Counting Real Conjugacy Classes In Some Finite Classical Groups, 2018 College of William and Mary

#### Counting Real Conjugacy Classes In Some Finite Classical Groups, Elena Amparo

*Undergraduate Honors Theses*

An element $g$ in a group $G$ is real if there exists $x\in G$ such that $xgx^{-1}=g^{-1}$. If $g$ is real then all elements in the conjugacy class of $g$ are real. In \cite{GS1} and \cite{GS2}, Gill and Singh showed that the number of real $\mathrm{GL}_n(q)$-conjugacy classes contained in $\mathrm{SL}_n(q)$ equals the number of real $\mathrm{PGL}_n(q)$-conjugacy classes when $q$ is even or $n$ is odd. In this paper, we use generating functions to show that the result is also true for odd $q ...

Supporting English Language Learners Inside The Mathematics Classroom: One Teacher’S Unique Perspective Working With Students During Their First Years In America, 2018 University of Nebraska - Lincoln

#### Supporting English Language Learners Inside The Mathematics Classroom: One Teacher’S Unique Perspective Working With Students During Their First Years In America, Amy Marie Fendrick

*Research and Evaluation in Literacy and Technology*

Reflecting upon my personal experiences teaching mathematics to English Language Learners (ELL) in a public high school in Lincoln, Nebraska, this essay largely focuses on the time I spent as the only Accelerated Math teacher in my school building. From 2012 – 2017, I taught three different subjects at this high school: Advanced Algebra, Algebra, and Accelerated Math. This essay highlights why I chose to become a math and ELL teacher, as well as the challenges, issues, struggles, and successes I experienced during my time teaching. I focus on the challenges I faced teaching students who did not share my native ...

Resolutions Of Finite Length Modules Over Complete Intersections, 2018 University of Nebraska-Lincoln

#### Resolutions Of Finite Length Modules Over Complete Intersections, Seth Lindokken

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

The structure of free resolutions of finite length modules over regular local rings has long been a topic of interest in commutative algebra. Conjectures by Buchsbaum-Eisenbud-Horrocks and Avramov-Buchweitz predict that in this setting the minimal free resolution of the residue field should give, in some sense, the smallest possible free resolution of a finite length module. Results of Tate and Shamash describing the minimal free resolution of the residue field over a local hypersurface ring, together with the theory of matrix factorizations developed by Eisenbud and Eisenbud-Peeva, suggest analogous lower bounds for the size of free resolutions of finite length ...

Nonassociative Right Hoops, 2018 Chapman University

#### Nonassociative Right Hoops, Peter Jipsen, Michael Kinyon

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

The class of nonassociative right hoops, or narhoops for short, is defined as a subclass of right-residuated magmas, and is shown to be a variety. These algebras generalize both right quasigroups and right hoops, and we characterize the subvarieties in which the operation x ^^ y = (x/y)y is associative and/or commutative. Narhoops with a left unit are proved to be integral if and only if ^ is commutative, and their congruences are determined by the equivalence class of the left unit. We also prove that the four identities defining narhoops are independent.

Strongly Real Conjugacy Classes In Unitary Groups Over Fields Of Even Characteristic, 2018 College of William and Mary

#### Strongly Real Conjugacy Classes In Unitary Groups Over Fields Of Even Characteristic, Tanner N. Carawan

*Undergraduate Honors Theses*

An element $g$ of a group $G$ is called strongly real if there is an $s$ in $G$ such that $s^2 = 1$ and $sgs^{-1} = g^{-1}$. It is a fact that if $g$ in $G$ is strongly real, then every element in its conjugacy class is strongly real. Thus we can classify each conjugacy class as strongly real or not strongly real. Gates, Singh, and Vinroot have classified the strongly real conjugacy classes of U$(n, q^2)$ in the case that $q$ is odd. Vinroot and Schaeffer Fry have classified some of the conjugacy classes of U ...

Tp Matrices And Tp Completability, 2018 College of William and Mary

#### Tp Matrices And Tp Completability, Duo Wang

*Undergraduate Honors Theses*

A matrix is called totally nonnegative (TN) if the determinant of

every square submatrix is nonnegative and totally positive (TP)

if the determinant of every square submatrix is positive. The TP

(TN) completion problem asks which partial matrices have a TP

(TN) completion. In this paper, several new TP-completable pat-

terns in 3-by-n matrices are identied. The relationship between

expansion and completability is developed based on the prior re-

sults about single unspecied entry. These results extend our un-

derstanding of TP-completable patterns. A new Ratio Theorem

related to TP-completability is introduced in this paper, and it can

possibly be ...

