Tag: mathematics
All the articles with the tag "mathematics".
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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.
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Pendulum period
Posted on:The pendulum period grows with amplitude; 2π√(L/g) is only the zeroth term. The exact period is an elliptic integral, and Gauss's arithmetic–geometric mean evaluates it in five iterations.
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The leading digit of a power of two
Posted on:Thirty of the first hundred powers of two begin with 1. The limiting frequency of leading digit d is log10(1 + 1/d), by Weyl equidistribution; scale invariance extends the law to Benford.
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Percolation
Posted on:Keep each edge of a grid with probability p and ask whether a path crosses the grid. The answer flips at a sharp threshold, exactly 1/2 on the square lattice by self-duality.
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The curve of fastest descent
Posted on:The wire carrying a sliding bead between two points in the least time is a cycloid arc, not the straight line. Optics finds it, convexity proves it, and the same arc is the tautochrone.
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Euler's identity
Posted on:e^{iπ} + 1 = 0 is the θ = π case of Euler's formula e^{iθ} = cos θ + i sin θ, the point on the unit circle at angle θ. Three derivations: geometric, as a limit, and by power series.
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Shattering and the VC dimension
Posted on:The VC dimension of a class of yes/no rules is the largest set of points it can label in every way. Past it, the labelings stop doubling, and that single integer governs learnability.
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The Hanson–Wright inequality
Posted on:A quadratic form in independent random variables concentrates around the trace: a Gaussian tail near the mean via the Frobenius norm, an exponential tail further out via the operator norm.