# Volume 66, № 8, 2014

### On Some Ramanujan Identities for the Ratios of Eta-Functions

Bhargava S., Rajanna K. R., Vasuki K. R.

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1011–1028

We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook.

### Exponentially Convergent Method for the First-Order Differential Equation in a Banach Space with Integral Nonlocal Condition

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1029–1040

For the first-order differential equation with unbounded operator coefficient in a Banach space, we study the nonlocal problem with integral condition. An exponentially convergent algorithm for the numerical solution of this problem is proposed and justified under the assumption that the operator coefficient *A* is strongly positive and certain existence and uniqueness conditions are satisfied. The algorithm is based on the representations of operator functions via the Dunford–Cauchy integral along a hyperbola covering the spectrum of *A* and the quadrature formula containing a small number of resolvents. The efficiency of the proposed algorithm is illustrated by several examples.

### On Removable Sets for Degenerated Elliptic Equations

Bayramova N. Q., Gadjiev T. S.

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1041–1057

We establish necessary and sufficient conditions of removability of compact sets.

### History of the Appearance of Infinite-Dimensional Analysis and its Development in Ukraine

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1058–1073

We present a brief survey of the development of functional analysis in Ukraine and the problems of infinite-dimensional analysis posed and solved for thousands of years, which laid the foundations of this branch of mathematics.

### Necessary and Sufficient Conditions for the Solvability of Linear Boundary-Value Problems for the Fredholm Integrodifferential Equations

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1074–1091

We propose a method for the investigation and solution of linear boundary-value problems for the Fredholm integrodifferential equations based on the partition of the interval and introduction of additional parameters. Every partition of the interval is associated with a homogeneous Fredholm integral equation of the second kind. The definition of regular partitions is presented. It is shown that the set of regular partitions is nonempty. A criterion for the solvability of the analyzed problem is established and an algorithm for finding its solutions is constructed.

### On One Convolution Equation in the Theory of Filtration of Random Processes

Barsegyan A. G., Engibaryan N. B.

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1092–1105

We study the problems of analytic theory and the numerical-analytic solution of the integral convolution equation of the second kind $$ \begin{array}{cc}\hfill {\varepsilon}^2f(x)+{\displaystyle \underset{0}{\overset{r}{\int }}K\left(x-t\right)f(t)dt=g(x),}\hfill & \hfill x\in \left[0,r\right)\hfill \end{array}, $$ where $$ \begin{array}{cccc}\hfill \varepsilon >0,\hfill & \hfill r\le \infty, \hfill & \hfill K\in {L}_1\left(-\infty, \infty \right),\hfill & \hfill K(x)={\displaystyle \underset{a}{\overset{b}{\int }}{e}^{-\left|x\right|s}d\sigma (s)\ge 0.}\hfill \end{array} $$ The factorization approach is used and developed. The key role in this approach is played by the V. Ambartsumyan nonlinear equation.

### Boundary Versions of the Worpitzky Theorem for Two-Dimensional Continued Fractions

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1106–1116

For a two-dimensional continued fraction another generalization of the Worpitzky theorem is proved and the limit sets are proposed for Worpitzky-like theorems in the case where the element sets of the twodimensional continued fraction are replaced by their boundaries.

### Trigonometric Approximations and Kolmogorov Widths of Anisotropic Besov Classes of Periodic Functions of Several Variables

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1117–1132

We describe the Besov anisotropic spaces of periodic functions of several variables in terms of the decomposition representation and establish the exact-order estimates of the Kolmogorov widths and trigonometric approximations of functions from unit balls of these spaces in the spaces *L* _{ q } *.*

### Fractional Calculus of a Unified Mittag-Leffler Function

Nathwani B. V., Prajapati J. C.

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1133–1145

The main aim of the paper is to introduce an operator in the space of Lebesgue measurable real or complex functions *L*(*a, b*)*.* Some properties of the Riemann–Liouville fractional integrals and differential operators associated with the function *E* _{ α,β,λ,μ,ρ,p } ^{ γ,δ } (*cz*; *s, r*) are studied and the integral representations are obtained. Some properties of a special case of this function are also studied by the means of fractional calculus.

### CLT-Groups with Hall S-Quasinormally Embedded Subgroups

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1146–1152

A subgroup *H* of a finite group *G* is said to be Hall S-quasinormally embedded in *G* if *H* is a Hall subgroup of the S-quasinormal closure *H* ^{ SQG } *.* We study finite groups *G* containing a Hall S-quasinormally embedded subgroup of index *p* ^{ n } for each prime power divisor *p* ^{ n } of the order of *G.*