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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/3
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Publication Metadata only 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; 113760Optimization 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.Publication Metadata only Spectral singularities and CPA-Laser action in a weakly nonlinear PT-Symmetric bilayer slab(Wiley, 2014) NA; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We study optical spectral singularities of a weakly nonlinear PT-symmetric bilinear planar slab of optically active material. In particular, we derive the lasing threshold condition and calculate the laser output intensity. These reveal the following unexpected features of the system: (1) for the case that the real part of the refractive index of the layers are equal to unity, the presence of the lossy layer decreases the threshold gain; (2) for the more commonly encountered situations when -1 is much larger than the magnitude of the imaginary part of the refractive index, the threshold gain coefficient is a function of that has a local minimum. The latter is in sharp contrast to the threshold gain coefficient of a homogeneous slab of gain material which is a decreasing function of . We use these results to comment on the effect of nonlinearity on the prospects of using this system as a coherence perfect absorption-laser.Publication Metadata only Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity(Springer, 2011) N/A; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev's formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.Publication Metadata only Extension of one-dimensional proximity regions to higher dimensions(Elsevier, 2010) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/AProximity regions (and maps) are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which gave rise to class cover catch digraph (CCCD) and was applied to pattern classification. In this article, we note some appealing properties of the spherical proximity map in compact intervals on the real line, thereby introduce the mechanism and guidelines for defining new proximity maps in higher dimensions. For non-spherical PCDs, Delaunay tessellation (triangulation in the real plane) is used to partition the region of interest in higher dimensions. We also introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of these maps and the resulting PCDs. We provide the distribution of graph invariants, namely, domination number and relative density, of the PCDs and characterize the geometry invariance of the distribution of these graph invariants for uniform data and provide some newly defined proximity maps in higher dimensions as illustrative examples. (C) 2010 Elsevier B.V. All rights reserved.Publication Metadata only 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/AThe 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.Publication Metadata only Computation of the canonical lifting via division polynomials(Union Matematica Argentina, 2015) Department of Mathematics; Erdoğan, Altan; Teaching Faculty; Department of Mathematics; College of Sciences; N/AWe study the canonical lifting of ordinary elliptic curves over the ring of Witt vectors. We prove that the canonical lifting is compatible with the base field of the given ordinary elliptic curve which was first proved in Finotti, J. Number Theory 130 (2010), 620-638. We also give some results about division polynomials of elliptic curves de fined over the ring of Witt vectors.