In article <9RWVj.40168$yq6.23736@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
[
> > > Generally my argument takes the pattern: "For any given m, and for
all
> > > intervals of length y-x+1..."
]
> > I read this to say that m is fixed and all intervals have the same
> > length. If that is so then all you need to do is say, somewhere near
> > the start, "Let m and a be positive integers." If you do that,
> > subsequently you will only need to reference the starting points of
> > your intervals - the intervals will be [x, x + a] (or maybe
> > [x, x + a + 1] if you prefer) for your various x's. With m and a fixed
> > your M_1 could be instead M_x which would convey all the needed
> > information.
>
> But m is an integer that I have defined as a number of primes, such that
> P(m) contains the first m primes.
I understand; does it change? Could you, for example, do everything you
want to do with, say, m = 6?
> And a is their product.
No, according to you that product is a(m). Anyway, pick your letter.
Could you do everything you want to do with all intervals having
length, say, 4?
In article <OiXVj.38779$815.16814@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> > I still think it is a mistake to have the exceptional first column. If
> > you really, really need it you could do something like
> > "Let p(2) < ... < p(m + 1) be the first m primes and let p(1) = 1. Let
> > x = x(1) < ... < x(n) = y (or x(n) = x + a - see below) be
consecutive
> > integers. Let M_1 be the m + 1 by n matrix (m(i,j)) where m(i,j) = 1
if
> > p(i) divides x(j) and m(i,j) = 0 otherwise."
>
>
> I'm afraid don't understand. The columns in M_(1) in which I want
there to
> be m entries are the columns indexed by zero and multiples of a. I don't
see
> how your expressions above say this.
Do you mean _factors_ of a? Having the first column be all 1's is an
exception to the rule defining the other entries of M_1 unless you
cheat a little as I did above.
[...]
> The
> problem is, you say matrices but I am tryuing to impress upon you that I
am
> constructing arrays, and the property determining the entry is not
> necessarily one of divisibility.
What do you mean by "array"? How does a matrix differ from an array?
We've got a language problem I think. According to you, your M_1 very
much depends on divisibility. I never have understood your "array" with
the black and white entries.
> Incidentally, how would I express the sum of all the entries, of value
1,
> in the n-th column, for an array in which the number of such entries in
a
> column is not determined by divisibility? Following my definition "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]", I had originally imagined it might be (in TeX) "\sum N(n)"; but
we
> are back to the same problem: N pertains to the matrix M_1, but I am
> thinking of an array. How might I re-jig my definitions and/or mode of
> notation?
For a matrix, "column sums" would be understood. You "arrays"
apparently have columns but I am frankly mystified by what you have in
mind.
A nice numerical example illustrating M_1 and your "black/white array"
would be very welcome at this point.
--
Paul Sperry
Columbia, SC (USA)
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