From ef97e28852771ffc63dd632780a55fce2bd043f4 Mon Sep 17 00:00:00 2001 From: KirisameVanilla <118162831+KirisameVanilla@users.noreply.github.com> Date: Tue, 2 Sep 2025 13:51:07 +0800 Subject: [PATCH] =?UTF-8?q?feat:=20make=20=E4=BF=A1=E5=AE=89=E6=95=B0?= =?UTF-8?q?=E5=AD=A6=E5=9F=BA=E7=A1=802=20paper?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../信息安全数学基础（二）/2024秋期末考试答案.md | 123 +++++++++++++++++ .../24FallFinalExam/2024秋期末考试答案.md | 67 ---------- .../24FallFinalExam/2024秋期末考试（部分）题目.md | 19 --- .../形式语言与自动机理论/index.md | 5 +- package-lock.json | 124 +++++++++--------- 5 files changed, 187 insertions(+), 151 deletions(-) create mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md diff --git a/docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md b/docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md new file mode 100644 index 0000000..8dba8c4 --- /dev/null +++ b/docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md @@ -0,0 +1,123 @@ +--- +title: 2024Fall_期末 +category: + - 软件工程学院 + - 课程资料 +tag: + - 试卷 +author: + - タクヤマ +--- + +## 说明 + +本学期（2024年秋季学期）的期末试卷难度**相当**低，本人评价为有手就行，故没有特意记忆所有题目，而是挑出一些有代表性的题目以供参考 + +我们这里再提供一道填空题，一道选择题，以显示这张卷子的**诚意** +
    +
  1. + + 设 $G$ 是一个群，对于 $\forall a,b \in G$， $a*b=b*a$， 则 $G$ 是 ____ 群 +
    2. +
  2. + + 设 $R$ 是一个环，对于 $\forall a,b \in R$， $a+b=b+a$， 则 $R$ 是？ + A. 整环 B. 交换环 C. 含幺环 D. 环 +
    3. +
+ +很大程度上，老师的出题和给分是十分宽容的，请大家好好学习这门课程，**数学并不是妖魔鬼怪！** + +## 题目 + +> **1. 设 $G=\{a,b,c,e\}$ 是一个群， $H=\{a,e\}$ 是 $G$ 的子群，写出 $H$ 的所有左陪集** + +
+ +解： + +将 $G$ 中元素 $g$ 各个代入，计算 $gH$ + +- $g = e$： + $eH = H$ + +- $g = a$： + $aH = \{ a \cdot a, a \cdot e \} = \{ a^2, a \}$，由于 $G$ 是群，且 $H$ 是子群，所以 $a^2$ 必须是 $G$ 中元素。 + 故 $a^2 = e$，则： $aH = \{ e, a \} = H$ + +- $g = b$： + $bH = \{ b \cdot a, b \cdot e \} = \{ ba, b \}$， + 假设 $ba = c$ ，则： + $bH = \{ c, b \}$ + +- $g = c$： + $cH = \{ c \cdot a, c \cdot e \} = \{ ca, c \}$， + 假设 $ca = b$ ，则： + $cH = \{ b, c \} = bH$ + +综上所述，H的所有子陪集是 $\{ e, a \}, \{ b, c \}$ + +
+ +*** + +> **2. 写出 $R=Z/6Z$ 的所有零因子** + +
+ +解： + + $R=Z/6Z \cong Z_6$，我们只要考虑 $Z_6$ 上的性质： + 显然有 $2\*3 \equiv 0 \pmod{6}, 4*3 \equiv 0 \pmod{6}$，所以零因子是2，3，4 + +
+ +*** + +> **3. 定义在有限域 $F_{17}$ 上的椭圆曲线 $E: x^3 + 2x + 3 = y^2$ 上有点 $P(2, 7), Q(11, 8)$ , 计算 $P+Q$ , $2P$** + +
+ +解： + + 直接计算即可，我们这里直接给出答案： $P+Q = (8, 15)$ , $2P = (14, 15)$ + +
