Mathematics Commons^™

Area And Volume Where Do The Formulas Come From?, Roger Yarnell

Masters Essays

No abstract provided.

Sisteme Vibrante Trilobice, Florentin Smarandache, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Algorithmic Foundations Of Heuristic Search Using Higher-Order Polygon Inequalities, Newton Henry Campbell Jr.

CCAC Theses and Dissertations

The shortest path problem in graphs is both a classic combinatorial optimization problem and a practical problem that admits many applications. Techniques for preprocessing a graph are useful for reducing shortest path query times. This dissertation studies the foundations of a class of algorithms that use preprocessed landmark information and the triangle inequality to guide A* search in graphs. A new heuristic is presented for solving shortest path queries that enables the use of higher order polygon inequalities. We demonstrate this capability by leveraging distance information from two landmarks when visiting a vertex as opposed to the common single landmark …

Analytic Geometry And Calculus I, Shaun V. Ault, Sudhir Goel

Mathematics Grants Collections

This Grants Collection for Analytic Geometry and Calculus I was created under a Round Two ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Differential Geometry: Curvature And Holonomy, Austin Christian

Math Theses

We develop the basic language of differential geometry, including smooth manifolds, bundles, and differential forms. Once this background is established, we explore parallelism in smooth manifolds -- in particular, in Riemannian manifolds -- and conclude by presenting a proof of the Ambrose-Singer theorem, which relates parallelism (holonomy) to curvature in principal bundles.

Schwarzschild Spacetime And Friedmann-Lemaitre-Robertson-Walker Cosmology, Zachary Cohen

Honors Scholar Theses

The advent of General Relativity via Einstein's field equations revolutionized our understanding of gravity in our solar system and universe. The idea of General Relativity posits that gravity is entirely due to the geometry of the universe -- that is, the mass distribution throughout the universe results in the "curving" of spacetime, which gives us the physics we see on a large scale. In the framework of General Relativity, we find that the universe behaves differently than was predicted in the model of gravitation developed by Newton. We will derive the general relativistic model for a simple system near a …

Identifying Similar Polygons: Comparing Prospective Teachers’ Routines With A Mathematician’S, Sasha Wang

Mathematics Faculty Publications and Presentations

This paper reports two prospective teachers' and a mathematician's ways of identifying similar triangle and hexagons through the analysis of routines, a characteristic of geometric discourse. The findings show that visual recognition was a common approach for the mathematician as wells as the two prospective teachers. However, when asked for justification, their routines of identifying similar polygons diverged. The paper also discusses the implication of classroom discourse practices to enhance prospective teachers' communication and reasoning skills while learning geometric concepts such as similarity.

Probleme De Geometrie Și Trigonometrie, Compilate Și Rezolvate, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

255 Compiled And Solved Problems In Geometry And Trigonometry, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

How Do They Know It Is A Parallelogram? Analysing Geometric Discourse At Van Hiele Level 3, Sasha Wang, Margaret Kinzel

Mathematics Faculty Publications and Presentations

In this article, we introduce Sfard's discursive framework and use it to investigate prospective teachers' geometric discourse in the context of quadrilaterals. In particular, we focus on describing and analysing two participants' use of mathematical words and substantiation routines related to parallelograms and their properties at van Hiele level 3 thinking. Our findings suggest that a single van Hiele level of thinking encompasses a range of complexity of reasoning and differences in discourse and thus a deeper investigation of students' mathematical thinking within assigned van Hiele levels is warranted.

A Kleinian Approach To Fundamental Regions, Joshua L. Hidalgo

Electronic Theses, Projects, and Dissertations

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be …

A Volume Bound For Montesinos Links, Kathleen Arvella Finlinson

Theses and Dissertations

The hyperbolic volume of a knot complement is a topological knot invariant. Futer, Kalfagianni, and Purcell have estimated the volumes of Montesinos link complements for Montesinos links with at least three positive tangles. Here we extend their results to all hyperbolic Montesinos links.

Propeller, Joel Kahn

The Transdisciplinary STEAM+ Journal

This image is based on several different algorithms interconnected within a single program in the language BASIC-256. The fundamental structure involves a tightly wound spiral working outwards from the center of the image. As the spiral is drawn, different values of red, green and blue are modified through separate but related processes, producing the changing appearance. Algebra, trigonometry, geometry, and analytic geometry are all utilized in overlapping ways within the program. As with many works of algorithmic art, small changes in the program can produce dramatic alterations of the visual output, which makes lots of variations possible.

