User contributions for Etone

25 May 2025

• 11:1711:17, 25 May 2025 diff hist +473 组合数学 (Spring 2025) →Lecture Notes

21 April 2025

• 08:0208:02, 21 April 2025 diff hist +18,939 N 组合数学 (Fall 2025)/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
• 08:0108:01, 21 April 2025 diff hist +158 组合数学 (Spring 2025) →Lecture Notes
• 07:5507:55, 21 April 2025 diff hist +153 概率论与数理统计 (Spring 2025) →Lectures

15 April 2025

• 12:0412:04, 15 April 2025 diff hist +36 概率论与数理统计 (Spring 2025) →Lectures
• 12:0212:02, 15 April 2025 diff hist +2 概率论与数理统计 (Spring 2025)/An exercise on induced distribution No edit summary current
• 12:0112:01, 15 April 2025 diff hist +2,228 N 概率论与数理统计 (Spring 2025)/An exercise on induced distribution Created page with "考虑如下问题：单位圆中独立均匀分布的两点<math>A</math>和<math>B</math>，令线段<math>AB</math>中点为<math>C</math>。问<math>C</math>的概率密度是什么？ 对此，我们做出如下猜测：单位圆中任意点<math>c</math>的概率密度应该正比于如下区域的面积——该区域中所有点及其关于<math>c</math>的对称点均在单位圆中，即<math>c</math>的概率密度正比于如下点集的面积 :<math>\begin{..."
• 11:4011:40, 15 April 2025 diff hist +82 概率论与数理统计 (Spring 2025) →Lectures

13 April 2025

• 04:4204:42, 13 April 2025 diff hist +27,151 N 组合数学 (Fall 2025)/The probabilistic method Created page with "== The Probabilistic Method == The probabilistic method provides another way of proving the existence of objects: instead of explicitly constructing an object, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. The basic principle of the probabilistic method is very simple, and can be stated in intuitive ways: *If an object chosen randomly from a universe satisfies a property with posi..." current
• 04:4204:42, 13 April 2025 diff hist +157 组合数学 (Spring 2025) →Lecture Notes
• 04:4104:41, 13 April 2025 diff hist +268 概率论与数理统计 (Spring 2025) →Lectures

9 April 2025

• 06:5806:58, 9 April 2025 diff hist 0 组合数学 (Spring 2025) →Announcement

1 April 2025

• 10:3310:33, 1 April 2025 diff hist +5,139 N 概率论与数理统计 (Spring 2025)/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
• 10:3310:33, 1 April 2025 diff hist +7,005 N 概率论与数理统计 (Spring 2025)/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
• 10:3210:32, 1 April 2025 diff hist +8,968 N 概率论与数理统计 (Spring 2025)/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
• 10:3210:32, 1 April 2025 diff hist +325 概率论与数理统计 (Spring 2025) →Lectures

26 March 2025

• 09:5809:58, 26 March 2025 diff hist +14,440 N 组合数学 (Fall 2025)/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
• 09:5709:57, 26 March 2025 diff hist +151 组合数学 (Spring 2025) →Lecture Notes
• 09:5609:56, 26 March 2025 diff hist +17,231 N 组合数学 (Fall 2025)/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
• 09:5609:56, 26 March 2025 diff hist +141 组合数学 (Spring 2025) →Lecture Notes
• 09:5509:55, 26 March 2025 diff hist +189 概率论与数理统计 (Spring 2025) →Lectures

21 March 2025

• 10:0310:03, 21 March 2025 diff hist 0 概率论与数理统计 (Spring 2025)/Average-case analysis of QuickSort →快速排序算法的平均复杂度分析 I（基于全期望法则） current
• 10:0310:03, 21 March 2025 diff hist +11,838 N 概率论与数理统计 (Spring 2025)/Average-case analysis of QuickSort Created page with "[http://en.wikipedia.org/wiki/Quicksort '''快速排序'''（'''Quicksort'''）]是由Tony Hoare发现的排序算法。该算法的伪代码描述如下（为方便起见，假设数组元素互不相同——更一般情况的分析易推广得到）： '''''QSort'''''(A): 输入A[1...n]是存有n个不同数字的数组 if n>1 then '''pivot''' = A[1]; 将A中<pivot的元素存于数组L，将A中>pivot的元素存于数组R; \\保持内部元素之..."
• 10:0210:02, 21 March 2025 diff hist +421 概率论与数理统计 (Spring 2025) →Lectures
• 09:5809:58, 21 March 2025 diff hist +19,696 N 组合数学 (Fall 2025)/Pólya's theory of counting Created page with "== Groups == A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisfies the following axioms: * ''closure'': <math>\forall g,h\in G, g\cdot h \in G</math>; * ''associativity'': <math>\forall f,g,h\in G, f\cdot(g\cdot h)=(f\cdot g)\cdot h</math>; * ''identity'': there exists a special element <math>e\in G</math>, called the '''identity''', such that <math>e\cdot g=g</math> for any <math>g\in G</math>; * ''inverse''..." current

