Cyclic order

This methodology, as in the cases of rearrangements and of plausible disconnections, does not meet the criterium of maximum simplicity, since the reconnection represents an increase of the "cyclic order" and therefore of the complexity of the molecule. 

As we have seen in some of the examples already commented on in the preceding Chapters, the same "heuristic principles" which apply to acyclic compounds may also be applied, in principle, to cyclic compounds. The bond disconnections lead to acyclic systems or, at least, to a reduction in the cyclic order. 

The "cyclic order" Cjj of a molecule -or of the corresponding graph- is equal to... 

If n is the cyclic order, as we have already defined it (see Chapter 6), it is possible to demonstrate that the maximum number of rings is 2" - 1, and that there exist, as a minimum, n primary rings. 

Cyclic order 13 - 11 -t- 1 = 3, minimum number of primary rings. Actual number of primary rings = 4 ... 

Each graph has a certain cyclic order n. This order is equal to the number of edges minus the number of vertices plus one3). The maximum number of circuits in the graph is equal to 2 — 1. 

In the real M-cyclic molecule, the cyclic order is equal to the number of bonds minus the number of atoms plus one (33). 

Information may be stored in the architecture of the receptor, in its binding sites and in the ligand layer surrounding bound <7 it is read out at the rate of formation and dissociation of the supermolecule. In addition to size and shape, a receptor is characterized by the dimensionality, the connectivity and the cyclic order of its structural graph these features have been used to define a ligand structural index L ... 

A cryptate effect is observed for anion complexes as is the case for cation complexes (Section 2.3). In general, an increase in cyclic order from acyclic to macro-cyclic to macrobicyclic significantly increases the stability and selectivity of the anion complexes formed by polyammonium ligands. 

Figure 4.1(b)). One can see in Figure 4.1(c) that reflection in a plane normal to the axis of rotation does not change the direction of rotation, but that it is reversed (Figure 4.1 (d)) on reflection in a plane that contains the axis of rotation. Specification of a rotation requires a statement about both the axis of rotation and the amount of rotation. We define infinitesimal rotations about the axes OX, OY, and OZ by (note the cyclic order)... 

The synthesis of macropolycyclic cryptands generally involves stepwise, straightforward pathways (18, 20, 33) based on the successive construction of systems of increasing cyclic order macrocyclic, macrobicyclic, and so on. Newkome has recently reported a satisfactory quaternization-dealkylation procedure, facilitating the synthesis of 8 (34). Unlike the synthetic approaches to simple crown ethers (10,... 

By replacement of just one oxygen atom in 5 by a methylene group, affinity of the corresponding cryptand 15 to NH4+ decreases by a factor of ca. 100 (NH4+ c 15 log /Ca = 4.3). The macrobicycle 16 has almost completely lost the complexation ability and the selectivity of 5 (NH4+ c 16 log /Ca = 1.7). This dramatic effect results from the removal of one bridge of 5, i.e., from a decrease in cyclic order from the tricyclic to the bicyclic ring system, demonstrating the importance of the spherical macrotricyclic structure for the binding properties of 5. 

In the second order determinants the numbers are in cyclic order 123, hence the factor of 02 in the second line is given a negative sign. The determinant of the third order has thus been reduced to three determinants of the second order, each multiplied by a single element of the third order determinant taken from a column in which no element of the second order determinant occurs. 

The target of differentiation is implied by the parentheses, and the indices i, j, and k are in cyclic order. We may express the rotational measuring vectors as explicit functions of the nuclear positions. As an example, we obtain from Eq. (140)... 

CYCLOPS Cyclically-ordered phase-sequence (for suppressing quadrature artefacts) 3.2.5... 

The sequence of I s occurring in eeich of the four 3-1 symbols on the right-hand side of Eq. (23) is easily remembered Each 3-1 contains an 1 from each column in the 6-1. One 3-1 uses the top row of the 6-1 and the others contain one 1 from the top row and two from the bottom row, in cyclic order. 

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