Fock asymptotics in the problem of diffraction by a moving smooth contour

In the paper the asymptotics are obtained of the solution of the diffraction problem in the Fock region for the incidence of a nonstationary modulated wave on a smooth moving contour. The formulas obtained generalize the solution of V. A. Fock to the stationary problem in a neighborhood of a point of tangency of an ray to the case of a nonstationary wave and a moving scatterer. A new characteristic of the problem is introduced the effective radius of curvature of the contour, depending on the geometric and kinematic properties of the scatterer. The boundaries are established of the applicability of the asymptotic formulas.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 195, pp. 69–81, 1991.In conclusion I want to thank the participants of the seminars of LGU and LOMI on the diffraction and propagation of waves for useful discussions.     

Short-wave asymptotics of the current in the problem of diffraction by nonplanar screens

A method is proposed for finding the short-wave asymptotics of the current arising oa a non-planar screen when a wave of geometrical optics is incident on it. The leading terms of the asymptotic expressions both for primary and multiple diffraction are obtained in the two-dimensional case for the Dirichlet problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 172–182, 1975.The author is grateful to I. A. Molotkov for his attention to the work and for useful suggestions in the writing of the paper.     

Artificial boundary conditions for finding surface waves in the problem of diffraction by a periodic boundary

Transparent artificial boundary conditions and an algorithm for computing the augmented scattering matrix are proposed for finding surface waves in a prescribed range of decay rates. An infinite-dimensional fictitious scattering operator is constructed that determines all waves decaying exponentially with distance from a periodic obstacle.     

On the asymptotics of a Robin eigenvalue problem

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to ?∞ as the perturbation goes to zero. We prove in this case that Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criterion to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter.     

Binomial asymptotics of the spectrum of a boundary-value problem

Functional Analysis and Its Applications -     

Improved impedance conditions for a thin layer problem in a nonsmooth domain

In this article, we consider the model problem of the Laplace equation in a domain with a thin layer on a part of its boundary. The singularities appearing where boundary conditions change deteriorate the efficiency of the classical impedance condition used to replace the layer. Modified impedance conditions are proposed, which lead to some improvements in the error estimates.     

An asymptotics of the Pauli problem in thermodynamics

A generalization of the quasithermodynamic theory of fluctuations is suggested. Motivated by an analogy with quantum mechanics, we state a Pauli problem concerning a reconstruction of a “wavefunction” based on the probability densities of the fluctuations of intensive and extensive thermodynamic quantities. The paper contains a theorem about the existence of approximate solutions of the problem mentioned in the co-called “subtropical” case.     

Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide

This work concerns the numerical finite element computation, in the frequency domain, of the diffracted wave produced by a defect (crack, inclusion, perturbation of the boundaries, etc.) located in a 3D infinite elastic waveguide. The objective is to use modal representations to build transparent conditions on some artificial boundaries of the computational domain. This cannot be achieved in a classical way, due to non-standard properties of elastic modes. However, a biorthogonality relation allows us to build an operator, relating hybrid displacement/stress vectors. An original mixed formulation is then derived and implemented, whose unknowns are the displacement field in the bounded domain and the normal component of the normal stresses on the artificial boundaries. Numerical validations are presented in the 2D case.     

Asymptotic approximation for the solution to a semilinear parabolic problem in a thin star‐shaped junction

A semilinear parabolic problem is considered in a thin 3‐D star‐shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter The purpose is to study the asymptotic behavior of the solution u_ε as ε→0, ie, when the star‐shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors and in nonlinear perturbed Robin boundary conditions. We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters {α_i}and {β_i}. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε→0. Namely, in each case, we derive the limit problem (ε=0)on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex, and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.     

Weak asymptotics in the 3-dim Frobenius problem

We consider the Frobenius problem for the 3-dim numerical semigroup $\mathsf{S}(d_{1},d_{2}, d_{3})$ generated by three positive integers d ₁, d ₂, d ₃ with gcd?(d ₁,d ₂,d ₃)=1 and calculate weak (Cesaro) asymptotics of its conductor C(d ₁,d ₂,d ₃) and the fraction p(d ₁,d ₂,d ₃) of the segment [0;C(d ₁,d ₂,d ₃)?1] occupied by the semigroup. Four conjectures #?1999-8, #?1999-9, #?1999-10 and #?2003-5 which were posed by V.I. Arnold (Arnold’s Problems, pp. 129–130, 163, 2004) and devoted to the statistics of the m-dim semigroups, m≥3, are refuted for the case m=3.     

Scattering characteristics of a problem of diffraction by a wedge and by a screen

In the case of the scattering problem on a wedge and on a screen, for a certain class of boundary conditions, one constructs explicitly the wave operators and one establishes their completeness. It is shown that a modified scattering matrix (including additionally the reflection operator) is a unitary operator with a pure point spectrum. In the case of a screen, the standard S-matrix is unitary. For Dirichlet (Neumann) boundary conditions, the S-matrix is reduced explicitly to a diagonal form. The spectrum of the S-matrix is simple, absolutely continuous, filling the lower (upper) semicircumference.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 193–197, 1982.