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02:57, 24 April 2024 Etone talk contribs created page 组合数学 (Fall 2024)/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...")
03:14, 22 April 2024 Etone talk contribs created page 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...")
03:13, 22 April 2024 Etone talk contribs created page 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...")
03:12, 22 April 2024 Etone talk contribs created page 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...")
13:27, 9 April 2024 Etone talk contribs created page 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...")
13:26, 9 April 2024 Etone talk contribs created page 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...")
08:41, 8 April 2024 Etone talk contribs created page 概率论与数理统计 (Spring 2024)/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...")
04:27, 27 March 2024 Etone talk contribs created page 组合数学 (Fall 2024)/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''...")
02:21, 20 March 2024 Etone talk contribs created page Assignment 1, Spring 2024 (Created page with "==Question #1== 请使用多年（>20年）的NCEP/NCAR月平均再分析资料，画出各季节（至少画出冬夏两季）纬向平均温度场、纬向平均纬向风场的高度－纬度剖面分布，并简述其分布特征和季节变化特征。 ==Question #2== 请使用多年（>20年）的NCEP/NCAR月平均再分析资料，画出各季节（至少画出冬夏两季）温度场、纬向风场在各高度（850、500、100 hPa，对于温度场请再画...")
15:55, 19 March 2024 Etone talk contribs created page General Circulation(Spring 2024) (Created page with "{{Infobox |name = Infobox |bodystyle = |title = 大气环流 General Circulation of the Atmosphere |titlestyle = |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 =Instructor |label1 = |data1 = |header2 = |label2 = |data2 = 张洋 |header3 = |label3 = Email |data3 = yangzhang@nju.edu.cn |header4 = |label4= office |data4= 仙林大气楼 B410 |header5 = Class |label5 = |data5 = |hea...")
09:51, 19 March 2024 Etone talk contribs created page 组合数学 (Fall 2024)/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...")
15:51, 18 March 2024 Etone talk contribs created page 概率论与数理统计 (Spring 2024)/Karger's min-cut algorithm (Created page with "Let <math>G(V, E)</math> be an undirected graph. A subset <math>C\subseteq E</math> of edges is a '''cut''' of graph <math>G</math> if <math>G</math> becomes ''disconnected'' after deleting all edges in <math>C</math>. Let <math>\{S,T\}</math> be a '''bipartition''' of <math>V</math> into nonempty subsets <math>S,T\subseteq V</math>, where <math>S\cap T=\emptyset</math> and <math>S\cup T=V</math>. A cut <math>C</math> is specified by this bipartition as :<math>C=E(S,T...")
02:58, 12 March 2024 Etone talk contribs deleted page 量子计算的算法与物理实现研讨会 (content was: "==时间地点 == *''' 2019年10月19、20日: 9 am--5 pm.''' * 南京大学计算机科学与技术系（南京大学仙林校区常州楼）111报告厅 [http://map.baidu.com/?l=&s=s%26wd%3D南京仙林大道163号++南京大学（仙林校区）计算机系楼在线地图] == 日程安...", and the only contributor was "[[Special:Contributions/imported>Qinminglong|imported>Qinminglong]]" ([[User talk:imported>Qinminglong|talk]]))
02:56, 12 March 2024 Etone talk contribs created page Quantum (Created page with "= 量子算法与物理实现研讨会= ==时间地点 == *''' 2019年10月19、20日: 9 am--5 pm.''' * 南京大学计算机科学与技术系（南京大学仙林校区常州楼）111报告厅 [http://map.baidu.com/?l=&s=s%26wd%3D南京仙林大道163号++南京大学（仙林校区）计算机系楼在线地图] ==联系人与联系方式 == * 姚鹏晖 pyao@nju.edu.cn * 张利剑 lijian.zhang@nju.edu.cn ==会议主旨...")
02:56, 12 March 2024 Etone talk contribs deleted page Quantum (content was: "= 量子算法与物理实现研讨会= ==时间地点 == *''' 2019年10月19、20日: 9 am--5 pm.''' * 南京大学计算机科学与技术系（南京大学仙林校区常州楼）111报告厅 [http://map.baidu.com/?l=&s=s%26wd%3D南京仙林大道163号++南京大学（仙林校区）计...", and the only contributor was "[[Special:Contributions/imported>Qinminglong|imported>Qinminglong]]" ([[User talk:imported>Qinminglong|talk]]))
11:41, 6 March 2024 Etone talk contribs created page 组合数学 (Fall 2024)/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...")
