HDTenEightyP/NetNews huggingface dataset:
From: victor@watson.ibm.com (Victor Miller)
Subject: Re: measures of the `size' of infinite sets
Date: Wed, 9 Sep 1992 21:46:26 GMT
Reply-To: victor@watson.ibm.com
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
	<BuBtsw.52B@unx.sas.com>
Organization: IBM, T.J. Watson Research Center

>>>>> On Wed, 9 Sep 1992 19:50:56 GMT, sasrdt@shewhart.unx.sas.com (Randall D. Tobias) said:


Randy> In article <1992Sep9.170045.1617@cs.rose-hulman.edu>, brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock) writes:
Randy> |>
Randy> |> In article <1992Sep9.042345.7472@galois.mit.edu> jbaez@riesz.mit.edu (John C.
Randy> |> Baez) writes:
Randy> |>
Randy> |> [a discussion of a natural way to calculate the density of a subset of
Randy> |> integers]
Randy> |>
Randy> |> > The set of integers that are multiples of 29 has density 1/29.
Randy> |> > The set of integers that aren't has density 28/29.
Randy> |> >
Randy> |> > Exercise to see if you get it: figure out, or guess, the density of the
Randy> |> > prime numbers.

Randy> What's the density of the set of prime powers?  Ie. how many Galois
Randy> fields are there?

The answer is 0 (now that's not what you expected is it).  More
precisely, the number of prime powers less than x is asympototic to x/log x.
Randy> -- 

Randy> Randy Tobias          SAS Institute Inc.     sasrdt@unx.sas.com
Randy> (919) 677-8000 x7933  SAS Campus Dr.         72450.2545@compuserve.com
Randy> (919) 677-8123 (Fax)  Cary, NC   27512-8000  norat@aol.com

Randy>  ... just my $(-exp(2*sqrt(-1)*arcos(0))/(((2**(2 + 1)) - 1)**2 + 1)).

--
		Victor S. Miller
		Bitnet: VICTOR at WATSON
		Internet: victor@watson.ibm.com
		IBM, TJ Watson Research Center
		"Great artists steal; lesser artists borrow" Igor Stravinsky
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Newsgroups: sci.math
Subject: Re: curves of degree 2
Date: 10 Sep 92 23:28:06 GMT
Organization: U of Cambridge, England

In article <KOSOWSKY.92Sep10133936@hall.harvard.edu>, kosowsky@hall.harvard.edu (Jeffrey J. Kosowsky) writes:

> Can anyone supply a proof for the following problem:
> 
> Given any 5 points in the plane, there exists a curve of degree 2
> passing thru all 5 points.
> 
> ie: There exist constants a,b,c,d,e,f such that the curve
> 	ax^2 + by^2 + cxy + dx +ey +f = 0 
> passes thru all 5 points.

The statement that the curve passes through (x,y) is a linear relation
between the six coefficients a,b,c,d,e,f. The statement that it passes
through the five given (x,y) is five linear relations between the six
coefficients a,b,c,d,e,f. In other words it says that an appropriate
linear map from R^6 to R^5 takes (a,b,c,d,e,f) to 0.
The rank of that map is at most 5, so its nullity (dimension of kernel)
is at least 6-5=1, so there is at least a 1-dimensional subspace of
(a,b,c,d,e,f) making all those things true. In particular there is a
sextuple other than (0,0,0,0,0,0) that works. This gives the required
curve.

If the points are (in some appropriate sense) in general position
(i.e. if the relations mentioned above are linearly independent)
then there is exactly one curve of dergee [at most] 2 passing through
the given points.

Incidentally, all the above is just a slightly highbrow way of saying: 
Six degrees of freedom, five conditions, therefore one degree of freedom
subject to those conditions.

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
From: ralf@ark.abg.sub.org (Ralf Stephan)
Newsgroups: sci.math
Subject: Re: Einstein's Erdos number
Date: 04 Sep 92 11:19
Organization: Private Coherent 4.0.1 site Augsburg/Bavaria/Germany

niall@nmsu.edu (Niall Graham) writes:
> 
> Correct.   Einstein's co-authors were 
> Ehrenfest, Lorentz, Minkowski, Somerfeld, Straus, Weyl.

and Rosen, Podolsky ?

--ralf
Newsgroups: sci.math
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: Proof of God's Existence
Date: 3 Sep 92 18:43:24 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <17ui6kINNsft@matt.ksu.ksu.edu> bubai@matt.ksu.ksu.edu (P.Chatterjee) writes:
> I was told by somebody that there is a mathematical proof of God's existence;
>was wondering if somebody could shed some light on the same.

