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.
<<> > 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')|".
<<If you want R(n,n') to be a set you are going to have to say what its
elements are. May 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? 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
positive
entry in the n-th column.
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|.
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.
Let K = {(n,n') : (n.n') is in [x,y] \times [x',y'], |N(n)| > |N(n')| >0}.
Let R = {R(n,n') : (n,n') in K, R(n,n') = I(n) - I(n') and \sum(|R(n,n')|
=
min(sum(|R(n,n')|)}.
Let S = {S(n,n') : (n.n') is in [x,y] \times [x',y'], |N(n')| > |N(n)| |>
0,
S(n,n') = I(n') - I(n)}.
Let V(n,n') = {v : (n,n') is in [x,y] \times [x',y'], |N(n)|=0 and
|N(n')|=1}
Let r and s be members of R(n,n') and S(n,n') respectively.
Let W(n,n') = {w : (r,s) or (r,v) and r, s and v are unique members of R,
S
and V(n,n') respectively}.
Let |N_{1}(r,n)| be the value |N(n)| for the column in which r is found."
I wonder, have I constructed a set of pairs (n,n') for which each value of
n-x and likewise of n'-x', among all such for members of R, is unique, and
no value of n-x or n'-x' among all such for S will be found for a member
of
R?
Have I successfully constructed pairs (r,s) and (r,v) as the members of
W(n,n')?
<<> 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 trust I will be able to use the letter "i" elsewhere in my paper,
to indicate an arbitrary integer, in a completely different context?
With thanks.
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