Group cyclic groups

Other techniques that work well on small computers are based on the molecules topology or indices from graph theory. These fields of mathematics classify and quantify systems of interconnected points, which correspond well to atoms and bonds between them. Indices can be defined to quantify whether the system is linear or has many cyclic groups or cross links. Properties can be empirically fitted to these indices. Topological and group theory indices are also combined with group additivity techniques or used as QSPR descriptors. 

TABLE ni--88. 4-POSmON WITH A HETIERO-CYCLIC GROUP... 

Section 19 15 An intramolecular esterification can occur when a molecule contains both a hydroxyl and a carboxyl group Cyclic esters are called lactones and are most stable when the nng is five or six membered... 

When the sequence rules permit alternatives, preference for lower-numbered locants and for inclusion in the principal chain is allotted as follows in the order stated Z over E groups and cis over trans cyclic groups. If a choice is still not attained, then the lower-numbered locant for such a preferred... 

For the case = 1, the symmetry group of a chiral nanotube specified by (n,m) is a cyclic group of order N given by... 

Applying the above symmetry formulation to armchair (n = m) and zigzag (m = 0) nanotubes, we find that such nanotubes have a symmetry group given by the product of the cyclic group and Cj , where 2n consists of only two symmetry operations the identity, and a rotation by 2ir/2n about the tube axis followed by a translation by T/2. Armchair and zig-... 

E the cyclic group of order and degree 5, generated by cyclic permutations of s objects ... 

If the cyclic group 2 is associated with the coronas of order k k 1, 2,. ..) then two graphs which are congruent according to the definition of Sec. 34 are called "planar nondistinct", while jf they are noncongruent in this sense they are called "planar distinct". 

Intramolecular addition of amines to carbonyl groups (cyclic imines)... 

Second, the symmetry properties of one of the processes (the Berry step) are analysed. The operators associated with it are shown to commute with the elements of a cyclic group of order ten. Because of the structure of the multiplication table, the same is true for the operators associated with the other stereoisomerization processes. The solution of the rate equations for any process are derived from these properties (Sections IV and V). 

Three independent P-N bond 137 lengths. OPO = 102.7°, exo-cyclic groups twisted at 48° to average plane of phosphazene ring... 

As a second example, consider the group formed by the elements 1, r, —1, -i, where i2 = -1. These elements are developed by the operations r , where n is an integer. If the law of combination is ordinary multiplication, the multiplication table for this cyclic group can be developed (Table 1). This... 

Though these substances are not monocyclic like benzene—and Hiickel s rule should not, strictly, apply to them—the introduction of the transannular bond, that makes them bi- and tricyclic, respectively, seems to cause relatively little perturbation, so far as delocalisation of the n electrons over the cyclic group of ten or fourteen carbon atoms is concerned. 

Some polysiloxanes with cyclic groups in the backbone... 

Alkyl radicals attached to benzene or another cyclic group in the molecule of an organic compound. 

There are two general classes of naturally-occurring antibiotics which influence the transport of alkali metal cations through natural and artificial membranes. The first category contains neutral macrocyclic species which usually bind potassium selectively over sodium. The second (non-cyclic) group contains monobasic acid functions which help render the alkaline metal complexes insoluble in water but soluble in non-polar solvents (Lauger, 1972 Painter Pressman, 1982). The present discussion will be restricted to (cyclic) examples from the first class. 

Mathematically this group is known as a cyclic group of order n, generated by the element c, and consisting of the elements e, c, c2,. ..,cn-1. The multiplication table of C3 is... 

We will first consider an abelian group, namely the cyclic group Zra. The pi s forming a regular representation commute with each other and... 

Figure 6. Formfactor K(r) and nearest neighbour spacing distribution P(s) for (a) the pi s form the regular representation of the cyclic group Z24 (b) pi s represent the symmetric group. S, (c) a random set of pi s without symmetries. The dashed curve in (b) labeled red. Poisson corresponds to a distribution of degenerate levels being Poisson distributed otherwise.

---