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Polishing up II
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Jack
Jun 25, 2012, 6:33:14 AM6/25/12
to
Once again I have got this issue of wanting to say two things in one (and
ideally I want to be able to reference the assertion, all in one), namely
that f(x)/g(x) is strictly decreasing and approaches 1.
Currently I am writing
'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
or
'lim_(x-->oo) f(x)/g(x) = 1
with f(x)/g(x) >1'.
I don't know if this is the best phrasing; I have read that the notation I
am using here is known as asymptotic notation but my functions are
monotonic -- the curves they produce are smooth. Any views?
Also, I have changed the sentence
<<For x<y in mathbb{N}, [x,y] = {n n mathbb{N} : x <= n <= y} >>
given to me by Paul, to <<all intervals are be taken to be intervals of
integers>>. I wonder whether this is OK, given that Paul's version might be
seen to be doing something more, namely indicating that the square brackets
denote endpoints. Is the open/-closed-bounded notation using round and
square brackets respectively, sufficiently standard that I don't need to
worry?
With thanks.
ideally I want to be able to reference the assertion, all in one), namely
that f(x)/g(x) is strictly decreasing and approaches 1.
Currently I am writing
'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
or
'lim_(x-->oo) f(x)/g(x) = 1
with f(x)/g(x) >1'.
I don't know if this is the best phrasing; I have read that the notation I
am using here is known as asymptotic notation but my functions are
monotonic -- the curves they produce are smooth. Any views?
Also, I have changed the sentence
<<For x<y in mathbb{N}, [x,y] = {n n mathbb{N} : x <= n <= y} >>
given to me by Paul, to <<all intervals are be taken to be intervals of
integers>>. I wonder whether this is OK, given that Paul's version might be
seen to be doing something more, namely indicating that the square brackets
denote endpoints. Is the open/-closed-bounded notation using round and
square brackets respectively, sufficiently standard that I don't need to
worry?
With thanks.
William Elliot
Jun 25, 2012, 7:03:53 AM6/25/12
to
On Mon, 25 Jun 2012, Jack wrote:
> Once again I have got this issue of wanting to say two things in one (and
> ideally I want to be able to reference the assertion, all in one), namely
> that f(x)/g(x) is strictly decreasing and approaches 1.
f(x) strictly decreases to 1 as x -> oo.
> Once again I have got this issue of wanting to say two things in one (and
> ideally I want to be able to reference the assertion, all in one), namely
> that f(x)/g(x) is strictly decreasing and approaches 1.
> Currently I am writing
> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
spell it all out. f is a monotonically decreasing function
with lim(x->oo) f(x) = r.
> or
> 'lim_(x-->oo) f(x)/g(x) = 1
> with f(x)/g(x) >1'.
> I don't know if this is the best phrasing; I have read that the notation I
> am using here is known as asymptotic notation but my functions are
> monotonic -- the curves they produce are smooth. Any views?
> Also, I have changed the sentence <<For x<y in mathbb{N}, [x,y] = {n n
> mathbb{N} : x <= n <= y} >> given to me by Paul, to <<all intervals are
> be taken to be intervals of integers>>. I wonder whether this is OK,
> given that Paul's version might be seen to be doing something more,
> namely indicating that the square brackets denote endpoints. Is the
> open/-closed-bounded notation using round and square brackets
> respectively, sufficiently standard that I don't need to worry?
{n n > mathbb{N} : x <= n <= y}
In Europe, they use ]a,b[ for (a,b).
When restriced to integers only
one is needed for (a,b) = [a-1, b+1].
Jack
Jun 25, 2012, 7:15:44 AM6/25/12
to
"William Elliot" <ma...@panix.com> wrote in message
news:Pine.NEB.4.64.12...@panix3.panix.com...
> On Mon, 25 Jun 2012, Jack wrote:
>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>
> f(x) strictly decreases to 1 as x -> oo.
>
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>
> I suppose. So when defining f(x) ~ r, don't be glib,
> spell it all out. f is a monotonically decreasing function
> with lim(x->oo) f(x) = r.
Or f(x)/g(x) is a monotonically decreasing function with <...>?