10:39, 27 February 2024 Etone talk contribs created page 组合数学 (Fall 2024)/Basic enumeration (Created page with "== Basic Enumeration == The three basic rules for enumeration are: *'''The sum rule''': for any '''''disjoint''''' finite sets <math>S</math> and <math>T</math>, the cardinality of the union <math>|S\cup T|=|S|+|T|</math>. *'''The product rule''': for any finite sets <math>S</math> and <math>T</math>, the cardinality of the Cartesian product <math>|S\times T|=|S|\cdot|T|</math>. *'''The bijection rule''': if there exists a bijection between finite sets <math>S</math> a...")
07:40, 18 February 2024 Etone talk contribs created page 组合数学 (Spring 2024)/Course materials (Created page with "== Textbooks == {|border="2" cellspacing="4" cellpadding="3" rules="all" style="margin:1em 1em 1em 0; border:solid 1px #AAAAAA; border-collapse:collapse;empty-cells:show;" |border|100px |width="100%"| :J. H. van Lint and R. M. Wilson. ''A course in Combinatorics, 2nd Edition.'' Cambridge University Press, 2001. |- |border|100px|| :S. Jukna. ''Extremal Combinatorics: With Applications in Computer Science, 2nd Edition...")
07:38, 18 February 2024 Etone talk contribs created page 组合数学 (Spring 2024) (Created page with "{{Infobox |name = Infobox |bodystyle = |title = 组合数学 Combinatorics |titlestyle = |image = |imagestyle = |caption = |captionstyle = |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 =Instructor |label1 = |data1 = |header2 = |label2 = |data2 = 尹一通 |header3 = |label3 = Email |data3 = yinyt@nju.edu.cn |header4 = |label4= office |data4=...")
07:33, 18 February 2024 Etone talk contribs created page 概率论与数理统计 (Spring 2024) (Created page with "{{Infobox |name = Infobox |bodystyle = |title = '''概率论与数理统计''' '''Probability Theory''' & '''Mathematical Statistics''' |titlestyle = |image = |imagestyle = |caption = |captionstyle = |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 =Instructor |label1 = |data1 = |header2 = |label2 = |data2 = '''尹一通''' |header3 = |label3 = Em...")
16:33, 29 November 2023 Etone talk contribs created page Assignment 4, Fall 2023 (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...")
06:12, 6 November 2023 Etone talk contribs created page Assignment 3, Fall 2023 (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 \phi -\frac{1}{2})+\Delta_v(\dfra...")
06:05, 6 November 2023 Etone talk contribs created page Namelist Assignment 2 2023 (Created page with "姓名学号魏斯诚 201830012 潘云逸 201830049 李宜卓 201830062 王若骥 201830143 吴佳根 201830196 王奕 201840022 张顺伟 201840065 范禹韬 201840173 谭金林 201840222 张宁萌 201840254 武越 201840285 谢卫宏 201850142 艾真珍 502023280001 曹珩 502023280003 岑琦胜 502023280004 陈石 502023280006 成柯蒙 502023280007 丁思聪 502023280010 樊浩然 502023280011 费照轩 502023...")
12:34, 24 October 2023 Etone talk contribs created page Namelist Assignment 1 2023 (Created page with "姓名学号魏斯诚 201830012 潘云逸 201830049 李宜卓 201830062 王若骥 201830143 吴佳根 201830196 王奕 201840022 张顺伟 201840065 范禹韬 201840173 谭金林 201840222 张宁萌 201840254 武越 201840285 谢卫宏 201850142 樊浩然 501023280011 温康 501023280041 艾真珍 502023280001 曹珩 502023280003 岑琦胜 502023280004 陈石 502023280006 成柯蒙 502023280007 丁思聪 502023280010...")
07:51, 21 October 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Dimension Reduction (Created page with "= Metric Embedding= A '''metric space''' is a pair <math>(X,d)</math>, where <math>X</math> is a set and <math>d</math> is a '''metric''' (or '''distance''') on <math>X</math>, i.e., a function :<math>d:X^2\to\mathbb{R}_{\ge 0}</math> such that for any <math>x,y,z\in X</math>, the following axioms hold: # (identity of indiscernibles) <math>d(x,y)=0\Leftrightarrow x=y</math> # (symmetry) <math>d(x,y)=d(y,x)</math> # (triangle inequality) <math>d(x,z)\le d(x,y)+d(y,z)</ma...")
07:38, 15 October 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Hashing and Sketching (Created page with "=Balls into Bins= Consider throwing <math>m</math> balls into <math>n</math> bins uniformly and independently at random. This is equivalent to a random mapping <math>f:[m]\to[n]</math>. Needless to say, random mapping is an important random model and may have many applications in Computer Science, e.g. hashing. We are concerned with the following three questions regarding the balls into bins model: * birthday problem: the probability that every bin contains at most one...")
