Plane projections

From (A.81), /3T, = k, and this equation implies that the yield surface in stress space is a circular cylinder of radius k, shown in a FI plane projection in Fig. 5.7(a). The corresponding yield surface in strain space may be obtained by inserting the deviatoric stress relation (5.86) into the yield function (5.92)... 

Plane projection of the slructure or/ B gH2]. The two decaborane units are fused at the S(7 > and 6(6 ) positions to give a non rTJmetric structure with Cj symmetry... 

Localized 3-centre bond formalism can readily be used to rationalize the structure and bonding in most of the non-c/r>5r>-boranes. This is illustrated for some typical nido- and arachno-horanes in the following plane-projection diagrams which use an obvious symbolism for nonnal 2-centre bonds B-B O—O, B-Hc O— , (t = terminal). 

The size of a spherical particle is readily expressed in terms of its diameter. With asymmetrical particles, an equivalent spherical diameter is used to relate the size of the particle to the diameter of a perfect sphere having the same surface area (surface diameter, ds), the same volume (volume diameter, dv), or the same observed area in its most stable plane (projected diameter, dp) [46], The size may also be expressed using the Stokes diameter, dst, which describes an equivalent sphere undergoing sedimentation at the same rate as the sample particle. Obviously, the type of diameter reflects the method and equipment employed in determining the particle size. Since any collection of particles is usually polydisperse (as opposed to a monodisperse sample in which particles are fairly uniform in size), it is necessary to know not only the mean size of the particles, but also the particle size distribution. 

The unit cell of cellulose from Chaetomorpha melagonium is monoclinic, with a = 16.43 A (1.643 nm), b(fiber axis) = 10.33 A (1.033 nm), c = 15.70 A (1.570 nm), and /3 = 96.97°. In base-plane projection, each of the Meyer-Misch subcells that make up the super-lattice are identical. All equatorial reflections can be indexed by using a one-chain unit-cell, meaning that every single chain has... 

First of all, we analyzed the 2-dimensional structure projected on the ab-plane in order to confirm the result of the analysis by electron diffraction. The 17 independent reflections on the equator were used (R=35%, B=0.075nm (isotropic)). The molecular conformation of the p-form was adopted because it lowered R. The mutual positions and orientations of molecular chains are almost identical to those analyzed by electron diffraction. In the 3-dimensional analysis, therefore, this structure in the aft-plane projection was basically fixed. 

Fig. 8 Form II unit cell packing of cocrystal 20 21, showing potential cleavage planes projected horizontally [50]...

Fig. 13 Form I crystal packing of caffeine glutaric acid (35 36), as viewed down the length of the hydrogen-bonded combs, showing potential non-polar cleavage plane, projected vertically [59]...

A two-dimensional projection of the clouds of data is made with the two axes of maximum inertia. In this plane, projected treatment points are clustered for each treatment. 

A plane projection of the crystal cell of cellulose (a) and of cellulose trinitrate b) is outlined by the schemes in Fig. 84. Much interest was aroused in the results... 

It is interesting to consider the shapes of the subharmonic trajectories that lock on the torus in the various entrainment regions of order p/q. The subharmonic period 4 at the 4/3 resonance horn is, for example, a three-peaked oscillation in time [Fig. 7(a)] and has three closed loops in its phase-plane projection [Fig. 7(b)], while the subharmonic period 4 at the 4/ 1 resonance is a single-peaked, single-loop oscillation [Figs. 7(d) and 7(e)]. A subharmonic period 2 at the 2/3 resonance is also included in Figs. 7(g) and 7(h). Multipeaked oscillations observed in chemical systems (Scheintuch and Schmitz, 1977 Flytzani-Stephanopoulos et al., 1980) may thus result from the interaction of frequencies of local oscillators. Such trajectories are the nonlin-... 

