Physical Sciences and Mathematics Commons^™

Solid Angle Measure Approximation Methods For Polyhedral Cones, Allison Fitisone

Theses and Dissertations--Mathematics

Polyhedral cones are of interest in many fields, like geometry and optimization. A simple, yet fundamental question we may ask about a cone is how large it is. As cones are unbounded, we consider their solid angle measure: the proportion of space that they occupy. Beyond dimension three, definitive formulas for this measure are unknown. Consequently, devising methods to estimate this quantity is imperative. In this dissertation, we endeavor to enhance our understanding of solid angle measures and provide valuable insights into the efficacy of various approximation techniques.

Ribando and Aomoto independently discovered a Taylor series formula for solid angle …

Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

Theses and Dissertations--Mathematics

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.

Bicategorical Character Theory, Travis Wheeler

Theses and Dissertations--Mathematics

In 2007, Nora Ganter and Mikhail Kapranov defined the categorical trace, which they used to define the categorical character of a 2-representation. In 2008, Kate Ponto defined a shadow functor for bicategories. With the shadow functor, Dr. Ponto defined the bicategorical trace, which is a generalization of the symmetric monoidal trace for bicategories. How are these two notions of trace related to one another? We’ve used bicategorical traces to define a character theory for 2-representations, and the categorical character is an example.

Computational Methods For Oi-Modules, Michael Morrow

Theses and Dissertations--Mathematics

Computational commutative algebra has become an increasingly popular area of research. Central to the theory is the notion of a Gröbner basis, which may be thought of as a nonlinear generalization of Gaussian elimination. In 2019, Nagel and Römer introduced FI- and OI-modules over FI- and OI-algebras, which provide a framework for studying sequences of related modules defined over sequences of related polynomial rings. In particular, they laid the foundations of a theory of Gröbner bases for certain classes of OI-modules. In this dissertation we develop an OI-analog of Buchberger's algorithm in order to compute such Gröbner bases, as well …

Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson

Theses and Dissertations--Mathematics

An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher kby constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and Grassmannian …

Adams Operations On The Burnside Ring From Power Operations, Lewis Dominguez

Theses and Dissertations--Mathematics

Topology furnishes us with many commutative rings associated to finite groups. These include the complex representation ring, the Burnside ring, and the G-equivariant K-theory of a space. Often, these admit additional structure in the form of natural operations on the ring, such as power operations, symmetric powers, and Adams operations. We will discuss two ways of constructing Adams operations. The goal of this work is to understand these in the case of the Burnside ring.

Properties Of Skew-Polynomial Rings And Skew-Cyclic Codes, Kathryn Hechtel

Theses and Dissertations--Mathematics

A skew-polynomial ring is a polynomial ring over a field, with one indeterminate x, where one must apply an automorphism to commute coefficients with x. It was first introduced by Ore in 1933 and since the 1980s has been used to study skew-cyclic codes. In this thesis, we present some properties of skew-polynomial rings and some new constructions of skew-cyclic codes. The dimension of a skew-cyclic code depends on the degree of its generating skew polynomial. However, due to the skew-multiplication rule, the degree of a skew polynomial can be smaller than its number of roots and hence tricky to …

Dirichlet Problems In Perforated Domains, Robert Righi

Theses and Dissertations--Mathematics

We establish W^1,p estimates for solutions u_ε to the Laplace equation with Dirichlet boundary conditions in a bounded C¹ domain Ω_{ε, η} perforated by small holes in ℝ^d. The bounding constants will depend explicitly on epsilon and eta, where epsilon is the order of the minimal distance between holes, and eta denotes the ratio between the size of the holes and epsilon. The proof relies on a large-scale L^p estimate for ∇u_ε, whose proof is divided into two main parts. First, we show that solutions of an intermediate problem for a …

Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson

Theses and Dissertations--Mathematics

Much of algebraic geometry is the study of curves. One tool we use to study curves is whether they can be embedded in a K3 surface or not. If the Wahl map is surjective on a curve, that curve cannot be embedded in a K3 surface. Therefore, studying if the Wahl map is surjective for a particular curve gives us more insight into the properties of that curve. We simplify this problem by converting graph curves to dual graphs. Then the information for graphs can be used to study the underlying curves. We will discuss conditions for the Wahl map …

Asymptotic Behaviour Of Hyperbolic Partial Differential Equations, Shi-Zhuo Looi

Theses and Dissertations--Mathematics

We investigate the asymptotic behaviour of solutions to a range of linear and nonlinear hyperbolic equations on asymptotically flat spacetimes. We develop a comprehensive framework for the analysis of pointwise decay of linear and nonlinear wave equations on asymptotically flat manifolds of three space dimensions that are allowed to be time-varying or nonstationary, including quasilinear wave equations. The Minkowski space and time-varying perturbations thereof are included among these spacetimes. A result on scattering for a nonlinear wave equation with finite-energy solutions on nonstationary spacetimes is presented. This work was motivated in part by the investigation of more precise asymptotic behaviour …

Methods Of Computing Graph Gonalities, Noah Speeter

Theses and Dissertations--Mathematics

Chip firing is a category of games played on graphs. The gonality of a graph tells us how many chips are needed to win one variation of the chip firing game. The focus of this dissertation is to provide a variety of new strategies to compute the gonality of various graph families. One family of graphs which this dissertation is particularly interested in is rook graphs. Rook graphs are the Cartesian product of two or more complete graphs and we prove that the gonality of two dimensional rook graphs is the expected value of (n − 1)m where n is …

Geometric And Combinatorial Properties Of Lattice Polytopes Defined From Graphs, Kaitlin Bruegge

Theses and Dissertations--Mathematics

Polytopes are geometric objects that generalize polygons in the plane and polyhedra in 3-dimensional space. Of particular interest in geometric combinatorics are families of lattice polytopes defined from combinatorial objects, such as graphs. In particular, this dissertation studies symmetric edge polytopes (SEPs), defined from simple undirected graphs. In 2019, Higashitani, Jochemko, and Michalek gave a combinatorial description of the hyperplanes that support facets of a symmetric edge polytope in terms of certain labelings of the underlying graph.
Using this framework, we explore the number of facets that can be attained by the symmetric edge polytopes for graphs with certain structure. …

Q-Polymatroids And Their Application To Rank-Metric Codes., Benjamin Jany

Theses and Dissertations--Mathematics

Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and …

Geometry Of Pipe Dream Complexes, Benjamin Reese

Theses and Dissertations--Mathematics

In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, we show its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v > w in the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of …

Toric Bundles As Mori Dream Spaces, Courtney George

Theses and Dissertations--Mathematics

A projective, normal variety is called a Mori dream space when its Cox ring is finitely generated. These spaces are desirable to have, as they behave nicely under the Minimal Model Program, but no complete classification of them yet exists. Some early work identified that all toric varieties are examples of Mori dream spaces, as their Cox rings are polynomial rings. Therefore, a natural next step is to investigate projectivized toric vector bundles. These spaces still carry much of the combinatorial data as toric varieties, but have more variable behavior that means that they aren't as straightforward as Mori dream …

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 $\mpi_{k\rho} H_{Q_8}\ulZ$.

Normalization Techniques For Sequential And Graphical Data, Cole Pospisil

Theses and Dissertations--Mathematics

Normalization methods have proven to be an invaluable tool in the training of deep neural networks. In particular, Layer and Batch Normalization are commonly used to mitigate the risks of exploding and vanishing gradients. This work presents two methods which are related to these normalization techniques. The first method is Batch Normalized Preconditioning (BNP) for recurrent neural networks (RNN) and graph convolutional networks (GCN). BNP has been suggested as a technique for Fully Connected and Convolutional networks for achieving similar performance benefits to Batch Normalization by controlling the condition number of the Hessian through preconditioning on the gradients. We extend …

