Multichannel algorithm based on generalized positional numeration system

This report is devoted to introduction in multichannel algorithm based on generalized numeration notations (GPN). The internal, external and mixed account are entered. The concept of the GPN and it’s classification as decomposition of an integer on composed of integers is discussed. Realization of multichannel algorithm on the basis of GPN is introduced. In particular, some properties of Fibonacci multichannel algorithm are discussed. A basis of this article is simple arithmetic equality decomposition of an integer on composed of integers. Using this decomposition the interrelation of the various parties of the science is shown. The plan of this article is following. In the beginning communication of decomposition of an integer on composed of integers with classification of the generalized positional notations (GPN) is considered. Then concepts of the internal, external and mixed account are entered. Bit transformation of the initial alphabet to set of other alphabets is discussed, using the decomposition of an integer specified above. Realization of multichannel algorithm on the basis of GPN is considered. In particular, some properties Fibonacci multichannel algorithm are discussed. At the beginning let’s give the formulation of a problem decomposition of an integer on composed of integers. Procedure easy enough, but demanding detailed elaboration. One of the first specifications concerns quantities composed. Really, at decomposition of integer N on n composed there is finite set of decisions. In Table 1 at small values of number possible variants are shown all. From the commutativity of addition in Tables 2,3 are given independent variants of decisions for next values of number N. Also last line of each table value of product of all composed is given, and its maximal size is allocated by more dark background of a corresponding cell. Integ er N \ composed 1 2 3 4 5 1 1 1 1 2 1 1 2 3 1 1 2 1 1 2 3 4 1 1 1 2 2 3 1 1 1 2 1 2 1 2 1 1 3 2 1 1 2 1 1 4 3 2 1 1 2 3 1 2 1 1 1 2 1 1 3 1 2 1 1 1 3 2 1 2 1 1 1 2 1 1 1 4 1 2 1 1 1 1 5 1 Product 1 1 2 2 1 3 4 3 2 2 2 1 4 6 6 4 3 4 3 4 4 3 2 2 2 2 1 Table 1. Every possible variants of decomposition of integer N on n composed. From the resulted tables it is visible, that such decomposition are similar to record of number in the certain notation with the basis not above value (N-n). Such sensation is not casual. Therefore the process of the account should be considered in more details and attentively. The account has arisen from practical activities of the person on ordering and the account of subjects. Therefore let's choose N objects, which nature is not important at present, and we shall order them. In other words, let's them arrange physically, for example, from itself forward one behind another. Establishing isomorphism or unequivocal conformity between the given objects and numbers or a numerical axis, there is a following picture. The choice of one any object will be a choice of any number. Hence, touching all our objects, we actually recalculate them. We shall name such account internal because of a choice as unit of the account one object. Integer N \ composed 6 7 1 1 2 3 1 1 2 1 1 1 1 1 2 3 1 1 1 2 1 1 1 1 1 1 1 2 5 4 3 1 2 2 1 1 1 1 6 5 4 1 2 3 2 1 1 2 1 1 1 1 3 4 3 2 1 2 1 1 5 4 3 3 1 2 2 1 1 1 1 4 3 2 1 1 4 3 2 1 2 1 1 5 2 1 3 2 1 1 6 1 2 1 7 1 Product 5 8 9 4 6 8 3 4 2 1 6 10 12 5 8 9 12 4 6 8 3 4 2 1 Table 2. Independent variants of decomposition N=6 and N=7 on n composed. Integer N \ composed 8 1 1 2 3 4 1 1 1 2 2 1 1 1 1 2 1 1 1 1 1 1 1 2 7 6 5 4 1 2 3 2 3 1 1 1 2 2 1 1 1 1 1 1 1 3 6 5 4 4 3 1 2 3 2 2 1 1 2 1 1 1 1 4 5 4 3 3 2 1 2 2 1 1 1 1 5 4 3 2 1 2 1 1 6 3 2 1 1 7 2 1 Product 7 12 15 16 6 10 12 16 18 5 8 9 12 16 4 6 8 3 4 2 1 Table 3. Independent variants of decomposition N=8 on n composed. However it is possible to act in another way. Having chosen for a unit of measure all set from ours N objects, it is possible to receive or empty set ((N+1) a state) or all set ((N+2) a state), i. e. there are only two values or states. Let's name such account external as it is the account not inside of initial set, and actually the account of the set and external in relation to objects. Usually in practice the given two accounts mix. Record of number is carried out by means of symbols of any alphabet where almost always include zero as ((N+1) a state. Thus, as a result of the described procedure of the account by means of N objects it is possible to describe all (N+2) states. Let's complicate our analysis [1]. We shall consider a situation when we have taken some such ordered sets or, on the contrary (that is in common equivalent), have divided initial set A on n subsets n A generally with unequal quantity of objects. i. e. in language of sets it we shall express so ∑ = n n A A . (1) In each its elements n A are ordered by means of the alphabet n α , and their total or capacity (power) of the alphabet n α is n n p P = ) (α . Hence, any set of single element of each set can be written down as follows: { } 1 2 1 1 α α α α α α K K m n n n m