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Consider a class $ \mathcal{C} $ of groups containing at least one nontrivial group and closed under subgroups, extensions, and Cartesian products of the form $ \prod\nolimits_{y\in Y}X_{y} $ , where $ X,Y\in{\mathcal{C}} $ and $ X_{y} $ is an isomorphic copy of $ X $ for each $ y\in Y $ . Given an arbitrary graph of groups, we obtain necessary and sufficient conditions for its fundamental group $ G $ to be a residually $ \mathcal{C} $ -group thus generalizing Shirvani’s conditions for the residual finiteness of $ G $ .

Article simulates the accumulation of financial resources for pension provision of citizens of the Russian Federation for the period from 1992 to 2020 in the case of using in practice the system of pension provision of citizens based on personalized pension accounts instead of the existing system of pension provision of citizens of Russia. All the components of the economic effect of using personalized pension accounts are described in detail and illustrated by calculations.

The proposed social technologies for financing enterprises, based on economic and mathematical models, algorithms and software, are a permanent mechanism for increasing the income of working citizens, developing enterprises, increasing tax revenues and social payments. The use of social technologies for financing enterprises allows with a 5% increase in revenue over 50 years to increase the wages of employees by 63%, which will amount to almost 104 trillion rubles in Russia; to increase deductions to the development fund by 72%, in which, first of all, the owners of enterprises are interested, since this ensures the growth of their income and the possibility of constant modernization and renewal of technological equipment, the release of new competitive products; to reduce to zero deductions to the Federal Fund of Compulsory Medical Insurance (FCHIF) from wages above 76 rubles, which is 955% higher than the average salary in Russia as of January 20. At the same time, despite the reduction in the rate of contributions to the FCHIF, the amount of deductions to the state in the form of income tax, income tax and in the FCHIF increases.

This article analyzes the effectiveness of various beta coefficient modifications in forecasting on the Russian stock market. Objective: To test the hypothesis of the superiority of modified beta coefficients in forecast accuracy. Methods: Calculating and comparing the stock returns of ten companies that play a key role in the Russian economy based on three beta coefficient modifications: classic, Monkhouse, and two beta. The criteria were used for verifying sample homogeneity in order to test the hypothesis that modified beta coefficients produce more accurate forecasting. Results: The analysis showed that, in most cases, none of the three models obtained relevant results for the Russian companies. None of the modifications tested can be recognized with high confidence as more accurate than traditional beta coefficients. Conclusions: In the contemporary Russian stock market, the CAPM does not allow one to produce satisfactory forecasts on stock returns.

The paper presents a project for organizing the transition of Russian citizens to the financing of an old-age labor pension using personalized pension accounts. On the basis of the simulation model of the financial provision of the old-age labor pension, a simulation of the accumulation of financial resources in the accounts of citizens was performed, proving the effectiveness of the development. It is shown that due to the funds accumulated on a personalized pension account, the amount of monthly pension provision of the employee increases annually, which encourages citizens to increase labor productivity, and the state-to reduce contributions to the Pension Fund of the Russian Federation, which reduces the cost of production at all enterprises in Russia and contributes to their accelerated development.

The proposed technology of financing enterprises and the Russian economy, harmoniously combining the interests of working citizens, owners and the state, makes it possible, at quite achievable rates of gross domestic product growth (enterprise revenue) by 3.5% per year, to ensure a 46.6% increase in wages of working citizens over 5 years, which will practically end poverty. To increase contributions to the development fund for 5 years by 25%, which the owners of enterprises and the entire workforce are interested in, since this ensures the growth of their incomes and the possibility of constant modernization and updating of technological equipment and the release of new competitive products. Increase in 5 years (despite a gradual decrease to 14.51% of contributions to the Pension Fund RF) the amount of funds received by budgets of all levels by 22%, which will allow the state to solve many social problems.

The article simulates the accumulation of financial resources on medical savings accounts (MSA) of working and non-working citizens of the Russian Federation for the period from 1994 to 2020 in the case of using in practice the system of compulsory medical insurance of citizens based on MSA instead of the existing system of financing domestic health care. Described and illustrated by calculations of all the elements of economic effect from the use of MSA.

Given a root class $$\mathscr{C}$$ of groups and a tree product P in which each amalgamated subgroup lies in the centers of the corresponding vertex groups, we point out certain sufficient conditions for the $$\mathscr{C}$$-residuality of P. In particular, we show that the tree product of solvable groups of bounded derived length with central amalgamated subgroups is residually solvable.

Let $${\cal K}$$ be a root class of groups closed under taking quotient groups, G be a free product of groups A and B with amalgamated subgroups H and K. Let also H be normal in A, K be normal in B, and AutG(H) denote the set of automorphisms of H induced by all inner automorphisms of G. We prove a criterion for G to be residually a $${\cal K}$$-group provided AutG(H) is an abelian group or it satisfies some other conditions. We apply this result in the cases when A and B are bounded nilpotent groups or A/H, B/K ∈ $${\cal K}$$.

