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14:08, 29 November 2024 ‎数据科学基础 (Fall 2024)/Problem Set 6 (hist | edit) ‎[8,428 bytes] ‎Liumingmou (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 *没有条件的同学可以用纸笔完成作业之后拍照。 * '''under construction''' == Assumption throughout Problem Set 6 == Without further notice, we are working on probability space <math>(\Omega,\mathcal{F},\Pr)</math>. Without further notice, we assume that the expectation of random variables are well-d...")
04:31, 29 November 2024 ‎数据科学基础 (Fall 2024)/OST and applications (hist | edit) ‎[18,855 bytes] ‎Liumingmou (talk | contribs) (Created page with "=可选停时定理 (OST)= '''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（...")
08:23, 25 November 2024 ‎高级算法 (Fall 2024)/第二次作业提交名单 (hist | edit) ‎[1,246 bytes] ‎Zhangyiyao (talk | contribs) (Created page with " 如有错漏请邮件联系助教. <center> {| class="wikitable" |- ! 学号 !! 姓名 |- | 211220076 || 杨景越 |- | 211220152 || 吴振宇 |- | 211250001 || 鞠哲 |- | 211250044 || 朱家辰 |- | 211250182 || 胡皓明 |- | 211502020 || 周相羽 |- | 211820202 || 彭钰朝 |- | 211830008 || 缪天顺 |- | 221220002 || 沈均文 |- | 221502010 || 梁志浩 |- | 221840186 || 陈端锐 |- | 221840188 || 谭泽晖 |- | 502023330043 || 吕铸恒 |- | 50202...") Tag: Visual edit: Switched
04:15, 22 November 2024 ‎数据科学基础 (Fall 2024)/Entropy and volume of Hamming balls (hist | edit) ‎[6,589 bytes] ‎Liumingmou (talk | contribs) (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;" | :'...")
04:15, 22 November 2024 ‎数据科学基础 (Fall 2024)/Hoeffding's lemma (hist | edit) ‎[4,639 bytes] ‎Liumingmou (talk | contribs) (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...")
12:56, 18 November 2024 ‎数据科学基础 (Fall 2024)/第四次作业提交名单 (hist | edit) ‎[1,510 bytes] ‎Liumingmou (talk | contribs) (Created page with "<center> {| class="wikitable" |- ! 学号 |- | 221850088 |- | 221850088 |- | 221870109 |- | 221900009 |- | 221900024 |- | 221900028 |- | 221900029 |- | 221900029 |- | 221900036 |- | 221900038 |- | 221900038 |- | 221900042 |- | 221900046 |- | 221900055 |- | 221900068 |- | 221900073 |- | 221900075 |- | 221900083 |- | 221900084 |- | 221900085 |- | 221900091 |- | 221900124 |- | 221900128 |- | 221900140 |- | 221900159 |- | 221900159 |- | 221900168 |- | 221900183 |- | 2219002...")
13:41, 13 November 2024 ‎数据科学基础 (Fall 2024)/Problem Set 5 (hist | edit) ‎[4,769 bytes] ‎Liumingmou (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 *没有条件的同学可以用纸笔完成作业之后拍照。 == Assumption throughout Problem Set 5 == Without further notice, we are working on probability space <math>(\Omega,\mathcal{F},\Pr)</math>. Without further notice, we assume that the expectation of random variables are well-defined. == Problems (...")
14:38, 10 November 2024 ‎Assignment 3, Fall 2024 (hist | edit) ‎[1,093 bytes] ‎Etone (talk | contribs) (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...")
02:40, 9 November 2024 ‎高级算法 (Fall 2024)/Problem Set 2 (hist | edit) ‎[9,114 bytes] ‎Zhangyiyao (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 == Problem 1 == '''(a.)''' Let <math> M_1 = (E_1, \mathcal{I}_1) </math> and <math> M_2 = (E_2, \mathcal{I}_2) </math> be two matroids with <math> E_1 \cap E_2 = \emptyset </math>. Define <math> \mathcal{I}_1 \oplus \mathcal{I}_2 = \{ A_1 \cup A_2 : A_1 \in \mathcal{I}_1, A_2 \in \mathcal{I}_2 \} </math>. Prove that <math> M := (...")
