Month: October 2016

Arnold传略+Arnold’s Article on Teaching Math

I used Arnold's book for my first ODE class, and it was a blast. I later on learned more about this mathematician and find his philosophy of doing math is really inspiring. Here is a biography of this mathematician I found online. Written in Chinese. P.S. I'm engrossed by Arnold's way of doing math (geometry, … Continue reading Arnold传略+Arnold’s Article on Teaching Math

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Creative Ways of Communicating Research

I'm part of LYNX under the Stanford Undergraduate Research Journal this year. Here's a useful piece we received as new writers: Awesome Examples Writing Quanta Magazine (here) Nautilus (here) Asian Scientist Magazine (here) Philosophy Now (here) Podcast / Radio / Audio Radiolab (here) Cold Spring Harbor Labs Oral History Collection (here) Philosophy Bites (here) Social Science … Continue reading Creative Ways of Communicating Research

Bolloba’s First Theorem

From Don Knuth‘s Theory Lunch Talk at Stanford on Oct 6. I found a blog post talking about the exact same thing, so won’t repeat. The idea behind the proof of this rather combinatorial theorem is interesting. The main idea is to construct a nice mapping between a permutation of (1,2, …., n) and a pair of (finite) sets: (A, B) based on the elements inside the sets.

You can try to figure out and prove when the equality holds by yourself as an exercise.

When is this Theorem useful? For example, in making representative families <– so that we have better memory performance. Think about it!411ei42toel-_sx328_bo1204203200_

Too lazy D:< Plzzzz let me know if you made a nice drawing for the proof of the theorem (Shouldn’t be too hard) so that I can include your image here rather than this textbook cover.

 

Uniformly at Random

The following theorem of Bollobas is an important result from extremal set theory.  It has apparently been rediscovered several times by others.  It implies certain classical results, such as Sperner’s Theorem, and has some interesting extensions.

Theorem (Bollobas).  Let $latex A_1, ldots, A_m$ and $latex B_1, ldots, B_m$ be sequences of $latex a$-element and $latex b$-element sets respectively.  Suppose that $latex A_i cap B_i = emptyset$ and $latex A_i cap B_j neq emptyset$ whenever $latex i neq j$.  Then $latex m leq {a+b choose a}$.

Proof.  Let $latex X$ be the union of all the $latex A_i$’s and $latex B_i$’s.  Consider an arbitrary permutation $latex pi$ of $latex X$.  We claim that there is at most one pair $latex (A_i, B_i)$ such that all of the elements of $latex A_i$ precede those of $latex B_i$ in the ordering $latex pi$.  Suppose to the contrary that there are two…

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A Solution to High-Energy Quantum Gravity: AdS/CFT + Quantum Error Correcting Code = ???!

I went to a talk given by Daniel Harlow at Stanford last Thursday (2016 Oct) on Emergent Locality and Gravity & Quantum Error Correction. He started from summarizing the existing attempts to  solve the short distance quantum gravity problem including string theory and the attempt to quantize the metric as a local field. Then he … Continue reading A Solution to High-Energy Quantum Gravity: AdS/CFT + Quantum Error Correcting Code = ???!