe closed subsets of the real line P ⊂ R for which there exists a continuous function from P onto P 2 , called Peano function. Our characterization of those sets is based on the number of connected components of P. We also include a few remarks on compact subsets of R 2 admitting Peano functions, expressed in terms of connectedness and local connectedness.

The Relationship Among Math Anxiety, Mathematical Performance, And Math Education In Undergraduate Nursing Students, Joshua D. Beall, Troy Roebuck, Paul Penkalsky

Williams Honors College, Honors Research Projects

Although nurses spend up to 40% of their day calculating and administering medication doses, undergraduate nursing students often perform poorly on nursing math exams. The purpose of this study was (a) to examine the relationship among mathematical education, performance, and anxiety and (b) to compare the mathematical education, performance, and anxiety in sophomore and senior baccalaureate nursing students at a public university in the Midwest. This cross-sectional, descriptive study was guided by Bandura's self-efficacy theory. Math performance was measured with an 11-item math instrument, math education was measured with number of math courses, and math anxiety was measured with Fennema–Sherman …

Parameters Estimation Of Material Constitutive Models Using Optimization Algorithms, Kiswendsida Jules Kere

Williams Honors College, Honors Research Projects

Optimization Algorithms are very useful for solving engineering problems. Indeed, optimization algorithms can be used to optimize engineering designs in terms of safety and economy. Understanding the proprieties of materials in engineering designs is very important in order to make designs safe. Materials are not really perfectly homogeneous and there are heterogeneous distributions in most materials. In this paper, Self-OPTIM which is an inverse constitutive parameter identification framework will be used to identify parameters of a linear elastic material constitutive model. Data for Self-OPTIM will be obtained using ABAQUS simulation of a dog-bone uniaxial test. Optimization Algorithms will be used …

A Thorough And Accessible Proof Of The Erdo˝S-Kac Theorem Following Granville And Soundararajan, Thomas John Scheithauer

Online Theses and Dissertations

The Erd\H{o}-Kac Theorem states that, as $n$ tends to infinity, the distribution of $\omega(n)$, the number of distinct prime divisors of $n$, becomes normally distributed with mean and variance $\log\log n$. Granville and Soundararajan gave a proof of the Erd\H{o}s-Kac Theorem that avoided many of the specialized techniques present in several earlier approaches. This thesis includes considerable detail, prerequisite theorems, and instructive background in order to provide a self-contained exposition of the proof of Granville and Soundararajan.

General Factoring Algorithms For Polynomials Over Finite Fields, Wade Combs

Online Theses and Dissertations

In this paper, we generate algorithms for factoring polynomials with coefficients in finite fields. In particular, we develop one deterministic algorithm due to Elwyn Berlekamp and one probabilistic algorithm due to David Cantor and Hans Zassenhaus. While some authors present versions of the algorithms that can only factor polynomials of a certain form, the algorithms we give are able to factor any polynomial over any finite field. Hence, the algorithms we give are the most general algorithms available for this factorization problem. After formulating the algorithms, we look at various ways they can be applied to more specialized inquiries. For …

Existence Of Positive Solutions To A Family Of Fractional Two-Point Boundary Value Problems, Christina Ashley Hollon

Online Theses and Dissertations

In this paper we will consider an nth order fractional boundary value problem with boundary conditions that include a fractional derivative at 1. We will develop properties of the Green's Function for this boundary value problem and use these properties along with the Contraction Mapping Principle, and the Schuader's, Krasnozel'skii's, and Legget-Williams fixed point theorems to prove the existence of positive solutions under different conditions.

Some Properties Of The Exchange Operator With Respect To Structured Matrices Defined By Indefinite Scalar Product Spaces, Hanz Martin C. Cheng, Roden Jason David

Mathematics Faculty Publications

The properties of the exchange operator on some types of matrices are explored in this paper. In particular, the properties of exc(A,p,q), where A is a given structured matrix of size (p+q)Ã(p+q) and exc : M ÃNÃN â M is the exchange operator are studied. This paper is a generalization of one of the results in [N.J. Higham. J-orthogonal matrices: Properties and generation. SIAM Review, 45:504â519, 2003.].

