Applied Mathematics Commons^™

Strength Of Lime Stabilized Pavement Materials: Possible Theoretical Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

When building a road, it is often necessary to strengthen the underlying soil layer. This strengthening is usually done by adding lime. There are empirical formulas that describe how the resulting strength depends on the amount of added lime. In this paper, we provide a theoretical explanation for these empirical formulas.

Fusion Of Probabilistic Knowledge As Foundation For Sliced-Normal Approach, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, it turns out to be efficient to use Sliced-Normal multi-D distributions, i.e., distributions for which the logarithm of the probability density function (pdf) is a polynomial -- -- to be more precise, it is a sum of squares of several polynomials. This class is a natural extension of normal distributions, i.e., distributions for which the logarithm of the pdf is a quadratic polynomial.

In this paper, we provide a possible theoretical explanation for this empirical success.

Expected Resurgences And Symbolic Powers Of Ideals, Eloísa Grifo, Craig Huneke, Vivek Mukundan

Department of Mathematics: Faculty Publications

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves

Mathematical Modelling Of The Tuberculosis Epidemiology, Ally Yeketi Ayinla

Student Works (2020-2029)

This project analyses the tuberculosis (TB) epidemic mathematically using compartmental modelling approach. Three models are presented to discuss drug susceptible and multi-drug resistant TB. The first model presented has 4 compartments; susceptible, exposed, infectious and recovered. The relevance of the exposed class in managing TB is analysed and found to be useful in delaying the eventual onset of the infection. Compared to previous researches, our results significantly show that when efforts are made such that no infected individual bypasses the exposed class and progresses directly to the infectious, the TB epidemic is successfully combatted. Also, the model is used to …

An Ultra-Sparse Approximation Of Kinetic Solutions To Spatially Homogeneous Flows Of Non-Continuum Gas, Alexander Alekseenko, Amy Grandilli, Aihua W. Wood

Faculty Publications

We consider a compact approximation of the kinetic velocity distribution function by a sum of isotropic Gaussian densities in the problem of spatially homogeneous relaxation. Derivatives of the macroscopic parameters of the approximating Gaussians are obtained as solutions to a linear least squares problem derived from the Boltzmann equation with full collision integral. Our model performs well for flows obtained by mixing upstream and downstream conditions of normal shock wave with Mach number 3. The model was applied to explore the process of approaching equilibrium in a spatially homogeneous flow of gas. Convergence of solutions with respect to the model …

A Sequential Partial Information Bomber‐Defender Shooting Problem, Krishna Kalyanam, David W. Casbeer, Meir Pachter

Faculty Publications

No abstract provided.

An Essay On Proof, Conviction, And Explanation: Multiple Representation Systems In Combinatorics, Elise Nicole Lockwood, John Caughman, Keith Weber

Mathematics and Statistics Faculty Publications and Presentations

There is a longstanding conversation in the mathematics education literature about proofs that explain versus proofs that only convince. In this essay, we offer a characterization of explanatory proofs with three goals in mind. We first propose a theory of explanatory proofs for mathematics education in terms of representation systems. Then, we illustrate these ideas in terms of combinatorial proofs, focusing on binomial identities. Finally, we leverage our theory to explain audience-dependent and audience-invariant aspects of explanatory proof. Throughout, we use the context of combinatorics to emphasize points and to offer examples of proofs that can be explanatory or only …

Using Differential Equations To Model Predator-Prey Relations As Part Of Scudem Modeling Challenge, Zachary Fralish, Bernard Tyson Iii, Anthony Stefan

Rose-Hulman Undergraduate Mathematics Journal

Differential equation modeling challenges provide students with an opportunity to improve their mathematical capabilities, critical thinking skills, and communication abilities through researching and presenting on a differential equations model. This article functions to display an archetype summary of an undergraduate student team’s response to a provided prompt. Specifically, the provided mathematical model estimates how certain stimuli from a predator are accumulated to trigger a neural response in a prey. Furthermore, it tracks the propagation of the resultant action potential and the physical flight of the prey from the predator through the analysis of larval zebrafish as a model organism. This …

A Game-Theoretic Model Of Monkeypox To Assess Vaccination Strategies, Sri Vibhaav Bankuru, Samuel Kossol, William Hou, Parsa Mahmoudi, Jan Rychtář, Dewey Taylor

