一、填空题 （共 1 题, 每小题 5 分，共 20 分， 请把答案直接填写在答题纸上）
正实数 $x, y, z$ 满足 $x+2 y^2+4 x^2 y^2 z^2=8$, 则 $\log _4 x+\log _2 y+\log _8 z$ 的最大值为
二、解答题 （ 共 20 题，满分 80 分，解答过程应写出必要的文字说明、证明过程或演算步骤 ）
设 $d \geq 0$ 是整数, $V$ 是 $2 d+1$ 维复线性空间, 有一组基
$$
\left\{v_1, v_2, \cdots, v_{2 d+1}\right\} \text {. }
$$
对任一整数 $j\left(0 \leq j \leq \frac{d}{2}\right)$, 记 $U_j$ 是
$$
v_{2 j+1}, v_{2 j+3}, \cdots, v_{2 d-2 j+1}
$$
生成的子空间. 定义线性变换 $f: V \rightarrow V$ 为
$$
f\left(v_i\right)=\frac{(i-1)(2 d+2-i)}{2} v_{i-1}+\frac{1}{2} v_{i+1}, 1 \leq i \leq 2 d+1 .
$$
这里我们约定 $v_0=v_{2 d+2}=0$.
(1) 证明: $f$ 的全部特征值为 $-d,-d+1, \cdots, d$.
(2) 记 $W$ 是从属于特征值 $-d+2 k(0 \leq k \leq d)$ 的 $f$ 的特征子空间的和. 求 $W \cap U_0$ 的维数.
(3) 对任一整数 $j\left(1 \leq j \leq \frac{d}{2}\right)$, 求 $W \cap U_j$ 的维数.
双曲线 $\Gamma: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ 的左右顶点 $A, B$ 的距离为 4 ，
$M, N$ 是 $\Gamma$ 右支上不重合的两动点且满足 $k_{B N}+2 k_{A M}=0$ ( $k_{A M}, k_{B N}$ 是相应直线的斜率). 求动直线 $M N$ 经过的定点的坐标.
Let $G$ be a finite group.
(1) Let $K$ be a field. Show that $G$ has a finite-dimensional fait hful $K$-linear representation.
(2) Show that $G$ has a faithful one-dimensional complex repr esentation if and only if $G$ is cyclic.
(3) Assume moreover that $G$ is commutative. Let $n \geq 1$ be an integer. Show that $G$ has a faithful $n$-dimensional complex re presentation if and only if $G$ can be generated by $n$ elements.
(4) Classify all finite groups having a faithful 2-dimensional re al representation.
Let $n \geq 1$ be an integer. Let $A$ be a discrete valuation ring wit h $K$ its field of fractions and $\pi \in A$ a uniformizer. For $\lambda=\left(\lambda_1, \cdots, \lambda_n\right) \in \mathbb{Z}^n$ write
$$
D_\lambda=\operatorname{diag}\left(\pi^{\lambda_1}, \cdots, \pi^{\lambda_n}\right)=\left(\begin{array}{cccc}
\pi^{\lambda_1} & 0 & \cdots & 0 \\
0 & \pi^{\lambda_2} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \pi^{\lambda_n}
\end{array}\right) \in \mathrm{GL}_n(K) .
$$
Show that, for $\lambda, \mu \in \mathbb{Z}^n$, the following intersection inside $\mathrm{GL}_n(K)$
$$
\mathrm{GL}_n(A) \cdot D_\mu \cdot \mathrm{GL}_n(A) \bigcap U(K) \cdot D_\lambda
$$
is non-empty if and only if $\lambda_{\text {dom }} \leq \mu_{\text {dom }}$. Here
- $\mathrm{GL}_n(K)$ (resp. $\mathrm{GL}_n(A)$ ) is the group of invertible $n \times n \mathrm{sq}$ uare matrices with coefficients in $K$ (resp.in $A$ ), and $U(K) \subset \mathrm{GL}_n(K)$ is the standard unipotent subgroup, that $\mathrm{i}$ $\mathrm{s}$, the subgroup of upper triangular matrices with coefficients 1 on the diagonal.
