The article considers the problem of estimating solutions and inverse matrices of systems of linear equations with a circulant matrix in the p -norm, 1 ⩽ p < ∞ . An estimate for a diagonally dominant circulant matrix is obtained. Based on this result and the idea of decomposing a matrix into a product of matrices associated with the decomposition of a characteristic polynomial, an estimate for the general circulant matrix is proposed.

A modification of the quadratic interpolation method for finding the root of a continuous function is proposed. Two quadratic interpolation polynomials are simultaneously constructed. It is shown that if the third derivative of the original function does not change sign on the considered interval of localization of the required root, then the root lies between the roots of the quadratic functions. This allows one to substantially narrow the localization interval and reduce the number of steps to calculate the root with a given accuracy. The proposed modification of the quadratic interpolation method is used in the problem of calculating isolines when modeling the hill diagram of hydraulic turbines.

In the presented chapter, the preparation of an earthquake on the example of the Kronotsky event that occurred on 1997-12-05 with a magnitude Mc = 7.7 is considered from the standpoint of nonequilibrium thermodynamics, in which the evolution of systems is due to self-organization processes. With this approach, the lithosphere is an open nonlinear system in which, due to internal dissipation and the coordinated interaction of its elements, a self-organization process can occur, leading the system to a critical state. In this case, the scales of the connection between different parts of the nonlinear structure change, that is, the scales of temporal and spatial correlation change. However, the methods of seismological monitoring of the stress-strain geoenvironment can be expanded if, for its study, the method of calculating the probability distribution of earthquakes for various random events is used, based on the axiomatic approach of Kolmogorov A.N., applied to the catalog of Kamchatka earthquakes. This makes it possible to follow the dynamics of correlated spatial and temporal changes in the probability distribution of random variables for weak earthquakes preceding a strong event using probabilistic methods.

Anomalous changes in the parameters characterizing the state of the ionospheric layers E and F, observed before the onset of seismic events, can be considered as possible ionospheric precursors of these earthquakes. In this work, according to the data obtained at the ionospheric station of vertical radio sounding, located in Paratunka ( 52.97^∘ N, 158.24^∘ E), an analysis of ionospheric disturbances recorded before the onset of earthquakes with magnitude M≥ 5.0 that occurred in the Kamchatka region was carried out. Anomalous values of the limiting reflection frequency foEs, the screening frequency fbEs and the virtual height h'Es of the sporadic Es layer, as well as the critical frequency foF2 of the F2 layer and the virtual height h'F of the F layer in the absence of geomagnetic disturbances were considered as possible ionospheric precursors of earthquakes. The predictive efficiency of the anomalous behavior of the complex of considered ionospheric parameters for earthquakes with M≥ 5.0 is estimated.

Существует прямая зависимость между солнечной активностью и процессами в магнитосфере, ионосфере и литосфере Земли. Однако существует и обратная связь, определяющая влияние литосферных процессов сейсмоактивных регионов на физические процессы, протекающие в верхних геосферных оболочках. В сейсмоактивных регионах любые аномальные изменения в поведении параметров ионосферы, которые формируются на фоне регулярного суточного изменения характеристик ионосферы, обусловленного влиянием Солнца, могут дать информацию о процессах подготовки землетрясений. При этом каждый сейсмоактивный регион характеризуется своими, наиболее информативными особенностями (аномалиями) в поведении параметров ионосферы, которые могут быть отождествлены с предвестниками землетрясений. В работе рассмотрены ионосферные возмущения, которые регистрировались в ходе вертикального радиозондирования ионосферы за несколько суток до наступления землетрясений с магнитудой M ≥ 6.0 в Камчатском регионе. Для этих ионосферных возмущений был проведен статистический анализ их прогностической эффективно- сти для землетрясений с магнитудой M ≥ 6.0 за период 2013–2021 гг. Рассмотрены варианты комплексов ионосферных параметров, имеющих наилучшую прогностическую эффективность, которые могут быть использованы для создания методики краткосрочного прогноза сильных землетрясений. Проведен ретроспективный анализ достоверности, надежности и эффективности прогноза землетрясений с магнитудой M ≥ 6.0 на основе рассматриваемых комплексов ионосферных параметров. There is a direct relationship between solar activity and processes in the magnetosphere, ionosphere and lithosphere of the Earth. However, there is also a feedback that determines the influence of lithospheric processes in seismically active regions on the physical processes occurring in the upper geospheric shells. In seismically active regions, any anomalous changes in the behavior of the ionospheric parameters that form against the background of a regular daily change in the characteristics of the ionosphere due to the influence of the Sun can provide information about the processes of earthquake preparation. At the same time, each seismically active region is characterized by its own most informative features (anomalies) in the behavior of ionospheric parameters, which can be identified with earthquake precursors. The paper considers ionospheric disturbances that were recorded in the course of vertical radio sounding of the ionosphere several days before the onset of earthquakes with magnitude M\geq≥6.0 in the Kamchatka region. For these ionospheric disturbances, a statistical analysis of their predictive effectiveness for earthquakes with magnitude M\geq≥6.0 was carried out for the period 2013–2021. Variants of complexes of ionospheric parameters with the best predictive efficiency, which can be used to create a technique for short-term prediction of strong earthquakes, are considered. A retrospective analysis of efficiency of the forecast of earthquakes with magnitude M 6:0 based on the considered complexes of ionospheric parameters was carried out.

