In article <Mksfk.20$dl5.11@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
> wrote:
> Paul,
>
> One chief reason why it looks right that, in my notation, I retain
the
> full x,y in eg. N(x,y,t_j) instead of using only x is that on occasions
I
> replace x and y with specific prescribed values.
I presume x and y come from [x, y]. Since you now say y - x is
variable, y is not determined by x, so it appears you _must_ include y.
> It is has become an issue
> as to how much emphasis I should place on consistency of style; I feel I
> oughtn't say N(x,t_J) one moment and N(1,a(J),t_J) the next, when 1 and
a(J)
> obviously replace the x and the merely implicit y.
That is correct; you ought not. My suggestion on omitting the y was
based on my apparently mistaken understanding that y - x was constant
throughout.
> This has become a broader issue because when I define my value z as
(1/2)y
> I then can, more justifiably, deploy only a z and not a full 1,y
reference.
Brian has discussed the z = (1/2)y business. I'll add that introducing
a new symbol "z" for the reader to remember when "y/2" (or even
(y + 1)/2) works just as well is probably counterproductive.
> But then some of my references, as a matter of consistency (in view of
how I
> have originally defined the set or function, eg. as N(x,y,t), I make the
1,z
> reference while in other, simultaneous references to sets defined after
I
> had defined z, I use only the z, but the reference is to exactly the
same
> interval. It looks awkward.
I'm not at all sure what you are trying to say above but, now, with
lengths being variable, _any_ reference to an interval _must_ reference
_both_ end points.
I'm guessing N(x,y,t) is some sort of function - it is probably a
surrogate for N([x,y],t). In any case, if it is a function, you are
obliged to give its domain and codomain; i.e. N : ? -> ??.
In regard to a question in another thread, a common notation for the
even non-negative integers is 2|N; the odd ones are |N - 2|N.
I've included below where we are so far.
*******
Let |N be the set of non-negative integers.
The notation [x, y] will be taken to mean x <= y in |N and
[x, y] = {n in |N : x <= n <= y}
DEFINITION 1.1 For a function t : |N -> |N define a function
c_t : |N -> |N by c_t(n) = (1/2)*t(n)*(t(n) - 1).
DEFINITION 2. For a set of primes J define t_J : |N -> |N by
t_J(n) = |{p in J : p divides n}|.
REMARK. For convenience of notation we will write c(n, J) instead of
c_(t_J)(n).
DEFINITION 3. For t : |N - |N define s_I(t) = sum(t(n) : n is in I);
define s'_I(t) = sum(t(n) : n is in I and t(n) > 1)
********
I've corrected the typos, replaced the reference to interval I by [x,
y] and changed c(J, n) to c(n, J).
Regarding the latter change, c_(t_j)(n) is read "c sub t sub J of n"
with "J" preceding "n"; that's why I originally wrote c(J, n).
I would much rather proceed from here rather than skip around with all
sorts of undefined things being referenced.
Feel free to make up your own definitions etc.; in fact, I would prefer
that you did.
--
Paul Sperry
Columbia, SC (USA)
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