Geometry and Topology Commons^™

Slices Of C_2, Klein-4, And Quaternionic Eilenberg-Mac Lane Spectra, Carissa Slone

Theses and Dissertations--Mathematics

We provide the slice (co)towers of \(\Si{V} H_{C_2}\ul M\) for a variety of \(C_2\)-representations \(V\) and \(C_2\)-Mackey functors \(\ul M\). We also determine a characterization of all 2-slices of equivariant spectra over the Klein four-group \(C_2\times C_2\). We then describe all slices of integral suspensions of the equivariant Eilenberg-MacLane spectrum \(H\ulZ\) for the constant Mackey functor over \(C_2\times C_2\). Additionally, we compute the slices and slice spectral sequence of integral suspensions of $H\ulZ$ for the group of equivariance $Q_8$. Along the way, we compute the Mackey functors \(\mpi_{k\rho} H_{K_4}\ulZ\) and $\mpi_{k\rho} H_{Q_8}\ulZ$.

Bicategorical Traces And Cotraces, Justin Barhite

Theses and Dissertations--Mathematics

Familiar constructions like the trace of a matrix and the Euler characteristic of a closed smooth manifold are generalized by a notion of trace of an endomorphism of a dualizable object in a bicategory equipped with a piece of additional structure called a shadow functor. Another example of this bicategorical trace, in the form of maps between Hochschild homology of bimodules, appears in a 1987 paper by Joseph Lipman, alongside a more mysterious ”cotrace” map involving Hochschild cohomology. Putting this cotrace on the same category-theoretic footing as the trace has led us to propose a ”bicategorical cotrace” in a closed …

Geometric Dissections, Daniel Robert Martin

MSU Graduate Theses

In the study of geometry, the notion of dissection and its mechanics are occasionally over-looked. We consider and trace the history and theorems surrounding geometric dissections in both recreational and academic mathematics. We explore the important advancements in this particular topic from antiquity through the nineteenth and early twentieth centuries. We conclude with an exploration of the Banach-Tarski paradox

Another Angle On Perspective, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Higher Spanier Groups, Johnny Aceti

West Chester University Master’s Theses

When non-trivial local structures are present in a topological space X, a common ap- proach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the n-th ˇCech homotopy group ˇπn(X, x0) under the canonical homomorphism Ψn : πn(X, x0) → ˇπn(X, x0). The subgroup ker Ψn is the obstruc- tion to this tactic as it consists of precisely those elements of πn(X, x0), which cannont be detected by polyhedral approximations to X. In this paper we present a definition of higher dimensional analouges of Thick Spanier groups use …

Finite Matroidal Spaces And Matrological Spaces, Ziyad M. Hamad

Graduate Theses, Dissertations, and Problem Reports

The purpose of this thesis is to present new different spaces as attempts to generalize the concept of topological vector spaces. A topological vector space, a well-known concept in mathematics, is a vector space over a field \mathbb{F} with a topology that makes the addition and scalar multiplication operations of the vector space continuous functions. The field \mathbb{F} is usually \mathbb{R} or \mathbb{C} with their standard topologies. Since every vector space is a finitary matroid, we define two spaces called finite matroidal spaces and matrological spaces by replacing the linear structure of the topological vector space with a finitary matroidal …

On The Uniqueness Of Continuation Of A Partially Defined Metric, Evgeniy Petrov

Theory and Applications of Graphs

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let G=G(V,E) be an undirected graph with the set of vertices V and the set of edges E. A necessary and sufficient condition under which the weight w : E → R⁺ on the graph G has a unique continuation to a metric d : V x V → R⁺ is found.

Apparent Contours For Piecewise Smooth Surfaces, Sarah Marie Jackman

UNF Graduate Theses and Dissertations

The set of points on an embedded surface $M$ that are tangent to a set viewing direction $\mathbf{v}$ is called the contour generator of $M$. The projection of those points to an image plane is called a surface's apparent contour. Apparent contours hold certain properties that allow for reconstruction of the original surface using only the information of the apparent contour. In this paper, we explore the structure of the apparent contour through contact classes and singularity types. Additionally we examine the properties of apparent contours that allow for 3 dimensional reconstruction. Our goal is to extend the properties of …

Multi-Trace Matrix Models From Noncommutative Geometry, Hamed Hessam

Electronic Thesis and Dissertation Repository

Dirac ensembles are finite dimensional real spectral triples where the Dirac operator is allowed to vary within a suitable family of operators and is assumed to be random. The Dirac operator plays the role of a metric on a manifold in the noncommutative geometry context of spectral triples. Thus, integration over the set of Dirac operators within a Dirac ensemble, a crucial aspect of a theory of quantum gravity, is a noncommutative analog of integration over metrics.

Dirac ensembles are closely related to random matrix ensembles. In order to determine properties of specific Dirac ensembles, we use techniques from random …

(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 …