Sentence examples for in the smooth case from inspiring English sources

For (d=2), we require smoothness of (F_{12}), (g^{jk}) and V marginally larger than (mathscr {C}^2) to recover the same remainder estimate as in the smooth case, but there is a twist: unless the smoothness is (mathscr {C}^3), a correction term needs to be included.

Because of this, to get as sharp asymptotics as in the smooth case, we need to request more smoothness than in Sect. 4.

In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case.

Results are similar to those in the smooth case.

4.5, which was indeed a theorem (Proposition 4.11) in the smooth case.

For more detail on proper actions in the smooth case we refer the reader to monographs [54] and [92].

We have begin{aligned} N^2+2N<frac{ n-1)(n-2)}{2}+2quad hbox {for}, Nge 5. end{aligned} (4.3)Thus, the group dimension range that arises in the complex case, for (Nge 5) lies strictly below the dimension range investigated in the classical smooth case and therefore is not in any way covered by the results discussed in Sect.

Unfortunately, this intuitive picture, while it can be made precise easily in the smooth boundary case, does not really apply to the general case.

Hence, what we do here is to present alternative characterizations of gradient flows in the smooth Euclidean case, which can be used as a definition of gradient flow in the metric case, since all the quantities which are involved have their metric counterpart.

On the contrary to what happened in the non-smooth case, here the gradient descent idea does not attain a rate that matches the lower bound ( v.s. ).

But in the compactly supported, non smooth, case the singularities at the boundary of the support produce resonances at distance ( M h log frac{1}{h} ) where M depends on the regularity of the potential at the boundary of the support.