Mathematics Commons^™

Topology And Ecology: Deducing States Of The Upper Mississippi River System, Killian Davis

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

ℂ-Motivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka

Mathematics Faculty Research Publications

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations.We compute the Steenrod algebra in this context, and we construct a “motivic modular forms” spectrum over ℂ.

On 𝜃- -Closed Sets And 𝜃- -Continuous Functlons, Amin Hamoud Saif, Nahid Mohammed Al-Showhati

Hadhramout University Journal of Natural & Applied Sciences

In topological spaces, the class of 𝜃-closed sets and 𝜃-continuous function have been introduced by Velicko and Fomin respectively. The purpose of this paper is to introduce and study these notions in grill topological spaces by giving the new classes of 𝜃- -closed sets and 𝜃- -continuous functions in grill topological space.

Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

A hypersurface M in Rⁿ or Sⁿ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in Rⁿ or Sⁿ can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective …

Clifford Harmonics, Samuel L. Hosmer

Dissertations, Theses, and Capstone Projects

In 1980 Michelsohn defined a differential operator on sections of the complex Clifford bundle over a compact Kähler manifold M. This operator is a differential and its Laplacian agrees with the Laplacian of the Dolbeault operator on forms through a natural identification of differential forms with sections of the Clifford bundle. Relaxing the condition that M be Kähler, we introduce two differential operators on sections of the complex Clifford bundle over a compact almost Hermitian manifold which naturally generalize the one introduced by Michelsohn. We show surprising Kähler- like symmetries of the kernel of the Laplacians of these operators in …

A Geometric Model For Real And Complex Differential K-Theory, Matthew T. Cushman

Dissertations, Theses, and Capstone Projects

We construct a differential-geometric model for real and complex differential K-theory based on a smooth manifold model for the K-theory spectra defined by Behrens using spaces of Clifford module extensions. After writing representative differential forms for the universal Pontryagin and Chern characters we transgress these forms to all the spaces of the spectra and use them to define an abelian group structure on maps up to an equivalence relation that refines homotopy. Finally we define the differential K-theory functors and verify the axioms of Bunke-Schick for a differential cohomology theory.

Representing The Derivative Of Trace Of Holonomy, Jeffrey Peter Kroll

Dissertations, Theses, and Capstone Projects

Trace of holonomy around a fixed loop defines a function on the space of unitary connections on a hermitian vector bundle over a Riemannian manifold. Using the derivative of trace of holonomy, the loop, and a flat unitary connection, a functional is defined on the vector space of twisted degree 1 cohomology classes with coefficients in skew-hermitian bundle endomorphisms. It is shown that this functional is obtained by pairing elements of cohomology with a degree 1 homology class built directly from the loop and equipped with a flat section obtained from the variation of holonomy around the loop. When the …

Differentiability Of The Liouville Map Via Geodesic Currents, Xinlong Dong

Dissertations, Theses, and Capstone Projects

For a conformally hyperbolic Riemann surface, the Teichmüller space is the space of quasiconformal maps factored by an equivalence relation, and it is a complex Banach manifold. The space of geodesic currents endowed with the uniform weak* topology is a subset of a Fréchet space of Hölder distributions. We introduce an appropriate topology on the space of Hölder distributions and this new topology coincides with the uniform weak* topology on the space of geodesic currents. The Liouville map of the Teichmüller space becomes differentiable in the Fréchet sense. In particular, the derivative of Liouville currents exists and belongs to the …

From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar

Mathematics Faculty Research Publications

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. …

Topology Optimization Of 2d Structures With Multiple Displacement Constraints, Patricio Uarac Pinto

International Programs

The use of topology optimization in the design process in Civil Engineering can lead to innovative building shapes that not only fulfill structural requirements but also open new opportunities for arquitectonics.

Maximal Spacelike Surfaces In A Certain Homogeneous Lorentzian 3-Manifold, Sungwook Lee

Faculty Publications

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds, which includes Minkowski 3-space and anti-de Sitter 3-space, is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula for maximal spacelike surfaces in the homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of maximal spacelike surfaces and its harmonicity are discussed.

