Home – Home – Jornals – Russian Geology and Geophysics 2023 number 6

Petrographic and geochemical studies of igneous and volcaniclastic rocks of the middle part of the Pronchishchev Ridge and the Terpei lowland provided new data on their composition and structure. The subdivision of the Permian-Lower Triassic strata enclosing igneous and volcanicalstic bodies has been clarified by lithological/stratigraphic methods. The Lower Triassic Ulakhan-Yuryakh, Chekanovsky, Ystanakh, and Pastakh formations have been traced for the first time in the ridge area. It is shown that the studied dolerite body and adjacent breccias in the field of middle-upper Permian rocks are part of the volcanic pipe. The tuffite sand gritstones in the field of Lower Triassic rocks are parts of an eroded tephra ring around this diatreme. The confinedness of the tuffite bodies to the Ystanakh Formation makes it possible to determine the age of the pipe as the beginning of the late Olenekian. The obtained ⁴⁰Ar/³⁹Ar dates for the dolerites of the volcanic pipe and one of the dikes of the Pronchishchev dike belt indicate the repeated tectonomagmatic activity of the studied area in the Triassic.

We document the evolution of A-type granitoid magmatism during the Cryogenian tectonic transformation of the Yenisei Ridge from a postcollisional mode to the early stage of development of an active continental margin. We illustrate the A-type granitoid magmatism evolution in a model for the emplacement and cooling of the intrusions of the Strelka pluton, reflecting the final stage of magmatism during the formation of the postcollisional Glushikha complex (719-702 Ma). These processes took place at the same time as the formation of mantle, mantle-crustal and crustal rocks of the Tatarka complex (711-683 Ma), including the Yagodka pluton A-type granites (711-705 Ma) during the early stage of active continental margin development. During this period of tectonic transition, both convergent events involved the continuous formation of felsic intrusions corresponding to oxidized A-type granites.

The paper presents data on the composition and age of mafic rocks of the shoshonitic series in the Irkut block of the Sharyzhalgai uplift (southwest of the Siberian craton). According to the U-Pb dating of magmatic zircon, monzodiorites of the Poludennyi massif and gabbro-dolerites in the endo- and exocontact zones of the Toisuk pluton formed 1.87 and 1.86-1.85 Ga, respectively. The intrusion of mafic magmas and their underplating into the basement of the crust under postcollisional extension resulted in the near-coeval mafic and granitoid magmatism in the Irkut block between 1.87 and 1.84 Ga. The Paleoproterozoic mafic associations belong to the shoshonitic series and are characterized by enrichment in incompatible elements, including Zr, and low negative ε_Nd(T) values. These geochemical and isotopic characteristics point to the magma derivation from a long-lived enriched mantle source, such as the subcontinental lithospheric mantle. The crystallization of zircon from the last portions of the evolved mafic melt is evidenced by low zirconium saturation temperatures (710-965 °C) and zircon enrichment in U and Th with increasing Th/U, reflecting the increase in the concentrations of highly incompatible elements in the residual melt.

The Precambrian-Cambrian transition is one of the most fundamental evolutionary turnovers in the Earth’s history. However, in many cases precise identification and correlation of this crucial boundary in lithologically contrasting sections is complicated, even if the whole spectrum of biostratigraphic and chemostratigraphic methods is applied. The Precambrian-Cambrian transitional strata of the Igarka Uplift (Sukharikha Formation) perfectly illustrate this problem. This unit has arguably one of the most detailed carbonate carbon isotope curves for this time interval (at least within the Siberian Platform). However, an extremely poor paleontological record previously reported from the Sukharikha Formation makes identification of major stratigraphic boundaries highly debatable. We present a detailed study of the Sukharikha and overlying Krasnyi Porog formations in three sections (Sukharikha River, Kulyumbe River, and Khantaisko-Sukhotungusskaya-1 well). Our chemostratigraphic and biostratigraphic data provide a correlation basis for these sections and identify the Cambrian boundary in terms of both the International Chronostratigraphic Chart and General Stratigraphic Scale of Russia. We show that lithologic boundaries and levels of the local first appearance of Tommotian small skeletal fossils are diachronous within the Igarka paleobasin. The latter is putatively caused by paleoecology of the early Cambrian biota and by taphonomic factors. Our data specify the location of the Tommotian Stage (sensu lato)/Cambrian Stage 2 boundary in other key Precambrian-Cambrian transitional sections of the Siberian Platform.

