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Mathematical retroreflectors
Plakhov, AlexanderRetroreflectors 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...
Spinning rough disc moving in a rarefied medium
Plakhov, Alexander; Tchemisova, Tatiana; Gouveia, Paulo D.F.We study the Magnus effect: deflection of the trajectory of a spinning body moving in a gas. It is well known that in rarefied gases, the inverse Magnus effect takes place, which means that the transversal component of the force acting on the body has opposite signs in sparse and relatively dense gases. The existing works derive the inverse effect from nonelastic interaction of gas particles with the body. We p...
Billiard scattering on rough sets: two-dimensional case
Plakhov, AlexanderThe 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.
A stochastic approximation algorithm with multiplicative step size modification
Plakhov, Alexander; Cruz, João Pedro Antunes Ferreira daAn algorithm of searching a zero of an unknown function $\vphi : \, \R \to \R$ is considered: $\, x_{t} = x_{t-1} - \gamma_{t-1} y_t$,\, $t=1,\ 2,\ldots$, where $y_t = \varphi(x_{t-1}) + \xi_t$ is the value of $\vphi$ measured at $x_{t-1}$ and $\xi_t$ is the measurement error. The step sizes $\gam_t > 0$ are modified in the course of the algorithm according to the rule: $\, \gamma_t = \min\{u\, \gamma_{t-1},\...
On the two-dimensional rotational body of maximal Newtonian resistance
Gouveia, Paulo D.F.; Plakhov, Alexander; Torres, Delfim F.M.We investigate, by means of computer simulations, shapes of nonconvex bodies that maximize resistance to their motion through a rarefied medium, considering that the 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, which improves previous results.
Two-dimensional body of maximum mean resistance
Gouveia, Paulo D.F.; Plakhov, Alexander; Torres, Delfim F.M.A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, [Alexander Yu. Plakhov, Paulo D.F. Gouveia, Problems of maximal mean resistance on the plane, Nonlinearity, 20 (2007), 2271–2287], shapes of nonconvex bodies were sought which would maximize the ...
Uma forma bidimensional que maximiza a resistência aerodinâmica newtoniana
Gouveia, Paulo D.F.; Plakhov, Alexander; Torres, Delfim F.M.Um corpo bidimensional, apresentando um ligeiro movimento rotacional, desloca-se num meio rarefeito de partículas que colidem com ele de uma forma perfeitamente elástica. Em investigações que os dois primeiros autores realizaram anteriormente [Plakhov and Gouveia, 2007], procuraram-se formas de corpos que maximizassem a força de travagem do meio ao seu movimento. Dando continuidade a esse estudo, encetam-se ago...
Problems of maximal mean resistance on the plane
Plakhov, Alexander; Gouveia, Paulo D.F.A two-dimensional body moves through a rarefied medium; the collisions of the medium particles with the body are absolutely elastic.The body performs both translational and slow rotational motion. It is required to select the body, from a given class of bodies, such that the average force of resistance of the medium to its motion is maximal. There are presented numerical and analytical results concerning this p...
Bodies of maximal aerodynamic resistance on the plane
Plakhov, Alexander; Gouveia, Paulo D.F.A two-dimensional body moves through a rarefied medium; the collisions of the medium particles with the body are absolutely elastic. The body performs both translational and slow rotational motion. It is required to select the body, from a given class of bodies, such that the average force of resistance of the medium to its motion is maximal. There are presented numerical and analytical results concerning this ...
Uma forma bidimensional que maximiza a resistência aerodinâmica newtoniana
Gouveia, Paulo D.F.; Plakhov, Alexander; Torres, Delfim F.M.In a previous work [18, 19] it is investigated, by means of computational simulations, shapes of nonconvex bodies that maximize resistance to its motion on a rare ed medium, considering that bodies are moving forward and at the same time slowly rotating. Here the previous results are improved: we obtain a two-dimensional geometric shape that confers to the body a resistance very close to the supremum value (R =...
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