Nonstationary interference wave fields of SV type in a free elastic layer and the problem of ultrasonic simulation of plane seismic fields by using plate models

The problem of quantitatively studying wave fields in a free elastic layer excited by a nonstationary point source is discussed. Much attention is given to the low-frequency part of the Fourier spectrum. The roots of the dispersion equation of the layer are studied comprehensively, and on this basis efficient methods of quantitatively estimating interference wave modes are developed. Particular attention is given to the ultralow-frequency band of the spectrum. This allows us to make progress toward the mathematical justification of methods of seismic modeling of plane wave fields on plate models. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 121–239. Translated by N. S. Zabavnikova.     

Structure of the frequency spectrum of nonaxisymmetric vibrations of a transversally isotropic hollow ball

We consider nonaxisymmetric vibrations of a hollow sphere made of a spherically Isotropic material. The original boundary-value problem is reduced to a system of ordinary differential equations by introducing new unknown junctions and expanding the sought solution in a series in spherical functions. The nonaxisymmetric problem of vibrations of a hollow ball decomposes into two independent problems that describe modes of first and second kind. The frequency spectrum for modes of first and second kind is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 32–35, 1986.     

On the relation between hypercomplex solutions and special unitary groups

In the example of the nonlinear Klein-Gordon equation, we demonstrate that even-indexed hypergeometric solutions admit matrix representations that can be associated with special unitary groups. For the index 2 in particular, this correspondence is shown to be1 :1. For the odd index 3, we show that no anticommuting matrices exist in the class of unitary anti-Hermitian matrices. We also show that the solutions obtained describe going around the potential barrier in the electron-proton transport problem. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 29–33, October, 1997.     

Combinatorial properties of rational representations of the groupGL(n, ℂ)

A generalized Robinson-Schensted-Knuth correspondence related to the decomposition of representations of GL(n, ℂ) in mixed tensor spaces is constructed. Bibliography: 5 titles. To Olga Alexandrovna Ladyzhenskaya on the occasion of her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 83–90. Translated by N. B. Lebedinskaya.     

AdS3/CFT2 on a Torus in the Sum over Geometries

We investigate the AdS₃/CFT₂ correspondence for the Euclidean AdS₃ space compactified on a solid torus with the CFT field on the regularizing boundary surface in the bulk. Correlation functions corresponding to the bulk theory at a finite temperature tend to the standard CFT correlation functions in the limit of removed regularization. In the sum over geometries in both the regular and the ℤ _N orbifold cases, the two-point correlation function for massless modes transforms into a finite sum of products of the conformal-anticonformal CFT Green's functions up to divergent terms proportional to the volume of the SL(2, ℤ)/ℤ group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 17–30, January, 2006.     

Applied aspects of studying the roots of dispersion equations on nonprincipal sheets of a complex plane

In quantitatively estimating interference wave fields by methods of function theory, the initial integration path is subjected to deformation in the complex (ζ) plane. As a result of this deformation, sometimes the contour λ may go into other sheets of the Riemann (ζ) surface. This forces us to study the integrands and, in particular, the dispersion equation of the problem on the other sheets of the (ζ) plane. Similar questions are discussed for the case where a layer SH is in contact with two elastic half-spaces. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 62–90. Translated by T. N. Surkova.     

The Euler problem in solid body dynamics and the Jacobi problem about geodesics on an ellipsoid are not topologically conjugate

Two integrable problems are considered: the geodesic flow of an ellipsoid (the Jacobi problem) and the rotation of a solid about its center of mass (the Euler problem). It is proved that transforming the dynamical system of the Euler problem into the dynamical system of the Jacobi problem by a continuous change of coordinates is impossible. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 252–258, February, 1997. Translated by M. A. Shishkova     

Chiral bosonization ofU A (1)-currents and the energy-momentum tensor in quantum chromodynamics

A generalization of the chiral lagrangian is obtained by extended low-temperature bosonization of vector, axial vector, scalar, and pseudoscalar U(3) quark currents. Estimates for its structural constants are given in accordance with the developed AANN model of QCD-bosonization. Following the ideology of chiral perturbation theory, we transform the bosonized chiral Lagrangian into the phenomenological form. Relations of Zweig-rule type are established. This enables us to express the new structural constants in terms of the given constants. The correspondence between the predictions obtained by the low-temperature bosonozation method and available phenomenological estimates are discussed. Bibliography: 9 titles. In memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 55–67. Translated by V. A. Andrianov.     

