Wood's anomalies in the scattering problem of a plane wave on a smooth periodic boundary

A series of multiple diffracted fields in a high-frequency asymptotic expansion of the solution to the problem of grazing scattering of a plane wave by a smooth periodic boundary is summed in the case of appearance of Wood's anomalies. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 87–100, 1990. Translated by V. V. Zalipaev.     

Diffraction of a plane wave by a step discontinuity in an impedance plane

In the present paper, the high-frequency diffraction of a plane wave by a right-angle step discontinuity in an impedance plane is analyzed with the help of the uniform geometric theory of diffraction, beginning with the Maluzhenets solution. The principal term of an asymptotic solution of the problem, which is uniform with respect to the angle of incidence of the plane wave and the angle of observation, is derived. The excitation of primary and multiply diffracted fields radiated from the upper edge, as well as surface waves, is considered (the lower edge does not radiate cylindrical or surface waves owing to a right-angle step). For simplicity, the details of computation are given here for a right-angle step discontinuity. A similar procedure is applied to other examples with more complicated geometry. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 97–108. Translated by V. V. Zalipaev.     

On free boundary problems with moving contact points for the stationary two-dimensional Navier-Stokes equations

Solvability of the problem of slow drying of a plane capillary in the classical setting (i. e., with the adherence condition on a rigid wall) is established. The proof is based on a detailed study of the asymptotics of the solution near a point of contact of the free boundary with a moving wall, including estimates of the coefficients in well known asymptotic formulas. It is shown that the only value of the contact angle admitting a solution of the problem with finite energy dissipation equals π. Bibliography: 18 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 179–205. Translated by E. V. Frolova.     

Shortwave scattering by a diffraction echelette-grating

The two-dimensional problem of the scattering of a plane wave by a periodic perfectly conducting grating, an echelette with a right angle, is considered in the case of a high-frequency approximation (the wavelength is assumed to be small as compared with the period of the grating). The situation where the incident plane wave glides along one of the faces of a wedge is discussed. A ray-optical solution of the problem (a shortwave asymptotic result) is derived by the method of summing multiple diffracted fields, which is well known in the geometric theory of diffraction. The main result of this paper consists of obtaining simple formulas for the efficiency e_n of a diffraction order with maximal value of e_n, derived in the shortwave approximation. Numerical results are presented, and important optical properties obtained by asymptotic analysis are described. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 109–135. Translated by V. V. Zalipaev.     

Monodromy and irreducibility criteria with algorithmic applications in characteristic zero

Consider a projective algebraic variety V, which is the set of all common zeros of homogeneous polynomials of degrees less than d in n + 1 variables over a field of characteristic zero. We suggest an algorithm that decides whether two (or more) given points of V belong to the same irreducible component of V. We also show how to construct, for each s < n, an (s + 1)-dimensional plane in the projective space such that the intersection of every irreducible component of dimension n — s of V with the constructed plane is transversal and is an irreducible curve. These algorithms are deterministic and polynomial in dⁿ and the input size. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 130–152.This revised version was published online in April 2005 with a corrected cover date and article title.     

Smoothness near the boundary of the solutions of the elliptic Bellman equations

One considers Bellman's elliptic equation with constant coefficients and zero boundary values on a plane part of the boundary. In this case one gives a simplified proof of N. V. Krylov's result regarding the boundary estimates of the Hölder constants of the second derivatives of the solutions of the Bellman equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 150–154, 1985.     

Some special solutions of the Helmholtz equation

Functions are constructed which describe a cylindrical wave diverging from the origin and tangent to a semi-infinite plane wave. The use of such functions is illustrated by the example of the problem of diffraction of a plane wave by a semi-infinite screen.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 197–202, 1975.     

Search for shortest path around semialgebraic obstacles in the plane

An algorithm is described which reduces in polynomial time the problem of constructing a shortest path (between two points) around semialgebraic obstacles in the plane to the problem of constructing a path of smallest weight in a graph the weights of whose vertices are integrals of positive algebraic functions (without singularities). As consequences of this construction one can get algorithms of polynomial complexity for constructing approximations to shortest paths. One such algorithm is given in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 163–173, 1991.     

Singularities of weak solutions of elliptic equations with discontinuous leading coefficients.

For the weak solutions of second-order equations in a plane bounded domain, one obtains the asymptotics and estimates of the solutions near the interior angular points of the line of discontinuity of the coefficients. It is established that the number of terms of the asymptotics, not in the Sobolev classW ₂ ² , can be arbitrarily large.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 47, pp. 166–169, 1974.     

