## Number 1081, Issue 68, 2013

### Ця сторінка Українською / This page in Ukrainian

Nguyen Van Quynh, About one property of the function $\|x-y\|^{2-m}$, P. 4-9.

The kernel $h_m(x-y)=\|x-y\|^{2-m}$ is important in the theory of subharmonic functions in the space $\mathbb{R}^m (m\ge3)$. For any $y\in\mathbb{R}^m$ we consider the kernel $h_m(x-y)$ as an element of the spaces $L_p(\gamma,\mathbb{R}^m)$. In this article we give a sufficient condition on a measure $\gamma$ the function $h_m(x-y)\in L_p(\gamma,\mathbb{R}^m)$ to be uniformly continious in the parameter $y$ in $\mathbb{R}^m$. We give examples of measures $\gamma$, which satisfy this condition.

Keywords: (m - 1)-dimensional Haysdorff measure, uniform continuity.

A.G. Kostianko, The limit set of the Henstock-Kurzweil integral sums of a vector-valued function, P. 10-20.

We introduce the notion of the limit set $I_{H-K}(f)$ of the Henstock-Kurzweil integral sums of a function $f: [0, 1]\to X$, where $X$ is a Banach space, and study its properties. In particular, we construct an example of function $f$, which is not integrable, but its limit set consists exactly of one point. We find sufficient conditions that guarantee the convexity of the limit set.

Keywords: Henstock-Kurzweil integral, Banach space, limit set of integral sums.

E.V. Massalitina, The Goursat problem with impulse perturbations, P. 21-32.

An estimate for the function of two variables satisfying the Goursat problem with the data on the characteristics and receives a pulse perturbation at defined curves.

Keywords: integral inequalities, hyperbolic equation, impulse perturbation.

V.A. Marchenko, Symmetrically-spectral operators on $\ell_p$ $(1\leq p< \infty)$ and $c_0$ spaces, P. 33-44.

We introduce the concept of symmetrically-spectral operator which generalizes the concept of Riesz-spectral operator to the case of Banach spaces with symmetric basis. We obtain the theorem on the main properties of symmetrically-spectral operators on $\ell_p$ $(1\leq p< \infty)$ and $c_0$ spaces.

Keywords: symmetrically-spectral operator, symmetric basis, $C_0$-semigroup.

T.A. Shtefan, H.V. Velichko, Research the potential energy of forming a rectangular plate, which pushes convex stamp, P. 45-53.

We consider the stationary problem of spatial deformation cuboid stamp. The influence of geometrical and mechanical parameters of the slab on the behavior of the potential energy function. Numerical results are shown for the stamp which surface is sinusoidal in two dimensions.

Keywords: plate, stamp, spatial deformation, the fourth hypothesis is strength, potential energy, the zone of plasticity.

V.N. Syrovatskyi, Functional models of commutative systems of operators close to the unitary, P. 54-66.

Operators were found, that define main commutative system of linear limited operators $T_1, T_2$ for unital metric knot. This operators are defined in terms of functional model that was built for system $T_1, T_2$.

Keywords: model, commutative system of operators.

V.I. Korobov, A.V. Lutsenko, Exponential stabilization of a class of nonlinear systems with constrained control, P. 67-75.

This paper studies the problem of robust exponential stabilization of a class of nonlinear controlled systems containing uncertainty and nonlinear depending on the control. Sufficient conditions for the robust stabilization are obtained and regulators implementing robust stabilization are synthesized. Numerical examples are given.

Keywords: exponential stabilization, robustness.

V.A. Skoryk, On a set of positional controls which solve the global synthesis problem for a linear equation in Hilbert spaces, P. 76-83.

On the basis of the method of the controllability functional it is shown that for a linear equation with a bounded skew self-adjoint operator in Hilbert spaces any non-increasing non-negative on the non-negative semiaxis function, which has a certain number of points of decreasing, and one has a negative derivative on some interval, generates a positional control, which solve the problem of the global synthesis.

Keywords: problem of global synthesis, Hilbert space, linear equation.

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