2022

Problem 1

Consider the following multiple conditions on $x,y,z\in \mathbb{R}$.

$$\{\begin{array}{l}0<z-xy<1\\ 0<z-(x+y{)}^2<-xy\end{array}$$

Let $\Omega$ be the set of points $(x,y)$ for which at least one $z$ exists satisfying the above conditions. Note that the set $\Omega$ can be seen in the three-dimensional Cartesian coordinate system as the orthogonal projection of points $(x,y,z)$ satisfying the above conditions onto the $xy$-plane. Answer the following questions.

1. Find the inequalities on $x$ and $y$ representing $\Omega$.

2. Draw a figure of $\Omega$ in the $xy$-plane. If the boundary of $\Omega$ intersects with the $x$-axis or the $y$-axis, write down the coordinates at each intersection.

3. The curved segments of the boundary of $\Omega$ correspond to the linear transformation of arcs of the unit circle with a matrix $\mathbf{X}$. Find one such $\mathbf{X}$. Note that the point $(1,0)$ on the unit circle must be transformed to a point where the curvature is maximized in the curved segments.

4. Calculate the determinant of $\mathbf{X}$ found in (3).

5. Calculate the area of the set $\Omega$. Note that the absolute value of the determinant of a matrix is the area scale factor of the transformation with that matrix.

---

考虑以下对 $x,y,z\in \mathbb{R}$ 的多个条件。

$$\{\begin{array}{l}0<z-xy<1\\ 0<z-(x+y{)}^2<-xy\end{array}$$

设 $\Omega$ 为至少存在一个 $z$ 满足上述条件的点 $(x,y)$ 的集合。注意，集合 $\Omega$ 可以看作是在三维笛卡尔坐标系中满足上述条件的点 $(x,y,z)$ 投影到 $xy$ 平面上的正交投影。回答以下问题。

1. 找出表示 $\Omega$ 的 $x$ 和 $y$ 的不等式。

2. 在 $xy$ 平面中绘制 $\Omega$ 的图形。如果 $\Omega$ 的边界与 $x$ 轴或 $y$ 轴相交，写下每个交点的坐标。

3. $\Omega$ 边界的曲线段对应于单位圆的弧线在矩阵 $\mathbf{X}$ 下的线性变换。找出一个这样的 $\mathbf{X}$。注意，单位圆上的点 $(1,0)$ 必须被变换到曲线段中曲率最大的点。

4. 计算在（3）中找到的 $\mathbf{X}$ 的行列式。

5. 计算集合 $\Omega$ 的面积。注意，矩阵行列式的绝对值是该矩阵变换的面积缩放因子。

---

Problem 2

Consider the following integral $I_n(\alpha )$ for $\alpha \ge 1$ and $n>0$.

$$I_n(\alpha )={\int }_{\frac{1}{n}}^n\frac{f(\alpha x)-f(x)}{x}dx$$

Assume that a real-valued function $f(x)$ is continuous and differentiable on $x\ge 0$, its derivative is continuous, and ${\lim }_{x\to \infty }f(x)=0$. Answer the following questions.

1. Define $J_n(\alpha )=\frac{d{I}_n(\alpha )}{d\alpha }$. Show that $J_n(\alpha )=\frac{1}{\alpha }\left(f(\alpha n)-f\left(\frac{\alpha }{n}\right)\right)$.

   You can use the fact that the integration and the differentiation commute in this context.

2. Define $I(\alpha )={\lim }_{n\to \infty }{I}_n(\alpha )$. Show that ${\lim }_{n\to \infty }{J}_n(\beta )$ exists for any $\beta \in [1,\alpha ]$ and it uniformly converges on $[1,\alpha ]$, and show that

   $$I(\alpha )={\int }_1^{\alpha }\left(\underset{n\to \infty }{\lim }{J}_n(\beta )\right)d\beta .$$

