I learned that the adjugate is the transpose of the matrix with the minors with the appropriate sign, that as we all know, alternates along rows and columns, corresponding to each element of the matrix on which the adjugate is taken. The matrix, multiplied with its adjugate, in fact, yields the determinant of that matrix, times the identity of course, to matrix it. Note that the diagonal elements of its result is exactly what one gets from applying the minors algorithm for calculating the determinant along each row. The other terms vanish. There are $n(n-1)$ of them, where $n$ is the number of rows (and columns) of the (square) matrix. They are, for each column of the adjugate and each column of it not equal to the current column, the sum of each entry in the column times the minor (with sign applied) determined by the removal of the other selected column (constant throughout the sum) and the row of the current entry. In the permutation expansion of this summation, each element has a (unique) sister element, with the sisterhood relation symmetric, determined by taking the entry of the adjugate matrix in the same column as the non minor element to which the permutation belongs and retrieving in the permutation expansion of the element times minor for that element the permutation, the product representing which contains the exact same terms of the matrix. Note that shift in position of the swapped element in the minor matrix is one less than that in the adjugate matrix. Thus, the signs of the permutations cancel. From this, we arrive at that the entire sum of entry times corresponding minor across the column is zero.

A corollary of this is that $\mathrm{adj}(\mathbf{AB}) = \mathrm{adj}(\mathbf{B})\mathrm{adj}(\mathbf{A})$.

## More math

Last night, I learned, once more, the definition of absolute continuity. Formally, a function $f : X \to Y$‘s being absolutely continuous is its for any $\epsilon > 0$, having a $\delta > 0$ such that for any finite number of pairs of points $(x_k, y_k)$ with $\sum |x_k - y_k| < \delta$ implies $\sum |f(x_k) - f(y_k)| < \epsilon$. It is stronger than uniform continuity, a special case of it. I saw that it implied almost everywhere differentiability and is intimately related to the Radon-Nikodym derivative. A canonical example of a function not absolute continuous but uniformly continuous, to my learning last night afterwards, is the Cantor function, this wacky function still to be understood by myself.

I have no textbook on this or on anything measure theoretic, and though I could learn it from reading online, I thought I might as well buy a hard copy of Rudin that I can scribble over to assist my learning of this core material, as I do with the math textbooks I own. Then, it occurred to me to consult my math PhD student friend Oleg Olegovich on this, which I did through Skype this morning.

He explained very articulately absolute continuity as a statement on bounded variation. It’s like you take any set of measure less than $\delta$ and the total variation of that function on that set is no more than $\epsilon$. It is a guarantee of a stronger degree of tightness of the function than uniform continuity, which is violated by functions such as $x^2$ on reals, the continuity requirements of which increases indefinitely as one goes to infinity and is thereby not uniformly continuous.

Our conversation then drifted to some lighter topics, lasting in aggregate almost 2 hours. We talked jokingly about IQ and cultures and politics and national and ethnic stereotypes. In the end, he told me that введите общение meant “input message”, in the imperative, and gave me a helping hand with the plural genitive conjugation, specifically for “советские коммунистические песни”. Earlier this week, he asked me how to go about learning Chinese, for which I gave no good answer. I did, on this occasion, tell him that with all the assistance he’s provided me with my Russian learning, I could do reciprocally for Chinese, and then the two of us would become like Москва-Пекин, the lullaby of which I sang to him for laughs.

Back to math, he gave me the problem of proving that for any group $G$, a subgroup $H$ of index $p$, the smallest prime divisor of $|G|$, is normal. The proof is quite tricky. Note that the action of $G$ on $G / H$ induces a map $\rho : G \to S_p$, the kernel of which we call $N$. The image’s order, as a subgroup of $S_p$ must divide $p!$, and as an isomorphism of a quotient group of $G$ must divide $n$. Here is where the smallest prime divisor hypothesis is used. The greatest common divisor of $n$ and $p!$ cannot not $p$ or not $1$. It can’t be $1$ because not everything in $G$ is a self map on $H$. $N \leq H$ as everything in $N$ must take $H$ to itself, which only holds for elements of $H$. By that, $[G:N] \geq [G:H] = p$ which means $N = H$. The desired result thus follows from $NgH = gH$ for all $g \in G$.