>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>
> f(x) strictly decreases to 1 as x -> oo.
>
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>
> I suppose. So when defining f(x) ~ r, don't be glib,
> spell it all out. f is a monotonically decreasing function
> with lim(x->oo) f(x) = r.
>
>> or
>> 'lim_(x-->oo) f(x)/g(x) = 1
>> with f(x)/g(x) >1'.
>
> No, that's saying something different.
write above is not in any way unacceptable? (I can see it almost looks like
a contradiction when both the "= 1" and the " >1" are used.)
>> I don't know if this is the best phrasing; I have read that the notation
>> I
>> am using here is known as asymptotic notation but my functions are
>> monotonic -- the curves they produce are smooth. Any views?
>
> Yes, asymptotic and monotone are different.
>
of '~', which I'm told is asymptotic notation, is not right? (Or not ideal?)
>> Also, I have changed the sentence <<For x<y in mathbb{N}, [x,y] = {n n
>> mathbb{N} : x <= n <= y} >> given to me by Paul, to <<all intervals are
>> be taken to be intervals of integers>>. I wonder whether this is OK,
>> given that Paul's version might be seen to be doing something more,
>> namely indicating that the square brackets denote endpoints. Is the
>> open/-closed-bounded notation using round and square brackets
>> respectively, sufficiently standard that I don't need to worry?
>
> {n n > mathbb{N} : x <= n <= y}
>
> is a typo and skip the TeX.
>
> In Europe, they use ]a,b[ for (a,b).
>
> When restriced to integers only
> one is needed for (a,b) = [a-1, b+1].
nuissance as it'll mean some of my equations won't fit on the line.
It sounds as though you're also saying that I *do* need to spell out in my
intro that square brackets denote closed-boundedness at an endpoint, and
round brackets open-boundedness (assuming I do indeed use round brackets).
With thanks.
Jack
Jun 25, 2012, 7:19:39 AM6/25/12
to
> On Mon, 25 Jun 2012, Jack wrote:
>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>
> f(x) strictly decreases to 1 as x -> oo.
>
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>
> I suppose. So when defining f(x) ~ r, don't be glib,
> spell it all out. f is a monotonically decreasing function
> with lim(x->oo) f(x) = r.
BTW isn't it -- at least in one sense -- stronger to say 'strictly
>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>
> f(x) strictly decreases to 1 as x -> oo.
>
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>
> I suppose. So when defining f(x) ~ r, don't be glib,
> spell it all out. f is a monotonically decreasing function
> with lim(x->oo) f(x) = r.
decreasing' than 'monotonically decreasing'? Ideally I would like to say
both.
Frederick Williams
Jun 25, 2012, 11:48:00 AM6/25/12
to
Jack wrote:
>
> Once again I have got this issue of wanting to say two things in one (and
> ideally I want to be able to reference the assertion, all in one), namely
> that f(x)/g(x) is strictly decreasing and approaches 1.
> Currently I am writing
> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
> or
> 'lim_(x-->oo) f(x)/g(x) = 1
> with f(x)/g(x) >1'.
You could write 'f(x)/g(x) diagonal arrow pointing from top left to
>
> Once again I have got this issue of wanting to say two things in one (and
> ideally I want to be able to reference the assertion, all in one), namely
> that f(x)/g(x) is strictly decreasing and approaches 1.
> Currently I am writing
> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
> or
> 'lim_(x-->oo) f(x)/g(x) = 1
> with f(x)/g(x) >1'.
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
$\searrow$.
> I don't know if this is the best phrasing; I have read that the notation I
> am using here is known as asymptotic notation but my functions are
> monotonic -- the curves they produce are smooth. Any views?
functions that are monotonic or smooth. That isn't so.
> Also, I have changed the sentence
> <<For x<y in mathbb{N}, [x,y] = {n n mathbb{N} : x <= n <= y} >>
> given to me by Paul, to <<all intervals are be taken to be intervals of
> integers>>. I wonder whether this is OK, given that Paul's version might be
> seen to be doing something more, namely indicating that the square brackets
> denote endpoints. Is the open/-closed-bounded notation using round and
> square brackets respectively, sufficiently standard that I don't need to
> worry?
sentence is 'all intervals are intervals of natural numbers.'
