Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus.

This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo's derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry.

Medical Applications of Nonadditive Entropies.

The Boltzmann-Gibbs additive entropy SBG=-k∑ipilnpi and associated statistical mechanics were generalized in 1988 into nonadditive entropy Sq=k1-∑ipiqq-1 and nonextensive statistical mechanics, respectively. Since then, a plethora of medical applications have emerged. In the present review, we illustrate them by briefly presenting image and signal processings, tissue radiation responses, and modeling of disease kinetics, such as for the COVID-19 pandemic.

Glueball Dark Matter Revisited.

We revisit the possibility that dark matter is composed of stable scalar glueballs of a confining dark SU(3) gauge theory coupled only to gravity. The relic abundance of dark glueballs is studied for the first time in a thermal effective theory accounting for strong-coupling dynamics. An important ingredient of our analysis is the use of an effective potential for glueballs that is fitted by lattice simulations. We predict the relic abundance to be in the range 0.12ζ_{T}^{-3}Λ/(137.9 eV)≲Ωh^{2}≲0.12ζ_{T}^{-3}Λ/(82.7 eV), with Λ being the confinement scale, ζ_{T} the visible-to-dark sector temperature ratio, and the uncertainty is coming from the fit to lattice data. This prediction is an order of magnitude smaller than the existing glueball abundance results in the literature. Our framework can be easily generalized to different gauge groups and modified cosmological histories paving the way toward consistent exploration of strongly coupled dark sectors and their cosmological implications.