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Notations, new Motion

The text of a new Motion is attached. The following mail exchange gives the rationale.

Best regards
Ulrich



On March 26, 2012 Ulrich Kulisch wrote:

Dear all,

There are some discrepancies in the notations of Drafts 4.02 and 4.04 and I think
we should straighten these out before less suited denotations spread. Let me briefly
comment on the history of these notations.
The real numbers R are defined as conditionally complete, linearly ordered field.
Conditionally complete means: Every bounded subset has an infimum and a supremum.
Every conditionally completely ordered set can be completed by joining a least and a
greatest element. In case of the real numbers R these are −∞ and +∞. However, these
new elements are not real numbers. For instance ∞−∞ not= 0, or ∞/∞ not= 1. I think
there was general agreement that the completion should be expressed by overlining the
R. So \overline{R} := R ∪ {−∞,+∞}.
Since the early days of interval arithmetic the set of nonempty, closed and bounded
real intervals has been denoted by IR. The ordering of the set {IR, ⊆} also is only
conditionally complete. For every bounded subset the infimum is the intersection and
the supremum is the interval hull. Completion of {IR, ⊆} brings the empty set and
unbounded intervals into the game. In my book (2008) and in the paper I prepared
for the proceedings of the Dagstuhl meeting (January 2008) I denoted the completed
set by (IR). This was critisized within P1788. Then I suggested writing JR for the
completed set. After some discussion I think we all agreed indicating the completion
again by overlining the set IR. In \overline{IR} the empty set is the least element. However, the empty set is not an interval arithmetically. As −∞ and +∞ are not real numbers the empty set does not follow conventional rules of interval arithmetic, for instance, ∅ · 0 not= 0.
For consistency the same scheme of denotations should be kept for the subsets
representable on computers. This leads to the following denotations:
R               the set of real numbers.
\overline{R}    \overline{R} := R ∪ {−∞,+∞}.
IR              the set of nonempty, closed and bounded real intervals.
\overline{IR}   the set of closed real intervals, including unbounded intervals and the empty set.
F               the set of (finite) floating-point numbers representable in some floating-point format.
\overline{F}    \overline{F} := F ∪ {−∞,+∞}.
IF              the intervals of IR whose bounds are in F.
\overline{IF}   the intervals of \overline{IR} whose bounds are in \overline{F} and the empty set.

Best regards
Ulrich

On April 4, 2012 Ralph Baker Kearfott wrote:

Ulrich et al:

We can take your notaional comments into consideration when writing the actual standard text. Also, it is in general a good idea to use our agreed upon notation in position papers. However, I view motion 31, as a position paper, as providing guidance for actual writing the standard text, so the meaning in motion 31 is what is of primary importance. That is, even when Motion 31 passes, we can still use your notation in the text.

Best regards, Baker

On April 4, 2012 Ulrich Kulisch wrote:

Baker,

it is not my notation. I think about two years ago we agreed upon the notation. I believe it is important that we settle this question now and not at the end of the standardization process. Many of us are still writing papers and even books on interval arithmetic and on applications. Standardizing the denotation of basic concepts is an essential contribution to the understanding and communication within P1788 and in the interval community.
If Motion 31 passes I really do not know which notation I should use in a paper I just have under preparation.

Should there still be disagreement upon the notation of the basic sets we should settle the question by another motion.

Best regards
Ulrich 


-- 
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik (IANM2)
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch

Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
www.kit.edu
www.math.kit.edu/ianm2/~kulisch/

KIT - Universität des Landes Baden-Württemberg und nationales Großforschungszentrum in der Helmholtz-Gemeinschaft

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