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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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Publication Open Access Stickelberger elements and Kolyvagin systems(Duke University Press (DUP), 2011) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of SciencesIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r >1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.Publication Open Access Perturbative analysis of spectral singularities and their optical realizations(American Physical Society (APS), 2012) Department of Mathematics; Mostafazadeh, Ali; Rostamzadeh, Saber; Faculty Member; Department of Mathematics; College of Sciences; 4231; N/AWe develop a perturbative method of computing spectral singularities of a Schrodinger operator defined by a general complex potential that vanishes outside a closed interval. These can be realized as zero-width resonances in optical gain media and correspond to a lasing effect that occurs at the threshold gain. Their time-reversed copies yield coherent perfect absorption of light that is also known as antilasing. We use our general results to establish the exactness of the nth-order perturbation theory for an arbitrary complex potential consisting of n delta functions, obtain an exact expression for the transfer matrix of these potentials, and examine spectral singularities of complex barrier potentials of arbitrary shape. In the context of optical spectral singularities, these correspond to inhomogeneous gain media.Publication Open Access Density of a random interval catch digraph family and its use for testing uniformity(National Statistical Institute (NSI), 2016) Department of Mathematics; Ceyhan, Elvan; Undergraduate Student; Faculty Member; Department of Mathematics; College of SciencesWe consider (arc) density of a parameterized interval catch digraph (ICD) family with random vertices residing on the real line. The ICDs are random digraphs where randomness lies in the vertices and are defined with two parameters, a centrality parameter and an expansion parameter, hence they will be referred as central similarity ICDs (CS-ICDs). We show that arc density of CS-ICDs is a U-statistic for vertices being from a wide family of distributions with support on the real line, and provide the asymptotic (normal) distribution for the (interiors of) entire ranges of centrality and expansion parameters for one dimensional uniform data. We also determine the optimal parameter values at which the rate of convergence (to normality) is fastest. We use arc density of CS-ICDs for testing uniformity of one dimensional data, and compare its performance with arc density of another ICD family and two other tests in literature (namely, Kolmogorov-Smirnov test and Neyman’s smooth test of uniformity) in terms of empirical size and power. We show that tests based on ICDs have better power performance for certain alternatives (that are symmetric around the middle of the support of the data).Publication Open Access Examples of area-minimizing surfaces in 3-manifolds(Oxford University Press (OUP), 2012) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of SciencesIn this paper, we give some examples of area-minimizing surfaces to clarify some wellknown features of these surfaces in more general settings. The first example is about Meeks–Yau’s result on the embeddedness of the solution to the Plateau problem. We construct an example of a simple closed curve in R3 which lies in the boundary of a mean convex domain in R3, but the area-minimizing disk in R3 bounding this curve is not embedded. Our second example shows that White’s boundary decomposition theorem does not extend when the ambient space has nontrivial homology. Our last examples show that there are properly embedded absolutely area-minimizing surfaces in a mean convex 3-manifold M such that, while their boundaries are disjoint, they intersect each other nontrivially, unlike the area-minimizing disks case.Publication Open Access The 2nd symposium on multiscale, multiphase, multiphysics and turbulent flow simulations(American Institute of Physics (AIP) Publishing, 2018) Çelebi, Serdar; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131Publication Open Access Blowing up light: a nonlinear amplification scheme for electromagnetic waves(Optical Society of America (OSA), 2018) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Ghaemidizicheh, Hamed; Hajizadeh, Sasan; Faculty Member; PhD Student; Department of Mathematics; Department of Physics; Graduate School of Sciences and Engineering; 4231; N/A; N/AWe use the blow-up solutions of nonlinear Helmholtz equations to introduce a nonlinear resonance effect that is capable of amplifying electromagnetic waves of a particular intensity. To achieve this, we propose a scattering setup consisting of a Kerr slab with a negative (defocusing) Kerr constant placed to the left of a linear slab in such a way that a left-incident coherent transverse electric wave with a specific