The Newcomb-Benford Law in Its Relation to Some Common Distributions

An often reported, but nevertheless persistently striking observation, formalized as the Newcomb-Benford law (NBL), is that the frequencies with which the leading digits of numbers occur in a large variety of data are far away from being uniform. Most spectacular seems to be the fact that in many data the leading digit 1 occurs in nearly one third of all cases. Explanations for this uneven distribution of the leading digits were, among others, scale- and base-invariance. Little attention, however, found the interrelation between the distribution of the significant digits and the distribution of the observed variable. It is shown here by simulation that long right-tailed distributions of a random variable are compatible with the NBL, and that for distributions of the ratio of two random variables the fit generally improves. Distributions not putting most mass on small values of the random variable (e.g. symmetric distributions) fail to fit. Hence, the validity of the NBL needs the predominance of small values and, when thinking of real-world data, a majority of small entities. Analyses of data on stock prices, the areas and numbers of inhabitants of countries, and the starting page numbers of papers from a bibliography sustain this conclusion. In all, these findings may help to understand the mechanisms behind the NBL and the conditions needed for its validity. That this law is not only of scientific interest per se, but that, in addition, it has also substantial implications can be seen from those fields where it was suggested to be put into practice. These fields reach from the detection of irregularities in data (e.g. economic fraud) to optimizing the architecture of computers regarding number representation, storage, and round-off errors

Calcul du rabattement des nappes aquifères

The calculation of aquifer drawdown. A battery of seepage wells arranged along a curved or broken line can be replaced by a similarly aligned seepage ditch. The error thus introduced is small, except in the immediate vicinity of a well, and can be estimated. The two cases to be considered are uniform drawdown and uniform discharge conditions. Explicit formulae are given for open polygonal or circular lines, and sites near a lake or river. The equation for the equipotential lines is also given.Une batterie de puits filtrants disposés le long d'une ligne courbe ou brisée, peut être remplacée par une tranchée filtrante ayant la forme de cette ligne. L'erreur commise est faible, sauf au voisinage immédiat d'un puits, et peut être estimée. On peut distinguer le cas d'un rabattement uniforme et d'un débit uniforme. Des formules explicites sont données dans le cas de lignes polygonales ouvertes ou circulaires, et près d'un lac ou fleuve. L'équation des lignes isopiestiques est donnée.Irmay S. Calcul du rabattement des nappes aquifères. In: L'hydraulique souterraine. Compte rendu des sixièmes journées de l'hydraulique, Nancy, 28-30 juin 1960. Tome 1, 1961

Letter dated 12 June 1967 from Shragga Irmay at Haifa, Israel, to Lorenzo A. Richards

Letter dated 12 June 1967 from Shragga Irmay at the Israel Institute of Technology (Technon) at Haifa, Israel, to Lorenzo A. Richards, immediately after the Six Day Wa