Mathematics Commons^™

On The Set Chromatic Number Of The Join And Comb Product Of Graphs, Bryan Ceasar L. Felipe, Agnes Garciano, Mark Anthony C. Tolentino

Mathematics Faculty Publications

A vertex coloring c : V(G) → of a non-trivial connected graph G is called a set coloring if NC(u) ≠ NC(v) for any pair of adjacent vertices u and v. Here, NC(x) denotes the set of colors assigned to vertices adjacent to x. The set chromatic number of G, denoted by χs (G), is defined as the fewest number of colors needed to construct a set coloring of G. In this paper, we study the set chromatic number in relation to two graph operations: …

Sigma Chromatic Number Of Graph Coronas Involving Complete Graphs, Agnes Garciano, Maria Czarina T. Lagura, Reginaldo M. Marcelo

Mathematics Faculty Publications

Let c : V(G) → be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic …

The N-Integral, Abraham P. Racca, Emmanuel A. Cabral

Mathematics Faculty Publications

In this paper, we introduced a Henstock-type integral named $N$-integral of a real valued function $f$ on a closed and bounded interval $[a,b]$. The $N$-integrable functions lie entirely between Riemann integrable functions and Henstock integrable functions. It was shown that for a Henstock integrable function $f$ on $[a,b]$ the following are equivalent: \begin{enumerate} \item[$(1)$] The function $f$ is $N$-integrable; \item[$(2)$] There exists a null set $S$ for which given $\epsilon >0$ there exists a gauge $\delta$ such that for any $\delta$-fine partial division $D=\{(\xi,[u,v])\}$ of $[a,b]$ we have \[(\phi_S(D)\cap \Gamma_{\epsilon})\sum |f(v)-f(u)||v-u|<\epsilon\] where $\phi_S(D)=\{(\xi,[u,v])\in D:\xi \notin S\}$ and \[\Gamma_{\epsilon}=\{(\xi,[u,v]): |f(v)-f(u)|\geq \epsilon\}\] \end{enumerate} and \begin{enumerate} \item[$(3)$] The function $f$ is continuous almost everywhere. \end{enumerate} A characterization of continuous almost everywhere functions was also given.

Designing Mobile Apps To Promote Numeracy And Statistical Reasoning, Ma. Louise Antonette N. De Las Peñas, Mark L. Loyola, Jumela F. Sarmiento, Mark Anthony C. Tolentino, Debbie Marie Versoza

Mathematics Faculty Publications

Developing countries typically do not perform well in international benchmarks of mathematics achievement. This may be partially explained by students’ immersion in classrooms characterized by superficial strategies or rote-learning methods. This paper reports on the design of mobile applications (apps) developed by the authors as part of an ongoing project funded by a national government agency and intended to promote structural thinking and statistical reasoning. It describes the general features of the apps, as well as the pedagogical principles upon which the apps’ designs were anchored on. These principles are grounded on research and established practices on number sense and …

Using Mechanics Of A Double Pendulum To Maximize Sport Performance, Heston Neal

Bridges: A Journal of Student Research

Sports have developed over time into an entire industry from the children who grow up playing sports to the professional players that fans love to watch. Because all sports involve movements of the human body, a bio-mechanical system, these movements may be modeled with mathematics and physics. Specifically, the sports of golf, tennis, and baseball can use the concepts of a double pendulum to maximize sport performance. The double pendulum model is applied to these sports by making the arm one pendulum and the club, racquet, or bat the second pendulum. A good understanding of the mechanics of a double …

On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers

HMC Senior Theses

Interpolation Jack polynomials are certain symmetric polynomials in N variables with coefficients that are rational functions in another parameter k, indexed by partitions of length at most N. Introduced first in 1996 by F. Knop and S. Sahi, and later studied extensively by Sahi, Knop-Sahi, and Okounkov-Olshanski, they have interesting connections to the representation theory of Lie algebras. Given an interpolation Jack polynomial we would like to differentiate it with respect to the variable k and write the result as a linear combination of other interpolation Jack polynomials where the coefficients are again rational functions in k. In this …