The paper presents the results of the study of organic geochemistry of carbonate rocks, silicites, and calcareous mudstones of the Vendian (Ediacaran) Khatyspyt Formation in the northeast of the Siberian Platform. The distribution of dispersed organic matter (OM) is shown to be uneven. The formation comprises carbonate-dominated intervals depleted in OM, with total organic carbon content (TOC) < 0.1%, or weakly enriched in OM (TOC up to 0.4%), interstratified with carbonate-shale, carbonate-shale-siliceous, carbonate-siliceous and siliceous intervals enriched in OM (TOC up to 1-4%). There is also a thin-layered calcareous mudstones anomalously enriched in OM (black shales with TOC at 10%) cropping out in the Khorbusuonka River valley (unit thickness 10 cm). The redox conditions in the water column varied during deposition of the Khatyspyt Formation. Certain intervals characterize a stratified water column and euxinic bottom water conditions. During the study, the assessment was conducted of the OM catagenesis and the generative potential of the Khatyspyt Formation. Based on the results of geochemical study and the synthesis of information on the geological structure of sedimentary succession in the northeastern Siberian Platform, the Khatyspyt Formation has been evaluated as one of the hydrocarbon sources for the territory.

We present results of measurements of the physical properties (thermal conductivity, porosity, permeability, and density) of 65 air-dry sedimentary-rock samples from the cores of six deep wells drilled in the Lena-Anabar interfluve. The rocks are compact low-porosity, almost impermeable siltstones, sandstones, and dolomites mainly of Paleozoic and Precambrian ages. Correlations of thermal conductivity with porosity and bulk density have been established. The available information about the thermal conductivity of rocks as well as the production thermograms recorded after drilling made it possible to estimate the geothermal gradient and heat flow ( q ) for the Ust’-Olenekskaya-2370, Charchykskaya-1, Khastakhskaya-930, and D’yappal’skaya-1 wells. The gradient was calculated from the temperature values at the lower boundary of the permafrost (0 ºC) and at the bottom-hole. The determined heat flow varies from 37 to 70 mW/m². These q estimates are consistent with the available data on the distribution of heat flow in the north of the Siberian Platform. The proposed method for heat flow estimation is worthy of use in other northern regions of Siberia for obtaining more geothermal data.

The work considered an integrated approach to the study of deep-lying Riphean strata of the Buolkalakh-Lena region, which have certain prospects in terms of oil and gas potential. The method of revision and synthesis of acoustic and density logs is described in detail using the example of wells drilled in 80-90 for the purpose of correct reference to seismic sections. It has been shown that the Khaipakh, Debengdin, Arymas, Kyutingdinsky, and Sygynakhtakh formations of the Early and Middle Riphean eras overlook the pre-Permian surface in the middle part of the Lena-Anabar trough. The region also developed older Riphean deposits that are not exposed on the Olenek uplift and are not studied by deep drilling. The prospects for the oil and gas content of Riphean strata in the region are mainly associated with carbonate reservoirs formed in the top of the Riphean sequence and cropping out on the pre-Permian erosive surface, the most promising rocks are those constituting large carbonate platforms that were partially eroded and subjected to processes of hypergenesis and karst formation.

To study the effect of tilted uniaxial electrical anisotropy parameters on electric and electromagnetic logging signals, their numerical simulation and comparative analysis in homogeneous and layered models have been performed. An algorithm for three-dimensional numerical simulation of resistivity logging signals in spatially heterogeneous models with tilted principal axes of the electrical conductivity tensor, based on the finite-element method, was used in the calculations. The obtained results indicate the potential for a more accurate estimation of electrophysical properties of fractured carbonate reservoirs of the pre-Jurassic basement of West Siberia, which are characterized by the tilt of the electrical conductivity tensor principal axes.

