User hu yi chen - MathOverflow

Weierstrass transform in complex variable

2012-08-28T03:54:57Z

The usual Weierstrass transform of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as: $$e^{D^2/2}f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-yD}f(x)e^{-y^2/2} dy=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x-y)e^{-y^2/2}dy$$ where $D=\frac{d}{dx}$.

Now if $D$ is with respect to complex variable $z$, how will the Weierstrass transform be different from the one above?

The explicit homomorphism to SL(2,$\mathbb{C}$)

2011-05-15T08:14:24Z

So I have already know that the fundamental group of figure-eight knot is given as: $G=(a,d\mid ada^{-1}da=dad^{-1}ad)$, where a,d are the two generators, then I have searched many books, but didn't find one explicit homomorphism $\rho$ that sends $G$ to $SL_{2}(\mathbb{C})$, the only thing they mentioned is that this map should send

$\rho$(a)=$\left(\begin{array}{rr} 1&1 \\ 0&1 \\ \end{array}\right)$ and $\rho$(d)=$\left(\begin{array}{rr} 1&0 \\ -\omega&1 \\ \end{array}\right)$

So I am wondering what will the explicit homomorphism be? Does this has anything to do with the character of the representation?

pull back of a connection 1-form on principal G-bundle

2011-04-29T05:53:54Z

Let $(P,\pi,M)$ be a principal $G$-bundle, and $A \in \Omega^1(P,\mathfrak g)$ is a connection 1-form, and $f:N \rightarrow M$ be a smooth map between manifolds. Then how to rigorously show that the pullback $f^* A$ is again a connection 1-form on $f^*P$

Horizontal distribution of principal G bundle

2011-04-25T03:10:08Z

If we consider $S^{2n-1}$->$\mathbb{CP}^{n-1}$ as a $S^1$ principal bundle, and given a connection 1-form as $C=\frac{1}{2\pi}\Sigma_i(x_i dx_i-y_i dy_i)$ (where $(x_1,y_1,...,x_{2n},y_{2n})$ are coordinates on $S^{2n-1}$). I think the horizontal distribution is the set of points $(x_1,y_1,...,x_{2n},y_{2n})$ and any real multiply of that point, which is just $\mathbb{RP}^{2n-2}$, is that right?

Handle slides homeomorphism

2011-04-20T07:28:54Z

For handle decomposition of surface, suppose I have a twisted 1-handle(twisted only once) adjacent to an isolated pair of linked handles, is the handle slide operation enough to move the previous one to become 3 twisted 1-handle(each is the same as the previous twisted 1-handle), which proves that these two are homeomorphic to each other?

Bergman kernel and Riemann mapping theorem

2011-04-19T17:05:25Z

Actually this is a problem I find on mathstackexchange, I find it interesting but I couldn't solve myself. So I'd post it here.

Suppose D is a simply connected domain whose boundary $\partial D$ is a smooth curve. Let $f(z)=f(z;\zeta)$ that maps $D$ onto the interior of unit circle such that $f(\zeta)$=0 and $f'(\zeta)$ is real and nonzero. How can we show that if the Bergman kernel $K(\zeta,z)=\frac{f'(z)^* f'(\zeta)}{\pi}$, use the fact that for any smooth complex function $f$ defined on domain $D$:

$\int \int_D \partial_{z^* } f(z,z^*)dxdy=\frac{1}{2i}\oint_{\partial_D} f(z,z^*)dz$

and the residue theorem to get that

$g(\zeta)=\int \int_D K(\zeta,z)g(z)dxdy$

$g$ is analytic on $D$

Whitehead doubles of any knots

2011-02-10T05:16:14Z

I was curious about the fact that the Whitehead doubles of all knots have Alexander polynomial equal to 1, which is the same as a unknot. How to prove this?

Why is it so hard to implement Haken's Algorithm for knot theory?

2011-01-25T19:00:55Z

Why is it so hard to implement Haken's Algorithm for recognizing whether a knot is unknotted? (Is there a computer implementation of this algorithm?)

A problem seeking for algorithm

2009-11-25T22:10:09Z

Now there are n independent projects, each project is composed by several steps, each step is labeled, there are k workers are going to work on these projects. Assume that each person can do only one step, and each step can only be done by a specific worker(it will be declared in input file). Each project has a order for its steps, for example, workers can operate the second step only if the first step has been done. when one person finished one step, then he can be assigned to another step immediately.

Task:Input the number of projects, and the number of steps of each project, and the worker corresponds to each step, and how much time it will take to finish the step. Try to find an algorithm to calculate the optimized time to finish all projects. Example inputs:

2 3(2 projects, 3 workers)

3 1 2 2 3 3 5(for project 1, there are 3 steps, the first will be done by worker 1 takes 2 hours, the second will be done by worker 2 by 3 hours, the third will be done by worker 3 by 5 hours)

2 2 3 3 2(for project 2, there are 2 steps, the first will be done by worker 2 by 3 hours, the second will be done by worker 3 by 2 hours)

Output 11 ( the minimized hours to finish the whole project)

Comment by Hu Yi Chen

2011-05-15T08:09:54Z

lift to the first fundamental group

Comment by Hu Yi Chen

2011-04-29T07:02:42Z

Which textbook are you referring to?

Comment by Hu Yi Chen

2011-04-22T06:06:17Z

you mean that all angles at intersection point are preserved?

Comment by Hu Yi Chen

2011-04-20T19:24:11Z

Any reference I can look at? There are too few materials online about this topic.