Regularities Of Distribution Research Articles

Session V:

In this study, work on modelling uncertainty in the tensile properties of two-dimensional plain woven silicon carbide fibre-reinforced silicon carbide matrix (2D SiC/SiC) composite and B-basis values of tensile strength was carried out in the following four aspects. First, the uncertainty law and formation mechanism of the tensile properties of 2D SiC/SiC composites were researched through microstructural analysis. Subsequently, three distribution function models were established for the tensile properties of 2D SiC/SiC composites with Weibull, normal, and log-normal distribution functions. The development of three functional models was based on 30 groups of valid tensile data, linear regression methods, and Kolmogorov hypothesis testing methods. Based on the application of the model validation method, the accuracy of the three functional models was quantitatively evaluated by employing the area index of model validation. B-basis values for the tensile strength of 2D SiC/SiC composites were calculated and finally discussed for three distribution functions. The results indicate that the distribution regularity of tensile properties of 2D SiC/SiC composites could be characterized by using the three function models, and the Weibull distribution function was the most accurate. Therefore, it was preferable to calculate the B-basis values by using the Weibull distribution, with a B-basis value of 266 MPa.

Droplet Collection Efficiency Regularity of NACA0012 Airfoil Based on the Eulerian Method

Obtaining droplet collection efficiency (DCE) is vital for icing numerical simulations. A droplet impact model based on the Eulerian two-phase flow model was established in this work to analyze the influence of different incoming flow parameters on DCE. Then, the mathematical regularity of the DCE distribution was explored using the parametric description, and the functional relation of the regularity between the incoming condition and the DCE distribution was accomplished. The results showed that the calculated DCE of NACA0012 and the two-dimensional cylinder matched well with the test data. The incoming flow velocity and droplet size, rather than the liquid water content (LWC) of the incoming flow, affected the collection efficiency significantly. The DCE distribution under different conditions was normalized to the same mathematical form with the peak collection efficiency and maximum position. The mathematical form had a highly normal distribution, which was expressed with good regularity of the Reynolds number and droplet diameter. Regularity can greatly reduce the amount of calculation and improve the calculation accuracy, which plays a positive role in the numerical prediction of the icing problem.

Forms of presence, interrelation and genetic significance of Al and Li impurities in quartz of gold deposits of the Darasunsky ore field (Eastern Transbaikalia, Russia)

Research subject. The distribution regularities of Al and Li impurities in gold-ore quartz. Materials and methods. The quartz of the Darasun, Teremkinskoye and Talatuy gold deposits of the Darasun ore field was studied. The gross contents of Al and Li impurities in quartz were determined by the LA-ICP-MS method; substitutional Al impurity concentrations were studied by the EPR method. The forms of Al impurity in quartz were determined based on the results of studying its behavior during material recrystallization. The genetic significance of Al and Li impurities in quartz was estimated taking into account the genetic information obtained during the study of the distribution of substitutional Al and Ti impurity concentrations. Results. It was found that Al is present in quartz in two main forms, i. e., as a substitutional Al impurity and Al complexes localized in the areas of high mineral defectiveness. Li+ ions are located in the structural channels of the mineral, serving as compensating ions for both Al impurity forms. The composition of Al complexes is assumed to include three Al3+ ions and one H+ or Li+ ion. Two stages of quartz recrystallization occurring at different temperatures of mineral formation were identified. The first, low-temperature stage leads to quartz enrichment with substitutional Al impurities. The second, high-temperature stage causes the decomposition of Al complexes. The recrystallization stages can be identified by the type of relationship between the gross concentrations of Al and Li. The increased content of Al impurity in ore quartz was found to be related to the presence of a large number of Al complexes. An assumption is made that these complexes formed during mineral crystallization from solutions with a high content of metal ions. Conclusions. The results obtained indicate that high Al impurity concentrations can serve as a genetic sign of ore quartz. At the same time, the decomposition of Al complexes during quartz recrystallization should be taken into account. A method for estimating the initial concentration of Al complexes is proposed, which is a more reliable genetic indicator.

