Geometry Across The Curriculum, 2020 California State University, San Bernardino

#### Geometry Across The Curriculum, Corey Dunn

*Q2S Enhancing Pedagogy*

This project is designed for a multicalculus class already familiar with computing the arc length of a parameterized curve in space. The activity asks the student to first recall basic facts about arc length, and then introduces the notion of measuring lengths of vectors differently, depending on where their initial point is. This is a foundational concept in metric differential geometry, and, this activity attempts to motivate this generalization of computing lengths of vectors through this arc length activity. The activity concludes with a short discussion of basic concepts of Lorentzian geometry, including the idea that lightlike vectors have length ...

Algebra I Topics Using Geogebra, 2020 John Carroll University

#### Algebra I Topics Using Geogebra, Matthew Rancourt

*Masters Essays*

No abstract provided.

Simultaneous Zeros Of A System Of Two Quadratic Forms, 2020 University of Kentucky

#### Simultaneous Zeros Of A System Of Two Quadratic Forms, Nandita Sahajpal

*Theses and Dissertations--Mathematics*

In this dissertation we investigate the existence of a nontrivial solution to a system of two quadratic forms over local fields and global fields. We specifically study a system of two quadratic forms over an arbitrary number field. The questions that are of particular interest are:

- How many variables are necessary to guarantee a nontrivial zero to a system of two quadratic forms over a global field or a local field? In other words, what is the
*u*-invariant of a pair of quadratic forms over any global or local field? - What is the relation between
*u*-invariants of a ...

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 ...

Patterns, Symmetries, And Mathematical Structures In The Arts, 2020 Georgia Southern University

#### Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach

*Honors College Theses*

Mathematics is a discipline of academia that can be found everywhere in the world around us. Mathematicians and scientists are not the only people who need to be proficient in numbers. Those involved in social sciences and even the arts can benefit from a background in math. In fact, connections between mathematics and various forms of art have been discovered since as early as the fourth century BC. In this thesis we will study such connections and related concepts in mathematics, dances, and music.

Stationary Distribution Of Recombination On 4x4 Grid Graph As It Relates To Gerrymandering, 2020 Claremont Colleges

#### Stationary Distribution Of Recombination On 4x4 Grid Graph As It Relates To Gerrymandering, Camryn Hollarsmith

*Scripps Senior Theses*

A gerrymandered political districting plan is used to benefit a group seeking to elect more of their own officials into office. This practice happens at the city, county and state level. A gerrymandered plan can be strategically designed based on partisanship, race, and other factors. Gerrymandering poses a contradiction to the idea of “one person, one vote” ruled by the United States Supreme Court case Reynolds v. Sims (1964) because it values one demographic’s votes more than another’s, thus creating an unfair advantage and compromising American democracy. To prevent the practice of gerrymandering, we must know how to ...

The Subconstituent Algebra Of A Hypercube, 2020 University of North Florida

#### The Subconstituent Algebra Of A Hypercube, Jared B. Billet

*UNF Graduate Theses and Dissertations*

We study the hypercube and the associated subconstituent algebra. Let Q_D denote the hypercube with dimension D and let X denote the vertex set of Q_D. Fix a vertex x in X. We denote by A the adjacency matrix of Q_D and by A* = A*(x) the diagonal matrix with yy-entry equal to D − 2i, where i is the distance between x and y. The subconstitutent algebra T = T(x) of Q_D with respect to x is generated by A and A* . We show that A 2A* − 2AA*A + A*A 2 = 4A* A*2A − 2A*AA* + AA*2 = 4A ...

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 ...

Gröbner Bases And Systems Of Polynomial Equations, 2020 Minnesota State University, Mankato

#### Gröbner Bases And Systems Of Polynomial Equations, Rachel Holmes

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

This paper will explore the use and construction of Gröbner bases through Buchberger's algorithm. Specifically, applications of such bases for solving systems of polynomial equations will be discussed. Furthermore, we relate many concepts in commutative algebra to ideas in computational algebraic geometry.

