Physical Sciences and Mathematics Commons^™

Bounds For The Augmented Zagreb Index, Ren Qingcuo, Li Wen, Suonan Renqian, Yang Chenxu

Theory and Applications of Graphs

The augmented Zagreb index (AZI for short) of a graph G, introduced by Furtula et al. in 2010, is defined as AZI(G)= Σ v_iv_j ∈ E(G)} (d(v_i)d(v_j)} {d(v_i)+d(v_j)-2)³, where E(G) is the edge set of G, and d(v_i) denotes the degree of the vertex v_i. In this paper, we give some new bounds on general connected graphs, molecular trees and triangle-free graphs.

New Diagonal Graph Ramsey Numbers Of Unicyclic Graphs, Richard M. Low, Ardak Kapbasov

Theory and Applications of Graphs

Grossman conjectured that R(G, G) = 2 ⋅ |V(G)| - 1, for all simple connected unicyclic graphs G of odd girth and |V(G)| ≥ 4. In this note, we prove his conjecture for various classes of G containing a triangle. In addition, new diagonal graph Ramsey numbers are calculated for some classes of simple connected unicyclic graphs of even girth.

Ts-Reconfiguration Of K-Path Vertex Covers In Caterpillars For K \Geq 4, Duc A. Hoang

Theory and Applications of Graphs

A k-path vertex cover (k-PVC) of a graph G is a vertex subset I such that each path on k vertices in G contains at least one member of I. Imagine that a token is placed on each vertex of a k-PVC. Given two k-PVCs I, J of a graph G, thek-Path Vertex Cover Reconfiguration (k-PVCR)} under Token Sliding (TS) problem asks if there is a sequence of k-PVCs between I and J where each intermediate member is obtained from its predecessor by sliding a token from some …

Ramsey Numbers For Connected 2-Colorings Of Complete Graphs, Mark Budden

Theory and Applications of Graphs

In 1978, David Sumner introduced a variation of Ramsey numbers by restricting to 2-colorings in which the subgraphs spanned by edges in each color are connected. This paper continues the study of connected Ramsey numbers, including the evaluation of several cases of trees versus complete graphs.

Over-Determined Control Systems On Time Scales, Danian Castillo, Drew Barnes, Nick Wintz

2023 Student Academic Showcase

Control theory is a branch of mathematics focused on observing or controlling a process governed by a dynamic equation. We use state-space notation to represent all meaningful information about our process. This means our processes are expressed in vector form. Typically, the state matrix is square. In this project, we consider a control system where the corresponding state is over-determined, meaning there are more rows than columns. In addition, our state equation is on a time scale T, which allows us to consider discrete, continuous, or hybrid measurements. Here, we offer two methods to solve the dynamic system. Finally, we …

Fast Computation Of Friction Stir Welding Process With Model Order Reduction And Machine Learning, Joshua Kay

Student Research Symposium

Joining aluminum alloys and other metal workpieces is a typical manufacturing process, which can be accomplished by various welding techniques. Friction stir welding (FSW), a relatively new technology patented in 1991, has many advantages over conventional welding processes. For the FSW process, it is desirable to determine an optimal set of parameters to avoid product defects in the joints. Modeling and simulating the FSW process is computationally expensive which creates a bottleneck in searching for optimal operating parameters. Reduced order modeling techniques will be used to significantly reduce computation time for FSW models while maintaining accuracy. In addition, machine learning …

How Daily Movement Impacts Elementary Students In Math, Lesley Mcmahan Lawson

Morehead State Theses and Dissertations

A capstone submitted in partial fulfillment of the requirements for the degree of Doctor of Education in the Ernst and Sara Lane Volgenau College of Education at Morehead State University by Lesley McMahan Lawson on April 11, 2023.

