Paul,
<<I wouldn't think so - at any time you can set P to be anything you
want. You could write t(n,P(m)) or t(n,J) if you want a subset J of
P(m) or t(n,{3, 11, 29}) if you wanted to for some reason.>>
That's precisely what I had been doing in any case; surely, in doing such,
I
wasn't being presumptuous?
<<Tell me again what you _want_ to do.>>
Take a finite set B whose members are all the occupied matrix components
in
a binary matrix. I wish to establish, without imposing any determining
criterion as to the distribution of such members, basic principles about
the
value o(x,y) in this matrix, when specific criteria are held fixed and
various other stipulations are made*. Then, I want to apply these
principles
to different [x,y] intervals in the matrix M_1.
[*see private email.]
<<I would have to know _exactly_ what you mean by "array" - that has
never been nailed down. A couple of exchanges ago you gave an example:
4 7 2 0 1 0 5 8 11 1
1 2 3 4 5 6 7 8 9 10
I don't know if you intended this to be an "array" or not. What it _is_
is a 2 x 10 matrix. It could more easily be a 10-tuple:
let S = (4, 7, 2, 0, 1, 0, 5, 8, 11, 1). Then. you have all the
information of your table. You can easily refer to the entries of S :
S(9) = 11 for example.>>
Recall that I had said "the column numbers, which you could call values of
n, are given as the bottom row and arbitrary column sums are the upper
row".
<<I really _don't_ know how you intend to use M_1 (nor why you subscript
it) nor how to use "arrays". If fact you have said that M_1 will
ultimately disappear.>>
It is merely to effect a grafting process in my method. As far as I can
see
I *need* the array, so naturally I use, in association with it, the
concept
of a matrix in order to make a smooth conceptual transition. The reason I
use the array is down to an issue you bring up later in your reply, and
about which I have expressed my own continuing concerns. It is that, in a
histogram, there are only two values relevant to my method: column-index
number and value on the vertical axis. But I wish to construct sets in
which
one unit -- one occupied matrix component -- is paired with another. So to
answer your question as to what I mean by an array, I mean a binary matrix
in which there is not necessarily any determining criterion imposed upon
the
distribution of occupied matrix components.
<<No, it has got to be a cross. In most Math typesetting packages it is
neither "x" nor "X" but a "cross" - a "+" rotated 45 degrees. You _can_
say [x,y] x [x',y'] - in fact I do so below.>>
This is most interesting. I had never before encountered this. Indeed the
professional mathematician I had mentioned to you previously had told me
it
was a multiplication symbol, and I guessed that this was attributable to
there being (y-x+1)^2 pair combinations. (But strangely he had used an 'x'
when normally he would use a * for multiplication.) You couldn't give me
the
TeX coding for it, could you? (If it's a + turned through 45 degrees I
fear
it'll look almost indistinguishable from \times....)
<<As you've described it, R(n,n') is merely a number - not even a set
much less a set of sets.>>
As I have mentioned above, this is precisely what has been worrying me all
along. If you could help me overcome this problem, such that, in my
initial
definitions, I can form purely algebraic sets whose members can be
interpreted as occupied matrix components, that would be splendid.
<<> and sum_R(n,n') is as low a value as it possibly can be given [x,y]
and
> [x',y'].
You can't sum a single number - at least there is no point in doing so.>>
Sorry, what I meant was, sum_#R(n,n') in [x,y]x[x',y']. I keep omitting
that
darn '#' sign!!
<<I think maybe this is something like what you are trying to do:
Let K = {(n,n') : (n.n') is in [x,y] x [x',y'] and
t(n,P(m)) > t(n',P(m))}.
[ Read this "K is set the of all pairs (n,n') which belong to [x,y] x
[x'.y'] (that is, n is in [x,y] and n' is in [x',y']) and
t(n,P(m)) > t(n',P(m))" ]. >>
Does this mean #K=(y-x+1)^2? I only need a maximum of y-x+1 pairs, of
course.
BTW I want another stipulation, which is that t(n',P(m))>0.
<<You could then Let R = {R(n,n') : (n,n') in K}.Then you could add up or
minimize the elements of R. Equivalently you could form
sum(R(n,n') : (n,n') in K) or min(R(n,n') : (n,n') in K).>>
This looks to be the kind of thing I am after. I want the minimum possible
value of sum_#R(n,n'), and a set of sets R(n,n') so that I have got a set
comprising every (n,n') for which there is a value #R(n,n').
Then, I want a set of sets S(n',n'), S(n,n') being a set comprising
t(n')-t(n) members (each member being an occupied matrix component), such
that (n,n') is in [x,y]x[x',y'] but neither n nor n' are members of a pair
R(n,n'), and t(n')-t(n) >1.
BTW does 'min' stand for 'minimum' or 'minus', or something else?.
<<How about an example; say m = 3 with [2,7] and [5,10]?>>
I don't quite understand this; I think it will end up with me naively
expressing something that, by some misinterpretation, serves only to muddy
the waters; but I hope you can understand what I'm saying at in my
previous
couple of paragraphs.
With many thanks once more.
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