Fiber-wise birational correspondences

We prove that each fiber-wise birational correspondence between smooth fibrations into (Fano) hypersurfaces, which is biregular on a generic fiber, is a fiber-wise isomorphism. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 120–130, July, 2000.     

Describing the symmetry of one-dimensional figures by power sums of point coordinates

The article examines the problem of describing the symmetry of a one-dimensional figure by power sums of the coordinates of its points. In case ofn points of unit mass, the figure symmetry requires [(n+1)/2] values of odd power sums; in case of massive points,n values are required in general. A one-to-one correspondence is established between the coordinates ofn points of the figure and the values of power sums to order (2n−1) inclusive. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 106–115, 1999.     

Concentration of the point spectrum on the continuous one in problems of linear water-wave theory

For the linear theory of water waves, we find out families of submerged or surface-piercing bodies in an infinite three-dimensional canal, which depend on a small parameter ε > 0 and have the following property: for any positive d and natural J, there exists ε(d, J) > 0 such that, for ε ∈ (0, ε(d, J)], the segment [0, d] of the continuous spectrum of the problem contains at least J eigenvalues. These eigenvalues are associated with trapped modes, i.e., solutions of the homogeneous problem, which decay exponentially at infinity and possess finite energy. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 98–126.     

More about the embedding of a cyclic extension into a cyclic one

Relationship between the concordance condition and the solvability of the problem of embedding a cyclic Galois extension into another cyclic Galois extension is investigated. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 293–299.     

A correspondence between the AKNS hierarchy and the JM hierarchy

Reference [1] presented a gauge transformation between thex parts of the AKNS eigenvalue problem and those of the JM (Jaulent-Miodek) eigenvalue problem. In this paper we discuss the correspondence between thet parts of the AKNS eigenvalue problem and thet parts of the JM eigenvalue prohlem under the gauge transformation, and give a correspondence between the AKNS hierarchy and the JM hierarchy and also three types of Darboux transformation for the JM hierarchy.Project supported by the Science Fund of the Ministry of Education.     

L p -theory for the Stefan problem

Local solvability of the one-phase Stefan problem is established in anisotropic Sobolev spaces. There is no loss of regularity. Hanzawa transformation of the Stefan problem to a problem in a domain with a fixed boundary is modified. Bibliography: 27 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 299–323. Translated by E. V. Frolova.     

Studying the properties of carbon nanotubes with the functional integration method

We obtain an exact, closed, self-consistent system of equations for describing nanotubes that takes electron and oscillation subsystems in the collective variables into account. Collective excitations in nanotubes are described by the quantum numbers n, m, and k, where n − 1 is the number of radial modes, m is the number of azimuthal modes, and k − 1 is the number of longitudinal modes of the wave function. The results obtained approximate the experimental data better than those obtained by the method of linear combinations of atomic orbitals. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 127–144, October, 2006.     

Free boundary surface bifurcation in the phase transition problem of elasticity

The paper is concerns the phase transition problem in an elastic medium lying in a one-parameter force field. The stability and bifurcation problem for the interface between the two phases is studied. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 169–200. Translated by N. A. Karazeeva.     

Propagation of waves along a slightly bent piezoelectric rod

The waveguide modes of a curved piezoelectric rod are considered. The propagation of waves along the rod is studied by the ray method. The transport equations that describe the change in the intensity along the rays are solved. Bibliography:10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 191–202. Translated by V. V. Lukyanov     

Influence of viscosity on the structure of shock waves in the MKdV model

A “non-self-adjoint” integrable MKdV model with boundary conditions of “step-like” type is considered. The time-asymptotic behavior of the solution for the Cauchy problem is obtained. A similar problem for the dissipative perturbation of this problem is discussed. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 37–42. Translated by R. F. Bikbaev.     

Attenuation of the interference slow wave in a cracked layer

The attenuation of the interference slow wave propagating in a cracked fluid-filled layer sandwiched between two elastic half-spaces is evaluated. It is proposed that the attenuation is induced by the inelasticity of the fluid. It is shown that the attenuation of the wave may be very small if the thickness of the layer is much less than the wavelength. This explains the possibility of observing the slow wave in real field seismic experiments. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 153–160. Translated by P. V. Krauklis.