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A user with 1,119 edits. Account created on 30 August 2022.
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29 May 2024

28 May 2024

26 May 2024

  • 02:5902:59, 26 May 2024 diff hist +18,855‎ N 概率论与数理统计 (Spring 2024)/OST and applications ‎ Created page with "=可选停时定理 (OST)= '''可选停时定理''' ('''Optional Stopping Theorem''', '''OST''')，有事也被称为'''鞅停时定理''' ('''Martingale Stopping Theorem''')、'''可选抽样定理''' ('''Optional Sampling Theorem''') 等，是约瑟夫·杜布 ([https://en.wikipedia.org/wiki/Joseph_L._Doob Joseph Doob]) 发现的关于鞅的停时的刻画定理。 首先定义鞅 (martingale)。这是一类由公平赌博定义的随机过程。 {{Theorem|定义（..." current
  • 02:5902:59, 26 May 2024 diff hist +4,639‎ N 概率论与数理统计 (Spring 2024)/Hoeffding's lemma ‎ Created page with "霍夫丁引理（Hoeffding's lemma）在霍夫丁不等式（[https://en.wikipedia.org/wiki/Hoeffding%27s_inequality Hoeffding's inequality]）的证明中，扮演着关键角色。该引理陈述如下： {{Theorem|霍夫丁引理| :若随机变量 <math>Y</math> 满足 <math>\mathbb{E}[Y]=0</math> 且存在实数 <math>a,b\in\mathbb{R}</math> 使得几乎必然地 (a.s.) <math>a\le Y\le b</math>，则对于任意 <math>\lambda\in\mathbb{R}</math>，都有 ::<math>\ma..." current
  • 02:5802:58, 26 May 2024 diff hist +576‎ 概率论与数理统计 (Spring 2024) ‎ →‎Lectures

20 May 2024

19 May 2024

14 May 2024

  • 09:3809:38, 14 May 2024 diff hist +18,939‎ N 组合数学 (Fall 2024)/Extremal graph theory ‎ Created page with "== Forbidden Cliques == Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can have, if <math>G</math> has some property?" === Mantel's theorem === We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</math>. {{Theorem|Theorem (Mantel 1907)| :Suppose <math>G(V,E)</math> is graph on <math>n</math> vertice without triangles. Then <m..." current
  • 09:3809:38, 14 May 2024 diff hist +158‎ 组合数学 (Spring 2024) ‎ →‎Lecture Notes

30 April 2024

24 April 2024

22 April 2024

  • 03:1403:14, 22 April 2024 diff hist +5,139‎ N 概率论与数理统计 (Spring 2024)/Weierstrass Approximation Theorem ‎ Created page with "[https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem '''魏尔施特拉斯逼近定理''']（'''Weierstrass approximation theorem'''）陈述了这样一个事实：闭区间上的连续函数总可以用多项式一致逼近。 {{Theorem|魏尔施特拉斯逼近定理| :设 <math>f:[a,b]\to\mathbb{R}</math> 为定义在实数区间 <math>[a,b]</math> 上的连续实值函数。对每个 <math>\epsilon>0</math>，存在一个多项式 <math>p</math> 使得对..." current
  • 03:1303:13, 22 April 2024 diff hist +7,005‎ N 概率论与数理统计 (Spring 2024)/Threshold of k-clique in random graph ‎ Created page with "在 Erdős-Rényi 随机图模型 <math>G(n,p)</math> 中，一个随机无向图 <math>G</math> 以如下的方式生成：图 <math>G</math> 包含 <math>n</math> 个顶点，每一对顶点之间都独立同地以概率 <math>p</math> 连一条无向边。如此生成的随机图记为 <math>G\sim G(n,p)</math>。 固定整数 <math>k\ge 3</math>，考虑随机图 <math>G\sim G(n,p)</math> 包含 <math>K_k</math>（<math>k</math>-团，<math>k</math>-clique）子图..." current
  • 03:1203:12, 22 April 2024 diff hist +8,968‎ N 概率论与数理统计 (Spring 2024)/Two-point sampling ‎ Created page with "= 利用线性同余方程构造两两独立的随机变量 = 令<math>p</math>为一质数。考虑模<math>p</math>余数构成的集合<math>[p]=\{0,1,\ldots,p-1\}=\mathbb{Z}_p</math>。众所周知，当<math>p</math>为质数时，<math>\mathbb{Z}_p</math>为对模<math>p</math>加法和乘法运算闭合的'''有限域'''。 我们现在构造一系列值域为<math>[p]</math>的'''两两独立'''（'''pairwise Independent'''）且'''均匀分布'''（'''uniforml..." current
  • 03:1203:12, 22 April 2024 diff hist +325‎ 概率论与数理统计 (Spring 2024) ‎ →‎Lectures

15 April 2024

12 April 2024

9 April 2024

  • 13:2713:27, 9 April 2024 diff hist +14,440‎ N 组合数学 (Fall 2024)/Existence problems ‎ Created page with "== Existence by Counting == === Shannon's circuit lower bound=== This is a fundamental problem in in Computer Science. A '''boolean function''' is a function in the form <math>f:\{0,1\}^n\rightarrow \{0,1\}</math>. [http://en.wikipedia.org/wiki/Boolean_circuit Boolean circuit] is a mathematical model of computation. Formally, a boolean circuit is a directed acyclic graph. Nodes with indegree zero are input nodes, labeled <math>x_1, x_2, \ldots , x_n</math>. A circuit h..." current
  • 13:2613:26, 9 April 2024 diff hist +17,231‎ N 组合数学 (Fall 2024)/Cayley's formula ‎ Created page with "== Cayley's Formula == We now present a theorem of the number of labeled trees on a fixed number of vertices. It is due to [http://en.wikipedia.org/wiki/Arthur_Cayley Cayley] in 1889. The theorem is often referred by the name [http://en.wikipedia.org/wiki/Cayley's_formula Cayley's formula]. {{Theorem|Cayley's formula for trees| : There are <math>n^{n-2}</math> different trees on <math>n</math> distinct vertices. }} The theorem has several proofs, including the bijectio..." current
  • 13:2613:26, 9 April 2024 diff hist +87‎ 组合数学 (Spring 2024) ‎ →‎Lecture Notes
  • 13:2413:24, 9 April 2024 diff hist +80‎ 组合数学 (Spring 2024) ‎ →‎Lecture Notes

8 April 2024

27 March 2024

26 March 2024

25 March 2024

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