Tag: mathematics
All the articles with the tag "mathematics".
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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.
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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.