16 March 2025

• 09:3109:31, 16 March 2025 diff hist +165 组合数学 (Spring 2025) →Lecture Notes

12 March 2025

• 16:0716:07, 12 March 2025 diff hist +33,970 N 组合数学 (Fall 2025)/Sieve methods Created page with "== Principle of Inclusion-Exclusion == Let <math>A</math> and <math>B</math> be two finite sets. The cardinality of their union is :<math>|A\cup B|=|A|+|B|-{\color{Blue}|A\cap B|}</math>. For three sets <math>A</math>, <math>B</math>, and <math>C</math>, the cardinality of the union of these three sets is computed as :<math>|A\cup B\cup C|=|A|+|B|+|C|-{\color{Blue}|A\cap B|}-{\color{Blue}|A\cap C|}-{\color{Blue}|B\cap C|}+{\color{Red}|A\cap B\cap C|}</math>. This is illu..." current
• 16:0716:07, 12 March 2025 diff hist +125 组合数学 (Spring 2025) →Lecture Notes

5 March 2025

• 05:4505:45, 5 March 2025 diff hist +179 概率论与数理统计 (Spring 2025) →Lectures

3 March 2025

• 07:3607:36, 3 March 2025 diff hist +72 组合数学 (Spring 2025) →Lecture Notes
• 07:3107:31, 3 March 2025 diff hist +32 概率论与数理统计 (Spring 2025)/Entropy and volume of Hamming balls No edit summary current
• 07:3007:30, 3 March 2025 diff hist +217 概率论与数理统计 (Spring 2025)/Entropy and volume of Hamming balls No edit summary
• 07:2707:27, 3 March 2025 diff hist +2,803 N 概率论与数理统计 (Spring 2025)/Entropy and volume of Hamming balls Created page with "在求解抛掷公平硬币（fair coin）的尾概率时，我们经常会需要分析如下二项式系数求和： :<math>\sum_{k=0}^r{n\choose k}</math>，对于某个<math>1\le r\le n</math> 这其实等价与求一个 <math>n</math> 维汉明空间中半径为 <math>r</math> 的球的体积。 :{|border="2" width="100%" cellspacing="4" cellpadding="3" rules="all" style="margin:1em 1em 1em 0; border:solid 1px #AAAAAA; border-collapse:collapse;empty-cells:show;" | :'..."
• 07:2607:26, 3 March 2025 diff hist +118 概率论与数理统计 (Spring 2025) →Lectures

25 February 2025

• 12:0912:09, 25 February 2025 diff hist +25,549 N 组合数学 (Fall 2025)/Generating functions Created page with "== Generating Functions == In Stanley's magnificent book ''Enumerative Combinatorics'', he comments the generating function as "the most useful but most difficult to understand method (for counting)". The solution to a counting problem is usually represented as some <math>a_n</math> depending a parameter <math>n</math>. Sometimes this <math>a_n</math> is called a ''counting function'' as it is a function of the parameter <math>n</math>. <math>a_n</math> can also be tre..." current
• 12:0912:09, 25 February 2025 diff hist +88 组合数学 (Spring 2025) →Lecture Notes

19 February 2025

• 08:5808:58, 19 February 2025 diff hist +69 组合数学 (Spring 2025) →Lecture Notes
• 08:5408:54, 19 February 2025 diff hist +77 Main Page →Home Pages for Courses and Seminars
• 08:5108:51, 19 February 2025 diff hist +125 概率论与数理统计 (Spring 2025) →Lectures

18 February 2025

• 08:5008:50, 18 February 2025 diff hist +75 Main Page →Home Pages for Courses and Seminars

16 February 2025

• 12:2012:20, 16 February 2025 diff hist 0 概率论与数理统计 (Spring 2025) →Lectures
• 12:1512:15, 16 February 2025 diff hist −75 Main Page →Home Pages for Courses and Seminars

20 December 2024

• 08:1708:17, 20 December 2024 diff hist 0 高级算法 (Fall 2024) →Lecture Notes
• 08:1608:16, 20 December 2024 diff hist +301 高级算法 (Fall 2024) →Lecture Notes

9 December 2024

• 01:4801:48, 9 December 2024 diff hist +2,459 N Assignment 4, Fall 2024 Created page with "在第四章中，我们从准地转近似下的纬向平均风场、温度场的趋势方程出发，定义了E-P通量。但是该定义下的E-P通量并没有考虑到大气湿过程的影响。如果从第三章介绍的水汽方程出发，我们可以按照以下步骤定义出一个包含大气大尺度运动中湿过程作用的广义的E-P通量。 1）在准地转近似下，如果我们按照对热力学方程的简化方法，将比湿（specific humidity..." current
• 01:4801:48, 9 December 2024 diff hist +54 General Circulation(Fall 2024) →Assignments current

10 November 2024

• 14:3814:38, 10 November 2024 diff hist +1,093 N Assignment 3, Fall 2024 Created page with "Held-Hou(1980) 讨论了当外部强迫的经向分布呈二次勒让德多项式，即<math>\dfrac{\Theta_E(\phi,z)}{\Theta_o}=1-\dfrac{2}{3}\Delta_H P_2(\sin \phi)+\Delta_v(\dfrac{z}{H}-\dfrac{1}{2})</math>的情况下，哈德莱环流内的风场、温度场、环流的空间范围等将怎样随纬度和外力强迫的强度而变化。 如果将外力强迫的空间分布改为<math>\dfrac{\Theta_E(\phi,z)}{\Theta_o}=1-\Delta_H(\sin^3 \phi - \dfrac{1}{4})+\Delta_v..." current
• 14:3714:37, 10 November 2024 diff hist +59 General Circulation(Fall 2024) →Assignments

21 October 2024

• 23:5923:59, 21 October 2024 diff hist +1 Namelist Assignment 1 2024 No edit summary current
• 23:5923:59, 21 October 2024 diff hist +1 Namelist Assignment 1 2024 No edit summary