05:25, 4 November 2024 ‎数据科学基础 (Fall 2024)/第三次作业提交名单 (hist | edit) ‎[1,525 bytes] ‎Liumingmou (talk | contribs) (Created page with "<center> {| class="wikitable" |- ! 学号 |- | 221850088 |- | 221870109 |- | 221900009 |- | 221900009 |- | 221900028 |- | 221900029 |- | 221900036 |- | 221900038 |- | 221900042 |- | 221900046 |- | 221900055 |- | 221900068 |- | 221900073 |- | 221900075 |- | 221900083 |- | 221900084 |- | 221900085 |- | 221900091 |- | 221900124 |- | 221900128 |- | 221900140 |- | 221900159 |- | 221900168 |- | 221900183 |- | 221900187 |- | 221900208 |- | 221900209 |- | 221900210 |- | 2219002...")
18:20, 25 October 2024 ‎数据科学基础 (Fall 2024)/第二次作业提交名单 (hist | edit) ‎[1,750 bytes] ‎Liumingmou (talk | contribs) (Created page with "<center> {| class="wikitable" |- ! 学号 |- | 221850088 |- | 221870109 |- | 221900009 |- | 221900024 |- | 221900028 |- | 221900029 |- | 221900036 |- | 221900038 |- | 221900042 |- | 221900046 |- | 221900055 |- | 221900068 |- | 221900073 |- | 221900075 |- | 221900076 |- | 221900083 |- | 221900084 |- | 221900085 |- | 221900091 |- | 221900124 |- | 221900128 |- | 221900140 |- | 221900159 |- | 221900168 |- | 221900175 |- | 221900183 |- | 221900187 |- | 221900208 |- | 2219002...")
14:15, 23 October 2024 ‎数据科学基础 (Fall 2024)/Problem Set 4 (hist | edit) ‎[7,102 bytes] ‎Liumingmou (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 *没有条件的同学可以用纸笔完成作业之后拍照。 == Assumption throughout Problem Set 4 == Without further notice, we are working on probability space <math>(\Omega,\mathcal{F},\Pr)</math>. Without further notice, we assume that the expectation of random variables are well-defined. == Problem 1...")
23:58, 21 October 2024 ‎Namelist Assignment 1 2024 (hist | edit) ‎[719 bytes] ‎Etone (talk | contribs) (Created page with "姓名学号刘屹恒 211830104 袁世伟 211830113 刘倚帆 211830136 孙哲至 211840029 胡冰雁 211840036 罗常 211840226 陈晨 211840233 王道熙 211840324 鲍饴欣 502024280001 曹士豪 502024280003 冯亮 502024280007 顾也楠 502024280013 何鑫 502024280015 郎乐怡 502024280018 李子琦 502024280021 刘子辰 502024280024 潘玉洁 502024280028 宋辰 502024280030 唐永平 502024280032 王俊昌 502024280035...")
01:10, 21 October 2024 ‎Assignment 2, Fall 2024 (hist | edit) ‎[2,706 bytes] ‎Etone (talk | contribs) (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>, <...")
11:09, 17 October 2024 ‎数据科学基础 (Fall 2024)/Problem Set 3 (hist | edit) ‎[7,288 bytes] ‎Liumingmou (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 *没有条件的同学可以用纸笔完成作业之后拍照。 == Assumption throughout Problem Set 3 == Without further notice, we are working on probability space <math>(\Omega,\mathcal{F},\mathbf{Pr})</math>. Without further notice, we assume that the expectation of random variables are well-defined. == Pr...")
08:27, 14 October 2024 ‎高级算法 (Fall 2024)/第一次作业提交名单 (hist | edit) ‎[1,309 bytes] ‎Zhangyiyao (talk | contribs) (Created page with "如有错漏请邮件联系助教. <center> {| class="wikitable" |- ! 学号 !! 姓名 |- | 211220076 || 杨景越 |- | 211220152 || 吴振宇 |- | 211250001 || 鞠哲 |- | 211250044 || 朱家辰 |- | 211250182 || 胡皓明 |- | 211502020 || 周相羽 |- | 211820282 || 彭钰朝 |- | 211830008 || 缪天顺 |- | 221220002 || 沈均文 |- | 221502010 || 梁志浩 |- | 221840186 || 陈端锐 |- | 221840188 || 谭泽晖 |- | 502023330043 || 吕铸恒 |- | 502024...")
