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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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Publication Open Access Conditional law and occupation times of two-sided sticky Brownian motion(Elsevier, 2020) Department of Mathematics; Çağlar, Mine; Can, Buğra; Faculty Member; Department of Mathematics; College of Sciences; 105131; N/ASticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point for the process.Publication Open Access Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions(American Physical Society (APS), 2021) Loran, Farhang; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a 2 x 2 matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for delta-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.Publication Open Access Minimal number of singular fibers in a nonorientable Lefschetz fibration(Springer, 2022) Onaran, Sinem; Department of Mathematics; Özbağcı, Burak; Faculty Member; Department of Mathematics; College of Sciences; 29746We show that there exists an admissible nonorientable genus g Lefschetz fibration with only one singular fiber over a closed orientable surface of genus h if and only if g >= 4 and h >= 1.Publication Open Access Certifying global optimality for the L-infinity-norm computation of large-scale descriptor systems(Elsevier, 2020) Schwerdtner, P.; Voigt, M.; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760We present a method for the certification of algorithms that approximate the L-infinity or H-infinity-norm of transfer functions of large-scale (descriptor) systems. This certification is needed because such algorithms depend heavily on user input, and may converge to a local maximizer of the related singular value function leading to an incorrect value, much lower than the actual norm. Hence, we design an algorithm that determines whether a given value is less than the L-infinity-norm of the transfer function under consideration, and that does not require user input other than the system matrices. In the algorithm, we check whether a certain structured matrix pencil has any purely imaginary eigenvalues by repeatedly applying a structure-preserving shift-and-invert Arnoldi iteration combined with an appropriate shifting strategy. Our algorithm consists of two stages. First, an interval on the imaginary axis which may contain imaginary eigenvalues is determined. Then, in the second stage, a shift is chosen on this interval and the eigenvalues closest to this shift are computed. If none of these eigenvalues is purely imaginary, then an imaginary interval around the shift of appropriate length is removed such that two subintervals remain. This second stage is then repeated on the remaining two subintervals until either a purely imaginary eigenvalue is found or no critical subintervals are left. We show the effectiveness of our method by testing it without any parameter adaptation on a benchmark collection of large-scale systems.Publication Open Access Embedding partial Latin squares in Latin squares with many mutually orthogonal mates(Elsevier, 2020) Donovan, Diane; Grannell, Mike; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions is that, if N(n) denotes the size of the largest set of MOLS of order n, then N(n2)⩾N(n)+2. In particular, it follows that N(576)⩾9, improving the previously known lower bound N(576)⩾8.Publication Open Access Maximum drawdown and drawdown duration of spectrally negative Lévy processes decomposed at extremes(Springer Nature, 2021) Vardar-Acar, Ceren; Avram, Florin; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131Path decomposition is performed to characterize the law of the pre-/post-supremum, post-infimum and the intermediate processes of a spectrally negative Lévy process taken up to an independent exponential time T. As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre-/post-supremum, post-infimum processes and the intermediate processes are obtained together with the law of drawdown durations.Publication Open Access Approximation of stability radii for large-scale dissipative Hamiltonian systems(Springer, 2020) Aliyev, Nicat; Mehrmann, Volker; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760A linear time-invariant dissipative Hamiltonian (DH) system (x) over dot = (J-R)Qx, with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625-1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + B Delta C-H for given matrices B, C, and another with respect to Hermitian perturbations in the form R + B Delta B-H, Delta = Delta(H). We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake.Publication Open Access Finite-parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities(Wiley, 2021) Department of Mathematics; Kalantarov, Varga; Gümüş, Serap; Faculty Member; Department of Mathematics; Graduate School of Sciences and Engineering; College of Sciences; 117655; N/AWe study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. We investigate both equations by employing controllers based on finitely many Fourier modes and the latter equation by employing finitely many volume elements. To ensure global exponential stabilization, we have provided sufficient conditions on the control parameters for each problem. We also show that solutions of the controlled equations are steering a concrete solution of the non-controlled system as t -> infinity with an exponential decay rate.Publication Open Access On open books for nonorientable 3-manifolds(Springer, 2021) Department of Mathematics; Özbağcı, Burak; Faculty Member; Department of Mathematics; College of Sciences; 29746We show that the monodromy of Klassen's genus two open book for P-2 x S-1 is the Y-homeomorphism of Lickorish, which is also known as the crosscap slide. Similarly, we show that S-2 (x) over tilde S-1 admits a genus two open book whose monodromy is the crosscap transposition. Moreover, we show that each of P(2)xS(1) and S-2 (x) over tilde S-1 admits infinitely many isomorphic genus two open books whose monodromies are mutually nonisotopic. Furthermore, we include a simple observation about the stable equivalence classes of open books for P-2 x S-1 and S-2 (x) over tilde S-1. Finally, we formulate a version of Stallings' theorem about the Murasugi sum of open books, without imposing any orientability assumption on the pages.Publication Open Access Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications(Institute of Physics (IOP) Publishing, 2022) Loran, Farhang; Department of Physics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Physics; Department of Mathematics; College of Sciences; 4231In two dimensions, the standard treatment of the scattering problem for a delta-function potential, v(r) = 3 delta(r), leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant 3. Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors' screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal.