Physical Sciences and Mathematics Commons^™

Homomorphism Obstructions For Satellite Maps, Allison N. Miller

Mathematics & Statistics Faculty Works

A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S³. This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides ± 1. We then provide subtle examples of satellite maps which map arbitrarily deep into the n-solvable filtration …

Benford’S Law And Its Applications To Accounting, Lucy Wilson

Mathematics Student Projects

In this paper, we introduce the concept of Benford’s Law, which is the mathematical observation that in many naturally occurring numerical datasets, the leading digits are not evenly distributed. We actually find that smaller leading digits appear more often than larger ones. We will explore when and why this phenomenon occurs and how we can use statistical tests to determine how well a dataset conforms to Benford’s Law. We will also see how Benford’s Law can be used by auditors in the accounting field to catch potential fraud.

The Traveling Salesman Problem At Taylor University, Jonathan Jinoo Pawley

Mathematics Student Projects

What is the shortest route to walk to every residence hall on campus, beginning and ending with the same hall? This question can be considered by applying the Traveling Salesman Problem, an easy to understand yet hard to solve problem in the realm of discrete combinatorial optimization. The Traveling Salesman Problem is useful as an introduction to optimization problems, and it also has immensely practical applications. This paper will serve as an introduction to the computational difficulty of the Traveling Salesman Problem and will also explore various approximation algorithms. We will subsequently apply our new understanding of the theory to …

Time As A Line: Helping Children Make Abstract Ideas Concrete, Rachel Mae Stenner

WWU Honors College Senior Projects

This is a math education project that included research, a lesson plan, and actual in the classroom work with students. Under the advisement of Dr. Rebecca Borowski, I looked into how time, an abstract idea, is taught to young children who are just starting to learn what measurement is, and examined how teachers can better teach time as a more concrete topic. This focused on the idea of turning the abstract time concepts that are thrown at children into the more abstract ideas of both circular and then linear number lines, using physical materials to help guide the process.

Math 115: College Algebra For Pre-Calculus, Seth Lehman

Open Educational Resources

OER course syllabus for Math 115, College Algebra, at Queens College

Just-In-Accuracy: Mobile Approach To Uncertainty, Martine Ceberio, Christoph Q. Lauter, Vladik Kreinovich

Departmental Technical Reports (CS)

To make a mobile device last longer, we need to limit computations to a bare minimum. One way to do that, in complex control and decision making problems, is to limit precision with which we do computations, i.e., limit the number of bits in the numbers' representation. A problem is that often, we do not know with what precision should we do computations to get the desired accuracy of the result. What we propose is to first do computations with very low precision, then, based on these computations, estimate what precision is needed to achieve the given accuracy, and then …

The Encyclopedia Of Neutrosophic Researchers, 5th Volume, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy.

In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements.

There are about 7,000 neutrosophic researchers, within 89 countries around the …

Rigid Body Constrained Motion Optimization And Control On Lie Groups And Their Tangent Bundles, Brennan S. Mccann

Doctoral Dissertations and Master's Theses

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive …

Elementary Mathematics Curriculum: State Policy, Covid-19, And Teachers’ Control, Mona Baniahmadi, Bima Sapkota, Amy M. Olson

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In the U.S., state guidance to schools in response to the COVID-19 pandemic was politicized. We used state-level political affiliation to explore whether access to curricular resources differed pre-pandemic or during pandemic remote teaching and teachers' reported control over curricular resources during pandemic teaching. We found that pre-pandemic the percentage of teachers in Republican states reported higher levels of resources overall, and use of core and teacher-created curricular resources in particular. They also reported having greater control over their curricular decision-making during the pandemic. There were no state-level differences in teachers’ level of preparation for pandemic teaching, but teachers in …

Semidefinite Programming Bounds For Distance Distribution Of Spherical Codes, Oleg R. Musin

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We present an extension of known semidefinite and linear programming upper bounds for spherical codes. We apply the main result for the distance distribution of a spherical code and show that this method can work effectively In particular, we get a shorter solution to the kissing number problem in dimension 4.