--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.
Frederick Williams
Jun 25, 2012, 11:52:29 AM6/25/12
to
Jack wrote:
>
> Or f(x)/g(x) is a monotonically decreasing function with <...>?
'f(x)/g(x) decreases monotonically to L as x increases' is ok.
>
> Or f(x)/g(x) is a monotonically decreasing function with <...>?
> [...] (I can see it almost looks like
> a contradiction when both the "= 1" and the " >1" are used.)
'limit of blah = L' doesn't require blah to ever equal L.
>
> You're sayng I'm best advised to avoid using open-bounded notation?
Frederick Williams
Jun 25, 2012, 11:54:09 AM6/25/12
to
Jack wrote:
>
> BTW isn't it -- at least in one sense -- stronger to say 'strictly
> decreasing' than 'monotonically decreasing'? Ideally I would like to say
> both.
If one _is_ stronger than the other, just state the stronger. The
>
> BTW isn't it -- at least in one sense -- stronger to say 'strictly
> decreasing' than 'monotonically decreasing'? Ideally I would like to say
> both.
definitions should reveal which (if either) is the stronger.
William Elliot
Jun 25, 2012, 10:30:59 PM6/25/12
to
On Mon, 25 Jun 2012, Jack wrote:
>
> "William Elliot" <ma...@panix.com> wrote in message
> news:Pine.NEB.4.64.12...@panix3.panix.com...
> > On Mon, 25 Jun 2012, Jack wrote:
> >
> >> Once again I have got this issue of wanting to say two things in one (and
> >> ideally I want to be able to reference the assertion, all in one), namely
> >> that f(x)/g(x) is strictly decreasing and approaches 1.
> >
> > f(x) strictly decreases to 1 as x -> oo.
> >
> >> Currently I am writing
> >> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
> >
> > I suppose. So when defining f(x) ~ r, don't be glib,
> > spell it all out. f is a monotonically decreasing function
> > with lim(x->oo) f(x) = r.
>
Since you've pointed out that ~ is used for "asymototic to",
>
> "William Elliot" <ma...@panix.com> wrote in message
> news:Pine.NEB.4.64.12...@panix3.panix.com...
> > On Mon, 25 Jun 2012, Jack wrote:
> >
> >> Once again I have got this issue of wanting to say two things in one (and
> >> ideally I want to be able to reference the assertion, all in one), namely
> >> that f(x)/g(x) is strictly decreasing and approaches 1.
> >
> > f(x) strictly decreases to 1 as x -> oo.
> >
> >> Currently I am writing
> >> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
> >
> > I suppose. So when defining f(x) ~ r, don't be glib,
> > spell it all out. f is a monotonically decreasing function
> > with lim(x->oo) f(x) = r.
>
don't use it for the above definition. Perhaps you could
write "f(x) down to r" for the defintion which could be
rendered as a one space down arrow in TeX, but don't
use it here.
Jack
Jun 26, 2012, 5:50:29 AM6/26/12
to
>> You're sayng I'm best advised to avoid using open-bounded notation?
>
> Just take Paul's advice that you mention in your OP.
The only thing is that when I go on to say 'let I be an interval', the idea
>
> Just take Paul's advice that you mention in your OP.
that I is an interval of integers is not covered by Paul's sentence; he
later advised me that it wasn't.
When you say
"One short sentence takes care of [n,m], [n,m), (n,m] and (n,m). That
sentence is 'all intervals are intervals of natural numbers.'" this is what
I have got now except that I say 'integers'. Am still concerned that if I
say 'let [x,y] be an interval' it might not necessarily be assumed that x <=
y.
I think, due to the ambiguity of the use of round brackets (eg. with their
also beig used for ordered pairs) I'll go for the [x+1,y], [x,y-1] option,
unless you object.
Cheers.
Jack
Jun 26, 2012, 5:53:26 AM6/26/12
to
news:Pine.NEB.4.64.12...@panix2.panix.com...
sign is said to be asymptotic notation. If, as Frederick says, it can also
be used for monotonic functions, then I'm happy with that.
