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Larry Riddle wrote:
> Let f be a continuous function defined on [0,1]. Define g by
> 
> g(x) = 1/2 * (f(x) + f(1-x))
> 
> Then g is symmetric about the line x = 1/2. My question is, will g have a
> smaller arc length than f, with the lengths being the same only if g = f? I
> believe the answer is yes. It is easy to verify if f is a piecewise linear
> function over a uniform partition with an even number of subintervals. But
> what about the more general case? I'd be satisfied with a proof for the
> case when f is a polynomial.
Surely if f is nice enough (in particular, if it's a polynomial) then
the PL approximation you get from a fine enough partition will have
arc-length arbitrarily close to that of f, and the same will be true
for g. That will give you everything apart from "equal only if...".
But we can do this pretty easily by hand, at least if f is differentiable.
Arc length of f is integral of sqrt(1+f'^2) dx, and similarly for
arc length of g. I claim that at each pair of symmetrical points
of the interval, the integrand for g is <= the average of those
for f, with equality iff the values of f' at those points are
equal in magnitude, opposite in sign.
Indeed, let a,b be those values. Then we want to show
        sqrt(1+(a-b)^2/4) <= (sqrt(1+a^2)+sqrt(1+b^2))/2
    <=> 2sqrt(4+(a-b)^2)  <= sqrt(1+a^2) + sqrt(1+b^2)
    <=> 4+(a-b)^2         <= 1+a^2 + 1+b^2 + 2sqrt((1+a^2)(1+b^2))
    <=> 1-ab              <= sqrt((1+a^2)(1+b^2)).
If a,b have the same sign then this is trivial. Otherwise, change
terminology so that b (WLOG the negative one) becomes -b. So now
a,b are positive and we're considering
        1+ab         <= sqrt((1+a^2)(1+b^2))
    <=> 1+2ab+a^2b^2 <= 1 + a^2 + b^2 + a^2b^2
    <=> 2ab          <= a^2 + b^2
which I hope I don't have to tell you how to prove. :-)
So, the inequality holds; it's strict unless f'(x)=-f'(1-x) at all
points, and (integrating out from x=1/2) this easily implies that
f is symmetrical. And of course it's implied by f being symmetrical.
What if f isn't differentiable? Well, in that case what does "arc
length" mean? At any rate, if f is piecewise differentiable we can
just make arbitrarily small corrections at the joins to make it
differentiable, and the above will still go through.
We're done.
-- 
Gareth McCaughan       Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@dpmms.cam.ac.uk  Cambridge University, England.

For elements of a Hilbert space F, we use the letter f with or without
a suffix, e.g., f, f1, f2, f3, ..., f', f'', f''', ...
For elements of a Hilbert space G, we use the letter g with or without
a suffix, e.g., g, g1, g2, g3, ..., g', g'', g''', ...
Form the set product F X G.  Pick two ordered pairs from it, e.g.,
(f1, g1) and (f2, g2).
Certainly F X G can be turned into a Hilbert space:
  Inner Product of (f1, g1) & (f2, g2) = (f1|f2) + (g1|g2)
where
	(f1|f2) = the inner product of f1 and f2 in F
	(g1|g2) = the inner product of g1 and g2 in G
But consider the following expressions:
    (f1|f2) + i (g1|g2)
or
    (f1|f2) + Z (g1|g2)
where i is the square root of -1 and Z is any complex constant with a
nonzero imaginary part.
What are the properties of these "psuedo-inner products"?
In particular is there an analog of the projection operator?  Is there
an analog of the theorem that a closed convex set in a Hilbert space
possesses a distinguished element of minimum norm?
Thanks for reading this and for any information you can provide.
Ananth
-- 
Ananth
ananth@iastate.edu

In article ,
lriddle@ness.agnesscott.edu (Larry Riddle) wrote:
> Let f be a continuous function defined on [0,1]. Define g by
> 
> g(x) = 1/2 * (f(x) + f(1-x))
> 
> Then g is symmetric about the line x = 1/2. My question is, will g have a
> smaller arc length than f, with the lengths being the same only if g = f? I
> believe the answer is yes. It is easy to verify if f is a piecewise linear
> function over a uniform partition with an even number of subintervals. But
> what about the more general case? I'd be satisfied with a proof for the
> case when f is a polynomial.
> 
Hi Larry...
If I understand it right, then something more general should be
true.  If f and h are two functions defined on [0,1], and
    g(x) = (1/2)*(f(x)+h(x)),   is their average,
then the square of the arclength of the graph of g is less than
or equal to the average of the squares of the arclengths of
the graphs of f and h.  This will follow from the inequality
(on each subinterval) for affine functions, which is just
the convexity of the function x^2.  The case of equality should
also work (something like f = constant+h), but that may
require some estimates.
-- 
Gerald A. Edgar                   edgar@math.ohio-state.edu

