Physical Sciences and Mathematics Commons^™

Applying The Sir Model: Can Students Advise The Mayor Of A Small Community?, Carrin Goosen, Mark I. Nelson, Mahime Watanabe

CODEE Journal

This is an account of a modelling scenario that uses the sir epidemic model. It was used in a third year applied mathematics subject. All students were enrolled in a mathematics degree of some type. Students are presented with the results of a test carried out on 100 individuals in a community containing 3000 people. From this they determined the number of infectious and recovered individuals in the population. Given the per capita recovery rate and making a suitable assumption about the number of infectious individuals at the start of the epidemic, they then estimate the infectious contact rate and …

Differential Equations For A Changing World:How To Engage Students In Learning And Applying Differential Equations, Biyong Luo

CODEE Journal

In this article, I share my decade-long experience teaching an intensive five-week summer Differential Equation course covering complex topics and tips for creating an interactive and supportive learning environment to optimize student engagement. This article provides my detailed approach to planning and teaching an asynchronous course with rigor and flexibility for each student. An interactive teaching approach and variety of learning activities will augment students’ mathematical fluency and appreciation of the importance of differential equations in modeling a wide variety of real-world situations with special attention to ways differential equations can be relevant to creating public policy.

Modeling Aircraft Takeoffs, Catherine Cavagnaro

CODEE Journal

Real-world applications can demonstrate how mathematical models describe and provide insight into familiar physical systems. In this paper, we apply techniques from a first-semester differential equations course that shed light on a problem from aviation. In particular, we construct several differential equations that model the distance that an aircraft requires to become airborne. A popular thumb rule that pilots have used for decades appears to emanate from one of these models. We will see that this rule does not follow from a representative model and suggest a better method of ensuring safety during takeoff. Aircraft safety is definitely a matter …

Raising Student Awareness Of Environmental Issues Via Writing Assignments With Differential Equations, Michelle L. Ghrist

CODEE Journal

In this paper, I discuss two environmentally-focused writing assignments that I developed and implemented in recent integral calculus and differential equations courses. These models of carbon storage and PCB’s in a river provide interesting applications of one-compartment mixing problems. The assignments were intended to focus student attention on sustainability concerns while also developing other essential skills. I discuss these assignments and their effect on my students’ technical writing and environmental awareness. Detailed introductory instructions and mostly complete solutions to these assignments appear in the appendices, to include sample student work.

Ode Models Of Wealth Concentration And Taxation, Bruce Boghosian, Christoph Borgers

CODEE Journal

We refer to an individual holding a non-negligible fraction of the country’s total wealth as an oligarch. We explain how a model due to Boghosian et al. can be used to explore the effects of taxation on the emergence of oligarchs. The model suggests that oligarchs will emerge when wealth taxation is below a certain threshold, not when it is above the threshold. The underlying mechanism is a transcritical bifurcation. The model also suggests that taxation of income and capital gains alone cannot prevent the emergence of oligarchs. We suggest several opportunities for students to explore modifications of the model.

Using A Sand Tank Groundwater Model To Investigate A Groundwater Flow Model, Christopher Evrard, Callie Johnson, Michael A. Karls, Nicole Regnier

CODEE Journal

A Sand Tank Groundwater Model is a tabletop physical model constructed of plexiglass and filled with sand that is typically used to illustrate how groundwater water flows through an aquifer, how water wells work, and the effects of contaminants introduced into an aquifer. Mathematically groundwater flow through an aquifer can be modeled with the heat equation. We will show how a Sand Tank Groundwater Model can be used to simulate groundwater flow through an aquifer with a no flow boundary condition.

