Physical Sciences and Mathematics Commons^™

Galois 2-Extensions, Masoud Ataei Jaliseh

Electronic Thesis and Dissertation Repository

The inverse Galois problem is a major question in mathematics. For a given base field and a given finite group $G$, one would like to list all Galois extensions $L/F$ such that the Galois group of $L/F$ is $G$.

In this work we shall solve this problem for all fields $F$, and for group $G$ of unipotent $4 \times 4$ matrices over $\mathbb{F}_2$. We also list all $16$ $U_4 (\mathbb{F}_2)$-extensions of $\mathbb{Q}_2$. The importance of these results is that they answer the inverse Galois problem in some specific cases.

This is joint work with J\'an Min\'a\v{c} and Nguyen Duy T\^an.

A Study Of Green’S Relations On Algebraic Semigroups, Allen O'Hara

Electronic Thesis and Dissertation Repository

The purpose of this work is to enhance the understanding regular algebraic semigroups by considering the structural influence of Green's relations. There will be three chief topics of discussion.

• Green's relations and the Adherence order on reductive monoids
• Renner’s conjecture on regular irreducible semigroups with zero
• a Green’s relation inspired construction of regular algebraic semigroups

Primarily, we will explore the combinatorial and geometric nature of reductive monoids with zero. Such monoids have a decomposition in terms of a Borel subgroup, called the Bruhat decomposition, which produces a finite monoid, R, the Renner monoid. We will explore the structure of …

Fourier Inequalities In Lorentz And Lebesgue Spaces, Javad Rastegari Koopaei

Electronic Thesis and Dissertation Repository

Mapping properties of the Fourier transform between weighted Lebesgue and Lorentz spaces are studied. These are generalizations to Hausdorff-Young and Pitt’s inequalities. The boundedness of the Fourier transform on $R^n$ as a map between Lorentz spaces leads to weighted Lebesgue inequalities for the Fourier transform on $R^n$ .

A major part of the work is on Fourier coefficients. Several different sufficient conditions and necessary conditions for the boundedness of Fourier transform on unit circle, viewed as a map between Lorentz $\Lambda$ and $\Gamma$ spaces are established. For a large range of Lorentz indices, necessary and sufficient conditions for boundedness are …

Combinatorial Polynomial Identity Theory, Mayada Khalil Shahada

Electronic Thesis and Dissertation Repository

This dissertation consists of two parts. Part I examines certain Burnside-type conditions on the multiplicative semigroup of an (associative unital) algebra $A$.

A semigroup $S$ is called $n$-collapsing if, for every $a_1,\ldots, a_n \in S$, there exist functions $f\neq g$ on the set $\{1,2,\ldots,n\}$ such that \begin{center} $s_{f(1)}\cdots s_{f(n)} = s_{g(1)}\cdots s_{g(n)}$. \end{center} If $f$ and $g$ can be chosen independently of the choice of $s_1,\ldots,s_n$, then $S$ satisfies a semigroup identity. A semigroup $S$ is called $n$-rewritable if $f$ and $g$ can be taken to be permutations. Semple and Shalev extended Zelmanov's Fields Medal writing solution of the Restricted …

Quantization Of Two Types Of Multisymplectic Manifolds, Baran Serajelahi

Electronic Thesis and Dissertation Repository

This thesis is concerned with quantization of two types of multisymplectic manifolds that have multisymplectic forms coming from a Kahler form. In chapter 2 we show that in both cases they can be quantized using Berezin-Toeplitz quantization and that the quantizations have reasonable semiclassical properties.

In the last chapter of this work, we obtain two additional results. The first concerns the deformation quantization of the (2n-1)-plectic structure that we examine in chapter 2, we make the first step toward the definition of a star product on the Nambu-Poisson algebra (C^{\infty}(M),{.,...,.}). The second result concerns the algebraic properties of the generalized …

Algorithms To Compute Characteristic Classes, Martin Helmer

Electronic Thesis and Dissertation Repository

In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases).

We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and …

Combinatorial Techniques In The Galois Theory Of P-Extensions, Michael Rogelstad

Electronic Thesis and Dissertation Repository

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of p-extensions is an important step toward a solution. We illustrate several techniques for counting Galois p-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the F_p-dimension of an n-th …

On Spectral Invariants Of Dirac Operators On Noncommutative Tori And Curvature Of The Determinant Line Bundle For The Noncommutative Two Torus, Ali Fathi Baghbadorani

Electronic Thesis and Dissertation Repository

We extend the canonical trace of Kontsevich and Vishik to the algebra of non-integer order classical pseudodifferntial operators on noncommutative tori. We consider the spin spectral triple on noncommutative tori and prove the regularity of eta function at zero for the family of operators $e^{th/2}De^{th/2}$ and the couple Dirac operator $D+A$ on noncommutative $3$-torus. Next, we consider the conformal variations of $\eta_{D}(0)$ and we show that the spectral value $\eta_D(0)$ is a conformal invariant of noncommutative $3$-torus. Next, we study the conformal variation of $\zeta'_{|D|}(0)$ and show that this quantity is also a conformal invariant of odd noncommutative tori. This …

Inclusions Among Mixed-Norm Lebesgue Spaces, Wayne R. Grey

Electronic Thesis and Dissertation Repository

A mixed L^P norm of a function on a product space is the

result of successive classical L^p norms in each variable,

potentially with a different exponent for each. Conditions to

determine when one mixed norm space is contained in another are

produced, generalizing the known conditions for embeddings

of L^p spaces.

The two-variable problem (with four L^p exponents, two for

each mixed norm) is studied extensively. The problem's ``unpermuted"

case simply reduces to a question of L^p embeddings. The other,

``permuted" case further divides, depending on the values of the

L^p exponents. Often, …

Determination Of Lie Superalgebras Of Supersymmetries Of Super Differential Equations, Xuan Liu

Electronic Thesis and Dissertation Repository

Superspaces are an extension of classical spaces that include certain (non-commutative) supervariables. Super differential equations are differential equations defined on superspaces, which arise in certain popular mathematical physics models. Supersymmetries of such models are superspace transformations which leave their sets of solutions invariant. They are important generalization of classical Lie symmetry groups of differential equations.

In this thesis, we consider finite-dimensional Lie supersymmetry groups of super differential equations. Such supergroups are locally uniquely determined by their associated Lie superalgebras, and in particular by the structure constants of those algebras. The main work of this thesis is providing an algorithmic method …

Rationality Of The Spectral Action For Robertson-Walker Metrics And The Geometry Of The Determinant Line Bundle For The Noncommutative Two Torus, Asghar Ghorbanpour

Electronic Thesis and Dissertation Repository

In noncommutative geometry, the geometry of a space is given via a spectral triple $(\mathcal{A,H},D)$. Geometric information, in this approach, is encoded in the spectrum of $D$ and to extract them, one should study spectral functions such as the heat trace $\Tr (e^{-tD^2})$, the spectral zeta function $\Tr(|D|^{-s})$ and the spectral action functional, $\Tr f(D/\Lambda)$.

The main focus of this thesis is on the methods and tools that can be used to extract the spectral information. Applying the pseudodifferential calculus and the heat trace techniques, in addition to computing the newer terms, we prove the rationality of the spectral action …