Papers by Alexander Plakhov
Retroreflectors are optical devices that reverse the direction of incident beams of light. Here w... more Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object notched angle is a new one; a proof of its retroreflectivity is given.
Discrete and Continuous Dynamical Systems, 2013
We study the problem of invisibility for bodies with a mirror surface in the framework of geometr... more We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1, 10], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.
We study the problem of invisibility for bodies with a mirror surface in the framework of geometr... more We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1, 10], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.
Arnold mathematical journal, Jan 17, 2022
This is a collection of problems composed by some participants of the workshop "Differential Geom... more This is a collection of problems composed by some participants of the workshop "Differential Geometry, Billiards, and Geometric Optics" that took place at CIRM on October 4-8, 2021.
Dynamics of a Pendulum in a Rarefied Flow
Regular & chaotic dynamics/Regular and chaotic dynamics, 2024
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Sep 1, 2017
This work is related to billiards and their applications in geometric optics. It is known that pe... more This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is, therefore, natural to search for bodies that are, in a sense, close to invisible. We introduce a visibility index of a body measuring the mean angle of deviation of incident light rays, and derive a lower estimate for this index. This estimate is a function of the body's volume and of the minimal radius of a ball containing the body. This result is far from being final and opens a possibility for further research.
Optimal impulse control of a SIR epidemic
Optimal Control Applications & Methods, Nov 27, 2019
SummaryBased on our recent results on the optimal impulse control, we solve explicitly an optimal... more SummaryBased on our recent results on the optimal impulse control, we solve explicitly an optimal isolation problem for a specific SIR (susceptible‐infective‐removed) epidemic model describing, eg, the spread of AIDS.
Journal | MESA, Nov 25, 2010
We consider a rarefied medium in R d , d ≥ 2 consisting of non-interacting point masses moving at... more We consider a rarefied medium in R d , d ≥ 2 consisting of non-interacting point masses moving at unit velocity in all directions. Given the density of velocity distribution, one easily calculates the pressure created by the medium in any direction. We then consider the inverse problem: given the pressure distribution f : S d−1 → R + , determine the density ρ : S d−1 → R +. Assuming that the reflection of medium particles by obstacles is elastic, we show that the solution for the inverse problem is generally non-unique, derive exact inversion formulas, and state necessary and sufficient conditions for existence of a solution. We also present arguments indicating that the inversion is typically unique in the case of non-elastic reflection, and derive exact inversion formulas in a special case of such reflection.
Archives of Mechanics, Jan 9, 2009
Magnus effect consists in deflection of the trajectory of a rotating body moving in a gas. It is ... more Magnus effect consists in deflection of the trajectory of a rotating body moving in a gas. It is a direct consequence of the interaction between the body surface and the gas particles. In this paper, we study the so-called inverse Magnus effect which can be observed in rarefied gases. We restrict ourselves to the two-dimensional case, namely a spinning disc moving through a sparse zero-temperature medium. We consider general non-elastic interaction between the disc and the particles depending on the incidence angle. We give a classification of auxiliary parameters with respect to possible dynamical response. In the absence of other forces, three kinds of trajectories are possible: (i) a converging spiral, (ii) a curve converging to a straight line and (iii) a circumference, the case intermediate between the two first ones. A specific 2-D parameter space has been introduced to provide respective classification.
Communications in computer and information science, 2015
Here we study retroreflectors based on specular reflections. Two kinds of asymptotically perfect ... more Here we study retroreflectors based on specular reflections. Two kinds of asymptotically perfect specular retroreflectors in two dimensions, Notched angle and Tube, are known at present. We conduct comparative study of their efficiency, assuming that the reflection coefficient is slightly less than 1. We also compare their efficiency with the one of the retroreflector Square corner (the 2D analogue of the well-known and widely used Cube corner). The study is partly analytic and partly uses numerical simulations. We conclude that the retro-reflectivity ratio of Notched angle is normally much greater than those of Tube and the Square corner. Additionally, simple Notched angle shapes are constructed, whose efficiency is significantly higher than that of the Square corner.
The Magnus Effect and the Dynamics of a Rough Disc
Springer eBooks, 2012
In this chapter we are concerned with the Magnus effect: the phenomenon governing the deflection ... more In this chapter we are concerned with the Magnus effect: the phenomenon governing the deflection of the trajectory of a spinning body (for example, a golf ball or a soccer ball). Surprisingly enough, in highly rarefied media (on Mars or in the thin atmosphere at a height corresponding to low Earth orbits: between 100 and 1,000 km) the inverse effect takes place; this means that the trajectory deflection has opposite signs in sparse and in dense media.
arXiv (Cornell University), May 19, 2010
Retroreflectors are optical devices that reverse the direction of incident beams of light. Here w... more Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object notched angle is a new one; a proof of its retroreflectivity is given.
Notation and Synopsis of Main Results
Springer eBooks, 2012
In this chapter, we introduce the main mathematical notation that will be used throughout the boo... more In this chapter, we introduce the main mathematical notation that will be used throughout the book and state the main results of the book. The proofs of these results are given in Chaps. 2 –9.
Journal De Physique I, Feb 1, 1994
arXiv (Cornell University), Jul 28, 2011
We study the problem of invisibility for bodies with a mirror surface in the framework of geometr... more We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1, 10], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.
This is a collection of open problems from workshop "Differential Geometry, Billiards, and Geomet... more This is a collection of open problems from workshop "Differential Geometry, Billiards, and Geometric Optics" at CIRM on October 4-8, 2021.
arXiv (Cornell University), Mar 14, 2020
In this paper we prove a somewhat stronger result. Namely, there exists a solution $u$ possessing... more In this paper we prove a somewhat stronger result. Namely, there exists a solution $u$ possessing the following property. If $\mathcal{U} \subset \Omega$ is open and all points of the surface graph$(u\rfloor_\mathcal{U})$ are regular, then this surface does not contain extreme points of the convex body $C_u = \{ (x,y,z) :\, (x,y) \in \Omega,\ 0 \le z \le u(x,y) \}$. As a consequence, we have $C_u = \text{Conv} (\overline{\text{Sing$C_u$}})$, where Sing$C_u$ denotes the set of singular points of $\partial C_u$. We prove a similar result for a generalized Newton's problem.
arXiv (Cornell University), Aug 31, 2008
A body in a parallel flow of non-interacting particles is considered. We introduce the notions of... more A body in a parallel flow of non-interacting particles is considered. We introduce the notions of a body of zero resistance, a body that leaves no trace, and a transparent body, and prove that all such kinds of bodies do exist.
arXiv (Cornell University), Apr 17, 2008
We study the Magnus effect: deflection of the trajectory of a spinning body moving in a medium. T... more We study the Magnus effect: deflection of the trajectory of a spinning body moving in a medium. The body is rough; that is, there are small cavities on its surface. We concentrate on the extreme case of rare medium, where mutual interaction of the medium particles is neglected and reflections of the particles from the body's surface are
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Papers by Alexander Plakhov