# 数学概论 Introductory Mathematics ECON10061

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An elliptical platform is designed to have a surface area of 725 square feet. To give the desired form, the major axis must be $1 \frac{1}{2}$ times as long as the minor axis. Determine the dimensions of the major and minor axes.
Let $x=$ minor axis
\begin{aligned} &b=0.5 x \ &a=0.75 x \end{aligned}
$$1.5 x=\text { major axis }$$

\begin{aligned}
725 \mathrm{sq} \mathrm{} \mathrm{ft} &=3.1416(0.75 x)(0.5 x) \
725 \mathrm{sq} \mathrm{} \mathrm{ft} &=1.1781 x^{2} \
x^{2} & \approx 615.3977 \mathrm{sq} \mathrm{ft} \
x &=24.807 \mathrm{ft} \
\text { Minor axis } & \approx 24.8 \mathrm{ft} \text { Ans } \
\text { Major axis } & \approx 1.5(24.807 \mathrm{ft})=37.2 \mathrm{ft} \text { Ans }
\end{aligned}

## ECON10061 COURSE NOTES ：

Find the area of the outside circle.
$A_{B}=0.7854 d^{2}$
$A_{B} \Rightarrow(0.7854)(4.00 \mathrm{in})^{2} \approx 12.5664 \mathrm{sq}$ in
Find the area of the hole.
$A_{B} \approx(0.7854)(3.40 \mathrm{in})^{2} \approx 9.079224 \mathrm{sq}$ in
Find the cross-sectional area.
$12.5664 \mathrm{sq}$ in $-9.079224 \mathrm{sq}$ in $\approx 3.487176 \mathrm{sq}$ in
Find the volume.
$V=3.487176 \mathrm{sq}$ in $\times 50.0$ in $\approx 174 \mathrm{cu}$ in Ans

# 数学概论| Introductory Mathematics 代写 MT1001

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Proof:
For simplicity let us consider the Lindbladian $\mathcal{L}$ associated with an element $r=\sum_{g \in \mathcal{G}} c_{g} U_{g} \in \mathcal{A}$ such that $|r|{2}:=\sum{g \in \mathcal{G}}\left|c_{g}\right||g|^{2}<\infty$. Here $\mathcal{L}$ takes the form

Denoting these two bounded derivations $\left[r_{k}^{}, .\right]$ and $\left[., r_{k}\right]$ on $\mathcal{A}$ by $\delta_{k}^{\dagger}$ and $\delta_{k}$ respectively, $\mathcal{L}(x)=\frac{1}{2} \sum_{k \in \mathbb{Z}^{d}} \delta_{k}^{\dagger}(x) r_{k}+r_{k}^{} \delta_{k}(x)$.
In order to prove (i), for $x \in \mathcal{C}^{1}(\mathcal{A})$, let us estimate the norm of $\mathcal{L}(x)$ :
$$|\mathcal{L}(x)| \leq \frac{1}{2} \sum_{k \in \mathbb{Z}^{d}}\left|\delta_{k}^{\dagger}(x) r_{k}+r_{k}^{*} \delta_{k}(x)\right|$$

## MT1001 COURSE NOTES ：

Summing over $\alpha^{\prime}$, it follows that
$$\left|\left(x_{n}\right)\right|_{1} \leq\left|(\lambda-\Gamma)^{-1}\right||y|_{1}<\infty$$ and hence $x_{n} \in \mathcal{C}^{1}(\mathcal{A})$. Now setting $y_{n}=(\mathcal{L}-\lambda)\left(x_{n}\right)$, we have $$\left|y_{n}-y\right|=\left|\left(\mathcal{L}-\mathcal{L}^{(n)}\right) x_{n}\right|=\sum_{|k|>n} \mathcal{L}{k}\left(x{n}\right)$$