The Local Lifting Problem For Curves With Quaternion Actions, 2022 The Graduate Center, City University of New York

#### The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell

*Dissertations, Theses, and Capstone Projects*

The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be ...

Reduction Of L-Functions Of Elliptic Curves Modulo Integers, 2022 The University of Western Ontario

#### Reduction Of L-Functions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau

*Electronic Thesis and Dissertation Repository*

Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T ...

The Design And Implementation Of A High-Performance Polynomial System Solver, 2022 The University of Western Ontario

#### The Design And Implementation Of A High-Performance Polynomial System Solver, Alexander Brandt

*Electronic Thesis and Dissertation Repository*

This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation.

Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical ...

Identifying Trace Affine Linear Sets Using Homotopy Continuation, 2022 Clemson University

#### Identifying Trace Affine Linear Sets Using Homotopy Continuation, Julianne Mckay

*All Theses*

We investigate how the coefficients of a sparse polynomial system influence the sum, or the trace, of its solutions. We discuss an extension of the classical trace test in numerical algebraic geometry to sparse polynomial systems. Two known methods for identifying a trace affine linear subset of the support of a sparse polynomial system use sparse resultants and polyhedral geometry, respectively. We introduce a new approach which provides more precise classifications of trace affine linear sets than was previously known. For this new approach, we developed software in Macaulay2.

Recognizing And Reducing Ambiguity In Mathematics Curriculum And Relations Of Θ-Functions In Genus One And Two: A Geometric Perspective, 2022 Utah State University

#### Recognizing And Reducing Ambiguity In Mathematics Curriculum And Relations Of Θ-Functions In Genus One And Two: A Geometric Perspective, Shantel Spatig

*All Graduate Plan B and other Reports*

Anxiety and mathematics come hand in hand for many individuals. This is due, in

part, to the fact that the only experience they have with mathematics is what some

mathematics educators refer to as "schoolmath," which uses a somewhat different

language than real mathematics. The language of schoolmath can cause individu-

als to have confusion and develop misconceptions related to several mathematical

concepts. One such concept is a fraction. In chapter one of this report, one possible

reason for this is discussed and a possible solution is purposed.

In chapter three of this report, genus-two curves admitting an elliptic involution ...

Academic Hats And Ice Cream: Two Optimization Problems, 2022 Moscow Power Engineering Institute (National Research University)

#### Academic Hats And Ice Cream: Two Optimization Problems, Valery F. Ochkov, Yulia V. Chudova

*Journal of Humanistic Mathematics*

This article describes the use of computer software to optimize the design of an academic hat and an ice cream cone!

Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, 2022 East Tennessee State University

#### Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, Bill Hemphill

*STEM Guitar Project’s BBT Acoustic Kit*

This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are

(a) a one-piece base for the standard kit's (Martin-style) bracing,

(b) 277 Ladder-style bracing, and

(c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.

The CADD data set for each of the three (3) top bracing designs includes

(a) a nominal 24" x 18" x 3mm (0.118") Baltic birch plywood laser layout of

(1) the one-piece base with slots,

(2) pre-radiused and pre-scalloped vertical braces with tabs to ensure proper orientation and alignment, and

(3) various gages and jigs and

(b) a nominal 15" x 20" marking template.

The 'provided as is" CADD data is formatted for use on a Universal Laser Systems (ULS) laser cutter digital (CNC) device. Each CADD drawing is also provided in two (2) formats: Autodesk AutoCAD 2007 .DWG and .DXF R12. Users should modify and adapt the CADD data as required to fit their equipment. This CADD data set is released and distributed under a Creative Commons license; users are also encouraged to make changes o the data and share (with attribution) their designs with the worldwide acoustic guitar building community.

Bbt Side Mold Assy, 2022 East Tennessee State University

#### Bbt Side Mold Assy, Bill Hemphill

*STEM Guitar Project’s BBT Acoustic Kit*

This electronic document file set covers the design and fabrication information of the ETSU Guitar Building Project’s BBT (OM-sized) Side Mold Assy for use with the STEM Guitar Project’s standard acoustic guitar kit. The extended 'as built' data set contains an overview file and companion video, the 'parent' CADD drawing, CADD data for laser etching and cutting a drill &/or layout template, CADD drawings in AutoCAD .DWG and .DXF R12 formats of the centerline tool paths for creating the mold assembly pieces on an AXYZ CNC router, and support documentation for CAM applications including router bit specifications, feeds ...

Unomaha Problem Of The Week (2021-2022 Edition), 2022 University of Nebraska at Omaha

#### Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

*Student Research and Creative Activity Fair*

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding ...

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, 2022 The Graduate Center, City University of New York

#### The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

*Dissertations, Theses, and Capstone Projects*

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal ...

On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, 2022 University of Rochester

#### On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi

*Rose-Hulman Undergraduate Mathematics Journal*

It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of ...

The Examination Of The Arithmetic Surface (3, 5) Over Q, 2022 California State University - San Bernardino

#### The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

*Electronic Theses, Projects, and Dissertations*

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i^{2} = 3, j^{2} = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL_{2}(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This list of ...

