Geometry and Topology Commons^™

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 …

Geometry In Spectral Triples: Immersions And Fermionic Fuzzy Geometries, Luuk S. Verhoeven

Electronic Thesis and Dissertation Repository

We investigate the metric nature of spectral triples in two ways.

Given an oriented Riemannian embedding i:X->Y of codimension 1 we construct a family of unbounded KK-cycles i!(epsilon), each of which represents the shriek class of i in KK-theory. These unbounded KK-cycles are further equipped with connections, allowing for the explicit computation of the products of i! with the spectral triple of Y at the unbounded level. In the limit epsilon to 0 the product of these unbounded KK-cycles with the canonical spectral triple for Y admits an asymptotic expansion. The divergent part of this expansion is known and …

Generating Polynomials Of Exponential Random Graphs, Mohabat Tarkeshian

Electronic Thesis and Dissertation Repository

The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM).

In the past, determining when a probability distribution has strong …

One Theorem, Two Ways: A Case Study In Geometric Techniques, John B. Little

Journal of Humanistic Mathematics

If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, and K on ΓA cutting those sides in same ratios:

AH : HB = BΘ : ΘΓ = ΓK : KA,

then Pappus of Alexandria proved that the triangles ABΓ and HΘK have the same centroid (center of mass). We present two proofs of this result: an English translation of Pappus's original synthetic proof and a modern algebraic proof making use of Cartesian coordinates and vector concepts. Comparing the two methods, we can see that while the algebraic …

Stability Of Cauchy's Equation On Δ+., Holden Wells

Electronic Theses and Dissertations

The most famous functional equation f(x+y)=f(x)+f(y) known as Cauchy's equation due to its appearance in the seminal analysis text Cours d'Analyse (Cauchy 1821), was used to understand fundamental aspects of the real numbers and the importance of regularity assumptions in mathematical analysis. Since then, the equation has been abstracted and examined in many contexts. One such examination, introduced by Stanislaw Ulam and furthered by Donald Hyers, was that of stability. Hyers demonstrated that Cauchy's equation exhibited stability over Banach Spaces in the following sense: functions that approximately satisfy Cauchy's equation are approximated with the same level of error by functions …

The G_2-Hitchin Component Of Triangle Groups: Dimension And Integer Points, Hannah E. Downs

Doctoral Dissertations

The image of $\PSL(2,\reals)$ under the irreducible representation into $\PSL(7,\reals)$ is contained in the split real form $G_{2}^{4,3}$ of the exceptional Lie group $G_{2}$. This irreducible representation therefore gives a representation $\rho$ of a hyperbolic triangle group $\Gamma(p,q,r)$ into $G_{2}^{4,3}$, and the \textit{Hitchin component} of the representation variety $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ is the component of $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ containing $\rho$.

This thesis is in two parts: (i) we give a simple, elementary proof of a formula for the dimension of this Hitchin component, this formula having been obtained earlier in [Alessandrini et al.], \citep{Alessandrini2023}, as part of a wider investigation using Higgs bundle techniques, …

Topological Data Analysis Of Convolutional Neural Networks Using Depthwise Separable Convolutions, Eliot Courtois

Dissertations

In this dissertation, we present our contribution to a growing body of work combining the fields of Topological Data Analysis (TDA) and machine learning. The object of our analysis is the Convolutional Neural Network, or CNN, a predictive model with a large number of parameters organized using a grid-like geometry. This geometry is engineered to resemble patches of pixels in an image, and thus CNNs are a conventional choice for an image-classifying model.

CNNs belong to a larger class of neural network models, which, starting at a random initialization state, undergo a gradual fitting (or training) process, often a …

Internal Yoneda Ext Groups, Central H-Spaces, And Banded Types, Jarl Gunnar Taxerås Flaten

Electronic Thesis and Dissertation Repository

We develop topics in synthetic homotopy theory using the language of homotopy type theory, and study their semantic counterparts in an ∞-topos. Specifically, we study Grothendieck categories and Yoneda Ext groups in this setting, as well as a novel class of central H-spaces along with their associated bands. The former are fundamental notions from homological algebra that support important computations in traditional homotopy theory. We develop these tools with the goal of supporting similar computations in our setting. In contrast, our results about central H-spaces and bands are new, even when interpreted into the ∞-topos of spaces.

In Chapter …

Jones Polynomial Obstructions For Positivity Of Knots, Lizzie Buchanan

Dartmouth College Ph.D Dissertations

The fundamental problem in knot theory is distinguishing one knot from another. We accomplish this by looking at knot invariants. One such invariant is positivity. A knot is positive if it has a diagram in which all crossings are positive. A knot is almost-positive if it does not have a diagram where all crossings are positive, but it does have a diagram in which all but one crossings are positive. Given a knot with an almost-positive diagram, it is in general very hard to determine whether it might also have a positive diagram. This work provides positivity obstructions for three …

(R1986) Neutrosophic Soft Contra E-Continuous Maps, Contra E-Irresolute Maps And Application Using Distance Measure, P. Revathi, K. Chitirakala, A. Vadivel

Applications and Applied Mathematics: An International Journal (AAM)

We introduce and investigate neutrosophic soft contra e-continuous maps and contra e-irresolute maps in neutrosophic soft topological spaces with examples. Also, neutrosophic soft contra econtinuous maps are compared with neutrosophic soft continuous maps, δ-continuous maps, δ- semi continuous maps, δ-pre continuous maps and e∗ continuous maps in neutrosophic soft topological spaces. We derive some useful results and properties related to them. An application in decision making problem using distance measure is given. An example of a candidate selection from a company interview is formulated as neutrosophic soft model problem and the hamming distance measure is applied to calculate the distance …

