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List of abbreviations: N This edit created a new page (also see list of new pages) m This is a minor edit b This edit was performed by a bot (±123) The page size changed by this number of bytes 26 March 2025 N 09:58 组合数学 (Fall 2025)/Existence problems diffhist +14,440 Etone talk contribs (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}^nrightarrow {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...") 09:57 组合数学 (Spring 2025) 2 changes history +292 [Etone (2×)] 09:57 (cur | prev) +151 Etone talk contribs (→Lecture Notes) 09:56 (cur | prev) +141 Etone talk contribs (→Lecture Notes) N 09:56 组合数学 (Fall 2025)/Cayley's formula diffhist +17,231 Etone talk contribs (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...") 09:55 概率论与数理统计 (Spring 2025) diffhist +189 Etone talk contribs (→Lectures) 21 March 2025 11:50 高级算法 (Spring 2025) diffhist +84 Liumingmou talk contribs (→通知)N 10:03 概率论与数理统计 (Spring 2025)/Average-case analysis of QuickSort 2 changes history +11,838 [Etone (2×)] 10:03 (cur | prev) 0 Etone talk contribs (→快速排序算法的平均复杂度分析 I（基于全期望法则）)N 10:03 (cur | prev) +11,838 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...") 10:02 概率论与数理统计 (Spring 2025) diffhist +421 Etone talk contribs (→Lectures) N 09:58 组合数学 (Fall 2025)/Pólya's theory of counting diffhist +19,696 Etone talk contribs (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,hin G, gcdot h in G</math>; * ''associativity'': <math>forall f,g,hin G, fcdot(gcdot h)=(fcdot g)cdot h</math>; * ''identity'': there exists a special element <math>ein G</math>, called the '''identity''', such that <math>ecdot g=g</math> for any <math>gin G</math>; * ''inverse''...") 20 March 2025 19 March 2025 18 March 2025 16 March 2025 14 March 2025 13 March 2025相关知识
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最后，要有责任心。及时清洁宠物的粪便，不可随意丢弃宠物及宠物尸体。 的翻译是： 中文翻译英文意思，翻译英语
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