	Someone wrote:
> Let f be a continuous function defined on [0,1].
and asked something about the arc length of f.
	In article ,
		Gareth McCaughan   answered:
> What if f isn't differentiable? Well, in that case what does
> "arc length" mean?
	Arc length isn't normally defined in terms of derivatives.
Let 0 = x1 < x2 < x3 < . . . < xn = 1, and look at the arc length of
the peicewise linear approximation to f that agrees with f at those n
points and whose graph is a straight line on the intervals between
adjacent points in this list.  The arc length of the peicewise linear
approximation is defined in the obvious way, and then the arc length of
f is defined as the smallest number that is not exceeded by the arc
length of any such peicewise linear approximation.  This always exists,
either as a finite real number or infinity.  There's no mention of
differentiability in the definition.  _If_ f happens to be
differentiable, or even peicewise differentiable, then the arc length
is the integral of sqrt(1+f'(x)^2) dx.  But if f is not differentiable,
I think you can still get all this as a Riemann-Stieltjes integral.
	Mike Hardy
Michael Hardy
hardy@stat.umn.edu

In 
Larry Riddle  asks:
> Let f be a continuous function defined on [0,1]. Define g by
>
> g(x) = 1/2 * (f(x) + f(1-x))
>
> ...will g have a smaller arc length than f? 
Let  h(x) = f(1-x).  It is enough to show that twice the
arclength of  g  on the interval  [0,1/2]  is less than
the sum of the arclengths of  f  and  h  on  [0,1/2].
Now  g = 1/2 * (f + h)  on this interval. If  f  is 
differentiable, just observe that
     2\sqrt{1+g'^2} <= \sqrt{1+f'^2}+\sqrt{1+h'^2},
so the corresponding inequality for the integrals shows 
that the answer is yes.
.

lueder wrote:
> 
> I am searching for an publication on the calculation of all rooted
> subtrees of a given rooted tree with the same root than the given rooted
> tree. Can anybody help me?
> 
> Thanks for help.
> 
> Arndt Lueder
> --
> ____________________________________________
>              Arndt Lueder
> Otto-von-Guerike-University of Magdeburg
>         PF 4120  39016 Magdeburg
>   e-mail:  arndt@hamlet.et.uni-magdeburg.de
>    http: //www.et.uni-magdeburg.de/~arndt/
> ____________________________________________
Take a look at "A Gray Code for the Ideals of a Forest Poset"
by Frank Ruskey and Y. Koda, Journal of Algorithms, 15 (1993)
324-340, which contains an algorithm for listing all subtrees
of a labelled rooted tree.  Each subtree differs from the next
by the addition or deletion of a single node, and the algorithm
is implemented so that only constant computation is done in
going from one subtree to the next.  Note that the labels matter;
if you desire all non-isomorphic rooted subtrees of a given
unlabelled rooted tree, then the problem is harder and I know
of no paper that addresses the listing problem --- although the
enumeration (ie counting) problem has most likely been
addressed. 
-- 
Frank Ruskey                     e-mail: fruskey@csr.uvic.ca
Dept. of Computer Science        fax:    604-721-7292
University of Victoria           office: 604-721-7232
Victoria, B.C. V8W 3P6 CANADA    WWW: http://www.csc.uvic.ca/~fruskey

[A complimentary Cc of this posting was sent to Ananth Sethuraman
],
who wrote in article <5b383m$16c@news.iastate.edu>:
> For elements of a Hilbert space F, we use the letter f with or without
> a suffix, e.g., f, f1, f2, f3, ..., f', f'', f''', ...
> 
> For elements of a Hilbert space G, we use the letter g with or without
> a suffix, e.g., g, g1, g2, g3, ..., g', g'', g''', ...
> 
> Form the set product F X G.  Pick two ordered pairs from it, e.g.,
> (f1, g1) and (f2, g2).
> 
> Certainly F X G can be turned into a Hilbert space:
>   Inner Product of (f1, g1) & (f2, g2) = (f1|f2) + (g1|g2)
> where
> 	(f1|f2) = the inner product of f1 and f2 in F
> 	(g1|g2) = the inner product of g1 and g2 in G
> 
> But consider the following expressions:
>     (f1|f2) + i (g1|g2)
> or
>     (f1|f2) + Z (g1|g2)
> where i is the square root of -1 and Z is any complex constant with a
> nonzero imaginary part.
> 
> What are the properties of these "psuedo-inner products"?
> 
> In particular is there an analog of the projection operator?  Is there
> an analog of the theorem that a closed convex set in a Hilbert space
> possesses a distinguished element of minimum norm?
I think it would be quite instructive to study the case of
1-dimensional H and G first ;-). All the "nasty" properties appear in
this case too.
Ilya

