On the adjugate

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}).

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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.

好搞笑的比喻

刚才看了一个李敖的视频,关于台湾的军费。在此,李敖提到台湾的从美而来的武器装备,非免费提供,而是购买的。评论里,有人将此形容为:美国摆明了“让狗看门,不但让狗自己买狗粮,还要拔狗毛抽狗血”。一看到我就哈哈大笑,描述的太好了!

这让我回想起我上小学中学,好几次有美国同学争论is Taiwan part of China,一般最终得到结论都是不是。不用说,他们所想的,无论如何,都不会改变事实,所以这种争论是毫无意义的,尤其在他们和当时的我对与历史客观具体事实的无知的情况下。基本在那儿,小学中学的历史课都是垃圾,尤其是在美国,因为老师水平一般不会太高,经常还会很差,比如在美国,好多历史老师会自以为是地将自己的主观偏见施加在学生上。孩子们都想得很简单,什么东西都用好与坏衡量,我也是。现在长大了,就知道好坏正邪非客观存在,但赢者输者是有的,无庸隐讳,败到台湾的蒋介石国民党就是极度的输者。

哪儿都有被洗脑的,洗脑定义为相信客观错误的过于某度,人。我自己小时候,如大多孩子一样,也是处于很洗脑的状态,毕竟小学老师讲的好多都是扯淡,加上父母也会讲圣诞老人之类的。理想的是,一个人随着智力成熟会多看,高质量的,文献,独立思考,客观严谨判断,从而将脑子里的洗脑物清洗掉,排泄掉,同时也克制控制自己的情绪,不允其干涉人的理性思维。在这一点,本人是有了大的进步,而且我相信它的导数,随着时间或年龄,现在还是正的。当然,我一定是有目前必未知的可改进的地方,未知前加必道理同Dunning-Kruger。高智商的人洗脑率及度当然是更低,可是有不少,说明智商不足于防御洗脑。谈一个很典型的例子,就是自由与民主,或资本主义或共产主义,这类词。任何科学人都应该知道一个人是否是某某主义者,非二进制变量,其甚为复杂,但我是看到好多有科学能力的人却说起我相信我们的民主和自由这种从科学角度空白的话,直接一点还不如说我拥护西方民主主义制度罢了。我也看到很多高智商的人未调查而出论,批评自己不理解的,而看上去还是相信自己所言的。这都是缺乏克制力的表现,因为有克制力的人是会把事情搞透了在张嘴。我看到过数学学得很好的香港人将中共称为独裁,将毛泽东视为杀人猛兽,并且坚定此未有根据的观点。我看到过台湾人指责简体字及其它宏观上微不足道之事为表示对异岸之心逆。我看到过俄罗斯人排斥一切苏联的,似乎认为九十年代的预期寿命下降,人口负增长,难得积累的工业制造业毁于寡头流氓的自由的俄罗斯好于往年的岁月。在这方面,据我所看到了,大陆人还真的不错,由于对于外来的信息相对比较开放,不像好多港台同胞或美国人会将任意来自中国政府的信息拒绝为洗脑或政治宣传。

对于台湾人的政治心态,我真的不想说的太负面。我对台湾其实知道的很少,从来没去过。我认识一些台湾人,他们都很有能力,品德也很好。不过,台湾会让我想到他们曾经禁的一位作家的代表性故事之一的主要人物。我所看到的许多台湾人,及香港人,的心态近于那个人物象征的心态。

老代中国科学家与诗词

读关于中国老一代科学家的资料,很难不观察到他们都有很好的语文修养,将写诗作为他们的业余爱好。几个月前,我看到了杨振宁吹捧陈省身的一首诗,为:

天衣岂无缝
匠心剪接成
浑然归一体
广邃妙绝伦
造化爱几何
四力纤维能
千古寸心事
欧高黎嘉陈

这被我翻译成英文为:

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

今天我又看到,来自这里,关于彭桓武与陈能宽长达十年的诗缘。此中,对我印象较深的是:

亭亭铁塔矗秋空,
六亿人民愿望同。
不是工农兵协力,
焉能数理化成功。

第一句代表的当然是中国第一颗,在秋天,放在铁塔上实验的,原子弹,此为当时中国所有人的共同愿望。后两句又,符合共产党对于无产阶级之重视,强调了工农兵的重要性。这首诗很符合当时的国情,强调知识分子为人民服务,而非摆着架子,瞧不起底层人民。六亿人民与工农兵又让我想起在毛泽东的《送瘟神》“六亿人民尽舜尧”那一句,所表达的观念相似。

如何解释之?我想是大科学家,按照在Steve Hsu的博客上在关于g的讨论中用的语言,V都很高与中国老的科举的那一套结合所产生的自然结果。

 

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.

China_IQ_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_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.

苏联歌曲

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。