这是一份leeds利兹大学MATH5743M01作业代写的成功案例
An iterative descent algorithm for solving
$$
C^{}=\min {C} \sum{k=1}^{K} N_{k} \sum_{C(i)=k}\left|x_{i}-\bar{x}{k}\right|^{2} $$ can be obtained by noting that for any set of observations $S$ $$ \bar{x}{S}=\operatorname{argmin}{m} \sum{i \in S}\left|x_{i}-m\right|^{2} .
$$
Hence we can obtain $C^{}$ by solving the enlarged optimization problem
MATH5743M01 COURSE NOTES ：
$\rho>0$ is the (constant) density of the fluid and $\Omega$ is some element of $C^{2}(I, \mathbb{R}) \cap$ $C(\bar{I}, \mathbb{R})$. Here $I:=\left(R_{1}, R_{2}\right)$. Note that
$$
v_{0}=\nabla \times\left(0,0, \psi_{0}\right)
$$
where
$$
\psi_{0}(x, y, z):=-\int_{R_{1}}^{\sqrt{x^{2}+y^{2}}} r Q(r) d r
$$
for all $(x, y, z) \in U \times \mathbb{R}$ and that the vorticity $\omega_{0}:=\nabla \times \nu_{0}$ is given by
$$
\omega_{0}(x, y, z)=\left(0,0,-\left(\Delta \psi_{0}\right)(x, y, z)\right)=\left(0,0, \omega_{0}\left(\sqrt{x^{2}+y^{2}}\right)\right)
$$
for all $(x, y, z) \in U \times \mathbb{R}$. Here $\omega_{b e}: I \rightarrow \mathbb{R}$ is defined by
$$
\omega_{Q}(r):=r \Omega^{\prime}(r)+2 \Omega(r)
$$
for all $r \in I$. In the vorticity formulation the governing equation for reduced small axial variations of such a $\omega_{0}$ of the form
$$
(0,0, \omega(r, z) \exp (i m \varphi))
$$
