Vol. 25 (2019), no. 4

A function is called a Schur function if it is holomorphic in the open unit disk and bounded by one. In the paper, the Schur parameters of polynomial Schur functions are characterized.

The (main) spectral problem for a star graph with three edges composed of Stieltjes strings is considered with the Dirichlet conditions at the pendant vertices. In addition we consider the Dirichlet-Neumann problem on the first edge (Problem 2) and the Dirichlet-Dirichlet problem on the union of the second and the third strings (Problem 3). It is shown that the spectrum of the main problem interlace (in a non-strict sense) with the union of spectra of Problems 2 and 3. The inverse problem lies in recovering the masses of the beads (point masses) and the lengths of the intervals between them using the spectra of the main problem and of Problems 2 and 3. Conditions on three sequences of numbers are proposed sufficient to be the spectra of the main problem and of Problems 2 and 3, respectively.

We discuss the spectral problem for limit distributions of conflict dynamical systems on spaces subjected to fractal divisions. Conditions ensuring the existence of the point spectrum are established in two cases, the repulsive and the attractive interactions between the opponents. A key demand is the strategy of priority in a single region.

We showed that for each function $f(t)$, which is not equal to zero almost everywhere in the Lebesgue measurable set, functionals $I\left(f,z\right)=\int_A{{|f(t)|}^z dt}$ as functions of a complex variable $z=p+iy$ are continuous on the domain and analytic on a set of all inner points of this domain. The functions $I(f,p)$ as functions of a real variable $ p $ are strictly convex downward and log-convex on the domain. We proved that functionals $n(f,p)$ as functions of a real variable $p$ are analytic at all inner points of the interval, in which the function $n(f,p)\neq 0$ except the point $p=0$, continuous and strictly increasing on this interval.

In the present paper we introduce and study the topological entropy of non-autonomous dynamical systems and define the non-autonomous dynamical system on Lie groups and manifolds. Our main purpose is to estimate the topological entropy of the non-autonomous dynamical system on Lie groups. We show that the topological entropy of the non-autonomous dynamical system on Lie groups and induced Lie algebra are equal under topological conjugacy, and a method to estimate the topological entropy of non-autonomous systems on Lie groups is given. To illustrate our results, some examples are presented. Finally some discussions and comments about positive entropy on nil-manifold Lie groups for non-autonomous systems are presented.

This paper is devoted to an investigation of the relationship between the essential approximate point spectrum (respectively, the essential defect spectrum) of a sequ\-ence of closed linear operators $(T_n)_{n\in\mathbb{N}}$ on a Banach space $X$, and the essential approximate point spectrum (respectively, the essential defect spectrum) of a linear operator $T$ on $X$, where $(T_n)_{n\in\mathbb{N}}$ converges to $T$, in the case of convergence in generalized sense as well as in the case of the convergence compactly

We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums of Weyl-Nagy classes $W^r_{\beta,p}, 1\le p\le\infty,$ for rapidly growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform metric. We obtain similar estimates for approximations of the classes $W^r_{\beta,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.