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Arnold Neumaier wrote:
Nate Hayes wrote:Arnold Neumaier wrote:Nate Hayes wrote:John Pryce wrote:On 18 Sep 2010, at 23:15, Nate Hayes wrote:I speak against this. Ulrich's interior is better.Note that the topological interior, i.e., "proper subset," is already expressed efficiently in terms of Ulrich's relations for intervals A,B:( A \subset B ) and not ( A == B )Doesn't that make [2,3] interior to [1,3]?Well, it seems weird to me too, but there it is. You're an expert on quantified statements. Isn't it inescapable from the definition "B is a neighbourhood of each a in A" (eqn1)?I don't see Entire should be interior to Entire.As you are so fond of quoting from George Corliss: if it even seems weird to you -- a seasoned mathemetician -- then "God help the casual user!"The thing about definitions grounded in standard theory (and this theory has been around for roughly 100 years) is that, compared with ad-hoc definitions, you KNOW they can't lead to inconsistencies -- assuming math itself is consistent.This is a reason I think P1788 might want to investigate sticking to compact intervals, instead.... which if you remember was one of my first choices.Unbounded intervals are essential for applications to general global optimization and nonlinear systems solving, where for complex modelsone often doesn't know in advance bounds on all variables.Yes, being able to initialize such unknown variables appropriately is important for these applications. I agree. I don't suggest P1788 should not provide a mechanism for this.Keep in mind, Ian McKintosh has suggested replacing unbounded closed intervals with "overflown" compact intervals. In practice, there is not really a difference between the two. For example, +INF and -INF are replaced by +OVR and -OVR in all compuations, where OVR is some unknown finite real number larger than the magnitude of the largest floating-point number. In this way, intervals such as [1,+OVR] remain compact intervals and are not unbounded like the interval [1,+INF]. Arithmetic operations on the endpoints of compact overflown intervals is the same as in Motion 5, i.e., (-OVR) + (-OVR) = -OVR, 0 * OVR = 0, etc.But then OVR has a different meaning in different places, which is unacceptable from a mathematical point of view. It is just INF in duisguise, and it hides mathematical propoerties of the limit underthe carpet. One cannot use it in a mathematical argument.... To argue for existence, one _needs_ in certain case the unbounded case.Mathematical programming had once a tradition for using big M (which is about the same as your OVR), and it is now deprecated (though still used by a few who don't like unbounded variables) since it behaves poorly not only mathematically but also algorithmically.
I would like to see examples specific to intervals and OVR why this should be the case.
So this approach seems to retain all the important advantages of unbounded intervals without actually introducing the complications that arise from them. To the extent I can see, it has big potential to simplify P1788 a great deal, without hurting or compromising the important needs of global optimization and nonlinear programming application you mention.It is an artificial device without a proper mathematical support. There are verygood reasons why mathematics has developed all thesetopological concepts, sincve they are _exactly_ right in the circumstances.
No doubt.But IEEE 1788 is a standard for computing as much as it is a standard for mathematics. Therefore it requires an interdisciplinary approach.
I think it is a pity you should be so quick to judge the possible outcome of such an investigation.
Nate Hayes