# Volume 69, № 4, 2017

### Existence of the solitary traveling waves for a system of nonlinearly coupled oscillators on the 2d -lattice

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 435-444

We consider a system of differential equations that describes the dynamics of an infinite system of nonlinearly coupled nonlinear oscillators on the 2d-lattice. By the method of critical points, we obtain a result on existence of the solitary traveling waves.

### On the relationship between the multiplicities of eigenvalues in finite- and infinite-dimensional problems on graphs

Boyko O. P., Martinyuk O. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 445-455

It is shown that some results concerning the multiplicities of eigenvalues of the spectral problem that describes small transverse vibrations of a star graph of Stieltjes strings and the multiplicities of the eigenvalues of tree-patterned matrices can be used for the description of possible multiplicities of normal eigenvalues (bound states) of the Sturm – Liouville operator on a star graph.

### Existence theorems for multidimensional generalized moment representations

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 456-465

The conditions of existence of multidimensional generalized moment representations are established.

### Convergence of Fourier series of functions $\text{Lip} 1$ with respect to general orthonormal systems

Gogoladze L., Tsagareishvili V.

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 466-477

We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) $\{ \varphi_n(x)\}$ in order that the Fourier series in this system for any function from the class $\mathrm{L}\mathrm{i}\mathrm{p} 1$ be convergent almost everywhere on $[0, 1]$. It is shown that the obtained conditions are best possible in a certain sense.

### Spaces of smooth and generalized vectors of the generator of an analytic semigroup and their applications

Gorbachuk M. L., Gorbachuk V. M.

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 478-509

For a strongly continuous analytic semigroup $\{ e^{tA}\}_{t\geq 0}$ of linear operators in a Banach space $B$ we investigate some locally convex spaces of smooth and generalized vectors of its generator $A$, as well as the extensions and restrictions of this semigroup to these spaces. We extend Lagrange’s result on the representation of a translation group in the form of exponential series to the case of these semigroups and solve the Hille problem on description of the set of all vectors $x \in B$ for which there exists $$\mathrm{l}\mathrm{i}\mathrm{m}_{n\rightarrow \infty }\biggl( I + \frac{tA}n \biggr)^n x$$ and this limit coincides with etAx. Moreover, we present a short survey of particular problems whose solutions are necessary for the introduction of the above-mentioned spaces, namely, the description of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator; the representation of solutions to operator-differential equations on an open interval and the analysis of their boundary values, and the existence of solutions to an abstract Cauchy problem in various classes of analytic vector-valued functions.

### Approximation of functions from the classes $W_{β}^r H^{α }$ by Weierstrass integrals

Hrabova U. Z., Kalchuk I. V., Stepanyuk T. A.

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 510-519

We investigate the asymptotic behavior of the least upper bounds of the approximations of functions from the classes $W_{β}^r H^{α }$ by Weierstrass integrals in the uniform metric.

### Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit evaluation

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 520-529

We study the ratios of parameters for Ramanujan’s function $χ(q)$ and their explicit values.

### Problem with integral conditions in the time variable for Sobolevtype system of equations with constant coefficients

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 530-549

In a domain obtained as a Cartesian product of an interval $[0, T]$ and the space $R^p, p \in N$, for a system of equations (with constant coefficients) unsolved with respect to the highest time derivative, we study a problem with integral conditions in the time variable in the class of functions almost periodic in the space variables. A criterion of uniqueness and sufficient conditions for the existence of the solution of this problem in different functional spaces are established. We use the metric approach to solve the problem of small denominators encountered in the construction of the solution.

### Balancing polynomials and their derivatives

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 550-564

We study the generalization of balancing numbers with a new sequence of numbers called $k$-balancing numbers. Moreover, by using the Binet formula for $k$-balancing numbers, we obtain the identities including the generating function of these numbers. In addition, the properties of divisibility of these numbers are investigated. Further, balancing polynomials that are natural extensions of the $k$-balancing numbers are introduced and some relations for the derivatives of these polynomials in the form of convolution are also proved.

### Affine curvature of plane geodesic lines on affine hypersurfaces

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 565-574

We establish a necessary and sufficient condition for a geodesic line on a nondegenerate hypersurface to be a plane curve. We deduce a formula for the affine curvature of a plane geodesic line on the affine hypersurface in terms of the affine fundamental form and the shape operator. We present the definition of transverse curvature and determine some of its elementary properties.

### Myroslav L’vovych Horbachuk

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 575