Количественные характеристики гидробионтов и молоди рыб шельфовой зоны Северного Сахалина (обзор)

The article presents a review of published materials and results of specialized ecological and fishery research for the period 1950-2021, containing data on abundance and biomass of the main groups of hydrobionts (phytoplankton, zooplankton, zoobenthos), larvae and young fish in different shelf areas of northern and northeastern Sakhalin Island. Regularities of distribution of different plankton groups by shelf areas and by different water, horizons are considered. Quantitative indicators of systematic groups of hydrobionts and juvenile fish of different shelf areas are compared. The work summarizes the materials on the Zapadno-Shmidtovsky and Deryuginsky license areas, Yuzhno-Kirinsky hydrocarbon field and others.

Hydrological bases of water resources management of the rivers of Northern Kazakhstan

It is well known that directional global and regional climatic changes have been taking place in recent decades, in the considered area, where the underground feeding of rivers is insignificant, the river flow has decreased. In this regard, the river flow rate and total water resources of the river basins of Northern Kazakhstan are estimated in several ways: for a multi-year period, for the modern period, as well as for the period preceding it, a significant part of which is characterized by conditionally natural (not significantly transformed) river flow. Flow characteristics necessary for practice have been obtained: distribution parameters and probability values. Territorial peculiarities and regularities of annual flow distribution have been analyzed. The regional dependence of the annual flow on the determining physical and geographical factors has been derived. Appropriate recommendations for calculating the flow of unstudied rivers are given. The obtained results can be used in the development of specific principles of integrated water resources management of the considered territory, and the long-term forecasting of water resources under changing climate and changes in anthropogenic pressure on the water bodies of Northern Kazakhstan.

Temperature Distribution Regularity and Dynamic Evolution of Spontaneous Combustion Coal Gangue Dump: Case Study of Yinying Coal Mine in Shanxi, China

The serious environmental pollution caused by the spontaneous combustion of coal gangue has become a problem which cannot be ignored in the world mining industry. It is urgently necessary to clarify the law of the temperature distribution of spontaneous combustion in gangue dumps and to grasp its future dynamic evolution of spontaneous combustion. In this study, the internal temperature of the second platform of the Yinying coal mine gangue dump was monitored on the basis of the self-developed wireless temperature monitoring system. Its temperature distribution was analysed, and areas of low (<80 °C), medium (80~280 °C), and high temperature (>280 °C) were delimited. The finite element method was also used to simulate its internal temperature development of 1–5 years. The results show that: (1) The high temperature area is mainly distributed on the side close to the slope. In the area 3 m deep, the high temperature started to propagate quickly. At a depth of 4 m, medium and high temperature represented 90% of the platform’s total surface area. At 6 m deep, temperature peaked at 667 °C. (2) The conduction of the internal temperature of the spontaneous combustion of gangue discharge is a non-linear conduction, and the conduction of heat in the horizontal direction is lower than the vertical direction. (3) During the spontaneous combustion of gangue discharge over the next five years, the overall temperature increase is faster at first, then it decreases and eventually stabilises. The high-temperature zones extend 10 m from the slope to the interior in five years, and the high-temperature zones are oval-shaped. This study provides a theoretical reference for the prevention and control of spontaneous combustion of coal gangue dumps.

Regularity of the depth distribution of earthquakes on the sharur-ordubad fault

Regularity of the depth distribution of earthquakes on the sharur-ordubad fault

A study of electrothermomechanical converter for technological purposes with nonlinear changes in the loading and cooling medium

Heavy temperature processes with high-temperature loads require optimisation of technological processes, ensuring high reliability and combining rotating parts of electric machines with actuators to achieve greater efficiency of electromechanical converters. The research aims to provide a theoretical justification and experimental confirmation of the effect of higher harmonics in the air gap under nonlinear changes in the temperature of the medium. The research is based on the basic principles of electrodynamics, heat and mass transfer, mathematical modelling by the finite element method, and experimental verification of multi-physical parameters. Based on the analysis of the differential equation for determining the increase in the surface temperature of a ferromagnetic rotor under conditions of nonlinear temperature changes in the environment surrounding the electromechanical converter, the regularities of the formation of the free and forced components of the instantaneous temperature values of the massive rotor are established. Depending on the mode of interaction between the load-cooling medium and the electromechanical part of the screw units, kinematic diagrams of single-mass and two-mass systems with variable or constant moments of inertia and stiffness were formed. According to the size of the electromagnetic system of the experimental sample, a mathematical model for studying thermal and electromagnetic processes was built. The regularities of the spatial distribution of the temperature of the screw electromechanical unit are determined. The spectra of higher harmonics of voltage and current in the frequency range from 0 to 50 kHz have been experimentally determined, which confirms the presence of the effect of generating higher harmonics when the temperature of the medium surrounding the rotor screw changes. The detected harmonic spectrum affects both the formation of the dynamics of the rotating system and additional thermal power while increasing the overall efficiency of the screw electromechanical converter. The practical value of the obtained results lies in the possibility of predicting the optimal indicators of interrelated electromagnetic and heat exchange processes in screw electromechanical converters for technological purposes

