62 research outputs found
"Minesweeper" and spectrum of discrete Laplacians
The paper is devoted to a problem inspired by the "Minesweeper" computer
game. It is shown that certain configurations of open cells guarantee the
existence and the uniqueness of solution. Mathematically the problem is reduced
to some spectral properties of discrete differential operators. It is shown how
the uniqueness can be used to create a new game which preserves the spirit of
"Minesweeper" but does not require a computer.Comment: We add consideration of tables based on the triangle tiling of the
plane. Its paper version encounters situations typical for the computer
"Minesweeper" gam
Sharp weyl law for signed counting function of positive interior transmission eigenvalues
We consider the interior transmission eigenvalue (ITE) problem that arises when scattering by inhomogeneous media is studied. The ITE problem is not self-adjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (as they approach ). We obtain a Weyl-type formula for the counting function of positive ITEs, which are taken together with the ascribed signs. The results are applicable to the case when the medium contains an unpenetrable obstacle
Bounds on positive interior transmission eigenvalues
The paper contains lower bounds on the counting function of the positive
eigenvalues of the interior transmission problem when the latter is elliptic.
In particular, these bounds justify the existence of an infinite set of
interior transmission eigenvalues and provide asymptotic estimates from above
on the counting function for the large values of the wave number. They also
lead to certain important upper estimates on the first few interior
transmission eigenvalues. We consider the classical transmission problem as
well as the case when the inhomogeneous medium contains an obstacle.Comment: We corrected inaccuracies cost by the wrong sign in the Green formula
(17). In particular, the sign in the definition of \sigma was change
Resonance regimes of scattering by small bodies with impedance boundary conditions
The paper concerns scattering of plane waves by a bounded obstacle with
complex valued impedance boundary conditions. We study the spectrum of the
Neumann-to-Dirichlet operator for small wave numbers and long wave asymptotic
behavior of the solutions of the scattering problem. The study includes the
case when is an eigenvalue or a resonance. The transformation from the
impedance to the Dirichlet boundary condition as impedance grows is described.
A relation between poles and zeroes of the scattering matrix in the non-self
adjoint case is established. The results are applied to a problem of scattering
by an obstacle with a springy coating. The paper describes the dependence of
the impedance on the properties of the material, that is on forces due to the
deviation of the boundary of the obstacle from the equilibrium position
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