Recent publications

The permutizer PG(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_G(H)$$\end{document} of a subgroup H in a finite group G is the subgroup generated by all cyclic subgroups of G which permute with H. A subgroup H of G is subpermuteral in G, if H=G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=G$$\end{document} or there is a subgroup chain H=H0≤H1≤…≤Hn-1≤Hn=G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=H_0\,\le \, H_1\, \le \, \ldots \, \le \, H_{n-1}\, \le \, H_n=G$$\end{document} such that Hi+1=PHi+1(Hi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{i+1}=P_{H_{i+1}}(H_{i})$$\end{document} for every i∈{0,1,…,n-1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{0,1,\ldots ,n-1\}$$\end{document}, and H is strongly subpermuteral in G if H is subpermuteral in every subgroup L such that H≤L≤G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\le L\le G$$\end{document}. We investigate groups with subpermuteral and strongly subpermuteral subgroups. New characterisations of supersoluble, w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {w}$$\end{document}-supersoluble (groups in which all metanilpotent subgroups are supersoluble) and v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {v}$$\end{document}-supersoluble (groups in which all subgroups with nilpotent derived subgroups are supersoluble) groups are obtained. We describe the structure of groups with all Sylow subgroups subpermuteral and also groups with all Sylow normalizers subpermuteral.

In this paper, 𝐺 is a finite group and 𝜎 a partition of the set of all primes ℙ, that is, σ = { σ i ∣ i ∈ I } , where P = ⋃ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j . If 𝑛 is an integer, we write σ ( n ) = { σ i ∣ σ i ∩ π ( n ) ≠ ∅ } and σ ( G ) = σ ( | G | ) . A group 𝐺 is said to be 𝜎-primary if 𝐺 is a σ i -group for some i = i ( G ) and 𝜎-soluble if every chief factor of 𝐺 is 𝜎-primary. We say that 𝐺 is a 𝜎-tower group if either G = 1 or 𝐺 has a normal series 1 = G 0 < G 1 < ⋯ < G t - 1 < G t = G such that G i / G i - 1 is a σ i -group, σ i ∈ σ ( G ) , and G / G i and G i - 1 are σ i ′ -groups for all i = 1 , … , t . A subgroup 𝐴 of 𝐺 is said to be 𝜎-subnormal in 𝐺 if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A t = G such that either A i - 1 ⊴ A i or A i / ( A i - 1 ) A i is 𝜎-primary for all i = 1 , … , t . In this paper, answering to Question 4.8 in [A. N. Skiba, On 𝜎-subnormal and 𝜎-permutable subgroups of finite groups, J. Algebra 436 (2015), 1–16], we prove that a 𝜎-soluble group G ≠ 1 with | σ ( G ) | = n is a 𝜎-tower group if each of its ( n + 1 ) -maximal subgroups is 𝜎-subnormal in 𝐺.

The eruption of the Cumbre Vieja volcano on the island of La Palma (Canary Islands, Spain) began on September 19, 2021 and ended on December 13, 2021. It lasted continuously for 85 days with short periods of calm when lava did not exit the cone of the volcano. Vast amounts of volcanic material, including ash and gases, were emitted into the environment. This research focuses on these emissions. The main objective is to use available open-source data to examine the impact on regional and local air quality. Data from the following sources were used: 1) Copernicus Atmosphere Monitoring Service (CAMS) data was used to track the transfer of volcanic SO2 in the troposphere in early October over long distances from the source of the eruption, including Western and Eastern Europe, across the Atlantic Ocean and the Caribbean; 2) Data from ground monitoring stations measured the concentrations of SO2 and PM10 near the source; 3) AErosol RObotic NETwork (AERONET) data from the La Palma station that showed high Aerosol Optical Depth (AOD) values (over 0.4) during the active phase of emissions on September 24 and 28, as well as on October 3; 4) Ångström Exponent (AE) values indicated the presence of particles of different sizes. On September 24, high AE values (>1.5), showed the presence of fine-mode fraction scattering aerosols such as sulfates; 5) Cloud Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data additionally confirmed the presence of sulfate and dust aerosols in the atmosphere over the region. However, the influence of Saharan dust on the atmosphere of the entire region could not be excluded. This research helps forecast air pollution resulting from large-scale volcanic eruptions and associated health risks to humans.

