# Volume 58, № 7, 2006

### Extremal problems of nonoverlapping domains with free poles on a circle

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 867–886

Let $α_1, α_2 > 0$ and let $r(B, a)$ be the interior radius of the domain $B$ lying in the extended complex plane $\overline{ℂ}$ relative to the point $a ∈ B$. In terms of quadratic differentials, we give a complete description of extremal configurations in the problem of maximization of the functional $\left( {\frac{{r(B_1 ,a_1 ) r(B_3 ,a_3 )}}{{\left| {a_1 - a_3 } \right|^2 }}} \right)^{\alpha _1 } \left( {\frac{{r(B_2 ,a_2 ) r(B_4 ,a_4 )}}{{\left| {a_2 - a_4 } \right|^2 }}} \right)^{\alpha _2 }$ defined on all collections consisting of points $a_1, a_2, a_3, a_4 ∈ \{z ∈ ℂ: |z| = 1\}$ and pairwise-disjoint domains $B_1, B_2, B_3, B_4 ⊂ \overline{ℂ}$ such that $a_1 ∈ B_1, a_1 ∈ B_2, a_3 ∈ B_3, and a_4 ∈ B_4$.

### Compatibly bi-Hamiltonian superconformal analogs of Lax-integrable nonlinear dynamical systems

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 887–900

Compatibly bi-Hamiltonian superanalogs of the known Lax-integrable nonlinear dynamical systems are obtained by using a relation for the Casimir functionals of central extensions of the Lie algebra of superconformal even vector fields and its adjoint semidirect sum.

### Asymptotic behavior of unbounded solutions of essentially nonlinear second-order differential equations. II

Evtukhov V. M., Kas'yanova V. A.

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 901–921

We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type.

### Two-boundary problems for a Poisson process with exponentially distributed component

Kadankov V. F., Kadankova T. V.

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 922–953

For a Poisson process with exponentially distributed negative component, we obtain integral transforms of the joint distribution of the time of the first exit from an interval and the value of the jump over the boundary at exit time and the joint distribution of the time of the first hit of the interval and the value of the process at this time. On the exponentially distributed time interval, we obtain distributions of the total sojourn time of the process in the interval, the joint distribution of the supremum, infimum, and value of the process, the joint distribution of the number of upward and downward crossings of the interval, and generators of the joint distribution of the number of hits of the interval and the number of jumps over the interval.

### Self-stochasticity phenomenon in dynamical systems generated by difference equations with continuous argument

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 954–975

For dynamical systems generated by the difference equations *x*(*t*+1) = *f*(*x*(*t*)) with continuous time (*f* is a continuous mapping of an interval onto itself), we present a mathematical substantiation of the self-stochasticity phenomenon, according to which an attractor of a deterministic system contains random functions.

### On Gibbs quantum and classical particle systems with three-body forces

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 976–996

For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit.

### Natural boundary of random Dirichlet series

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 997–1005

For the random Dirichlet series $$\sum\limits_{n = 0}^\infty {X_n (\omega )e^{ - s\lambda _n } } (s = \sigma + it \in \mathbb{C}, 0 = \lambda _0 < \lambda _n \uparrow \infty )$$ whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit conditions for the line of convergence to be its natural boundary a.s.

### Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 1006–1008

We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in $R^d$. We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the iterated logarithm for martingales with arbitrary matrix normalization.