- I have a question about notation. I'm trying to formulate an approach
to the sieve of Eratosthenes while working into the model principles
that are not exclusively applicable to the the sieve of Eratosthenes.
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 (and if so, in
what form?) 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
I want to be able to say, 'in this case, the black cells exhibit the
divisibility distribution found for members of J over [x,y], .... and
in this case, they do not'. It strikes me that if it is indeed
possible, I won't need a completely different set of definitions for
my array M_[x,y] as I have for my matrix M_1.
I was thinking that, for a value of t, the notation would be
t(n,M_[x,y],J). But someone (unlike myself, a trained mathematician)
was telling me that this mixture of arguments is something that is
just not done....
With thanks in advance.