Re: Motion P1788/M0034.01:Notation-- discussion period begins

["Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>]

Ulrich, P1788,

Attached is my feedback on Motion 34.

Sincerely,

Nate

----- Original Message ----- 
From: "Ralph Baker Kearfott" <rbk@xxxxxxxxxxxx>
To: <owner-stds-1788@xxxxxxxxxxxxxxxxx>
Cc: "stds-1788" <stds-1788@xxxxxxxxxxxxxxxxx>
Sent: Monday, April 16, 2012 4:41 PM
Subject: Motion P1788/M0034.01:Notation-- discussion period begins


> P-1788:
>
> Since Motion 34 has been made by Ulrich Kulisch and
> seconded by Bo Einarsson, the discussion period now
> begins, and will end after Monday, May 7, 2012.
> The motion is attached (and some rationale / discussion
> is appended).
>
> Discussion on this motion will proceed according to the rules for
> position papers.
>
> Juergen:  Please place the motion and associated information
>            in the appropriate place on the web page, as
>            you have aptly done in the past.
>
> Acting secretary:  Please record the transaction in the minutes.
>
> As usual, please contact me if you need the password to the private
> area of the P-1788 web site.
>
> Best regards,
>
> Baker (acting as chair, P-1788)
>
> On 04/16/2012 10:13 AM, Ulrich Kulisch wrote:
> > 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
> >
> > --
> > 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
>
>
> --
>
> ---------------------------------------------------------------
> Ralph Baker Kearfott,   rbk@xxxxxxxxxxxxx   (337) 482-5346 (fax)
> (337) 482-5270 (work)                     (337) 993-1827 (home)
> URL: http://interval.louisiana.edu/kearfott.html
> Department of Mathematics, University of Louisiana at Lafayette
> (Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
> Box 4-1010, Lafayette, LA 70504-1010, USA
> ---------------------------------------------------------------

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