Evaluation of the mineral profile of wild plant raw materials and products of its processing for the creation of functional food products

Vegetable raw materials and their processed products are a source of necessary biologically active nutrients and are widely used in food production when creating functional and specialized nutrition products. Studies of the mineral profile of wild cranberry berries (Vaccinium oxycoccos) growing in the Republic of Karelia and berry processing products (juice and cake) with different methods of processing raw materials have been carried out. Studies of the mineral composition of fresh cranberry berries and their processed products obtained by various methods showed a high content of essential macro- and microelements in fresh berries, juices and cakes and allowed us to establish the regularity of migration and distribution of micronutrients depending on the method of pre-processing of raw materials. The data obtained during the experiment allow us to judge the high mineral profile of wild cranberry berries and its processed products and the content of certain elements (in particular phosphorus and manganese) in a physiologically significant amount, which explains the prospect of using products of processing wild berry raw materials to enrich food with micronutrients. Studies of the mineral composition of edible packaging films based on concentrated cranberry juice have shown a higher mineral profile of all samples of films with the addition of juice obtained with pretreatment of berry raw materials with steam, regardless of the type of structure-forming matrix, which is due to a higher content of concentrated juice solids. It has been established that the pre-processing of berry raw materials with steam before extracting the juice, as well as its subsequent concentration, allows processing raw materials with maximum preservation of essential components with a high degree of biological activity.)

The Distribution Regularity and Flotation Study of Niobium-Bearing Minerals in Baiyun Obo

The characteristics of Baiyun Obo niobium-bearing minerals are complex physicochemical properties that make the beneficiation of niobium minerals extremely difficult. In this paper, X-ray diffraction, X-ray fluorescence and mineral liberation analyzer (MLA) systems were used to study the niobium occurrence state and distribution of niobium-bearing minerals in the samples from Baiyun Obo. The results show that the chemical and mineral compositions of the sample are complex, with a Nb2O5 grading of 0.24%. There are many kinds of niobium minerals, including ilmenorutile, nioboaeschynite-Nd, baotite, latrappite, euxenite-Y, fergusonite and columbite-Mn, and the highest mass fraction of 0.55% is achieved with Nb in nioboaeschynite-Nd, followed by the mass fraction of ilmenorutile (0.33%). All of the niobium-containing minerals demonstrate a low degree of dissociation. Flotation experiments explored the optimal flotation conditions for HOBA (1-hydroxyoctyl-1,1-bisphosphonic acid) as a flotation collector for Baiyun Obo niobium minerals, which is able to increase the grade of Nb2O5 in the concentrate to 1.31%. The optimal use conditions of the reagent are pH 3.5–4.5, and the amount of the collector is 1000 g/t. By further optimizing the beneficiation process and reagent system, ilmenorutile and nioboaeschynite-Nd were significantly enriched in the concentrate, which suggested that HOBA can efficiently increase the grade of Nb2O5 in the concentrate.

Analysis of Conflict Distribution Characteristics between Motor Vehicles and E-Bikes at Roundabouts

Roundabouts are a common traffic infrastructure, which are supposed to facilitate safe and smooth traffic flow. Electric bikes as a common traffic tool in a lot of cities in China play an important role in relieving traffic congestion due to the rapid increase of motor vehicles on roads. However, compared with cyclists, e-bikers are more vulnerable because of their higher speed when colliding with motor vehicles. In this research, the spatial-temporal distribution characteristics of conflicts between motor vehicles and electric bikes (e-bikes) at signalized roundabouts are explored. First, the time distance method was used to identify conflicts, and the time to collision (TTC) was selected as the discrimination index we proposed for two representative conflict types. On conflict heat maps and conflict distribution during one signal cycle, we found a series of spatial-temporal conflict distribution regularities. Spatially, the proposed two representative types of conflicts were mainly distributed at exit areas and near the outermost circulatory lanes. Conflict Type 1 was mainly distributed around the outermost motor vehicle lanes, and Type 2 was mainly distributed inside the outermost vehicle lanes and behind the second stop line. In time span, both types of conflicts showed rapid increase before reaching a peak at 30 percentile green time and then decreasing gradually after that peak. Type 1 presented a sharp increasing range during 0–10 percentile green time, while the sharp increasing range for Type 2 presented during the 10–20 percentile. The conclusions developed by this article could provide a theoretical basis for improving traffic safety at roundabouts.

