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In this paper, the scaling-law vector calculus, which is related to the connection between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The Gauss-Ostrogradsky-like theorem, Stokes-like theorem, Green-like theorem, and Green-like identities are considered in the sense of the scaling-law vector calculus. The Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.
All previous experiments in open turbulent flows (e.g. downstream of grids, jet and atmospheric boundary layer) have produced quantitatively consistent values for the scaling exponents of velocity structure functions. The only measurement in closed t
We generalize Huberman-Rudnick universal scaling law for all periodic windows of the logistic map and show the robustness of $q$-Gaussian probability distributions in the vicinity of chaos threshold. Our scaling relation is universal for the self-sim
Understanding the behaviour of topologically ordered lattice systems at finite temperature is a way of assessing their potential as fault-tolerant quantum memories. We compute the natural extension of the topological entanglement entropy for T > 0, n
We show how the periodicity of 180^{o} domains as a function of crystal thickness scales with the thickness of the domain walls both for ferroelectric and for ferromagnetic materials. We derive an analytical expression for the universal scaling facto
We present a version of Kleibers scaling law for fetus and placenta.