There are many examples in history of proofs of the existence of God. Most
recently Mortimer Adler has patched up a combination of the cosmological
argument and the ontological argument. See his book _How to think about God_.

Many philosophers believe that Immanuel Kant disposed of the possibility of
proving God, and haven't really thought about it since.

A good place to get an appreciation of the theological import of existence
proofs of God is in the books of Hans Kung _On Being a Christian_ and _Does
God Exist?_ They are supposed to be companion books, but _Does God Exist?_
comes a bit after _On Being a Christian_. In _On Being a Christian_, Kung
presents a kind of argument for existence, but in _Does God Exist?_ he has
backed away from it. Most of _Does God Exist?_ is an examination of the
philosophical criticism of theism.

There is a range of informed opinion on the question of the possibility and
meaning of proving existence of God, and there are very well established
theistic traditions (most Protestant Christians) which do not believe that
there is any possibility of proving existence of God in any meaningful way.
(In fact this is the essential point at the core of Karl Barth's theology.)

Russell and Hardy were hard-core atheists but first class mathematicians, so
it stands to reason that they never saw a mathematical proof of existence of
God. Needless to say this hasn't prevented many other prominent mathematicians
from being theists. From this division of opinion, it follows that if there
were a modern mathematician's existence proof for God, it would be controversial
enough that you'd know about it.

Were there to be discovered such a proof, only certain kinds of mathematicians
would be liable to be convinced by it. It is unlikely that a constructivist or
a formalist could believe such a proof, etc.


Later,
Andrew Mullhaupt
From: gm115@cus.cam.ac.uk (Gabor Megyesi)
Newsgroups: sci.math
Subject: Re: Help: inscribed circle of three circles
Date: 8 Sep 92 19:55:18 GMT
Organization: U of Cambridge, England

The classical geometric solution is the following. By increasing or
decreasing the radii by the same amount, you can assume that two circles
intersect, except when there is obviously no solution, i.e., the first
circle contains the second, the second contains the third. Inverting them
with respect to the point of intersection, these two circles get mapped
into straight lines, and then the problem is reduced to solving a quadratic
equation. It has a nice geometric solution too, I just cannot do it off the
top of my head. Do everything backwards, and then you get what you want. It
is going to be a horrible mess if you write it out in explicit formulae,
but you cannot expect anything better with so many parameters.

Gabor
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: measures of the `size' of infinite sets
Date: 13 Sep 92 23:23:17 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <18nem0INN2vg@function.mps.ohio-state.edu> edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:
>Here is an exercise (alleged to work, I haven't done it...):
>Take the front page of the New York Times.  Write down in decimal form
>all of the numbers mentioned.  Then about 30% of the numbers will
>begin with the digit '1'.

I claim this doesn't do - but I won't spoil the problem yet.

>Or, take an atlas, and look up the lengths of the rivers (in km).
>About 30% of them begin with the digit '1'.  Repeat, using miles.
>Still 30%.

This might do better. I gave an undergraduate lecture on this one when
I was at U.B. The lecture following me in the series (by Pierre Gaillard)
was a very interesting consideration of the Buffon needle problem, and
how it is nearly certain that early investigators who claimed to compute
pi by this method cheated, and we know this because they claimed that
they were able to get too many digits of pi.

Putting these two problems together, you get the question, _how well should
the river measurements agree to the '30%' theory?_ (i.e. is measuring rivers
a 'good' way to compute '30%'?)

Suppose that instead of rivers, we consider lengths of coastlines (in the
sense of L. F. Richardson's classic "what is the length of...") is there a
'30%' theory for coastlines?

Later,
Andrew Mullhaupt
Newsgroups: sci.math.research
From: dgc@math.ucla.edu (David G. Cantor)
Subject: pari-gp
Summary: pari-gp for MSDOS
X-Submissions-To: sci-math-research@uiuc.edu
Organization: UCLA Mathematics Department
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 14 Sep 1992 19:15:03 GMT
Keywords: pari MSDOS

I have placed a completely unsupported version of pari-gp for MSDOS in
the directory pub/pari.  This can be obtained by anonymous ftp from
math.ucla.edu

David G. Cantor
Department of Mathematics
University of California
Los Angeles, CA 90024-1555
Internet:  dgc@math.ucla.edu



Newsgroups: sci.math.research
From: martin@math.rwth-aachen.de (  Martin Schoenert)
Subject: Re: Group Theory Reference needed.
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 15 Sep 1992 09:16:51 GMT

In his article of 14-Sep-92 Jack McKinney writes:

         I am looking for a reference that contains a list of all the groups
    of all orders up to a given limit [i.e., a book can only be so large.]

