[r-t] Proofs, etc

[Robin Woolley]

I plead guilty to the following things, but I'm not going to apologise for
1, 3 or 4:

1. Using the word group in the sense of 'lead head group' - it may not be
matthematical, but it is 'custom & practice'. It is used, for example, in CC
pubs. as in the 'Universal Compositions' on pp 2 and 4.

2. Using the J = '12' when I should have said J = 12436587. This can be
found in Hodgson's paper (1962/75).

3. Confining my remark to regular palindromic methods - but at least that's
almost all of them. Non-regular or non-palindromes are just more tedious to
deal with.

4. Saying 'operating on' when I meant 'inv(B) left multiplies A' - but I
thought this was obvious anyway given its normal English meaning. Actually,
I can't see the problem here since both A & B are permutations (in the true
sense of the word) so one starts with A and permutes it by inv(B), so for
'operating on' read 'permuting' - chirality is then irrelevant.

As I mentioned before, it takes 13 pages or so to get through the pre-amble.
One doesn't want to post too much at once.

Falseness groups aren't groups either - they're not closed for one thing -
but they're still referred to as that.

As a thought - is change ringing the earliest group theory? It predates
Lagrange by 100 years or more.

As Richard Smith points out, since one lead can generate 8 ratios based on
X, there will be a possibility of 56 leads false against the plain course of
a given method since the course has seven leads. Usually, it is not as bad
as this as B falsesness (typified by 24365) consists of 7 pairs (one in- and
one out-of-course, the members of the pair being related by 13254768. Only
N, O, T and e have the full 28 pairs.

In fact, the analysis I use, using J = 12436587, gives the correct results
for an eighths place variation (e.g., Primrose cf Cambridge) both as to
falseness and incidence (if place bells are quoted, rather than positions in
the course).  For clarity, J = 12436587 or J = 13254768 are usable for all
regular methods but the appropriate lead-head type must be used in irregular
methods.
(All lead ends are, whether regular or irregular, self-inverse, however -
the reason is obvious).

When Royal is considered, the maximum false lead set size is 72 and, as can
be seen by inspection , but also shown algebraically, we have the situation
where the in - course tenors together members of falseness groups B & D
always appear together. The Exercise has been fortunate that a more radical
revision of the FCH groups has not been necessary as we go to stages higher
than major.

Once again, I can't see how any of the other treatments proposed deal with
incidence. For another treatment, why not look at, say RW77/923 and
RW80/259?

Finally, RPI Lewis says: 'Just goes to show that people who talk maths don't
have the monopoly on confusing people with undefined terms...' There is an
important difference here. On a Ringing Theory site, one would have thought
members are at least familiar with ringing terms. It may not be sensible to
talk about 'lead head groups' and 'falseness groups' when they are not, but
that's (ringing) life for you. We can't start every e-mail with a definition
of plain hunt, but it is justified to expect an explanation of M11 or
'geoetrics' which may only be familiar to those researching a particular
area. (Is the latter peculiar to Bangor?) To a first approximation, no-one
who
has done mathematics at university will have heard of them. But then, isn't
this where we came in on 16th September?

Best wishes,
Robin Woolley