Research on Interrupted Sampling Repeater Jamming Performance Based on Joint Frequency Shift/Phase Modulation.

Interrupted sampling repeater jamming (ISRJ) is a classic active coherent jamming. Due to its structural limitations, it has inherent defects such as a discontinuous time-frequency (TF) distribution, strong distribution regularity of pulse compression results, limited jamming amplitude, and strong false targets lagging behind the real target. These defects have not been fully resolved yet due to the limitation of the theoretical analysis system. Based on the analysis of the influence factors of ISRJ on the interference performance for linear-frequency-modulated (LFM) and phase-coded signals, this paper proposes an improved ISRJ method based on the joint subsection frequency shift and two-phase modulation. The coherent superposition of jamming signals at different positions for LFM signals is achieved by controlling the frequency shift matrix and the phase modulation parameters to form a strong pre-lead false target or multiple positions and ranges of blanket jamming areas. For the phase-coded signal, the pre-lead false targets are generated through code prediction and the two-phase modulation of the code sequence, resulting in similar noise interference. The simulation results show that this method can overcome the inherent defects of ISRJ.

Апостериорная оценка статистических показателей производственного травматизма и профессиональной заболеваемости работников строительной отрасли

The current level of injuries and occupational morbidity in the construction industry of the Russian Federation is assessed based on the application of a statistical research method, the essence of which is to provide comparative characteristics of a posteriori quantitative indicators. The ranking of harmful factors of the working environment and the labor process was carried out for reducing the degree of prevalence at the workplaces in construction: severity — chemical factor — biological factor — non-ionizing radiation — vibration (general and local) and tension — aerosols of predominantly fibrogenic action — ionizing radiation — microclimate — noise, air ultrasound, infrasound — light medium. The regularity in distribution of accidents in the studied time interval was revealed, which consists in reducing the frequency coefficient, and the risk of occupational injuries in the construction industry with persistently high rates of fatal injury to employees relative to the largest sectors of economic activity. It is established that the risk of injury for one employee during one year in the construction industry varies from 3.03·10–3 to 1.33·10–3, which is higher than the acceptable risk of 1·10–6. Thus, it is required to develop additional measures to reduce industrial injuries and the severity of the consequences of employee injury. Change in the frequency rate of occupational diseases in the studied time interval is uneven, the maximum was recorded in 2015. A survey was conducted among the employees of construction organizations in the Belgorod region, the problem was identified related to concealing the information about injuries (microtraumas, minor accidents). The implementation of the corrective and preventive measures presented in the work will allow to reduce the likelihood of identified hazards, the level of industrial injuries and occupational morbidity.

Anticancer Efficacy of Antibacterial Quinobenzothiazines

The antitumor potency of a series of designed and prepared antibacterial quinobenzothiazines was evaluated against different types of human cancer cell lines, such as glioblastoma SNB-19, lung adenocarcinoma A549 and breast cancer T47D, and the activities of the compounds were compared to cisplatin and doxorubicin. 9-Propoxy-5-methyl-12H-quino[3,4-b][1,4]benzo- thiazinium chloride (4a), 9-allyloxy-5-methyl-12H-quino[3,4-b][1,4]benzothiazinium chloride (4d) and 11-benzyloxy-5-methyl-12H-quino[3,4-b][1,4]benzothiazinium chloride (4l) were the most active compounds; their IC50 values against all three cell lines ranged from 5.3 to 9.3 µM. The effective derivatives showed no cytotoxic effect up to 100 µM on normal human dermal fibroblasts (NHDFs). To explore the structure–activity relationship, the effect of the type/nature and position of the substituents on the tetracyclic quinobenzothiazine system on the anticancer activity was investigated. Additionally, the receptor-dependent approach was used to specify the mutual ligand–enzyme (bio)compositions that might be potentially valid for the antitumor characteristics of new quinobenzothiazine derivatives. In particular, the molecular docking procedure was applied for the most potent agents against the human breast cancer line T47D in order to obtain comprehensive knowledge about the aromatase–inhibitor binding mode. The docking study revealed that some regularities in the spatial atomic distribution and nonbonding interactions (e.g., hydrophobic patterns) can be observed for the most active molecules. The surface of the electron-rich aromatic rings of 4d and 4l molecules could also contribute to π–π stacking interactions with protoporphyrin IX (HEM) as well as to the formation of π–cation interactions with the adjacent iron cofactor.

