Newsgroup sci.math.research 6677

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Several months ago I asked a question about groups generated by a
single conjugacy class (I was, and am, particularly interested in 
the braid group B_n).  In a follow-up, Greg Kuperberg wrote:
>I
>misread the question at the time, but to atone for my sins I went and
>asked Bill Thurston.  It has led to some interesting joint research.
>In general new research seems to be a remarkably easy prospect when
>talking to Thurston.
>
>Here is an argument that he found that addresses the original
>question.  Theorem:  Many elements g in the commutator subgroup have
>the property that the commutator length of g^n grows linearly in n.
>This implies that the conjugacy word length also grows linearly.
Partly in the hope that the difficulty of new research might
increase no more than linearly with talk-distance from Thurston,
I have another (clearly related) question.  As before, I will
phrase it generally, but my particular interest is in B_n.
Let G be a group.  Following Freudenthal (I think it was), define the 
genus of an element g of the commutator subgroup [G,G] to be the least 
k such that g is the product of k commutators aba^{-1}b^{-1} (i.e.,
the genus is what Kuperberg and Thurston have called the commutator
length, above; genus is a good name for this, since if one thinks of
[G,G] being represented by null-homologous loops in K(G,1), then 
the genus of such a loop is the least genus of a singular surface
bounded by the loop).  Question: given (say) a presentation of G
(for instance, the standard presentation of B_n), and an element
g of [G,G], how can one calculate the genus of g (or bound it above)?
Lee Rudolph

There is an open position for a Research Assistant at Lund University
in Computer Science, Numerical Analysis, Mathematical Statistics, or
Mathematics.  The position will be connected with the appropriate
department when awarded.  Preference is granted to female applicants.
The position generally entails 100% research under four years, with
the possibility of a two-year extension.  There is no requirement to
speak Swedish nor to have any experience within the Swedish university
system.  Generally applicants must have received a doctoral degree
_within the last five years_.  Salary is comparable to that of the
position of Lecturer.
The following is an informal translation of the official announcement:
------------------------------------------------------------------------
January 15, 1997
Lund University announces the following position:
Research Assistant intended for women in Applied Mathematics and
Computer Science
The position, with preference to female applicants, concerns
the subjects:
    Computer Science
    Numerical Analysis
    Mathematical Statistics
    Mathematics
Ref. nr: 16257
Employment Date: as soon as possible
Information: Andrzej Lingas (Computer Science) tel: +46-46-222 4519
             Gustaf S=F6derlind (Numerical Analysis) tel: +46-46-222-4909
             S=F8ren Asmussen (Mathematical Statistics) tel: +46-46-222-4=
747
             G=F6ran Wanby (Mathematics) tel: +46-46-222-8559
            =20
The qualifications for a research assistant position is as stated in
ch. 4 para. 13 in H=F6gskolef=F6rordningen (SFS 1993:100)
The basis for awarding the position of research assistant
is given in ch. 4 paras. 15 and 16 in H=F6gskolef=F6rordningen (SFS 1993:=
100)
The duration of the position has a time limit in accordance to
ch. 4 para 21:6 in H=F6gskolef=F6rordningen (SFS 1993:100)
Salary level is arranged according to individuals.  For this position,
settled guidelines are applied concerning salary placement.  Applicants
may specify salary requirements, if any.
This position is part of a university effort to obtain a more even
distribution of gender among the teaching positions at the university.
The application should include:
1. a signed, short written description of the applicant's scientific and
   pedagogical activities, including a research proposal -- 1 copy
2. a signed and attested summary of merits (curriculum vit=E6) and a
   list of scientific publications -- 1 copy
3. scientific work which the applicant wish to refer to
   (numbered according to the list of publications) -- 2 copies each
The application should be directed to:
  Lunds universitet
  Matematisk-naturvetenskaplig fakultet
and should be addressed to:
  Registrator
  Lunds universitet
  Box 117
  S-221 00 Lund
The application must arrive no later than Wednesday February 5, 1997.
The reference number (16257) must be stated in the application.
--=20
Kurt Swanson, Department of Computer Science, Lund University.
Kurt.Swanson@dna.lth.se (http://www.dna.lth.se/home/kurt/)

