In article <Tk7fk.9977$015.2830@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> Paul,
>
> > Let |N be the set of non-negative integers.
> > Let I be a finite interval of integers.
> >
> > DEFINITION 1. For a function t : |N -> N define
> > c_t(n) = (1/2)*t(n)*(t(n) - 1).
>
> Will I need to say anything about the integer n? Or even the standard
|N? Or
> can I take their meanings as understood?
It never hurts to nail things down. If |N is not explicitly given there
may be a question of whether or not 0 is included.
It certainly wouldn't be amiss to make the definition explicit:
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).
> > REMARK. For convenience of notation we will write c(J, n) instead of
> > c_(t_J)(n).
>
> I trust it's OK to use, instead, my old c(n,J)....
Sure.
> >> In particular I am wavering in the phrase ".for each of at least |J|
> values
> >> of n over [x,y]."
> >
> > Should be "in".
>
> I wonder if you know any general rules on this that I should adhere
to...? I
> can't work out why, in a different expression, you corrected me to
'_over_
> [x,y]'.
"Over" is more informal. "The sum of t(n) over [x,y]" , "the sum of
t(n) for all n in [x,y]" and "sum(t(n) : n is in [x,y])" all say the
same thing the first is most informal the last, most formal. Formality
is a good thing.
> I got this further issue. Brian set me up with a definition which goes
'We
> will find it convenient to define auxilliary functions in the following
way:
> given (g,n) in G there is a unique (s,r) in G', such that h(g,n)=(s,r),
and
> we set h_1(g,n)=s and h_2(g,n)=r'. Can I take this to be a definition
that
> holds in my reference to s and r subsequently, or do I have to go
through
> the process of saying what they are every time I make a reference to
them?
> If so, what about my later use of r to denote memebrs of R?
It would appear that you are free to recycle "r" and "s".
> More to the
> point, if I define j(x,y,t) to denote a specific thing, can I casually
say,
> later on, "take any integers i and j", without any reference to j(x,y,t)
> being meant?
In general, no. I have no context but "j" cannot be simultaneously the
name of a function and the name of an integer.
--
Paul Sperry
Columbia, SC (USA)
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