Screw electromechanical hydrolyzer for processing poultry by-products

The use of keratin-containing products in various industries, including agriculture, cosmetics, and biomedicine requires solving the urgent problem of improving the methods and means of hydrothermal hydrolysis of keratin waste. The purpose of this work was to study the screw electromechanical hydrolyzer to provide modes of efficient conversion of keratin waste into a useful product with added value. The research was based on the basic principles of electrodynamics, heat and mass transfer, mathematical modeling by the finite element method, and experimental verification of the multiphysical parameters of the electromechanical hydrolyzer. The experimental model of the electromechanical hydrolyzer was used to determine the optimal conditions for hydrothermal hydrolysis of keratin waste in terms of thermal conditions, raw material transportation, pressure, and magnetic field effects. As a result of the processing, bulk hydrolyzed brown feather flour with a specific odor was obtained. According to the size of the electromagnetic system of the experimental sample, a mathematical model for studying thermal and electromagnetic processes was built. The regularities of the spatial distribution of magnetic induction for the cross-section, axial distribution on the surface of the screw, and the upper faces of the screw winding of an electromechanical hydrolyzer were determined. Three-dimensional images of the heating of the screw and the screw electromechanical hydrolyzer itself were obtained. By comparing the results of mathematical modeling and empirical studies, the adequacy of the mathematical model was confirmed. The results of the research can find application in the field of animal husbandry in the production of feed additives of increased digestibility for the needs of feeding agricultural animals and poultry

Systematic Design and Study of Star-like Polymeric Prodrug Unimolecular Micelles β-CD-P[CL-co-(ACL-g-DOX)-SS-MPEG]21 by DPD Simulations.

Unimolecular micelles composed of a single polymeric molecule have recently attracted significant attention in anti-cancer drug delivery due to their high thermodynamic stability and small particle sizes. Applying the prodrug strategy to unimolecular micelles may provide superior nano-drug carriers with simultaneous high stability, low drug leakage, and well-drug loading capacity. However, the formation mechanism of the unimolecular prodrug micelles, the superiority of the prodrug strategy, as well as the prodrug controlled release mechanism were scantily understood at the mesoscopic scale. In this work, dissipative particle dynamics mesoscopic simulations were employed to investigate the self-assembly behavior, formation conditions, drug distribution regularities, and the prodrug release process of the star-like polymeric prodrug unimolecular micelles formed by β-CD-P[CL-co-(ACL-g-DOX)-SS-MPEG]21. A special bond-breaking script was used to accomplish the bond-breaking simulation of the grafted DOX bonds and the disulfide bonds. Results showed that to form well monodispersed and superior DOX-loaded unimolecular micelles, the polymer concentration should be well controlled at low volume fractions (≤10.59%), and the detailed molecular structure of the polymer was suggested as β-cyclodextrin-P[caprolactone-co-(amino caprolactone-g-doxorubicin)-disulfide-methyl polyethylene glycol]21) (β-CD-P[CL30-co-(ACL-g-DOX)8-SS-MPEG49]21). By comparison with the DOX physically loaded micelles, it was found that the prodrug unimolecular micelles with DOX grafted on the polymer displayed no drug leakage and superior drug loading capacity. Simulations on the prodrug release process showed that the prodrug unimolecular micelles assembled by β-CD-P[CL30-co-(ACL-g-DOX)8-SS-MPEG49]21 would provide good dual pH/reduction-responsive DOX release performance.

