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In addition to the Gnevyshev-Ohl rule (GOR), the relation of the odd cycle with the subsequent even one in the 22-year Hale solar cycle was found. It is shown that 3 years before the 11-year minimum $m$, the value of the relative sunspot number SN in an odd cycle is closely related to the value of the maximum in the next even cycle (correlation coefficient $\rho=0.94$), and the same relation of an odd cycle with the previous even one is weaker. Like GOR, cycles are linked in pairs, but opposite to the Rule. Based on this result, we propose to use SN$_{m-3}$ on the descending phase of the previous odd cycle as a precursor of the subsequent EVEN cycle (Figure 3a) -- a precursor called MI3E. For the prediction of an odd cycle or a prediction without consideration of parity (as in the article by Braj\v{s}a et al., 2022), this method gives less reliable results. To predict the amplitude of an ODD cycle, we propose to use the precursor of the seventh year to its maximum $M$ MA7O -- SN$_{M-7}$ on the descending phase of the previous even cycle (Figure 3b). It turned out that in this case, we can also predict the years near the maximum with a high correlation coefficient ($\rho=0.90{-}0.94$). Thus, the proposed approaches allow us to predict cycles of different parity. According to our prediction, the current solar Cycle 25 in 2023 will reach a maximum of 154 units with a prediction error of $\pm25$ (68% confidence) and $\pm53$ (95% confidence). In 2024, SN will be almost as high as in 2023 -- 147 units, so with smaller time averaging scales, the maximum will fall at the end of 2023., Comment: 12 pages, 6 figures; submitted to Solar Physics

Synthetic powder, ore samples and mineral single crystals of goethite ($\alpha-$FeOOH) were investigated with polarized Raman spectroscopy at temperatures from 300 K to 473 K. The symmetry of the vibrational modes, observed in different scattering configurations, was determined unequivocally. The assignment of the Raman-active modes to definite atomic vibrations is supported by two types of lattice-dynamical calculations: empirical shell-model and $\textit{ab initio}$ DFT. The temperature dependencies of the lineshape parameters of some Raman-active vibrations near to the antiferromagnetic-paramagnetic phase transition infers for a significant spin-lattice coupling in this compound. The most informative in this aspect is the $B_{3g}$ phonon at 387 cm$^{-1}$, which overlays a broad scattering background and displays a pronounced asymmetric Fano-lineshape. The asymmetry increases in the paramagnetic state above the Neel temperature ($T_{\text{N}}$ = 393 K) indicating a strong interaction of this mode with the underlying excitation continuum. The origin of the excitation background is discussed in light of our experimental results and the existing data for the magnetic structure and transport properties of $\alpha-$FeOOH. We rationalize that, most probably, the background stems from magnetic excitations, and the asymmetric shape of the $B_{3g}$ phonon is a result of a linear spin-phonon coupling of this mode with the Fe-O1-Fe spin dimers. Another mechanism, a phonon interaction with thermally activated charge carriers above above $T_{\text{N}}$, is also considered.

Two important sources of information about sunspots in the Maunder minimum are the Sp\"orer catalog and observations of the Paris observatory, which cover in total the last quarter of the 17th and the first two decades of the 18th century. These data, in particular, contain information about sunspot latitudes. As we showed in previous papers, dispersions of sunspot latitude distributions are tightly related to sunspot indices, so we can estimate the level of solar activity in this epoch by a method which is not based on direct calculation of sunspots and is weakly affected by loss of observational data. The latitude distributions of sunspots in the time of transition from the Maunder minimum to the common regime of solar activity proved to be wide enough. It gives evidences in favor of, first, not very low cycle No. -3 (1712-1723) with the Wolf number in maximum W=$100\pm50$, and, second, nonzero activity in the maximum of cycle No. -4 (1700-1711) W=$60\pm45$. Therefore, the latitude distributions in the end of the Maunder minimum are in better agreement with the traditional Wolf number and new revisited indices of activity SN and GN than with the GSN; the latter provide much lower level of activity in this epoch., Comment: 7 pages, 4 figures; submitted to "Geomagnetism and Aeronomy"

We study and compare characteristics of sunspot group latitude distribution in two catalogs: the extended Greenwich (1874--2014) and Schwabe ones (1825--1867). We demonstrate that both datasets reveal similar links between latitude and amplitude characteristics of the 11-year cycle: the latitude dispersion correlates with the current activity and the mean latitude of sunspots in the cycle's maximum is proportional to its amplitude, It agrees with conclusions that we made in previous papers for the Greenwich catalog. We show that the latitude properties of sunspot distribution are much more stable against loss of observational data than traditional amplitude indices of activity. Therefore, the found links can be used for estimates of quality of observations and independent normalizing of activity levels in a gappy pre-Greenwich data. We demonstrate it using the Schwabe catalog as an example. In addition, we show that the first part of the Schwabe data probably contains errors in determination of sunspot latitudes that lead to overestimation of the sunspot latitude dispersions., Comment: 10 pages, 6 figures; will be published in Geomagnetism and Aeronomy