+ +*** + +> **4. 求域 $F_{16}=F_2[x]/(x^4+x^3+1)$的一个生成元 $g(x)$，并用 $g(x)$ 的幂表示 $F_{16}$ 中的所有非零元** + +
+ +解： + + 这时候有同学要问了，生成元怎么找啊？其实很简单，直接验证就可以了，（出于强大的直觉和观察力）我们在这里直接验证 $x$ : + + | $x^n$ | $x^n\pmod{x^4+x^3+1}$ | + | -------- | --------------------- | + | $x^0$ | $1$ | + | $x^1$ | $x$ | + | $x^2$ | $x^2$ | + | $x^3$ | $x^3$ | + | $x^4$ | $x^3 + 1$ | + | $x^5$ | $x^3 + x + 1$ | + | $x^6$ | $x^3 + x^2 + x + 1$ | + | $x^7$ | $1 + x^2 + x$ | + | $x^8$ | $x^2 + x + 1$ | + | $x^9$ | $x^3 + x^2 + x$ | + | $x^{10}$ | $x^3 + x^2 + x + 1$ | + | $x^{11}$ | $x^3 + x + 1$ | + | $x^{12}$ | $x^3 + 1$ | + | $x^{13}$ | $x^3$ | + | $x^{14}$ | $x^2$ | + +如上表，由 $x$ 生成的15个非零元素互不相等，所以 $x$ 确实是生成元，非零元的表示如表中所示 + +
+ +*** + +> **5. 证明：有限环的特征一定不为0** + +证明：请参考课件Chap7.pdf中27-28页 diff --git a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md b/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md deleted file mode 100644 index 5641ec1..0000000 --- a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md +++ /dev/null @@ -1,67 +0,0 @@ -# 答案 - -## - -### 1.设 $G=\{a,b,c,e\}$ 是一个群， $H=\{a,e\}$ 是 $G$ 的子群，写出 $H$ 的所有左陪集 - -解： - 将 $G$ 中元素 $g$ 各个代入，计算 $gH$ - -- $g = e$： - $eH = H$ - -- $g = a$： - $aH = \{ a \cdot a, a \cdot e \} = \{ a^2, a \}$，由于 $G$ 是群，且 $H$ 是子群，所以 $a^2$ 必须是 $G$ 中元素。 - 故 $a^2 = e$，则： $aH = \{ e, a \} = H$ - -- $g = b$： - $bH = \{ b \cdot a, b \cdot e \} = \{ ba, b \}$， - 假设 $ba = c$ ，则： - $bH = \{ c, b \}$ - -- $g = c$： - $cH = \{ c \cdot a, c \cdot e \} = \{ ca, c \}$， - 假设 $ca = b$ ，则： - $cH = \{ b, c \} = bH$ - -综上所述，H的所有子陪集是 $\{ e, a \}, \{ b, c \}$ - -**2.写出 $R=Z/6Z$ 的所有零因子** - -解： - $R=Z/6Z \cong Z_6$，我们只要考虑 $Z_6$ 上的性质： - 显然有 $2\*3 \equiv 0 \pmod{6}, 4*3 \equiv 0 \pmod{6}$，所以零因子是2，3，4 - -**3.定义在有限域 $F_{17}$ 上的椭圆曲线 $E: x^3 + 2x + 3 = y^2$ 上有点 $P(2, 7), Q(11, 8)$ , 计算 $P+Q$ , $2P$** - -解： - 直接计算即可，我们这里直接给出答案： $P+Q = (8, 15)$ , $2P = (14, 15)$ - -**4.求域 $F_{16}=F_2[x]/(x^4+x^3+1)$的一个生成元 $g(x)$，并用 $g(x)$ 的幂表示 $F_{16}$ 中的所有非零元** - -解： - 这时候有同学要问了，生成元怎么找啊？其实很简单，直接验证就可以了，（出于强大的直觉和观察力）我们在这里直接验证 $x$ : - - | $x^n$ | $x^n\pmod{x^4+x^3+1}$ | - |------|----------| - | $x^0$ | $1$ | - | $x^1$ | $x$ | - | $x^2$ | $x^2$ | - | $x^3$ | $x^3$ | - | $x^4$ | $x^3 + 1$ | - | $x^5$ | $x^3 + x + 1$ | - | $x^6$ | $x^3 + x^2 + x + 1$ | - | $x^7$ | $1 + x^2 + x$ | - | $x^8$ | $x^2 + x + 1$ | - | $x^9$ | $x^3 + x^2 + x$ | - | $x^{10}$ | $x^3 + x^2 + x + 1$ | - | $x^{11}$ | $x^3 + x + 1$ | - | $x^{12}$ | $x^3 + 1$ | - | $x^{13}$ | $x^3$ | - | $x^{14}$ | $x^2$ | - -如上表，由 $x$ 生成的15个非零元素互不相等，所以 $x$ 确实是生成元，非零元的表示如表中所示 - -### 5.