College Algebra Through Problem Solving (2018 Edition), 2018 CUNY Queensborough Community College

#### College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Masklanko, Ewa Dabkowska

*Open Educational Resources*

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

On The Density Of The Odd Values Of The Partition Function, 2018 Michigan Technological University

#### On The Density Of The Odd Values Of The Partition Function, Samuel Judge

*Dissertations, Master's Theses and Master's Reports*

The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities ...

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, 2018 Murray State University

#### Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, William Atkins

*Murray State Theses and Dissertations*

We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as ﬁlling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is associated with ...

Lattices Of Supercharacter Theories, 2018 University of Colorado at Boulder

#### Lattices Of Supercharacter Theories, Jonathan P. Lamar

*Mathematics Graduate Theses & Dissertations*

The set of supercharacter theories of a finite group forms a lattice under a natural partial order. An active area of research in the study of supercharacter theories is the classification of this lattice for various families of groups. One other active area of research is the formation of Hopf structures from compatible supercharacter theories over indexed families of groups. This thesis therefore has two goals. First, we will classify the supercharacter theory lattice of the dihedral groups *D*_{2n} in terms of their cyclic subgroups of rotations, as well as for some semidirect products of the form ℤ_{n} ⋊ ℤ_{p ...}

The Relationship Between K-Forcing And K-Power Domination, 2018 Texas State University

#### The Relationship Between K-Forcing And K-Power Domination, Daniela Ferrero, Leslie Hogben, Franklin H.J. Kenter, Michael Young

*Mathematics Publications*

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. The concept of k-power domination was introduced by Chang et al. (2012) as a generalization of power domination and standard graph domination. Independently, k-forcing was defined by Amos et al. (2015) to generalize zero forcing. In this paper, we combine the study of k-forcing and k-power domination, providing a new approach to analyze both ...

Quiver Varieties And Crystals In Symmetrizable Type Via Modulated Graphs, 2018 University of Sydney, Australia

#### Quiver Varieties And Crystals In Symmetrizable Type Via Modulated Graphs, Vinoth Nandakumar, Peter Tingley

*Mathematics and Statistics: Faculty Publications and Other Works*

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac–Moody algebra. The underlying set consists of the irreducible components of Lusztig’s quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac–Moody algebras by replacing Lusztig’s preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.

Ancient Cultures + High School Algebra = A Diverse Mathematical Approach, 2018 John Carroll University

#### Ancient Cultures + High School Algebra = A Diverse Mathematical Approach, Laryssa Byndas

*Masters Essays*

No abstract provided.

Group Rings, 2018 John Carroll University

Determinantal Conditions On Integer Splines, 2018 Bard College

#### Determinantal Conditions On Integer Splines, Kathryn Elizabeth Blaine

*Senior Projects Fall 2018*

In this project, we work with integer splines on graphs with positive integer edge labels. We focus on graphs that are (m, n)-cycles for some natural numbers m, n, specifically the diamond graph, which consists of two triangles joined at an edge. We extend previous research on integer splines over the diamond graph. In particular, we prove that a set of splines on the diamond graph forms a basis if and only if it satisfies a certain determinantal criterion.

An Implementation Of The Solution To The Conjugacy Problem On Thompson's Group V, 2018 Bard College

#### An Implementation Of The Solution To The Conjugacy Problem On Thompson's Group V, Rachel K. Nalecz

*Senior Projects Spring 2018*

We describe an implementation of the solution to the conjugacy problem in Thompson's group V as presented by James Belk and Francesco Matucci in 2013. Thompson's group V is an infinite finitely presented group whose elements are complete binary prefix replacement maps. From these we can construct closed abstract strand diagrams, which are certain directed graphs with a rotation system and an associated cohomology class. The algorithm checks for conjugacy by constructing and comparing these graphs together with their cohomology classes. We provide a complete outline of our solution algorithm, as well as a description of the data ...

A Relation Between Mirkovic-Vilonen Cycles And Modules Over Preprojective Algebra Of Dynkin Quiver Of Type Ade, 2018 University of Massachusetts Amherst

#### A Relation Between Mirkovic-Vilonen Cycles And Modules Over Preprojective Algebra Of Dynkin Quiver Of Type Ade, Zhijie Dong

*Doctoral Dissertations*

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian to a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. The relation between this conjecture and the reduceness ...