Rough Numbers And Variations On The Erdős--Kac Theorem, Kai Fan

Dartmouth College Ph.D Dissertations

The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory.

In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y>=1. Denoting by Phi(x,y) the count of these numbers up to any given x>=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)<0.6x/\log y holds uniformly for all 3<=y<=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges.

In the second part, we turn …

Figured Worlds Of Women Mathematics Education Scholars, Lili Zhou, Ricki L. Geller-Mckee, Brooke Max, Hyunyi Jung, Bima Sapkota, Jill Newton, Lindsay M. Keazer

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Drawing on the concept of figured worlds (Holland et al., 1998), this project focuses on addressing, responding to, and understanding the self within the figured world of the mathematics education community. Specifically, we examine a group of women with diverse backgrounds in terms of race, class, and cultural contexts, who are engaged in various roles as mathematics education scholars, including teachers, teacher educators, and researchers. Using a dialogical self approach, we facilitate both internal and external discourses, exploring personal histories, narratives, and the development of evolving identities. Our findings reveal that culture and social positions, such as gender, class, and …

On Explicit Soliton Solutions And Blow-Up For Coupled Variable Coefficient Nonlinear Schrödinger Equations, Jose Escorcia, Erwin Suazo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This work is concerned with the study of explicit solutions for a generalized coupled nonlinear Schrödinger equations (NLS) system with variable coefficients. Indeed, we show, employing similarity transformations, the existence of Rogue wave and dark-bright soliton like-solutions for such a generalized NLS system, provided the coefficients satisfy a Riccati system. As a result of the multiparameter solution of the Riccati system, the nonlinear dynamics of the solution can be controlled. Finite-time singular solutions in the L∞ norm for the generalized coupled NLS system are presented explicitly. Finally, an n-dimensional transformation between a variable coefficient NLS coupled system and a constant …

Convergence Of The Two-Point Modulus-Based Matrix Splitting Iteration Method, Ximing Fang, Ze Gu, Zhijun Qiao

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we discuss the convergence of the two-point modulus-based matrix splitting iteration method for solving the linear complementarity problem. Some convergence conditions are presented from the spectral radius and the matrix norm when the system matrix is a -matrix. Besides, the quasi-optimal cases of the method are studied. Numerical experiments are provided to show the presented results.

A Second Homotopy Group For Digital Images, Gregory Lupton, Oleg R. Musin, Nicholas A. Scoville, P. Christopher Staecker, Jonathan Treviño-Marroquín

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere.

Rogue Waves In The Massive Thirring Model, Junchao Chen, Bo Yang, Bao-Feng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index- and two dimension-ones are proved to be consistent by only one constraint relation on parameters of tau-functions of the KP-Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled …

K-Distinct Lattice Paths, Eric J. Yager, Marcus Engstrom

Rose-Hulman Undergraduate Mathematics Journal

Lattice paths can be used to model scheduling and routing problems, and, therefore, identifying maximum sets of k-distinct paths is of general interest. We extend the work previously done by Gillman et. al. to determine the order of a maximum set of k-distinct lattice paths. In particular, we disprove a conjecture by Gillman that a greedy algorithm gives this maximum order and also refine an upper bound given by Brewer et. al. We illustrate that brute force is an inefficient method to determine the maximum order, as it has time complexity O(n^k).

Excursions In Vector Calculus, Diego Castano

Mathematics Colloquium Series

Vector calculus is an invaluable tool in much of physics – electromagnetism is a prime example. The use of vector calculus is highlighted in an exploration of the concept of inductance and a reconsideration of its calculation. A form of the standard equation for inductance that is more versatile is derived and applied in some examples.

