# Volume 66, № 6, 2014

### Trigonometric Widths of the Nikol’skii–Besov Classes in the Lebesgue Space with Mixed Norm

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 723–732

We establish exact-order estimates for the trigonometric widths of the Nikol’skii–Besov classes of periodic functions of many variables in the Lebesgue space with mixed norm.

### The Dirichlet Problem with Laplacian with Respect to a Measure in the Hilbert Space

Bogdanskii Yu. V., Sanzharevskii Ya. Yu.

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 733–739

We study the Dirichlet problem for a specified class of elliptic equations in a region of the Hilbert space consistent with a given Borel measure.

### Jackson-Type Inequalities for the Special Moduli of Continuity on the Entire Real Axis and the Exact Values of Mean $ν$ - Widths for the Classes of Functions in the Space $L_2 (ℝ)$

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 740–766

The exact values of constants are obtained in the space $L_2 (ℝ)$ for the Jackson-type inequalities for special moduli of continuity of the $k$ th order defined by the Steklov operator $S_h (f)$ instead of the translation operator $T_h (f)$ in the case of approximation by entire functions of exponential type $σ ∈ (0,∞)$. The exact values of the mean $ν$-widths (linear, Bernstein, and Kolmogorov) are also obtained for the classes of functions defined by the indicated characteristic of smoothness.

### $I-n$-Coherent Rings, $I-n$-Semihereditary Rings, and $I$-Regular Rings

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 767–786

Let $R$ be a ring, let $I$ be an ideal of $R$, and let $n$ be a fixed positive integer. We define and study $I-n$-injective modules and $I-n$-flat modules. Moreover, we define and study left $I-n$-coherent rings, left $I-n$-semihereditary rings, and $I$-regular rings. By using the concepts of $I-n$-injectivity and $I-n$-flatness of modules, we also present some characterizations of the left $I-n$-coherent rings, left $I-n$-semihereditary rings, and $I$-regular rings.

### Asymptotic Expansion of the Moments of Correlogram Estimator for the Random-Noise Covariance Function in the Nonlinear Regression Model

Ivanov O. V., Moskvychova K. K.

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 787–805

We establish asymptotic expansions of the bias, mean-square deviation, and variance for the correlogram estimator of the unknown covariance function of a Gaussian stationary random noise in the nonlinear regression model with continuous time.

### Estimation of the Remainder for the Interpolation Continued *C*-Fraction

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 806–814

We estimate the remainder of the interpolation continued *C*-fraction.

### Sharpening of the Explicit Lower Bounds for the Order of Elements in Finite Field Extensions Based on Cyclotomic Polynomials

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 815–825

We explicitly construct elements of high multiplicative order in any extensions of finite fields based on cyclotomic polynomials.

### Some Approximation Properties of Szasz–Mirakyan–Bernstein Operators of the Chlodovsky Type

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 826–834

We motivate a new sequence of positive linear operators by means of the Chlodovsky-type Szasz–Mirakyan–Bernstein operators and investigate some approximation properties of these operators in the space of continuous functions defined on the right semiaxis. We also find the order of this approximation by using the modulus of continuity and present the Voronovskaya-type theorem.

### Approximations by Fourier Sums on the Sets $L^{ψ} L^{P(∙)}$

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 835–846

We study some problems of imbedding of the sets of $ψ$-integrals of the functions $f \in L^{p(∙)}$ and determine the orders of approximations of functions from these sets by Fourier’s sums.

### Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 847–852

We study the well-posedness of the inverse problem of determination of the coefficient of a minor term of a semilinear parabolic equation in the presence of a nonlinear boundary condition. The additional condition is given in the nonlocal integral form. A uniqueness theorem and a “conditional” stability theorem are proved.

### On Invariant Subspaces in Weighted Hardy Spaces

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 853–857

We consider the description of translation invariant subspaces of a weighted Hardy space in the half plane. The obtained result includes the Beurling–Lax theorem for the Hardy space as a special case. We discuss the problem of generalization of the definition of inner function.

### $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 858–864

We study the $p$-nilpotency of a group such that every maximal subgroup of its Sylow $p$-subgroups is $s$-conditionally permutable for some prime $p$. By using the classification of finite simple groups, we get interesting new results and generalize some earlier results.