[[ I got your e-mail and I'm willing to hang in for a while at least;
it may sometimes take me a while to respond. Maybe Brian, if he has
recovered from the end of term agonies, would be willing to join in
again.]]
In article <b4DTj.24339$jn6.18319@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> 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.
You can restrict J as needed - for example p(t) could be the m-th prime.
> I must also say that in the column indexed by zero, there are #J
occupied
> matrix components.
Since the first column is _always_ all 1's it seemed better to handle
that as a special case rather than mess up a clean definition of
M(J,C).
>
> > 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.
[...]
> >
> 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.
Well, there's the rub: what _is_ the issue you have? In particular, it
is im****tant (for me) to know if you are intending the "matrix" as
merely a data structure rather than a mathematical object; i.e. are you
planning on doing any matrix algebra?
> 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].>>
These are OK - I understand what they say. You might (or might not)
find it more convenient to let your intervals be [x, x + k].
> 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.
Is what you are after the number of distinct prime divisors of an
integer n?
The phrase "the number of black cells is not necessarily determined by
factorisation" is bothersome. What _does_ determine the number of black
cells? For that matter, why the very unmathematical "black" and
"white"?
> 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.
I can't comment on this since I don't know what the objective is.
Observe however that the sequences J and C can be anything you want
them to be within the given constraints - J could be all primes less
than sqrt(n) and C could be the single integer n, for example. A fairly
common usage is something like "Let M(J,C) = (m(i,j))" and then use
m(i,j) to refer to the (i,j) entry of M(J,C).
What is your ultimate goal? Just an outline or abstract would help a
lot. It is looking more and more like it is some sort of integer
factoring algorithm perhaps using some sort of sieve.
--
Paul Sperry
Columbia, SC (USA)
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