Notes on Yukari's Blog

Computer Networking — Chapter 1: Computer Networks and the Internet

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 1 of Computer Networking: A Top-Down Approach (7th Edition). Covers the Internet overview, network edge, network core, delay/loss/throughput, protocol layers, and network security.

Computer Networking — Chapter 2: Application Layer

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 2 of Computer Networking: A Top-Down Approach (7th Edition). Covers HTTP, email, DNS, P2P, CDN, video streaming, and socket programming.

Computer Networking — Chapter 3: Transport Layer

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 3 of Computer Networking: A Top-Down Approach (7th Edition). Covers transport-layer services, UDP, reliable data transfer, TCP, and congestion control.

Computer Networking — Chapter 4: The Network Layer: Data Plane

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 4 of Computer Networking: A Top-Down Approach (7th Edition). Covers network layer data plane functions: forwarding, router architecture, IP protocol, addressing, NAT, IPv6, and SDN generalized forwarding.

Computer Networking — Chapter 5: The Network Layer: Control Plane

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 5 of Computer Networking: A Top-Down Approach (7th Edition). Covers routing algorithms, OSPF, BGP, SDN control plane, ICMP, and SNMP.

Computer Networking — Chapter 6: The Link Layer and LANs

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 6 of Computer Networking: A Top-Down Approach (7th Edition). Covers link-layer services, error detection, multiple access protocols, Ethernet, switches, VLANs, MPLS, data center networking.

Computer Networking — Chapter 7: Wireless and Mobile Networks

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 7 of Computer Networking: A Top-Down Approach (7th Edition). Covers wireless links, WiFi (802.11), cellular networks, mobility management, and Mobile IP.

Computer Networking — Chapter 8: Security in Computer Networks

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 8 of Computer Networking: A Top-Down Approach (7th Edition). Covers cryptography, message integrity, authentication, SSL, IPsec, wireless security, and firewalls.

Computer Networking — Chapter 9: Multimedia Networking

Sat, 27 Jun 2026 00:00:00 +0000

Notes on Chapter 9 of Computer Networking: A Top-Down Approach (7th Edition). Covers multimedia applications, streaming video, VoIP, RTP, SIP, Diffserv, Intserv, and RSVP.

Game of Chess

Thu, 25 Jun 2026 00:00:00 +0000

Notes on Chess from a Game Theory perspective.

Manifold

Wed, 03 Jun 2026 00:00:00 +0000

Reference: Introduction to Smooth Manifolds by John M. Lee.

A topological space $M$ is a topological manifold of dimension $n$ if $M$ is locally Euclidean of dimension $n$, where locally Euclidean means that for every point $p \in M$, there exists an open neighborhood $U$ of $p$ in $M$ and a homeomorphism $\varphi: U \to B^n$, then we obtain a set of such charts $\{(U_\alpha, \varphi_\alpha)\}$ called a topological atlas on $M$.

分析力学

Wed, 03 Jun 2026 00:00:00 +0000

Euler-Lagrange 方程
$$\dfrac {\mathrm d}{\mathrm dt} \dfrac {\partial L}{\partial \dot q_i} - \dfrac {\partial L}{\partial q_i} = 0.$$
正则动量
$$p_i = \dfrac{\partial L}{\partial \dot q_i},\quad \dot p_i = \dfrac {\partial L}{\partial q_i}.$$
Hamiltonian，可以对于正则 Lagrangian 定义
$$H(q,p,t) = \sum_i p_i \dot q_i - L(q, \dot q, t)=\sum_i p_i \dot q_i - L(q, p, t).$$
Hamilton 方程
$$\dot q_i = \dfrac{\partial H}{\partial p_i}, \quad \dot p_i = -\dfrac{\partial H}{\partial q_i},\quad \dfrac{\partial H}{\partial t} = -\dfrac{\partial L}{\partial t}.$$
相空间作用量原理*

Sobolev 空间和弱导数

Mon, 01 Jun 2026 00:00:00 +0000

$W^{m,p}$

回忆广义函数空间 $\mathcal D(\Omega)$ 上的对偶元素构造。给定任意 $f\in L_{\mathrm{loc}}^1(\Omega)$，存在线性映射

$$T:\mathcal D(\Omega)\to \mathcal D^*(\Omega),\quad \varphi\mapsto\left(T_f:\varphi\mapsto \int_{\Omega} f\varphi\mathrm dx\right).$$

在 Lebesgue 意义下（默认）是单射：

$T$ 是单射：对任意 $f,g\in L_{\mathrm{loc}}^1(\Omega)$，如果 $T_f=T_g$，则 $f=g$.

