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From: achille on 12 Aug 2010 05:38
On Aug 12, 3:55 pm, achille <achille_...(a)yahoo.com.hk> wrote:
> 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

A more interesting example,

f(z) = \sum{k=1..oo} 2^(k-2) { 1 - sqrt(1+ z^2/3^k) }

At z = 0, the value inside each {..} is either 0 or 2.
As a result, the branch is uniquely identified by the
value of f(0) \in N ;-)
From: master1729 on 12 Aug 2010 14:32
> 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!

when you walk :

you head is up.

your feet are down.

left is where your big toe on the left foot is on the right.

right is where your big toe on the right foot is on the left.

>
> (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.

yes thats what i mean.

initial branch has no predecessor.

all branches have successor.

no branches intersect.


>
> 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

tommy1729
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