Oscillation theory

G. S. Ladde et ai. Oscillation Theory of Differential Equations with Deviating Arguments (1987)... 

L. H. Erbe et ai. Oscillation Theory for Functional Differential Equations (1995)... 

Lamoureux G, Roux B (2003) Modeling induced polarization with classical Drude oscillators theory and molecular dynamics simulation algorithm. J Chem Phys 119(6) 3025-3039... 

The modelling of micro- and macro- processes with the methods of classic oscillation theory. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

A simple application of the multiple-oscillator theory is to fit measured reflectance data for MgO in the Reststrahlen region. In Section 9.1 we considered the electronic excitations of MgO, whereas we now turn our attention to its lattice vibrations. A glance at the far-infrared reflectance spectrum of MgO in Fig. 9.7 shows that it does not completely exhibit one-oscillator behavior there is an additional shoulder on the high-frequency side of the main reflectance peak, which signals a weaker, but still appreciable, second oscillator. The solid curves in Fig. 9.7 show the results of a two-oscillator calculation using (9.25) the reflectance data were taken from Jasperse et al. (1966), who give the following parameters for MgO at 295°K ... 

The observed complex CD of the antibiotic rifamycin chromophore was simulated (250-190 nm) by means of coupled oscillator theory, from coupling of the long-axis polarized aromatic transition with dienone transition312. 

For many years no attempt was made to determine the absolute stereochemistry of transition metal complexes, although Kuhn (8) calculated the absolute configuration of (—)n-[Co(C204)3]3- from its anomalous optical rotatory dispersion, using classical coupled oscillator theory. He later (9) extended his theory to other tris-chelated cobalt complexes such as (+)n-[Co(en)3]3+. However, in 1955 Saito (10) showed by anomalous X-ray dispersion that Kuhn s suggested configuration for (+)n-[Co(en)3]3+ was incorrect. 

Both vibrational and rotovibrational relaxation can be described analyti-caDy as multiplicative stochastic processes. For these processes, RMT is equivalent to the stochastic Liouville equation of Kubo, with the added feature that RMT takes into account the back-reaction from the molecule imder consideration on the thermal bath. The stochastic Liouville equation has been used successfully to describe decoupling in the transient field-on condition and the effect of preparation on decay. When dealing with liquid-state molecular dynamics, RMT provides a rigorous justification for itinerant oscillator theory, widely applied to experimental data by Evans and coworkers. This implies analytically that decoupling effects should be exhibited in molecular liquids treated with strong fields. In the absence of experimental data, the computer runs described earlier amount to an independent means of verifying Grigolini s predictions. In this context note that the simulation of Oxtoby and coworkers are semistochastic and serve a similar purpose. 

Current opinion favors the big bang theory over other creation theories, although many controversial points are yet to be explained. Other theories, such as the steady-state or oscillating theories, have their advocates, and the creation of the universe is certain to remain a source of controversy and study. 

This has led to the postulate of total or partial bond rupture of the acyl radical occurring simultaneously with the peroxide bond rupture, the latter hssion being aided by the exothermicity of the former. This is the so called multiple critical oscillator theory. However, the kinetic data in the gas phase show activation energies which are independent of the nature of the alkyl group R. The critical gas phase studies, such as a comparison of the activation energies in the series. 

Facts that have been brought to light since the suggestions of Kekule, in the study of tautomerism and desmotropism, give strong support to this oscillation theory. 

Since elastic oscillation theory confines itself, normally, to small deformations only, the last term in relation (13.7) can be neglected as a small quantity of the second order, compared to the first two terms. As a result, the matrix of the deformation tensor takes the form... 

The surface forces act on the surface elements conceived inside an elastic body or on its boundary. Let us imagine a flat infinitesimally small site ds at some point in an elastic body, and let us draw a unit normal n towards it, taking some direction of that normal as positive. If there is an elastic stress in the body, the parts of the body positioned on the different sides of the site nds will act on one another with a certain force. Let us denote by d the force acting on the site nds. Within the framework of elastic oscillation theory, this force is linearly proportional to the site nds ... 

Kupradze, V. D., 1950, Boundary problems of the oscillation theory and integral equations (in Russian) Gostekhteoizdat, Moscow, Leningrad, 280 pp. 

The amenability of the n-paraffins to treatment by simple coupled-oscillator theory was demonstrated by Snyder (32,33) i.e., he was able to show that the frequencies of the components of CHg rocking motion in each of the n-paraffins, n-CggH g through CggHjg, correlate with a... 

CD exciton chirality method 11 the most simple and reliable method applicable to a variety of natural products, because the exciton-coupled CD is based on the coupled oscillator theory and the mechanism of this method has already been established as will be briefly explained in the following sections. Therefore, numerical calculations using a computer are not necessary. 

De Voe calculation 2 13 a simple method based on the coupled oscillator theory, which is applicable to more complex chiral molecules composed of two or more groups. This method needs numerical calculations using a computer. Some examples are listed in the section of applications. 

Davydov splitting for exciton resonance in anthracene, and for the first time obtained reasonable agreement with available experimental data. He used a dipole approximation for the intermolecular interaction and the only ingredients in his theory were the resonance frequencies and oscillator strength. In contrast to quantum theory described in this chapter the classical dipole theory does not take into account the contribution of the nondipole interaction, which are important in the majority of solids. It is clear that also multiexciton states including states with few quantum of excitations on the same molecule (what is forbidden for the two-level model) in classical harmonic oscillator theory contribute to the energy of excitons. However, in the framework of the classical theory it is impossible to develop the estimation of corrections which we discussed here. 

Migulin VV (1988) Fundamentals of oscillation theory. Nauka, Moscow... 

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