On the stability of uniformly rotating liquid in a weak magnetic field

We analyze the simplest free boundary problem of magnetohydrodynamics governing the evolution of an isolated mass of a viscous incompressible liquid in the presence of the magnetic field. The motion of the liquid is governed by the Navier–Stokes equations, and for the magnetic field we have the Maxwell equations with an excluded displacement current. The magnetic field should be determined not only in the domain filled with the liquid, but also in the surrounding vacuum region. On the free boundary of the liquid standard jump conditions for the magnetic field are prescribed, as well as kinematic and dynamic boundary conditions, where the magnetic stress tensor is taken into account. We prove that the solution corresponding to a rigid rotation of the fluid and to zero magnetic field is stable if the functional of potential energy has a positive second variation. Bibliography: 11 titles.     

Class field theory for a product of curves over a local field

We prove that the kernel of the reciprocity map for a product of curves over a p-adic field with split semi-stable reduction is divisible. We also consider the K ₁ of a product of curves over a number field.      

Vacuum solutions in the Neveu-Schwarz field theory of a fermionic string and the zero-curvature representation in graded spaces

We demonstrate that vacuum solutions in the Neveu-Schwarz field theory of a fermionic string incorporating the GSO(−) sector are naturally related to the zero-curvature representation in a graded space of special form. We use the same representation to describe the equivalence of the cubic and nonpolynomial theories with the GSO(−) sector also taken into account. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 131–141, April, 2009.     

Vibrations of a conductive string in a nonstationary magnetic field under presence of two nonlinear factors

We consider vibrations of a conductive string with fixed ends in a magnetic field. Induction of the magnetic field is a preassigned function of time. Two nonlinear factors are taken into account simultaneously: the variation of string tension with displacement and the magnetostrictive effect. It is shown that, in the case of a periodic magnetic field, the nonlinear factors can compensate each other and the problem can be reduced to studying the linearized parametric vibrations.     

Quantum equations of motion for multimode laser generation with a spatial dependence of the atom interaction with the field taken into account

We derive equations of motion for the electromagnetic field operators a_q′ ⁺ a_q″ for a three-level multimode laser with a spatial dependence of the interaction of atoms with the field of a standing wave in a cavity taken into account. We calculate and analyze the dynamics of means of photon numbers in the field modes and of the correlation function of field modes. We explore the effect of intermode correlations on the dynamics of establishing stationary laser generation. We find that taking the spatial dependence of the interaction of atoms with the field and the intermode correlation into account in investigating the means of photon numbers leads to revealing new properties of laser generation, such as saturation of the laser radiation intensity in a single-mode regime and generation of short light pulses of side below-threshold modes with the amplitudes depending on the initial state of the field in a cavity.     

Magnetic field in a turbulent conducting fluid

The problem of magnetic field in conducting turbulent, incompressible fluid is considered. The velocity of the fluid is taken to be independent of the magnetic field and is described by a Gaussian field, ‘white noise’ in time with smooth space correlation. The main result is that no fast dynamo (by which is meant almost sure exponential growth of magnetic field) can exist for an incompressible fluid when the magnetic viscosity is positive. For d = 2, sharper results are obtained; the magnetic field dies out when the magnetic viscosity is strictly positive. Furthermore, when d = 2, existence and characterization of invariant measure are given for d = 2 when the magnetic viscosity is zero. The results are compared to those discussed by Baxendale and Rosovskii in [2]     

Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field

We present complete results devoted to the study of the equations of motion of a dynamically symmetric four-dimensional rigid body in a nonconservative force field. The form of the body is taken from the dynamics of real two- or three-dimensional rigid bodies interacting with a resisting medium according to the streamline flow around laws under which a non-conservative pair of forces acts on the body and forces the body center of mass to move rectilinearly and uniformly.     

