Forever young

• Notation
  • 1. Math Symbols
  • 2. Logic Symbols
  • 3. Computer Symbols
• Folders with clear dependencies
• 友链

Notation

1. Math Symbols

• Vector is denoted as bold letter, e.g. $\mathbf{a}$, Matrix is denoted as capital letter, e.g. $A$, $a_i$ means the $i$th element of $\mathbf{a}$, $a_{ij}$ means the $(i,j)$th element of $A$
• All vectors are column vectors
• $\mathbb{Z}$ is the set of integers.
• $\mathbb{Q}$ is the set of rational numbers.
• $\mathbb{R}$ is the set of real numbers.
• $\mathbb{C}$ is the set of complex numbers.
• $\mathbb{F}$ is a filtration.
• $\mathbb{P}$ is a probability measure.
• $\mathbb{E}$ is the expectation operator.
• $[n]$ := $\{1,2,\cdots ,n\}$
• ${\mathbb{Z}}_n$ := $\{0,1,\cdots ,n-1\}$
• ${\mathbb{Z}}_{\ge n}$ := $\{i\mid i\in \mathbb{Z}\wedge i\ge n\}$
• $\overline{\mathbb{R}}$ := $\mathbb{R}\cup \{-\infty ,+\infty \}$
• For $X\subseteq \overline{\mathbb{R}}$, $X^{+}$ := $\{x\in X|x>0\}$
• For $X\subseteq \overline{\mathbb{R}}$, $X^{\ast }$ := $\{x\in X|x\ge 0\}$
• For $\mathbf{a},\mathbf{b}\in {\overline{\mathbb{R}}}^n,\mathbf{a}\le \mathbf{b}$ := $\forall i,a_i\le b_i$
• Class of sets is denoted as calligraphic letter, e.g. $\mathcal{A}$, $\mathcal{B}$, $\mathcal{F}$
• $\mathcal{P}(X)$ is the power set of $X$
• $⨄$ represents the union of disjoint sets.
• $A^{\text{\#}}$ is the adjoint matrix of $A$
• $(X,\tau )$ is a topological space, briefly written as $X$
• $\mathcal{B}(X):=\sigma (\tau )$, where $(X,\tau )$ is a topological space
• $\wedge$ represents the and for logic, min for function
• $\vee$ represents the or for logic, max for function
• $A:=B$ 表示 $A$ 被定义为 $B$
• $A=:B$ 表示 $A$ 被记作 $B$

2. Logic Symbols

• Little letters (e.g. $p,q,r$) are used to represent propostional variables
• $\neg$ is used to represent negation
• $\wedge$ is used to represent conjunction
• $\vee$ is used to represent disjunction
• $\to$ is used to represent implication
• $⟺$ is used to represent equivalence
• $\mathcal{A}⟹\mathcal{B}$ means $(\mathcal{A}\to \mathcal{B})$ is True
• $\mathcal{A}⟺\mathcal{B}$ means $((\mathcal{A}\to \mathcal{B})\wedge (\mathcal{B}\to \mathcal{A}))$ is True
• ${\mathcal{L}}_0$ is used to represent a zero-order language
• $\mathcal{L}$ is used to represent a first-order language
• mathscr upper letters (e.g. $\mathcal{A},\mathcal{B},\mathcal{C}$) are used to represent the formulas (well-formed formulas,wfs) of a language $\mathcal{L}$ and ${\mathcal{L}}_0$

3. Computer Symbols

• Local network is denoted as capital letter, e.g. $A$,$B$,$C$
• Host in local network is denoted as small letter, e.g. $a$,$b$,$c$ located in local network $A$,$B$,$C$
• Public ip is denoted as Greek letter, e.g. $\alpha$,$\gamma$,$\beta$
• Hosts in the same local network are distinguished by subscripts, e.g. $a_1$,$a_2$,$a_5$ located in local network $A$.

