數學有何用
陳宜良
應用數學講座教授
中央大學數學系
數學有何用
• 數學是追求樣式規律性與本質性的學科 • 數學是理性思維的工具 – 探索未知 – 解決現實問題 – 開創未來世界需要數學
一個探索未知的例子
• Tycho Brahe (第谷·∙布拉赫1546-‐1601) – 一組改變歷史的數據 • Johannes Kepler (刻卜勒1571-‐1630) – 一個追求真理的探索 • Isaac Newton(牛頓1642-‐1727) – 千古謎題破解日,萬有引力發現時探索未知
Geocentric Model (地心說)
• In the 4th century BC, two influen>al Greek philosophers, Plato and his student Aristotle, wrote works based on the geocentric model. According to Plato, the Earth was a sphere,
sta>onary at the center of the universe. The stars and planets were carried around the Earth on
spheres or circles, arranged in the order
(outwards from the center): Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, fixed stars, with the fixed stars located on the celes>al sphere.
• In the Ptolemaic system, each planet is moved by a system of two spheres:
one called its deferent, the others, its epicycle
• The deferent is a circle whose center point, called the eccentric and marked in the diagram with an X, is removed from the Earth. The original
purpose of the eccentric was to account for the differences of the lengths of the seasons (autumn is the shortest by a week or so), by placing the
Earth away from the center of rota>on of the rest of the universe.
• Another sphere, the epicycle(周轉圓), is
embedded inside the deferent sphere and is represented by the smaller doTed line to the right. A given planet then moves around the epicycle at the same >me the epicycle moves along the path marked by the deferent
hTp://en.wikipedia.org/wiki/Tycho_Brahe
Tycho Brahe was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observa>ons.
• Tycho realized that progress in astronomy required systema>c, rigorous observa>on, night aZer night, using the most accurate
instruments obtainable. This program became his life's work. Tycho improved and enlarged exis>ng instruments, and built en>rely new ones.
• The king of Denmark gave Tycho tremendous resources: an island with many families on it, and money to build an observatory. (One
es>mate is that this was 10% of the gross na>onal product at the >me!)
• Tycho Brahe was granted an estate on the island of Hven and the funding to build the
Uraniborg, an early research ins>tute, where he built large astronomical instruments and took many careful measurements, and later
Stjerneborg, underground, when he
discovered that his instruments in the former were not sufficiently steady.
• The observatory had a large mural quadrant affixed to a north-‐south wall, used to measure the al>tude of stars as they passed the
meridian. This, along with many other
instruments of the observatory, was depicted and described in detail in Brahe's 1598 book
Accuracy
• He aspired to a level of accuracy in his es>mated posi>ons of celes>al bodies of being consistently within 1 arcminute of their real celes>al
loca>ons, and also claimed to have achieved this level. But in fact many of the stellar posi>ons in his star catalogues were less accurate than that. The median errors for the stellar posi>ons in his final published catalog were about 1'.5, indica>ng that only half of the entries were more accurate than that, with an overall mean error in each
• AZer disagreements with the new Danish king Chris>an IV in 1597, he was invited by the
Bohemian king and Holy Roman emperor
Rudolph II to Prague, where he became the official imperial astronomer. He built the new observatory at Benátky nad Jizerou. There,
from 1600 un>l his death in 1601, he was assisted by Johannes Kepler who later used Tycho's astronomical data to develop
Eduard Ender's portrait of 1855 showing Tycho demonstra>ng a celes>al globe to the Emperor Rudolph II in Prague.
刻卜勒的探索
• hTp://www.keplersdiscovery.com/ • 刻卜勒簡介
• 刻卜勒的新天文學
• 從刻卜勒到牛頓 -‐-‐千古謎題破解日,萬有引 力發現時 項武義‧張海潮 • 牛頓萬有引力 Isaac Newton(1642~1727),英國 的數學家及物理學家,微積分主要締 造者,萬有引力理論的發明者。
• Tycho’s observa>ons were accurate enough
for Kepler to discover that the planets moved in ellip>c orbits, and his other laws, which
gave Newton the clues he needed to establish universal inverse-‐square gravita>on.
