Mathematics Commons^™

Proper 3-Colorings Of Cycles And Hypercubes, Emily Cairncross

Honors Papers

In this paper, we look at two families of graphs, cycles and hypercubes, and compare how their sets of proper 3-colorings differ as the graphs get arbitrarily large. In particular, we find the probability of pairs of vertices at various distances being the same color in order to understand the range and scale of interactions between them. As we look at larger and larger cycles, larger and larger hypercubes, patterns begin to emerge. While the colors of vertices fixed fractions of the cycle away from each other are independent, a random 3-coloring of the hypercube is essentially a 2-coloring. This …

Get The News Out Loudly And Quickly: Modeling The Influence Of The Media On Limiting Infectious Disease, Anna Mummert, Howard Weiss

Mathematics Faculty Research

During outbreaks of infectious diseases with high morbidity and mortality, individuals closely follow media reports of the outbreak. Many will attempt to minimize contacts with other individuals in order to protect themselves from infection and possibly death. This process is called social distancing. Social distancing strategies include restricting socializing and travel, and using barrier protections. We use modeling to show that for short-term outbreaks, social distancing can have a large influence on reducing outbreak morbidity and mortality. In particular, public health agencies working together with the media can significantly reduce the severity of an outbreak by providing timely accounts of …

Modeling The Global Plastic Pollution In Our Oceans, Anna Fateiger

Honors Program Theses and Projects

Plastic is everywhere—from our plastic bottles and straws to the inside of our phones and the clothes we wear every day. Its widespread use has left a legacy of trash, with large amounts of plastic spilling from landfills into oceans. The accumulation of plastic debris in our oceans has severely affected marine life and has even entered into the human food chain. In this project, we created a mathematical model to estimate global plastic waste-generation and ocean runoff using existing data from 1980 to 2015. Using a dynamic system, we calculated the amount of plastic that ends up in landfills …

Bayesian Models For Repeated Measures Data Using Markov Chain Monte Carlo Methods, Yuanzhi Li

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Bayesian models for repeated measures data are fitted to three different data an analysis projects. Markov Chain Monte Carlo (MCMC) methodology is applied to each case with Gibbs sampling and/or an adaptive Metropolis-Hastings (MH) algorithm used to simulate the posterior distribution of parameters. We implement a Bayesian model with different variance-covariance structures to an audit fee data set. Block structures and linear models for variances are used to examine the linear trend and different behaviors before and after regulatory change during year 2004-2005. We proposed a Bayesian hierarchical model with latent teacher effects, to determine whether teacher professional development (PD) …

A Perspective On Multiple Waves Of Influenza Pandemics, Anna Mummert, Howard Weiss, Li-Pong Long, Jose´ M. Amigo´, Xiu-Feng Wan

Mathematics Faculty Research

Background: A striking characteristic of the past four influenza pandemic outbreaks in the United States has been the multiple waves of infections. However, the mechanisms responsible for the multiple waves of influenza or other acute infectious diseases are uncertain. Understanding these mechanisms could provide knowledge for health authorities to develop and implement prevention and control strategies. Materials and Methods: We exhibit five distinct mechanisms, each of which can generate two waves of infections for an acute infectious disease. The first two mechanisms capture changes in virus transmissibility and behavioral changes. The third mechanism involves population heterogeneity (e.g., demography, geography), where …

Measures Of Centrality Based On The Spectrum Of The Laplacian, Scott D. Pauls, Daniel Remondini

Dartmouth Scholarship

We introduce a family of new centralities, the k-spectral centralities. k-Spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, k-spectral centralities have various interpretations in terms of spectrally determined information.

We explore this centrality in the context of several examples. While for sparse unweighted net- works 1-spectral centrality behaves similarly to other standard centralities, for dense weighted net- works they show different properties. In summary, the k-spectral centralities provide a novel and useful measurement of relevance (for single network elements as well as whole subnetworks) …

Homogenization Of Large-Scale Movement Models In Ecology With Application To The Spread Of Chronic Wasting Disease In Mule Deer, Martha J. Garlick

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Chronic wasting disease (CWD) is a transmissible spongiform encephalopathy (TSE) that affects deer, elk, and moose. TSEs are prion diseases which include mad cow disease and scrapie in sheep and goats. The disease agent is a misshapen protein called a prion, which causes lesions in the brain, and to date, there is no cure. CWD is a slow-developing, fatal disease, which is rare in the free-ranging mule deer population of Utah. Infected deer shed prions into the environment through saliva, feces, and decaying carcasses. These prions remain infective in soils for many years and healthy deer may contract CWD by …

Colonel Blotto Games And Lancaster's Equations: A Novel Military Modeling Combination, Andrew Collins, Patrick T. Hester

VMASC Publications

Military strategists face a difficult task when engaged in a battle against an adversarial force. They have to predict both what tactics their opponent will employ and the outcomes of any resultant conflicts in order to make the best decision about their actions. Game theory has been the dominant technique used by analysts to investigate the possible actions that an enemy will employ. Traditional game theory can be augmented by use of Lanchester equations, a set of differential equations used to determine the outcome of a conflict. This paper demonstrates a novel combination of game theory and Lanchester equations using …

