Equation (5.8) tends to predict vapor loads slightly higher than those predicted by the full multicomponent form of the Underwood equation. The important thing, however, is not the absolute value but the relative values of the alternative sequences. Porter and Momoh have demonstrated that the rank order of total vapor load follows the rank order of total cost. [Pg.137]

Although they are small in absolute value, these fluctuations must be taken into account in the various commercial transactions related to storage and distribution of gasoline. [Pg.188]

Because of the double sound path involved in PE measurements of the back wall echo, we approximate the corresponding attenuation at a certain frequency to be twice as large as the attenuation that would be obtained by an ordinary TT measurement. We propose to use the logarithm of the absolute value of the Fourier transform of the back wall echo as input data, i.e... [Pg.889]

A measurement procedure has been developed that allows to determine the mass of the inclusions as well as their locations with respect to radius, angle, and depth (2). For the depth determination use is made of the approximate 1/R dependence of the magnetic field strength from the distance R to the inclusion When in a first measurement at a small lift off an inclusion is detected, the measurement is repeated at an increased lift off From the signal ratio the depth can be calculated or seen from a diagram like fig. 5a which was generated experimentally. After that, from calibration curves like fig. 5b the absolute value of the signal leads to the mass of the inclusion. [Pg.989]

Finally, in the case of solids, there is the difficulty that surface atoms and molecules differ in their properties from one location to another. The discussion in Section VII-4 made clear the variety of surface heterogeneities possible in the case of a solid. Those measurements that depend on the state of surface atoms or molecules will generally be influenced differently by such heterogeneities. Different methods of measuring surface area will thus often not only give different absolute values, but may also give different relative values for a series of solids. [Pg.574]

In general the width of the coexistence line (Ap, Ax, or AM) is proportional to an order parameter s, and its absolute value may be written as... [Pg.639]

On short length scales the coarse-grained description breaks down, because the fluctuations which build up the (smooth) intrinsic profile and the fluctuations of the local interface position are strongly coupled and camiot be distinguished. The effective interface Flamiltonian can describe the properties only on length scales large compared with the width w of the intrinsic profile. The absolute value of the cut-off is difficult... [Pg.2373]

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. [Pg.96]

The question of determination of the phase of a field (classical or quantal, as of a wave function) from the modulus (absolute value) of the field along a real parameter (for which alone experimental determination is possible) is known as the phase problem [28]. (True also in crystallography.) The reciprocal relations derived in Section III represent a formal scheme for the determination of phase given the modulus, and vice versa. The physical basis of these singular integral relations was described in [147] and in several companion articles in that volume a more recent account can be found in [148]. Thus, the reciprocal relations in the time domain provide, under certain conditions of analyticity, solutions to the phase problem. For electromagnetic fields, these were derived in [120,149,150] and reviewed in [28,148]. Matter or Schrodinger waves were... [Pg.104]

The evolution of the nuclear wavepacket is also traced by a number of snapshots of the absolute values of the wavepacket, again integrating over the... [Pg.261]

diagonal elements that have to be negative and it is the third one that must be positive. Note that these D terms are reasonably close to fulfilling the expected features just mentioned ... [Pg.711]

For each combination of atoms i.j, k, and I, c is defined by Eq. (29), where X , y,. and Zj are the coordinates of atom j in Cartesian space defined in such a way that atom i is at position (0, 0, 0), atomj lies on the positive side of the x-axis, and atom k lies on the xy-plaiic and has a positive y-coordinate. On the right-hand side of Eq. (29), the numerator represents the volume of a rectangular prism with edges % , y ., and Zi, while the denominator is proportional to the surface of the same solid. If X . y ., or 2 has a very small absolute value, the set of four atoms is deviating only slightly from an achiral situation. This is reflected in c, which would then take a small absolute value the value of c is conformation-dependent because it is a function of the 3D atomic coordinates. [Pg.424]

The value of the correlation coefficient ranges from r = -1 to r = +1. In those cases where r =l, the data are completely correlated, either positively or negatively (see Figure 9-4). The smaller the absolute value of r, the lower is... [Pg.444]

Set this threshold Lo a small positive constant (the default value is 10 Hartrcc), Tli is tli resh old is used by HyperCh cm to igu ore all two-cicetron repulsion in tegrals with an absolute value less th an th is value. Tli is option controls the performance of the SCF itera-lious and the accuracy of the wave function and energies since it can decrease the number of ealeulatcd Iwo-elcclrou integrals. [Pg.113]

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