Space curvature

It is proposed to use the terms world space and aether synonymously and to reserve space and vacuum for empty three-dimensional space. Curvature of world space may refer to any of its unknown number of dimensions. It... 

FbwO) is the Fourier transformation of effective beam width as a function of spatial frequency / Fuff) is the MTF of the XRll. Because of the XRll windows curvature, projection data must be transformed to obtain uniform pixel spacing, described by Errors in object centre... 

Butterfly Valves These valves (Fig. 10-155) occupy less space in the line than any other valves. Relatively tight sealing without excessive operating torque and seat wear is accomphshed by a variety of methods, such as resilient seats, piston rings on the disk, and inclining the stem to limit contact between the portions of disk closest to the stem and the body seat to a few degrees of curvature. 

Vertical pre.s.sure leaf filters. These filters have vertical, paraUel, rec tangular leaves mounted in an upright cylindrical pressure tank. The leaves usually are of such different widths as to aUowthem to conform to the curvature of the tank and to fill it without waste space. The leaves often rest on a filtrate manifold, the connec tion being sealed by an O ring, so that they can be lifted individuaUy from the top of the fil-... 

One further effect of the formation of bands of electron energy in solids is that the effective mass of elecuons is dependent on the shape of the E-k curve. If dris is the parabolic shape of the classical free electron tlreoty, the effective mass is the same as tire mass of the free electron in space, but as tlris departs from the parabolic shape the effective mass varies, depending on the curvature of tire E-k curve. From the dehnition of E in terms of k, it follows that the mass is related to the second derivative of E widr respect to k tlrus... 

It has been shown that there is a two-dimensional cut of the PES such that the MEP lies completely within it. The coordinates in this cut are 4, and a linear combination of qs-q-j. This cut is presented in fig. 64, along with the MEP. Motion along the reaction path is adiabatic with respect to the fast coordinates q -q and nonadiabatic in the space of the slow coordinates q -qi-Nevertheless, since the MEP has a small curvature, the deviation of the extremal trajectory from it is small. This small curvature approximation has been intensively used earlier [Skodje et al. 1981 Truhlar et al. 1982], in particular for calculating tunneling splittings in (HF)2- The rate constant of reaction (6.45a) found in this way is characterized by the values T<. = 20-25 K, = 10 -10 s , = 1-4 kcal/mol above T, which compare well with the experiment. 

The average intercylinder spacing, which depends somewhat on the diameter, can be derived from the oo.l reflexions in the diffraction pattern, using the ho.o (or hh.o) spacing of graphite for internal calibrations since the latter seems to be independent of curvature. 

Another phase which has attracted recent interest is the gyroid phase, a bicontinuous ordered phase with cubic symmetry (space group Ia3d, cf. Fig. 2 (d) [10]). It consists of two interwoven but unconnected bicontinuous networks. The amphiphile sheets have a mean curvature which is close to constant and intermediate between that of the usually neighboring lamellar and hexagonal phases. The gyroid phase was first identified in lipid/ water mixtures [11], and has been found in many related systems since then, among other, in copolymer blends [12]. 

This variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal. 

The competition between the polar and steric dipoles of molecules may also lead to internal frustration. In this case, the local energetically ideal configuration cannot be extended to the whole space, but tends to be accomodated by the appearance of a periodic array of defects. For example, the presence of the strong steric dipole at the head of a molecule forming bilayers will induce local curvature. As the size of the curved areas increases, an increase in the corresponding elastic energy makes energetically preferable the... 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

Definitions of flame parameters in channels. D, distance between channel walls effective in flame quenching (quenching distance). D, flame width dead space R, radius of curvature of the flame. 

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