Results On Nonorientable Surfaces For Knots And 2-Knots, Vincent Longo

Department of Mathematics: Dissertations, Theses, and Student Research

A classical knot is a smooth embedding of the circle into the 3-sphere. We can also consider embeddings of arbitrary surfaces (possibly nonorientable) into a 4-manifold, called knotted surfaces. In this thesis, we give an introduction to some of the basics of the studies of classical knots and knotted surfaces, then present some results about nonorientable surfaces bounded by classical knots and embeddings of nonorientable knotted surfaces. First, we generalize a result of Satoh about connected sums of projective planes and twist spun knots. Specifically, we will show that for any odd natural n, the connected sum of the n-twist …

Manifold Learning With Tensorial Network Laplacians, Scott Sanders

Electronic Theses and Dissertations

The interdisciplinary field of machine learning studies algorithms in which functionality is dependent on data sets. This data is often treated as a matrix, and a variety of mathematical methods have been developed to glean information from this data structure such as matrix decomposition. The Laplacian matrix, for example, is commonly used to reconstruct networks, and the eigenpairs of this matrix are used in matrix decomposition. Moreover, concepts such as SVD matrix factorization are closely connected to manifold learning, a subfield of machine learning that assumes the observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. Since …

Multilateration Index., Chip Lynch

Electronic Theses and Dissertations

We present an alternative method for pre-processing and storing point data, particularly for Geospatial points, by storing multilateration distances to fixed points rather than coordinates such as Latitude and Longitude. We explore the use of this data to improve query performance for some distance related queries such as nearest neighbor and query-within-radius (i.e. “find all points in a set P within distance d of query point q”). Further, we discuss the problem of “Network Adequacy” common to medical and communications businesses, to analyze questions such as “are at least 90% of patients living within 50 miles of a covered emergency …

Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv

Graduate Theses/Dissertations

Finding rational points that satisfy functions known as elliptic curves induces a finitely-generated abelian group. Such functions are powerful tools that were used to solve Fermat's Last Theorem and are used in cryptography to send private keys over public systems. Elliptic curves are also useful in factoring and determining primality.

Towards The Homotopy Type Of The Morse Complex, Connor Donovan

Mathematics Summer Fellows

Mathematicians have long been interested in studying the properties of simplicial complexes. In 1998, Robin Forman developed gradient vector fields as a tool to study these complexes. Having gradient vector fields to study these simplicial complexes, in 2005, Chari and Joswig discovered the Morse complex, a complex consisting of all gradient vector fields on a fixed complex. Although the Morse complex has been studied since 2005, there is little information regarding its homotopy type for different simplicial complexes. Pursuing our curiosity of the topic, we extend a result by Ayala et. al., stating that the pure Morse complex of a …

The Degeneration Of The Hilbert Metric On Ideal Pants And Its Application To Entropy, Marianne Debrito, Andrew Nguyen, Marisa O'Gara

Rose-Hulman Undergraduate Mathematics Journal

Entropy is a single value that captures the complexity of a group action on a metric space. We are interested in the entropies of a family of ideal pants groups $\Gamma_T$, represented by projective reflection matrices depending on a real parameter $T > 0$. These groups act on convex sets $\Omega_{\Gamma_T}$ which form a metric space with the Hilbert metric. It is known that entropy of $\Gamma_T$ takes values in the interval $\left(\frac{1}{2},1\right]$; however, it has not been proven whether $\frac{1}{2}$ is the sharp lower bound. Using Python programming, we generate approximations of tilings of the convex set in the projective …

Knot Theory In Virtual Reality, Donald Lee Price

Masters Theses & Specialist Projects

Throughout the study of Knot Theory, there have been several programmatic solutions to common problems or questions. These solutions have included software to draw knots, software to identify knots, or online databases to look up pre-computed data about knots. We introduce a novel prototype of software used to study knots and links by using Virtual Reality. This software can allow researchers to draw links in 3D, run physics simulations on them, and identify them. This technique has not yet been rigorously explored and we believe it will be of great interest to Knot Theory researchers. The computer code is written …

An Equivalence Between Contact Gluing Maps In Sutured Floer Homology: A Conjecture Of Zarev, Charles Ryan Leigon

LSU Doctoral Dissertations

We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description in bordered sutured Floer homology. In particular, we establish Zarev's conjecture that his gluing map on sutured Floer homology is equivalent, in the appropriate sense, to the contact gluing map. This further solidifies the relationship between bordered Floer theory and contact geometry.

