Several Issues Concerning the Corruption Crimes of Private Entrepreneurs: Based on Empirical Research

Studies have shown that top leaders and department heads of privately-owned enterprises differ significantly in the types of corruption crimes and the links where crimes are committed. Therefore, corruption crimes of private entrepreneurs should be countered with the methods well targeted at the subjects of crime. In addition, penalties do not fit crimes by the conviction and penalty imposition for corruption crimes, which seriously undermine the effect of criminal penalties on crime prevention. Therefore, the sentencing range should be elaborated and other measures should be taken so that crimes and penalties will be balanced

Universal chaotic dynamics from Krylov space

Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.Comment: 40+26 pages, 19 figures, major revision, published versio