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## By the theory of the electron waves and the discrete electron- wave interaction in the stopbands of the slow-wave systems

The role of the discrete description of electron-wave interaction in high-power traveling wave tubes is shown. Deals with the development of this trend on the basis of the theory of interaction of the di?erence equation excitation waveguides. The validity of the linear theory and developed a universal characteristic equation of electron waves, using the di?erence equation excitation waveguides in stopbands of slow-wave systems.

## Nonlinear systems with fast and slow motions. The change of the probability distribution of fast motions influenced by slow ones

The in?uence of slow processes (random or regular) on the probability distribution of fast random processes is considered. We show that such in?uence is universal for all random processes, and in some cases this universality is of the multifractal character. As an example we consider stochastic resonance.

## Does god dice?

Authors used the simple mathematical models as a base for the discussion of the evolution of human society.

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## Radiative processes, radiation instability and chaos in the radiation formed by relativistic beams moving in three-dimensional (two-dimensional) space-periodic structures (natural and photonic crystals)

We review the results of studies of spontaneous and stimulated emission of relativistic particles in natural and photonic crystals.We consider the di?raction of electromagnetic waves in a crystal, and the resonance and parametric (quasi-Cherenkov) X-ray radiation, the radiation in the channeling of relativistic particles in crystals, di?raction radiation in conditions of channeling, di?raction radiation of a relativistic oscillator, induced radiation in multidimensional space-periodic resonators (natural or arti?cial (electromagnetic, photonic) crystals).

## Nonlinear dynamics and acoustic signals generated by periodic impacts of corundum probe on the solid surface

Experimental and theoretical study of nonlinear dynamics and acoustic signals generated by periodic impacts of corundum probe on the solid surface are conducted. In the work two models are considered for the description of experiments: the analytical model based on the laws of conservation of energy and momentum; the model based on the numerical solution of the nonlinear equation of probe motion. It is shown that the acoustic signal amplitude increases in direct proportion to the oscillations probe amplitude.

## Dynamics of roller domains at parametric excitation of capillary waves in rectangular geometry boundary

The work presents the results of experimental investigation of roller domains parametrically excited by the capillary waves. Domains rollers were oriented parallel to the di?erent borders of the rectangular cell and perpendicular to each other. Found that depending on the initial and boundary conditions on the edges of the cell can emerge two-dimensional domains of di?erent forms. The dynamics of the domain is determined by the movement of their fronts. A model is proposed to explain the observed phenomena, numerical calculations by which agree well with experiment.

## Solution of two-dimensional self-organized critical manna model

We propose a full solution for Manna model – two-dimensional conservative sandpile model with the rules of grains redistribution isotropic at the average. Indices of the probability distributions of avalanches main characteristics (size, area, perimeter, duration, topplings multiplicity) are determined for this model both from theory and from simulations. The solution bases on the spatiotemporal decomposition of avalanches described in terms of toppling layers and waves.

## Mathematical models of the world-system evolution

We propose new mathematical models of the evolution of the human society based on the synergistic approach. They describe the dynamics of the indicators of the major integral development of the World-System such as the total population and the level of the technological development. Our models capture the basic laws of the space and temporal development of the society. They indicate the hyperbolic growth of the population that agrees with the demographical data and the cyclic dynamics.

## Subharmonic resonance in a system of two dissipative coupled van der pol oscillators with external force

The problem of the excitation of two coupled oscillators is discussed in the case of the simple subharmonic resonance between the external force and eigen-frequencies of the oscillators. The corresponded phase equation is obtained.

## Sergey p. Kurdyumov and his evolutionary model of dynamics of complex systems

Sergei P. Kurdyumov (1928–2004) and his distinguished contribution in the development of the modern interdisciplinary theory and methodology of study of complex self-organizing systems, i.e. synergetics, is under consideration in the article. The matter of a mathematical model of evolutionary dynamics of complex systems elaborated by him is demonstrated. The nonlinear equation of heat conductivity serves as a basis of the model.