Geometry and Topology Commons^™

(R1518) The Dual Spherical Curves And Surfaces In Terms Of Vectorial Moments, Süleyman Şenyurt, Abdussamet Çalışkan

Applications and Applied Mathematics: An International Journal (AAM)

In the article, the parametric expressions of the dual ruled surfaces are expressed in terms of the vectorial moments of the Frenet vectors. The integral invariants of these surfaces are calculated. It is seen that the dual parts of these invariants can be stated by the real terms. Finally, we present examples of the ruled surfaces with bases such as helix and Viviani’s curves.

(R1960) Connectedness And Compactness In Fuzzy Nano Topological Spaces Via Fuzzy Nano Z Open Sets, R. Thangammal, M. Saraswathi, A. Vadivel, C. John Sundar

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study the notion of fuzzy nano Z connected spaces, fuzzy nano Z disconnected spaces, fuzzy nano Z compact spaces and fuzzy nano Z separated sets in fuzzy nano topological spaces. We also give some properties and theorems of such concepts with connectedness and compactness in fuzzy nano topological spaces.

Manufacturability And Analysis Of Topologically Optimized Continuous Fiber Reinforced Composites, Jesus A. Ferrand

Doctoral Dissertations and Master's Theses

Researchers are unlocking the potential of Continuous Fiber Reinforced Composites for producing components with greater strength-to-weight ratios than state of the art metal alloys and unidirectional composites. The key is the emerging technology of topology optimization and advances in additive manufacturing. Topology optimization can fine tune component geometry and fiber placement all while satisfying stress constraints. However, the technology cannot yet robustly guarantee manufacturability. For this reason, substantial post-processing of an optimized design consisting of manual fiber replacement and subsequent Finite Element Analysis (FEA) is still required.

To automate this post-processing in two dimensions, two (2) algorithms were developed. The …

A Representation For Cmc 1 Surfaces In H^3 Using Two Pairs Of Spinors, Tetsuya Nakamura

Doctoral Dissertations

For Bryant's representation $\Phi\colon \widetilde{M} \rightarrow \SL_2(\C)$ of a constant mean curvature (CMC) $1$ surface $f\colon M\rightarrow \Hyp^3$ in the $3$-dimensional hyperbolic space $\Hyp^3$, we will give a formula expressed only by the global $\tbinom{P}{Q}$ and local $\tbinom{p}{q}$ spinors and their derivatives. We will see that this formula is derived from the Klein correspondence, understanding $\Phi$ as a null curve immersion into a $3$-dimensional quadric. We will show that, if $f$ is a CMC $1$ surface with smooth ends modeled on a compact Riemann surface, the linear change of $\tbinom{P}{Q}\oplus \tbinom{p}{-q}$ by some $\Sp(\C^4)$ matrices gives rise to a transformtion …

P-36 The Delta-Crossing Number For Links, Zachary Duah

Celebration of Research and Creative Scholarship

An m-component link is an embedding of m circles into 3-dimensional space; a 1-component link is called a knot. The diagram for a link may be drawn so that all crossings occur within delta tangles, collections of three crossings as appear in a delta move. The delta crossing number is defined to be the minimal number of delta tangles in such a diagram. The delta crossing number has been well-studied for knots but not for links with multiple components. Using bounds we determine the delta crossing number for several 2-component links with up to 8 crossings as well as for …

P-37 Self And Mixed Delta Moves On Algebraically Split Links, Justyce Goode, Davielle Smith, Yamil Kas-Danouche, Devin Garcia, Anthony Bosman

Celebration of Research and Creative Scholarship

A link is an embedding of circles into 3-dimensional space. A Delta-move is a local move on a link diagram. The Delta-Gordian distance between links measures the minimum number of Delta-moves needed to move between link diagrams. We place restrictions on the Delta-move by either requiring the move to only involve a single component of the link, called a self Delta-move, or multiple components of the link, called a mixed Delta-move. We prove a number of results on how (mixed/self) Delta-moves relate to classical link invariants including the Arf invariant and crossing number. This allows us to produce a graph …

Classifications Of Dupin Hypersurfaces In Lie Sphere Geometry, Thomas E. Cecil

Mathematics Department Faculty Scholarship

This is a survey of local and global classification results concerning Dupin hypersurfaces in Sⁿ (or Rⁿ) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of Sⁿ (or Rⁿ), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

Automorphism-Preserving Color Substitutions On Profinite Graphs, Michal Cizek

Electronic Thesis and Dissertation Repository

Profinite groups are topological groups which are known to be Galois groups. Their free product was extensively studied by Luis Ribes and Pavel Zaleskii using the notion of a profinite graph and having profinite groups act freely on such graphs. This thesis explores a different approach to study profinite groups using profinite graphs and that is with the notion of automorphisms and colors. It contains a generalization to profinite graphs of the theorem of Frucht (1939) that shows that every finite group is a group of automorphisms of a finite connected graph, and establishes a profinite analog of the theorem …

On The Thom Isomorphism For Groupoid-Equivariant Representable K-Theory, Zachary J. Garvey

Dartmouth College Ph.D Dissertations

This thesis proves a general Thom Isomorphism in groupoid-equivariant KK-theory. Through formalizing a certain pushforward functor, we contextualize the Thom isomorphism to groupoid-equivariant representable K-theory with various support conditions. Additionally, we explicitly verify that a Thom class, determined by pullback of the Bott element via a generalized groupoid homomorphism, coincides with a Thom class defined via equivariant spinor bundles and Clifford multiplication. The tools developed in this thesis are then used to generalize a particularly interesting equivalence of two Thom isomorphisms on TX, for a Riemannian G-manifold X.

