207 research outputs found
Two-dimensional body of maximum mean resistance
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, Plakhov &
Gouveia (2007, Nonlinearity, 20), shapes of nonconvex bodies were sought which
would maximize the braking force of the medium on their movement. Giving
continuity to this study, new investigations have been undertaken which
culminate in an outcome which represents a large qualitative advance relative
to that which was achieved earlier. This result, now presented, consists of a
two-dimensional shape which confers on the body a resistance which is very
close to its theoretical supremum value. But its interest does not lie solely
in the maximization of Newtonian resistance; on regarding its characteristics,
other areas of application are seen to begin to appear which are thought to be
capable of having great utility. The optimal shape which has been encountered
resulted from numerical studies, thus it is the object of additional study of
an analytical nature, where it proves some important properties which explain
in great part its effectiveness.Comment: Accepted (April 16, 2009) for publication in the journal "Applied
Mathematics and Computation
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-
Optimal control of Newton-type problems of minimal resistance
We address Newton-type problems of minimal resistance from an optimal control perspective. It is proven that for Newton-type problems the Pontryagin maximum principle is a necessary and sufficient condition. Solutions are then computed for concrete situations, including the new case when the flux of particles is non-parallel
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
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