2016

Problem 1

The tribonacci numbers $\{T_n\}$ are defined for non-negative integers $n$ as follows.

$$\{\begin{array}{l}{T}_0=T_1=0\\ T_2=1\\ T_{n+3}=T_{n+2}+T_{n+1}+T_n(n\ge 0).\end{array}$$

Answer the following questions.

1. Find the matrix $\mathbf{A}$ that satisfies Eq. (1.1) for all non-negative integers $n$.

$$\left(\begin{array}{c}{T}_{n+3}\\ T_{n+2}\\ T_{n+1}\end{array}\right)=\mathbf{A}\left(\begin{array}{c}{T}_{n+2}\\ T_{n+1}\\ T_n\end{array}\right).\text{\hspace{1em}(1.1)}$$

2. Find the rank and the characteristic equation, i.e., the equation that eigenvalues satisfy, of the matrix $\mathbf{A}$.

3. Let ${\lambda }_1,{\lambda }_2,{\lambda }_3$ denote the eigenvalues of the matrix $\mathbf{A}$. Express an eigenvector corresponding to each of the eigenvalues using ${\lambda }_1,{\lambda }_2,{\lambda }_3$.

4. Prove that the matrix $\mathbf{A}$ has only one real number eigenvalue. Letting ${\lambda }_1$ correspond to this eigenvalue, prove that $1<{\lambda }_1<2$.

5. Prove that $T_n$ can be expressed as $T_n=c_1{\lambda }_1^n+c_2{\lambda }_2^n+c_3{\lambda }_3^n$ using constant complex numbers $c_1,c_2,c_3$. You do not need to find values of $c_1,c_2,c_3$ explicitly.

6. Prove ${\lim }_{n\to \infty }\frac{T_{n+1}}{T_n}={\lambda }_1$.

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三斐波那契数列 $\{T_n\}$ 定义为对非负整数 $n$ 如下。

$$\{\begin{array}{l}{T}_0=T_1=0\\ T_2=1\\ T_{n+3}=T_{n+2}+T_{n+1}+T_n(n\ge 0).\end{array}$$

回答以下问题。

1. 求满足方程 (1.1) 对所有非负整数 $n$ 成立的矩阵 $\mathbf{A}$。

$$\left(\begin{array}{c}{T}_{n+3}\\ T_{n+2}\\ T_{n+1}\end{array}\right)=\mathbf{A}\left(\begin{array}{c}{T}_{n+2}\\ T_{n+1}\\ T_n\end{array}\right).\text{\hspace{1em}(1.1)}$$

2. 求矩阵 $\mathbf{A}$ 的秩和特征方程，即特征值满足的方程。

3. 设 ${\lambda }_1,{\lambda }_2,{\lambda }_3$ 表示矩阵 $\mathbf{A}$ 的特征值。用 ${\lambda }_1,{\lambda }_2,{\lambda }_3$ 表示对应于每个特征值的特征向量。

4. 证明矩阵 $\mathbf{A}$ 只有一个实数特征值。令 ${\lambda }_1$ 对应于该特征值，证明 $1<{\lambda }_1<2$。

5. 证明 $T_n$ 可以表示为 $T_n=c_1{\lambda }_1^n+c_2{\lambda }_2^n+c_3{\lambda }_3^n$，其中 $c_1,c_2,c_3$ 为常量复数。你不需要明确找到 $c_1,c_2,c_3$ 的值。

6. 证明 ${\lim }_{n\to \infty }\frac{T_{n+1}}{T_n}={\lambda }_1$。

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Problem 2

Consider a twice differentiable function $y(x)$ in an $xy$ plane which connects two points $A(-1,2)$ and $B(1,2)$. Let $S$ be the outer surface area of the cylindrical object created by rotation of the curve $y(x)$ about the $x$ axis. Answer the following questions.

1. Prove that the surface area $S$ is given by

$$S=2\pi {\int }_{-1}^{1}F(y,y^{\prime })\mathrm{d}x,$$ $$F(y,y^{\prime })=y\sqrt{1+(y^{\prime }{)}^2},$$

where $y^{\prime }=\frac{\mathrm{d}y}{\mathrm{d}x}$.

2. Let the curve $y(x)$ satisfy the following Euler–Lagrange equation for arbitrary $x$:

$$\frac{\partial F}{\partial y}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial F}{\partial y^{\prime }}=0.$$

Considering Eq. (2.3) along with $\frac{\mathrm{d}F}{\mathrm{d}x}$, prove that the following relation holds:

$$F-y^{\prime }\frac{\partial F}{\partial y^{\prime }}=c.$$

Here $c$ is a constant.

3. Express a differential equation satisfied by the curve $y(x)$ using $y,y^{\prime },c$.

4. Represent the curve $y(x)$ as a function of $x$ and $c$. Obtain an equation which should be satisfied by the constant $c$.

