Geometry and Topology Commons^™

Echolocation On Manifolds, Kerong Wang

Honors Theses

We consider the question asked by Wyman and Xi [WX23]: ``Can you hear your location on a manifold?” In other words, can you locate a unique point x on a manifold, up to symmetry, if you know the Laplacian eigenvalues and eigenfunctions of the manifold? In [WX23], Wyman and Xi showed that echolocation holds on one- and two-dimensional rectangles with Dirichlet boundary conditions using the pointwise Weyl counting function. They also showed echolocation holds on ellipsoids using Gaussian curvature.

In this thesis, we provide full details for Wyman and Xi's proof for one- and two-dimensional rectangles and we show that …

Extending Natural Mates In Euclidean 3-Space And Applications To Bertrand Pairs, Yun Myung Oh, Alexander Navarro

Faculty Publications

In Euclidean 3-space, a family of curves, the co-successor, is motivated and then introduced in relation to the natural mate. A complete characterization of co-successors is proved, followed by an application of the co-successor towards describing Bertrand curves and their mates.

Legendrian Torus And Cable Links, Jennifer Dalton,, John B. Etnyre, Lisa Traynor

Mathematics Faculty Research and Scholarship

We give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give the first family of links that are non-destabilizable but do not have maximal Thurston-Bennequin invariant and observe a curious distribution of Legendrian torus knots that can be realized as the components of a Legendrian torus link. This classification of Legendrian torus links leads to a classification of transversal torus links.

We also give a classification of Legendrian and transversal cable links of …

Frieze And Tiling Groups In The Lorentz-Minkowski Plane, Michael O. Lynch

Honors Undergraduate Theses

In this thesis, there is a presentation of the isometries from the Lorentz-Minkowski Plane and a solution to the Frieze Patterns. There is a suggestion for a solution for the Tiling Patterns. Since the construction of these mathematical structures is well understood in the Euclidean plane, one can follow a similar approach to the construction of such objects to find the unique number of groups that describe all possible frieze patterns while there is a suggestion of the number for the tiling case. There is a reflection of these results in a computational and cosmological context.

Farey Recursion And Hyperbolic Dehn Filling, Jose Ebenezer Martinez

Graduate Student Theses, Dissertations, & Professional Papers

In this work, we present a solution to William Thurston's edge gluing equations for Dehn fillings of hyperbolic 3-manifolds. This is done for triangulations that involve the layered solid torus. Our approach uses Farey recursive functions, and we present a Farey recursive function that provides a solution to the gluing equations for any hyperbolic Dehn filling admitting a triangulation by the layered solid torus. We provide examples that demonstrate our solution for multiple 3-manifolds, and study the roots of the corresponding Farey recursive polynomials. As an additional application of our solution, we provide a formula for the complex length of …

Id Numbers Of Lobster Graphs, Mark Anthony C. Tolentino, Luis Silvestre Jr, Richwell T. Chan Sim, Amir Jann Erikson Diga, Althea Julia R. Loyola

Mathematics Faculty Publications

No abstract provided.

Shadow Hands, John Adam

Mathematics & Statistics Faculty Publications

The article titled "Shadow hands" discusses a statement made by authors Dave Lynch and Bill Livingston in their book "Color and Light in Nature." The authors state that when you cast a shadow with your hand onto a piece of paper, the sharpness of the shadow's edge diminishes as you raise your hand above the surface. This fuzzy outer edge of the shadow is called the penumbra, and its width divided by its distance from your hand is always a constant fraction of about 1/112. The article poses two questions related to this statement and asks readers to explain it …

Shadow Hands: Solutions For Fermi Questions, September 2024, John Adam

Mathematics & Statistics Faculty Publications

The article titled "Shadow hands: Solutions for Fermi Questions, September 2024" discusses the enigmatic statement made by authors Dave Lynch and Bill Livingston in their book "Color and Light in Nature." The statement explains that the fuzzy outer edge of a shadow, known as the penumbra, is always a constant fraction of about 1/112 of its width divided by its distance from the object casting the shadow. The article provides a solution to Question 1, which asks for an explanation of this statement using elementary geometry. It also offers a solution to Question 2, which asks why the shadow of …

Super Hiper Funcion Y Super Hiper Estructura Y Sus Correspondientes Super Hiper Funcion Neutrosofica Y Super Hiper Estructura Neutrosofica, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

El n-ésimo Conjunto Potencia de un Conjunto {o Pn(S)} describe mejor nuestro mundo real, porque un sistema S (que puede ser una empresa, institución, asociación, país, sociedad, conjunto de objetos/plantas/animales/seres, conjunto de conceptos/ideas/proposiciones, etc.) está formado por subsistemas, que a su vez están formados por sub-subsistemas, y así sucesivamente. Demostramos que la Super Hiper Función es una generalización de la Función clásica, Super Función y la Hiper Función. Y el Super Hiper Álgebra, Super Hiper Gráfico son parte de la Super Hiper Estructura. Casi todas las estructuras en nuestro mundo real son Super Hiper Estructuras Neutrosóficas ya que tienen …

Q-Rung Neutrosophic Sets And Topological Spaces, Michael Gr. Voskoglou, Florentin Smarandache, Mona Mohamed

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of q-rung orthopair neutrosophic set is introduced in this paper and fundamental properties of it are studied. Also the ordinary notion of topological space is extended to q-rung orthopair neutrosophic environment, as well as the fundamental concepts of convergence, continuity, compactness and Hausdorff topological space. All these generalizations are illustrated by suitable examples.

