Re: Motion 13
Vincent Lefevre schrieb:
On 2010-05-03 08:53:40 +0200, Ulrich Kulisch wrote:
I think the problem with the empty set should be further discussed.
I agree.
I have great sympathy with Dan's mail of April 22. I had very
similar thoughts when I was working on my book. I studied modells
representing the empty set by a tuple aiming to avoid the necessity
of frequent checking for it.
Dan:
[+oo, -oo] * [+oo, -oo] = [-oo, +oo] while [NaN, NaN] * [NaN, NaN] =
[NaN, NaN]. So it seems that
[NaN, NaN] is better suited to represent the empty set.
In a mail of April 22 Arnold Neumaier
wrote:
"This is just to let you know of Siegfried Rump's decision to remove
empty intervals from Intlab Version 6;".
I conclude from this decision that there are hardly any applications
where the empty set occurs as an operand in an operation or a comparison.
Therefore my question: DOES ANYBODY HAVE AN IMPOTANT APPLICATION WHERE
THE EMPTY SET APPEARS AS AN OPERAND IN AN OPERATION OR A COMPARISON?
In a mail of April 23 Juergen wrote:
"I strongly propose to stay with our clear math. model and not to
sacrifice the empty set for efficiency."
I agree to a certain extent.
Intlab is a software package. So frequent checking for the empty set is
time consuming.
This would not be the case if the checking is done in hardware. Here it
could be executed in parallel and simultaneously with the operation. So
it just would need a little silicon.
I think that the empty set should be regarded as some kind of
exception. But this isn't obvious.
I agree with that.
Some mathematical properties
(such as the subset inclusion yielding a lattice) would be true
only when the empty set is considered, while for others (ordering
between intervals yielding a lattice) it would be the opposite.
I have no problem with that.
This could be a possible use for the empty set:
Occasionally it happens in programming that a variable is used before a
definite value has been assigned to it. This is a terrible error in
interval arithmetic and should be avoided by checking for it. A solution
could be to set every variable at definition to be the empty set.
Best regards
Ulrich
--
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik (IANM2)
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch
Telefon: +49 721 608-2680
Fax: +49 721 608-6679
E-Mail: ulrich.kulisch@xxxxxxx
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KIT - Universität des Landes Baden-Württemberg und nationales Großforschungszentrum in der Helmholtz-Gemeinschaft