The methodology for studying the theme "Recursive solving of the problem using the "divide & conquer" method" in preparing schoolchildren for Olympiads in informatics is illustrated in the article by solving the "Head on shoulders" problem. The solution of this problem includes several algorithms (subtasks) considered sequentially, namely: merge sort, counting the number of inversions in a permutation, the number of pairs of intersecting segments. The study of the method is based on the sequential solution of these subtasks. For each subtask, the following materials are given: the formulation of the subtask, the idea of a solution with a proposal to come up with an implementation on their own, the solution in the Pascal programming language. Distance learning system DL.GSU. BY is the eﬀective technical base for teaching. The system allows to oﬀer for a student a formulation of the task; to submit the solution for review; to get a verdict from the system — a correct or incorrect solution; for incorrect solution, the number of the test on which the solution did not pass is indicated. A student can take a test (input and output data), on which his solution did not pass, fgure out what the error is in his program, correct and send the solution again.

Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and let Π be a nonempty subset of the set σ. A set ℋ of subgroups of a finite group G is said to be a complete Hall Π-set of G if every member of ℋ is a Hall σi-subgroup of G for some σi ∈ Π and ℋ contains exactly one Hall σi-subgroup of G for every σi ∈ Π such that σi ∩ π(G) ≠ ∅. A subgroup A of G is called (i) ℋG-permutable if AHx = HxA for H ∈ ℋ and x ∈ G; (ii) Π-permutable in G if A is ℋG-permutable for some complete Hall Π-set ℋ of G. We study the influence of Π-permutable subgroups on the structure of G. In particular, we prove that if π=∪σi∈Πσi and G = AB, where A and B are ℋG-permutable π-separable (resp., π-closed) subgroups of G, then G is also π-separable (resp., π-closed). Some known results are generalized.

A new cyanobacterial species of the genus Desmonostoc is described according to the International Code of Nomenclature for algae, fungi, and plants. The new species Desmonostoc caucasicum sp. nov. is recorded in the mountain meadow subalpine soil from the Greater Caucasus, Russia. The analysis is based on morphological characters, the 16S rRNA gene phylogenetic analysis, and ITS secondary structure. Desmonostoc caucasicum differs from the other species of the genus in colony morphology, size of vegetative cells and heterocytes, and habitat type. The 16S rRNA gene sequence of the new strain displayed 95.3%-97.9% similarities to other species of the genus Desmonostoc. The phylogeny inferred by maximum likelihood and Bayesian inference placed D. caucasicum in the Desmonostoc clade, within the Nostocaceae. The novel strain formed an independent lineage within the clade. The D1-D1ʹ, Box-B and V3 helices obtained from the 16S-23S ITS region didn't fit those of any described species of Desmonostoc. Amplification of a fragment of the mcy gene involved in microcystin biosynthesis from D. caucasicum confirmed that it has this genetic determinant. An integrative taxonomic approach, based on morphological, 16S rRNA and mcy genes molecular analyses, ITS secondary structure, along with ecological data, is used to delimit this new species.

Fano resonance is based on plasmonic metasurfaces and has many applications in all kinds of fields. In this paper, we propose an independently switchable double-layer raster structure based on graphene. Depending on the highly adjustable nature of graphene, the Fermi energy level can be adjusted to control the Fano resonance at different wavelengths. The equivalent resonator coupling mode method is used to simulate the Fano resonance, and the transmission spectrum fits well. Functional switch at different wavelengths can be achieved using Fano resonance technology. The simulation obtained a fantastic group refractive index of the designed structure, indicating that there is a possibility to apply it in slow light. The effect of the environmental refractive index on sensing performance was studied and we found the structure has great potential in making high-sensitivity sensors. To sum up, it is hoped that this structure can make a great contribution to the manufacture of integrated optics.

Let G be a finite group and F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}$$\end{document} a class of groups. A subgroup A of G is said to be F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}$$\end{document}-normal in G if either A⊴G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\trianglelefteq G$$\end{document} or AG≠AG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{G}\ne A^{G}$$\end{document} and every chief factor H/K of G between AG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{G}$$\end{document} and AG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{G}$$\end{document} is F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}$$\end{document}-central in G, that is, (H/K)⋊(G/CG(H/K))∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H/K)\rtimes (G/C_{G}(H/K)) \in {\mathfrak {F}}$$\end{document}. We say that a subgroup H of G is F∧sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}\wedge sn$$\end{document}-embedded if for some F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}$$\end{document}-normal subgroup A and subnormal subgroup B of G we have H=A∩B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=A\cap B$$\end{document}. In this paper, we analyze the influence of F∧sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}\wedge sn$$\end{document}-embedded subgroups on the structure of a group. In particular, we prove the following generalization of Schenkman’s Theorem on the nilpotent residual of subnormal subgroups: If F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}$$\end{document} is a hereditary saturated formation containing all nilpotent groups and S is an F∧sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}\wedge sn$$\end{document}-embedded subgroup of G, such that ZF(E)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{\mathfrak {F}}(E)=1$$\end{document} for every subgroup E of G containing S, then CG(SF)≤SF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{G}(S^{{\mathfrak {F}}})\le S^{\mathfrak {F}}$$\end{document}.