解決現實生活中的問題
數學與真實世界連結
• 可以讓我們理性分析生活周遭所發生的事 • 可以防災 • 可以讓生活便利 • 可以增進效率 • 可以理性的判斷與協助決策 • 可以協助解決問題第一類的例子
• 柯南: 死亡時間的推斷
柯南如何判斷受害者死亡時間
• 一屍體在6:30 AM 被發現,屍體溫 度為 20◦. 7:30 AM 時,屍體溫度為 18◦. 當時室溫為 16◦ 。人體溫約為 37◦. 柯南怎麼估計 受害者死亡時間 呢?數學建模
• 設屍體溫度在時刻 t 時為Y(t) • 室溫為 K • 模型: • 由Y 在兩個時間點的值推斷 與初期始時間Y (t +
t)
Y (t)
t
= ↵ (K
Y (t))
↵
Euler 方法
Leonhard Euler
1707-‐1783
馬爾薩斯
(Malthus)人口論
• 馬爾薩斯Malthus人口論指出人口呈幾何成 長,糧食受限於耕地面積的限制,終究會 達到飽和,而有限的土地分配給呈幾何成 長的人口,終究不足分配,若土地可以買 賣,則其價值也會愈來愈高。 • 這個理論也可以解釋貴族體制崩解的原因。Thomas Robert Malthus (1766-‐1834)
人口論的數學模型
(Malthus, 1798)
• 人類社會的人口若不作適當控制,終會 被細菌或戰爭所摧毀。
P (t +
t) = P (t)(1 + r t)
環境承載有上限
• 環境承載量R := r/
P (t +
t)
P (t)
t
= rP (t)
P (t)
2= rP (t)
✓
1
P (t)
R
◆
P (t)
t
= rP (t)
✓
1
P (t)
R
◆
Pierre Verhulst (1804-‐1849數值方法
: 尤拉法
• 初始值 • 迭代公式: • 終點值P (0)
P (N t)
for n = 0, . . . , N
1
P (t)
! R as t ! 1.
P ((n + 1) t) = (1 + r t)P (n t) ✓ 1 P (n t) R ◆漁業
養殖業: 永續經營
• 設每單位時間漁獲量為 C • 問題: C 應選擇何值時可以永續經營?P (t)
t
= rP (t)
✓
1
P (t)
R
◆
C
永續的社會
• 數學建模可以讓我們理性分析生活周遭所發生 的事。 – 人口成長 – 糧食丶水的需求 – 都市擴張 – 房價高漲 • 法國經濟學家Thomas PikeTy 近日出了一本有 名的書: 廿一世紀資本論書中有許多數據值得數學建模與並做更進一步的分析。
第二類例子
• 亞瑪遜書局如何推薦書
• 影視租賃公司如何推薦影帶
• 歐巴馬團隊如何估計社群政黨取向
亞馬遜如何推薦書
• 亞馬遜網上書店 Amazon.com 是美國最大的一家 網路電子商務公司,是網路上最早開始經營電子 商務的公司之一,亞馬遜書店成立於1995年,一 開始只經營網路的書籍銷售業務,現在則擴及了 範圍相當廣的其他產品,包括了DVD、音樂光碟、 電腦、軟體、電視遊戲、電子產品、衣服、傢具 等等。 • 亞馬遜會根據你購買或評比過的書籍,推薦你相 關的書籍Ra>ng by correla>on-‐weights
Nerlix Problem
• Nerlix是一家美國公司,提供北美地區綫上 播放DVD的出租業務。 • 2011年4月,Nerlix宣布在美有2360萬用戶, 而全世界則超過2600萬,可提供10萬種DVD 選擇。 • Nerlix會根據用戶訂閱過或評比過的資料提 供推薦影片。
Nerlix
獎
• Nerlix為追求最佳collabora>ve filtering算法 的獎,獎金一百萬美金,由Nerlix公司提供。 • 比賽方式是所提供的方法要比該公司用戶 的方法(Cinematch)好10%. 度量方法是RMSE (root-‐mean-‐square-‐error).Nerlix
問題
• Data format: <user, movie, date of grade, grade> • Training data: Training set (99,072,112 ra>ngs not
including the probe set, 100,480,507 including the probe set)
• Quiz set (1,408,342 ra>ngs), used to calculate leaderboard scores
• Test set (1,408,342 ra>ngs), used to calculate compe>>on scores
• On September 18, 2009, Nerlix announced team
"BellKor's Pragma>c Chaos" as the prize winner (a Test RMSE of 0.8567)
A Matrix Comple>on Approach
• Given less than 1% of movie ra>ngs • Goal: predict missing ra>ngs
第三類例子
How google works
• 爬挖資料 (Crawling)
• 建立索引 (Indexing) 60 trillion pages
How google works
1. Lars Elden, Matrix Methods in Data Mining and PaTern Recogni>on, SIAM 2. PageRank, Wiki
Two problems
• We may get stuck at some page
Challenges
• 60 Trillions of individual pages
• Size of stochas>c matrix can be hundreds of thousands
第四類問題
• 壓縮感知 • 影像除躁 • 影像填補
How to reduce imaging >me for MRI
• Mo>va>on from medical imaging
– Low dose
– Fast imaging – High resolu>on
MRI images
Basic Principles of Nuclear Magne>c
Resonance
• Atoms with odd number of protons and/or neutrons possess nuclear spin angular momentum S
