Centroidal Voronoi Tessellations With Few Generator Points, 2018 Bard College

#### Centroidal Voronoi Tessellations With Few Generator Points, Kirill Shakhnovskiy

*Senior Projects Spring 2018*

A Voronoi tessellation with $n$ generator points is the partitioning of a bounded region in $\rr^2$ into polygons such that every point in a given polygon is closer to its generator point than to any other generator point. A centroidal Voronoi tessellation (CVT) is a Voronoi tessellation where each polygon’s generator point is also its center of mass. In this project I will demonstrate what kinds of CVTs can exists within specific parameters, such as a square or rectangular region, and a set number generator points. I will also prove that the examples I present are the only ...

On The Landscape Of Random Tropical Polynomials, 2018 Claremont Colleges

#### On The Landscape Of Random Tropical Polynomials, Christopher Hoyt

*HMC Senior Theses*

Tropical polynomials are similar to classical polynomials, however addition and multiplication are replaced with tropical addition (minimums) and tropical multiplication (addition). Within this new construction, polynomials become piecewise linear curves with interesting behavior. All tropical polynomials are piecewise linear curves, and each linear component uniquely corresponds to a particular monomial. In addition, certain monomial in the tropical polynomial can be trivial due to the fact that tropical addition is the minimum operator. Therefore, it makes sense to consider a graph of connectivity of the monomials for any given tropical polynomial. We investigate tropical polynomials where all coefficients are chosen from ...

An Incidence Approach To The Distinct Distances Problem, 2018 Claremont Colleges

#### An Incidence Approach To The Distinct Distances Problem, Bryce Mclaughlin

*HMC Senior Theses*

In 1946, Erdös posed the distinct distances problem, which asks for the minimum number of distinct distances that any set of n points in the real plane must realize. Erdös showed that any point set must realize at least &Omega(n^{1/2}) distances, but could only provide a construction which offered &Omega(n/&radic(log(n)))$ distances. He conjectured that the actual minimum number of distances was &Omega(n^{1-&epsilon}) for any &epsilon > 0, but that sublinear constructions were possible. This lower bound has been improved over the years, but Erdös' conjecture seemed to hold until in 2010 ...

Geometric Serendipity, 2018 Virginia Commonwealth University

#### Geometric Serendipity, Dakota Becker

*Auctus: The Journal of Undergraduate Research and Creative Scholarship*

The central focus of my practice is the serendipitous exploration into geometry, symmetry, design, and color. I have found more and more that the affinity I have for hard-edge geometric abstraction is a deeper reflection of the way in which I process my thoughts and surroundings. In the past year, I have sought to challenge myself by questioning the core of my practice and pushing it to go beyond its individual elements. In this way, I seek to create work that is more than its parts. As a result, I have become more purposeful with my designs and push both ...

Framed Sheaves On A Quadric Surface, 2018 University of Massachusetts Amherst

#### Framed Sheaves On A Quadric Surface, Nguyen Thuc Huy Le

*Doctoral Dissertations*

We study framed sheaves on a smooth quadric surface and conjecture that the moduli of such framed sheaves admits a twistor deformation similar to one studied in the paper "Brill-Noether duality for moduli spaces of sheaves on K3 surfaces" by Markman.

A Journey To The Adic World, 2018 Georgia Southern University

#### A Journey To The Adic World, Fayadh Kadhem

*Electronic Theses and Dissertations*

The first idea of this research was to study a topic that is related to both Algebra and Topology and explore a tool that connects them together. That was the entrance for me to the “adic world”. What was needed were some important concepts from Algebra and Topology, and so they are treated in the first two chapters.

The reader is assumed to be familiar with Abstract Algebra and Topology, especially with Ring theory and basics of Point-set Topology.

The thesis consists of a motivation and four chapters, the third and the fourth being the main ones. In the third ...