05:58, 11 October 2023 Etone talk contribs created page Assignment 2, Fall 2023 (Created page with "请在题目一和题目二中任选一题作为本章的作业题目。另外的一题可作为选做题。 ==Question 1== 假设在大气层顶（TOA），在多年全年平均的情况下，入射的太阳辐射随纬度的分布满足 <math>Q=Q_o \cdot s(x)</math>, <math>s(x)=s_o \cdot P_o(x)+ s_2 \cdot P_2(x)</math>，其中，<math>P_o(x)=1</math>, <math>P_2(x)=\frac{1}{2}(3x^2-1)</math>, <math>s_o=1</math>, <math>s_2=-0.473</ma...")
10:53, 10 October 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Concentration of measure (Created page with "=Chernoff Bound= Suppose that we have a fair coin. If we toss it once, then the outcome is completely unpredictable. But if we toss it, say for 1000 times, then the number of HEADs is very likely to be around 500. This phenomenon, as illustrated in the following figure, is called the '''concentration''' of measure. The Chernoff bound is an inequality that characterizes the concentration phenomenon for the sum of independent trials. File:Coinflip.png|border|450px|cent...")
10:53, 10 October 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Conditional expectations (Created page with "= Conditional Expectations = The '''conditional expectation''' of a random variable <math>Y</math> with respect to an event <math>\mathcal{E}</math> is defined by :<math> \mathbf{E}[Y\mid \mathcal{E}]=\sum_{y}y\Pr[Y=y\mid\mathcal{E}]. </math> In particular, if the event <math>\mathcal{E}</math> is <math>X=a</math>, the conditional expectation :<math> \mathbf{E}[Y\mid X=a] </math> defines a function :<math> f(a)=\mathbf{E}[Y\mid X=a]. </math> Thus, <math>\mathbf{E}[Y\mid...")
10:52, 4 October 2023 Etone talk contribs created page Assignment 1, Fall 2023 (Created page with "==Question #1== 请使用多年（>20年）的NCEP/NCAR月平均再分析资料，画出各季节（至少画出冬夏两季）纬向平均温度场、纬向平均纬向风场的高度－纬度剖面分布，并简述其分布特征和季节变化特征。 ==Question #2== 请使用多年（>20年）的NCEP/NCAR月平均再分析资料，画出各季节（至少画出冬夏两季）温度场、纬向风场在各高度（850、500、100 hPa，对于温度场请再画...")
16:39, 26 September 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Basic deviation inequalities (Created page with "=Markov's Inequality= One of the most natural information about a random variable is its expectation, which is the first moment of the random variable. Markov's inequality draws a tail bound for a random variable from its expectation. {{Theorem |Theorem (Markov's Inequality)| :Let <math>X</math> be a random variable assuming only nonnegative values. Then, for all <math>t>0</math>, ::<math>\begin{align} \Pr[X\ge t]\le \frac{\mathbf{E}[X]}{t}. \end{align}</math> }} {{Proo...")
16:38, 26 September 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Limited independence (Created page with "= <math>k</math>-wise independence = Recall the definition of independence between events: {{Theorem |Definition (Independent events)| :Events <math>\mathcal{E}_1, \mathcal{E}_2, \ldots, \mathcal{E}_n</math> are '''mutually independent''' if, for any subset <math>I\subseteq\{1,2,\ldots,n\}</math>, ::<math>\begin{align} \Pr\left[\bigwedge_{i\in I}\mathcal{E}_i\right] &= \prod_{i\in I}\Pr[\mathcal{E}_i]. \end{align}</math> }} Similarly, we can define independence between...")
10:58, 19 September 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Finite Field Basics (Created page with "=Field= Let <math>S</math> be a set, '''closed''' under binary operations <math>+</math> (addition) and <math>\cdot</math> (multiplication). It gives us the following algebraic structures if the corresponding set of axioms are satisfied. {|class="wikitable" !colspan="7"|Structures !Axioms !Operations |- |rowspan="9" style="background-color:#ffffcc;text-align:center;"|'''''field''''' |rowspan="8" style="background-color:#ffffcc;text-align:center;"|'''''commutative rin...")