证明：有限环的特征一定不为0 - -证明：请参考课件Chap7.pdf中27-28页 diff --git a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md b/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md deleted file mode 100644 index 4499a66..0000000 --- a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md +++ /dev/null @@ -1,19 +0,0 @@ -## 说明： -本学期（2024年秋季学期）的期末试卷难度**相当**低，本人评价为有手就行，故没有特意记忆所有题目，而是挑出一些有代表性的题目以供参考 - -
    -
  1. 设 $G=\{a,b,c,e\}$ 是一个群， $H=\{a,e\}$ 是 $G$ 的子群，写出 $H$ 的所有左陪集
    2. -
  2. 写出 $R=Z/6Z$ 的所有零因子
    3. -
  3. 定义在有限域 $F_{17}$ 上的椭圆曲线 $E: x^3 + 2x + 3 = y^2$ 上有点 $P(2, 7), Q(11, 8)$ , 计算 $P+Q$ , $2P$
    4. -
  4. 求域 $F_{16}=F_2[x]/(x^4+x^3+1)$的一个生成元 $g(x)$，并用 $g(x)$ 的幂表示 $F_{16}$ 中的所有非零元
    5. -
  5. 证明：有限环的特征一定不为0
    6. -
- -我们这里再提供一道填空题，一道选择题，以显示这张卷子的**诚意** -
    -
  1. 设 $G$ 是一个群，对于 $\forall a,b \in G$， $a*b=b*a$， 则 $G$ 是 ____ 群
    2. -
  2. 设 $R$ 是一个环，对于 $\forall a,b \in R$， $a+b=b+a$， 则 $R$ 是？ - A. 整环 B.交换环 C.含幺环 D.环
    3. -
- -很大程度上，老师的出题和给分是十分宽容的，请大家好好学习这门课程，**数学并不是妖魔鬼怪！** diff --git a/docs/courses/软件工程学院/形式语言与自动机理论/index.md b/docs/courses/软件工程学院/形式语言与自动机理论/index.md index e248e6d..ccbb1c4 100644 --- a/docs/courses/软件工程学院/形式语言与自动机理论/index.md +++ b/docs/courses/软件工程学院/形式语言与自动机理论/index.md @@ -1,5 +1,5 @@ --- -title: 信息安全数学基础（二） +title: 形式语言与自动机理论 category: - 软件工程学院 - 课程评价 @@ -9,5 +9,4 @@ tag: - 抽象代数 --- -### 这门课是抽象代数，但是并没有涉及到抽象代数真正复杂的部分，较为浅尝辄止，所以课程本身的学习难度并不高。 -### 同样地，这门课实际考试不难，老师会捞，请大家不要害怕。 + diff --git a/package-lock.json b/package-lock.json index b42a27d..bb8546b 100644 --- a/package-lock.json +++ b/package-lock.json @@ -1846,76 +1846,76 @@ ] }, "node_modules/@shikijs/core": { - "version": "3.12.0", - "resolved": "https://registry.npmjs.org/@shikijs/core/-/core-3.12.0.tgz", - "integrity": "sha512-rPfCBd6gHIKBPpf2hKKWn2ISPSrmRKAFi+bYDjvZHpzs3zlksWvEwaF3Z4jnvW+xHxSRef7qDooIJkY0RpA9EA==", + "version": "3.12.1", + "resolved": "https://registry.npmjs.org/@shikijs/core/-/core-3.12.1.tgz", + "integrity": "sha512-j9+UDQ6M50xvaSR/e9lg212H0Fqxy3lYd39Q6YITYQxfrb5VYNUKPLZp4PN9f+YmRcdpyNAm3obn/tIZ2WkUWg==", "dev": true, "license": "MIT", "dependencies": { - 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