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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/3

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Now showing 1 - 9 of 9
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    A support function based algorithm for optimization with eigenvalue constraints
    (Siam Publications, 2017) N/A; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and Hermitian matrix-valued function. We propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally. the quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of Hermitian matrices. We establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. the convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue. We illustrate the practical applicability of the algorithm on the pseudospectral functions.
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    On the anticyclotomic Iwasawa theory of CM forms at supersingular primes
    (European Mathematical Soc, 2015) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    In this paper, we study the anticyclotomic Iwasawa theory of a CM form f of even weight w >= 2 at a supersingular prime, generalizing the results in weight 2, due to Agboola and Howard. In due course, we are naturally lead to a conjecture on universal norms that generalizes a theorem of Perrin-Riou and Berger and another that generalizes a conjecture of Rubin (the latter seems linked to the local divisibility of Heegner points). Assuming the truth of these conjectures, we establish a formula for the variation of the sizes of the Selmer groups attached to the central critical twist of f as one climbs up the anticyclotomic tower. We also prove a statement which may be regarded as a form of the anticyclotomic main conjecture (without p-adic L-functions) for the central critical twist of f.
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    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; N/A
    We show that for a generic simple closed curve Gamma in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exists a unique least area plane Sigma in X such that partial derivative(infinity)Sigma = Gamma. This result has interesting topological applications for constructions of canonical 2-dimensional objects in Gromov hyperbolic 3-manifolds.
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    Simulation and characterization of multi-class spatial patterns from stochastic point processes of randomness, clustering and regularity
    (Springer, 2014) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Spatial pattern analysis of data from multiple classes (i.e., multi-class data) has important implications. We investigate the resulting patterns when classes are generated from various spatial point processes. Our null pattern is that the nearest neighbor probabilities being proportional to class frequencies in the multi-class setting. In the two-class case, the deviations are mainly in two opposite directions, namely, segregation and association of the classes. But for three or more classes, the classes might exhibit mixed patterns, in which one pair exhibiting segregation, while another pair exhibiting association or complete spatial randomness independence. To detect deviations from the null case, we employ tests based on nearest neighbor contingency tables (NNCTs), as NNCT methods can provide an omnibus test and post-hoc tests after a significant omnibus test in a multi-class setting. In particular, for analyzing these multi-class patterns (mixed or not), we use an omnibus overall test based on NNCTs. After the overall test, the pairwise interactions are analyzed by the post-hoc cell-specific tests based on NNCTs. We propose various parameterizations of the segregation and association alternatives, list some appealing properties of these patterns, and propose three processes for the two-class association pattern. We also consider various clustering and regularity patterns to determine which one(s) cause segregation from or association with a class from a homogeneous Poisson process and from other processes as well. We perform an extensive Monte Carlo simulation study to investigate the newly proposed association patterns and to understand which stochastic processes might result in segregation or association. The methodology is illustrated on two real life data sets from plant ecology.
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    Edge density of new graph types based on a random digraph family
    (Elsevier Science Bv, 2016) Department of Mathematics; Ceyhan, Elvan; Undergraduate Student; Department of Mathematics; College of Sciences; N/A
    We consider two types of graphs based on a family of proximity catch digraphs (PCDs) and study their edge density. in particular, the PCDs we use are a parameterized digraph family called proportional-edge (PE) PCDs and the two associated graph types are the "underlying graphs" and the newly introduced "reflexivity graphs" based on the PE-PCDs. these graphs are extensions of random geometric graphs where distance is replaced with a dissimilarity measure and the threshold is not fixed but depends on the location of the points. PCDs and the associated graphs are constructed based on data points from two classes, say X and y, where one class (say class X) forms the vertices of the PCD and the Delaunay tessellation of the other class (i.e., class y) yields the (Delaunay) cells which serve as the support of class X points. We demonstrate that edge density of these graphs is a U-statistic, hence obtain the asymptotic normality of it for data from any distribution that satisfies mild regulatory conditions. the rate of convergence to asymptotic normality is sharper for the edge density of the reflexivity and underlying graphs compared to the arc density of the PE-PCDs. for uniform data in Euclidean plane where Delaunay cells are triangles, we demonstrate that the distribution of the edge density is geometry invariant (i.e., independent of the shape of the triangular support). We compute the explicit forms of the asymptotic normal distribution for uniform data in one Delaunay triangle in the Euclidean plane utilizing this geometry invariance property. We also provide various versions of edge density in the multiple triangle case. the approach presented here can also be extended for application to data in higher dimensions.
