In article <ukfYj.27164$EH5.11372@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> Paul,
>
>
>
> <<From the start, you have distinguished between matrices and arrays but
> you have never been willing to tell me what you thought an array
_was_.>>
>
> Have said at least three times that I thought an array was a matrix in
which
> there was no prescribed determinant of the distribution of occupied
matrix
> components. In future I shall try to avoid the term.
Good plan. Protect my delicate sensibilities :-)
> <<> > This isn't at all clear; sum(R(n,n') : (n,n') in K) is just a
number.
> >
> >
> > What's the problem with that? BTW note I write sum_#R(n,n'); each
value of
> > #R(n,n') is t(n,m)-t(n',m).
>
> Frankly, I thought that was a typo. I guessed you were using the pound
> sign to indicate the size of a set. But "sum_#R(n,n')" says literally
> the word "sum" subcripted by the size of R(n,n').>>
>
> My friend the professional mathematician always used that form of
> expression. I'll put what I meant in TeX coding: "\sum |R(n,n')|".
Well, that's a little better. I didn't realize the first try was
"TeX-like". It would have produced an upper case sigma with a "#"
underneath followed by |R(n,n')| on the same line as the sigma. The
second is the same without the "#". What you need is something like
sum(|R(n,n')|; n in [x,y]) i.e. tell us _which |R(n,n')|'s are being
added up.
> <<If you want R(n,n') to be a set you are going to have to say what its
> elements are. May[be] some thing like the set of primes that divide n
but
> don't divide n' or something like that.>>
>
> I see where the confusion has arisen. I once spoke, to the mathematician
I
> have mentioned, of a set A that has subsets B and C, such that members
of C
> are members of A that are not in B. He replied, 'So C is A-B'. Hence my
use
> of the minus sign as a way of defining sets. I wonder, could you tell me
if
> his reply was mathematically valid?
Yes, it is fine: for sets A and B, A - B is the set of elements of A
which do not belong to B. (I prefer A \ B but never mind that.)
What would make sense is if your t(n,m) was the _set_ of primes less
than or equal to m which are factors of n. You could then reference
|t(n,m)| if you wanted the count of those primes i.e. get your original
t(n,m).
With the new t(n,m) you could let R(n,n') = t(n,m) - t(n',m) = the set
of primes in P(m) that are factors of n but not of n' or
R(n,n') = |t(n,m) - t(n',m)| = how _many_ such primes there are or
R(n.n') = |t(n,m)| - |t(n',m)| = how many more (or less) such primes
divide n that divide n'. Which, if any, of those you want I don't know.
By the way, with the new t(n,m), t(n,m) > 0 makes sense (or at least it
will be understood); it says at least one prime in P(m) is a factor of
m. Similarly t(n,m) > t(n',m) makes sense; it says every prime in P(m)
which is a factor of n' is also a factor of n.
> Assuming it is, I shall have another
> shot, if I may, at laying out my definitions; I'd be really grateful if
you
> could suggest any possible improvements:
>
> "We shall be constructing a binary matrix, M, in which there is no fixed
> determinant of the distribution of positive entries, and in which the
column
> sum for the n-th column is equal to the greatest integer indexing [a row
in which there exists] positive entry in the n-th column.
We shall be constructing a binary r by s matrix M such that
sum(M(i,n); i = 1 .. r) = max(i : M(i,n) =/= 0). [Here, =/= means "not
equal (remember M is binary), M(i,j) is the entry in the i-th row and
j-th column.]
[Although it is very common, I dislike the editorial "we". I prefer the
passive "A matrix M will be constructed ...." It's a matter of taste.]
> Let P(m) be the first m primes.
>
> Let J be a subset of P(m).
>
> Given an interval [x,y] of integers, define N(n) to be the set of
positive
> entries in a column indexed by an integer n in [x,y], N(x,y) to be the
set
> of positive entries in [x,y], and N(x,y,|J|) to be the set of positive
> entries in [x,y] such that the greatest value of |N(n)| in [x,y] is |J|.
Is the matrix still M? You've got problems : N(n) = {1} or N(n) = {}.
Moreover what if M only has 10 columns and [x,y] = [100,110]?
I was guessing you meant x >= 0. No? If so N(x,y) = y - x + 1 or (in
case x = 0) y-x.
|N(n)| is either 0 or 1. What do you _really_ want N(n) to be? Do you
want maybe N(n) = {i: M(i,n) =/= 0}? Oops, I see below that that is
I(n) (I think.) Until I'm confident I know what you want N(n) to be,
I'll skip the rest of your definitions.
Suppose M turned out to be
1 0 1 1
0 1 1 0
0 0 1 1
1 1 1 0
What would you want N(1), N(2), N(3) and N(4) to be?
> Let I(n) be the set of integers i indexing all rows for which there is a
> positive entry in the n-th column in M, such that the i-th positive
entry in
> the n-th column is indexed by i.
[...]
> <<> What about something like t(n) and, for the other construct,
> > t_{\alpha}(n,P), instead?
>
> That would be OK - why not just a different letter?>>
>
>
>
> Over my full work, I have well and truly run out of letters of the Roman
> alphabet and am fast moving through Greek letters. I was advised that
this
> meant that my definitions were "not sufficiently modular", whatever that
> means.
I don't know what that means either. However asking a reader to keep
track of that many definitions/functions is asking a lot. If you only
use something once or twice there is probably no reason to attach a
symbol to it.
> I trust I will be able to use the letter "i" elsewhere in my paper,
> to indicate an arbitrary integer, in a completely different context?
Sure.
--
Paul Sperry
Columbia, SC (USA)
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