Wave field in the caustic shadow in a neighborhood of an inflection point of the boundary

The problem of the propagation of whispering gallery waves in a neighborhood of an inflection point of the boundary is considered. It is shown that a caustic shadow zone occurs away from the boundary along the normal. The asymptotics of the wave field in the caustic shadow are obtained, and their geometric interpretation in terms of complex rays is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 246–260, 1979.     

Four-momentum of the field of a point charge in nonlinear electrodynamics

We prove that the four-momentum of the electromagnetic field of a point charge is a four-vector if the field Lagrangian is nonlinear (with respect to field invariants) and the field mass is finite. We define the class of Lagrangians leading to a bound on the field mass.     

The resistive coated sphere in the presence of a point generated wave field

We consider the low‐frequency scattering problem of a point source generated incident field by a small penetrable sphere. The sphere, which is also lossy, contains in its interior a co‐ecentric spherical core on the boundary of which an impedance boundary condition is satisfied. An appropriate modification of the incident wave field allows for the reduction of the solution to the corresponding scattering problem of plane wave incidence, by moving the point source to infinity. For the near field, we obtain the low‐frequency coefficients of the zeroth and the first order. This was done with the help of the corresponding solution for the hard core problem and an appropriate use of linearity with respect to the Robin parameter. In the far field, we derive the leading non‐vanishing terms for the normalized scattering amplitude and the scattering cross‐section, which are both of the second order, as well as for the absorption cross‐section, which is of the zeroth order. The special cases of a lossy or a lossless penetrable sphere, of a resistive sphere, and of a hard sphere are recovered by an appropriate choice of the physical or the geometrical parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

A comparison of the eigenvalues of the Dirac and Laplace operators on a two-dimensional torus

We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples. Received: 10 December 1998     

Refined asymptotic formulas for eigenvalues and eigenfunctions of the Dirac system

Doklady Mathematics -     

Heat-transfer near the stagnation point of a body of revolution in the presence of a magnetic field

Zusammenfassung Die affine Lösung vonNeuringer undMcIlroy [1] für die ebene, laminare Staupunktströmung einer elektrisch leitenden Flüssigkeit in einem magnetichen Feld wird auf den rotationssymmetrischen Fall erweitert. Die gemäss dem Jouleschen Gesetz umgesetzte Wärme sowie die Wärmezufuhr durch Reibung werden dabei ohne Vernachlässigung berücksichtigt.Die Randbedingungen werden neu formuliert, und Lösungen werden erhalten in der Gestalt von gewöhnlichen Differentialgleichungen, die numerisch integriert werden.Die Grössenordnung der auftretenden Parameter wird abgeschätzt in Fällen, welche physikalisches Interesse haben. In den meisten Fällen ist es möglich, eine Grenzschicht-Annäherung einzuführen. Die Resultate hängen dann im wesentlichen nur von einem einzigen Parameter ab. Die Geschwindigkeitsverteilung, das magnetische Feld und der Wärmeübergang an der Körperoberfläche werden berechnet.     

Electromagnetic field in a neighborhood of a focus

The hf wave field near a focus in an inhomogeneous medium is described by a recurrent system of equations the solutions of which are matched with various incoming and corresponding outgoing waves. In particular, a homocentric diaphragm ray solution is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 126–136, 1976.     

On the eigenvalues of a class of saddle point matrices

We study spectral properties of a class of block 2 × 2 matrices that arise in the solution of saddle point problems. These matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system. We give conditions for having a (positive) real spectrum and for ensuring diagonalizability of the matrix. In particular, we show that these properties hold for the discrete Stokes operator, and we discuss the implications of our characterization for augmented Lagrangian formulations, for Krylov subspace solvers and for certain types of preconditioners. The work of this author was supported in part by the National Science Foundation grant DMS-0207599 Revision dated 5 December 2005.     

Semiclassical approximation of the Dirac equation in a central field

We develop a semiclassical approximation of the Dirac equation in a central field with a Coulomb asymptotic behavior. We obtain relativistic semiclassical scattering phases, energy levels of hydrogen-like ions, and a semiclassical expression for the multiplicative constant in the asymptotic expansion of the wave function of the valence electron in a relativistic multiply charged ion, which plays an important role in quantum defect theory.     

Universal dynamics in a neighborhood of a generic elliptic periodic point

We show that a generic area-preserving two-dimensional map with an elliptic periodic point is C ^ω -universal, i.e., its renormalized iterates are dense in the set of all real-analytic symplectic maps of a two-dimensional disk. The results naturally extend onto Hamiltonian and volume-preserving flows.     

Theory of quasielastic atomic reactions in the presence of an alternating electric field

We propose a variant of the theory of quasielastic (e, 2e) atomic reactions in the presence of a standing electromagnetic wave constructed in analogy with the stationary case. Along the way, we formulate mathematical problems that must be solved to justify this theory rigorously.     

Self-action of a point charge in classical electrodynamics

A direct calculation demonstrates that the causal Green function for classical equations of an electromagnetic field contains an additional singular term cancelling the divergence in the self-action of a point charge. Thus, the problem of mass renormalization is avoided. An exact relativistic expression for the self-action force is presented as a sum of two terms. The first one gives the radiation damping and the second one describes the electromagnetic component of the particle momentum depending on its velocity and acceleration. Accordingly, the work of the force also consists of two terms: the radiation energy and the electromagnetic component of the particle energy. To perform the calculations, we have to extend the radial spherical coordinate in the -function argument to negative values.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 256–269, November, 1995.     

Regularity of the free boundary in a neighborhood of a contact point in the obstacle problem

The obstacle problem for an arbitrary linear elliptic equation is considered. The regularity of the boundary of the noncoincident set is studied in a neighborhood of a contact points of the boundary of a domain. The C1-regularity of the boundary and the continuity up to contact points of the second-order derivatives of the solution are proved. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 101–104.