Applied Mathematics Commons^™

Mathematical Modeling Of Blood Coagulation, Joana L. Perdomo

HMC Senior Theses

Blood coagulation is a series of biochemical reactions that take place to form a blood clot. Abnormalities in coagulation, such as under-clotting or over- clotting, can lead to significant blood loss, cardiac arrest, damage to vital organs, or even death. Thus, understanding quantitatively how blood coagulation works is important in informing clinical decisions about treating deficiencies and disorders. Quantifying blood coagulation is possible through mathematical modeling. This review presents different mathematical models that have been developed in the past 30 years to describe the biochemistry, biophysics, and clinical applications of blood coagulation research. This review includes the strengths and limitations …

4. Dragging Along, Ruth Dover

Differential Equations

More information on air drag.

The Kretschmann Scalar, Charles G. Torre

How to... in 10 minutes or less

On a pseudo-Riemannian manifold with metric g, the "Kretschmann scalar" is a quadratic scalar invariant of the Riemann R tensor of g, defined by contracting all indices with g. In this worksheet we show how to calculate the Kretschmann scalar from a metric.

Steady State Solutions For A System Of Partial Differential Equations Arising From Crime Modeling, Bo Li

HMC Senior Theses

I consider a model for the control of criminality in cities. The model was developed during my REU at UCLA. The model is a system of partial differential equations that simulates the behavior of criminals and where they may accumulate, hot spots. I have proved a prior bounds for the partial differential equations in both one-dimensional and higher dimensional case, which proves the attractiveness and density of criminals in the given area will not be unlimitedly high. In addition, I have found some local bifurcation points in the model.

Topological Data Analysis For Systems Of Coupled Oscillators, Alec Dunton

HMC Senior Theses

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a …

1-65-S-Algal Blooms: Algal Blooms Threatening Lake Chapala, R. Corban Harwood

Faculty Publications - Department of Mathematics

This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, starting in 1994. After reading a summary of articles written on the incidents, students are guided through the process of creating a first order differential equation from a verbal model of the factors and analyze the nonautonomous ODE using direction field, parameter evaluation, and exact solution computation to fully describe the population behavior. Students are expected to be familiar with the separable method and direction fields. Students will learn building and improving a model from qualitative descriptions, nondimensionalization, evaluating parameters, and how to use DFIELD software to …

Oscillation-Free Method For Semilinear Diffusion Equations Under Noisy Initial Conditions, R. Corban Harwood, Likun Zhang, V. S. Manoranjan

Faculty Publications - Department of Mathematics

Noise in initial conditions from measurement errors can create unwanted oscillations which propagate in numerical solutions. We present a technique of prohibiting such oscillation errors when solving initial-boundary-value problems of semilinear diffusion equations. Symmetric Strang splitting is applied to the equation for solving the linear diffusion and nonlinear remainder separately. An oscillation-free scheme is developed for overcoming any oscillatory behavior when numerically solving the linear diffusion portion. To demonstrate the ills of stable oscillations, we compare our method using a weighted implicit Euler scheme to the Crank-Nicolson method. The oscillation-free feature and stability of our method are analyzed through a …

Logistics Of Mathematical Modeling-Focused Projects, R. Corban Harwood

Faculty Publications - Department of Mathematics

Projects provide tangible connections to course content and can motivate students to learn at a deeper level. This article focuses on the implementation of projects in both lower and upper division math courses which develop and analyze mathematical models of a problem based upon known data and real-life situations. Logistical pitfalls and insights are highlighted as well as several key implementation resources. Student feedback demonstrate a positive correlation between the use of projects and an enhanced understanding of the course topics when the impact of logistics is reduced. Best practices learned over the years are given along with example project …

Image Harvest: An Open-Source Platform For High-Throughput Plant Image Processing And Analysis, Avi C. Knecht, Malachy T. Campbell, Adam Caprez, David R. Swanson, Harkamal Walia

Holland Computing Center: Faculty Publications

High-throughput plant phenotyping is an effective approach to bridge the genotype-to-phenotype gap in crops. Phenomics experiments typically result in large-scale image datasets, which are not amenable for processing on desktop computers, thus creating a bottleneck in the image-analysis pipeline. Here, we present an open-source, flexible image-analysis framework, called Image Harvest (IH), for processing images originating from high-throughput plant phenotyping platforms. Image Harvest is developed to perform parallel processing on computing grids and provides an integrated feature for metadata extraction from large-scale file organization. Moreover, the integration of IH with the Open Science Grid provides academic researchers with the computational resources …

A Model For Spheroid Versus Monolayer Response Of Sk-N-Sh Neuroblastoma Cells To Treatment With 15-Deoxy-Pgj2, Dorothy I. Wallace, Ann Dunham, Paula X. Chen, Michelle Chen, Milan Huynh, Evan Rheingold, Olivia F. Prosper

Mathematics Faculty Publications

Researchers have observed that response of tumor cells to treatment varies depending on whether the cells are grown in monolayer, as in vitro spheroids or in vivo. This study uses data from the literature on monolayer treatment of SK-N-SH neuroblastoma cells with 15-deoxy-PGJ₂ and couples it with data on growth rates for untreated SK-N-SH neuroblastoma cells grown as multicellular spheroids. A linear model is constructed for untreated and treated monolayer data sets, which is tuned to growth, death, and cell cycle data for the monolayer case for both control and treatment with 15-deoxy-PGJ₂. The monolayer …

Multiple Problem-Solving Strategies Provide Insight Into Students’ Understanding Of Open-Ended Linear Programming Problems, Marla A. Sole

Publications and Research

Open-ended questions that can be solved using different strategies help students learn and integrate content, and provide teachers with greater insights into students’ unique capabilities and levels of understanding. This article provides a problem that was modified to allow for multiple approaches. Students tended to employ high-powered, complex, familiar solution strategies rather than simpler, more intuitive strategies, which suggests that students might need more experience working with informal solution methods. During the semester, by incorporating open-ended questions, I gained valuable feedback, was able to better model real-world problems, challenge students with different abilities, and strengthen students’ problem solving skills.

