# Volume 56, № 6, 2004

### On the 70th anniversary of the Institute of Mathematics of the Ukrainian National Academy of Sciences

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 723–736

### Gluing of quasisymmetric imbeddings in the problem of quasiconformal extension

Aseev V.V., Sychev A.V., Tetenov A. V.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 737–744

We prove a general theorem on gluing of quasisymmetric imbeddings. Using this theorem, we solve the problem of quasiconformal extension from a one-parameter family of quasiconformal triangles of a special type that do not have a general estimate of quasiconformal convexity.

### Approximation of sine-shaped functions by constants in the spaces $L_p,\; p < 1$

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 745–762

We investigate the best approximations of sine-shaped functions by constants in the spaces $L_p$ for $p < 1$. In particular, we find the best approximation of perfect Euler splines by constants in the spaces Lp for certain $p∈(0,1)$.

### Marcinkiewicz-type strong means of Fourier—Laplace series

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 763–773

We obtain estimates for Marcinkiewicz-type strong means of the Fourier—Laplace series of continuous functions in terms of the best approximations.

### On locally perturbed equilibrium distribution functions

Malyshev D. V., Malyshev P. V.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 774–781

We construct a new class of locally perturbed equilibrium distribution functions for which local (in time) solutions of the BBGKY equations can be extended onto the entire time axis.

### On the identities in algebras generated by linearly connected idempotents

Rabanovych V. I., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 782–795

We investigate the problem of the existence of polynomial identities (PI) in algebras generated by idempotents whose linear combination is equal to identity. In the case where the number of idempotents is greater than or equal to five, we prove that these algebras are not *PI*-algebras. In the case of four idempotents, in order that an algebra be a *PI*-algebra, it is necessary and sufficient that the sum of the coefficients of the linear combination be equal to two. In this case, these algebras are *F* _{4}-algebras.

### Singular locally scalar representations of quivers in Hilbert spaces and separating functions

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 796–809

We consider locally scalar representations of extended Dynkin graphs in Hilbert spaces. The relation between these representations and the function ρ( *n* ) = 1 + ( *n* − 1 ) / ( *n* + 1 ) is established. We construct a family of separating functions that generalize the function ρ and play a similar role in a broader class of graphs.

### Long-range order in linear ferromagnetic oscillator systems. Strong pair quadratic n-n potential

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 810–817

Long-range order is proved to exist for lattice linear oscillator systems with ferromagnetic potential energy containing a term with strong nearest-neighbor (n-n) quadratic pair potential. A contour bound and a generalized Peierls argument are used in the proof.

### A necessary condition for the regularity of a boundary point for degenerating parabolic equations with measurable coefficients

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 818–836

We prove a necessary condition for the regularity of a point on a cylindrical boundary for solutions of second-order quasilinear parabolic equations of divergent form whose coefficients have a superlinear growth relative to derivatives with respect to space variables. This condition coincides with the sufficient condition proved earlier by the author. Thus, we establish a criterion for the regularity of a boundary point similar to the well-known Wiener criterion for the Laplace equation.

### On the existence of solutions of operator differential equations

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 837–850

We consider nonlinear equations of parabolic type in reflexive Banach spaces. We present sufficient conditions for the existence of solutions of these equations. We use methods for the investigation of problems with operators of pseudomonotone (on a subspace) type. In addition, a sufficient criterion in the Sobolev space L_{ p }(0, *T*; *W* _{ p } ^{1} (Ω)∩*L* _{2} (0, *T*; L_{2}(Ω)) is considered for the case where an operator introduced with the use of functional coefficients belongs to a given class. We also show that it is possible to weaken the classical condition of coerciveness.

### On zeros, singular boundary functions, and modules of angular boundary values for one class of functions analytic in a half-plane

Sharan V.L., Vynnyts’kyi B. V.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 851–856

We obtain the description of the zeros, singular boundary functions, and modules of angular boundary values of the functions $f \neq 0$ which are analytic in the half-plane $C_{+} = \{ z : \Re z > 0 \}$ and satisfy the condition $$( \forall \varepsilon > 0 ) ( \exists c_1 > 0 ) (\forall z \in \mathbb{Ñ}_{+} ): | f ( z ) | \leq c_1 \exp ( (\sigma + \varepsilon) | z \eta ( | z | ) ), $$, where $0 \leq \sigma < +\infty$ is a given number and $\eta$ is a positive function continuously differentiable on $[0; +\infty$ and such that $t\eta'(t)/\eta(t) \rightarrow 0$ as $t \rightarrow + \infty$/

### On the smoothness of the generalized solution of a parabolic system in domains with conic points on the boundary

Nguyen Manh Hung, Pham Trieu Duong

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 857–864

We consider the first initial boundary-value problems for parabolic systems in domains with conic points on a boundary. We study the smoothness of their solutions with respect to spatial variables.