Hyperbolicity Equations For Knot Complements, Christopher Martin Jacinto

Theses Digitization Project

This study analyzes Carlo Petronio's paper, An Algorithm Producing Hyperbolicity Equations for a Link Complement in S³. Using the figure eight knot as an example, we will explain how Petronio's algorithm was able to decompose the knot complement of an alternating knot into tetrahedra. Then, using the vertex invariants of these tetrahedra, we will explain how Petronio was able to create hyperbolicity equations.

Elementary College Geometry, Henry Africk

Open Educational Resources

This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra.

The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem-proving or formal methods of reasoning. However the topics are …

Prospective Teachers’ Learning In Geometry: Changes In Discourse And Thinking, Sasha Wang

Mathematics Faculty Publications and Presentations

This study investigates changes in prospective teachers’ levels of geometric thinking and the development of their geometric discourses in the classification of quadrilaterals. To examine prospective teachers’ thinking about geometry, this study connects Sfard’s discursive framework to another, namely the van Hiele theory. Findings of the study reveal discursive similarities and differences in participants’ geometric discourses within the same van Hiele level, as well as changes in geometric discourse as a result of changes in levels of geometric thinking. The study also investigates the usefulness of a discursive framework in providing rich descriptions of prospective teachers’ thinking processes.

Degree Constrained Triangulation, Roshan Gyawali

UNLV Theses, Dissertations, Professional Papers, and Capstones

Triangulation of simple polygons or sets of points in two dimensions is a widely investigated problem in computational geometry. Some researchers have considered variations of triangulation problems that include minimum weight triangulation, de-launay triangulation and triangulation refinement. In this thesis we consider a constrained version of the triangulation problem that asks for triangulating a given domain (polygon or point sites) so that the resulting triangulation has an increased number of even degree vertices. This problem is called Degree Constrained Triangulation (DCT). We propose four algorithms to solve DCT problems. We also present experimental results based on the implementation of the …

Modeling Spatial Uncertainties In Geospatial Data Fusion And Mining, Boris Kovalerchuk, Leonid Perlovsky, Michael Kovalerchuk

All Faculty Scholarship for the College of the Sciences

Geospatial data analysis relies on Spatial Data Fusion and Mining (SDFM), which heavily depend on topology and geometry of spatial objects. Capturing and representing geometric characteristics such as orientation, shape, proximity, similarity, and their measurement are of the highest interest in SDFM. Representation of uncertain and dynamically changing topological structure of spatial objects including social and communication networks, roads and waterways under the influence of noise, obstacles, temporary loss of communication, and other factors. is another challenge. Spatial distribution of the dynamic network is a complex and dynamic mixture of its topology and geometry. Historically, separation of topology and geometry …

Some Aspects Of Toric Topology., Soumen Sarkar Dr.

Doctoral Theses

The main goal of this thesis is to study the topology of torus actions on manifolds and orbifolds. In algebraic geometry actions of the torus (C * ) n on algebraic varieties with nice properties produce bridges between geometry and combinatorics see [Dan78], [Oda88] and [Ful93]. We see a similar bridge called moment map for Hamiltonian action of compact torus on symplectic manifolds see [Aud91] and [Gui94]. In particular whenever the manifold is compact the image of moment map is a simple polytope, the orbit space of the action. A topological counterpart called quasitoric manifolds, a class of topological manifolds …

The Geometry Of Homological Triangles, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

This book is addressed to students, professors and researchers of geometry, who will find herein many interesting and original results. The originality of the book The Geometry of Homological Triangles consists in using the homology of triangles as a “filter” through which remarkable notions and theorems from the geometry of the triangle are unitarily passed. Our research is structured in seven chapters, the first four are dedicated to the homology of the triangles while the last ones to their applications. In the first chapter one proves the theorem of homological triangles (Desargues, 1636), one survey the remarkable pairs of homological …

Vortex Patterns Beyond Hypergeometric, Andrei Ludu

Publications

We prove that loop vortices are created by a point-like magnetic dipole in an infinite superconductor space. The geometry of the vortex system is obtained through analytic solutions of the linearized Ginzburg-Landau equation described in terms of Heun functions, generalizing the traditional hypergeometric behavior of such magnetic singularity.