We consider the inverse kinematic problem of seismics (IKPS) with internal sources.It consists in determining the velocities of the longitudinal and transverse waves by the traveltimes from earthquake sources in the focal zone to a group of seismic stations. We propose analgorithm for the numerical solution of the problem which bases on the eikonal equation and thetechnology of smoothing multidimensional splines, which give an approximation of the velocitystructure of the focal zone. The paper presents some theoretical results that substantiate thealgorithm for solving the problem by approximation methods on using smoothing withmultidimensional splines from data on irregular grids. We describe the results of the numericalsolution of the problem, the calculations with real data on earthquakes in the focal zone, and givethe estimates of the velocity and elastic parameters of a medium.

Рассмотрена новая микросхема приемопередатчика для интерфейса RS 485 компании АО «ПКК Миландр» – К5559ИН86SI. Приведена информация о характеристиках и особенностях данной микросхемы.

Существующие в настоящий момент методы определения периодов повышения сейсмической активности на Камчатке [7] на основе анализа только ионосферных прогностических признаков не дают ответа на вопросы ни о силе события, ни о месте его возникновения. Возникает необходимость объединения методов мониторинга сейсмического режима на основе анализа каталога землетрясений (среднесрочный прогноз) и геофизического мониторинга на основе изучения ионосферных параметров (краткосрочный прогноз) с целью разработки методов предупреждения о грозящей опасности, определяя возможную область и период ожидания землетрясения. В данной работе произведены прогностических оценок вероятности, возможной области ожидания и временного периода наступления сильных землетрясений с энергетическим классом KS≥13.5 (M≥6.0) в Камчатском регионе за 01.01.2019‒01.06.2021 гг. на основе совместного анализа аномальных значений сейсмического прогностического признака, полученного на основе вероятностной модели сейсмического режима, и комплекса ионосферных предвестников.

Согласно данным последнего долгосрочного сейсмического прогноза [9] наиболее вероятным местом следующего сильнейшего (M ≥ 7.7) землетрясения для всей Курило-Камчатской дуги является протяженная сейсмическая брешь в районе Авачинского залива и южной части Камчатки. Неглубокое землетрясение с M ≥ 8.0 в районе Авачинского залива способно вызвать в г. Петропавловске-Камчатском и расположенных поблизости городов Елизово и Вилючинск сотрясения до девяти баллов и привести к гибели нескольких десятков тысяч человек. По этой причине достаточно надежный вероятностный краткосрочный прогноз такого землетрясения рассматривается авторами как наиболее актуальная и ответственная задача для настоящего момента.

We consider the so-called simplest formula for local approximation by polynomial splines of order n (Schoenberg splines). The spline itself and all derivatives except that of the highest order, approximate a given function and its corresponding derivatives with the second order. We show that the jump of the highest derivative of order n - 1; i. e., the value of discontinuity, divided by the meshsize, approximates the nth derivative of the original function. We found an asymptotic expansion of the jump.

The experience of short-term forecasting of Kamchatka earthquakes based on complex well measurements at the Petropavlovsk-Kamchatsky geody-namic polygon (PGP) shows that, as a rule, the preparation of strong Kamchatka earthquakes is fairly reliable in the medium-term time scale (months or years be-fore the earthquake). However the determination of the stage beginning imme-diately preceding an earthquake (weeks or days before the main event) is a very difficult task. At present time, the solution of this problem is largely associated with the involvement in the preparation of forecast conclusions of data from continuous monitoring of the ionosphere, carried out by ground-based means of vertical radiosonding and measurements of total electronic content (TEC) using the global navigation satellite system GLONASS and GPS. This is due to the fact that significant changes in a number of ionospheric parameters occur mainly 1-5 days before the Kamchatka earthquakes. The results of the compar-ison of the data of daily monitoring of the ionosphere, including information on TEC, with the data integrated downhole measurements showed a rather high correlation of occurrence of anomalies in the ionosphere before strong earth-quakes with changes in the complex parameters in borehole measurements. As one example, the report presents the results of ionospheric and borehole monitoring obtained in the time neighborhood of the strong (MW= 7.5) the earth-quake that occurred on March 25, 2020 in the area of the Northern Kurils. The results show a high correlation between changes in the specific electrical resis-tivity of the Geospace in the area of the PGP with variations in the TEC and the formation of a number of other anomalies in the ionosphere a few days be-fore the earthquake. These results indicate that it is possible to determine fairly reliably the beginning of the final stage of preparation for a strong earthquake. Currently, methods based on atmospheric parameters monitoring are used quite successfully for predictive estimates of the epicenter and magnitude of an earth-quake: the method of chemical potential corrections for measurements at an altitude of ∼ 100 m, as well as data from measurements of outgoing long-wave infrared radiation (OLR) at the level of the upper edge of clouds (heights of 10 -15 km).