Lattice Reduction Algorithms, 2022 California State University, San Bernardino

#### Lattice Reduction Algorithms, Juan Ortega

*Electronic Theses, Projects, and Dissertations*

The purpose of this thesis is to propose and analyze an algorithm that follows

similar steps of Guassian Lattice Reduction Algorithm in two-dimensions and applying

them to three-dimensions. We start off by discussing the importance of cryptography in

our day to day lives. Then we dive into some linear algebra and discuss specific topics that

will later help us in understanding lattice reduction algorithms. We discuss two lattice

problems: the shortest vector problem and the closest vector problem. Then we introduce

two types of lattice reduction algorithms: Guassian Lattice Reduction in two-dimensions

and the LLL Algortihm. We illustrate how both ...

Towards A Generalization Of Fulton's Intersection Multiplicity Algorithm, 2022 The University of Western Ontario

#### Towards A Generalization Of Fulton's Intersection Multiplicity Algorithm, Ryan Sandford

*Electronic Thesis and Dissertation Repository*

In this manuscript we generalize Fulton's bivariate intersection multiplicity algorithm to a partial intersection multiplicity algorithm in the n-variate setting. We extend this generalization of Fulton's algorithm to work at any point, rational or not, using the theory of regular chains. We implement these algorithms in Maple and provide experimental testing. The results indicate the proposed algorithm often outperforms the existing standard basis-free intersection multiplicity algorithm in Maple, typically by one to two orders of magnitude. Moreover, we also provide some examples where the proposed algorithm outperforms intersection multiplicity algorithms which rely on standard bases, indicating the proposed ...

On The Geometry Of Multi-Affine Polynomials, 2022 The University of Western Ontario

#### On The Geometry Of Multi-Affine Polynomials, Junquan Xiao

*Electronic Thesis and Dissertation Repository*

This work investigates several geometric properties of the solutions of the multi-affine polynomials. Chapters 1, 2 discuss two different notions of invariant circles. Chapter 3 gives several loci of polynomials of degree three. A locus of a complex polynomial p(z) is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The study of such objects allows one to improve upon know results about the location of zeros and critical points of complex polynomials, see for example [22] and [24]. A complex polynomial has many loci. It ...

Kissing The Archimedeans, 2022 Northern Michigan University

#### Kissing The Archimedeans, Anthony Webb

*All NMU Master's Theses*

In this paper the three dimensional kissing problem will be related to the Platonic and Archimedean solids. On each polyhedra presented their vertices will have spheres expanding such that the center of each of these outer spheres are the vertices of the polyhedron, and these outer spheres will continue to expand until they become tangent to each other. The ratio will be found between the radius of each outer sphere, and the radius of an inner sphere such that each inner sphere's center is the circumcenter of the polyhedron, and the inner sphere is tangent to each outer sphere ...

Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, 2022 University of Massachusetts Amherst

#### Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, Arie A. Stern Gonzalez

*Doctoral Dissertations*

In this thesis we study anticanonical models of smoothings of cyclic quotient singularities. Given a surface cyclic quotient singularity $Q\in Y$, it is an open problem to determine all smoothings of $Y$ that admit an anticanonical model and to compute it. In \cite{HTU}, Hacking, Tevelev and Urz\'ua studied certain irreducible components of the versal deformation space of $Y$, and within these components, they found one parameter smoothings $\Y \to \A^1$ that admit an anticanonical model and proved that they have canonical singularities. Moreover, they compute explicitly the anticanonical models that have terminal singularities using Mori's ...

Tropical Geometry Of T-Varieties With Applications To Algebraic Statistics, 2022 University of Kentucky

#### 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 ...

Equisingular Approximation Of Analytic Germs, 2021 The University of Western Ontario

#### Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel

*Electronic Thesis and Dissertation Repository*

This thesis deals with the problem of approximating germs of real or complex analytic spaces by Nash or algebraic germs. In particular, we investigate the problem of approximating analytic germs in various ways while preserving the Hilbert-Samuel function, which is of importance in the resolution of singularities. We first show that analytic germs that are complete intersections can be arbitrarily closely approximated by algebraic germs which are complete intersections with the same Hilbert-Samuel function. We then show that analytic germs whose local rings are Cohen-Macaulay can be arbitrarily closely approximated by Nash germs whose local rings are Cohen- Macaulay and ...

Acceleration Skinning: Kinematics-Driven Cartoon Effects For Articulated Characters, 2021 Clemson University

#### Acceleration Skinning: Kinematics-Driven Cartoon Effects For Articulated Characters, Niranjan Kalyanasundaram

*All Theses*

Secondary effects are key to adding fluidity and style to animation. This thesis introduces the idea of “Acceleration Skinning” following a recent well-received technique, Velocity Skinning, to automatically create secondary motion in character animation by modifying the standard pipeline for skeletal rig skinning. These effects, which animators may refer to as squash and stretch or drag, attempt to create an illusion of inertia. In this thesis, I extend the Velocity Skinning technique to include acceleration for creating a wider gamut of cartoon effects. I explore three new deformers that make use of this Acceleration Skinning framework: followthrough, centripetal stretch, and ...