Paul,
>
> How about "We shall construct binary matrices whose first column
> consists only of 1's and whose other entries are determined by certain
> divisibility properties"? It is OK to be a little vague at this stage.
>
> I still think it is a mistake to have the exceptional first column.
By an exceptional first column, do you mean that it will be taken as
given,
without any statement from myself, that the first column is all ones? 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.
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."
>
> If you want to get fancy, there is a function div(d,n) which is 1 if d
> divides n and 0 otherwise. Mathematica implements it with the function
> "divides"; I think it is fairly obscure but you do have
> m(i,j) = div(p(i),x(j)) which is kind of pretty.
>
>> > Your paragraph quoted above is a little too vague for me to be of any
>> > help. Are you going to change m's or intervals or both or none of the
>> > above. How many changes at one time are you going to want to make?
>> >
>>
>> 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. Also, if you prefer, m(i,j) = div(p(j),(x - 1) + j).
>
I hoped what I had said in this paragraph <<I have two distinct
objectives.
The first is to show what happens when certain values are held fixed, and
some other set of values change. For this I am constructing an array. In
establi****ng those principles, I do not want the complication of there
being
any prescribed
determinant of the number of black cells in a column.
The second objective is to apply the findings of the above to sieves as
defined in my matrix M_1, in particular with regard to such values as
t(n,m). The value of t(n,m) is not necessarily the number of distinct
factors of an integer; it's the number of its distinct factors that are in
J>> would enable you to show me just what it is I need to construct. My
ultimate question is one of notation. I don't know how it has to be done.
What did you make of this <<I was thinking that maybe I could make my
array one in which the numbers of black cells in a column over the first
(a(m)+y-x ) columns are precisely equal to the values found for the matrix
we are defining. That way I could make my reference something like
t(n,m,[1,a(m)+y-x]) to indicate that n in [x,y] is a column in that
interval>>? Would that be acceptable?
With thanks.
|