In article ,
Lou Pecora  wrote:
>At present I am using an old version of Tinkham's book on Group Theory and
>Quantum Mechanics.  Is there another (better?) book to serve as an
>introduction to applications of group theory and group representations? 
>I'd like something on the grad level with good coverage on the basic math
>that's need, but lots of applications, too (Condensed matter, chemistry,
>etc.).  Should cover point and space groups (definitely), the symmetric
>group (maybe), and, perhaps, some introduction to Lie Groups (nothing deep
>here).
I almost hesitate to recommend it, for fear it might be too simplistic, but 
Groups, Representations and Physics by HF Jones is a book I found useful, 
if quite simple-minded. Very definitely a book for the beginner, and more 
concerned with introducing representation theory than with applications. 
Good on the basic representation theory of finite groups, but not much 
systematic study of space groups. Had a reasonable chapter on molecular 
vibrations and some good stuff on Hartree-Fock if I remember correctly. A 
book to avoid is Cornwell's Group Theory in Physics. If you didn't like 
Hammermesh, you'll hate this. 
B
-- 
Kitty (bje1001@cam.ac.uk) Girton College, Cambridge, UK
C! N* F+ O(b+) G+ A++++  http://bust.web.site/ Tel: 328943
"Oh the prawns, the eels" - A testimony to the decline of seafood in Rome.

Lou Pecora (pecora@zoltar.nrl.navy.mil) wrote:
: At present I am using an old version of Tinkham's book on Group Theory and
: Quantum Mechanics.  Is there another (better?) book to serve as an
: introduction to applications of group theory and group representations? 
: I'd like something on the grad level with good coverage on the basic math
: that's need, but lots of applications, too (Condensed matter, chemistry,
: etc.).  Should cover point and space groups (definitely), the symmetric
: group (maybe), and, perhaps, some introduction to Lie Groups (nothing deep
: here).
BTW I have checked out Hammermesh and hate it.  Thanks for any suggestions.
: Lou Pecora
: code 6343
: Naval Research Lab
: Washington  DC  20375
: USA
:  == My views are not those of the U.S. Navy. ==
: ------------------------------------------------------------
:   Check out the 4th Experimental Chaos Conference Home Page:
:   http://natasha.umsl.edu/Exp_Chaos4/
: ------------------------------------------------------------
Try the book by Jean-Pierre Serre on group representations. It's supposed to
be tailormade for applications to quantum chemistry.
Good luck.
Jasho.

In article <5avmv7$cqs@geraldo.cc.utexas.edu>,
Jeriad Zoghby  wrote:
>Help with Dirichlet Distributions:
>I am looking for a text or article which discusses 
>some of the properties of the multivariate ordered 
>dirichlet distribution.  Any suggestions would be great.
>Thanks, Jeriad
Dirichlet distributions are the multivariate version of
the univariate Beta distribution.  But I have no idea 
what you mean by an ordered Dirichlet distribution.
The n-variate distribution is the distribution of the
n+1 positive random variables X_0, ..., X_n, whose sum
is 1, with parameters a_i > 0 whose sum is b, and with
the density
	\Gamma(b) \prod ((x_i)^{a_i - 1}/Gamma(a_i)) 
over the n-dimensional space of positive x's with the 
sum of all n+1 being 1.  Any n of the n+1 can be used.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (317)494-6054   FAX: (317)494-0558

It seems to me that a continuous function can have infinite arc
length.  We construct the function as the sum of an infinite series.
Each term of the series is a triangular sawtooth.  The height of the
n th sawtooth is 2^{-n}.  This guarantees that the sum of the series
is continuous because it is a uniformly convergent series of
continuous functions.  We make the frequencies of the successive
sawtooths increase at such a rate that the arc length of each
dominates the sum of the arc lengths of all preceding terms by a
factor of (say) 2.  Thus the arc lengths of the successive partial
sums go to infinity.  By any reasonable definition of arc length,
the arc length of the sum should be infinite.
This makes it hard to give a meaning to the proposed theorem for
arbitrary continuous functions.
-- 
John McCarthy, Computer Science Department, Stanford, CA 94305
http://www-formal.stanford.edu/jmc/progress/
He who refuses to do arithmetic is doomed to talk nonsense.

