What are the vertical bars "|" I see everywhere in the article? For example "| have N | components"? I'm using Chrome. -Andriy
This wikipedia entry reads well and is written by experts. Nonetheless, the authors may still want to improve it further, because occasionally it sounds anachronistic. The following are optional pointers.
I propose to change `DAEs are not ODEs' to something like `DAEs generalize ODEs'. (Would you say that incompressible Navier-Stokes is `not a time-dependent PDE system?' Also, the list of things that are not ODEs does include the kitchen sink.)
-The "DAEs are not ODEs" line is a classic one and was originally aimed at the numerical ODE community. We concur that it is not helpful for the more general audience of Scholarpedia. We have dropped that line and now talk of DAEs as generalizations of ODEs with additional properties.
Also, and involving a more complex change, it is unnatural to say that the equation x = 1 requires initial conditions. Rather, after index reduction by differentiation one obtains an ODE on an *algebraic invariant* (see Hairer, Lubich and Wanner's book for a more precise terminology). The invariant equations complement an *independent* set (typically fewer than N) of initial or side conditions.
-Many mathematical definitions get a bit strange when special cases are considered. Note that that initial conditions are sometimes not viewed as requirements but rather as the set of values that are consitent. In that case a 0-dimensional set of initial conditions is quite natural. Also there are circumstances, such as in control theory, where the degrees of freedom lie in a subspace called the zero dynamics. Here a zero dimensional subspace is natural and is what is called a flat system. So the current terminolgy is not unreasonable.
-We debated at great length among ourselves on how much detail to include. As we understand the charge this article is aimed at an extremely broad audience. Our experience in communicating with such audiences is that every new term increases the difficulty. Accordingly we deliberately did not go into the more technical discussions of manifolds and invariants. We leave that to the interested reader who wants to go further.
The sentence listing different index definitions is awkward.
Incidentally, there is a recent book by R. Riaza on DAEs and the folks in Berlin are planning another.
-At least one of the authors has read the Riaza book, and in fact, was one of the reviewers. We deliberately kept the bib very small and aimed at the specific points that we wished to make. As we understand it, once accepted this article can be edited by others. in particular, readers can try and add additional references they consider important at that point. We imagine approving such additions unless they are unrelated or full of errors. The Riaza book, in particular, would be approved.