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

A one-phase Stefan problem with size-dependent thermal conductivity

In this paper a one-phase Stefan problem with size-dependent thermal conductivity is analysed. Approximate solutions to the problem are found via perturbation and numerical methods, and compared to the Neumann solution for the equivalent Stefan problem with constant conductivity. We find that the size-dependant thermal conductivity, relevant in the context of solidification at the nanoscale, slows down the solidification process. A small time asymptotic analysis reveals that the position of the solidification front in this regime behaves linearly with time, in contrast to the Neumann solution characterized by a square root of time proportionality. This has an important physical consequence, namely the speed of the front predicted by size-dependant conductivity model is finite while the Neumann solution predicts an infinite and, thus, unrealistic speed as $t\rightarrow0$

Substrate melting during laser heating of nanoscale metal films

We investigate heat transfer mechanisms relevant to metal films of nanoscale thickness deposited on a silicon (Si) substrate coated by a silicon oxide (SiO2) layer and exposed to laser irradiation. Such a setup is commonly used in the experiments exploring self and directed assembly of metal films that melt when irradiated by laser and then evolve on time scale measured in nanoseconds. We show that in a common experimental setting, not only the metal but also the SiO2 layer may melt. Our study of the effect of the laser parameters, including energy density and pulse duration, shows that melting of the substrate occurs on spatial and temporal scales that are of experimental relevance. Furthermore, we discuss how the thicknesses of metal and of the substrate itself influence the maximum depth and liquid lifetime of the melted SiO2 layer. In particular, we find that there is a minimum thickness of SiO2 layer for whichthis layer melts and furthermore, the melting occurs only for metal films of thickness in a specified range. In the experiments, substrate melting is of practical importance since it may significantly modify the evo-lution of the deposited nanoscale metal films or other geometries on nanoscale.Peer ReviewedPostprint (published version

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)

Solution method for the time-fractional hyperbolic heat equation

In this article, we propose a method to solve the time-fractional hyperbolic heat equation. We first formulate a boundary value problem for the standard hyperbolic heat equation in a finite domain and provide an analytical solution by means of separation of variables and Fourier series. Then, we consider the same boundary value problem for the fractional hyperbolic heat equation. The fractional problem is solved using three different definitions of the fractional derivative: the Caputo fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio and the Atangana–Baleanu. A closed form of the solution is provided for each case. Finally, we compare the solutions of the fractional and the standard problem and show numerically that the solution of the standard hyperbolic heat equation can be retrieved from the solution of the fractional equation in the limit ¿¿2, where ¿ represents the exponent of the fractional derivativePeer ReviewedPostprint (author's final draft