Second Announcement & Call for Registration
APPLIED NONLINEAR MATHEMATICS
1997 EPSRC SPRING SCHOOL 
The 6th Annual EPSRC UK Postgraduate Applied Nonlinear Mathematics Spring
School will be held at Loughborough University, UK from 14--18 April 1997.
The deadline for applications is 28 February 1997. For a Registration Form
see our web page:
http://info.Lboro.ac.uk/departments/ma/events/anmspring/index.html
The main lecturers, Professor James Yorke, Maryland USA, and Professor
Kurt Wiesenfeld, Georgia Tech USA, will each give five lectures at a level
suitable for students with a wide range of mathematical backgrounds.
Professor Yorke will give lectures on ``Chaotic Dynamics'' and Professor
Wiesenfeld will lecture on ``Coupled Nonlinear Systems''.
Other lectures will be given to introduce applications or provide
additional viewpoints. Those speaking will be
Professor R S MacKay
Professor Sir M V Berry
Professor D Broomhead
Professor P V E McClintock & Dr N Stein
Dr G King
Dr P C Bressloff
Professor M J Kearney.
Participants are encouraged to prepare a poster of their research work,
and there will be a session reserved for their display.
There are a number of places on the course reserved for students supported
by UK Research Council funding. These students may be eligible for full
funding by EPSRC of their accommodation and subsistence costs whilst on
the course. Travel costs should be met by the student's own institution.
A small number of bursaries are available for other postgraduate students,
from funds made available by the London Mathematical Society. Individuals
should apply for these by contacting the organisers directly.
Organiser:
Dr P C Bressloff
ANM Spring School 1997
Department of Mathematical Sciences
Loughborough University
Leics, LE11 3TU, UK
Tel: 01509 223188 
Fax: 01509 223969
Email: anm-springschool@Lboro.ac.uk 
For further information and Registration form see our web page:
http://info.Lboro.ac.uk/departments/ma/events/anmspring/index.html
-- 
Dr A H Osbaldestin
Department of Mathematical Sciences
Loughborough University
Loughborough
Leics. LE11 3TU
UK
E-mail: A.H.Osbaldestin@Lboro.ac.uk
Tel: +44 (0)1509-223189
Secretary: +44 (0)1509-223181
Fax: +44 (0)1509-223969
WWW: http://info.lboro.ac.uk/departments/ma/staff/aho/index.html

In article <32D39E85.41C6@damtp.cam.ac.uk>,
Tom Chou   wrote:
>Hello,
>
>I have an infinite, real, nonsymmetric square matrix 
>of which I want to find the lowest 10 or so eigenvalues. 
>I am taking larger and larger truncations and seeing if the eigenvalues
>converge. I am using balancing, then reduction to Hessenberg form, then 
>use a QR algorithm as described in Numerical Recipes. 
>
>However, for my particular matrix, I find that the eigenvalues don't
>quite converge at 40 X 40, where the algorithm uses too 
>many interations and exits (the lowest eigenval. changes by ~5% in going
>from 20 X 20 to 40 X 40). . Looking at the qualitative trends, I figure
>I need about a 400 X 400 truncation in the worst cases. 
>
>I think the problem is that the off diagonals get very large
>numerically. The matrix elements go as n^2*m^3, so numerically 
>get very large as one goes down the diagonal (~n^5) or, far away from
>the diagonals.
>
>
>My questions are:
>
>(1) Are there analytical bounds on how large a matrix I 
>need to take for a required accuracy in the lowest few
>eigenvalues? Where can I find theories about the convergence of
>the eigenvalues as the matrix is taken to be larger and larger?
>
>(2) What codes should I use? Can I simply reset the 
>number of iterations in the Numerical Recipes routines
>without catastrophic consequences? Are there other 
>routines/packages suited for this kind of matrix?
>
>(3)  Now suppose that each matrix element now depends on a parameter,
>s. I want to plot the eigenvalues as a function of s. Are there 
>theorems which can say when or when not any eigenvalues are degenerate?
>Or in particular, whether the lowest eigenvalue for one values of 
>s=s0 can become larger that say the 2nd largest at s=s0 
>at a different value s=s1? Is it possible to say that the lowest
>eigenval. is ALWAYS lower than the second lowest, for all s in 
>some range?
>
>This problem is related to band structure/floquet matrics.
>Any suggestions on where to look for the answers will be greatly 
>appreciated.
>
>Thx,
>
>Tom
>
It is not necessarily true that the eigenvalues will converge; you need
assumptions such as 'Hilbert Schmidt' operator.
There are several references: a classical reference is the book by
Kantorovitch and Akilov - Functional Analysis - see Chapter 14.
There are several theorems there that show when and how to guarantee the
convergence. Another book is the book by Krasnoselkii - Approximate
Solutions of Operator Equations.
-- 
||Henry Wolkowicz                |Fax:   (519) 725-5441
||University of Waterloo         |Tel:   (519) 888-4567, 1+ext. 5589
||Dept of Comb and Opt           |email:  henry@orion.math.uwaterloo.ca
||Waterloo, Ont. CANADA N2L 3G1  |URL: http://orion.math.uwaterloo.ca/~hwolkowi

