Mathematics Commons^™

Dimension Theory Of Conformal Iterated Function Systems, Sharon Sneha Spaulding

Honors Scholar Theses

This thesis is an expository investigation of the conformal iterated function system (CIFS) approach to fractals and their dimension theory. Conformal maps distort regions, subject to certain constraints, in a controlled way. Let $\mathcal{S} = (X, E, \{\phi_e\}_{e \in E})$ be an iterated function system where $X$ is a compact metric space, $E$ is a countable index set, and $\{\phi_e\}_{e \in E}$ is a family of injective and uniformly contracting maps. If the family of maps $\{\phi_e\}_{e \in E}$ is also conformal and satisfies the open set condition, then the distortion properties of conformal maps can be extended to the …

Quantum Dimension Polynomials: A Networked-Numbers Game Approach, Nicholas Gaubatz

Honors College Theses

The Networked-Numbers Game--a mathematical "game'' played on a simple graph--is incredibly accessible and yet surprisingly rich in content. The Game is known to contain deep connections to the finite-dimensional simple Lie algebras over the complex numbers. On the other hand, Quantum Dimension Polynomials (QDPs)--enumerative expressions traditionally understood through root systems--corresponding to the above Lie algebras are complicated to derive and often inaccessible to undergraduates. In this thesis, the Networked-Numbers Game is defined and some known properties are presented. Next, the significance of the QDPs as a method to count combinatorially interesting structures is relayed. Ultimately, a novel closed-form expression of …

A Priori And A Posteriori Error Estimates For The Quad-Curl Eigenvalue Problem, Lixiu Wang, Qian Zhang, Jiguang Sun, Zhimin Zhang

Michigan Tech Publications, Part 1

In this paper, we consider a priori and a posteriori error estimates of the H(curl2)-conforming finite element when solving the quad-curl eigenvalue problem. An a priori estimate of eigenvalues with convergence order 2(s 1) is obtained if the corresponding eigenvector u a Hs 1(Ω) and-u a Hs(Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the errors of eigenvectors in the energy norm and upper bounds for the errors of eigenvalues. Numerical examples are presented for validation.

Waring Rank And Apolarity Of Some Symmetric Polynomials, Max Brian Sullivan

Boise State University Theses and Dissertations

We examine lower bounds for the Waring rank for certain types of symmetric polynomials. The first are Schur polynomials, a symmetric polynomial indexed by integer partitions. We prove some results about the Waring rank of certain types of Schur polynomials, based on their integer partition. We also make some observations about the Waring rank in general for Schur polynomials, based on the shape of their Semistandard Young Tableaux. The second type of polynomials we refer to as a Power of a Fermat-type polynomial, or a PFT polynomial. This is a Fermat type (or power sum) polynomial over n variables with …

Symmetric Presentations Of Finite Groups And Related Topics, Samar Mikhail Kasouha

Electronic Theses, Projects, and Dissertations

A progenitor is an infinite semi-direct product of the form m∗n : N, where N ≤ Sn and m∗n : N is a free product of n copies of a cyclic group of order m. A progenitor of this type, in particular 2∗n : N, gives finite non-abelian simple groups and groups involving these, including alternating groups, classical groups, and the sporadic group. We have conducted a systematic search of finite homomorphic images of numerous progenitors. In this thesis we have presented original symmetric presentations of the sporadic simple groups, M12, J1 as homomorphic images of the progenitor 2∗12 : …

Dynamical Systems Analysis In Adaptive And Metapopulation Ecology With Applications To Conservation Management, Guenchik Grosklos

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The ability for a species to persist largely relies on how well they adapt to the environment and their interactions with local and global communities. Specifically, if adaptation occurs quickly enough or nearby communities sufficiently promote growth rates, populations at risk of extinction may persist. In this dissertation, we first develop a method that estimates and compares rates of change in time series data of population densities and measurable traits (phenotypes). Additionally, we compare between genetic (evolutionary) and non-genetic (plastic) trait change to determine whether phenotypes change faster when driven by evolutionary or plastic change. We then focus on metapopulation …