• Associated with S is a magne>c dipole moment
• Magne>c dipole moment rotates under external magne>c field, exhibit magne>c resonance phenomena
• The varia>on of rota>on of spins generate magne>c fluxes and can be recorded
• Hydrogen H+ atoms are abundant in biological specimens
MRI:
use magne>c fields to perform
• Relaxa>on: Main field B0
• Excita>on: Radio Frequency (RF) field B1
Principle of MRI
Compressive sensing
E. Candes, J. Romberg, T. Tao 2006 David Donoho 2006
Compressive sensing
71
70% Salt-and-Pepper
Noise
Chan, Ho, Nikolova L1
影像除躁
Denoising
73
Chan, Ho, Nikolova
70% Salt-and-Pepper
Noise
影像填補(Image Inpainting) “Image Inpain+ng : An Overview”, Guillermo Sapiro “Fast Digital Image Inpain+ng”, Manuel M. Oliveira, Brian Bowen, Richard McKenna and Yu-‐Sung Chang
Some Applied Mathema>cians in Image Processing
Image Processing
Compressive Sensing (2004)
• Dv
David Donoho Emmanual Candes Terence Chi-‐Shen Tao
Candes, Romberg, Tao, Robust uncertainty principles: Exact signal reconstruc>on from highly incomplete frequency informa>on, IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 2, FEBRUARY 2006 cited 4894
DL Donoho, Compressed sensing, Informa>on Theory, IEEE Transac>ons on, 2006 , cited 6716
數學有何用
• 數學是理性思維的工具 – 探索未知 – 解決現實問題 – 開創未來世界需要數學
未來趨勢 未來挑戰
• 全球暖化 • ⾃自動化的時代 • 都市化 • 資訊的世界 • ... • 能源危機 • ⾼高失業率 • 貧富懸殊
因應未來挑戰
vs 開創未來世界
我的一個經驗
• 1979年我在IBM的Watson Research Center 做
summer student
• 我的mentor: Williard Miranker (1956 NYU Ph.D)
• 我兩周內把他給我的問題做完了, 之後他帶我參觀很多部 門 – 在空間設計上,部門之間沒有隔閡,全部都連通,吃飯丶喝下午 茶也都在一起 – Miranker因此和很多部門的人都熟識,知道很多部門的前緣研 究, 包括語音部門,programming language, 還有一些物理實驗 的部門 等 – 其中最特別的是未來學的部門: 他們常與科學家與工程師互 動,探 討未來科技或社會的走向,或者是想像一種新的發明,並 激盪出 可行的方案
我的一個經驗
• 過去30年許多知名的科技發展都出自此,例如
DES(Data EncrypOon Standard)加密演算、經典 的電腦程式語言: FORTRAN(Formula TranslaOon System)、本 華·∙·∙曼德博(Benoît B. Mandelbrot)
的報告書中發表了碎形 (Fractal)、磁性碟片儲
存(硬碟)、用單一個電晶體即 可記憶一個Bit的
動態RAM(Dynamic Random Access
Memory,DRAM)、精簡指令集電腦(RISC)架構、
以及 關聯式資料庫等。IBM研究院在物理科學
上也有所貢獻, 包括掃描隧道顯微鏡(簡稱:STM)
以及高溫超導等,此 兩項成就都獲得了諾貝爾
創造未來的一個設計
• 這是創造未來的一個設計。 • 進行前緣研究的專家們在開放的環境下, 彼 此自然認識 • 有機會激盪丶藴釀丶成熟思想。看到跨領 域研究自然產生。第二個例子
: 誘發動機
• 2012年加州理工學院(Caltech) 的新生訓練 • Adam Steltzner演講: 他是噴射引擎實驗室 (JPL)的登陸火星探測車(好奇號,Curiosity Rover)的計畫主持人hTp:// www.youtube.com/watc?v=l9P9JNwwiMY • 講題: How the Trip to Mars has Changed Us• 他小時候對星空的好奇,對人類登陸月球 的興奮。
• 對探索火星的奧祕感到無比的興奮
• 這是一個追求真理的好奇之旅,而這個探測
車有一 輛小汽車那麼大,要零失誤地在7分
鐘內自大氣層 外登陸火星地表,這是一項非
常不容易的任務, 他說明許多可能的困難及
好奇與探索
• 而就在他演講的一刻,同步放映了探測車自 火星地表傳來的影像,映在三層樓高的大銀 幕 上,學生仿如身歷其境,既震撼又感動。這 是一 埸主題為好奇丶夢想丶挑戰未來的大 型表演,無疑的它是一埸振奮新生的成功表 演。
This scene from the panoramic camera (Pancam) on NASA's Mars Explora>on Rover Opportunity looks back toward part of the west rim of Endeavour Crater that the Rover drove along, heading southward, during the summer of 2014.