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    Big Heegner point Kolyvagin system for a family of modular forms
    (Springer International Publishing Ag, 2014) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    The principal goal of this paper is to develop Kolyvagin's descent to apply with the big Heegner point Euler system constructed by Howard for the big Galois representation attached to a Hida family of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.
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    Generic uniqueness of area minimizing disks for extreme curves
    (Johns Hopkins Univ Press, 2010) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We show that for a generic nullhomotopic simple closed curve Gamma in the boundary of a compact, orientable. mean convex 3-manifold M with H-2(M, Z) = 0. there is a unique area minimizing disk D embedded in M with partial derivative D = Gamma. We also show that the same is true for nullhomologous curves in the absolutely area minimizing surface case.
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    Cell-specific and post-hoc spatial clustering tests based on nearest neighbor contingency tables
    (Korean Statistical Soc, 2017) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Spatial clustering patterns in a multi-class setting such as segregation and association between classes have important implications in various fields, e.g., in ecology, and can be tested using nearest neighbor contingency tables (NNCTs). a NNCT is constructed based on the types of the nearest neighbor (NN) pairs and their frequencies. We survey the cell-specific (or pairwise) and overall segregation tests based on NNCTs in literature and introduce new ones and determine their asymptotic distributions. We demonstrate that cell-specific tests enjoy asymptotic normality, while overall tests have chi-square distributions asymptotically. Some of the overall tests are confounded by the unstable generalized inverse of the rank-deficient covariance matrix. To overcome this problem, we propose rank-based corrections for the overall tests to stabilize their behavior. We also perform an extensive' Monte Carlo simulation study to compare the finite sample performance of the tests in terms of empirical size and power based on the asymptotic and Monte Carlo critical values and determine the tests that have the best size and power performance and are robust to differences in relative abundances (of the classes). in addition to the cell-specific tests, we discuss one(-class)-versus-rest type of tests as post-hoc,tests after a significant overall test. We also introduce the concepts of total, strong, and partial segregatioN/Association to differentiate different levels of these patterns. We compare the new tests with the existing NNCT-tests in literature with simulations and illustrate the tests on an ecological data set. (C) 2016 the Korean Statistical Society. Published by Elsevier B.V. all rights reserved.
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    Comparison of relative density of two random geometric digraph families in testing spatial clustering
    (Springer, 2014) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We compare the performance of relative densities of two parameterized random geometric digraph families called proximity catch digraphs (PCDs) in testing bivariate spatial patterns. These PCD families are proportional edge (PE) and central similarity (CS) PCDs and are defined with proximity regions based on relative positions of data points from two classes. The relative densities of these PCDs were previously used as statistics for testing segregation and association patterns against complete spatial randomness. The relative density of a digraph, D, with n vertices (i.e., with order n) represents the ratio of the number of arcs in D to the number of arcs in the complete symmetric digraph of the same order. When scaled properly, the relative density of a PCD is a U-statistic; hence, it has asymptotic normality by the standard central limit theory of U-statistics. The PE- and CS-PCDs are defined with an expansion parameter that determines the size or measure of the associated proximity regions. In this article, we extend the distribution of the relative density of CS-PCDs for expansion parameter being larger than one, and compare finite sample performance of the tests by Monte Carlo simulations and asymptotic performance by Pitman asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each alternative in finite samples and in the limit as the sample size tending to infinity. As a result of our comparisons, we demonstrate that in terms of empirical power (i.e., for finite samples) relative density of CS-PCD has better performance (which occurs for expansion parameter values larger than one) for the segregation alternative, while relative density of PE-PCD has better performance for the association alternative. The methods are also illustrated in a real-life data set from plant ecology.