User contributions for Etone
Jump to navigation
Jump to search
6 June 2025
- 02:3102:31, 6 June 2025 diff hist +34,076 N 组合数学 (Fall 2025)/Matching theory Created page with "== Systems of Distinct Representatives (SDR)== A '''system of distinct representatives (SDR)''' (also called a '''transversal''') for a sequence of (not necessarily distinct) sets <math>S_1,S_2,\ldots,S_m</math> is a sequence of <font color=red>''distinct''</font> elements <math>x_1,x_2,\ldots,x_m</math> such that <math>x_i\in S_i</math> for all <math>i=1,2,\ldots,m</math>. === Hall's marriage theorem === If the sets <math>S_1,S_2,\ldots,S_m</math> have a system of dist..." current
- 02:3002:30, 6 June 2025 diff hist +141 组合数学 (Spring 2025) →Lecture Notes
25 May 2025
- 11:2211:22, 25 May 2025 diff hist 0 组合数学 (Spring 2025) →Lecture Notes
- 11:1911:19, 25 May 2025 diff hist +26,029 N 组合数学 (Fall 2025)/Ramsey theory Created page with "== Ramsey's Theorem == === Ramsey's theorem for graph === {{Theorem|Ramsey's Theorem| :Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :If <math>n\ge R(k,\ell)</math>, for any coloring of edges of <math>K_n</math> with two colors red and blue, there exists a red <math>K_k</math> or a blue <math>K_\ell</math>. }} {{Proof| We show that <math>R(k,\ell)</math> is finite by induction on <math>k+\ell</math>. For the..." current
- 11:1911:19, 25 May 2025 diff hist +137 组合数学 (Spring 2025) →Lecture Notes
- 11:1811:18, 25 May 2025 diff hist +51,460 N 组合数学 (Fall 2025)/Extremal set theory Created page with "== Sunflowers == An set system is a '''sunflower''' if all its member sets intersect at the same set of elements. {{Theorem|Definition (sunflower)| : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size <math>r</math> with a '''core''' <math>C\subseteq X</math> if ::<math>\forall S,T\in\mathcal{F}</math> that <math>S\neq T</math>, <math>S\cap T=C</math>. }} Note that we do not require the core to be nonempty, thus a family of disjoint sets is..." current
- 11:1711:17, 25 May 2025 diff hist 0 组合数学 (Spring 2025) →Lecture Notes
- 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
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
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
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
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