Spherical 3-space

Such a solution was found by Willem de Sitter on assuming a fourdimensional spherical space-time with constant curvature, rather than a spherical 3-space. He started with hve-dimensional Minkowski space to dehne the hypersurface... 

Within a spherical space of radius a, by definition Qy = 0, so that the value of potential of the ionic atmosphere here is constant and equal to that at point r = a ... 

The factor C k of B(R) is often called the /th component of the Racah spherical tensor of rank k the three tensor components of rank 1 may be considered unit basis vectors spanning the (spherical) space. 

Press ( 3) makes a similar approximation, comparing the closed system available energy of a volume of radiation in a cone compared to that in a full spherical space. His results translated into flux terms indicate that approximately 38% of the available energy flux in a solar solid angle is retained if the energy is uniformly scattered over a full 4vsolid angle. 

Recognizing that the substitutents residing on a tetra-substituted carbon atom are spherically spaced equidistant from one another, if all substituents are connected with lines, a tetrahedron is formed. Furthermore, as shown in Figure 4.1, if all four substituents are unique, they can be arranged in two configurations where the two molecules are mirror images and not superimposable. Because these two molecules are identical in composition but not configuration in three-dimensional space, they are referred to as... 

Often ions may rotate within the crystal and, if the rotation is completely free, the spinning ion will occupy a spherical space of diameter equal to the maximum cross section of the ion. The nitrate, ammonium, and cyanide ions may rotate in favorable cases, but such rotation becomes inhibited at lower temperatures. Thus, the high-temperature forms of both ammonium nitrate and cesium cyanide have the same structure as cesium chloride. 

These definitions are for lolol hemispherical properties, since G represents the radiation flux incident on tlic surface from all directions over the hemi spherical space and over all wavelengths. Thus, a, p, and t are the average properties of a medium for all directions and all wavelengths. However, like einissivity, these properties can also be defined for a specific wavelength and/or direction. For example, the spectral directional absorptivity and spectral directional reflectivity of a surface are defined, respectively, as the absorbed and leflected fractions of the intensity of radiation incident at a specified wavelength in a specified direction as... 

S electron An electron that occupies spherical space about the nucleus of an atom. 

The square root for 2 serves to spread the ball density though the spherical space and to suppress the tight packing for small r. (As a reminder, to convert from 0, (p, r to X, y, z use a spherical transformation x = r sin 0 cos rp, y = r sin 0 sin tp, z = r cos 0.) Each successive triplet overlaps, so that each point has the opportunity to become an r, 0, or tp. (The advantage over a Cartesian system is discussed later in this chapter.)... 

In equation 6.24, integration for all possible directions O over the entire spherical space has been performed. For polychromatic radiation, integration over the wavelength range of interest must be carried out. In the elementary volume of radiation absorption, for single photon absorption, energy is absorbed according to... 

A spherical space colony proposed by Gerald O Neill (Princeton University) has a diameter of 1.00 km. How many grams of N2 are needed to fill the interior of the colony at one atmosphere and 20 °C (room temperature) ... 

Solubility parameters are determined by Hansen s iteration method from the 3D solubility parameters of the solvents in which the polymer is miscible [9,10]. In this method, solubility of the polymer in various solvents is initially examined. The plots of the 3D solubility parameters of the solvents (available in the literature [8]) give a 3D spherical space, called Hansen s space . If the distance between any two points is measured by a computation method, then the straight line connecting the two points situated at the longest distance will represent the diameter of the sphere, and the center of the sphere will represent the 3D solubility parameter of the polymer. 

It is now recognized that, when hydrocarbon molecules are dissolved in water, a structure like an iceberg may be formed around the solute molecules[l]. The structure resembles that of a gas hydrate where the gas molecule is encaged in a nearly spherical space formed by hydrogen bonded water molecules. Structures of various gas hydrates have so far been extensively investigated with X-ray diffraction techniques[2-U], but a study of the iceberg structures formed in solution is very difficult because of quite low solubilities of hydrocarbons in water. 

Coefficient a usually depends on the difference in temperature between the body and its environment. In the microcalorimeter described by Czarnota et al. [7], free convection occurs in the spherical space around the calorimeter, and as a result the coefficient a is defined by... 

The inverse square law states that sound held intensity drops roughly 6 dB for each doubling of distance in spherical space. This is shown by... 

Spherical space is an ideal, ohen referred to as free space and limited in practice to elevated sound sources in outdoor settings and in the characteristics of anechoic chambers. 

Here, instead of considering shell elements, we "sample" the spherical space at the spherical surfaces in Fig. 3.8 at r, r2. .. We then... 

Pardon [4] used local surgery theory to extend the work of Madsen, Thomas and Wall on the classification of free acti of finite groups on spheres ("the topological spherical space form problem") to the classification of free actions of finit groups on manifolds which are 3p-homology spheres. 

Invariant knots and surgery in codimension Proc. 1970 ICM, Nice, Vol. 2, 99-112 (1971) (11 Smooth spherical space forms... 

Spherical space forms in the period dimensic prepr int... 

High technology proliferated in the cold war. The US pavilion at the 1967 World Exposition in Montreal comprised a three-quarters spherical space frame twenty storeys high. 

Steady-State Concentration Profile in Spherical Space Problem... 

Figure 15.9 The microphase-separated structures for cylinder-forming diblock copolymers confined in a spherical space. Chen et al. [61]. Reproduced with permission of American Chemical Society.

A wide variety of focal liver lesions can be diagnosed by ultrasound, notably cysts, for which it is the most specific and sensitive test (Gaines and Sampson 1989). They are seen as echo-free spherical spaces with thin, smooth walls and a characteristic band of brighter liver distally, caused by the lower attenuation of ultrasound by their fluid compared to the liver (Fig. 1.4) (Bryant et al. 2004). The same appearance characterises the individual cysts of dominant polycystic disease except that they may be very numerous (Kuni et al. 1978). The lesions themselves and the heterogeneous liver texture that results from the numerous bands of increased sound transmission make the detection of co-existent liver disease difficult or impossible. Similarly, haemorrhage into a cyst or superinfection are not usually detectable with ultrasound. 

In the converse situation free of gravity, a drop assumes a perfectly spherical shape. At one point, the U.S. Space program tested this idea with the solidification of ball bearings from molten metal drops in microgravity conditions. 

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

Figure Bl.9.5. Geometrical relations between the Cartesian coordmates in real space, the spherical polar coordinates and the cylindrical polar coordinates.

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