This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers' equation in the limit of vanishing viscosity.It is well known that Burgers turbulence with a power law energy spectrum E 0 (k) ∼ |k| n has a self-similar regime of evolution. For n < 1 this regime is characterised by an integral scale L(t) ∼ t 2/(3+n) , which increases with the time due to the multiple mergings of the shocks, and therefore, the energy of a random wave decays more slowly than the energy of a periodic signal. In this paper a deterministic model of turbulence-like evolution is considered. We construct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wavenumbers of this function form a "Weierstrass spectrum", which accumulates at the origin in geometric progression. "Reverse" sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in Burgers turbulence. Combining these two ideas allows us to obtain an exact analytical solution for the 1 velocity field. We also notice that such multiscale waves may be constructed for multidimensional Burgers' equation.This solution has scaling exponent h = −(1 + n)/2 and its evolution in time is selfsimilar with logarithmic periodicity and with the same average law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-like structures. This model also describes important properties of the Burgers turbulence such as the self-preservation of the evolution of large scale structures in the presence of small scales perturbations.PACS 43.25.Cb,47.27.Eq.

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The social-network formation and analysis is nowadays one of objects that are in a focus of intensive research. The objective of the paper is to suggest the perspective of representing social networks as graphs, with the application of the graph theory to problems connected with studying the network-like structures and to study spreading activation algorithm for reasons of analyzing these structures. The paper presents the process of modeling multidimensional networks by means of directed graphs with several characteristics. The paper also demonstrates using Spreading Activation algorithm as a good method for analyzing multidimensional network with the main focus on recommender systems. The experiments showed that the choice of parameters of the algorithm is crucial, that some kind of constraint should be included and that the algorithm is able to provide a stable environment for simulations with networks.

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Here we study numerically the structure of directed state transition graphs for several types of finite-state devices representing morphology of 16 languages. In all numerical experiments we have found that the distribution of incoming and outcoming links is highly skewed and is modeled well by the power law, not by the Poisson distribution typical for classical random graphs. Studied for three languages, distribution of nodes according to the traffic they experience during corpora processing obeys the power law as well. Traffic and out-degree are the parameters, which affect performance of finite-state devices. We discuss how specific properties of power law, like distribution of these parameters (coexistence of small number of "hubs" with large number of "small events"), can be exploited for efficient computer implementation of finite-state devices used in morphology.