Probability estimate and the optimal text size

Reliable language corpus implies a text sample of size n that provides stable probability distributions of linguistic phenomena. The question is what is the minimal (i.e. the optimal) text size at which probabilities of linguistic phenomena become stable. Specifically, we were interested in probabilities of grammatical forms. We started with an a priori assumption that text size of 1.000.000 words is sufficient to provide stable probability distributions. Text of this size we treated as a "quasi-population". Probability distribution derived from the "quasi-population" was then correlated with probability distribution obtained on a minimal sample size (32 items) for a given linguistic category (e.g. nouns). Correlation coefficient was treated as a measure of similarity between the two probability distributions. The minimal sample was increased by geometrical progression, up to the size where correlation between distribution derived from the quasi-population and the one derived from an increased sample reached its maximum (r=1). Optimal sample size was established for grammatical forms of nouns, adjectives and verbs. General formalism is proposed that allows estimate of an optimal sample size from minimal sample (i.e. 32 items).

FRACTIONAL DIFFUSIVE WAVES

By fractional diffusive waves we mean the solutions of the so-called time-fractional diffusion-wave equation. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with a fractional derivative of order β∈(0,2) and is expected to govern evolution processes intermediate between diffusion and wave propagation when β∈(1,2). Here it is shown to govern the propagation of stress waves in viscoelastic media which, by exhibiting a power law creep, are of relevance in acoustics and seismology since their quality factor turns out to be independent of frequency. The fundamental solutions for the Cauchy and signaling problems are expressed in terms of entire functions (of Wright type) in the similarity variable. Their behaviors turn out to be intermediate between those found in the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. Furthermore, their scaling properties and the relations with some stable probability distributions are outlined.

Seismic pulse propagation with constant Q and stable probability distributions

The one-dimensional propagation of seismic waves with constant Q is shown to be governed by an evolution equation of fractional order in time, which interpolates the heat equation and the wave equation. The fundamental solutions for the Cauchy and Signalling problems are expressed in terms of entire functions (of Wright type) in the similarity variable and their behaviours turn out to be intermediate between those for the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. In view of the small dissipation exhibited by the seismic pulses, the nearly elastic limit is considered. Furthermore, the fundamental solutions for the Cauchy and Signalling problems are shown to be related to stable probability distributions with an index of stability determined by the order of the fractional time derivative in the evolution equation.