A Graphical User Interface Using Spatiotemporal Interpolation To Determine Fine Particulate Matter Values In The United States, Kelly M. Entrekin

Honors College Theses

Fine particulate matter or PM2.5 can be described as a pollution particle that has a diameter of 2.5 micrometers or smaller. These pollution particle values are measured by monitoring sites installed across the United States throughout the year. While these values are helpful, a lot of areas are not accounted for as scientists are not able to measure all of the United States. Some of these unmeasured regions could be reaching high PM2.5 values over time without being aware of it. These high values can be dangerous by causing or worsening health conditions, such as cardiovascular and lung diseases. Within …

Multilevel Optimization With Dropout For Neural Networks, Gary Joseph Saavedra

Mathematics & Statistics ETDs

Large neural networks have become ubiquitous in machine learning. Despite their widespread use, the optimization process for training a neural network remains com-putationally expensive and does not necessarily create networks that generalize well to unseen data. In addition, the difficulty of training increases as the size of the neural network grows. In this thesis, we introduce the novel MGDrop and SMGDrop algorithms which use a multigrid optimization scheme with a dropout coarsening operator to train neural networks. In contrast to other standard neural network training schemes, MGDrop explicitly utilizes information from smaller sub-networks which act as approximations of the full …

Nash Blowups Of Toric Varieties In Prime Characteristic, Daniel Duarte, Jack Jeffries, Luis Núñez-Betancourt

Department of Mathematics: Faculty Publications

We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.

Matroid Generalizations Of Some Graph Results, Cameron Crenshaw

LSU Doctoral Dissertations

The edges of a graph have natural cyclic orderings. We investigate the matroids for which a similar cyclic ordering of the circuits is possible. A full characterization of the non-binary matroids with this property is given. Evidence of the difficulty of this problem for binary matroids is presented, along with a partial result for binary orderable matroids.

For a graph G, the ratio of |E(G)| to the minimum degree of G has a natural lower bound. For a matroid M that is representable over a finite field, we generalize this to a lower bound on …

Arma Model Development And Analysis For Global Temperature Uncertainty, Mahmud Hasan, Gauree Wathodkar, Mathias Muia

Faculty and Student Publications

Temperature uncertainty models for land and sea surfaces can be developed based on statistical methods. In this paper, we developed a novel time-series temperature uncertainty model, which is the autoregressive moving average (ARMA) (1,1) model. The model was developed for an observed annual mean temperature anomaly X(t), which is a combination of a true (latent) global anomaly Y(t) for a year (t) and normal variable w(t). The uncertainty is taken as the variance of w(t), which was divided into land surface temperature (LST) uncertainty, sea surface temperature (SST) uncertainty, and the corresponding source of uncertainty. The ARMA …

Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson

LSU Doctoral Dissertations

We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an …

Commutators On Fock Spaces, Daniel Alpay, Paula Cerejeiras, Uwe Kähler, Trevor Kling

Mathematics, Physics, and Computer Science Faculty Articles and Research

Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.

Classification Of Seven-Dimensional Solvable Lie Algebras With A5,2 And A5,3 Nilradicals, Robert Dolan

Honors Projects

This paper provides a classification of seven-dimensional indecomposable solvable Lie algebras over the reals for which the nilradical is isomorphic to A_5,2 and A_5,3. We follow a technique that was first introduced by Mubarakzyanov.

Applying Hallgren’S Algorithm For Solving Pell’S Equation To Finding The Irrational Slope Of The Launch Of A Billiard Ball, Sangheon Choi

Mathematical Sciences Technical Reports (MSTR)

This thesis is an exploration of Quantum Computing applied to Pell’s equation in an attempt to find solutions to the Billiard Ball Problem. Pell’s equation is a Diophantine equation in the form of x² − ny² = 1, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. We will be applying Hallgren’s algorithm for finding irrational periods in functions, in the context of billiard balls and their movement on a friction-less unit square billiard table. Our central research question has been the following: Given the cutting sequence of the billiard …

A New Approach To Proper Orthogonal Decomposition With Difference Quotients, Sarah Locke Eskew, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

