A Proof of First Digit Law from Laplace Transform

The first digit law, also known as Benford’s law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form log(1 + 1/d), where d= 1, 2,..., 9.Such a law has been elusive for over 100 years because it has been obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. It is revealed that the first digit law originates from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.

Ultra-high energy cosmic neutrinos from gamma-ray bursts

Based on the recent association of IceCube TeV and PeV neutrino events with gamma-ray bursts(GRBs)by considering the Lorentz violation of neutrinos,we provide a new estimate on the GRB neutrino flux with a more significant result compared to the previous constraint by the IceCube Collaboration.Among these 24 neutrino“shower”events above 60 TeV,12 events are associated with GRBs.Such a result is compatible with the prediction from GRB fireball models.Analysis of track events provides a consistent result with the shower events to associate high energy cosmic neutrinos with GRBs under the same Lorentz violation features of neutrinos.We also make a background estimation and reveal GRBs as a significant source for the ultra-high energy IceCube neutrino events.Our work supports the Lorentz violation and CPT-violation of neutrinos,indicating new physics beyond relativity.

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.

Light speed variation from active galactic nuclei

According to Einstein’s relativity,the speed of light is a constant in free space.However,it is speculated from quantum gravity that the light speed may receive a correction in powers of E=EPl,where E is the energy of the photon and EPl?■≈1.22×10^19 GeV is the Planck energy.Therefore the light speed might be energy dependent due to the Lorentz invariance violation(see,e.g.,Refs.[1–6]).Such speed variation is very tiny and can hardly be detected on the earth.It was suggested by Amelino-Camelia et al.[7]that photons from gamma-ray bursts(GRBs)can be used to detect the light speed variation.

LHAASO discovery of highest-energy photons towards new physics

Ultrahigh-energy(UHE)photons up to 1.4 peta-electronvolts have been observed by new cosmic-ray telescope in China[1]–a hint that Lorentz invariance might break down at the Planck-scale level[2].As one of the most sensitive gamma-ray detector arrays currently operating in the UHE band,the Large High Altitude Air Shower Observatory(LHAASO)recently reported large numbers of gamma-ray photons with energies larger than 100 TeV from twelve cosmic accelerators within the Milky Way[1],including the most high-energetic c-ray detected at about 1:4 PeV.The observation of such high-energy lights from space can be used not only to investigate the long-lasting puzzle of the origin of UHE cosmic rays and their acceleration mechanism in the extreme regions of the Universe,but can provide also the unique and crucial opportunity in testing fundamental concepts of physics,such as one basic symmetry underlying Einstein’s relativity–Lorentz invariance[2].