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From: Deane Yang (deane_yang_at_[hidden])
Date: 20051012 09:21:55
Jeff Garland wrote:
> On Mon, 10 Oct 2005 23:22:24 0400, Deane Yang wrote
>> Jeff Garland wrote:
>>> Wow, that's amazing. Replace quantity_vector with time_duration and
>>> quantity_point with time_point and you have the core concepts behind
>>> datetime. And the 32 versus zero  seems an awful lot like an epoch_time
>>> adjustment. So I suppose I'm saying I believe there is something fundamental
>>> here :)
>>>
>> Yes, there is.
>>
>> If you dig back in the archives, I tried to explain it all before,
>> but obviously not very well.
> (stuff deleted)
>
> Sorry if I missed or didn't respond to your comments the first time.
Definitely understandable.
> However,
> I now have reasons to be more interested in the theoretical foundations.
> Specifically, the LWG committee is rightly an intense group that needs to
> ensure the quality and correctness of things that get added to the standard 
> having additional theoretical foundations for the datetime concepts can't
> hurt. Any chance you can point me to some sort of references (preferably
> readable to folks that have only taken ~15 college math courses ;)
>
Here's a brief summary of the idea (assuming that you learned about
vector spaces (http://en.wikipedia.org/wiki/Vector_space) in linear
algebra) (In anything you read about vector and affine spaces, if they
say "field F", just assume that F is the set of real numbers):
A "dimension" or, as Matt calls it, a "point quantity" or, as you call
it the set of "time_points", without choosing any particular unit,
corresponds to the mathematical concept of an "affine line" or an
"1dimensional affine space" A (see
http://en.wikipedia.org/wiki/Affine_space).
Associated with any "point quantity" or "dimension" or the set of
"time_points" is the set of all possible differences between two points.
Matt calls this a "vector quantity"; you call it the set of
"time_durations"; mathematicians call it the (1dimensional) vector
space (http://en.wikipedia.org/wiki/Vector_space) associated with the
affine space.
Assigning (relative) units to a vector quantity corresponds to fixing a
nontrivial realvalued linear function f: R > V on the vector space V.
Think of V as a straight line with the "zero vector" marked and the map
f as like putting a ruler on top of the straight line, assigning a real
number to each point on the line. Notice, however, that you have to line
up the ruler so that the 0 on the ruler is placed against the
zerovector. So your only flexibility in what you do is stretching or
shrinking the ruler.
Assigning (absolute) units to a point quantity corresponds to fixing a
nontrivial affine map
(http://en.wikipedia.org/wiki/Affine_transformation) f: R > A, where A
is the set of all possible values of the point quantity. Think of A as a
competely blank straight line and the map f as putting a ruler on top
of the line. This time, since the line is blank, you can slide the ruler
along the line and put the 0 on the ruler against any point on A. So you
have more flexibility with your units; you can stretch, shrink, or slide
the ruler.
Unit casting corresponds to the situation where you have two different
rulers you want to use, and you need to convert a number on one ruler
into the corresponding number on the other. For vector quantities, you
just need to multiply the number by a fixed proportionality constant.
For point quantities (temperature is the best example here), you might
also need to do an addition or subtraction, too.
You can always turn any vector space V into an affine space, by simply
"forgetting" where the zero vector is. You can always turn any affine
space into a vector space V by arbitrarily choosing a point in A and
declaring it to be the zero vector.
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