Sentence examples for smooth case from inspiring English sources

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

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.

Results are similar to those in the smooth case.

The achieved Nusselt number was about 1.2 times greater than 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].

The way evidence fell together seamlessly made detectives past and present marvel and reminisce about the smooth cases they had solved.

It is found to be more precise than the stochastic Galerkin method for smooth cases but above all for discontinuous cases.

Within these models, token-identity theories do not back up strong or perfectly smooth cases of reduction, but they nevertheless support reductions that are stronger than those that are merely extreme cases of replacement, not involving correction in any robust sense.

The non-smooth case, ν = 0.5, is illustrated in Fig. 8.

end{aligned}This last condition, written as begin{aligned} frac{x^tau _{k+1}-x^tau _k}{tau }=- nabla Fleft( x^tau _{k+1}right) end{aligned}is exactly the discrete-time implicit Euler scheme for (x'=-nabla F x))! (note that in the convex non-smooth case this becomes ( frac{x^tau _{k+1}-x^tau _k}{tau }in - partial F(x^tau _{k+1}))).