Mathematics | Open Access Articles | Digital Commons Network™

Model Selection Through Cross-Validation For Supervised Learning Tasks With Manifold Data, Derek Brown

The Journal of Purdue Undergraduate Research

No abstract provided.

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Covariant Anyons Via Mackey Machinery, Radhakrishnan Balu

Journal of Stochastic Analysis

No abstract provided.

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Rad-⊕-Supplemented Semimodules Over Semirings, Ahmed H. Alwan

Al-Bahir Journal for Engineering and Pure Sciences

. In this paper, Rad-⊕-supplemented semimodules are defined as generalization of ⊕-supplemented semimodules. Let R be a semiring. An R-semimodule A is called a Rad-⊕-supplemented semimodule, if each subsemimodule of A has a Rad-supplement which is a direct summand of A. Here, we investigate some properties of these semimodules and generalize some results on Rad-⊕-supplemented modules to semimodules. We prove that any finite direct sum of Rad-⊕-supplemented semimodules is Rad-⊕-supplemented. Also, we prove that if A is a subtractive semimodule with (D₃) then A is Rad-⊕-supplemented if and only if every direct summand to A is …

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Construction Of Quot-Schemes, Majid Dehghani

Electronic Theses and Dissertations

The Quot Scheme is a construction representing parameter spaces for quotient objects of sheaves or coherent modules over a scheme. It encapsulates families of quotients by fixing a certain quotient's structure. The Hilbert Scheme, a specific type of Quot Scheme, focuses on parameterizing subschemes of a fixed projective space by fixing their Hilbert polynomials. After recalling the basic concepts of the theory, we explain the Grothendieck’s Quot scheme construction and its Grassmannian embedding. Then we continue to an explicit construction of Quot scheme in the case of graded modules over graded rings.

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Special Issue On Public Policy: Front Matter

CODEE Journal

The Front Matter contains the Editor-in-Chief's Foreword, a Dedicatory by Associate Editor Douglas Meade, a Preface by the Special Editors Bev West and Samer Habre, and the Table of Contents.

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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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On A Class Of James-Stein’S Estimators In High-Dimensional Data, Arash Aghaei Foroushani

Electronic Theses and Dissertations

In this thesis, we consider the estimation problem of the mean matrix of a multivariate normal distribution in high-dimensional data. Building upon the groundwork laid by Chételat and Wells (2012), we extend their method to the cases where the parameter is the mean matrix of a matrix normal distribution. In particular, we propose a novel class of James-Stein’s estimators for the mean matrix of a multivariate normal distribution with an unknown row covariance matrix and independent columns. Given a realistic assumption, we establish that our proposed estimator outperforms the classical maximum likelihood estimator (MLE) in the context of high-dimensional data. …

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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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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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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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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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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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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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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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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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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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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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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 …

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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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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 …

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From Normal Distribution To What? How To Best Describe Distributions With Known Skewness, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we only have partial information about the probability distribution -- e.g., all we know is its few moments. In such situations, it is desirable to select one of the possible probability distributions. A natural way to select a distribution from a given class of distributions is the maximum entropy approach. For the case when we know the first two moments, this approach selects the normal distribution. However, when we also know the third central moment -- corresponding to skewness -- a direct application of this approach does not work. Instead, practitioners use several heuristic techniques, techniques …

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Facilitating Mathematics And Computer Science Connections: A Cross-Curricular Approach, Kimberly E. Beck, Jessica F. Shumway, Umar Shehzad, Jody Clarke-Midura, Mimi Recker

Publications

In the United States, school curricula are often created and taught with distinct boundaries between disciplines. This division between curricular areas may serve as a hindrance to students' long-term learning and their ability to generalize. In contrast, cross-curricular pedagogy provides a way for students to think beyond the classroom walls and make important connections across disciplines. The purpose of this paper is a theoretical reflection on our use of Expansive Framing in our design of lessons across learning environments within the school. We provide a narrative account of our early work in using this theoretical framework to co-plan and enact …

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The Independence Polynomial Of A Graph At −1, Phoebe Rose Zielonka

Theses, Dissertations and Culminating Projects

No abstract provided.

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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.

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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.

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The Law Of The Iterated Logarithm For Lp-Norms Of Kernel Estimators Of Cumulative Distribution Functions, Fuxia Cheng

Faculty Publications – Mathematics

In this paper, we consider the strong convergence of Lp-norms (p ≥ 1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF.

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Full-Text Articles in Mathematics

Model Selection Through Cross-Validation For Supervised Learning Tasks With Manifold Data, Derek Brown

The Journal of Purdue Undergraduate Research

Covariant Anyons Via Mackey Machinery, Radhakrishnan Balu

Journal of Stochastic Analysis

Rad-⊕-Supplemented Semimodules Over Semirings, Ahmed H. Alwan

Al-Bahir Journal for Engineering and Pure Sciences

Construction Of Quot-Schemes, Majid Dehghani

Electronic Theses and Dissertations

Special Issue On Public Policy: Front Matter

CODEE Journal

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

On A Class Of James-Stein’S Estimators In High-Dimensional Data, Arash Aghaei Foroushani

Electronic Theses and Dissertations

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

CODEE Journal

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

CODEE Journal

Blue Whale And Krill Populations Modeling, Li Zhang

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

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

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

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

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

CODEE Journal

Modeling Aircraft Takeoffs, Catherine Cavagnaro

CODEE Journal

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

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, Part 2

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

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

Departmental Technical Reports (CS)

From Normal Distribution To What? How To Best Describe Distributions With Known Skewness, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Facilitating Mathematics And Computer Science Connections: A Cross-Curricular Approach, Kimberly E. Beck, Jessica F. Shumway, Umar Shehzad, Jody Clarke-Midura, Mimi Recker

Publications

The Independence Polynomial Of A Graph At −1, Phoebe Rose Zielonka

Theses, Dissertations and Culminating Projects

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

Physics: Faculty Publications and Other Works

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

Honors Theses and Capstones

The Law Of The Iterated Logarithm For Lp-Norms Of Kernel Estimators Of Cumulative Distribution Functions, Fuxia Cheng

Faculty Publications – Mathematics