Geometry and Topology Commons^™

Conventions, Definitions, Identities, And Other Useful Formulae, Robert Mcnees

Robert A McNees IV

As the name suggests, these notes contain a summary of important conventions, definitions, identities, and various formulas that I often refer to. They may prove useful for researchers working in General Relativity, Supergravity, String Theory, Cosmology, and related areas.

Geometric Deformations Of Sodalite Frameworks, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

In mathematical crystallography and computational materials science, it is important to infer flexibility properties of framework materials from their geometric representation. We study combinatorial, geometric and kinematic properties for frameworks modeled on sodalite.

A Simplification Of Inclusion-Exclusion Via Minimal Complexes, Andrew J. Brandt

Summer Research

This poster discusses the discovery and use of previously unproved methods for solving counting problems using the fundamental ideas of the inclusion exclusion-principle and the Euler characteristic. While both methods use a weighted version of the Euler characteristic to determine the order of a union of finite sets, the first method can be used with contractible, planar graphs whereas the second method generalizes this idea to multi-dimensional complexes and their minimal complexes. These methods seem to be promising in their applications to subjects such as homology theory, Betti numbers, and abstract tubes.

A Special Tribute To Martin Gardner, Jeremiah Farrell

Scholarship and Professional Work - LAS

There are exactly 12 different letters in the phrase GATHERING FOR MARTIN GARDNER. We use each of the 12 letters three times each in 18 different two-letter words that are to be placed on the nodes of the graph so adjoining nodes have a letter in common.

The Kretschmann Scalar, Charles G. Torre

How to... in 10 minutes or less

On a pseudo-Riemannian manifold with metric g, the "Kretschmann scalar" is a quadratic scalar invariant of the Riemann R tensor of g, defined by contracting all indices with g. In this worksheet we show how to calculate the Kretschmann scalar from a metric.

Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden

Faculty Publications

In this paper, we show that any nonarithmetic hyperbolic 2-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of 3-manifolds with nontrivial JSJ-decomposition and rank-two fundamental groups. We also show that the only commensurable hyperbolic 2-bridge link complements are the figure-eight knot complement and the 6²₂ link complement. Our work requires a careful analysis of the tilings of R² that come from lifting the canonical triangulations of …

Nuclear Space Facts, Strange And Plain, Jeremy Becnel, Ambar Sengupta

Faculty Publications

We present a scenic but practical guide through nuclear spaces and their dual spaces, examining useful, unexpected, and often unfamiliar results both for nuclear spaces and their strong and weak duals.

Orthogonal Projections Of Lattice Stick Knots, Margaret Marie Allardice

Senior Projects Spring 2016

A lattice stick knot is a closed curve in R³ composed of finitely many line segments, sticks, that lie parallel to the three coordinate axes in R³, such that the line segments meet at points in the 3-dimensional integer lattice. The lattice stick number of a knot is the minimal number of sticks required to realize that knot as a lattice stick knot. A right angle lattice projection is a projection of a knot in R³onto the plane such that the edges of the projection lie parallel to the two coordinate axes in the plane, …

Trilobic Vibrant Systems, Florentin Smarandache, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Alexander And Writhe Polynomials For Virtual Knots, Blake Mellor

Mathematics Faculty Works

We give a new interpretation of the Alexander polynomial Δ₀ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, Δ₀ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …

Geometry Of Cubic Polynomials, Xavier Boesken

Mathematics

No abstract provided.

Some Convergence Properties Of Minkowski Functionals Given By Polytopes, Jesse Moeller

Dissertations and Theses @ UNI

In this work we investigate the behavior of the Minkowski Functionals admitted by a sequence of sets which converge to the unit ball ‘from the inside’. We begin in R 2 and use this example to build intuition as we extend to the more general R n case. We prove, in the penultimate chapter, that convergence ‘from the inside’ in this setting is equivalent to two other characterizations of the convergence: a geometric characterization which has to do with the sizes of the faces of each polytope in the sequence converging to zero, and the convergence of the Minkowski functionals …

Locally Anisotropic Toposes, Jonathon Funk, Pieter Hofstra

Publications and Research

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.

Growth Conditions For Uniqueness Of Smooth Positive Solutions To An Elliptic Model, Joon Hyuk Kang

Faculty Publications

The uniqueness of positive solution to the elliptic model

∆u + u[a + g(u, v)] = 0 in Ω, ∆v + v[a + h(u, v)] = 0 in Ω, u = v = 0 on ∂Ω,

were investigated.

In Search Of A Class Of Representatives For Su-Cobordism Using The Witten Genus, John E. Mosley

Theses and Dissertations--Mathematics

In algebraic topology, we work to classify objects. My research aims to build a better understanding of one important notion of classification of differentiable manifolds called cobordism. Cobordism is an equivalence relation, and the equivalence classes in cobordism form a graded ring, with operations disjoint union and Cartesian product. My dissertation studies this graded ring in two ways:

1. by attempting to find preferred class representatives for each class in the ring.

2. by computing the image of the ring under an interesting ring homomorphism called the Witten Genus.

Exploration Of Curvature Through Physical Materials, Lucinda-Joi Chu-Ketterer

Pitzer Senior Theses

Parametric equations are commonly used to describe surfaces. Looking at parametric equations does not provide tangible information about an object. Thus through the use of physical materials, an understanding of the limitations of the materials allows someone to gain a broader understanding of the surface. A M$\ddot{o}$bius strip and Figure 8 Klein bottle were created through knitting due to the precision and steady increase in curvature allowed through knitting. A more standard Klein bottle was created through crochet due to the ease in creating quick increases in curvature. Both methods demonstrate the change in curvature for both surfaces where the …

Classification Of Compact 2-Manifolds, George H. Winslow

Theses and Dissertations

It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then observe that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we …

Algorithmic Foundations Of Heuristic Search Using Higher-Order Polygon Inequalities, Newton Henry Campbell Jr.

CCE Theses and Dissertations

The shortest path problem in graphs is both a classic combinatorial optimization problem and a practical problem that admits many applications. Techniques for preprocessing a graph are useful for reducing shortest path query times. This dissertation studies the foundations of a class of algorithms that use preprocessed landmark information and the triangle inequality to guide A* search in graphs. A new heuristic is presented for solving shortest path queries that enables the use of higher order polygon inequalities. We demonstrate this capability by leveraging distance information from two landmarks when visiting a vertex as opposed to the common single landmark …

New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe

Williams Honors College, Honors Research Projects

The balanced minimal evolution (BME) polytope arises from the study of phylogenetic trees in biology. It is a geometric structure which has a variant for each natural number n. The main application of this polytope is that we can use linear programming with it in order to determine the most likely phylogenetic tree for a given genetic data set. In this paper, we explore the geometric and combinatorial structure of the BME polytope. Background information will be covered, highlighting some points from previous research, and a new result on the structure of the BME polytope will be given.