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From: quasi on 11 Aug 2010 18:34
On Wed, 11 Aug 2010 16:16:47 -0500, Robert Israel
<israel(a)math.MyUniversitysInitials.ca> wrote:

>master1729 <tommy1729(a)gmail.com> writes:
>
>> > master1729 <tommy1729(a)gmail.com> writes:
>> >
>> > > what is the simplest example for a coo function
>> > that has oo
>> > > non-intersecting branches labeled by the positive
>> > integers ?
>> > >
>> > > so we have non-intersecting branches :
>> > >
>> > > branch 0 , branch 1 , branch 2 , ...
>> > >
>> > > by analogue the logaritm has branches labeled by
>> > the integers :
>> > >
>> > > .. branch -1 , branch 0 , branch 1 , branch 2 , ...
>> >
>> > The logarithm with branches relabelled.
>> > --
>> > Robert Israel
>> > israel(a)math.MyUniversitysInitials.ca
>> > Department of Mathematics
>> > http://www.math.ubc.ca/~israel
>> > University of British Columbia Vancouver,
>> > BC, Canada
>>
>> funny and sad.
>>
>> i knew you were gonna say that.
>>
>> but thats not what i meant.
>>
>> im looking for a function where you cannot go a branch downward from branch
>> 0. but infinite branches upward.
>
>Which way is up in this context?

So sad -- you don't even know which way is up!

(just kidding)

On a more serious note, while I know very little about this topic, let
me try to give a possible interpretation of tommy's question.

The two-way progression of branches surely has a natural topological
interpretation where the graph (surface?) associated with each branch
is strictly between the graphs of 2 other branches.

A one-way progression would have an initial branch which would not be
(topologically) strictly between 2 other branches. In other words, the
initial branch would have a successor but no predecessor, whereas all
other branches would have both a predecessor and a successor.

Is there such a one-way progression (subject to tommy's specified
conditions)?

Perhaps a two-way progression can be converted to a one-way
progression by just "squaring it" (in some sense)?

quasi
From: achille on 11 Aug 2010 23:07
On Aug 12, 6:34 am, quasi <qu...(a)null.set> wrote:
> On Wed, 11 Aug 2010 16:16:47 -0500, Robert Israel
>
> A one-way progression would have an initial branch which would not be
> (topologically) strictly between 2 other branches. In other words, the
> initial branch would have a successor but no predecessor, whereas all
> other branches would have both a predecessor and a successor.
>
> Is there such a one-way progression (subject to tommy's specified
> conditions)?
>

Isn't 'failure of monodromy' forms a group, the
monodromy group, on spaces with a base point?
0,1,2,3 doesn't have any natural group structure ????
From: quasi on 12 Aug 2010 00:41
On Wed, 11 Aug 2010 20:07:32 -0700 (PDT), achille
<achille_hui(a)yahoo.com.hk> wrote:

>On Aug 12, 6:34 am, quasi <qu...(a)null.set> wrote:
>> On Wed, 11 Aug 2010 16:16:47 -0500, Robert Israel
>>
>> A one-way progression would have an initial branch which would not be
>> (topologically) strictly between 2 other branches. In other words, the
>> initial branch would have a successor but no predecessor, whereas all
>> other branches would have both a predecessor and a successor.
>>
>> Is there such a one-way progression (subject to tommy's specified
>> conditions)?
>>
>
>Isn't 'failure of monodromy' forms a group, the
>monodromy group, on spaces with a base point?
>0,1,2,3 doesn't have any natural group structure ????

Although I offered a possible interpretation, my knowledge in this
area is almost zero, so if what you're saying is that the idea makes
no sense, I'm not at all surprised.

quasi
From: achille on 12 Aug 2010 01:59
On Aug 12, 12:41 pm, quasi <qu...(a)null.set> wrote:
>
> Although I offered a possible interpretation, my knowledge in this
> area is almost zero, so if what you're saying is that the idea makes
> no sense, I'm not at all surprised.
>
> quasi

I just want to point out any 'natural' labeling of the branches
should reflect the group structure of underlying monodromies.

For finitely many singularities/branch cuts, this doesn't seem
possible. However, it just struck me one can naturally label the
elements of Z_2[x] by N through binary representation of positve
numbers. Perhaps we can start with infintely many square root type
of branch cuts, one at each natural number and glue them togather...

BTW, my knowledge in this area is less than almost zero ;-p
From: achille on 12 Aug 2010 03:55
On Aug 12, 1:59 pm, achille <achille_...(a)yahoo.com.hk> wrote:
>
> I just want to point out any 'natural' labeling of the branches
> should reflect the group structure of underlying monodromies.
>
> For finitely many singularities/branch cuts, this doesn't seem
> possible. However, it just struck me one can naturally label the
> elements of Z_2[x] by N through binary representation of positve
> numbers. Perhaps we can start with infintely many square root type
> of branch cuts, one at each natural number and glue them togather...

Hmm...

\sum_{n=1..oo} sqrt( (z+n)^2 + 1 ) / n^3

with square root type branch cuts along [n+i,n-i] for
n = -1,-2,...-oo seem to work. Though it might not be
what is in tommy's mind ;-p



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