The Hyperboloid Model Of Hyperbolic Geometry, Zachery S. Lane Solheim

EWU Masters Thesis Collection

"The main goal of this thesis is to introduce and develop the hyperboloid model of Hyperbolic Geometry. In order to do that, some time is spent on Neutral Geometry as well as Euclidean Geometry; these are used to build several models of Hyperbolic Geometry. At this point the hyperboloid model is introduced, related to the other models visited, and developed using some concepts from physics as aids. After the development of the hyperboloid model, Fuchsian groups are briefly discussed and the more familiar models of Hyperbolic Geometry are further investigated"--Document.

Quantum Stochastic Flows: Trotter Product Formula, Dilations And Quantum Brownian Motion., Biswarup Das Dr.

Doctoral Theses

Motivated by the major role played by probabilistic models in many areas of science, several quantum (i.e. non-commutative) generalizations of classical probability have been attempted by a number of mathematicians. The pioneering works of K.R. Parthasarathy, L. Accardi, R.L. Hudson, P.A. Meyer and others led to the development of one such non-commutative model called ‘quantum probability’ which has a very rich theory of quantum stochastic calculus a la Hudson and Parthasarathy. Within the framework of quantum stochastic calculus, the ‘grand design’ that engages us is the canonical construction and study of ∗-homomorphic flows (jt)t≥0 on a given C ∗ or …

Curd, Spencer, 1788-1832 (Sc 966), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Finding aid and full text (click on "Additional Files" below) for Manuscripts Small Collection 966. Ciphering book (94 p.), focusing chiefly on surveying and navigating rules for solving problems as well as giving specific problems, of Spencer Curd, Logan County, Kentucky, with accompanying genealogical data.

A Space-Filling, Nonregular Tetrahedron, Margaret Cagle, Joyce Frost, Christine Latulippe, Darryl H. Yong

All HMC Faculty Publications and Research

This activity is an investigation of a special nonregular tetrahedron that can be arranged to fill space without leaving any internal gaps in the same way that certain planar figures tessellate the plane. These tetrahedra can be connected together with hinges to make fun and interesting puzzles. More background information can be found in the paper "An Amazing, Space-Filling, Non-Regular Tetrahedron" by Joyce Frost and Peg Cagle, published by the IAS/Park City Mathematics Institute (available at mathforum.org/pcmi/hstp/resources/dodeca/).

Excess Porteous, Coherent Porteous, And The Hyperelliptic Locus In M3, Thomas S. Bleier

Mathematics - Dissertations

In the moduli space of curves of genus 3, the locus of hyperelliptic curves forms a divisor, that is a closed subscheme of codimension 1. J. Harris and I. Morrison compute an expression for the class of this divisor, in the Chow ring of the moduli space, using a map of vector bundles and by applying the Thom-Porteous formula to determine an expression for a certain degeneracy locus of this map. One would like to extend their idea in order to compute an expression for the divisor associated to the closure of the hyperelliptic locus, in the Chow ring of …

A Locus Construction In The Hyperbolic Plane For Elliptic Curves With Cross-Ratio On The Unit Circle, Lyudmila Shved

Theses Digitization Project

This project demonstrates how an elliptic curve f defined by invariance under two involutions can be represented by the locus of circumcenters of isosceles triangles in the hyperbolic plane, using inversive model.

Constructible Numbers: Euclid And Beyond, Joshua Scott Marcy

Theses Digitization Project

The purpose of this project is to demonstrate first why trisection for an arbitrary angle is impossible with compass and straightedge and second how trisection does become possible if a marked ruler is used instead.

Geodesics Of Surface Of Revolution, Wenli Chang

Theses Digitization Project

The purpose of this project was to study the differential geometry of curves and surfaces in three-dimensional Euclidean space. Some important concepts such as, Curvature, Fundamental Form, Christoffel symbols, and Geodesic Curvature and equations are explored.

Problems With And Without … Problems!, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This book is addressed to College honor students, researchers, and professors. It contains 136 original problems published by the author in various scientific journals around the world. The problems could be used to preparing for courses, exams, and Olympiads in mathematics. Many of these have a generalized form. For each problem we provide a detailed solution.

I was a professeur coopérant between 1982-1984, teaching mathematics in French language at Lycée Sidi EL Hassan Lyoussi in Sefrou, Province de Fès, Morocco. I used many of these problems for selecting and training, together with other Moroccan professors, in Rabat city, of the …