The Grothendieck Biography Project
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The problem "Light Intensity on a Cylinder" is taken from the collection "50 Problems Anthology" that I put together a few years ago, a selection of the best problems from the 6 collections published over the decade.
Anyone who sends in a complete solution to this problem can receive, through E-Mail, an html version of my translation of the first 150 pages of Alexandre Grothendieck's "Recoltes et Semailles"
Let H be a cylinder of infinite length, with its base centered on the origin, extending to the left on the x-axis, and with circumference C = 1. A source of light is placed on the upper generator line z=0, at distance L from the origin. At the origin, on this same generator, there is an observer O.
The intensity at the light source is some constant K. Light rays are constrained to move on the surface of the cylinder, the intensity of a beam of light along a geodesic diminishing as the inverse square of the distance from the source.
Problem
Calculate the total intensity received by the observer O. Show that, at great distances, the intensity diminishes (asymptotically) as O (1/L), the inverse of the length.
Either mail the solution to:
Roy Lisker
8 Liberty Street
Middletown, CT 06457
or send it as an attachment (preferably .pdf) through E-mail torlisker@yahoo.com
To buy a copy of this translation, send $22 to the above address
- The complete text of Recoltes et Semailles, thanks to the tremendous labor of Alain Herreman, can be read in French on the Grothendieck Circle website.Go to:
Grothendieck Circle
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- Alexandre Grothendieck's Favorite Ballad, arranged for voice and guitar, 1972, by Roy Lisker: Ma Belle Si Tu Voulais
- The complete text of Recoltes et Semailles, thanks to the tremendous labor of Alain Herreman, can be read in French on the Grothendieck Circle website.Go to: