The Hermitian Null-Range Of A Matrix Over A Finite Field, 2018 University of Trento

#### The Hermitian Null-Range Of A Matrix Over A Finite Field, Edoardo Ballico

*Electronic Journal of Linear Algebra*

Let $q$ be a prime power. For $u=(u_1,\dots ,u_n), v=(v_1,\dots ,v_n)\in \mathbb {F} _{q^2}^n$, let $\langle u,v\rangle := \sum _{i=1}^{n} u_i^qv_i$ be the Hermitian form of $\mathbb {F} _{q^2}^n$. Fix an $n\times n$ matrix $M$ over $\mathbb {F} _{q^2}$. In this paper, it is considered the case $k=0$ of the set $\mathrm{Num} _k(M):= \{\langle u,Mu\rangle \mid u\in \mathbb {F} _{q^2}^n, \langle u,u\rangle =k\}$. When $M$ has coefficients in $\mathbb {F ...

The Properties Of Partial Trace And Block Trace Operators Of Partitioned Matrices, 2018 Poznań University Of Technology

#### The Properties Of Partial Trace And Block Trace Operators Of Partitioned Matrices, Katarzyna Filipiak, Daniel Klein, Erika Vojtková

*Electronic Journal of Linear Algebra*

The aim of this paper is to give the properties of two linear operators defined on non-square partitioned matrix: the partial trace operator and the block trace operator. The conditions for symmetry, nonnegativity, and positive-definiteness are given, as well as the relations between partial trace and block trace operators with standard trace, vectorizing and the Kronecker product operators. Both partial trace as well as block trace operators can be widely used in statistics, for example in the estimation of unknown parameters under the multi-level multivariate models or in the theory of experiments for the determination of an optimal designs under ...

Preface: International Conference On Matrix Analysis And Its Applications -- Mattriad 2017, 2018 University of Tampere

#### Preface: International Conference On Matrix Analysis And Its Applications -- Mattriad 2017, Oskar Maria Baksalary, Natalia Bebiano, Heike Fassbender, Simo Puntanen

*Electronic Journal of Linear Algebra*

No abstract provided.

Norm Inequalities Related To Clarkson Inequalities, 2018 University of Jordan

#### Norm Inequalities Related To Clarkson Inequalities, Fadi Alrimawi, Omar Hirzallah, Fuad Kittaneh

*Electronic Journal of Linear Algebra*

Let $A$ and $B$ be $n\times n$ matrices. It is shown that if $p=2$, $4\leq p<\infty$, or $2

Bounds For The Completely Positive Rank Of A Symmetric Matrix Over A Tropical Semiring, 2018 University of Ljubljana

#### Bounds For The Completely Positive Rank Of A Symmetric Matrix Over A Tropical Semiring, David Dolžan, Polona Oblak

*Electronic Journal of Linear Algebra*

In this paper, an upper bound for the CP-rank of a matrix over a tropical semiring is obtained, according to the vertex clique cover of the graph prescribed by the positions of zero entries in the matrix. The graphs that beget the matrices with the lowest possible CP-ranks are studied, and it is proved that any such graph must have its diameter equal to $2$.

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 ...

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 ...

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 ...

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 ...

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

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 ...

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.

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 ...

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.