Folders with clear dependencies

dependency_graph

graph TD
1["数理逻辑/一阶逻辑"] --> 0["数理逻辑/一阶逻辑演算"]
2["数理逻辑/命题逻辑演算"] --> 1["数理逻辑/一阶逻辑"]
3["数理逻辑/命题逻辑"] --> 2["数理逻辑/命题逻辑演算"]
5["朴素集合论/naive-set-theory-2"] --> 3["数理逻辑/命题逻辑"]
5["朴素集合论/naive-set-theory-2"] --> 6["点集拓扑/拓扑空间"]
7["朴素集合论/zorn_lemma"] --> 6["点集拓扑/拓扑空间"]
9["点集拓扑/子空间"] --> 8["点集拓扑/连续映射"]
6["点集拓扑/拓扑空间"] --> 9["点集拓扑/子空间"]
12["数学分析/real-number"] --> 11["点集拓扑/度量空间"]
6["点集拓扑/拓扑空间"] --> 11["点集拓扑/度量空间"]
8["点集拓扑/连续映射"] --> 13["点集拓扑/紧空间"]
15["数学分析/metric-space"] --> 14["点集拓扑/product-metric-space"]
6["点集拓扑/拓扑空间"] --> 14["点集拓扑/product-metric-space"]
8["点集拓扑/连续映射"] --> 16["点集拓扑/连通空间"]
6["点集拓扑/拓扑空间"] --> 17["点集拓扑/乘积空间"]
6["点集拓扑/拓扑空间"] --> 18["点集拓扑/序拓扑"]
20["抽象代数/群论(循环群)"] --> 19["抽象代数/环论(有限环)"]
22["抽象代数/环论(一)"] --> 21["抽象代数/环论(二)"]
23["抽象代数/同态映射"] --> 21["抽象代数/环论(二)"]
19["抽象代数/环论(有限环)"] --> 24["抽象代数/环论(杂)"]
25["抽象代数/环论(多项式环)"] --> 24["抽象代数/环论(杂)"]
27["朴素集合论/naive-set-theory-1"] --> 26["抽象代数/二元运算"]
29["抽象代数/群论(二)"] --> 28["抽象代数/群论(杂)"]
20["抽象代数/群论(循环群)"] --> 28["抽象代数/群论(杂)"]
31["抽象代数/域论(域拓展)"] --> 30["抽象代数/形式导数"]
32["抽象代数/群论(一)"] --> 29["抽象代数/群论(二)"]
23["抽象代数/同态映射"] --> 29["抽象代数/群论(二)"]
33["抽象代数/群论(幂运算)"] --> 20["抽象代数/群论(循环群)"]
34["朴素集合论/初等数论"] --> 20["抽象代数/群论(循环群)"]
36["数学分析/实数构造/整数"] --> 35["抽象代数/环论(幂运算)"]
29["抽象代数/群论(二)"] --> 37["抽象代数/群论(群作用)"]
38["抽象代数/群论(置换群)"] --> 37["抽象代数/群论(群作用)"]
21["抽象代数/环论(二)"] --> 39["抽象代数/环论(商域)"]
32["抽象代数/群论(一)"] --> 22["抽象代数/环论(一)"]
29["抽象代数/群论(二)"] --> 40["抽象代数/群论(高等群论)"]
42["抽象代数/域论(有限域)"] --> 41["抽象代数/Galois"]
43["数学分析/实数构造/自然数"] --> 32["抽象代数/群论(一)"]
26["抽象代数/二元运算"] --> 32["抽象代数/群论(一)"]
29["抽象代数/群论(二)"] --> 44["抽象代数/向量空间"]
21["抽象代数/环论(二)"] --> 44["抽象代数/向量空间"]
22["抽象代数/环论(一)"] --> 46["抽象代数/环论(有序环)"]
21["抽象代数/环论(二)"] --> 25["抽象代数/环论(多项式环)"]
24["抽象代数/环论(杂)"] --> 31["抽象代数/域论(域拓展)"]
44["抽象代数/向量空间"] --> 31["抽象代数/域论(域拓展)"]
30["抽象代数/形式导数"] --> 42["抽象代数/域论(有限域)"]
36["数学分析/实数构造/整数"] --> 33["抽象代数/群论(幂运算)"]
29["抽象代数/群论(二)"] --> 47["抽象代数/群论(群列)"]
26["抽象代数/二元运算"] --> 23["抽象代数/同态映射"]
34["朴素集合论/初等数论"] --> 38["抽象代数/群论(置换群)"]
8["点集拓扑/连续映射"] --> 49["流形/prepare"]
9["点集拓扑/子空间"] --> 50["流形/smooth-manifold"]
37["抽象代数/群论(群作用)"] --> 52["机器学习/SE(3)"]
27["朴素集合论/naive-set-theory-1"] --> 7["朴素集合论/zorn_lemma"]
36["数学分析/实数构造/整数"] --> 34["朴素集合论/初等数论"]
34["朴素集合论/初等数论"] --> 5["朴素集合论/naive-set-theory-2"]
44["抽象代数/向量空间"] --> 55["复分析/derivative"]
60["数学分析/非负级数"] --> 59["数学分析/级数"]
44["抽象代数/向量空间"] --> 61["数学分析/derivate"]
62["数学分析/funct-limit"] --> 61["数学分析/derivate"]
12["数学分析/real-number"] --> 63["数学分析/converge"]
