Applied Mathematics Commons^™

Reward For Good Performance Works Better Than Punishment For Mistakes: Economic Explanation, Olga Kosheleva, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

How should we stimulate people to make them perform better? How should we stimulate students to make them study better? Many experiments have shown that reward for good performance works better than punishment for mistakes. In this paper, we provide a possible theoretical explanation for this empirical fact.

Hydrodynamic Instability Simulations Using Front-Tracking With Higher-Order Splitting Methods, Dillon Trinh

Mathematical Sciences Undergraduate Honors Theses

The Rayleigh-Taylor Instability (RTI) is an instability that occurs at the interface of a lighter density fluid pushing onto a higher density fluid in constant or time-dependent accelerations. The Richtmyer-Meshkov Instability (RMI) occurs when two fluids of different densities are separated by a perturbed interface that is accelerated impulsively, usually by a shock wave. When the shock wave is applied, the less dense fluid will penetrate the denser fluid, forming a characteristic bubble feature in the displacement of the fluid. The displacement will initially obey a linear growth model, but as time progresses, a nonlinear model is required. Numerical studies …

Why Most Empirical Distributions Are Few-Modal, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, any non-negative function can serve as a probability density function -- provided that it adds up to 1. All kinds of processes are possible, so it seems reasonable to expect that observed probability density functions are random with respect to some appropriate probability measure on the set of all such functions -- and for all such measures, similarly to the simplest case of random walk, almost all functions have infinitely many local maxima and minima. However, in practice, most empirical distributions have only a few local maxima and minima -- often one (unimodal distribution), sometimes two (bimodal), and, …

Economics Of Reciprocity And Temptation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguyen Ngoc Thach

Departmental Technical Reports (CS)

Behavioral economics has shown that in many situations, people's behavior differs from what is predicted by simple traditional utility-maximization economic models. It is therefore desirable to be able to accurately describe people's actual behavior. In some cases, the difference from the traditional models is caused by bounded rationality -- our limited ability to process information and to come up with a truly optimal solutions. In such cases, predicting people's behavior is difficult. In other cases, however, people actually optimize -- but the actual expression for utility is more complicated than in the traditional models. In such case, it is, in …

A Novel Mathematical Model Of The Trojan Y-Chromosome Strategy With Optimal Control, Christopher Turner

Electronic Theses and Dissertations

Invasive species are a prevalent problem all over the world. Controlling and eradicating an invasive species is an even more diffcult problem. The Trojan Y Chromosome (TYC) eradication strategy is one control method. This method alters the female to male sex ratio by introducing sex reversed males called supermales. These sex reversed males can only produce male progeny. Mathematical models of this strategy have shown that a population can be driven to extinction with a continuous supply of these sex reversed males. There are many different mathematical models of this strategy, but most have serious flaws, such as negative solutions …

Control Of Secondary Extinctions In Stochastic Food Webs, Dunia M. Fernandez

Theses, Dissertations and Culminating Projects

Studies on both model-based and empirical food webs have shown that per- turbations to an ecological community can cause a species to go extinct, often resulting in the loss of additional species in a cascade of secondary extinctions. These eects can seriously debilitate a food web and threaten the existence of an ecosystem. Here, we consider niche model-based food webs with internal noise and investigate the eects of a control on a secondary extinction cas- cade triggered by a noise-induced extinction. We show that the forced removal of a nonbasal species immediately after a primary extinction can extend the mean …

Towards Gross-Pitaevskiian Description Of Solar System & Galaxies, Florentin Smarandache, Victor Christianto, Yunita Umniyati

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we argue that Gross-Pitaevskii model can be a more complete description of both solar system and spiral galaxies, especially taking into account the nature of chirality and vortices in galaxies. We also hope to bring out some correspondence among existing models, e.g., the topological vortex approach, Burgers equation in the light of KAM theory, and the Cantorian Navier-Stokes approach. We hope further investigation can be done around this line of approach.

Joint Inversion Of Gpr And Er Data, Diego Domenzain

Boise State University Theses and Dissertations

Imaging the subsurface can shed knowledge on important processes needed in a modern day human's life such as ground-water exploration, water resource monitoring, contaminant and hazard mitigation, geothermal energy exploration and carbon dioxide storage. As computing power expands, it is becoming ever more feasible to increase the physical complexity of Earth's exploration methods, and hence enhance our understanding of the subsurface.