incidence angle and intensity realizes a blow-up solution of the corresponding Helmholtz equation whenever its wavenumber k takes a certain critical value, k(*). For k = k(*), the solution blows up at the right-hand boundary of the Kerr slab. For k < k(*), the setup defines a scattering system with a transmission coefficient that diverges as (k - k(*))(-4) for k -> k(*). By tuning the distance between the slabs, we can use this setup to amplify coherent waves with a wavelength in an extremely narrow spectral band. For nearby wavelengths, the setup serves as a filter. Our analysis makes use of a nonlinear generalization of the transfer matrix of the scattering theory as well as properties of unidirectionally invisible potentials.Publication Open Access Parameterization of submesoscale eddy-rich flows using a stochastic velocity model(American Meteorological Society (AMS), 2006) Özgökmen, Tamay M.; Piterbarg, Leonid I.; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131In light of the recent high-resolution radar data of surface velocity, which have revealed submesoscale eddies between the Florida Current and the coast, an objective method of detecting eddies and estimating their parameters such as center coordinates, size, and intensity is suggested. The obtained statistics are used to parametrically represent the birth-death process of eddies filling up the observation area via a model stochastic velocity field known as Cinlar flow. It appears that the suggested approach leads to a reasonable parameterization of this process for potential future use in OGCMs or coastal models.Publication Open Access Singularity links with exotic Stein fillings(Worldwide Center of Mathematics, 2014) Akhmedov, Anar; Department of Mathematics; Özbağcı, Burak; Faculty Member; Department of Mathematics; College of Sciences; 29746In [4], it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are singularity links. In addition, we prove that the contact structures induced by the Stein fillings are the canonical contact structures on these singularity links. As a consequence, we verify a prediction of András Némethi by providing examples of isolated complex surface singularities whose links with their canonical contact structures admitting infinitely many exotic simply-connected Stein fillings. Moreover, for infinitely many of these contact singularity links and for each positive integer n, we also construct an infinite family of exotic Stein fillings with fixed fundamental group ? ? ?n.Publication Open Access Nonlinear spectral singularities for confined nonlinearities(American Physical Society (APS), 2013) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We introduce a notion of spectral singularity that applies for a general class of nonlinear Schrodinger operators involving a confined nonlinearity. The presence of the nonlinearity does not break the parity-reflection symmetry of spectral singularities but makes them amplitude dependent. Nonlinear spectral singularities are, therefore, associated with a resonance effect that produces amplified waves with a specific amplitude-wavelength profile. We explore the consequences of this phenomenon for a complex delta-function potential that is subject to a general confined nonlinearity.Publication Open Access Certifying global optimality for the L-infinity-norm computation of large-scale descriptor systems(Elsevier, 2020) Schwerdtner, P.; Voigt, M.; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760We present a method for the certification of algorithms that approximate the L-infinity or H-infinity-norm of transfer functions of large-scale (descriptor) systems. This certification is needed because such algorithms depend heavily on user input, and may converge to a local maximizer of the related singular value function leading to an incorrect value, much lower than the actual norm. Hence, we design an algorithm that determines whether a given value is less than the L-infinity-norm of the transfer function under consideration, and that does not require user input other than the system matrices. In the algorithm, we check whether a certain structured matrix pencil has any purely imaginary eigenvalues by repeatedly applying a structure-preserving shift-and-invert Arnoldi iteration combined with an appropriate shifting strategy. Our algorithm consists of two stages. First, an interval on the imaginary axis which may contain imaginary eigenvalues is determined. Then, in the second stage, a shift is chosen on this interval and the eigenvalues closest to this shift are computed. If none of these eigenvalues is purely imaginary, then an imaginary interval around the shift of appropriate length is removed such that two subintervals remain. This second stage is then repeated on the remaining two subintervals until either a purely imaginary eigenvalue is found or no critical subintervals are left. We show the effectiveness of our method by testing it without any parameter adaptation on a benchmark collection of large-scale systems.