Putting Fürer's Algorithm Into Practice With The Bpas Library, 2018 The University of Western Ontario

#### Putting Fürer's Algorithm Into Practice With The Bpas Library, Linxiao Wang

*Electronic Thesis and Dissertation Repository*

Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as other disciplines. In 1971, Schönhage and Strassen designed an algorithm that improved the multiplication time for two integers of at most n bits to O(log n log log n). In 2007, Martin Fürer presented a new algorithm that runs in O (n log n · 2 ^O(log* n)) , where log*n is the iterated logarithm of n. We explain how we can put Fürer’s ideas into practice for multiplying polynomials over a prime field Z/pZ, which characteristic is a Generalized ...

Upper Bound For The Number Of Distinct Eigenvalues Of A Perturbed Matrix, 2018 Sookmyung Women's University,

#### Upper Bound For The Number Of Distinct Eigenvalues Of A Perturbed Matrix, Sunyo Moon, Seungkook Park

*Electronic Journal of Linear Algebra*

In 2016, Farrell presented an upper bound for the number of distinct eigenvalues of a perturbed matrix. Xu (2017), and Wang and Wu (2016) introduced upper bounds which are sharper than Farrell's bound. In this paper, the upper bounds given by Xu, and Wang and Wu are improved.

Range-Compatible Homomorphisms Over The Field With Two Elements, 2018 Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles

#### Range-Compatible Homomorphisms Over The Field With Two Elements, Clément De Seguins Pazzis

*Electronic Journal of Linear Algebra*

Let U and V be finite-dimensional vector spaces over a field K, and S be a linear subspace of the space L(U, V ) of all linear operators from U to V. A map F : S → V is called range-compatible when F(s) ∈ Im s for all s ∈ S. Previous work has classified all the range-compatible group homomorphisms provided that codimL(U,V )S ≤ 2 dim V − 3, except in the special case when K has only two elements and codimL(U,V )S = 2 dim V − 3. This article gives a thorough treatment of that special case. The results ...

Linear Algebra (Ung), 2018 University of North Georgia

#### Linear Algebra (Ung), Hashim Saber, Beata Hebda, Piotr Hebda, Benkam Bobga

*Mathematics Grants Collections*

This Grants Collection for Linear Algebra was created under a Round Seven 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

Potential Stability Of Matrix Sign Patterns, 2018 College of William and Mary

#### Potential Stability Of Matrix Sign Patterns, Christopher Hambric

*Undergraduate Honors Theses*

The topic of matrix stability is very important for determining the stability of solutions to systems of differential equations. We examine several problems in the field of matrix stability, including minimal conditions for a $7\times7$ matrix sign pattern to be potentially stable, and applications of sign patterns to the study of Turing instability in the $3\times3$ case. Furthermore, some of our work serves as a model for a new method of approaching similar problems in the future.

Implementation And Analysis Of The Nonlinear Decomposition Attack On Polycyclic Groups, 2018 College of William and Mary

#### Implementation And Analysis Of The Nonlinear Decomposition Attack On Polycyclic Groups, Yoongbok Lee

*Undergraduate Honors Theses*

Around two years ago, Roman'kov introduced a new type of attack called the nonlinear decomposition attack on groups with solvable membership search problem. To analyze the precise efficiency of the algorithm, we implemented the algorithm on two protocols: semidirect product protocol and Ko-Lee protocol. Because polycyclic groups were suggested as possible platform groups in the semidirect product protocol and polycyclic groups have a solvable membership search problem, we used poly- cyclic groups as the platform group to test the attack. While the complexity could vary regarding many different factors within the group, there was always at least one exponential ...

An Investigation Into The Properties Of Quaternions: Their Origin, Basic Properties, Functional Analysis, And Algebraic Characteristics, 2018 John Carroll University

#### An Investigation Into The Properties Of Quaternions: Their Origin, Basic Properties, Functional Analysis, And Algebraic Characteristics, James Miller

*Masters Essays*

No abstract provided.