The results for the total multi-photon decay rates of the 3p and 4s levels of hydrogen, presented by D. Solovyev and L. Labzowsky within the cascade approximation, are revisited. The corrected results for certain decay channels differ from original ones of those authors sometimes by order of magnitude. Some aspects with respect to the cosmological recombination process are clarified., Comment: That is a longer version. A shorter version is accepted by PRA. In v2 references are updated

Based on the extended Greenwich-NOAA/USAF catalogue of sunspot groups it is demonstrated that the parameters describing the latitudinal width of the sunspot generating zone (SGZ) are closely related to the current level of solar activity, and the growth of the activity leads to the expansion of SGZ. The ratio of the sunspot number to the width of SGZ shows saturation at a certain level of the sunspot number, and above this level the increase of the activity takes place mostly due to the expansion of SGZ. It is shown that the mean latitudes of sunspots can be reconstructed from the amplitudes of solar activity. Using the obtained relations and the group sunspot numbers by Hoyt and Schatten (1998), the latitude distribution of sunspot groups ("the Maunder butterfly diagram") for the 18th and the first half of the 19th centuries is reconstructed and compared with historical sunspot observations., Comment: 16 pages, 11 figures; accepted by Solar Physics; the final publication will be available at www.springerlink.com

The complete contribution of diagrams with the light-by-light scattering to the Lamb shift is found for muonic hydrogen, deuterium and helium ion. The results are obtained in the static muon approximation and a part of the paper is devoted to the verification of this approximation and analysis of its uncertainty., Comment: 6 pages, 5 figures; cross-refs upgraded

Contributions to the energy levels in light muonic atoms and, in particular, to the Lamb shift fall into a few well-distinguished classes. The related diagrams are calculated using different approaches. In particular, there is a specific kind of non-relativistic contributions. Here we consider such corrections to the Lamb shift in order $\alpha^5m_\mu$. These contributions are due to free vacuum polarization loops as well as to various effects of light-by-light scattering. The closed loop in the related diagrams is an electronic one, which allows a non-relativistic consideration of the muon. Both kinds of contributions have been known for a while, however, the results obtained up to date are only partial ones. We complete a calculation of the $\alpha^5m_\mu$ contributions for muonic hydrogen. The results are also adjusted for muonic deuterium and muonic helium ion., Comment: 4 pages, 3 figures; cross-refs upgraded

The lattice dynamics of the ferromagnetic spinel CuCr2Se4 (Tc = 430K) was studied experimentally by measuring the Raman spectra and theoretically by calculation of zone center phonon frequencies within a shell model. All Raman allowed modes (A1g + E_g + 3F2g) were identified and assigned to specific atomic motions. The relative intensity of the Raman lines varies strongly with excitation photon energy between 1.58 and 2.71 eV, but no significant phonon anomalies are observed near Tc., Comment: 6 pages, 7 figures, 5 tables

The Raman spectra of single crystals of NiFe2O4 were studied in various scattering configurations in close comparison with the corresponding spectra of Ni0.7Zn0.3Fe2O4 and Fe3O4. The number of experimentally observed Raman modes exceeds significantly that expected for a normal spinel structure and the polarization properties of most of the Raman lines provide evidence for a microscopic symmetry lower than that given by the Fd-3m space group. We argue that the experimental results can be explained by considering the short range 1:1 ordering of Ni2+ and Fe3+ at the B-sites of inverse spinel structure, most probably of tetragonal P4_122/P4_322 symmetry., Comment: 10 pages, 5 figures, 6 tables

In the comment we discuss an additional term to the $\alpha^3$ correction to the Lamb shift, which was missed in the calculation by Kinoshita and Nio (Phys. Rev. Lett. 82, 3240 (1999)).

The correction to the wave function of the ground state in a hydrogen-like atom due to an external homogenous magnetic field is found exactly in the parameter $Z\alpha$. The $j=1/2$ projection of the correction to the wave function of the $ns_{1/2}$ state due to the external homogeneous magnetic field is found for arbitrary $n$. The $j=3/2$ projection of the correction to the wave function of the $ns_{1/2}$ state due to the nuclear magnetic moment is also found for arbitrary $n$. Using these results, we have calculated the shielding of the nuclear magnetic moment by the $ns_{1/2}$ electron., Comment: 15 pages

The Uehling correction to the energy levels is presented in terms of the hypergeometric functions 2F1. This presentation allows to derived various asymptotics and approximations. Further applications of this method to other atomic characteristics are also considered.