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              EUROCONFERENCES IN MATHEMATICS ON CRETE
    The Department of Mathematics of the University of Crete
announces the 1997 conferences of the series Euroconferences in 
Mathematics on Crete, sponsored by the Training and Mobility of 
Researchers Programme of the Commission of the European Union.
22-28 June 1997   DIRICHLET FORMS AND THEIR APPLICATIONS IN GEOMETRY 
                  AND STOCHASTICS
    Organizers: J.Jost (Leipzig, Germany), U.Mosco (Rome, Italy), 
K.T.Sturm (Erlangen, Germany)
    Main speakers: J.Jost (Leipzig, Germany), W.Kendall (Warwick, 
United Kingdom), U.Mosco (Rome, Italy), M.Roeckner (Bielefeld, 
Germany), K.T.Sturm (Erlangen, Germany)
29 June - 5 July 1997   NONLINEAR DISPERSIVE WAVES: THEORY AND 
                        APPLICATIONS
    Organizers: V.A.Dougalis (Athens, Greece), A.S.Fokas (London, 
United Kingdom)
    Main speakers: J.L.Bona (Austin, U.S.A.), D.Crighton (Cambridge, 
United Kingdom), A.Its (Purdue, U.S.A.), J.-C.Saut (Paris-Sud, 
France), V.E.Zakharov (Moscow, Russia)
    The conferences will take place at the Anogia Academic Village, 
a conference center located at the traditional Cretan village of 
Anogia on the slopes of the mountain Ida. Anogia is located at an 
elevation of 750 m, about 45 minutes by car from Heraklion, the 
largest city of Crete, and about half an hour from the closest coast.
The living expenses (accommodation plus meals) per day for a person 
are estimated at about 33 ECU in a double room or 43 ECU in a single 
room. The registration fee amounts to 250 ECU. 
    The Training and Mobility of Researchers Programme financially 
supports young researchers from the countries of the European 
Economic Area to enable them to attend the conferences. There will be
also some limited funds from other sources available to support 
participants not belonging to the above group. Support can cover 
(all or certain) travel, living and registration expenses.  For 
information please contact the local co-ordinator of the conference 
series indicated below. 
    The conference series will continue in summer 1998 with the 
following three conferences:
    Groups of finite Morley rank (A.Borovik, Manchester, United 
Kingdom), Galois representations in arithmetic geometry (M.Taylor, 
Manchester, United Kingdom), Front propagation (P.L.Lions, 
Paris-Dauphine, France).
    The topics of the conferences, which will follow in the next 
years, will be decided by the international scientific committee 
consisting of: H. Abels (Bielefeld, Germany), H. Bauer (Erlangen, 
Germany), C. Dafermos (Brown University, USA), O. Kegel (Freiburg, 
Germany), S. Papadopoulou (Crete, Greece), V. Thomee (Goeteborg, 
Sweden), A. Wilkie (Oxford, United Kingdom). The next meeting of 
the committee will be in fall 1997. Suggestions for topics for 
future conferences should be sent to the local co-ordinator of the 
series.
    For additional information please contact the local co-ordinator:
            Susanna Papadopoulou
            Department of Mathematics
            University of Crete
            Heraklion, Crete, GREECE
            Fax-Nr.: 81-234516
            e-mail: souzana@math.uch.gr
    or, for the conferences of 1997:
     V.A.Dougalis
     Mathematics Department
     University of Athens
     Panepistimiopolis
     15784 Zografou
     GREECE
     e-mail: doug@eudoxos.dm.uoa.gr
     K.T.Sturm
     Mathematisches Institut
     Universitaet Erlangen-Nuernberg
     Bismarckstrasse 1 1/2
     91054 Erlangen
     GERMANY
     e-mail: sturm@mi.uni-erlangen.de
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Robert Israel  wrote in article <32DFE4D2.2781E494@math.ubc.ca>...
| U Sharan wrote:
| 
| > Given that the ith digit of pi is unknown to a person, is it equally
| > likely for that digit to be any of {0,1,...,9} ?
[...]
| A better way to put the question that I think you really
| want to ask is the following: let N(n,d) be the number of
| times the ith digit of pi is d for 1 <= i <= n.  Does
| N(n,d)/n -> 1/10 as n -> infinity?
| 
| Nobody has any idea how to prove this, but AFAIK the 
| statistical evidence, based on the digits that have 
| been computed so far (the record is over 6 billion) 
| indicates that it is true.
There is a nice discussion of this and other "statistical" 
properties of pi's decimal digits at:
http://www.ast.univie.ac.at/~wasi/PI/pi_normal.html
--
Robert E Sawyer 
soen@pacbell.net 

