Book V Of The Mathematical Collection Of Pappus Of Alexandria, Translated By John B. Little, 2024 College of the Holy Cross

#### Book V Of The Mathematical Collection Of Pappus Of Alexandria, Translated By John B. Little, Pappus Of Alexandria, John B. Little

*Holy Cross Bookshelf*

John B. Little is the translator.

Book V of the Mathematical Collection is addressed to a certain Megethion, about whom we know nothing else. From the context he may have been a student or patron of Pappus in Alexandria. In a heading at the start, Pappus says that the general theme will be comparisons between different geometric figures. The overall structure brings interesting relations and connections to the fore. The book opens with a very well-known and charming discussion of how the importance of such comparisons can be seen by considering the structures built by non-human creatures such as bees. …

Counting The Classes Of Projectively-Equivalent Pentagons On Finite Projective Planes Of Prime Order, 2024 Rose-Hulman Institute of Technology

#### Counting The Classes Of Projectively-Equivalent Pentagons On Finite Projective Planes Of Prime Order, Maxwell Hosler

*Rose-Hulman Undergraduate Mathematics Journal*

In this paper, we examine the number of equivalence classes of pentagons on finite projective planes of prime order under projective transformations. We are interested in those pentagons in general position, meaning that no three vertices are collinear. We consider those planes which can be constructed from finite fields of prime order, and use algebraic techniques to characterize them by their symmetries. We are able to construct a unique representative for each pentagon class with nontrivial symmetries. We can then leverage this fact to count classes of pentagons in general. We discover that there are (1/10)((*p*+3)(*p*-3)+4 …

Nano Topology And Decision Making In Medical Applications, 2024 Tanta University - Faculty of Engineering

#### Nano Topology And Decision Making In Medical Applications, Samir Mukhtar, Mohamed Shokry, Manar Omran

*Journal of Engineering Research*

**Nano Topology** **is one of the essential topics that receive special attention from some athletes in the field of General Topology, Operations Research, and Computer Science, because it has a vital role in the generalizing most of the various mathematical concepts. Recently, many efforts have been made to study many types of Nano Topology, as the previous studies lacked real applications in Engineering, Medicine, Pharmacy, and Social Sciences. In this paper, we present some different applications of these studies. The paper is divided into two parts: Firstly, we study the theory of The Nano Topology and investigate its relation with …**

New Operation Defined Over Dual-Hesitant Fuzzy Set And Its Application In Diagnostics In Medicine, 2024 Tanta University - Faculty of Engineering

#### New Operation Defined Over Dual-Hesitant Fuzzy Set And Its Application In Diagnostics In Medicine, Manar Mohamed Omran, Reham Abdel-Aziz Abo-Khadra

*Journal of Engineering Research*

In recent decades, several types of sets, such as fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, type 2 fuzzy sets, type *n* fuzzy sets, and hesitant fuzzy sets, have been introduced and investigated widely. In this paper, we propose dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multi-sets as special cases. Then we investigate the basic operations and properties of DHFSs. We also discuss the relationships among the sets mentioned above, and then propose an extension principle of DHFSs. Additionally, we give an example to illustrate …

Twisted Alexander Polynomials And Ptolemy Varieties Of Knots And Surface Bundles, 2024 The Graduate Center, City University of New York

#### Twisted Alexander Polynomials And Ptolemy Varieties Of Knots And Surface Bundles, Michael R. Marinelli

*Dissertations, Theses, and Capstone Projects*

The first focus of this dissertation is to compute Ptolemy varieties for triangulations of two infinite families of manifolds. Given an ideal triangulation of a cusped manifold, one can compute the Ptolemy variety and using it, obtain parabolic representations of the fundamental group. We compute certain obstruction classes for these manifolds, which are necessary to obtain the discrete faithful representation. This leads to our second focus of the dissertation, the twisted Alexander polynomial. The twisted Alexander polynomial (TAP) is a variation of the classical Alexander polynomial twisted by a representation of the fundamental group into a linear group. It was …

Categorical Chain Conditions For Étale Groupoid Algebras, 2024 The Graduate Center, City University of New York

#### Categorical Chain Conditions For Étale Groupoid Algebras, Sunil Philip

*Dissertations, Theses, and Capstone Projects*

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G, Steinberg defined an étale groupoid algebra RG. These étale groupoid algebras generalize various algebras, including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. In this work, we characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, thereby generalizing existing results for Leavitt path algebras and introducing new results for inverse …

New Class Function In Dual Soft Topological Space, 2024 Ministry of Education, Directorate of Educational Babylon, Hilla, Iraq,

#### New Class Function In Dual Soft Topological Space, Maryam Adnan Al-Ethary, Maryam Sabbeh Al-Rubaiea, Mohammed H. O. Ajam

*Al-Bahir Journal for Engineering and Pure Sciences*

In this paper we introduce a new class of maps in the dual Soft topological space and study some of its basic properties and relations among them, then we study and mapping.