Positive Solutions To Semilinear Elliptic Equations With Logistic-Type Nonlinearities And Harvesting In Exterior Domains, Eric Jameson

UNLV Theses, Dissertations, Professional Papers, and Capstones

Existing results provide the existence of positive solutions to a class of semilinear elliptic PDEs with logistic-type nonlinearities and harvesting terms both in RN and in bounded domains U ⊂ RN with N ≥ 3, when the carrying capacity of the environment is not constant. We consider these same equations in the exterior domain Ω, defined as the complement of the closed unit ball in RN , N ≥ 3, now with a Dirichlet boundary condition. We first show that the existing techniques forsolving these equations in the whole space RN can be applied to the exterior domain with some …

A Survey Of The Br´Ezis-Nirenberg Problem And Related Theorems, Edward Huynh

UNLV Theses, Dissertations, Professional Papers, and Capstones

Nonlinear elliptic partial differential equations on bounded domains arise in several different areas of mathematics that include geometry, mathematical physics, and the calculus of variations. The Br ́ezis-Nirenberg problem is concerned with a boundary-value problem that is intimately connected to the existence of positive solutions of the Yamabe problem, of non-minimal solutions to Yang-Mills functionals, and of extremal functions to several important inequalities. Results on existence and uniqueness have been obtained in cases when the exponent is sub-critical, but such results have not been obtained when the exponent is critical due to a lack of compactness. The earliest results obtained …

Some Graph Laplacians And Variational Methods Applied To Partial Differential Equations On Graphs, Daniel Anthony Corral

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this dissertation we will be examining partial differential equations on graphs. We start by presenting some basic graph theory topics and graph Laplacians with some minor original results. We move on to computing original Jost graph Laplacians of friendly labelings of various finite graphs. We then continue on to a host of original variational problems on a finite graph. The first variational problem is an original basic minimization problem. Next, we use the Lagrange multiplier approach to the Kazdan-Warner equation on a finite graph, our original results generalize those of Dr. Grigor’yan, Dr. Yang, and Dr. Lin. Then we …

Independent Dominating Sets In Unicyclic Graphs, Rachel Lopez

Theses, Dissertations and Culminating Projects

Wilf found the maximum number of independent dominating sets of a tree using algebraic methods, while Sagan gave an elementary proof. In this thesis, we maximize the number of independent dominating sets of unicyclic graphs, giving a new proof of a result of Jou and Chang. In our proof, we are able to reduce the problem to finding independent dominating sets of single-legged caterpillar graphs. We also study the number of single-legged caterpillar graphs, both oriented and unoriented, which are related to the Fibonacci Sequence. Finally, this thesis also examines the domination ratio in unicyclic graphs. The domination ratio is …

Modeling Functions Into An Angular Displacement Of An Elastic Pendulum, Brenda Lee Garcia

Theses and Dissertations

In this thesis we study the relation between analytic signals and a variety of pendulum systems. The representation of a signal as a pair of time varying amplitude and phase has been well studied and often related to linear mass spring systems. The differential equations describing pendulum systems are nonlinear and we provide analytical and numerical results regarding interpretation about the amplitude and the phase of signals in different pendulum settings. We report an explicit solution of the Elastic Pendulum problem in the case of linear phase. We develop an experimental procedure to piece-wise approximate bounded functions on a partition …

Sangaku In Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid, Nathan Hartmann

Honors College Theses

When the country of Japan was closed from the rest of the world from 1603 until
1867 during the Edo period, the field of mathematics developed in a different way
from how it developed in the rest of the world. One way we see this development
is through the sangaku, the thousands of geometric problems hung in various Shinto and Buddhist temples throughout the country. Written on wooden tablets by people from numerous walks of life, all these problems hold true within Euclidean geometry. During the 1800s, while Japan was still closed, non-Euclidean geometries began to develop across the …

Visions Of Science: An Art Historical Exploration Of Medieval Scientific Manuscripts, Olivia Brock

