Shape geometric description

The partial model Geometrical Modeling Objects [418] is concerned with the geometrical description of the structural modeling objects. It provides concepts to represent surfaces, shapes, and spatial coordinate systems. 

The important qualitative features of the various geometric descriptions of the void space are that the topologies of the internal surface of the faujasite represent an interlinked three dimensional network for diffusion, i.e., if a molecule is of appropriate size to diffuse throughout the ftamework, it can move from any initial site to any other site on the internal surface. The important quantitative features of the void space are related to size and shape features of diffusing species relauve to the size/shape features of the void space. The interactions of size/shape features of the zeolite with size/shape features of the adsorbed molecules are critical in determining the rotational and diffusional characteristics of species within the internal surface. 

The individual particles of which we have spoken seem, in many cases, amenable to a relatively simple geometric description. In solution, however, particles may floe or aggregate due to random particle-particle and particle-floc collisions, and generally complex shapes arise that belie the much simpler shape of the original particle. Figure 1.3.5 from Weitz Oliveria (1984) shows in two-dimensional projection an irreversible aggregate of uniform-size, spherical gold particles with diameter 15 nm. 

Geometrical Description of the Flank Shape (Without Cutting Forces)... 

Morphology. The morphology of a crystal is the geometric description of its outward appearance in terms of the crystal faces (i.e., size, shape, and angular relationships of the facets—see Figure 11). 

A wide range of fibres of different mechanical, physical and chemical properties have been considered and used for reinforcement of cementitious matrices, as outlined in Chapter 1. The fibre-reinforcing array can assume various geometries and in characterizing its nature two levels of geometrical description must be considered (i) the shapes of the individual fibres and (ii) their dispersion in the cementitious matrices (Figure 2.1) [7]. 

Fig.l Gas penetration in a clip-shaped square tube. Fig.2 Geometrical description of a clip-shape square (Hsu,1995) tube.

The present model takes into account how capillary, friction and gravity forces affect the flow development. The parameters which influence the flow mechanism are evaluated. In the frame of the quasi-one-dimensional model the theoretical description of the phenomena is based on the assumption of uniform parameter distribution over the cross-section of the liquid and vapor flows. With this approximation, the mass, thermal and momentum equations for the average parameters are used. These equations allow one to determine the velocity, pressure and temperature distributions along the capillary axis, the shape of the interface surface for various geometrical and regime parameters, as well as the influence of physical properties of the liquid and vapor, micro-channel size, initial temperature of the cooling liquid, wall heat flux and gravity on the flow and heat transfer characteristics. 

The optimized geometries of (4.55a)-(4.55d) are shown in Fig. 4.17 and selected geometrical parameters are summarized in Table 4.13. The molecular shapes and NBO descriptors (not presented) generally agree with the idealized Lewis-like sd/x picture for Os(CH2)2, HW(CH2)(CH), and W(CH2)3. However, theC—W—C angle (42°) in the ground state of W(CH)2 is much smaller than expected for idealized sd1 geometry, and the optimal NBO description corresponds to a metallacyclopropene,... 

A soliton is a solitary wave that preserves its shape and speed in a collision with another solitary wave [12,13]. Soliton solutions to differential equations require complete integrability and integrable systems conserve geometric features related to symmetry. Unlike the equations of motion for conventional Maxwell theory, which are solutions of U(l) symmetry systems, solitons are solutions of SU(2) symmetry systems. These notions of group symmetry are more fundamental than differential equation descriptions. Therefore, although a complete exposition is beyond the scope of the present review, we develop some basic concepts in order to place differential equation descriptions within the context of group theory. 

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

There has been a decisive evolution in the treatment of steric effects in heteroaromatic chemistry. The quantitative estimation of the role of steric strain in reactivity was first made mostly with the help of linear free energy relationships. This method remains easy and helpful, but the basic observation is that the description of a substituent by only one parameter, whatever its empirical or geometrical origin, will describe the total bulk of the substituent and not its conformationally dependent shape. A better knowledge of static and dynamic stereochemistry has helped greatly in understanding not only intramolecular but also intermolecular steric effects associated with rates and equilibria. Quantum and molecular mechanics calculations will certainly be used in the future to a greater extent. 

The geometric shape of the walls is characterized by a selected CLD too, see Fig. 4. Here, the shape is more complex and a description exclusively based on one random variable only (like wall thickness), the size distribution of which could be estimated then, is not simple. 

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