This paper approaches the problem of signal denoising in time-variable noise conditions. Non-stationary noise results in variable degradation of the signal’s useful information content over time. In order to maximize the correct recovery of the useful part of the signal, this paper proposes a denoising method that uses a criterion based on amplitude segmentation and local Rényi entropy estimation which are limited over short time blocks of the signal spectrogram. Local estimation of the signal features reduces the denoising problem to the stationary noise case. Results, presented for synthetic and real data, show consistently better performance gained by the proposed adaptive method compared to denoising driven by global criteria.

We study the asymptotic behaviour on a finite interval of a class of linear nonautonomous singular differential equations in Banach space by the nonintegrability of the first derivative of its solutions. According to these results, the nonrectifiable attractivity on a finite interval of the zero solution of the two-dimensional linear integrable differential systems with singular matrix-elements is characterized.

Fractal nests are sets defined as unions of unit $n$-spheres scaled by a sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise the concept to subsets of such spheres and find the formulas for their box counting dimensions. We introduce some novel classes of parameterised fractal nests and apply these results to compute the dimensions with respect to these parameters. We also show that these dimensions can be seen numerically. These results motivate further research that may explain the unintuitive behaviour of box counting dimensions for nest-type fractals, and in general the class of sets where the box-counting dimension differs from the Hausdorff dimension.