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St – ELEMENTS AND PROGRAMMING OPERATORS
24.01.2023 10:48
[1. Информационные системы и технологии]
Автор: Illia Vitaliovich Danilishyn, student, Sumy State University
Introduction: There is a need to develop an instrumental mathematical base for new technologies. The task of the work is to develop new approaches for this through the introduction of new concepts and methods.
St - elements
Definition 1. The set of elements {а}=(а1,а2,...,аn) at one point x of space X we shall call St – element, and such a point in space is called capacity of the St – element. We shall denote Stx{a} .
Definition 2. An ordered set of elements at one point in space is called an ordered St – element.
It is possible to Stx{a} correspond to the set of elements {а}, and to the ordered St - element - a vector, a matrix, a tensor, a directed segment in the case when the totality of elements is understood as a set of elements in a segment.
It is allowed to add St – elements: Stx{a} +Stx{b} =Stx{a}∪{b}.
Self-capacity
Definition 3. The self-capacity A of the first type is the capacity containing itself as an element. Denote S1fA.
Definition 4. The self-capacity of the second type is the capacity that contains the program that allows it to be generated. Let's denote S2fA. An example of self-capacity of the first type is a self-set containing itself. An example of self-capacity of the second type is a living organism, since it contains a program: DNA, RNA.
Definition 5. Partial self-capacity of the third type is called self-capacity, which contains itself in part or contains a program that allows it to be generated partially. Let us denote S3f.
Connection of St – elements with self-capacities.
For example, Sf{R}g{R} is the self-capacity of the second type if g{R} it is a program capable of generating {R}.
Consider a third type of self-capacity. For example, based on Stx{a}, where {а}=(а1,а2,...,аn), i.e. n - elements at one point, it is possible to consider the self-capacity S3f with m elements and from {а}, at m<n, which is formed by the form:
wmn=(m,(n,1)) (1)
that is, only m elements are located in the structure Stx{a}.
Self-capacities of the third type can be formed for any other structure, not necessarily St, only through the obligatory reduction in the number of elements in the structure. In particular, using the form
wm1 ...mn =(m1,(m2,(...(mn,1)...))) (2)
Structures more complex than S3f can be introduced.
Mathematics itself
Consider first the arithmetic of St:
Simultaneous addition of a set of elements {а}=(а1,а2,...,аn) are realized by Stx{a+}.
By analogy, for simultaneous multiplication: Stx{a*}.
Similarly for simultaneous execution of various operations: Stx{aq} , where {q}=(q1,q2,..,qn). qi an operation, i = 1,…,n.
Similarly, for the simultaneous execution of various operators: Stx{Fа}, where {F} = (F1,F2,...,Fn). Fi is an operator, i = 1,…,n.
The arithmetic itself for self-capacities will be similar: addition - S1f{a+}, (or S3fx{a+} for the third type), multiplication S1f{а*}, (S3fx{a*}).
Similarly with different operations: S1f{аq}, (S3fx{aq}), and with different operators: S1f{Fа}, (S3fx{Fa}).
Operator itself.
Definition: An operator that transforms Stx{a} into any Sifх{b} =2,3 ; where{b}⊂{a}; is the operator itself.
Example. The operator includes the set itself.
Lim-itself.
Lim St
S3f– software operators will differ only in that aggregates {а},{p},{B},{f} will be formed from corresponding St program operators in form (1) for more complex operators in form (2).
Quite interesting is the OS (operating system), the principles and modes of operation of the computer for this programming. But this is already the material of the next articles.
Conclusions: New concepts and new processing methods of information based on them and new software operators were introduced. Further development is associated with changing the structure of the arithmetic-logical device, the corresponding software and application for new technologies, in the light of the new approach.
REFERENCES:
1. Kantor G.(1914) Fundamentals of the general doctrine of diversity. New ideas in mathematics, 6 (in Russian).
___________________________
Scientific supervisor: Vladimir Nikolaevich Pasynkov, Ph.D., Associate Professor, National Metallurgical Academy of Ukraine
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