Applied Mathematics Commons^™

Why Filtering Out Higher Harmonics Makes It Easier To Carry A Tune, Griselda Acosta, Eric Freudenthal, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent patent shows that filtering out higher harmonics helps people sing in-tune. In this paper, we use the general signal processing ideas to explain this empirical phenomenon. We also show that filtering out higher harmonics is the optimal way of increasing the signal-to-noise ratio -- and thus, of making it easier for people to recognize when they are signing out of tune.

Epicycles Are Almost As Good As Trigonometric Series: General System-Based Analysis, Griselda Acosta, Eric Smith, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To adequately describe the planets' motion, ancient astronomers used epicycles, when a planet makes a circular motion not around the Earth, but around a special auxiliary point which, in turn, performs a circular motion around the Earth -- or around a second auxiliary point which, in turns, rotates around the Earth, etc. Standard textbooks malign this approach by calling it bad science, but in reality, this is, in effect, trigonometric (Fourier) series -- an extremely useful tool in science and engineering. It should be mentioned, however, that the epicycles are almost as good as trigonometric series -- in the sense …

Common Sense Addition Explained By Hurwicz Optimism-Pessimism Criterion, Bibek Aryal, Laxman Bokati, Karla Godinez, Shammir Ibarra, Heyi Liu, Bofei Wang, Vladik Kreinovich

Departmental Technical Reports (CS)

If we place a can of coke that weigh 0.35 kg into a car that weighs 1 ton = 1000 kg, what will be the resulting weight of the car? Mathematics says 1000.35 kg, but common sense says 1 ton. In this paper, we show that this common sense answer can be explained by the Hurwicz optimism-pessimism criterion of decision making under interval uncertainty.

Why Area Under The Curve In Hypothesis Testing?, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

To compare two different hypothesis testing techniques, researchers use the following heuristic idea: for each technique, they form a curve describing how the probabilities of type I and type II errors are related for this technique, and then compare areas under the resulting curves. In this paper, we provide a justification for this heuristic idea.

If Space-Time Is Discrete, We May Be Able To Solve Np-Hard Problems In Polynomial Time, Ricardo Alvarez, Nick Sims, Christian Servin, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional physics assumes that space and time are continuous. However, this reasonable model leads to some serious problems. One the approaches that physicists follow to solve these problems is to assume that the space-time is actually discrete. In this paper, we analyze possible computational consequences of this discreteness. It turns out that in a discrete space-time, we may be able to solve NP-hard problems in polynomial time.

A Natural Explanation For The Minimum Entropy Production Principle, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well known that, according to the second law of thermodynamics, the entropy of a closed system increases (or at least stays the same). In many situations, this increase is the smallest possible. The corresponding minimum entropy production principle was first formulated and explained by a future Nobelist Ilya Prigogine. Since then, many possible explanations of this principle appeared, but all of them are very technical, based on complex analysis of differential equations describing the system's dynamics. Since this phenomenon is ubiquitous for many systems, it is desirable to look for a general system-based explanation, explanation that would not …

Avoiding Einstein-Podolsky-Rosen (Epr) Paradox: Towards A More Physically Adequate Description Of A Quantum State, Joseph Bernal, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The famous EPR paradox shows that if we describe quantum particles in the usual way -- by their wave functions -- then we get the following seeming contradiction. If we entangle the states of the two particles, then move them far away from each other, and measure the state of the first particle, then the state of the second particle immediately changes -- which contradicts to special relativity, according to which such immediate-action-at-a-distance is not possible. It is known that, from the physical viewpoint, this is not a real paradox: if we measure any property of the second particle, the …

When Revolutions Succeed? 80/20 Rule And 7 Plus Minus 2 Law Explain The 3.5% Rule, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A statistical analysis of hundreds of successful and unsuccessful revolution attempts led historian to a very unexpected conclusion: that most attempts involving at least 3.5% of the population succeeded, while most attempts that involved a smaller portion of the population failed. In this paper, we show that this unexpected threshold can be explained based on the other two known rules of human behavior: the 80/20 rule (20% of the people drink 80% of the beer) and 7 plus minus 2 law according to which we naturally divide everything into 7 plus minus 2 classes.

How To Make Decisions: Consider Multiple Scenarios, Consult Experts, Play Down Emotions -- Quantitative Explanation Of Commonsense Ideas, Julio Urenda, Francis Biney, Marco Cardiel, Perla De La O, Anthony Desarmier, Noa Dodson, Taylor Dodson, Sebastian Gonzalez, Laura Hinojos, Jorge Huerta, Ryan Jones, Oliver Martinez, Carlos A. Saldaña Matamoros, Manuel Muñoz, Vladik Kreinovich

Departmental Technical Reports (CS)

There are a lot of commonsense advices in decision making: e.g., we should consider multiple scenarios, we should consult experts, we should play down emotions. Many of these advices come supported by a surprisingly consistent quantitative evidence. In this paper, on the example of the above advices, we provide a theoretical explanations for these quantitative facts.