In this paper, an attempt is made to combine seismological and ionospheric predictive signs formed on the eve of earthquakes in order to develop a methodology for assessing the region and the time period of waiting for strong Kamchatka earthquakes with energy class KS ≥ 13:5 (M ≥ 6:0). The seismological parameter determined on the basis of the probabilistic model of the seismicity of Kamchatka region and allowing to estimate the region and the probability of occurrence of a strong earthquake in it is proposed as a mediumterm precursor. The complex of ionospheric disturbances is considered as a short-term precursor with a waiting period of up to 5 days for earthquakes. The estimation of their prognostic efficiency for the considered precursors is carried out by methods A. A. Gusev and G. M. Molchan, and their joint analysis was performed on the eve of earthquakes with KS ≥ 13:5 (M ≥ 6:0), that occurred in the period 2009–2018.

The dynamics of invariant magnetic surfaces is considered with examples of axially symmetric magnetic systems. The inner trap (magnetic mirror) and the outer trap (dipole) are distinguished in a system formed by two coils with current. The dynamics of noninteracting particles moving in the plane of the magnetic equator is considered in the drift approximation for the perturbation vector satisfying the third invariant conservation condition. The results are extended to the equator of the geomagnetic trap, in which the external perturbing field is given by the ring current. It is shown that the real effect of the ring current on plasma dynamics in the bounded magnetosphere of the Earth in the approximation of constancy of the third invariant differs from conventional model calculations based on dipole representations of the geomagnetic field. In this case, it is not the magnetic flux that makes practical sense but only its change, which is equal to the difference between the final and initial fluxes calculated by the parameters of the real field. The minimum values of the energies of charged particles in the radiation belts required to satisfy the third condition of adiabatic invariant conservation are estimated for the main phase of a particular magnetic storm.

We consider the problem on shape-preserving interpolation by classical cubic splines. Namely, we consider conditions guaranteeing that, for a positive function (or a function whose kth derivative is positive), the cubic spline (respectively, its kth derivative) is positive. We present a survey of known results, completely describe the cases in which boundary conditions are formulated in terms of the first derivative, and obtain similar results for the second derivative. We discuss in detail mathematical methods for obtaining sufficient conditions for shape-preserving interpolation. We also develop such methods, which allows us to obtain general conditions for a spline and its derivative to be positive. We prove that, for a strictly positive function (or a function whose derivative is positive), it is possible to find an interpolant of the same sign as the initial function (respectively, its derivative) by thickening the mesh.

A method is presented for short-term prediction of strong earthquakes, in which the precursors are considered the excess of current values of f oF2 critical frequency of the ionospheric F2 layer over the median values in periods of perturbed state of the magnetosphere, the appearance of ionospheric disturbances: K-layer, Es-spread F-spread, the stratification of the F2 layer, Es is the r type. As predicted earthquakes were considered earthquakes with magnitudes M ≥ 5:0. Assessment of the effectiveness of the forecast was carried out in the spring and autumn periods for 2013–2017 according to the methods of A. A. Gusev and G. M. Molchan. It is shown that the method under consideration has the best prognostic efficiency for seismic events with M ≥ 6:5 magnitude.

The paper presents the results of investigation of cyclone possible effect on ionosphere parameters. Monitoring of ionosphere state has been carried out by automatic sounding applying low orbital navigational spacecrafts in the conditions of low seismicity. Receiving stations were locating in meridian direction of Kamchatka peninsula in Paratunka, Mil'kovo and Esso villages. It has been shown that during nighttime cyclone passage (according to local time) on 18.01., 03.02. and 17.02.2013 over Kamchatka, synchronous increase of electron concentration occurred in the region over its epicenter in comparison to the next days without cyclones. With the increase of the latitude, i.e. further from the center of the cyclone, electron concentration decreased in comparison to calm days.

A method of the operational forecast of strong earthquakes is considered. Exceedance of the current values of ionospheric critical frequency f 0 F 2 over the median values during the disturbed state of the magnetosphere is considered to be a precursor. The reliability, validity and the efficiency of the precursor were evaluated by A.A. Gusev's and G.M. Molchan's methods for the period of 2009-2016. Earthquakes with the magnitude of M ≥ 5.0 occurring within the zone of precursor appearance determined by Dobrovol'skiy's radius were considered as forecasting earthquakes. It was shown that the precursor has the best predictive efficiency for the seismic events with the magnitude of M ≥ 6.5.