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Full Issue - Engaging The World: Differential Equations Can Influence Public Policies

CODEE Journal

This is the full issue (front matter and all papers) of the Third CODEE Special Issue, with the theme, "Engaging the World: Differential Equations can Influence Public Policies."

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Nonlinear Filtering Of Classical And Quantum Spin Systems, Sivaguru S. Sritharan, Saba Mudaliar

Journal of Stochastic Analysis

No abstract provided.

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On The Singular Pebbling Number Of A Graph, Harmony R. Morris

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we define a new parameter of a connected graph as a spin-off of the pebbling number (which is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble). This new parameter is the singular pebbling number, the smallest t such that a player can be given any configuration of at least t pebbles and any target vertex and can successfully move pebbles so that exactly one pebble ends on the target vertex. We also prove that the singular pebbling number of any graph on 3 or more vertices is equal …

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Internal Parametricity, Without An Interval, Thorsten Altenkirch, Yorgo Chamoun, Ambrus Kaposi, Michael Shulman

Mathematics: Faculty Scholarship

Parametricity is a property of the syntax of type theory implying, e.g., that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven externally, and does not hold internally. Internalising it is difficult because once there is a term witnessing parametricity, it also has to be parametric itself and this results in the appearance of higher dimensional cubes. In previous theories with internal parametricity, either an explicit syntax for higher cubes is present or the theory is extended with a new sort for the interval. In this paper we present a type theory …

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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.

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Blue Whale And Krill Populations Modeling, Li Zhang

CODEE Journal

We present an intriguing topic in an undergraduate mathematical modeling course where predator-prey models are taught to our students. We describe modeling activities and the use of technology that can be implemented in teaching this topic. Through modeling activities, students are expected to use the numerical and graphical methods to observe the qualitative long-term behavior of predator and prey populations. Although there are other choices of predators and prey, we find that using blue whales and krill as predator and prey, respectively, would be most beneficial in strengthening our students' awareness of protecting endangered species and its impact on climate …

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Nonlinear Dynamics Of Mountain Pine Beetle Populations: Discussion Of Forestry Policy, A Survey Of Existing Mathematical Models, And Code Base Demonstration, Scott A. Strong, Maya Maes-Johnson

CODEE Journal

This article presents existing mathematical models associated with mountain pine beetle populations in lodgepole pine forests, whose reproductive cycle requires the destruction of colonized host trees, decreasing timber availability/quality, and providing fuel sources for wildfires. With the existence of a positive-feedback loop with environmental warming, the need for intervention and management is clear. However, the legislative responses to the focusing events from our 2000-2010 North American epidemics are characterized as under-leveraged. While the reasons for this are multifaceted, increasing the capacity of STEM-informed individuals to take part in quantitative modeling of the underlying ecosystem generates awareness and provides pathways connecting …

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Solar Panels, Euler’S Method And Community-Based Projects: Connecting Differential Equations With Climate Change, Victor J. Donnay

CODEE Journal

How does mathematics connect with the search for solutions to the climate emergency? One simple connection, which can be explored in an introductory differential equations course, can be found by analyzing the energy generated by solar panels or wind turbines. The power generated by these devices is typically recorded at standard time intervals producing a data set which gives a discrete approximation to the power function $P(t)$. Using numerical techniques such as Euler’s method, one can determine the energy generated. Here we describe how we introduce the topic of solar power, apply Euler’s method to determine the energy generated, and …

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To Open Or Not To Open: Developing A Covid-19 Model Specific To Small Residential Campuses, Christina Joy Edholm, Maryann Hohn, Nicole Lee Falicov, Emily Lee, Lily Natasha Wartman, Ami Radunskaya

CODEE Journal

In May 2020, administrators of residential colleges struggled with the decision of whether or not to open their campuses in the Fall semester of 2020. To help guide this decision, we formulated an ODE model capturing the dynamics of the spread of COVID-19 on a residential campus. In order to provide as much information as possible for administrators, the model accounts for the different behaviors, susceptibility, and risks in the various sub-populations that make up the campus community. In particular, we start with a traditional SEIR model and add compartments representing relevant variables, such as quarantine compartments and a hospitalized …

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Fitting A Covid-19 Model Incorporating Senses Of Safety And Caution To Local Data From Spartanburg County, South Carolina, D. Chloe Griffin, Amanda Mangum

CODEE Journal

Common mechanistic models include Susceptible-Infected-Removed (SIR) and Susceptible-Exposed-Infected-Removed (SEIR) models. These models in their basic forms have generally failed to capture the nature of the COVID-19 pandemic's multiple waves and do not take into account public policies such as social distancing, mask mandates, and the ``Stay-at-Home'' orders implemented in early 2020. While the Susceptible-Vaccinated-Infected-Recovered-Deceased (SVIRD) model only adds two more compartments to the SIR model, the inclusion of time-dependent parameters allows for the model to better capture the first two waves of the COVID-19 pandemic when surveillance testing was common practice for a large portion of the population. We find …

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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.