In a Recent Work (Koc Et Al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the Authors Showed that Including Difference Quotients (DQs) is Necessary in Order to Prove Optimal Pointwise in Time Error Bounds for Proper Orthogonal Decomposition (POD) Reduced Order Models of the Heat Equation. in This Work, We Introduce a New Approach to Including DQs in the POD Procedure. Instead of Computing the POD Modes using All of the Snapshot Data and DQs, We Only Use the First Snapshot Along with All of the DQs and Special POD Weights. We Show that This Approach Retains All of the …

Rank-Based Inference For Survey Sampling Data, Akim Adekpedjou, Huybrechts F. Bindele

Mathematics and Statistics Faculty Research & Creative Works

For regression models where data are obtained from sampling surveies, the statistical analysis is often based on approaches that are either non-robust or inefficient. The handling of survey data requires more appropriate techniques, as the classical methods usually result in biased and inefficient estimates of the underlying model parameters. This article is concerned with the development of a new approach of obtaining robust and efficient estimates of regression model parameters when dealing with survey sampling data. Asymptotic properties of such estimators are established under mild regularity conditions. To demonstrate the performance of the proposed method, Monte Carlo simulation experiments are …

Numerical Range Of Strictly Triangular Matrices Over Finite Fields, Ariel Russell

Mathematics Student Projects

In this paper, we investigate the numerical range of matrices over finite fields, particularly triangular matrices. We conjecture that all strictly triangular matrices over finite fields of dimension 3 or greater have a numerical range encompassing the entirety of the finite field. We use both algebraic and computational methods to support this claim, making some concrete progress towards the algebraic proof. Further, we conjecture that all matrices over finite fields have a numerical range falling into one of five potential categories, providing an extensive appendix of randomly generated computational examples which seems to support this conjecture.

Everything Is A Matter Of Degree: The Main Idea Behind Fuzzy Logic Is Useful In Geosciences And In Authorship, Christian Servin, Aaron Velasco, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

This paper presents two applications of the general principle -- the everything is a matter of degree -- the principle that underlies fuzzy techniques. The first -- qualitative -- application helps explain the fact that while most earthquakes occur close to faults (borders between tectonic plates or terranes), earthquakes have also been observed in areas which are far away from the known faults. The second -- more quantitative -- application is to the problem of which of the collaborators should be listed as authors and which should be simply thanked in the paper. We argue that the best answer to …

Foundations Of Neural Networks Explain The Empirical Success Of The "Surrogate" Approach To Ordinal Regression -- And Recommend What Next, Salvador Robles, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Recently, a new efficient semi-heuristic statistical method -- called Surrogate Approach -- has been proposed for dealing with regression problems. How can we explain this empirical success? And since this method is only an approximation to reality, what can we recommend if there is a need for a more accurate approximation? In this paper, we show that this empirical success can be explained by the same arguments that explain the empirical success of neural networks -- and these arguments can also provide us with possible more general techniques (that will hopefully lead to more accurate approximation to real-life phenomena).

Integrity First, Service Before Self, And Excellence: Core Values Of Us Air Force Naturally Follow From Decision Theory, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

By analyzing data both from peace time and from war time, the US Air Force came with three principles that determine success: integrity, service before self, and excellent. We show that these three principles naturally follow from decision theory, a theory that describes how a rational person should make decisions.