On Representations Of The Jacobi Group And Differential Equations, 2018 University of North Florida

#### On Representations Of The Jacobi Group And Differential Equations, Benjamin Webster

*UNF Graduate Theses and Dissertations*

In PDEs with nontrivial Lie symmetry algebras, the Lie symmetry naturally yield Fourier and Laplace transforms of fundamental solutions. Applying this fact we discuss the semidirect product of the metaplectic group and the Heisenberg group, then induce a representation our group and use it to investigate the invariant solutions of a general differential equation of the form .

Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, 2017 University of Nebraska-Lincoln

#### Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, Solomon Akesseh

*Dissertations, Theses, and Student Research Papers in Mathematics*

Fat points and their ideals have stimulated a lot of research but this dissertation concerns itself with aspects of only two of them, broadly categorized here as, the ideal containments and polynomial interpolation problems.

Ein-Lazarsfeld-Smith and Hochster-Huneke cumulatively showed that for all ideals I in k[**P**^{n}], I^{(mn)} ⊆ I^{m} for all m ∈ N. Over the projective plane, we obtain I^{(4)}< ⊆ I^{2}. Huneke asked whether it was the case that I^{(3)} ⊆ I^{2}. Dumnicki, Szemberg and Tutaj-Gasinska show that if I is the saturated homogeneous radical ideal of the 12 points of the Hesse configuration, then ...

Descartes Comes Out Of The Closet, 2017 Vassar College

#### Descartes Comes Out Of The Closet, Nora E. Culik

*Journal of Humanistic Mathematics*

While “Descartes Comes Out of the Closet” is ostensibly about a young woman’s journey to Paris, the descriptive detail borrows language and images from Cartesian coordinate geometry, dualistic philosophy, neuroanatomy (the pineal), and projections of three dimensions onto planes. This mathematical universe is counterpointed in the natural language of the suppressed love story that locates the real in the human. Thus, at the heart of the story is the tension between competing notions of mathematics, i.e., as either an independent realm apart from history or as a culturally produced and historical set of practices. Of course, the central ...

College Algebra, Trigonometry, And Precalculus (Clayton), 2017 Clayton State University

#### College Algebra, Trigonometry, And Precalculus (Clayton), Chaogui Zhang, Scott Bailey, Billie May, Jelinda Spotorno, Kara Mullen

*Mathematics Grants Collections*

This Grants Collection for College Algebra, Trigonometry, and Precalculus was created under a Round Five ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

- Linked Syllabus
- Initial Proposal
- Final Report

Counting Rational Points, Integral Points, Fields, And Hypersurfaces, 2017 The Graduate Center, City University of New York

#### Counting Rational Points, Integral Points, Fields, And Hypersurfaces, Joseph Gunther

*All Dissertations, Theses, and Capstone Projects*

This thesis comes in four parts, which can be read independently of each other.

In the first chapter, we prove a generalization of Poonen's finite field Bertini theorem, and use this to show that the obvious obstruction to embedding a curve in some smooth surface is the only obstruction over perfect fields, extending a result of Altman and Kleiman. We also prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

In the second chapter, for a fixed base curve over a finite field of characteristic at least 5 ...

Klein Four Actions On Graphs And Sets, 2017 Gettysburg College

#### Klein Four Actions On Graphs And Sets, Darren B. Glass

*Math Faculty Publications*

We consider how a standard theorem in algebraic geometry relating properties of a curve with a (ℤ/2ℤ)2-action to the properties of its quotients generalizes to results about sets and graphs that admit (ℤ/2ℤ)2-actions.

Cox Processes For Visual Object Counting, 2017 Portland State University

#### Cox Processes For Visual Object Counting, Yongming Ma

*Student Research Symposium*

We present a model that utilizes Cox processes and CNN classifiers in order to count the number of instances of an object in an image. Poisson processes are well suited to events that occur randomly in space, like the location of objects in an image, as well as to the task of counting. Mixed Poisson processes also offer increased flexibility, however they do not easily scale with image size: they typically require O(n3) computation time and O(n2) storage, where n is the number of pixels. To mitigate this problem, we employ Kronecker algebra which takes advantage of the ...

