A concise proof of Benford's law.

This article presents a concise proof of the famous Benford's law when the distribution has a Riemann integrable probability density function and provides a criterion to judge whether a distribution obeys the law. The proof is intuitive and elegant, accessible to anyone with basic knowledge of calculus, revealing that the law originates from the basic property of human number system. The criterion can bring great convenience to the field of fraud detection.

Analyzing Transverse Momentum Spectra of Pions, Kaons and Protons in p-p, p-A and A-A Collisions via the Blast-Wave Model with Fluctuations.

The transverse momentum spectra of different types of particles, π±, K±, p and p¯, produced at mid-(pseudo)rapidity in different centrality lead-lead (Pb-Pb) collisions at 2.76 TeV; proton-lead (p-Pb) collisions at 5.02 TeV; xenon-xenon (Xe-Xe) collisions at 5.44 TeV; and proton-proton (p-p) collisions at 0.9, 2.76, 5.02, 7 and 13 TeV, were analyzed by the blast-wave model with fluctuations. With the experimental data measured by the ALICE and CMS Collaborations at the Large Hadron Collider (LHC), the kinetic freeze-out temperature, transverse flow velocity and proper time were extracted from fitting the transverse momentum spectra. In nucleus-nucleus (A-A) and proton-nucleus (p-A) collisions, the three parameters decrease with the decrease of event centrality from central to peripheral, indicating higher degrees of excitation, quicker expansion velocities and longer evolution times for central collisions. In p-p collisions, the kinetic freeze-out temperature is nearly invariant with the increase of energy, though the transverse flow velocity and proper time increase slightly, in the considered energy range.

First-digit law in nonextensive statistics.

Nonextensive statistics, characterized by a nonextensive parameter q, is a promising and practically useful generalization of the Boltzmann statistics to describe power-law behaviors from physical and social observations. We here explore the unevenness of the first-digit distribution of nonextensive statistics analytically and numerically. We find that the first-digit distribution follows Benford's law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature. The fluctuation decreases when q increases, and the result converges to Benford's law exactly as q approaches 2. The relevant regularities between nonextensive statistics and Benford's law are also presented and discussed.