"Paul Sperry" <plsperry@[EMAIL PROTECTED]
> wrote in message
news:110720082303066299%plsperry@[EMAIL PROTECTED]
> 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.
I can't say that I have a clear view of what the answer to my question is.
I
am quite happy with the second of my definitions; it was written out for
me
by a mathematician. The first, I lay down as 'Fix t to be the function
such
that t: N --> N is arbitrary'; but doubtless this is inadequate...?
I am just concerned about the other definitions that are dependent upon t.
Cheers.
|