> On Mon, 25 Jun 2012, Jack wrote:
>
>>
>> "William Elliot" <ma...@panix.com> wrote in message
>> news:Pine.NEB.4.64.12...@panix3.panix.com...
>> > On Mon, 25 Jun 2012, Jack wrote:
>> >
>> >> Once again I have got this issue of wanting to say two things in one
>> >> (and
>> >> ideally I want to be able to reference the assertion, all in one),
>> >> namely
>> >> that f(x)/g(x) is strictly decreasing and approaches 1.
>> >
>> > f(x) strictly decreases to 1 as x -> oo.
>> >
>> >> Currently I am writing
>> >> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>> >
>> > I suppose. So when defining f(x) ~ r, don't be glib,
>> > spell it all out. f is a monotonically decreasing function
>> > with lim(x->oo) f(x) = r.
>>
> Since you've pointed out that ~ is used for "asymototic to",
> don't use it for the above definition.
It's not a definition, but a result. I am just rather confused that the ~
>
>>
>> "William Elliot" <ma...@panix.com> wrote in message
>> news:Pine.NEB.4.64.12...@panix3.panix.com...
>> > On Mon, 25 Jun 2012, Jack wrote:
>> >
>> >> Once again I have got this issue of wanting to say two things in one
>> >> (and
>> >> ideally I want to be able to reference the assertion, all in one),
>> >> namely
>> >> that f(x)/g(x) is strictly decreasing and approaches 1.
>> >
>> > f(x) strictly decreases to 1 as x -> oo.
>> >
>> >> Currently I am writing
>> >> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>> >
>> > I suppose. So when defining f(x) ~ r, don't be glib,
>> > spell it all out. f is a monotonically decreasing function
>> > with lim(x->oo) f(x) = r.
>>
> Since you've pointed out that ~ is used for "asymototic to",
> don't use it for the above definition.
sign is said to be asymptotic notation. If, as Frederick says, it can also
be used for monotonic functions, then I'm happy with that.
Jack
Jun 26, 2012, 5:58:50 AM6/26/12
to
> On Mon, 25 Jun 2012, Jack wrote:
>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>
> f(x) strictly decreases to 1 as x -> oo.
>
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>
> I suppose. So when defining f(x) ~ r, don't be glib,
> spell it all out. f is a monotonically decreasing function
> with lim(x->oo) f(x) = r.
Would you object to my using the term 'converges to'?
>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>
> f(x) strictly decreases to 1 as x -> oo.
>
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>
> I suppose. So when defining f(x) ~ r, don't be glib,
> spell it all out. f is a monotonically decreasing function
> with lim(x->oo) f(x) = r.
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.
Jack
Jun 26, 2012, 6:06:22 AM6/26/12
to
"Frederick Williams" <freddyw...@btinternet.com> wrote in message
news:4FE8882F...@btinternet.com...
> Jack wrote:
>>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>> or
>> 'lim_(x-->oo) f(x)/g(x) = 1
>> with f(x)/g(x) >1'.
>
> 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?
>>
>> Once again I have got this issue of wanting to say two things in one (and
>> ideally I want to be able to reference the assertion, all in one), namely
>> that f(x)/g(x) is strictly decreasing and approaches 1.
>> Currently I am writing
>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>> or
>> 'lim_(x-->oo) f(x)/g(x) = 1
>> with f(x)/g(x) >1'.
>
> 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
> $\searrow$.
>
So if I was talking of an upward approach, would it be $\nwarrow$?
Still seems to be no way to overcome the clumsiness of spelling out
'f(x)/g(x) \searrow 1 is a monotonic convergence'.
Jack
Jun 26, 2012, 6:30:57 AM6/26/12
to
"Jack" <no1em...@hotmail.com> wrote in message
news:xQfGr.351877$pD7....@fx21.am4...
>
> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> news:4FE8882F...@btinternet.com...
>> Jack wrote:
>>>
>>> Once again I have got this issue of wanting to say two things in one
>>> (and
>>> ideally I want to be able to reference the assertion, all in one),
>>> namely
>>> that f(x)/g(x) is strictly decreasing and approaches 1.
