Paul,
Tahnks again. Comments embedded.
>>
>> 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".
OK, understood.
>> > 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?
For my purposes this would not make any difference. But perhaps I would
nevertheless be well advised to order the sets (and perhaps you could give
me some prompts on that subject...? Would be well received!)
Also, I
> don't think mathematicians would be happy indexing rows and columns by
> anything except 1, 2, 3, ... ( I know _I_ would not.)
>
Hmmm... The one I had been communicating with seemed OK with it. But how I
could get round it, I don't know. The 1,2,3... will in turn just be
indexing
2, 3,5....etc.
>> 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.
>
I had thought it was like a proper subset except that the members could be
exactly the same as the set itself. So if J is a general subset of P(m),
it
may (or may not) be the same as P(m). That, at least, is how I wanted to
define J.
> 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)
Quite right - I meant #J.
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.
Sounds absolutely perfect.
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
Certainly does.
I suggest that you start over.
What do I need to say?
> 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.
But (naive that I am!) I didn't think I had any...
Best regards.
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