(2008). Derivatives and Eulerian Numbers. The American Mathematical Monthly: Vol. 115, No. 5, pp. 458-460.

In the present paper, we investigate the Gompertz function, which is commonly used, mostly as diffusion model, in economics and management. Our approach is based on indicating in a given time series, presumably with a Gompertz trend, some characteristic points corresponding to zeroes of successive derivatives of this function. This allows us to predict the saturation level of a phenomenon under investigation, by using only the early values of the time series. We also give an example of applications of this method.

In the present paper, we model the cumulative number of persons, reported to be infected with COVID-19 virus, by a sum of several logistic functions (the so-called multilogistic function). We introduce logistic wavelets and describe their properties in terms of Eulerian numbers. Moreover, we implement the logistic wavelets into Matlab’s Wavelet Toolbox and then we use the continuous wavelet transform (CWT) to estimate the parameters of the approximating multilogistic function. Using the examples of several countries, we show that this method is effective as a method of fitting a curve to existing data. However, it also has a predictive value, and, in particular, allows for an early assessment of the size of the emerging new wave of the epidemic, thus it can be used as an early warning method.

In the present paper we propose a new proof of the Grosset–Veselov formula connecting one-soliton solution of the Korteweg–de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.