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 $\mathbf{a},\mathbf{b}\in {\overline{\mathbb{R}}}^n,\mathbf{a}<\mathbf{b}$ := $\forall i,a_i\le b_i$
• 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

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
• $\leftrightarrow$ 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["测度论与概率论/拓扑空间"] --> 0["流形/smooth-manifold"]
2["测度论与概率论/relative-topology"] --> 0["流形/smooth-manifold"]
1["测度论与概率论/拓扑空间"] --> 3["流形/prepare"]
2["测度论与概率论/relative-topology"] --> 3["流形/prepare"]
4["测度论与概率论/homeomorphism"] --> 3["流形/prepare"]
6["朴素集合论/naive-set-theory-1"] --> 5["朴素集合论/zorn_lemma"]
8["数学分析/实数构造/整数"] --> 7["朴素集合论/初等数论"]
6["朴素集合论/naive-set-theory-1"] --> 9["朴素集合论/naive-set-theory-2"]
8["数学分析/实数构造/整数"] --> 9["朴素集合论/naive-set-theory-2"]
7["朴素集合论/初等数论"] --> 9["朴素集合论/naive-set-theory-2"]
11["数理逻辑/命题逻辑"] --> 10["数理逻辑/命题逻辑演算"]
11["数理逻辑/命题逻辑"] --> 12["数理逻辑/一阶逻辑"]
10["数理逻辑/命题逻辑演算"] --> 12["数理逻辑/一阶逻辑"]
9["朴素集合论/naive-set-theory-2"] --> 11["数理逻辑/命题逻辑"]
12["数理逻辑/一阶逻辑"] --> 13["数理逻辑/一阶逻辑演算"]
15["抽象代数/群论(一)"] --> 14["抽象代数/群论(置换群)"]
7["朴素集合论/初等数论"] --> 14["抽象代数/群论(置换群)"]
15["抽象代数/群论(一)"] --> 16["抽象代数/群论(二)"]
17["抽象代数/同态映射"] --> 16["抽象代数/群论(二)"]
19["抽象代数/环论(一)"] --> 18["抽象代数/环论(有序环)"]
15["抽象代数/群论(一)"] --> 17["抽象代数/同态映射"]
15["抽象代数/群论(一)"] --> 20["抽象代数/群论(幂运算)"]
8["数学分析/实数构造/整数"] --> 20["抽象代数/群论(幂运算)"]
19["抽象代数/环论(一)"] --> 21["抽象代数/环论(商域)"]
19["抽象代数/环论(一)"] --> 22["抽象代数/环论(幂运算)"]
18["抽象代数/环论(有序环)"] --> 22["抽象代数/环论(幂运算)"]
8["数学分析/实数构造/整数"] --> 22["抽象代数/环论(幂运算)"]
15["抽象代数/群论(一)"] --> 23["抽象代数/群论(循环群)"]
20["抽象代数/群论(幂运算)"] --> 23["抽象代数/群论(循环群)"]
7["朴素集合论/初等数论"] --> 23["抽象代数/群论(循环群)"]
17["抽象代数/同态映射"] --> 23["抽象代数/群论(循环群)"]
7["朴素集合论/初等数论"] --> 24["抽象代数/环论(杂)"]
25["抽象代数/环论(三)"] --> 24["抽象代数/环论(杂)"]
18["抽象代数/环论(有序环)"] --> 24["抽象代数/环论(杂)"]
26["抽象代数/环论(多项式环)"] --> 24["抽象代数/环论(杂)"]
15["抽象代数/群论(一)"] --> 19["抽象代数/环论(一)"]
16["抽象代数/群论(二)"] --> 19["抽象代数/环论(一)"]
27["抽象代数/环论(二)"] --> 25["抽象代数/环论(三)"]
7["朴素集合论/初等数论"] --> 25["抽象代数/环论(三)"]
23["抽象代数/群论(循环群)"] --> 25["抽象代数/环论(三)"]
19["抽象代数/环论(一)"] --> 27["抽象代数/环论(二)"]
17["抽象代数/同态映射"] --> 27["抽象代数/环论(二)"]
16["抽象代数/群论(二)"] --> 28["抽象代数/群论(高等群论)"]
16["抽象代数/群论(二)"] --> 29["抽象代数/群论(杂)"]
23["抽象代数/群论(循环群)"] --> 29["抽象代数/群论(杂)"]
19["抽象代数/环论(一)"] --> 26["抽象代数/环论(多项式环)"]
30["同态映射"] --> 26["抽象代数/环论(多项式环)"]
6["朴素集合论/naive-set-theory-1"] --> 15["抽象代数/群论(一)"]
31["数学分析/实数构造/自然数"] --> 15["抽象代数/群论(一)"]
16["抽象代数/群论(二)"] --> 32["抽象代数/群论(群作用)"]
14["抽象代数/群论(置换群)"] --> 32["抽象代数/群论(群作用)"]
34["数学分析/converge"] --> 33["数学分析/funct-limit"]
36["数学分析/real-number"] --> 35["数学分析/complex-number"]
37["数学分析/向量空间"] --> 35["数学分析/complex-number"]
39["抽象代数/向量空间"] --> 38["数学分析/derivate"]
33["数学分析/funct-limit"] --> 38["数学分析/derivate"]
