Some 2-Categorical Aspects In Physics, 2016 The Graduate Center, City University of New York

#### Some 2-Categorical Aspects In Physics, Arthur Parzygnat

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description ...

A Geometric Model Of Twisted Differential K-Theory, 2016 The Graduate Center, City University of New York

#### A Geometric Model Of Twisted Differential K-Theory, Byung Do Park

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

We construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. We use smooth U(1)-gerbes with connection as differential twists and twisted vector bundles with connection as cycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. We also construct an odd twisted Chern character of a twisted vector bundle with an automorphism. In addition to our geometric model of twisted differential K-theory, we introduce a smooth variant of the Hopkins-Singer model of differential K-theory. We prove that our model ...

The Fourth Movement Of György Ligeti's Piano Concerto: Investigating The Musical-Mathematical Connection, 2016 The Graduate Center, City University of New York

#### The Fourth Movement Of György Ligeti's Piano Concerto: Investigating The Musical-Mathematical Connection, Cynthia L. Wong

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

This interdisciplinary study explores musical-mathematical analogies in the fourth movement of Ligeti’s *Piano Concerto.* Its aim is to connect musical analysis with the piece’s mathematical inspiration. For this purpose, the dissertation is divided into two sections. Part I (Chapters 1-2) provides musical and mathematical context, including an explanation of ideas related to Ligeti’s mathematical inspiration. Part II (Chapters 3-5) delves into an analysis of the rhythm, form, melody / motive, and harmony. Appendix A is a reduced score of the entire movement, labeled according to my analysis.

On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, 2016 The Graduate Center, City University of New York

#### On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form ...

On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, 2016 The University of Western Ontario

#### On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, Sajad Sadeghi

*Electronic Thesis and Dissertation Repository*

In the first part of this thesis, a noncommutative analogue of Gross' logarithmic Sobolev inequality for the noncommutative 2-torus is investigated. More precisely, Weissler's result on the logarithmic Sobolev inequality for the unit circle is used to propose that the logarithmic Sobolev inequality for a positive element $a= \sum a_{m,n} U^{m} V^{n} $ of the noncommutative 2-torus should be of the form $$\tau(a^{2} \log a)\leqslant \underset{(m,n)\in \mathbb{Z}^{2}}{\sum} (\vert m\vert + \vert n\vert) \vert a_{m,n} \vert ^{2} + \tau (a^{2})\log ( \tau (a^2))^{1 ...

Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, 2016 The University of Western Ontario

#### Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, Josue Rosario-Ortega

*Electronic Thesis and Dissertation Repository*

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this thesis we use elliptic theory for edge- degenerate differential operators on singular manifolds to study general deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

Bridge Spectra Of Cables Of 2-Bridge Knots, 2016 University of Nebraska-Lincoln

#### Bridge Spectra Of Cables Of 2-Bridge Knots, Nicholas John Owad

*Dissertations, Theses, and Student Research Papers in Mathematics*

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Advisors: Mark Brittenham and Susan Hermiller

Introduction To Classical Field Theory, 2016 Department of Physics, Utah State University

#### Introduction To Classical Field Theory, Charles G. Torre

*All Complete Monographs*

This is an introduction to classical field theory. Topics treated include: Klein-Gordon field, electromagnetic field, scalar electrodynamics, Dirac field, Yang-Mills field, gravitational field, Noether theorems relating symmetries and conservation laws, spontaneous symmetry breaking, Lagrangian and Hamiltonian formalisms.

Patterns Formed By Coins, 2016 Formlabs

#### Patterns Formed By Coins, Andrey M. Mishchenko

*Journal of Humanistic Mathematics*

This article is a gentle introduction to the mathematical area known as *circle packing*, the study of the kinds of patterns that can be formed by configurations of non- overlapping circles. The first half of the article is an exposition of the two most important facts about circle packings, (1) that essentially whatever pattern we ask for, we may always arrange circles in that pattern, and (2) that under simple conditions on the pattern, there is an essentially unique arrangement of circles in that pattern. In the second half of the article, we consider related questions, but where we allow ...

Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, 2016 Smith College

#### Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, Joseph O'Rourke

*Faculty Publications*

A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut is infinitesimally opened by the curvature at the vertices along the path. It is shown that a class of planar, triangulated convex domains always have a radially monotone spanning forest, a forest that can be found by an essentially greedy algorithm. This algorithm can be mimicked in 3D and applied to polyhedra inscribed in a sphere. Although the algorithm does ...

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, 2016 Rose-Hulman Institute of Technology

#### Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

Two Riemann surfaces *S*_{1} and *S*_{2} with conformal *G*-actions have topologically equivalent actions if there is a homeomorphism *h :* *S _{1} -> S_{2} *which intertwines the actions. A weaker equivalence may be defined by comparing the representations of

*G*on the spaces of holomorphic

*q-*differentials

*H*and

^{q}(S_{1})*H*In this note we study the differences between topological equivalence and

^{q}(S_{2}).*H*equivalence of prime cyclic actions, where

^{q}*S*and

_{1}/G*S*have genus zero.

