Tag: mathematics
All the articles with the tag "mathematics".
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Voronoi tessellations and Lloyd's algorithm
Posted on:Generators partition the plane into cells, each closer to its generator than to any other. Lloyd's algorithm moves each generator to its cell's centroid and converges to k-means.
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Pólya's recurrence theorem
Posted on:Simple random walk returns to the origin with probability one in one and two dimensions, but not in three. The transition reduces to the convergence of a single series.
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High-dimensional Gaussians live on a sphere
Posted on:The bell-curve picture says Gaussian samples live near the mean. In high dimensions almost all the mass lies in a thin shell at radius √d: density and mass are not the same thing.
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Central Limit Theorem - why sums become Gaussian
Posted on:Adding random variables convolves their densities, and convolution smooths. A geometric look at the central limit theorem, with a slider over the number of summands.
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Why hypercubes look spiky
Posted on:Counting the corners of an n-cube by Hamming weight gives a binomial distribution; plotted as a cross-section, it recovers the spiky cube of the textbooks.
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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.