On the two-dimensional rotational body of maximal Newtonian resistance

We investigate, by means of computer simulations, shapes of nonconvex bodies that maximize resistance to their motion through a rarefied medium, considering that bodies are moving forward and at the same time slowly rotating. A two-dimensional geometric shape that confers to the body a resistance very close to the theoretical supremum value is obtained, improving previous results.Comment: This is a preprint version of the paper published in J. Math. Sci. (N. Y.), Vol. 161, no. 6, 2009, 811--819. DOI:10.1007/s10958-009-9602-

The problem of camouflaging via mirror reflections

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

Mathematical retroreflectors

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.Comment: 32 pages, 19 figure

Billiards, scattering by rough obstacles, and optimal mass transportation

This article presents a brief exposition of recent results of the author on billiard scattering by rough obstacles. We define the notion of a rough body and give a characterization of scattering by rough bodies. Then we define the resistance of a rough body; it can be interpreted as the aerodynamic resistance of the somersaulting body moving through a rarefied medium. We solve the problems of maximum and minimum resistance for rough bodies (more precisely, for bodies obtained by roughening a prescribed convex set) in arbitrary dimension. Surprisingly, these problems are reduced to special problems of optimal mass transportation on the sphere

Retroreflecting curves in nonstandard analysis

We present a direct construction of retroreflecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class C(1), except for a hyper-finite set of values, such that the probability of a particle being reflected from the curve with the velocity opposite to the velocity of incidence, is infinitely close to 1. The constructed curves are of two kinds: a curve infinitely close to a straight line and a curve infinitely close to the boundary of a bounded convex set. We shall see that the latter curve is a solution of the problem: find the curve of maximum resistance infinitely close to a given curve.CEOCFCTFEDER/POCT

Comment on "functions and domains having minimal resistance under a single-impact assumption"

Recently Comte and Lachand-Robert [SIAM J. Math. Anal., 34 (2002), pp. 101-120] stated a very interesting and actual problem of minimizing mean specific resistance of infinite surfaces in a parallel flow of noninteracting point particles. They also constructed surfaces having resistance 0.593 and proved that they are minimizers. Unfortunately, their proof is incorrect. In this comment we provide a counterexample showing that the least value of resistance is not attained and is less than 0.581 (but greater than or equal to 0.5). Therefore, the problem remains open. Copyright by SIAM

Billiard scattering on rough sets: two-dimensional case

The notion of a rough two-dimensional (convex) body is introduced, and to each rough body there is assigned a measure on T3 describing billiard scattering on the body. The main result is characterization of the set of measures generated by rough bodies. This result can be used to solve various problems of least aerodynamical resistance

Optimization in the Natural Sciences: 30th Euro Mini-Conference, EmC-ONS 2014, Aveiro, Portugal, February 5-9, 2014: revised selected papers

This book constitutes the refereed proceedings of the 30th Euro Mini-Conference, EmC-ONS 2014, held in Aveiro, Portugal, in February 2014. The 13 revised full papers presented were carefully reviewed and selected from 70 submissions. The papers are organized in topical sections on dynamical systems; optimization and applications; modeling and statistical techniques for data analysis

Invisibility in billiards

The question of invisibility for bodies with mirror surface is studied in the framework of geometrical optics. We construct bodies that are invisible/have zero resistance in two mutually orthogonal directions, and prove that there do not exist bodies which are invisible/have zero resistance in all possible directions of incidence