“A pure mathematician, when stuck on the problem under study, often decides to narrow the problem further and so avoid the obstruction. An applied mathematician interprets being stuck as an indication that it is time to learn more mathematics and find better tools”
— Distinguished differential geometer Eugenio Calabi
“Mathematician is a person who turns coffee into Theorems” – Erdös Paul , the Hungarian Mathematician who had no home, no wife, travelled around the world to visit his math collaborators proving 1000+ theorems. 数学界的”丐帮洪七公”! 他是华罗庚的好友, 和 陈省身 同年得 数学Wolf Prize.
A mathematician is a person who can find analogies between theorems;
A better mathematician is one who can see analogies between proofs
The best mathematician can notice analogies between theories.
One can imagine that the ultimate mathematician is one who can see analogies between analogies.
Applications of Algorithm:
1) Economic : “Stable Marriage” Algorithm (2012 Nobel Economic)
◇ Boston Medical Students Matching Hospitals
◇ Schools & Students’ Choice
◇ Racial Segregation Housing Policy
2) Evolution: Genetics “Mixability” thru the algorithm “Sex” (mix of genes past down from father and mother to their children).
“God did not create species. He designed the Algorithm for creating them.”
3) Multiplicative Weights Update Algorithm: choose the best expert (C1, C2, …Cj…)
Cj = Cj (1 + 0.00001*$)
Weight = 0.000001
$ = profit or loss made by the Financial expert
2016 = 999+999+9+9
The Axioms for a fair voting system (eg. Political Election) :
◇ Outcome decided not by one ‘dictator’, but more than one person.
◇ If everyone votes that A is better than B, then A will be ranked higher than B in the final result.
3. Independence of irrelevant alternatives
◇ The ranking of A relative to B should not be affected by someone changing their mind about C.
Arrow’s Theorem says that if there are more than 2 people (parties) to vote for, then there is no fair voting system.
Most democratic voting systems violate the 3rd axiom (independence of irrelevant alternatives).