Center, geometric

As discussed in detail in [10], equivalent results are not obtained with these three unitary transformations. A principal difference between the U, V, and B results is the phase of the wave function after being h ansported around a closed loop C, centered on the z axis parallel to but not in the (x, y) plane. The pertm bative wave functions obtained from U(9, <])) or B(0, <()) are, as seen from Eq. (26a) or (26c), single-valued when transported around C that is ( 3 )(r Ro) 3< (r R )) = 1, where Ro = Rn denote the beginning and end of this loop. This is a necessary condition for Berry s geometric phase theorem [22] to hold. On the other hand, the perturbative wave functions obtained from V(0, <])) in Eq. (26b) are not single valued when transported around C. 

The angles ot, p, and x relate to the orientation of the dipole nionient vectors. The geonieti y of interaction between two bonds is given in Fig. 4-16, where r is the distance between the centers of the bonds. It is noteworthy that only the bond moments need be read in for the calculation because all geometr ic features (angles, etc.) can be calculated from the atomic coordinates. A default value of 1.0 for dielectric constant of the medium would normally be expected for calculating str uctures of isolated molecules in a vacuum, but the actual default value has been increased 1.5 to account for some intramolecular dipole moment interaction. A dielectric constant other than the default value can be entered for calculations in which the presence of solvent molecules is assumed, but it is not a simple matter to know what the effective dipole moment of the solvent molecules actually is in the immediate vicinity of the solute molecule. It is probably wrong to assume that the effective dipole moment is the same as it is in the bulk pure solvent. The molecular dipole moment (File 4-3) is the vector sum of the individual dipole moments within the molecule. 

Budgeting. These changes in the storage and retrieval of chemical information requite that Hbraries and information centers now consider not only what should be purchased but also what monies should be allocated for the purchase of information in nonprint formats such as CD-ROMs (compact disk read-only memory) and on-line databases. Coupled with this is budgeting for the cost of hardware and software to enable the rapid and cost-effective deHvery of needed information (15). The geometric increase in sources, both printed and on-line, has increased the role of information speciaHst as an expert in the deHvery of chemical information. Retrieval from increasingly diverse and complex sources becomes the paramount issue for searchers of chemical Hterature in the 1990s. 

When additional substituents ate bonded to other ahcycHc carbons, geometric isomers result. Table 2 fists primary (1°), secondary (2°), and tertiary (3°) amine derivatives of cyclohexane and includes CAS Registry Numbers for cis and trans isomers of the 2-, 3-, and 4-methylcyclohexylamines in addition to identification of the isomer mixtures usually sold commercially. For the 1,2- and 1,3-isomers, the racemic mixture of optical isomers is specified ultimate identification by CAS Registry Number is fisted for the (+) and (—) enantiomers of /n t-2-methylcyclohexylamine. The 1,4-isomer has a plane of symmetry and hence no chiral centers and no stereoisomers. The methylcyclohexylamine geometric isomers have different physical properties and are interconvertible by dehydrogenation—hydrogenation through the imine. 

The force exerted on a submerged planar surface of area A is given by F = p A where p is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example. Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. 

The normal operating position of a shaft inside a bearing is shovvm in Fig, 29-64, It can be seen that, due to radial forces, the geometric center of the shaft does not coincide with the one of the bearing. This displacement creates a vv edge, vvFich combined with the shaft motion, forces the oil into a continiioiislv decreasing area, and a... 

The rolling elements are installed in a cage that performs the ei v important role of reducing the friction inside bearings. The conditions without a cage are shown in Fig, 29-70, It can be demonstrated that the rotational speed of a rolling element around its geometric center is ... 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

An uneven distribution of mass about the geometric axis of the system. This distribution causes the center of mass to be different from the center of rotation. 

Static eccentricities are amplified due to rotation of the shaft about its geometric center. 

If supported by journal bearings, the shaft may describe an orbit so that the axis of rotation itself rotates about the geometric center of the bearings. 

The previous equation shows that when lu < uj ,8r is positive. Thus, when operating below the critical speed, the system rotates with the center of mass on the outside of the geometric center. Operating above the critical speed (lu > LUn), the shaft deflection 8r tends to infinity. Actually, this vibration is damped by outside forces. For very high speeds (lu >> LUn), the amplitude 8r equals —e, meaning that the disc rotates about its center of gravity. 

Unbalance. This stimulus is caused by material imperfections, tolerances, etc. The mass center of gravity is different from the geometric case, leading to a centrifugal force acting on the system. 

The motor magnetic center must be within /u inch of the motor .s geometric center. 

The stereochemistiy of reactions involving substituted alkenyl free radicals indicates that radicals formed at trigonal centers rapidly undergo interconversion with the geometric isomer. Reactions proceeding through alkenyl radical intermediates usually give rise to the same mixture from both the E- and the Z-precursor ... 

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