Phase of oscillator

As already mentioned, electronically resonant, two-pulse impulsive Raman scattering (RISRS) has recently been perfonned on a number of dyes [124]. The main difference between resonant and nom-esonant ISRS is that the beats occur in the absorption of tlie probe rather than the spectral redistribution of the probe pulse energy [124]. These beats are out of phase with respect to the beats that occur in nonresonant ISRS (cosinelike rather tlian sinelike). RISRS has also been shown to have the phase of oscillation depend on the detuning from electronic resonance and it has been shown to be sensitive to the vibrational dynamics in both the ground and excited electronic states [122. 124]. 

The molecular assays in Clk"mAic2As bom fide rhythms with a predominant effect on circadian rhythm amplitude and no more than a modest effect on phase or period. With circadian per and tim enhancers, we observed reduced enhancer activity and a reduced cycling amplitude in a Clk" background, consistent with the role of Clk in regulating these enhancers. Nonetheless, the phase of oscillating bioluminescence is similar to that of wild-type flies. The presence of molecular rhythms contrasts with the absence of detectable behavioural rhythms. We favour the notion that this reflects a level or amplitude reduction below a critical threshold for behavioural rhythmicity. The absence of anticipation of light—dark transitions makes it very unlikely that an effect restricted to the lateral neurons — the absence of the neuropeptide PDF, for example — is primarily responsible for the behavioural phenotypes. This is also because LD behavioural rhythms are largely normal in flies devoid of PDF or the pacemaker lateral neurons (Renn et al 1999). However, we cannot exclude the possibility of selective effects of Clk" on other behaviourally relevant neurons. 

In this case we again have ft = Q and 7 = 7x — turning angle a around the vector J equals zero. 

Examples, as described by Eqs. (4.41), (4.42) and (4.43) show what kind of information one may obtain directly by registering oscillations in the fluorescence decay. These are the lifetime r = T-1 of the state, the factors affecting it, the precession frequency uij/ and, consequently, the value of the Lande factor gj>, as well as its sign (by the initial phase of oscillations), and, finally, the degree of polarization V. A favorable condition for registration should be the validity of T = 2tt/(Qu>ji) [Pg.135]

Effects on oscillatory behavior of the treatment of the binary oxide-metal catalyst bed in different gases are presented in Fig.5. If the binary catalyst was treated in inert gas, the sharp increase of temperature begins immediately after the reactants are supplied to the reactor, and then the process proceeds in the regular oscillatory manner, despite a phase of oscillation in which the flow of reactants was switched to the inert gas flow. 

AT = (T - Tmin), Tmin " temperature in the reactor in the absence of reaction or in a low-activity phase of oscillations k - heat dissipation coefficient ... 

Temperatures and concentrations of products in different phases of oscillation and in... 

Yellow (blue) colour corresponds to an excess of the Ce(IV) ions while the colourless (red) solution corresponds to an excess of the Ce(III) ions. Changes in concentrations may be followed directly or measured poten-tiometrically (Br, Ce(IV)/Ce(III)) or colorimetrically (without an addition of ferroin) — Ce(IV) absorbs radiation of a wavelength about 340 nm. The observations, and particularly the quantitative measurements, allows us to distinguish four fundamental phases of oscillations of concentrations, see Fig. 92. 

Here, 0, represents the phase of oscillator i, which is coupled with strength e to a set of nearest neighbors Ni in a one- or two-dimensional lattice. The natural frequencies u>i are fixed in time, uncorrelated and taken from a distribution p u>). A scaling of time and a transformation into a rotating reference frame can always be applied so that e = 1 and the ensemble mean frequency lJ is equal to zero. We refer to the variance = var(wi) of the random frequencies as the disorder of the medium. 

In the general case the phase of oscillations of the surface tension and the phase of periodical alterations of the surface area do not coincide. Then it is convenient to use complex numbers in order to describe the oscillations, and to consider two components of the surface elasticity, the surface storage modulus z, and the surface loss modulus ct... 

The variation in time of concentrations of species in an oscillatory reaction can be stopped temporarily by application of a pulse perturbation of a species to the system. If the system is close to a supercritical Hopf bifurcation, an interpretation of the addition leading to quenching of oscillations provides information on nonessential species, and on the Jacobian matrix. In quenching experiments, the phase of oscillation at which the perturbation is added and the amount of perturbant added are varied for each perturbing... 

A pulsed perturbation of one species at a time is applied to an oscillatory system, often but not necessarily near a supercritial Hopf bifurcation. The phase of oscillation at which the perturbation is added and the amount of perturbant added are varied. It is determined whether each perturbation results in an advance or delay of the next oscillatory peak relative to the period of the unperturbed autonomous system. If we let t denote the time of the nth reference event (an easily followed feature of the oscillations, such as a sharp rise or fall in the concentration of a monitored species), the period of the unperturbed limit cycle is given by Tq = t - r i. A pulse perturbation is applied at some time ipert, between t and r +i, from which we define a phase of perturbation as y>pert = - r i)/ro. Another reference event of the same type... 

Suppose we have oscillatory reagent in a tall cylinder and we have arranged that the period is everywhere the same (= T) but that the phase of oscillation varies linearly from top to bottom. By this we understand... 

Figure 5.9 Schematic diagram of a quenching experiment in phase space. F is the steady state the circle is the limit cycle, with / the point at which the system is perturbed. The quenching vector q shifts the state from / to P, which lies on the stable manifold along t. The system then moves to F. The vector d shows the result of a dilution. The radius r varies with the phase of oscillation. (Adapted from Hynne and Sorensen, 1987.)...

When an AFM tip strikes the smface of the studied biopolymer, energy is transferred to the siffface. As a consequence, the cantilever s phase of oscillation will lag behind the driving signal by an amount that depends on the level of interaction with the sample. Energy can be lost due to inelastic processes, capillary forces, and hydrophobic forces that arise dming the tip-sample interaction [56,57] such energy loss is depicted by the phase lag, and is an... 

The condition (i) implies that the modulus VFi vanishes at the centre of the spiral. Thus the centre of the spiral is a singular point of the structure as the phase of oscillations cannot be defined at this point. The latter is called a topological defect and possesses a topological charge equal to m. In the sequel, we consider only spirals for which m = lorm = -l, i.e., one-arm spirals, respectively left-handed or right-handed. 

For the majority of problems it is important to know the dependence of oscillations of different points of media at a given instant This dependence can be considered as determined if the amplitudes and phases of oscillation are known. For transverse waves it is also necessary to know the polarization. For a plane one-dimensional polarized wave it is sufficient to have an expression defining the displacement of any wave point of (x,i) with the coordinate X in the instant of time t. Such an expression is called an equation of wave. 

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