A Coherent Proof Of Mac Lane's Coherence Theorem, 2020 Claremont Colleges

#### A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo

*HMC Senior Theses*

Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a ...

On The Mysteries Of Interpolation Jack Polynomials, 2020 Claremont Colleges

#### On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers

*HMC Senior Theses*

Interpolation Jack polynomials are certain symmetric polynomials in* N* variables with coefficients that are rational functions in another parameter *k,* indexed by partitions of length at most *N*. Introduced first in 1996 by F. Knop and S. Sahi, and later studied extensively by Sahi, Knop-Sahi, and Okounkov-Olshanski, they have interesting connections to the representation theory of Lie algebras. Given an interpolation Jack polynomial we would like to differentiate it with respect to the variable *k* and write the result as a linear combination of other interpolation Jack polynomials where the coefficients are again rational functions in *k.* In this thesis ...

On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, 2020 Bucknell University

#### On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank

*Faculty Journal Articles*

We determine the ranks of string C-group representations of 4-dimensional projective symplectic groups over a finite field, and comment on those of higher-dimensional symplectic and orthogonal groups.

Exact Sequences Of Inner Automorphisms Of Tensors, 2020 Bucknell University

#### Exact Sequences Of Inner Automorphisms Of Tensors, Peter A. Brooksbank

*Faculty Journal Articles*

We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics.

Orthogonal Groups In Characteristic 2 Acting On Polytopes Of High Rank, 2020 University of Auckland

#### Orthogonal Groups In Characteristic 2 Acting On Polytopes Of High Rank, Peter A. Brooksbank, Dimitri Leemans, John T. Ferrara

*Faculty Journal Articles*

No abstract provided.

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 ...

Invariant Sum Defined In Terms Of Complex Multivariate Polynomial Given Degree, 2019 Illinois Mathematics and Science Academy

#### Invariant Sum Defined In Terms Of Complex Multivariate Polynomial Given Degree, Matthew Niemiro '20

*Exemplary Student Work*

We use a generalized version of arithmetic progressions to obtain a non- trivial everywhere-zero sum in terms of a complex univariate polynomial and its degree. We then remark on its generalization to multivariate polynomials.

Eventually Positive Matrices And Tree Sign Patterns, 2019 William & Mary

#### Eventually Positive Matrices And Tree Sign Patterns, Madellyne Waugh

*Undergraduate Honors Theses*

A n-by-n matrix A is said to be eventually positive if there is a power k such that $A^k$ is entrywise positive, and all subsequent powers are also entrywise positive. Here we provide an expression for the smallest such exponent of a 2-by-2 eventually positive matrix in terms of its entries; we also show that if the graph of an eventually positive matrix is a tree, then the positive part of that matrix must be primitive.

Groups Satisfying The Converse To Lagrange's Theorem, 2019 Missouri State University

#### Groups Satisfying The Converse To Lagrange's Theorem, Jonah N. Henry

*MSU Graduate Theses*

Lagrange’s theorem, which is taught early on in group theory courses, states that the order of a subgroup must divide the order of the group which contains it. In this thesis, we consider the converse to this statement. A group satisfying the converse to Lagrange’s theorem is called a CLT group. We begin with results that help show that a group is CLT, and explore basic CLT groups with examples. We then give the conditions to guarantee either CLT is satisﬁed or a non-CLT group exists for more advanced cases. Additionally, we show that CLT groups are properly ...

Testing Isomorphism Of Graded Algebras, 2019 Colorado State University - Fort Collins

#### Testing Isomorphism Of Graded Algebras, Peter A. Brooksbank, James B. Wilson, Eamonn A. O'Brien

*Faculty Journal Articles*

We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma.

Rank Reduction Of String C-Group Representations, 2019 Bucknell University

#### Rank Reduction Of String C-Group Representations, Peter A. Brooksbank, Dimitri Leemans

*Faculty Journal Articles*

We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207–3222] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on d-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group Alt(11)—the only known group having “rank gaps ...