To perform simple cWrite the relevant mathematical arguments in a precise and lucid fashion.

这是一份Bath巴斯大学MA50251作业代写的成功案

For a nonnegative integer $m$, we denote by $C^{m}(\mathcal{O})$ the set of all $m$-times continuously differentiable real-valued functions in $\mathcal{O}$, and by $C_{0}^{m}(\mathcal{O})$ the subspace of $C^{m}(\mathcal{O})$ consisting of those functions which have compact supports in O. For $u \in C^{m}(\mathcal{O})$ and $1 \leq p<\infty$, we define
$$
|u|_{m, p}=\left(\int_{\mathcal{O}} \sum_{|\alpha| \leq m}\left|D^{\alpha} u(x)\right|^{p} d x\right)^{1 / p}
$$
and for $p=2, u, v \in C^{m}(\mathcal{O})$,
$$
\langle u, v\rangle_{m, 2}=\int_{\mathcal{O}} \sum_{|\alpha| \leq m} D^{\alpha} u(x) \cdot D^{\alpha} v(x) d x .
$$

MA50251 COURSE NOTES ：

We start our discussion by considering the following linear system on the $n$-dimensional Euclidean space $\mathbf{R}^{n}$ :
$$
\frac{d X_{t}\left(x_{0}\right)}{d t}=A X_{t}\left(x_{0}\right), \quad X_{0}\left(x_{0}\right)=x_{0} \in \mathbf{R}^{n},
$$
where $A$ is some $n \times n$ constant matrix. Clearly, the equation has a unique solution which is given by
$$
X_{t}\left(x_{0}\right)=e^{A t} x_{0}
$$