Ufa Mathematical Journal

In memory of Mukminov Farit Khamsaevich

V.F. Vil'danova — 2025-08-13

Dimensions of Riesz products and pluriharmonic measures

E.S. Doubtsov — 2025-08-13

On the unit sphere in $\mathbb{C}^n$, $n\geq 2$, we consider the Riesz products generated by the Ryll — Wojtaszczyk polynomials. We obtain the lower bound for the energy dimension of such Riesz products. The obtained inequality implies immediately an estimate for the Hausdorff dimension of the considered products. This results is also obtained in another way, by means of known one–dimensional estimates and decomposition into slice–products. These decompositions are employed for sharp estimating of the Hausdorff dimension of pluriharmonic measures on an $n$–dimensional torus, $n\geq 2$.

On degenerate solutions of second order elliptic equations in plane

A.B. Zaitsev — 2025-08-13

In the work we study conditions, under which a solution to a second order partial differential equation in the unit disk on the plane degenerates. We prove that each degenerate solution is either a polynomial of degree at most $2$ or a linear combination of a constant and the logarithm of a fractional–rational expression. In proof of the main result we use the Taylor series expansion of the degenerate solution of the equation at an arbitrary point and study the dependence of coefficients of resulting series on the coefficients at the lower powers of the same series.

Semianalytic approximation of normal derivative of double layer potential near and at boundary of two–dimensional domain

D.Yu. Ivanov — 2025-08-13

The normal derivatives (ND) of the double layer potential (DLP) are defined on a boundary of a domain by hyper--singular integrals. This is why, it is impossible to calculate ND DLP with a satisfactory accuracy either on the boundary or in its vicinity using traditional quadrature formulas, which allow one to calculate ND DLP with a good accuracy at a sufficient distance from the boundary. In the present paper, we obtain semi–analytical approximations of ND DLP for the two–dimensional Laplace equation, which uniformly converge with an almost cubic velocity in a closed near--boundary domain that includes the boundary. For this purpose, we use exact integration over the smooth component of the distance function near the observation point, an additive–multiplicative method for extracting a singularity, and a piecewise quadratic interpolation of slowly varying functions. We provide the results of calculating ND DLP in a closed near–boundary domain of a unit circle, which confirm the uniform almost cubic convergence of the proposed approximations.

On recovering problem for Sturm — Liouville operator with two frozen arguments

M.A. Kuznetsova — 2025-08-13

Inverse spectral problems consist in recovering operators by their spectral characteristics. The problem of recovering the Sturm — Liouville operator with one frozen argument by one spectrum was considered earlier in works by various authors. In this paper, we study the uniqueness of recovering the operator with two frozen arguments and different coefficients $p$, $q$ by the spectra of two boundary value problems. This case is significantly more difficult than the case of one frozen argument since the operator is no longer a one–dimensional perturbation. We prove that the operator with two frozen arguments can not be recovered by two spectra in the general case. For the unique recovery, one has to impose some conditions on the coefficients. We assume that the coefficients $p$ and $q$ are zero on some interval and prove the uniqueness theorem. We also obtain formulas for regularized traces of two spectra. The result is formulated in terms of the convergence of a certain series, which allows us to avoid smoothness conditions for the coefficients.

On orbits in $ \mathbb C^4 $ of 7–dimensional Lie algebras possessing two Abelian subalgebras

A.V. Loboda — 2025-08-13

The paper focuses on the problem on description of holomorphically homogeneous real hypersurfaces of multidimensional complex spaces based on the properties of the Lie algebras and their nilpotent and Abelian subalgebras corresponding to these manifolds. Using classifications of a large family of 7–dimensional solvable non–decomposable Lie algebras, earlier we studied the orbits of algebras with ``strong'' commutative properties. In particular, it was established that a 7–dimensional Lie algebra with an Abelian subalgebra of dimension 5 admits no Levi nondegenerate orbits in the space $\mathbb C^4.$

In the present paper we study all 82 types of solvable non–decomposable 7–dimensional Lie algebras, which have exactly two 4–dimensional Abelian subalgebras and a 6–dimensional nilradical. We prove that for 75 types of algebras, any 7–dimensional orbit in $ \mathbb C^4 $ is either Levi–degenerate or can be reduced to a tubular manifold by a holomorphic transformation. We provide all (up to local holomorphic coordinate transformations) realizations of 7 exceptional types of abstract Lie algebras as algebras of holomorphic vector fields in $ \mathbb C^4.$For most of these realizations, we give coordinate descriptions of orbits, which are holomorphically homogeneous nondegenerate real hypersurfaces in this space.

On homoclinic points and topological entropy of continuous maps on one-dimensional ramified continua

E.N. Makhrova — 2025-08-13

Let $X$ be a dendroid, $f:X\to X$ be a continuous map, $p$ be a periodic point of $f$ and let $x$ be a homoclinic point in $X$ to the periodic point $p$. We study the properties of the homoclinic point $x$ and the unstable manifold of the point $p$. We investigate the local structure of $X$ under which the existence of a homoclinic point implies the positive topological entropy of $f$. We also present differences in the properties of homoclinic points and the unstable manifolds of periodic points for continuous maps defined on dendroids, dendrites and finite trees.

On multiple interpolation of periodic complex-valued functions

A.I. Fedotov — 2025-08-13

We obtain fully constructive results on construction of trigonometric interpolation polynomials with multiple nodes. We construct polynomials interpolating periodic complex-valued functions of a real variable. The polynomials are represented in general form and in the form of expansions over fundamental polynomials. We provide examples and discuss unresolved problems.

Multiplicity of solutions for resonant discrete $2n$-th order periodic boundary value problem

O. Hammouti — 2025-08-13

We examine a class periodic boundary value problems for a discrete equation of order $2n$. We demonstrate the existence of multiple solutions by using the critical point theory and variational methods. Additionally, we consider two examples, in which we discuss the fundamental characteristics of the multiplicity of solutions.

Dynamical systems of quadratic operators on set of idempotent measures

I.T. Juraev — 2025-08-13

We consider quadratic operators, which map the $n$–dimensional simplex of idempotent measures into itself. We introduce the concept of Volterra quadratic operator on the simplex of idempotent measures and provide some general properties of such operators.

We also consdier a special Volterra quadratic operator and provide a comprehensive analysis of the dynamical system generated by this operator. Moreover, the dynamical systems generated by general Volterra operators defined on 2 and 3–dimensional simplices of idempotent measures are studied. For each case, we find fixed points and limits of trajectories.

On basic summability in $\mathbb{R}$

B. Sarić — 2025-08-13

The paper deals with the concept of basic summability of residue function of interval function, which is a synonym for its differential form. As one comprehensive concept, it includes not only all known concepts of integrability, such as Newton's, generalized Riemann and generalized Riemann — Stieltjes integrability, but also arithmetic series.