# Volume 58, № 5, 2006

### Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 579–596

We show how the use of nonlinear symmetries of higher-order derivatives allows one to study the regularity of solutions of nonlinear differential equations in the case where the classical Cauchy-Liouville-Picard scheme is not applicable. In particular, we obtain nonlinear estimates for the boundedness and continuity of variations with respect to initial data and discuss their applications to the dynamics of unbounded lattice Gibbs models.

### Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables

Babenko V. F., Churilova M. S.

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 597–606

We investigate the correlation between the constants $K(ℝ^n)$ and $K(T^n)$, where $$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \right\|_{L_p (G^n )}^{\mu _0 } \prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )}^{\mu _i } }}$$ is the exact constant in a Kolmogorov-type inequality, $ℝ$ is the real straight line, $T = [0,2π],\; L^l_{p, p} (G^n)$ is the set of functions $ƒ ∈ L_p (G^n)$ such that the partial derivative $D_i^{l_i } f(x)$ belongs to $L_p (G^n), i = \overline {1,n}, 1 ≤ p ≤ ∞, l ∈ ℕ^n, α ∈ ℕ_0^n = (ℕ ∪ 〈0〉)^n, D^{α} f$ is the mixed derivative of a function $ƒ, 0 < µi < 1, i = \overline {0,n},$ and $∑_{i=0}^n µ_i = 1$. If $G^n = ℝ$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i),\; µ_i = α_i/l_i,\; i = \overline {1,n}$ if $G^n = T^n$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i) − ∑_{i=0}^n (λ/l_i),\; µ_i = α_i/ l_i + λ/l_i , i= \overline {1,n},\; λ ≥ 0$. We prove that, for $λ = 0$, the equality $K(ℝ^n) = K(T^n)$ is true.

### Finite *A*-groups with complementable nonmetacyclic subgroups

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 607–615

We study groups *G*, satisfying the following conditions:

1)*G* — is a finite soluble group with nontrivial prime-power metacyclic second commutator subgroup;

2)all Sylow subgroups of *G* are elementary Abelian.

We describe the structure of these groups with complemented nonmetacyclic subgroup.

### Piecewise-continuous Riemann boundary-value problem on a rectifiable curve

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 616–628

We extend classes of closed rectifiable Jordan curves and given functions in the theory of the piecewise-continuous Riemann boundary-value problem and the characteristic singular integral equation with Cauchy kernel related to this problem.

### Partial asymptotic stability of abstract differential equations

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 629–637

We consider the problem of partial asymptotic stability with respect to a continuous functional for a class of abstract dynamical processes with multivalued solutions on a metric space. This class of processes includes finite-and infinite-dimensional dynamical systems, differential inclusions, and delay equations. We prove a generalization of the Barbashin-Krasovskii theorem and the LaSalle invariance principle under the conditions of the existence of a continuous Lyapunov functional. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the partial stability of continuous semigroups in a Banach space.

### Continuity of certain pseudodifferential operators in spaces of generalized smoothness

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 638–652

We investigate the continuity of a pseudodifferential operator in some spaces of generalized smoothness. Some properties of spaces of generalized smoothness and generalized Lipschitz spaces are established.

### One moment estimate for the supremum of normalized sums in the law of the iterated logarithm

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 653–665

For a sequence of independent random elements in a Banach space, we obtain an upper bound for moments of the supremum of normalized sums in the law of the iterated logarithm by using an estimate for moments in the law of large numbers. An example of their application to the law of the iterated logarithm in Banach lattices is given.

### Nonexistence of some T-factorizations of order 12

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 666–674

We investigate the Beineke problem of the existence of *T*-factorizations of complete graphs and prove several theorems on the existence of *T*-factorizations. Using these theorems, we establish the nonexistence of *T*-factorizations for 32 nonisomorphic admissible trees of order 12.

### Symplectic method for the construction of ergodic measures on invariant submanifolds of nonautonomous hamiltonian systems: Lagrangian manifolds, their structure, and mather homologies

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 675–691

We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog of a Mather-type ?-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems.

### Approximation characteristics of the classes $B_{p,θ}^{Ω}$ of periodic functions of many variables

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 692–704

We obtain exact order estimates for the approximation of the classes $B_{p,θ}^{Ω}$ of periodic functions of many variables in the space $L_q$ by using operators of orthogonal projection and linear operators satisfying certain conditions.

### On iteration stability of the Birkhoff center with respect to power 2

Polulyakh E. O., Vlasenko I. Yu.

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 705–707

It is proved that the Birkhoff center of a homeomorphism on an arbitrary metric space coincides with the Birkhoff center of its power 2.

### Operational queueing system in the scheme of diffusion approximation with evolution averaging

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 708–714

We consider an operational queuing system of the type $[SM | M | \infty]^N$ in the scheme of diffusion approximation. The queueing system is described by a semi-Markov random evolution.

### On generalized solutions of differential equations with operator coefficients

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 715–720

We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients.