Physical Sciences and Mathematics Commons^™

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 …

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 …