Single scale factor for the universe from the creation of radiation and matter till the present

A scheme for incorporating the creation of radiation and matter into the cosmological evolution is introduced so that it becomes possible to merge the times before and after the creation of radiation and matter in a single scale factor in Robertson-Walker metric. This scheme is illustrated through a toy model that has the prospect of constituting a basis for a realistic model.Comment: Minor typos are corrected, an acknowledgment is added, to be published in The European Physical Journal

Circular chromatic numbers of a class of distance graphs

AbstractSuppose m,k,s are positive integers with m>sk. Let Dm,k,s denote the set {1,2,…,m}⧹{k,2k,…,sk}. The distance graph G(Z,Dm,k,s) has as vertex set all integers Z and edges connecting i and j whenever |i−j|∈Dm,k,s. This paper determines the circular chromatic number of all the distance graphs G(Z,Dm,k,s)