User contributions for Etone

11:3611:36, 13 April 2023 diff hist +17,231‎ N 组合数学 (Fall 2023)/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
11:3611:36, 13 April 2023 diff hist +80‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes
03:2603:26, 13 April 2023 diff hist 0‎ 概率论与数理统计 (Spring 2023)/Average-case analysis of QuickSort ‎ →‎快速排序算法的平均复杂度分析 II（基于期望的线性） current

14:2514:25, 29 March 2023 diff hist +19,696‎ N 组合数学 (Fall 2023)/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
14:2514:25, 29 March 2023 diff hist +105‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes
14:2414:24, 29 March 2023 diff hist −33‎ 概率论与数理统计 (Spring 2023)/Problem Set 2 ‎No edit summary

12:4612:46, 16 March 2023 diff hist +772‎ 组合数学 (Spring 2023) ‎ →‎Concepts
12:4512:45, 16 March 2023 diff hist +33,970‎ N 组合数学 (Fall 2023)/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
12:4512:45, 16 March 2023 diff hist +68‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes

08:3108:31, 3 March 2023 diff hist +24,916‎ N 组合数学 (Fall 2023)/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..."
08:3008:30, 3 March 2023 diff hist +165‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes

13:2913:29, 16 February 2023 diff hist +75‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes
13:2313:23, 16 February 2023 diff hist +853‎ 组合数学 (Spring 2023) ‎ →‎Concepts
13:2213:22, 16 February 2023 diff hist +39,712‎ N 组合数学 (Fall 2023)/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..." current
13:2113:21, 16 February 2023 diff hist +76‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes
10:2610:26, 16 February 2023 diff hist 0‎ N File:Stanley-2e.jpg ‎No edit summary current
10:2610:26, 16 February 2023 diff hist +3‎ 组合数学 (Spring 2023)/Course materials ‎ →‎References current

15:2315:23, 29 November 2022 diff hist +2,459‎ N Assignment 4, Fall 2022 ‎ 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
15:2215:22, 29 November 2022 diff hist +54‎ General Circulation(Fall 2022) ‎ →‎Assignments

05:5105:51, 23 November 2022 diff hist +822‎ N Namelist Assignment 3 2022 ‎ Created page with "姓名学号周榆杰 191870288 罗健峰 191830098 欧阳玥芸 191830111 代欣怡 652022280002 丁宇轩 502022280004 窦吉斌 502022280006 方旭 502022280008 付凯杰 502022280009 何世诚 502022280011 胡书挥 502022280012 李晨曦 652022280005 李惠林 502022280016 李颜昀 502022280018 李颖 502022280019 林齐颢 502022280021 刘洪江 502022280024 刘芷含 502022280026 闵隽玉 502022280028 倪乘岚 5020222..." current
05:5105:51, 23 November 2022 diff hist +55‎ General Circulation(Fall 2022) ‎ →‎Assignments

15:4515:45, 19 November 2022 diff hist +33,826‎ N 高级算法 (Fall 2022)/Greedy and Local Search ‎ Created page with "= Scheduling = We consider the following scheduling problem: * There are <math>n</math> '''jobs''' to be processed. * There are <math>m</math> identical parallel '''machines'''. Each machine can start processing jobs at time 0 and can process at most one job at a time. * Each job <math>j=1,2,\ldots, n</math> must be processed on one of these machines for <math>p_j</math> time units without interruption. <math>p_j</math> is called the '''processing time''' of job <math>j..." current
15:4215:42, 19 November 2022 diff hist +522‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes

11:2311:23, 9 November 2022 diff hist +130‎ General Circulation(Fall 2022) ‎ →‎Course Slides
11:2211:22, 9 November 2022 diff hist +69‎ General Circulation(Fall 2022) ‎ →‎Announcement
11:1911:19, 9 November 2022 diff hist 0‎ Namelist Assignment 2 2022 ‎No edit summary current
11:1711:17, 9 November 2022 diff hist +822‎ N Namelist Assignment 2 2022 ‎ Created page with "姓名学号周榆杰 191870288 罗健峰 191830098 欧阳玥芸 191830111 代欣怡 652022280002 丁宇轩 502022280004 窦吉斌 502022280006 付凯杰 502022280009 何世诚 502022280011 胡书挥 502022280012 李晨曦 652022280005 李惠林 502022280016 李颜昀 502022280018 李颖 502022280019 林齐颢 502022280021 刘洪江 502022280024 刘芷含 502022280026 闵隽玉 502022280028 倪乘岚 502022280030 孙楚 502022..."
11:1711:17, 9 November 2022 diff hist 0‎ General Circulation(Fall 2022) ‎ →‎Assignments
11:1611:16, 9 November 2022 diff hist +55‎ General Circulation(Fall 2022) ‎ →‎Assignments

