Digital Commons Network^™

A Short Solution Of The Kissing Number Problem In Dimension Three, Alexey Glazyrin

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We give a short solution of the kissing number problem in dimension three.

Ramanujan–Sato Series For 1/Π, Timothy Huber, Daniel Schultz, Dongxi Ye

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We compute Ramanujan–Sato series systematically in terms of Thompson series and their modular equations. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.

P-Adic Cellular Neural Networks: Applications To Image Processing, B. A. Zambrano-Luna, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The p-adic cellular neural networks (CNNs) are mathematical generalizations of the neural networks introduced by Chua and Yang in the 80s. In this work we present two new types of -adic CNNs that can perform computations with real data, and whose dynamics can be understood almost completely. The first type of networks are edge detectors for grayscale images. The stationary states of these networks are organized hierarchically in a lattice structure. The dynamics of any of these networks consists of transitions toward some minimal state in the lattice. The second type is a new class of reaction–diffusion networks. We investigate …

The Reproducing Kernel Method For Solving Integro-Differential And Volterra Integral Equations, Khulood Gamal Qaid

Theses

Integro-differential equations are a class of mathematical equations that involve both derivatives and integrals. They have applications in a wide range of fields, including physics, engineering, finance, and biology such as the spread of diseases, population dynamics, and the behavior of financial markets. The study of these equations requires advanced mathematical techniques, including functional analysis, approximation methods, and numerical analysis. They are a rich area of research with many open questions and challenges.
In this thesis, we will develop and implement the reproducing kernel method to solve a class of integro-differential and Volterra integral equations. We discuss both cases when …

Enots Wolley Variations And Related Sequences, Nathan Myles Nichols

Master's Theses (2009 -)

The Enots Wolley sequence is a lexicographically earliest sequence (LES) that is closely related to the Yellowstone sequence. It is an open conjecture by N. J. Sloane that every number with at least two distinct prime factors appears as a term of the Enots Wolley sequence. In this thesis, this conjecture is proved for a variation of the Enots Wolley sequence that operates on the binary representation of a positive integer rather than the prime factorization. The methods used are then applied to prove some new properties of the prime factorization Enots Wolley sequence.

An Exploration Of Computational Text Analysis Of Co-Design Discourse In A Research-Practice Partnership, Mei Tan, Victor R. Lee

Publications

In combination with contextualized human interpretation, computational text analysis offers a quantitative approach to interrogating the nature of participation and social positioning in discourse. Using meeting transcript data from the development of a co-design research-practice partnership, we examine the roles and forms of participation that contribute to an effective collaboration between a multileveled school system and researcher partners. We apply computational methods to explore the language of co-design and multi-stakeholder perspectives in support of educational improvement science efforts and our theoretical understanding of partnership roles. Results indicate participation patterns align with documented roles in co- design partnerships and highlight the …

Geometry And Coding: Introducing An Interactive And Integrated Mathematics-Computer Science Unit, Kimberly Beck, Jessica F. Shumway

Publications

As part of a collaborative project between Utah State University, the Cache County School District, and Stanford, instructional units were designed for fifth-grade students. These units integrated math concepts of geometrical shapes and computer science concepts of sequences, conditionals, and loops. One component of the unit was implemented in math classrooms by math teachers, and the other component was implemented in computer labs. This presentation will focus on the math unit as presented at the National Council of Teachers of Mathematics (NCTM-V).

Counting Elliptic Curves With A Cyclic M-Isogeny Over Q, Grant S. Molnar

Dartmouth College Ph.D Dissertations

Using methods from analytic number theory, for m > 5 and for m = 4, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic m-isogeny up to quadratic twist over the rational numbers. For m > 5, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic m-isogeny up to isomorphism over the rational numbers.

Irregular Domination In Graphs, Caryn Mays

Dissertations

Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters …

Extreme Covering Systems, Primes Plus Squarefrees, And Lattice Points Close To A Helix, Jack Robert Dalton

Theses and Dissertations

This dissertation considers three different topics.

In the first part, we prove that if the least modulus of a distinct covering system is 4, its largest modulus is at least 60; also, if the least modulus is 3, the least common multiple of the moduli is at least 120; finally, if the least modulus is 4, the least common multiple of the moduli is at least 360. The constants 60, 120, and 360 are best possible, they cannot be replaced by larger constants. We also show that there do not exist distinct covering systems with all of the moduli in …

Causality: Hypergraphs, Matter Of Degree, Foundations Of Cosmology, Cliff Joslyn, Andres Ortiz-Muñoz, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The notion of causality is very important in many applications areas. Because of this importance, there are several formalizations of this notion in physics and in AI. Most of these definitions describe causality as a crisp ("yes"-"no") relation between two events or two processes -- cause and effect. However, such descriptions do not fully capture the intuitive idea of causality: first, often, several conditions are needed to be present for an effect to occur, and, second, the effect is often a matter of degree. In this paper, we show how to modify the current description of causality so as to …