Cauchy Extension Of Mα-Integral And Absolute Mα-Integrability, Ian June L. Garces, Abraham P. Racca

Mathematics Faculty Publications

Park, Ryu, and Lee recently introduced a Henstock-type integral, which lies between the Mcshane and the Henstock integrals. This paper proves the closure property of this new integral under Cauchy extension, and presents a characterization on absolute Mα-integrability

Computing The Metric Dimension Of Truncated Wheels, Ian June L. Garces, Jose B. Rosario

Mathematics Faculty Publications

For an ordered subset W = {w1, w2, w3, . . . , wk} of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v|W) = (d(v, w1), d(v, w2), d(v, w3), . . . , d(v, wk)). The set W is called a resolving set of G if r(u|W) = r(v|W) implies u = v. The metric dimension of G, denoted by β(G), is the minimum cardinality of a resolving set of G. Let n ≥ 3 be an integer. A truncated wheel, denoted …

On Completely K-Magic Regular Graphs, Arnold A. Eniego, Ian June L. Garces

Mathematics Faculty Publications

Let k be a positive integer. A graph G = (V (G), E(G)) is said to be k-magic if there is a function (or edge labeling) ` : E(G) → Zk \ {0}, where Z1 = Z, such that the induced function (or vertex labeling) ` + : V (G) → Zk, defined by ` +(v) = P uv∈E(G) `(uv), is a constant map, where the sum is taken in Zk. We say that G is c-sum k-magic if ` +(v) = c for all v ∈ V (G). The set of all c ∈ Zk such that G is …

A Simpler Proof For The Ε-Δ Characterization Of Baire Class One Functions, Emmanuel A. Cabral, Jonald P. Fenecios

Mathematics Faculty Publications

We offer a new and simpler proof of a recent ϵ-δ characterization of Baire class one functions using a theorem by Henri Lebesgue. The proof is more elementary in the sense that it does not use the Baire Category Theorem. Furthermore, the proof requires only that the domain and range be separable metric spaces instead of Polish spaces.

The Forbidden Number Of A Knot, Alissa S. Crans, Blake Mellor, Sandy Ganzell

Mathematics, Statistics and Data Science Faculty Works

Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the forbid- den number. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.

A Stochastic Model For Microbial Fermentation Process Under Gaussian White Noise Environment, Yanping Ma

Mathematics, Statistics and Data Science Faculty Works

In this paper, we propose a stochastic model for the microbial fermentation process under the framework of white noise analysis, where Gaussian white noises are used to model the environmental noises and the specific growth rate is driven by Gaussian white noises. In order to keep the regularity of the terminal time, the adjustment factors are added in the volatility coefficients of the stochastic model. Then we prove some fundamental properties of the stochastic model: the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on the initial values.

Solving The Ko Labyrinth, Alissa Crans, Robert J. Rovetti, Jessica Vega

Mathematics, Statistics and Data Science Faculty Works

The KO Labyrinth is a colorful spherical puzzle with 26 chambers, some of which can be connected via holes through which a small ball can pass when the chambers are aligned correctly. The puzzle can be realigned by performing physical rotations of the sphere in the same way one manipulates a Rubik’s Cube, which alters the configuration of the puzzle. The goal is to navigate the ball from the entrance chamber to the exit chamber. We find the shortest path through the puzzle using Dijkstra’s algorithm and explore questions related to connectivity of puzzle with the adjacency matrix, distance matrix, …

Enhanced Lasso Recovery On Graph, Xavier Bresson, Thomas Laurent, James Von Brecht

Mathematics, Statistics and Data Science Faculty Works

This work aims at recovering signals that are sparse on graphs. Compressed sensing offers techniques for signal recovery from a few linear measurements and graph Fourier analysis provides a signal representation on graph. In this paper, we leverage these two frameworks to introduce a new Lasso recovery algorithm on graphs. More precisely, we present a non-convex, non-smooth algorithm that outperforms the standard convex Lasso technique. We carry out numerical experiments on three benchmark graph datasets.

Second Hamiltonian Cycles In Claw-Free Graphs, Hossein Esfandiari, Colton Magnant, Pouria Salehi Nowbandegani, Shirdareh Haghighi

Theory & Applications of Graphs

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sheehan’s conjecture holds for claw-free graphs whose order is not divisible by 6. In addition, we believe that the structure that we introduce can be useful for further studies on claw-free graphs.

Bounds For The Zero Forcing Number Of Graphs With Large Girth, Randy Davila, Franklin Kenter

Theory & Applications of Graphs

The zero-forcing number, Ζ(G) is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the graph. A simple lower bound is δ ≤ Ζ(G) where δ is the minimum degree. An improvement of this bound is provided in the case that G has girth of at least 5. In particular, it is shown that 2δ − 2 ≤ Ζ(G) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Lastly, a conjecture is made regarding a lower bound for Ζ(G) as a …

Scheduling N Burgers For A K-Burger Grill: Chromatic Numbers With Restrictions, Peter Johnson, Xiaoya Zha