- for $\alpha=\left(a_1, \cdots, a_n\right)$ and $\beta=\left(b_1, \cdots, b_n\right)$ two elements $\mathrm{i}$ $\mathrm{n} \mathbb{Z}^n$, we write $\alpha \leq \beta$ if
$$
\sum_{i=1}^k a_i \leq \sum_{i=1}^k b_i, \quad \text { for any } 1 \leq k \leq n
$$
and if $\sum_{i=1}^n a_i=\sum_{i=1}^n b_i$. Write also $\alpha_{\text {dom }}:=\left(a_1^{\prime}, \cdots, a_n^{\prime}\right)$ with $a_1^{\prime}, \cdots, a_n^{\prime}$ an arrangement of $a_1, \cdots, a_n$ such that
$$
a_1^{\prime} \geq a_2^{\prime} \geq \cdots \geq a_n^{\prime}
$$
Let $k$ be an imperfect field of characteristic $p>0$. Let $a \in k \backslash k^p$.
(1) Show that the polynomial $X^p-a \in k[X]$ is irreducible.
(2) Let $A=k[X] /\left(X^{p^2}-a X^p\right)$. Compute $A_{\text {red }}$, the quotien $\mathrm{t}$ of $A$ by its nilpotent radical.
Let $k$ be a field of character $p>0$ and consider $k(t) \subset k((t))$.
(1) Show that the field extension $k((t)) / k(t)$ is transcendent al, i.e., $k((t))$ contains at least one transcendental element ov er $k(t)$. (Hint: don't spend too much time on this question.)
(2) Let $\alpha \in k[[t]]$ which is transcendental over $k(t)$, and write $\beta=\alpha^p \in k[[t]]$. Let $K=k(t, \beta)$ and $L=k(t, \alpha)$ : both are subfields of $k((t))$. Let $A=k[[t]] \cap k(t, \beta)$. Determine the i ntegral closure $B$ of $A$ in $L$, and show that $A$ and $B$ are two discrete valuation rings.
(3) Is $B$ finitely generated over $A$ as a module? Justify your as sertion.
Let $p$ be a prime number. Consider $\mathbb{Z}_p$ (resp. $\mathbb{Q}_p$ ) the ring of $p$ adic integers (resp. field of $p$-adic numbers). Clearly $\mathbb{Z} \subset \mathbb{Z}_p$.
(1) Show that the set $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, and deduce that a ma $\mathrm{p} f: \mathbb{Z} \rightarrow \mathbb{Q}_p$ can be extended to a continuous function on $\mathbb{Z}_p$ if and only if $f$ is uniformly continuous, i.e., for any $\varepsilon>0$. there exists some integer $N>0$ so that $|f(n)-f(m)| < \varepsilon$ for any integers $n, m \in \mathbb{Z}$ with $m \equiv n\left(\bmod p^N\right)$.
(2) Let $a \in \mathbb{Q}_p \backslash\{0\}$. Under what condition on $a$, the map
$$
f: \mathbb{Z} \rightarrow \mathbb{Q}_p, \quad n \mapsto a^n
$$
can be extended to a continuous function over $\mathbb{Z}_p$ ? Justify yo ur assertion.
(3) Assume that the condition in (2) is fulfilled. Can we even $\mathrm{e}$ xtend the function $f$ in (2) to a continuous map
$$
a^x: \mathbb{Q}_p \rightarrow \mathbb{Q}_p,
$$
so that $a^{x+y}=a^x a^y$ for any $x, y \in \mathbb{Q}_p$ ? Justify your assertio n.
