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 its 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.
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.
\ composed 6 7
3 2 1 7 2 Product 7 12 15 16 6 10 12 16 18 5 8 9 12 16 4 6 8 3 4 2 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
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
. Hence, any set of single element of each set can be written down as follows:
For reception of new qualitative result, we shall do following procedure. We shall order both objects inside of subsets, and the chosen subsets. Objects can be presented physically located not only further or more close (ordering of objects inside of a subset), but also placed on the right or to the left of itself as representatives of different subsets (ordering of subsets). In other words, we have bidimentional space of ordering of objects. For the description of a maximum quantity of states we shall enter a following rule-conformity. Let (N+2) a state of the previous subset (as the subset) will be unit of a following subset. It allows at transition from one subset to another to change unit of the account for number of objects of the previous subset plus one (due to ( N+1) or a zero state). Thus, choosing on one representative from each subset, we have an opportunity to describe the quantity of states much more exceeding total of objects. Representation of this number in the form of decomposition on n composed, each of which can be various, actually corresponds to the division of initial set described above from N objects on n the ordered subsets. In other words, we have the certain system of the account or notation. We shall name its generalized notation (GPN). All discussed in literature GPN are described in our approach. If to choose accordingly m A as m b , ! m or m n C 1 -, we shall receive b -radix polynomial, factorial and binomial GPN. As example GPN we shall offer in a role of radix the generalized k -deformed (
In other words, there is the connection of GPN classification with decomposition A P on sum of composed n A P . In this direction there is an interesting problem about the most effective decomposition of such number. In another way it can be formulated as follows -at the fixed value of number N to find its such decomposition on composed that their product was maximal. In our case to it will correspond such GPN which will describe the maximal number of states at the fixed value of all objects. At splitting into two numbers composed by effective decomposition there will be its splitting half-and-half. Hence, the most effective GPN will be GPN with equal number of objects which will describe to us corresponding number of states. If to extrapolate the given result on the general case the choice of mankind as daily GPN as GPN with equal quantity of objects in subsets does not look casual. Detailed consideration of a situation in case of n composed for greater and any values of number N complex enough will not be resulted here again. As small acknowledgement told above it is possible to consider the result given in tables 1-3.
Let’s return a little back
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