Geometry and Topology Commons^™

Torsors Over Simplicial Schemes, Alexander S. Rolle

Electronic Thesis and Dissertation Repository

Let X be a simplicial object in a small Grothendieck site C, and let G be a sheaf of groups on C. We define a notion of G-torsor over X, generalizing a definition of Gillet, and prove that there is a bijection between the set of isomorphism classes of G-torsors over X, and the set of maps in the homotopy category of simplicial presheaves on C, with respect to the local weak equivalences, from X to BG. We prove basic results about the resulting non-abelian cohomology invariant, including an exact sequence associated to a central extension of sheaves of groups, …

Optimal Inequalities For Submanifolds Of An Indefinite Space Form, Rıfat Güneş, Mehmet Gülbahar, Sadık Keleş, Erol Kılıç

Applications and Applied Mathematics: An International Journal (AAM)

Optimal inequalities involving the scalar curvature, the mean curvature vector and the second fundamental form for pseudo Riemannian submanifolds are proved and the equality cases of these inequalities are discussed. These results are studied for submanifolds of various indefinite contact space forms.

On Integrability Of Semi-Invariant Submanifolds Of Trans-Sasakian Finsler Manifolds, Nesrin Caliskan, Ayse F. Saglamer, Nurten Kilic

Applications and Applied Mathematics: An International Journal (AAM)

We introduce the notion of trans-Sasakian Finsler manifold, then study semi-invariant submanifold F^m = (Y, Y', F) of a trans-Sasakian Finsler manifold F^{2n+1} + (bar-Y, bar-Y', bar-F) and we discuss the integrability conditions of the distrubitions of the semi-invariant submanifolds of the trans-Sasakian Finsler manifold.

A Comparison Of Original And Inverse Motion In Minkowski Plane, Kemal Eren, Soley Ersoy

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we investigate the inflection circle, circling-point curve, and center-point curve for the original and inverse motion of Minkowski planes, and we also deal with their degenerate cases individually. For this purpose, we consider the trajectory of origin with respect to the instantaneous invariants of Bottema in Minkowski plane by using hyperbolic numbers. Finally, we give the geometric interpretation of the circling-point and center-point curves by comparing the original and inverse motion in Minkowski planes.

The T3,T4-Conjecture For Links, Katie Tucker

Department of Mathematics: Dissertations, Theses, and Student Research

An oriented n-component link is a smooth embedding of n oriented copies of S¹ into S³. A diagram of an oriented link is a projection of a link onto R² such that there are no triple intersections, with notation at double intersections to indicate under and over strands and arrows on strands to indicate orientation. A local move on an oriented link is a regional change of a diagram where one tangle is replaced with another in a way that preserves orientation. We investigate the local moves t₃ and t₄, which are …

Merge Trees In Discrete Morse Theory, Benjamin Johnson

Mathematics Summer Fellows

The field of topological data analysis seeks to use techniques in topology to study large data sets. The hope is that rather than single quantities that summarize the data, such as mean or standard deviation, information about the data can be learned by studying the overall ``shape” of the data. One way to summarize this data is through a merge tree. Merge trees can be thought of as keeping track of certain clusters of data and determining when they merge together. In this paper, we will study merge trees induced by a discrete Morse function on a tree. Under a …

Mathematics And Programming Exercises For Educational Robot Navigation, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

This paper points students towards ideas they can use towards developing a convenient library for robot navigation, with examples based on Botball primitives, and points educators towards mathematics and programming exercises they can suggest to students, especially advanced high school students.

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 …

Localization Theory In An Infinity Topos, Marco Vergura

Electronic Thesis and Dissertation Repository

We develop the theory of reflective subfibrations on an ∞-topos E. A reflective subfibration L on E is a pullback-compatible assignment of a reflective subcategory D_X ⊆ E/X with associated localization functor L_X, for every X in E. Reflective subfibrations abound in homotopy theory, albeit often disguised, e.g., as stable factorization systems. The added properties of a reflective subfibration L on E compared to a mere reflective subcategory of E are crucial for most of our results. For example, we can prove that L-local maps (i.e., those maps p in D_X for some X in E) admit a classifying map. …

Dehn Functions Of Bestvina-Brady Groups, Yu-Chan Chang

LSU Doctoral Dissertations

In this dissertation, we prove that if the flag complex on a finite simplicial graph is a 2-dimensional triangulated disk, then the Dehn function of the associated Bestvina--Brady group depends on the maximal dimension of the simplices in the interior of the flag complex. We also give some examples where the flag complex on a finite simplicial graph is not 2-dimensional, and we establish a lower bound for the Dehn function of the associated Bestvina--Brady group.

Euler Entertainments, Jeremiah Farrell, Karen Farrell

Jeremiah Farrell

No abstract provided.

The White Rabbit 12-Puzzle, Chris Morgan, Jeremiah Farrell

Jeremiah Farrell

Martin Gardner's fondness for the characters and themes of Lewis Carroll's "Alice" is well-known and to honor Gardner we offer two word puzzles to be played on the 12-node diagram of the WHITE RABBIT.

Kate Jones – A Tribute, Karen Farrell, Jeremiah Farrell

Jeremiah Farrell

Kate is also an accomplished recreational mathematician and poet. To try to match in a small way her creative ability, we offer three puzzle-games in her honor: O'BEIRNE's TRI-HEX, PAPPUS and "KATe JONES". These three are specific examples of (9,3) symmetric configurations. More generally an (n,r) configuration is a collection of n "points"and n "lines" subject to the following requirements:

Rl: Any two points belong to at most one line.

R2: Each line has r points, and each point belongs to r lines.

Martin Gardner Puzzle-Games, Stephen Bloom, Lacey Echols, Jeremiah Farrell, Shannon Lieb

Jeremiah Farrell

No abstract provided.

Paul Swinford – A Tribute, Jeremiah Farrell

Jeremiah Farrell

No abstract provided.

The Jin And Jang Of Quantum Physics Truth Tables, Shannon Lieb, Jeremiah Farrell

Jeremiah Farrell

No abstract provided.

The Tea Party, Stephen Bloom, Jeremiah Farrell

Jeremiah Farrell

No abstract provided.

A Gathering For Gardner Puzzle-Game, Jeremiah Farrell, Chris Morgan

Jeremiah Farrell

Each different letter of "GATHERING FOR GARDNER" is used exactly three times in the following words: DIE, FAD, FIT, FOG, GIN, HAG, HER, HOD, NOR, RAT, TEN.

Alice In Wonderland For G4g13, Jeremiah Farrell, Emmanuelle Malte Salvatore, Todd Wilk Estroff

Jeremiah Farrell

Each of the ten different letters in the title is used exactly three times to form the words in the circles. Martin Gardner's famous work The Annotated Alice was first published in 1960 and we honor him in this essay.

Paul Swinford – A Tribute, Jeremiah Farrell

Jeremiah Farrell

No abstract provided.