随机算法 (Fall 2011)/Graph Coloring and 组合数学 (Fall 2011): Difference between pages
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|title = 组合数学 | |||
Combinatorics | |||
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|header1 =Instructor | |||
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|data2 = 尹一通 | |||
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|label3 = Email | |||
|data3 = yitong.yin@gmail.com yinyt@nju.edu.cn yinyt@lamda.nju.edu.cn | |||
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|label4= office | |||
|data4= TBA | |||
|header5 = Class | |||
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|label6 = Class meetings | |||
|data6 = TBA <br>TBA | |||
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|label7 = Place | |||
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|label8 = Office hours | |||
|data8 = TBA | |||
|header9 = Textbook | |||
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|header10 = | |||
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|data10 = ''van Lint and Wilson,'' <br> A course in Combinatorics, 2nd Ed, <br> Cambridge Univ Press, 2001. | |||
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This is the page for the class ''Combinatorics'' for the Fall 2011 semester. Students who take this class should check this page periodically for content updates and new announcements. | |||
= Announcement = | |||
= Course info = | |||
* '''Instructor ''': 尹一通 | |||
:*email: yitong.yin@gmail.com, yinyt@nju.edu.cn, | |||
:*office: | |||
* '''Teaching fellow''': | |||
:*email: | |||
* '''Class meeting''': | |||
* '''Office hour''': | |||
= Syllabus = | |||
=== 先修课程 Prerequisites === | |||
* 离散数学(Discrete Mathematics) | |||
* 线性代数(Linear Algebra) | |||
* 概率论(Probability Theory) | |||
=== Course materials === | |||
* [[组合数学 (Fall 2011)/Course materials|教材和参考书清单]] | |||
=== 成绩 Grades === | |||
* 课程成绩:本课程将会有六次作业和一次期末考试。最终成绩将由平时作业成绩和期末考试成绩综合得出。 | |||
* 迟交:如果有特殊的理由,无法按时完成作业,请提前联系授课老师,给出正当理由。否则迟交的作业将不被接受。 | |||
=== <font color=red> 学术诚信 Academic Integrity </font>=== | |||
学术诚信是所有从事学术活动的学生和学者最基本的职业道德底线,本课程将不遗余力的维护学术诚信规范,违反这一底线的行为将不会被容忍。 | |||
作业完成的原则:署你名字的工作必须由你完成。允许讨论,但作业必须独立完成,并在作业中列出所有参与讨论的人。不允许其他任何形式的合作——尤其是与已经完成作业的同学“讨论”。 | |||
本课程将对剽窃行为采取零容忍的态度。在完成作业过程中,对他人工作(出版物、互联网资料、其他人的作业等)直接的文本抄袭和对关键思想、关键元素的抄袭,按照 [http://www.acm.org/publications/policies/plagiarism_policy ACM Policy on Plagiarism]的解释,都将视为剽窃。剽窃者成绩将被取消。如果发现互相抄袭行为,<font color=red> 抄袭和被抄袭双方的成绩都将被取消</font>。因此请主动防止自己的作业被他人抄袭。 | |||
学术诚信影响学生个人的品行,也关乎整个教育系统的正常运转。为了一点分数而做出学术不端的行为,不仅使自己沦为一个欺骗者,也使他人的诚实努力失去意义。让我们一起努力维护一个诚信的环境。 | |||
= Assignments = | |||
= Lecture Notes = | |||
# [[组合数学 (Fall 2011)/Basic enumeration|Basic enumeration]] | |||
# [[组合数学 (Fall 2011)/Generating functions|Generating functions]] | |||
# [[组合数学 (Fall 2011)/Sieve methods|Sieve methods]] | |||
# [[组合数学 (Fall 2011)/Pólya's theory of counting|Pólya's theory of counting]] | |||
# [[组合数学 (Fall 2011)/Existential proofs|Existential proofs]] | |||
# [[组合数学 (Fall 2011)/Discrete probability|Discrete probability]] | |||
# [[组合数学 (Fall 2011)/The probabilistic method|The probabilistic method]] | |||
# [[组合数学 (Fall 2011)/Extremal graph theory| Extremal graph theory]] | |||
# [[组合数学 (Fall 2011)/Matching theory|Matching theory]] | |||
# [[组合数学 (Fall 2011)/Flow and matching | Flow and matching]] | |||
# [[组合数学 (Fall 2011)/Optimization|Optimization]] | |||
# [[组合数学 (Fall 2011)/Matroid|Matroid]] | |||
# [[组合数学 (Fall 2011)/Extremal set theory|Extremal set theory]] | |||
# [[组合数学 (Fall 2011)/Extremal set theory II|Extremal set theory II]] | |||
# [[组合数学 (Fall 2011)/Ramsey theory|Ramsey theory]] | |||
# [[组合数学 (Fall 2011)/The Szemeredi regularity lemma|The Szemeredi regularity lemma]] | |||
= | = Concepts = | ||