Wormholes, Superfast Computations, And Selivanov's Theorem, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While modern computers are fast, there are still many practical problems that require even faster computers. It turns out that on the fundamental level, one of the main factors limiting computation speed is the fact that, according to modern physics, the speed of all processes is limited by the speed of light. Good news is that while the corresponding limitation is very severe in Euclidean geometry, it can be more relaxed in (at least some) non-Euclidean spaces, and, according to modern physics, the physical space is not Euclidean. The differences from Euclidean character are especially large on micro-level, where quantum …

What Do Goedel's Theorem And Arrow's Theorem Have In Common: A Possible Answer To Arrow's Question, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Kenneth Arrow, the renowned author of the Impossibility Theorem that explains the difficulty of group decision making, noticed that there is some commonsense similarity between his result and Goedel's theorem about incompleteness of axiomatic systems. Arrow asked if it is possible to describe this similarity in more precise terms. In this paper, we make the first step towards this description. We show that in both cases, the impossibility result disappears if we take into account probabilities. Namely, we take into account that we can consider probabilistic situations, that we can make probabilistic conclusions, and that we can make probabilistic decisions …

K-8 Preservice Teachers’ Statistical Thinking When Determining Best Measure Of Center, Ha Nguyen, Eryn M. Stehr Maher, Gregory Chamblee, Sharon Taylor

Department of Mathematical Sciences Faculty Publications

The purpose of this study was to determine K-8 preservice teacher (PST) candidates’ statistical thinking when selecting the best center representation for the given data. Forty-four PSTs enrolled in a Statistics and Probability for K-8 Teachers course in a university located in the southeastern region of the United States were asked to complete a 2007 National Assessment of Educational Progress test item. All 44 PSTs’ data were qualitatively analyzed for correctness and statistical thinking strategies used. Findings were that most PSTs either incorrectly selected the mean, rather than median, as the best measure of center for the given data or …

Extremal Absorbing Sets In Low-Density Parity-Check Codes, Emily Mcmillon, Allison Beemer, Christine A. Kelley

Department of Mathematics: Faculty Publications

Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of b for a given a for which an (a, b)-absorbing set may exist. We identify certain cases of extremal …

A Change-Point Analysis Of Air Pollution Levels In Silao, Mexico And Fresno, California, Rachael Goodwin

WWU Honors College Senior Projects

We analyzed PM10 levels in the city of Silao, Mexico, as well as PM2.5 and PM10 levels in Fresno, California to determine if there was a shift in air pollution levels in either location. A change point based analysis was used to determine if there was a shift in air pollution levels. In the city of Silao, there was a significant increase in PM10 levels, but there was no significant change in Fresno for either pollutant.

Structure Of Extremal Unit Distance Graphs, Kaylee Weatherspoon

Senior Theses

This thesis begins with a selective overview of problems in geometric graph theory, a rapidly evolving subfield of discrete mathematics. We then narrow our focus to the study of unit-distance graphs, Euclidean coloring problems, rigidity theory and the interplay among these topics. After expounding on the limitations we face when attempting to characterize finite, separable edge-maximal unit-distance graphs, we engage an interesting Diophantine problem arising in this endeavor. Finally, we present a novel subclass of finite, separable edge-maximal unit distance graphs obtained as part of the author's undergraduate research experience.

Regularity Theory Of Quasilinear Ellipitic And Parabolic Equations In The Heisenberg Group, Luca Capogna, Giovanna Citti, Xiao Zhong

Mathematics Sciences: Faculty Publications

This note provides a succinct survey of the existing literature concerning the H¨older regularity for the gradient of weak solutions of PDEs of the form modeled on the p-Laplacian in a domain Ω in the Heisenberg group Hⁿ, with 1 ≤ p < ∞, and of its parabolic counterpart. We present some open problems and outline some of the difficulties they present.

Using Visual Imagery To Develop Multiplication Fact Strategies, Gina Kling

Dissertations

The learning of basic facts, or the sums and products of numbers 0–10 and their related differences and quotients, has always been a high priority for elementary school teachers. While memorization of basic facts has been a hallmark of elementary school, current recommendations focus on a more nuanced development of fluency with these facts. Fluency is characterized by the ability to demonstrate flexibility, accuracy, efficiency, and appropriate strategy use. Despite recommendations to focus on strategy use, there is insufficient information on instructional approaches that are effective for developing strategies, particularly for multiplication facts. Using visual imagery with dot patterns has …