Towards Decision Making Under Interval Uncertainty, Juan A. Lopez, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, we need to make a decision. In many cases, we know the optimal decision in situations when we know the exact value of the corresponding quantity x. However, often, we do not know the exact value of this quantity, we only know the bounds on the value x -- i.e., we know the interval containing $x$. In this case, we need to select a decision corresponding to some value from this interval. The selected value will, in general, be different from the actual (unknown) value of this quantity. As a result, the quality of our decision …

Conflict Situations Are Inevitable When There Are Many Participants: A Proof Based On The Analysis Of Aumann-Shapley Value, Sofia Holguin, Vladik Kreinovich

Departmental Technical Reports (CS)

When collaboration of several people results in a business success, an important issue is how to fairly divide the gain between the participants. In principle, the solution to this problem is known since the 1950s: natural fairness requirements lead to the so-called Shapley value. However, the computation of Shapley value requires that we can estimate, for each subset of the set of all participants, how much gain they would have gained if they worked together without others. It is possible to perform such estimates when we have a small group of participants, but for a big company with thousands of …

People Prefer More Information About Uncertainty, But Perform Worse When Given This Information: An Explanation Of The Paradoxical Phenomenon, Jieqiong Zhao, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In a recent experiment, decision makers were asked whether they would prefer having more information about the corresponding situation. They confirmed this preference, and such information was provided to them. However, strangely, the decisions of those who received this information were worse than the decisions of the control group -- that did not get this information. In this paper, we provide an explanation for this paradoxical situation.

Low-Probability High-Impact Events Are Even More Important Than It Is Usually Assumed, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A large proportion of undesirable events like earthquakes, floods, tornados occur in zones where these events are frequent. However, a significant number of such events occur in other zones, where such events are rare. For example, while most major earthquakes occur in a vicinity of major faults, i.e., on the border between two tectonic plates, some strong earthquakes also occur inside plates. We want to mitigate all undesirable events, but our resources are limited. So, to allocate these resources, we need to decide which ones are more important. For this decision, a natural idea is to use the product of …

How People Make Decisions Based On Prior Experience: Formulas Of Instance-Based Learning Theory (Ilbt) Follow From Scale Invariance, Palvi Aggarwal, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To better understand human behavior, we need to understand how people make decisions, how people select one of possible actions. This selection is usually based on predicting consequences of different actions, and these predictions are, in their turn, based on the past experience. For example, consequences that occur more frequently in the past are viewed as more probable. However, this is not just about frequency: recent observations are usually given more weight that past ones. Researchers have discovered semi-empirical formulas that describe our predictions reasonably well; these formulas form the basis of the Instance-Based Learning Theory (ILBT). In this paper, …

Zonality In Graphs, Andrew Bowling

Dissertations

Graph labeling and coloring are among the most popular areas of graph theory due to both the mathematical beauty of these subjects as well as their fascinating applications. While the topic of labeling vertices and edges of graphs has existed for over a century, it was not until 1966 when Alexander Rosa introduced a labeling, later called a graceful labeling, that brought the area of graph labeling to the forefront in graph theory. The subject of graph colorings, on the other hand, goes back to 1852 when the young British mathematician Francis Guthrie observed that the countries in a map …

Discovering Dune: Essays On Frank Herbert’S Epic Saga., Edited By Dominic J. Nardi And N. Trevor Brierly, G. Connor Salter

Mythlore: A Journal of J.R.R. Tolkien, C.S. Lewis, Charles Williams, and Mythopoeic Literature

G. Connor Salter reviews Discovering Dune: Essays on Frank Herbert’s Epic Saga, edited by Dominic J. Nardi and N. Trevor Brierly, considering its new contributions to studies of Frank Herbert's work. Essays included fit into four categories (Politics and Power, History and Religion, Biology and Ecology, and Philosophy, Choice and Ethics) and range from Herbert's use of ecology in Dune to how game theory may help explain certain characters' apparent ability to see the future. Discovering Dune also includes an appendix which contains the only up-to-date bibliography of Herbert's work (primary and secondary sources).

Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren

Senior Honors Theses

It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that …

Valuation Of Asian Options In A High Volatility Market With Jumps, Zeeshan Khalid

Theses

The evaluation of financial derivatives represents a central part of financial risk management. There are many types of derivatives among other path-dependent options. In this study, we aim at valuing Asian options. They are path dependent and have several benefits. For instance, their values are habitually lower than European options. Also, an Asian option on a commodity drops the risk value close to maturity. Though, the disadvantage is that they are in general difficult to value since the distribution of the payoff is usually unknown. It is agreed in the literature that a stochastic process with a jumps model for …

A Support Theorem For A Wave Equation, Aysha Khaled Alshamsi

Theses

It is well known that the fundamental solution to the classical wave equation Δ𝑢 (𝑥, 𝑡) − ∂_𝑡𝑡𝑢(𝑥,𝑡) = 0 is supported on the light cone {(𝑥, 𝑡) ∈ ℝ^𝑛× ℝ : ||𝑥|| = |𝑡|} if and only if the dimension 𝑛 is odd and ≥ 3. Because we are living in a 3-dimensional world we can hear each other clearly; One has a pure propagator without residual waves. In this thesis we consider the wave equation

2||𝑥||Δ_𝑘𝑢_𝑘(𝑥, 𝑡) − ∂_𝑡𝑡𝑢(𝑥,𝑡) = 0, (𝑥, 𝑡) ∈ ℝ^𝑛 × ℝ …

Number Theoretic Arithmetic Functions And Dirichlet Series, Ivan V. Morozov

Publications and Research

In this study, we will study number theoretic functions and their associated Dirichlet series. This study lay the foundation for deep research that has applications in cryptography and theoretical studies. Our work will expand known results and venture into the complex plane.

