Numerically statistical investigation of the partly super-exponential growth rate in the COVID-19 pandemic (throughout the world)

A number of particles in a multiplying medium under rather general conditions is asymptotically exponential with respect to time t with the parameter λ, i.e., with the index of power {\lambda t}. If the medium is random, then the parameter λ is the random variable. To estimate the temporal asymptotics of the mean particles number (via the medium realizations), it is possible to average the exponential function via the corresponding distribution. Assuming that this distribution is Gaussian, the super-exponential estimate of the mean particle number could be obtained and expressed by the exponent with the index of power {t{\rm E}\lambda+t^{2}{\rm D}\frac{\lambda}{2}}. The application of this new formula to investigation of the COVID-19 pandemic is performed.

SPATIAL ANALYTICITY PROPERTIES OF NONLINEAR WAVES

In this paper, we study spatial analyticity properties of two classes of equations modeling unidirectional waves in nonlinear, dispersive media, namely KdV-type equations and BBM-type equations. The commentary begins with KdV-type equations and the observation that, for a class of such equations, boundedness of a solution suffices to maintain analyticity and so loss of analyticity detects loss of L∞-regularity. For a larger class of KdV-type equations, the same conclusion is valid provided that L∞-boundedness of a solution is replaced by [Formula: see text]-boundedness. It is also shown that these nonlinear dispersive wave equations are amenable to Gevrey-class analysis based on the boundedness of a Sobolev norm. This analysis yields an explicit lower bound on the possible rate of decrease in time of the uniform radius of analyticity of a solution in terms of the assumed Sobolev bound and the Gevrey-norm of the initial data. Attention is then shifted to BBM-type equations. It is shown that, regardless of the strength of the nonlinearity, a solution starting in a Gevrey space remains in this class for all time. Moreover, a lower bound on the possible rate of decrease in time of the uniform analyticity radius has temporal asymptotics that are independent of the degree of the nonlinearity, and so apparently determined in the main by the dispersion.