具体数学Concrete Mathematics

赵启阳

2023年4月19日

北京航空航天大学计算机学院

body warming……

let’s start from Paul Erdös, something about our Course & Textbook

The Man Who Loved Only Numbers• Name: Paul Erdös• Sex: Male• Life: 1913-1996• Nationality: Hungarian• Vocation: Mathematician• Interesting Activities/Speech: 我们数学家都有些疯狂……

1970 年，宣称自己 25 亿岁了。（ Why ：小的时候，科学家说地球有20 亿岁， now ，他们又宣称为 45 亿岁， so… ）

我终于不再愈变愈蠢了。（自撰墓志铭）

abbrName and E-Numbers• 50- years old: abbrName = NULL;• 50+ years old: abbrName = abbrName + P. G. O. M.; //Poor Great Old

Man• 60 years old: abbrName = abbrName + L. D.; // Living Dead• 65 years old: abbrName = abbrName + A. D.; //Archeological Discovery• 70 years old: abbrName = abbrName + L. D.; //Legal Death• 75 years old: abbrName = abbrName + C. D. ; //Counting Death• So at last, Paul Erdös is a

P. G. O. M. L. D. A. D. L. D. (An abbr.!? Sure?)• A famous number, E-Number, how far are you away from Paul Erdös,

mathematically. 0 : You are Paul Erdös himself!1 : Cooperate with Paul Erdös!2 : Cooperate with E-N-1 men!

…………∞ : Nothing but a freshman! (We’re here )

E-Number picture• E-Number project on http://www.oakland.edu/enp

Books related to Erdös

• Proofs from THE BOOK, by Martin Aigner, Günter M. Ziegler and Karl H. Hofmann (Hardcover - Nov 23, 2009)

翻译版：《数学天书中的证明》，冯荣权、宋春伟、宗传明译，高等教育出版社， 2009 年。

1 、 Erdös designed and guided writing the book.2 、 GOD is called SF (Supreme Fascist, 超级法西斯 ). 3 、 SF hide all beautiful proofs and solutions in THE BOOK.• Concrete Mathematics, by Graham, Knuth and Patashnik.

1 、 Our Textbook!!! Paul provided lots of foundational works.2 、 Many beautiful Models and Solutions are in OUR BOOK.

Erdös and Authors of CM

• Person E-Number• Graham 1 (A chief coauthor of Erdös.)Married with a Chinese American 金芳蓉前国际耍球协会主席• Knuth 2 (Knuth- Graham- Erdös.)author of The Art of Computer ProgrammingTuring Award in 1974• Patashnik 2 (Patashnik-Graham- Erdös.)I don’t know much more than CM about Pata.

OK, let’s have a look on CM course……

Outline

• Course Description• Why deserve learning?• Topics in the Class• Text Book and Other Materials• Evaluation

Course Description• Concrete mathematics is a blending of continuous and discrete

mathematics. See,Chap. 3 – Integer Functions 实数的整数部分运算Chap. 9 – Asymptotics 离散到连续的渐进

• It introduces the mathematics that supports the analysis of algorithms, modeling probems in real world. See,

Chap. 1 – Recurrence 递归的计数Chap. 2 – Sum 各种求和，用于算法复杂度计算等Chap. 6 – Special Numbers 叠卡片问题中的调和数列

• The goal is for the student to have mathematical skills to solve complex problems, and to discover subtle patterns in data.

Chap. 7 – Generating Functions 用于概率计算的母函数Chap. 8 – Discrete Probability 离散问题概率 ( 最有趣的

一章 )

Why deserve learning?

• The content is not taught in other courses, but, is usually used in Computer Science, especially designing algorithms and estimating problem complexity.

• The course involves lots of Mathematical Subject, Tools and Conclusions. Good choice to review them.

• A start-point and preparation to advanced CS.• Help to decide which problem to improve, which

problem to avoid (especially reply your BOSS!!!)• Helpful when seeking jobs of top IT companies (See B.

Gates’ words on the back-cover of TAOCP by D. E. Knuth)

Topics in the Class

• Chap.1 Recurrent Problems (Full)• Chap. 2 Sums (Full)• Chap. 3 Integer Functions (Full)• Chap. 6 Special Numbers (6.3, 6.4)• Chap. 7 Generating Functions (7.1, 7.5 optional)• Chap. 8 Discrete Probabilities (Full)

Text Book

• Concrete Mathematics—A foundation For Computer Science, GRAHAM, KNUTH, PATASHNIK, ADDISON WESLEY, 1994

• 影印版：“具体数学—计算机科学基础” ，机械工业出版社， 2002

• 翻译版 ( 第 1 版 ) ：“具体数学—计算机科学基础”，西安电子科技大学出版社，庄心谷译， 1992(available on Internet)

Other Materials

• The Art of Computer Programming, Vol. I, II, III, IV, Donald. E. Knuth

The so-called Bible in Computer ScienceNeed tough effort to finish reading

• The Introduction to Algorithms• Combinatorial Mathematics, Bruldi• The Man Who loved only Numbers – Paul

Erdös( 中译本，上海世纪出版集团 )

EvaluationYour scores are composed of 2 parts:• 50% : Solve my problems (maybe not in Textbook!!!).

Submit your elec. answers to me before each deadline.You might write in English for extra 10 bonus, or Chinese.Be sure your job not be copied by others, or you got only 0.

• 50% : Open testYou might bring your textbooks in the test.Test would be hold in 13th week.

Moreover……

• 3~5 exercises would be recommended for you each time (So sadly………)

• They are all in exercises with the solutions in Appendix A (Great!)

• I’d like to choose 1~3 guys to explain the whole thing to all of us, in the first 10~15 MINs on next time.

• Any changes and/or discuss are welcome.

Course Contact Info

Website: gait.buaa.edu.cn/~zqy/cmaths• Important notices, useful links would be published

here.

Mail Box: cmaths@126.com• Do not hesitate to contact me on any CM problems,

any style.• Be sure to submit your solutions to me, with your

name and NO. Anonymous for 0.