Approximate 2-Dimensional Pexider Quadratic Functional Equations In Fuzzy Normed Spaces And Topological Vector Space, Mohammad A. Abolfathi, Ali Ebadian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we prove the Hyers-Ulam stability of the 2-dimensional Pexider quadratic functional equation in fuzzy normed spaces. Moreover, we prove the Hyers-Ulam stability of this functional equation, where f, g are functions defined on an abelian group with values in a topological vector space.

On Digital Metric Space Satisfying Certain Rational Inequalities, Krati Shukla

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we have established some new results by extending some existing theorems in the setting of Digital Metric Space. We also proved some results in Digital Metric Space which were established earlier in the context of Complete Metric Space by different authors.

Hierarchical Hyperbolicity Of Graph Products And Graph Braid Groups, Daniel James Solomon Berlyne

Dissertations, Theses, and Capstone Projects

This thesis comprises three original contributions by the author concerning hierarchical hyperbolicity, a coarse geometric tool developed by Behrstock, Hagen, and Sisto to provide a common framework for studying aspects of non-positive curvature in a wide variety of groups and spaces.

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this to answer two questions of Genevois about the electrification of a graph product of finite groups. We also answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on a graph product forms a …

Visualizing Geometric Structures On Topological Surfaces, Andrea Clark

All NMU Master's Theses

We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the …

Sobolev Inequalities And Riemannian Manifolds, John Reever

Honors Scholar Theses

Sobolev inequalities, named after Sergei Lvovich Sobolev, relate norms in Sobolev spaces and give insight to how Sobolev spaces are embedded within each other. This thesis begins with an overview of Lebesgue and Sobolev spaces, leading into an introduction to Sobolev inequalities. Soon thereafter, we consider the behavior of Sobolev inequalities on Riemannian manifolds. We discuss how Sobolev inequalities are used to construct isoperimetric inequalities and bound volume growth, and how Sobolev inequalities imply families of other Sobolev inequalities. We then delve into the usefulness of Sobolev inequalities in determining the geometry of a manifold, such as how they can …

Evaluating The Historical Accuracy Of Blackwork Embroidery With Fractal Analysis, Rhiannon Cire

Undergraduate Theses and Capstone Projects

The intricate monochromatic embroidery that graced the collars and cuffs of Renaissance nobility and domestic materials from that era has been little studied beyond the historical costuming and crafting communities. This style, known as blackwork, for it was traditionally done in black silk on white linen, exemplifies how complex and visually-appealing designs can arise from repetition of simple forms, often demonstrating the fractal property of self-similarity. Though most blackwork patterns are not true fractals, fractal analysis offers a means of objectively quantifying their complexity and new lens through which to examine this embroidery technique. The purpose of this study was …

Solving Parabolic Interface Problems With A Finite Element Method, Henry Brown

Mathematics Student Work

Partial differential equations (PDEs) dominate mathematical models given their effectiveness and accuracy at modeling the physical realities which govern the world. Though we have these powerful tools, analytic solutions can only be found in the simplest of cases due to the complexity of PDE models. Thus, efficient and accurate computational methods are needed to approximate solutions to PDE models. One class of these methods are finite element methods which can be used domain to provide close approximations to the PDE model in a finite domain. In this presentation, we discuss the use of a Discontinuous Galerkin (DG) Finite Element Methods …

New Characterizations Of Reproducing Kernel Hilbert Spaces And Applications To Metric Geometry, Daniel Alpay, Palle E. T. Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

Translation Of: Isoparametrische Hyperflächen In Sphären, Math. Ann. 251, 57–71 (1980) By Hans Friedrich Münzner., Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

This is an English translation of the article "Isoparametrische Hyperflächen in Sphären" by Hans Friedrich Münzner, which was originally published in Math. Ann. 251, 57–71 (1980).

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in German. All references should be made to the original paper. I want to thank Cristina Ballantine for her help with the translation.

Translation Of: Isoparametrische Hyperflächen In Sphären Ii. Über Die Zerlegung Der Sphäre In Ballbündel, Math. Ann. 256, 215–232 (1981) By Hans Friedrich Münzner., Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

This is an English translation of the article "Isoparametrische Hyperflächen in Sphären II. Über die Zerlegung der Sphäre in Ballbündel" by Hans Friedrich Münzner, which was originally published in Math. Ann. 256, 215–232 (1981).

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in German. All references should be made to the original paper. I want to thank Cristina Ballantine for her help with the translation.

On The Menger And Almost Menger Properties In Locales, Tilahun Bayih, Themba Dube, Oghenetega Ighedo

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober T_D-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.