Modeling and analysis of the states of objects and subjects of monitoring by means of secure sensor networks and IoT networks with a supercomputer

The paper proposes an information technology for evidence-based monitoring of the states of remote and mobile objects and subjects. The proposed method for the effective implementation of long-term monitoring of a large number of objects based on modeling information states of objects by means of aperture or zone control of changes in selected indicators and calculated signal characteristics. Taking into account the minimization of computations with performance-limited processor facilities of the object systems of secure wireless monitoring networks at the places of introduction of monitoring signals, it is proposed to form logical and statistical information models of the behavior of objects that correspond to the current functional and operating states of objects of long-term monitoring. To identify the most informative signals and characteristics of the states of objects, it is proposed to calculate and analyze the relative and normalized indicators and characteristics of signals. Information technology is focused on long-term monitoring of objects and subjects in various spheres of human activity.

Combinatorial configurations, fractals, fractal dimension of combinatorial sets

Combinatorial configurations and their sets are considered. The definitions of these objects are given, recurrent combinatorial operators are introduced, with the help of which they are formed, and rules are formulated according to which their sets are ordered. The property of periodicity, which takes place in the generation of combinatorial configurations, is described. It follows from the recurrent way of their formation and ordering. The fractal structure of combinatorial sets is formed due to the described rules, in which the property of periodicity is used. Analysis of these structures shows that they are self-similar, both finite and infinite, which is characteristic of fractals. Their fractal dimension is introduced, which follows from the rules of generating combinatorial configurations and corresponds to the number of these objects in their set.

Differential-difference method with approximation of the inverse operator

The problem of finding an approximate solution of a nonlinear equation with operator decomposition is considered. For equations of this type, a nonlinear operator can be represented as the sum of two operators – differentiable and nondifferentiable. For numerical solving such an equation, a differential-difference method, which contains the sum of the derivative of the differentiable part and the divided difference of the nondifferentiable part of the nonlinear operator, is proposed. Also, the proposed iterative process does not require finding the inverse operator. Instead of inverting the operator, its one-step approximation is used. The analysis of the local convergence of the method under the Lipschitz condition for the first-order divided differences and the bounded second derivative is carried out and the order of convergence is established.

Maximum Penalty Function in Linear Programming

A linear program can be equivalently reformulated as an unconstrained nonsmooth minimization problem, whose objective is the sum of the original objective and a penalty function with a sufficiently large penalty parameter. The article presents two methods for choosing this parameter. The first one applies to linear programs with usual linear inequality constraints. Then, we use a corresponding theorem by N.Z. Shor on the equivalence of a convex program to an unconstrained nonsmooth minimization problem. The second method is for linear programs of a special type. This means that all inequalities are of the form that a linear expression on the left-hand side is less or equal to a positive constant on the right-hand side. For this special type, we use a corresponding theorem of B.N. Pshenichny on establishing a penalty parameter for convex programs. For differently sized linear programs of the special type, we demonstrate that suitable penalty parameters can be computed by a procedure in GNU Octave based on GLPK software.

Some aspects of extrapolation based on interpolation polynomials

The problem of extrapolation on the basis of interpolation polynomials is considered in the paper. A simple computational procedure is proposed to find the predicted value for a polynomial of any degree under conditions of a uniform grid. An algorithm for determining the best polynomial for extrapolation is proposed. To construction of integral transformation for operator of equation of convective diffusion under mixed boundary conditions.

Adaptive algorithms for researching problems in a variable computer environment

The paper considers tools for studying computer models of problems in modeling physical and technical processes. Adaptive algorithms for studying structural and mathematical properties and solving problems in a variable computer environment are presented. The proposed innovative functionality is integrated into the intelligent computer mathematics system.

Multi-bit subtraction operation in a parallel computational model

The paper proposes a new method for implementing the parallel multidigit subtraction. The paper provides an analysis on the basis of which it is possible to predict carry signs between words and between groups of words into which multidigit numbers are split on the substracting. The analysis is presented in the form of a lemma. The paper presents an iterative bitwise operation for correcting carry signs for each word in a group of words. An algorithm for implementation the substraction operation using k processors is proposed.

Optimization model in the flexible manufacture

Flexible manufacturing shop is examined. Multilevel control system of the shop is built. The demands to the system construction are listed. The existing input information is insufficient for building of the inventory model, for choice and realization of the optimal model of the manufacture. By this reason the situation of the uncertainties is arise. For the purpose to find needed data and to receive necessary knowledge the following intelligence procedures are suggested: ”Pendulum” – for stock foundation and for construction of the inventory model, “Symmetry in the arithmetical progression” – to define the values of the coefficients with unknown quantities in the aim function of the optimal task. With the help of the procedure “Pendulum” the equal corteges of the quantum of time are building. Cortege of the quantum of time is the unit of the stock and the direction for selection of the kind of the optimization model. The optimization model of location is suggested to use. Building of the indicated models (they are small dimension and equal dimension) give the opportunity to organize the parallel calculations.

Theorem about the sufficient condition for the equality of the estimates of the OLS and Aitken of the leading coefficient of quadratic regression

At the paper in the case of heteroscedastic independent deviations a quadratic regression model is studied. A theorem is formulated that gives a sufficient condition on the variance of deviations for the coincidence of Aitken's estimate of the leading regression coefficient with his estimation of the OLS in the case of an odd number of observation points and a bisymmetric covariance matrix. On the basis of this theorem, in some cases examples of non-unit covariance matrices are constructed for which the indicated estimations of the leading coefficient of the quadratic regression coindcide.

Regularization methods for ill-posed problems of general physics

On the example of a specific physical problem of noise reduction associated with losses, dark counts, and background radiation, a summary of methods for regularizing ill-posed problems is given in the statistics of photocounts of quantum light. The mathematical formulation of the problem is presented by an operator equation of the first kind. The operator is generated by a matrix with countable elements. In the sense of Hadamard, the problem of reconstructing the number of photons of quantum light is due to the compactness of the operator of the mathematical model. A rigorous definition of a regularizing operator (regularizer) is given. The problem of stable approximation to the exact solution of the operator equation with inaccurately given initial data can be overcome by one of the most well-known regularization methods, the theoretical foundations of which were laid in the works of A.N. Tikhonov. The selection of an important class of regularizing algorithms is based on the construction of a parametric family of functions that are Borel measurable on the semiaxis and satisfy some additional conditions. The set of regularizers in this family includes most of the known regularization methods. The main ones are given in the work.