Mathematics and Applied Mathematics Publications

Monkeypox (MPX) is a zoonotic disease similar to smallpox. Its fatality rate is about 11% and it is endemic to the Central and West African countries. In this paper, we analyze a compartmental model of MPX dynamics. Our goal is to see whether MPX can be controlled and eradicated by voluntary vaccinations. We show that there are three equilibria—disease free, fully endemic and previously neglected semi-endemic (with disease existing only among humans). The existence of semi-endemic equilibrium has severe implications should the MPX virus mutate to increased viral fitness in humans. We find that MPX is controllable and can be …

The N-Integral, Abraham P. Racca, Emmanuel A. Cabral

Mathematics Faculty Publications

In this paper, we introduced a Henstock-type integral named $N$-integral of a real valued function $f$ on a closed and bounded interval $[a,b]$. The $N$-integrable functions lie entirely between Riemann integrable functions and Henstock integrable functions. It was shown that for a Henstock integrable function $f$ on $[a,b]$ the following are equivalent: \begin{enumerate} \item[$(1)$] The function $f$ is $N$-integrable; \item[$(2)$] There exists a null set $S$ for which given $\epsilon >0$ there exists a gauge $\delta$ such that for any $\delta$-fine partial division $D=\{(\xi,[u,v])\}$ of $[a,b]$ we have \[(\phi_S(D)\cap \Gamma_{\epsilon})\sum |f(v)-f(u)||v-u|<\epsilon\] where $\phi_S(D)=\{(\xi,[u,v])\in D:\xi \notin S\}$ and \[\Gamma_{\epsilon}=\{(\xi,[u,v]): |f(v)-f(u)|\geq \epsilon\}\] \end{enumerate} and \begin{enumerate} \item[$(3)$] The function $f$ is continuous almost everywhere. \end{enumerate} A characterization of continuous almost everywhere functions was also given.

Mathematical Modeling Of Diabetic Foot Ulcers Using Optimal Design And Mixed-Modeling Techniques, Michael Belcher

Mahurin Honors College Capstone Experience/Thesis Projects

A mathematical model for the healing response of diabetic foot ulcers was developed using averaged data (Krishna et al., 2015). The model contains four major factors in the healing of wounds using four separate differential equations with 12 parameters. The four differential equations describe the interactions between matrix metalloproteinases (MMP-1), tissue inhibitors of matrix metalloproteinases (TIMP-1), the extracellular matrix (ECM) of the skin, and the fibroblasts, which produce these proteins. Recently, our research group obtained the individual patient data that comprised the averaged data. The research group has since taken several approaches to analyze the model with the individual …

Variational Analysis Of Composite Optimization, Ashkan Mohammadi

Wayne State University Dissertations

The dissertation is devoted to the study of the first- and second-order variational analysis of the composite functions with applications to composite optimization. By considering a fairly general composite optimization problem, our analysis covers numerous classes of optimization problems such as constrained optimization; in particular, nonlinear programming, second-order cone programming and semidefinite programming(SDP). Beside constrained optimization problems our framework covers many important composite optimization problems such as the extended nonlinear programming and eigenvalue optimization problem. In first-order analysis we develop the exact first-order calculus via both subderivative and subdifferential. For the second-order part we develop calculus rules via second-order subderivative …

Use Of Kalman Filtering In State And Parameter Estimation Of Diabetes Models, Cassidy Le

HMC Senior Theses

Diabetes continues to affect many lives every year, putting those affected by it at higher risk of serious health issues. Despite many efforts, there currently is no cure for diabetes. Nevertheless, researchers continue to study diabetes in hopes of understanding the disease and how it affects people, creating mathematical models to simulate the onset and progression of diabetes. Recent research by David J. Albers, Matthew E. Levine, Andrew Stuart, Lena Mamykina, Bruce Gluckman, and George Hripcsak¹ has suggested that these models can be furthered through the use of Data Assimilation, a regression method that synchronizes a model with a …

On Von Neumann Equivalence And Group Approximation Properties, Ishan Ishan

Department of Mathematics: Faculty Publications

The notion of von Neumann equivalence (vNE), which encapsulates both measure equivalence and W* -equivalence, was introduced recently by Ishan, Peterson, and Ruth (2019). They have shown that many analytic properties, such as amenability, property (T), the Haagerup property, and proper proximality are preserved under von Neumann equivalence. In this article, we expand on the list of properties that are stable under von Neumann equivalence, and prove that weak amenability, weak Haagerup property, and the approximation property (AP) are von Neumann equivalence invariants. In particular, we get that AP is stable under measure equivalence. Furthermore, our techniques give an alternate …

