Math Forum / Mathematics / Research / November 2007

A Limit-Question
Consider the sets of positive even numbers E, of prime numbers P, of
Ulam's lucky numbers L, and of tetration numbers T
E = { 2, 4, 6, ... }

This is a minor unsolved quaternion problem.
Tian, Yongge, "The equations ax - xb = c, ax - x*b = c, and x*ax = b
in quaternions," 2004, Southeast
Asian Bulletin of Mathematics 28, 343-362.

Theorem: Every partially ordered set can be extended to a total order.
The easy proof I've found of this uses the axiom of choice (well,
specifically, Zorn's Lemma.)
Is it equivalent to the axiom of choice?

Define the (integer) tetration (="powertower of integer height") as
  Tb(x,h) = b^b^b^...^b^x            with h-fold repetion of base-parameter b
for height h=0
  Tb(x,0) = x

For f(x)=1/Abs(x)^a, the integral from -1 to 1 diverges for a>=1 and
converges for a<1. Observe that for a<1 the derivative become infinite
at 0. Is this observation part of some more general theorem? E.g. if
at a single point f becomes infinite and its derivative is also

I have a weighted undirected graph. I need to partition it in two
disjoint parts (blue and red nodes) so that the sum of the weights of
blue-red edges is minimal.
Does this problem have been studied in optimization and graph theory?

I would like to re_present the following question for
a possible discussion(I had posted it some yearS ago)
Thanks

Could some kind soul please point me to a proof of the result
that the only complete archimedean fields are R and C.
By an archimedean field I mean a field with an archimedean valuation.
By complete I mean complete with respect to this valuation.

Some results which appear to be in the space half between number theory and
statistics.
Trends in Euler's phi function.
Enjoy:

I am wondering if any of the following concepts has a standard name in
the graph-theoretic literature.
- a graph G all whose edges are part of some triangle.
- a graph constructed this way: its vertices are the triangles in G.

My earlier question about up-dated accounts of Emmy Noether's own
version(s) of the Noether conservation theorem(s) gets a thorough
answer (with appreciative historical remarks on Lie and Noether) in
Peter Olver's book Applications of Lie Groups to Differential

For the generalised hypergeometric function denoted by
F_pq(a_1,a_2...a_p;b_1,b_2...b_q;z), under what condition(s) does the
function converge for z=1? Is this condition some sort of region of
convergence in the (p+q)-dimensional space spanned by the p a's and

It is possible to prove that:
-i = h( e^pi/2) where h(z)  is a power tower function , infinite
times.
Proof:

Is there a way to find the (real valued) nxn matrix A whose k'th
compound
matrix is given? I am interested in the regime where k lies
between 1 and n/2.