In article <YihTj.12036$_f5.2485@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> Paul,
> Many thanks. Comments embedded.
>
> >> I'm wondering if there is anything especially problematic about the
> >> wording
> >> of the questions I have posed on this newsgroup. I re-write one of
them,
> >> below, hoping this time it's all fully clear:
> >>
> >> Take x and y to be integers.
> >> If we have a set P(m) comprising the first m primes and a
> >> general subset J of of P(m), then I am constructing a matrix M_1 in
> >> which the members of J index the rows. Let the columns number f. Let
p be
> >> a
> >> member of J. The matrix components, of which in any subinterval [x,y]
of
> >> [0,f-1], there are (y-x+1)*J, are either occupied or unoccupied. If n
is
> >> an
> >> integer indexing a column, the occupied matrix components are given
by
> >> (p|n), save for zero which contains #J occupied matrix components.
The
> >> number of occupied matrix components in the n-th column is given by
> >> g(n,M_1).
> >
> > Let's see if I understand all this. I'll start over.
> >
> > *****
> >
> > Let P(m) be the m-th prime and let p(1) < p(2) < ... < p(t) <= P(m) be
> > primes.
>
>
> P(m) is a set comprising the first m primes. So if m=3, the members of
P(m)
> are 2,3 and 5.
Yes, but I intended p(1) < p(2) < ... < p(t) <= P(m) to replace your
set "J".
> > Let f be an integer and let 1 <= x(1) < x(2) < ... x(r) <= f be
> > integers and let M(x,y) be a t by r matrix.
....and I intended 1 <= x(1) < x(2) < ... x(r) <= f to replace your "any
subinterval [x,y] of [0,f-1]".
In both cases, I don't think mere _subsets_ will do for your intended
purposes - you need to order the elements of those subsets. What if
your P(m) was {2, 3, 5, 7} and you picked J to be 7, 2, 5? Also, I
don't think mathematicians would be happy indexing rows and columns by
anything except 1, 2, 3, ... ( I know _I_ would not.)
> I don't know about t by r. The matrix has #J rows, and J is a general
subset
> of P(m). It has f columns.
I don't think we know what a _general_ subset is.
With my setup, t, is the size of your J. From your "there are
(y-x+1)*J, are either occupied or unoccupied" I gathered you wanted #J
(= | J | usually) rows and y - x + 1 columns. I just replaced
y - x + 1 by r.
> The (i, j) entry of M(x,y)
> > is said to be "occupied" if p(i) divides x(j); otherwise, the entry is
> > "unoccupied".
>
>
> Yes.
>
> > *******
> >
> > I don't know where all of this is heading but, if you can, I'd
> > recommend that "occupied" entries have value 1 and "unoccupied"
entries
> > have value 0. Then your g(n, M_1) is just the sum of the entries in
the
> > n-th column.
> >
>
> Good thinking. I'm much looking forward to any further help.
> With Regards.
If ones and zeros are OK with you then the definition of M becomes
cleaner: M_(i,j) = 1 if p(i) divides x(j) and M_(i,j) = 0 otherwise. If
that works for you it can have many advantages for you. For example
M_(i,j)*d can "keep" or "reject" d depending on whether p(i) divides
x(j). Also there is quite a lot known (but not by me) about matrices
with only zeros an ones ( "binary" or "(0,1)" matrices).
I snipped the material about black/white entries because it was not
immediately apparent how the new matrix differed from M - you may also
want zeros and ones for these matrices too.
If all of this meets with your approval I suggest that you start over.
You want your notation to be absolutely precise - better too many
details than too few. If you are going to use notation like M_[x,y] for
a matrix, you don't want there to be any choice about what the matrix
is - no undefined entries.
--
Paul Sperry
Columbia, SC (USA)
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