* [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient] | |||
* [http://en.wikipedia.org/wiki/Composition_(number_theory) Composition of a number] | |||
* [http://en.wikipedia.org/wiki/Combination#Number_of_combinations_with_repetition Combinations with repetition], [http://en.wikipedia.org/wiki/Multiset_coefficient#Multiset_coefficients <math>k</math>-multisets on a set] | |||
* [http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling number of the second kind] | |||
* [http://en.wikipedia.org/wiki/Partition_(number_theory) Partition of a number] | |||
* [http://en.wikipedia.org/wiki/Twelvefold_way The twelvefold way] | |||
* [http://en.wikipedia.org/wiki/Partition_(number_theory)#Ferrers_diagram Ferrers diagram] (and the MathWorld [http://mathworld.wolfram.com/FerrersDiagram.html link]) | |||
* [http://en.wikipedia.org/wiki/Inclusion-exclusion_principle The principle of inclusion-exclusion] (and more generally the [http://en.wikipedia.org/wiki/Sieve_theory sieve method]) | |||
* [http://en.wikipedia.org/wiki/Derangement Derangement], and [http://en.wikipedia.org/wiki/M%C3%A9nage_problem Problème des ménages] | |||
* [http://en.wikipedia.org/wiki/Generating_function Generating function] and [http://en.wikipedia.org/wiki/Formal_power_series formal power series] | |||
* [http://en.wikipedia.org/wiki/Fibonacci_number Fibonacci number] | |||
* [http://en.wikipedia.org/wiki/Binomial_series Newton's formula] | |||
* [http://en.wikipedia.org/wiki/Catalan_number Catalan number] | |||
* [http://en.wikipedia.org/wiki/Double_counting_(proof_technique) Double counting] and the [http://en.wikipedia.org/wiki/Handshaking_lemma handshaking lemma] | |||
* [http://en.wikipedia.org/wiki/Cayley's_formula Cayley's formula] | |||
* [http://en.wikipedia.org/wiki/Pigeonhole_principle Pigeonhole principle] | |||
:* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem Erdős–Szekeres theorem] | |||
:* [http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem Dirichlet's approximation theorem] | |||
* [http://en.wikipedia.org/wiki/Probabilistic_method The Probabilistic Method] | |||
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model Erdős–Rényi model for random graphs] | |||
* [http://en.wikipedia.org/wiki/Graph_property Graph property] | |||
* Some graph parameters: [http://en.wikipedia.org/wiki/Girth_(graph_theory) girth <math>g(G)</math>], [http://mathworld.wolfram.com/ChromaticNumber.html chromatic number <math>\chi(G)</math>], [http://mathworld.wolfram.com/IndependenceNumber.html Independence number <math>\alpha(G)</math>], [http://mathworld.wolfram.com/CliqueNumber.html clique number <math>\omega(G)</math>] | |||
* [http://en.wikipedia.org/wiki/Extremal_graph_theory Extremal graph theory] | |||
* [http://en.wikipedia.org/wiki/Turan_theorem Turán's theorem], [http://en.wikipedia.org/wiki/Tur%C3%A1n_graph Turán graph] | |||
* Two analytic inequalities: | |||
:*[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality Cauchy–Schwarz inequality] | |||
:* the [http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means inequality of arithmetic and geometric means] | |||
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem Erdős–Stone theorem] (fundamental theorem of extremal graph theory) | |||
