# Volume 15, № 3, 1963

### Dmitry Alexandrovich Grave (on the centenary of his birth)

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 235-239

### Silov $р$-subgroups of an even symmetrical group

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 240-249

A plexus of substitution groups was used to describe all Silov p-subgroups of an even symmetrical group. The power of a set of isomorphics proved equal to a continuum. Each of them is characterized by certain invariants — namely, by some whole p-adic number and a certain invariant which is a finite or countable cardinal number.

### Dispersion ratios and analytical properties of partial wave amplitudes in perturbation theory

Chernikov N. A., Liu Yi - Chen, Logunov A. A., Todorov I. Т.

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 250-276

The analytical properties of elastic scattering amplitudes are studied in the frame of perturbation theory. Dispersion relations are derived for fixed t (or s), as well as for fixed cos 8. From these latter dispersion relations spectral representations are obtained for the partial wave amplitudes in the case of elastic scattering. The analytical properties of the nN partial wave amplitudes are also studied.

### On an ordinary linear differential equation of higher order

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 277-289

In this article which is a detailed exposition of our notes (7) and (8), we consider the differential equation (7) in which At(p) are polynomials in a complex parameter, these being uniquely determined by the condition that the equation is satisfied by the family (depending on p) of the functions (1) For non - singular values of p we build all linearly independent solutions (19) of the equation (7) and find for them integral representations (27) and (28), with the help of which we get addition theorems of type (40) and (41). If m = 2 we get the known results of the Bessel functions theory.

### Silov $p$-subgroups of orthogonal and simplex groups of a hyperbolic space

Chernobylskaya E. N., Tsikunоv I. K.

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 290-298

The authors consider the construction of Silov $p$-subgroups of a group of isometrics (linear transformations of a space preserving the metric) of a bilinear-metric space with orthogonal or simplex metrics, consisting of an
orthogonal sum of two-dimensional subspaces of the form
$$V_{2n} = \langle N_1, M_1\rangle \perp \langle N_2, M_2\rangle \perp ...\perp \langle N_n, M_n\rangle ,$$
where $\langle N_i, M_i \rangle $ is a linear shell of the vectors $N_i$ and $M_i$, $N_i^2 = M_i^2 = 0,\; N_i \cdot M_i = 1$ in the sense defined in the $V_{2n}$ scalar product (metrics).

The results are formulated as the following theorems.

Theorem 1. In order that subgroup $S$ of a group of isometrics of space $V_{2n}$ should be a Silov $p$-subgroup, the following conditions are necessary and sufficient:

a) subgroup $S$ must have an invariant maximal isotropic subspace $U_n \subset V_{2n}$

b) subspace $S$ must induce in $U_n$ a Silov $p$-subgroup of its full linear group.

Theorem 2. The Silov $p$-subgroup $S$ of a group of isometrics of space $V_{2n}$ is a semidirect product of the Silov $p$-subgroup of the full linear group of the vector space $U_n$ and the normal divisor $H$, which in the orthogonal metrics is isomorphic to the additive group of obliquely symmetric matrices of order n, in simplex metrics — to the additive group of symmetrical matrices of order $n$.
b) subspace $S$ must induce in $U_n$ a Silov $p$-subgroup of its full linear group.

Theorem 2. The Silov $p$-subgroup $S$ of a group of isometrics of space $V_{2n}$ is a semidirect product of the Silov $p$-subgroup of the full linear group of the vector space $U_n$ and the normal divisor $H$, which in the orthogonal metrics is isomorphic to the additive group of obliquely symmetric matrices of order n, in simplex metrics — to the additive group of symmetrical matrices of order $n$.

### Alexandr Yulyevich Ishlinsky (on the 50th anniversary of his birth)

Putуatа Т. V., Savin G. N., Sokolov Yu. D.

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 299-302

### On the operational theory of I. Z. Shtokalo in the solution of linear differential equations

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 303-305

### Motion of an artificial satellite of the earth relative to the centre of mass

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 305-309

### On an approximate solution of nonlinear operator equations by Y. D. Sokolov's method

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 309-314

### Construction by electrical analogy of a potential stream, allowing for its compressibility, around a Krylov profile

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 314-320

### Feinman's integrals and Poincare's method

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 320-312

### On the convergence of iterational processes on determining the constants of the Christoffel—Schwarz integral

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 321-327

### On periodic solutions of differential equations with undifferentiable right parts

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 328-332

### Construction of solutions for systems subjected to the action of external periodic forces explicitly depending on the time

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 332-228

### Investigation of the dynamic stability of a cylindrical shell under the action of axial periodic forces of high frequency

Ukr. Mat. Zh. - 1963. - 15, № 3. - pp. 338-343