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 …

A Scattering Result For The Fifth-Order Kp-Ii Equation, Camille Schuetz

Theses and Dissertations--Mathematics

We will prove scattering for the fifth-order Kadomtsev-Petviashvilli II (fifth-order KP-II) equation. The fifth-order KP-II equation is an example of a nonlinear dispersive equation which takes the form $u_t=Lu + NL(u)$ where $L$ is a linear differential operator and $NL$ is a nonlinear operator. One looks for solutions $u(t)$ in a space $C(\R,X)$ where $X$ is a Banach space. For a nonlinear dispersive differential equation, the associated linear problem is $v_t=Lv$. A solution $u(t)$ of the nonlinear equation is said to scatter if as $t \to \infty$, the solution $u(t)$ approaches a solution $v(t)$ to the linear problem in the …

Lattice Minors And Eulerian Posets, William Gustafson

Theses and Dissertations--Mathematics

We study a partial ordering on pairings called the uncrossing poset, which first appeared in the literature in connection with a certain stratified space of planar electrical networks. We begin by examining some of the relationships between the uncrossing poset and Catalan combinatorics, and then proceed to study the structure of lower intervals. We characterize the lower intervals in the uncrossing poset that are isomorphic to the face lattice of a cube. Moving up in complexity certain lower intervals are isomorphic to the poset of simple vertex labeled minors of an associated graph.

Inspired by this structure, we define a …

Novel Architectures And Optimization Algorithms For Training Neural Networks And Applications, Vasily I. Zadorozhnyy

Theses and Dissertations--Mathematics

The two main areas of Deep Learning are Unsupervised and Supervised Learning. Unsupervised Learning studies a class of data processing problems in which only descriptions of objects are known, without label information. Generative Adversarial Networks (GANs) have become among the most widely used unsupervised neural net models. GAN combines two neural nets, generative and discriminative, that work simultaneously. We introduce a new family of discriminator loss functions that adopts a weighted sum of real and fake parts, which we call adaptive weighted loss functions. Using the gradient information, we can adaptively choose weights to train a discriminator in the direction …

Geometric Properties Of Weighted Projective Space Simplices, Derek Hanely

Theses and Dissertations--Mathematics

A long-standing conjecture in geometric combinatorics entails the interplay between three properties of lattice polytopes: reflexivity, the integer decomposition property (IDP), and the unimodality of Ehrhart h*-vectors. Motivated by this conjecture, this dissertation explores geometric properties of weighted projective space simplices, a class of lattice simplices related to weighted projective spaces.

In the first part of this dissertation, we are interested in which IDP reflexive lattice polytopes admit regular unimodular triangulations. Let Delta_(1,q)denote the simplex corresponding to the associated weighted projective space whose weights are given by the vector (1,q). Focusing on the case where Delta …

Batch Normalization Preconditioning For Neural Network Training, Susanna Luisa Gertrude Lange

Theses and Dissertations--Mathematics

Batch normalization (BN) is a popular and ubiquitous method in deep learning that has been shown to decrease training time and improve generalization performance of neural networks. Despite its success, BN is not theoretically well understood. It is not suitable for use with very small mini-batch sizes or online learning. In this work, we propose a new method called Batch Normalization Preconditioning (BNP). Instead of applying normalization explicitly through a batch normalization layer as is done in BN, BNP applies normalization by conditioning the parameter gradients directly during training. This is designed to improve the Hessian matrix of the loss …

On Flow Polytopes, Nu-Associahedra, And The Subdivision Algebra, Matias Von Bell

Theses and Dissertations--Mathematics

This dissertation studies the geometry and combinatorics related to a flow polytope F_car(ν) constructed from a lattice path ν, whose volume is given by the ν-Catalan numbers. It begins with a study of the ν-associahedron introduced by Ceballos, Padrol, and Sarmiento in 2019, but from the perspective of Schröder combinatorics. Some classical results for Schröder paths are extended to the ν-setting, and insights into the geometry of the ν-associahedron are obtained by describing its face poset with two ν-Schröder objects. The ν-associahedron is then shown to be dual to a framed triangulation of F_car(ν), which is a …

