Math Forum / Mathematics / Research / February 2007

if f and g are two smooth functions from a smooth manifold M^n->S^p,
and we have the condition
that |f-g|<2 then f and g are smoothly homotopic. I think I understand
that in this case f and g

There's a theorem one often runs across in physics, but I'm not sure of
the precise statement of it. The theorem says something along the lines
of,
Let M be a Calabi-Yau n-fold such that the holonomy group fills out

Consider the following conditions on a commutative ring A.
(1) A is the product of finitely many quotients of PIDs.
(2) The ideals of A are principal.
(3) The image of any A-bilinear form is an ideal.

Fourier transform of a unilateral function f(x), e.g. f(x)= 0 for x<0,
yields a complex-valued F(y) with symmetrical real part and
antisymmetrical imaginary part. Also vice versa. For physical
implications cf.

What is a homotopy coherent functor? A simple reference will do.

Let a be an irrational number. Is the sequence ap, p prime,
dense, or even uniformally distributed, mod 1?

i am quite curious to know the existing results on relation between GL_2 and the group generated by elementary matrices for an arbitrary ring R.
thanks

I am trying to prove that for bounded X and Y,
E{Y E[X| F ]}= E{X E[Y| F ]}
where F is a sigma-field
Shall I try to prove the statement with X and Y indicator variables first? Then, combinations of indicator variables? Then make the transition to variables X and Y by a convergence ...

I discovered the following identity:
Let s(n,k) be the Stirling Numbers of the first kind,
and |s(n,k)| the unsigned Stirling Numbers of the first kind.
Then

let d be an integer divisor of the integer n. We say that d is an exact divisor of n if  gcd(d, n/d)  = 1 i.e. if d and n/d are relatively prime. My question is related to the characteristic function of the exact divisors F(d,n) such that:
F(d,n) = 1 if d exact divisor of n, 0 ...

It's easy to calculate the Fourier transform of
1/[ exp(x) + exp(-x) ]
--- in fact sech is its own Fourier transform (ignoring 2 pi-like
multiples).  I'm wondering whether anyone knows if there's an easy way

Let X be a surface such that the rank of the neron severi group of X is equal
to dim(H^2(Z,X)).
This implies that p_g(X):=H^2(O_x,X) is 0.
Can anybody explain why this implication holds?

Can someone please point me to some notes/links on surface flows which
are not surface diffusion? I came across some work by Cahn-Elliott and
Novick and by Cahn-Taylor on some mean curvature flow for surface
diffusion.

Hi,
I'm looking for information on something like an inner convex hull computation problem.
I want to compute the largest convex set in a subset S of R^d. The common convex hull problem determines the smallest convex set in R^d containing a subset S of R^d.
So what about the ...

Lets say one has a set of n points that form the vertices of a convex
polygon.
Is there any analytical way to find, or approximate, the interior point
of the polygon which maximizes the minimal distance to the vertices?