Outer and Inner Perimeters

In a tuple coronoid, the perimeter length n can be split into n for the outer perimeter and (wj ) ) ( 5 )2 perimeters of the g corona holes so that 

Correspondingly we may split the invariants 5 and tj which also pertain to the perimeters. 

The invariants of the outer perimeter obey the relations for benzenoids, which may be obtained from the appropriate functions of Table 1 as special cases for = 0. (It is mentioned by passing that n in that case corresponds to the invariant often denoted by in benzenoids and referred to as the number of external vertices, equal to the number of external edges or the unique perimeter length.) Now we have 

When it comes to the inner perimeters it is expedient to invoke the interpretation of the corona holes as benzenoids. The benzenoid which corresponds to a corona hole is just defined by the inner perimeter of the hole as its unique perimeter. Let the numbers of external vertices of degree two and of degree three on the perimeters of these benzenoids be identified by the symbols SgO, SgO,. , s o and 2° respectively. Now we have 

It is implied that the above equations (11) — (15) are valid individually for every corona hole. Let us now take the summations over all k and introduce the notations 

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There is a close relationship between polycyclic compounds such as coronene 81198) or Staab s kekulene 82199) and the macrocyclic cyclophanes such as 83200). They can be described as perimeter structures in which an outer and inner perimeter can be considered. 

Figure 23.17 A phospholipid has a polar head and two nonpolar tails. The membranes of living cells are formed by a double layer of lipids, called a bilayer. The polar heads are on the outer and inner perimeter of the membrane and the tails are on the inside of the bilayer. 

In the case of special degenerate single coronoids, as well as in single (nondegenerate) coronoids, one speaks about the outer and inner perimeter, viz. C and C", respectively, but now some edge(s) belong to both C fl C" 0. 

Having the graph—theoretical planarity in mind, it is clear that the outer and inner perimeters of a single corohebcene always can be identified. However, an identification of the corona hole with a benzenoid may be obscured. Therefore we must rephrase the requirement that the corona hole should have a size of at least two hexagons (cf. Sect. 2.1). The following formulation is valid for corofusenes any inner perimeter should consist of at least ten edges (and vertices). 

Case 1. If r = then obviously both the outer and inner perimeter are M-alternating cycles. 

Three Kekule structures are indicated, each by their n/2 = 20 double bonds. In the structures (a) and (b), r= n 2 = 18. The outer and inner perimeters are alternating cycles in both these structures. This condition (Case 1) does not exclude the possibility for aromatic sextets to be present, as is demonstrated here in (a) there is no aromatic sextet, but in (b) there are three (marked by small circles). In the structure (c), r = 16 and therefore r < n /2 (Case 2). This condition prescribes with certainty the presence of an aromatic sextet there are four of them in the structure at hand. 

In both subcases (Subcase 3.1 and Subcase 3.2) it was demonstrated that a coronoid G is generated from G by a normal tearing down so that both the outer and inner perimeters of G are M —alternating cycles for some Kekule structure M of G. Repeat the argument for G. Eventually one will arrive at a coronoid with a hexagon of mode L2 or A2, whereby the case is reduced to Case 1. Thus the proof of sufficiency is completed. 

Nevertheless, the coronoid of this example is not HED because U Kg does not constitute the total of 82688 Kekule structures for this system in fact, the formidable amount of 74624 Kekule structures do not belong to K U Kg. One of them is shown in the right-hand drawing above. This Kekule structure was selected among those where both the outer and inner perimeter are alternating cycles, confirming that the system is a regular coronoid. 

The outer and inner tubes extend from separate stationary tube sheets. The process fluid is heated or cooled by heat transfer to/from the outer tube s outside surface. The overall heat transfer coefficient for the O.D. of the inner tube is found in the same manner as for the double-pipe exchanger. The equivalent diameter of the annulus uses the perimeter of the O.D. of the inner tube and the I.D. of the inner tube. Kem presents calculation details. 

The poly hex under consideration (Fig. 6) is embeddable in the hexagonal lattice, and it has overlapping edges or, more precisely, two overlapping hexagons. Hence it is reasonable to consider this system as helicenic, and that has actually been done previously (Randid, Nikolid and Trinajstid 1988 Randic, Gimarc et al. 1989 Cyvin BN, Brunvoll and Cyvin 1992b). However, cyclohelicene is not a corohelicene, and of course not a coronoid. It is true that two perimeters are present, but they are symmetrically equivalent none of them qualifies to be identified especially as the outer— or inner perimeter. There are two enantiomers of the structure in question (Fig. 6). 

Outdoor grounds must have barriers and access points arranged in the same strategy as discussed previously for main buildings. However, with outdoor grounds such as athletic fields, barriers are used at outer perimeters and inner perimeters. Outer perimeters have much looser controls because of limited resources. Nevertheless, strategic barriers are the main tool to funnel different types of traffic to the access points of the inner perimeter. 

The hydraulic mean diameter, dm, is defined as four times the cross-sectional area divided by the wetted perimeter. Equation 3.69 gives the value dm for an annulus of outer radius r and inner radius r, as ... 

The hydrocarbon which corresponds to a polyhex under consideration, has the chemical formula C H. Here the number of carbon atoms (n) corresponds to the number of vertices (see the above listing). The number of hydrogens (5) is equivalent to the number of secondary carbon atoms and corresponds to the number of vertices of degree two. These vertices are exclusively on the perimeters (inner and outer) of the polyhex. The total number of tertiary carbon atoms on the perimeters is t (see above). The total number of the boundary vertices (on the perimeter), viz. is also the total number of edges on the perimeters. This number (n is the combined perimeter length (i.e. the sum of the lengths for the outer and all the inner perimeters). 

It is especially interesting to compare these two equations, viz. (18) and (19), or alternatively (15) and (14), with eqns. (9) and (10), respectively. These relations emphasize the different properties of an outer perimeter from those of an inner perimeter. This feature has been described in detail by Hall (1988). Also Polansky and Rouvray (1977) have offered an elaborate treatment of the perimeter lengths of coronoids. 

For the outer perimeter, eqns. (3.9) and (3.10) are immediately applicable to single coronoids for the inner perimeter one has... 

Consider now a single coronoid G and assume that A is a set of a nonadjacent vertices, all of them on the outer or on the inner perimeter of G. The same restriction is imposed on A as above. 

Ap. Therefore G is disconnected. This implies that w is on one of the perimeters of G. Since e and e are on the outer perimeter of G by assumption, the inner perimeter of G entirely... 

The overall philosophy is to establish outer perimeters and an inner perimeter. It is possible to need multiple inner perimeters. Outer perimeters and strategic barriers are used in conjunction to guide foot traffic and vehicular traffic to central areas. The students or others used to assist in parking then take over. The strategic barriers also keep traffic from inadvertently going into areas where access is not desired. 

POSTEC - unaxial research tester In the POSTEC - uniaxial tester, discussed by Maltby and Enstad (1993), the sample is confined in a cylindrical die and wrapped in a flexible membrane which is stretched between the outer periphery of the piston and the inner perimeter of the lower part of the die. Since the membrane is stretched and in contact with the wall and powder, the sample is compacted homogeneously thus the wall friction between the specimen and the die is reduced. Comparison of the POSTEC uniaxial tester with a biaxial and Jenike-type shear eell testers, with the standardised CRM-116 limestone powder, indicated that the fc values obtained with the POSTEC are slightly less than those obtained with Jenike-type shear cells and a biaxial tester. Since the total time for... 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

Annulene and dehydro[22]annulene are also diatropic. A dehydroben-zo[22]annulene has been prepared that has eight C=C units, is planar and possesses a weak induced ring current. In the latter compound there are 13 outer protons at 6.25-8.45 8 and 7 inner protons at 0.70-3.45 5. Some aromatic bridged [22]annu-lenes are also known. The [26]annulene has not yet been prepared, but several dehydro[26]annulenes are aromatic.Furthermore, the dianion of 1,3,7,9,13,15, 19,21-octadehydro[24]annulene is another 26-electron system that is aromatic. Ojima and co-workers prepared bridged dehydro derivatives of [26], [30], and [34]annulenes. All of these are diatropic. The same workers prepared a bridged tetradehydro[38]annulene, which showed no ring current. On the other hand, the dianion of the cyclophane 89 also has 38 perimeter electrons, and this species is diatropic. ... 

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