Theory & Applications of Graphs

The chromatic number has a well-known interpretation in the area of scheduling. If the vertices of a finite, simple graph are committees, and adjacency of two committees indicates that they must never be in session simultaneously, then the chromatic number of the graph is the smallest number of hours during which the committees/vertices of the graph may all have properly scheduled meetings of one continuous hour each. Slivnik [3] showed that the fractional chromatic number can be similarly characterized. In that characterization, the meetings are allowed to be broken into a finite number of disjoint intervals. Here we consider chromatic …

Path-Tables Of Trees: A Survey And Some New Results, Kevin Asciak

Theory & Applications of Graphs

The (vertex) path-table of a tree Τ contains quantitative information about the paths in Τ. The entry (i,j) of this table gives the number of paths of length j passing through vertex ^vi. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree Τ does not determine Τ uniquely. We shall show that in trees, the number of paths passing through …

Connection And Separation In Hypergraphs, Mohammad A. Bahmanian, Mateja Sajna

Theory & Applications of Graphs

In this paper we study various fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. We prove a number of new results involving these concepts. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph.

Modeling And Computations Of Cellular Dynamics Using Complex-Fluid Models, Jia Zhao

Theses and Dissertations

Cells are fundamental units in all living organisms as all living organisms are made up of cells of different varieties. The study of cells is therefore an essential part of research in life science. Cells can be classified into two basic types: prokaryotic cells and eukaryotic cells. One typical organisms of prokaryotes is bacterium. And eukaryotes mainly consist of animal cells. In this thesis, we focus on developing predictive models mathematically to study bacteria colonies and animal cell mitotic dynamics.

Instead of living alone, bacteria usually survive in a biofilm, which is a microorganism where bacteria stick together by extracellular …

Toward The Combinatorial Limit Theory Of Free Words, Danny Rorabaugh

Theses and Dissertations

Free words are elements of a free monoid, generated over an alphabet via the binary operation of concatenation. Casually speaking, a free word is a finite string of letters. Henceforth, we simply refer to them as words. Motivated by recent advances in the combinatorial limit theory of graphs–notably those involving flag algebras, graph homomorphisms, and graphons–we investigate the extremal and asymptotic theory of pattern containment and avoidance in words.

Word V is a factor of word W provided V occurs as consecutive letters within W. W is an instance of V provided there exists a nonerasing monoid homomorphsism phi with …

Trees, Partitions, And Other Combinatorial Structures, Heather Christina Smith

Theses and Dissertations

This dissertation contains work on three main topics.

Chapters 1 through 4 provide complexity results for the single cut-or-join model for genome rearrangement. Genomes will be represented by binary strings. Let S be a finite collection of binary strings, each of the same length. Define M to be the collection of medians – binary strings μ which minimize Sigma v belongs to S H(μ,v) where H is the Hamming distance. For any non-negative function f(x), define Z(f(x), S) to be (Sigma μ belongs to M) (Pi v belongs to S)f(H(μ,v)). We study the complexity of calculating Z(f(x), S), with respect …

Avoiding Doubled Words In Strings Of Symbols, Michael Lane

Theses and Dissertations

A word on the n-letter alphabet is a finite length string of symbols formed from a set of n letters. A word is doubled if every letter that appears in the word appears at least twice. A word w avoids a word u if there is no non-erasing homomorphism h (a map that respects concatenation) such that h (u) is a subword of w. Finally, a word w is n-avoidable if there is an infinite list of words on the n-letter alphabet that avoid w. In 1906, Thue showed that the simplest doubled word, namely xx, is 3-avoidable. In 1984, …

Commutator Studies In Pursuit Of Finite Basis Results, Nathan E. Faulkner

Theses and Dissertations

Several new results of a general algebraic scope are developed in an effort to build tools for use in finite basis proofs. Many recent finite basis theorems have involved assumption of a finite residual bound, with the broadest result concerning varieties with a difference term (Kearnes, Szendrei, and Willard (2013+)). However, in varieties with a difference term, the finite residual bound hypothesis is known to strongly limit the degree of nilpotence observable in a variety, while, on the other hand, there is another, older series of results in which nilpotence plays a key role (beginning with those of Lyndon (1952) …

The Packing Chromatic Number Of Random D-Regular Graphs, Ann Wells Clifton

Theses and Dissertations

Let G = (V (G),E(G)) be a simple graph of order n and let i be a positive integer. X_i superset V (G) is called an i-packing if vertices in X_i are pairwise distance more than i apart. A packing coloring of G is a partition X = {X₁,X₂,X₃, . . . ,X_k} of V (G) such that each color class X_i is an i-packing. The minimum order k of a packing coloring is called the packing chromatic number of G, denoted by X_p(G). Let G_{n,d …}