On Mon 2006/01/16 12:11:04 -0000, Michael Deckers wrote
in a message to: LEAPSECS@ROM.USNO.NAVY.MIL
     "

>   UTC (since 1972) is a disseminated timescale that is equal to TAI
>   except for additional date and time codes transmitted with the
>   signal. These codes indicate the values of TAI - DTAI for each
>   full minute of TAI - DTAI.
>
>   In a reading of UTC, (the most recent values of) these date and
>   time codes can be taken as is so as to yield a reading that
>   approximates UT1. The codes may also be used to derive the value of
>   DTAI (from a table of leap seconds since 1972), and thus, to yield
>   a reading of TAI. When a timestamp is characterised as UTC
>   (rather than TAI), then the first type of reading is implied.
>
>   In order to ensure a unique derivation of TAI from a recorded
>   reading of UTC in the vicinity of a positive leap second (where
>   DTAI jumps up by 1 s and the value of TAI for a given value of
>   TAI - DTAI is not unique), the UTC reading that corresponds to the
>   earlier TAI value shall be recorded with a second field >= 60 s,
>   and the other UTC reading, with a second field < 60 s.
>
>   Michael Deckers (still trying to understand the topology you referred to)

I think we are basically on the same wavelength, though I would turn
some of your sentences around.

The way UTC is disseminated is not directly relevant to the discussion,
and I don't think I said anything about topology.  A better way to think
about it would be from the combinatorics point of view.

Enumerative combinatorics is basically concerned with counting things,
usually counting them two ways and drawing conclusions from that.  Use
of mixed-radix numbers is one of the strategies used in combinatorics,
e.g. see http://en.wikipedia.org/wiki/Factoradic.

The representation of UTC as a variable-mixed-radix number is a clever
way (possibly too clever) of counting seconds of TAI in a way that
allows interpretation as either TAI or, approximately, UT1.

The UT1 (or time_t) interpretation is obtained by treating UTC's
representation (label, notation) at face value, as though it was an
ordinary sexagesimal number.  UT1 so derived does have a discontinuity
when leap seconds are inserted and presumably this is what leads people
to say, misleadingly, that UTC itself is "discontinuous".

On the other hand, TAI is recovered when the variable radix of the
seconds field is taken into account.  However, since they occur
irregularly, this interpretation requires a table containing the full
history of changes in the radix.  Alternately, the cumulative effect of
the table for epochs since the last leap second is conveniently
summarized in a number that we call DTAI, usually referred to as
TAI-UTC, where "UTC" here is understood to be the usual mixed-but-
fixed-radix sexagesimal representation.

So there is plenty of scope for confusion.  However, think back to the
graph of age vs UTC that I described at the start of this discussion and
particularly that each instant of the continuous time axis has a unique
UTC representation.

Mark Calabretta
ATNF

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