Homomorphic

We are particularly concerned with isomorphisms and homomorphisms, in which one of the groups mvolved is a matrix group. In this circumstance the matrix group is said to be a repre.sentation of the other group. The elements of a matrix group are square matrices, all of the same dimension. The successive application of two... 

Note 1. To maintain homomorphic relationships between classes of sugars, the (potential) aldehyde group of a uronic acid is regarded as the principal function for numbering and naming (see 2-Carb-2.2.1 and 2-Carb-22). 

The (non-detachable) prefix carba- signifies replacement of a heteroatom by carbon in general natural product nomenclature [26], and may be applied to replacement of the hemiacetal ring oxygen in carbohydrates if there is a desire to stress homomorphic relationships. If the original heteroatom is unnumbered, the new carbon atom is assigned the locant of the non-anomeric adjacent skeletal atom, with suffix a . 

Structures in which the linking glycosidic oxygen is replaced by -CH2- may be named by use of the replacement prefix carba- (c/. 2-Carb-34.2) for emphasis of homomorphic relationships. The oxygen replaced is given the locant of the carbon atom to which it is attached in the residue with the lower roman numeral (cited as superscript) (cf 2-Carb-37.2), with suffix a1. 

Note. These representations do not follow the recommendations for choice of main chain given in 2-Carb-37.3. Such deviations are common in depicting series of naturally occurring oligosaccharides where it is desirable to show homomorphic relationships. 

If/(z) possesses a derivative at z and at every point in some neighborhood ofz0, then/(z) is said to be analytic or homomorphic at z0. If the... 

P is a homomorphic image of P if there is a function h from the addresses... 

DEFINITION A flow diagram P is a graph homomorphic image of flow diagram P under graph homomorphism h, if h is a function from the address set of P ... 

Both possibilities are shown. Scheme P is a strong homomorphic image of each of the two resulting schemes all three schemes are strongly equivalent. 

There are two possible approaches. In one approach we in effect build the execution sequence tree for P. We start with node (1,0) labelled START. A node (k,r) will be at level r of the new tree-like structure and be labelled with the instruction named by k. Suppose statement k in P is connected by an arrow (with or without a label) to statement p in P and that we have constructed node (k,r) in P to date. If (k,r) has no ancestor of the form (p,r ), r < r, place node (p,r+l) labelled by statement p on the tree, with an arrow from (k,r) to (p,r+l) which contains any label on the arrow from k to p. If there is already an ancestor (p,rT), r < r, of (k,r) on the tree, then do not create (p,r+l) but instead add an arrow from (k,r) back to (p,r ) containing any label also on the arrow from k to p. If P has N statements, this process must terminate in a scheme P with at most N levels. Clearly P is tree-like and is strongly equivalent to P. This transformation is global and structure preserving. In fact P is a strong homomorphic image of P under the homomorphism h taking each (k,r>) back into k. ... 

An alternative method of proof uses repeated applications of a local transformation, the duplicate operation, which also preserves graph homomorphic images. Call a direct connection in P from n to m anomalous if n m and n... 

CLAIM Given a flow diagram P we can construct a flow diagram P such that P is a graph homomorphic image of P and a(P ) = 0. ... 

The scheme in Example IV-2 is a graph homomorphic image of this scheme. 

The groups G and G are said to be homomorphic to each other with a one-to-n correspondence, or that there is a homomorphism between G and G. ... 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

The group T of matrices is homomorphic to G and the matrices T can be characterized by the same parameters as used for characterizing the elements of G. It is implied that as the values of the parameters change continuously from those for x to those for x in the parameter space, the corresponding matrix r( r) goes continuously to T(a ). 

This result amounts to a 1 to 2 homomorphic mapping of the unitary group SU(2) onto the rotation group. From (28) it follows that the two unitary matrices... 

---