In article <czRdk.60924$7v1.18433@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> "Paul Sperry" <plsperry@[EMAIL PROTECTED]
> wrote in message
> news:110720081503413540%plsperry@[EMAIL PROTECTED]
> >
> >
> >
> > In article <8tMdk.2$Ek.1@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
> wrote:
> >
> > For completeness:
> > Call an _H-Gram_ a 5-tuple H = (L, R, u, v, w) where L and R are sets
> > and u, v and w are functions; u : L -> |N; v : R -> |N ;
> > w : L x R -> |N.
> >
> >
> >> I have got two definitions of t;
> >
> > That is forbidden.
> >
>
> That's what I had thought. But then Brian said, "Do you need to discuss
> *simultaneously* a function t_J : N --> N that is defined in terms of
> divisibility by members of a set J of primes *and* a completely
arbitrary
> function t : N --> N? If not, then of course you can call
> them both 't'; whyever not?!"
What Brian is saying and what I am saying is that you cannot have two
_concurrent_ definitions of t.
> Am really confused (not least by the irate tone of everyone's replies!).
I'm not irate - terse, perhaps, but not irate. If I become irate I'll
try to make sure you are in no doubt.
> The t I was employing is exactly the same t as I have been using in
every
> post on this NG in which I have used it, even in the construction of a
> matrix that you and I carried out.
That's the problem. You have a particular t in mind and plan to use
that t throughout whatever it is that you are doing. You cannot do that
and _also_ say that t is arbitrary. By the way, in mathematics,
"arbitrary' means "reader's choice".
> I can't say I understood anything of what you said about H-Grams.
Since I don't know where you are going or how you intend to get there
all I could do is try to create a setting or template which is general
enough to accommodate some of the various anticipated twists and turns.
Here is a previous example.
Let d be in |N; for a in |N let
I_a = {a, a + 1, a + 2, ..., a + d}.
Let P bet a set of primes. Let u and v be the identity functions for P
and I_a; that is, for all p in P, u(p) = p and, for all n in I_a, let
v(n) = n. Define w : P x I_a -> |N by w(p, n) = 1 if p is a divisor of
n and w(p, n) = 0 otherwise. Let H_1 = (P, I_a, u, v, w).
Here is my thinking. In this example, u and v really add nothing but
there may come a time when they _are_ of interest. I used "w" instead
of "t" because I worried that "t" came with some baggage due to
previous conversations.
I used I_a instead of [x, y] for two reasons: The "x" is somehow "too
variable" for my tastes and since you have said all your intervals have
the same length, the "y" is redundant.
I defined "w" the way I did because then you can create a "count the
divisors" function with sum(w(p, n) : p in P). If you are never going
to be using simultaneously two different sets of primes you can call
that function "t" if you want - otherwise, as Brian suggested, t_P
would be better.
My overall idea and the genesis of the name was to give you a rigorous
setting for all that "histogram" stuff you were trying to do. I'll
confess that I wasn't paying close attention. My third example was a
stab at showing how that might work.
My intention is to act only as an editor/referee. The whole H_Gram
business was just an attempt to give you a leg up - feel free to ignore
it. I _do_ think you should start over.
--
Paul Sperry
Columbia, SC (USA)
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