19:04, 11 October 2024 ‎数据科学基础 (Fall 2024)/第一次作业提交名单 (hist | edit) ‎[2,155 bytes] ‎Liumingmou (talk | contribs) (Created page with "<center> {| class="wikitable" |- ! 学号 |- | 231880099 |- | 221900009 |- | 231880087 |- | 231880426 |- | 221900075 |- | 221900324 |- | 231880433 |- | 221900426 |- | 221900068 |- | 221900328 |- | 221900230 |- | 231880367 |- | 221900091 |- | 221900036 |- | 221900320 |- | 221900323 |- | 221900450 |- | 231880199 |- | 221900085 |- | 221900455 |- | 231880410 |- | 221900042 |- | 221900311 |- | 221900309 |- | 221900325 |- | 221900263 |- | 231880185 |- | 231880292 |- | 2318802...")
16:55, 10 October 2024 ‎数据科学基础 (Fall 2024)/Problem Set 2 (hist | edit) ‎[6,234 bytes] ‎Liumingmou (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 *没有条件的同学可以用纸笔完成作业之后拍照。 <h2 id="assumption"> Assumption throughout Problem Set 2</h2> Without further notice, we are working on probability space <math>(\Omega,\mathcal{F},\mathbf{Pr})</math>. Without further notice, we assume that the expectation of random variables are well-d...")
06:31, 30 September 2024 ‎数据科学基础 (Fall 2024)/Average-case analysis of QuickSort (hist | edit) ‎[11,829 bytes] ‎Etone (talk | contribs) (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...")
16:19, 29 September 2024 ‎Assignment 1, Fall 2024 (hist | edit) ‎[2,971 bytes] ‎Etone (talk | contribs) (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...")
16:13, 29 September 2024 ‎General Circulation(Fall 2024) (hist | edit) ‎[8,503 bytes] ‎Etone (talk | contribs) (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...")
06:38, 27 September 2024 ‎高级算法 (Fall 2024)/Problem Set 1 (hist | edit) ‎[8,456 bytes] ‎Kvrmnks (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 == Problem 1 (Min-cut/Max-cut) == == Problem 2 (Fingerprinting) == == Problem 3 (Hashing) == == Problem 4 (Concentration of measure) == == Problem 5 (Dimension reduction) ==")
04:40, 27 September 2024 ‎高级算法 (Fall 2024)/Dimension Reduction (hist | edit) ‎[16,764 bytes] ‎Etone (talk | contribs) (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...")
07:39, 22 September 2024 ‎数据科学基础 (Fall 2024)/Volume of Hamming balls (hist | edit) ‎[2,803 bytes] ‎Etone (talk | contribs) (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;" | :'...")
14:51, 19 September 2024 ‎数据科学基础 (Fall 2024)/Problem Set 1 (hist | edit) ‎[12,987 bytes] ‎Liumingmou (talk | contribs) (Created page with "*每道题目的解答都要有完整的解题过程，中英文不限。 *我们推荐大家使用LaTeX, markdown等对作业进行排版。 <h2 id="problem-1">Problem 1 (Probability space)</h2> <ul> <li>[De Morgan’s Laws] Let <math>\{A_i\}</math> be a collection of sets. Prove <math>\left(\bigcup_iA_i\right)^c=\bigcap_iA_i^c</math>.</li> <li>[<math>\sigma</math>-algebra] Let <math>A</math> and <math>B</math> belong to some <math>\s...")
11:32, 19 September 2024 ‎高级算法 (Fall 2024)/Conditional expectations (hist | edit) ‎[10,715 bytes] ‎Etone (talk | contribs) (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...")
11:31, 19 September 2024 ‎高级算法 (Fall 2024)/Concentration of measure (hist | edit) ‎[41,013 bytes] ‎Etone (talk | contribs) (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...")
02:45, 16 September 2024 ‎高级算法 (Fall 2024)/Basic deviation inequalities (hist | edit) ‎[5,755 bytes] ‎Etone (talk | contribs) (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...")
02:44, 16 September 2024 ‎高级算法 (Fall 2024)/Limited independence (hist | edit) ‎[15,812 bytes] ‎Etone (talk | contribs) (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...")
02:44, 16 September 2024 ‎高级算法 (Fall 2024)/Hashing and Sketching (hist | edit) ‎[49,292 bytes] ‎Etone (talk | contribs) (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...")
11:22, 12 September 2024 ‎高级算法 (Fall 2024)/Finite Field Basics (hist | edit) ‎[6,210 bytes] ‎Etone (talk | contribs) (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...")
11:21, 12 September 2024 ‎高级算法 (Fall 2024)/Fingerprinting (hist | edit) ‎[38,283 bytes] ‎Etone (talk | contribs) (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...")

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