Allee Effects Introduced By Density Dependent Phenology., Timothy James Pervenecki

Electronic Theses and Dissertations

We consider both the nonspatial model and spatial model of a species that gives birth to eggs at the end of the year. It is assumed that the timing of emergence from eggs is controled by phenology, which is density dependent. In general, the solution maps for our models are implicit; When the solution map is explicit, it is extremely complex and it is easier to work with the implicit map. We derive integral conditions for which the nonspatial model exhibits strong Allee effect. We also provide a necessary condition and a sufficient condition for the existence of positive equilibrium …

Algebraic Properties Of Neural Codes., Katie C. Christensen

Electronic Theses and Dissertations

The neural rings and ideals as algebraic tools for analyzing the intrinsic structure of neural codes were introduced by C. Curto, V. Itskov, A. Veliz-Cuba, and N. Youngs in 2013. Since then they have been investigated in several papers, including the 2017 paper by S. G\"unt\"urk\"un, J. Jeffries, and J. Sun, in which the notion of polarization of neural ideals was introduced. We extend their ideas by introducing the polarization of motifs and neural codes, and show that these notions have very nice properties which allow the studying of the intrinsic structure of neural codes of length $n$ via the …

Radial Basis Function Finite Difference Approximations Of The Laplace-Beltrami Operator, Sage Byron Shaw

Boise State University Theses and Dissertations

Partial differential equations (PDEs) are used throughout science and engineering for modeling various phenomena. Solutions to PDEs cannot generally be represented analytically, and therefore must be approximated using numerical techniques; this is especially true for geometrically complex domains. Radial basis function generated finite differences (RBF-FD) is a recently developed mesh-free method for numerically solving PDEs that is robust, accurate, computationally efficient, and geometrically flexible. In the past seven years, RBF-FD methods have been developed for solving PDEs on surfaces, which have applications in biology, chemistry, geophysics, and computer graphics. These methods are advantageous, as they are mesh-free, operate on arbitrary …

Krylov Subspace Spectral Methods With Non-Homogenous Boundary Conditions, Abbie Hendley

Master's Theses

For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent …

Characterizing Majority Rule On Various Discrete Models Of Consensus., Trevor Leach

Electronic Theses and Dissertations

In any social structure, there is often a need to reach decisions, not only within a group but between groups as well, sometimes even urgently so. Each of the individuals constituting these groups has their own preference for the decision to be made. We will discuss the problem of aggregating individual preferences into a collective preference and under what conditions we are required to select a collective majority. In this dissertation we will look at three models of consensus and show the conditions vary based on the model.

An Information Theory Model For Optimizing Quantitative Magnetic Resonance Imaging Acquisitions, Drew Mitchell

Dissertations and Theses (Open Access)

Quantitative magnetic resonance imaging (qMRI) is a powerful group of imaging techniques with a growing number of clinical applications, including synthetic image generation in post-processing, automatic segmentation, and diagnosis of disease from quantitative parameter values. Currently, acquisition parameter selection is performed empirically for quantitative MRI. Tuning parameters for different scan times, tissues, and resolutions requires some measure of trial and error. There is an opportunity to quantitatively optimize these acquisition parameters in order to maximize image quality and the reliability of the previously mentioned methods which follow image acquisition.

The objective of this work is to introduce and evaluate a …

An Application Of Conformal Mapping To The Boundary Element Method For Unconfined Steady Seepage With A Phreatic Surface, Jorge Eduardo Reyes

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this thesis, numerical results using the Boundary Element Method (BEM) for groundwater flow in a domain with a boundary that contains numerous singularities with a phreatic surface are developed. The flow in the domain is modeled using Darcy’s law for a homogeneous isotropic porous medium. The boundary conditions are a combination of Dirichlet and Neumann with the phreatic surface having both boundary conditions. Exact solutions by Conformal Mapping for simplified domains with the same singularity as the original domain allow for modifications to the BEM resulting in an improvement to the numerical solution.

An iterative process is used to …

Smoothness Of Defining Functions And The Diederich-Fornæss Index, Felita Nadia Humes

Graduate Theses and Dissertations

Let Ω ⊂ Cn be a smooth, bounded, pseudoconvex domain, and let M ⊂ ∂Ω be a complex submanifold with rectifiable boundary. In 2017, Harrington studied the equation dM A = α ̃ on M, where α ̃ is D’Angelo’s 1-form and A is real. In this thesis, we will study a non-pseudoconvex example in which M has a non-rectifiable boundary. In spite of the lack of topological obstructions on the boundary, there are no continuous solutions to dM A = α ̃.

One-Note-Samba Approach To Cosmology, Florentin Smarandache, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

Inspired by One Note Samba, a standard jazz repertoire, we present an outline of Bose-Einstein Condensate Cosmology. Although this approach seems awkward and a bit off the wall at first glance, it is not impossible to connect altogether BEC, Scalar Field Cosmology and Feshbach Resonance with Ermakov-Pinney equation. We also briefly discuss possible link with our previous paper which describes Newtonian Universe with Vortex in terms of Ermakov equation.

Trefftz Finite Elements On Curvilinear Polygons, Akash Anand, Jeffrey S. Ovall, Samuel E. Reynolds, Steffen Weisser

Mathematics and Statistics Faculty Publications and Presentations

We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be "polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local …

Combating Tuberculosis: Using Time-Dependent Sensitivity Analysis To Develop Strategies For Treatment And Prevention, Kendall B. Clark, Mayleen Cortez, Cristian Hernandez, Beth E. Thomas, Allison L. Lewis

Spora: A Journal of Biomathematics

Although many organizations throughout the world have worked tirelessly to control tuberculosis (TB) epidemics, no country has yet been able to eradicate the disease completely. We present two compartmental models representing the spread of a TB epidemic through a population. The first is a general TB model; the second is an adaptation for regions in which HIV is prevalent, accounting for the effects of TB/HIV co-infection. Using active subspaces, we conduct time-dependent sensitivity analysis on both models to explore the significance of certain parameters with respect to the spread of TB. We use the results of this sensitivity analysis to …