Newsgroup sci.math.research 6127

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On 26 Oct 1996, paolo cascini wrote:
> I'm not English so you have to soory my language...
> 
> can you find two space that are diffeomorph of class Cn for each n, but 
> that are not diffepmorph of class c-inf.
> 
> 
> 
I don't know what you mean by space.
If space means C-infinity manifold then it's true that two C-infinity 
manifolds which are C-1 diffeomorphic are also C-infinity diffeomorphic 
(you can prove that the space of C-infinity functions is dense is the 
space of C-1 functions with the C-1 Whitney strong topology. You can also 
prove that the C-1 diffeomorphisms are an open subset of the set of C-1 
functions (I'm using manifolds without boundary). So if the set of C-1 
diffeomorphisms is not empty, it contains a C-infinity function - which is 
therefore a C-infinity diffeomorphism).
I think you can find the proofs in the book Hirsch (I think it's called 
Differential Topology, or something like that).
Daniel Victor Tausk - tausk@ime.usp.br

Given an undirected graph G, let G' be the graph whose set of vertices are the 
edges of G, and whose adjacency matrix is defined so that if e_1 and e_2 are 
vertices of G' (i.e. edges of G), then e_1 and e_2 are adjacent if and only if 
they have a vertex of G in common.
Does this construction of G' have a name?
The results of the construction look rather pretty for regular trees.  If G is 
a binary tree (i.e. a tree each of whose vertices has degree 3), then G' looks
like G except that each vertex is replaced by a triangle.  Has anyone made use
of such G' for G a regular tree for any purpose (e.g. analysis on graphs)?
Is there a name associated with such things?
Alexander R. Pruss
Dept. of Philosophy
Univ. of Pittsburgh
	pruss+@pitt.edu

In my last mail, the correct definition for r(A,P) is 
r(A,P)=A+A^q+.....+A^(q^m-1)  Mod P 
where  q=2^n and m is the degree of P.
Sorry about that.
-Amr

Alexander R Pruss (pruss+@pitt.edu) wrote:
: Given an undirected graph G, let G' be the graph whose set of vertices are the 
: edges of G, and whose adjacency matrix is defined so that if e_1 and e_2 are 
: vertices of G' (i.e. edges of G), then e_1 and e_2 are adjacent if and only if 
: they have a vertex of G in common.
: Does this construction of G' have a name?
O.K., already two people have answered that G' is the line graph L(G) of G.
Thank you kindly!
By the way, does anyone know what geometric significance the line graph
of a complete graph on n points has?  If n=4, then the line graph is an
octahedron.  Is there a more general nice geometric meaning?
Alex Pruss.

My putative logic is: f is in L^2, and has an inverse g, 
presumably in L^2. Then g is in L^1. And we know that f(g) = I = g(f), 
which must hold even when in L^1. 
Is this true? 
If not, are there any tricks in deducing that f is invertible in L^1,
given that it is invertible in L^2?
Thanks, Lones
  .-.     .-.     .-.     .-.     .-.     .-.    
 / L \ O / N \ E / S \   / S \ M / I \ T / H \   
/     `-'     `-'     `-'     `-'     `-'     ` 
 Lones Smith, Economics Department, M.I.T., E52-252C, Cambridge MA 02139
 (617)-253-0914 (work)  253-6915 (fax)   lones@lones.mit.edu 

In number theory one is typically interested in continued fractions
of the form a_0 + 1/(a_1 + 1/(a_2 + 1/...)).  I am looking for an
account of the basic theory of continued fractions of the more general
form a_0 + b_1/(a_1 + b_2/(a_2 + ...)).  There's lots of literature on
continued fractions of this form when the a's and b's are functions,
but I'm interested in the case where the a's and b's are integers.
I have found it surprisingly hard to find a good discussion of this.
Any help would be greatly appreciated.
-- 
Tim Chow       tchow@umich.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949

pruss+@pitt.edu (Alexander R Pruss) writes:
>Given an undirected graph G, let G' be the graph whose set of vertices are the 
>edges of G, and whose adjacency matrix is defined so that if e_1 and e_2 are 
>vertices of G' (i.e. edges of G), then e_1 and e_2 are adjacent if and only if 
>they have a vertex of G in common.
>Does this construction of G' have a name?
	This is just the linegraph of the graph...
	Linegraphs have been pretty heavily studied and there 
	are a variety of characterizations and known properties. Any
	elementary graph theory text will define linegraph by about 
	page 5.
	Here's a linegraph problem for you... 
	Start with a tree T, and form the numerical sequence
		|L(T)|, |L(L(T))|, |L(L(L(T)))|... 
	(where |L(T)| is the number of vertices in the linegraph of T)
	Can two non-ismorphic trees give you the same sequence of numbers..
	Cheers
	Gordon
-- 
Gordon Royle ---- gordon@cs.uwa.edu.au
Visit http://www.cs.uwa.edu.au/~gordon
--

This posting is a completion to the posting of Gerry Myerson
(gerry@mpce.mq.edu.au) with two purposes:
1.  correction of my former statement concerning the proof of Curtiss    and
2.  explaining my strong interest in the proof for Erdoes' and Grahams
    theorem.
Gerry mentioned a "former posting" saying that the proof of Curtiss is wrong.
This posting was submitted to this newsgroup by me in July. In the meantime I
found out that the proof is definitely right (but obviously difficult to
understand). (Sorry for that.)
But this mistake (of me) led to a generalization of Curtiss' theorem that has
not been published elsewhere yet: The best underapproximation of a rational
a/b is given by the greedy algorithm at least for the case b = -1 mod a. (In
the case a=b=1 this is Curtiss' theorem; in the case a=1 this is a statement
proved by Erdoes in 1950.)
I was also able to prove that the methods Curtiss invented work if and only
if b = -1 mod a. But there seem to be a lot more rationals for which the
underapproximation is given by the greedy algorithm. While trying to
characterize these rationals I came across the theorem of Erdoes and Graham,
which I mentioned in my first posting on this subject.
Any hint on the proof of this theorem would be helpful.
Stefan Bartels

In article <55547r$e8t@usenet.srv.cis.pitt.edu>, pruss+@pitt.edu
(Alexander R Pruss) wrote:
> Given an undirected graph G, let G' be the graph whose set of vertices
are the 
> edges of G, and whose adjacency matrix is defined so that if e_1 and e_2 are 
> vertices of G' (i.e. edges of G), then e_1 and e_2 are adjacent if and
only if 
> they have a vertex of G in common.
> 
> Does this construction of G' have a name?
It's called the line graph of G, commonly denoted L(G).
michael
brundage@ipac.caltech.edu

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