The Fundamental Groupoid In Discrete Homotopy Theory, 2024 Western University

#### The Fundamental Groupoid In Discrete Homotopy Theory, Udit Ajit Mavinkurve

*Electronic Thesis and Dissertation Repository*

Discrete homotopy theory is a homotopy theory designed for studying graphs and for detecting combinatorial, rather than topological, “holes”. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis.

In this thesis, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us …

Exploring Intraplate Seismicity In The Midwest, 2024 University of Nebraska-Lincoln

#### Exploring Intraplate Seismicity In The Midwest, Alexa Fernández

*Department of Earth and Atmospheric Sciences: Dissertations, Theses, and Student Research*

Intraplate seismicity represents a notable occurrence within the stable North American Craton. This research explores the potential sources of stresses that could reactivate older faults and influence seismic activity within this region. Among these sources, the enduring impact of the last glacial period is considered, which includes continued glacial isostatic adjustments (GIA). During GIA the lithosphere rebounds due to the retreating ice, and the forebulge caused by far-field flexure in response to the glacial load, collapses. This results in significant faulting, fracturing, and seismic activity associated with the deglaciation phase. The adjustment of the lithosphere manifests as both near surface …

Convex Ancient Solutions To Anisotropic Curve Shortening Flow, 2024 University of Tennessee, Knoxville

#### Convex Ancient Solutions To Anisotropic Curve Shortening Flow, Benjamin Richards

*Doctoral Dissertations*

We construct ancient solutions to Anisotropic Curve Shortening Flow, including a

noncompact translator and compact solution that lives in a slab. We then show that

these are the unique ancient solutions that exist in a slab of a given width.

The Geometry Of Ancient Solutions To Curvature Flows, 2024 University of Tennessee, Knoxville

#### The Geometry Of Ancient Solutions To Curvature Flows, Sathyanarayanan Rengaswami

*Doctoral Dissertations*

Following the tremendous success of the mean curvature flow, other variants such as the Gauss curvature flow, inverse mean curvature flow have been investigated in great detail, leading to interesting applications to other fields including partial differential equations, convex geometry etc. This calls for an investigation of curvature flow as a general phenomenon. While basic existence and uniqueness results, roundness estimates etc have been obtained, there isn't a substantial body of work that addresses the geometry of solutions of curvature flows and their relation to the choice of speed function used. It is therefore interesting to investigate curvature flows as …

Analytic Properties Of Quantum States On Manifolds, 2024 The University of Western Ontario

#### Analytic Properties Of Quantum States On Manifolds, Manimugdha Saikia

*Electronic Thesis and Dissertation Repository*

The principal objective of this study is to investigate how the Kahler geometry of a classical phase space influences the quantum information aspects of the quantum Hilbert space obtained from geometric quantization and vice versa. We associated states with subsets of a product of two integral Kahler manifolds using a quantum line bundle in a particular manner. We proved that the states associated this way are separable when the subset is a finite union of products. We presented an asymptotic result for the average entropy over all the pure states on the Hilbert space H^{0}(M_{1},L_{1 …}

Circling The Square: Computing Radical Two, 2024 Liberty University

#### Circling The Square: Computing Radical Two, Isaiah Mellace, Joshua Kroeker

*NEXUS: The Liberty Journal of Interdisciplinary Studies*

Discoveries of equations for irrational numbers are not new. From Newton’s Method to Taylor Series,there are many ways to calculate the square root of two to arbitrary precision. The following method is similar in this way, but it is also a fascinating derivation from geometry that has applications to other irrationals. Additionally, the equation derived has some properties that may lead to fast computation. The first part of this paper is dedicated to deriving the equation, and the second is focused on computer science implementations and optimizations.