The computer system GAP contains a collection of all solvable  groups  of
size up to 100.  The groups are represented by so called power-commutator
presentations.  This is  the reason why  we  didn't include A(5) for  the
collection of  *all* groups of size up to 100; A(5) cannot be represented
this way.  GAP is able to compute, among many other things, the character
tables of these groups in a moderate amount of time.

More information  about  GAP can be obtained by  getting 'pub/gap/README'
via anonymous 'ftp' to 'samson.math.rwth-aachen.de' (137.226.152.6).

The collection is a compilation of the data in

    Marshall Hall, Jr. and James K. Senior
    The Groups of Order 2^n (n <= 6)
    The Macmillan Company, New York, 1964
        [the groups of size 64]

    Reinhard Laue
    Zur Konstruktion und Klassifikation endlicher aufl"osbarer Gruppen
    vol. 9 of Bayreuther Mathematische Schriften, Uni. Bayreuth, 1982
        [the groups of size 96]

    Joachim Neub"user
    Die Untergruppenverb"ande der Gruppen der Ordnungen <= 100
     mit Ausnahme der Ordnungen 64 und 96
    Habilitationsschrift, Uni. Kiel, 1967
        [other sizes]

Martin.

--
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany

Newsgroups: sci.math.research
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Subject: Re: dynamical systems and (perhaps) algebraic geometry
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Illinois at Urbana
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 8 Sep 1992 19:22:29 GMT

In <Bu9Ax6.H59@ens-lyon.fr> koiran@ens.ens-lyon.fr (Pascal Koiran) writes:

>Hello,

>Can someone provide a solution to the following problem, or give references
>to related work in the litterature ?

>Let f:[0,1]^n-->[0,1]^n (n>1) be a function of the form
>f(x)=(P_1(x),P_2(x),...,P_n(x)) where the P_i's are polynomials.
>Let's consider the iterates of f. Is the following situation possible ?

>for all k, there exists an infinite number of k-cycles (by k-cycle, I mean
>a cycle of length exactly k).

P_1(x) = g(x_1) = 4 x_1 (1-x_1); P_i(x) = x_i (2<=i<=n) should work, as:

g has k-cycles for all k (well-known result; e.g. (1-cos(2 pi/(2^k-1)))/2
is in a k-cycle of g).

If a_1,a_2,...,a_k is a k-cycle of g, then (a_1,x_2,...,x_n),
(a_2,x_2,...,x_n), ..., (a_k,x_2,...,x_n) is a k-cycle of f.



--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
216-5888@mcimail.com 70707.453@compuserve.com arthur@pnet01.cts.com (personal)
My opinions are my own, and do not represent those of my employer.
My interaction with our news system is unstable; please mail anything important.

Newsgroups: sci.math.research
From: koiran@ens.ens-lyon.fr (Pascal Koiran)
Subject: dynamical systems and (perhaps) algebraic geometry
Reply-To: koiran@ens.ens-lyon.fr
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Ecole Normale Superieure de Lyon
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 8 Sep 1992 11:07:54 GMT

Hello,

Can someone provide a solution to the following problem, or give references
to related work in the litterature ?

Let f:[0,1]^n-->[0,1]^n (n>1) be a function of the form
f(x)=(P_1(x),P_2(x),...,P_n(x)) where the P_i's are polynomials.
Let's consider the iterates of f. Is the following situation possible ?

for all k, there exists an infinite number of k-cycles (by k-cycle, I mean
a cycle of length exactly k).

Generalize to rational functions instead of polynomials, and to analytic 
functions.

Thanks,

Pascal.


Newsgroups: sci.math.research
From: ahulpke@math.rwth-aachen.de (Alexander Hulpke)
Subject: Re: Group Theory Reference needed.
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 15 Sep 1992 07:47:22 GMT

J. McKinney asked for a reference to a list of all groups up to a given
limit. This question has several answers:

Listing *all* groups of a given order can be very tedious since there might
be lots of abelian groups of a given order, that can be obtained easily via
the canonical decomposition in elementary divisors. Thus most references
leave out abelian groups.

A standard reference (that should be aviable easily) is
   Coxeter, Moser: Generators and relations for discrete groups, 
   Ergebnisse der Mathematik und ihrer Grenzgebiete Bd. 14, Springer
that contains for example a list of all non abelian groups up to order 32.

