In article <znCXj.16559$iD4.12587@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
[...]
> <<Tell me again what you _want_ to do.>>
>
>
>
> Take a finite set B whose members are all the occupied matrix components
in
> a binary matrix. I wish to establish, without imposing any determining
> criterion as to the distribution of such members, basic principles about
the
> value o(x,y) in this matrix, when specific criteria are held fixed and
> various other stipulations are made*. Then, I want to apply these
principles
> to different [x,y] intervals in the matrix M_1.
> [*see private email.]
All your apostrophes appeared as commas - that didn't help.
See <http://en.wikipedia.org/wiki/Matrix_%28mathematics%29>;
scroll
down a little to the mathematical definition (you'll also see the cross
product there).
See also <http://mathworld.wolfram.com/Array.html>.
You'll see why I am having so much trouble understanding what you are
getting at - the two references are speaking my language; unfortunately
you aren't.
[...]
> <<No, it has got to be a cross. In most Math typesetting packages it is
> neither "x" nor "X" but a "cross" - a "+" rotated 45 degrees. You _can_
> say [x,y] x [x',y'] - in fact I do so below.>>
>
> This is most interesting. I had never before encountered this. Indeed
the
> professional mathematician I had mentioned to you previously had told me
it
> was a multiplication symbol, and I guessed that this was attributable to
> there being (y-x+1)^2 pair combinations. (But strangely he had used an
'x'
> when normally he would use a * for multiplication.) You couldn't give me
the
> TeX coding for it, could you? (If it's a + turned through 45 degrees I
fear
> it'll look almost indistinguishable from \times....)
It _is_ \times; it is not a special symbol.
> <<As you've described it, R(n,n') is merely a number - not even a set
> much less a set of sets.>>
>
> As I have mentioned above, this is precisely what has been worrying me
all
> along. If you could help me overcome this problem, such that, in my
initial
> definitions, I can form purely algebraic sets whose members can be
> interpreted as occupied matrix components, that would be splendid.
>
> <<> and sum_R(n,n') is as low a value as it possibly can be given [x,y]
and
> > [x',y'].
>
> You can't sum a single number - at least there is no point in doing
so.>>
>
> Sorry, what I meant was, sum_#R(n,n') in [x,y]x[x',y']. I keep omitting
that
> darn '#' sign!!
Not only did you forget it, you put "#" in the wrong place (I think).
How about if we use the more or less universal |A| for the number of
elements in A a.k.a the cardinality of A.
For integers n and n', R(n,n') = t(n,P(m))-t(n',P(m)).
[New K] Let K = {(n,n') : (n.n') is in [x,y] x [x',y'] and
t(n,P(m)) > t(n',P(m)) > 0}.
[...]
> Does this mean #K=(y-x+1)^2? I only need a maximum of y-x+1 pairs, of
> course.
I don't know. It is not immediately clear to me either way but then I'm
a long way from being a number theorist.
> BTW I want another stipulation, which is that t(n',P(m))>0.
Fixed - see above.
> <<You could then Let R = {R(n,n') : (n,n') in K}.Then you could add up
or
> minimize the elements of R. Equivalently you could form
> sum(R(n,n') : (n,n') in K) or min(R(n,n') : (n,n') in K).>>
>
> This looks to be the kind of thing I am after. I want the minimum
possible
> value of sum_#R(n,n'),
This isn't at all clear; sum(R(n,n') : (n,n') in K) is just a number.
> and a set of sets R(n,n') so that I have got a set
> comprising every (n,n') for which there is a value #R(n,n').
Saying #R(n,n') is like saying #2.
> Then, I want a set of sets S(n',n'), S(n,n') being a set comprising
> t(n')-t(n) members (each member being an occupied matrix component),
such
> that (n,n') is in [x,y]x[x',y'] but neither n nor n' are members of a
pair
> R(n,n'), and t(n')-t(n) >1.
Wow - I'm mystified. To answer one of your questions: it is not a good
idea to use both t(n) and t(n,P) even though context may make things
clear. Many of us consider "t" to be the name of the function so you
have two different functions with the same name.
> BTW does 'min' stand for 'minimum' or 'minus', or something else?.
Minimum.
> <<How about an example; say m = 3 with [2,7] and [5,10]?>>
>
> I don't quite understand this; I think it will end up with me naively
> expressing something that, by some misinterpretation, serves only to
muddy
> the waters; but I hope you can understand what I'm saying at in my
previous
> couple of paragraphs.
The waters, in regard to matrices and arrays, could hardly be muddier.
I haven't a clue as to what you are trying to do. Since, as near as I
can tell, what you have done so far involves a set of primes and two
intervals, you should be able to do the calculations in this very
specific case so that I (and maybe you) would have a better idea of
what is going on.
I'll go farther: if you can't calculate, for a particular small
example, everything you've talked about so far then you've got serious
problems.
--
Paul Sperry
Columbia, SC (USA)
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