Are All Weakly Convex And Decomposable Polyhedral Surfaces Infinitesimally Rigid?, 2024 University of Bonn, Germany

#### Are All Weakly Convex And Decomposable Polyhedral Surfaces Infinitesimally Rigid?, Jilly Kevo

*Rose-Hulman Undergraduate Mathematics Journal*

It is conjectured that all decomposable (that is, interior can be triangulated without adding new vertices) polyhedra with vertices in convex position are infinitesimally rigid and only recently has it been shown that this is indeed true under an additional assumption of codecomposability (that is, the interior of the difference between the convex hull and the polyhedron itself can be triangulated without adding new vertices). One major set of tools for studying infinitesimal rigidity happens to be the (negative) Hessian M_{T} of the discrete Hilbert-Einstein functional. Besides its theoretical importance, it provides the necessary machinery to tackle the problem …

Khovanov Homology And Legendrian Simple Knots, 2024 Dartmouth College

#### Khovanov Homology And Legendrian Simple Knots, Ryan J. Maguire

*Dartmouth College Ph.D Dissertations*

The Jones polynomial and Khovanov homology are powerful invariants in knot theory. Their computations are known to be NP-Hard and it can be quite a challenge to directly compute either of them for a general knot. We develop explicit algorithms for the Jones polynomial and discuss the implementation of an algorithm for Khovanov homology. Using this we tabulate the invariants for millions of knots, generate statistics on them, and formulate conjectures for Legendrian and transversely simple knots.

#### A Thesis, Or Digressions On Sculptural Practice: In Which, Concepts & Influences Thereof Are Explained, Set Forth, Catalogued, Or Divulged By Way Of Commentaries To A Poem, First Conceived By The Artist, Fed Through Chatg.P.T., And Re-Edited By The Artist, To Which Are Added, Annotated References, Impressions And Ruminations Thereof, Also Including Private Thoughts & Personal Accounts Of The Artist, Jaimie An

*Masters Theses*

This thesis is an exercise in, perhaps a futile, attempt to trace just some of the ideas, stories, and musings I might meander through in my process. It’s not quite a map, nor is it a neat catalogue; it is a haphazard collection of tickets and receipts from a travel abroad, carelessly tossed in a carry-on, only to be stashed upon returning home. These ideas are derived from much greater thinkers and authors than myself; I am a mere collector or a translator, if that, and not a very good one, for much is lost. I do not claim comprehensive …

Parabolic And Non-Parabolic Surfaces With Small Or Large End Spaces Via Fenchel-Nielsen Parameters, 2024 The Graduate Center, City University of New York

#### Parabolic And Non-Parabolic Surfaces With Small Or Large End Spaces Via Fenchel-Nielsen Parameters, Michael Antony Pandazis

*Dissertations, Theses, and Capstone Projects*

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that determine whether or not a surface X is parabolic. Fix a geodesic pants decomposition of a surface and call the boundary geodesics in the decomposition cuffs. For a zero or half-twist flute surface, we prove that parabolicity is equivalent to the surface having a covering group of the first kind. Using that result, we give necessary and sufficient conditions on the Fenchel-Nielsen parameters of a half-twist flute surface X with increasing cuff lengths such that X is parabolic. As an application, we determine whether or not each …

Higher Diffeology Theory, 2024 The Graduate Center, City University of New York

#### Higher Diffeology Theory, Emilio Minichiello

*Dissertations, Theses, and Capstone Projects*

Finite dimensional smooth manifolds have been studied for hundreds of years, and a massive theory has been built around them. However, modern mathematicians and physicists are commonly dealing with objects outside the purview of classical differential geometry, such as orbifolds and loop spaces. Diffeology is a new framework for dealing with such generalized smooth spaces. This theory (whose development started in earnest in the 1980s) has started to catch on amongst the wider mathematical community, thanks to its simplicity and power, but it is not the only approach to dealing with generalized smooth spaces. Higher topos theory is another such …

Hyperbolic Groups And The Word Problem, 2024 California Polytechnic State University, San Luis Obispo

#### Hyperbolic Groups And The Word Problem, David Wu

*Master's Theses*

Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.

Dehn's Problems And Geometric Group Theory, 2024 California Polytechnic State University, San Luis Obispo

#### Dehn's Problems And Geometric Group Theory, Noelle Labrie

*Master's Theses*

In 1911, mathematician Max Dehn posed three decision problems for finitely

presented groups that have remained central to the study of combinatorial

group theory. His work provided the foundation for geometric group theory,

which aims to analyze groups using the topological and geometric properties

of the spaces they act on. In this thesis, we study group actions on Cayley

graphs and the Farey tree. We prove that a group has a solvable word problem

if and only if its associated Cayley graph is constructible. Moreover, we prove

that a group is finitely generated if and only if it acts geometrically …