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# 数学代写|黎曼曲面代写Riemann surface代考|MAT565 This is equal to zero for odd values of k, but nonzero for even values.

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## 数学代写|黎曼曲面代写Riemann surface代考|This is equal to zero for odd values of k, but nonzero for even values.

This is equal to zero for odd values of $k$, but nonzero for even values. If $b_{i_{r_r i_r}}$ is chosen generically with respect to $a$, then the term $f_0$ does not vanish. Therefore the asymptotic expansion is nonzero. This completes the proof of the theorem.

Remark: It is evident from this proposition that for $S \ell(2)$, the asymptotic expansion can be obtained by continuing along a geodesic path. Thus the $S \ell(2)$ case is fully covered by the known method of matched asymptotic expansions. The points $s_1, \ldots, s_n$ encountered above are the “turning points”.

As a corollary of 11.6 in the $S \ell(2)$ case, we obtain a description of the convex hull of the exponents $\mathcal{H}$ (defined in $\S 2$ ). Fix a generic choice of one-form $a$. Consider the resulting function $g: Z \rightarrow \mathbf{C}$ as a branched covering. Given points $P$ and $Q$ in $Z$, there is a path $\gamma$ between them with the following properties:

$g_*(\gamma)$ is a union of line segments between points $s_0=g(P), s_1, \ldots, s_r=g(Q)$; the points $s_i$ are the images of branch points; and the path $\gamma$ changes branches at each corner between segments. Such a path $\gamma$ may be obtained by choosing a geodesic with respect to the pull-back of the euclidean metric on $\mathbf{C}$.

## 数学代写|黎曼曲面代写Riemann surface代考|Example: some S/(3) connections

The special behaviour shown by the above example lies in marked contrast to the case of matrices of higher rank. When there is more than one function $g$, it is no longer possible to choose one path which will exhibit the expansion. We will use the criterion of Proposition 11.5 to construct a $3 \times 3$ example where $\xi<\xi_{\text {path }}$

Let $A=\left(\begin{array}{ccc}a_1 & 0 & 0 \ 0 & a_2 & 0 \ 0 & 0 & -a_1-a_2\end{array}\right)$. Let $U$ be a neighborhood in the Riemann surface $S$, with a local coordinate $z: U \rightarrow \mathbf{C}$ mapping $U$ isomorphically to the disc of radius 5 at the origin. By choosing $a_1$ and $a_2$ appropriately, and adjusting the local coordinate, we may assume that
$$a_1=i C_1 d z$$

$$a_2=2(z-i) d z+u(z) d z$$
where $C_1$ is a large positive real constant, and $|u(z)| \leq c_2$ where $c_2>0$ is small. Note that $a_2$ has a single zero in $U$, located in $|z-i| \leq c_2$.

## 数学代写|黎曼曲面代写Riemann surface代考|This is equal to zero for odd values of $k$, but nonzero for even values.

$g_*(\gamma)$ 是点之间线段的并集 $s_0=g(P), s_1, \ldots, s_r=g(Q)$; 积分 $s_i$ 是分支点的图像; 和路径 $\gamma$ 更改线段之间每 个角的分支。这样的路径 $\gamma$ 可以通过选择相对于欧几里德度量的回调的测地线来获得C.

## MATLAB代写

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