Integrating External Controls By Regression Calibration For Genome-Wide Association Study, Lirong Zhu, Shijia Yan, Xuewei Cao, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications, Part 2

Genome-wide association studies (GWAS) have successfully revealed many disease-associated genetic variants. For a case-control study, the adequate power of an association test can be achieved with a large sample size, although genotyping large samples is expensive. A cost-effective strategy to boost power is to integrate external control samples with publicly available genotyped data. However, the naive integration of external controls may inflate the type I error rates if ignoring the systematic differences (batch effect) between studies, such as the differences in sequencing platforms, genotype-calling procedures, population stratification, and so forth. To account for the batch effect, we propose an approach …

Time Scale Theory On Stability Of Explicit And Implicit Discrete Epidemic Models: Applications To Swine Flu Outbreak, Gülşah Yeni, Elvan Akın, Naveen K. Vaidya

Mathematics and Statistics Faculty Research & Creative Works

Time scales theory has been in use since the 1980s with many applications. Only very recently, it has been used to describe within-host and between-hosts dynamics of infectious diseases. In this study, we present explicit and implicit discrete epidemic models motivated by the time scales modeling approach. We use these models to formulate the basic reproduction number, which determines whether an outbreak occurs, or the disease dies out. We discuss the stability of the disease-free and endemic equilibrium points using the linearization method and Lyapunov function. Furthermore, we apply our models to swine flu outbreak data to demonstrate that the …

On A Multivalued Prescribed Mean Curvature Problem And Inclusions Defined On Dual Spaces, Vy Khoi Le

Mathematics and Statistics Faculty Research & Creative Works

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, …

Integrable Symplectic Maps With A Polygon Tessellation, T. Zolkin, Y. Kharkov, S. Nagaitsev

Physics Faculty Publications

Identifying integrable dynamics remains a formidable challenge, and despite centuries of research, only a handful of examples are known to date. In this article, we explore a distinct form of area-preserving (symplectic) mappings derived from the stroboscopic Poincaré cross section of a kicked rotator—an oscillator subjected to an external force periodically switched on in short pulses. The significance of this class of problems extends to various applications in physics and mathematics, including particle accelerators, crystallography, and studies of chaos. Notably, Suris's theorem constrains the integrability within this category of mappings, outlining potential scenarios with analytic invariants of motion. In this …

Recommendations To Internal Auditors Regarding The Auditing And Attestation Of Mathematical Programming Models, Jose Rincón, Greg Akai, Daryl Ono

LMU Librarian Publications & Presentations

Mathematical programming planning models increase operational efficiency and minimize operating costs, but the underlying mathematics generally is complex. Combinatorial optimization is technically sophisticated which requires a strong quantitative background to successfully implement. Most internal auditors will not have the technical training to critically assess the underlying mathematics of mathematical programming planning models, but the internal auditor can still provide insight and attestation which can increase the efficiency of mathematical programming planning models.

Every Feasibly Computable Reals-To-Reals Function Is Feasibly Uniformly Continuous, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that every computable function is continuous; moreover, it is computably continuous in the sense that for every ε > 0, we can compute δ > 0 such that δ-close inputs lead to ε-close outputs. It is also known that not all functions which are, in principle, computable, can actually be computed: indeed, the computation sometimes requires more time than the lifetime of the Universe. A natural question is thus: can the above known result about computable continuity of computable functions be extended to the case when we limit ourselves to feasible computations? In this paper, we prove that this …

A Spiral Workbook For Discrete Mathematics 2nd Edition, Harris Kwong

Milne Open Textbooks

This updated text covers the standard topics in a sophomore-level course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions, relations, and elementary combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a final polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: many topics are revisited multiple times, sometimes from a different perspective or at a higher level of complexity. The goal is …

Mixed Mechanisms Of Multi-Site Phosphorylation, Suha Jayyousi Dajani

Graduate Research Theses & Dissertations

Multi-site phosphorylation is an important mechanism in cell biology that regulates protein function and activity. It also plays a critical role in a wide variety of cellular processes that control intra-cellular signaling. Studies on mono- and dual-site phosphorylation have been conducted theoretically and experimentally by researchers. However, there is little research on triple-site and multi-site mixed mechanism phosphorylation.

The aim of this research is to study and identify the number of positive steady states (multistationarity) in a mathematical model of a triple-site mixed mechanism, processive and distributive, phosphorylation network. This research is accomplished by means of ordinary differential equations, Jacobian …

Conventions, Definitions, Identities, And Other Useful Formulae, Robert A. Mcnees Iv

Physics: Faculty Publications and Other Works

As the name suggests, these notes contain a summary of important conventions, definitions, identities, and various formulas that I often refer to. They may prove useful for researchers working in General Relativity, Supergravity, String Theory, Cosmology, and related areas.