High Order Explicit Semi-Lagrangian Method For The Solution Of Lagrangian Transport And Stochastic Differential Equations, Hareshram Natarajan

CGU Theses & Dissertations

A semi-Lagrangian method is developed for the solution of Lagrangian transport equations and stochastic differential equations that is consistent with Discontinuous Spectral Element Method (DSEM) approximations of Eulerian conservation laws. The method extends the favorable properties of DSEM that include its high-order accuracy, its local and boundary fitted properties and its high performance on parallel platforms for the concurrent semi-Lagrangian and Eulerian solution of a class of time-dependent problems that can be described by coupled Eulerian-Lagrangian formulations. Such formulations include the probabilistic models used for the simulation of chemically reacting turbulent flows or particle-laden flows. Motivated by the high-fidelity simulation …

Evaluating An Ordinal Output Using Data Modeling, Algorithmic Modeling, And Numerical Analysis, Martin Keagan Wynne Brown

Murray State Theses and Dissertations

Data and algorithmic modeling are two different approaches used in predictive analytics. The models discussed from these two approaches include the proportional odds logit model (POLR), the vector generalized linear model (VGLM), the classification and regression tree model (CART), and the random forests model (RF). Patterns in the data were analyzed using trigonometric polynomial approximations and Fast Fourier Transforms. Predictive modeling is used frequently in statistics and data science to find the relationship between the explanatory (input) variables and a response (output) variable. Both approaches prove advantageous in different cases depending on the data set. In our case, the data …

Nanomagnetic Resonance Imaging (Nano-Mri) Gives Personalized Medicine A New Perspective, Lorenzo Rosa, Jonathan Blackledge, Albert Boretti

Books/Book chapters

This chapter provides a brief overview of molecular imaging techniques and its present and future potential in personalized medicine, with special a focus on the magnetic resonance imaging (MRI) approach. It discusses the current techniques that allow for the in vivo visualization of molecular processes at the nanoscale resolution (nano-MRI). Nano-MRI is progressing rapidly thanks to the work of a very small but extremely brilliant community of experts. This paper is not intended to be a comprehensive review of nano-MRI written for these experts, but rather a concise description of the present achievements for a much broader audience of medical …

Agent-Based Modeling Of Locust Foraging And Social Behavior, Hannah Larson

HMC Senior Theses

Locust swarms contain millions of individuals and are a threat to agriculture on four continents. At low densities, locusts are solitary foragers; however, when crowded, they undergo an epigenetic phase change to a gregarious state in which they are attracted to other locusts. It is believed that this is an evolutionary adaptation that optimizes the seeking of resources. We have developed an agent-based model based on the solitary-gregarious transition and foraging behaviors due to hunger levels. A novel feature of our model is that it treats food resources as a dynamic variable in the environment. We discuss how social interaction …

A Framework Of Multi-Dimensional And Multi-Scale Modeling With Applications, Zilong Li

Doctoral Dissertations

In this dissertation, a framework for multi-dimensional and multi-scale modeling is proposed. The essential idea is based on oriented space curves, which can be represented as a 3D slender object or 1D step parameters. SMILES and Masks provide functionalities that extend slender objects into branched and other objects. We treat the conversion between 1D, 2D, 3D, and 4D representations as data unification. A mathematical analysis of different methods applied to helices (a special type of space curves) is also provided. Computational implementation utilizes Model-ViewController design principles to integrate data unification with graphical visualizations to create a dashboard. Applications of multi-dimensional …

Association Rules Patterns Discovery From Mixed Data, Welendawa Acharige Charith A. Elson

College of Graduate Studies: Theses & Dissertations

Finding Association Rules has been a popular unsupervised learning technique for dis covering interesting patterns in commercial data for well over two decades. The method seeks groups of data attributes and their values where their probability density of these attributesattherespectivevaluesismaximized. Therearecurrentlywell-establishedmeth ods for tackling this problem for data with categorical (discrete) attributes. However, for the cases of data with continuous variables, the techniques are largely focusing on cate gorizing continuous variables into intervals of interest and then relying on the categorical data methods to address the problem. We address the problem of finding association rules patterns in mixed data by …