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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/3
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Publication Metadata only Stabilization of solutions of marine riser equations(Wiley, 2024) Ahmedov, S. Z.; Namazov, A. A.; Department of Mathematics; Kalantarov, Varga; Department of Mathematics; College of SciencesWe study the problem of stabilization to zero stationary state of nonlinear fourth-order wave equation with nonlinear damping term modelling dynamics of marine riser by feedback control terms that employ finitely many Fourier modes. Additionally, we demonstrate that the corresponding equation with linear damping, which represents the dynamics of pipes conveying fluids, can be exponentially stabilized by a feedback controller employing a finite number of Fourier modes.Publication Metadata only Exactness of the first born approximation in electromagnetic scattering(Oxford Univ Press Inc, 2024) Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Department of Mathematics; College of SciencesFor the scattering of plane electromagnetic waves by a general possibly anisotropic stationary linear medium in three dimensions, we give a condition on the permittivity and permeability tensors of the medium under which the first Born approximation yields the exact expression for the scattered wave whenever the incident wavenumber k does not exceed a preassigned value alpha. We also show that under this condition the medium is omnidirectionally invisible for k <= alpha/2, i.e. it displays broadband invisibility regardless of the polarization of the incident wave.Publication Metadata only A characterization of heaviness in terms of relative symplectic cohomology(Wiley, 2024) Mak, Cheuk Yu; Sun, Yuhan; Department of Mathematics; Varolgüneş, Umut; Department of Mathematics; College of SciencesFor a compact subset K$K$ of a closed symplectic manifold (M,omega)$(M, \omega)$, we prove that K$K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is nonzero. As an application, we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results is also included.Publication Metadata only Stopping levels for a spectrally negative Markov additive process(Springer Science and Business Media Deutschland GmbH, 2024) Vardar-Acar, C.; Department of Mathematics; Çağlar, Mine; Department of Mathematics; College of SciencesThe optimal stopping problem for pricing Russian options in finance requires taking the supremum of the discounted reward function over all finite stopping times. We assume the logarithm of the asset price is a spectrally negative Markov additive process with finitely many regimes. The reward function is given by the exponential of the running supremum of the price process. Previous work on Russian optimal stopping problem suggests that the optimal stopping time would be an upcrossing time of the drawdown at a certain level for each regime. We derive explicit formulas for identifying the stopping levels and computing the corresponding value functions through a recursive algorithm. A numerical is provided for finding these stopping levels and their value functions.Publication Metadata only Counterexamples to regularity of Mane projections in the theory of attractors(Institute of Physics (IOP) Publishing, 2013) Eden, Alp; Zelik, Sergey V.; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least C-1-smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a C-1-smooth inertial manifold may not exist. on the other hand, since an attractor usually has finite fractal dimension, by Mane's theorem it projects bijectively and Holder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mane projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness.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 Continuous dependence for the convective brinkman–forchheimer equations(Taylor & Francis, 2005) Çelebi, A.O.; Ugurlu, D.; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655In this article, we have considered the convective Brinkman–Forchheimer equations with Dirichlet's boundary conditions. The continuous dependence of solutions on the Forchheimer coefficient in H 1 norm is proved.Publication Metadata only Quantitative analysis of structural alterations in the choroid of patients with active Behçet uveitis(Lippincott Williams and Wilkins (LWW), 2018) Oray, Merih; Herbort, Carl P.; Akman, Mehmet; Tugal-Tutkun, Ilknur; Department of Mathematics; N/A; N/A; N/A; N/A; Mengi, Emre; Önal, Sumru; Uludağ, Günay; Metin, Mustafa Mert; Akbay, Aylin Koç; Faculty Member; Other; Doctor; Undergraduate Student; Doctor; Department of Mathematics; College of Sciences; School of Medicine; N/A; School of Medicine; N/A; N/A; N/A; Koç University Hospital; N/A; Koç University Hospital; 113760; 52359; N/A; N/A; N/APurpose: To quantitatively analyze in vivo morphology of subfoveal choroid during an acute attack of Behcet uveitis. Methods: In this prospective study, 28 patients with Behcet uveitis of <= 4-year duration, and 28 control subjects underwent enhanced depth imaging optical coherence tomography. A novel custom software was used to calculate choroidal stroma-to-choroidal vessel lumen ratio. Subfoveal choroidal thickness was measured at fovea and 750 mu m nasal, temporal, superior, and inferior to fovea. Patients underwent fluorescein angiography and indocyanine green angiography. Receiver operating characteristic curve and area under the curve were computed for central foveal thickness. The eye with a higher Behcet disease ocular attack score 24 was studied. The main outcome measures were choroidal stromato-choroidal vessel lumen ratio and choroidal thickness. Results: The mean total Behcet disease ocular attack score 24, fluorescein angiography, and indocyanine green angiography scores were 7.42 +/- 4.10, 17.42 +/- 6.03, and 0.66 +/- 0.73, respectively. Choroidal stroma-to-choroidal vessel lumen ratio was significantly higher in patients (0.413 +/- 0.056 vs. 0.351 +/- 0.063, P = 0.003). There were no significant differences in subfoveal choroidal thickness between patients and control subjects. Choroidal stroma-tochoroidal vessel lumen ratio correlated with retinal vascular staining and leakage score of fluorescein angiography (r = 0.300, P = 0.036). Central foveal thickness was significantly increased in patients (352.750 +/- 107.134 mu m vs. 263.500 +/- 20.819 p.m, P < 0.001). Central foveal thickness showed significant correlations with logarithm of minimum angle of resolution vision, Behcet disease ocular attack score 24, total fluorescein angiography score, retinal vascular staining and/or leakage and capillary leakage scores of fluorescein angiography, and total indocyanine green angiography score. At 275 mu m cutoff, diagnostic sensitivity and specificity of central foveal thickness for acute Behcet uveitis were 89% and 72%, respectively (area under the curve = 0.902; 95% CI = 0.826-0.978, P < 0.001). Conclusion: There was choroidal stromal expansion which was not associated with thickening of the choroid. Central foveal thickness may be used as a noninvasive measure to assess inflammatory activity in early Behcet uveitis.Publication Metadata only 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/AIn 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.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.