In article <wOyek.209518$312.194887@[EMAIL PROTECTED]
>, Jack
<jj@[EMAIL PROTECTED]
> wrote:
> Paul,
>
> >> Well, I've got, for example, N(x,y), which is the sum of t(n) -- or,
> >> later,
> >> t_J(n) --
> >
> > You mean t = t_J is a particular case. You should probably "tag" with
> > "t": like N(x, y, t).
>
> So maybe it would be N(x, y, t) for the arbitrary case and N(x, y,
t_{J}(n))
> for the non-arbitrary?
Well, close anyway. Here's what I would do:
For interval {a, a + 1, ..., a + d} and function f : |N -> |N define
N(a,f) = sum(f(n) : n = a, ..., a + d). That would be the _general_
case. For the particular case using t_J I would, without comment, write
N(a, t_J).
> What I thought would help most is if I could say that t(n) is a value a,
and
> if a is arbitrary it is denoted by A, so we have N(x,y,A) and c(n,A),
and if
> a is determined by divisibility, it's N(x,y,J) and c(n,J). How does that
> sound?
Frankly, it sounds awful. For a _fixed_ n you could write t(n) = a. For
example if t(30) = 3 then a = 3 but, unless I've badly misunderstood,
N(x,y,3) makes no sense. Renaming A = 3 doesn't help.
I suppose you could introduce notation N(x,y,J) = N(x,y,t_J) but I
wouldn't recommend it - you don't want to do very much of that sort of
thing.
I wish you would eliminate the word "arbitrary" from your vocabulary -
you have been terribly abusing it.
Two minute lecture on functions:
\begin{lecture}
Let A = {1, 2, 3, 4}; let B = {a, b, c, d}. Here is a function from A
to B: {(1, a), (2, c), (3, a), (4, d)}.
We know it is a function from A to B because
(i) it is a subset of A x B;
(ii) every element of A is paired with an element of B;
(iii) no (single) element of A is paired with more than one element of
B.
Rather than to need to keep writing out the set, we'll give it a name:
Bob.
So we have Bob : A -> B.
Now Bob(1) = a is signified by the fact that (1, a) belongs to Bob.
Similarly Bob(2) = c, Bob(3) = a and Bob(4) = d.
\end{lecture}
When you are working with functions, please think of Bob.
You could write N(x,y,Bob) but not N(x,y,Bob(2)) or even N(x,y,Bob(n)).
> If OK, how would I write out my initial definition/function?
>
>
> >
> >> in a given interval [x,y];
> >
> > _over_ [x, y]. (If you still have all intervals of the same length,
you
> > would be doing everybody a favor if you dropped the redundant "y".)
>
> Good thought.
If all intervals are to have length d and you want the interval that
starts at x then for [x, y] you could use I_x as I have done previously
or [x] or <x> (but not {x}).
[...]
--
Paul Sperry
Columbia, SC (USA)
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