_{2}/GThe Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, 2016 New Mexico State University

#### The Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, Jerry Lodder

*Geometry*

No abstract provided.

Nédélec Spaces In Affine Coordinates, 2016 Portland State University

#### Nédélec Spaces In Affine Coordinates, Jay Gopalakrishnan, Luis E. García-Castillo, Leszek Demkowicz

*Jay Gopalakrishnan*

In this note, we provide a conveniently implementable basis for simplicial Nédélec spaces of any order in any space dimension. The main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex.

Nédélec Spaces In Affine Coordinates, 2016 Portland State University

#### Nédélec Spaces In Affine Coordinates, Jay Gopalakrishnan, Luis E. García-Castillo, Leszek Demkowicz

*Jay Gopalakrishnan*

In this note, we provide a conveniently implementable basis for simplicial Nédélec spaces of any order in any space dimension. The main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex.

Octahedral Dice, 2016 Butler University

#### Octahedral Dice, Todd Estroff, Jeremiah Farrell

*Jeremiah Farrell*

All five Platonic solids have been used as random number generators in games involving chance with the cube being the most popular. Martin Gardenr, in his article on dice (MG 1977) remarks: "Why cubical?... It is the easiest to make, its six sides accomodate a set of numbers neither too large nor too small, and it rolls easily enough but not too easily."

Gardner adds that the octahedron has been the next most popular as a randomizer. We offer here several problems and games using octahedral dice. The first two are extensions from Gardner's article. All answers will be ...

The Magic Octagon, 2016 Butler University

#### The Magic Octagon, Jeremiah Farrell, Tom Rodgers

*Jeremiah Farrell*

The black nodes mark the corners of an octagon and each of these nodes in connected to four others by lines. The (rather hard) puzzle is to assign the sixteen numbers 0 through 15 to each of the sixteen lines so that each black node has a sum of 30 when the line numbers leading into it are added.

The word version of the puzzle was described in the article "Most-Perfect Word Magic", Oscar Thumpbindle, *Word Ways* Vol. 40(4). Nov. 2007.

Some Curious Cut-Ups, 2016 Butler University

#### Some Curious Cut-Ups, Jeremiah Farrell, Ivan Moscovich

*Jeremiah Farrell*

We have noticed a certain kind of n-gon dissection into triangles that has a wonderful property of interest to most puzzlists. Namely that any two triangles have at least one edge in common yet no two triangles need be congruent. In an informal poll of specialists at a recent convention, none of them saw immediately how this could be accomplished. But in fact it is very straightforward.

Quaternion Algebras And Hyperbolic 3-Manifolds, 2016 Graduate Center, City University of New York

#### Quaternion Algebras And Hyperbolic 3-Manifolds, Joseph Quinn

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

I use a classical idea of Macfarlane to obtain a complex quaternion model for hyperbolic 3-space and its group of orientation-preserving isometries, analogous to Hamilton’s famous result on Euclidean rotations. I generalize this to quaternion models over number ﬁelds for the action of Kleinian groups on hyperbolic 3-space, using arithmetic invariants of the corresponding hyperbolic 3-manifolds. The class of manifolds to which this technique applies includes all cusped arithmetic manifolds and inﬁnitely many commensurability classes of cusped non-arithmetic, compact arithmetic, and compact non-arithmetic manifolds. I obtain analogous results for actions of Fuchsian groups on the hyperbolic plane. I develop ...

Cohomology Of Certain Polyhedral Product Spaces, 2016 Graduate Center, City University of New York

#### Cohomology Of Certain Polyhedral Product Spaces, Elizabeth A. Vidaurre

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

The study of torus actions led to the discovery of moment-angle complexes and their generalization, polyhedral product spaces. Polyhedral products are constructed from a simplicial complex. This thesis focuses on computing the cohomology of polyhedral products given by two different classes of simplicial complexes: polyhedral joins (composed simplicial complexes) and $n$-gons. A homological decomposition of a polyhedral product developed by Bahri, Bendersky, Cohen and Gitler is used to derive a formula for the case of polyhedral joins. Moreover, methods from and results by Cai will be used to give a full description of the non-trivial cup products in a ...

The Conway Polynomial And Amphicheiral Knots, 2016 University of Tennessee - Knoxville

#### The Conway Polynomial And Amphicheiral Knots, Vajira Asanka Manathunga

*Doctoral Dissertations*

The Conant's conjecture [7] which has foundation on the Conway polynomial and Vassiliev invariants is the main theme of this research. The Conant's conjecture claim that the Conway polynomial of amphicheiral knots split over integer modulo 4 space. We prove Conant's conjecture for amphicheiral knots coming from braid closure in certain way. We give several counter examples to a conjecture of A. Stoimenow [32] regarding the leading coefficient of the Conway polynomial. We also construct integer bases for chord diagrams up to order 7 and up to order 6 for Vassiliev invariants. Finally we develop a method ...