1. Concrete Mathematics 演算法的數學分析 Ming-Jer Tsai

2. Outline Text Book The Topics in the Class Course Description Evaluation

3. Text Book “ Concrete Mathematics — A foundation For Computer Science, ” GRAHAM, KNUTH, PATASHNIK, ADDISON WESLEY

4. Topics in the Class Recurrent Problems Sums Integer Functions Number Theory Binomial Coefficients Generating Functions

5. Recurrent Problems The Josephus Problem: We start with n people numbered 1 to n around a circle, and we eliminate every second remaining person until only one survives.

6. The Josephus Problem Recurrent Relation: Closed Form: (An expression for a quantity f(n) which can be computed using at most a fixed number of “ well known ” standard operations.)

7. Sums Sum: Multiple Sum:

8. Integer Functions Floor/Ceiling Sum: Floor/Ceiling Recurrence:

9. Number Theory Theory:

10. Number Theory Theory:

11. Number Theory Applications :

12. Binomial Coefficients Sum of Binomial Coefficients: Sum of Ratios of Binomial Coefficients: Sum of Products of Binomial Coefficients:

13. Generating Functions

14. Course Description Concrete mathematics is a blending of continuous and discrete mathematics. It introduces the mathematics that supports the analysis of algorithms. The goal is for the student to have mathematical skills to solve complex problems, and to discover subtle patterns in data.

15. Evaluation 2 Mid-term Exams (40%) 1 Final (25%) 10 Quizs (20%) Discussion (15%)