10:2310:23, 1 November 2022 diff hist +198‎ General Circulation(Fall 2022) ‎ →‎Announcement
10:1010:10, 1 November 2022 diff hist +1,093‎ N Assignment 3, Fall 2022 ‎ 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
10:0910:09, 1 November 2022 diff hist −11‎ General Circulation(Fall 2022) ‎ →‎Assignments
10:0810:08, 1 November 2022 diff hist +59‎ General Circulation(Fall 2022) ‎ →‎Assignments
10:0710:07, 1 November 2022 diff hist +59‎ General Circulation(Fall 2022) ‎ →‎Assignments

04:5504:55, 27 October 2022 diff hist +22‎ Namelist Assignment 1 2022 ‎No edit summary current
02:4002:40, 27 October 2022 diff hist +33‎ Namelist Assignment 1 2022 ‎No edit summary
02:3902:39, 27 October 2022 diff hist +789‎ N Namelist Assignment 1 2022 ‎ Created page with "姓名学号周榆杰 191870288 罗健峰 191830098 欧阳玥芸 191830111 孙森灿 191830136 代欣怡 652022280002 丁宇轩 502022280004 窦吉斌 502022280006 付凯杰 502022280009 何世诚 502022280011 胡书挥 502022280012 李晨曦 652022280005 李惠林 502022280016 李颜昀 502022280018 李颖 502022280019 林齐颢 502022280021 刘洪江 502022280024 刘芷含 502022280026 闵隽玉 502022280028 倪乘岚 502022280030 孙楚 5020..."
02:3102:31, 27 October 2022 diff hist +56‎ General Circulation(Fall 2022) ‎ →‎Assignments

09:3209:32, 24 October 2022 diff hist +16,764‎ N 高级算法 (Fall 2022)/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..." current
09:3209:32, 24 October 2022 diff hist 0‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
09:3209:32, 24 October 2022 diff hist +523‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes

12:5312:53, 11 October 2022 diff hist +10,715‎ N 高级算法 (Fall 2022)/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..." current
12:5312:53, 11 October 2022 diff hist +41,013‎ N 高级算法 (Fall 2022)/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..." current
12:5212:52, 11 October 2022 diff hist +415‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
05:5405:54, 11 October 2022 diff hist −216‎ 高级算法 (Fall 2021) ‎ →‎Assignments current
05:5305:53, 11 October 2022 diff hist −66‎ 高级算法 (Fall 2021) ‎ →‎Assignments

02:5502:55, 5 October 2022 diff hist +1‎ General Circulation(Fall 2022) ‎ →‎Assignments
02:5502:55, 5 October 2022 diff hist +16‎ General Circulation(Fall 2022) ‎ →‎Assignments
02:5202:52, 5 October 2022 diff hist +2‎ Assignment 2, Fall 2022 ‎ →‎Question 2 current
02:5102:51, 5 October 2022 diff hist 0‎ Assignment 2, Fall 2022 ‎No edit summary
02:5102:51, 5 October 2022 diff hist +2,704‎ N Assignment 2, Fall 2022 ‎ 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>\begin{align} Q&=Q_o \cdot s(x), \\ s(x)&=s_o \cdot P_o(x)+ s_2 \cdot P_2(x), \end{align}</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>, <..."
02:4902:49, 5 October 2022 diff hist +66‎ General Circulation(Fall 2022) ‎ →‎Assignments
02:4702:47, 5 October 2022 diff hist −99‎ Main Page ‎No edit summary

