By Theorem 1 of [14], we know that if admits a weakly sequentially continuous duality mapping, then satisfies Opial's condition, and is smooth; for the details, see [14].
If X admits a weakly sequentially continuous duality mapping, then X satisfies Opial's condition, and X is smooth, for the details, see [16].
By [25, Theorem 1], it is well known that if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial's condition, and E is smooth.
The GRP is a special Cauchy problem for a hyperbolic system with source terms whose initial condition is piecewise smooth.
For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time.
At the beginning of the process (t = 0), in liquid conditions, the cell surface is smooth (Fig. 5a H), showing up dark in the contrast of the AFM phase image (Fig. 5a P), indicating soft material.
Treating the solid contact as a layer of vanishingly small thickness within which the solution is smooth yields jump conditions that connect the states across the contact, as well as a prescription that allows the contribution of the nozzling terms to be computed unambiguously.
The Avalon is smooth, smooth, smooth.
"Slow is smooth, smooth is fast".
