[r-t] RE: Various
pabs at cantab.net
Sun Nov 21 12:48:55 UTC 2004
Robin Woolley <robin at robinw.org.uk> wrote at 09:02:00 on Sun, 21 Nov
>I think that the use of 'countable' means that there is a simple
>algorithm to count the members of a set.
A set is countable if there is a mapping from the integers onto the set
(i.e. without missing any). All finite sets are countable.
>Looking at extensions themselves. Classical extension had (2n-3) possible
>paths below (worst case scenario) and 2(n-2) possible paths above, giving a
>maximum set size of 2(n-2)(2n-3), which is of the order of n^2. Quantum
>extension (extension by modes) has (n-1)(n-2) both above and below which is
>of the order of n^4. It's interesting to see the increase in the limit of
>possible extension paths quantified. Classical extension of minor can be
>carried out by hand on the back of the RW address label.
You forget that we can now extend by more than two stages, giving
n(n-1)(n-2)/4 constructions each above and below (for even n). In most
practical cases the number of distinct constructions will be much less.
>As regards a program to do this, is it more efficient to generate the 400
>extensions of any minor parent, pick off the ones with (say) pb lead ends
>and then apply the restrictions, or apply the restrictions then generate the
Generate the distinct constructions, including any restrictions, then
check for pb lead ends (don't forget that you have to do this at an
infinite number of stages).
>Here's a case of 'uncountably finite' in that there's no algorithm to give
>the number of true extensions, but it is finite.
How about 'generate all extensions, then count them'.
>Back to the old topic:
>Part of the MC's justification for (G)B1 is "This new restriction gives us
>the confidence to introduce several new constructions without the risk that
>they will increase the number of unsatisfactory extensions". If an
>extension is 'unsatisfactory', which I take to mean almost quite, but not
>totally, unlike the parent, would any ringers wish to disregard it as an
>extension anyway? (Or am I being naive on this?) Taking Roker as my usual
>example, the two extensions to royal aren't particularly similar to the
London Major isn't particularly similar to the parent - nowhere in the
Minor is there an equivalent to turning round in 4ths below the treble.
>On families, Sir Martin Rees, the Astronomer Royal, reminds us that, whilst
>there is an infinite set of plane regular figures, three dimensions give us,
>as pointed out already, just five regular solids. In four dimensions there
>are six, but just three at all possible higher stages.
So these three are extensions - the others don't extend!
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