卷 71, 编号 2 (2016)

A problem of a dynamically balanced asymmetric ball with a rotor rolling over a rough horizontal plane is considered. Earlier, A. Y. Moskvin constructed bifurcation diagrams of the momentum map and bifurcation complexes in order to study the dynamics of the system and to obtain singular solutions. A natural development of this research is a fine Liouville analysis of the system. The first step in this direction is presented in the paper, namely, we verify the non-degeneracy of singularities and describe the Liouville foliation in a neighborhood of singular points of the momentum map.

The almost everywhere convergence condition similar to the Menchoff-Rademacher condition is obtained for functional series with some weak analogue of the orthogonality property. As a corollary, results related to almost everywhere convergence of series with respect to Riesz systems, Hilbert and Bessel systems, and frames are obtained.

The case of weakly supercritical branching random walks is considered. A theorem on asymptotic behavior of the eigenvalue of the operator defining the process is obtained for this case. Analogues of the theorems on asymptotic behavior of the Green function under large deviations of a branching random walk and asymptotic behavior of the spread front of population of particles are established for the case of a simple symmetric branching random walk over a many-dimensional lattice. The constants for these theorems are exactly determined in terms of parameters of walking and branching.

A parametric family of Lipschitz homeomorphisms of a compact metric space continuously dependent on some parameter is studied. A family such that the topological entropy of homeomorphism considered as a function of the parameter does not belong to the first Baire class is constructed.

The complexity of realization of Boolean functions by circuits of functional elements in the basis consisting of all characteristic functions of antichains over a Boolean cube is studied. It is proved that the complexity of realization of an n-variable parity function is \(\left\lfloor {\frac{{n + 1}}{2}} \right\rfloor \) and the complexity of its negation equals the complexity of the n-variable majority function which is \(\left\lceil {\frac{{n + 1}}{2}} \right\rceil \).