Atomic functions: Generalization to the multivariable case and promising applications

Main trends in the theory of atomic functions are outlined. These functions have been studied for more than 35 years and gave rise to new research areas in mathematical analysis, approximation theory, numerical methods, etc. Atomic functions are infinitely differentiable solutions with compact support to special functional-differential equations. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 155–177, November–December 2007.     

Relative rate of change of a function

A quantity is examined that characterizes the relative deviation of a derivative of a function from a specified reference value. This quantity is shown to be interpreted as a derivative of a function in some field constructed using a new operation. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 160–165, July–August, 1999.     

Nonlinear Recursive Regression Transformers: Dynamic Systems and Optimization

An approximation-prediction problem for functions represented by empirical data is analyzed. A class of functions as predictors, the so-called RFT-transformers, based on the least-square and superposition methods is proposed. Special classes of dynamic systems with delay are introduced and analyzed to obtain classical results for calculation of gradients. These results are used to optimize RFT-transformers. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 58–68, May–June 2005.     

Robust dissipativity of discrete systems and its investigation with the help of a sequence of sets of Lyapunov functions

A generalized concept of robust dissipativity is introduced and theorems on the analysis of this property are formulated and proved with the help of the Lyapunov functions method. It is proposed to use a specially constructed sequence of sets of Lyapunov functions that can allow one to improve the initial estimate of the boundary set of a dissipative system up to the establishment of the asymptotic robust stability property in the limiting case. An example of investigating the dissipativity of a linear discrete system with unknown parameters under additive perturbation is given. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 13–23, March–April 2008.     

Boolean problem of sequential minimization of moduli of linear functions and stability theorems

A multicriterion Boolean problem of lexicographic optimization with partial criteria represented by moduli (absolute values) of linear functions is considered. Five types of stability of a set of lexicographic optima under “small” variations in the parameters of a vector criterion are investigated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 178–187, September–October 2007.     

Associativity in combination of belief functions; a derivation of minC combination

 The nature of a contradiction (conflict) between two belief functions is investigated. Alternative ways of distributing the contradiction among nonempty subsets of frame of discernment are studied. The paper employes a new approach to understanding contradictions and introduces an original notion of potential contradiction. A method of an associative combination of generalized belief functions – minC combination and its derivation – is presented as part of the new approach. A proportionalization of generalized results is suggested as well. RID="*" ID="*" Support by Grant No. 1030803 of the GA AV ČR is acknowledged. I am grateful to Philippe Smets for a fruitful discussion on the topic.     

A new approach to the decomposition of boolean functions. 4. non-disjoint decomposition: the method of <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>-partitions

A set-theoretical approach to the non-disjoint decomposition of different forms of representation of Boolean functions of n variables is considered. This approach is based on the method of p,q-partitions of conjuncterms and on the concept of a decomposition clone. Two techniques of searching for some non-disjoint functional decomposition are described. Theorems on the non-disjoint decomposition of full and partial functions and their systems are formulated. The proposed approach is illustrated by examples. Parts I, II, and III of this article are published in No. 5 (2001), No. 1 (2002), and No. 2 (2007). Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 15-41, May–June 2009.     

Atomic radial basis functions in numerical algorithms for solving boundary-value problems for the Laplace equation

A numerical method for solution of boundary-value problems of mathematical physics is described that is based on the use of radial atomic basis functions. Atomic functions are compactly supported solutions of functional-differential equations of special form. The convergence of this numerical method is investigated for the case of using an atomic function in solving the Dirichlet boundary-value problem for the Laplace equation. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 165–178, July–August 2008.     

Accelerated simulation of failure probability of a system consisting of elements with considerably differing reliability

The failure probability of a system in a given time interval is investigated. Failure and repair distribution functions are assumed to be of a general form. A new accelerated simulation method is proposed that makes it possible to construct estimates with bounded relative root-mean-square errors for systems consisting of components with frequency-different reliability. A numerical example is given. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 48–58, November–December, 1999.     

