arXiv subjects

E. Lakshtanov

Publications and source records attributed to E. Lakshtanov.

4 recordsLinked to original sources

arXiv2012

Classification of Singularities and Bifurcations of Critical Points of Even Functions

Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic value of the parameter are also studied. A determinant for singularity types and conditions for versal deformations are given in terms of partial derivatives (not requiring a preliminary reduction to a canonical form).

math.DG↗

arXiv2011

Finiteness of the playing time in 'Beggar-my-neighbour' card game

It is proved that in card games similar to 'Beggar-my-neighbour' the mathematical expectation of the playing time is finite, provided that the player who starts the round is determined randomly and the deck is shuffled when the trick is added. The result holds for the generic setting of the game.

math.PR↗

arXiv2010

High Frequency Scattering by a Classically Invisible Body

We consider a polyhedron with zero classical resistance, i.e., a polyhedron invisible to an observer viewing only the paths of geometrical optics rays. The corresponding problem of scattering of plane waves by the polyhedron is studied. The quasiclassical approximation is obtained and justified in the case of impedance boundary conditions with a non zero absorbing part. It is shown that the total momentum transmitted to the obstacle vanishes when the frequency $k$ goes to infinity, and that the total cross section oscillates at high frequencies. When the impedance $λ_0$ is real (i. e., there is no absorption), it is shown that there exists a sequence of frequencies $k_n$ such that the averages in the impedance of the total cross section over shrinking intervals around $λ_0 $ go to zero as $k_n \to \infty$.

math-ph↗

arXiv2009

Semiclassical limit of the scattering cross section as a distribution

We consider quantum scattering from a compactly supported potential $q$. The semiclassical limit amounts to letting the wavenumber $k \to \infty$ while rescaling the potential as $k^2 q$ (alternatively, one can scale Planck's constant $\hbar \searrow 0$). It is well-known that, under appropriate conditions, for $\om \in \bbS_{n-1}$ such that there is exactly one outgoing ray with direction $\om$ (in the sense of geometric optics), the differential scattering cross section $|f(\om,k)|^{2}$ tends to the classical differential cross section $|f_{cl}(\om)|^2$ as $k \uparrow \infty$. It is also clear that the same can not be true if there is more than one outgoing ray with direction $\om$ or for \emph{nonregular} directions (including the forward direction $θ_0$). However, based on physical intuition, one could conjecture $|f|^2 \to |f_{cl}|^2 + σ_{cl} δ_{θ_0}$ where $|f_{cl}|^2$ is the classical cross section and $δ_{θ_0}$ is the Dirac measure supported at the forward direction $θ_0$. The aim of this paper is to prove this conjecture.

math-ph↗