Undergraduate Honors Capstone Projects

The late 16th century saw a new movement in the world of science to push scientific ideas and practice out of academia and into the hands of the layman. No longer were scholars the sole proprietors of science –everyday laborers, craftsman, and artists now had practical scientific principles at their fingertips that they could incorporate into their professions. This new spread of science was facilitated in several ways, including the publication of books incorporating detailed, explanatory images, new utilitarian instruments, and public lectures. Though science was disseminated through a variety of means, I have been particularly interested in the ways …

Minimal Inscribed Polyforms, Jack Hanke

Honors Scholar Theses

A polyomino of size n is constructed by joining n unit squares together by their edge to form a shape in the plane. This thesis will first examine the formal definition of a polyomino and the common equivalence classes polyominos are enumerated under. We then turn to polyomino families, and provide exact enumeration results for certain families, including the minimal inscribed polyominos. Next we will generalize polyominos to polyforms, and provide novel formulae for polyform analogues of minimal inscribed polyominos. Finally, we discuss some further questions concerning minimal inscribed polyforms.

A Decomposition Formula For The Multi-Soliton Solutions To The 'Good' Boussinesq Equation, Aldo Gonzalez

Theses and Dissertations

In this thesis, we relate multi-soliton waves generated by the 'good' Boussinesq equation to the distribution functions in the classical linear Schrödinger equation. The linear Schrödinger equation describes the distribution of a particle or particles in a particular environment. The Schrödinger equation is linear, the superposition principle of the solutions, especially the eigenfunctions is nonlinear and we will show that we may observe similar behavior in the solutions of the Boussinesq equations for soliton waves. The work extends the study of two-soliton solutions to the Boussinesq equation to the case of three-soliton solutions. …

Iterated Rascal Triangles, Jena M. Gregory

Theses and Dissertations

We introduce a sequence of number triangles, {Ri} i=0 infty , such that the entries of each share a common generalized recurrence relation. R1 is the Rascal triangle and as i grows large, Ri becomes Pascal's triangle. For all i, we provide a combinatorial interpretation and find closed-term formulas for the entries of Ri . Our proofs rely on generating functions and other combinatorial arguments.

Pentagonal Tilings Of The Plane, Ariana T. Hinojosa

Theses and Dissertations

Tilings are mathematical objects that allow us to use geometry to visualize interaction between objects as well as to create artistic realization of mathematical objects in the plane and in the space.

We will focus on tilings of the plane that use only one type of convex pentagonal tile each, the pentagonal tilings. There are fifteen types of pentagonal tiles, with each containing their own set of restrictions. The main result of this thesis is an interactive realization of all fifteen types of pentagonal tiles using GeoGebra.

A Gpu Accelerated Genetic Algorithm For The Construction Of Hadamard Matrices, Raven I. Ruiz

Theses and Dissertations

Hadamard matrices are square matrices with +1 and -1 entries and with columns that are mutually orthogonal. The applications include signal processing and quantum computing. There are several methods for constructing Hadamard matrices of order 2k for every positive integer k. The Hadamard conjecture proposes that there are also Hadamard matrices of order 4k for every positive integer k. We use a genetic algorithm to construct (search for) Hadamard Matrices. The initial population of random matrices is generated to have a balanced number of +1 and -1 entries in each column. Several fitness functions are implemented exploiting …

Boundary Feedback Control Of The 3d Navier-Stokes Equations, Camille Renee Vasquez

Theses and Dissertations

We present a boundary feedback stabilization of the parabolic steady state profile of the incompressible Navier-Stokes Equations in a three-dimensional channel flow. The decentralized, static boundary feedback control laws are derived using Lyapunov technique. While the theoretical results are limited to stability enhancement for small Reynolds numbers, extensive numerical simulations and visualizations demonstrate the effectiveness of the proposed feedback law even in cases when the uncontrolled flow is turbulent.

Homomorphic Images And Related Topics, Alejandro Martinez

Electronic Theses, Projects, and Dissertations

In this thesis, we have demonstrated our method of writing symmetric presentations of permutation progenitors, finding monomial representations and symmetric presentations of monomial progenitors. We have also explained how various types of additional relations are found. We have discovered original symmetric presentations and original constructions of numerous groups.