Symmetrie

Friedrich Jaecker studied with György Ligeti at the Hochschüle für Musik in Cologne. Jaecker is deeply expert in the music of Feldman and Scelsi, both of whom influence his music. American composer and instrument builder Harry Partch developed a system of just intonation, in which the exact tuning of spectral sounds and their inversions are made possible. In the composition Harrys Dream, each of the thirty-three musicians is responsible for a single note to be played on a tuned wine glass. Each glass tuned precisely to the (microtonal) cent helps to build a tremendous glass harp, in which the listener is enveloped by a mass of sounds, moving from dense to transparent, from fast to slow.

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 föund 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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This introduction to Linear Algebra and Group Theory will, it is hoped, serve as a stepping stone to students of Physics, in particular of Theoretical Physics and possibly also of Mathematics, who wish to pur- sue advanced study and research. The first few chapters on elementary Group Theory and Linear Vector Spaces are included to make the book self-contained and mayaIso be used as instructional material for grad- uate and under-graduate classes. Since there are several good books on application of Group Theory to Physical Problems, the emphasis here is almost entirely on the theory which is presented with enough care and detail to enable the student to acquire a reasonably sound grasp of the fundamentals.