sci.math.research Google Group

Nine papers published by Geometry & Topology Publications

2010-03-14T02:24:30Z

Seven papers have been published by Algebraic & Geometric Topology
(1) Algebraic & Geometric Topology 10 (2010) 525-530
Faithfulness of a functor of Quillen
by William G Dwyer, Andrei Radulescu-Banu and Sebastian Thomas
URL: [link]
DOI: 10.2140/agt.2010.10.525

Re: decomposing rationals

2010-03-12T22:06:54Z

If a and b are rational,
this becomes an integer linear programming problem in two variables.
Such things are solvable in polynomial time.
I don't remember details.
I think that you can solve the linear relaxation
and plug away with Gomory fractional cuts.
Selecting the approximations could be tricky.

Re: minimum height of a 0-1 simplex

2010-03-12T19:13:15Z

The above is roughly 1/n, but there are worse examples:
e_j for j in 1..k
k
SUM e_j + e_L for L in k+1..n
j=1
k n
The vertices satisfy SUM x_j + (1-k) SUM x_j = 1
j=1 j=k+1
For k=2n/3, the distance to zero is roughly sqrt(27/(4n**3)).

Call for Papers Reminder (extended): The World Congress on Engineering WCE 2010

2010-03-12T10:42:22Z

CFP: The World Congress on Engineering WCE 2010
WCE 2010: London, U.K., 30 June - 2 July, 2010
[link]
Draft Paper Submission Deadline (extended): 18 March, 2010
The WCE 2010 is organized by International Association of Engineers
(IAENG), a non-profit international association for the engineers and

Re: minimum height of a 0-1 simplex

2010-03-10T19:48:09Z

That is what I get, also.
The base satisfies the linear equation
n-1
SUM x[j] + (2-n)*x[n] = 1
j=1
Lower bound, anyone?

Re: minimum height of a 0-1 simplex

2010-03-10T12:37:27Z

I'm not sure what this means,
but I think that it is not true even in three dimension.
that a minimal-height-defining line segment of a
tetrahedron lies within the tetrahedron.
Consider a tetrahedron with the following vertices:
(0, 0, 0), (3, 2, 0), (3, -2, 0) and (4, 0, 1).
Its minimal-height-defining segment is (4, 0, 1)_(4, 0, 0).

Re: minimum height of a 0-1 simplex

2010-03-10T12:37:27Z

Yes, but it seems I made a mistake there, as the height-defining
segment
does not always lie inside the simplex.
However, it does so, if the simplex has the property that all angles
between
edges are at most \pi/2.
Now for a simplex with vertices in {0,1}^n this condition is
satisfied!
By symmetry, it is enough to check this at the vertex zero.

Re: minimum height of a 0-1 simplex

2010-03-10T12:37:29Z

no, is not.
For example, consider the simplex spanned by zero, e=e_1+...+e_n, e_1,
e_2,...,e_{n-1},
where e_1,...,e_n is the standard basis.
A calculation shows that the vertex at zero has height equal to 1/
sqrt(n-1+(n-2)^2)
which for n>3 is less that 1/sqrt(n).

Whitehead Groups of Short Exact Sequence

2010-03-10T12:37:28Z

Hello, all!
The Whitehead group of a group G, Wh(G), is a functor from the category
of (finitely presented?) groups to the category of Abelian groups. A
reference is _A Course in Simple Homotopy Theory_ by Cohen.
If I have a short exact sequence of finitely presented groups 1 -> K -> G
-> Q -> 1, and I take the Whitehead torsion of the sequence, forming Wh

Re: minimum height of a 0-1 simplex

2010-03-09T18:12:37Z

Excellent news. Thanks much.
I'm having trouble following the proof, though.
I'll get back to it when I'm not supposed to be working.
The crucial point that I didn't think of.
Once I can prove that both ends of the height-defining segment are in
n-cube,
the rest is easy.

Re: Do free abelian groups embed into connected semisimple complex lie groups?

2010-03-09T18:12:36Z

Any such Lie group contains a complex torus isomorphic to C^*.
This group contains a subgroup isomorphic to the additive group R
of reals. Now let n be a natural number and choose v_1,...,v_n in R
which are linearly independent over Q.
Then the group Z v_1 + ... + Z v_n is free abelian of rank n and

Re: minimum height of a 0-1 simplex

2010-03-09T16:26:40Z

The lower bound is 1/sqrt(n).
Proof as follows. Let e_1,...,e_n be the standard basis of R^n.
Let s be a simplex of dimension n in R^n.
The minimal height is taken at a vertex v_1 which lies opposite to a
face F of maximal area.
(This is so since the volume of the simplex is height times area(F)
times dimension factor.)

minimum height of a 0-1 simplex

2010-03-08T00:16:08Z

What is the minimum height of a full dimensional n-simplex whose
vertices are members {0, 1}**n ?
Equivalently, what is the minimum non-zero distance to the origin from
a hyperplane defined by members of {0, 1}**n ?
The answer is at most 1/sqrt(n) .
The corner simplex provides an example.
A smaller answer might be possible if the simplex is oblique enough

Re: delta function as a continuous linear map?

2010-03-06T19:04:28Z

Oh now I see. Thanks G.A and Dan for pointing out the caveat in my
argument.

Ten papers published by GT Publications

2010-03-04T12:35:16Z

Five papers have been published by Algebraic & Geometric Topology
(1) Algebraic & Geometric Topology 10 (2010) 315-342
An involution on the K-theory of bimonoidal categories with anti-involution
by Birgit Richter
URL: [link]
DOI: 10.2140/agt.2010.10.315