The Unknown Unknown in the Logic of the Mind

The Unknown Unknown in the Logic of the Mind

https://m.youtube.com/watch?v=HYsv5DRl8L4 Second talk by Misha Gromov at Stanford. Nov 11. I really enjoyed the talk, truly inspiring. He walked through different scientific fields including biology, linguistics, and psychology across several centuries, analyzing how people studied science and gave his insights (from the perspective of a mathematician I think). -worth noticing how rephrasing a finding in … Continue reading The Unknown Unknown in the Logic of the Mind

Scalar Curvature, Convex Polyhedra & Differential Operators

Prof Misha Gromov gave two lectures at Stanford on Nov 10 and Nov 11. Here is a brief summary of his Nov 10 lecture: Scalar curvature influences topology and asymptotic geometry of Riemannian manifolds, it is associated with the Einstein equation, it controls the Hamilton-Ricci flow. Yet, the geometric meaning of the scalar curvature remains … Continue reading Scalar Curvature, Convex Polyhedra & Differential Operators

A Spectral Analysis of Moore Graphs

Math ∩ Programming

For fixed integers $latex r > 0$, and odd $latex g$, a Moore graph is an $latex r$-regular graph of girth $latex g$ which has the minimum number of vertices $latex n$ among all such graphs with the same regularity and girth.

(Recall, A the girth of a graph is the length of its shortest cycle, and it’s regular if all its vertices have the same degree)

Problem (Hoffman-Singleton): Find a useful constraint on the relationship between $latex n$ and $latex r$ for Moore graphs of girth $latex 5$ and degree $latex r$.

Note: Excluding trivial Moore graphs with girth $latex g=3$ and degree $latex r=2$, there are only two known Moore graphs: (a) the Petersen graph and (b) this crazy graph:

hoffman_singleton_graph_circle2

The solution to the problem shows that there are only a few cases left to check.

Solution: It is easy to show that the minimum number of vertices of a…

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Non-Measuable Ghost: Persi’s Halloween Talk

It was a ghostful Monday--a Monday full of non-measurable ghosts. You've probably heard of this one sphere cut into five pieces becomes two spheres story. This Monday Professor Persi Diaconis gave a public Halloween lecture about these ghosts. I would love to share an amazing proof that shows 0.5=0.99 or 0.999 or 0.9999. Well, with … Continue reading Non-Measuable Ghost: Persi’s Halloween Talk

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

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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