Powers of sets in free groups

We prove that |A^n| > c_n |A|^{[\frac{n+1}{2}]} for any finite subset A of a free group if A contains at least two noncommuting elements, where c_n>0 are constants not depending on A. Simple examples show that the order of these estimates are the best possible for each n>0.Comment: 3 page

Control of the nonlinear frequency shift for the spin-transfer nanooscillator using a bias magnetic field

We investigated the possibilities of controlling the nonlinear frequency shift of the magnetization oscillations in a spin-transfer nanoscillator by varying the magnitude and direction of the bias magnetic field. We considered both isotropic ferromagnetic materials and crystals with uniaxial and cubic crystallographic anisotropies. We have shown that achieving a zero nonlinear frequency shift is possible with a certain orientation of the bias magnetic field vector. The results of the theoretical analysis based on the method of Hamiltonian formalism are in good agreement with the micromagnetic simulations. Our research reveals the way to control the frequency tuning of a spin transfer nanoscillator, which is crucial for spintronic signal generation devices.Comment: 6 pages, 3 figure

Noise Properties of Two Mutually Coupled Spin-Transfer Nanooscillators in the Phase Locking Regime

Introduction. Today, many research endeavors are devoted to the miniaturization of microwave sources. One of the promising approaches is the use of magnetic nanostructures (spintronics elements), providing a wide range of frequency tuning and low power consumption. The main disadvantage of spintronics generators (spintransfer nanoscillators ‒ STNO) is a low output power of generated oscillations (tens of nanowatts and less). A possible solution is to sum up the power of many STNOs in a mutual synchronization mode.Aim. The investigation of noise properties of two connected STNOs with identical and non-identical parameters in a phase synchronization mode.Materials and methods. A model was developed of two STNOs interconnected by spin waves taking into account thermal noises. Spectral power densities of the amplitude and phase noise were obtained by the method of effective linearization.Results. Dependencies were obtained in a general form for attenuation coefficients of the amplitude and phase fluctuations of noise sources for each STNO. Three cases of synchronization were considered: completely identical STNOs, two identical STNOs but with different oscillation frequencies, and two non-identical STNOs, differing in an allowance of self-excitation by frequencies and amplitudes of the oscillations. It was possible to obtain a gain in the amplitude and phase noise for two identical STNOs. In this case, an increase in the allowance of self-excitation led to a decrease in the level of phase and amplitude noise.Conclusion. This analysis of the attenuation coefficients for non-identical STNOs demonstrates the possibility of improving the noise properties of each of the generators. In this case, the best noise value is obtained for an STNO with greater stability in a stand-alone mode.Introduction. Today, many research endeavors are devoted to the miniaturization of microwave sources. One of the promising approaches is the use of magnetic nanostructures (spintronics elements), providing a wide range of frequency tuning and low power consumption. The main disadvantage of spintronics generators (spintransfer nanoscillators ‒ STNO) is a low output power of generated oscillations (tens of nanowatts and less). A possible solution is to sum up the power of many STNOs in a mutual synchronization mode.Aim. The investigation of noise properties of two connected STNOs with identical and non-identical parameters in a phase synchronization mode.Materials and methods. A model was developed of two STNOs interconnected by spin waves taking into account thermal noises. Spectral power densities of the amplitude and phase noise were obtained by the method of effective linearization.Results. Dependencies were obtained in a general form for attenuation coefficients of the amplitude and phase fluctuations of noise sources for each STNO. Three cases of synchronization were considered: completely identical STNOs, two identical STNOs but with different oscillation frequencies, and two non-identical STNOs, differing in an allowance of self-excitation by frequencies and amplitudes of the oscillations. It was possible to obtain a gain in the amplitude and phase noise for two identical STNOs. In this case, an increase in the allowance of self-excitation led to a decrease in the level of phase and amplitude noise.Conclusion. This analysis of the attenuation coefficients for non-identical STNOs demonstrates the possibility of improving the noise properties of each of the generators. In this case, the best noise value is obtained for an STNO with greater stability in a stand-alone mode

Complex Formation of Copper(II) with 2,2′-Biimidazolyl

The acid-base properties of 2,2′-biimidazolyl (2,2′-BiIm) are studied using methods of pH-potentiometry and spectrophotometry. In aqueous solutions, 2,2′-BiIm shows a tendency to self-associate. The characteristics of complex formation of Cu(II) with 2,2′-biimidazolyl are determined in a wide range of pH and reagent concentrations using potentiometric titration. EPR, and NMR. In solutions of pH ranging from 2 to 5, the specific feature of complexation between copper(II) and 2,2′-BiIm is a formation, in an excess of the ligand, of stable binuclear complexes, in which the 2,2′-BiIm molecules act as bridging Iigands

Modeling of the thermal regime of a mobile electric vehicle charge unit

Rechargeable batteries are an important element in mobile electric vehicle charging systems. Batteries have an operating temperature range, and outside of this range the performance will be reduced or the equipment will cease to function. It is problematic to check this situation experimentally due to the number of batteries used (1200 pieces), so it is necessary to simulate the thermal operating conditions beforehand. The simulation has been performed in Comsol and a preliminary confirmation of the operability of the batteries in the mobile electric vehicle charging unit has been obtained

ARTIFICIAL BOG OAK

The wood is bog oak – the exclusive material used to create artistic products of high value, which explains the numerous studies in the field of the same material in an industrial environment. The proposed technology of heat treatment allows to obtain wood similar in appearance with wood stained oak and with improved physical and mechanical properties.Представленная работа выполнялась при поддержке гранта Президента Российской Федерации для государственной поддержки молодых российских ученых – докторов наук (МД-5596.2016.8)