15:5615:56, 3 October 2022 diff hist −1‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
15:5515:55, 3 October 2022 diff hist −5‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
15:5415:54, 3 October 2022 diff hist +217‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
15:5115:51, 3 October 2022 diff hist +97‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
15:4915:49, 3 October 2022 diff hist +5,755‎ N 高级算法 (Fall 2022)/Basic tail 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..." current
15:4815:48, 3 October 2022 diff hist +25,717‎ N 高级算法 (Fall 2022)/Hashing and Sketching ‎ Created page with "=Distinct Elements= Consider the following problem of '''counting distinct elements''': Suppose that <math>\Omega</math> is a sufficiently large universe. *'''Input:''' a sequence of (not necessarily distinct) elements <math>x_1,x_2,\ldots,x_n\in\Omega</math>; *'''Output:''' an estimation of the total number of distinct elements <math>z=|\{x_1,x_2,\ldots,x_n\}|</math>. A straightforward way of solving this problem is to maintain a dictionary data structure, which costs..." current
15:4715:47, 3 October 2022 diff hist +1‎ 高级算法 (Fall 2022)/Limited independence ‎ →‎2-universal hash families current
15:4515:45, 3 October 2022 diff hist +8,181‎ 高级算法 (Fall 2022)/Balls into bins ‎No edit summary current
15:4515:45, 3 October 2022 diff hist +1‎ 高级算法 (Fall 2022)/Balls into bins ‎ →‎2-universal hash families
15:3915:39, 3 October 2022 diff hist +13,598‎ 高级算法 (Fall 2022)/Balls into bins ‎No edit summary
15:3815:38, 3 October 2022 diff hist +5,762‎ 高级算法 (Fall 2022)/Balls into bins ‎ →‎Coupon Collector
15:3615:36, 3 October 2022 diff hist +157‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes

08:5408:54, 26 September 2022 diff hist +175‎ General Circulation(Fall 2022) ‎ →‎Assignments
08:5308:53, 26 September 2022 diff hist 0‎ General Circulation(Fall 2022) ‎ →‎Assignments
08:3308:33, 26 September 2022 diff hist +407‎ General Circulation(Fall 2022) ‎ →‎Course Slides
08:2808:28, 26 September 2022 diff hist +3‎ Assignment 1, Fall 2022 ‎ →‎Bonus Question
08:2808:28, 26 September 2022 diff hist +228‎ Assignment 1, Fall 2022 ‎No edit summary
08:1308:13, 26 September 2022 diff hist +6‎ Assignment 1, Fall 2022 ‎ →‎On the netcdf file
08:0908:09, 26 September 2022 diff hist −17‎ Assignment 1, Fall 2022 ‎ →‎下载区
08:0808:08, 26 September 2022 diff hist +2,560‎ N Assignment 1, Fall 2022 ‎ 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..."
08:0708:07, 26 September 2022 diff hist −24‎ General Circulation(Fall 2021) ‎No edit summary current
08:0508:05, 26 September 2022 diff hist +5,196‎ General Circulation(Fall 2022) ‎No edit summary
08:0408:04, 26 September 2022 diff hist +1,062‎ General Circulation(Fall 2022) ‎No edit summary
08:0308:03, 26 September 2022 diff hist +2,249‎ N General Circulation(Fall 2022) ‎ 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..."
08:0208:02, 26 September 2022 diff hist +99‎ Main Page ‎No edit summary Tag: Visual edit: Switched

11:4811:48, 21 September 2022 diff hist +15,811‎ N 高级算法 (Fall 2022)/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..."
11:4811:48, 21 September 2022 diff hist +75‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
11:4711:47, 21 September 2022 diff hist +11,376‎ N 高级算法 (Fall 2022)/Balls into bins ‎ 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..."
11:4711:47, 21 September 2022 diff hist +61‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
11:4611:46, 21 September 2022 diff hist 0‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
11:4611:46, 21 September 2022 diff hist −122‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
11:4611:46, 21 September 2022 diff hist +371‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes

08:0708:07, 13 September 2022 diff hist −71‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
05:4305:43, 13 September 2022 diff hist +6,210‎ N 高级算法 (Fall 2022)/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..." current
05:4305:43, 13 September 2022 diff hist +30,540‎ N 高级算法 (Fall 2022)/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..." current
05:4205:42, 13 September 2022 diff hist +73‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
05:4205:42, 13 September 2022 diff hist +134‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes

07:3107:31, 5 September 2022 diff hist +17,328‎ N 高级算法 (Fall 2022)/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..." current
07:3107:31, 5 September 2022 diff hist +71‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes
07:3107:31, 5 September 2022 diff hist +44,285‎ N 高级算法 (Fall 2022)/Min-Cut and Max-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..." current
07:3007:30, 5 September 2022 diff hist +126‎ 高级算法 (Fall 2022) ‎ →‎Lecture Notes

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