Math Forum / Mathematics / Research / April 2009

Let D be a smooth bounded convex domain in R^2, and
u be strongly convex (its Hessian matrix is positive definite) smooth
function defiend in D. Suppose
\sum u_{ij}x_i x_j <[d(x)]^{-3/2}

Does anybody have a reference to the non-trivial half of following
theorem, which I have seen quoted but never proved: "A metric space X
is compact if and only if every metric space homeomorphic to X is
complete"?

I am looking for research work and/or published algorithms and
software for the following problem:
  Given a set of intervals, I = I[0} ... I[n-1}, I[k] = [lower bound,
upper bound] (closed interval)

Let M be a 3-manifold, possibly with boundary. Consider Morse
functions on M to R or S^1. Any two Morse functions can be described
up to isotopy preserving the levels of critical points and
reparametrization of the target space by specifying the order in which

Someone I met has posted pictures of an alleged Chinese neutron bomb test
at:
ftp://ftp.aaone.dlinkddns.com/pub/Pictures%20and%20Videos/China_neutron_bomb_test/
A preliminary analysis given here:

Cayley's Theorem says that any group can be embedded into a symmetric
group. I have a "dual" question:
Which groups arise as quotients of symmetric groups?
More precisely: For any set X let Sym(X) be the group of bijections

Good day,
I am looking for a good reference about differential rings in the
following sence:
d:R->R is a differential if

The ralations between genus of and  the polynormial of order n? if the
order 3 means genus less than 2?

Consider a power series \sum a_n x^n that is convergent for all real
x, thus defining a function f: R \to R.
Are there criteria to decide whether f is bounded (which e.g. is the
case for a_n = (-1)^n/(2n)!) ?

Given two subsets A, B of a subgroup G, write AB for {xy : x in A, y
in B}.
Let H_1, H_2 be subgroups of a group G.
Suppose first (for the sake of exposition) that we are dealing with an

I have come across a transcendental equation which can be solved via
functional iteration, and appears to have a unique fixed point. The
technical analysis of this function is, however, rather complicated,
and so, I am wondering if anyone has come across it before.

 it is well known, that for A a noetherian local ring with residue
field k the following statements are equivalent for a module M *finitely
generated over A*
i) M is a free A-Module

Dehn's lemma can be phrased as saying that if we have an S^1 K in S^3
which bounds an immersion f:D^2->S^3 s.t. f^-1(K)=\partial D^2, then K
bounds an embedded D^2.
Is the analogous statement true for higher dimensions (in the smooth

I have the following question: I believe that for any natural D\ge3
\frac{2\times3^{D}(2\times3^{D}-3)}{2D+1} and \frac{(2\times3^D-3)}{D}
are not integer expressions simultaneously.
How can I prove it? I have tried to prove it with no luck so far. I am

Hello newsgroup!
Assume I have a sequence of N points on the sphere, say given by
(x,y,z) coordinates.
I want to find a Möbius transformation that "centers" these points,