Geometry Of Linear Subspace Arrangements With Connections To Matroid Theory, 2020 University of Kentucky

#### Geometry Of Linear Subspace Arrangements With Connections To Matroid Theory, William Trok

*Theses and Dissertations--Mathematics*

This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond's about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points.

We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface'' …

Harmonic Morphisms With One-Dimensional Fibres And Milnor Fibrations, 2020 University of North Florida

#### Harmonic Morphisms With One-Dimensional Fibres And Milnor Fibrations, Murphy Griffin

*UNF Graduate Theses and Dissertations*

We study a problem at the intersection of harmonic morphisms and real analytic Milnor fibrations. Baird and Ou establish that a harmonic morphism from G: \mathbb{R}^m \setminus V_G \rightarrow \mathbb{R}^n\setminus \{0\} defined by homogeneous polynomials of order p retracts to a harmonic morphism \psi|: S^{m-1} \setminus K_\epsilon \rightarrow S^{n-1} that induces a Milnor fibration over the sphere. In seeking to relax the homogeneity assumption on the map G, we determine that the only harmonic morphism $\varphi: \mathbb{R}^m \setminus V_G \rightarrow S^{m-1}\K_\epsilon$ that preserves \arg G is radial projection. Due to this limitation, we confirm Baird and Ou's result, yet establish …

Codes, Cryptography, And The Mceliece Cryptosystem, 2020 Liberty University

#### Codes, Cryptography, And The Mceliece Cryptosystem, Bethany Matsick

*Senior Honors Theses*

Over the past several decades, technology has continued to develop at an incredible rate, and the importance of properly securing information has increased significantly. While a variety of encryption schemes currently exist for this purpose, a number of them rely on problems, such as integer factorization, that are not resistant to quantum algorithms. With the reality of quantum computers approaching, it is critical that a quantum-resistant method of protecting information is found. After developing the proper background, we evaluate the potential of the McEliece cryptosystem for use in the post-quantum era by examining families of algebraic geometry codes that allow …

Phylogenetic Networks And Functions That Relate Them, 2020 The University of Akron

#### Phylogenetic Networks And Functions That Relate Them, Drew Scalzo

*Williams Honors College, Honors Research Projects*

Phylogenetic Networks are defined to be simple connected graphs with exactly n labeled nodes of degree one, called leaves, and where all other unlabeled nodes have a degree of at least three. These structures assist us with analyzing ancestral history, and its close relative - phylogenetic trees - garner the same visualization, but without the graph being forced to be connected. In this paper, we examine the various characteristics of Phylogenetic Networks and functions that take these networks as inputs, and convert them to more complex or simpler structures. Furthermore, we look at the nature of functions as they relate …

Heat Kernel Voting With Geometric Invariants, 2020 Minnesota State University, Mankato

#### Heat Kernel Voting With Geometric Invariants, Alexander Harr

*All Graduate Theses, Dissertations, and Other Capstone Projects*

Here we provide a method for comparing geometric objects. Two objects of interest are embedded into an infinite dimensional Hilbert space using their Laplacian eigenvalues and eigenfunctions, truncated to a finite dimensional Euclidean space, where correspondences between the objects are searched for and voted on. To simplify correspondence finding, we propose using several geometric invariants to reduce the necessary computations. This method improves on voting methods by identifying isometric regions including shapes of genus greater than 0 and dimension greater than 3, as well as almost retaining isometry.

Scrollar Invariants Of Tropical Chains Of Loops, 2020 University of Kentucky

#### Scrollar Invariants Of Tropical Chains Of Loops, Kalila Joelle Sawyer

*Theses and Dissertations--Mathematics*

We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and compute these invariants for a much-studied family of tropical curves. Our examples highlight many parallels between the classical and tropical theories, but also point to some substantive distinctions.

Algebraic And Geometric Properties Of Hierarchical Models, 2020 University of Kentucky

#### Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj

*Theses and Dissertations--Mathematics*

In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models.

Model Selection And Experimental Design Of Biological Networks With Algebraic Geometry, 2019 Southern Methodist University

#### Model Selection And Experimental Design Of Biological Networks With Algebraic Geometry, Anyu Zhang

*Mathematics Theses and Dissertations*

Model selection based on experimental data is an essential challenge in biological data science. In decades, the volume of biological data from varied sources, including laboratory experiments, field observations, and patient health records has seen an unprecedented increase. Mainly when collecting data is expensive or time-consuming, as it is often in the case with clinical trials and biomolecular experiments, the problem of selecting information-rich data becomes crucial for creating relevant models.

Motivated by certain geometric relationships between data, we partitioned input data sets, especially data sets that correspond to a unique basis, into equivalence classes with the same basis to …

Albert Forms, Quaternions, Schubert Varieties & Embeddability, 2019 The University of Western Ontario

#### Albert Forms, Quaternions, Schubert Varieties & Embeddability, Jasmin Omanovic

*Electronic Thesis and Dissertation Repository*

The origin of embedding problems can be understood as an effort to find some minimal datum which describes certain algebraic or geometric objects. In the algebraic theory of quadratic forms, Pfister forms are studied for a litany of powerful properties and representations which make them particularly interesting to study in terms of embeddability. A generalization of these properties is captured by the study of central simple algebras carrying involutions, where we may characterize the involution by the existence of particular elements in the algebra. Extending this idea even further, embeddings are just flags in the Grassmannian, meaning that their study …

Algebraic Methods For Proving Geometric Theorems, 2019 California State University, San Bernardino