Structure Of Extremal Unit Distance Graphs, Kaylee Weatherspoon

Senior Theses

This thesis begins with a selective overview of problems in geometric graph theory, a rapidly evolving subfield of discrete mathematics. We then narrow our focus to the study of unit-distance graphs, Euclidean coloring problems, rigidity theory and the interplay among these topics. After expounding on the limitations we face when attempting to characterize finite, separable edge-maximal unit-distance graphs, we engage an interesting Diophantine problem arising in this endeavor. Finally, we present a novel subclass of finite, separable edge-maximal unit distance graphs obtained as part of the author's undergraduate research experience.

Mth 50 Syllabus, Koby Kohulan

Open Educational Resources

No abstract provided.

Dynamic Equations, Control Problems On Time Scales, And Chaotic Systems, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

The unification of integral and differential calculus with the calculus of finite differences has been rendered possible by providing a formal structure to study hybrid discrete-continuous dynamical systems besides offering applications in diverse fields that require simultaneous modeling of discrete and continuous data concerning dynamic equations on time scales. Therefore, the theory of time scales provides a unification between the calculus of the theory of difference equations with the theory of differential equations. In addition, it has become possible to examine diverse application problems more precisely by the use of dynamical systems on time scales whose calculus is made up …

Rational Functions Of Degree Five That Permute The Projective Line Over A Finite Field, Christopher Sze

USF Tampa Graduate Theses and Dissertations

Rational functions over a finite field Fq induce mappings from the projective line P1(Fq) to itself. Rational functions that permute the projective line are called permutation rational functions (PRs). The notion of permutation rational functions is a natural extension of the permutation polynomials which have been studied for over a century. Recently, PRs of degrees up to four have been determined. This dissertation is a project aimed at determining PRs of degree five.

Rational functions of degree five (excluding those that are equivalent to polynomials) are divided into five cases according to the factorization of their denominators. Our main results …

Matrix Models Of 2d Critical Phenomena, Nathan Hayford

USF Tampa Graduate Theses and Dissertations

The 2D Ising model has played an important role in the theory of phase transitions, as one of only ahandful of exactly solvable models in statistical mechanics. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to quantum gravity, the planar Ising model still had an elegant solution. The methods used by V. Kazakov and his collaborators involved the method of orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of the phase transition has been …

Recovering Generators Of Principal Ideals Using Subfield Structure And Applications To Cryptography, William Youmans

USF Tampa Graduate Theses and Dissertations

The principal ideal problem (PIP) is the problem of determining if a given ideal of a number field is principal, and if so, of finding a generator.Algorithms for resolving the PIP can be efficiently adapted to solve many hard problems in algebraic number theory, such as the computation of the class group, unit group, or $S$-unit group of a number field. The PIP is also connected to the search for approximate short vectors, known as the $\gamma$-Shortest Vector Problem ($\gamma$-SVP), in certain structured lattices called ideal lattices, which are prevalent in cryptography. We present an algorithm for resolving the PIP …

Data-Driven Learning Algorithm Via Densely-Defined Multiplication Operators And Occupation Kernels., John Kyei

USF Tampa Graduate Theses and Dissertations

Consider a nonautonomous nonlinear evolution $\dot{x}=f(x,t,\mu)$, where the vector $x(t) \in \mathbb{R}^n$ represents the state of the dynamical system at time $t$, $\mu$ contains system parameters, and $f(\cdot)$ represents a dynamic constraint. In most practical applications, the nonlinear dynamic constraint $f$ is unknown analytically. The problem of approximating $f$ directly from data measurements generated by the system is a main goal of this manuscript. In the postulates of the Nonlinear Autoregressive (NAR) framework, we show that the problem of approximating $f$ can be studied through symbols of densely defined multiplication operators over a Reproducing Kernel Hilbert Spaces (RKHS). In this …

Continuous Semi-Supervised Nonnegative Matrix Factorization, Michael R. Lindstrom, Xiaofu Ding, Feng Liu, Anand Somayajula, Deanna Needell

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Nonnegative matrix factorization can be used to automatically detect topics within a corpus in an unsupervised fashion. The technique amounts to an approximation of a nonnegative matrix as the product of two nonnegative matrices of lower rank. In certain applications it is desirable to extract topics and use them to predict quantitative outcomes. In this paper, we show Nonnegative Matrix Factorization can be combined with regression on a continuous response variable by minimizing a penalty function that adds a weighted regression error to a matrix factorization error. We show theoretically that as the weighting increases, the regression error in training …