Let $$\mathcal{C}$$ be a root class of groups and $$\mathcal{\pi}_{1}(\mathcal{G})$$ be the fundamental group of a graph $$\mathcal{G}$$ of groups. We prove that if $$\mathcal{G}$$ has a finite number of edges and there exists a homomorphism of $$\mathcal{\pi}_{1}(\mathcal{G})$$ onto a group of $$\mathcal{C}$$ acting injectively on all the edge subgroups, then $$\mathcal{\pi}_{1}(\mathcal{G})$$ is residually a $$\mathcal{C}$$ -group. The main result of the paper is that the inverse statement is not true for many root classes of groups. The proof of this result is based on the criterion for the fundamental group of a graph of isomorphic groups to be residually a $$\mathcal{C}$$ -group, which is of independent interest.

We introduce the construction of a generalized direct product associated with a graph of groups and prove two sufficient conditions for its existence. These results are applied to obtain some sufficient conditions for an HNN-extension with central associated subgroups to be residually a C-group where C is a root class of groups. In particular, it is proved that an HNN-extension of a solvable group with central associated subgroups is residually solvable.

The article discloses a system of financing health care in Germany. The features and advantages of competing sickness funds (funds) and organizations of interaction between the state by associations of doctors and state non-profit and commercial hospitals are shown.

The paper shows the principles of organizing the transition of Russian citizens to medical savings accounts (MSA). A simulation of the accumulation of financial resources on the MSA of all categories of citizens of the Russian Federation was performed, taking into account the need for a significant increase in the salary of medical personnel, namely: in accordance with the “may” Decrees of 2012, V. V. Putin, to bring the salary of doctors to a level twice higher than the average for the region by 2018. It is proved that for all categories of Russian citizens, including citizens, have chronic diseases, the funds accumulated on their personal MSA, will be sufficient to cover the costs of medical care given the salaries of medical personnel established in the “may” decrees of the President of the Russian Federation.

The article proves that it is impossible to achieve a significant increase in the salary of medical personnel within the existing system of healthcare financing of the Russian Federation, namely, in accordance with the “may” Decrees of 2012 by V. V. Putin, to bring the salary of doctors to a level twice higher than the average for the region by 2018. It is proved that to achieve the necessary results in terms of increasing doctors’ salaries and motivating citizens to increase labor productivity and a healthy lifestyle, it is necessary to transfer the system of healthcare financing within the framework of obligatory medical insurance to medical savings accounts.

The article analyzes the financing and organization of medical services for the population in Singapore and Russia. The main difference is revealed, which is that in Singapore funds transferred within the framework of compulsory health insurance by enterprises for medical care of citizen’s workings for them go not to funds and insurance companies, as in Russia, but to personal medical savings accounts of citizens. Interest is charged on them, and these funds are inherited, which motivates citizens to increase labor productivity and a healthy lifestyle.

The article shows the effectiveness of 24-hour outpatient CT-centers for the treatment of COVID-19. A comparative analysis of the expenditure of mandatory health insurance funds for the provision of outpatient and inpatient medical care to the population of Moscow was carried out, and the economic effect of transferring all patients with coronavirus with a non-severe form (light and medium) from round-the-clock hospitals to home treatment was estimated. It is proved that at present, the most important thing for defeating the pandemic is the gradual, taking into account all existing restrictions (distance between citizens, hand disinfection, temperature measurement, sanitary treatment of premises and vehicles, testing for coronavirus and antibodies, etc.), the resumption of work of all enterprises, organizations and self-employed citizens.

The article defines the costs and losses of the state and working citizens of Russia to combat the new COVID- 19 coronavirus infection in the treatment of COVID-19 patients at home, inpatient treatment without the use of artificial ventilation, and inpatient treatment if artificial ventilation is necessary. A marketing research of prices for the COVID-19 test in Moscow was carried out. Gross domestic product (GDP) losses by differ industry branches from the forced downtime caused by the fight against the spread of the new COVID-19 coronavirus infection are calculated, which amounted to 5117,2 billion rubles. The daily losses of GDP in the amount of 83,9 billion rubles, the Federal budget in the amount of 15,5 billion rubles and wages of employees, equal to 39,2 billion rubles. The necessity of returning to normal life of citizens and enterprises of Russia is proved.

Suppose that $G$ is a group, $H$ and $K$ are proper isomorphic central subgroups of $G$, and $\mathfrak{G}$ is an HNN-extension of $G$ with the associated subgroups $H$ and $K$. We prove necessary and sufficient conditions for $\mathfrak{G}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for each $y \in Y$., Comment: 25 pages; the English version of the previously published Russian original

Gyrotrons for plasma fusion installations operated at frequencies 40–170 GHz with an output power up to 1 MW are presented. Some novel ideas are proposed to enhance gyrotron output energy, efficiency, frequency spectrum. In the collaboration between the Institute of Applied Physics and industrial company GYCOM Ltd the series of gyrotrons for plasma installations has been realized. The brightest examples are 170 GHz gyrotrons for ITER, long-pulse 140 GHz gyrotrons developed for EAST and KSTAR installations, 82.6 GHz for T-15MD tokamak, 28 GHz and 45 GHz complex for modern electron cyclotron resonance ion sources (ECRIS) facilities. Some steps were done in development of 230–700 GHz gyrotrons for future plasma installations and for plasma diagnostics.

A 45-GHz gyrotron-based microwave oscillator setup has been purposefully developed for powering of a superconducting electron cyclotron resonance (ECR) ion source. The setup provides a smoothly regulated output power in the 0.1–20-kW range (up to 26 kW in manual mode) with an efficiency of up to 50% and collector depression in both continuous wave (CW) and pulse modes. Pulse mode specifications: duration 5–200 ms, leading edge less than ${2}~\mu \text{s}$ , trailing edge less than ${1}~\mu \text{s}$ , repetition rate 1–10 Hz, and triggered operation using an external reference oscillator. The set of high-voltage power supplies (HVPSs) is equipped with a fast electronic protective system to ensure trouble-free operation of the gyrotron. The functional structure of the setup corresponds to the general design method for feedback-controlled microwave sources. The feedback loop consists of a power meter (or a detector unit, to the user’s choice), a control system, and an anode HVPS, the regulated voltage of which controls the output power of the gyrotron. The feedback characteristic time is less than 1 s. Feedback control is provided for both CW and pulse modes. The setup includes a unique quasi-optical line that transmits the microwave beam through a 2-m air gap, as well as a synthesized mode converter from the Gaussian beam to the TE01 or TE11 mode (to the user’s choice) installed on a 300-kV platform. During the tests, the developed setup proved to be a reliable and user-friendly tool applicable for long-term powering of the fourth-generation ECR ion sources.

Let G be amultiple HNN-extension of a group A, and let all its associated subgroups be properly contained in some locally nilpotent subgroup of A. We prove that if G is residually nilpotent, then all the associated subgroups are p′-isolated in A for some prime p. Moreover, if q is a prime such that G is residually a q′-torsion-free nilpotent group, then p = q.

We obtain a generalization of Shirvani’s results on necessary conditions of residual finiteness of generalized free products and HNN-extensions of groups to the case of residually C groups, where C is an arbitrary class of groups.

The article presents an analysis of revenues and expenditures for medical care for non-working, working citizens and the lowest-paid category of working citizens in all regions of the Russian Federation and Russia as a whole. The technical possibility and expediency of introducing a new breakthrough system for financing domestic health care based on medical savings accounts (MSA) based on the criterion of not reducing the amount of funding for medical services per one resident of the subject of the Russian Federation is justified and proved. It is analyzed and shown that the funds accumulated on the MSA in accordance with the new scheme for financing health care in the Russian Federation are sufficient to cover the average cost of medical care for citizens in all regions of the Russian Federation. The economic effect of the introduction of these accounts in all subjects of the Russian Federation is estimated, which will amount to 378 billion rubles in the first year, of which 292 billion rubles are funds on medical savings accounts of citizens, 18.2 billion rubles are reducing the costs of territorial budgets and compulsory health insurance funds for pensioners and 68.6 billion rubles are reduction of expenses of territorial budgets and compulsory health insurance funds at the expense of working citizens exit from “shadow”, in five years the economic effect will make already 2.9 trillion rubles. The main components of the economic effect are the same as for the first year of medical savings accounts introduction. Funds for medical savings accounts can be used only for medical care. Therefore, about 146 billion rubles of 292 billion rubles citizens will spend on their better medical care. These funds will go to medical organizations (hospitals and polyclinics). The funds saved by the territorial budgets and funds of the compulsory health insurance, it is advisable to direct to research, development and manufacture of competitive medical equipment. Thus, the demand for medical equipment, provided with funding from medical organizations and the population, will increase by 8 times and will amount to approximately 1.240 trillion rubles and the budget funds, which should be directed to the research, development and manufacture of medical equipment, will grow by about 5 times and will amount to 420 billion rubles.