3. Obtain $I(\alpha)$. 4. Calculate the following integral. Note that $p > q > 0$.

\int_{0}^{\infty} \frac{e^{-px} \cos(px) - e^{-qx} \cos(qx)}{x} , dx

--- 考虑以下积分 $I_n(\alpha)$ 其中 $\alpha \geq 1$ 且 $n > 0$。

I_n(\alpha) = \int_{\frac{1}{n}}^{n} \frac{f(\alpha x) - f(x)}{x} , dx

假设一个实值函数 $f(x)$ 在 $x \geq 0$ 上是连续且可微的，其导数是连续的，并且 $\lim_{x \to \infty} f(x) = 0$。回答以下问题。 1. 定义 $J_n(\alpha) = \frac{d I_n(\alpha)}{d \alpha}$。证明 $J_n(\alpha) = \frac{1}{\alpha} \left( f(\alpha n) - f\left(\frac{\alpha}{n}\right) \right)$。 在这种情况下，你可以使用积分和微分可交换的事实。 2. 定义 $I(\alpha) = \lim_{n \to \infty} I_n(\alpha)$。证明 $\lim_{n \to \infty} J_n(\beta)$ 对于任何 $\beta \in [1, \alpha]$ 存在并且在 $[1, \alpha]$ 上一致收敛，并证明

I(\alpha) = \int_{1}^{\alpha} \left( \lim_{n \to \infty} J_n(\beta) \right) d\beta.

3. 求 $I(\alpha)$。 4. 计算以下积分。注意 $p > q > 0$。

\int_{0}^{\infty} \frac{e^{-px} \cos(px) - e^{-qx} \cos(qx)}{x} , dx

--- ## Problem 3 Consider a region $R$ defined by $0 < x < 1$ and $0 < y < 1$ in the $xy$-plane. We randomly select a point on $R$ and refer to the selected point as $A$. We assume that $A$ is uniformly distributed on $R$. Let $AB$ be a perpendicular line from $A$ to the $y$-axis and $AC$ be a perpendicular line from $A$ to the $x$-axis as shown in the figure. We call rectangle $OCAB$ as "the rectangle of $A$", where $O$ denotes the origin. Let $S$ be a random variable representing the area of the rectangle of $A$. Answer the following questions. 1. Calculate the expectation value of $S$. 2. Calculate the probability that $S \leq r$ holds, where $0 < r < 1$. 3. Calculate the probability density function of $S$. Again consider the region $R$. Let $n$ be a positive integer. We select $n$ points on $R$ and refer to the selected points as $A_1, A_2, \ldots, A_n$. We assume that each of the points is uniformly distributed on $R$, and $A_i$ and $A_j$ for $i \neq j$ are selected independently. Answer the following question. 4. Let $S_i$ be a random variable representing the area of the rectangle of $A_i$. Let $Z$ be a random variable which is the minimum of $S_1, S_2, \ldots, S_n$. Calculate the probability density function of $Z$. --- ![[Pasted image 20240627144917.png]] --- 考虑一个区域 $R$，定义为 $0 < x < 1$ 和 $0 < y < 1$ 在 $xy$ 平面上。我们随机选择 $R$ 上的一个点，并将其称为 $A$。我们假设 $A$ 在 $R$ 上均匀分布。令 $AB$ 为从 $A$ 到 $y$ 轴的垂线，$AC$ 为从 $A$ 到 $x$ 轴的垂线，如图所示。我们称矩形 $OCAB$ 为“$A$ 的矩形”，其中 $O$ 表示原点。令 $S$ 为表示 $A$ 的矩形面积的随机变量。回答以下问题。 1. 计算 $S$ 的期望值。 2. 计算 $S \leq r$ 的概率，其中 $0 < r < 1$。 3. 计算 $S$ 的概率密度函数。 再次考虑区域 $R$。令 $n$ 为正整数。我们在 $R$ 上选择 $n$ 个点，并将选定的点称为 $A_1, A_2, \ldots, A_n$。我们假设每个点在 $R$ 上均匀分布，并且 $A_i$ 和 $A_j$ 对于 $i \neq j$ 是独立选择的。回答以下问题。 4. 令 $S_i$ 为表示 $A_i$ 的矩形面积的随机变量。令 $Z$ 为 $S_1, S_2, \ldots, S_n$ 的最小值的随机变量。计算 $Z$ 的概率密度函数。