Utilizing Graph Thickness Heuristics On The Earth-Moon Problem, Robert C. Weaver

Rose-Hulman Undergraduate Mathematics Journal

This paper utilizes heuristic algorithms for determining graph thickness in order to attempt to find a 10-chromatic thickness-2 graph. Doing so would eliminate 9 colors as a potential solution to the Earth-moon Problem. An empirical analysis of the algorithms made by the author are provided. Additionally, the paper lists various graphs that may or nearly have a thickness of 2, which may be solutions if one can find two planar subgraphs that partition all of the graph’s edges.

Using Assessments To Promote Growth Mindset In College Algebra, Hannah M. Lewis, Kady Schneiter, David Lane Tait

Mathematics and Statistics Faculty Publications

Scientific evidence highlights the positive impact of a growth mindset on student achievement. Students with a growth mindset view errors and obstacles as opportunities for growth and welcome challenges and the opportunity to learn from their mistakes. Much has been written about promoting growth mindset through lectures and attitudes, however, assessments can also be an important avenue for encouraging a growth mindset in students. In this paper, we describe how we used assessments to promote growth mindset in a college algebra class. In the sections that follow, we discuss the need for these assessments and the principles that underly their …

Study Of Multilayer Flow Of Two Immiscible Nanofluids In A Duct With Viscous Dissipation, Jawali C. Umavathi, Mahanthesh Basavarajappa

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Numerical simulations for the mixed convective multilayer flow of two different immiscible nanofluids in a duct with viscous heating effects were performed in this study. The left and right faces of the duct are maintained to be isothermal, while other side faces are insulated. The mathematical governing system for each layer consists of an incompressibility condition equation, the Navier–Stokes momentum equation, and the conservation of energy equation. At the interface of the immiscible layer, the continuity of velocity, shear stress, temperature, and heat flux are considered. The dimensionless equations governing each layer were numerically integrated using the finite difference method …

Adjunction Greatest Element To Ordered Hypersemigroups, Niovi Kehayopulu

Turkish Journal of Mathematics

As a continuation of the paper "Adjunction Identity to Hypersemigroup" in Turk J Math 2022; 46 (7): 2834--2853, it has been proved here that the adjunction of a greatest element to an ordered hypersemigroup is actually an embedding problem. The concept of pseudoideal has been introduced and has been proved that for each ordered hypersemigroup $S$ an ordered hypersemigroup $V$ having a greatest element ($poe$-hypersemigroup) can be constructed in such a way that there exists a pseudoideal $T$ of $S$ such that $S$ is isomorphic to $T$. If $S$ does not have a greatest element, then this can be regarded …

On The Eigenstructure Of The $Q$-Durrmeyer Operators, Övgü Gürel Yilmaz

Turkish Journal of Mathematics

The purpose of this paper is to establish the eigenvalues and the eigenfunctions of both the $q$-Durrmeyer operators $D_{n,q}$ and the limit $q$-Durrmeyer operators $D_{\infty,q}$ introduced by V. Gupta in the case 0<$q$<1. All moments for $D_{n,q}$ and $D_{\infty,q}$ are provided. The coefficients for the eigenfunctions of the operators are explicitly derived and the eigenfunctions of these operators are illustrated by graphical examples.

Higher Topological Complexity Of A Map, Cesar Augusto Ipanaque Zapata, Jesús González

Turkish Journal of Mathematics

The higher topological complexity of a space $X$, $\text{TC}_r(X)$, $r=2,3,\ldots$, and the topological complexity of a map $f$, $\text{TC}(f)$, have been introduced by Rudyak and Pavesiç, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map $f$, $\text{TC}_{r,s}(f)$, for $1\leq s\leq r\geq2$, which simultaneously extends Rudyak's and Pavesiç notions. Our unified concept is relevant in the $r$-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in $s$ prescribed stages of the motion task. We study the homotopy …

Operators Affiliated To Banach Lattice Properties And Their Enveloping Norms, Eduard Emelyanov, Svetlana Gorokhova