Nearly 20 years ago demand had arisen in the world practice of plasma investigations with gyrotron usage - to have a source of long-pulse microwave radiation with step-tunable frequency at megawatt-power level. Transparency bands of typical modern 1.8-mm single disc diamond output windows are distant at step of ~35GHz. Attempts to build the reliable broad band output window unit capable to provide long-pulse gyrotron multi-frequency operation with much smaller steps has not succeeded yet. In the same time 10 dual-frequency 140/105GHz long-pulse gyrotrons using single disc diamond window successfully run now at megawatt-power level providing ECRH at ASDEX Upgrade and KSTAR facilities.

A brief summary of current developments in Russia of megawatt power gyrotrons is presented. The gyrotron developed are applied in experiments at several modern tokamaks. New demands to gyrotron parameters and possible ways to reach them are discussed.

Given a nonempty set π of primes, call a nilpotent group π-bounded whenever it has a central series whose every factor F is such that: In every quotient group of F all primary components of the torsion subgroup corresponding to the numbers in π are finite. We establish that if G is a residually π-bounded torsion-free nilpotent group, while a subgroup H of G has finite Hirsh–Zaitsev rank then H is π’-isolated in G if and only if H is separable in G in the class of all finite nilpotent π-groups. By way of example, we apply the results to study the root-class residuality of the free product of two groups with amalgamation.

A 250 GHz continuous-wave (CW) gyrotron has been developed at the IAP RAS jointly with GYCOM Ltd., as a prototype of the microwave source for the envisaged prospective nuclear fusion power plants (DEMO). The main applications of such a tube are electron cyclotron resonance heating and electron cyclotron resonance current drive of magnetically confined plasma as well as its diagnostics based on collective Thomson scattering in various reactors for controlled thermonuclear fusion (e.g., tokamaks and stellarators). The results of the preliminary experimental tests in a pulsed mode of operation are presented. The microwave power of up to 330 kW with an efficiency of 30% without collector depression was obtained. At an accelerating voltage of 55 kV and an electron beam current of 12.5 A (which corresponds to the design parameters for CW operation), the measured output power was about 200 kW. The TEM00 mode content evaluated at the tube output is not less than 98.6%.

We study experimentally the automated microwave complex for microwave spectroscopy and diagnostics of various media, which was developed at the Institute of Applied Physics of the Russian Academy of Sciences in cooperation with GYCOM Ltd. on the basis of a gyrotron with a frequency of 263 GHz and operated at the first gyrofrequency harmonic. In the process of the experiments, a controllable output power of 0.1–1 kW was achieved with an efficiency of up to 17% in the continuous-wave generation regime. The measured radiation spectrum with a relative width of about 10 −6 and the frequency values measured at various parameters of the device are presented. The results of measuring the parameters of the wave beam, which was formed by a built-in quasioptical converter, as well as the data obtained by measuring the heat loss in the cavity and the vacuum output window are analyzed.

Given a class K of groups, we prove that the free product of a K -group A and a residually K -group B with amalgamated subgroup which is a retract of B is a residually K -group. We also obtain a sufficient condition for the root-class residuality of a generalized free product of two residually K -groups with amalgamated subgroup which is a retract of one of the factors.

Let C be an arbitrary class of groups which has the root property, consists of finite groups only, and contains at least one nonidentity group. It is proved that every extension of a free group by a C-group is conjugacy C-separable. It is also proved that, if G is a free product of two conjugacy C-separable groups with finite amalgamated subgroup or an HNN-extension of a conjugacy C-separable group with finite associated subgroups, then the group G is residually C if and only if it is conjugacy C-separable.

An experimental pressure field in operating clearance and flowrate parameters analysis for different eccentricities, inlet oil pressures, and journal rotating velocities are presented for a radial multilobe journal bearing. Lubricating film breakdown zones in bearing oil ducts caused by complicated construction and journal rotation are observed.

Design peculiarities and operational parameters of modern Russian gyrotrons developed for 105-170 GHz frequency range with pulse duration up to 3600 s and output power of 1-1.5MW at efficiency over 50 % (CPD collector) are presented.

A brief review of research activity on gyrotron traveling-wave tube amplifiers (gyro-TWTs) with helically corrugated waveguides developing by the Institute of Applied Physics and GYCOM (Nizhny Novgorod, Russia) will be presented.

Development of the gyrotron system for ITER is based on solution of many very difficult scientific and engineering problems. At present the main purpose of the system modification is an enhancement of the system reliability and implementation of all gyrotron systems into ITER machine and its control and safety system. Recently development of ITER gyrotron system in Russia came into completion phase.

The paper presents development status of a gyrotron setup for ITER. Twenty four such setups are to be used for heating and current drive in ITER plasma. The setup includes a gyrotron in a superconducting magnet and several tens of other components. Recently the main required gyrotron parameters were demonstrated. In the last year tests confirmed 95% reliability of gyrotron operation. Now a prototype of complete gyrotron setup is assembled and it is under thorough testing.