Integrating Non-Euclidean Geometry Into High School, 2017 Loyola Marymount University

#### Integrating Non-Euclidean Geometry Into High School, John Buda

*Honors Thesis*

The purpose of this project is to provide the framework for integrating the study of non-Euclidean geometry into a high school math class in such a way that both aligns with the Common Core State Standards and makes use of research-based practices to enhance the learning of traditional geometry. Traditionally, Euclidean geometry has been the only strand of geometry taught in high schools, even though mathematicians have developed several other strands. The non-Euclidean geometry that I focus on in this project is what is known as taxicab geometry. With the Common Core Standards for Math Practice pushing students to “model ...

Beurling-Lax Type Theorems In The Complex And Quaternionic Setting, 2017 Chapman University

#### Beurling-Lax Type Theorems In The Complex And Quaternionic Setting, Daniel Alpay, Irene Sabadini

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

We give a generalization of the Beurling–Lax theorem both in the complex and quaternionic settings. We consider in the first case functions meromorphic in the right complex half-plane, and functions slice hypermeromorphic in the right quaternionic half-space in the second case. In both settings we also discuss a unified framework, which includes both the disk and the half-plane for the complex case and the open unit ball and the half-space in the quaternionic setting.

Student-Created Test Sheets, 2017 Bowling Green State University

#### Student-Created Test Sheets, Samuel Laderach

*Honors Projects*

Assessment plays a necessary role in the high school mathematics classroom, and testing is a major part of assessment. Students often struggle with mathematics tests and examinations due to math and test anxiety, a lack of student learning, and insufficient and inefficient student preparation. Practice tests, teacher-created review sheets, and student-created test sheets are ways in which teachers can help increase student performance, while ridding these detrimental factors. Student-created test sheets appear to be the most efficient strategy, and this research study examines the effects of their use in a high school mathematics classroom.

On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, 2017 The University of Western Ontario

#### On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, Nadia Alluhaibi

*Electronic Thesis and Dissertation Repository*

The objective of the study is to investigate the behaviour of the inner products of vector-valued Poincare series, for large weight, associated to submanifolds of a quotient of the complex unit ball and how vector-valued automorphic forms could be constructed via Poincare series. In addition, it provides a proof of that vector-valued Poincare series on an irreducible bounded symmetric domain span the space of vector-valued automorphic forms.

Polygons, Pillars And Pavilions: Discovering Connections Between Geometry And Architecture, 2017 University High School

#### Polygons, Pillars And Pavilions: Discovering Connections Between Geometry And Architecture, Sean Patrick Madden

*Journal of Catholic Education*

Crowning the second semester of geometry, taught within a Catholic middle school, the author's students explored connections between the geometry of regular polygons and architecture of local buildings. They went on to explore how these principles apply famous buildings around the world such as the monuments of Washington, D.C. and the elliptical piazza of Saint Peter's Basilica at Vatican City within Rome, Italy.

Classification Of Rectifying Space-Like Submanifolds In Pseudo-Euclidean Spaces, 2017 Michigan State University

#### Classification Of Rectifying Space-Like Submanifolds In Pseudo-Euclidean Spaces, Bang Yen Chan, Yun Oh

*Faculty Publications*

The notions of rectifying subspaces and of rectifying submanifolds were introduced in [B.-Y. Chen, Int. Electron. J. Geom 9 (2016), no. 2, 1–8]. More precisely, a submanifold in a Euclidean m-space Em is called a rectifying submanifold if its position vector field always lies in its rectifying subspace. Several fundamental properties and classification of rectifying submanifolds in Euclidean space were obtained in [B.-Y. Chen, op. cit.]. In this present article, we extend the results in [B.-Y. Chen, op. cit.] to rectifying space- like submanifolds in a pseudo-Euclidean space with arbitrary codimension. In particular, we completely classify ...

Session A-3: Three-Act Math Tasks, 2017 Illinois Mathematics and Science Academy

#### Session A-3: Three-Act Math Tasks, Lindsey Herlehy

*Professional Learning Day*

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.