We use non-invasive geophysical active source methods that rely on electromagnetic fields to probe the depths of the Earth. In particular, we use Ground penetrating radar (GPR) and Electrical resistivity (ER). Both methods are sensitive to electrical conductivity while …

Using Modern Portfolio Theory To Analyze Virgil's Aeneid (Or Any Other Poem), David Patterson

Master's Theses

This paper demonstrates that it is possible to use mathematics to study literature as it has been used to study the social sciences. By focusing on mathematically defining economic and literary terms, it can be shown that the underlying mathematical structure behind key concepts in economics and literature are analogous. This opens the possibility of applying economic models in literature. Specifically, it is demonstrated that the economic mathematical model of modern portfolio theory can answer long standing questions around the Roman epic Aeneid by Virgil. The poet died before completing his poem. The relative completeness of the books of the …

Homotopy Analysis Method For Nonlinear Ordinary Eigenvalue Problems, Subagya Perera

Master's Theses

In this thesis, we solve nonlinear differential equations by the homotopy analysis method (HAM), which is a semi-analytic method first introduced by Shijun Liao in 1992. The modified HAM can be viewed as a more generalized method that encloses many perturbation and non-perturbation methods. It is different from perturbation or other analytical methods in that it allows considerable freedomformanyvariables. Using the modified HAM, especially zero-order and higher-order deformation equations, we solve a nonlinear initial value problem and a nonlinear eigenvalue problem. We adjust the convergence region of a solution by modifying auxiliary parameter values. The results converge in very few …

A Time Integration Method Of Approximate Particular Solutions For Nonlinear Ordinary Differential Equations, Cyril Ocloo

Master's Theses

We consider a time-dependent method which is coupled with the method of approximate particular solutions (MAPS) of Delta-shaped basis functions and the method of fundamental solutions (MFS) to solve nonlinear ordinary differential equations. Firstly, we convert a nonlinear problem into a sequence of time-dependent non-homogeneous boundary value problems through a fictitious time integration method. The superposition principle is applied to split the numerical solution at each time step into an approximate particular solution and a homogeneous solution. Delta-shaped basis functions are used to provide an approximation of the source function at each time step. The purpose of this is to …

Probabilistic Analysis Of Revenues In Online Games, Nishchal Sapkota

Honors Theses

Online games are captivating and engage users across the world. Some game formats maintain a pseudo-currency to give incentive to the players to play the game in search of rewards as set by the game provider. We model a multi-stage online game and predict how much revenue game providers obtain per game. We compare the revenues generated from different tournament formats to find the one with the maximum per-game revenue for the provider. We have also found the limiting value of the revenue as the game provider increases the number of stages.

Our methods are based on concepts of the …

The Long Time Behavior Of The Predator-Prey Model With Holling Type Iii, Regen S. Mcgee

Honors Theses

In this paper, the classical Lotka-Volterra model is expanded based on functional response of Holling type III to analyze a dynamical predator-prey relationship with hunting cooperation (a) and the Allee effect among predators. The stability of equilibrium solutions was first analyzed by deriving a Jacobian matrix from partial derivatives of our model. Newly derived eigenvalues are then used to determine the stability. The viability of the model is then demonstrated by using MATLAB. The numerical results show a clear Allee effect and a variety of possible phenomena related to stability when carrying capacity (k) is varied. Two different types of …

Automatic Numerical Methods For Enhancement Of Blurred Text-Images Via Optimization And Nonlinear Diffusion, Aaditya Kharel

Honors Theses

In this paper, we propose an automatic numerical method for solving a nonlinear partialdifferential- equation (PDE) based image-processing model. The Perona-Malik diffusion equation (PME) accounts for both forward and backward diffusion regimes so as to perform simultaneous denoising and deblurring depending on the value of the gradient. One of the limitations of this equation is that a large value of the gradient for backward diffusion can lead to singularity formation or staircasing. Guidotti-Kim-Lambers (GKL) came up with a bound for backward diffusion to prevent staircasing, where the backward diffusion is only limited to a specific range beyond which backward diffusion …

Twisted Central Configurations Of The Eight-Body Problem, Gokul Bhusal

Honors Theses

The N-body problem qualifies as the problem of the twenty-first century because of its fundamental importance and difficulty to solve [1]. A number of great mathematicians and physicists have tried but failed to come up with the general solution of the problem. Due to the complexity of the problem, even a partial result will help us in the understanding of the N-body problem. Central configurations play a ‘central’ role in the understanding of the N-body problem. The well known Euler and Lagrangian solutions are both generated from three-body central configurations. The existence and classifications of central configurations have attracted number …

Stochastic Modeling Of Zoonotic Disease, Sausan Odatalla

Theses, Dissertations and Culminating Projects

We provide an overview of the mathematical modeling of deterministic and stochastic infectious disease models. These models enable one to understand the outbreak, spread, and extinction of disease. We then focus on stochastic models with a disease reservoir to understand outbreak vulnerability for zoonotic diseases such as Ebola Virus Disease (EVD). Numerical results from a more complicated EVD model are compared with the theoretical results of a simplified stochastic SISk model. We also demonstrate the effect that vaccine has on outbreak vulnerability in a population that is connected to a disease reservoir.

Mathematical Modeling Of A Variable Mass Rocket’S Dynamics Using The Differential Transform Method, Ashwyn Sam

Honors Theses

In this paper, the mathematical modelling of a rocket with varying mass is investigated to construct a function that can describe the velocity and position of the rocket as a function of time. This research is geared more towards small scale rockets where the nonlinear drag term is of great interest to the underlying dynamics of the rocket. A simple force balance on the rocket using Newton’s second law of motion yields a Riccati differential equation for which the solution yields the velocity of the rocket at any given time. This solution can then be integrated with respect to time …

Hadamard Well-Posedness For Two Nonlinear Structure Acoustic Models, Andrew Becklin

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation focuses on the Hadamard well-posedness of two nonlinear structure acoustic models, each consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ strongly coupled with a Berger plate equation acting only on a flat portion of the boundary of $\Omega$. In each case, the PDE is of the following form: \begin{align*} \begin{cases} u_{tt}-\Delta u +g_1(u_t)=f(u) &\text{ in } \Omega \times (0,T),\\[1mm] w_{tt}+\Delta^2w+g_2(w_t)+u_t|_{\Gamma}=h(w)&\text{ in }\Gamma\times(0,T),\\[1mm] u=0&\text{ on }\Gamma_0\times(0,T),\\[1mm] \partial_\nu u=w_t&\text{ on }\Gamma\times(0,T),\\[1mm] w=\partial_{\nu_\Gamma}w=0&\text{ on }\partial\Gamma\times(0,T),\\[1mm] (u(0),u_t(0))=(u_0,u_1),\hspace{5mm}(w(0),w_t(0))=(w_0,w_1), \end{cases} \end{align*} where the initial data reside in the finite energy space, i.e., $$(u_0, u_1)\in H^1_{\Gamma_0}(\Omega) \times L^2(\Omega) \, \text{ …

A Mathematical Development Of Minimal Surface Theory: From Soap Films To Black Holes, Timothy Pitts

Theses and Dissertations

Minimal surfaces are a special subset of surfaces that have gone through a long and extensive development and have also led to many fruitful findings in mathematics. Several periods that are key to the progression of the theory are coined as Golden Ages for the field’s development. Here, a historical and mathematical development of minimal surface theory is presented that spans from its inception in the late 18th century to the present day. Along with the development, there is an emphasis on showing connections of minimal surfaces to various natural phenomena that occur such as soap films, black holes, biological …

Constraining Neutron Star Nuclear Equations Of State Based On Observational Data, Alexander Clevinger

Undergraduate Honors Thesis Projects

This project analyzes recent observational data of neutron stars. It uses this to data to constrain nuclear equations of state proposed by Oter. et al. based on the maximum masses proposed by these equations of state. I do this by using numerical integration of the Tolman-Oppenheimer-Volkov equation to provide equilibrium states for each proposed EoS.

Critical Elliptic Boundary Value Problems With Singular Trudinger-Moser Nonlinearities, Shiqiu Fu

Theses and Dissertations

In this dissertation, we prove the existence of solutions for two classes of eliptic problems that are critical with respect to singular Trudinger-Moser embedding. The proofs are based on compactness and regularity arguments.

A Computational Investigation Of The Biomechanics For Platelets Aggregation, Ghadah Mohammed Alhawael

Theses and Dissertations

The proximal cause of most heart attacks and many strokes is the rapid formation of a blood clot (thrombus) in response to the rupture or erosion of an arterial atherosclerotic plaque. The formation of a thrombus in arteries is a very complex process whose workings are subjects of intense research. In this dissertation, we investigate the biomechanics of platelet aggregation in large arteries using a two-phase continuum computational model. The model tracks the number densities of various platelet populations, the concentration of one platelet-activating chemical, as well as the number densities of inter-platelet bonds. Through the formation of elastic bonds, …

Optimal Control Of Coefficients For The Second Order Parabolic Free Boundary Problems, Ali Hagverdiyev

Theses and Dissertations

Dissertation aims to analyze inverse Stefan type free boundary problem for the second order parabolic PDE with unknown parameters based on the additional information given in the form of the distribution of the solution of the PDE and the position of the free boundary at the final moment. This type of ill-posed inverse free boundary problems arise in many applications such as biomedical engineering problem about the laser ablation of biomedical tissues, in-flight ice accretion modeling in aerospace industry, and various phase transition processes in thermophysics and fluid mechanics. The set of unknown parameters include a space-time dependent diffusion, convection …

