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Polishing up II
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More options Jun 26 2012, 1:21 pm
Newsgroups: alt.algebra.help
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Tue, 26 Jun 2012 18:21:26 +0100
Local: Tues, Jun 26 2012 1:21 pm
Subject: Re: Polishing up II

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.'  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.

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More options Jun 26 2012, 1:35 pm
Newsgroups: alt.algebra.help
From: "Jack" <no1email...@hotmail.com>
Date: Tue, 26 Jun 2012 18:35:19 +0100
Local: Tues, Jun 26 2012 1:35 pm
Subject: Re: Polishing up II

"Frederick Williams" <freddywilli...@btinternet.com> wrote in message

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.

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More options Jun 26 2012, 1:39 pm
Newsgroups: alt.algebra.help
From: "Jack" <no1email...@hotmail.com>
Date: Tue, 26 Jun 2012 18:39:19 +0100
Local: Tues, Jun 26 2012 1:39 pm
Subject: Re: Polishing up II

"Frederick Williams" <freddywilli...@btinternet.com> wrote in message

news:4FE9EF5C.DE25D420@btinternet.com...

> 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.'

>> 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?

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More options Jun 26 2012, 1:41 pm
Newsgroups: alt.algebra.help
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Tue, 26 Jun 2012 18:41:28 +0100
Local: Tues, Jun 26 2012 1:41 pm
Subject: Re: Polishing up II

Jack wrote:

> [...] 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?

--
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More options Jun 26 2012, 1:46 pm
Newsgroups: alt.algebra.help
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Tue, 26 Jun 2012 18:46:01 +0100
Local: Tues, Jun 26 2012 1:46 pm
Subject: Re: Polishing up II

I don't get 'approach'.  Such-and-such approaches so-and-so.
--
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More options Jun 26 2012, 2:21 pm
Newsgroups: alt.algebra.help
From: "Jack" <no1email...@hotmail.com>
Date: Tue, 26 Jun 2012 19:21:45 +0100
Local: Tues, Jun 26 2012 2:21 pm
Subject: Re: Polishing up II

"Frederick Williams" <freddywilli...@btinternet.com> wrote in message

news:4FE9F448.219D8B57@btinternet.com...

> Jack wrote:

>> [...] 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....'

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More options Jun 26 2012, 2:24 pm
Newsgroups: alt.algebra.help
From: "Jack" <no1email...@hotmail.com>
Date: Tue, 26 Jun 2012 19:24:31 +0100
Local: Tues, Jun 26 2012 2:24 pm
Subject: Re: Polishing up II

"Frederick Williams" <freddywilli...@btinternet.com> wrote in message

news:4FE9F559.7E42531F@btinternet.com...

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!

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More options Jun 26 2012, 2:25 pm
Newsgroups: alt.algebra.help
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Tue, 26 Jun 2012 19:25:58 +0100
Local: Tues, Jun 26 2012 2:25 pm
Subject: Re: Polishing up II

You ought to be able to label plain English just as you can formulae.

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More options Jun 26 2012, 2:27 pm
Newsgroups: alt.algebra.help
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Tue, 26 Jun 2012 19:27:27 +0100
Local: Tues, Jun 26 2012 2:27 pm
Subject: Re: Polishing up II

Jack wrote:
> Jeeez, I can't believe there is no standard way to say the simple things I
> am trying to say!

As I suggested: 'f(x)/g(x) decreases monotonically to 1 as x -> oo'.

--
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More options Jun 26 2012, 2:46 pm
Newsgroups: alt.algebra.help
From: "Jack" <no1email...@hotmail.com>
Date: Tue, 26 Jun 2012 19:46:10 +0100
Local: Tues, Jun 26 2012 2:46 pm
Subject: Re: Polishing up II

"Frederick Williams" <freddywilli...@btinternet.com> wrote in message

news:4FE9FF0F.6E6B3250@btinternet.com...

> Jack wrote:

>> Jeeez, I can't believe there is no standard way to say the simple things
>> I
>> am trying to say!

> As I suggested: 'f(x)/g(x) decreases monotonically to 1 as x -> oo'.

Yeh but it's not just monotonic, it's *strictly decreasing*.

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More options Jun 26 2012, 3:11 pm
Newsgroups: alt.algebra.help
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Tue, 26 Jun 2012 20:11:48 +0100
Local: Tues, Jun 26 2012 3:11 pm
Subject: Re: Polishing up II

Jack wrote:

> "Frederick Williams" <freddywilli...@btinternet.com> wrote in message
> news:4FE9FF0F.6E6B3250@btinternet.com...
> > Jack wrote:

> >> Jeeez, I can't believe there is no standard way to say the simple things
> >> I
> >> am trying to say!

> > As I suggested: 'f(x)/g(x) decreases monotonically to 1 as x -> oo'.

> Yeh but it's not just monotonic, it's *strictly decreasing*.

f(x)/g(x) decreases strictly to 1 as x -> oo.

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More options Jun 26 2012, 3:43 pm
Newsgroups: alt.algebra.help
From: "Jack" <no1email...@hotmail.com>
Date: Tue, 26 Jun 2012 20:43:10 +0100
Local: Tues, Jun 26 2012 3:43 pm
Subject: Re: Polishing up II

"Frederick Williams" <freddywilli...@btinternet.com> wrote in message

news:4FEA0974.C34B0429@btinternet.com...

Or maybe f(x)/g(x) is strictly decreasing to 1?
Are you sure I can just stick a label against this, in the middle of a load
of text?

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More options Jun 26 2012, 3:53 pm
Newsgroups: alt.algebra.help
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Tue, 26 Jun 2012 20:53:12 +0100
Local: Tues, Jun 26 2012 3:53 pm
Subject: Re: Polishing up II

If it's clear what x is doing.

> Are you sure I can just stick a label against this, in the middle of a load
> of text?

Yes.

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More options Jun 26 2012, 9:51 pm
Newsgroups: alt.algebra.help
From: William Elliot <ma...@panix.com>
Date: Tue, 26 Jun 2012 18:51:35 -0700
Local: Tues, Jun 26 2012 9:51 pm
Subject: Re: Polishing up II

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.

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More options Jun 26 2012, 9:54 pm
Newsgroups: alt.algebra.help
From: William Elliot <ma...@panix.com>
Date: Tue, 26 Jun 2012 18:54:14 -0700
Local: Tues, Jun 26 2012 9:54 pm
Subject: Re: Polishing up II

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.'

> Does this entirely preclude that for some x,

> f(x)/g(x) <= f(x+1)/g(x+1)?

"Tends to one from above" needs defining.

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More options Jun 28 2012, 11:44 am
Newsgroups: alt.algebra.help
From: "Jack" <no1email...@hotmail.com>
Date: Thu, 28 Jun 2012 16:44:51 +0100
Local: Thurs, Jun 28 2012 11:44 am
Subject: Re: Polishing up II

>> Are you sure I can just stick a label against this, in the middle of a
>> 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.

Thanks.

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More options Jun 28 2012, 12:28 pm
Newsgroups: alt.algebra.help
From: Jussi Piitulainen <jpiit...@ling.helsinki.fi>
Date: 28 Jun 2012 19:28:58 +0300
Local: Thurs, Jun 28 2012 12:28 pm
Subject: Re: Polishing up II

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.)