# Volume 61, № 12, 2009

### Monogenic functions in a biharmonic algebra

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1587-1596

We present a constructive description of monogenic functions that take values in a commutative biharmonic algebra by using analytic functions of complex variables. We establish an isomorphism between algebras of monogenic functions defined in different biharmonic planes. It is proved that every biharmonic function in a bounded simply connected domain is the first component of a certain monogenic function defined in the corresponding domain of a biharmonic plane.

### Asymptotic representations of solutions of essentially nonlinear two-dimensional systems of ordinary differential equations

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1597-1611

We establish asymptotic representations for one class of solutions of two-dimensional systems of ordinary differential equations that are more general than systems of the Emden–Fowler type.

### Structure of finite groups with S-quasinormal third maximal subgroups

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1630-1639

We study finite groups whose 3-maximal subgroups are permutable with all Sylow subgroups.

### Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1640-1659

We consider asymptotic expansions of solutions of the first initial boundary-value problem for strong Schrödinger systems near a conical point of the boundary of a domain.

### On an invariant on isometric immersions into spaces of constant sectional curvature

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1660-1704

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups $G$ with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of $G$ is not symmetric, then there are no local isometric immersions of $G$ into $Q_{c^4}$.

### On simple $n$-tuples of subspaces of a Hilbert space

Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1668-1703

This survey is devoted to the structure of “simple” systems $S = (H;H_1,…,H_n)$ of subspaces $H_i,\; i = 1,…, n,$ of a Hilbert space $H$, i.e., $n$-tuples of subspaces such that, for each pair of subspaces $H_i$ and $H_j$, the angle $0 < θ_{ij} ≤ π/2$ between them is fixed. We give a description of “simple” systems of subspaces in the case where the labeled graphs naturally associated with these systems are trees or unicyclic graphs and also in the case where all subspaces are one-dimensional. If the cyclic range of a graph is greater than or equal to two, then the problem of description of all systems of this type up to unitary equivalence is *-wild.

### On the automorphism of some classes of groups

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1704-1712

We study two classes of 2-generated nilpotent groups of nilpotency class 2 and compute the order of their automorphism groups.

### Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1713–1719

We show that the derivative of an arbitrary rational function $R$ of degree $n$ that increases on the segment $[−1, 1]$ satisfies the following equality for all $0 < ε < 1$ and $p, q > 1$: $$∥R′∥_{L_p[−1+ε,1−ε]} ≤ C⋅9^{n(1−1/p)}ε^{1/p−1/q−1}∥R∥_{L_q[−1,1]},$$ where the constant $C$ depends only on $p$ and $p$. The degree of a rational function $R(x) = P(x)/Q(x)$ is understood as the largest degree among the degrees of the polynomials $P$ and $Q$.