这等价于说明 $T_f=0$，即

$$\langle T_f,\varphi\rangle = \int_{\Omega} f\varphi\mathrm dx=0,\quad \forall \varphi\in \mathcal D(\Omega),$$

可以推出 $f=0$，这是变分法基本引理的 Lebesgue 版本。考虑截断空间和 Friedrichs 磨光核，则

$$f*\rho_\varepsilon(x)=\int_{\Omega} f(y)\rho_\varepsilon(x-y)\mathrm dy=\langle T_f,\rho_\varepsilon(x-\cdot)\rangle=0$$

接下来只需证明，对于任意有界开集 $U\subseteq\Omega$，有 $\|f\|_{L^1(U)}=0$，这是因为

$$f(x)-f*\rho_\varepsilon(x)=\int_{\Omega} (f(x)-f(x-y))\rho_\varepsilon(y)\mathrm dy,$$

因此有 Minkowski 不等式

$$\|f-f*\rho_\varepsilon\|_{L^1(U)}\leq \int_{\Omega} \|f(\cdot)-f(\cdot-y)\|_{L^1(U)}\rho_\varepsilon(y)\mathrm dy\leq \sup_{|y|\leq \varepsilon} \|f(\cdot)-f(\cdot-y)\|_{L^1(U)}.$$

由 $L^1$ 的平移连续性，取 $\varepsilon\to 0$，结合 $f*\rho_\varepsilon=0$ 即可。

我们还需要指出，$L^p(\Omega)\subseteq L_{\mathrm{loc}}^1(\Omega)$，对任意 $1\leq p\leq \infty$，这是因为 Hölder 不等式

调和函数与极值原理

Mon, 01 Jun 2026 00:00:00 +0000

极值原理

弱极值原理

强极值原理

调和函数

设 $\Omega \subseteq \mathbb R^d$ 是开区域，如果 $u\in C^2(\Omega)$ 满足 $\Delta u=0$，则称 $u$ 是 $\Omega$ 上的调和函数，记作 $u\in \mathcal H(\Omega)$。调和函数是 Laplace 方程的解，继承其性质。

球面平均性

球面平均性：如果 $u\in \mathcal H(\Omega)$，则对任意 $r\in (0,R]$ 和 $x\in \Omega$ 满足 $B_R(x)\subseteq \Omega$，有
$$u(x)=\frac 1{|\partial B_r(0)|}\int_{\partial B_r(x)} u(y)\mathrm dS=\dfrac 1{|B_r(0)|}\int_{B_r(x)} u(y)\mathrm dy=: h(x;r).$$
反过来，如果 $u\in C^2(\Omega)$ 在 $x\in \Omega$ 满足上述球面平均性，则 $\Delta u(x)=0$.

从连续性出发

$$u(x)=\lim_{r\to 0}\frac 1{|\partial B_r(0)|}\int_{\partial B_r(x)} u(y)\mathrm dS=\lim_{r\to 0}\dfrac 1{|\partial B_1(0)|}\int_{\partial B_1(0)} u(x+ry)\mathrm dS=:\lim_{r\to 0} h(x;r).$$

考虑含参积分

位势方程与 Green 公式

Wed, 27 May 2026 00:00:00 +0000

位势方程

一些基本的结果：

基本解：在 $\mathbb R^n$ 上，位势方程 $-\Delta u=f(x)$ 的基本解 $-\Delta E=\delta(x)$ 满足
$$E(x)=\begin{dcases} -\frac{1}{2\pi}\log|x|,&n=2,\\[14pt] \frac{1}{(n-2)S_{n-1}|x|^{n-2}},&n\geq 3, \end{dcases}$$

尽管基本解 $E$ 在 $x=0$ 处有奇点，但通过球坐标变换

$$\int_{|x|<\varepsilon} E(x)\mathrm dx=\int_0^\varepsilon\int_{\Theta}E(r)J(\Theta)r^{n-1}\mathrm d\Theta\mathrm dr\to 0,\quad \varepsilon\to 0,$$

其中 $J(\Theta)$ 是球坐标变换的 Jacobian 的角度部分。因此 $E\in L^1_{\mathrm{loc}}(\mathbb R^n)$；另外

$$\nabla E(x)=-\dfrac 1{S_{n-1}}\dfrac x{|x|^n},\quad \forall n\geq 2,$$

这推出 $\nabla E\in L^1_{\mathrm{loc}}(\mathbb R^n)$，因为

$$\int_{|x|<\varepsilon} |\nabla E(x)|\mathrm dx=\int_0^\varepsilon\int_{\Theta}\dfrac {J(\Theta)}{S_{n-1}}\mathrm d\Theta\mathrm dr=\int_0^\varepsilon\mathrm dr=\varepsilon,$$

如果我们进一步求二阶微分

$$|\nabla^2 E(x)|=\dfrac 1{S_{n-1}}\left|\dfrac n{|x|^n}\dfrac {xx^T}{|x|^2}-\dfrac 1{|x|^n}I\right|\sim \dfrac 1{|x|^n},$$

这推出 $\nabla^2 E\notin L^1_{\mathrm{loc}}(\mathbb R^n)$，因为

无界区域的热传导方程

Sat, 16 May 2026 00:00:00 +0000