# Algebra Commons™

## All Articles in Algebra

1,083 full-text articles. Page 2 of 41.

2019 Montclair State University

#### One Teacher's Transformation Of Practice Through The Development Of Covariational Thinking And Reasoning In Algebra : A Self-Study, Jacqueline Dauplaise

##### Theses, Dissertations and Culminating Projects

CCSSM (2010) describes quantitative reasoning as expertise that mathematics educators should seek to develop in their students. Researchers must then understand how to develop covariational reasoning. The problem is that researchers draw from students’ dialogue as the data for understanding quantitative relationships. As a result, the researcher can only conceive the students’ reasoning. The objective of using the self-study research methodology is to examine and improve existing teaching practices. To improve my practice, I reflected upon the implementation of my algebra curriculum through a hermeneutics cycle of my personal history and living educational theory. The critical friend provoked through dialogues ...

2019 The Graduate Center, City University of New York

#### A Differential Algebra Approach To Commuting Polynomial Vector Fields And To Parameter Identifiability In Ode Models, Peter Thompson

##### All Dissertations, Theses, and Capstone Projects

In the first part, we study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field. One motivating factor is that we can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. We first show that a linear vector field admits a full complement of commuting vector fields. Then we study a type of planar vector field for which there exists an upper bound on the degree of a ...

2019 The Graduate Center, City University of New York

#### Analysis Of A Group Of Automorphisms Of A Free Group As A Platform For Conjugacy-Based Group Cryptography, Pavel Shostak

##### All Dissertations, Theses, and Capstone Projects

Let F be a finitely generated free group and Aut(F) its group of automorphisms.

In this monograph we discuss potential uses of Aut(F) in group-based cryptography.

Our main focus is on using Aut(F) as a platform group for the Anshel-Anshel-Goldfeld protocol, Ko-Lee protocol, and other protocols based on different versions of the conjugacy search problem or decomposition problem, such as Shpilrain-Ushakov protocol.

We attack the Anshel-Anshel-Goldfeld and Ko-Lee protocols by adapting the existing types of the length-based attack to the specifics of Aut(F). We also present our own version of the length-based attack that significantly increases ...

Integrating Mathematics And Educational Robotics: Simple Motion Planning, 2019 Loyola University Chicago

#### Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg

##### George K. Thiruvathukal

This paper shows how students can be guided to integrate elementary mathematical analyses with motion planning for typical educational robots. Rather than using calculus as in comprehensive works on motion planning, we show students can achieve interesting results using just simple linear regression tools and trigonometric analyses. Experiments with one robotics platform show that use of these tools can lead to passable navigation through dead reckoning even if students have limited experience with use of sensors, programming, and mathematics.

Vector Spaces Of Generalized Linearizations For Rectangular Matrix Polynomials, 2019 Indian Institute of Technology Guwahati

#### Vector Spaces Of Generalized Linearizations For Rectangular Matrix Polynomials, Biswajit Das, Shreemayee Bora

##### Electronic Journal of Linear Algebra

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when ...

Potentially Eventually Positive 2-Generalized Star Sign Patterns, 2019 Huaiyin Institute of Technology