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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 …

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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.

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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 …

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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.

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

Michigan Tech Publications

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 …

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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 …

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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, …

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Pre-Calculus: Thinking Deeply About Simple Things, Jacob Carter

Graduate Research Showcase

“Pre-Calculus: Thinking Deeply About Simple Things” is a research-based creative endeavor focused on designing a high-school pre-calculus course. This course aims to foster deep, meaningful thinking, as well as an appreciation of the values of diversity, equity, and inclusion in the math classroom. The course leverages students’ funds of knowledge to employ culturally responsive teaching methods to connect mathematical concepts to the students’ backgrounds, interests, and real-life situations. This course also integrates social-emotional learning to create an engaging and supportive learning environment for all students. By combining Peter Liljedahl’s “Building Thinking Classroom in Mathematics” approach with problem-based learning, the course …

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Lipschitz Stability For Impulsive Riemann–Liouville Fractional Differential Equations, Martin Bohner, Snezhana Hristova

Mathematics and Statistics Faculty Research & Creative Works

Initial and impulsive conditions for initial value problems of systems of nonlinear impulsive Riemann–Liouville fractional differential equations are introduced. The case when the lower limit of the fractional derivative is changed at each time point of the impulses is studied. In the case studied, the solution has a singularity at the initial time and at any point of the impulses. This leads to the need to appropriately generalize the classical concept of Lipschitz stability. Two derivative types of Lyapunov functions are utilized in order to deduce sufficient conditions for the new stability concept. Three examples are provided for illustration purpose …

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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

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On A Fully Coupled Nonlocal Multipoint Boundary Value Problem For A Dual Hybrid System Of Nonlinear Q -Fractional Differential Equations, Ahmed Alsaedi, Martin Bohner, Bashir Ahmad, Boshra Alharbi

Mathematics and Statistics Faculty Research & Creative Works

A new class of nonlocal multipoint boundary value problems involving a dual hybrid system of nonlinear Riemann-Liouville-type q-fractional differential equations is studied in this paper. Existence and uniqueness results for the given problem are derived by applying the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle. Examples are presented for illustrating the obtained results. The work established in this paper is a useful contribution to the existing literature on q-fractional differential equations. Some interesting special cases are also discussed.

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Critical Point Approaches To Nonlinear Square Root Laplacian Equations, Martin Bohner, Giuseppe Caristi, Shapour Heidarkhani, Amjad Salari

Mathematics and Statistics Faculty Research & Creative Works

This work is devoted to the study of multiplicity results of solutions for a class of nonlinear equations involving the square root of the Laplacian. Indeed, we will use variational methods for smooth functionals, defined on reflexive Banach spaces, in order to achieve the existence of at least three solutions for the equations. Moreover, assuming that the nonlinear terms are nonnegative, we will prove that the solutions are nonnegative. Finally, by presenting an example, we will ensure the applicability of our results.

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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.

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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 …

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Computing The Roots Of Twisting Sheaves Over The Projective Line Arising From Monodromy Representations, Diego Yepez

Wayne State University Dissertations

Given a monodromy representation of the projective line minus a finite number of points, one can extend the resulting vector bundle with connection map canonically to a vector bundle with logarithmic connection map over all of the projective line. Now, since vector bundles split as twisting sheaves over the projective line, the focus of this work regards knowing the exact decomposition.

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Additive Energies Of Subsets Of Discrete Cubes, Xuancheng Shao

Mathematics Faculty Publications

For a positive integer n ≥ 2, define tn to be the smallest number such that the additive energy E (A) of any subset A ⊂ {0, 1, · · · , n − 1}d and any d is at most |A|tn . Trivially, we have tn ≤ 3 and tn ≥ 3 − logn 3n3 2n3 + n by considering A = {0, 1, · · · , n − 1}d. In this note, we investigate the behaviour of tn for large n and obtain the following non-trivial bounds: 3 − (1 + on→∞(1)) logn 3√3 4 ≤ tn …

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A Limit Order Book Model For High Frequency Trading With Rough Volatility, Yun S. Chen-Shue

