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Roger Penrose's solution of the illumination problem using elliptical arcs (blue) and straight line segments (green), with 3 positions of the single light source (red spot). The purple crosses are the foci of the larger arcs. Lit and unlit regions are shown in yellow and grey respectively.
Assume two inertial reference frames (t, x, y, z) and (t′, x′, y′, z′), and two points P 1, P 2, the Lorentz group is the set of all the transformations between the two reference frames that preserve the speed of light propagating between the two points:
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A common and simple version of this problem asks to measure a time of 45 seconds using only two fuses that each burn for a minute. The assumptions of the problem are usually specified in a way that prevents measuring out 3/4 of the length of one fuse and burning it end-to-end, for instance by stating that the fuses burn unevenly along their length.
A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation. However, since we want to construct examples for linear representations - where groups act on vector spaces instead of on arbitrary finite sets - we have to proceed in a different way.
The Ages of Three Children puzzle (sometimes referred to as the Census-Taker Problem [1]) is a logical puzzle in number theory which on first inspection seems to have insufficient information to solve. However, with closer examination and persistence by the solver, the question reveals its hidden mathematical clues, especially when the solver ...
The light seen is approximately the integral of all emission along the line of sight modulated by the optical depth to the viewer (i.e. 1/e times the emission at 1 optical depth, 1/e 2 times the emission at 2 optical depths, etc.). Near the center of the star, optical depth is effectively infinite, causing approximately constant brightness.
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