Mathematics Commons^™

Improved Definition Of Nonstandard Neutrosophic Logic And Introduction To Neutrosophic Hyperreals (Third Version), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

On the third version of this response-paper to Imamura’s criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995-1998) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy – otherwise they would be inapplicable. We point out several errors and false statements by Imamura with respect to the inf/sup of nonstandard …

Shape Resonances Of The Transverse Magnetic Mode In A Spherically Stratified Medium, Umaporn Nuntaplook, John A. Adam

Mathematics & Statistics Faculty Publications

Although morphology-dependent resonances (MDRs) have been studied for decades, it is interesting to note that TM resonances have not been as widely investigated as those of the TE mode. Nevertheless, the formers are also worthy of additional study. Even though the TE and TM mode resonances can be generated using the same technique, their properties (such as the additional sharp peak in the source function at the particle surface) are quite distinct. We present the derivation of the resonance formulations for TM mode for both increasing and decreasing piecewise-constant refractive index profiles in a two-layer model of a sphere embedded …

Bootcmatch: A Software Package For Bootstrap Amg Based On Graphweighted Matching, Pasqua D'Ambra, Salvatore Filipone, Panayot S. Vassilevski

Mathematics and Statistics Faculty Publications and Presentations

This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch, which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the …

On Saturation Numbers Of Ramsey-Minimal Graphs, Hunter M. Davenport

Honors Undergraduate Theses

Dating back to the 1930's, Ramsey theory still intrigues many who study combinatorics. Roughly put, it makes the profound assertion that complete disorder is impossible. One view of this problem is in edge-colorings of complete graphs. For forbidden graphs H1,...,Hk and a graph G, we write G "arrows" (H₁,...,H_k) if every k-edge-coloring of G contains a monochromatic copy of H_i in color i for some i=1,2,...,k. If c is a (red, blue)-edge-coloring of G, we say c is a bad coloring if G contains no red K₃or blue K …

I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, Jason L. Greuling

Honors Undergraduate Theses

Research has shown that a frame for an n-dimensional real Hilbert space offers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and sufficient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space does not ensure that its dilation will offer phase retrieval. In this thesis, we will explore and provide what necessary and sufficient …

Some Aggregation Operators For Bipolar-Valued Hesitant Fuzzy Information, Florentin Smarandache, Tahir Mahmood, Kifayat Ullah, Qaisar Khan

Branch Mathematics and Statistics Faculty and Staff Publications

In this article we define some aggregation operators for bipolar-valued hesitant fuzzy sets. These operations include bipolar-valued hesitant fuzzy ordered weighted averaging (BPVHFOWA) operator, bipolar-valued hesitant fuzzy ordered weighted geometric (BPVHFOWG) operator and their generalized forms. We also define hybrid aggregation operators and their generalized forms and solved a decision-making problem on these operation.

Existence Of Global Solutions For Nonlinear Magnetohydrodynamics With Finite Larmor Radius Corrections, Fariha Elsrrawi, Harumi Hattori

Faculty & Staff Scholarship

We discuss the existence of global solutions to the magnetohydrodynamics (MHD) equations, where the effects of finite Larmor radius corrections are taken into account. Unlike the usual MHD, the pressure is a tensor and it depends on not only the density but also the magnetic field. We show the existence of global solutions by the energy methods. Our techniques of proof are based on the existence of local solution by semigroups theory and a priori estimate.

Lipschitz Restrictions Of Continuous Functions And A Simple Construction Of Ulam-Zahorski C1 Interpolation, Krzysztof Ciesielski

Faculty & Staff Scholarship

We present a simple argument that for every continuous function f : R → R its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the C 1 free interpolation theorem, that for every continuous function f : R → R there exists a continuously differentiable function g : R → R which agrees with f on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with the Lebesgue measure theory.