Cayley Graphs Of Psl(2) Over Finite Commutative Rings, 2018 Western Kentucky University

#### Cayley Graphs Of Psl(2) Over Finite Commutative Rings, Kathleen Bell

*Masters Theses & Specialist Projects*

Hadwiger's conjecture is one of the deepest open questions in graph theory, and Cayley graphs are an applicable and useful subtopic of algebra.

Chapter 1 will introduce Hadwiger's conjecture and Cayley graphs, providing a summary of background information on those topics, and continuing by introducing our problem. Chapter 2 will provide necessary definitions. Chapter 3 will give a brief survey of background information and of the existing literature on Hadwiger's conjecture, Hamiltonicity, and the isoperimetric number; in this chapter we will explore what cases are already shown and what the most recent results are. Chapter 4 will ...

Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, 2018 Rose-Hulman Institute of Technology

#### Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

Let *S* be a Riemann surface and *G* a large subgroup of* Aut(S)* (*Aut(S)* may be unknown). We are particularly interested in regular *n*-gonal surfaces, i.e., the quotient surface *S/G* (and hence *S/Aut(S)*) has genus zero. For various *H *the ramification information of the branched coverings *S/K -> S/H* may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of *Aut(S)*. The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy ...

The Hafnian And A Commutative Analogue Of The Grassmann Algebra, 2018 Department of Mathematics, Komi Science Centre UD RAS

#### The Hafnian And A Commutative Analogue Of The Grassmann Algebra, Dmitry Efimov

*Electronic Journal of Linear Algebra*

A close relationship between the determinant, the pfaffian, and the Grassmann algebra is well-known. In this paper, a similar relation between the permanent, the hafnian, and a commutative analogue of the Grassmann algebra is described. Using the latter, some new properties of the hafnian are proved.

Extremal Copositive Matrices With Zero Supports Of Cardinality N-2, 2018 Laboratoire Jean Kuntzmann / CNRS

#### Extremal Copositive Matrices With Zero Supports Of Cardinality N-2, Roland Hildebrand

*Electronic Journal of Linear Algebra*

Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that $\Supp v \subset \Supp u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In ...

Monomial Progenitors And Related Topics, 2018 California State University - San Bernardino

#### Monomial Progenitors And Related Topics, Madai Obaid Alnominy

*Electronic Theses, Projects, and Dissertations*

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M_{11}, HS × D_{5}, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L_{2}(149) as homomorphic images of the monomial progenitors 11*^{4} :_{m} (5 :4), 5*^{6 } :_{m} S_{5} and 149*^{2 } :_{m } D_{37}. We have also discovered 2^{4} : S_{3} × C_{2}, 2 ...

Progenitors, Symmetric Presentations And Constructions, 2018 California State University - San Bernardino

#### Progenitors, Symmetric Presentations And Constructions, Diana Aguirre

*Electronic Theses, Projects, and Dissertations*

Abstract

In this project, we searched for new constructions and symmetric presentations of important groups, nonabelian simple groups, their automorphism groups, or groups that have these as their factor groups. My target nonabelian simple groups included sporadic groups, linear groups, and alternating groups. In addition, we discovered finite groups as homomorphic images of progenitors and proved some of their isomorphism type and original symmetric presentations. In this thesis we found original symmeric presentations of M12, J1 and the simplectic groups S(4,4) and S(3,4) on various con- trol groups. Using the technique of double coset enumeration we ...

Progenitors, Symmetric Presentations, And Related Topics, 2018 California State University-San Bernardino

#### Progenitors, Symmetric Presentations, And Related Topics, Joana Viridiana Luna

*Electronic Theses, Projects, and Dissertations*

Abstract

A progenitor developed by Robert T. Curtis is a type of infinite groups formed by the semi-direct product of a free group m∗n and a transitive permutation group of degree n. To produce finite homomorphic images we had to add relations to the progenitor of the form 2∗n : N. In this thesis we have investigated several permutations progenitors and monomials, 2∗12 : S4, 2∗12 : S4 × 2, 2∗13 : (13 : 4), 2∗30 : ((2• : 3) : 5), 2∗13 :13,2∗13 :(13:2),2∗13 :(13:S3),53∗2 :m (13:4),7∗8 :m (32 :8 ...