# Volume 69, № 3, 2017

### On principal ideal multiplication modules

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 291-299

Let $R$ be a commutative ring with identity and let $M$ be a unitary $R$-module. A submodule $N$ of $M$ is said to be a multiple of $M$ if $N = rM$ for some $r \in R$. If every submodule of $M$ is a multiple of $M$, then $M$ is said to be a principal ideal multiplication module. We characterize principal ideal multiplication modules and generalize some results from [Azizi A. Principal ideal multiplication modules // Algebra Colloq. – 2008. – 15. – P. 637 – 648].

### On total moduli of continuity for $2\pi$-periodic functions of two variables in the space $L_{2,2}$

Vakarchuk M. B., Vakarchuk S. B.

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 300-310

The description of the total moduli of continuity of $2\pi$ -periodic functions of two variables are obtained in the space $L_{2,2}$. The proposed description can be regarded as an extension of the famous results by O. V. Besov, S. B. Stechkin, V. A.Yudin on the moduli of continuity in $L_{2}$ in the two-dimensional case.

### Approximate solutions of the Boltzmann equation with infinitely many modes

Gordevskii V. D., Gukalov A. A.

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 311-323

For the nonlinear kinetic Boltzmann equation in the case of a model of hard spheres, we construct an approximate solution in the form of a series of Maxwellian distributions with coefficient functions of time and the space coordinate. We establish the sufficient conditions for the coefficient functions and the values of hydrodynamic parameters appearing in the distribution that enable us to make the analyzed deviation arbitrarily small.

### $A$-cluster points via ideals

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 324-331

Following the line of the recent work by Sava¸s et al., we apply the notion of ideals to $A$-statistical cluster points. We get necessary conditions for the two matrices to be equivalent in a sense of $A^I$ -statistical convergence. In addition, we use Kolk’s idea to define and study $B^I$ -statistical convergence.

### On conditionless bases of the kernels generated by differential equations of the second order

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 332-347

We establish necessary and sufficient conditions for a system of functions generated by differential equations of the second order to be a basis. Our method is based on the application of the Muckenhoupt condition.

### Fundamental solution of the Cauchy problem for the Shilov-type parabolic systems with coefficients of bounded smoothness

Litovchenko V. A., Unguryan G.M.

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 348-364

Under the condition of minimal smoothness of the coefficients, we construct the fundamental solution of the Cauchy problem and study the principal properties of this solution for a special class of linear parabolic systems with bounded variable coefficients covering the class of Shilov-type parabolic systems of nonnegative kind.

### Systems parabolic in Petrovskii's sense in Hörmander spaces

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 365-380

We study a general parabolic initial-boundary-value problem for systems parabolic in Petrovskii’s sense with zero initial Cauchy data in some anisotropic H¨ormander inner-product spaces.We prove that the operators corresponding to this problem are isomorphisms between the appropriate H¨ormander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions of the problem. We also obtain new sufficient conditions of continuity for the generalized partial derivatives of a given order of a chosen component of the solution.

### Bessel functions of two complex mutually conjugated variables and their application in boundary-value problems of mathematical physics

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 381-396

We formulate boundary-value problems for the eigenvalues and eigenfunctions of the Helmholtz equation in simply connected domains by using two complex mutually conjugated variables. The systems of eigenfunctions of these problems are orthogonal in the domain. They are formed by Bessel functions of complex variables and the powers of conformal mappings of the analyzed domains onto a circle. The boundary-value problems for the main equations of mathematical physics are formulated in an infinite cylinder with the use of complex and time variables. The solutions of the boundaryvalue problems are obtained in the form of series in the systems of eigenfunctions. The Cauchy problem for the main equations of mathematical physics with three independent variables is also considered.

### On the relationships between the norms of operators with endpoint singularities in Lebesgue and Hölder spaces with weight

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 397-406

We select a special class of operators with endpoint singularities was found. For these operators we establish inequalities connecting the norms in Lebesgue spaces with weight and in H¨older spaces with weight. We describe specific types of operators satisfying the conditions of the main theorem on the relationship between the norms. These results can be used to study the operators acting on H¨older spaces with weight on the basis of the well-known results for operators acting on Lebesgue spaces with weight.

### New fractional integral inequalities for differentiable convex functions and their applications

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 407-425

We establish some new fractional integral inequalities for differentiable convex functions and give several applications for the Beta-function.

### Directional logarithmic derivative and the distribution of zeros of an entire function of bounded $L$-index in the direction

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 426-432

We establish new criteria of boundedness of the $L$-index in the direction for entire functions in $C^n$. These criteria are formulated as estimate of the maximum modulus via the minimum modulus on a circle and describe the distribution of their zeros and the behavior of the directional logarithmic derivative. In this way, we prove Hypotheses 1 and 2 from the article [Bandura A. I., Skaskiv O. B. Open problems for entire functions of bounded index in direction // Mat. Stud. – 2015. – 43, № 1. – P. 103 – 109]. The obtained results are also new for the entire functions of bounded index in $C$. They improve the known results by M. N. Sheremeta, A. D. Kuzyk, and G. H. Fricke.