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Publication Metadata only A bias phenomenon on the behavior of Dedekind sums(Int Press Boston, Inc, 2008) Xiong, Maosheng; Zaharescu, Alexandru; Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803In this paper we present a bias phenomenon on the behavior of Dedekind sums at visible points in a dilated region. Our results indicate that in more than three quarters of the time the Dedekind sum increases as one moves from one visible point to the next.Publication Metadata only A characterization of the invertible measures(Polish Acad Sciences Inst Mathematics, 2007) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of Sciences; N/ALet G be a locally compact abelian group and M(G) its measure algebra. Two measures mu and lambda are said to be equivalent if there exists an invertible measure pi such that pi * mu = lambda. The main result of this note is the following: A measure mu is invertible iff vertical bar(mu) over cap vertical bar >= epsilon on (G) over cap for some epsilon > 0 and mu is equivalent to a measure lambda of the form lambda = a + theta, where a is an element of L-1(G) and theta is an element of M(G) is an idempotent measure.Publication Open Access A class of Banach algebras whose duals have the Schur property(TÜBİTAK, 1999) Mustafayev, H.; Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of SciencesCall a commutative Banach algebra A a γ-algebra if it contains a bounded group Λ such that aco(Λ) contains a multiple of the unit ball of A. In this paper, first by exhibiting several concrete examples, we show that the class of γ-algebras is quite rich. Then, for a γ-algebra A, we prove that A* has the Schur property iff the Gelfand spectrum Σ of A is scattered iff A* = ap(A) iff A* = Span(Σ).Publication Metadata only A class of banach algebras whose duals have the schur property(Scientific and Technical research Council of Turkey - TUBITAK/Türkiye Bilimsel ve Teknik Araştırma Kurumu, 1999) Mustafayev, Heybetkulu; Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of Sciences; N/ACall a commutative Banach algebra A a γ-algebra if it contains a bounded group Λ such that aco(Λ) contains a multiple of the unit ball of A. In this paper, first by exhibiting several concrete examples, we show that the class of γ-algebras is quite rich. Then, for a γ-algebra A, we prove that A* has the Schur property iff the Gelfand spectrum Σ of A is scattered iff A* = ap(A) iff A* = Span(Σ).Publication Metadata only A dynamical formulation of one-dimensional scattering theory and its applications in optics(Academic Press Inc Elsevier Science, 2014) NA; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrodinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.Publication Metadata only A faster simulation method for the stochastic response of hysteretic structures subject to earthquakes(Elsevier, 1996) Çakmak, Ahmet Ş.; Nielsen, Søren R.K.; Department of Mathematics; Köylüoğlu, Hasan Uğur; Teaching Faculty; Department of Mathematics; College of Sciences; N/AA semi-analytical forward-difference Monte Carlo simulation procedure is proposed for the determination of the lower order statistical moments and the joint probability density function of the stochastic response of hysteretic non-linear multi-degree-of-freedom structural systems subject to nonstationary gaussian white noise excitation, as an alternative to conventional direct simulation methods. The method generalizes the so-called Ermak-Allen algorithm developed for simulation applications in molecular dynamics to structural hysteretic systems. The proposed simulation procedure rely on an assumption of local gaussianity during each time step. This assumption is tantamount to various linearizations of the equations of motion. The procedure then applies an analytical convolution of the excitation process, hereby reducing the generation of stochastic processes and numerical integration to the generation of random vectors only. Such a treatment offers higher rates of convergence, faster speed and higher accuracy. The procedure has been compared to the direct Monte Carlo simulation procedure, which uses a fourth-order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process. The considered system was a multi-dimenensional hysteretic shear frame, where the constitutive equation of the hysteretic shear forces are described by a bilinear hysteretic model. The comparisons show that significant savings in computer time and accuracy can be achieved.Publication Metadata only A Hamiltonian formulation of the Pais-Uhlenbeck oscillator that yields a stable and unitary quantum system(Elsevier Science Bv, 2010) NA; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 105131We offer a new Hamiltonian formulation of the classical Pais-Uhlenbeck oscillator and consider its canonical quantization. We show that for the non-degenerate case where the frequencies differ, the quantum Hamiltonian operator is a Hermitian operator with a positive spectrum, i.e., the quantum system is both stable and unitary. Furthermore it yields the classical Pais-Uhlenbeck oscillator in the classical limit. A consistent description of the degenerate case based on a Hamiltonian that is quadratic in momenta requires its analytic continuation into a complex Hamiltonian system possessing a generalized PT-symmetry (an involutive antilinear symmetry). We devise a real description of this complex system, derive an integral of motion for it, and explore its quantization.Publication Metadata only A new class of adiabatic cyclic states and geometric phases for non-Hermitian Hamiltonians(Elsevier Science Bv, 1999) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(O). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field.Publication Metadata only A note on the geometry of M-3 and symplectic structures on S-1 x M-3(Akademiai Kiado, 2007) Department of Mathematics; Etgü, Tolga; Faculty Member; Department of Mathematics; College of Sciences; 16206We investigate the relationship between the geometry of a closed, oriented 3-manifold M and the symplectic structures on S-1 x M. In most cases the existence of a symplectic structure on S-1 x M and Thurston's geometrization conjecture imply the existence of a geometric structure on M. This observation together with the existence of geometric structures on most 3-manifolds which fiber over the circle suggests a different approach to the problem of finding a fibration of a 3-manifold over the circle in case its product with the circle admits a symplectic structure.Publication Open Access A subspace framework for H-infinity-norm minimization(Society for Industrial and Applied Mathematics (SIAM), 2020) Aliyev, Nicat; Benner, Peter; Voigt, Matthias; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760We deal with the minimization of the H-infinity-norm of the transfer function of a parameter-dependent descriptor system over the set of admissible parameter values. Subspace frameworks are proposed for such minimization problems where the involved systems are of large order. The proposed algorithms are greedy interpolatary approaches inspired by our recent work [Aliyev et al., SIAM J. Matrix Anal. Appl., 38 (2017), pp. 1496-1516] for the computation of the H-infinity-norm. In this work, we minimize the H-infinity-norm of a reduced-order parameter-dependent system obtained by two-sided restrictions onto certain subspaces. Then we expand the subspaces so that Hermite interpolation properties hold between the full and reduced-order system at the optimal parameter value for the reduced-order system. We formally establish the superlinear convergence of the subspace frameworks under some smoothness and nondegeneracy assumptions. The fast convergence of the proposed frameworks in practice is illustrated by several large-scale systems.