Applied Mathematics Commons^™

The Mathematics Behind Illusion, Kouassi Adou

All Zyzzogeton Presentations

Historically, research on optical, or visual illusions has belonged mainly to the field of psychology. However, in the 1980s, Professor Kokichi Sugihara, Meiji University, Japan, introduced a mathematical approach to design and classify 3-dimensional optical illusions. This presentation provides a sample of the mathematics behind some types of visual illusions.

The Game Of Life On The Hyperbolic Plane, Yuncong Gu

Mathematical Sciences Technical Reports (MSTR)

In this paper, we work on the Game of Life on the hyperbolic plane. We are interested in different tessellations on the hyperbolic plane and different Game of Life rules. First, we show the exponential growth of polygons on the pentagon tessellation. Moreover, we find that the Group of 3 can keep the boundary of a set not getting smaller. We generalize the existence of still lifes by computer simulations. Also, we will prove some propositions of still lifes and cycles. There exists a still life under rules B1, B2, and S3.

How To Estimate The Stiffness Of The Multi-Layer Road Based On Properties Of Layers: Symmetry-Based Explanation For Odemark's Equation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

When we design a road, we would like to check that the current design provides the pavement with sufficient stiffness to withstand traffic loads and climatic conditions. For this purpose, we need to estimate the stiffness of the road based on stiffness and thickness of its different layers. There exists a semi-empirical formula for this estimation. In this paper, we show that this formula can be explained by natural scale-invariance requirements.

How The Proportion Of People Who Agree To Perform A Task Depends On The Stimulus: A Theoretical Explanation Of The Empirical Formula, Laxman Bokati, Vladik Kreinovich, Doan Thanh Ha

Departmental Technical Reports (CS)

For each task, the larger the stimulus, the larger proportion of people agree to perform this task. In many economic situations, it is important to know how much stimulus we need to offer so that a sufficient proportion of the people will agree to perform the needed task. There is an empirical formula describing how this proportion increases as we increase the amount of stimulus. However, this empirical formula lacks a convincing theoretical explanation, as a result of which practitioners are somewhat reluctant to use it. In this paper, we provide a theoretical explanation for this empirical formula, thus making …

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.

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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{ …

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.

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 …

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.

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.