Interpretation of magnetotelluric data within the North Caucasus marginal massif, the uplift of the Main Range, and the Stavropol Arch, performed on the basis of one- and two-dimensional inversions of the impedance tensor components and three-dimensional mathematical modeling, makes it possible to compile the starting and test models that are necessary for a three-dimensional inversion. It is shown by the three-dimensional inversion of all the impedance tensor components, carried out on a test three-dimensional model, that it is possible to estimate the parameters of conducting model blocks. The three-dimensional inversion of all the experimental impedance matrix components, performed with account for the results obtained on the test three-dimensional model, significantly corrects the parameters of the conducting blocks identified at the stages of one- and two-dimensional inversions, as well as the three-dimensional mathematical modeling of magnetotelluric fields in the central part of the North Caucasus. In the resulting three-dimensional geoelectrical model of the region, the position of low-resistance blocks correlates with the location of suture zones, deep faults, volcanic chambers, and domains with converted earthquake waves, whose velocities are reduced and whose absorption is increased. This can be explained by the dependence of the conductivity of crustal blocks on water saturation. Earthquake hypocenters are grouped near low-resistance anomalies.

The perturbation theory is applied to calculation of the vibrational energy levels of isotopically substituted molecules. Within the Born-Oppenheimer approximation, the isotopic shifts of energy levels are represented in the form of power series of small parameters, i.e., the relative change in the masses of substituted atoms. The coefficients of the series are functions of the molecular constants of only one modification of the isotopic family of a molecule. This makes it possible, having determined these coefficients either from calculation or on the basis of a semi-empirical approach from experimental data, to calculate isotopic shifts for any isotopologues, including short-lived ones. As an example, the isotopic shifts of the levels of unstable water isotopologues Н₂^ХО, Х = 13-15 and 19-24, with a half-life of more than 1 ms are calculated. The levels calculated are compared with the results of variational calculations with a high-precision function of the intramolecular potential energy.

Absorption spectrum of water vapor highly enriched with ¹⁷O (more than 80%) was recorded using a Bruker IFS 125M Fourier spectrometer in the 7900-9500 cm^-1 range at room temperature. The spectrum was recorded at a pressure of about 24 mbar with a spectral resolution of 0.02 cm^-1. About 6300 lines with a minimal intensity value of 5.0 × 10^-27 cm/molecule are found in the recorded spectrum. 4835 recorded lines are assigned to 5185 transitions of five water isotopologues (H₂¹⁶O, H₂¹⁷O, H₂¹⁸O, HD¹⁶O, and HD¹⁷O). H₂¹⁷O lines are assigned to 14 vibration bands. Most of them are the lines of the ν₂ + 2ν₃, 3ν₂ + ν₃, ν₁ + ν₂ + ν₃, ν₁ + 3ν₂, and 2ν₁ + ν₂ bands. The lines of the ν₁ + 2ν₂ + ν₃ - ν₂, 2ν₁ + ν₃ - ν₂, and 2ν₁ + 2ν₂ - ν₂ hot bands are assigned for the first time. The assigned lines allow us to determine 153 new vibration-rotation energies of nine vibrational states of the H₂¹⁷O molecule and 22 energies of two states of the HD¹⁷O molecule. The data obtained are compared with the results of previous studies, the HITRAN2020 spectroscopic database, and W2020 list.

Using numerical finite-difference time-domain (FDTD) calculations, we simulate and examine the absorption dynamics of the near-infrared optical radiation in a spherical microcapsule surrounded by solid nanoparticles of different optical properties (metal, biocompatible dielectric). A model microcapsule resembles a microcontainer used in modern bio- and medical technologies for targeted delivery of therapeutic nanodoses of drugs to the desired region of the biological tissues. We show that due to light scattering on nanoparticles, the optical field superlocalization in the "hot regions" on the microcapsule surface take place. The three-fold light absorption enhancement can be achieved due to the addition of buffer nanoparticles.

