Numerical analysis of geoelectric fields in a medium with faults

An efficient method of electromagnetic field calculation is applied to the model of a multilayer nonhomogeneous medium with an arbitrary number of vertical faults. The model simulates a medium that includes the asthenosphere, the crustal layer, and the surface sedimentary layer, all joined by conducting faults. The numerical results are calculated and the effect of faults is assessed. This research was partially supported by the Russian Foundation for Basic Research (grant No. 96-05-64340) and by the Interuniversity Scientific Program “Russian Universities: Basic Research.” Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 85–93, 1998.     

Integral characteristics in the inverse problem of magnetotelluric sounding

This article examines some aspects of the one-dimensional inverse problem of magnetotelluric sounding. A uniqueness theorem is proved in the presence ofS-surfaces. A numerical algorithm based on transformation formulas is proposed. This research was partially supported by Russian Foundation for Basic Research (grant No. 96-01-00410) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

Solving elliptical equations with high-contrast coefficients in an unbounded region

An approximation model is proposed for an elliptical equation with complex rapidly varying coefficients. An efficient numerical method is developed and implemented. A problem of geoelectricity requiring solution of an equation in this setting is investigated. This research was partially supported by the Russian Foundation for Basic Research (grant No. 96-05-64340) and by the Interuniversity Scientific Program “Russian Universities: Basic Research.” Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 37–45, 1998.     

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 105–118, April, 1999.     

Junctions of singularly degenerating domains with different limit dimensions 1

The asymptotic series for solutions of the mixed boundary-value problem for the Poisson equation in a domain, which is a junction of singularly degenerating domains, are constructed. In this paper, which is the first part of the publication, the three-dimensional problem (“wheel hub with spokes”) and the analogous two-dimensional problems are considered. The methods of matched and compound asymptotic expansions are used. It is shown that a special self-adjoint extension of the operator of the limit problem in the “hub” supplied by the straight-line segments (“limits of spokes”) can be chosen as an asymptotical model of the problem in question; the extension parameters are to be some integral characteristics of the boundary-layer problems. Bibliography: 39 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 18, pp. 3–78, 1995.     

Quantum Backreaction (Casimir) Effect II. Scalar and Electromagnetic Fields

Casimir effect in most general terms may be understood as a backreaction of a quantum system causing an adiabatic change of the external conditions under which it is placed. This paper is the second installment of a work scrutinizing this effect with the use of algebraic methods in quantum theory. The general scheme worked out in the first part is applied here to the discussion of particular models. We consider models of the quantum scalar field subject to external interaction with “softened” Dirichlet or Neumann boundary conditions on two parallel planes. We show that the case of electromagnetic field with softened perfect conductor conditions on the planes may be reduced to the other two. The “softening” is implemented on the level of the dynamics, and is not imposed ad hoc, as is usual in most treatments, on the level of observables. We calculate formulas for the backreaction energy in these models. We find that the common belief that for electromagnetic field the backreaction force tends to the strict Casimir formula in the limit of “removed cutoff” is not confirmed by our strict analysis. The formula is model dependent and the Casimir value is merely a term in the asymptotic expansion of the formula in inverse powers of the distance of the planes. Typical behaviour of the energy for large separation of the plates in the class of models considered is a quadratic fall-of. Depending on the details of the “softening” of the boundary conditions the backreaction force may become repulsive for large separations. Communicated by Klaus Fredenhagen submitted 9/09/04, accepted 1/07/05     

The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of “volume” type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic “matrix” material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a “naive” continuous model. The present paper continues along these lines, exploring “normal boundary component controllability” criterion for determining achievable configurations for the controlled system in the two-dimensional case. Connections with conformal mapping lead to affirmative results for approximate controllability in this sense and Fourier series techniques provide exact controllability results for the case wherein the domain of the uncontrolled system is a two-dimensional disk.      

A PDE PROBLEM ARISING FROM CALCULATION OF THE MODEL FOR CONTINUOUS CASTING OF STEEL

This paper deals with the relationship between a certain three-dimensional elliptic problem and a two-dimensional parabolic problem, which arises from numerical calculation in the process of the continuous casting of steel.     

