Narrow band approximation

Comparing the two graphs in Fig. S, it is obvious that the behavior, in the case of the wide band, where the oscillations are relatively small in magnitude, differs qualitatively from the rapid and large variations found for the narrow band. Approximate methods can be used to explain these and other differences. 

It is also possible to increase the interpolation degree without going all the way, by using narrow-banded approximations to the inverse of E. 

In general, the smaller the diameter of the vessel, the further the distribution of stresses in tlie circumferential direction. In small diameter vessels, the longitudinal stresses are confined to a narrow band (approximately 2 in. for a 24-in.-diameter vessel). The opposite becomes true for laiger-diameter vessels or larger R, /t ratios. 

First Mechanism. The first mechanism is simply that a freak wave can be caused by a linear superposition of waves. In this case, the probability distribution for wave height in the limit of the narrow-band approximation obeys a Rayleigh dis-tribution ° corrections due to finite spectral band width have been obtained. Wave crest statistics can be obtained by using a second-order theory. The probability distribution for wave crests has been found in a narrow-band approximation and for finite band width. This mechanism is theoretically well established and relatively easy to follow the basic theory will be reviewed in Sec. 6.3. 

In the narrow-band approximation, the wave height H = 2A. The exceedance probability Ph H) for wave height now follows by integrating (6.10) from H to oo ... 

To summarize, quasi-resonant four-wave interactions introduce deviations from linear statistics for surface elevation in particular, for weakly nonlinear, narrow-banded and long-crested wave trains, the kurtosis evolves according to Eq. (6.19). In the narrow-band approximation, the kurtosis is related to the BFl [defined in Eq. (6.25)] the tail of the wave height distribution depends on the kurtosis/BFl and it increases as the kurtosis increases. Finally, the maximum wave height distribution depends on the number of waves in the wave train (record length) and on the kurtosis of the nonlinear wave field, as shown in Eq. (6.30). 

Bell Syst Tech J 23 282-332 Rychlik I (1993) On the narrow-band approximation for expected fatigue damage. Probab Eng Mech 8 1-4 Shinozuka M, Jan CM (1972) Digital simulation of random processes and its applications. J Sound Vib 25 111-128... 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

The conformation of HO(SiPh2) OH species (n = 4, 5, or 7) has also been investigated in solution by IR spectroscopy. The intramolecular -interaction in the heptasilane is seen in CC14 solution as a relatively narrow band at 3605 cm-1, a shift of 80 cm-1 from the approximate... 

The quantity N is approximately constant for different bands or peaks in a chromatogram for a given set of operating conditions (a particular column and mobile phase, with fixed mobile-phase velocity, and temperature). Hence N is a useful measure of column efficiency the relative ability of a given column to provide narrow bands (small values of tw) and improved separations. 

In view of the results discussed in the previous section, it is not surprising that, in finding suitable approximations , a critical factor is bandwidth. We consider first an approximation suitable for a very narrow band. Since the difference between energy levels in the band must be small, Gi(f — ) in (25) can be approximated by setting = ej (for all k), so that... 

A full analysis of the SOA suggests that it should be restricted to narrow bands or to high-velocity atoms, but that, within these regimes, it works well. The complementary approximation for broad bands and low velocity atoms is the wide-band approximation (WB A) The essence of the WB A lies in the... 

Figure 2.21 Bragg plane tilt aberration, (a) Diffracting planes parallel, diffraction occurs simultaneously over the whole height of the beam, (b) Diffracting planes skewed, diffraction only takes place over a narrow band, (c) As the crystal is rotated to measure the rocking curve, the band moves up or down the crystal. The integrated intensity remains approximately the same as in case (a) but the peak intensity decreases and the width increases...

Little is known about the fluorescence of the chla spectral forms. It was recently suggested, on the basis of gaussian curve analysis combined with band calculations, that each of the spectral forms of PSII antenna has a separate emission, with Stokes shifts between 2nm and 3nm [133]. These values are much smaller than those for chla in non-polar solvents (6-8 nm). This is due to the narrow band widths of the spectral forms, as the shift is determined by the absorption band width for thermally relaxed excited states [157]. The fluorescence rate constants are expected to be rather similar for the different forms as their gaussian band widths are similar [71], It is thought that the fluorescence yields are also probably rather similar as the emission of the sj tral forms is closely approximated by a Boltzmann distribution at room temperature for both LHCII and total PSII antenna [71, 133]. 

Narrow band peaking at 440 nm with decay time of approximately 1 ps is evidently connected with Eu " center substituting for Ca (Fig. 4.59d). 

Laser-induced time-resolved luminescence spectra of natural emeralds also demonstrate l -lines of Cr at 680 and 684 nm accompanied by a narrow band peaking at 715 nm, which have similar decay times of approximately 55ps(Fig. 4.53). 

Figure 4.37a represents the time-resolved luminescence spectrum of a hydrozincite under 266 nm laser excitation. A relatively broad band is detected at 430 nm, which is responsible for the well-known blue hydrozindte luminescence. Its spectral position and decay time of approximately 700 ns are typical for Eu luminescence. However, the excitation spectrum of this band consists of one narrow band at 240 nm (Fig. 4.37b), which does not correspond to an Eu " excitation spectrum. Two bands usually characterize the latter with relatively small Stokes shifts of 30-50 nm caused by crystal field splitting of the 4/ 5d-levels. Moreover, the measured Eu concentrations in the hydrozincite samples under investigation are very low (less than 0.5 ppm) and they do not correlate with the intensity of the blue luminescence, i.e. the band at 430 nm. 

In the case of narrow bands - and this will be the case of hybridized 5 f bands when 5 f electrons are itinerant - an approximate treatment has to be done. Kubo and Obata have studied the case of transition metals in the tight binding approximation. The narrow band susceptibility is the sum of 4 terms... 

As far as the controls are concerned, we here consider time-continuous modulation of the system Hamiltonian, which allows for vastly more freedom compared to control that is restricted to stroboscopic pulses as in DD [42, 55, 91]. We do not rely on rapidly changing control fields that are required to approximate stroboscopic a -pulses. These features allow efficient optimization under energy constraint. On the other hand, the generation of a sequence of well-defined pulses may be preferable experimentally. We may choose the pulse timings and/or areas as continuous control parameters and optimize them with respect to a given bath spectrum. Hence, our approach encompasses both pulsed and continuous modulation as special cases. The same approach can also be applied to map out the bath spectrum by measuring the coherence decay rate for a narrow-band modulation centered at different frequencies [117]. 

Much of this book is concerned with the properties of narrow bands to which the tight-binding approximation is appropriate. In this case, if the band is half full or nearly so, the short-range repulsion between the electrons may have very important effects on the properties of the electrons in the bands, producing magnetic moments and non-conducting properties. These are a major theme of this book. At this point we introduce the Hubbard intra-atomic energy ... 

Results on Unheated Samples. The results obtained in Southampton for the set of British samples are shown in Figures 2 and 3. In Figure 2 the spin concentration of each maceral is plotted against the carbon content (daf) of the associated vitrinite, and the points for each set of macerals from one coal are joined by a vertical line. It can be seen that the vitrinite series forms a well-defined narrow band which curves upwards sharply at about 90% carbon, and the data resemble closely those presented earlier by Austen and Ingram for whole coals. The values of exinites form a wider, approximately horizontal band lying a little below the vitrinite band, while the fusinite data appear to vary at random but lie consistently well above the vitrinite band and are appreciably higher than vitrinites of the same carbon content (90-92%) would be. 

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