Chemical processes are generally modeled through monoscale approaches, which, while not adequate, satisfy a useful role in product-process design.
To prove the existence of a common coupled fixed point for such pair of operators, it is assumed for operators to satisfy a useful condensing and contractive conditions (conditions ((C_{1})) and ((C_{2})) hereafter) involving the two-dimensional setting.
The rate-to-rate synchronization function τ(t) satisfies several useful constraints, that are.
Then we prove that Green's function satisfies the useful properties.
This answer may not satisfy or be useful to or be informative for the speaker who poses (46), but it counts as an answer despite these purely pragmatic failings, according to Braun's theory.
Furthermore, we show how even a thin graded coating can be useful to satisfy the requirements of a specific application without having to make an entire wall with graded properties.
Summing up, an EM approach on Markov models provides an accurate 4-taxon tree reconstruction method suited for data not known to satisfy homogeneity and very useful as input of quartet-based methods, specially for largely divergent trees.
In solving boundary value problems, use of basis functions that independently satisfy the boundary conditions are useful.
I suspect he already knew the answer was yes, or at least he would have done if he'd ever heard the intro music to Streets of Rage 2, which more than comfortably satisfies the most useful definition of art I've come across, formulated by Scottish author Alasdair Gray: anything that stimulates people into "thinking about their feelings, and feeling about their thoughts".
If students are interested enough to ask, I don't assume they're asking because they're nosy, but that it satisfies some more useful level of curiosity.
