In article <TXdXj.11805$iD4.5714@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> Paul,
> It looks like we're making good progress. Comments embedded.
>
> > I've paraphrased your definitions below. I saw nothing wrong with them
> > the way they were; my changes just make typing them easier and give
you
> > a little more flexibility if you ever want it.
> >
> > <<Let P(m) be the set of the first m prime numbers.
> > Let a(m) be their product.
> >
> > My changes start here.
> >
> > Let P be a set of primes and let I be a set of positive integers.
> > [Anytime you want to you can insert the statement "Let P = P(m) and
> > I = [x,y]" or words to that effect.]
> >
> > For an integer n let t(n,P) be the number of members of P that divide
n.
>
>
> But, im****tantly, we are not just taking an arbitrary set of primes. In
P(m)
> we are taking those primes that divide multiples, of value less than the
> square of (y-x+1). Surely it's easier just to begin with a set P(m) and
> define a subset, J?
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.
> > Define c(I,P) to be the sum of (1/2)t(n,P)(t(n,P)-1) for n in I.
> > [Or, sum((1/2)t(n,P)(t(n,P)-1); n in I)]
> >
> > Define T(I,P) to be the set n in I for which t(n,P) > 1.
> >
> > Define o(I,P) to be the number of n in I for which t(n,P) > 0.
> >
> > Define N(I,P) to be the sum of t(n,P) for n in I.
> > [Or, sum(t(n,P); n in I)] >>
> >
> > As I said, once you say P = P(m) and I = [x,y] these are exactly what
> > you had - just easier to type.
>
>
> If we go back to the problem that I'm really keen to resolve, which is
how I
> use both an array and the matrix M_1 in the way I have already
discussed,
> then what do you think I should do?
I don't know. Tell me again what you _want_ to do.
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.
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.
I wondering if you wouldn't be better off to forget about matrices and
"arrays" - at least for the time being - and just work with your
function definitions. I haven't seen any indication that you are going
to use any of the vast body of machinery that applies to matrices.
[...]
> > There is something called the Cartesian (or "cross") product of sets.
> > (Forgive me if you already know all about this.)
>
> I have heard mere mention of something along these lines.
>
> If A and B are sets
> > then A x B is the set of all ordered pairs (a,b) where a is in A and b
> > is in B. Read A x B as "A cross B". "Ordered" means, for example that
> > (2,3) is different from (3,2).
I'm going to break your next two paragraphs into pieces so It'll
probably look awful.
> So I can say [x,y]*[x',y']?
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.
> I wonder if you could fill me in with a precise
> wording, given how I have described by objectives - that would be much
> appreciated.
> In making my pairs (n,n'), I am trying to construct a set of sets
R(n,n')
> such that R(n,n') = t(n,m)-t(n',m), and, for all R(n,n'),
t(n,m)-t(n',m)>0,
As you've described it, R(n,n') is merely a number - not even a set
much less a set of sets.
> 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.
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))" ].
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).
I don't know if you really want to do either of those. By the way, I'm
going to cling stubbornly to _my_ notation.
How about an example; say m = 3 with [2,7] and [5,10]?
[...]
--
Paul Sperry
Columbia, SC (USA)
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