Mathematical Methods For Biosensor Models, Qi Wang

Doctoral

A biosensor is defined as a compact analytical device incorporating a biological sensing element integrated within a physico-chemical transducer whose aim is to produce optical or electronic signals proportional to the concentration of an analyte in a sample. Biosensors offer enormous potential to detect a wide range of analytes in health care, the food industry, environmental monitoring, security and defence. The beneficial impact on society as a result of the availability of such systems is immense, therefore investigating any strategy that could reduce development times and costs and reveal alternative designs is of utmost importance. In particular, mathematical modelling and …

Periodic Traveling Waves In Sirs Endemic Models, Tong Li, Yi Li, Herbert W. Hethcote

Yi Li

Mathematical models are used to determine if infection wave fronts could occur by traveling geographically in a loop around a region or continent. These infection wave fronts arise by Hopf bifurcation for some spatial models for infectious disease transmission with distributed-contacts. Periodic traveling waves are shown to exist for the spatial analog of the SIRS endemic model, in which the temporary immunity is described by a delay, but they do not exist in a similar spatial SIRS endemic model without a delay. Specifically, we found that the ratio of the delay ω in the recovered class and the average infectious …

Periodic Traveling Waves In Sirs Endemic Models, Tong Li, Yi Li, Herbert W. Hethcote

Mathematics and Statistics Faculty Publications

Mathematical models are used to determine if infection wave fronts could occur by traveling geographically in a loop around a region or continent. These infection wave fronts arise by Hopf bifurcation for some spatial models for infectious disease transmission with distributed-contacts. Periodic traveling waves are shown to exist for the spatial analog of the SIRS endemic model, in which the temporary immunity is described by a delay, but they do not exist in a similar spatial SIRS endemic model without a delay. Specifically, we found that the ratio of the delay ω in the recovered class and the average infectious …

Logistic Models With Missing Categorical Covariates, Jeremiah Rounds

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

We present an EM based solution to missing categorical covariates in Binomial models with logit links using an assumption that experimental units are drawn from a Multinomial population of infinite size. We further address the problem of separation of points inducing large variances on parameter estimates by the use of a novel score-modification based on Firth's bias-reduction score-modification. We simulate to address questions about estimate bias, distribution, and appropriate parameter coverage by Wald intervals.

Mixtures Of Truncated Normal Data Assimilation Models For Parameter Estimation And Prediction In Hydrological Systems, Darl D. Flake Ii

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Physical models in the hydrological sciences are often calibrated using methods that do not formally quantify uncertainty in the model parameters. Additionally, many competing hydrological models exist and are used to model the same processes. Considering existing mechanistic models of rainfall-run off in a statistical context can assist hydrologists in understanding the true physical process taking place. This paper introduces a data assimilation mixture model of runoff that yields statistical estimates of hydrological mode l parameters and predictions. This statistical model incorporates two commonly used hydrological models, each with strengths and weaknesses. The mixture framework allows comparisons between models as …

Studies In Free Module And It's Basis, Hsu-Chia Chen

Theses Digitization Project

The purpose of this project was to study some basic properties of free modules over a ring. A module with a basis is called a free module and a free module over a division ring (or field) is called a vector space. We show every vector has a basis and any two bases of a vector space have same cardinality. However, a free module over an arbitrary ring (with identity) does not have this property.

Macroscopic Consequences Of Calcium Signaling In Microdomains: A First-Passage-Time Approach, Robert Rovetti

Mathematics, Statistics and Data Science Faculty Works

Calcium (Ca) plays an important role in regulating various cellular processes. In a variety of cell types, Ca signaling occurs within microdomains where channels deliver localized pulses of Ca activating a nearby collection of Ca-sensitive receptors. The small number of channels involved ensures that the signaling process is stochastic. The aggregate response of several thousand of these microdomains yields a whole-cell response which dictates the cell behavior. Here, we study the statistical properties of a population of these microdomains in response to a trigger signal. We use a first-passage-time approach to show analytically how Ca release in the whole cell …

Stochastic Modeling Of Retail Mortgage Loans Based On Past Due, Prepaid, And Default States, Chang Liu

Doctoral Dissertations

Stochastic models were developed that provide important measures related to retail mortgages and credit cards for the management of a bank. Based on Markov theory, two models were developed that predict mortgage portfolio size and expected duration of stay in each of the states, which are defined according to the criteria of Basel Accord II and the Federal Reserve Bank. Also, to facilitate comparisons among different types of credit products and different time periods, a model was developed to generate a health index for a retail mortgage. This model could be easily extended, using multivariate regression or multivariate time series …

Black-Scholes And Extended Black-Scholes Models: A Comparative Statistical Analysis, Bradley Thomas Bush

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Much research has been done on options pricing. Black and Scholes [12] set the benchmark in 1973 with their model for arbitrage-free, risk-neutral options valuation. Arbitrage-free refers to a market environment where prices are such that trading opportunities with no risk do not exist and risk-neutral commodities earn a risk free interest rate. Since then the literature has seen a multitude of models improving the fit of the traditional Black -Scholes (BS) model. A brief overview of options and these models is given. A derivation and discussion of BS is followed by a derivation and discussion of the Extended Black-Scholes …

Special Classification Models For Lichens In The Pacific Northwest, Janeen Ardito

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

A common problem in ecological studies is that of determining where to look for rare species. This paper shows how statistical models, such as classification trees, may be used to assist in the design of probability-based surveys for rare species using information on more abundant species that are associated with the rare species. This model assisted approach to survey design involves first building models for the more abundant species. The models are then used to determine stratifications for the rare species that are associated with the more abundant species. The goal of this approach is to increase the number of …