FIGURE 7 Typical shapes of subharmonic trajectories. A subharmonic period 4 within the 4/3 resonance horn is a three-peaked oscillation in time (a), has three loops in its phase plane projection (b), and four loops in its x-cos 0 projection (c) (Brusselator, a = 0.0072, o = 4/3). The subharmonic period 4 within the 4/1 resonance horn has one loop in its phase plane projection (e), four loops in the x-cos projection (f) and is a one-peaked oscillation in time (d). Stroboscopic points are denoted by O. Try to imagine them winding around the doughnut in three-dimensional space An interesting shape shows up at the period 2 resonance in the 2/3 resonance horn (surface model >/aio = 2/3, alao = 0.1, o0 = 0.001) (g, h). These shapes are comparatively simple because of the shape of the unperturbed limit cycle which for all cases was a simple closed curve. 

Fig. 6 Basal plane projection of the structure of the (NH NaAiQ, series showing the network of hydrogen bonding N-H—7i(6 6) interactions between the Na(NH3)+ and C units (only one of eight possible orientations of Na(NH3)+ is shown)...

For the knot plane projection with defined passages, the following Reidemeister theorem is valid [39] different knots (or links) are topologically isomorphic to each other if they can be transformed continuously into one another by means of a sequence of simple local Reidemeister moves of types 1, 2 and 3 (see Fig. 9). Two knots are called regular isotopic if they are isomorphic with respect to the last two types of moves (2 and 3) if they are isomorphic with respect to all types of Reidemeister moves, they are called ambient isotopic. As can be seen from Fig. 9, a Reidemeister move of type 1 leads to the cusp creation on chain projection. At the same time, it is noteworthy that all real 3D-knots (links) are of ambient isotopy. 

FIGURE 2. Projection of the deduced Fermi surface of T12201 onto the ab-plane. The magnitude of the c-axis warping has been increased three-fold to emphasise the 8 loci where kz dispersion vanishes. The Brillouin zone has also been simplified as to accommodate the full aZ>-plane projection. 

Figure 8-39 Flat Plane Projection of the Location of a Colloidal Particle Subject to Brownian Movement. Source From H. Schubert, Food Particle Technology. Part 1 Properties of Particles and Particulate Food Systems, J. Food Eng., Vol. 6, pp. 1-32,1987, Elsevier Applied Science Publishers, Ltd.

The two relatively simple ALPOs that have been studied in most detail are A1P04-5 and VPI-5. The framework (0 0 1) plane projections of their structures are shown in Figure 12.10. 

Special cases are discussed in some detail in the literature [112,197,198], where the shape representation P is chosen as a space curve representing a protein backbone and the topological descriptors Fj(s) on the local tangent plane projections are either graphs or knots defined by the crossing pattern on the planar projection at each tangent plane T(s) of the sphere S. 

Figure 7.12 A D-PCS cubic membrane in glandular cells of a marine catfish, which exhibit a similar lattice to the PLB. (a) Overview showing the unusually large extension of this cubic membrane, (b) Higher magnification of a detail of (a) which shows, among more complex lattice planes, projections along the [100], the [111] and the [311] directions. See Fig. 7.1 for the corresponding computer generated prqjectians. Reproduced from [55], v/ith permission.

If, instead of a monoclinic crystal, we considered a tetragonal crystal having a fourfold axis along c and therefore perpendicular to the ab plane, then the plane projection would also have fourfold symmetry. So too would the corresponding reflections on the MO zone of reciprocal space have a fourfold distribution ... 

This description corresponds to a plane projection of the apparatus. In a three-dimensional perspective, all of the diffracted beams converge to F if and only if the sample is placed on a section of the toms generated by the rotation of the arc AC around the line SF and, of course, if the source is punctual. 

Figure 3. Stereographic projection. Pole P of the crystallographic plane projects to P on the projection plane (Ref. 14).

Fig. 1. Cross section view of aMo03 (100) and (120) planes (projection of the lattice on the (001) plane) (from [7]).

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