Problem of the motion of two compressible fluids separated by a closed free interface

We consider the problem of the simultaneous evolution for two barotropic capillary viscous compressible fluids occupying the space ℝ³ and separated by a closed free interface. Under some restrictions on the viscosities of the liquids, the local (in time) unique solvability of this problem is obtained in the Sobolev-Slobodetskii spaces. After the passage to Lagrangian coordinates it is possible to exclude the fluid density from the system of equations. The proof of the existence theorem for a nonlinear, noncoercive initial boundary-value problem is based on the method of successive approximations and on an explicit solution of a model linear problem with a plane interface between the liquids. The restrictions on the viscosities mentioned above appear in the intermediate estimation of this explicit solution in the Sobolev spaces with an exponential weight. Bibliography: 8 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 000, 1997, pp. 61–86. Translated by I. V. Denisova.     

Inverse problems of the linear theory of radiation transport in a plane layer of a medium scattering and absorbing radiation

An analytical method of investigating inverse problems of linear transport theory in a plane layer is proposed in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 123–133, 1981.     

Solutions of nonlinear equations concentrated near curves in the plane

A method of successive approximations is considered which makes it possible to construct formal solutions of some nonlinear equations concentrated near curves in the plane.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 84–90, 1979.In conclusion, the author thanks V. M. Babich, A. Kachalov, and V. Nomofilov for fruitful discussions and valuable suggestions.     

Numerical Study of Properties of Quasilocal Plane Waves of the Modal Type in the Case of a Thin Low-Velocity Layer that is in Contact with an Elastic Half-Space

Numerical study of properties of quasilocal plane waves of the modal type propagating deep into a medium is carried out with the example of the model of a low-velocity elastic layer in the case of rigid contact with the underlying half-space. It is established that the genesis of these waves is closely related to singular complex roots of the dispersion equation of the problem. Eighteen variants of the model differing by relative parameters of the problem, which have a physical meaning, are considered. For every variant, seismograms of modal and body waves are computed and a comparison of them in intensity is carried out. Bibliography: 4 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 182–196.     

The cohomology ring of the colored braid group

The cohomology ring is obtained for the space of ordered sets of n different points of a plane.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 227–231, February, 1969.The author thanks V. P. Palamodov and D. B. Fuks for useful discussions.     

Computation of the second term of the series of the ray method in the dimension two and a half

An algorithm for computing the second term of the series of the ray method suggested in this paper is specified for the case of the dimension two and a half. By this is meant that the elastic paramenters of a medium depend on x, z only and the source and observation points are placed in a plane orthogonal to the y axis. Thus, to compute the wave field we need to take into account the rays which are placed in the plane indicated. This enables us to simplify the algorithm. We discuss the algorithm in detail for the case of a point source. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp 94–108 Translated by N. Ya. Kirpichnikova.     

Boundary-value problem for the two-dimensional elliptic sine-Gordon equation and its application to the theory of the stationary Josephson effect

The boundary-value problem for a half plane for the elliptic sine-Gordon equation is solved using the inverse scattering method. The solution is applied to the theory of the stationary Josephson effect.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 53–62, 1990.     

A nonuniform estimate of the remainder in the central limit theorem

In this note, a refinement and generalization of one result of V. V. Petrov are obtained. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 238–241.     

Nonstability of the inversion of the radon transform

Several examples of the distributions on the plane for which the distance in variation between their projections on an arbitrary one-dimensional direction is less than or equal to ε, but the uniform distance between their two-dimensional distribution functions is equal to 1/2, are constructed. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 76–85. Translated by the author.     

Almost isometric operators: Functional model,invariant subspaces,commutant

A new function model for an arbitrary bounded operator on a Hilbert space is constructed. This model generalizes the model of Sz.-Nagy and Foiaş for contractions and seems to be useful for operators close to an isometry (in a sense). All the model spaces are Hilbert spaces, but instead of dilation a generalization of it is used. The model admits a symmetry with respect to the map z→1/z of the complex plane. In terms of the model the question of lifting the commutant is investigated, a relationship between invariant subspaces of a unitary operator is established, and the characteristic function of the model operator is calculated. Some other problems are solved as well. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 95–116. Translated by V. Vasyunin.     

Tightenable curves and the Möbius theorem on three points of inflection

Conclusion The following conjecture generalizes the above methods, and its validity would imply that each curve on the projective plane that can be obtained from a line by a homotopy without dangerous self-tangencies must contain at least three points of inflection. Partially supported by the Russian Foundation for Basic Research (project No. 96-01-01104). M. V. Lomonosov Moscow State University, Department of Mechanics and Mathematics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 32, No. 1, pp. 29–39, January–March, 1998. Translated by A. I. Shtern