Let $n \geq 1$ be an integer and write $\Phi_n(X)$ the $n$-th cyclotomi c polynomial, that is, the minimal polynomial of a primitive $n$ th root of unity in $\mathbb{C}$ over $\mathbb{Q}$. Write also $\varphi(n)=\operatorname{deg}\left(\Phi_n(X)\right)$
(1) Let $q$ be a power of a prime number such that $(q, n)=1$. Show that $\Phi_n$, viewed as an element in $\mathbb{F}_q[X]$, can be decom posed as a product of $\varphi(n) / d$ irreducible polynomials of deg ree $d$, with $d$ the order of $q$ in the multiplicative group $(\mathbb{Z} / n \mathbb{Z})^{\times}$.
(2) From now on, assume $n=2^{r+1}$ for some integer $r \geq 1$. L et $\zeta=\zeta_n$ be a primitive $n$-th root of unity and $K=\mathbb{Q}[\zeta]$. Let $p$ be a prime with $p \equiv-3(\bmod 8)$.
(a) For $x, y \in K=\mathbb{Q}[\zeta]$, define
$$
(x, y):=\sum_\tau \tau(x) \cdot \overline{\tau(y)}
$$
where $\tau$ runs through all the embeddings $K \hookrightarrow \mathbb{C}$ of $K$ into the field $\mathbb{C}$ of complex numbers. Write $K_{\mathbb{R}}=K \otimes_{\mathbb{Q}} \mathbb{R}$, and we use the same notation to denote the (à priori $\mathbb{C}$-valued) bi linear form on $K_{\mathbb{R}}$ obtained by extension of scalars. Show tha $\mathrm{t}(\cdot, \cdot)$ gives an inner product on $K_{\mathbb{R}}$ and for $0 \leq i, j < 2^r$,
$$
\left(\zeta^i, \zeta^j\right)= \begin{cases}2^r, & \text { if } i=j \\ 0, & \text { otherwise }\end{cases}
$$
In particular, we obtain an Euclidean space $K_{\mathbb{R}}$ and $\left(\zeta^i / \sqrt{2^r}\right)_{0 \leq i < 2^r}$ is an orthonormal basis.
(b) Decompose $p \mathcal{O}_K$ into a product of prime ideals.
(c) Let $\mathfrak{p} \subset \mathcal{O}_K$ be a prime ideal of $\mathcal{O}_K$ containing $p$. Show th at for every $\alpha \in \mathfrak{p},|\alpha|^2 \in 2^r p \mathbb{Z}$, and compute the length of $\mathrm{t}$ he shortest non-zero vector in the prime ideal $\mathfrak{p} \subset K_{\mathbb{R}}$.
Let $A \in \mathbb{R}^{n \times n}$ be a non-singular matrix. Let $u, v \in \mathbb{R}^n$ be col umn vectors. Define the rank 1 perturbation $\hat{A}=A+u v^T$.
(a) Derive a necessary and sufficient condition for $\hat{A}$ to be inv ertible.
(b) Let $x, z$ and $b$ be column vectors in $\mathbb{R}^n$. Suppose one can solve $A z=b$ with $\mathcal{O}(n)$ floating-point operations (flops). Un der conditions derived in (a), design an algorithm to solve $\hat{A} x=b$ with $\mathcal{O}(n)$ flops, and provide justification for your an swer.
Consider the integral
$$
\int_0^{\infty} f(x) \mathrm{d} x
$$
where $f$ is continuous, $f^{\prime}(0) \neq 0$, and $f(x)$ decays like $x^{-1-\alpha}$ with $\alpha>0$ in the limit $x \rightarrow \infty$.
(a) Suppose you apply the equispaced composite trapezoid $\mathrm{r}$ ule with $n$ subintervals to approximate
$$
\int_0^L f(x) \mathrm{d} x
$$
What is the asymptotic error formula for the error in the limit $n \rightarrow \infty$ with $L$ fixed?
(b) Suppose you consider the quadrature from (a) to be an ap proximation to the full integral from 0 to $\infty$. How should $L$ in crease with $n$ to optimize the asymptotic rate of total error $d$ ecay? What is the rate of error decrease with this choice of $L$ ?
