Paul,
> As near as I can tell, it boils down to this:
>
> Let J be a sequence of primes p(1) < p(2) < ... < p(t);
> Let C be a sequence of integers x(1) < x(2) < ... < x(r).
> Define M(J,C) to be the t x r matrix whose (i,j) entry is 1 if p(i)
> divides x(j) and is 0 otherwise.
>
So I don't need to mention m? Ultimately I will have to say that what I
am
callingt the m-th prime is proximate to, but less than, the square root of
y-x.
I must also say that in the column indexed by zero, there are #J occupied
matrix components.
> For example let J be 2, 5, 11 let C be 4, 15, 16, 22 then M(J,C) is
> 1 0 1 1
> 0 1 0 0
> 0 0 0 1.
>
> If this is really what you intended, you might want to rewrite the rest
> of it in this sprit.
>
> I forgot to mention that a binary matrix is the adjacency matrix for a
> simple non-directed graph. That is, your whole setup could be in terms
> of graphs instead of matrices. A quick look at Wikipedia might tell you
> if that would be of any use to you.
>
> Incidentally, the matrix multiplication (1, 1, ..., 1)*M(J,C) will give
> you, all at once, the number of non-zero entries in each column of
> M(J,C).
>
My matrix will become redundant, and can be replaced by ordinary algebraic
expressions. I am only mentioning it because it's the best way I can see
to
explain the issue I have.
Currently, I have got the following:
<<Let P(m) be the set of the first m prime numbers.
Let a(m) be their product. For an integer n let t(n,m)
be the number of members of P(m) that divide n (no
multiplicity counted, so t(9,2) = 1, for example).
Given an interval [x,y] of integers,
define c(x,y,m) to be the sum of
(1/2)t(n,m)(t(n,m)-1) for n in [x,y].
Given an interval [x,y] of integers, define
T(x,y,m) to be the set n in [x,y] for which
t(n,m) > 1.
Given an interval [x,y] of integers, define
o(x,y,m) to be the number of n in [x,y] for
which t(n,m) > 0.
Given an interval [x,y] of integers, define
N(x,y,m) to be the sum of t(n,m) for n in [x,y].>>
Now, the definitions above pertain to the case where the number of
occupied
matrix components in the n-th column, given by t(n,m), is determined by
factorisation. However, I intend to construct an array, with cells that
are
either black or white, in which the number of black cells is not
necessarily
determined by factorisation. Does this mean that I will have to introduce
a
whole lot more definitions, using different letters of the alphabet, to
use
the same concepts -- number of black cells in a column etc. -- as given
above? Apparently I can't use the notation t(n,M_[x,y]) for number of
black
cells in a column because t wasn't defined for the array.
I was thinking that maybe I could make my array one in which the numbers
of
black cells in a column over the first (a(m)+y-x ) columns are precisely
equal to the values found for the matrix we are defining. That way I could
make my reference something like t(n,m,[1,a(m)+y-x]) to indicate that n in
[x,y] is a column in that interval, or t(n,m,a(m)+y-x],f) to indicate
that
it isn't (f being the number of coulmns in the array for which [x,y[ and
[x',y'] are subintervals.
With Thanks.
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