(R1957) Some Types Of Continuous Function Via N-Neutrosophic Crisp Topological Spaces, A. Vadivel, C. John Sundar

Applications and Applied Mathematics: An International Journal (AAM)

The aim of this article is to introduced a new type of continuous functions such as N-neutrosophic crisp gamma continuous and weakly N-neutrosophic crisp gamma continuous functions in a N-neutrosophic crisp topological space and also discuss a relation between them in a N-neutrosophic crisp topological spaces. We also investigate some of their properties in N-neutrosophic crisp gamma continuous function via N-neutrosophic crisp topological spaces. Further, a contra part of continuity called N-neutrosophic crisp gamma-contra continuous map in a N-neutrosophic crisp topology is also initiated. Finally, an application based on neutrosophic score function of medical diagnosis is examined with graphical representation.

(R1977) On Geometry Of Equiform Smarandache Ruled Surfaces Via Equiform Frame In Minkowski 3-Space, Emad Solouma

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, some geometric properties of equiform Smarandache ruled surfaces in Minkowski space E₁³ using an equiform frame are investigated. Also, we give the sufficient conditions that make these surfaces are equiform developable and equiform minimal related to the equiform curvatures and when the equiform base curve contained in a plane or general helix. Finally, we provide an example, such as these surfaces.

(R2026) Special Smarandache Ruled Surfaces According To Flc Frame In E^3, Süleyman Şenyurt, Kebire Hilal Ayvacı, Davut Canlı

Applications and Applied Mathematics: An International Journal (AAM)

In this study, we introduce some special ruled surfaces according to the Flc frame of a given polynomial curve. We name these ruled surfaces as TD₂, TD₁ ve D₂D₁ Smarandache ruled surfaces and provide their characteristics such as Gauss and mean curvatures in order to specify their developability and minimality conditions. Moreover, we examine the conditions if the parametric curves of the surfaces are asymptotic, geodesic or curvature line. Such conditions are also argued in terms of the developability and minimality conditions. Finally, we give an example and picture the corresponding graphs of ruled …

Groups Of Non Positive Curvature And The Word Problem, Zoe Nepsa

Master's Theses

Given a group $\Gamma$ with presentation $\relgroup{\scr{\scr{A}}}{\scr{R}}$, a natural question, known as the word problem, is how does one decide whether or not two words in the free group, $F(\scr{\scr{A}})$, represent the same element in $\Gamma$. In this thesis, we study certain aspects of geometric group theory, especially ideas published by Gromov in the late 1980's. We show there exists a quasi-isometry between the group equipped with the word metric, and the space it acts on. Then, we develop the notion of a CAT(0) space and study groups which act properly and cocompactly by isometries on these spaces, such groups …

(R1965) Some More Properties On Generalized Double Fuzzy Z Alpha Open Sets, K. Jayapandian, A. Saivarajan, O. Uma Maheswari, J. Sathiyaraj

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a new class of sets termed as double fuzzy generalized Z alpha closed sets and double fuzzy generalized Z alpha open sets are introduced with the help of double fuzzy Z alpha open and double fuzzy Z alpha closed sets, respectively. Using these sets double fuzzy generalized Z alpha border, double fuzzy generalized Z alpha exterior and double fuzzy generalized Z alpha frontier of a fuzzy set in double fuzzy topological spaces are introduced. Also, the topological properties and characterizations of these sets and operators are studied. Furthermore, suitable examples have been provided to illustrate the theory.

An Explicit Construction Of Sheaves In Context, Tyler A. Bryson

Dissertations, Theses, and Capstone Projects

This document details the body of theory necessary to explicitly construct sheaves of sets on a site together with the development of supporting material necessary to connect sheaf theory with the wider mathematical contexts in which it is applied. Of particular interest is a novel presentation of the plus construction suitable for direct application to a site without first passing to the generated grothendieck topology.

Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, Jonathan Toledo

Dissertations, Theses, and Capstone Projects

Bousfield and Kan traditionally formulated their homotopy spectral sequence over a simplicial set X resolved with respect to a ring R. By considering an adequate category of ring spectra, one can take a ring spectrum E, create from it a functor of a triple on the category of simplicial sets, and build a cosimplicial simplicial set EX. The homotopy spectral sequence can then be formed over such cosimplicial spaces by a similar construction to the original. Pairings can be established on these spectral sequences, and, for nice enough spaces, these pairings on the E₂-terms coincide with certain …

Quantifying Separability In Limit Groups, Keino Brown

Dissertations, Theses, and Capstone Projects

We show that for any finitely generated non-abelian subgroup H of a limit group L, there exists a finite-index subgroup K which is fully residually H. This generalizes the result of Wilton that limit groups admit local retractions. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes results of Louder, …

A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić

Mathematics, Physics, and Computer Science Faculty Articles and Research

For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x₀, y₀) such that A nontrivially intersects curves y − y0 = a(x −x0)^β for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take β = 1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can …

Motion Planning Algorithm In A Y-Graph, David Baldi

Rose-Hulman Undergraduate Mathematics Journal

We present an explicit algorithm for two robots to move autonomously and without collisions on a track shaped like the letter Y. Configuration spaces are of practical relevance in designing safe control schemes for automated guided vehicles. The topological complexity of a configuration space is the minimal number of continuous instructions required to move robots between any initial configuration to any final one without collisions. Using techniques from topological robotics, we calculate the topological complexity of two robots moving on a Y-track and exhibit an optimal algorithm realizing this exact number of instructions given by the topological complexity.