On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acting on a regular rooted tree acts absolutely non-freely on the boundary of the tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II₁-factor representations, and infinitely many continuous ergodic invariant random subgroups.     

Parametric analysis of the oscillatory solutions to stochastic differential equations with the Wiener and Poisson components by the Monte Carlo method

Using the Monte Carlo method, we address the influence of the Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs). For the linear and Van der Pol oscillators, we study the accuracy of estimates of the functionals of numerical solutions to SDEs obtained by the generalized explicit Euler method. For a linear oscillator, we obtain the exact analytical expressions for the mathematical expectation and the variance of the SDE solution. These expressions allow us to investigate the dependence of the accuracy of estimates of the solution moments on the values of SDE parameters, the size of meshsize, and the ensemble of simulated trajectories of the solution. For the Van der Pol oscillator, we study the dependence of the frequency and the damping rate of the oscillations of the mathematical expectation of SDE solution on the values of parameters of the Poisson component. The results of the numerical experiments are presented.     

On disjunctions of algebraic sets in completely simple semigroups

A semigroup S is called an equational domain if any finite union of algebraic sets over S is algebraic. We give some necessary and su?cient conditions for a completely simple semigroup to be an equational domain.     

On Toughness and Hamiltonicity of 2K2‐Free Graphs

The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields components. Determining toughness is an NP‐hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K₂‐free graphs, that is, graphs that do not contain two vertex‐disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvátal's toughness conjecture is true for 2K₂‐free graphs.     

The efficiency of ultrasonic oscillations transfer into the load

The results of ultrasonic action to the substances have been presented. It is examined, the correlation between the electrical parameters of ultrasonic equipment and acoustic performances of the ultrasonic field in treating the medium, the efficiency of ultrasonic technological facility, and the peculiarities of oscillations introduced into the load under cavitation development. The correlation between the acoustic powers of oscillations securing the needed level of cavitation and desired technological effect, and the electrical parameters of the ultrasonic facility, first of all, the power, is established. The peculiarities of cavitation development in liquids with different physical-chemical properties (including the molten low-melting metals) have been studied, and the acoustic power of oscillations introduced into the load under input variation of electric power to the generator has been also estimated.     

N‐Methylbenzoazacrown ethers with the nitrogen atom conjugated with the benzene ring: the improved synthesis and the reasons for the high stability of complexes with metal and ammonium cations

An improved method for the synthesis of formyl derivatives of N‐methylbenzoazacrown ethers is proposed. They are prepared in up to 68% yields over fewer steps and with a much shorter time required for the last step. The stability constants of complexes formed by N‐methylbenzoazacrown ethers and their structural analogs with alkali metal, alkaline‐earth metal and ammonium cations were determined by ¹H NMR titration in CD₃CN. High stability of complexes of N‐methyl derivatives of benzoazacrown ethers is demonstrated, comparable with or even exceeding the stability of benzocrown‐ether complexes and markedly exceeding the stability of complexes of phenylazacrown ethers with the same macrocycle size. The structures of azacrown ethers and their complexes with Ba(ClO₄)₂ were studied by X‐ray diffraction. A high degree of pre‐organization of N‐methylbenzoazacrown ethers toward the formation of complexes with metal and ammonium cations was noted, which is due to the clear‐cut pyramidal geometry of the nitrogen atom and the orientation of the lone electron pairs (LEPs) of most heteroatoms towards the centre of the macroheterocycle. Copyright © 2009 John Wiley & Sons, Ltd.     

Structure and magnetic properties of Cu-Fe fiber composites obtained using packet hydrostatic extrusion

The structure and magnetic properties of Cu-Fe fiber composites prepared by packet hydrostatic extrusion are studied. The number of armco-iron fibers in the copper matrix attained n ∼ 8 × 10¹⁰, and rated diameter d of fibers varied from 2 mm to 3 nm. A correlation is observed between the magnetization hysteresis curves and low-frequency magnetic susceptibility obtained on samples with various values of d. It is shown that the magnetic properties of composites in the submicrometer range of d values are satisfactorily described in the magnetization reversal theory for small ferromagnetic particles.