Target Control Of Networked Systems, Isaac S. Klickstein

Mechanical Engineering ETDs

The control of complex networks is an emerging field yet it has already garnered interest from across the scientific disciplines, from robotics to sociology. It has quickly been noticed that many of the classical techniques from controls engineering, while applicable, are not as illuminating as they were for single systems of relatively small dimension. Instead, properties borrowed from graph theory provide equivalent but more practical conditions to guarantee controllability, reachability, observability, and other typical properties of interest to the controls engineer when dealing with large networked systems. This manuscript covers three topics investigated in detail by the author: (i) the …

Symbolic Construction Of Matrix Functions In A Numerical Environment, Evan D. Butterworth

Honors College Theses

Within the field of Computational Science, the importance of programs and tools involving systems of differential equations cannot be overemphasized. Many industrial sites, such as nuclear power facilities, are unable to safely operate without these systems. This research explores and studies matrix differential equations and their applications to real computing structures. Through the use of software such as MatLab, I have constructed a toolbox, or collection, of programs that will allow any user to easily calculate a variety of matrix functions. The first tool in this collection is a program that computes the matrix exponential, famously studied and presented by …

A Study Of Addiction: The Opioid Epidemic, An Analysis At The State And County Level, Jamey Van Dyke

Undergraduate Theses

Addictive diseases such as those stemming from the use of alcohol, cocaine and opioids lead to serious negative consequences at both the individual and societal level. Over the last two decades, there has been a significant increase in opioid prescriptions and addiction. The potential for addiction is related to factors that include genetics, prescriber behavior, user behavior and characteristics, in addition to environmental and systemic determinants. One measure of the seriousness of the opioid epidemic is the number of overdose deaths. In 2017, drug overdoses killed over seventy thousand Americans, and overdose deaths are projected to increase in the future. …

358— Hybridization Of Particle Swam Optimization And Pattern Search Algorithms With Application, Eric Koessler

GREAT Day Posters

We test three methods of hybridizing Particle Swarm Optimization (PSO) and Pattern Search (PS) to improve the global minima, speed, and robustness. All methods let PSO run first followed by PS. The first method lets PSO use a large number of particles for a limited number of iterations. The second method lets PSO run normally until tolerance is reached. The third method lets PSO run normally until the average particle distance from the global best location is within a threshold. Numerical results using non-differentiable test functions reveal that all three methods improve the global minima and robustness versus PSO, while …

483— Effectiveness Of Mmr Vaccination In Orthodox Jewish Neighborhoods, Meenu Mundackal

GREAT Day Posters

Measles is a highly contagious disease, where large outbreaks arise by direct contact between susceptible (unvaccinated) and infectious individuals. Many Orthodox Jewish neighborhoods were affected by measles from 2018-2019. To quantify the vaccination effort on this susceptible population, a retrospective analysis was used to study the NYC and Rockland County populations using a differential equations model. A subsequent model, known as a realistically-structured network model, studied only the NYC population, in relation to typical household size. Vaccination strategies were applied to three cohorts: unvaccinated family members, members with 1 prior MMR dose, and members with 2 prior MMR doses. The …

465— Modeling Vaccine Efficacy For Tuberculosis In A Prison Population, Kaitlyn Mundackal

GREAT Day Posters

Tuberculosis is a highly contagious disease and is particularly problematic in confined communities such as prisons. I simulated how Tuberculosis moves through a prison population and tested how much vaccination effort is needed to control its spread. To explore this, I tested adding ever increasing numbers of randomly placed edges in a network and determined the size of the largest component. Afterwards, I removed edges in the model using two different methods, one illustrating if the edges were removed randomly and the other starting with prisoners that had the most connections, to simulate the effect of vaccination. My results show …

484— Modeling Social Distancing Methods And Their Effectiveness In Combating The Spread Of Ebola, Rachel Fair

GREAT Day Posters

Ebola Virus Disease (EVD) is a rare but severe disease that is transmitted among humans through direct-contact with, and close proximity to, infected bodily fluids. From 2014-16, West Africa experienced the largest Ebola outbreak ever recorded, infecting over 28,000 people, and killing over 11,000. Although the symptoms of EVD are treatable, the disease can be extremely deadly, with an average of 50% EVD cases resulting in fatality. In areas where healthcare is scarce and vaccinations are not readily available, the practices of social distancing and self-quarantining have been shown to be highly effective in combating the spread of EVD. To …