好奇與探索
• 你對什麼好奇呢?
• 你做過什麼探索呢?
• 你的探索有用到數學嗎?
未來與過去
: 一個思維模式的差異
• 西方的電影: 有很多Science Fic>ons, 災難片, 很 多是以未來為背景 • 大陸的電影: 宮廷片,背景是過去,現代片 • 港片:古裝,現代片 • 台片:現代片東西文化面對過去與未來的差異
• 西方文化對未知事務的好奇,對未來的夢想, 對求新求變的渴望,對奇發異想的包容甚至 賞識, 對科技發展的高視野,這些都是推動創 新的文化元素 • 東方文化較為現世,同時也有太多過去的包 袱。 在思維上較少向未來看,或幻想未來的 世界。開創未來
• 轉個念,多向未來看看吧。
高中生數學建模競賽
• Cornell University has a couple of sites for high school students • hTp://www.math.cornell.edu/~mec/
• hTp://www.math.cornell.edu/~numb3rs/
• COMAP has High school Math Contest for Modeling • hTp://www.comap.com/highschool/contests/h
• SIAM M3 challenge
• hTp://m3challenge.siam.org/
• WPI CIMS Industrial Mathema>cs Project for High School Students • hTps://www.wpi.edu/academics/math/CIMS/IMPHSS/
大學生數學建模競賽
• COMAP MCM/ICM • hTp://www.comap.com/undergraduate/contests/ • UCLA RIPS • hTp://www.ipam.ucla.edu/programs/student-‐research-‐ programs/research-‐in-‐industrial-‐projects-‐students-‐ rips-‐2014/?tab=overview• WPI CIMS REU
• hTps://www.wpi.edu/academics/math/CIMS/REU/ • NCSU REU • hTps://www.math.ncsu.edu/REU/ •
Lunch Crunch: Can nutriGous be
affordable and delicious?
• First Lady Michelle Obama spearheaded an ini>a>ve on good nutri>on that led to passage of the Healthy, Hunger-‐Free Kids Act of 2010.
Implementa>on of the act, however, revealed the compe>ng
preferences of the school lunch program’s three major stakeholders.
Students care most about taste and quan>ty; school districts are
concerned about affordability; and the federal government, which provides financial support, wants to promote lifelong healthy ea>ng habits.
• Schools have seen the cost of offering lunch go up (since healthier foods are oZen more expensive), while par>cipa>on goes down
(students are less sa>sfied with school lunch, either because it doesn’t taste as good or
it isn’t filling enough), causing a fiscal crisis for some school districts1. • The USDA has asked your consul>ng firm to provide a report with
mathema>cally founded insights into the problem; you should address at minimum the following three concerns.
• 1. You are what you eat? Students’ caloric needs at lunch depend on how ac>ve they are, whether they eat breakfast, and a host of other factors. Develop a
mathema>cal model that takes as input a student’s individual aTributes, and outputs the number of calories that a student with those aTributes should eat at lunch.
• 2. One size doesn’t necessarily fit all. The guidelines dictated by the Healthy, Hunger-‐Free Kids Act of 2010 are based on mee>ng the needs of an “average student”2. However, mee>ng the average need may not necessarily be the right amount for many students. Now that you’ve iden>fied aTributes that affect caloric needs at lunch, create a model to determine the distribu>on of U.S. high school students among each of these categories. If every student eats the standard school lunch, what percentage of students will have their caloric needs met at lunch?
• 3. There’s no such thing as a free lunch. A sample school district has a weekly budget of $6 per student
for the purchase of food only. Leverage math modeling to develop a lunch plan (using food categories) that stays within the budget, meets the nutri>onal
standards and appeals to students. What changes would you make if your budget was increased by $1?
You may want to take into account how your model could be applied across different geographic and/or socio-‐economic regions.Your
report to the USDA should include a one-‐page summary of your findings.
You may find the following websites helpful: hTp://www.globalrph.com/es>mated_energy_requirement.htm hTp://www.cdc .gov/mmwr/pdf/ss/ss5905.pdf hTp://www.cdc.gov/growthcharts/charts.htm#Set3 hTp://www.amstat.org/censusatschool/about.cfm Winning team
如何做好準備
• 數學物理兼備• 學會寫程式 (matlab)