# Algebra Commons™

## All Articles in Algebra

527 full-text articles. Page 1 of 17.

On The Characterization And Parametrization Of Strong Linearizations Of Polynomial Matrices, 2016 Alexander Technological Education Institute of Thessaloniki

#### On The Characterization And Parametrization Of Strong Linearizations Of Polynomial Matrices, Efstathios Antoniou, Stavros Vologiannidis

##### Electronic Journal of Linear Algebra

In the present note, a new characterization of strong linearizations, corresponding to a given regular polynomial matrix, is presented. A linearization of a regular polynomial matrix is a matrix pencil which captures the finite spectral structure of the original matrix, while a strong linearization is one incorporating its structure at infinity along with the finite one. In this respect, linearizations serve as a tool for the study of spectral problems where polynomial matrices are involved. In view of their applications, many linearization techniques have been developed by several authors in the recent years. In this note, a unifying approach is ...

The Inverse Along An Element In Rings, 2016 Universidad Politecnica de Valencia

#### The Inverse Along An Element In Rings, Julio Benitez, Enrico Boasso

##### Electronic Journal of Linear Algebra

Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.

Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, 2016 University of Houston

#### Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta

##### STAR (STEM Teacher and Researcher) Presentations

With advances in technology in brighter sources and larger and faster detectors, the amount of data generated at national user facilities such as SLAC is increasing exponentially. Humans have a superb ability to recognize patterns in complex and noisy data and therefore, data is still curated and analyzed by humans. However, a human brain is unable to keep up with the accelerated pace of data generation, and as a consequence, the rate of new discoveries hasn't kept pace with the rate of data creation. Therefore, new procedures to quickly assess and analyze the data are needed. Machine learning approaches ...

On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, 2016 The Graduate Center, City University of New York

#### On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form ...

Some 2-Categorical Aspects In Physics, 2016 The Graduate Center, City University of New York

#### Some 2-Categorical Aspects In Physics, Arthur Parzygnat

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description ...

Generalizations Of The Cauchy And Fujiwara Bounds For Products Of Zeros Of A Polynomial, 2016 University of Guelph

#### Generalizations Of The Cauchy And Fujiwara Bounds For Products Of Zeros Of A Polynomial, Rajesh Pereira, Mohammad Ali Vali

##### Electronic Journal of Linear Algebra

The Cauchy bound is one of the best known upper bounds for the modulus of the zeros of a polynomial. The Fujiwara bound is another useful upper bound for the modulus of the zeros of a polynomial. In this paper, compound matrices are used to derive a generalization of both the Cauchy bound and the Fujiwara bound. This generalization yields upper bounds for the modulus of the product of $m$ zeros of the polynomial.

A Survey Of Graphs Of Minimum Order With Given Automorphism Group, 2016 University of Texas at Tyler

#### A Survey Of Graphs Of Minimum Order With Given Automorphism Group, Jessica Alyse Woodruff

##### Math Theses

We survey vertex minimal graphs with prescribed automorphism group. Whenever possible, we also investigate the construction of such minimal graphs, confirm minimality, and prove a given graph has the correct automorphism group.

2016 University of Nebraska - Lincoln

#### Homological Characterizations Of Quasi-Complete Intersections, Jason M. Lutz

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

Let R be a commutative ring, (f) an ideal of R, and E = K(f; R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T, the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.

Advisers: Luchezar L. Avramov and Srikanth B. Iyengar

2016 Rose-Hulman Institute of Technology

#### Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

##### Mathematical Sciences Technical Reports (MSTR)

Two Riemann surfaces S1 and S2 with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S1 -> S2 which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials Hq(S1) and Hq(S2). In this note we study the differences between topological equivalence and Hq equivalence of prime cyclic actions, where S1/G and S2/G have genus zero.

Kemeny's Constant And An Analogue Of Braess' Paradox For Trees, 2016 University of Manitoba

#### Kemeny's Constant And An Analogue Of Braess' Paradox For Trees, Steve Kirkland, Ze Zeng

##### Electronic Journal of Linear Algebra

Given an irreducible stochastic matrix M, Kemeny’s constant K(M) measures the expected time for the corresponding Markov chain to transition from any given initial state to a randomly chosen final state. A combinatorially based expression for K(M) is provided in terms of the weights of certain directed forests in a directed graph associated with M, yielding a particularly simple expression in the special case that M is the transition matrix for a random walk on a tree. An analogue of Braess’ paradox is investigated, whereby inserting an edge into an undirected graph can increase the value of ...

#### Richard Dedekind And The Creation Of An Ideal: Early Developments In Ring Theory, Janet Heine Barnett

##### Abstract Algebra

No abstract provided.

2016 SUNY Brockport

#### Rearranging Algebraic Equations Using Electrical Circuit Applications: A Unit Plan Aligned To The New York State Common Core Learning Standards, Susan L. Sommers

##### Education and Human Development Master's Theses

As a response to both the implementation of the Common Core State Standards (CCSS) and a recent approval of a change by the New York State Board of Regents to allow multiple pathways for graduation, this curriculum project, which will be referred to as a unit plan throughout the paper, was designed to meet the need for more units of study that apply mathematical modeling in algebra to real world situations that allow students to explore applications of mathematics in careers. The unit plan on rearranging algebraic equations using electrical circuit applications is aligned to the New York State Common ...