Second Announcement
                   PACIFIC NORTHWEST GEOMETRY SEMINAR
                          1997 Winter Meeting
                Mathematical Sciences Research Institute
                              Berkeley, CA
                       Saturday, February 8, 1997
SCHEDULE:
---------
9:00    Coffee 
9:30    Jerry Marsden (Dept. of Control & Dynamical Systems, Cal Tech)
         
        Geometry and Classical Mechanics 
11:00   Jeonghyeong Park (Honam Univ., S. Korea, & Max-Planck Institute)
        
        The spectral geometry of Hermitian submersions 
12:00   Lunch
2:30    Bill Thurston (MSRI & UC Davis)
        
        Foliations and Geometry
4:00    Rick Schoen (Stanford)
        
        Least volume Lagrangian cycles 
All talks will be in the lecture hall at MSRI.
------------------------------------------------------------------------
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schiller c wrote:
> 
> In string theory,  the so-called K3 manifolds are a central ingredient.
> However, it is difficult to find papers explaining how to visualise them. Is
> it possible to get an idea on how they ``look'' - their shape, for example?
> 
> Is it possible at all to visualize them in 3d or 4d somehow? Is there a 3d or
> 4d object - a knot or a link or some other object - which shares at least some
> properties - perhaps some topological ones - with these manifolds?
> 
This depends on how you want to see them.
All K3s are diffeomorphic, but the complex structure differs.
Examples are the surfaces of degree 4 in P^3 (i.e. in complex
3-space). You can look at the real points.
Or: take the double cover of P^2 branched along a sextic.
Take e.g. a complete 4-gon (ie 6 lines joining four points),
take the double cover and desingularise. 
Jan.
--
email: stevens@math.chalmers.se
Matematiska Institutionen
Chalmers Tekniska H"ogskola
SE 412 96 G"oteborg
Sweden

Can anyone give me a lead to the fast hankel transform?
I know it exists but cannot find it anywhere
email reply appreciated
Thank you for reading this! (and many more thanks if you can reply!)

