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 ...

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 ...

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.

Homological Characterizations Of Quasi-Complete Intersections, 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

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, 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 *S*_{1} and *S*_{2} with conformal *G*-actions have topologically equivalent actions if there is a homeomorphism *h :* *S _{1} -> S_{2} *which intertwines the actions. A weaker equivalence may be defined by comparing the representations of

*G*on the spaces of holomorphic

*q-*differentials

*H*and

^{q}(S_{1})*H*In this note we study the differences between topological equivalence and

^{q}(S_{2}).*H*equivalence of prime cyclic actions, where

^{q}*S*and

_{1}/G*S*have genus zero.

_{2}/GKemeny'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, 2016 Colorado State University-Pueblo

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

*Abstract Algebra*

No abstract provided.

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

Polynomial Extension Operators. Part I, 2016 University of Texas at Austin

#### Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

*Jay Gopalakrishnan*

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H^{½}(∂K) into H¹(K), for any tetrahedron K.

A Mixed Method For Axisymmetric Div-Curl Systems, 2016 Portland State University

#### A Mixed Method For Axisymmetric Div-Curl Systems, Dylan M. Copeland, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

Polynomial Extension Operators. Part I, 2016 University of Texas at Austin

#### Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

*Jay Gopalakrishnan*

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H^{½}(∂K) into H¹(K), for any tetrahedron K.

A Mixed Method For Axisymmetric Div-Curl Systems, 2016 Portland State University

#### A Mixed Method For Axisymmetric Div-Curl Systems, Dylan M. Copeland, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

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.