In article <pafWj.134$Pp2.31@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> > Do you mean _factors_ of a?
>
> No, I meant *multiples* of a(m). Or a(J) when I am using the subset of
m, J.
> The product of all primes in J is a column in which there are J entries.
> Same with the column indexed by twice the product, etc. The fact that
these
> columns are found exclusively at intervals of a(m)+1 is the reason I
want
> one such at the column indexed by zero.
What we have here is a failure to communicate.
You started with
>>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).
>>
>>I am also constructing an array M_[x,y] comprising black cells and
>>white cells. The number of black cells in a column is given by
>>t(n,M_[x,y]). My question is, will it be acceptable in a
>>mathematical paper
>>to devise a form of notation referencing an interval [x,y] in
>>such a way
>>that the values of t(n,M_[x,y]) over [x,y] are precisely those
>>given by
>>g(n,M_1) for an interval [x,y] in M_1? (i.e. if u divides v,
>>and u is a
>>member of J, then there is a single black cell on the u-th row
>>and in the
>>v-th column. Otherwise, there is none such, save in the
>>column indexed by zero.)
I eventually proposed
>Let P(m) be the m-th prime and let p(1) < p(2) < ... < p(t) <= P(m) be
>primes.
>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. The (i, j) entry of M(x,y)
>is said to be "occupied" if p(i) divides x(j); otherwise, the entry is
>"unoccupied".
And then
> 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.
Which you seemed to like.
I even gave an example
>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) [i.e M_1] to be the t x r matrix whose (i,j) entry is 1
if p(i)
>divides x(j) and is 0 otherwise.
>
>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.
Which you didn't object to.
_Now_ you say
>>No, I meant *multiples* of a(m) [the product of the first m primes].
>>Or a(J) when I am using the subset of [P(m)], J.
>>The product of all primes in J is a column in which there are
>> entries.
>>Same with the column indexed by twice the product, etc. The fact
>>that these
>>columns are found exclusively at intervals of [a(m)] is the
>>reason I want
>>one such at the column indexed by zero.
Well, I'm lost. How does this conform to your previous definition of
M_1? For that matter what do you mean by "The product of all primes in
J is a column in which there are J entries"?
In an effort to clear things up I asked for an example. I _still_ would
like to see an example with actual numbers and matrices/arrays.
You still haven't said what you mean by "array" - it is not a commonly
used mathematical term.
I have a vague hunch that what you want is something like what
_programmers_ call an array of arrays.
Here's an example - the numbers have nothing to do with your problem.
It is sort of suggested by your mention of "histogram".
A := [U, V, W, X]
U := [1, 2, 3]
V := [4, 5, 6, 7]
W := [8, 9]
X := [ 10, 11, 12]
A is an array (of arrays); U, V, W and X are arrays.
A[0] = U; A[2] = W; A[2][1] = A[2, 1] = W[1] = 9.
Schematically (row dominant) A is
1 2 3
4 5 6 7
8 9
10 11 12
--
Paul Sperry
Columbia, SC (USA)
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