## Volume 81, 2015

M.V. Soloviova, Bishop-Phelps-Bollobas modulus of a uniformly non-square Banach space, P. 4-9.

Chica, Kadets, Martin and Soloviova demonstrated recently that the Bishop-Phelps-Bollobas modulus $\Phi^S_X$ of a Banach spaces $X$ can be estimated from above through the parameter of uniform non-squareness $\alpha(X)$: $\Phi^S_X(\varepsilon) \leq \sqrt{2\varepsilon}\,\sqrt{1-\frac{1}{3}\alpha(X)}$. In this short note we demonstrate that the right-hand side in the above theorem cannot be substituted by anything smaller than $\sqrt{2\varepsilon}\,\sqrt{1-\alpha(X)}$.

Keywords: Bishop-Phelps theorem; uniformly non-square spaces.

V.A. Rezunenko, The field of vertical electric dipole placed above the spiral conductive unclosed sphere, P. 10-19.

The problem of the electromagnetic field an open spiral conductive sphere is analyzing. The method of regularization of operator tasks is applied. The integral equations with a weak singularity in the kernel is used. The infinite system of algebraic equations of type II with a compact operator in $\ell_2$ is received. Some properties of electromagnetic fields are studied.

Keywords: spiral conductive sphere, vertical dipole, compact operator.

N.F. Patsegon, S.I. Potseluiev, The volumetric parametric resonance in magnetizable medium, P. 20-35.

The stability of magnetizable medium stationary states at parametric excitation of a magnetic field is studied. Parameters of exited acoustic wave and the influence of oscillating magnetic field on the dispersion of sound and its propagation velocity are determined using asymptotic and numerical methods.

Keywords: parametric resonance, oscillating magnetic field.

M.O. Bebiya, Global synthesis of bounded controls for systems with power nonlinearity, P. 36-51.

In this work we consider the problem of global bounded control synthesis for a nonlinear system with uncontrollable first approximation. A class of bounded controls that steer the system from any initial state to the origin in some finite time is constructed based on the controllability function method.

Keywords: synthesis problem, finite-time stabilization, nonlinear systems, controllability function method.

A.V. Rezounenko, On the history of our journal: 50th anniversary or 137th ordinary year of publication? P. 52-59.

The evolution of titles of mathematical journal Visnyk of Kharkiv University during period 1879-2015 is presented. This evolution reflects the changes of names of the University where the journal was published. We also discover and discuss several ways of numbering during the history and list names of members of Editorial Board during last 50 years.

Keywords: Visnyk, mathematical journal, Kharkiv University.

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