By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE(n,0)$, which can be consider as a notion of curvature fo
r graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincare inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is finite, which original is a conjecture of Yau on Riemannian manifold proved by Colding and Minicozzi. Under the assumption of positive curvature on graphs, we derive the Bonnet-Myers type theorem that the diameter of graphs is finite and bounded above in terms of the positive curvature by proving some Log Sobolev inequalities.
From recent high resolution observations obtained with the Swedish 1 m Solar Telescope in La Palma, we detect swaying motions of individual filament threads in the plane of the sky. The oscillatory character of these motions are comparable with oscil
latory Doppler signals obtained from corresponding filament threads. Simultaneous recordings of motions in the line of sight and in the plane of the sky give information about the orientation of the oscillatory plane. These oscillations are interpreted in the context of the magnetohydrodynamic theory. Kink magnetohydrodynamic waves supported by the thread body are proposed as an explanation of the observed thread oscillations. On the basis of this interpretation and by means of seismological arguments, we give an estimation of the thread Alfven speed and magnetic field strength by means of seismological arguments.