Development And Separation Of Forced Convective Flow, Adefemi Sunmonu

Publications and Research

Singularities are considered in the solution of the laminar bound- ary - layer equation at a position of separation. The works of Howarth (1938). Goldstein (1948), Stewartson (1958), Terrill (1960) and Akin- relere [(1981), (1982)] are reviewed to fully establish the existence of singularity in the incompressible boundary layer at separation for both the velocity and thermal fields. A ow at a large Reynold's number along an immersed solid surface around which bounda y layer is formed through which the velocity rises rapidly from zero at the surface to its value in the main stream is considered. It is found …

Optimal Control And Its Application To The Life-Cycle Savings Problem, Tracy A. Taylor

Theses and Dissertations

Throughout the course of this thesis, we give an introduction to optimal control theory and its necessary conditions, prove Pontryagin's Maximum Principle, and present the life-cycle saving under uncertain lifetime optimal control problem. We present a very involved sensitivity analysis that determines how a change in the initial wealth, discount factor, or relative risk aversion coefficient may affect the model the terminal depletion of wealth time, optimal consumption path, and optimal accumulation of wealth path. Through simulation of the life-cycle saving under uncertain lifetime model, we are not only able to present the model dynamics through time, but also to …

Homogenization Of Stokes Systems With Periodic Coefficients, Shu Gu

Theses and Dissertations--Mathematics

In this dissertation we study the quantitative theory in homogenization of Stokes systems. We study uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L^∞ estimates for the pressure as well as Liouville property for solutions in ℝ^d. We are able to obtain the boundary W^{1,p} estimates in a bounded C¹ domain for any 1 < p < ∞. We also study the convergence rates in L² and H¹ of Dirichlet and Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.

Black-Scholes Equation And Heat Equation, Charles D. Joyner

Honors College Theses

First, we present and define the Black-Scholes equation which is used to model assets on the stock market. After that, we derive the heat equation that describes how the temperature increases through a homogeneous material. Finally, we detail how the two equations are related. We introduce and relate the Black-Scholes equation and Heat Equation.

The Mathematics Of Scientific Research: Scientometrics, Citation Metrics, And Impact Factors, Clayton Hayes

Library Scholarly Publications

This talk will consist of an introduction to the rationale behind the establishment of the field of Scientometrics, which attempts to measure and analyze the body of research done in the sciences. This is primarily through analyzing the connections between scientific journals, articles, and authors. I will cover the foundational works of Price and Garfield in the fields of citation networks and impact factor, respectively, and will give some historical context for these developments. I will also speak briefly on more recent developments in these areas, such as the Eigenfactor, author citation metrics, and (time permitting) alternative metrics.

Audience: …

An Analysis Of The Impact Of The Csu Peer Leader Program On Student Performance In 2014-2015, Harrison E. Sharitt

Theses and Dissertations

In efforts to retain students and increase student performance, Columbus State implemented a Peer Leader program. With access to archival data on the CSU Peer Leader Program, the goal of this project was to determine if the program at CSU had a significant impact on students’ course performance. The impact of the program was assessed by performing multiple regression analyses and ANOVA. Preliminary results showed that Columbus State’s Peer Leader program had a small, but statistically significant, positive impact on students’ final course grades.

Pricing Of Geometric Asian Options In General Affine Stochastic Volatility Models, Johannes Ruppert

Masters Theses

"In this thesis, we look at the general affine pricing model introduced in [11]. This model allows to price geometric Asian options, which are of big interest due to their lower volatility in comparison to, for example, European options. Because of their structure and in order to be able to price these options, we look at the basic theory of Lévy processes and stochastic calculus. Furthermore, we provide the detailed description of the parameters of the pricing formulas for some popular specific single-factor stochastic volatility models with jumps and generalize the approach of [11] to multi-factor models"--Abstract, page iii.

Controllability Of Neutral Stochastic Integro-Differential Evolution Equations Driven By A Fractional Brownian Motion, El Hassan Lakhel, Mark A. Mckibben

Mathematics Faculty Publications

We establish sufficient conditions for the controllability of a certain class of neutral stochastic functional integro-differential evolution equations in Hilbert spaces. The results are obtained using semigroup theory, resolvent operators and a fixed-point technique. An application to neutral partial integro-differential stochastic equations perturbed by fractional Brownian motion is given.

A Discontinuous Galerkin Method For Unsteady Two-Dimensional Convective Flows, Andreas C. Aristotelous, N. C. Papanicolaou

Mathematics Faculty Publications

We develop a High-Order Symmetric Interior Penalty (SIP) Discontinuous Galerkin (DG) Finite Element Method (FEM) to investigate two-dimensional in space natural convective flows in a vertical cavity. The physical problem is modeled by a coupled nonlinear system of partial differential equations and admits various solutions including stable and unstable modes in the form of traveling and/or standing waves, depending on the governing parameters. These flows are characterized by steep boundary and internal layers which evolve with time and can be well-resolved by high-order methods that also are adept to adaptive meshing. The standard no-slip boundary conditions which apply on the …