63["数学分析/converge"] --> 60["数学分析/非负级数"]
63["数学分析/converge"] --> 62["数学分析/funct-limit"]
12["数学分析/real-number"] --> 65["数学分析/complex-number"]
44["抽象代数/向量空间"] --> 65["数学分析/complex-number"]
66["数学分析/实数构造/实数"] -.-> 12["数学分析/real-number"]
69["计算机/docker"] -.-> 68["计算机/easyconnect"]
71["linear-algebra/matrix"] --> 70["linear-algebra/rank"]
22["抽象代数/环论(一)"] --> 71["linear-algebra/matrix"]
71["linear-algebra/matrix"] --> 73["linear-algebra/diagonalization"]
71["linear-algebra/matrix"] --> 74["linear-algebra/norm"]
76["故事/纯粹理性批判"] --> 75["故事/固执"]
78["故事/价格"] --> 77["故事/资本"]
80["故事/安全"] --> 79["故事/狡猾的小说家"]
76["故事/纯粹理性批判"] --> 81["故事/实践理性批判"]
82["故事/时间"] --> 76["故事/纯粹理性批判"]
81["故事/实践理性批判"] --> 80["故事/安全"]
81["故事/实践理性批判"] --> 83["故事/拥有"]
81["故事/实践理性批判"] --> 84["故事/爱"]
85["故事/货币"] --> 78["故事/价格"]
80["故事/安全"] --> 87["故事/安全与道德"]
84["故事/爱"] --> 88["故事/爱与幸福"]
82["故事/时间"] --> 85["故事/货币"]
81["故事/实践理性批判"] --> 89["故事/判断力批判"]
6["点集拓扑/拓扑空间"] --> 90["测度论与概率论/class-of-sets"]
92["测度论与概率论/almost-everywhere"] --> 91["测度论与概率论/Lebesgue-integral"]
90["测度论与概率论/class-of-sets"] --> 93["测度论与概率论/measurable-mapping"]
95["测度论与概率论/measure"] --> 94["测度论与概率论/L-S-measure"]
93["测度论与概率论/measurable-mapping"] --> 96["测度论与概率论/measurable-func"]
63["数学分析/converge"] --> 96["测度论与概率论/measurable-func"]
90["测度论与概率论/class-of-sets"] --> 97["测度论与概率论/semi-ring"]
91["测度论与概率论/Lebesgue-integral"] --> 98["测度论与概率论/三大积分收敛定理"]
96["测度论与概率论/measurable-func"] --> 92["测度论与概率论/almost-everywhere"]
96["测度论与概率论/measurable-func"] --> 99["测度论与概率论/random_variable"]
90["测度论与概率论/class-of-sets"] --> 95["测度论与概率论/measure"]
94["测度论与概率论/L-S-measure"] --> 101["测度论与概率论/L-measure"]
90["测度论与概率论/class-of-sets"] --> 102["测度论与概率论/Borel-R"]
18["点集拓扑/序拓扑"] --> 102["测度论与概率论/Borel-R"]
104["原始数据"] --> 103["机器学习/pytorch/torch.utils.data"]
105["batch_size"] --> 103["机器学习/pytorch/torch.utils.data"]
106["drop_last"] --> 103["机器学习/pytorch/torch.utils.data"]
107["shuffle"] --> 103["机器学习/pytorch/torch.utils.data"]
108["sampler"] --> 103["机器学习/pytorch/torch.utils.data"]
109["batch_sampler"] --> 103["机器学习/pytorch/torch.utils.data"]
110["batched data"] --> 103["机器学习/pytorch/torch.utils.data"]
111["数学分析/实数构造/有理数"] --> 66["数学分析/实数构造/实数"]
36["数学分析/实数构造/整数"] --> 111["数学分析/实数构造/有理数"]
39["抽象代数/环论(商域)"] --> 111["数学分析/实数构造/有理数"]
27["朴素集合论/naive-set-theory-1"] --> 43["数学分析/实数构造/自然数"]
21["抽象代数/环论(二)"] --> 36["数学分析/实数构造/整数"]
46["抽象代数/环论(有序环)"] --> 36["数学分析/实数构造/整数"]

指向原始笔记的链接

• linear-algebra
• 数学分析
• 抽象代数
• 数理逻辑
• 朴素集合论
• 测度论与概率论
• 点集拓扑

友链

友链

名称	链接
virgiling	https://virgiling.wiki/

指向原始笔记的链接