41["数学分析/metric-space"] --> 40["数学分析/complex-seq-series"]
1["测度论与概率论/拓扑空间"] --> 40["数学分析/complex-seq-series"]
34["数学分析/converge"] --> 40["数学分析/complex-seq-series"]
42["数学分析/series"] --> 40["数学分析/complex-seq-series"]
36["数学分析/real-number"] --> 41["数学分析/metric-space"]
31["数学分析/实数构造/自然数"] --> 8["数学分析/实数构造/整数"]
17["抽象代数/同态映射"] --> 8["数学分析/实数构造/整数"]
15["抽象代数/群论(一)"] --> 8["数学分析/实数构造/整数"]
19["抽象代数/环论(一)"] --> 8["数学分析/实数构造/整数"]
18["抽象代数/环论(有序环)"] --> 8["数学分析/实数构造/整数"]
8["数学分析/实数构造/整数"] --> 43["数学分析/实数构造/有理数"]
21["抽象代数/环论(商域)"] --> 43["数学分析/实数构造/有理数"]
18["抽象代数/环论(有序环)"] --> 43["数学分析/实数构造/有理数"]
6["朴素集合论/naive-set-theory-1"] --> 31["数学分析/实数构造/自然数"]
43["数学分析/实数构造/有理数"] --> 44["数学分析/实数构造/实数"]
18["抽象代数/环论(有序环)"] --> 44["数学分析/实数构造/实数"]
36["数学分析/real-number"] --> 34["数学分析/converge"]
34["数学分析/converge"] --> 42["数学分析/series"]
31["数学分析/实数构造/自然数"] --> 36["数学分析/real-number"]
8["数学分析/实数构造/整数"] --> 36["数学分析/real-number"]
43["数学分析/实数构造/有理数"] --> 36["数学分析/real-number"]
44["数学分析/实数构造/实数"] --> 36["数学分析/real-number"]
39["抽象代数/向量空间"] --> 45["复分析/derivative"]
47["linear-algebra/matrix"] --> 46["linear-algebra/rank"]
19["抽象代数/环论(一)"] --> 47["linear-algebra/matrix"]
47["linear-algebra/matrix"] --> 48["linear-algebra/diagonalization"]
47["linear-algebra/matrix"] --> 49["linear-algebra/norm"]
9["朴素集合论/naive-set-theory-2"] --> 50["测度论与概率论/class-of-sets"]
1["测度论与概率论/拓扑空间"] --> 50["测度论与概率论/class-of-sets"]
52["测度论与概率论/measurable-mapping"] --> 51["测度论与概率论/measurable-func"]
34["数学分析/converge"] --> 51["测度论与概率论/measurable-func"]
54["测度论与概率论/measure"] --> 53["测度论与概率论/L-S-measure"]
6["朴素集合论/naive-set-theory-1"] --> 55["测度论与概率论/order-topology"]
1["测度论与概率论/拓扑空间"] --> 55["测度论与概率论/order-topology"]
51["测度论与概率论/measurable-func"] --> 56["测度论与概率论/random_variable"]
50["测度论与概率论/class-of-sets"] --> 57["测度论与概率论/Borel-R"]
1["测度论与概率论/拓扑空间"] --> 4["测度论与概率论/homeomorphism"]
50["测度论与概率论/class-of-sets"] --> 58["测度论与概率论/semi-ring"]
1["测度论与概率论/拓扑空间"] --> 2["测度论与概率论/relative-topology"]
53["测度论与概率论/L-S-measure"] --> 59["测度论与概率论/L-measure"]
51["测度论与概率论/measurable-func"] --> 60["测度论与概率论/almost-everywhere"]
50["测度论与概率论/class-of-sets"] --> 52["测度论与概率论/measurable-mapping"]
41["数学分析/metric-space"] --> 1["测度论与概率论/拓扑空间"]
9["朴素集合论/naive-set-theory-2"] --> 1["测度论与概率论/拓扑空间"]
5["朴素集合论/zorn_lemma"] --> 1["测度论与概率论/拓扑空间"]
50["测度论与概率论/class-of-sets"] --> 54["测度论与概率论/measure"]
41["数学分析/metric-space"] --> 61["测度论与概率论/product-metric-space"]
1["测度论与概率论/拓扑空间"] --> 61["测度论与概率论/product-metric-space"]
51["测度论与概率论/measurable-func"] --> 62["测度论与概率论/Lebesgue-integral"]
60["测度论与概率论/almost-everywhere"] --> 62["测度论与概率论/Lebesgue-integral"]

指向原始笔记的链接

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

友链

友链

• virgiling

指向原始笔记的链接