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## Analytic Form (analytic + form)
## Selected Abstracts## Positive width function and energy indeterminacies in ammonia molecule INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 8 2007Theodosios G. DouvropoulosA recently published methodology based on the semiclassical path integral theory was applied in a double well structure and gave the analytic form of the system's Green's function. This type of potential can describe the ammonia molecule as far as the motion of the nitrogen atom perpendicular to the hydrogen plane is discussed. Because of the fact that a double well describes a bound system and correspondingly stationary states (constructed by the symmetric and antisymmetric superposition of the eigenstates of the two unperturbed wells), it was expected that the energy spectrum would be real, in a form of doublets due to the splitting effect that takes place. However, the result was a pair of complex poles, which had a clearly positive imaginary part for each member. The present work explains the role of the imaginary parts of the complex poles as the decay rate of quantities defined as the energy indeterminacies, which are directly related to the fact that energy is not well determined in a classically forbidden region of motion. These quantities come as a function of (d,)/dE, which is the derivative of the classical action inside the potential barrier, with respect to energy. The major contribution comes from the turning points, and then the imaginary parts are responsible, not only for the conservation of energy, but for the correct sign of time as well. In this way, a different approach for the tunneling process is adopted, in which the entry or exit of the particle from the potential barrier takes place inside a neighborhood of the turning point, as though the latter was broadened and fluctuating. The magnitude of the previously mentioned decay rate is equal to ,/,, where , is the frequency of the classical oscillations inside one well. In contrast, the inversion frequency is generated by the part of the complex pole that is unrelated to (d,)/dE and is much smaller in magnitude than the classical frequency, since it is given as ,/, exp(,,). In this way, the period of the energy fluctuations is much smaller than the internal period of the system produced by the oscillating communication of the two classically allowed regions of motion. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source] ## Semiclassical path integral theory of a double-well potential in an electric field INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 5 2006Theodosios G. DouvropoulosAbstract A recently published methodology based on semiclassical path integral (SCPI) theory was implemented in the case of a model of a double-well potential perturbed by a static electric field, with application to the inversion frequency of NH3. This model was chosen as an idealized case for testing of the present approach, as well as for quantum mechanical models that might be applied in the future. The calculations were concerned with the variation of the frequency of inversion as a function of field strength, F, and of distance, xf (from the symmetric point xo = 0), where the field is "felt." It is found that this variation occurs sharply in very small regions of values of these parameters, and the system switches from internal oscillation to diffusion to the continuum. The fact that the theory is in analytic form allows the extraction of results and conclusions not only at the full SCPI level, but also at the Jeffreys,Wentzel,Kramers,Brillouin (JWKB) level. Comparison shows that the discrepancy sets in as the field strength increases, in accordance with the well-known limitations of the JWKB method regarding its dependence on the degree of variation of the potential as a function of position. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] ## Numerical nonlinear observers using pseudo-Newton-type solvers INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 17 2008Shigeru HanbaArticle first published online: 12 DEC 200Abstract In constructing a globally convergent numerical nonlinear observer of Newton-type for a continuous-time nonlinear system, a globally convergent nonlinear equation solver with a guaranteed rate of convergence is necessary. In particular, the solver should be Jacobian free, because an analytic form of the state transition map of the nonlinear system is generally unavailable. In this paper, two Jacobian-free nonlinear equation solvers of pseudo-Newton type that fulfill these requirements are proposed. One of them is based on the finite difference approximation of the Jacobian with variable step size together with the line search. The other uses a similar idea, but the estimate of the Jacobian is mostly updated through a BFGS-type law. Then, by using these solvers, globally stable numerical nonlinear observers are constructed. Numerical results are included to illustrate the effectiveness of the proposed methods. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Constructive model predictive control for constrained nonlinear systems OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 6 2008De-Feng HeAbstract This paper develops a new model predictive control (MPC) design for stabilization of continuous-time nonlinear systems subject to state and input constraints. The key idea is to construct an analytic form of the controller with some undetermined parameters and to calculate the parameters by minimizing online a performance index. By using the method of control Lyapunov functions (CLFs), we construct an appropriate variation on Sontag's formula, with one degree of freedom reflecting ,decay rate' of CLFs. Moreover, the constructed univariate control law is used to characterize the terminal region that guarantees the feasibility of the optimal control problem. Provided that the initial feasibility of the optimization problem is satisfied, the stability of the control scheme can be guaranteed. An example is given to illustrate the application of the constructive MPC design. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Thinking inside the box: Novel linear scaling algorithm for Coulomb potential evaluation INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 4 2006David C. ThompsonAbstract Beginning with the Poisson equation, and expanding the electronic potential in terms of sine functions, the natural orbitals for describing the particle-in-a-box problem, we find that simple analytic forms can be found for the evaluation of the Coulomb energy for both the interacting and non-interacting system of N -electrons in a box. This method is reminiscent of fast-Fourier transform and scales linearly. To improve the usefulness of this result, we generalize the idea by considering a molecular system, embedded in a box, within which we determine the electrostatic potential, in the same manner as that described for our model systems. Within this general formalism, we consider both periodic and aperiodic recipes with specific application to systems described using Gaussian orbitals; although in principle the method is seen to be completely general. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] ## Shape, shear and flexion: an analytic flexion formalism for realistic mass profiles, MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2009P. D. LaskyABSTRACT Flexion is a non-linear gravitational lensing effect that arises from gradients in the convergence and shear across an image. We derive a formalism that describes non-linear gravitational lensing by a circularly symmetric lens in the thin-lens approximation. This provides us with relatively simple expressions for first- and second-flexion in terms of only the surface density and projected mass distribution of the lens. We give details of exact lens models, in particular providing flexion calculations for a Sérsic-law profile, which has become increasingly popular over recent years. We further provide a single resource for the analytic forms of convergence, shear, first- and second-flexion for the following mass distributions: a point mass, singular isothermal sphere (SIS); Navarro,Frenk,White (NFW) profile; Sérsic-law profile. We quantitatively compare these mass distributions and show that the convergence and first-flexion are better indicators of the Sérsic shape parameter, while for the concentration of NFW profiles the shear and second-flexion terms are preferred. [source] ## Projected properties of triaxial modified Hubble mass models MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2000D. K. ChakrabortyThe projected properties of a triaxial generalization of the modified Hubble mass model are investigated. The projected surface density can be evaluated analytically, allowing us to investigate its properties in analytic forms. The profiles of axis ratio and position angle of the major axis of constant density elliptical contours, as a function of viewing angles, can be compared with observations. [source] |