## General considerations

Both general properties of positive matrices and general statistical considerations favor sharper results than the ancestral distributions taken from de Bruijn matrices seem to justify. The reasons lie in the difference between positive matrices for which strict conclusions are possible, and nonnegative matrices, for which they

mayhold; and also in the strongly asymmetrical frequency distribution.So, in spite of the encouragement given by matrix theory and Eq. 22, it is necessary to look elsewhere for actual proofs. An ideal theoretic result, that is, one which depends on collective properties and not on individual matrices themselves, has been published by Michel Dubois-Violette and Alain Rouet [35]; similar reasoning had already been used by Hedlund [7], was recounted by Nasu [8], and was probably the basis of Amoroso and Patt's assertion [22] regarding the balancing of counterimages (they cited an unpublished laboratory report).

Nasu has shown [36] that the argumemts, ostensibly valid for de Bruijn diagrams, actually apply to any graphs for which the numbers of incoming and outgoing nodes are themselves balanced. In all cases, the important connection lies between the variance and ; otherwise understood as the fact that when every configuration has at least one counterimage, the number of counterimages is uniform.

Harold V. McIntosh

E-mail:mcintosh@servidor.unam.mx