>>> Currently I am writing
>>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>>> or
>>> 'lim_(x-->oo) f(x)/g(x) = 1
>>> with f(x)/g(x) >1'.
>>
>> 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?
> So if I was talking of an upward approach, would it be $\nwarrow$?
>
Ah, of course, it would be $\nearrow$!
> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> news:4FE8882F...@btinternet.com...
>> Jack wrote:
>>>
>>> Once again I have got this issue of wanting to say two things in one
>>> (and
>>> ideally I want to be able to reference the assertion, all in one),
>>> namely
>>> that f(x)/g(x) is strictly decreasing and approaches 1.
>>> Currently I am writing
>>> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
>>> or
>>> 'lim_(x-->oo) f(x)/g(x) = 1
>>> with f(x)/g(x) >1'.
>>
>> 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?
> So if I was talking of an upward approach, would it be $\nwarrow$?
>
Jack
Jun 26, 2012, 6:32:31 AM6/26/12
to
news:tJfGr.304511$9s1.2...@fx31.am4...
Now I've writtem
"f(x)/g(x) \searrow 1
is a strictly decreasing approach."
Sound OK?
"f(x)/g(x) \searrow 1
is a strictly decreasing approach."
Sound OK?
Jack
Jun 26, 2012, 6:34:00 AM6/26/12
to
news:FBfGr.586673$kS.3...@fx17.am4...
I've now got
"Throughout this paper, |N will be the set of non-negative integers and all
intervals are to be taken to be nonempty subsets, [x,y], of |N".
( |N is a script N).
Does that sound OK?
Cheers.
"Throughout this paper, |N will be the set of non-negative integers and all
intervals are to be taken to be nonempty subsets, [x,y], of |N".
( |N is a script N).
Does that sound OK?
Cheers.
Frederick Williams
Jun 26, 2012, 8:59:30 AM6/26/12
to
Jack wrote:
>
> [...] Am still concerned that if I
> I think, due to the ambiguity of the use of round brackets (eg. with their
> also beig used for ordered pairs) I'll go for the [x+1,y], [x,y-1] option,
> unless you object.
I don't object at all. Other possibilities are: ]x,y[ for (x,y), and,
if you use (x,y) for an ordered pair, use <x,y> for the ordered pair
instead and use (x,y) for the open interval.
>
> [...] Am still concerned that if I
> say 'let [x,y] be an interval' it might not necessarily be assumed that x <=
> y.
It must be, if not-(x<=y) then [x,y] wouldn't be an interval.
> y.
> I think, due to the ambiguity of the use of round brackets (eg. with their
> also beig used for ordered pairs) I'll go for the [x+1,y], [x,y-1] option,
> unless you object.
if you use (x,y) for an ordered pair, use <x,y> for the ordered pair
instead and use (x,y) for the open interval.
Frederick Williams
Jun 26, 2012, 9:01:56 AM6/26/12
to
Jack wrote:
>
>
> "Throughout this paper, |N will be the set of non-negative integers and all
> intervals are to be taken to be nonempty subsets, [x,y], of |N".
I'd omit ", [x,y],". I thought some of your intervals had one or more
>
>
> "Throughout this paper, |N will be the set of non-negative integers and all
> intervals are to be taken to be nonempty subsets, [x,y], of |N".
open end-points. If not, what's the worry?
> ( |N is a script N).
Frederick Williams
Jun 26, 2012, 9:03:51 AM6/26/12
to
Jack wrote:
>
> It's not a definition, but a result. I am just rather confused that the ~
> sign is said to be asymptotic notation. If, as Frederick says, it can also
> be used for monotonic functions, then I'm happy with that.
What are you saying I have said? I hope all I said in connection with
>
> It's not a definition, but a result. I am just rather confused that the ~
> sign is said to be asymptotic notation. If, as Frederick says, it can also
> be used for monotonic functions, then I'm happy with that.
this is that if two functions are asymptotic to one another, they can be
monotonic.