Ground-based MW spectrometers play an important role in the O₃ regional monitoring. They measure the downward thermal radiation in various ozone absorption lines. Using numerical simulation, the information content of typical ground-based measurements of downward MW thermal radiation in the ozone absorption line at 110.836 GHz, the errors in the ozone vertical profiles at altitudes of 20-60 km, and the influence on them of a priori information about the solution and type of the inverse operator used in the regression method for solution of the inverse problem are studied.

The observations of stratospheric aerosol layers at the lidar stations of Roshydromet in Sankt-Peterburg (59.9° N), Obninsk (55.1° N), and Znamensk (48.6° N) some months after the explosive eruption of the submarine Hunga Tonga (21° S, 175° W) volcano in January 2022 are presented. Aerosol layers at 20-24 km altitudes are observed in April, May, and October with a backscattering ratio of 1.2-1.4. The degree of depolarization at 532 nm and the ratio of the backscattering coefficients at 355 and 532 nm wavelengths show that aerosol consists of submicron spherical particles. The observation results are compared with known data on Hunga Tonga aerosol propagation.

The results of measurements of atmospheric parameters (level of turbulence and average wind speed) along an extended path are discussed. These measurements were carried out to determine the effective average values of atmospheric turbulence parameters along the path of optical radiation propagation. The measurements were carried out in August 2022 using an AMK-03 acoustic weather station. It is shown that the AMK-03 weather station, with a certified measurement technique, provides reliable information on the level of turbulence and on the wind speed. The measurement data of the turbulence level at different points along the atmospheric path are correctly recalculated into the values of the coherence length (the Fried parameter) for an optical wave of an arbitrary length and different divergence by the formulas of the wave propagation theory. This makes it possible to compare local acoustic and optical path measurements of the turbulence level.

We quantified the seasonal/regional patterns and changes in cyclone activity in the atmosphere of the Northern Hemisphere and determined the contribution of extratropical cyclones to the formation of corresponding patterns and precipitation changes. It is established that the contribution of extratropical cyclones to the total amount of precipitation exceeds 60% on the whole; for regions with a high cyclone occurrence, it attains 75% in winter and 65% in summer. The strongest contribution is related to intense cyclones: 60% in winter and 35% in summer.

The combined effect of wind and lake topographic characteristics on the temperature distribution and autumnal thermal bar evolution on an example of Lake Baikal was studied by numerical modeling. It was determined that under meteorological conditions in November 1-30, 2015, the thermal bar behavior at an initial stage of its development on the Boldakov River - Maloye More Strait cross-section was more sensitive to the wind action than that on the Srednyaya arm (Selenga River Delta) - the Buguldeika River cross-section. However, faster water cooling and thermal front propagation take place in the Selenga shallow water basin.

An impact of uncertainties of the atmospheric gases absorption line parameters in the modern spectroscopic databases on the longwave fluxes simulation in the atmosphere is estimated. The mass calculations of downward and upward IR fluxes are carried out for meteorological conditions observed in summer months in the Lower Volga Region and winter months in Novosibirsk and for average zonal meteorological models. The radiative fluxes and cooling rates at different levels of the atmosphere calculated with use of new versions of HITRAN and GEISA spectroscopic databases and its previous versions are compared. It is shown that the difference in absorption line parameters in the spectroscopic databases leads to an error less than 0.7 W/m² (0.3%) in the simulated integral fluxes in the 0-3000 cm^-1 region, at that the relative differences in the spectral fluxes calculated with moderate spectral resolution (20 cm^-1) are up to 10%. The atmospheric gases and spectral intervals contributing more to the errors in the IR fluxes simulations due to uncertainties of initial spectroscopic information are revealed.

The seasonal and daily variations in the radiation extinction coefficient due to midges (RECM) in Western Siberia are calculated for June, July, August, and September 2018. The average RECM values in the visible and infrared spectral regions were ~ 0.06 km^-1 in June, ~ 0.05 km^-1 in July, ~ 0.04 km^-1 in August, and ~ 0.02 km^-1 in September. There are two maxima in the daily variations in June, July, August, and September. The first maximum falls at 9 a.m. The second maximum is more pronounced and is observed at 17 p.m. The RECM is minimal in nighttime and at noon. In addition, a different daily behavior is characteristic for certain days of September, with a maximum in daytime and a minimum in nighttime.