We introduce and study a special class of Hausdorff operators, we call this class “Hausdorff-Zhu operators”, on Banach spaces of analytic functions in the unit disc. Conditions of boundedness, compactness, and nuclearity of these operators are given. A special attention is paid to particular but important cases of analytic symbols and radial symbols. In particular it is shown that Hausdorff-Zhu operators with analytic symbols are at most two-dimensional and their spectra are computed.

Найдены приближенные аналитические решения уравнения Логунова-Тавхелидзе в случае потенциала, который в одномерном релятивистском конфигурационном представлении имеет вид, аналогичный потенциалу нерелятивистского гармонического осциллятора в координатном представлении. Волновые функции получены в импульсном и в релятивистском конфигурационном представлениях. Приближенные значения энергии релятивистского гармонического осциллятора являются корнями трансцендентных уравнений. Волновые функции в релятивистском конфигурационном представлении имеют дополнительные нули по сравнению с волновыми функциями соответствующих состояний нерелятивистского гармонического осциллятора в координатном представлении.

Amorphous carbon (a-C) coatings have numerous practical applications as antifriction or hard coatings. However, they are incredibly fragile and not compatible with many substrates. Therefore it is necessary to manage and improve the mechanical properties of a-C coatings, which depend on the properties and composition of the intermediate sublayer deposited between the substrate and coating. In this study, we proposed to control the structural and phase composition and, therefore, properties and practical applicability of a-C coatings deposited by pulsed cathodic arc by altering the chemical composition of the titanium nitride (TiN) sublayer deposited by direct current arc evaporation. We studied the effect of highly reactive and chemically neutral alloying elements on the coating structure and properties using aluminium and copper as typical examples. The results of Raman and X-ray photoelectron spectroscopies, atomic force microscopy, and mechanical testings suggested that highly reactive alloying elements led to an increase of the hardness and elastic modulus of the bilayered coating while alloying with copper resulted in a reduced friction coefficient. We also demonstrated that the introduction of aluminium in TiN sublayers increased the strength of the bilayer coatings and the crack resistance of the upper a-C layer. As a result, we achieved a significant level of control of the mechanical properties of the a-C coatings, which could be of interest in mechanical engineering or metallurgy for targeted control of the surface properties of products through the application of these coatings with expected and controllable mechanical properties.

The possibility of using a conducting double DNA-like helix as the basis of an electromagnetic wave polarizer, which converts an incident linearly polarized wave into a reflected wave with circular polarization, has been shown. A high-frequency resonance is studied, at which the wavelength of the incident radiation is approximately equal to the length of a helical turn. The simulation of a double DNA-like helix has been carried out. The electric currents arising in the helical strands under waves with circular polarization at high-frequency resonance have been analyzed. Fundamentally different behavior of the double DNA-like helix concerning waves with right-hand or left-hand circular polarization has been established, which can be called the effect of polarization selectivity. This effect is manifested in the fact that a double DNA-like helix at high-frequency resonance can create a reflected wave having only one sign of circular polarization. The electric vector of the reflected wave produces a turn in space with the opposite winding direction compared to the double helix. These studies also highlight the electromagnetic forces of interaction between helical strands. The equilibrium of the double DNA-like helix has been studied, including as an element of metamaterials and as an object with a high potential for use in nanotechnology.

We answer the question 17.56 of the Kourovka Notebook posed by V.N. Knyagina and V.S. Monakhov about subgroups of finite groups permuting with its Schmidt subgroups. We characterise the Schmidt subgroups of a finite group with trivial Frattini subgroup that permute with its Schmidt subgroups.