Turkish Journal of Mathematics

Several recent papers were devoted to various modifications of limited, Grothendieck, and Dunford-Pettis operators, etc., through involving the Banach lattice structure. In the present paper, it is shown that many of these operators appear as operators affiliated to well-known properties of Banach lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck property, the property (d), the sequential w$^\ast$-continuity of the lattice operations, etc. We also introduce new classes of operators such as the s-GPP-operators, s-BDP-operators, and bi-sP-operators. It is proved that the spaces consisting of regular versions of the above-mentioned operators are all the Banach spaces. The domination problem …

Extended Calculus On ${\Cal O}({\Mathbb C}_{H}^{1\Vert1})$, Sali̇h Çeli̇k

Turkish Journal of Mathematics

We give an extended calculus over the function algebra on $h$-deformed superplane. For this, we extend the $(h_1,h_2)$-deformed differential calculus on the $h$-deformed superplane by adding inner derivations. We reformulate the results with an $R$-matrix and present the tensor product realization of the wedge product. We also discuss Cartan calculus via a contraction.

Fractional Semilinear Neumann Problem With Critical Nonlinearity, Zhenfeng Jin, Hongrui Sun

Turkish Journal of Mathematics

In this paper, we consider the following critical fractional semilinear Neumann problem \begin{equation*} \begin{cases} (-\Delta)^{1/2}u+\lambda u=u^{\frac{n+1}{n-1}},~u>0\quad&\, \mathrm{in}\ \Omega,\\ \partial_\nu{u}=0 &\mathrm{on}\ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^n~(n\geq5)$ is a smooth bounded domain, $\lambda>0$ and $\nu$ is the outward unit normal to $\partial\Omega$. We prove that there exists a constant $\lambda_0>0$ such that the above problem admits a minimal energy solution for $\lambda<\lambda_0$. Moreover, if $\Omega$ is convex, we show that this solution is constant for sufficiently small $\lambda$.

On The Weak And Strong Solutions Of The Velocity-Vorticity Model Of The $G$-Navier-Stokes Equations, Özge Kazar, Meryem Kaya

Turkish Journal of Mathematics

In this work, we consider a velocity-vorticity formulation for the $g$-Navier-Stokes equations. The system is constructed by combining the velocity-pressure system which is included by using the rotational formulation of the nonlinearity and the vorticity equation for the $g$ -Navier-Stokes equations. We prove the existence and uniqueness of weak and strong solutions of this system with the periodic boundary conditions.

Liouville-Type Theorem For One-Dimensional Porous Medium Systems With Sources, Anh Tuan Duong

Turkish Journal of Mathematics

In this paper, we are concerned with the one-dimensional porous medium system with sources \begin{align*} \begin{cases}u_t-( u^m)_{xx} =a_{11}u^{p}+a_{12} u^rv^{r+m}, (x,t)\in J\times I\subset\mathbb{R}\times \mathbb{R}\\ v_t-(v^m)_{xx} =a_{21} u^{r+m}v^{r}+a_{22}v^{p},\;(x,t)\in J\times I\subset \mathbb{R}\times \mathbb{R}, \end{cases} \end{align*} where $p=2r+m$, $m>1$, $r>0$. Under the conditions $a_{12}\geq 0, a_{21}\geq 0$, $a_{11}>0$, and $a_{22}>0$, we prove that the system does not possess any nontrivial nonnegative weak solution.

On The Monoid Of Partial Isometries Of A Cycle Graph, Vitor H. Fernandes, Tania Paulista

Turkish Journal of Mathematics

In this paper we consider the monoid $DPC_n$ of all partial isometries of an $n$-cycle graph $C_n$. We show that $DPC_n$ is the submonoid of the monoid of all oriented partial permutations on an $n$-chain whose elements are precisely all restrictions of a dihedral group of order $2n$. Our main aim is to exhibit a presentation of $DPC_n$. We also describe Green's relations of $DPC_n$ and calculate its cardinality and rank.