We present a mathematical model of the process of laser-induced fluorescence of phosphorus oxide (PO) molecules. Based on the model, the dependences of the fluorescence intensity of PO molecules on the energy and time parameters of the exciting laser radiation are derived. It has been established that the dependence of the PO fluorescence signal on the energy density of the exciting radiation has the form of a saturation curve, and the dependence on the pulse duration under real atmospheric conditions has a local maximum. It is shown that the optimal pulse duration decreases with the energy density of the exciting radiation.

A mobile ozone lidar for sounding at wavelengths of 299 and 341 nm was designed and put into operation. The lidar is capable of covering the altitude range from 0.1 to 12 km with a spatial resolution of lidar signals from 1.5 to 150 m. The specification of the mobile lidar is given; the results of a field experiment on laser sounding of the atmosphere in Tomsk are presented. Echo signals received and the ozone profiles retrieved confirm the information content of lidar measurements in the troposphere. The results of lidar and satellite (MetOp) measurements are compared.

We compare the main characteristics of the Arctic polar vortex obtained from the NASA GSFC data (zonal mean wind at 60° N, mean temperature in the region 60-90° N) and by the vortex delineation method using geopotential (mean wind speed along the vortex edge, mean temperature inside the vortex) on the example of three largest Arctic ozone depletion events and on average over 1979-2021. The mean wind speed along the vortex edge according to the delineation method is on average two times higher than the zonal mean wind at 60° N and is 37.3 ± 5.6 and 58.9 ± 13.1 m/s in January at the 50 and 10 hPa levels, respectively. The mean temperature inside the vortex according to the delineation method is on average 5 °C lower than the mean temperature in the region 60-90° N in the lower stratosphere. The quantitative characteristics obtained expand the understanding of the Arctic polar vortex dynamics in the lower stratosphere.

This paper presents a three-dimensional direct numerical simulation of the chaotic dynamics of the free surface of a dielectric liquid placed in an external tangential electric field. The physical model includes the effects of energy pumping (external force), energy dissipation (viscosity), and surface tension. As the external electric field strength increases, a transition from the turbulence of dispersive capillary waves (at zero field) to anisotropic electrohydrodynamic wave turbulence is observed. In the strong field limit, where the fluid motion becomes highly anisotropic, a cascade of small-scale capillary waves is formed that propagates perpendicular to the external field direction. In this regime of motion, a new turbulence spectrum occurs, which differs from the classical spectrum of capillary turbulence.

The possibility of changing the morphology of the silicon surface at certain parameters of the glow discharge is shown. It has been established that oxidation is the main process influencing the surface morphology during glow discharge plasma treatment. As a result of processing in the investigated range of parameters, various stages of the surface oxidation process are observed: the formation of a uniform oxide layer, the formation of nano- and microstructures of silicon oxide. It is shown that these processes lead to surface modification, which acquires stable hydrophilic and superhydrophilic properties.

Predicted numerical distributions of the vibrational temperature of molecular oxygen behind a reflected shock wave are compared with experimental measurements in a shock tube. The computations are performed with the use of five two-temperature models (those developed by Park and Kuznetsov, β-model, and models of Marrone and Treanor and by Macheret and Fridman), five dissociation constants, and three variants of the source term that describes the rate of change of the vibrational energy due to chemical reactions. The Landau-Teller model is used to calculate the rate of translational-vibrational energy transfer, and the time of vibrational relaxation is calculated by the Millikan-White formula with allowance for Park's high-temperature correction. The numerical and experimental results are found to be in reasonable agreement. The greatest difference between the numerical and experimental data is observed in the region of relaxation of the shock wave incident onto the wall.