The V1-Periodic Region In Complex Motivic Ext And A Real Motivic V1-Selfmap, Ang Li

Theses and Dissertations--Mathematics

My thesis work consists of two main projects with some connections. In the first project we establish a v₁ periodicity theorem in Ext over the complex motivic Steenrod algebra. The element h₁ of Ext, which detects the homotopy class \eta in the motivic Adams spectral sequence, is non-nilpotent and therefore generates h₁-towers. Our result is that, apart from these h₁-towers, v₁ periodicity operators give isomorphisms in a range near the top of the Adams chart. This result generalizes well-known classical behavior.

In the second project we consider a nontrivial action of C_{2 …}

An Integrated Computational Pipeline To Construct Patient-Specific Cancer Models, Daniel Plaugher

Theses and Dissertations--Mathematics

Precision oncology largely involves tumor genomics to guide therapy protocols. Yet, it is well known that many commonly mutated genes cannot be easily targeted. Even when genes can be targeted, resistance to therapy is quite common. A rising field with promising results is that of mathematical biology, where in silico models are often used for the discovery of general principles and novel hypotheses that can guide the development of new treatments. A major goal is that eventually in silico models will accurately predict clinically relevant endpoints and find optimal control interventions to stop (or reverse) disease progression. Thus, it is …

Inverse Boundary Value Problems For Polyharmonic Operators With Non-Smooth Coefficients, Landon Gauthier

Theses and Dissertations--Mathematics

We consider inverse boundary problems for polyharmonic operators and in particular, the problem of recovering the coefficients of terms up to order one. The main interest of our result is that it further relaxes the regularity required to establish uniqueness. The proof relies on an averaging technique introduced by Haberman and Tataru for the study of an inverse boundary value problem for a second order operator.

Tropical Geometry Of T-Varieties With Applications To Algebraic Statistics, Joseph Cummings

Theses and Dissertations--Mathematics

Varieties with group action have been of interest to algebraic geometers for centuries. In particular, toric varieties have proven useful both theoretically and in practical applications. A rich theory blending algebraic geometry and polyhedral geometry has been developed for T-varieties which are natural generalizations of toric varieties. The first results discussed in this dissertation study the relationship between torus actions and the well-poised property. In particular, I show that the well-poised property is preserved under a geometric invariant theory quotient by a (quasi-)torus. Conversely, I argue that T-varieties built on a well-poised base preserve the well-poised property when the base …

Maximums Of Total Betti Numbers In Hilbert Families, Jay White

Theses and Dissertations--Mathematics

Fix a family of ideals in a polynomial ring and consider the problem of finding a single ideal in the family that has Betti numbers that are greater than or equal to the Betti numbers of every ideal in the family. Or decide if this special ideal even exists. Bigatti, Hulett, and Pardue showed that if we take the ideals with a fixed Hilbert function, there is such an ideal: the lexsegment ideal. Caviglia and Murai proved that if we take the saturated ideals with a fixed Hilbert polynomial, there is also such an ideal. We present a generalization of …

The Plus-Minus Davenport Constant Of Finite Abelian Groups, Darleen S. Perez-Lavin

Theses and Dissertations--Mathematics

Let G be a finite abelian group, written additively. The Davenport constant, D(G), is the smallest positive number s such that any subset of the group G, with cardinality at least s, contains a non-trivial zero-subsum. We focus on a variation of the Davenport constant where we allow addition and subtraction in the non-trivial zero-subsum. This constant is called the plus-minus Davenport constant, D_±(G). In the early 2000’s, Marchan, Ordaz, and Schmid proved that if the cardinality of G is less than or equal to 100, then the D_±(G) …