#### Potentially Eventually Positive 2-Generalized Star Sign Patterns, Yu Ber-Lin, Ting-Zhu Huang, Xu Sanzhang

A sign pattern is a matrix whose entries belong to the set $\{+, -, 0\}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually exponentially positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a nonnegative integer $t_ ... An Anatomical And Functional Analysis Of Digital Arteries, 2019 University of Lynchburg #### An Anatomical And Functional Analysis Of Digital Arteries, Katie Highsmith ##### Student Scholar Showcase Blood flow to the tissue of the hands and digits is efficiently regulated by vasoconstriction and vasodilation. Through a series of cadaveric dissection, we examined arteries in the hands and digits, including ulnar artery, radial artery, palmar arteries, and digital arteries, for their distribution (branching) patterns and morphological parameters (e.g., thickness, length between branches, external and internal diameters). Using data directly collected from three female cadavers as input variables to our mathematical model, we simulated vasoconstriction (-20% and -10% diameter) and vasodilation (+10% and +20 diameter) to evaluate the extent of changes in blood volume and flow within the ... 2019 University of Lynchburg #### Group Theoretical Analysis Of Arbitrarily Large, Colored Square Grids, Brett Ehrman ##### Student Scholar Showcase In this research, we examine n x n grids whose individual squares are each colored with one of k distinct colors. We seek a general formula for the number of colored grids that are distinct up to rotations, reflections, and color reversals. We examine the problem using a group theoretical approach. We define a specific group action that allows us to incorporate Burnside’s Lemma, which leads us to the desired general results Greatest Common Divisor: Algorithm And Proof, 2019 University of St. Thomas - Houston #### Greatest Common Divisor: Algorithm And Proof, Mary K. Flagg ##### Number Theory No abstract provided. Partial Difference Sets In Nonabelian Groups And Strongly Regular Cayley Graphs, 2019 William & Mary #### Partial Difference Sets In Nonabelian Groups And Strongly Regular Cayley Graphs, Gabrielle Tauscheck ##### Undergraduate Honors Theses A regular graph$\Gamma$with$v$vertices and valency$k$is said to be a$(v,k,\lambda, \mu)$-strongly regular graph if any two adjacent vertices are both joined to exactly$\lambda$other vertices and two nonadjacent vertices are both joined to exactly$\mu$other vertices. Let$G$be a group of order$v$and$D$a$k$-element subset of$G$. Then$D$is called a$(v,k,\lambda,\mu)$-partial difference set if for every nonidentity element$g$of$D$, the equation$d_1d_2^{-1}=g$has exactly$\lambda$solutions$(d_1,d_2) \in D \times D$; and for ... 2019 Central Washington University #### How To Calculate Pi: Buffon's Needle (Non-Calculus Version), Dominic Klyve ##### Pre-calculus and Trigonometry No abstract provided. Integrating Mathematics And Educational Robotics: Simple Motion Planning, 2019 Loyola University Chicago #### Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg ##### Computer Science: Faculty Publications and Other Works This paper shows how students can be guided to integrate elementary mathematical analyses with motion planning for typical educational robots. Rather than using calculus as in comprehensive works on motion planning, we show students can achieve interesting results using just simple linear regression tools and trigonometric analyses. Experiments with one robotics platform show that use of these tools can lead to passable navigation through dead reckoning even if students have limited experience with use of sensors, programming, and mathematics. Monoidal Supercategories And Superadjunction, 2019 University of Ottawa #### Monoidal Supercategories And Superadjunction, Dene Lepine ##### Rose-Hulman Undergraduate Mathematics Journal We define the notion of superadjunction in the context of supercategories. In particular, we give definitions in terms of counit-unit superadjunctions and hom-space superadjunctions, and prove that these two definitions are equivalent. These results generalize well-known statements in the non-super setting. In the super setting, they formalize some notions that have recently appeared in the literature. We conclude with a brief discussion of superadjunction in the language of string diagrams. Strengthening Relationships Between Neural Ideals And Receptive Fields, 2019 Texas A&M University #### Strengthening Relationships Between Neural Ideals And Receptive Fields, Angelique Morvant ##### Rose-Hulman Undergraduate Mathematics Journal Neural codes are collections of binary vectors that represent the firing patterns of neurons. The information given by a neural code C can be represented by its neural ideal JC. In turn, the polynomials in JC can be used to determine the relationships among the receptive fields of the neurons. In a paper by Curto et al., three such relationships, known as the Type 1-3 relations, were linked to the neural ideal by three if-and-only-if statements. Later, Garcia et al. discovered the Type 4-6 relations. These new relations differed from the first three in that they were related ... Dissertation_Davis.Pdf, 2019 University of Kentucky #### Dissertation_Davis.Pdf, Brian Davis ##### brian davis Simplices are the simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields. In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincare series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of the ... Enhanced Koszulity In Galois Cohomology, 2019 The University of Western Ontario #### Enhanced Koszulity In Galois Cohomology, Marina Palaisti ##### Electronic Thesis and Dissertation Repository Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro-$p$-groups, Galois ... Parametric Natura Morta, 2019 Independent researcher, Palermo, Italy #### Parametric Natura Morta, Maria C. Mannone ##### The STEAM Journal Parametric equations can also be used to draw fruits, shells, and a cornucopia of a mathematical still life. Simple mathematics allows the creation of a variety of shapes and visual artworks, and it can also constitute a pedagogical tool for students. Diagonal Sums Of Doubly Substochastic Matrices, 2019 Georgian Court University #### Diagonal Sums Of Doubly Substochastic Matrices, Lei Cao, Zhi Chen, Xuefeng Duan, Selcuk Koyuncu, Huilan Li ##### Electronic Journal of Linear Algebra Let$\Omega_n$denote the convex polytope of all$n\times n$doubly stochastic matrices, and$\omega_{n}$denote the convex polytope of all$n\times n$doubly substochastic matrices. For a matrix$A\in\omega_n$, define the sub-defect of$A$to be the smallest integer$k$such that there exists an$(n+k)\times(n+k)$doubly stochastic matrix containing$A$as a submatrix. Let$\omega_{n,k}$denote the subset of$\omega_n$which contains all doubly substochastic matrices with sub-defect$k$. For$\pi$a permutation of symmetric group of degree$n$, the sequence of elements$a_{1\pi(1 ...

In-Sphere Property And Reverse Inequalities For Matrix Means, 2019 Ton Duc Thang University

#### In-Sphere Property And Reverse Inequalities For Matrix Means, Trung Hoa Dinh, Tin-Yau Tam, Bich Khue T Vo

##### Electronic Journal of Linear Algebra

The in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained.

Surjective Additive Rank-1 Preservers On Hessenberg Matrices, 2019 Walailak University

#### Surjective Additive Rank-1 Preservers On Hessenberg Matrices, Prathomjit Khachorncharoenkul, Sajee Pianskool

##### Electronic Journal of Linear Algebra

Let $H_{n}(\mathbb{F})$ be the space of all $n\times n$ upper Hessenberg matrices over a field~$\mathbb{F}$, where $n$ is a positive integer greater than two. In this paper, surjective additive maps preserving rank-$1$ on $H_{n}(\mathbb{F})$ are characterized.