10:57, 19 September 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Fingerprinting (Created page with "=Polynomial Identity Testing (PIT) = The '''Polynomial Identity Testing (PIT)''' is such a problem: given as input two polynomials, determine whether they are identical. It plays a fundamental role in ''Identity Testing'' problems. First, let's consider the univariate ("one variable") case: * '''Input:''' two polynomials <math>f, g\in\mathbb{F}[x]</math> of degree <math>d</math>. * Determine whether <math>f\equiv g</math> (<math>f</math> and <math>g</math> are identica...")
01:25, 14 September 2023 Etone talk contribs created page General Circulation(Fall 2023) (Created page with "{{Infobox |name = Infobox |bodystyle = |title = 大气环流 General Circulation of the Atmosphere |titlestyle = |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 =Instructor |label1 = |data1 = |header2 = |label2 = |data2 = 张洋 |header3 = |label3 = Email |data3 = yangzhang@nju.edu.cn |header4 = |label4= office |data4= 仙林大气楼 B410 |header5 = Class |label5 = |data5 = |hea...")
10:36, 12 September 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Min Cut, Max Cut, and Spectral Cut (Created page with "= Graph Cut = Let <math>G(V, E)</math> be an undirected graph. A subset <math>C\subseteq E</math> of edges is a '''cut''' of graph <math>G</math> if <math>G</math> becomes ''disconnected'' after deleting all edges in <math>C</math>. Let <math>\{S,T\}</math> be a '''bipartition''' of <math>V</math> into nonempty subsets <math>S,T\subseteq V</math>, where <math>S\cap T=\emptyset</math> and <math>S\cup T=V</math>. A cut <math>C</math> is specified by this bipartition as...")
10:34, 12 September 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Probability Basics (Created page with "=Probability Space= The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one of the greatest mathematician of the 20th century, who advanced various very different fields of mathematics. {{Theorem|Definition (Probability Space)| A '''probability space''' is a triple <math>(\Omega,\Sigma,\Pr)</math>. *<math>\Omega</math> is a set, called the '''sample space'''. *<math>\Sigma\subseteq 2^{\Omega}</math> is the...")
10:33, 12 September 2023 Etone talk contribs created page 高级算法 (Fall 2023)/Min Cut, Min Cut, and Spectral Cut (Created page with "= Graph Cut = Let <math>G(V, E)</math> be an undirected graph. A subset <math>C\subseteq E</math> of edges is a '''cut''' of graph <math>G</math> if <math>G</math> becomes ''disconnected'' after deleting all edges in <math>C</math>. Let <math>\{S,T\}</math> be a '''bipartition''' of <math>V</math> into nonempty subsets <math>S,T\subseteq V</math>, where <math>S\cap T=\emptyset</math> and <math>S\cup T=V</math>. A cut <math>C</math> is specified by this bipartition as...")
11:44, 10 September 2023 Etone talk contribs created page 高级算法 (Fall 2023) / Course materials (Created page with "= Course textbooks = {|border="2" cellspacing="4" cellpadding="3" rules="all" style="margin:1em 1em 1em 0; border:solid 1px #AAAAAA; border-collapse:collapse;empty-cells:show;" |border|100px |width="100%"| :Rajeev Motwani and Prabhakar Raghavan. :'''''Randomized Algorithms'''''. :Cambridge University Press, 1995. |- |border|100px|| : Vijay Vazirani. :'''''Approximation Algorithms'''''. :S...")
14:01, 3 June 2023 Etone talk contribs created page 概率论与数理统计 (Spring 2023)/OST and applications (Created page with "=可选停时定理= '''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| :令 <ma...")
11:44, 3 June 2023 Etone talk contribs created page 组合数学 (Fall 2023)/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 ''distinct'' 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...")
06:39, 26 May 2023 Etone talk contribs created page 组合数学 (Fall 2023)/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...")
20:08, 24 May 2023 Etone talk contribs created page 概率论与数理统计 (Spring 2023)/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...")
12:27, 24 May 2023 Etone talk contribs created page 概率论与数理统计 (Spring 2023)/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_{1\le k\le 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;...")
12:14, 21 May 2023 Etone talk contribs created page 组合数学 (Fall 2023)/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...")
08:55, 4 May 2023 Etone talk contribs created page 组合数学 (Fall 2023)/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...")
15:53, 27 April 2023 Etone talk contribs created page 组合数学 (Fall 2023)/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...")
12:56, 18 April 2023 Etone talk contribs created page 组合数学 (Fall 2023)/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...")
11:44, 18 April 2023 Etone talk contribs created page 概率论与数理统计 (Spring 2023)/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）子图概...")
18:17, 17 April 2023 Etone talk contribs created page 概率论与数理统计 (Spring 2023)/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：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> 使得对于 <math>[a,b]</math>...")

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