# Volume 62, № 12, 2010

### The order law of large numbers of the Marcinkiewicz - Zygmund

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1587-1597

The Marcinkiewicz - Zygmund order law of large numbers is established for random variables in Banach lattices. Similar results are obtained also for the maximum scheme.

### Generalized separation of variables and exact solutions of nonlinear equations

Barannyk A. F., Barannyk T. A., Yuryk I. I.

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1598 - 1609

We consider the generalized procedure of separation of variables of the nonlinear hyperbolic-type equations and the Korteweg - de Vries-type equations. We construct a wide class of exact solutions of these equations which cannot be obtained with the use of the S. Lie method and the method of conditional symmetries.

### On a continued fraction of order twelve

Kahtan Abdulrawf A. A., Sathish Kumar C., Sharath G., Vasuki K. R.

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1609 - 1619

We present some new relations between a continued fraction $U(q)$ of order twelve (which is established by M. S. M. Naika et al.) and $U(q^n)$ for $n = 7,9,11\;\text{and}\; 13$.

### The best uniform approximation in the metric space of continuous maps with compact convex images

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1620 - 1633

For the problem of the best uniform approximation of a continuous map with compact convex images by sets of other continuous maps with compact convex images, we establish necessary and sufficient conditions and the criterion for an extremal element, which is a generalization of the classical Kolmogorov criterion for the polynomial of best approximation.

### Sard’s theorem for mappings between Fréchet manifolds

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1634–1641

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let $M$ (respectively, $N$) be a bounded Fréchet manifold with compatible metric $d_M$ (respectively, $d_N$ ) modeled on Fréchet spaces $E$ (respectively, $F$) with standard metrics. Let $f : M → N$ be an $MC^k$ -Lipschitz–Fredholm map with $k > \max \{\text{Ind}\; f, 0\}$. Then the set of regular values of $f$ is residual in $N$.

### Estimation of a distribution function by an indirect sample

Babilua P., Nadaraya E., Sokhadze G. A.

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1642–1658

The problem of estimation of a distribution function is considered in the case where the observer has access only to a part of the indicator random values. Some basic asymptotic properties of the constructed estimates are studied. The limit theorems are proved for continuous functionals related to the estimation of $F^n(x)$ in the space $C[a,\; 1 - a], 0 < a < 1/2$.

### Riemann boundary-value problem on an open rectifiable Jordan curve. II

Kud'yavina Yu. V., Plaksa S. A.

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1659–1671

The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.

### Approximation of Poisson integrals by de la Valleé-Poussin sums in uniform and integral metrics

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1672–1686

On the classes of Poisson integrals of functions belonging to the unit balls of the spaces $L_s , 1 ≤ s ≤ ∞$, we establish asymptotic equalities for upper bounds of approximations by de la Vallée-Poussin sums in the uniform metric. Asymptotic equalities are also obtained for the case of approximation by de la Vallée-Poussin sums in the metrics of the spaces $L_s , 1 ≤ s ≤ ∞$, on the classes of Poisson integrals of functions belonging to the unit ball of the space $L_1$.

### On lattice oscillator-type Kirkwood - Salsburg equation with attractive manybody potentials

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1687–1704

We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials.

### Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1705–1714

We present conditions under which a linear homogeneous second-order equation is nonoscillatory on a semiaxis and conditions under which its solutions have infinitely many zeros.

### Elliptic equation with singular potential

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1715 – 1723

We consider the following problem of finding a nonnegative function $u(x)$ in a ball $B = B(O, R) ⊂ R^n,\; n ≥ 3:$ $$−Δu=V(x)u,u|∂B=ϕ(x),$$ where $Δ$ is the Laplace operator, $x = (x 1, x 2,…, x n )$, and $∂B$ is the boundary of the ball $B$. It is assumed that $0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B)$, and $φ(x)$ is continuous on $∂B$. We study the behavior of nonnegative solutions of this problem and prove that there exists a constant $C_{*} (n) = (n − 2)^2/4$ such that if $V_0 (x) = \frac{c}{|x|^2}, then, for $0 ≤ c ≤ $C_{*} (n)$ and $V(x) ≤ V_0 (x)$ in the ball $B$, this problem has a nonnegative solution for any nonnegative continuous boundary function $φ(x) ∈ L_1(∂B)$, whereas, for $c > C_{*} (n)$ and $V(x) ≥ V_0(x)$, the ball $B$ does not contain nonnegative solutions if $φ(x) > 0$.

### Index of volume 62 of „Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1724 - 1728