Physical Sciences and Mathematics Commons^™

Development Of Anatomical And Functional Magnetic Resonance Imaging Measures Of Alzheimer Disease, Samaneh Kazemifar

Electronic Thesis and Dissertation Repository

Alzheimer disease is considered to be a progressive neurodegenerative condition, clinically characterized by cognitive dysfunction and memory impairments. Incorporating imaging biomarkers in the early diagnosis and monitoring of disease progression is increasingly important in the evaluation of novel treatments. The purpose of the work in this thesis was to develop and evaluate novel structural and functional biomarkers of disease to improve Alzheimer disease diagnosis and treatment monitoring. Our overarching hypothesis is that magnetic resonance imaging methods that sensitively measure brain structure and functional impairment have the potential to identify people with Alzheimer’s disease prior to the onset of cognitive decline. …

Computation Of Real Radical Ideals By Semidefinite Programming And Iterative Methods, Fei Wang

Electronic Thesis and Dissertation Repository

Systems of polynomial equations with approximate real coefficients arise frequently as models in applications in science and engineering. In the case of a system with finitely many real solutions (the $0$ dimensional case), an equivalent system generates the so-called real radical ideal of the system. In this case the equivalent real radical system has only real (i.e., no non-real) roots and no multiple roots. Such systems have obvious advantages in applications, including not having to deal with a potentially large number of non-physical complex roots, or with the ill-conditioning associated with roots with multiplicity. There is a corresponding, but more …

On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, Sajad Sadeghi

Electronic Thesis and Dissertation Repository

In the first part of this thesis, a noncommutative analogue of Gross' logarithmic Sobolev inequality for the noncommutative 2-torus is investigated. More precisely, Weissler's result on the logarithmic Sobolev inequality for the unit circle is used to propose that the logarithmic Sobolev inequality for a positive element $a= \sum a_{m,n} U^{m} V^{n} $ of the noncommutative 2-torus should be of the form $$\tau(a^{2} \log a)\leqslant \underset{(m,n)\in \mathbb{Z}^{2}}{\sum} (\vert m\vert + \vert n\vert) \vert a_{m,n} \vert ^{2} + \tau (a^{2})\log ( \tau (a^2))^{1/2},$$ where $\tau$ is the normalized positive faithful trace of the noncommutative 2-torus. A possible approach to prove this …

K-Theory Of Root Stacks And Its Application To Equivariant K-Theory, Ivan Kobyzev

Electronic Thesis and Dissertation Repository

We give a definition of a root stack and describe its most basic properties. Then we recall the necessary background (Abhyankar’s lemma, Chevalley-Shephard-Todd theorem, Luna’s etale slice theorem) and prove that under some conditions a quotient stack is a root stack. Then we compute G-theory and K-theory of a root stack. These results are used to formulate the theorem on equivariant algebraic K-theory of schemes.

Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, Josue Rosario-Ortega

Electronic Thesis and Dissertation Repository

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this thesis we use elliptic theory for edge- degenerate differential operators on singular manifolds to study general deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

Uniform Approximation On Riemann Surfaces, Fatemeh Sharifi

Electronic Thesis and Dissertation Repository

This thesis consists of three contributions to the theory of complex approximation on

Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is usually not possible to approximate f uniformly by functions holomorphic on all of R. Firstly, we show, however, that for every open Riemann surface R and every closed subset E of R; there is closed subset F of E, which approximates E extremely well, such that every function holomorphic on F can be approximated much …

Gauss-Bonnet-Chern Type Theorem For The Noncommutative Four-Sphere, Mitsuru Wilson

Electronic Thesis and Dissertation Repository

We introduce a pseudo-Riemannian calculus of modules over noncommutative al- gebras in order to investigate to what extent the differential geometry of classical Riemannian manifolds can be extended to a noncommutative setting. In this frame- work, it is possible to prove an analogue of the Levi-Civita theorem. It states that there exists at most one connection, which satisfies torsion-free condition and metric compatibility condition, on a given smooth manifold with fixed metric. More signif- icantly, the corresponding curvature operator has the same symmetry properties as the classical curvature tensors. We consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and the …