Graduate Thesis and Dissertation 2023-2024

We introduce a financial model for limit order book with two main features: First, the limit orders and market orders for the given asset both appear and interact with each other. Second, the high frequency trading (HFT, for short) activities are allowed and described by the scaling limit of nearly-unstable multi-dimensional Hawkes processes with power law decay. The model eventually becomes a stochastic partial differential equation (SPDE, for short) with the diffusion coefficient determined by a Volterra integral equation governed by a Hawkes process, whose Hurst exponent is less than 1/2, which makes the volatility path of the stochastic PDE …

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On The Transmuted Distributions; Properties And Application, Jacob D. Kretzer

Theses, Dissertations and Capstones

The transmuted distributions first appeared in (2007) after Shaw and Buckley constructed a quadratic rank transmutation map (QRTM), G(u) = (1 + λ)u − λu², as a transformation of a cumulative distribution function of a random variable X, to generate the transmuted-X distribution. In (2017), Jayakumar & Babu defined the T -transmuted-X family of distributions by incorporating a transmuted-X into a transformer-transformed class of distributions (Aljarrah et al., 2014). This thesis surveys the main properties of the transmuted-X distribution, such as density shapes, moments, and entropy. Detailed attention will be given …

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

Undergraduate Research (Journal)

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 …

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Full-Text Articles in Entire DC Network

Full Issue - Engaging The World: Differential Equations Can Influence Public Policies

CODEE Journal

Nonlinear Filtering Of Classical And Quantum Spin Systems, Sivaguru S. Sritharan, Saba Mudaliar

Journal of Stochastic Analysis

On The Singular Pebbling Number Of A Graph, Harmony R. Morris

Rose-Hulman Undergraduate Mathematics Journal

Internal Parametricity, Without An Interval, Thorsten Altenkirch, Yorgo Chamoun, Ambrus Kaposi, Michael Shulman

Mathematics: Faculty Scholarship

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

CODEE Journal

Blue Whale And Krill Populations Modeling, Li Zhang

CODEE Journal

Nonlinear Dynamics Of Mountain Pine Beetle Populations: Discussion Of Forestry Policy, A Survey Of Existing Mathematical Models, And Code Base Demonstration, Scott A. Strong, Maya Maes-Johnson

CODEE Journal

Solar Panels, Euler’S Method And Community-Based Projects: Connecting Differential Equations With Climate Change, Victor J. Donnay

CODEE Journal

To Open Or Not To Open: Developing A Covid-19 Model Specific To Small Residential Campuses, Christina Joy Edholm, Maryann Hohn, Nicole Lee Falicov, Emily Lee, Lily Natasha Wartman, Ami Radunskaya

CODEE Journal

Fitting A Covid-19 Model Incorporating Senses Of Safety And Caution To Local Data From Spartanburg County, South Carolina, D. Chloe Griffin, Amanda Mangum

CODEE Journal

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

CODEE Journal

Modeling Aircraft Takeoffs, Catherine Cavagnaro

CODEE Journal

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

CODEE Journal

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

CODEE Journal

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

CODEE Journal

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

Michigan Tech Publications

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

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

Mathematics and Statistics Faculty Research & Creative Works

Pre-Calculus: Thinking Deeply About Simple Things, Jacob Carter

Graduate Research Showcase

Lipschitz Stability For Impulsive Riemann–Liouville Fractional Differential Equations, Martin Bohner, Snezhana Hristova

Mathematics and Statistics Faculty Research & Creative Works

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

On A Fully Coupled Nonlocal Multipoint Boundary Value Problem For A Dual Hybrid System Of Nonlinear Q -Fractional Differential Equations, Ahmed Alsaedi, Martin Bohner, Bashir Ahmad, Boshra Alharbi

Mathematics and Statistics Faculty Research & Creative Works

Critical Point Approaches To Nonlinear Square Root Laplacian Equations, Martin Bohner, Giuseppe Caristi, Shapour Heidarkhani, Amjad Salari

Mathematics and Statistics Faculty Research & Creative Works

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

Mathematics and Statistics Faculty Research & Creative Works

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

Theses, Dissertations and Capstones

Computing The Roots Of Twisting Sheaves Over The Projective Line Arising From Monodromy Representations, Diego Yepez

Wayne State University Dissertations

Additive Energies Of Subsets Of Discrete Cubes, Xuancheng Shao

Mathematics Faculty Publications

A Limit Order Book Model For High Frequency Trading With Rough Volatility, Yun S. Chen-Shue

Graduate Thesis and Dissertation 2023-2024

On The Transmuted Distributions; Properties And Application, Jacob D. Kretzer

Theses, Dissertations and Capstones

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

Undergraduate Research (Journal)