Catch Me If You Can: Using A Threshold Model To Simulate Support For Presidential Candidates In The Invisible Primary, Elizabeth A. Stiles, Colin D. Swearingen, Linda Seiter, Brendan Foreman

2020 Faculty Bibliography

The invisible primary is an important time in United States Presidential primary politics as candidates gain momentum for their campaigns before they compete formally in the first state caucus (Iowa) and primaries (e.g. New Hampshire). This critical period has not been possible to observe, hence the name. However, by simulating networks of primary followers, we can explicate hypotheses for how messages travel through networks to affect voter preferences. To do so, we use a threshold model to drive our simulated network analysis testing spread of public support for candidates in invisible primaries. We assign voter thresholds for candidates and vary …

Support For The Habitat Amount Hypothesis From A Global Synthesis Of Species Density Studies, James I. Watling, Rebecca Fang

2020 Faculty Bibliography

Decades of research suggest that species richness depends on spatial characteristics of habitat patches, especially their size and isolation. In contrast, the habitat amount hypothesis predicts that (1) species richness in plots of fixed size (species density) is more strongly and positively related to the amount of habitat around the plot than to patch size or isolation; (2) habitat amount better predicts species density than patch size and isolation combined, (3) there is no effect of habitat fragmentation per se on species density and (4) patch size and isolation effects do not become stronger with declining habitat amount. Data on …

A Tutorial Of Bland Altman Analysis In A Bayesian Framework, Krissina M. Alari, Steven B. Kim, Jeffrey O. Wand

Mathematics and Statistics Faculty Publications and Presentations

There are two schools of thought in statistical analysis, frequentist, and Bayesian. Though the two approaches produce similar estimations and predictions in large-sample studies, their interpretations are different. Bland Altman analysis is a statistical method that is widely used for comparing two methods of measurement. It was originally proposed under a frequentist framework, and it has not been used under a Bayesian framework despite the growing popularity of Bayesian analysis. It seems that the mathematical and computational complexity narrows access to Bayesian Bland Altman analysis. In this article, we provide a tutorial of Bayesian Bland Altman analysis. One approach we …

Where The Wild Knots Are, Forest Kobayashi

HMC Senior Theses

The new work in this document can be broken down into two main parts. In the first, we introduce a formalism for viewing the signed Gauss code for virtual knots in terms of an action of the symmetric group on a countable set. This is achieved by creating a "standard unknot" whose diagram contains countably-many crossings, and then representing tame knots in terms of the action of permutations with finite support. We present some preliminary computational results regarding the group operation given by this encoding, but do not explore it in detail. To make the encoding above formal, we require …

Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly

Electronic Theses and Dissertations

Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based …

Existence And Multiplicity Of Weak Solutions For A Neumann Elliptic Problem With -Laplacian, Martin Bohner, Giuseppe Caristi, Fariba Gharehgazlouei, Shapour Heidarkhani

Mathematics and Statistics Faculty Research & Creative Works

We are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic -Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti-Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to …

The Mobius Function And Mobius Inversion, Carl Lienert

Number Theory

No abstract provided.

Use Of Kalman Filtering In State And Parameter Estimation Of Diabetes Models, Cassidy Le

HMC Senior Theses

Diabetes continues to affect many lives every year, putting those affected by it at higher risk of serious health issues. Despite many efforts, there currently is no cure for diabetes. Nevertheless, researchers continue to study diabetes in hopes of understanding the disease and how it affects people, creating mathematical models to simulate the onset and progression of diabetes. Recent research by David J. Albers, Matthew E. Levine, Andrew Stuart, Lena Mamykina, Bruce Gluckman, and George Hripcsak¹ has suggested that these models can be furthered through the use of Data Assimilation, a regression method that synchronizes a model with a …

Pascal's Mystic Hexagon In Tropical Geometry, Hanna Hoffman

HMC Senior Theses

Pascal's mystic hexagon is a theorem from projective geometry. Given six points in the projective plane, we can construct three points by extending opposite sides of the hexagon. These three points are collinear if and only if the six original points lie on a nondegenerate conic. We attempt to prove this theorem in the tropical plane.

