Fourier component

If the surface tension is a fiinction of position, then there is an additional temi, da/dx, to the right-hand side in the last equation. From the above description it can be shown drat the equation of motion for the Fourier component of the broken synnnetry variable is... 

Clip acts in phase (the same Fourier component) with the first action of cii to produce a polarization that is anti-Stokes shifted from oi (see fV (E) and IFj (F) of figure B 1.3.2(b)). For the case of CSRS the third field action has frequency CO2 and acts in phase with the earlier action of CO2 (W (C) and IFj (D) of figure Bl.3.2 (b). Unlike the Class I spectroscopies, no fields in CARS or CSRS (or any homodyne detected Class II spectroscopies) are in quadrature at the polarization level. Since homodyne detected CRS is governed by the modulus square of hs lineshape is not a synmretric lineshape like those in the Class I... 

The Fourier sum, involving the three dimensional FFT, does not currently run efficiently on more than perhaps eight processors in a network-of-workstations environment. On a more tightly coupled machine such as the Cray T3D/T3E, we obtain reasonable efficiency on 16 processors, as shown in Fig. 5. Our initial production implementation was targeted for a small workstation cluster, so we only parallelized the real-space part, relegating the Fourier component to serial evaluation on the master processor. By Amdahl s principle, the 16% of the work attributable to the serially computed Fourier sum limits our potential speedup on 8 processors to 6.25, a number we are able to approach quite closely. 

Example This example of an HN-C(O) amide torsion uses the AMBER force field. The Fourier component with a periodicity of one (n = 1) also has a phase shift of 0 degrees. This component shows a maximum at a dihedral angle of 0 degrees and minima at both -180 and 180 degrees. The potential uses another Fourier component with a periodicity of two (n = 2). 

The relative sizes of the potential barriers indicate that the V2 force constant is larger than the Vj constant. The phase shift is 180 degrees for the Fourier component with a two-fold barrier. Minima occur at -180, 0, and 180 degrees and maxima at -90 and 90... 

For small p the contribution of paths with large x (n 0) to the partition function Z is suppressed because they are associated with large kinetic-energy terms proportional to v . That is why the partition function actually becomes the integral over the zeroth Fourier component Xq. It is therefore plausible to conjecture that the quantum corrections to the classical TST formula (3.49a) may be incorporated by replacing Z by... 

At temperatures above there is no instanton, and escape out of the initial well is accounted for by the static solution Q = Q with the action S ff = PVo (where Vq is the adiabatic barrier height here) which does not depend on friction. This follows from the fact that the zero Fourier component of K x) equals zero and hence the dissipative term in (5.38) vanishes if Q = constant. The dissipative effects come about only through the prefactor which arises from small fluctuations around the static solution. Decomposing the trajectory into Fourier series. 

Figures 6.4 shows some of the variety of possible shapes of P f) for elementary rules shown in the figures are the power spectra for rules Rll, R56, R150 and R200. The plots were generated for lattice size N = 2048, ignoring the first 15 transient steps and averaging a total of 20 runs. Also, since there are only N data points but 2N real Fourier components, half of the components are redundant. Thus, only the first half of the components are shown (see [H89b] or [H87] for a complete set of power spectra).

Thus the mutual intensity at the observer is the Fourier transform of the source. This is a special case of the van Cittert-Zernike theorem. The mutual intensity is translation invariant or homogeneous, i.e., it depends only on the separation of Pi and P2. The intensity at the observer is simply / = J. Measuring the mutual intensity will give Fourier components of the object. 

We assume that the perturbation operator V(t, e) can be expanded in a sum over Fourier components as... 

Multi-photon processes involve higher Fourier components of the electron density. For example, the density fluctuation caused by two photons with frequencies and ui2 can be described by =... 

TOWARDS THE HYDRODYNAMIC LIMIT STRUCTURE FACTORS AND SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms of the spatial Fourier ( ) components of the density, particle currents, stress, and energy fluxes. The time correlation function of those Fourier components detail the decay of density, current, and fluctuation on the length scale of the Ijk. 

Fourier transformation of Rf pulses (which are in the time domain) produces frequency-domain components. If the pulse is long, then the Fourier components will appear over a narrow frequency range (Fig. 1.24) but if the pulse is narrow, the Fourier components will be spread over a wide range (Fig. 1.25). The time-domain signals and the corresponding frequency-domain partners constitute Fourier pairs. 

Figure 1.24 Fourier components of a long Rf pulse ( soft pulse) are spread over a relatively narrow frequency range. (Reprinted from S. W. Homans, A dictionary of concepts in NMR, copyright 1990, pp. 127-129, by permission of Oxford University Press, Walton Street, Oxford 0X2 6DP, U.K.)...

The wave vectors k can be expressed in terms of any basis vectors we choose. At the moment there is neither a direct nor a reciprocal lattice. Using (II.3a) in (II. 1) we see that the Fourier components of two indistinguishable densities can differ only by a phase factor ... 

A related concept is that of phase function (t>g(k) which relates the Fourier components of... 

If g is an element of the point group of the material meaning that p(r) and p(gr) are indistinguishable for all elements g in that group, corresponding Fourier components can differ only by a phase factor ... 

If gk = k, (II.7) implies that either the Fourier component vanishes or the phase funetion(t>g(k) is an integer or zero. Another way of expressing that important result is to say, that given a phase function ()>g(k), those wave vectors k, for which that function is not... 

Here u j(k K) is the Fourier component of the square of the absolute value of the Bloeh... 

Using (III. 16) we can also write this Fourier component in terms of the momentum space orbitals as... 

The symmetry properties of the density show up experimentally as properties of its Fourier components p. If those components vanish except when the wave vector k equals one of the lattice vectors K of a certain reciprocal lattice, the general plane wave expansion of the density,... 

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