#### Algebraic Methods For Proving Geometric Theorems, Lynn Redman

*Electronic Theses, Projects, and Dissertations*

Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal …

Local And Global Color Symmetries Of A Symmetrical Pattern, 2019 Ateneo de Manila University

#### Local And Global Color Symmetries Of A Symmetrical Pattern, Ma. Louise Antonette N. De Las Peñas, Agatha Kristel Abila, Eduard C. Taganap

*Mathematics Faculty Publications*

This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern P consisting of disjoint congruent symmetric motifs. The pattern P has local symmetries that are not necessarily contained in its global symmetry group G. The usual approach in color symmetry theory is to arrive at perfect colorings of P ignoring local symmetries and considering only elements of G. A framework is presented to systematically arrive at what Roth [Geom. Dedicata (1984), 17, 99–108] defined as a coordinated coloring of P, a coloring that is perfect and transitive under G, satisfying the condition that the coloring …

Essential Dimension Of Parabolic Bundles, 2019 The University of Western Ontario

#### Essential Dimension Of Parabolic Bundles, Dinesh Valluri

*Electronic Thesis and Dissertation Repository*

Essential dimension of a geometric object is roughly the number of algebraically independent parameters needed to define the object. In this thesis we give upper bounds for the essential dimension of parabolic bundles over a non-singular curve X of genus g greater than or equal to 2 using Borne's correspondence between parabolic bundles on a curve and vector bundles on a root stack. This is a generalization of the work of Biswas, Dhillon and Hoffmann on the essential dimension of vector bundles, by following their method for curves and adapting it to root stacks. In this process, we invoke the …

Effective Statistical Energy Function Based Protein Un/Structure Prediction, 2019 University of New Orleans

#### Effective Statistical Energy Function Based Protein Un/Structure Prediction, Avdesh Mishra

*University of New Orleans Theses and Dissertations*

Proteins are an important component of living organisms, composed of one or more polypeptide chains, each containing hundreds or even thousands of amino acids of 20 standard types. The structure of a protein from the sequence determines crucial functions of proteins such as initiating metabolic reactions, DNA replication, cell signaling, and transporting molecules. In the past, proteins were considered to always have a well-defined stable shape (structured proteins), however, it has recently been shown that there exist intrinsically disordered proteins (IDPs), which lack a fixed or ordered 3D structure, have dynamic characteristics and therefore, exist in multiple states. Based on …

Topology And Dynamics Of Gene Regulatory Networks: A Meta-Analysis, 2019 Iowa State University

#### Topology And Dynamics Of Gene Regulatory Networks: A Meta-Analysis, Claus Kadelka

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Unifications Of Pythagorean Triple Schema, 2019 East Tennessee State University

#### Unifications Of Pythagorean Triple Schema, Emily Hammes

*Undergraduate Honors Theses*

Euclid’s Method of finding Pythagorean triples is a commonly accepted and applied technique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid’s Method.

On The Complexity Of Computing Galois Groups Of Differential Equations, 2019 The Graduate Center, City University of New York

#### On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

*Dissertations, Theses, and Capstone Projects*

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential …

Data Parsing For Optimized Molecular Geometry Calculations, 2019 Stephen F Austin State University

#### Data Parsing For Optimized Molecular Geometry Calculations, Luke Rens

*Undergraduate Research Conference*

The purpose of this project is to optimize and streamline to process of using ADF and ReaxFF. There is no efficient way to effectively add constraints to a compound and run it through ADF, take the ADF output and create a file that can be run through Reaxff, then take that Reaxff output and come to conclusions on it. To streamline this process, scripts were developed using Python to parse information out of data generated by ADF.

Enhanced Koszulity In Galois Cohomology, 2019 The University of Western Ontario

#### Enhanced Koszulity In Galois Cohomology, Marina Palaisti

*Electronic Thesis and Dissertation Repository*

Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro-$p$-groups, Galois cohomology is …

Dense Geometry Of Music And Visual Arts: Vanishing Points, Continuous Tonnetz, And Theremin Performance, 2019 Independent researcher, Palermo, Italy

#### Dense Geometry Of Music And Visual Arts: Vanishing Points, Continuous Tonnetz, And Theremin Performance, Maria Mannone, Irene Iaccarino, Rosanna Iembo

*The STEAM Journal*

The dualism between continuous and discrete is relevant in music theory as well as in performance practice of musical instruments. Geometry has been used since longtime to represent relationships between notes and chords in tonal system. Moreover, in the field of mathematics itself, it has been shown that the continuity of real numbers can arise from geometrical observations and reasoning. Here, we consider a geometrical approach to generalize representations used in music theory introducing continuous pitch. Such a theoretical framework can be applied to instrument playing where continuous pitch can be naturally performed. Geometry and visual representations of concepts of …

Radiation Effect On Mixed Convection Flow Of Nanofluid Between Two Concentric Cylinders With Hall And Ion-Slip Effects, 2019 Vaagdevi College of Engineering

#### Radiation Effect On Mixed Convection Flow Of Nanofluid Between Two Concentric Cylinders With Hall And Ion-Slip Effects, Md. Shafeeurrahman, D. Srinivasacharya

*Applications and Applied Mathematics: An International Journal (AAM)*

This paper analyzes the effects of thermal radiation, Hall and ion slip parameter on mixed convective nanofluid flow in an annuli between two concentric cylinders in the existence of strong magnetic field. The nonlinear governing equations are non-dimensionalized and then solved by using homotopy analysis method. The influence of radiation, magnetic, Hall and ion slip parameters on the velocity, temperature, nanoparticle concentration, Nusselt number and nanoparticle Sherwood number are investigated and represented graphically.