Relative convexity

Connectivity may be defined as the maximal number of particles (branches, cell walls, and so on) that may be cut without separating the structure. For several porous foods, such as foamed chocolate bar, honeycomb chocolate bar, chocolate muffin, marshmallow, and strawberry mousse, air cell connectivity was defined as a measure of the relative convexity or concavity of the total slid surface (Lim and Barigou, 2004). Concavity indicates connectivity, whereas convexity indicates isolated disconnected structures. Through the image analysis, authors compare the solid area and perimeter before and after an image dilation operation and calculate the index of connectivity as the following ... 

Within the general scheme of relative convexity, the conventional, ordinary local convexity is obtained as a special, degenerate case of relative local convexity, with a tangent sphere of infinite radius as reference, that is, with a tangent plane of reference curvature b = 0. 

The numerical value of the reference curvature b can be specified in absolute units or in units scaled relative to the size of the object G(a). If absolute units are used, then a relative convexity characterization of G(a) involves size information if an object G(a) is scaled twofold, then its shape remains the same, but with respect to a fixed, nonzero b value a different relative convexity characterization is obtained. That is, the pattern of relative shape domains Do(b)> D (b), and D2(b) defined with respect to some fixed, nonzero reference curvature value b (b K)) is size-dependent. On the other hand, if the reference curvature b is specified with respect to units proportional to the size of G(a), then a simple. scaling of the object does not alter the pattern of relative shape domains with respect to the scaled reference curvature b. In this case, the shape characterization is size-invariant, that is, a "pure" shape characterization is obtained. 

A natural, size-independent relative convexity characterization is obtained if the relative curvature parameter b is scaled by a size parameter of the object G(a),... 

Consider a MIDCO G(a) and a choice for the curvature parameter b, and assume that the shape domains of relative convexity of G(a) have been determined. By using an appropriate neighbor relation to describe the mutual arrangements of the domains along the MIDCO surface G(a), the corresponding shape matrix s(a,b) and the associated shape graph g (a,b) can be defined [109,110,158,193]. 

Figure 5.9 The construction of Shape Globe Invariance Maps (SGIM s), of MIDCO relative convexity shape domain patterns for two reference curvature values, b = 0 and b < 0.

A practical implementation of the above approach is the following a global shape property of the molecule is assigned to each point of the sphere S, followed by the determination of those domains of S where this shape property is invariant. A pair of examples is shown in Figure 5.9, where the shape globe invariance domains of a MIDCO surface for two relative convexity shape domain partitionings (P) with respect to two reference curvatures, b = 0, and b < 0, are given. As... 

Mezey, P.G. (1988a). Global and Local Relative Convexity and Oriented Relative Convexity Application to Molecular Shapes in External Fields. J.Math.Chem.,2,325. 

Mezey, P.G. (1988a) Global and local relative convexity and oriented relative convexity application to molecular shapes in external fields. /. Math. Chem., 2, 325. 

Subsequently, these topological methods have been adopted and modified to the significantly simpler, three-dimensional molecular shape problem, where the shape of the molecule is the quantum mechanical shape of the electron density cloud [13-19], This has led to the development of the shape group methods, where the ranks of homology groups describing relative convexity domains of the complete set of all isodensity surfaces of the molecule, the so-called Shape Group Betti numbers, provided a detailed, numerical shape code for the quantum chemical electron density [13-19]. 

On the local level, shape complementarity implies matches between locally concave and locally convex domains, as well as matches between properly placed saddle-type domains, where a directional convex-concave match is important. Replacing the simple D (K,a) notation, the more elaborate notation D b),i(K,a) is used sometimes when studying the complementarity of local shape domains, where the notation includes the relative convexity specification fi(b). This quantity takes values... 

All these hypothetical steps can be traced in the obolellates, which evolved simple articulatory structures soon after the development of hemiperipheral growth in the dorsal valve. Together with the closely placed ventral diductors of the obolellates, which operated a low angle to the commissural plane, so too did the articulatory structures work mostly to prevent anterior shifts of the dorsal valve. Another distinctive trend in obolel-late evolution was the maintenance of the position of the diductor muscles sub-parallel to the commissural plane. With an increased relative convexity of both valves, one result was the evolution of high ventral muscle platforms in naukatides. 

In side-bending and rotation dysfunctions, one side of the head is relatively convex, the other relatively flat. Table 104-3 lists the important findings. 

Both Type III and Type V isotherms are characterized by convexity towards the relative pressure axis, commencing at the origin. In Ty )e III isotherms the convexity persists throughout their course (Fig. 5.1(a), whereas in Type V isotherms there is a point of inflection at fairly high relative pressure, often 0-5 or even higher, so that the isotherm bends over and reaches a plateau DE in the multilayer region of the isotherm (cf. Fig. 5.1 (b)) sometimes there is a final upward sweep near saturation pressure (see DE in Fig. 5.1(b)) attributable to adsorption in coarse mesopores and macropores. 

The weakness of the adsorbent-adsorbate forces will cause the uptake at low relative pressures to be small but once a molecule has become adsorbed, the adsorbate-adsorbate forces will promote the adsorption of further molecules—a cooperative process—so that the isotherms will become convex to the pressure axis. 

A perforated plate can be flat, concave, convex, or double-dished. The main advantages of the perforated plate are that it is simple, inexpensive, easy to modify, and easy to clean. The disadvantages of a perforated plate are the possibiUty of soflds leaking, ie, weeping through it into the plenum lower turndown capabiUty than other distributors the requirement of a peripheral seal and a relatively high pressure drop requited for good distribution. 

All points on the two tangents HRi, HR2, to the curve of solutions represent heterogeneous systems composed of solid hydrate in contact with solutions. If the curve between Ri and R2 is convex the heterogeneous systems are stable, and inversely. At a given temperature and pressure the hydrate can be in equilibrium with two liquid phases of different composition, one containing relatively more, the other relatively less, salt than the hydrate. With rise of temperature the form of the curve and the altitude of H change ... 

Modified Temp. Conv . Butan-2-ol convex lO Relative... 

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