Frederick Williams
Jun 26, 2012, 9:08:27 AM6/26/12
to
Jack wrote:
>
> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> news:4FE8882F...@btinternet.com...
> > Jack wrote:
> >>
> >> Once again I have got this issue of wanting to say two things in one (and
> >> ideally I want to be able to reference the assertion, all in one), namely
> >> that f(x)/g(x) is strictly decreasing and approaches 1.
> >> Currently I am writing
> >> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
> >> or
> >> 'lim_(x-->oo) f(x)/g(x) = 1
> >> with f(x)/g(x) >1'.
> >
> > 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.
>
> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> news:4FE8882F...@btinternet.com...
> > Jack wrote:
> >>
> >> Once again I have got this issue of wanting to say two things in one (and
> >> ideally I want to be able to reference the assertion, all in one), namely
> >> that f(x)/g(x) is strictly decreasing and approaches 1.
> >> Currently I am writing
> >> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'
> >> or
> >> 'lim_(x-->oo) f(x)/g(x) = 1
> >> with f(x)/g(x) >1'.
> >
> > 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?
> So if I was talking of an upward approach, would it be $\nwarrow$?
> Still seems to be no way to overcome the clumsiness of spelling out
> 'f(x)/g(x) \searrow 1 is a monotonic convergence'.
is obvious, you can just write 'f/g decreases to 1'.
William Elliot mentions the down-pointing arrow (not sloping) that too
is ok.
Jack
Jun 26, 2012, 12:28:44 PM6/26/12
to
news:4FE9B2C4...@btinternet.com...
I thought that [x ,y] would be a useful inclusion because I might be asking
the reader to ditch the very concept of an interval and instead just think
of any old subset of |N whenever I say 'interval'.
> Jack wrote:
>>
>>
>> "Throughout this paper, |N will be the set of non-negative integers and
>> all
>> intervals are to be taken to be nonempty subsets, [x,y], of |N".
>
> I'd omit ", [x,y],".
Have changed them all to closed.
>>
>>
>> "Throughout this paper, |N will be the set of non-negative integers and
>> all
>> intervals are to be taken to be nonempty subsets, [x,y], of |N".
>
> I'd omit ", [x,y],".
I thought that [x ,y] would be a useful inclusion because I might be asking
the reader to ditch the very concept of an interval and instead just think
of any old subset of |N whenever I say 'interval'.
Jack
Jun 26, 2012, 12:35:35 PM6/26/12
to
>> > 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.
>> > 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.
>
Jack
Jun 26, 2012, 12:39:25 PM6/26/12
to
> 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,
> bottom right 1 as x --> oo'; which is read as 'f(x)/g(x) tends to one
> from above as x tends to infinity.'
f(x)/g(x) <= f(x+1)/g(x+1)?
Frederick Williams
Jun 26, 2012, 12:58:22 PM6/26/12
to
Jack wrote:
> I thought that [x ,y] would be a useful inclusion because I might be asking
> the reader to ditch the very concept of an interval and instead just think
> of any old subset of |N whenever I say 'interval'.
Oh, don't do that. If you mean 'subset of |N' then that's what you
> I thought that [x ,y] would be a useful inclusion because I might be asking
> the reader to ditch the very concept of an interval and instead just think
> of any old subset of |N whenever I say 'interval'.
should say. 'Interval' is entirely different.
Jack
Jun 26, 2012, 1:05:11 PM6/26/12
to
news:4FE9EA2E...@btinternet.com...
going about changing the idea of what is meant by 'interval', which of
course I am not :-).
> Jack wrote:
>
>> I thought that [x ,y] would be a useful inclusion because I might be
>> asking
>> the reader to ditch the very concept of an interval and instead just
>> think
>> of any old subset of |N whenever I say 'interval'.
>
> Oh, don't do that. If you mean 'subset of |N' then that's what you
> should say. 'Interval' is entirely different.
>
Yeh I know but the reader might think I am sufficiently eccentric as to be
>
>> I thought that [x ,y] would be a useful inclusion because I might be
>> asking
>> the reader to ditch the very concept of an interval and instead just
>> think
>> of any old subset of |N whenever I say 'interval'.