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考虑一个二维平面上的二阶可微函数 $y(x)$，它连接了两个点 $A(-1,2)$ 和 $B(1,2)$。设 $S$ 为通过旋转曲线 $y(x)$ 绕 $x$ 轴所形成的圆柱体的外表面积。回答以下问题。

1. 证明表面积 $S$ 由下式给出：

$$S=2\pi {\int }_{-1}^{1}F(y,y^{\prime })\mathrm{d}x,$$ $$F(y,y^{\prime })=y\sqrt{1+(y^{\prime }{)}^2},$$

其中 $y^{\prime }=\frac{\mathrm{d}y}{\mathrm{d}x}$。

2. 令曲线 $y(x)$ 满足以下任意 $x$ 的欧拉-拉格朗日方程：

$$\frac{\partial F}{\partial y}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial F}{\partial y^{\prime }}=0。$$

结合方程 (2.3) 和 $\frac{\mathrm{d}F}{\mathrm{d}x}$，证明以下关系成立：

$$F-y^{\prime }\frac{\partial F}{\partial y^{\prime }}=c。$$

其中 $c$ 为常数。

3. 用 $y,y^{\prime },c$ 表示曲线 $y(x)$ 满足的微分方程。

4. 将曲线 $y(x)$ 表示为 $x$ 和 $c$ 的函数。求出常数 $c$ 应满足的方程。

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Problem 3

Answer the following questions.

1. Calculate the number of possible ways to distribute $n$ equivalent balls to $r$ distinguishable boxes such that each box contains at least one ball, where $n\ge 1$ and $1\le r\le n$.

   Next, consider placing $n$ black balls and $m$ white balls in a line uniformly at random. A run is defined to be a succession of the same color. Let $r$ be the number of runs of black balls and $s$ be the number of runs of white balls. Assume that $n\ge 1$, $m\ge 1$, $1\le r\le n$, and $1\le s\le m$. For example, the sequence

$$\bullet \bullet \circ \circ \circ \bullet \bullet \bullet \circ \circ \bullet$$

corresponds to $r=3$ and $s=2$.

2. Calculate the total number of arrangements when we do not distinguish among balls of the same color.

3. Calculate the probability $P(r,s)$ that the number of runs of black balls is $r$ and the number of runs of white balls is $s$.

4. Calculate the probability $P(r)$ that the number of runs of black balls is $r$.

5. Using $(1+x{)}^n(1+x{)}^m=(1+x{)}^{n+m}$, show that the following equations hold.

$$\sum _{\ell =0}^{\min \{n,m\}}\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)=\left(\genfrac{}{}{0ex}{}{n+m}{m}\right)\text{\hspace{1em}(3.1)}$$ $$\sum _{\ell =0}^{\min \{n-1,m\}}\left(\genfrac{}{}{0ex}{}{n}{\ell +1}\right)\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)=\left(\genfrac{}{}{0ex}{}{n+m}{m+1}\right)\text{\hspace{1em}(3.2)}$$

6. Calculate the expected value $E(r)$ and the variance $V(r)$ of $r$. Calculate ${\lim }_{N\to \infty }\frac{E(r)}{N}$ and ${\lim }_{N\to \infty }\frac{V(r)}{N}$ supposing that $N=n+m$ and ${\lim }_{N\to \infty }\frac{n}{N}=\lambda$, where $\lambda$ is a real constant.

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回答以下问题。

1. 计算将 $n$ 个相同的球分配到 $r$ 个可区分的盒子中的可能方式的数量，使得每个盒子中至少包含一个球，其中 $n\ge 1$，且 $1\le r\le n$。

   接下来，考虑将 $n$ 个黑球和 $m$ 个白球随机均匀地排列在一条线上。一个连续相同颜色的序列称为一个 “run”。令 $r$ 表示黑球的 “run” 的数量，$s$ 表示白球的 “run” 的数量。假设 $n\ge 1$，$m\ge 1$，$1\le r\le n$，且 $1\le s\le m$。例如，序列

$$\bullet \bullet \circ \circ \circ \bullet \bullet \bullet \circ \circ \bullet$$

对应于 $r=3$ 和 $s=2$。

2. 计算当我们不区分相同颜色的球时的排列总数。

3. 计算概率 $P(r,s)$，即黑球的 “run” 的数量为 $r$，白球的 “run” 的数量为 $s$。

4. 计算概率 $P(r)$，即黑球的 “run” 的数量为 $r$。

5. 使用 $(1+x{)}^n(1+x{)}^m=(1+x{)}^{n+m}$，证明以下等式成立。

$$\sum _{\ell =0}^{\min \{n,m\}}\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)=\left(\genfrac{}{}{0ex}{}{n+m}{m}\right)\text{\hspace{1em}(3.1)}$$ $$\sum _{\ell =0}^{\min \{n-1,m\}}\left(\genfrac{}{}{0ex}{}{n}{\ell +1}\right)\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)=\left(\genfrac{}{}{0ex}{}{n+m}{m+1}\right)\text{\hspace{1em}(3.2)}$$

6. 计算 $r$ 的期望值 $E(r)$ 和方差 $V(r)$。计算 ${\lim }_{N\to \infty }\frac{E(r)}{N}$ 和 ${\lim }_{N\to \infty }\frac{V(r)}{N}$，假设 $N=n+m$，且 ${\lim }_{N\to \infty }\frac{n}{N}=\lambda$，其中 $\lambda$ 为常数。