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Kuznetsov L.A., Voronin A.S. Information model for analysis of correspondence between empirical sets

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Kuznetsov L.A., Voronin A.S. Information model for analysis of correspondence between empirical sets

The paper deals with a problem which can be stated as follow: assume that Xn is a set of input variables and Ym is a set of output variables. A certain subset Y+ is specified in set Y. It is necessary to determine such subset X+ on X, which would be the best the corespondent to subset Y+ in a sense.

Subsets X+ and Y+ are the areas of a certain form, specified in the space of input and output variables. They can be given as:

1) rectangular areas: yi'ii", i=1…m, xi'ii", i=1…n

2) multidimensional elliptic areas (ellipsoids).

The case when X+ and Y+ have the rectangular form was already been dealt with in the literature on the subject[1]. This work is concerned with a case when X+ and(or) Y+ is shaped like a multidimensional elliptic area. A method of identification of correspondence is proposed using criterion of amount of mutual information as the base. An algorithm of identification is given.

The experiments have demonstrate that the value of the amount of information for the elliptical areas is approximately 10-30 % more than for rectangular area with the same data. Consequently, it is possible to make a conclusion that the use of methods of identification correspondence with elliptic areas can be justified in situations in which the duration of the identification process is not a critical value and high-performance computers are used for identification.


  1. Kuznetsov L.A. Identification of Subsets Correspondence of Two Spaces/ Proc. Inter. AMSE Confer. “Information & Systems method for Engineering Problems”, Dec 28-30,1993, Malta/AMSE Press, V.2, pp.3-9.


In this article the algorithm of generation of the arithmetic formulas on the basis of the offered formula is suggested. The arithmetic formula is considered as connection of one parameter with other different parameters through arithmetic operations.

The special oriented cyclic graphs for preservation of the information about the simple arithmetic formulas are offered. Every such of the graphs allows generating all formulas, which may be obtained from an initial formula. Algorithm of formation of the graph for the formulas being a superposition of the considered simple formulas also is offered. The vertices of such graph are described as oriented cyclic subgraph, which has three vertices.

In the article it is offered use of the doubly linked circular lists, which elements are linked circular lists of three nodes, for computer realization above described graphs. Such representation of the formulas allows generating all formulas, which may be obtained from initial formula, or to define the formula for calculation of necessary parameter.

The offered representation of the formulas further allows using the universal program for calculation of the formulas. Such program generates any necessary formula for definition of necessary parameter on the basis of the given arithmetic formula.

There is a computer realization of the offered algorithm.


It is useful for many reseaches and industry to use resources of powerful mainframes. It is fruitful to apply interactive and remote access to powerful computing servers hosted on the Internet. Wide spreading approaches, such as based on telnet, are oriented on usage of mainframe by computer specialists and not suppose wide access of common user to computational resources.

There is proposed a general model of remote computing on the Internet in [1]. This model is based on the concept of computational proxy servers concept and visual object-oriented interfaces with natural metaphors. The aid of the model is making easy access to powerful computational resources for everybody. According to this model, web interface could be used to interact with computing proxy servers, which applies the supercomputer resources. This human-computer interaction allows to use computing resources without any knowledge of operating system.

Many mathematical functions require powerful resources, and it is fruitful to use computational proxy servers for for their computing. It is developed the software, which allow to incoporate a wide set of mathematical functions to computational proxy servers. The software provides effective computing, simplicity and comfort in use with the help of web interface.

Set of functions, realized on current proxy server base, includes more one hundred functions from various branches of mathematics, for example, linear algebra, differential equations, image processing.

This research is funded by the Russian Foundation of Basic Research (code 99-01-00468).


1. Prokhorov V.V. Technology of computing resources use on the Internet on the base of computational proxy servers// Algoritms and software of parallel calculations. Yekaterinburg: UB RAS, 1998. P. 256-267.


At a solution of many modern tasks the high-speed processing of data represented by multivariate bit maps is required. Research of mathematical base and methods of a solution of such tasks on the SIMD of a computer and systems has shown, that at constructing algorithms the special algebraic systems and structures, which are to the homomorphic classic Boolean algebra, and with other, take into account singularities of computations in multivariate processor spaces. For formal exposition, analysis and synthesis of algorithms for such processor spaces the application of a special algebraic system - S-algebra is expedient.

The carrier S-algebra is the set of Boolean matrixes ordered in computing space. As the metric of space the metric of the Moore is used.

The signature of S-algebra is made metric by independent operations of inversion, logical addition and multiplying and metric dependent operations of item permutation of units.

Definition 1. S-operation of the oriented shift Q N (A) the operation of mass permutation of units of a set A = { aij ; i=1,2,...,n , j=1,2,...,m,  = 1 , 2 ,..., k } in metric space on N of positions without a modification of their values by the rule:

aij = a pqP ( a i j ) , if 1  p  n , 1  q  m , differently 0.

N - coefficient of permutation, defined on sets {n} and {m} depending on modality of the operation of shift.

Definition 2. S-operation of inversion the operation of mass replacement of values of units of a set A on their additions without a modification of a location in metric space: : С = A ,

A = { ai j ; i = 1,...,n , j = 1,..., m } , ai j = a 1 i j ,a 2i ja 3i j .....a ki j .

Definition 3. S-operation of the disjunction the operation of mass logical addition of single units of two sets A = {a i j; i = 1,2..., n; j = 1,2..., m} and B = {b i j; i = 1,2..., n; j = 1,2..., m}, definite on the metric d1: C = A v B,

C = { c i j ; i= 1,2,...,n ; j = 1,2,...,m} , c i j = a i j v b i j = c1 i j ,c2 i j ,..., i j .

Definition 4.. S-conjunction operation the operation of mass logical multiplying of single units of two sets A = {a i j; i = 1,2..., n; j = 1,2..., m} and B = {b i j; i = 1,2..., n; j = 1,2..., m}, :

C = A  B, C = { c i j ; i = 1,2,...,n ; j = 1,2,...,m } , c i j = a i j  b i j = c1 i j ,c2 i j ,..., i j .

Definition 5. S-function of disjunctive testing Tv (A) the operation a computation of a Boolean function from (n x m) variables on a set A = { a i j ; i= 1,2,...,n ; j = 1,2,...,m} :

Tv ( A ) = a111 v a211 v ....v akmn = V ( a ki j ) .

Definition 6. S-function of conjunctive testing T& (A) the operation a computation of a Boolean function from (n x m) variables on a set A= { a i j ; i = 1,2,...,n ; j = 1,2,...,m} :

T ( A ) = a111  a211  ...  akmn =  ( a ki j).

Definition 7. The set A is named as logical S-unit , If T ( A )=1, and logical S- in zero ,

if TV ( A ) = 0.

Definition 8. The set A is named as logical S-function, if its each unit a a i j  is a Boolean function f ( x 1, x 2 ,...,x g ), which one possess the value of logical zero or logical unit on a gang of logical variables x 1, x 2 ,...,x g .

In the report it is exhibited, that the laws of S-algebra are distinct from the classic laws of the Boolean Algebra because of singular properties of operations of space permutation. However in that specific case, at an information of a potency of units of the carrier to 1 and usage of operations of space permutation of an order zero S - the algebra is to isomorphous classic algebra of a Blister.

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