>
> Oh, don't do that. If you mean 'subset of |N' then that's what you
> should say. 'Interval' is entirely different.
>
going about changing the idea of what is meant by 'interval', which of
course I am not :-).
Frederick Williams
Jun 26, 2012, 1:20:28 PM6/26/12
to
monotonically to 1 as x -> oo'.
Frederick Williams
Jun 26, 2012, 1:21:26 PM6/26/12
to
Jack
Jun 26, 2012, 1:35:19 PM6/26/12
to
news:4FE9EF96...@btinternet.com...
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.
> 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.
So by the use of \searrow, it *can* dip beneath? Oh, no!
>>
>> >> > 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.
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.
Jack
Jun 26, 2012, 1:39:19 PM6/26/12
to
news:4FE9EF5C...@btinternet.com...
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?
> 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
>>
>> > 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'.
>
f(x)/g(x) ~ 1
is a strictly and monotonically decreasing approach. Or
f(x)/g(x) ~ 1
is a strictly decreasing approach.
Frederick Williams
Jun 26, 2012, 1:41:28 PM6/26/12
to
Jack wrote:
>
> [...] I want to say f/g is
>
> [...] 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?
> symbols instead of words -- at least as far as possible.
Frederick Williams
Jun 26, 2012, 1:46:01 PM6/26/12
to
Jack
Jun 26, 2012, 2:21:45 PM6/26/12
to
news:4FE9F448...@btinternet.com...
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....'
> 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
>>
>> [...] 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?
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....'
Jack
Jun 26, 2012, 2:24:31 PM6/26/12
to
news:4FE9F559...@btinternet.com...
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!
> Jack wrote:
>>
>> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
>> news:4FE9EF5C...@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?
>
> 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
>>
>> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
>> news:4FE9EF5C...@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?
>
> I don't get 'approach'. Such-and-such approaches so-and-so.
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!
Frederick Williams
Jun 26, 2012, 2:25:58 PM6/26/12
to
Frederick Williams
Jun 26, 2012, 2:27:27 PM6/26/12
to
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'.
> Jeeez, I can't believe there is no standard way to say the simple things I
> am trying to say!
Jack
Jun 26, 2012, 2:46:10 PM6/26/12
to
news:4FE9FF0F...@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*.
>
>> 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'.
>
Frederick Williams
Jun 26, 2012, 3:11:48 PM6/26/12
to
Jack
Jun 26, 2012, 3:43:10 PM6/26/12
to
news:4FEA0974...@btinternet.com...
Are you sure I can just stick a label against this, in the middle of a load
of text?
> Jack wrote:
>>
>> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
>> news:4FE9FF0F...@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.
>
Or maybe f(x)/g(x) is strictly decreasing to 1?
>>
>> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
>> news:4FE9FF0F...@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.
>
Are you sure I can just stick a label against this, in the middle of a load
of text?
Frederick Williams
Jun 26, 2012, 3:53:12 PM6/26/12
to
Jack wrote:
>
> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> news:4FEA0974...@btinternet.com...
> > Jack wrote:
> >>
> >> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> >> news:4FE9FF0F...@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.
> >
>
> Or maybe f(x)/g(x) is strictly decreasing to 1?
If it's clear what x is doing.
>
> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> news:4FEA0974...@btinternet.com...
> > Jack wrote:
> >>
> >> "Frederick Williams" <freddyw...@btinternet.com> wrote in message
> >> news:4FE9FF0F...@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.
> >
>
> 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?
William Elliot
Jun 26, 2012, 9:51:35 PM6/26/12
to
> > 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
> > 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.
>
>
> "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.
> "f(x)/g(x) searrow 1
>
> is a strictly decreasing approach."
>
> Sound OK?
>
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.
William Elliot
Jun 26, 2012, 9:54:14 PM6/26/12
to
On Tue, 26 Jun 2012, Jack wrote:
Jack
Jun 28, 2012, 11:44:51 AM6/28/12
to
>> 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
>> load
>> of text?
>
> Yes.
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.
Jussi Piitulainen
Jun 28, 2012, 12:28:58 PM6/28/12
to
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.)
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