Tag: mathematics
All the articles with the tag "mathematics".
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Central Limit Theorem - why sums become Gaussian
Posted on:A geometric look at the central limit theorem. Adding random variables is convolving their densities. Convolution smooths. Watch a Bernoulli, a die roll, or a bimodal distribution become Gaussian as you slide the number of summands.
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Why hypercubes look spiky
Posted on:Counting the corners of an n-dimensional cube by Hamming weight gives a binomial distribution. Plot it as a vertical cross-section and you recover the spiky cube shape that high-dimensional textbooks love to draw.
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Heisenberg uncertainty as Fourier duality
Posted on:The textbook framing makes the uncertainty principle sound like a measurement artifact. It isn't. It is a property of any wave: a packet narrow in x has a Fourier decomposition spread in p, with the bound σx · σp ≥ ℏ/2 saturated by a Gaussian. This post derives that statement from photoelectric-effect-level physics.
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The 100 prisoners problem
Posted on:100 prisoners must each find their own number among 100 randomly filled boxes, opening at most 50 each. Random guessing succeeds with probability one in a nonillion. A particular cycle-following strategy succeeds about 31% of the time. The reason is the cycle structure of a random permutation.
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Marchenko-Pastur and the Wigner semicircle
Posted on:The eigenvalues of a large random matrix do not scatter around. They concentrate, as a histogram, on a deterministic shape. For sample covariance matrices the shape is Marchenko-Pastur; for symmetric matrices with i.i.d. entries it is the Wigner semicircle. Both shapes are computable, and they explain precisely why high-dimensional covariance estimation is biased.
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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 geometric reason is the Gaussian shell: noise pushes you outward, and pulling back is uniformly better. A precursor of ridge regression and most modern regularization.
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The Newton fractal
Posted on:Newton's method for finding roots of polynomials is a discrete dynamical system on the complex plane. Each starting point converges (almost always) to one of the roots. The basins of attraction have fractal boundaries, intricate to a degree the algebra of the polynomial gives no hint of.
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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.