To study the effect of laser surface remelting (LSR) on the organization and properties of a 304 stainless steel surface layer, the microscopic morphology, hardness, roughness, adhesion, and corrosion resistance of the remelted layer (RL) are examined by changing the laser scanning speed (LSS). The experimental results show that the LSR technique hardens the 304 stainless steel substrate surface with a substrate hardness of 185 HV, and the maximum hardness after remelting is 248.9 HV. With an increase in the LSS, the surface roughness gradually decreases, while the bonding force first increases and then decreases, with the maximum bonding force being 26.1 N. At the LSS of 20 mm/s, the phase distribution in the RL is more uniform. The maximum self-corrosion potential of the RL reaches -0.718V, and the self-corrosion current density is 3.872 A/cm². The surface properties of 304 stainless steel are improved by using LSR.

This paper presents the results of numerical analysis of the filtration equations for highly miscible liquids obtained by two-scale homogenization of the Navier-Stokes and Cahn-Hilliard equations for two-dimensional flows. It is shown that the permeability tensor is generally anisotropic. For one-dimensional flows, the miscibility dynamics is investigated, and it is shown that the displacement of one phase by injection of another phase can occur even in the absence of a pressure gradient in the sample.

Taking into account the degradation of the properties of materials, an analysis of the nonlinear processes of deformation and fracture in a preloaded three-dimensional body with a sharp concentrator in the zone of a dissimilar joint under impact was performed. A generalized mathematical model of nonlinear interrelated deformation and destruction of damaged polycrystalline media subjected to time-varying thermomechanical influences is presented. The strong nonlinearity of the model is due to large (finite) strains and the strain-rate dependent behavior of media with a variable microstructure. Taking into account the anisotropic hardening of media and the Bauschinger effect, the corresponding nonlinear boundary value problems are formulated and their solutions are obtained using efficient numerical methods. The viscosity of the medium and second-order gradients from the internal variables of the system were used as regulators of the correctness of the problem statement. Experimental data were used to test the model. Simulation results presented.

The strength of a rectangular plate with two opening-mode (mode I) edge cracks was studied using the Neuber-Novozhilov approach and a modified Leonov-Panasyuk-Dugdale model with nonzero prefracture zone. A coupled discrete-integral fracture criterion is used since the stress field in the vicinity of the crack tip has a singularity. At the tip of a real crack, the fracture criterion for ultimate strain is satisfied, and at the tip of the model crack, the criterion for normal stresses is satisfied. The constitutive equations for the analytical model are analyzed. Simple formulas for fracture load in quasi-brittle and quasi-ductile fracture are derived. Plate fracture curves are constructed for a plane stressed state.

Equations are obtained that describe the process of propagation of elastic waves in a collapsing solid. It is shown that the solution of the equation for the velocity potential of acoustic emission longitudinal waves can be represented as a superposition of the fields of an ensemble of radiating cavities - monopoles. The parameters of the ensemble of monopoles are selected in such a way that the structural characteristics of a medium with cavities differ slightly from those of a solid body.

The microstructure, surface quality, and thermal shielding properties of the coatings prepared by the methods of magnetron sputtering and electroplating are studied. The results show that the Cr-Ag coating prepared by means of magnetron sputtering is smoother, its porosity and emissivity are lower, and the hardness is higher than that of the coating prepared by means of electroplating.

An algorithm for solving an inverse problem of shaping of a ribbed panel made of the AK4-1 alloy (analog of the AA2018 and Al-Cu₂-Mg₂-Ni₁ alloys) at a temperature of 200 °С is proposed. The problem is reduced to solving auxiliary direct problems on the stress-strain states of the elements of the panel structure. The structural elements are taken to be T-beams subjected to pure bending. The resultant moment for obtaining a desired shape is calculated by the Nelder-Mead optimization method. Two methods of obtaining target-shaped articles are considered: deformation under the plastic flow condition and under the creep condition. The material models used in the study take into account the difference in the mechanical properties under tension and compression, as well as the presence of accumulated damage in the material. The calculations are performed with allowance for elastic spring unloading of the article after load removal.