* [http://en.wikipedia.org/wiki/Dirac's_theorem Dirac's theorem] | |||
* [http://en.wikipedia.org/wiki/Hall's_theorem Hall's theorem ] (the marriage theorem) | |||
* [http://en.wikipedia.org/wiki/Birkhoff-Von_Neumann_theorem Birkhoff-Von Neumann theorem] | |||
* [http://en.wikipedia.org/wiki/K%C3%B6nig's_theorem_(graph_theory) König-Egerváry theorem] | |||
* [http://en.wikipedia.org/wiki/Dilworth's_theorem Dilworth's theorem] | |||
* [http://en.wikipedia.org/wiki/Sperner_family Sperner system] | |||
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem] | |||
* [http://en.wikipedia.org/wiki/VC_dimension VC dimension] | |||
* [http://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem Kruskal–Katona theorem] | |||
* [http://en.wikipedia.org/wiki/Ramsey_theory Ramsey theory] | |||
:*[http://en.wikipedia.org/wiki/Ramsey's_theorem Ramsey's theorem] | |||
:*[http://en.wikipedia.org/wiki/Happy_Ending_problem Happy Ending problem] | |||
:*[http://en.wikipedia.org/wiki/Van_der_Waerden's_theorem Van der Waerden's theorem] | |||
:*[http://en.wikipedia.org/wiki/Hales-Jewett_theorem Hales–Jewett theorem] | |||
* [http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma Lovász local lemma] | |||
* [http://en.wikipedia.org/wiki/Combinatorial_optimization Combinatorial optimization] | |||
:* [http://en.wikipedia.org/wiki/Optimization_(mathematics) optimization] | |||
:* [http://en.wikipedia.org/wiki/Convex_combination convex combination], [http://en.wikipedia.org/wiki/Convex_set convex set], [http://en.wikipedia.org/wiki/Convex_function convex function] | |||
:* [http://en.wikipedia.org/wiki/Local_optimum local optimum] (see also [http://en.wikipedia.org/wiki/Maxima_and_minima maxima and minima]) | |||
* [http://en.wikipedia.org/wiki/Linear_programming Linear programming] | |||
:* [http://en.wikipedia.org/wiki/Linear_inequality linear constraint] | |||
:* [http://en.wikipedia.org/wiki/Hyperplane hyperplane], [http://en.wikipedia.org/wiki/Half_space halfspace], [http://en.wikipedia.org/wiki/Polyhedron polyhedron], [http://en.wikipedia.org/wiki/Convex_polytope convex polytope] | |||
:* [http://en.wikipedia.org/wiki/Simplex_algorithm the Simplex algorithm] | |||
* The [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut Theorem] | |||
:* [http://en.wikipedia.org/wiki/Maximum_flow_problem Maximum flow] | |||
:* [http://en.wikipedia.org/wiki/Minimum_cut minimum cut] | |||
* [http://en.wikipedia.org/wiki/Unimodular_matrix Unimodularity] | |||
* [http://en.wikipedia.org/wiki/Dual_linear_program Duality] | |||
:* [http://en.wikipedia.org/wiki/Linear_programming#Duality LP Duality] | |||
* [http://en.wikipedia.org/wiki/Matroid Matroid] | |||
:* [http://en.wikipedia.org/wiki/Weighted_matroid weighted matroid] and [http://en.wikipedia.org/wiki/Greedy_algorithm greedy algorithm] | |||
:* [http://en.wikipedia.org/wiki/Matroid_intersection Matroid intersection] | |||
* [http://en.wikipedia.org/wiki/Laplacian_matrix Laplacian] | |||
* [http://en.wikipedia.org/wiki/Algebraic_connectivity <math>\lambda_2</math> of a graph] and [http://en.wikipedia.org/wiki/Expander_graph#Cheeger_Inequalities Cheeger Inequalities] | |||
* [http://en.wikipedia.org/wiki/Expander_graph Expander graph] | |||
* [http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma Szemerédi regularity lemma] |
Revision as of 03:41, 17 August 2011
This is the page for the class Combinatorics for the Fall 2011 semester. Students who take this class should check this page periodically for content updates and new announcements.