Fundamentos De Topologias De Vanguardia (Fundamentals Of Vanguard Topologies), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Recientemente hemos encontrado nueve nuevas topologías: Topología No Estándar, Mayor Extensión de la Topología Real No Estándar, Topologías Débiles/Fuertes de Triplete Neutrosófico, Topologías Débiles/Fuertes de Triplete Neutrosófico Extendido, Topología de Dupla Neutrosófica, Topología de Dupla Neutrosófica Extendida, Topología de Multi Conjunto Neutrosófico, y recordamos y mejoramos las siete topologías previamente fundadas en los años (2019-2023), a saber: Topología Neutrosófica No Estándar, Neutro Topología, Anti Topología, Topología Neutrosófica Refinada, Topología Nítida Neutrosófica Refinada, Super Hiper Topología y Super Hiper Topología Neutrosófica. Se llaman topologías de vanguardia debido a sus formas innovadoras.

Two Applications Of The Concepts Of Pole And Polar With Respect To A Circle, Ion Pătrașcu, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this article, we present in a unified form the theoretical results regarding the concepts of pole and polar with respect to a circle, and as applications of these, we offer a demonstration of Newton’s theorem related to the circumscribed quadrilateral and a demonstration of a theorem usually derived as a particular case of Pascal’s theorem related to the inscribed hexagon.

Pappus Of Alexandria, Book Iii Of The Mathematical Collection, Pappus Of Alexandria, John B. Little

Holy Cross Bookshelf

John B. Little is the translator.

This is a translation of Book III of the Mathematical Collection by Pappus of Alexandria (ca. 290 - 350 CE) from the original Greek to English, following the edition of Friedrich Hultsch. While other books of the Mathematical Collection have been translated into English and short quotations from Book III have appeared in a number of places (see the Introduction), to my knowledge, no complete English translation of Book III has been published. Pappus was very influential as a sort of conduit between knowledge preserved from ancient Greek mathematics and European mathematicians in the …

Aspects Of Stochastic Geometric Mechanics In Molecular Biophysics, David Frost

All Dissertations

In confocal single-molecule FRET experiments, the joint distribution of FRET efficiency and donor lifetime distribution can reveal underlying molecular conformational dynamics via deviation from their theoretical Forster relationship. This shift is referred to as a dynamic shift. In this study, we investigate the influence of the free energy landscape in protein conformational dynamics on the dynamic shift by simulation of the associated continuum reaction coordinate Langevin dynamics, yielding a deeper understanding of the dynamic and structural information in the joint FRET efficiency and donor lifetime distribution. We develop novel Langevin models for the dye linker dynamics, including rotational dynamics, based …

The Construction Of Khovanov Homology, Shiaohan Liu

Master's Theses

Knot theory is a rich topic in topology that studies the how circles can be embedded in Euclidean 3-space. One of the main questions in knot theory is how to distinguish between different types of knots efficiently. One way to approach this problem is to study knot invariants, which are properties of knots that do not change under a standard set of deformations. We give a brief overview of basic knot theory, and examine a specific knot invariant known as Khovanov homology. Khovanov homology is a homological invariant that refines the Jones polynomial, another knot invariant that assigns a Laurent …

An Exposition Of The Curvature Of Warped Product Manifolds, Angelina Bisson

Electronic Theses, Projects, and Dissertations

The field of differential geometry is brimming with compelling objects, among which are warped products. These objects hold a prominent place in differential geometry and have been widely studied, as is evident in the literature. Warped products are topologically the same as the Cartesian product of two manifolds, but with distances in one of the factors in skewed. Our goal is to introduce warped product manifolds and to compute their curvature at any point. We follow recent literature and present a previously known result that classifies all flat warped products to find that there are flat examples of warped products …

Complex Dimensions Of 100 Different Sierpinski Carpet Modifications, Gregory Parker Leathrum

Master's Theses

We used Dr. M. L. Lapidus's Fractal Zeta Functions to analyze the complex fractal dimensions of 100 different modifications of the Sierpinski Carpet fractal construction. We will showcase the theorems that made calculations easier, as well as Desmos tools that helped in classifying the different fractals and computing their complex dimensions. We will also showcase all 100 of the Sierpinski Carpet modifications and their complex dimensions.

Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu

Rose-Hulman Undergraduate Mathematics Journal

In this manuscript, we present computational results approximating the Hausdorff dimension for the limit sets of complex Kleinian groups. We apply McMullen's eigenvalue algorithm \cite{mcmullen} in symmetric and non-symmetric examples of complex Kleinian groups, arising in both real and complex hyperbolic space. Numerical results are compared with asymptotic estimates in each case. Python code used to obtain all results and figures can be found at \url{https://github.com/WXML-HausDim/WXML-project}, all of which took only minutes to run on a personal computer.

A Natural Pseudometric On Homotopy Groups Of Metric Spaces, Jeremy Brazas, Paul Fabel

Mathematics Faculty Publications

For a path-connected metric space (X, d), the n-th homotopy group π n ( X) inherits a natural pseudometric from the n-th iterated loop space with the uniform metric. This pseudometric gives π n ( X) the structure of a topological group and when X is compact, the induced pseudometric topology is independent of the metric d. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on π n ( X). Our main result is that the pseudometric topology agrees with the shape topology on π n ( X) if X …

Elliptic Triangles Which Are Congruent To Their Polar Triangles, Jarrad S. Epkey, Morgan Nissen, Noelle K. Kaminski, Kelsey R. Hall, Nicholas Grabill

Rose-Hulman Undergraduate Mathematics Journal

We prove that an elliptic triangle is congruent to its polar triangle if and only if six specific Wallace-Simson lines of the triangle are concurrent. (If a point projected onto a triangle has the three feet of its projections collinear, that line is called a Wallace-Simson line.) These six lines would be concurrent at the orthocenter. The six lines come from projecting a vertex of either triangle onto the given triangle. We describe how to construct such triangles and a dozen Wallace-Simson lines.