In article <qBUWj.10642$iD4.6123@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> Paul,
[...]
> Indeed on my reader they don't line up properly; but can I take it there
are
> two X's at n=4 and n=8? In M_1, there would be one for each, as there is
> only one prime factor, 2. So in my definition <<For an integer n let
t(n,m)
> be the number of members of P(m) that divide n>>, would you be able to
> advise me on phrasing?
No, your t(n,m) is fine - my graph got garbled even worse than I
feared. Let me try it sideways.
30 | XXX
|
.
.
.
|
10 | XX
9 | X
8 | X
7 | X
6 | XX
5 | X
4 | X
3 | X
2 | X
1 |
_______________________
[...]
> > If I've finally managed to guess correctly, what you are really
> > interested in is t(n,m) and o(x,y,m) and the business with the
> > "matrices" and "arrays" is only a way to visualize the values of those
> > functions.
>
> Yes, that's just it.
>
> Also, it occurs to me that you are not using the words
> > "matrix" and "array" in a precise mathematical sense but rather a
loose
> > English language sense.
> >
> > Am I getting close?
>
> It might very well be the case. As I say, I feel my matrix M_1 will
become
> redundant. But, Paul, I hope we don't lose sight of what I am really
driving
> at, which is first and foremost an issue of notation. I want to be able
to
> use the concepts -- column sum etc. -- behind all those definitions that
I
> put forward - t(n,m), o(x,y,m) etc. not only for M_1 but also for the
kind
> of histogram/array we have discussed. I am very keen to know how to lay
out
> the definitions. I'm told I mustn't use the same definitions for two
> different arguments, which makes me think the whole thing is going to be
> inefficient in terms of use of letters of the alphabet. Perhaps it would
be
> best just to define a set, B, of entries, whose members are b, then to
use
> \sum_b(n) in place of t(n,m), 1/2*sum_b(n) *((sum_b(n))-1) for c(n,m)
etc..
> It's just it looks very ***bersome; and it will get considerably worse
when
> I start referring to the set T(x,y,m). Maybe, for my matrix as distinct
from
> my array/histogram, I simply use the same terms but with a subscript?
> I've got a hunch that you will object to my use of 'entries', above,
in
> relation to a histogram. Let's imagine what I term 'entries' as units in
a
> histogram bin. One thing that is crucial to my method is a system of
forming
> pairs of units, each pair comprising one unique unit from a bin in an
> interval [x,y] and one unique unit from a different bin in a different
> interval [x',y']. The reason I was reluctant immediately to construct
> histograms instead of what I have been referring to as 'arrays' was that
I
> guessed that amathematician would regard the bin value as just a number:
> it's presumably not to be regarded as a set of elements that can be
paired
> in the manner described. This is something that troubles me, as I don't
know
> the relevant conventions. My problem is, as ever, one of expressing my
final
> version in a professional manner.
I've paraphrased your definitions below. I saw nothing wrong with them
the way they were; my changes just make typing them easier and give you
a little more flexibility if you ever want it.
<<Let P(m) be the set of the first m prime numbers.
Let a(m) be their product.
My changes start here.
Let P be a set of primes and let I be a set of positive integers.
[Anytime you want to you can insert the statement "Let P = P(m) and
I = [x,y]" or words to that effect.]
For an integer n let t(n,P) be the number of members of P that divide n.
Define c(I,P) to be the sum of (1/2)t(n,P)(t(n,P)-1) for n in I.
[Or, sum((1/2)t(n,P)(t(n,P)-1); n in I)]
Define T(I,P) to be the set n in I for which t(n,P) > 1.
Define o(I,P) to be the number of n in I for which t(n,P) > 0.
Define N(I,P) to be the sum of t(n,P) for n in I.
[Or, sum(t(n,P); n in I)] >>
As I said, once you say P = P(m) and I = [x,y] these are exactly what
you had - just easier to type. If I weren't typing in ASCII, I would
probably subscript e.g. t_P(n). With proper mathematical notation using
sigmas and indices and such they would clean up very nicely and I don't
think anyone would object.
There is something called the Cartesian (or "cross") product of sets.
(Forgive me if you already know all about this.) If A and B are sets
then A x B is the set of all ordered pairs (a,b) where a is in A and b
is in B. Read A x B as "A cross B". "Ordered" means, for example that
(2,3) is different from (3,2).
So, if you wanted to pair up elements of I = [x,y] with elements of
I' = [x',y'], you would just write I x I'. [For example,
sum(t(i,P)*t(j,P); (i,j) in I x I') would sum over all possible
products with i in [x,y] and j in [x',y']. I don't know why you would
want to do that but you _could_.] If you only want some of the pairs
there is nothing wrong with using a subset of I x I'. For example you
could let B be the set of all elements (a,b) of I x I' such that a < b.
I didn't understand your use of "bin" so I just ignored it :-)
--
Paul Sperry
Columbia, SC (USA)
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