The papers comprising this collection are directly or indirectly
related to an important branch of mathematical physics - the
mathematical theory of wave propagation and diffraction. The paper
by V. M. Babich is concerned with the application of the
parabolic-equation method (of Academician V. A. Fok and M. A,
Leontovich) to the problem of the asymptotic behavior of eigenfunc
tions concentrated in a neighborhood of a closed geodesie in a
Riemannian space. The techniques used in this paper have been
foeund useful in solving certain problems in the theory of open
resonators. The topic of G. P. Astrakhantsev's paper is similar to
that of the paper by V. M. Babich. Here also the parabolic-equation
method is used to find the asymptotic solution of the elasticity
equations which describes Love waves concentrated in a neighborhood
of some surface ray. The paper of T. F. Pankratova is concerned
with finding the asymptotic behavior of th~ eigenfunc tions of the
Laplace operator from the exact solution for the surface of a
triaxial ellipsoid and the re gion exterior to it. The first three
papers of B. G. Nikolaev are somewhat apart from the central theme
of the col lection; they treat the integral transforms with respect
to associated Legendre functions of first kind and their
applications. Examples of such applications are the use of this
transform for the solution of integral equations with symmetrie
kernels and for the solution of certain problems in the theory of
electrical prospecting.
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