A note on the theorems of M. G. Krein and L. A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by M.G.Krein and recently generalized to matrix systems by L.A.Sakhnovich. We prove that the continuous analog of the adjoint polynomials converges in the upper half-plane in the case of L^2 coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szego-Kolmogorov-Krein condition is satisfied. Thus we point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results

First-passage times over moving boundaries for asymptotically stable walks

Let $\{S_n, n\geq1\}$ be a random walk wih independent and identically distributed increments and let $\{g_n,n\geq1\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\infty)$. Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\{c_n,n\geq1\}$ such that $S_n/c_n$ converges to a stable law. In this paper we determine the tail behaviour of $T_g$ for all oscillating asymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$. Furthermore, we prove that the rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the stable meander.Comment: 20 page

First-passage times over moving boundaries for asymptotically stable walks

Let $\{S_n, n\geq1\}$ be a random walk wih independent and identically distributed increments and let $\{g_n,n\geq1\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\infty)$. Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\{c_n,n\geq1\}$ such that $S_n/c_n$ converges to a stable law. In this paper we determine the tail behaviour of $T_g$ for all oscillating asymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$. Furthermore, we prove that the rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the stable meander.Comment: 20 page

Impact of renewable energy sources on relay protection operation

The current trend of any electric power system is the integration of renewable energy sources (RES). Mostly these are solar and wind power plants. The penetration of renewable energy leads to significant changes in the operating mode of the power system and, accordingly, affects the functioning of the relay protection and automation devices. In particular, the use of renewable energy can lead to a decrease in short-circuit currents and the insensitivity of the protection to this fault. The paper demonstrates the results of a study that confirmed this. In the paper analyzed the existing approaches to setting up relay protection and automation. Based on this analysis, conclusions are made

Nanostructural states in Nb-Al mechanocomposite after conbined deformation treatment

Nanostructural states were investigated, that were formed in Nb-Al system-based mechanocomposite after combined deformation treatment that includes mechanical activation in a planetary ball mill and subsequent consolidation by torsion under pressure on Bridgman anvils. The formation of the layered structure, consisting of Nb and Al nanobands with width from several to several tens of nanometers was revealed. The structural states with high elastic curvature of crystal lattice and high level of local internal stresses found in Nb and Al subgrains were investigated by transmission electron microscopy