User contributions for Etone

16 October 2025

• 04:0304:03, 16 October 2025 diff hist 0 高级算法 (Fall 2025) →Lecture Notes Tag: Reverted
• 04:0004:00, 16 October 2025 diff hist +322 高级算法 (Fall 2025) →Lecture Notes
• 03:5903:59, 16 October 2025 diff hist +30 高级算法 (Fall 2025) →Assignments

12 October 2025

• 09:2609:26, 12 October 2025 diff hist +16 General Circulation(Fall 2025) →Assignments

29 September 2025

• 03:4503:45, 29 September 2025 diff hist +16,764 N 高级算法 (Fall 2025)/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
• 03:4403:44, 29 September 2025 diff hist −151 高级算法 (Fall 2025) →Lecture Notes
• 03:4403:44, 29 September 2025 diff hist +504 高级算法 (Fall 2025) →Lecture Notes
• 03:4203:42, 29 September 2025 diff hist +14 高级算法 (Fall 2025) →Course info
• 03:0703:07, 29 September 2025 diff hist +2,522 N Assignment 2, Fall 2025 Created page with "<font color="red" size="2">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。</font> ==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..." current
• 03:0703:07, 29 September 2025 diff hist +66 General Circulation(Fall 2025) →Assignments

22 September 2025

• 14:3114:31, 22 September 2025 diff hist +2,971 N Assignment 1, Fall 2025 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..." current
• 14:3114:31, 22 September 2025 diff hist +265 General Circulation(Fall 2025) →Assignments
• 02:5002:50, 22 September 2025 diff hist +10,715 N 高级算法 (Fall 2025)/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
• 02:4902:49, 22 September 2025 diff hist +41,013 N 高级算法 (Fall 2025)/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
• 02:4902:49, 22 September 2025 diff hist +229 高级算法 (Fall 2025) →Lecture Notes

10 September 2025

• 10:4510:45, 10 September 2025 diff hist +256 高级算法 (Fall 2025) →Announcement

9 September 2025

• 05:0505:05, 9 September 2025 diff hist +5,755 N 高级算法 (Fall 2025)/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..." current
• 05:0405:04, 9 September 2025 diff hist +15,812 N 高级算法 (Fall 2025)/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..." current
• 05:0405:04, 9 September 2025 diff hist +49,292 N 高级算法 (Fall 2025)/Hashing and Sketching Created page with "=Balls into Bins= The following is the so-called balls into bins model. 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..." current
• 05:0305:03, 9 September 2025 diff hist +304 高级算法 (Fall 2025) →Lecture Notes

7 September 2025

• 13:1913:19, 7 September 2025 diff hist +6,210 N 高级算法 (Fall 2025)/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<br>rin..." current
• 13:1913:19, 7 September 2025 diff hist +38,283 N 高级算法 (Fall 2025)/Fingerprinting Created page with "= Checking Matrix Multiplication= thumb|360px|right|The evolution of time complexity <math>O(n^{\omega})</math> for matrix multiplication. Let <math>\mathbb{F}</math> be a feild (you may think of it as the filed <math>\mathbb{Q}</math> of rational numbers, or the finite field <math>\mathbb{Z}_p</math> of integers modulo prime <math>p</math>). We suppose that each field operation (addition, subtraction, multiplication, division) has u..." current
• 13:1513:15, 7 September 2025 diff hist +203 高级算法 (Fall 2025) →Lecture Notes
• 11:3711:37, 7 September 2025 diff hist −1 General Circulation(Fall 2025) →Course info
• 11:3611:36, 7 September 2025 diff hist 0 General Circulation(Fall 2025) No edit summary
• 11:3611:36, 7 September 2025 diff hist +8,044 N General Circulation(Fall 2025) Created page with "{{Infobox |name = Infobox |bodystyle = |title = 大气环流 <br> 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..."
• 11:3311:33, 7 September 2025 diff hist 0 Main Page No edit summary

26 August 2025

• 09:0809:08, 26 August 2025 diff hist +17,328 N 高级算法 (Fall 2025)/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
• 09:0809:08, 26 August 2025 diff hist +49,923 N 高级算法 (Fall 2025)/Min Cut, Max Cut, and Spectral Cut Created page with "= Graph Cut = Let <math>G(V, E)</math> be an undirected graph. 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 defined by a bipartition <math>\{S,T\}</math> of <math>V</math> as :<math>C=E(S,T)\,</math>, where <math>E(S,T)</math> denotes the set of "crossing edges" with one endpoint in each of <math>S</math> and..." current
• 09:0709:07, 26 August 2025 diff hist +227 高级算法 (Fall 2025) →Lecture Notes

25 August 2025

• 05:0105:01, 25 August 2025 diff hist +63 高级算法 (Fall 2025) →Course info
• 04:5904:59, 25 August 2025 diff hist 0 高级算法 (Fall 2025) →Course info
• 04:5804:58, 25 August 2025 diff hist +6 高级算法 (Fall 2025) →Course info

23 August 2025

• 05:3805:38, 23 August 2025 diff hist −75 Main Page →Home Pages for Courses and Seminars
• 05:3805:38, 23 August 2025 diff hist −83 Main Page →Home Pages for Courses and Seminars

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