(c) Make the following change of variable $x=\frac{L(1+y)}{1-y}$, $y=\frac{x-L}{x+L}$ in the original integral to obtain
$$
\int_{-1}^1 F_L(y) \mathrm{d} y
$$
Suppose you apply the equispaced composite trapezoid rule; what is the asymptotic error formula for fixed $L$ ?
(d) Depending on $\alpha$, which method - domain truncation of ch ange-of-variable - is preferable?
Consider the Chebyshev polynomial of the first kind
$$
T_n(x)=\cos (n \theta), \quad x=\cos (\theta), \quad x \in[-1,1] .
$$
The Chebyshev polynomials of the second kind are defined a $\mathrm{s}$
$$
U_n(x)=\frac{1}{n+1} T^{\prime}(x), \quad n \geq 0 .
$$
(a) Derive a recursive formula for computing $U_n(x)$ for all $n \geq 0$.
(b) Show that the Chebyshev polynomials of the second kind are orthogonal with respect to the inner product
$$
\langle f, g\rangle=\int_{-1}^1 f(x) g(x) \sqrt{1-x^2} \mathrm{~d} x .
$$
(c) Derive the 2-point Gaussian Quadrature rule for the integr al
$$
\int_{-1}^1 f(x) \sqrt{1-x^2} \mathrm{~d} x=\sum_{j=1}^3 w_j f\left(x_j\right) .
$$
Consider the boundary value problem
$$
-\frac{d}{d x}\left(a(x) \frac{d u}{d x}\right)=f(x), \quad u(0)=u(1)=0
$$
where $a(x)>\delta \geq 0$ is a bounded differentiable function in $[0,1]$. We assume that, although $a(x)$ is available, an expressi on for its derivative, $\frac{d a}{d x}$, is not available.
(a) Using finite differences and an equally spaced gird in $[0,1], x_l=h l, l=0, \cdots, n$ and $h=1 / n$, we discretize the ODE to obtain a linear system of equations, yielding an $O\left(h^2\right)$ approximation of the ODE. After the application of the boundary conditions, the resulting coefficient matrix of the li near system is an $(n-1) \times(n-1)$ tridiagonal matrix.
Provide a derivation and write down the resulting linear syste $\mathrm{m}$ (by giving the expressions of the elements).
(b) Utilizing all the information provided, find a disc in $\mathbb{C}$, the smaller the better, that is guaranteed to contain all the eigenv alues of the linear system constructed in part (a).
(a) Verify that the PDE
$$
u_t=u_{x x x}
$$
is well posed as an initial value problem.
(b) Consider solving it numerically using the scheme
$$
\frac{u(t+k, x)-u(t-k, x)}{2 k}=\frac{-\frac{1}{2} u(x-2 h, t)+u(x-h, t)-u(x+h, t)+\frac{1}{2} u(x+2 h, t)}{h} .
$$
Determine this scheme's stability condition.
Consider the diffusion equation
$$
\frac{\partial v}{\partial t}=\mu \frac{\partial^2 v}{\partial x^2}, \quad v(x, 0)=\phi(x), \quad \int_a^b v(x, t) \mathrm{d} x=0
$$
with $x \in[a, b]$ and periodic boundary conditions. The solutio $\mathrm{n}$ is to be approximated using the central difference operator $L$ for the 1D Laplacian.
$$
L v_m=\frac{v_{m+1}-2 v_m+v_{m-1}}{h^2}
$$
and the following two finite different approximations, (i) Forw ard-Euler
$$
v_{n+1}=v_n+\mu k L v_n,(1)
$$
and (ii) Crank-Nicolson
$$
v_{n+1}=v_n+\mu k\left(L v_n+L v_{n+1}\right)
$$
Throughout, consider $[a, b]=[0,2 \pi]$ and the finite differenc e stencil to have periodic boundary conditions on the spatial lattice $[0, h, 2 h, \cdots,(N-1) h]$ where $h=\frac{2 \pi}{N}$ and $N$ is ev en.