Characterizations Of Linear Mappings Through Zero Products Or Zero Jordan Products, 2016 East China University of Science and Technology

#### Characterizations Of Linear Mappings Through Zero Products Or Zero Jordan Products, Guangyu An, Jiankui Li

##### Electronic Journal of Linear Algebra

Let $\mathcal{A}$ be a unital algebra and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule. A characterization of generalized derivations and generalized Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$, through zero products or zero Jordan products, is given. Suppose that $\mathcal{M}$ is a unital left $\mathcal{A}$-module. It is investigated when a linear mapping from $\mathcal{A}$ into $\mathcal{M}$ is a Jordan left derivation under certain conditions. It is also studied whether an algebra with a nontrivial idempotent is zero Jordan product determined, and Jordan homomorphisms, Lie homomorphisms and Lie derivations on zero Jordan product ...

The Remedy That's Killing: Cuny, Laguardia, And The Fight For Better Math Policy, 2016 Graduate Center, City University of New York

#### The Remedy That's Killing: Cuny, Laguardia, And The Fight For Better Math Policy, Rachel A. Oppenheimer

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

Nationwide, there is a crisis in math learning and math achievement at all levels of education. Upwards of 80% of students who enter the City University of New York’s community colleges from New York City’s Department of Education high schools fail to meet college level math proficiencies and as a result, are funneled into the system’s remedial math system. Once placed into pre-college remedial arithmetic, pre-algebra, and elementary algebra courses, students fail at alarming rates and research indicates that students’ failure in remedial math has negative ripple effects on their persistence and degree completion. CUNY is not ...

P-Adic L-Functions And The Geometry Of Hida Families, 2016 Graduate Center, City University of New York

#### P-Adic L-Functions And The Geometry Of Hida Families, Joseph Kramer-Miller

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this talk we explain results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special \$L ...

Resolving Classes And Resolvable Spaces In Rational Homotopy Theory, 2016 Western Michigan University

#### Resolving Classes And Resolvable Spaces In Rational Homotopy Theory, Timothy L. Clark

##### Dissertations

A class of topological spaces is called a resolving class if it is closed under weak equivalences and homotopy limits. Letting R(A) denote the smallest resolving class containing a space A, we say X is A-resolvable if X is in R(A), which induces a partial order on spaces. These concepts are dual to the well-studied notions of closed class and cellular space, where the induced partial order is known as the Dror Farjoun Cellular Lattice. Progress has been made toward illuminating the structure of the Cellular Lattice. For example: Chachólski, Parent, and Stanley have shown that it ...

On The Reduction Of Matrix Polynomials To Hessenberg Form, 2016 Washington State University

#### On The Reduction Of Matrix Polynomials To Hessenberg Form, Thomas R. Cameron

##### Electronic Journal of Linear Algebra

It is well known that every real or complex square matrix is unitarily similar to an upper Hessenberg matrix. The purpose of this paper is to provide a constructive proof of the result that every square matrix polynomial can be reduced to an upper Hessenberg matrix, whose entries are rational functions and in special cases polynomials. It will be shown that the determinant is preserved under this transformation, and both the finite and infinite eigenvalues of the original matrix polynomial can be obtained from the upper Hessenberg matrix.

Totally Positive Density Matrices And Linear Preservers, 2016 University of Guelph

#### Totally Positive Density Matrices And Linear Preservers, David Kribs, Jeremy Levick, Rajesh Pereira

##### Electronic Journal of Linear Algebra

The intersection between the set of totally nonnegative matrices, which are of interest in many areas of matrix theory and its applications, and the set of density matrices, which provide the mathematical description of quantum states, are investigated. The single qubit case is characterized, and several equivalent conditions for a quantum channel to preserve the set in that case are given. Higher dimensional cases are also discussed.

The Distance Spectral Radius Of Graphs With Given Number Of Odd Vertices, 2016 South China Normal University

#### The Distance Spectral Radius Of Graphs With Given Number Of Odd Vertices, Hongying Lin, Bo Zhou

##### Electronic Journal of Linear Algebra

The graphs with smallest, respectively largest, distance spectral radius among the connected graphs, respectively trees with a given number of odd vertices, are determined. Also, the graphs with the largest distance spectral radius among the trees with a given number of vertices of degree 3, respectively given number of vertices of degree at least 3, are determined. Finally, the graphs with the second and third largest distance spectral radius among the trees with all odd vertices are determined.

2016 Portland State University

#### Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, Kyle Oddson

##### Student Research Symposium

Encoding Sudoku puzzles as partially colored graphs, we state and prove Akman’s theorem [1] regarding the associated partial chromatic polynomial [5]; we count the 4x4 sudoku boards, in total and fundamentally distinct; we count the diagonally distinct 4x4 sudoku boards; and we classify and enumerate the different structure types of 4x4 boards.