      I am looking for references for the theory of unbounded operators,
particularly selfadjoint operators and differential operators, on
real Hilbert spaces. The books I know all assume from the beginning that
everything is over the complex numbers. The sort of thing I am looking
for might include the spectral theorem for unbounded selfadjoint operators
on a real Hilbert space, and a theorem or lemma that says an unbounded
operator is selfadjoint if and only if its complexification is
selfadjoint. (It might not bother to prove this because it is easy, but
I would like to know a place where this is stated.)
	As far as I know, everything goes through very easily simply by
complexifying. I am fairly certain that this is true for the things I
actually need. I am primarily looking for a place to cite that says this,
and secondarily for any odd effects that might happen in the real case.
	Please respond by email to ncp@darkwing.uoregon.edu or by posting.
I will post a summary if there are three or more requests for one; otherwise
I will email summaries to those who ask.
						---Chris Phillips

Robert E Sawyer wrote:
> 
> Robert Israel  wrote in article <32DFE4D2.2781E494@math.ubc.ca>...
> | U Sharan wrote:
> |
> | > Given that the ith digit of pi is unknown to a person, is it equally
> | > likely for that digit to be any of {0,1,...,9} ?
> [...]
> | A better way to put the question that I think you really
> | want to ask is the following: let N(n,d) be the number of
> | times the ith digit of pi is d for 1 <= i <= n.  Does
> | N(n,d)/n -> 1/10 as n -> infinity?
> |
> | Nobody has any idea how to prove this, but AFAIK the
> | statistical evidence, based on the digits that have
> | been computed so far (the record is over 6 billion)
> | indicates that it is true.
> 
> There is a nice discussion of this and other "statistical"
> properties of pi's decimal digits at:
> 
> http://www.ast.univie.ac.at/~wasi/PI/pi_normal.html
> 
> --
> Robert E Sawyer
> soen@pacbell.net
I think I read a book on this subject and the followings are what I can
remember:
1) the term to describe the randomness of fractional digits of a real
numbers is normal   (and non-normal).
2) a non-normal number, e.g. 1/3= 0.3333333.... would have predictable
fractional digits    after certain digits, but for a normal number, like
pi, its fractional digits are not    predictable.
3) the overwhelming majority of real numbers are normal, thus the mane
normal.
4) I vaguely remember that Berel is guy who started this normality
business.
5) The normality of a number not only has to to with the frequency of
recurrence of a    sigle digit, but more importantly that of any
sequense of digits, e.g. 123456788888.    And there is a question of
whether a particular sequence of digits, say, a    sequense of
333333333..... (one billion of 3) would occur at all in a normal number.
6) Not much is known what gives rise to the phenomenon of normality, the
classification 
   of normal and non-normal dosen't fit into the better known
classifications such as    rational/irrational or
algebaric/transcendental.
-Yuntong Kuo
[ Moderator's note:
1) Predictable is not the right word.  All the digits of Pi are "predictable"
in the sense that there is an algorithm that calculates them.
2) The "guy" was Emile Borel.
3) The following terminology is often used: a number is said to be "simply 
normal" if in its decimal expansion all ten digits occur with equal 
frequency, and "normal" if all blocks of digits of the same length occur
with equal frequency.
4) According to the definition, it is clear that a block of a billion 3's,
or any other finite sequence, will occur in the expansion of any normal
number.  What's not clear is whether this will occur in a number that is
not known to be normal (e.g. Pi).
5) Rational numbers are always non-normal (though they may be simply normal).
No algebraic irrationals are known to be simply normal or known to 
be non-normal. 
]