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#ConradMitzenmacher-7
"A. Notation and Terminology
Throughout this paper, the phrase ›log-ratio‹ for a pair of positive real numbers refers to the ratio of their logarithms (to a common base, the choice of which cancels out)..
All logarithms without an indicated base
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"A. Notation and Terminology
Throughout this paper, the phrase ›log-ratio‹ for a pair of positive real numbers refers to the ratio of their logarithms (to a common base, the choice of which cancels out)..
All logarithms without an indicated base
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>are understood to be taken to the base 𝑒..
II. Review of Definitions and History
Our treatment here is based on a recent survey by Mitzenmacher [#Internet #Mathematics 1(2), 226‒251 (2004)], to which we refer the reader for more information.[ Footnote: For instance,
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II. Review of Definitions and History
Our treatment here is based on a recent survey by Mitzenmacher [#Internet #Mathematics 1(2), 226‒251 (2004)], to which we refer the reader for more information.[ Footnote: For instance,
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>this survey describes another argument that leads to a #PowerLaw of word frequency based on preferential attachment, originally due to Simon [Biometrika 42(3/4), 425–440 (1955)]. We do not present this argument here.]
In what follows, we let 𝑓ⱼ be the (asymptotic) fraction of>
In what follows, we let 𝑓ⱼ be the (asymptotic) fraction of>
#ConradMitzenmacher-6
"In this paper, we begin by reviewing the fascinating history of this fundamental problem. Then, we use methods from #ComplexAnalysis to prove that Miller’s random monkey experiment yields power laws for [the] rank-frequency distribution
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"In this paper, we begin by reviewing the fascinating history of this fundamental problem. Then, we use methods from #ComplexAnalysis to prove that Miller’s random monkey experiment yields power laws for [the] rank-frequency distribution
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>with probability assignments to keys satisfying a rationality assumption on log-ratios of pairs of probabilities.
We use analytic methods to establish a simple explicit power law in cases with rational log-ratios for pairs of probabilities; more specifically,
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We use analytic methods to establish a simple explicit power law in cases with rational log-ratios for pairs of probabilities; more specifically,
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>we use generalized Dirichlet series and an elementary identity established by means of Fourier series. Passing to a limit on these formulas predicts an analogous result in the remaining “irrational” cases, and this prediction agrees with an unpublished theorem..by Montgomery;
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