# Volume 66, № 3, 2014

### Imbedding Theorems in Metric Spaces $L_{ψ}$

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 291–301

Let $L_0 (T^m)$ be the set of periodic measurable real-valued functions of $m$ variables, let $ψ: R_+^1 → R_+^1$ be the continuity modulus, and let $${L}_{\psi}\left({T}^m\right)=\left\{f\in {L}_0\left({T}^m\right):{\left\Vert f\right\Vert}_{\psi }:={\displaystyle {\int}_{T^m}\psi \left(\left|f(x)\right|\right)dx<\infty}\right\}.$$ The relationship between the modulus of continuity of functions from $L_{ψ} (T^m)$ and the corresponding $K$-functionals is analyzed and sufficient conditions for the imbedding of the classes of functions $H_{ψ}^{ω} (T^m)$ into $L_q (T^m),\; q ∈ (0; 1]$, are obtained.

### Frequency of a Digit in the Representation of a Number and the Asymptotic Mean Value of the Digits

Klymchuk S. O., Makarchuk O. P., Pratsiovytyi M. V.

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 302–310

We study the relationship between the frequency of a ternary digit in a number and the asymptotic mean value of the digits. The conditions for the existence of the asymptotic mean of digits in a ternary number are established. We indicate an infinite everywhere dense set of numbers without frequency of digits but with the asymptotic mean of the digits.

### Functions and Vector Fields on $C(ℂP^N)$-Singular Manifolds

Libardi Alice Kimie Miwa, Sharko V. V.

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 311–315

Let $M^{2n+1}$ be a $C(ℂP^N)$ -singular manifold. We study functions and vector fields with isolated singularities on $M^{2n+1}$. A $C(ℂP^N)$ -singular manifold is obtained from a smooth manifold $M^{2n+1}$ with boundary in the form of a disjoint union of complex projective spaces $ℂP^n ∪ ℂP^n ∪ . . . ∪ ℂP^n$ with subsequent capture of a cone over each component of the boundary. Let $M^{2n+1}$ be a compact $C(ℂP^N)$ -singular manifold with k singular points. The Euler characteristic of $M^{2n+1}$ is equal to $X\left({M}^{2n+1}\right)=\frac{k\left(1-n\right)}{2}$. Let $M^{2n+1}$ be a $C(ℂP^n)$-singular manifold with singular points $m_1 , ... ,m_k$. Suppose that, on $M^{2n+1}$, there exists an almost smooth vector field $V(x)$ with finite number of zeros $m_1 , ... ,m_k , x_1 , ... ,x_l$. Then $X(M 2n + 1) = ∑_{i = 1}^l ind(x_i ) + ∑_{i = 1}^k ind(m_i )$.

### Splitting Obstruction Groups Along one-Sided Submanifolds

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 316–332

We construct new commutative diagrams of exact sequences which relate surgery and splitting obstruction groups for pairs of manifolds. The splitting and surgery obstruction groups are computed for pairs of manifolds and various geometric diagrams of groups corresponding to the problem of splitting along a one-sided submanifold of codimension 1.

### Inverse Problem for a Semilinear Ultraparabolic Equation with Unknown Right-Hand Side

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 333–348

The inverse problem of determination of a time-dependent multiplier of the right-hand side is studied for a semilinear ultraparabolic equation with integral overdetermination condition in a bounded domain. The conditions for the existence and uniqueness of solution of the posed problem are obtained.

### Constructive Characteristic of ho¨ Lder Classes and *M*-Term Approximations in the Multiple Haar Basis

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 349–360

In terms of the best polynomial approximations in the multiple Haar basis, we obtain a constructive characteristic of the Hölder classes *H* _{ p } ^{ α } of functions defined on the unit cube \( \mathbb{I} \) ^{ d } of the space ℝ^{ d } under the restriction \( 0<\alpha <\frac{1}{p}\le 1 \) . We also solve the problem of order estimates of the best *m*-term approximations in the Haar basis of classes *H* _{ p } ^{ α } in the Lebesgue spaces *L* _{ q }( \( \mathbb{I} \) ^{ d }).

### On Equicontinuous Families of Mappings Without Values in Variable Sets

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 361–370

The present paper is devoted to the study of the classes of mappings with unbounded characteristics of quasiconformality. We prove sufficient conditions for the equicontinuity of the families of these mappings that do not take values from a set *E* provided that a real-valued characteristic *c*(*E*) of these mappings has a lower bound of the form *c*(*E*) ≥ \( \delta \) , \( \delta \) \( \epsilon \) ℝ.

### Generalized Bombieri–Lagarias’ Theorem and Generalized Li’s Criterion with its Arithmetic Interpretation

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 371–383

We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums $$ {k}_n={\Sigma}_{\rho}\left(1-{\left(1-\frac{1}{\rho}\right)}^n\right) $$ over zeros of the Riemann xi-function and the derivatives $$ \begin{array}{ccc}\hfill {\lambda}_n\equiv \frac{1}{\left(n-1\right)!}\frac{d^n}{d{z}^n}{\left.\left({z}^{n-1} \ln \left(\xi (z)\right)\right)\right|}_{z=1},\hfill & \hfill \mathrm{where}\hfill & \hfill n=1,2,3,\dots, \hfill \end{array} $$ are nonnegative if and only if the Riemann hypothesis is true, can be generalized and the nonnegativity of certain derivatives of the Riemann xi-function estimated at an *arbitrary* real point *a*, except *a* = 1/2, can be used as a criterion equivalent to the Riemann hypothesis. Namely, we demonstrate that the sums $$ {k}_{n,a}={\Sigma}_{\rho}\left(1-{\left(\frac{\rho -a}{\rho +a-1}\right)}^n\right) $$ for any real *a* such that *a* < 1/2 are nonnegative if and only if the Riemann hypothesis is true (correspondingly, the same derivatives with *a* > 1/2 should be nonpositive). The arithmetic interpretation of the generalized Li’s criterion is given. Similarly to Li’s criterion, the theorem of Bombieri and Lagarias applied to certain multisets of complex numbers is also generalized along the same lines.

### Conditions for Almost Periodicity of Bounded Solutions of Nonlinear Differential Equations Unsolved with Respect to the Derivative

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 384–393

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in Banach spaces without using the *H*-classes of these equations.

### Inequalities for Eigenvalues of a System of Higher-Order Differential Equations

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 394–403

We establish some sharper inequalities for eigenvalues of a system of higher-order differential equations. Moreover, we present some sharper estimates for the upper bound of the (*k* +1)th eigenvalue and the gaps of its consecutive eigenvalues.

### Infinitely Many Fast Homoclinic Solutions for Some Second-Order Nonautonomous Systems

Luo Liping, Luo Zhenguo, Yang Liu

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 404–414

We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly improved.

### Structure of Finite-Dimensional Nodal Algebras

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 415–419

The structure of finite-dimensional nodal algebras over an arbitrary field is described.

### Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings

Al-Shaalan Khalid H., Kamal Ahmed A. M.

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 420–424

We give some classes of zero-symmetric 3-prime near-rings such that every member of these classes has no nonzero derivation. Moreover, we extend the concept of “3-prime” to subsets of near-rings and use it to generalize Theorem 1.1 due to Fong, Ke, and Wang concerning the transformation near-rings *M* _{ o }(*G*) by using a different technique and a simpler proof.

### Greatest common divisor of matrices one of which is a disappear matrix

Romaniv A. M., Shchedrik V. P.

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 425–430

We study the structure of the greatest common divisor of matrices one of which is a disappear matrix. In this connection, we indicate the Smith normal form and the transforming matrices of the left greatest common divisor.