> >> > You could write 'f(x)/g(x) diagonal arrow pointing from top left to
> >> > bottom right 1 as x --> oo'; which is read as 'f(x)/g(x) tends to one
> >> > from above as x tends to infinity.' The arrow in question is
> >> > $\searrow$.
> >> Does that arrow imply that the approach is *strictly* from above?
> > I don't know what that means.
> I meant that it never dips beneath the value it converges to.
Then say f/g decreases monotonically to 1.
-- The animated figures stand Adorning every public street And seem to breathe in stone, or Move their marble feet.
>> >> > You could write 'f(x)/g(x) diagonal arrow pointing from top left to
>> >> > bottom right 1 as x --> oo'; which is read as 'f(x)/g(x) tends to >> >> > one
>> >> > from above as x tends to infinity.' The arrow in question is
>> >> > $\searrow$.
>> >> Does that arrow imply that the approach is *strictly* from above?
>> > I don't know what that means.
>> I meant that it never dips beneath the value it converges to.
> Then say f/g decreases monotonically to 1.
So by the use of \searrow, it *can* dip beneath? Oh, no!
I want say more than just that it's monotonic; I want to say f/g is *strictly* decreasing to 1; and I really want to be able to say it using symbols instead of words -- at least as far as possible.
>> > You could write 'f(x)/g(x) diagonal arrow pointing from top left to
>> > bottom right 1 as x --> oo'; which is read as 'f(x)/g(x) tends to one
>> > from above as x tends to infinity.'
>> Does this entirely preclude that for some x,
>> f(x)/g(x) <= f(x+1)/g(x+1)?
> No, I don't think it does. You could say 'f(x)/g(x) decreases
> monotonically to 1 as x -> oo'.
I think I am going to have to say
f(x)/g(x) ~ 1
is a strictly and monotonically decreasing approach. Or
> [...] I want to say f/g is
> *strictly* decreasing to 1; and I really want to be able to say it using
> symbols instead of words -- at least as far as possible.
Why?
-- The animated figures stand Adorning every public street And seem to breathe in stone, or Move their marble feet.
> >> > You could write 'f(x)/g(x) diagonal arrow pointing from top left to
> >> > bottom right 1 as x --> oo'; which is read as 'f(x)/g(x) tends to one
> >> > from above as x tends to infinity.'
> >> Does this entirely preclude that for some x,
> >> f(x)/g(x) <= f(x+1)/g(x+1)?
> > No, I don't think it does. You could say 'f(x)/g(x) decreases
> > monotonically to 1 as x -> oo'.
> I think I am going to have to say
> f(x)/g(x) ~ 1
> is a strictly and monotonically decreasing approach. Or
> f(x)/g(x) ~ 1
> is a strictly decreasing approach.
> Are they OK?
I don't get 'approach'. Such-and-such approaches so-and-so.
-- The animated figures stand Adorning every public street And seem to breathe in stone, or Move their marble feet.
>> [...] I want to say f/g is
>> *strictly* decreasing to 1; and I really want to be able to say it using
>> symbols instead of words -- at least as far as possible.
> Why?
Again, it's because I can't find a way of referencing the expressions -- I mean the whole concept of what is happening in the convergence -- that doesn't look awkward. I can't just say 'Then it follows by that bit on the middle of page 15, that....'
>> >> > You could write 'f(x)/g(x) diagonal arrow pointing from top left to
>> >> > bottom right 1 as x --> oo'; which is read as 'f(x)/g(x) tends to >> >> > one
>> >> > from above as x tends to infinity.'
>> >> Does this entirely preclude that for some x,
>> >> f(x)/g(x) <= f(x+1)/g(x+1)?
>> > No, I don't think it does. You could say 'f(x)/g(x) decreases
>> > monotonically to 1 as x -> oo'.
>> I think I am going to have to say
>> f(x)/g(x) ~ 1
>> is a strictly and monotonically decreasing approach. Or
>> f(x)/g(x) ~ 1
>> is a strictly decreasing approach.
>> Are they OK?
> I don't get 'approach'. Such-and-such approaches so-and-so.
