離散專題(I) (Topics in Discrete Mathematics) Hung-Lin Fu (傅恆霖) Department of Applied Mathematics National Chiao Tung University Hsin Chu, Taiwan 30010

Discrete? The best known definition for “discrete” is “not continuous”. We can define “continuous functions” from a set into another set. Without a suitable domain, we go nowhere in defining such a function. In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.

Discrete Mathematics? Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.

Topics? Theoretical computer science (Applications) Information theory (Coding Theory and Cryptography Theory) Logic Set theory Combinatorial Designs (Many branches) Graph theory (Many branches) Probability (Discrete) Number theory (Integers)

More Topics Algebra and Linear Algebra (Applied) Calculus of finite differences, discrete calculus or discrete analysis Geometry (Finite Geometry and Discrete Geometry) Topology (Combinatorial Topology) Operations research (Combinatorial Optimization) Game theory, decision theory, utility theory, social choice theory (Combinatorial Optimization) Discretization Discrete analogues of continuous mathematics Hybrid discrete and continuous mathematics

And More Wireless Communication Small World Complex Networks Computational Molecular Biology Machine Learning Enumerative Combinatorial Theory

What to do? Clearly, there is no way that you can study all the topics mentioned above.

From the Basics! 你(妳)先要跑的快才決定要參加徑賽的那一個項目。 Prepare your chips before you play the game. We are in the department of Applied Mathematics not Mathematical Applications.

Unique Place to Start 唯一的組合數學組。 Always, you have something to learn in each semester: Graph Theory, Combinatorial Designs, Graph Algorithms, Cryptography, Probability, Computational Molecular Biology, …. You also have the freedom to learn whatever you like yourself. (共同指導。)

No Short Cut! Be Patient! Be Passionate! Be Positive! You have to work harder than the others in order to be better!

一分耕耘,一分收穫。 She was a student of mine and has been the only archer in Taiwan who was in our Olympic Team three times. She has shot over a million arrows in her shooting career.

About this course (離散專題) We have classes on Sep. 17th, 24th; Oct. 1st, 8th, 15th, 22nd, 29th; Nov. 5th, 19th, 26th; Dec. 3rd, 10th, 17th, 24th; Jan. 7th. Time: 13:20 – 15:10, Monday. You have to present a scheduled talk in English and the title of your talk can be chosen from the following list but one title will be presented by at most two of you (in next page). Let me know as soon as you decide which one you choose. You may use power point but your talk can not be shorter than 40 minutes.

List of titles Distance between unsigned permutations Distance between signed permutations Distance between arbitrary strings Distance between balanced strings Set systems and the syntenic distance Median and Halving Problems The large parsimony problem Ref. Combinatorics of Genome Rearrangements

Talks will be presented starting from the 15th of Oct. We shall have a couple of invited talks before that time. The order of presentation will follow the following the registration form 1 – 14: 0082202, 0182202, 9822804, 9922803, …. (You may come to check your order.)