Later on, I looked at some random linear algebra problems, such as proving that an invertible matrix $A$ is normal iff $A^*A^{-1}$ is unitary, and that the spectrum of $A^*$ is the complex conjugate of the spectrum of $A$, which can be shown via examination of $A^* - \lambda I$. Following that, I stumbled across some text involving minors of matrices, which reminded me of the definition of determinant, the most formal one of which is $\sum_{\sigma \in S_n}\mathrm{sgn}(\sigma)\prod_{i=1}^{n}a_{i,\sigma_{i}}$. In school though we learn its computation via minors with alternating signs as one goes along. Well, why not relate the two formulas.

In this computation, we are partitioning based on the element that $1$ or any specific element of $[n] = \{1, 2, \ldots, n\}$, with a corresponding row in the matrix, maps to. How is the sign determined for each? Why does it alternate. Well, with the mapping for $1$ already determined in each case, it remains to determine the mapping for the remainder, $2$ through $n$. There are $(n-1)!$ of them, from $\{2, 3, \ldots, n\}$ to $[n] \setminus \sigma_1$. If we were to treat $1$ through $i-1$ as shifted up by one so as to make it a self map on $\{2, 3, \ldots, n\}$ then each entry in the sum of the determinant of the minor would have its sign as the sign of the number of two cycles between consecutive elements (which generate the symmetric group). Following that, we’d need to shift back down $\{2, 3, \ldots, i\}$, the presentation of which, in generator decomposition, would be $(i\ i+1)(i-1\ i) \ldots (1\ 2)$, which has sign equal to the sign of $i$, which is one minus the column we’re at, thereby explaining why we alternate, starting with positive.

## 老代中国科学家与诗词

How not woven the fabric of the universe
Spliced with craft
Comes together as one
Wide and broad with unparalleled mystery
Nature loves geometry
Fiber bundles describe four forces
Long unsolved problems
Euclid Gauss Riemann Cartan Chern

## Some speculations on the positive eugenics effects on the far right tail of intelligence of the Chinese population of the imperial examination system

Over the past few months, I had read casually on the imperial examination system (科举) out of curiosity. My knowledge of it, the system that very much defined pre-modern Chinese society, is still very limited and vague, but now I at least know what 进士 and 秀才 are, along with some classical Chinese, background indispensable for understanding that system. I hope, if time permits, to learn more about this over the next year, on the side.

It has occurred to me that the imperial examination system, while doing much to prevent China from developing modern science as the West had for cultural reasons, did select for intelligence at the far tail. The reason is simple. The tests, which were very g-loaded, conferred those who scored highly on them wealth, position, and status that enabled them to have more children, and those from families who scored highly married those from similar families. Over time, there emerged an elite subpopulation with very high base genotypic IQ, one that results in those born from such families to regress not to the overall Chinese mean but to the high mean of that subpopulation. This is consistent with the fact that in the 20th century and probably even today, a disproportionately high percentage of top scholars, scientists, engineers, and even revolutionaries and political leaders of Chinese descent can be traced back to those elite 科举 families, based on the many examples I have seen. I’ll not give specific examples for now; they can easily be found by anyone who reads Chinese.

I will conclude with a note that is likely to be very relevant. Brian Bi, about a year ago, made this following IQ map of China by province.

You can also view it here.

First of all, the data may not be very accurate; I’ll have to check on its source. But for now, let’s assume that it is. Then, what’s most noticeable is the high average of Zhejiang, consistent with the number of mathematical and scientific geniuses of Zhejiangnese ancestry relative to the number of those with ancestry of other provinces, adjusted for province population of course. Examples are numerous: Shiing-Shen ChernWu WenjunFeng KangYitang Zhang, etc for math. There is also, in another field, Qian Xuesen. Too many to name. Brian Bi and I have wondered the cause of this. It is plausible that the aforementioned effect was much more pronounced in Zhejiang than in other provinces in China. Of course, there is a probably substantial environmental effect here too. So I guess to satisfy this curiosity, I might study some Zhejiangnese history as well.

Aside from prominence in science, Zhejiangnese are stereotyped in China for being really entrepreneurial. They are now one of the most prosperous provinces in China, needless to say. They are, to put it simply, a super breed among Chinese, to my superficial view.

