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Publication Metadata only Time-dependent diffeomorphisms as quantum canonical transformations and the time-dependent harmonic oscillator(Iop Publishing Ltd, 1998) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrodinger equation. A simple generalization of this result leads to the reduction of the Schrodinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive red solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.Publication Metadata only Interior energy focusing within an elasto-plastic material(Pergamon-Elsevier Science Ltd, 1996) Tadi, M; Rabitz, H; Kim, YS; Prevost, JH; McManus, JB; Department of Mathematics; Aşkar, Attila; Faculty Member; Department of Mathematics; College of Sciences; 178822In this paper we consider the problem of focusing acoustic energy within a subsurface volume in a solid by means of a designed surface pattern, and ask how that focusing is affected by plastic yield of the material. Surface force patterns that yield efficient subsurface acoustic focusing have been designed using optimal control theory, based on a linear elastic model of a solid. The acoustic waves generated by these forces then are propagated, via numerical algorithms, in a model solid that exhibits plastic yield. Numerical results indicate that as the amplitude of the force increases, yield begins to develop at the focus, with the formation of an expanding region of permanent plastic deformation. Despite the occurrence of yield near the focus, acoustic energy still can be delivered to the focal volume with good efficiency.Publication Metadata only Large deformations of helices(İstanbul Technical University (İTÜ) / İstanbul Teknik Üniversitesi (İTÜ), 1992) Aköz, Yalçın; Department of Mathematics; Aşkar, Attila; Faculty Member; Department of Mathematics; College of Sciences; 178822N/APublication Metadata only Generation of controlled acoustic-waves by optimal-design of surface loads with constrained forms(Pergamon-Elsevier Science Ltd, 1995) Kim, Ys; Rabitz, H; Tadi, M; Mcmanus, Jb; Department of Mathematics; Aşkar, Attila; Faculty Member; Department of Mathematics; College of Sciences; 178822Model calculations are presented for the optimal design of surface force patterns to generate acoustic waves that come to a focus within the bulk of a homogeneous elastic solid. The optimal design consists of achieving a high level of energy at the target at a prescribed time by applying a relatively minimal surface force while aiming for a minimal system disturbance away from the focal target. Such optimal designs were derived in an earlier paper, in which no restriction was imposed on the functional form of the applied stress. In this paper we examine the importance of the fine detail in the earlier derived forcing functions in achieving efficient acoustic focusing. We repeat the optimal design calculations with the surface stress constrained to be in the form of rings of variable radius, with cross sectional profiles made by the superposition of two Gaussians. The optimality conditions are secured via the conjugate gradient algorithm (CGA) and the mechanics of the elastic medium are treated by the finite element method along with using the half space Green's function matrix. We use a criterion for focusing efficiency of the ratio of acoustic energy in the target volume to the total work done on the surface, at a prescribed time. The calculations show the high levels of focusing efficiency derived in earlier work with unconstrained force patterns also can be achieved with constrained and simplified force patterns. This observation is encouraging in terms of the robustness of the optimal solution as well as the possibility of laboratory realizations of the designed force patterns for generating focused acoustic waves.Publication Metadata only Interval algebra to deal with pattern loading and structural uncertainties - Closure(Asce-Amer Soc Civil Engineers, 1997) Çakmak, Ahmet Ş.; Nielsen, Søren R.K.; Department of Mathematics; Köylüoğlu, Hasan Uğur; Teaching Faculty; Department of Mathematics; College of Sciences; N/AN/APublication Metadata only System dynamics and modified cumulant neglect closure schemes(Amer Soc Civil Engineers, 1996) Nielsen, Rune Kristian Lundedal; Department of Mathematics; Köylüoğlu, Hasan Uğur; Teaching Faculty; Department of Mathematics; College of Sciences; N/AN/APublication Metadata only Faster simulation methods for the non-stationary random vibrations of non-linear mdof systems(A A Balkema, 1995) Department of Mathematics; Department of Mathematics; N/A; N/A; Aşkar, Attila; Köylüoğlu, Hasan Uğur; Çakmak, Ayşe Selin; Nielsen, Susanne Ramtung; Faculty Member; Teaching Faculty; Other; Other; Department of Mathematics; College of Sciences; College of Sciences; N/A; N/A; 178822 N/A; N/A; N/AN/APublication Metadata only Alternative polynomial and holomorphic Dunford-Pettis properties(Scientific and Technical research Council of Turkey - TUBITAK/Türkiye Bilimsel ve Teknik Araştırma Kurumu, 1999) Department of Mathematics; Freedman, Walden; Faculty Member; Department of Mathematics; College of Sciences; N/AAlternatives to the Polynomial Dunford-Pettis property and the Holomorphic Dunford-Pettis property, called the PDP1 and HDP1 properties, respectively, are introduced. These are shown to be equivalent to the DP1 property, an alternative Dunford-Pettis property previously introduced by the author, thus mirroring the equivalence of the three original properties. © TÜBİTAK.Publication Metadata only Finite-element method for quantum scattering(Kluwer Academic Publ, 1993) Department of Mathematics; Aşkar, Attila; Faculty Member; Department of Mathematics; College of Sciences; 178822The finite element method is introduced and applied to quantum mechanical scattering problems. In this procedure the space is discretized on a grid with the unknown quantities being the wavefunction values. Local polynomials approximate the wave function and no global basis set expansion is required. The scattering solution is constructed by a suitable combination of independent standing wave solutions. These latter solutions are generated numerically by using real, not complex, arithmetic. A one-dimensional barrier crossing is studied as a first example to illustrate finite element discretization and the construction of the scattered wave forms in an uncomplicated situation. A two variable generalization is given next. The method is then sucessfully applied to a model collinear problem which is analytically soluble and to the collinear H + H2 system. Next, a three variable formulation of the co-planar A + BC system is discussed with specific reference to co-planar H + H2 . Some comments on the generalization of the technique complete the discussion.Publication Metadata only Quantitative study of laser beam propagation in a thermally loaded absorber(Optical Society of America, 1997) Department of Physics; Department of Mathematics; Department of Mathematics; Sennaroğlu, Alphan; Aşkar, Attila; Atay, Fatihcan; Faculty Member; Faculty Member; Faculty Member; Department of Physics; Department of Mathematics; College of Sciences; College of Sciences; College of Sciences; 23851; 178822; 253074The effect of thermal loading on the propagation of Gaussian laser beams in a solid-state absorber is modeled by a novel quantitative scheme. The zeroth-order Gaussian beam solution of the wave equation in a homogeneous, cylindrically symmetric absorbing medium is used as the source term in the heat equation to calculate the temperature field. Modifications in the beam parameters caused by the temperature dependence of the absorption coefficient and the index of refraction are then calculated as first-order corrections. The formulation identifies a dimensionless parameter that controls the strength of thermal effects. Numerical results that show the dependence of crystal transmission and the spatial beam spot-size variation on incident pump power are presented. In particular, the power transmission of the crystal is found to decrease with increasing incident power, and power-dependent thermal lensing is observed. The asymptotic behavior of the solutions yields explicit formulas for the focal length of the thermal lens and the power transmission of the crystal. These explicit formulas should prove useful as a rule of thumb for experimentalists.