Efficiency of a DC magnetic field used for braking the flow within a continuous casting mould

The continuous casting technology provides about 90 percent of the world steel production. The application of DC magnetic fields in form of so-called electromagnetic brakes is considered for an effective flow control with substantial capabilities to improve the product quality or to enhance the productivity of the process. The main effect of the DC magnetic field is supposed to result in a uniform reduction of the maximum velocities in the discharging jet from the submerged entry nozzle and to damp violent turbulent fluctuations. However, the electromagnetic braking of such highly turbulent and complex flows is complicated phenomenon and has not been understood fully until now. We present numerical and experimental investigations focusing on the fluid flow in the continuous casting mould under the influence of a transverse magnetic field. Numerical calculations were performed using the software package CFX with an implemented RANS-SST turbulence model. the non-isotropic nature of the MHD turbulence was taken into account by specific modifications of the turbulence model. Corresponding experimental investigations were carried out at the mock-up LIMMCAST at HZDR. The comparison between our numerical calculations and the experimental results display a very well agreement. An important outcome of this study is the feature that the magnetic field does not provide a continuous reduction of the velocity fluctuations at the nozzle port. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Supersymmetric electron in a Coulomb field

We analyze the motion of an electron in a Coulomb field using the Di Vecchia-Ravndal model for a supersymmetric Dirac particle. We find an exact analytic solution of the spatial part of the pseudoclassical Bargmann-Michel-Telegdi equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 75–80, April, 2000.     

Spectrally arbitrary, nonderogatory factorization over a general field

It is known that a nonsingular, nonscalar matrix A, over the complex field, may be factored as A=BC, in which the spectra of B and C are arbitrary, subject to detBdetC=detA, and that B and C may be taken to be nonderogatory. The purpose of this paper is to establish this result over a general field with at least four elements.     

Rayleigh-type isoperimetric inequality with a homogeneous magnetic field

We prove that the two dimensional free magnetic Schrödinger operator, with a fixed constant magnetic field and Dirichlet boundary conditions on a planar domain with a given area, attains its smallest possible eigenvalue if the domain is a disk. We also give some rough bounds on the lowest magnetic eigenvalue of the disk.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field

This work is a relatively final result in studying the equations of motion of a dynamically symmetric, four-dimensional rigid body in a nonconservative force field in two logically possible cases of its tensor of inertia. The form of the force field considered is taken from the dynamics of real three-dimensional rigid bodies interacting with a medium.     

Generalized power series on a non-Archimedean field

Power series with rational exponents on the real numbers field and the Levi-Civita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the Levi-Civita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.     

Degenerate plasma in a half-space under an external electric field

An analytic solution is found for the problem on the behavior of a collisional plasma in a variable external electric field. We elucidate the structure of the screened electric field and investigate the case where the frequency of the external field is close to the plasma oscillation frequency (resonance). We show that there are two near-surface layers where the behavior of the screened field differs essentially. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 487–502, June, 2006.     

Complexity of prime-dimensional sequences over a finite field

V.?I. Arnold has recently defined the complexity of a sequence of n zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complex sequences of elements of a finite field, whose dimension n is a prime number. We prove that, with n→∞, this property is inherent in almost all sequences, while the values of multiplicative functions possess this property with any n different from the characteristic of the field. We also describe the prime values of the parameter n which make the logarithmic function almost most complex. All these sequences reveal a stronger complexity; its algebraic sense is quite clear.     

Mahler measures in a cubic field

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E. This extends the result of Schinzel who proved the same statement for every real quadratic field E. A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.     

Estimation of a linear functional and extrapolation of a random field

We consider minimax estimation of a linear functional of a homogeneous random field. A linear optimal estimator of the functional is derived and the field achieving the minimax error of the functional estimate is determined.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 105–113, 1986.     

Ramification theory for varieties over a local field

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an ?-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic. We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin character for a group action with an isolated fixed point on a regular local ring, assuming the dimension is 2.     

Electroweak Nucleon Decays in a Superstrong Magnetic Field

We study the influence of a magnetic field on the electroweak processes of nucleon decay in a degenerate ideal gas of neutrons, protons, and electrons situated in an external superstrong constant and homogeneous magnetic field with effects due to the interaction of nucleon anomalous magnetic moments with the magnetic field taken into account. For different values of the chemical potentials of degenerate fermions, we obtain expressions for probabilities of electroweak processes, which are assumed to be responsible for the chemical equilibrium in the central domain of a neutron star with a frozen superstrong magnetic field. We show that the difference between the neutron decay probabilities in the presence of a magnetic field B ≪ 10¹⁷ G and without this field is completely determined by changing the phase volume of electron states. We discuss the process of proton decay into a neutron, positron, and neutrino. This process is energetically allowed only when the interaction of nucleon anomalous magnetic moments with a superstrong magnetic field is taken into account. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 108–122, October, 2005.