Announcement
Course info
- Instructor : 尹一通
- email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
- office:
- Teaching fellow:
- email:
- Class meeting:
- Office hour:
Syllabus
先修课程 Prerequisites
- 离散数学(Discrete Mathematics)
- 线性代数(Linear Algebra)
- 概率论(Probability Theory)
Course materials
成绩 Grades
- 课程成绩:本课程将会有六次作业和一次期末考试。最终成绩将由平时作业成绩和期末考试成绩综合得出。
- 迟交:如果有特殊的理由,无法按时完成作业,请提前联系授课老师,给出正当理由。否则迟交的作业将不被接受。
学术诚信 Academic Integrity
学术诚信是所有从事学术活动的学生和学者最基本的职业道德底线,本课程将不遗余力的维护学术诚信规范,违反这一底线的行为将不会被容忍。
作业完成的原则:署你名字的工作必须由你完成。允许讨论,但作业必须独立完成,并在作业中列出所有参与讨论的人。不允许其他任何形式的合作——尤其是与已经完成作业的同学“讨论”。
本课程将对剽窃行为采取零容忍的态度。在完成作业过程中,对他人工作(出版物、互联网资料、其他人的作业等)直接的文本抄袭和对关键思想、关键元素的抄袭,按照 ACM Policy on Plagiarism的解释,都将视为剽窃。剽窃者成绩将被取消。如果发现互相抄袭行为, 抄袭和被抄袭双方的成绩都将被取消。因此请主动防止自己的作业被他人抄袭。
学术诚信影响学生个人的品行,也关乎整个教育系统的正常运转。为了一点分数而做出学术不端的行为,不仅使自己沦为一个欺骗者,也使他人的诚实努力失去意义。让我们一起努力维护一个诚信的环境。
Assignments
Lecture Notes
- Basic enumeration
- Generating functions
- Sieve methods
- Pólya's theory of counting
- Existential proofs
- Discrete probability
- The probabilistic method
- Extremal graph theory
- Matching theory
- Flow and matching
- Optimization
- Matroid
- Extremal set theory
- Extremal set theory II
- Ramsey theory
- The Szemeredi regularity lemma
Concepts
- Binomial coefficient
- Composition of a number
- Combinations with repetition, [math]\displaystyle{ k }[/math]-multisets on a set
- Stirling number of the second kind
- Partition of a number
- The twelvefold way
- Ferrers diagram (and the MathWorld link)
- The principle of inclusion-exclusion (and more generally the sieve method)
- Derangement, and Problème des ménages
- Generating function and formal power series
- Fibonacci number
- Newton's formula
- Catalan number
- Double counting and the handshaking lemma
- Cayley's formula
- Pigeonhole principle
- The Probabilistic Method
- Erdős–Rényi model for random graphs
- Graph property
- Some graph parameters: girth [math]\displaystyle{ g(G) }[/math], chromatic number [math]\displaystyle{ \chi(G) }[/math], Independence number [math]\displaystyle{ \alpha(G) }[/math], clique number [math]\displaystyle{ \omega(G) }[/math]
- Extremal graph theory
- Turán's theorem, Turán graph
- Two analytic inequalities:
- Erdős–Stone theorem (fundamental theorem of extremal graph theory)
- Dirac's theorem
- Hall's theorem (the marriage theorem)
- Birkhoff-Von Neumann theorem
- König-Egerváry theorem
- Dilworth's theorem
- Sperner system
- Erdős–Ko–Rado theorem
- VC dimension
- Kruskal–Katona theorem
- Ramsey theory