The book  
                 LES R\'ESEAUX PARFAITS DES ESPACES EUCLIDIENS
                     (PERFECT LATTICES IN EUCLIDEAN SPACES) 
                                      by 
                                Jacques MARTINET
                        Professor, University of Bordeaux 
recently appeared. It is sold at the price of 385 FF, 
and can be ordered at 
	                Editions MASSON, 
	                5, rue Laromigui\`ere, 
	                F-75241 Paris cedex 05. 
                        *********************** 
Here is a translation into English of the back cover of the book, followed 
by the table of contents: 
                        *********************** 
This book is dedicated to beginning graduate (or to advanced undergraduate)  
students in mathematics or computer science, as well as to researchers. 
The reader is expected to be somewhat familiar with the basic techniques 
of algebra and Euclidean geometry that one can usually learn 
in graduate courses. 
To a given lattice in a Euclidean space in naturally attached a 
``regular'' sphere packing. To exhibit dense sphere packings is one 
of the main problems in the geometry of numbers. The property of 
perfection, the central topic of this book, is a property of 
a linear nature, which is fullfilled by all extreme lattices, 
those on which the density attains a local maximum. 
The book contains many (164) exercises. Two introductory chapters and 
four appendices will help the reader to master the language of the theory 
as well as the techniques from algebra which are needed. 
                        *********************** 
                    TABLE DES MATI\`ERES  --  CONTENT 
Introduction (Introduction)
Chapter I 
   G\'en\'eralit\'es sur les r\'eseaux 
   Generalities on lattices 
Chapter II 
   In\'egalit\'es g\'eom\'etriques 
   Geometrical inequalities 
Chapter III 
   Perfection et eutaxie 
   Perfection and eutaxy 
Chapter IV 
   Les r\'eseaux de racines  
   Root lattices 
Chapter V 
   R\'eseaux li\'es aux r\'eseaux de racines 
   Lattices related to root lattices 
Chapter VI 
   R\'eseaux parfaits de petitie dimension 
   Low-dimensional perfect lattices 
Chapter VII 
   L'algorithme de Vorono{\"\i} 
   The Voronoi algortithm 
Chapter VIII
   R\'eseaux hermitiens 
   Hermitian lattices 
Chapter IX 
   Les configurations de vecteurs minimaux 
   Configurations of minimal vectors 
Chapter X 
   Extr\'emalit\'e dans des familles de r\'eseaux 
   Extremality in families of lattices 
Chapter XI 
   Op\'erations de groupes 
   Group actions 
Chapter XII 
   Sections des r\'eseaux 
   Sections of lattices 
Chapter XIII 
   Extensions de l'algorithme de Vorono{\"\i} 
   Enlargements of the Voronoi algorithm 
Chapter XIV 
   Donn\'ees num\'eriques 
   Numerical data 
Appendix 1 
   Formes quadratiques et anneaux de Dedekind 
   Quadratic forms and Dedekind domains 
Appendix 2 
   Les groupes de quaternions 
   Quaternionic groups 
Appendix  3 
   Alg\`ebres semi-simples 
   Semi-simple algebras 
Appendix 4 
   Arithm\'etique dans les alg\`ebres semi-simples 
   Arithmetic in semi-simple algebras 
Bibliographie 
References 

Suppose that V(n,c) is the intersection of an n-sphere of radis c
(centered at 0), and the unit cube [-1,1]^n.  If 0<= a <=1, I'm
interested in a expression for
		Vol(V(n,a*sqrt(n)))
	         --------------
                     2^n
I would expect that this would be a nice function of a.  Does anyone
know a reference for this?
-- 
Victor S. Miller     | " ... Meanwhile, those of us who can compute can hardly
victor@ccr-p.ida.org | be expected to keep writing papers saying 'I can do the
CCR, Princeton, NJ   | following useless calculation in 2 seconds', and indeed
    08540 USA        | what editor would publish them?"  -- Oliver Atkin

	Maybe this one is pretty routine for measure-theorists:
	Suppose B is a subalgebra of a Boolean algebra A,
and m : B ---> [0,1] is a normalized measure.  ("Normalized" means
m(1)=1, "measure" is intended to imply that the values of m are
always non-negative and m is countably additive.)
Suppose
(1) x is a member of the interval [0,1], and
(2) p is a member of A but not of B, and
(3) for every q in B such that q or= p we have m(q) >or= x.
Let C be the smallest subalgebra of A that includes B and contains x.
Can we conclude that m can be extended to a measure on C satisfying the
constraint m(p)=x?
Can we conclude that m can be extended to a measure on all of A
satisfying the constraint m(p)=x?
Can we draw the same conclusion if we weaken countable additivity to
finite additivity?  And is the proof any more difficult, or is it the
same?
Finally, suppose instead of taking values in [0,1] the values are to be
in some linearly ordered cancellative semigroup.  (I'm not sure
_countable_ additivity makes sense then, so let's just assume finite
additivity.)  By "linearly ordered semigroup" I mean not just a
semigroup on which a linear ordering is defined, but also that the
linear ordering is compatible with the semigroup operation, meaning a
member of the semigroup can be added to both sides of an inequality
without changing its truth-value.  What are the answers to the questions
above then?  (I think if the linearly ordered semigroup is Archimedean,
then it's got to be (isomorphic to) a subsemigroup of the real numbers
with addition.  So the dropping of the assumption that that's which
semigroup it is just amounts to dropping the assumption of
Archimedeanism.)
	Mike Hardy
Michael Hardy
hardy@stat.umn.edu