Minimal Degrees Of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimates, Krzysztof Ciesielski

Faculty & Staff Scholarship

A rational function of the form x α1 1 x α2 2 ⋯x αn n x β1 1 +x β2 2 +⋯+x βn n is a Genocchi-Peano example, GPE, provided it is discontinuous, but its restriction to any hyperplane is continuous. We show that the minimal degree D(n) of a GPE of n-variables equals 2 ⌊ e 2 e2−1 n⌋ + 2i for some i ∈ {0, 1, 2}. We also investigate the minimal degree Db(n) of a bounded GPE of n-variables and note that D(n) ≤ Db(n) ≤ n(n + 1). Finding better bounds for numbers Db(n) remains an …

Bounded Point Derivations On Certain Function Spaces, Stephen Deterding

Theses and Dissertations--Mathematics

Let 𝑋 be a compact subset of the complex plane and denote by 𝑅^𝑝(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿^𝑝(𝑋) norm. We show that if a point 𝑥₀ admits a bounded point derivation on 𝑅^𝑝(𝑋) for 𝑝 > 2, then there is an approximate derivative at 𝑥₀. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for 𝑅(𝑋), the uniform closure of rational functions with poles off 𝑋. In addition, we show that if …

Polytopes Associated To Graph Laplacians, Marie Meyer

Theses and Dissertations--Mathematics

Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, P_G, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, P_D.

Basic properties of both families of simplices, P_G and P …

Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

Theses and Dissertations--Mathematics

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H¹-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, …

Alcoholism: A Mathematical Model With Media Awareness Campaigns, Erik H. Ander, Zeynep Teymuroglu

Undergraduate Mathematics Day: Past Content

In this paper, we study how media awareness campaigns influence the spread and persistence of drinking behavior in a community. Here, we present a compartmental population model with an additional differential equation to describe the dynamics of media awareness campaigns in combating problem drinking ([10], [12], [21]). Our model indicates a basic reproductive number, R0, where there exists an asymptotically stable drinking-free equilibrium if R0 < 1, and a unique endemic state, which appears to be stable when R0 > 1. We found that the following two components affect the basic reproductive number: the strength of peer influence of problem drinkers on susceptibles and the average overall time spent in the problem drinking environment. Furthermore, …

How One’S Risk Preferences Affect Their Investment Decisions, Kari Hayes, Anna Petrick

Undergraduate Mathematics Day: Past Content

The purpose of our project was to display how our personal risk preferences affect our investment decisions, if we invested on two assets: one risky asset (stock) and one risk-free asset (bank account). We considered the problem in both discrete and continuous case. In particular, the stock price follows a multinomial tree in the discrete case; and follows a Geometric Brownian motion in the continuous case. We then found the expected value of the stocks at varying times. By setting what we expect our bank account to be at those times equal to these expected values, we solved for the …

Finite Sum Representations Of Elements In R And R2, Lewis T. Dominguez, Rachelle R. Bouchat

Undergraduate Mathematics Day: Past Content

In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a master’s student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2.

Mathematics With Only Rods, Jianqiao Mao, Zheng Yang

Undergraduate Mathematics Day: Past Content

We discuss in this expository paper the rod system used in ancient China based on the mathematical classic work of Sun Zi, with a focus on application to solving systems of linear equations. The mathematics involved is authentic and beautiful, and we believe it is also of interest from historical, cultural, and pedagogical perspectives.

Generalized Catalan Numbers And Objects: X; Y Equivalence Classes And Polyominoes, Emily S. Dautenhahn, Hannah E. Pieper

Undergraduate Mathematics Day: Past Content

No abstract provided.

Baseball: Defense Or No?, Jacob D. Stemmerich

Undergraduate Mathematics Day: Past Content

Defense wins championships, or so they say. How do baseball organizations find the right defenders to win games? FanGraphs has published a series of metrics that teams throughout Major Leage Baseball use to quantify players’ fielding prowess. Baseball analysts use Wins Above Replacement, WAR, to predict who should be the league most valuable player, MVP. This uses defensive metrics to quantify how many runs the player produces when the team wins. The paper will discuss the metrics that already exist, and the technology that has been developed to analyze these metrics and other measurements of a player’s defensive skills.

Initial Value Problems For Caputo Fractional Differential Equations, Paul W. Eloe, Tyler Masthay

Mathematics Faculty Publications

Let n ≥ 1 denote an integer and let n - 1 < α ≤ n: We consider an initial value problem for a nonlinear Caputo fractional differential equation of order α and obtain results analogous to well known results for initial value problems for ordinary differential equations. These results include Picard’s existence and uniqueness theorem, Peano’s existence theorem, extendibility of solutions to the right, maximal intervals of existence, a Kamke type convergence theorem, and the continuous dependence of solutions on parameters. The nonlinear term is assumed to depend on higher order derivatives and solutions are obtained in the space of n - 1 times continuously differentiable functions.

Concavity In Fractional Calculus, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

No abstract provided.