If you would like to set the upper bound higher I strongly recommend
obtaining the group theoretical program GAP (aviable via anonymous ftp from

  samson.math.rwth-aachen.de
  dimacs.rutgers.edu
  math.ucla.edu
  pell.anu.edu.au

and various other servers), that contains (among others) all groups up to
order 100 and all 2-groups of order dividing 256.

Included are also character tables of a wide range of groups, including all
tables from the Cambridge ATLAS. The manual contains a more complete
bibliography of group lists.

The soon to be released (hopefully) version 3.2 will also include facilities
for computing the character table of a given finite group, which would allow
for a complete list of character tables of the groups up to order 100.

--
Alexander Hulpke,   Alexander.Hulpke@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany

Newsgroups: sci.math.research
From: werner@pell.anu.edu.au (Werner Nickel)
Subject: Re: Group Theory Reference needed.
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Australian National University
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 15 Sep 1992 19:54:34 GMT


A.D. Thomas, G.V. Wood: Group Tables
Shiva Publishing Limited, 1980

The book contains all groups of order less than or equal to 32
and various information about them including the character
tables for the non-abelian groups.

Werner Nickel
Mathematics Research Section
Australian National University

Newsgroups: sci.math.research
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Subject: Re: Better than Cauchy-Schwarz
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Illinois at Urbana
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 10 Sep 1992 22:44:02 GMT
Keywords: analysis cauchy-schwarz

In <18ltblINN8oq@darkstar.UCSC.EDU> sutin@helios.ucsc.edu (Brian Sutin) writes:

>I have a conjecture which I have been unable to find in the library, prove,
>or discover a conterexample:

>There exists some constant C > 0 such that for all non-negative functions
>f(x) on [0,inf] which satisfy the inequality

>INT{ f(x) x dx } <= C INT{ f(x) dx } ,

>the inequality

>INT{ f(x) dx }^2 <= INT{ f(x)^2 dx }

>also holds.  All integrals are over [0,inf] and converge.

Without loss of generality, we can assume INT { f(x) dx } = 1, as all the
inequalities scale.

Now we can rewrite the problem as:

INT {f(x) dx} = 1
INT {f(x) x dx} = A <= C
implies
min INT {f(x)^2 dx} >= 1

As a calculus of variations problem, we can (to find what f has the min
value) write

D[INT {f(x)^2 dx} - \lambda INT {f(x) dx} - \mu INT {f(x) x dx},f] =
  INT {2 f(x) df(x) dx} - \lambda INT {df(x) dx} - \mu {df(x) x dx} =
  INT {df(x) (2 f(x) - \lambda - \mu x) dx} >= 0,

for any df which is non-negative where f is 0.

Hence, f(x) > 0 -> f(x) = 1/2 (\lambda + \mu x)

Working over any interval [0,X] where X >= 3A, (to make it likely that
things converge and have limits), we find:

f_A(x) = (3A-x) (2/9 A^-2), for 0<=x<=3A, and 0 otherwise.

should be the extreme function.

Then, INT {f(x)^2 dx} = 4/(9 A)

In this case, we can verify directly that, if INT {g(x) dx} = 1, and
INT {g(x) x dx} = A, that

INT { g(x)^2 dx} = 
INT {(f_A(x)^2 + 2 f_A(x) (f_A(x) - g(x)) + (f_A(x)-g(x))^2) dx} =
INT {f_A(x)^2 dx} + 2 INT{f_A(x) (f_A(x) - g(x)) dx} + INT {(f_A(x)-g(x))^2) dx} =
     4/(9A)       +   (>= 0 by differencial approx.) +        >= 0
 
Hence, if C>= 9/4, the inequality holds.


--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
216-5888@mcimail.com 70707.453@compuserve.com arthur@pnet01.cts.com (personal)
My opinions are my own, and do not represent those of my employer.
My interaction with our news system is unstable; please mail anything important.

Newsgroups: sci.math.research
From: sutin@helios.ucsc.edu (Brian Sutin)
Subject: Better than Cauchy-Schwarz
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Lick Observatory UCSC
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 9 Sep 1992 22:20:37 GMT
Keywords: analysis cauchy-schwarz

I have a conjecture which I have been unable to find in the library, prove,
or discover a conterexample:

There exists some constant C > 0 such that for all non-negative functions
f(x) on [0,inf] which satisfy the inequality

INT{ f(x) x dx } <= C INT{ f(x) dx } ,

the inequality

INT{ f(x) dx }^2 <= INT{ f(x)^2 dx }

also holds.  All integrals are over [0,inf] and converge.

Physically, this means that for any positive distribution of mass f(x), if
the center of mass is less than C, then the mass is bounded by the L_2 norm.
The conclusion is similar to Cauchy-Schwarz, but with an interval of
integration longer than 1.

