### Article

## Universal Scaling of Thin Current Sheets

Thin current sheets (TCSs) with thicknesses about ion Larmor radii are widespread in space. It

is important to describe their equilibrium structure allowing them to store and then explosively release the

accumulated free energy. When ions are moving along quasi‐adiabatic trajectories while magnetized

electrons follow guiding center drift orbits, TCSs can be described within the framework of a hybrid

approach. The thickness of the embedded electron sheet remains uncertain because of the scale‐free

character of electron motion. In this work, we propose a novel analytical model of the multilayer TCS that

provides a universal expression describing the inner (embedded) electron sheet in dependence of TCS

characteristics. An unusual property of the embedded electron layer revealed in this analysis is the nonlinear

profile of the magnetic field in the inner layer: B(z) ~ z1/3, which conforms excellently with MAVEN

observations of 43 TCSs in the Martian magnetotail.

We outline the history and development of the theory of thin current sheets in a collisionless space plasma from the early ideas of V L Ginzburg and S I Syrovatskii to the present day. We review the key achievements of the quasi- adiabatic theory, which provided insight into the fine structure of thin current sheets and enabled a comparison with experi- ment. This comparison showed the quasi-adiabatic approach to be more effective than the classical MHD approximation. With the development of the quasi-adiabatic theory in the last two decades, the existence of a number of new thin current sheet features, such as multi-scaling, metastability, and em- bedding, has been predicted and subsequently confirmed in situ; the role of individual particle populations in the forma- tion of the current sheet fine structure has also been investi- gated. The role of nonadiabatic effects in accelerating plasma beamlets interacting with current sheets is examined. Asym- metry mechanisms in thin current sheets in the presence of a magnetic shear component are described. A study is carried out of current sheet self-organization processes leading to the formation of a shear magnetic component consistent with currents flowing in the plasma. It is demonstrated that the ongoing development of the theory of thin current structures is a logical continuation of Syrovatskii's and Ginzburg's ideas on cosmic rays and reconnected current sheets in the solar corona.

We consider two random walkers starting at the same time t = 0 from different points in space separated by a given distance R. We compute the average volume of the space visited by both walkers up to time t as a function of R and t and dimensionality of space d. For d < 4, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable R^2/t. We provide general integral formulas for scaling functions for arbitrary dimensionality d < 4. In contrast, we show that no scaling function exists for higher dimensionalities d more or equal to 4.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables