Narrow Results

We provide some special algorithms to construct the fractals which have not been constructed previously. Our generalized graphical representation can be applied to any alphabet with any cardinal number. It is noted that the Hao's frame representation and the Tino's generalization are the special cases of our result. Moreover, we will apply the recurrence formula to derive their characteristic equations and roots.

By using the frame representation method proposed by B. L. Hao et al. in 2000^1,2 and a method provided by P. Tino,^3,4 we will devise the generalized graphical representation for the strings defined on an alphabet Σ = {0, 1, …, n - 1} with cardinal n but all these graphs are still shown in a unit square, and generate its corresponding struture tables automatically. In particular, the words defined on Σ are mapped into n-adic numbers so that the index set Ind(B) can be computed. Thus, the fractals of the forbidden words can be naturally constructed. It has already been known that for biological sequences, the circular codes can be applied to some special bacteria and their forbidden words are the 5-necklaces. In this paper, we notice that the fractals of the forbidden words are particularly useful to represent the circular codes for some biological sequences.