Hard self-excitation

This analogy has been found to be very useful in investigating more complicated situations. For instance, in applications one often encounters the so-called phenomenon of a hard self-excitation. The... 

This phenomenon is often called "hard self-excitation" because there exists a self-excited (i.e. orbitally asymptotically stable) limit cycle, but to reach the self-excited oscillation requires a "hard" (i.e. finite) perturbation from the steady state. (In contrast, a "soft self-excitation" is illustrated in Fig. I.l.) There is some experimental indication of hard self-excitation in the Belousov-Zhabotinskii reaction. Notice in Fig. II. 1 that after a short induction period the oscillations appear suddenly with large amplitude. This is to be expected for hard self-excitation during the induction period the system is trapped in a locally stable steady state until the kinetic parameters change such that the steady state loses its stability and the system jumps to large amplitude stable oscillations. In the case of soft self-excitation it is expected that as the steady state loses stability, small amplitude stable oscillations first appear and then grow in size. 

We have seen that hard self-excitation is to be expected for h slightly less than 1.207. What can we expect over the whole range 0.5 < h < 1.207 To answer this question requires a rather delicate analysis of the nonlinear terms in Eq. (12) when p = p (Tyson, 1975). 

The conclusion is that system (12) exhibits hard self-excitation for all... 

Recalling the phenomenon of hard self-excitation in the Oregonator (Chapter III), we have every reason to suspect that, besides a stable homogeneous resting state, there exist stable rotating spiral waves. Furthermore, knowing the 27T/C0q-per iodic solution x(t) of the reaction equations as well as we do, we can immediately describe the spiral wave solution far from the origin ... 

Phenomenological quasiparticle model. Taking into account only the dominant contributions in (7), namely the quasiparticle contributions of the transverse gluons as well as the quark particle-excitations for Nj / 0, we arrive at the quasiparticle model [8], The dispersion relations can be even further simplified by their form at hard momenta, u2 h2 -rnf, where m.t gT are the asymptotic masses. With this approximation of the self-energies, the pressure reads in analogy to the scalar case... 

QUALITATIVE COMMENTS (with 300 mg) I would have liked to, and was expecting to, have an exciting visual day, but 1 seemed to be unable to escape self-analysis. At the peak of the experience I was quite intoxicated and hyper with energy, so that it was not hard to move around. I was quite restless. But I spent most of the day in considerable agony, attempting to break through without success. I... 

Touch more exciting Forget start of conversation Insights into others More subtle humor Ordinary social games hard to play Less noisy than when drunk Often forget to finish some task Easily sidetracked Spontaneous insights into self Harder to read... 

In this chapter, we discuss several approaches that have led from molecular entities to supramolecular soft and hard molecular architectures. Systems based on metal complexes with d and d electronic configuration forming assemblies such as micelles, vesicles, and gels, as well as crystalline structures, will be illustrated. The focus is on the role played by the metal complexes chemical structures as well as the choice of the intermolecular interactions in the ground and/or excited electronic states within the arrays. The selected examples, based on noncovalently linked luminescent systems, aim to the development of multifunctional assemblies, in which the self-organization generates new... 

Herein, we wdl discuss several approaches that have led from molecular entities to supramolecular soft and hard systems. In particular, we will show how the molecular structure can be modified to induce the controlled self-assembly of transition metal complexes into sophisticated photoactive arrays with imusual properties derived from the structiu-e of the metal complexes and their intermolecular interactions in the ground and/or excited electronic states within the assemblies. We will start with a survey of the photophysical properties of selected transition metal complexes, followed by an overview of the aggregation mechanism they can undergo to. We will focus our attention on soft assemblies... 

Infants with high intrauterine exposure had higher scores on the BNBAS excitabihty cluster than infants with low exposure. Infants with a high BNBAS excitability score had poorer tone and motor movement, were more irritable and hard to console, and had difficulties in self-quieting. 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

Whereas the distinction between collective and cooperative effects can appear artificial, it is obvious that, since optical responses are gs properties, their nonadditivity cannot be ascribed to the delocalized nature of excited states. On the other hand, static responses can be calculated from sum-over-state (SOS) expressions involving excited state energies and transition dipole moments [35]. And in fact tlie exciton model has been recently used by several authors to calculate and/or discuss linear and non-linear optical responses of mm [36, 37, 38, 39, 40, 41, 42]. But tlie excitonic model hardly accounts for cooperativity and one may ask if there is any link between collective effects related to the delocalized nature of exciton states and cooperative effects in the gs, related to the self-consistent dependence of tlie local molecular gs on the surrounding molecules. 

McCoy et a/. have proposed a method termed ERASER based on self-refocused and hard rectangular pulses. The scheme does not generate phase roll. Moy et alP have demonstrated the use of frequency shifted, self-refocused top hat pulses to observe amide resonances, that they termed selective-excitation-corrected spectroscopy (SelECSy). This method, although not providing exceptional water suppression, overcomes the dynamic range problem and produces uniform amide proton excitation... 

The molecule is not quite harmonic and its energy levels are more accurately given by the Dunham type expression given in Eq. (58). But both the cross- and the self-anhar-monicity terms, the x,/s, are smaller than the harmonic frequencies by more than an order of magnitude. What this means is that following the first time scale there is a quiescent period during which the system hardly explores its phase space. Figure 20 is a schematic illustration of this point. The very same point is made by the observations of nested spectra. By the end of the second period the volume sampled in phase space is still quite restricted, provided that, as we have tacitly assumed, the initial excitation was selective. 

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