A Note on Evolution of Quaternionic Curves in the Euclidean Space $\mathbb{R}^{4}$
In this paper, we investigate the evolution of quaternionic curve in Euclidean 4-space $\mathbb{R}^{4}$. We obtain evolution equations of the Frenet frame and curvatures. Then we give integrability conditions for the evolutions. Finally we give examples of evolution of curvatures.
Abstract In this paper, we study the long-time behavior of solutions for a non-autonomous strongly damped wave equation. We first prove the existence of a uniform attractor for the equation with a translation compact driving force and then obtain an upper estimate for the Kolmogorov e -entropy of the uniform attractor. Finally we obtain an upper bound of the fractal dimension of the uniform attractor with quasiperiodic force.