v You are you you are the only

v. You are you, you are the only one in the world just like you! v. Mathematics is mathematics, there is nothing else in the world just like mathematics! v. Mathematics is a part of “Life”.

What is Mathematics?

v Mathematics is the study of quantity, structure, space, relation, change, and various topics of pattern, form and entity. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere. Mathematicians formulate new conjectures and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

v 阿諾史瓦辛格: v v 白天多運動, 晚上做數學. Arnold Alois Schwarzenegger is an American and Austrian bodybuilder, actor, businessman, and politician, currently serving as the 38 th Governor of the state of California. Schwarzenegger began weight-training at 15. He was awarded the title of Mr. Universe at age 22 and went on to win the Mr. Olympia contest a total of seven times. Schwarzenegger has remained a prominent face in the bodybuilding sport long after his retirement, and has written several books and numerous articles on the sport.

懸賞 Problem: I have a favor number in my pocket and the number is in {1, 2, 3, …, 63}. Can you find the number by asking (me) as few queries (問題) as possible?

An Answer 1 17 33 49 3 19 35 51 5 21 37 53 7 23 39 55 9 25 41 57 11 27 43 59 13 29 45 61 15 31 47 63

2 18 34 50 3 19 35 51 6 22 38 54 7 23 39 55 10 26 42 58 11 27 43 59 14 30 46 62 15 31 47 63

4 20 36 52 5 21 37 53 6 22 38 54 7 23 39 55 12 28 44 60 13 29 45 61 14 30 46 62 15 31 47 63

8 24 40 56 9 25 41 57 10 26 42 58 11 27 43 59 12 28 44 60 13 29 45 61 14 30 46 62 15 31 47 63

16 24 48 56 17 25 49 57 18 26 50 58 19 27 51 59 20 28 52 60 21 29 53 61 22 30 54 62 23 31 55 63

32 40 48 56 33 41 49 57 34 42 50 58 35 43 51 59 36 44 52 60 37 45 53 61 38 46 54 62 39 47 55 63

Blood Testing AIDS Tests (World war II)

One by one

More positives (72 tests)

Test a group at one time

Red is positive (陽性)and Blue is negative (陰性)

Some group is negative!

They are done!

Save a lot of Money! (Three Positive Groups) Save 33 -3 times What’s next?

v以上資料係由貴校資 系教授 阮夙姿提供. v. Thanks. v. This is the so-called Group Testing.

Human Genome Project

去氧核糖核酸 v DNA: Deoxyribonucleic Acid v Nucleotide (核苷酸) Adenine (A) Thymine (T) Guanine (G) Cytosine (C)

Goals of HGP Identify all the approximate 30, 000 genes in human DNA. v Determine the sequences of the 3 billion chemical base pairs that make up human DNA. v Store this information in databases. v Improve tools for data analysis. v Transfer related technologies to the private sector. v Address the ethical, legal, and social issues (ELSI) that may arise from the project. v

Milestone 1990: Project initiated as joint effort of U. S. Department of Energy and the National Institutes of Health. v June 2000: Completion of a working draft of the entire human genome. v February 2001: Analyses of the working draft are published. v April 2003: HGP sequencing is completed and Project is declared finished two years ahead of schedule. (Why? ) v

群試理論的應用 v In screening clone library the goal is to determine which clones in the clone library hybridize with a given probe in an efficient fashion. A clone is said to be positive if it hybridize with the probe( 探針), and negative otherwise.

Shotgun Sequencing v Shotgun sequencing is a throughput technique resulting in the sequencing of a large number of bacterial genomes, mouse genomes and the celebrated human genomes. In all such projectss, we are left with a collection of contigs that for special reasons cannot be assembled with general assembly algorithms. Continued …

Random shotgun approach genomic segment cut many times at random (Shotgun) 6

v Whole-genome shotgun sequencing Short reads are obtained and covering the genome with redundancy and possible gaps. Circular genome 34

Reads are assembled into contigs with unknown relative placement.

Primers : (short) fragments of DNA characterizing ends of contigs.

A PCR (Polymerase Chain Reaction) reaction reveals if two primers are proximate (adjacent to the same gap). Multiplex PCR can treat multiple primers simultaneously and outputs if there is a pair of adjacent primers in the input set and even sometimes the number of such pairs.

v Two primers of each contig are “mixed together” Find a Hamiltonian cycle by PCRs!

v Primers are treated independently. Find a perfect matching by PCRs.

Goal v Our goal is to provide an experimental protocol that identifies all pairs of adjacent primers with as few PCRs (queries) (or multiplex PCRs respectively) as possible.

機率 v Everyone knows probability. v Everywhere we use probability. v This is an example why mathematics play an important role in our life. v So, you do know how to live with 數學。