Oscillatory function

The breakdown of a given signal into a sum of oscillatory functions is accomplished by application of Fourier series techniques or by Fourier transforms. For a periodic function F t) with a period t, a Fourier series may be expressed as... 

There have also been a number of simulations of more realistic models of polymers at surfaces [65-77], The behavior of these more realistic models of polymers is similar to that of the model systems discussed above with no real surprises. Of course, the use of realistic models allows a direct comparison with experiment. For example, surface forces apparatus measurements [78] show that in some branched alkanes the force is a monotonic rather than oscillatory function of the separation. This is a surprising result because these branched alkanes pack quite efficiently (in fact they crystallize under some conditions), and this would imply that the surface forces should be oscillatory. Several... 

Recently the wall-PRISM theory has been used to investigate the forces between hydrophobic surfaces immersed in polyelectrolyte solutions [98], Polyelectrolyte solutions display strong peaks at low wavevectors in the static structure factor, which is a manifestation of liquid-like order on long lengths-cales. Consequently, the force between surfaces confining polyelectrolyte solutions is an oscillatory function of their separation. The wall-PRISM theory predicts oscillatory forces in salt-free solutions with a period of oscillation that scales with concentration as p 1/3 and p 1/2 in dilute and semidilute solutions, respectively. This behavior is explained in terms of liquid-like ordering in the bulk solution which results in liquid-like layering when the solution is confined between surfaces. In the presence of added salt the theory predicts the possibility of a predominantly attractive force under some conditions. These predictions are in accord with available experiments [99,100]. 

Here, the sum is over all possible resonance vectors. Since the leading contribution to 0,(f) is co,t, it follows that near the particular resonance m of interest, other phase factors in Eq. (3.29) will be oscillatory functions of time while due to Eq. (3.27), exp(/m 0) will be slowly varying. Retaining only the one resonance term in Eq. (3.29),... 

In particular, a circular boundary (centered at x = 0, y = 0) for D is excluded. We also require that kR2/z 4ir, which ensures that the domain of integration includes a large number of maxima and minima of the oscillatory function exp[/k(r — z)]. If we use (3.32), then (3.30) becomes... 

In these last two examples of equations of motion, the objective is to determine functions of the form h = /(/) or x=g(t), respectively, which satisfy the appropriate differential equation. For example, the solution of the classical harmonic motion equation is an oscillatory function, x=g t), where g(f) = cos a>t, and a> defines the frequency of oscillation. This function is represented schematically in Figure 7.1 (see also Worked Problem 4.4). 

The monotonic decay of the polarization in the Schiby— Ruckenstein model (the main critique of the polarization models) is a consequence of the assumption of the homogeneous distribution of water molecules in the vicinity of the surface. However, when the water was assumed to be structured in icelike layers in the vicinity of the surface, the polarization became an oscillatory function of the distance from the interface.13 This result was due to the particular locations of the water molecules... 

The nature of the excitation has a profound influence on the subsequent relaxation of molecular Uquid systems, as the molecular dynamics simulations show. This influence can be exerted at field-on equiUbrium and in decay transients (the deexdtation effect). GrigoUni has shown that the effect of high-intensity excitation is to slow the time decay of the envelope of such oscillatory functions as the angular velocity autocorrelation function. The effect of high-intensity pulses is the same as that of ultrafast (subpicosecond laser) pulses. The computer simulation by Abbot and Oxtoby shows that... 

The relaxation time for a given polarization mode (v) and wave vector (ic ) is calculated by i) integrating the energy autocorrelation function [71], or ii) fitting the energy correlation function with an exponential [72] or damped oscillatory function [73], as shown in Fig. 3. For each wave vector, the anharmonic frequency can also be extracted. 

Remark In the FLR scheme an extrapolation method (e-algorithm) is used to eliminate the effects of integrand singularities. Although Ninomiya s method is an improved N-C automatic quadrature by virtue of three devices above, his method might be less effective, say for oscillatory functions or when the tolerance e is very small. 

Integrals of oscillatory functions (Problems 9,13,17 and 18) are difficult for adaptive routines. The present AQN9D copes with these problems better than other routines except for the Problems 13 and 17 with e = 10 . 

Stratification, as illustrated by the plots in Figs. 5.4-5.G, is due to constraints on the packing of molecules next to the substrate surface and is therefore largely determined by the repulsive part of the intermolecular potential [38). Stratification is observed even in the complete absence of intermolecular attractions, such as in the case of a hard-sphere fluid confined between planar hard walls [165-167]. For this system Evans et al. [168] demonstrated that, as a consequence of the damped oscillatory character of the local density in the vicinity of the walls, is itself a damped oscillatory function of s, if s is of the order of a few molecular diameters, which is confirmed by the plot in Fig. 5.3. 

The oscillatory dependence on the thickness (d) of the polymer layer arises from the proximity of the metallic mirror electrode (the cathode) the emitting oscillator interacts with the virtual image oscillator behind the mirror [36, 37]. Because the radiation from the emitter and the retarded radiation from the image oscillator interfere, the PL decay rate is an oscillatory function of the distance from the mirror. Consequently, for a thin film, the quantum yields for photoluminescence and electroluminescence are oscillatory functions of d. 

In addition to creep and stress relaxation experiments, another type of measurement is quite common. Here the stress or strain, instead of being a step function, is an oscillatory function with an angular frequency a. The standard unit of a is radians per second (rad/s).++ Dynamic modulus values measured using such perturbations are functions of a> rather than time. The problem of putting dynamic experiments on a quantitative level is only slightly more difficult than is the case with step-deformation experiments. 

This property is actually not surprising as the series limit is approached, the bound state wavefunction acquires more and more nodes, and tends to the oscillatory function of the continuum. The position of the nodes is related to the phase in the continuum, and we may expect that the two are connected, since the wavefunction at very high n must change smoothly into the free electron s wavefunction just above the series limit. In QDT, as for H, the wavefunction for r > ro preserves this... 

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