I thught it would be obvious that it's to one. Perhaps I should revert to the term 'convergence' instead? But then the preceding 'decreasing' won't look right.
Jeeez, I can't believe there is no standard way to say the simple things I am trying to say!
> >> [...] I want to say f/g is
> >> *strictly* decreasing to 1; and I really want to be able to say it using
> >> symbols instead of words -- at least as far as possible.
> > Why?
> Again, it's because I can't find a way of referencing the expressions -- I
> mean the whole concept of what is happening in the convergence -- that
> doesn't look awkward. I can't just say 'Then it follows by that bit on the
> middle of page 15, that....'
You ought to be able to label plain English just as you can formulae.
-- The animated figures stand Adorning every public street And seem to breathe in stone, or Move their marble feet.
On Tue, 26 Jun 2012, Jack wrote:
> "Jack" <no1email...@hotmail.com> wrote in message
> > Would you object to my using the term 'converges to'?
> > I am irked by the mixing of symbology with wording.
> > I write
> > "f(x) /g(x) ~ 1 (1)
> > and is strictly decreasing". Perhaps I'll try that downward-pointing > > arrow.
> Now I've written
> "f(x)/g(x) searrow 1
> is a strictly decreasing approach."
> Sound OK?
No, it stinks for being too terse. Use some words, verbal cheap skate.
Define "f(x) down to r" as "f(x) is strictly decreasing and lim(x->oo) = r.
Furthermore, don't use TeX in this newsgroup, it's not designed for it.
On Tue, 26 Jun 2012, Jack wrote:
> > You could write 'f(x)/g(x) diagonal arrow pointing from top left to
> > bottom right 1 as x --> oo'; which is read as 'f(x)/g(x) tends to one
> > from above as x tends to infinity.'
>> Are you sure I can just stick a label against this, in the middle of a >> load
>> of text?
> Yes.
How would I do this? If I just put '(18)' as text, it doesn't get recognised among the system's numbered equations.
And \label{18} doesn't seem to work outside an equation environment.
I can't say I much like the look of the label at the end of
"... so f(x)/g(x) is strictly and monotonically decreasing to one (18)." but if you say it's OK I'll go with it.
Jack writes:
> >> Are you sure I can just stick a label against this, in the middle
> >> of a load of text?
> > Yes.
> How would I do this? If I just put '(18)' as text, it doesn't get
> recognised among the system's numbered equations.
> And \label{18} doesn't seem to work outside an equation environment.
> I can't say I much like the look of the label at the end of
> "... so f(x)/g(x) is strictly and monotonically decreasing to one
> (18)." but if you say it's OK I'll go with it.
A usual way to label statements for further reference is to set them
aside as numbered theorems, lemmas, conjectures, propositions and
whatever. LaTeX has the means to set statements aside this way, number
them automatically just like it does for equations, with the ability
to refer to them with a symbolic label that is then replaced with the
actual number in the final document, just like with equations.
On to agonize about whether they should be called theorems or
propositions and how deep a thought needs to be before it can be so
labelled at all and when if ever it is appropriate to use words
instead of symbols, and is it really acceptable to set the theorem in
italics, and should the numbering be tied to section or chapter
numbers or not, and what if there is only one theorem in the whole
paper, should it still be numbered? (Do relax some.)
\nonstopmode\documentclass{article}
\newtheorem{thinko}{Thinko}
\begin{document}
(Assume enough material here that the wording of the following thinko
can be understood by the diligent follower.)
\begin{thinko}[the descent from above] Let $f$ and $g$ be mutually
discreet in an ordinary way. Then $f(x)/g(x)$ decreases to $1$
strictly monotonically and most probably without fail as $x$
increases without bound. \label{descent}
\end{thinko}
(assume much stream of consciousness here) by reference to Thinko
\ref{descent} here the labeling of thinkoes for later reference is
hereby demonstrated.
\end{document}
(You need not be so heavy-handed, but you seem to want something like
that. There are books that are essentially a sequence of such labeled
and numbered thoughts. Then there are other books that never use them,
and yet others that label the few central points in them.)