Numerical Studies Of Correlated Topological Systems, Rahul Soni

Doctoral Dissertations

In this thesis, we study the interplay of Hubbard U correlation and topological effects in two different bipartite lattices: the dice and the Lieb lattices. Both these lattices are unique as they contain a flat energy band at E = 0, even in the absence of Coulombic interaction. When interactions are introduced both these lattices display an unexpected multitude of topological phases in our U -λ phase diagram, where λ is the spin-orbit coupling strength. We also study ribbons of the dice lattice and observed that they qualitative display all properties of their two-dimensional counterpart. This includes flat bands near …

Development Of Graphical Models And Statistical Physics Motivated Approaches To Genomic Investigations, Yashwanth Lagisetty

Dissertations & Theses (Open Access)

Identifying genes involved in disease pathology has been a goal of genomic research since the early days of the field. However, as technology improves and the body of research grows, we are faced with more questions than answers. Among these is the pressing matter of our incomplete understanding of the genetic underpinnings of complex diseases. Many hypotheses offer explanations as to why direct and independent analyses of variants, as done in genome-wide association studies (GWAS), may not fully elucidate disease genetics. These range from pointing out flaws in statistical testing to invoking the complex dynamics of epigenetic processes. In the …

The Dope Distance Is Sic: A Stable, Informative, And Computable Metric On Ordered Merge Trees, Jose Arbelo, Antonio Delgado, Charley Kirk, Zach Schlamowitz

Mathematics Summer Fellows

When analyzing time series data, it is often of interest to categorize them based on how different they are. We define a new dissimilarity measure between time series: Dynamic Ordered Persistence Editing (DOPE). DOPE satisfies metric properties, is stable to noise, is as informative as alternative approaches, and efficiently computable. Satisfying these properties simultaneously makes DOPE of interest to both theoreticians and data scientists alike.

Left-Separation Of Ω1, Lukas Stuelke, Adrienne Stanley Ph.D.

Summer Undergraduate Research Program (SURP) Symposium

A topological space is left-separated if it can be well-ordered so that every initial segment is closed. Here, we show that all countable ordinal numbers are left-separated. We then prove that a similar method could not work for ω₁ , using the pressing-down lemma¹ . We finish by showing that a left-separating well-ordering on ω₁ necessarily leads to a contradiction.

Rendezvous Numbers Of Compact And Connected Spaces, Kevin Demler, Bill Wood Ph.D.

Summer Undergraduate Research Program (SURP) Symposium

The concept of a rendezvous number was originally developed by O. Gross in 1964, and was expanded upon greatly by J. Cleary, S. Morris, and D. Yost in 1986. This number exists for every metric space, yet very little is known about it, and it’s exact value for most spaces is not known. Furthermore, it’s exact value is difficult to calculate, and in most cases we can only find bounds for the value. We focused on their arguments using convexity and applied it to shapes in different metrics and graphs. Using sets of points that stood out (vertices, midpoints) as …

Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, Bill Hemphill

STEM Guitar Project’s BBT Acoustic Kit

This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are
(a) a one-piece base for the standard kit's (Martin-style) bracing,
(b) 277 Ladder-style bracing, and
(c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.

The CADD data set for each of the three (3) top bracing designs includes …

Finding Approximate Pythagorean Triples (And Applications To Lego Robot Building), Ronald I. Greenberg, Matthew Fahrenbacher, George K. Thiruvathukal

Computer Science: Faculty Publications and Other Works

This assignment combines programming and data analysis to determine good combinations of side lengths that approximately satisfy the Pythagorean Theorem for right triangles. This can be a standalone exercise using a wide variety of programming languages, but the results are useful for determining good ways to assemble LEGO pieces in robot construction, so the exercise can serve to integrate three different units of the Exploring Computer Science high school curriculum: "Programming", "Computing and Data Analysis", and "Robotics". Sample assignment handouts are provided for both Scratch and Java programmers. Ideas for several variants of the assignment are also provided.

On A Relation Between Ado And Links-Gould Invariants, Nurdin Takenov

LSU Doctoral Dissertations

In this thesis we consider two knot invariants: Akutsu-Deguchi-Ohtsuki(ADO) invariant and Links-Gould invariant. They both are based on Reshetikhin-Turaev construction and as such share a lot of similarities. Moreover, they are both related to the Alexander polynomial and may be considered generalizations of it. By experimentation we found that for many knots, the third order ADO invariant is a specialization of the Links-Gould invariant. The main result of the thesis is a proof of this relation for a large class of knots, specifically closures of braids with five strands.

General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul

Doctoral Dissertations

In this thesis we develop a formulation of general covariance, an essential property for many field theories on curved spacetimes, using the language of stacks and the Batalin-Vilkovisky formalism. We survey the theory of stacks, both from a global and formal perspective, and consider the key example in our work: the moduli stack of metrics modulo diffeomorphism. This is then coupled to the Batalin-Vilkovisky formalism–a formulation of field theory motivated by developments in derived geometry–to describe the associated equivariant observables of a theory and to recover and generalize results regarding current conservation.

The Local Cohomology Spectral Sequence For Topological Modular Forms, Robert Bruner, John Greenlees, John Rognes

Mathematics Faculty Research Publications

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald–Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.

Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

UNO Student Research and Creative Activity Fair

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding to …