# Volume 58, № 3, 2006

### Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

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

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 291–302

We obtain new exact inequalities of the form $$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$ for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$ for functions defined on the axis $R$ in the case where $$r = 2,\; k = 1,\; q ∈ [2,∞),\; p = ∞,\; s= 1,$$ and for functions of constant sign on $R$ or $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s = ∞$$ and in the case where $$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$

### On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$

Shchitov A. N., Vakarchuk S. B.

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 303-316

We consider and study properties of the smoothness characteristics $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$, of functions $f(x)$ that belong to the space $S^p,\quad 1 \leq p < \infty$, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and the exact values of the widths of the classes of functions defined by using $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$ are calculated.

### Conditions for the existence of bounded solutions of one class of nonlinear differential equations

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 317–325

For systems of nonlinear differential equations (*dx/dt*) = *A*(*x*)*x* + *f*(*t*) in a Banach space, we establish sufficient conditions for the existence of their solutions bounded on the entire real axis R.

### On the rate of convergence of a regular martingale related to a branching random walk

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 326–342

Let $\mathcal{M}_n,\quad n = 1, 2, ..., $ be a supercritical branching random walk in which a number of direct descendants of an individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mathcal{M}_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a nonnegative function which regularly varies at infinity, with exponent greater than —1. We present sufficient conditions of almost sure convergence of the series $\sum^{\infty}_{n=1}a(n)(W - W_n)$. We also establish a criteria of finiteness of $EW \ln^+Wa(ln+W)$ and $EW \ln^+|Z_{\infty}|a(ln+|Z_{\infty}|)$, where $Z_{\infty} = Q_1 + \sum^{\infty}_{n=2}M_1 ... M_nQ_{n+1}$ and $(M_n, Q_n)$ are independent identically distributed random vectors, not necessarily related to $\mathcal{M}_n$.

### Topological equivalence of functions on oriented surfaces

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 343–351

On closed oriented surfaces of genus *g* ? 1, we consider functions that possess only one saddle critical point in addition to local maxima and minima. We study the problem of the realization of these functions on surfaces and construct an invariant that distinguishes them. For surfaces of genus \(g = \frac{{n - 1}}{2}\), where *n* is a prime number, we calculate the number of topologically nonequivalent functions with one maximum and one minimum.

### Improved scales of spaces and elliptic boundary-value problems. II

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 352–370

We study improved scales of functional Hilbert spaces over $\mathbb{R}^n$ and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.

### Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions

Caraffini G. L., Petrina D. Ya.

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 371–380

The problem of the existence of solutions of the hierarchy for the sequence of correlation functions is investigated in the direct sum of spaces of summable functions. We prove the existence and uniqueness of solutions, which are represented through a semigroup of bounded strongly continuous operators. The infinitesimal generator of the semigroup coincides on a certain everywhere dense set with the operator on the right-hand side of the hierarchy. For initial data from this set, solutions are strong; for general initial data, they are generalized ones.

### Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 381–387

We determine a stationary measure for a process defined by a differential equation with phase space on the segment $[V_0 , V_1]$ and constant values of a vector field that depend on a controlling semi-Markov process with finite set of states.

### Long-range order in Gibbs lattice classical linear oscillator systems

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 388–405

The existence of the ferromagnetic long-range order (lro) is proved for Gibbs classical lattice systems of linear oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other (nonpair) potentials that are weak if they are not ferromagnetic. A generalized Peierls argument and two different contour bounds are our main tools.

### Asymptotically optimal estimators for moments of change

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 406–416

We consider the problem of finding asymptotically optimal estimators for many moments of change in the case of incomplete information on distributions. We prove that if the maximum-likelihood estimator is asymptotically optimal, then, under certain conditions, it preserves this property after the replacement of actual values by density estimators. We solve the problem for the case of one moment of change and generalize the results obtained to the case of several moments of change.

### Artinian rings with nilpotent adjoint group

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 417–426

Let $R$ be an Artinian ring (not necessarily with unit element), let $Z(R)$ be its center, and let $R ^{\circ}$ be the group of invertible elements of the ring $R$ with respect to the operation $a ∘ b = a + b + ab$. We prove that the adjoint group $R ^{\circ}$ is nilpotent and the set $Z (R) + R ^{\circ}$ generates $R$ as a ring if and only if $R$ is the direct sum of finitely many ideals each of which is either a nilpotent ring or a local ring with nilpotent multiplicative group.

### Asymptotic solutions of the Dirichlet problem for the heat equation with impulses

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 427–430

We propose an algorithm for the construction of asymptotic expansions for solutions of the Dirichlet problem for the heat equation with impulses.