We consider the most accurate tests of bound state QED, precision theory of simple atoms, related to the hyperfine splitting in light hydrogen-like atoms. We discuss the HFS interval of the 1s state in muonium and positronium and of the 2s state in hydrogen, deuterium and helium-3 ion. We summarize their QED theory and pay attention to involved effects of strong interactions. We also consider recent optical measurements of the 2s HFS interval in hydrogen and deuterium., Comment: presented at The International Workshop "e+e- collisions from phi to psi"

We consider the most accurate tests of bound state QED theory of the hyperfine splitting in two-body atoms related to the HFS interval of the 1s state in muonium and positronium and the 2s state in hydrogen, deuterium and the helium-3 ion. We summarize their QED theory and pay special attention to involved effects of strong interactions and to recent optical measurements of the 2s HFS interval in hydrogen and deuterium. We present results for specific ratios of the 1s-2s frequencies in hydrogen and deuterium which happen to be among the most accurately measured and calculated quantities., Comment: Presented at CPEM (Conference on Precision Electromagnetic Measurements, Torino, 2006)

Effects of vacuum polarization modify the energy levels in atoms with an orbiting particle heavier than an electron. The dominant effect is due to the Uehling potential. In this paper we consider the relativistic corrections to the energy levels caused by the Uehling potential and in particular the fine structure in muonic and antiprotonic atoms. We derive general expressions and consider in detail specific regions of parameters which allow simple asymptotic expansion. We take into account the recoil effects and anomalous magnetic moment in the case of an antiproton as the orbiting particle.

We consider a correction to energy levels in a pionic atom induced by the Uehling potential, i.e., by a free electron vacuum-polarization loop. The calculation is performed for circular states (l=n-1). The result is obtained in a closed analytic form as a function of $ Z\alpha$ and the pion-to-electron mass ratio. Certain asymptotics of the result are also presented.

On the base of the solar magnetic field measurements obtained in Stanford in 1976--2003 the properties of the cyclic evolution of the large-scale magnetic field are investigated. Some regularities are found in longitudinal and latitudinal evolution of the magnetic field in cycles 21, 22 and 23. The cyclic development of the large-scale magnetic field can be divided into two main phases. The phase I, which includes a period approximately from two years before and until three years after the maximum of the solar cycle, is studied in detail. It is found that before the reversal of the large-scale magnetic field the neutral line of the magnetic field in antipodal longitudinal intervals shifts from the equator to opposite directions in cycles 21 and 22, but not in cycle 23. During the sign reversal of the large-scale magnetic field in cycles 21 and 22 in the antipodal longitudinal intervals the magnetic field of opposite polarity is observed in all latitudes, thereby forming an equatorial dipole. After the magnetic field reversal a longitudinal oscillation of the magnetic neutral line with regard to the equator takes place, which has a period about 2 years and damps to the minimum of the 11-year cycle. The intervening longitudinal intervals of the large-scale magnetic field correspond to positions of the active longitudes of sunspot activity, thus indicating a close connection of the large-scale and the local magnetic fields. In evolution of the large-scale magnetic field a periodicity with period $1.23\pm0.16$ year is revealed, which is close to the period found by helioseismological methods in variations of the solar rotation near the tachocline., Comment: 15 pages, 8 figures

Energy levels of a two-body atomic system in an external homogeneous magnetic field can be presented in terms of magnetic moments of their components, however, those magnetic moments being related to bound particles differ from their free values. Study of bound g factors in simple atomic systems are now of interest because of a recent progress in experiments on medium Z ions and of a new generation of muonium experiments possible with upcoming intensive muon sources. We consider bound corrections to the g factors in several atomic systems, experimental data for which are available in literature: hydrogen, helium-3 ion, muonium, hydrogen-like ions with spinless nuclei with medium Z., Comment: presented at PSAS 2002 (St. Petersburg)

Global canonical transformations to free chiral fields are constructed for DG models minimally coupled to scalar fields. The boundary terms for such canonical transformations are shown to vanish in asymptotically static coordinates if there is no scalar field., Comment: 5 pages, LaTeX, Conf. Report, Dubna, July 13-17, 1998

Integrable models of 1+1 dimensional gravity coupled to scalar and vector fields are briefly reviewed. A new class of integrable models with nonminimal coupling to scalar fields is constructed and discussed., Comment: LaTeX, 8 pages, no figures. Talk given at the VIII International Conference on Symmetry Methods in Physics (Dubna 1997), to be published at Phys. of At. Nucl. 1998, vol. 61, #10