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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6

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Now showing 1 - 10 of 28
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    PublicationOpen Access
    Stickelberger elements and Kolyvagin systems
    (Duke University Press (DUP), 2011) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences
    In 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.
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    PublicationOpen 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 Sciences
    We 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).
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    PublicationOpen 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 Sciences
    In 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.
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    PublicationOpen Access
    New perspectives on the relevance of gravitation for the covariant description of electromagnetically polarizable media
    (Institute of Physics (IOP) Publishing, 2007) Gratus, J; Tucker, R.W.; Department of Physics; Dereli, Tekin; Faculty Member; Department of Physics; College of Sciences; 201358
    By recognizing that stress-energy-momentum tensors are fundamentally related to gravitation in spacetime it is argued that the classical electromagnetic properties of a simple polarizable medium may be parameterized in terms of a constitutive tensor whose properties can in principle be determined by experiments in non-inertial ( accelerating) frames and in the presence of weak but variable gravitational fields. After establishing some geometric notation, discussion is given to basic concepts of stress, energy and momentum in the vacuum where the useful notion of a drive form is introduced in order to associate the conservation of currents involving the flux of energy, momentum and angular momentum with spacetime isometries. The definition of the stress energy-momentum tensor is discussed with particular reference to its symmetry based on its role as a source of relativistic gravitation. General constitutive properties of material continua are formulated in terms of spacetime tensors including those that describe magneto-electric phenomena in moving media. This leads to a formulation of a self-adjoint constitutive tensor describing, in general, inhomogeneous, anisotropic, magneto-electric bulk matter in arbitrary motion. The question of an invariant characterization of intrinsically magneto-electric media is explored. An action principle is established to generate the phenomenological Maxwell system and the use of variational derivatives to calculate stress-energy-momentum tensors is discussed in some detail. The relation of this result to tensors proposed by Abraham and others is discussed in the concluding section where the relevance of the whole approach to experiments on matter in non-inertial environments with variable gravitational and electromagnetic fields is stressed.
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    PublicationOpen Access
    Foliations of hyperbolic space by constant mean curvature hypersurfaces
    (Oxford University Press (OUP), 2009) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of Sciences
    We show that the constant mean curvature hypersurfaces in Hn+1 spanning the boundary of a star-shaped C1,1 domain in Sn∞ (Hn+1) give a foliation of Hn+1. We also show that if is a closed codimension-1 C2,α submanifold in Sn∞ (Hn+1) bounding a unique constant mean curvature hypersurface H in Hn+1 with ∂∞ H = for any H ∈ (−1, 1), then the constant mean curvature hypersurfaces { H} foliate Hn+1.
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    PublicationOpen Access
    Number of Least area planes in gromov hyperbolic 3-spaces
    (American Mathematical Society (AMS), 2010) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of Sciences
    We show that for a generic simple closed curve Γ in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exists a unique least area plane Σ in X such that ∂∞Σ = Γ. This result has interesting topological applications for constructions of canonical 2-dimensional objects in Gromov hyperbolic 3-manifolds.
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    PublicationOpen Access
    Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems
    (Oxford University Press (OUP), 2017) Meerbergen, Karl; Michiels, Wim; Van Beeumen, Roel; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.
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    PublicationOpen Access
    Embedded plateau problem
    (American Mathematical Society (AMS), 2012) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of Sciences
    We show that if Gamma is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk Sigma in M with boundary Gamma such that Sigma minimizes the area among the embedded disks with boundary Gamma. Moreover, Sigma is smooth, minimal and embedded everywhere except where the boundary Gamma meets the interior of Sigma. The same result is also valid for homogeneously regular manifolds with sufficiently convex boundary.
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    PublicationOpen Access
    Main conjectures for CM fields and a Yager-type theorem for Rubin-Stark elements
    (Oxford University Press (OUP), 2014) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences
    In this article, we study the p-ordinary Iwasawa theory of the (conjectural) Rubin–Stark elements defined over abelian extensions of a CM field F and develop a rank-g Euler–Kolyvagin system machinery (where Graphic), refining and generalizing Perrin-Riou's theory and the author's prior work. This has several important arithmetic consequences: using the recent results of Hida and Hsieh on the CM main conjectures, we prove a natural extension of a theorem of Yager for the CM field F, where we relate the Rubin–Stark elements to the several-variable Katz p-adic L-function. Furthermore, beyond the cases covered by Hida and Hsieh, we are able to reduce the p-ordinary CM main conjectures to a local statement about the Rubin–Stark elements. We discuss applications of our results in the arithmetic of CM abelian varieties.
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    PublicationOpen Access
    Milnor fillable contact structures are universally tight
    (International Press Institute (IPI), 2010) Lekili, Yanki; Department of Mathematics; Özbağcı, Burak; Faculty Member; Department of Mathematics; College of Sciences; 29746
    We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations. This answers two questions of Etnyre in [12].