Actually, the problem I need to solve is much more complicated than this,
but I am hoping that I can learn something from the method of proof for
this, since it is very similar.

Brian Sutin				sutin@helios.ucsc.edu
Lick Observatory, UCSC			Santa Cruz, CA 95064
    Fred:   "May I rescue you?"
    Ginger: "No, thank you.  I prefer being in distress."

Newsgroups: sci.math.research
From: koiran@ens.ens-lyon.fr (Pascal Koiran)
Subject: decidability and dynamical systems
Reply-To: koiran@ens.ens-lyon.fr
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Ecole Normale Superieure de Lyon
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 8 Sep 1992 11:10:30 GMT

Hello,

Can someone provide a solution to the following problem, or give references
to related work in the litterature ?

Let f:[0,1]-->[0,1] be a continuous piecewise-linear function (with a finite
number of laps). The endpoints of the laps have rational coordinates.
Given such an f and a rational x \in [0,1], is it decidable whether a fixed
point is reached in finite time when f is iterated on x ?
In other words, is it decidable whether there exists t, f^t(x)=f^{t+1}(x) ?

One can ask the same question for other classes of non-linear functions,
such as polynomials, rational functions, ...

Thanks,

Pascal.



Newsgroups: sci.math.research
From: chernoff@garnet.Berkeley.EDU (Paul R. Chernoff)
Subject: Re: Exponential of an arbitrary mapping f: R^n -> R^n
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of California, Berkeley
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 7 Sep 1992 20:40:05 GMT

In article <1992Aug28.181224.14463@nas.nasa.gov> asimov@nas.nasa.gov (Daniel A. Asimov) writes:
>One way to generalize the exponential of a matrix to arbitrary
>continuous functions f: R^n -> R^n is as follows:
>
>exp(f): R^n -> R^n via
>
>	exp(f)(x) = x + f(x) + f(f(x))/2! + f(f(f(x)))/3! +...
>

----------------------------------------
Comment: this is an unusual generalization of the exponential to nonlinear 
functions f.  I'd be curious as to its motivation.

The usual idea, of course, is to regard exp(tA)x, where A is a linear map,
as the solution of the differential equation
              dy/dt = Ay(t),  y(0) = x.

The standard generalization is the solution of
              dy/dt = f(y(t)), y(0) = x.

There is no "explicit" form of the solution, but there is a highly developed
theory, e.g. the theory of nonlinear semigroups.

----


-- 
# Paul R. Chernoff                       chernoff@math.berkeley.edu     #
# Department of Mathematics              ucbvax!math!chernoff           #
# University of California               chernoff%math@ucbvax.bitnet    #
# Berkeley, CA  94720                                                   #

Newsgroups: sci.math.research
From: neilfp@astro.ocis.temple.edu (Neil F. Palumbo)
Subject: XAY+XB+CY+D=0, HELP
X-Newsreader: Tin 1.1 PL5
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Temple University
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 10 Sep 1992 13:49:10 GMT

I am trying to solve the matrix equation XAY+XB+CY+D=0 analytically. Anyone
have any ideas, suggestions, comments ...?

Newsgroups: sci.math.research
From: le@fuhainf.fernuni-hagen.de
Subject: Union of flat cells in R^3
X-Submissions-To: sci-math-research@uiuc.edu
Organization: FernUniversitaet Hagen
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 14 Sep 1992 10:10:25 GMT
Keywords: Topological embeddings.



I believe that the following statement is true, but I have
been unable to find it in the literature
(by the way, I know that \beta(3,3,3,0) is false).



Assume that D_1,...,D_n (for n >= 2) are locally flat 3-cells
in Euclidian 3-space R^3 such that
all the D_i's have exactly one point p in common; i.e.,

	D_i \cap D_j = Bd D_i \cap Bd D_j = {p}
		for all pairwise distinct i and j.

If for each \epsilon > 0 there exists a flat 2-sphere R (e.g.
R is homeomorphic to S^2 and flat) such that

	p is contained in the interior of R,
	the diameter of R is smaller than \epsilon, and
	the sphere R intersects each D_i in a simple closed curve;
	
then the cells D_1,...,D_n are simultaneously flat; e.g., there
exists a "flattening" homeomorphism h of R^3 such that h(D_i) is a
simplex with h(p) as a vertex (for i=1,...,n).




Any suggestions, hints are welcome.

Thanks in advance.

(Le, Computer Science Dep.)

le@fuhainf.fernuni-hagen.de

Newsgroups: sci.math.research