Classification In Supervised Statistical Learning With The New Weighted Newton-Raphson Method, Toma Debnath

Electronic Theses and Dissertations

In this thesis, the Weighted Newton-Raphson Method (WNRM), an innovative optimization technique, is introduced in statistical supervised learning for categorization and applied to a diabetes predictive model, to find maximum likelihood estimates. The iterative optimization method solves nonlinear systems of equations with singular Jacobian matrices and is a modification of the ordinary Newton-Raphson algorithm. The quadratic convergence of the WNRM, and high efficiency for optimizing nonlinear likelihood functions, whenever singularity in the Jacobians occur allow for an easy inclusion to classical categorization and generalized linear models such as the Logistic Regression model in supervised learning. The WNRM is thoroughly investigated …

Numerical Investigation And Statistical Analysis Of The Flow Patterns Behind Square Cylinders Arranged In A Staggered Configuration Utilizing The Lattice Boltzmann Method, M. Abid, N. Yasin, M. Saqlain, S. Ul-Islam, S. Ahmad

Mathematics & Statistics Faculty Publications

Flow past bluff bodies like square cylinders is important in engineering applications, but flow patterns behind staggered cylinder arrangements remain poorly understood. Existing studies have focused on tandem or side-by-side configurations, while offset orientations have received less attention. The aim of this paper is to numerically investigate flow dynamics and force characteristics behind two offset square cylinders using the single relaxation time lattice Boltzmann method. The effects of changing both the Reynolds number (Re = 1-150) and gap spacing ratio (g* = 0.5-5) between the cylinders are analyzed. Instantaneous vorticity contours, time histories of drag and lift coefficients, power spectra …

Ookami: An A64fx Computing Resource, A. C. Calder, E. Siegmann, C. Feldman, S. Chheda, Dennis C. Smolarski Sj, F. D. Swesty, A. Curtis, J. Dey, D. Carlson, B. Michalowicz, R. J. Harrison

Mathematics and Computer Science

We present a look at Ookami, a project providing community access to a testbed supercomputer with the ARM-based A64FX processors developed by a collaboration between RIKEN and Fujitsu and deployed in the Japanese supercomputer Fugaku. We provide an overview of the project and details of the hardware, and describe the user base and education/training program. We present highlights from previous performance studies of two astrophysical simulation codes and present a strong scaling study of a full 3D supernova simulation as an example of the the machine’s capability.

Existence Of Solutions By Coincidence Degree Theory For Hadamard Fractional Differential Equations At Resonance, Martin Bohner, Alexander Domoshnitsky, Seshadev Padhi, Satyam Narayan Srivastava

Mathematics and Statistics Faculty Research & Creative Works

Using the Coincidence Degree Theory of Mawhin and Constructing Appropriate Operators, We Investigate the Existence of Solutions to Hadamard Fractional Differential Equations (FRDEs) at Resonance

Developing Machine Learning And Time-Series Analysis Methods With Applications In Diverse Fields, Muhammed Aljifri

Theses and Dissertations

This dissertation introduces methodologies that combine machine learning models with time-series analysis to tackle data analysis challenges in varied fields. The first study enhances the traditional cumulative sum control charts with machine learning models to leverage their predictive power for better detection of process shifts, applying this advanced control chart to monitor hospital readmission rates. The second project develops multi-layer models for predicting chemical concentrations from ultraviolet-visible spectroscopy data, specifically addressing the challenge of analyzing chemicals with a wide range of concentrations. The third study presents a new method for detecting multiple changepoints in autocorrelated ordinal time series, using the …

Post Developmental Mathematics: Experiences In College Algebra For Stem Students, Maria Cruciani

Undergraduate Research

Students majoring in a STEM discipline whose sequence of collegiate mathematics begins at the developmental level follow a unique progression towards degree completion. With an elongated sequence of mathematics courses, these students have already had exposure to collegiate mathematics when enrolling in a college algebra course. A structured multiple case study provided a context for understanding students’ perceptions about how their developmental mathematics experiences may have influenced their experiences in college algebra. Qualitative data was gathered through interviews with three students who are majoring in a STEM field of study. The selected students had similar quantitative literacy expectations for their …

Edge Colored And Edge Ordered Graphs, Per Gustin Wagenius

Graduate College Dissertations and Theses

In this work, the current state of the field of edge-colored graphs is surveyed. Anew concept of unshrinkable edge colorings is introduced which is useful for rainbow subgraph problems and interesting in its own right. This concept is analyzed in some depth. Building upon the linear edge ordering described in a recent work from Gerbner, Methuku, Nagy, Pálvölgyi, Tardos, and Vizer, edge-ordering graphs with the cyclic group is introduced and some results are given on this and a related counting problem.