A new approach to the decomposition of Boolean functions by the method of <Emphasis Type="Italic">q</Emphasis>-partitions. III. Joint decomposition of a function system

The paper considers the set-theoretical approach to the joint decomposition of systems of Boolean functions of variables specified in different representation forms. The approach is based on the method of q-partitions of conjuncterms and concept of decomposition clones. Theorems on joint decomposition of a system of full and partial functions are formulated. The approach is illustrated by examples. Parts I and II of this article are published in No. 5 (2001) and No. 1 (2002). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 39–58, March–April 2007.     

An asymptotic estimate of the error of eigenvalues in the method of finite elements for the sturm-liouville problem

A scheme of the second order of accuracy for the Sturm—Liouville problem is constructed by the method of finite elements with the use of a special basic system of compact functions. The convergence of the method of finite elements is proved. An exact formula for estimating errors of eigenvalues is obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 28–36, March–Apni, 2000.     

Some stochastic Leontief-Ford models and their deterministic equivalents

Some stochastic Leontief-Ford models of ecological-economic interaction and their deterministic equivalents are proposed. A theory of stochastic variants of ecological-economic structural-type functions is developed. Appropriate algorithms are developed. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 3–10, May–June 2007.     

Optimal quadrature and cubature formulas for computing Fourier transform of finite functions of one class. Case of strong oscillation

The paper deals with quadrature and cubature formulas, optimal and nearly optimal in accuracy, used to evaluate integrals of quickly oscillating functions in a class of bounded functions with piecewise continuous first derivatives bounded by a constant in one-and multidimensional cases. Optimal estimates for the numerical integration error and upper bounds are obtained for these integrals in case of strong oscillation of subintegral functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 144–164, September–October 2007.     

Description of some classes of equiprobable Boolean functions with no zero Fourier coefficients

Theory of groups and methods of combinatorial analysis are used to obtain some classes of equiprobable Boolean functions with no zero Fourier coefficients. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 95–111, November–December 2006.     

Statement and solution of some problems on a mathematical safe

The problem on a mathematical safe is formulated and studied in terms of graph theory. The cases of simplest digraphs such as paths, contours, and doubly connected components are analyzed. A number of statements on the existence of solutions to these problems are proved. The results obtained are extended to the case of corresponding nondirected graphs. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 3–14, May–June 2006.     

Reducing models of general economic equilibrium to optimization problems

Some particular cases of the model of general economic equilibrium are considered. These models are reduced to problems of convex programming under the assumptions that the utility functions are concave and positive homogeneous and the production functions are concave. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 75–86, September–October, 1999.     

Orthogonal Transforms in Bases of Slant Step Functions. I. Constructing Complete Sets of Orthogonal Slant Step Functions

The author introduces a new class of step functions and defines them in terms of the Rademacher functions. Two complete systems of orthogonal slant step functions are constructed and their characteristics are analyzed. The relationship between the basis functions of these systems and the system of the Walsh functions is established and their orthonormality is proved. It is shown that the proposed systems of slant step functions can be efficiently used to code video signal. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 118–132, May–June 2005.     

Field interactions and distinctive features of development of devices for diagnostics and correction of states of biologically active points

Based on a model of field interactions of thermochemical potentials, the possibility of correlation between the conductance and other parameters of biologically active points (BAPs) is investigated. New methods are described for diagnostics and correction of physiological systems by mild actions on BAPs. These methods can be taken into account in developing new devices for reflexotherapy. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 157–165, March–April 2006.     

Foundations of programmology

An explication of the intuitive concept of programming is given in the form of explicative programming, which is a modern form of composition programming. It is based on a pragmatically finite system of principles. On this basis, an explication of the concept of composition is obtained. A hierarchical composition structure of programmology is constructed. Translated from Kibemetika i Sistemnyi Analiz, No. 1, pp. 35–57, January–February, 2000.     

Invariance theorems for a class of systems of random nonlinear equations over an arbitrary finite ring with left unity

A class of homogenous systems of random nonlinear equations over an arbitrary finite ring with left unity is considered. The author analyzes the invariance boundaries for limit factorial moments of nonzero solutions, the limit distribution of the number of nonzero solutions, and the geometrical structure of the set of nonzero solutions of the system as the number of unknowns tends to infinity. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 106–115, November–December 2008.