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 ...

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 ...

Decoding Book Barcode Images, 2018 Claremont McKenna College

#### Decoding Book Barcode Images, Yizhou Tao

*CMC Senior Theses*

This thesis investigated a method of barcode reconstruction to address the recovery of a blurred and convoluted one-dimensional barcode. There are a lot of types of barcodes used today, such as Code 39, Code 93, Code 128, etc. Our algorithm applies to the universal barcode, EAN 13. We extend the methodologies proposed by Iwen et al. (2013) in the journal article "A Symbol-Based Algorithm for Decoding barcodes." The algorithm proposed in the paper requires a signal measured by a laser scanner as an input. The observed signal is modeled as a true signal corrupted by a Gaussian convolution, additional noises ...

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 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 ...

Logic -> Proof -> Rest, 2018 The College of Wooster

#### Logic -> Proof -> Rest, Maxwell Taylor

*Senior Independent Study Theses*

REST is a common architecture for networked applications. Applications that adhere to the REST constraints enjoy significant scaling advantages over other architectures. But REST is not a panacea for the task of building correct software. Algebraic models of computation, particularly CSP, prove useful to describe the composition of applications using REST. CSP enables us to describe and verify the behavior of RESTful systems. The descriptions of each component can be used independently to verify that a system behaves as expected. This thesis demonstrates and develops CSP methodology to verify the behavior of RESTful applications.

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 ...

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.

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.

Group Rings, 2018 John Carroll University

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 .

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 ...

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.

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 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 ...

Noncommutative Reality-Based Algebras Of Rank 6, 2018 University of Regina

#### Noncommutative Reality-Based Algebras Of Rank 6, Allen Herman, Mikhael Muzychuk, Bangteng Xu

*EKU Faculty and Staff Scholarship*

We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150 ...

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 ...

Low Rank Perturbations Of Quaternion Matrices, 2017 TU Berlin

#### Low Rank Perturbations Of Quaternion Matrices, Christian Mehl, Andre C.M. Ran

*Electronic Journal of Linear Algebra*

Low rank perturbations of right eigenvalues of quaternion matrices are considered. For real and complex matrices it is well known that under a generic rank-$k$ perturbation the $k$ largest Jordan blocks of a given eigenvalue will disappear while additional smaller Jordan blocks will remain. In this paper, it is shown that the same is true for real eigenvalues of quaternion matrices, but for complex nonreal eigenvalues the situation is different: not only the largest $k$, but the largest $2k$ Jordan blocks of a given eigenvalue will disappear under generic quaternion perturbations of rank $k$. Special emphasis is also given ...

The General $\Phi$-Hermitian Solution To Mixed Pairs Of Quaternion Matrix Sylvester Equations, 2017 Auburn University

#### The General $\Phi$-Hermitian Solution To Mixed Pairs Of Quaternion Matrix Sylvester Equations, Zhuo-Heng He, Jianzhen Liu, Tin-Yau Tam

*Electronic Journal of Linear Algebra*

Let $\mathbb{H}^{m\times n}$ be the space of $m\times n$ matrices over $\mathbb{H}$, where $\mathbb{H}$ is the real quaternion algebra. Let $A_{\phi}$ be the $n\times m$ matrix obtained by applying $\phi$ entrywise to the transposed matrix $A^{T}$, where $A\in\mathbb{H}^{m\times n}$ and $\phi$ is a nonstandard involution of $\mathbb{H}$. In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix $A_{\phi}$ are given. Two systems of mixed pairs of quaternion matrix Sylvester equations $A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_ ...