The classical implicit function theorem states that if X, Y , and Z are Banach spaces and F is a continuously differentiable mapping from a neighborhood of a point (x, y) ∈ X × Y into Z such that F(x, y)=0 and Fy(x, y) is an invertible operator, then there exists a continuously differentiable mapping (implicit function) ϕ from a neighborhood of V of the point x into Y for which the equality F(x, ϕ(x)) = 0 holds for all x ∈ V . It turns out that, under these assumptions, more is true: there exists an implicit function not only for the mapping F, but also for all mappings that are close enough (in a certain sense) to F. Moreover, if a mapping close to F is continuously differentiable, then the corresponding implicit function is also continuously differentiable. Such properties are important for applications in situations where the source data is not precise or where an “intricate” mapping F is approximated by a simpler one.

An analytical model of the magnetic field over the Martian surface in the southwestern part of the Elysium Planitia is constructed using satellite data and modified S-approximations within the framework of the structural-parametric approach. Analytic downward continuations of the magnetic field of Mars at various altitudes, including the planet’s surface, are presented.

We consider a method for calculating buffer tanks for the temporary storage of wastewater containing pollutants of water user enterprises and algorithms for minimizing river pollution in accidents accompanied by an uncontrolled spill of contaminants. We propose a distributed control system structure for controlling discharges of enterprises-water users into a river basin

The problems of artificial intelligence, control and decision-making with incomplete or unreliable information include a wide class of problems of abductive explanation of the observed, including cause–effect problems. The study is devoted to the substantiation of the method of logical formation of hypotheses explaining the observed. Means of knowledge representation and derivation of hypotheses are proposed. A language possessing the property of substitutability is introduced. The properties of language and the calculi introduced in it provide hypothesizing by combining deduction and abduction. In contrast to the well-known logical methods of abduction, the proposed techniques make it possible to derive hypotheses (minorants) that are necessary and sufficient for a formal explanation of the observed. Based on minorants in combination with the basic theory of the subject area, reliable causes of the observed or relevant circumstances leading to these causes are formed. In this case, in situations with the availability of empirical data, these causes and circumstances can also be formed in plausible versions. Examples from technology and medicine are considered.

Abstract— In this report, we have considered the problem of detecting and identifying the types of changes in non-stationary time series that lead to the need to rebuild the models describing the process. We compare two approaches to the construction of algorithms for detecting changes in the monitoring mode: fluctuation tests for detecting structural changes and sequential detection algorithms. The sequential algorithms have optimal properties in the sense of the criterion for the fastest detection