User binzhou xia - MathOverflow

software for numerical constraint satisfaction problems

2012-07-21T10:42:59Z

Let $m$ be an even integer greater than $8$. Is there any software I can use to determine for some small $m$ whether the following constraints on $t_0,\ldots,t_{m-1}$ and $w$ have solutions?

$-\pi\leq t_r\leq\pi,\ r=0,\ldots,m-1,$

$0\leq w\leq\frac{1}{m},$

$\sum\limits_{j=1}^{2r-1}\cos(t_j+t_{2r-j})+\sum\limits_{j=2r+1}^{m-1}\cos(t_j+t_{2r+m-j})=2w\cos t_{2r},\ r=1,\ldots,\frac{m}{2}-1,$

$\sum\limits_{j=1}^{2r-1}\sin(t_j+t_{2r-j})+\sum\limits_{j=2r+1}^{m-1}\sin(t_j+t_{2r+m-j})=2w\sin t_{2r},\ r=1,\ldots,\frac{m}{2}-1,$

$t_r+t_{m-r}=0,\ r=1,\ldots,\frac{m}{2},$

$\cos(t_r+t_{\frac{m}{2}+r})=\cos(t_{2r}+rt_0),\ r=1,\ldots,\frac{m}{2}-1,$

$\sin(t_r+t_{\frac{m}{2}+r})=\sin(t_{2r}+rt_0),\ r=1,\ldots,\frac{m}{2}-1,$

$\cos(\frac{mt_0}{2})=1.$

bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2

2011-09-20T07:07:59Z

Are there any results about the bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2? More precisely, let $k$ be a positive integer and $m=4/k$. Write \begin{equation*} \sum\limits_{n=1}^{\infty}a_nq^n=\eta(k\tau)^2\eta(2k\tau)^{1+m}\eta(4k\tau)^{3-3m}\eta(8k\tau)^{2m-2}, \end{equation*} where $\eta(\tau)=q^{1/24}\prod\limits_{n=0}^{\infty}(1-q^n)$ is the Dedekind eta function with $q=e^{2\pi i\tau}$ and $Im \tau>0$. I want to know the bound for $a_{k^2+k+1}$.

A family of systems of multivariate quadratic polynomial equations

2011-07-15T07:53:17Z

Let $n$ be any integer divisible by and greater than 8. I have run into the question that whether there exist complex numbers $z_1,z_2,\ldots,z_{n-1}$ with modules $1$ satisfying the following quadratic equations, $\sum\limits_{r=1}^{2t-1}(-1)^rz_rz_{2t-r}+\sum\limits_{r=2t+1}^{n-1}(-1)^rz_rz_{n+2t-r}=\frac{2z_{2t}}{(n^2+n+1)^{\frac{1}{2}}},t=1,2,\ldots,\frac{n}{2}-1,$ $z_tz_{\frac{n}{2}+t}=z_{2t},t=1,2,\ldots,\frac{n}{2}-1,$ and $z_tz_{n-t}=(-1)^t,t=1,2,\ldots,\frac{n}{2}-1.$

What I seek for is a method which can deal with general $n$. So the Groebner basis theory seems hard to apply to this problem. Also, I wonder if translating the equations as well as the module constraints to quadratic equations of real variables by setting $z_j=x_j+iy_j$ is helpful.

Comment by Binzhou Xia

2012-07-22T13:55:37Z

Is the algorithm guaranteed to be convenrgent in theory? I mean does there exist some case such that the algorithm can not tell no matter what precision I make?

Comment by Binzhou Xia

2012-07-22T08:45:52Z

Yes, actually I'm considering how small can $w$ be.