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The Crab Nebula is a bright source of gamma-rays powered by the Crab Pulsar's rotational energy, through the formation and termination of a relativistic electron-positron wind. We report the detection of γ-rays from this source with energies from 5 × 10−4 to 1.1 petaelectronvolts (PeV), with a spectrum showing gradual steepening over three energy decades. The ultra-high-energy photons imply the presence of a PeV electron accelerator (a pevatron) in the nebula, with an acceleration rate exceeding 15% of the theoretical limit. We constrain the pevatron's size between 0.025 and 0.1 pc, and magnetic field ≈110 μG. The production rate of PeV electrons, 2.5 × 1036 erg s−1, constitutes 0.5% of the pulsar spin-down luminosity, although we cannot exclude a contribution of PeV protons to the production of the highest energy γ-rays.

This paper mainly discusses fractional differential approach to detecting textural features of digital image and its fractional differential filter. Firstly, both the geometric meaning and the kinetic physical meaning of fractional differential are clearly explained in view of information theory and kinetics, respectively. Secondly, it puts forward and discusses the definitions and theories of fractional stationary point, fractional equilibrium coefficient, fractional stable coefficient, and fractional grayscale co-occurrence matrix. At the same time, it particularly discusses fractional grayscale co-occurrence matrix approach to detecting textural features of digital image. Thirdly, it discusses in detail the structures and parameters of n×n any order fractional differential mask on negative x-coordinate, positive x-coordinate, negative y-coordinate, positive y-coordinate, left downward diagonal, left upward diagonal, right downward diagonal, and right upward diagonal, respectively. Furthermore, it discusses the numerical implementation algorithms of fractional differential mask for digital image. Lastly, based on the above-mentioned discussion, it puts forward and discusses the theory and implementation of fractional differential filter for digital image. Experiments show that the fractional differential-based image operator has excellent feedback for enhancing the textural details of rich-grained digital images. fractional stationary point, fractional equilibrium coefficient, fractional stable coefficient, fractional grayscale co-occurrence matrix, fractional differential mask