Using the Schwarz function method, we have obtained a new exact solution for the problem of capillary waves on a fluid sheet. The reliability of the solution was confirmed by the results of numerical verification of the main boundary equation. It was revealed that the pressure coefficient is the determining physical parameter. Its value makes it possible to predict that one should expect either the capillary wave of a certain type (symmetric or antisymmetric) or the disintegration of the fluid sheet into individual drops. It is shown that the maximum possible length of the capillary wave is approximately 1.6 times the thickness of the fluid sheet, regardless of the wave type. The question of reproducing the known exact solution of W. Kimmersley for this problem remains open.

The article continues the study of multiplicatively idempotent semirings with the annihilation condition. It is proven that for multiplicatively idempotent semirings with zero the annihilation condition is equivalent to the equalizing property (Theorem 1). New conditions are obtained (Rickart property, properties of a simple spectrum, and others) under which a multiplicatively idempotent semiring is isomorphic to the direct product of a Boolean ring and a generalized Boolean lattice (Theorems 2 and 3). Some other statements have also been proved, examples have been given, and explanatory remarks have been made.

Let $(x_n)$ be a sequence and $\rho\geq 1$. For two fixed sequences $n_1<n_2<n_3<\dots$, and $M$ define the oscillation operator

$$\mathcal{O}_\rho (x_n)=(\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\m\in M}}\left|x_m-x_{n_k}\right|^\rho)^{1/\rho}.$$

Let $(X,\mathscr{B} ,\mu , \tau)$ be a dynamical system with $(X,\mathscr{B} ,\mu )$ a probability space and $\tau$ a measurable, invertible, measure preserving point transformation from $X$ to itself.

Suppose that the sequence $(n_k)$ is a lacunary, and $M$ is any sequence of positive real numbers such that there exists an $\ell \in \mathbb{R}$ satisfying $\#\{m\in M:n_k\leq m<n_{k+1}\}\leq \ell$ for all $k\in \mathbb{N}$ to obtain the above mentioned results, where $\#$ denotes cardinality. Then the following results are proved in this article for $\rho\geq 2$.

(i) Define $\phi_n(x)=\dfrac{1}{n}\chi_{[0,n]}(x)$ on $\mathbb{R}$. Then there exists a constant $C>0$ such that $$\|\mathcal{O}_\rho (\phi_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$$

for all $f\in H^1(\mathbb{R})$.

(ii) Let $\displaystyle A_nf(x)=\frac{1}{n}\sum_{k=1}^nf(\tau^kx)$ be the usual ergodic averages in ergodic theory. Then $$\|\mathcal{O}_\rho (A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$$

for all $f\in H^1(X)$.

(iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{O}_\rho (A_nf)$ is integrable.

In the author's previously published article (S. Demir "Oscillation inequalities on real and ergodic $H^1$ spaces", Russ. Math. 67 (3), 42-52 (2023)) the above results have been obtained when both $(n_k)$ and $M$ are lacunary. Thus the results of this work extend those results to a nonlacunary sequence $M$ with a more general growth condition.

We find the necessary and sufficient conditions for an act $X$ over a trivial semigroup $\{e\}$ to be cantorian (or co-cantorian), i.e., for any act $Y$ the existence of injective (resp., surjective) homomorphisms $X \to Y$ and $Y \to X$ implies the isomorphism $X \cong Y$.

Let $\beta\geq\alpha>-1/2$ and $F$ be an even function of class $C^2(\mathbb{R})$. The paper studies the properties of solutions to the Cauchy problem

$$\frac{\partial^2U}{\partial x^2}+\frac{(2\alpha+1)}{x}\frac{\partial U}{\partial x}=   \frac{\partial^2U}{\partial t^2}+\frac{(2\beta+1)}{t} \frac{\partial U}{\partial t}, \quad x>0,\,\, t>0,$$

$$U(x,0)=F(x), \quad \frac{\partial U}{\partial t}(x,0)=0, \quad x\geq 0$$

related to the structure of the kernel of the operator

$$\mathcal{A}F(t)=\int\limits_{0}^{\pi}F(\sqrt{r^2+t^2-2rt\cos\theta})\sin^{2\alpha}\theta d\theta$$

for a fixed $r>0$. It is shown that the functions from $\mathrm{Ker}\, \mathcal{A}$ are uniquely determined by their values on $(0,r)$ and this interval cannot be replaced by the interval $(0,\rho)$ with $\rho<r$. A description of $\mathrm{Ker}\, \mathcal{A}$ is found in the form of series of normalized Bessel functions $j_\alpha(\lambda x)$, $\lambda\in\mathcal{N}_r$, where $\mathcal{N}_r=\{x>0: j_\alpha(rx)=0 \}$. With the help of these results, new uniqueness theorems for solutions to the indicated Cauchy problem are established, theorems on the representation of solutions satisfying the condition $U(\xi,t)=0$, $\xi\in E$, $t>0$ are obtained, where the set $E$ consists of one positive number or $E$ coincides with the set of positive zeros of the function $j_\alpha$, and a new theorem on two radii is proved.

It is known that the limit of a sequence of (quasi)conformal mappings is either a constant or a (quasi)conformal mapping. In this paper, we prove that in the case of Heisenberg-type Carnot groups, a similar property is valid for mappings that are quasiconformal in the mean, i.e., for homeomorphisms with finite distortion and a distortion function integrable to an appropriate degree. This result is applied to solving model problems of nonlinear elasticity theory on Carnot groups.

In this paper it is proved that there exist universal pairs $\big({\{\lambda_{k,s}\}_{_{k,s=0} }^{\infty}}\mathbf{,}E\big)$ in the sense of modification with respect to double multiplicative systems. A universal series with respect to the double Vilenkin system in the class of measurable functions of two variables is constructed.

In this paper we consider $3 \times 3$ operator matrix ${\mathcal A}_\mu$ with spectral parameter $\mu>0$ related with the Hamiltonian of a system with nonconserved and no more than three particles on a one-dimensional lattice. Essential and discrete spectra of the operator matrix ${\mathcal A}_\mu$ are described. It is established that the operator matrix ${\mathcal A}_\mu$ has at most four simple eigenvalues outside of the essential spectrum. Spectral estimates for the lower and upper bounds of the operator matrix ${\mathcal A}_\mu$ are obtained using cubic numerical range, Gershgorin enclosures and classical perturbation theory.