Tag: mathematics
All the articles with the tag "mathematics".
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The colors of a soap film
Posted on:A soap film reflects a spectrum that is a single cosine in the reciprocal of the wavelength, so its color is a fixed white minus the cosine transform of the CIE color-matching functions, evaluated at twice the film's optical thickness. The computed sequence agrees with Newton's table of thin-film colors, and the fade to white in thick films is the decay of the transform.
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The kaleidoscope in an elevator
Posted on:Two mirrors that meet at an angle place every image on a circle around the line where they meet. Elevator mirrors a tenth of a degree off parallel form a kaleidoscope with 3600 sectors centered almost a kilometer away, which bends the mirror tunnel and lets you read the tilt off a photograph.
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Gaps between random points
Posted on:Drop n points uniformly on a circle. The gaps are exchangeable, the expected largest gap is exactly Hₙ/n, about ln n times the average, and its exact distribution is an alternating sum due to Fisher. The same formula is the ln n load imbalance of consistent hashing.
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The Ehrenfest urn
Posted on:N numbered balls in two urns, one moved at random per step, is the random walk on the corners of the N-cube. Its equilibrium is the binomial, the mean imbalance decays like (1 − 2/N)^t, and the expected time to return to the all-in-one-urn state is exactly 2^N.
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The arcsine law
Posted on:In a fair coin-tossing game the fraction of time one player is ahead is not concentrated near one half. Its exact distribution is u₂ₖ u₂ₙ₋₂ₖ, the limit is the U-shaped arcsine density, and with probability 1/2 one player leads for more than 85% of the game.
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Johnson–Lindenstrauss lemma
Posted on:Any n points embed linearly into about ε⁻² log n dimensions with all pairwise distances preserved within 1 ± ε, whatever the source dimension: a chi-square tail bound plus a union bound.
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Kelly criterion
Posted on:How much of the bankroll to stake on a repeated favorable bet: maximizing the expected logarithm gives f* = p − q/b, with closed forms for the growth rate and the drawdowns.
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German tank problem
Posted on:Estimating a population size N from the largest of k random serial numbers: m(1 + 1/k) − 1 is unbiased with error of order N/k. Allied statisticians used it on German tank production.