On The K-Theory Of Generalized Bunce-Deddens Algebras, 2018 University of Colorado at Boulder

#### On The K-Theory Of Generalized Bunce-Deddens Algebras, Nathan Davidoff

*Mathematics Graduate Theses & Dissertations*

We consider a ℤ-action σ on a directed graph -- in particular a rooted tree *T* -- inherited from the odometer action. This induces a ℤ-action by automorphisms on *C**(*T*). We show that the resulting crossed product *C**(*T*) ⋊_{σ}ℤ is strongly Morita equivalent to the Bunce-Deddens algebra. The Pimsner-Voiculescu sequence allows us to reconstruct the *K*-theory for the Bunce-Deddens algebra in a new way using graph methods. We then extend to a ℤ^{k}-action σ̃ on a *k*-graph when *k* = 2, show that *C**(*T*_{1} ✕ *T*_{2})⋊_{σ}ℤ^{2} is strongly Morita equivalent to a generalized Bunce-Deddens ...

Extensions Of The Morse-Hedlund Theorem, 2018 Bucknell University

#### Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell

*Honors Theses*

Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...*ababababab*...". The Morse-Hedlund theorem says that a bi-infinite word *f* repeats itself, in at most *n* letters, if and only if the number of distinct subwords of length *n* is at most *n*. Using the example, "...*ababababab*...", there are 2 subwords of length 3, namely "*aba*" and "*bab*". Since 2 is less than 3, we must have that "...*ababababab*..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. Interestingly, there are many extensions of this theorem to multiple dimensions ...

On Spectral Theorem, 2018 Colby College

#### On Spectral Theorem, Muyuan Zhang

*Honors Theses*

There are many instances where the theory of eigenvalues and eigenvectors has its applications. However, Matrix theory, which usually deals with vector spaces with finite dimensions, also has its constraints. Spectral theory, on the other hand, generalizes the ideas of eigenvalues and eigenvectors and applies them to vector spaces with arbitrary dimensions. In the following chapters, we will learn the basics of spectral theory and in particular, we will focus on one of the most important theorems in spectral theory, namely the spectral theorem. There are many different formulations of the spectral theorem and they convey the "same" idea. In ...

Parametric Polynomials For Small Galois Groups, 2018 Colby College

#### Parametric Polynomials For Small Galois Groups, Claire Huang

*Honors Theses*

Galois theory, named after French mathematician Evariste Galois in 19th-century, is an important part of abstract algebra. It brings together many different branches of mathematics by providing connections among fields, polynomials, and groups.

Specifically, Galois theory allows us to attach a finite field extension with a finite group. We call such a group the Galois group of the finite field extension. A typical way to attain a finite field extension to compute the splitting field of some polynomial. So we can always start with a polynomial and find the finite group associate to the field extension on its splitting field ...

Applications Of Analysis To The Determination Of The Minimum Number Of Distinct Eigenvalues Of A Graph, 2018 Iowa State University

#### Applications Of Analysis To The Determination Of The Minimum Number Of Distinct Eigenvalues Of A Graph, Beth Bjorkman, Leslie Hogben, Scarlitte Ponce, Carolyn Reinhart, Theodore Tranel

*Mathematics Publications*

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues of several families of graphs and small graphs.

I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, 2018 University of Central Florida

#### I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, Jason L. Greuling

*Honors Undergraduate Theses*

Research has shown that a frame for an n-dimensional real Hilbert space oﬀers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and suﬃcient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space does not ensure that its dilation will oﬀer phase retrieval. In this thesis, we will explore and provide what necessary and suﬃcient ...

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 ...

Abelian Subalgebras Of Maximal Dimension In Euclidean Lie Algebras, 2018 Wilfrid Laurier University

#### Abelian Subalgebras Of Maximal Dimension In Euclidean Lie Algebras, Mark Curro

*Theses and Dissertations (Comprehensive)*

In this paper we define, discuss and prove the uniqueness of the abelian subalgebra of maximal dimension of the Euclidean Lie algebra. We also construct a family of maximal abelian subalgebras and prove that they are maximal.

Categories Of Residuated Lattices, 2018 University of Denver

#### Categories Of Residuated Lattices, Daniel Wesley Fussner

*Electronic Theses and Dissertations*

We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on ...