In this work, we study a new implicit method to compute the solutions of ill-posed linear operator equations of the first kind under the setting of compact operators. The regularization theory can be used to demonstrate the stability and convergence of this scheme. Furthermore, we obtain convergence results and effective stopping criteria according to Morozov's discrepancy principle. Numerical performances are calculated to show the validity of our implicit method and demonstrate its applicability to deblurring problems.

The system of linear acoustic equations is hyperbolic. It describes the process of the acoustic wave propagation in deformable media. An important property of the schemes used for the numerical solution is their high approximation order. This property allows one to simulate the perturbation propagation process over sufficiently large distances. Another important property is monotonicity of the schemes used, which prevents the appearance of non-physical solution oscillations. In this paper, we present linear quasi-monotone and hybrid grid-characteristic schemes for a linear transport equation and a one-dimensional acoustic system. They are constructed by a method of analysis in the space of unknown coefficients proposed by A.S. Kholodov and a grid-characteristic monotonicity criterion. Wide spatial stencils with five to seven nodes of the computational grid are considered. Reflection of a longitudinal wave with a sharp front from the interface between media with different parameters is used to compare the numerical solutions.

In this study, we introduce multi-step collocation methods (MSCM) for solving the Volterra integral equation (VIE) of the auto-convolution type such that without increasing the computational cost, the order of convergence of the proposed one-step collocation methods will be increased. A convergence analysis of the MSCM is investigated using the Peano theorems for interpolation and, finally, two numerical examples are introduced to clarify the significant advantage of the MSCM.

The demands of accuracy in measurements and engineering models today render the condition number of problems larger. While a corresponding increase in the precision of floating point numbers ensured a stable computing, the uncertainty in convergence when using residue as a stopping criterion has increased. We present an analysis of the uncertainty in convergence when using relative residue as a stopping criterion for iterative solution of linear systems, and the resulting over/under computation for a given tolerance in error. This shows that error estimation is significant for an efficient or accurate solution even when the condition number of the matrix is not large. An Ο(1) error estimator for iterations of the CG algorithm was proposed more than two decades ago. Recently, an Ο(κ²) error estimator was described for the GMRES algorithm which allows for non-symmetric linear systems as well, where κ is the iteration number. We suggest a minor modification in this GMRES error estimation for increased stability. In this work, we also propose an Ο(n) error estimator for A-norm and l₂-norm of the error vector in Bi-CG algorithm. The robust performance of these estimates as a stopping criterion results in increased savings and accuracy in computation, as condition number and size of problems increase.

An equilibrium problem of a two-dimensional body with a thin defect whose properties are characterized by a fracture parameter is considered. The problem is discretized, and an approximation accuracy theorem is proved. To solve the problem, a dual method based on a modified Lagrange functional is used. In computational experiments, when solving the direct problem, a generalized Newton's method is used with a step satisfying Armijo's condition.

The relation between complex matrices H and A, given by the equality H A = ĀH is called the pseudo-commutation. The set S_H of all A that pseudo-commute with a nonsingular n × n matrix H is called the pseudo-commutation class defined by H . Every class S_H is a subspace of the space M_n(C) interpreted as a real vector space of dimension 2n². Under the assumption dim_R S_H = n², we find a necessary and sufficient condition for the possibility to decomplexify all the matrices in S_H by one and the same similarity transformation.

In the article, formulas for exact calculation of the approximation error of multiple Itô stochastic integrals based on their orthogonal expansion are obtained. As an example, stochastic Itô integrals with multiplicities 2-4 are considered, which are used in the numerical methods for solving stochastic differential equations with orders of strong convergence 1-2.

The paper discusses the question of why intervals, which are the main object of Interval Analysis, have exactly the form that we know well and habitually use, and not some other. In particular, we investigate why traditional intervals are closed, i.e. contain their endpoints, and also what is wrong with an empty interval. A second question considered in the work is how expedient it is to expand the set of traditional intervals by some other objects. We show that improper («reversed») intervals and the arithmetic of such intervals (the Kaucher complete interval arithmetic) are very useful from many different points of view.