# Algebra Commons™

## All Articles in Algebra

880 full-text articles. Page 2 of 32.

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 M11, HS × D5, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L2(149) as homomorphic images of the monomial progenitors 11*4 :m (5 :4), 5*6 :m S5 and 149*2 :m D37. We have also discovered 24 : S3 × C2, 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 ...

Algebraic Methods For The Construction Of Algebraic-Difference Equations With Desired Behavior, 2018 Aristotle University of Thessaloniki

#### Algebraic Methods For The Construction Of Algebraic-Difference Equations With Desired Behavior, Lazaros Moysis, Nicholas Karampetakis

##### Electronic Journal of Linear Algebra

For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma$ denotes the shift forward operator and $A\left( \sigma \right)$ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right)$. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right)$, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the ...

Families Of Graphs With Maximum Nullity Equal To Zero Forcing Number, 2018 Iowa State University

#### Families Of Graphs With Maximum Nullity Equal To Zero Forcing Number, Joseph S. Alameda, Emelie Curl, Armando Grez, Leslie Hogben, O'Neill Kingston, Alex Schulte, Derek Young, Michael Young

##### Mathematics Publications

The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The ...

Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, 2018 University of Texas at El Paso

#### Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

##### Departmental Technical Reports (CS)

Traditionally, many ethnic groups had their own versions of arithmetic algorithms. Nowadays, most of these algorithms are studied mostly as pedagogical curiosities, as an interesting way to make arithmetic more exciting to the kids: by applying to their patriotic feelings -- if they are studying the algorithms traditionally used by their ethic group -- or simply to their sense of curiosity. Somewhat surprisingly, we show that one of these algorithms -- a traditional Italian multiplication algorithm -- is actually in some reasonable sense better than the algorithm that we all normally use -- namely, it is easier to parallelize.

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

#### College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Maslanko, 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.

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.

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.

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.

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 ...

The Fundamental Group And Knots, 2018 University of Redlands

#### The Fundamental Group And Knots, Hannah Michelle Solomon

This project will focus on studying the fundamental groups of topological spaces. The goal is to specifically use ideas from group theory to differentiate between different types of knots by using the fundamental group as a topological invariant. First, we aim to provide a background in topology, including introducing deformation retractions and homotopy types. We will then explore new algebraic concepts, such as free groups and group presentations. This will allow us to develop a general understanding of how to find the fundamental group of a topological space and how to use it to gain more insight into which spaces ...

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 ...

2018 University of Central Florida

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

##### Honors in the Major 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 ...

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 ...

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_ ...