Hilbert–Kunz multiplicity, McKay correspondence and good ideals¶in two-dimensional rational singularities

Hilbert–Kunz multiplicity is known to be a very mysterious invariant of a ring or an ideal. We will show a very beautiful formula on Hilbert–Kunz multiplicity for integrally closed ideals in two-dimensional Gorenstein rational singularities. In the proof, “McKay correspondence” and “Riemann–Roch formula” play essential roles. Also this formula gives a new significance to “good ideals”. Received: 25 October 2000     

CHEBYSHEV SPECTRAL-FINITE ELEMENT METHOD FOR INCOMPRESSIBLE FLUID FLOW

In this paper, we propose a mixed method for solving two-dimensional unsteady vorticity equations by using Chebyshev spectral-fiuite element approximation. The generalized stability and the optimal rate of convergence are proved. The numerical results show the advantages of such method. The technique in this paper is also useful for other nonlinear problems.     

Computable classes of constructivizations of 2-constructivizable models

It is known that a class of constructivizations of a nonautostable 2-constructivizable model is infinite. Whether that class is effectively infinite or at least noncomputable is still not known, though there are grounds to believe that it is effectively infinite. Here, we argue for the effective infinity of a class of constructivizations of a nonautostable 2-constructivizable model which is not 1-prime in any finite enrichment by constants. Supported by RFFR grant No. 99-01-00485, by the Federal Research Program (FRP) “Integration,” and by the Program “Universities of Russia. Fundamental Research”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 697–722, November–December, 1999.     

Relative Haag Duality for the Free Field in Fock Representation

We consider a natural generalization of Haag duality to the case in which the observable algebra is restricted to a subset of the space-time and is not irreducible: the commutant and the causal complement have to be considered relatively to the ambient space. We prove this relative form of Haag duality under quite general conditions for the free scalar and electromagnetic field of space dimension d ≥ 2 in the vacuum representation. Such property is interesting in view of a theory of superselection sectors for the electromagnetic field. Supported by the EU network “Quantum Spaces – Noncommutative Geometry” HPRN-CT-2002-00280. Submitted: August 1, 2006. Accepted: March 2, 2007.     

A matrix solution of the pentagon equation with anticommuting variables

We construct a solution of the pentagon equation with anticommuting variables on two-dimensional faces of tetrahedra. In this solution, matrix coordinates are assigned to tetrahedron vertices. Because matrix multiplication is noncommutative, this provides a “more quantum” topological field theory than in our previous works.     

The effect of a strong light field on the spectrum shift of a hydrogen-like atom

Calculation results are presented for the transformation of the spectra of a hydrogen-like atom in an external nonhomogeneous field in the axisymmetric case and in plane geometry. Numerical results reported for prototype problems demonstrate the efficiency of the proposed algorithms. This research was partially support by grant No. 5142 of the program “Russian Universities: Basic Research.” Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 76–84, 1998.     

Numerical integration of ordinary differential equations on manifolds

Summary This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.     

Inverse problem of magnetotelluric sounding in a medium with faults

This article considers a method for solving the direct and the inverse problems in the presence ofH-polarization for the fundamental three-layer model of deep magnetotelluric sounding with vertical tectonic faults and horizontal high-conductivityS-layers in a poorly conducting crystalline matrix. The conductivities may vanish on parts of the vertical cracks or the horizontalS-layers, i.e., in this modelS-layers and vertical faults may degenerate into an insulator. This research was partially supported by the Russian Foundation for Basic Research (grant No. 96-95-64340) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

“Almost quasistationary” approximation for the problem of solidification front stability

“Almost quasistationary” approximation is suggested for the investigation of the problem of solidification front stability. It is appropriate for the initial stage of the process when sizes of particles are sufficiently small. The cases of “sphere like” and “cylinder-like” nuclei are considered. Capillary forces are taken into account.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

An approach to matrix problems

A new general approach to the so-called “matrix problems” is given. With this approach, the “derivative” of a matrix problem is again a matrix problem of the same type.