Faculty of Physics
M.V.Lomonosov Moscow State University
Theoretical and mathematical physics

## A Comparative Analysis of Reduction Quality for Probabilistic and Possibilistic Measurement Models

### D.A. Balakin, Yu.P. Pyt'ev

Moscow University Physics Bulletin 2017. 72. N 2. P. 101

In this article, several known and new methods of solving the measurement data interpretation problem for probabilistic and possibilistic measurement models are compared and the dependency of their quality on the completeness and accuracy of the measurement model is analyzed. It is shown that optimal use of a researcher’s prior information about the measurement model allows one to significantly increase the accuracy of the interpretation of measurements. In some cases the error of possibilistic interpretation was less than that of probabilistic one, even though possibilistic interpretation minimizes the necessity of the error, rather than the mean squared error. This is due to the fact that prior information may be sufficient to model the input signal using a fuzzy vector, but insufficient to model it using a random vector.

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## Mathematical Modeling of Randomness and Fuzziness Phenomena in Scientific Studies. II. Applications

### Yu. P. Pyt'ev

Moscow University Physics Bulletin 2017. 72. N 2. P. 113

This paper considers some elements of the optimal fuzzy decision theory that are similar to the optimal statistical decision theory, in particular, the theory of optimal fuzzy identification and optimal fuzzy hypothesis testing, such as Neyman–Pearson statistical hypothesis testing and optimal fuzzy estimation along with a sequential fuzzy identification algorithm similar to the Wald sequential statistical criterion. Some elements of the fuzzy measuring and computing transducer theory and its applications in the problems of the analysis and interpretation of measurement experiment data are given.

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## Undulators and Generation of X-Ray Pulses in Free-Electron Lasers with Self-Amplified Spontaneous Emission

### K. V. Zhukovsky

Moscow University Physics Bulletin 2017. 72. N 2. P. 128

The results of theoretical examination and comparative analysis of synchrotron radiation sources (specifically, undulators and X-ray free-electron lasers (FELs)) are presented. The problem of generation of shorter radiation pulses is prioritized; undulator systems and their corresponding FELs, which are considered to be the most promising in terms of generation of high-frequency ultrashort pulses of such radiation (in particular, in the X-ray range) are studied. The possibility of generation of higher harmonics is explored. The advantages and disadvantages of single-pass (with no reflecting elements) and multi-pass (with mirrors) FEL lasing schemes are revealed. The potential to reduce the duration of laser pulses produced by undulators and FELs and use them as sources of femtosecond pulses is investigated. The prospects for further development of X-ray free-electron lasers and the ways to improve the quality of their radiation with the given parameters are discussed.

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Astronomy, astrophysics, and cosmology

## The Current Status of Research in Ultrahigh-Energy Cosmic Ray Physics: A Brief Review

### M.Yu. Zotov$^1$, O.E. Kalashev$^2$, M.S. Pshirkov$^{2,3}$

Moscow University Physics Bulletin 2017. 72. N 2. P. 144

The origin and nature of ultrahigh-energy cosmic rays (UHECRs, E > 1018 eV) is one of the most intriguing unsolved problems of modern astrophysics. This review is dedicated to the current status of research in this field. We describe the largest ongoing experiments carried out at the Pierre Auger Observatory and Telescope Array, at the first orbital detector of UHECRs, that is, TUS, and for the KLPVE and JEM-EUSO orbital telescopes, which are currently being developed. We discuss the latest results on the energy spectrum and mass composition of UHECRs and the relationship between UHECRs on the one hand and ultrahigh-energy neutrinos and photons on the other. Finally, we review the latest results on the anisotropy of the arrival directions of UHECRs, which is a crucially important area of research in the search for astrophysical sources of cosmic rays in the highest energy range.

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Theoretical and mathematical physics

## Analytic Solution of Transversal Oscillation of Quintic Non-linear Beam with Energy Balance Method and Global Residue Harmonic Balance Method

### M. Akbarzade , A. Farshidianfar

Moscow University Physics Bulletin 2017. 72. N 2. P. 157

Beams are very important element in structures because of its widespread usage in steel construction such as buildings, bridges, aircraft arm, aerospace vehicles and many other industrial usages. To attain a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. The main objective of present study is to obtain highly accurate analytical solutions for vibrations of a beam with quintic nonlinearity. Two new analytical methods are the improved energy balance method, and the global residue harmonic balance method. Subsequently, the analytical results are compared with the numerical solution of the exact equation in order to evaluate the correctness of the applied approaches.

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## A Numerical Solution of the Layer-to-Layer Transition Problem as a Waveguide Diffraction Problem

### A.L.Delitsyn

Moscow University Physics Bulletin 2017. 72. N 2. P. 163

Application of net-point methods to the solution of the layer-to-layer transition problem leads to the necessity to restrict the domain and formulate artificial boundary conditions. In the present work the introduction of an artificial coaxial for reducing the original problem to a waveguide diffraction problem is applied. The test results of the finite-element program that we developed, which demonstrate the high accuracy of the method, are considered. The investigated dependence of the solution on the artificial boundary location shows the rapid convergence of the method.

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## The Empirical Construction of Mathematical Models of Measuring and Optimal Computing Transducers

### D.A. Balakin

Moscow University Physics Bulletin 2017. 72. N 2. P. 168

—It follows from the theory of measuring-computing systems [1] that the requirements for a measuring transducer (MT) that converts an external influence into an electric signal in order to obtain the maximal interpretation accuracy are significantly different depending on how it is going to function: by itself or as a part of a measuring-computing transducer (MCT) as its component. In the first case, the maximal accuracy has to be provided by the MT and it is bounded by physical laws. In the second case, it has to be provided by the MCT, which is considered to be a measuring device for the same or a different purpose, in particular, that of the “perfect” device for a researcher. The accuracy of an MCT is determined by both the mathematical properties of its model and the “quality” of the algorithm that converts MT output into a form determined by the measurement objective that is achieved by a computing transducer (CT) as a component of the MCT. For any specific mathematical model of the MCT this algorithm has to provide the maximal quality of the MCT. An MT that is optimal for that purpose can often be different from an MT that is optimal by itself. As a rule, the exact mathematical model of the MT and, hence, the algorithm of the CT that is optimal for it, are not known to the researcher, but he can perform test measurements of known input signals that simulate the interaction of the MT and the measured object using the MT. The aim of this article is to use test measurements to synthesize both the response of an MT with an unknown model and the optimal interpretation of the measurement result, i.е., the output signal of the MCT. It is shown that even without knowing the exact MT model, but with the ability to perform test measurements on the same MT a researcher is able to synthesize both the MT response and the results of the interpretation of measurement results, both in an optimal way.

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## On the Properties of the Four-Momentum of an Isolated Charge + Field System

### M. B. Ependiev

Moscow University Physics Bulletin 2017. 72. N 2. P. 176

The question of the isolation of a charged particle along with its field such that the four-momentum of an entire system is a four-vector is raised. The relationships between the characteristics of a particle and a field upon such isolation have been obtained. In particular, it has been demonstrated that within the considered nonclassical models the particle mass should be greater than one-third of the field mass.

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## Radiative Effects in Quantum Electrodynamics with Lorentz Violation

### A.V. Borisov, T.G. Kiril'tseva

Moscow University Physics Bulletin 2017. 72. N 2. P. 182

We calculate one-loop mass and vertex (at zero momentum transfer) operators of an electron in a weak background tensor field that violates Lorentz invariance. The upper bounds on the background field strength are obtained using modern experimental values of the electron mass and charge.

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Chemical physics, physical kinetics, and plasma physics

## Synthesis, structural and dielectric properties of SrBi$_{2-x}$Sm$_{x}$Nb$_2$O$_9$

### Mohamed Afqir$^{1,2}$, A. Tachafine$^2$, D. Fasquelle$^2$, M. Elaatmani$^1$, J. C. Carru$^2$, A. Zegzouti$^1$, M. Daoud$^1$

Moscow University Physics Bulletin 2017. 72. N 2. P. 196

The SrBi$_{2-x}$Sm$_{x}$Nb$_2$O$_9$ (x=0, 0.4, 0.5) ceramics were synthesized by a conventional solid-state method. The X-ray diffraction shows an orthorhombic at room temperature. Dielectric constant, loss tangent and Ac conductivity of Sm-doped SrBi$_2$Nb$_2$O$_9$ were carried out, as well bel. A higher concentration of samarium (x= 0.4 and 0.5) doping leads to a significant reduction in peak dielectric constant and in Curie temperature. Rather relaxor type of behavior took place.

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Condensed matter physics

## Vacancy defects in delafossite CuAlO$_2$: First-principles calculations

### Mi Zhong$^{1,2}$, Qi-Jun Liu$^{1,2}$, Zhen Jiao$^{1,2}$, Fu-Sheng Liu$^{1,2}$, Zheng-Tang Liu$^3$

Moscow University Physics Bulletin 2017. 72. N 2. P. 191

Electronic properties and formation energies of vacancy defects in delafossite CuAlO2 have been investigated by using the first-principles density functional theory. The band structures and density of states of various vacancy defects have been obtained and analyzed. The results show that the VCu systems with different charge states influence the type of conductivity. The introduced vacancy defects enhance the hybridization between O-2p and Cu-3d states, which is good for p-type conductivity. The calculated formation energies indicate that the Cu vacancy is relatively easy to form and it trends to have positive charge.

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Physics of nuclei and elementary particles

## New Energy Estimates of Extensive Air Showers Using Signals Detected at Great Distances from the Shower Axis

### L.G. Dedenko$^{1,2}$, T.M. Roganova$^2$, G.F. Fedorova$^2$

Moscow University Physics Bulletin 2017. 72. N 2. P. 187

On the basis of detailed tables of signals from Yakutsk extensive air shower scintillation detectors that were calculated using the GEANT4 package, the numerical evaluation of the energy Eθ of inclined extensive air showers has been obtained using a pilot signal at a distance of 1000 m from the shower axis and the known zenith angle θ using the CORSIKA package. This original estimate was obtained without using the method of cutting the signal spectra according to lines of equal intensity. We also obtained the values of practical units of measure for signals in ground-based and underground detectors.

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