Excitation coefficients of whispering gallery waves in a neighborhood of a flat point of a concave boundary

A method is proposed for the computer calculation of the excitation coefficients of whispering gallery waves which arise beyond a flat point of a concave boundary when a single whispering gallery wave is incident on this point. Values of the moduli of these coefficients are presented for the incidence of the first and second whispering gallery waves. From the results obtained it follows that in both cases considered the major portion of the energy of the incident wave (27 and 23%, respectively) goes over into the first wave beyond the flat point of the boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 138–145, 1980.The authors are grateful to I. Pshenchik (Geophysical Institute, Prague) for his assistance and advice.     

Diffraction of whispering gallery waves by a conjunction line

A nonhomogeneous elastic body with stress-free boundary is considered. The boundary consists of two smooth cylindrical surfaces with continuous tangent plane. The curvature of the boundary is assumed to have a discontinuity on the conjunction line. The behavior of two kinds of transversal surface whispering gallery waves passing through the conjunction line is studied. For a wave of the first kind, corresponding to the Dirichlet boundary condition, the displacement vector is normal to the boundary. For a wave of the second kind, corresponding to the Neumann boundary condition, the displacement vector is tangent to the boundary and normal to the ray, which is similar to the case of Love waves.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 95–109.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Number of quasimodes of whispering gallery type

One presents a new two-scale expansion for the eigenfunctions of the whispering gallery type and the corresponding eigenvalues of the Laplace operator with the Dirichlet or Neumann condition in a plane domain.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 152–157, 1983.     

Radiation field of whispering gallery waves over a concave-convex boundary

In this work the wave field arising over a concave-convex reflecting boundary is studied in the Kirchhoff approximation. The field arises as a result of the incidence of whispering gallery waves on an inflection point of the boundary from the concave side. The shortwave asymptotics of the Kirchhoff integral are obtained which is expressed in terms of special functions in a neighborhood of the inflection point of the boundary and in a neighborhood of the tangent to the boundary at the inflection point. Diagrams are constructed that illustrate the behavior of the scattered field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 49–65, 1981.     

Uniform asymptotics of eigenfunctions of whispering gallery type

An application of the asymptotic method of uniform expansions is illustrated in finding the eigenfunctions of the Laplace operator that are concentrated near the boundary of a domain and satisfy Dirichlet or Neumann conditions on the boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 112–119, 1979.The author is grateful to V. M. Babich for valuable remarks and assistance.     

On whispering gallery wave propagation in a medium with a vertical separating boundary

The solution of the problem of transmission of a whispering gallery wave through the separating boundary is constructed in the parabolic equation approximation. Formulas for the transformation coefficients of the normal modes are constructed in the case of the Dirichlet and Neumann problems. Computational results are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 147–151, 1989.     

Whispering gallery waves in a neighborhood of a flat point of a concave boundary

The propagation of whispering gallery waves near a concave (from the side of the wave field) boundary having a flat point is studied. As in [4], the problem that arises for an equation of Schrodinger type is solved by the method of grids. The results of the computations are presented as shadow figures.     

Problem of whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary

The mathematical formulation of the problem on the propagation of whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary is given for the principal term of the short-wave asymptotics, and the uniqueness of the solution of this problem is demonstrated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 197–206, 1976.     

Correctness of the problem of whispering gallery waves in a neighborhood of points of zero curvature of the boundary

Problems of the propagation of whispering gallery waves in a neighborhood of boundary points where the curvature has a zero of multiplicity one to two are considered. The existence and uniqueness of a solution of these problems are established by means of wave operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 261–269, 1979.     

Numerical solution of the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary

A method of numerically solving the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary is described, and results are presented of calculating the wave field in the case of the first and second whispering gallery waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 207–219, 1976.     

The Dedekind-MacNeille completion as a reflector

We introduce a special type of order-preserving maps between quasiordered sets, the so-called cut-stable maps. These form the largest morphism class such that the corresponding category of quasiordered sets contains the category of complete lattices and complete homomorphisms as a full reflective subcategory, the reflector being given by the Dedekind-MacNeille completion (alias normal completion or completion by cuts). Suitable restriction of the object class leads to the category of separated quasiordered sets and its full reflective subcategory of completely distributive lattices. Similar reflections are obtained for continuous lattices, algebraic lattices, etc.     

The gallery distance of flags

An explicit formula for the gallery distance of two maximal flags in a vector space is given. The main tool of the proof is the Jordan-Hölder permutation. The result and its proof hold more generally for any semimodular lattice of finite height and with minor changes also for the distance of two chambers in the Bruhat-Tits building of the general linear group.     

The point source field near a concave boundary

A new representation for the field emitted by a short-wave source placed near a concave boundary is obtained by the hybrid method. A Dirichlet problem is considered. The representation obtained provides an efficient algorithm for numerical experiments. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 212–219.     

Comparison of the field of a whispering gallery wave computed by Kirchhoff's method and by the method of the parabolic equation in a neighborhood of zero curvature of the boundary

In the paper a comparison is made of results obtained by Kirchhoff 's method and by the method of the parabolic equation in solving the problem of whispering gallery waves propagating over a concave — convex boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 139–145, 1981.     

The dependency diagram of a linear programme

The Dependency Diagram of a Linear Programme (LP) shows how the successive inequalities of an LP depend on former inequalities, when variables are projected out by Fourier–Motzkin Elimination. It is also explained how redundant inequalities can be removed, using the method attributed to Chernikov and to Kohler. Some new results are given. The procedure also leads to a transparent explanation of Farkas’ Lemma, LP Duality, the dual form of Caratheodory’s Theorem as well as generating all vertices and extreme rays of the Dual Polytope.     

The affine deformation space of a rank two Schottky group: a picture gallery

The Margulis invariant of an affine hyperbolic element measures the signed Lorentzian displacement along the unique closed geodesic in its class. Given a group of affine transformations of Minkowski spacetime whose linear part is Schottky, the Margulis invariant is a useful tool in determining properness of its action. This paper, based on a presentation given at CGG IV Oostende 2005, describes the affine deformation space of a rank two Schottky group. Several illustrations are included, contrasting the difference between the case of a three-holed sphere and that of a one-holed torus. Paper based on a presentation given at CGG IV Oostende 2005. Attendance of the conference and the subsequent writing of this paper was made possible by an NSERC Discovery Grant.     

Optimization for products of concave functions

Ifh denotes the product of finitely many concave non-negative functions on a compact interval [a, b], then it is shown that there exist pointsα andβ witha≤α≤β≤b such thath is strictly increasing on [α, α), constant on (α, β), and strictly decreasing on (β, b]. This structure theorem leads to an extension of several classical optimization results for concave functions on convex sets to the case of products of concave functions.     

On a pull-back diagram for orthodox semigroups

In the present paper we deal with two problems concerning orthodox semigroups. M. Yamada raised the questions in [6] whether there exists an orthodox semigroup T with band of idempotents E and greatest inverse semigroup homomorphic image S for every band E and inverse semigroup S which have the property that is isomorphic to the semilattice of idempotents of S, and if T exists then whether it is always unique up to isomorphism. T. E. Hall [1] has published counter-examples in connection with both questions and, moreover, he has given a necessary and sufficient condition for existence. Now we prove a more effective necessary and sufficient condition for existence and deal with uniqueness, too. On the other hand, D. B. McAlister's theorem in [4] saying that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup is generalized for orthodox semigroups. The proofs of these results are based on a theorem concerning a special type of pullback diagrams. In verifying this theorem we make use of the results in [5] which we draw up in Section 1. The main theorems are stated in Section 2. For the undefined notions and notations the reader is referred to [2].