Microstructure evolution and mechanical properties in the depth direction of ultra-high power laser-arc hybrid weld joint of 316L stainless steel

Ultra-high power laser-arc hybrid welding (LAHW) provides a novel and effective method for joining thick-section with single-pass owing to the advantages of ultra-high penetration and efficiency. However, the evolution of microstructure and properties along penetration direction under ultra-high power LAHW (25 kW-level laser power) are still unclear. In this study, the distribution regularities of microstructure and properties in the depth direction of ultra-high power LAHW joint of 316L stainless steel are explored under the optimal weld formation. Results show that along the penetration direction, the cooling rate exhibits a significantly increasing trend by temperature field simulation. The macroscopic growth direction of δ dendrites changes from “Ʌ” to “→←” and “▪’’, and the microscopic dendrites changes from disordered coarse skeletal to directional fine skeletal and lathy morphology. Besides, the increase in cooling rate in the depth direction refines the grain size and reduces the texture composition types but increases the {001} 〈100〉 texture strength. Notably, “teardrop-like” boundary is formed in the lower layer and shrinkage cavities present in its central zone, the formation mechanism is investigated in detail. Finally, the tensile properties of joints in the depth direction exhibit a decreasing trend. The final fracture of the lower layer is in the weld center due to the existence of shrinkage cavities. This study points out the evolution of microstructure and mechanical properties in ultra-high power LAHW process, which provides a theoretical and experimental basis for regulating the performance of thick plates.

A probability distribution of dependencies in interlanguage

AbstractThe diversity of syntactic units in second language has attracted much scholarly attention. Most existing studies on syntactic diversity have focused on only a small number of syntactic structures, and it is difficult to find studies that consider the full range of syntactic dependencies present in the dataset. Based on a syntactic annotated interlanguage corpus that we constructed, this paper is a quantitative study of dependencies in English-speaking Chinese learners’ interlanguage across proficiency levels. We fit the frequency distributions of dependency type, word class (both as dependent and governor), verb as a governor, and noun as a dependent with a modified right-truncated Zipf-Alekseev distribution and Zipf’s law. Our findings show that: (1) from the mathematical model, interlanguage followed distributional regularities like natural languages in terms of the syntactic structure distribution; (2) most of the determination coefficients’R2were high, indicating that the investigated distributions in interlanguage fit the distributional law finding in natural languages. This also demonstrated that both interlanguages and natural languages consistently conform to the law of linguistic diversity and uniformity; (3) the dependency relation distribution parametersaandbmanifest the developmental trend of L2 learners’ proficiency levels, demonstrating that the parameters had universal applicability in reflecting interlanguage proficiency.

Research on the Spatial Perception of Stakeholders in Brownfield Redevelopment Based on Value Compatibility Analysis

Under the current urban renewal background, the subjective attitude of stakeholders directly affects the feasibility of planning projects in the development or protection activities related to brownfield redevelopment. It is key that the public effectively participates in planning and decision-making to explore the suitable expression method of public attitude. In this paper, Jigang, Jinan, Shandong Province is taken as an example. By using participatory mapping and semi-structural interviews, the landscape perceived value of 365 (342 valid) stakeholders in the original site of Jigang is investigated. By using hot spot analysis, correspondence analysis and compatibility index analysis, the spatial composition of public perceived landscape value, the correspondence between landscape value and land use and its compatibility with existing conservation and renewal schemes are revealed. On this basis, three types of brownfield land redevelopment attitude areas are identified. The results show that: 1. The attitude of Jinan Iron and Steel Group’s renewal based on the degree of compatibility is location-dependent, and the spatial difference analysis of this attitude provides more detailed data support for the protection and renewal design, planning management and conflict control; 2. The landscape perceived value of case stakeholders has regularity in spatial distribution and is related to a certain material landscape foundation (land use), which is beneficial in explaining the possible social phenomena caused by landscape change; 3. Participatory cartography combined with landscape value investigation provides an effective method for the study of perceived landscape. Through cartographic visualization, statistical analysis and index model construction, the spatial structure characteristics of perceived landscape value can be revealed. It can provide effective decision support for brownfield urban renewal projects, solve the problem that the current upper-level planning of such renewal is not matched with the actual demand, and improve the vitality of brownfield sites in the development area.

Regularization of distributions

Sufficient conditions are presented for the construction of a regularization of a distribution in the form $a(\sigma)f$, where $f$ is a distribution and $a(\sigma)$ is an infinitely differentiable function outside a closed set $N$ which has power-like singularities of derivatives on $N$. Applications of such regularizations to an effective construction of solutions of the equation $Pu=f$, where $P(\sigma)$ is a polynomial, are considered. Bibliography: 14 titles.