Log-log Convexity of Type-Token Growth in Zipf's Systems

It is traditionally assumed that Zipf's law implies the power-law growth of the number of different elements with the total number of elements in a system - the so-called Heaps' law. We show that a careful definition of Zipf's law leads to the violation of Heaps' law in random systems, and obtain alternative growth curves. These curves fulfill universal data collapses that only depend on the value of the Zipf's exponent. We observe that real books behave very much in the same way as random systems, despite the presence of burstiness in word occurrence. We advance an explanation for this unexpected correspondence

Energy conservation in the one-phase supercooled Stefan problem

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A one-phase reduction of the one-dimensional two-phase supercooled Stefan problem is developed. The standard reduction, employed by countless authors, does not conserve energy and a recent energy conserving form is valid in the limit of small ratio of solid to liquid conductivity. The present model assumes this ratio to be large and conserves energy for physically realistic parameter values. Results for three one-phase formulations are compared to the two-phase model for parameter values appropriate to supercooled salol (similar values apply to copper and gold) and water. The present model shows excellent agreement with the full two-phase model.Peer ReviewedPostprint (author's final draft

Large-scale analysis of Zipf's law in English texts

Despite being a paradigm of quantitative linguistics, Zipf's law for words suffers from three main problems: its formulation is ambiguous, its validity has not been tested rigorously from a statistical point of view, and it has not been confronted to a representatively large number of texts. So, we can summarize the current support of Zipf's law in texts as anecdotic. We try to solve these issues by studying three different versions of Zipf's law and fitting them to all available English texts in the Project Gutenberg database (consisting of more than 30000 texts). To do so we use state-of-the art tools in fitting and goodness-of-fit tests, carefully tailored to the peculiarities of text statistics. Remarkably, one of the three versions of Zipf's law, consisting of a pure power-law form in the complementary cumulative distribution function of word frequencies, is able to fit more than 40% of the texts in the database (at the 0.05 significance level), for the whole domain of frequencies (from 1 to the maximum value) and with only one free parameter (the exponent)

Exact derivation of a finite-size-scaling law and corrections to scaling in the geometric Galton-Watson process

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.Comment: 21 pages, 4 figure