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Articles 1 - 17 of 17
Full-Text Articles in Physical Sciences and Mathematics
Proving Dirichlet's Theorem On Arithmetic Progressions, Owen T. Abma
Proving Dirichlet's Theorem On Arithmetic Progressions, Owen T. Abma
Undergraduate Student Research Internships Conference
First proved by German mathematician Dirichlet in 1837, this important theorem states that for coprime integers a, m, there are an infinite number of primes p such that p = a (mod m). This is one of many extensions of Euclid’s theorem that there are infinitely many prime numbers. In this paper, we will formulate a rather elegant proof of Dirichlet’s theorem using ideas from complex analysis and group theory.
On The Geometry Of Multi-Affine Polynomials, Junquan Xiao
On The Geometry Of Multi-Affine Polynomials, Junquan Xiao
Electronic Thesis and Dissertation Repository
This work investigates several geometric properties of the solutions of the multi-affine polynomials. Chapters 1, 2 discuss two different notions of invariant circles. Chapter 3 gives several loci of polynomials of degree three. A locus of a complex polynomial p(z) is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The study of such objects allows one to improve upon know results about the location of zeros and critical points of complex polynomials, see for example [22] and [24]. A complex polynomial has many loci. It is …
Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel
Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel
Electronic Thesis and Dissertation Repository
This thesis deals with the problem of approximating germs of real or complex analytic spaces by Nash or algebraic germs. In particular, we investigate the problem of approximating analytic germs in various ways while preserving the Hilbert-Samuel function, which is of importance in the resolution of singularities. We first show that analytic germs that are complete intersections can be arbitrarily closely approximated by algebraic germs which are complete intersections with the same Hilbert-Samuel function. We then show that analytic germs whose local rings are Cohen-Macaulay can be arbitrarily closely approximated by Nash germs whose local rings are Cohen- Macaulay and …
Contemporary Mathematical Approaches To Computability Theory, Luis Guilherme Mazzali De Almeida
Contemporary Mathematical Approaches To Computability Theory, Luis Guilherme Mazzali De Almeida
Undergraduate Student Research Internships Conference
In this paper, I present an introduction to computability theory and adopt contemporary mathematical definitions of computable numbers and computable functions to prove important theorems in computability theory. I start by exploring the history of computability theory, as well as Turing Machines, undecidability, partial recursive functions, computable numbers, and computable real functions. I then prove important theorems in computability theory, such that the computable numbers form a field and that the computable real functions are continuous.
Polynomial And Rational Convexity Of Submanifolds Of Euclidean Complex Space, Octavian Mitrea
Polynomial And Rational Convexity Of Submanifolds Of Euclidean Complex Space, Octavian Mitrea
Electronic Thesis and Dissertation Repository
The goal of this dissertation is to prove two results which are essentially independent, but which do connect to each other via their direct applications to approximation theory, symplectic geometry, topology and Banach algebras. First we show that every smooth totally real compact surface in complex Euclidean space of dimension 2 with finitely many isolated singular points of the open Whitney umbrella type is locally polynomially convex. The second result is a characterization of the rational convexity of a general class of totally real compact immersions in complex Euclidean space of dimension n..
On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, Nadia Alluhaibi
On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, Nadia Alluhaibi
Electronic Thesis and Dissertation Repository
The objective of the study is to investigate the behaviour of the inner products of vector-valued Poincare series, for large weight, associated to submanifolds of a quotient of the complex unit ball and how vector-valued automorphic forms could be constructed via Poincare series. In addition, it provides a proof of that vector-valued Poincare series on an irreducible bounded symmetric domain span the space of vector-valued automorphic forms.
Development Of Anatomical And Functional Magnetic Resonance Imaging Measures Of Alzheimer Disease, Samaneh Kazemifar
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. …
On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, Sajad Sadeghi
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 …
Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, Josue Rosario-Ortega
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
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 …
Fourier Inequalities In Lorentz And Lebesgue Spaces, Javad Rastegari Koopaei
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
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
Inclusions Among Mixed-Norm Lebesgue Spaces, Wayne R. Grey
Electronic Thesis and Dissertation Repository
A mixed LP norm of a function on a product space is the
result of successive classical Lp 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 Lp spaces.
The two-variable problem (with four Lp exponents, two for
each mixed norm) is studied extensively. The problem's ``unpermuted"
case simply reduces to a question of Lp embeddings. The other,
``permuted" case further divides, depending on the values of the
Lp exponents. Often, …
Determination Of Lie Superalgebras Of Supersymmetries Of Super Differential Equations, Xuan Liu
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
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 …
A Convexity Theorem For Symplectomorphism Groups, Seyed Mehdi Mousavi
A Convexity Theorem For Symplectomorphism Groups, Seyed Mehdi Mousavi
Electronic Thesis and Dissertation Repository
In this thesis we study the existence of an infinite-dimensional analog of maximal torus in the symplectomorphism groups of toric manifolds. We also prove an infinite-dimensional version of Schur-Horn-Kostant convexity theorem. These results are extensions of the results of Bao-Raiu, Elhadrami, Bloch-Flachka-Ratiu and Bloch-El Hadrami-Flaschka-Raiu.
Holomorphic K-Differentials And Holomorphic Approximation On Open Riemann Surfaces, Nadya Askaripour
Holomorphic K-Differentials And Holomorphic Approximation On Open Riemann Surfaces, Nadya Askaripour
Electronic Thesis and Dissertation Repository
This thesis is of two parts: At the first part (Chapters 1 and 2) we study some spaces of holomorphic k-differentials on open Riemann surfaces, and obtain some observations about these spaces, then we obtain two main theorems about the kernel of Poincar\'e series map. In the second part (Chapters 3 and 4), we study holomorphic approximation on closed subsets of non-compact Riemann surfaces. We add a condition to the Extension Theorem and fixing its proof. Extension Theorem was first stated and proved by G. Schmieder, but there are few examples, where the theorem fails. That is slightly effecting a …