## intelligence, math, chess

I met up with Kolya, Austin, and Ethan today. We ate out at this Ramen place, where we chatted mostly about light math, mathematicians, and intelligence related topics. I remember telling Austin about how the brain doesn’t peak in many until mid-late 20s or so, with rapid growth spurts often occurring in one’s early and mid twenties. This is consistent with the precipitous dip in car accident rates at age 25-26, the age when the prefrontal cortex matures, according to various online sources I’ve seen. So if you are struggling with things and not past that age, you still have plenty of hope! As for math, I asked Austin, who is entering a math PhD program next year, about the proof of Rolle’s theorem, which an old friend had asked me about a week earlier. It goes as follows. The hypotheses are continuity from $[a,b] \subset R$ and differentiability on $(a,b)$. As for the extreme value, it can occur either at an endpoint, in which case the function is constant, or in between, in which case the left and right derivatives are less than or equal to zero and greater than or equal to zero, which combined necessitates a derivative of zero at that point.

Afterwards, we played some piano with some singing alongside, and following that, I played a game of chess with Ethan after he asked if I had a chessboard, which I did. Chess has basically not crossed my mind for almost 10 years, and I have probably not played a single game in the last 3 years or so. I would sometimes observe the live 2700 to see changes in rankings, but there was no actual chess content in my head whatsoever. I still have nostalgia for when I played in chess tournaments in 6th grade. I remember at the end, I had some state rating of only a little above 1200, having placed in the top 30 in the state tournament that year. Needless to say, the level of chess going on between those little kids, of which I was one, was quite low. I stopped in junior high as there was no chess club there. Nonetheless, I always had a mix of fascination and awe with the game. At that time, I was, to put it bluntly, quite clueless about it; I simply had not the requisite intellectual maturity.

I obviously lost to Ethan, who is rated at 2200 something, but to my great pleasure, I didn’t lose in a pathetic way. I was very calculating and rational in every move, to the extent that my level of knowledge and experience permitted. At the end, I lost with a reckless sacrifice where I forwent a minor piece for two of the pawns that covered his kingside castle, hoping to launch an attack. I did not calculate far enough and it was not successful, and seeing that all hope was lost, I resigned. The biggest contrast between this time and when I played long before was that I had much better positional sense, which I suspect is derived from a substantially higher level of qualitative reasoning, the aspect of intelligence captured by the verbal side of IQ tests, relative to before. I believe this because position is all about how different pieces to relate together and about thinking of the pieces in a high level of coherence. In every move, I took into account positional considerations. Unlike before, when I could make moves recklessly, without thorough calculation, I would think carefully on what exactly I gained from making such a move, as well as thinking how the opponent could respond. It is just like how in writing, every word you use must be there for a good reason, and how in social interaction, one needs the cognitive empathy to predict how the other party is likely to respond. I will not go much into the details of the game, which I am not confident I could easily reconstruct. I do remember it began with the Caro-Kann, the name of which I still remembered well, and that in the early middle game, it felt like there was little that could be done that made sense. Then, I had said, “I feel like I’m in a zugswang right now.” Ethan responded with a why, along with a remark this game appears more positional than tactical.

Back on the car, Ethan and Austin played some blindfold chess (or at least it seemed like that), which I don’t think I could do. Austin thought that blindfold chess required visual spatial, but I told him that chess viewed properly is not visual spatial at all. The state of a chess game in essence is entirely qualitative, representable as an array of states, each of which is empty or of some piece, along with states for castle and enpassant. The board is nothing but visual distraction. This is akin to how blindness did not interfere much with genius mathematicians like Euler and Pontryagin; the math exists independent of the visual representation through text.

While they were playing, I brought up Mikhail Botvinnik, who I was reading about on Wikipedia in both English and Russian (okay I still know very little of that, but enough to get *something* out of glancing through texts). He was a Soviet Jew who was one of the top chess players during the Stalinist era and a world champion. He characterized himself as “Jewish by blood, Russian by culture, and Soviet by upbringing.” On a victory in a great tournament in Nottingham, he sent an effusive telegram to Stalin. He also became a committed communist, whatever that means. Speaking of which, could it be a coincidence that both Kasparov and Fischer became political radicals notorious for opposing their home countries in an obnoxious manner, especially Fischer, who behaved as if he had developed some form of schizophrenia? Will Magnus Carlsen become the same? (I think not.) Anyhow, chess is a crazy world, with the people at the top most definitely not normal, and the politics, viewed superficially by me, not qualified to discuss the matter intelligently, can be intense as that pertaining to the Olympics, which can, as we all know, also go quite out of hand.

I’ll conclude by saying that if I were take up chess again, I could probably do much better than before with my much bigger brain, though of course, I have matters of higher priority. In any case, I’ll probably keep a casual interest in chess, and perhaps read more about the lives of and culture amongst the top players of the world, as well as studying the game itself.

## 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 $p_i$s divides and is thus equal to one of the $q_i$s. 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.