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Articles 1 - 30 of 38
Full-Text Articles in Physical Sciences and Mathematics
Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers
Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers
Milne Open Textbooks
Differential Calculus: From Practice to Theory covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the …
Spectral Sequences And Khovanov Homology, Zachary J. Winkeler
Spectral Sequences And Khovanov Homology, Zachary J. Winkeler
Dartmouth College Ph.D Dissertations
In this thesis, we will focus on two main topics; the common thread between both will be the existence of spectral sequences relating Khovanov homology to other knot invariants. Our first topic is an invariant MKh(L) for links in thickened disks with multiple punctures. This invariant is different from but inspired by both the Asaeda-Pryzytycki-Sikora (APS) homology and its specialization to links in the solid torus. Our theory will be constructed from a Z^n-filtration on the Khovanov complex, and as a result we will get various spectral sequences relating MKh(L) to Kh(L), AKh(L), and APS(L). Our …
How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli
How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli
The Review: A Journal of Undergraduate Student Research
The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph …
Exploration Of Piccirillo's Trick On Low Crossing Number Knots, Gabriel Adams
Exploration Of Piccirillo's Trick On Low Crossing Number Knots, Gabriel Adams
Honors Theses
Piccirillo recently discovered a process that can be applied to an unknotting number one knot to convert it into a different knot called a Piccirillo dual. Piccirillo duals have been shown to have the same n-trace and the same sliceness. However, exploration and knowledge of this process is limited. We were able to generate the Piccirillo duals for several low-crossing number knots. We offer the foundation for and explain how to follow the Piccirillo process and generate Piccirillo duals. This talk assumes little knowledge of knot theory and concisely gives newcomers a clear introduction to get started working with Piccirillo …
Stroke Clustering And Fitting In Vector Art, Khandokar Shakib
Stroke Clustering And Fitting In Vector Art, Khandokar Shakib
Senior Independent Study Theses
Vectorization of art involves turning free-hand drawings into vector graphics that can be further scaled and manipulated. In this paper, we explore the concept of vectorization of line drawings and study multiple approaches that attempt to achieve this in the most accurate way possible. We utilize a software called StrokeStrip to discuss the different mathematics behind the parameterization and fitting involved in the drawings.
Perceiving Mathematics And Art, Edmund Harriss
Perceiving Mathematics And Art, Edmund Harriss
Mic Lectures
Mathematics and art provide powerful lenses to perceive and understand the world, part of an ancient tradition whether it starts in the South Pacific with tapa cloth and wave maps for navigation or in Iceland with knitting patterns and sunstones. Edmund Harriss, an artist and assistant clinical professor of mathematics in the Fulbright College of Arts and Sciences, explores these connections in his Honors College Mic lecture.
Fern Or Fractal... Or Both?, Christina Babcock
Fern Or Fractal... Or Both?, Christina Babcock
Research and Scholarship Symposium Posters
Fractals are series of self similar sets and can be found in nature. After researching the Barnsley Fern and the iterated function systems using to create the fractal, I was able to apply what I learned to create a fractal shell. This was done using iterated function systems, matrices, random numbers, and Python coding.
Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly
Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly
Electronic Theses and Dissertations
Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based …
Properties Of Functionally Alexandroff Topologies And Their Lattice, Jacob Scott Menix
Properties Of Functionally Alexandroff Topologies And Their Lattice, Jacob Scott Menix
Masters Theses & Specialist Projects
This thesis explores functionally Alexandroff topologies and the order theory asso- ciated when considering the collection of such topologies on some set X. We present several theorems about the properties of these topologies as well as their partially ordered set.
The first chapter introduces functionally Alexandroff topologies and motivates why this work is of interest to topologists. This chapter explains the historical context of this relatively new type of topology and how this work relates to previous work in topology. Chapter 2 presents several theorems describing properties of functionally Alexandroff topologies ad presents a characterization for the functionally Alexandroff topologies …
Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg
Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg
George K. Thiruvathukal
This paper shows how students can be guided to integrate elementary mathematical analyses with motion planning for typical educational robots. Rather than using calculus as in comprehensive works on motion planning, we show students can achieve interesting results using just simple linear regression tools and trigonometric analyses. Experiments with one robotics platform show that use of these tools can lead to passable navigation through dead reckoning even if students have limited experience with use of sensors, programming, and mathematics.
Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg
Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg
Computer Science: Faculty Publications and Other Works
This paper shows how students can be guided to integrate elementary mathematical analyses with motion planning for typical educational robots. Rather than using calculus as in comprehensive works on motion planning, we show students can achieve interesting results using just simple linear regression tools and trigonometric analyses. Experiments with one robotics platform show that use of these tools can lead to passable navigation through dead reckoning even if students have limited experience with use of sensors, programming, and mathematics.
Barrier Graphs And Extremal Questions On Line, Ray, Segment, And Hyperplane Sensor Networks, Kirk Anthony Boyer
Barrier Graphs And Extremal Questions On Line, Ray, Segment, And Hyperplane Sensor Networks, Kirk Anthony Boyer
Electronic Theses and Dissertations
A sensor network is typically modeled as a collection of spatially distributed objects with the same shape, generally for the purpose of surveilling or protecting areas and locations. In this dissertation we address several questions relating to sensors with linear shapes: line, line segment, and rays in the plane, and hyperplanes in higher dimensions.
First we explore ray sensor networks in the plane, whose resilience is the number of sensors that must be crossed by an agent traveling between two known locations. The coverage of such a network is described by a particular tripartite graph, the barrier graph of the …
Applications Of Geometric And Spectral Methods In Graph Theory, Lauren Morey Nelsen
Applications Of Geometric And Spectral Methods In Graph Theory, Lauren Morey Nelsen
Electronic Theses and Dissertations
Networks, or graphs, are useful for studying many things in today’s world. Graphs can be used to represent connections on social media, transportation networks, or even the internet. Because of this, it’s helpful to study graphs and learn what we can say about the structure of a given graph or what properties it might have. This dissertation focuses on the use of the probabilistic method and spectral graph theory to understand the geometric structure of graphs and find structures in graphs. We will also discuss graph curvature and how curvature lower bounds can be used to give us information about …
Optimal Weak Parallelogram Constants For L-P Spaces, Raymond Cheng, Javad Mashreghi, William T. Ross
Optimal Weak Parallelogram Constants For L-P Spaces, Raymond Cheng, Javad Mashreghi, William T. Ross
Department of Math & Statistics Faculty Publications
Inspired by Clarkson's inequalities for L-p and continuing work from [5], this paper computes the optimal constant C in the weak parallelogram laws parallel to f + g parallel to(r )+ C parallel to f - g parallel to(r )= 2(r-1 )(parallel to f parallel to(r) + parallel to g parallel to(r)) for the L-p spaces, 1 < p < infinity.
Advanced Enrichment Topics In An Honors Geometry Course, Kayla Woods
Advanced Enrichment Topics In An Honors Geometry Course, Kayla Woods
Masters Essays
No abstract provided.
An Introduction To Topology For The High School Student, Nathaniel Ferron
An Introduction To Topology For The High School Student, Nathaniel Ferron
Masters Essays
No abstract provided.
Student-Created Test Sheets, Samuel Laderach
Student-Created Test Sheets, Samuel Laderach
Honors Projects
Assessment plays a necessary role in the high school mathematics classroom, and testing is a major part of assessment. Students often struggle with mathematics tests and examinations due to math and test anxiety, a lack of student learning, and insufficient and inefficient student preparation. Practice tests, teacher-created review sheets, and student-created test sheets are ways in which teachers can help increase student performance, while ridding these detrimental factors. Student-created test sheets appear to be the most efficient strategy, and this research study examines the effects of their use in a high school mathematics classroom.
Spectrally Similar Incommensurable 3-Manifolds, David Futer, Christian Millichap
Spectrally Similar Incommensurable 3-Manifolds, David Futer, Christian Millichap
Faculty Publications
Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds Nn and Nµn whose volume is approximately n and whose length spectra agree up to length n.
Both Nn and Nµn are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that …
Mutations And Short Geodesics In Hyperbolic 3-Manifolds, Christian Millichap
Mutations And Short Geodesics In Hyperbolic 3-Manifolds, Christian Millichap
Faculty Publications
In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.
The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least …
Area And Volume Where Do The Formulas Come From?, Roger Yarnell
Area And Volume Where Do The Formulas Come From?, Roger Yarnell
Masters Essays
No abstract provided.
Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden
Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden
Faculty Publications
In this paper, we show that any nonarithmetic hyperbolic 2-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of 3-manifolds with nontrivial JSJ-decomposition and rank-two fundamental groups. We also show that the only commensurable hyperbolic 2-bridge link complements are the figure-eight knot complement and the 622 link complement. Our work requires a careful analysis of the tilings of R2 that come from lifting the canonical triangulations of …
Growth Conditions For Uniqueness Of Smooth Positive Solutions To An Elliptic Model, Joon Hyuk Kang
Growth Conditions For Uniqueness Of Smooth Positive Solutions To An Elliptic Model, Joon Hyuk Kang
Faculty Publications
The uniqueness of positive solution to the elliptic model
∆u + u[a + g(u, v)] = 0 in Ω, ∆v + v[a + h(u, v)] = 0 in Ω, u = v = 0 on ∂Ω,
were investigated.
New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe
New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe
Williams Honors College, Honors Research Projects
The balanced minimal evolution (BME) polytope arises from the study of phylogenetic trees in biology. It is a geometric structure which has a variant for each natural number n. The main application of this polytope is that we can use linear programming with it in order to determine the most likely phylogenetic tree for a given genetic data set. In this paper, we explore the geometric and combinatorial structure of the BME polytope. Background information will be covered, highlighting some points from previous research, and a new result on the structure of the BME polytope will be given.
Sisteme Vibrante Trilobice, Florentin Smarandache, Mircea Eugen Selariu
Sisteme Vibrante Trilobice, Florentin Smarandache, Mircea Eugen Selariu
Branch Mathematics and Statistics Faculty and Staff Publications
No abstract provided.
Inside Out: Properties Of The Klein Bottle, Andrew Pogg, Jennifer Daigle, Deirdra Brown
Inside Out: Properties Of The Klein Bottle, Andrew Pogg, Jennifer Daigle, Deirdra Brown
Thinking Matters Symposium Archive
A Klein Bottle is a two-dimensional manifold in mathematics that, despite appearing like an ordinary bottle, is actually completely closed and completely open at the same time. The Klein Bottle, which can be represented in three dimensions with self-intersection, is a four dimensional object with no intersection of material. In this presentation we illustrate some topological properties of the Klein Bottle, use the Möbius Strip to help demonstrate the construction of the Klein Bottle, and use mathematical properties to show that the Klein Bottle intersection that appears in ℝ3 does not exist in ℝ4. Introduction: Topology
My Finite Field, Matthew Schroeder
My Finite Field, Matthew Schroeder
Journal of Humanistic Mathematics
A love poem written in the language of mathematics.
Factorial Growth Rates For The Number Of Hyperbolic 3-Manifolds Of A Given Volume, Christian Millichap
Factorial Growth Rates For The Number Of Hyperbolic 3-Manifolds Of A Given Volume, Christian Millichap
Faculty Publications
The work of Jørgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume v can grow at least factorially fast with v. A similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N(v) in terms of v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v …
The Efficacy Of Mathematics Education, Eric Geimer
The Efficacy Of Mathematics Education, Eric Geimer
The STEAM Journal
Evidence supports the notion that mathematics education in the United States is inadequate. There is also evidence that mathematics education deficiencies extend internationally. The worldwide mathematics education deficit appears large enough that improving student performance in this educational problem area could yield great economic benefit. To improve the efficacy of mathematics education, education’s root problems must first be understood. Often supposed educational root problems are considered and contrasted against potential deficiencies of mathematics methodologies and curricula that are based on mainstream educational philosophies. The educational philosophies utilized to form early-grade mathematics methodologies and related curricula are judged to be the …
Orderly Ε-Homotopies Of Discrete Chains, Alexander Thomas Happ
Orderly Ε-Homotopies Of Discrete Chains, Alexander Thomas Happ
Chancellor’s Honors Program Projects
No abstract provided.
King, John (Sc 594), Manuscripts & Folklife Archives
King, John (Sc 594), Manuscripts & Folklife Archives
MSS Finding Aids
Finding aid and full-text (click on "Additional Files" below) for Manuscripts Small Collection 594. Ciphering book of John King including mathematical exercises, numeration of money, simple and compound reduction, weights and measures, and word problems.