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We study the effect of the slab topology $T\times R\times R$ of the Universe on the form of gravitational potentials and forces created by point-like masses. We obtain two alternative forms of solutions: one is based on the Fourier series expansion of the delta function using the periodical property along the toroidal dimension, and another one is derived by direct summation of solutions of the Helmholtz equation for the source particle and all its images. The latter one takes the form of the sum of Yukawa-type potentials. We demonstrate that for the present Universe the latter solution is preferable for numerical calculations since it requires less terms of the series to achieve the necessary precision., This is a preprint of an article published in the European Physical Journal Plus. The final authenticated version is available online at: https://doi.org/10.1140/epjp/s13360-021-01139-y

Within the cosmic screening approach, we obtain the exact formulas for the velocity-independent gravitational potentials produced by matter in the form of discrete sources distributed in the open and closed Universes. These formulas demonstrate that spatial curvature of the Universe considerably affect the form of the potentials and forces. While in the open Universe the gravitational force undergoes exponential suppression at cosmological distances, in the closed Universe the force induced by an individual mass is equal to zero at the antipodal point with respect to this mass. The derived formulas are applicable for investigations of the motion of astrophysical objects (e.g., galaxies) in the open and closed Universes, and for simulations of the large scale structure formation., Comment: v2 (matching the publication in European Physical Journal C) = v1 + minor changes + new Refs.; 8 pages, 3 figures

Energy storage systems (ESS) have a wide spectrum of functions. They must provide power quality, shaving of load change, coordination of distributed power systems, bulk energy storage, and end-user reliability, e.g., uninterrupted power supply. In present paper the configuration and design of experimental ESS based on both Li-ion batteries and supercapacitors have been proposed. Such a hybrid energy storage system (HESS) includes three main components: Li-ion batteries, supercapacitors, and grid interconnection consisting of two invertors and control and monitoring system. Energy storage capacity of developed HESS prototype is 100 kWh, nominal power—100 kW, peak power—200 kW. HESS was created and tested within the experimental facility also including 1.5 MW gas turbine power plant, 200 kW controllable active and reactive loads, and control and measurement system. Experimental results showed that HESS successfully provides the following advantages: (i) suppression of voltage, current, and frequency disturbances in the grid; (ii) compensation of reactive power in the circuit; and (iii) uninterrupted power supply. Cost analysis of proposed hybrid system has also been carried out. In comparison with battery ESS without supercapacitors, HESS showed longer life time, lower cost, and higher peak power.

Let M be either the space of 2π-periodic functions Lp, where 1 ≤ p < ∞, or C; let ωr(f, h) be the continuity modulus of order r of the function f, and let $$D_{n,r,l} (f,x) = \frac{{( - 1)^{\tfrac{r}{2} + 1} 2}}{{C_r^{r/2} }}\int\limits_\mathbb{R} {\left\{ {\sum\limits_{k = 1}^{r/2} {( - 1)^{k + r/2} C_r^{k + r/2} f(x + kt)} } \right\}} V_{n,2l} (t) dt$$ , where $$V_{n,2l} (t) = \frac{{(2l - 1)!2^{2l - 1} }}{{\lambda _{2l} \pi (n + 1)^{2l - 1} }}\left( {\frac{{\sin \tfrac{{(n + 1)t}}{2}}}{t}} \right)^{2l} , \lambda _{2l} = \sum\limits_{k = 0}^{l - 1} {( - 1)^k C_{2l}^k (l - k)^{2l - 1} } $$ , be the generalized Jackson-Vallee-Poussin integral. Denote $$K_m (f) = K_{m,\varphi } (f) = \mathop {\sup }\limits_{0 < v < \infty } \frac{{\omega _m (f,v)}}{{\varphi (v)}}$$ . The paper studies the quantity Km(f − Dn,r,l(f)). The general results obtained are applicable to other approximation methods. Bibliography: 11 titles.

In the paper, a generalization of a known theorem by Hardy and Young is obtained; a formula interrelating the integral of a 2π-periodic function over the period with the integral over the entire axis is established; new approximation characteristics for functions belonging to saturation classes of continuity modules of different orders for the spaces Lp of periodic functions are provided, and some issues concerning approximation, in the uniform metric, of continuous periodic functions even with respect to each of their variables and having nonnegative Fourier coefficients are considered. Bibliography: 17 titles.

In approximation theory, a number of quantities are interrelated via orthogonal transformations. The revelation of these orthogonalities allows one to obtain useful relations and, in particular, to use known theorems on the Fourier coefficients. The paper considers this approach in application to the Fourier transforms of finite functions, to the best L2-approximations, and to trigonometric polynomials with coefficients in a vector space. Bibliography: 11 titles.

A non-linear gravitational model with a multidimensional geometry and quadratic scalar curvature is considered. For certain parameter ranges, the extra dimensions are stabilized if the internal spaces have negative constant curvature. As a consequence, the 4-dimensional effective cosmological constant as well as the bulk cosmological constant become negative. The homogeneous and isotropic external space is asymptotically AdS 4 . The connection between the D-dimensional and the 4-dimensional fundamental mass scales sets an additional restriction on the parameters of the considered non-linear models.

In this paper we study scalar perturbations of the metric for nonlinear $f(R)$ models. We consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. We investigate the astrophysical approach in the case of Minkowski spacetime background and two cases in the cosmological approach, the large scalaron mass approximation and the quasi-static approximation, getting explicit expressions for scalar perturbations for both these cases. In the most interesting quasi-static approximation, the scalar perturbation functions depend on both the nonlinearity function $f(R)$ and the scale factor $a$. Hence, we can study the dynamical behavior of the inhomogeneities (e.g., galaxies and dwarf galaxies) including into consideration their gravitational attraction and the cosmological expansion, and also taking into account the effects of nonlinearity. Our investigation is valid for functions $f(R)$ which have stable de Sitter points in future with respect to the present time, that is typical for the most popular $f(R)$ models., Comment: 9 pages, no figures

The Wheeler-De Witt equation for multidimensional cosmology withn spaces of constant curvature (n>1) is proposed. This equation has a conformally covariant form. Some integrable cases are considered.