Open Diameter Maps On Suspensions, Hussam Abobaker, Włodzimierz J. Charatonik, Robert Paul Roe

Mathematics and Statistics Faculty Research & Creative Works

It is shown that if X is a metric continuum, which admits an open diameter map, then the suspension of X, admits an open diameter map. As a corollary, we have that all spheres admit open diameter maps.

Educators’ Beliefs About Using Academic Acceleration With Gifted Math Students And Others: Barriers And Opportunities, Jason Gorgia

Theses, Dissertations and Capstones

This study examined the perceptions of educators (i.e., math teachers, administrators, and others) for insight into the absence of acceleration as a common pedagogical strategy in mathematics, despite longstanding research supporting the practice for students gifted in math and the interest frequently articulated by policymakers and educators in boosting American K-12 students’ math achievement. Educators from 48 states responded to scale-based and open-ended questions about math acceleration through an online survey where 713 of 818 respondents were teachers, balanced almost evenly among elementary, middle, and high schools, and among urban, suburban, and rural settings. The responses of teachers and non-teaching …

Persistent Relative Homology For Topological Data Analysis, Christian J. Lentz

Mathematics, Statistics, and Computer Science Honors Projects

A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via the language of persistent homology, which encodes features of interest as holes within a filtration of the data. The recently presented U-Match Decomposition places the standard persistence computation in a flexible form, allowing for straight-forward extensions of the algorithm to variations of persistent homology. We describe U-Match Decomposition in the context of persistent homology, and extend it to an algorithm for persistent relative homology, providing proofs for the correctness and stability of the presented algorithm.

Festival Of Research Abstracts, 2024, College Of Science And Mathematics, Wright State University

Festival of Research

The collection of abstracts accepted for the 2024 Festival of Research hosted by the Wright State University College of Science and Mathematics.

Counting Conjugates Of Colored Compositions, Jesus Omar Sistos Barron

Honors College Theses

The properties of n-color compositions have been studied parallel to those of regular compositions. The conjugate of a composition as defined by MacMahon, however, does not translate well to n-color compositions, and there is currently no established analogous concept. We propose a conjugation rule for cyclic n-color compositions. We also count the number of self-conjugates under these rules and establish a couple of connections between these and regular compositions.

Instances Of Undecidability In The Semigroup Word Problem, Timothy C. Grosky

Honors Theses and Capstones

We will examine the decidability of the word problem in semigroups, which is a yes/no question. We will examine tools that have been developed to help answer it, and then look at some examples where the word problem is decidable or undecidable.

Examining The Relationship Between Students’ Home Environment And Math Test Scores, Autumn Michlig

Master of Science in Mathematics

This research study examined the relationship between a student’s home environment and their STAR math score by investigating 9 home environment variables. It also examines the best model to predict STAR score from the 9 home environment variables while controlling for the covariate, prior STAR score. The results being that including the covariate in the 9 predictor model was a statistically significant improvement over not including the covariate. However, the best model to predict STAR score was a model that only included the covariate and one variable, number of people living in house(s). Furthermore, when investigating the 9 variables, it …

An Expository Analysis Of The Theorem: Every Integer Greater Than Two Is The Sum Of A Prime And A Square-Free Number, Kurt Mckenzie

Master of Science in Mathematics

The thesis focuses on a relatively recent theorem (2017) contributed by Adrian Dudek to the field of Number Theory. It states that every integer greater than two can be represented as a sum of a prime and a square-free number. This result immediately attracts attention since it bears similarities with the famous and not yet proved Goldbach’s Conjecture, namely that every even number larger than two can be represented as a sum of two primes. The expository analysis aims to understand the theorem in the general setting of the Additive Number Theory, the method of proof and previous results on …