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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 Decay and growth estimates for solutions of second-order and third-order differential-operator equations(Elsevier, 2013) Yilmaz, Y.; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655We obtained decay and growth estimates for solutions of second-order and third-order differential-operator equations in a Hilbert space. Applications to initial-boundary value problems for linear and nonlinear non-stationary partial differential equations modeling the strongly damped nonlinear improved Boussinesq equation, the dual-phase-lag heat conduction equations, the equation describing wave propagation in relaxing media, and the Moore-Gibson-Thompson equation are given.Publication Metadata only Symplectic fillings of lens spaces as Lefschetz fibrations(European Mathematical Society, 2016) Bhupal, M.; Department of Mathematics; Özbağcı, Burak; Faculty Member; Department of Mathematics; College of Sciences; 29746We construct a positive allowable Lefschetz fibration over the disk on any minimal (weak) symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.Publication Metadata only Existence of an attractor and determining modes for structurally damped nonlinear wave equations(Elsevier Science Bv, 2018) N/A; Department of Mathematics; Bilgin, Bilgesu Arif; Kalantarov, Varga; PhD Student; Faculty Member; Department of Mathematics; Graduate School of Sciences and Engineering; College of Sciences; N/A; 117655The paper is devoted to the study of asymptotic behavior as t -> +infinity of solutions of initial boundary value problem for structurally damped semi-linear wave equation partial derivative(2)(t)u(x, t) - Delta u(x, t)+gamma(-Delta)(theta)partial derivative(t) u(x,t) + f(u) = g(x), theta is an element of(0, 1), x is an element of Omega, t > 0 under homogeneous Dirichlet's boundary condition in a bounded domain Omega subset of R-3. We proved that the asymptotic behavior as t -> infinity of solutions of this problem is completely determined by dynamics of the first N Fourier modes, when N is large enough. We also proved that the semigroup generated by this problem when theta is an element of(1/2, 1) possesses an exponential attractor. (C) 2017 Elsevier B.V. All rights reserved.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 Generalized eigenvalue problems with specified eigenvalues(Oxford University Press (OUP), 2014) Kressner, Daniel; Nakic, Ivica; Truhar, Ninoslav; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582-601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithms.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 Balancing the inverted pendulum using position feedback(Pergamon-Elsevier Science Ltd, 1999) Department of Mathematics; Atay, Fatihcan; Faculty Member; Department of Mathematics; College of Sciences; 253074It is shown how to obtain asymptotic stability in second-order undamped systems using time-delay action in the feedback of position. The effect of the delay is similar to derivative feedback in modifying the behavior of the system. Results are given on the selection of the controller parameters both in the absence and the presence of additional delay ill the feedback path. The timelag position feedback is shown to compare favorably with the conventional PD controller in terms of stability. (C) 1999 Elsevier Science Ltd. All rights reserved.Publication Metadata only Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations(Academic Press Inc Elsevier Science, 2013) N/A; Department of Mathematics; Bilgin, Bilgesu Arif; Kalantarov, Varga; PhD Student; Faculty Member; Department of Mathematics; Graduate School of Sciences and Engineering; College of Sciences; N/A; 117655We obtain sufficient conditions on initial functions for which the initial boundary value problem for second order quasilinear strongly damped wave equations blow up in a finite time. (C) 2013 Elsevier Inc. All rights reserved.