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The differential eigenvalue problem governing eigenvibrations of an elastic bar with fixed first end and mechanical resonator attached to second end is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We introduce limit differential eigenvalue problems and derive the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as a resonator parameter tending to infinity. The original differential eigenvalue problem is approximated by the finite element method on a uniform mesh. Error estimates for approximate eigenvalues and eigenfunctions are established.

We survey the progress in data mining methods for spacecraft health monitoring. The main emphasis is placed on the analysis of telemetry data enabling the identification of spacecraft states that are atypical during normal operation and the prediction of possible failures in the operation of the spacecraft or its components. The main stages required for the creation of general-purpose spacecraft state monitoring systems are considered; methods for detecting anomalies in telemetry data taking into account the specific features of the spacecraft are presented in detail; and publications on this topic known to the authors are analyzed. Examples of the implementation of such systems in flight control centers of various countries are given. The promising areas of development of methods for analyzing the technical state of complex systems relevant for solving problems in space technology are discussed, and the main factors that hinder the development of machine learning methods for analyzing telemetry data are noted.

The electrodynamic characteristics (EDCs) of open resonators (ORs) formed by multifocal spherical and plane mirrors are studied. The following periodic structures are used on the plane mirrors: a single-row quarter-wavelength comb in which the lamella height is hs = λ/4 and two-row periodic structures (TRPSs) with hs = λ/2 (half-wavelength) or λ/4 < hs < λ/2 (intermediate) (λ is the wavelength). The corresponding EDCs are compared. It is shown that the OR with an intermediate TRPS exhibits a previously unknown feature of the distribution of the high-frequency field of the fundamental mode along the OR symmetry axis. Application of the OR with the intermediate TRPS in the orotron at hs/λ < 0.3–0.32 makes it possible to increase the efficiency of the electron-wave interaction and, hence, the efficiency and generated power. Supporting experimental and calculated results are presented.

The paper presents the results of studying the composition and phase transitions occurring at thermal treatment of tusk of Mammuthus primigenius by the thermogravimetric and X-ray analysis methods. The elemental analysis of the tusk sample shows that, in addition to elements standard for hydroxyapatite, magnesium ions are present, that is, hydroxyapatite in the studied samples is represented in the magnesium-substituted form Ca10 – xMgx(PO4)6(OH)2. It is revealed that thermal decomposition of mammoth tusk components proceeds in a wide temperature range accompanied by transformation of the mineral part from hydroxyapatite to whitlockite. It is established that this process is caused by the presence of magnesium ions.

This paper reports the results of satellite image and seismological data analysis obtained when considering the causes of a landslide on the Yelbashinskii dump at the Kolyvan anthracite deposit. The potential for determining the natural vibrations of large objects from low-frequency seismological records and monitoring thereof has been demonstrated.