Let $(x_n)$ be a sequence and $\{c_k\}\in \ell^\infty (\mathbb{Z})$ such that $\|c_k\|_{\ell^\infty}\leq 1$. Define
$$\mathcal{G}(x_n)=\sup_j\left|\sum_{k=0}^j c_k(x_{n_{k+1}}-x_{n_k})\right|.$$
Let now $(X,\beta ,\mu ,\tau )$ be an ergodic, measure preserving dynamical system with $(X,\beta ,\mu )$ a totally $\sigma$-finite measure space. Suppose that the sequence $(n_k)$ is lacunary. Then we prove the following results:
(i) Define $\phi_n(x)=\dfrac{1}{n}\chi_{[0,n]}(x)$ on $\mathbb{R}$. Then there exists a constant $C>0$ such that
$$\|\mathcal{G}(\phi_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$$
for all $f\in H^1(\mathbb{R})$,
(ii) Let
$$A_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f(\tau^kx)$$
be the usual ergodic averages in ergodic theory. Then
$$\|\mathcal{G}(A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$$
for all $f\in H^1(X)$,
(iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{G}(A_nf)$ is integrable.

It is established that any effectively separable many-sorted universal algebra has an enrichment that is the only (up to isomorphism) model constructed from constants for a suitable computably enumerable set of sentences

We obtain sharp inequalities between the best approximations of analytic in the unit disk functions by algebraic complex polynomials and the moduli of continuity of higher-order derivatives in the Bergman weighted space $\mathscr{B}_{2,\mu}$. Based on these inequalities, the exact values of some
known $n$-widths of classes of analytic in the unit disk functions are 
calculated.

For the equation
$
({\rm sign}\,y)|y|^{m}u_{xx}+u_{yy}+\alpha_{_{0}}|y|^{(m-2)/2}u_{x}+(\beta_
{0}/y)u_{y}=0,
$, considered in some unbounded mixed domain, uniqueness and existence theorems for a solution to the problem with the missing shift condition  on the boundary characteristics and an analogue of the  Frankl type condition on the interval of degeneracy of the equation are proved. 

In this paper, we consider the necessary and sufficient conditions for the subharmonicity of functions of two variables, representable as a product of two functions of one variable in the Cartesian coordinate system or in the polar coordinate system in domains on the plane. We establish a connection of such functions with functions that are convex with respect to solutions of second-order linear differential equations, i.e., convex with respect to two functions. 

In the paper, a generalized Lotka–Volterra – type system with switching is considered. The conditions for the ultimate boundedness of solutions and the permanence of the system are studied. With the aid of the direct Lyapunov method, the requirements for the switching law are established to guarantee the necessary dynamics of the system. An attractive compact invariant set is constructed in the phase space of the system, and a given region of attraction for this set is provided. A distinctive feature of the work is the use of a combination of two different Lyapunov functions, each of which plays its own special role in solving the problem. 

The problem of finding the supremums of the best simultaneous polynomial approximations of some classes of functions analytic in the unit disk and belonging to the Bergman space B₂ is considered. The indicated function classes are defined by the averaged values of the mth order moduli of continuity of the highest derivative bounded from above by some majorant Φ.

A general form of the equation of a curvilinear three-web admitting a one-parameter family of automorphisms (AW-webs) is found. It is proved that the trajectories of automorphisms of an AW-web are geodesics of its Chern connection. All AW-webs are found for which one of the covariant derivatives of curvature is zero.