Tag: mathematics
All the articles with the tag "mathematics".
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Heisenberg uncertainty as Fourier duality
Posted on:The uncertainty principle is not a measurement artifact but a property of any wave: a packet narrow in x is spread in p, and σx · σp ≥ ℏ/2 with equality exactly for Gaussians.
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The 100 prisoners problem
Posted on:100 prisoners each seek their own number in 100 boxes, opening at most 50. Random guessing succeeds once in a nonillion; following cycles succeeds about 31% of the time.
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Marchenko-Pastur and the Wigner semicircle
Posted on:The eigenvalue histogram of a large random matrix converges to a deterministic shape: Marchenko-Pastur for sample covariance matrices, the Wigner semicircle for symmetric i.i.d. entries.
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Stein's paradox
Posted on:In three or more dimensions the sample mean is dominated everywhere by a shrinkage estimator. The reason is the Gaussian shell, and the idea is a precursor of ridge regression.
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The Newton fractal
Posted on:Newton's method on a polynomial is a dynamical system on the complex plane: almost every start converges to some root, and the basins of attraction have fractal boundaries.
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Nearest neighbor breaks in high dimensions
Posted on:In high dimensions, all pairwise distances become essentially equal. Nearest and farthest neighbor are no longer meaningfully different. A short geometric tour of the curse of dimensionality.