An Exploration Of Combinatorial Interpretations For Fibonomial Coefficients, Richard Shapley

HMC Senior Theses

We can define Fibonomial coefficients as an analogue to binomial coefficients as F(n,k) = F_nF_n-1 … F­_n-k+1 / F­_kF_k-­1…F₁, where F_n represents the nth Fibonacci number. Like binomial coefficients, there are many identities for Fibonomial coefficients that have been proven algebraically. However, most of these identities have eluded combinatorial proofs.

Sagan and Savage (2010) first presented a combinatorial interpretation for these Fibonomial coefficients. More recently, Bennett et al. (2018) provided yet another interpretation, that is perhaps more tractable. However, there still has been little progress towards using these interpretations …

A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo

HMC Senior Theses

Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a precise, clear development of …

The Wedge Family Of The Cohomology Of The C-Motivic Steenrod Algebra, Hieu Trung Thai

Wayne State University Dissertations

Computing the stable homotopy groups of the sphere spectrum is one of the most important problems of stable homotopy theory. Focusing on the 2-complete stable homotopy groups instead of the integral homotopy groups, the Adams spectral sequence appears to be one of the most effective tools to compute the homotopy groups. The spectral sequence has been studied by J. F. Adams, M. Mahowald, M. Tangora, J. P. May and others.

In 1999, Morel and Voevodsky introduced motivic homotopy theory. One of its consequences is the realization that almost any object studied in classical algebraic topology could be given a motivic …

Variational Analysis In Second-Order Cone Programming And Applications, Hang Thi Van Nguyen

Wayne State University Dissertations

This dissertation conducts a second-order variational analysis for an important class on nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone. These second-order cone programs (SOCPs) are mathematically challenging due to the nonpolyhedrality of the underlying second-order cone while being important for various applications. The two main devices in our study are second epi-derivative and graphical derivative of the normal cone mapping which are proved to accumulate vital second-order information of functions/constraint systems under investigation. Our main contribution is threefold:

- proving the twice epi-differentiability of the indicator function of the second-order cone and of the augmented Lagrangian associated with …

The Chebyshev Difference Equation, Tom Cuchta

Mathematics Faculty Research

We define and investigate a new class of difference equations related to the classical Chebyshev differential equations of the first and second kind. The resulting “discrete Chebyshev polynomials” of the first and second kind have qualitatively similar properties to their continuous counterparts, including a representation by hypergeometric series, recurrence relations, and derivative relations.

A Description Of A Humans Knowledge Using Artificial Intelligence, Dj Price

Mahurin Honors College Capstone Experience/Thesis Projects

There currently does not exist a way to easily view the relationships between a collection of written items (e.g. sports articles, diary entries, research papers). In recent years, novel machine learning methods have been developed which are very good at extracting semantic relationships from large numbers of documents. One of them is the (unsupervised) machine learning model Doc2Vec which constructs vectors for documents. The research project detailed in this paper uses this and other already existing algorithms to analyze the relationship between pieces of text. We set forth a broader ambition for this project before discussing the use and need …

The Mathematics, Computer Science, And Data Science Student Research Showcase, Seton Hall University

Petersheim Academic Exposition

No abstract provided.

Petersheim Academic Exposition 2020 Math & Computer Science Abstracts, Seton Hall University

Petersheim Academic Exposition

No abstract provided.

Algebraic And Combinatorial Approaches For Counting Cycles Arising In Population Biology, Brian Chau

Honors Undergraduate Theses

Within population biology, models are often analyzed for the net reproduction number or other generalized target reproduction numbers, which describe the growth or decline of the population based on specific mechanisms. This is useful in determining the strength and efficiency of control measures for inhibiting or enhancing population growth. The literature contains many algebraic and combinatorial approaches for deriving the net reproduction number and generalized target reproduction numbers from digraphs and associated matrices. Finding, categorizing, and counting the permutations of disjoint cycles, or cycles unions is a requirement of the Cycle Union approach by Lewis et al. (2019). These cycles …

Preliminary Study Of Highway Pavement And Materials, Omer Eljairi

CGU Theses & Dissertations

This preliminary study covered (a) the effects of in-place air voids and other factors on fatigue cracking using Long-Term Pavement Performance data, (b) fracture properties of asphalt concrete in a semicircular bend (SCB) test using a noncontact camera and crosshead movement, and (c) hot applied modified-binder-chip-seal field performance in California. The objective is to improve pavement performance and life, establish a quality assurance/quality control (QA/QC) tests of fracture properties of asphalt mixtures, and save millions of dollars on maintenance and rehabilitation. Chapter 1 investigated the effect of in-place air voids (AV), asphalt content (AC), bulk-specific gravity (BSG), and maximum specific …

Comparative Survival Of Asian And White Metastatic Castration-Resistant Prostate Cancer Men Treated With Docetaxel, Susan Halabi, Sandipan Dutta, Catherine M. Tangen, Mark Rosenthal, Daniel P. Petrylak, Ian M. Thompson Jr., Kim N. Chi, Johann S. De Bono, John C. Araujo, Christopher Logothetis, Mario A. Eisenberger, David I. Quinn, Karim Fizazi, Michael J. Morris, Celestia S. Higano, Ian F. Tannock, Eric J. Small, William Kevin Kelly

Mathematics & Statistics Faculty Publications

There are few data regarding disparities in overall survival (OS) between Asian and white men with metastatic castration-resistant prostate cancer (mCRPC). We compared OS of Asian and white mCRPC men treated in phase III clinical trials with docetaxel and prednisone (DP) or a DP-containing regimen. Individual participant data from 8820 men with mCRPC randomly assigned on nine phase III trials to receive DP or a DP-containing regimen were combined. Men enrolled in these trials had a diagnosis of prostate adenocarcinoma. The median overall survival was 18.8 months (95% confidence interval [CI] = 17.4 to 22.1 months) and 21.2 months (95% …

Quantitative Analysis Of A Stochastic Seitr Epidemic Model With Multiple Stages Of Infection And Treatment, Olusegun M. Otunuga, Mobolaji O. Ogunsolu

Mathematics Faculty Research

We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered (SEITR) model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission, treatment and recovery rates. We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals. It is shown that the expected number of secondary infections produced (in the absence of noise) reduces as treatment is introduced into the population. By defining R_T,n and ℛ_T,n as the basic deterministic and stochastic reproduction …

Dynamic Gompertz Model, Tom Cuchta, Sabrina Streipert

Mathematics Faculty Research

After a brief introduction that includes some fundamentals of time scales, we lay the foundation for dynamic Gompertz models. We derive their unique solutions, present examples in the discrete, quantum, and mixed time scale settings, and we compare its behavior to the solution in the continuous time setting. A discussion of the results and open problems are addressed in the conclusion.

Gröbner Bases And Systems Of Polynomial Equations, Rachel Holmes

All Graduate Theses, Dissertations, and Other Capstone Projects

This paper will explore the use and construction of Gröbner bases through Buchberger's algorithm. Specifically, applications of such bases for solving systems of polynomial equations will be discussed. Furthermore, we relate many concepts in commutative algebra to ideas in computational algebraic geometry.

Investigating The Numerical Stability Of Using An Impedance Boundary Condition To Model Broadband Noise Scattering With Acoustic Liners, Michelle E. Rodio, Fang Q. Hu, Douglas M. Nark

Mathematics & Statistics Faculty Publications

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for achieving aircraft noise reduction and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model will be discussed in which the liner impedance …