Let 𝜎 be a partition of the set of all primes, and let 𝔉 denote a hereditary formation. We describe all formations 𝔉 for which the 𝔉-hypercenter and the intersection of weak 𝐾-𝔉-subnormalizers of all Sylow subgroups coincide in every finite group. In particular, the formation of all 𝜎-nilpotent groups has this property. With the help of our results, we solve a particular case of Shemetkov’s problem about the intersection of 𝔉-maximal subgroups and the 𝔉-hypercenter. As a corollary, we obtain Hall’s classical result about the hypercenter. We prove that the non-𝜎-nilpotent graph of a group is connected and its diameter is at most 3.

Fano resonance is based on plasmonic metasurfaces and has many applications in all kinds of fields. In this paper, we propose an independently switchable double-layer raster structure based on graphene. Depending on the highly adjustable nature of graphene, the Fermi energy level can be adjusted to control the Fano resonance at different wavelengths. The equivalent resonator coupling mode method is used to simulate the Fano resonance, and the transmission spectrum fits well. Functional switch at different wavelengths can be achieved using Fano resonance technology. The simulation obtained a fantastic group refractive index of the designed structure, indicating that it there is a possibility to apply it in slow light. The effect of the environmental refractive index on sensing performance was studied and we found the structure has great potential in making high-sensitivity sensors. To sum up, it is hoped that this structure can make a great contribution to the manufacture of integrated optics.

The design and modeling of a metasurface is carried out, which makes it possible to transform an incident linearly polarized electromagnetic wave into a transmitted wave with elliptical polarization close to circular. At the same time, the reflection coefficient of the wave is close to zero at the resonant frequency, since the metasurface is similar to the free space in its wave resistance. The resonant elements of the meta-surface (meta-atoms) are two-turn planar spirals with balanced dielectric and magnetic properties. Such spirals exhibit radically different properties with respect to waves with right and left circular polarization. The metasurface as a polarization converter has strong chiral properties, since it contains planar spirals of only one direction of twisting, and can be manufactured within the framework of printed circuit board technologies.

The paper presents a comparative analysis of the structure and mechanical properties of carbon coatings, highly alloyed with metals Ti, Zr, and Al. Coatings were deposited from combined flows of metal and carbon plasma with approximately the same mass content. The influence of the metal’s nature on the size of Csp2 clusters and the degree of ordering of the carbon matrix has been determined using Raman spectroscopy. The features of the morphology of coatings due to the formation of a carbide phase have been evaluated using scanning electron microscopy. It is shown that the values of the hardness, elastic modulus, and the coefficient of elastic recovery (η) of the coatings are also determined by the nature of the metal phase: a higher hardness is achieved when alloying with titanium, and the η - when alloying with aluminum.

Using a pulsed arc discharge, carbon coatings binary-doped with silicon and nitrogen are deposited on quartz and silicon substrates. The structure and phase composition of the coatings are studied by atomic force microscopy, Raman spectroscopy, and X-ray photoelectron spectroscopy. The optic band gap Eg and the refractive index of the coatings are determined depending on the alloying elements. The influence of nitrogen and silicon on the formation of the structure of the carbon matrix and the formation of chemical compounds in the coatings, leading to a change in the width optic band gap Eg, has been established. Changes in the roughness Ra and the size of the Csp2 cluster are shown in the case of binary doping with nitrogen and silicon.

The paper presents the calculation of the temperatures fields, created at different depths, via artificial neural networks and the finite element method during laser welding of steel 30XГCH2A. The training data array and the array data for testing neural networks were created using ANSYS. The paper studies the effect of neural network modeling parameters on the accuracy of determining temperatures in the zone of laser processing of structural steels.

Biotin film was prepared by low-energy electron beam deposition (LEBD). The molecular structure, chemical composition and micromorphology of the biotin film were investigated by ¹HNMR, FTIR, XPS, AFM and SEM. The results showed the molecular structure of a monolayer of biotin film is fully consistent with the molecular structure of the initial biotin powders. The contact angle test showed that the biotin film exhibit good hydrophilicity. The release kinetics of biotin film was tested by UV–Vis method. It was found that the film was almost completely released in about two weeks. The cell viability of MC3T3-E1 cells on the surface of the biotin film was attaining 100.54 ± 1.7% (P < 0.05), showing excellent biocompatibility and biosafety. Titanium implant with surface of biotin film was implanted into the femoral head of rabbits as experimental group. The animals were euthanized after four weeks. Compared with the control group, mature lamellar bone formation was observed with dense trabecular bone, and the expression of Coll-I, Runx2 and BMP-2 was better. The results showed that the repair effect of bone defect in the experimental group was excellent.

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