Heritability of BMI and strength measures

We all know that height is highly heritable (about 0.8 according to my memory). Less well known is that BMI and muscle strength are very heritable too, at 0.5-0.6, according to this. This is somewhat counterintuitive in that common sense in some sense tells us that BMI can be reduced by not eating unhealthy and exercising, and similarly, that everyone can grow muscle from working out with the correct form. Well, it appears that the ability to do so is highly heritable. It conforms to my observation of the sheer difficulty with which overweight people have with gastronomic self control and of the rarity with which physical clumsiness changes within people.

This reminds me of an analogous cognitive characteristic. On the cognitive end, a similarly high level of heritability for vocabulary is similarly counterintuitive (it was to me too, before I had developed some basic psychometric intuition). Isn’t vocabulary memorized, learned? Yes but some people learn it much more naturally than others, and actually retain it. There is, I hypothesize, some neurobiological mechanism that more or less determines the capacity for storing vocabulary. From my personal experience vocabulary is much more about recognition of analogies than about memorization. With the former, the latter is hardly needed to obtain a sizable vocabulary. I also believe analogies are universal, independent of the language in which they are represented; this is largely supported by the fact that the two languages I know well, English and Chinese, share more often than not the same analogies for different definitions of characters/words. A common linguistic construction of semantic units in both, for instance, is the extension of concrete to abstract. So different cultures, with different languages, more or less discovered the same analogies for construction of their respective languages. All this suggests some faculty in the brain, that one has to some degree or another, linked with general ability to discern analogies, that is manifested through vocabulary size, a direct consequence of that ability.

As an FYI, this is personal speculation of a quasi-educated nature. I say quasi because I am constructing an explanation most rational based on my personal knowledge, which is very limited, my having at the present moment not even passing knowledge of linguistics or neuroscience, which I hope to obtain on a gradual basis.

Second isomorphism theorem

This is copied from a Facebook chat message I had with someone a few weeks ago, with wordpress latex applied over the math:
A couple weeks ago, I learned the statement of the second isomorphism theorem, which states that given a subgroup S and normal subgroup N of G, SN is a subgroup of G and S \cap N is a normal subgroup of S, with SN / N isomorphic to S / (S \cap N).
Any element of SN / N can be represented as anN = aN for a \in S, where the n on the LHS is in N. A similar statement of representation via a(S \cap N), a \in S holds for S / (S \cap N). Define \phi: SN/N \to S / (S \cap N) with \phi(aN) = a(S \cap N), which is bijective. By normality, \phi(abN) = ab(S \cap N) = a(S \cap N)b(S \cap N) = \phi(aN)\phi(bN). Thus, \phi is an isomorphism. QED.

Proof of fundamental theorem of arithmetic

I read it a couple days ago and actually remembered it this time in a way that I will never forget it. It invokes Euclid’s lemma, which states that if p | ab for p prime, then p | a or p | b, which can be proved using Bezout’s lemma. For existence, it does induction on the number of factors, with 1 as the trivial base case. For the non base case, wherein our number is composite, apply the inductive hypothesis on the factors. For uniqueness, assume two distinct factorizations: p_1p_2\ldots p_n = q_1q_2\ldots q_n. By Euclid’s lemma, each of the p_is divides and is thus equal to one of the q_is. Keep invoking Euclid’s lemma, canceling out a prime factor on each iteration and eventually we must end with 1 = 1 in order for the two sides to be equal.


今天又听了诸多苏联歌曲,甚悦耳,大多来自于薛范的YouTube account。音乐与镜头配的好的不能再好了。记得曾经薛范在采访中说过:做他的工作需要音乐,俄文,和中文三项通,具备此条件的人并不多。

所听的,我能想到的有:москва майская, москва-пекин, прощание славянки, подмосковные вечера, марш защитников москвы, в путь, 等等。这些无论是其音乐还是其歌词艺术质量都很高。我认为一个人可以不喜欢这些歌代表的苏联的政治那一套,但是很难从一个纯粹艺术角度上低评它,就像一个天才可能是个asshole,但是这并不改变他是天才的现实。在这里边,对我印象比较深的是在《莫斯科-北京》里,还有,若我没判断错,郭沫若的镜头。另一位出现的我记得的好像是胡乔木,但这我不能太肯定。

A mechanics problem

Today, I solved one from the Cracking the AP Physics C Exam 2008 Edition, which I had bought in high school. It is


Questions (c), (d), and (e) are not included. They are to determine the relationship between the magnitude of the acceleration of the block and the linear acceleration of the cylinder, to determine the acceleration of the cylinder, and to determine the acceleration of the block respectively. It is obvious that from (b) that the magnitude of the acceleration of the block is twice the linear acceleration of the cylinder, and the answer to (e) can be derived from that of (d) via this. As for (d), I first solved it with a free body diagram and Newton’s laws. Then, my ultra gifted friend Brian Bi suggested using the Euler-Lagrange equations, which simplifies things greatly, to the extent that he characterized himself as having forgotten how to solve mechanics problems the way I had done (obviously not true, but you get what he’s saying). I was able to work it out this way shortly, getting the same answer.


Note: I left overlooked the 4 coefficient initially and only filled it in at the end. It is still omitted in some of the intermediate equations.

First blog post

I plan, hope to be writing out here a miscellany of thoughts and musings pertaining to my intellectual interests, which can be roughly divided into a primary one, which is pure math and other topics of a mathematical nature, and secondary ones, which, of the ones I can think of now, are Chinese language, language (in general), psychometrics, neuroscience, and stories of scientists and of unusual people of one form or another.