(a) Determine the order of accuracy of the central difference operator $L v$ is approximating the second derivative $v_{x x}$.
(b) Using $v_m^n=\sum_{l=0}^{N-1} \hat{v}_l^n \exp \left(-i \frac{2 \pi l m}{N}\right)$ give the updates $\hat{v}_l^{n+1}$ in terms of $\hat{v}_l^n$ for each of the methods, including the ca se $l=0$.
(c) Give the solution for $v_m^n$ for each method when the initial condition is $\phi(m \Delta x)=(-1)^m$.
(d) What are the stability constraints on the time step $k$ for ea ch of the methods, if any, in equation (1) and (2)? Show there are either no constraints or express them in the form $k \leq F(h, \mu)$
双曲线 $\Gamma: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ 的左右顶点 $A, B$ 的距离为 $4, M, N$ 是 $\Gamma$ 右支上不重合的两动点且满足 $k_{B N}+2 k_{A M}=0$ ( $k_{A M}, k_{B N}$ 是相应直线的斜率). 求动直线 $M N$ 经过的定点的坐标.
实数 $a, b, c$ 满足 $a b+b c+c a=44$, 求 $\left(a^2+4\right)\left(b^2+4\right)\left(c^2+4\right)$ 的最小值.
设 $m, n$ 是大于 2 的整数. 已知平面上的正 $n$ 边形区域 $T$ 包含某个边长为 1 的正 $m$ 边形区域. 求证: 该平面上的任意一个边长为 $\cos \frac{\pi}{[m, n]}$ 的正 $m$ 边形区域 $S$ 可以平移嵌入区域 $T$, 即存在向量 $\vec{\alpha}$, 满足区域 $S$ 中的每个点平移 $\vec{\alpha}$ 后都落在区域 $T$ 中.
注: 多边形区域包含内部和边界.
对互质的正整数 $a, b$, 用 $\left(a^{-1} \bmod b\right)$ 表示满足 $a m \equiv 1(\bmod b)$ 且 $0 \leq m < $ $b$ 的唯一整数 $m$.
(1) 求证: 对任意两两互质的正整数 $a, b, c, 1 < a < b < c$, 都有
$$
\left(a^{-1} \bmod b\right)+\left(b^{-1} \bmod c\right)+\left(c^{-1} \bmod a\right)>\sqrt{a} .
$$
(2) 求证: 对任意正整数 $M$, 存在两两互质的正整数 $a, b, c$, 满足 $M < a < b < $ $c$, 且
$$
\left(a^{-1} \bmod b\right)+\left(b^{-1} \bmod c\right)+\left(c^{-1} \bmod a\right) < 100 \sqrt{a} .
$$
设 $m, n$ 是自然数, $a_0, a_1, \cdots, a_m, b_0, b_1, \cdots, b_n$ 是非负实数. 对 $0 \leq k \leq$ $m+n$, 记 $c_k=\max _{i+j=k} a_i b_j$. 求证:
$$
\frac{c_0+c_1+\cdots+c_{m+n}}{m+n+1} \geq \frac{a_0+a_1+\cdots+a_m}{m+1} \cdot \frac{b_0+b_1+\cdots+b_n}{n+1} .
$$
若实数 $\tau$ 满足: 对任意正整数 $x, y, z$, 均有 $x^2+2 y^2+4 z^2+8 \geq 2 x(y+z+\tau),$
则称 $\tau$ 为 "平生数". 记最大的平生数为 $\tau_0$.
(1) 求 $\tau_0$ 的值;
(2) 求方程 $x^2+2 y^2+4 z^2+8=2 x\left(y+z+\tau_0\right)$ 的所有正整数解 $(x, y, z)$.
(董秋仙供题)