## On China

I’m talking to that 犹太IMO金牌 again. I first asked him if he knew the Riesz representation theorem, the statement of which I saw today. He said he used to. Then I brought up Shizuo Kakutani, who was quite a genius mathematician, who created some generalization of the aforementioned theorem or something like that. His daughter Michiko is also a distinguished writer. On that I said:

Lol I haven’t gotten to meet many Japanese
They don’t emigrate much nowadays, so patriotic
They’re so well organized and efficient
Produces lots of top mathematicians too

He responded with “china weak.” And “china deserved to get fucked by japan.”

On that, I was like:

Haha
China was super weak back then
Of course, the situation has reversed/is reversing
China is still behind Japan in many advanced areas, but it’s just a matter of time
Japan lost to America in WWII
China on the other hand could defeat America in the Korean War
Thanks to communist ideology

He said that “china did not defeat america.” I responded:

It was a stalemate whatever
But China proved it could get even with number one in the world
When she was still very behind
In any case, in the war in North Korea, America clearly lost, America had to flee
If China had better logistics and equipment probably could’ve taken over the entire Korean peninsula
Because of the Korean War, many of those top Chinese in STEM in America returned
There were negotiations as America knew if they let them return these people would serve their enemy
People contrast that to the brain drain after reforms
The younger generation of Chinese do not have the type of selfless patriotism that the older generation did
Lol you don’t like China
I think America lost its best chance to bring China down, that was during the 89 protests
That was actually kind of close
It’s quite remarkable that China recovered so well. When you’re down, it’s really hard to get back up.
In any case, by 1970s, people in China knew that the most difficult/critical period was past.
And that China had succeeded at it
It’s like earning money, the beginning is the hardest, once you’re rich and high up, it’s almost impossible to fail

He says: “fuck china. china anti human rights.” It’s funny how so many people say that, and I believe privately, or not so much, many in the world have a rather low opinion of China. Though I’m Chinese, I wouldn’t say I really care; it’s just a perception as far as I can tell, not something that can be objectively defined. When I grew up in America, I kept hearing this negative stuff about China and was wondering what was going on. Back a decade ago, China was much less developed than now, and perhaps because of that, the bashing sometimes feels to have subsided quite a bit now compared then, but maybe not, considering that even a guy like him will say that. Whether he genuinely believes it, that is another matter.

On this, I’ll give some of my thoughts. Recall that I said in my chat with him: “when you’re down, it’s really hard to get back up.” This is in general, it applies to individuals as well, with unemployment and such. In the context of the chat, I was referring to the century from 1850 to 1950, when China kept being beaten and made little progress when the rest of the world was advancing rapidly, including China’s foe from the East, Japan. Back then, many intellectuals desperate believed China to be hopeless and on that, even advocated the abolishment of Chinese characters. I believe China was very fortunate to get out of that, as it could have easily been much worse. The international situation, in particular the world’s having been exhausted after WWII destruction, gave China the opportunity to win the civil war, ending a century of violent internal strife that had severely hampered development. The Korean War did much to help Chinese regain their confidence. It proved Chinese military ability for the first time in modern history, much needed at the time, and America blundered by letting China do so. The 1950s was a golden period for China, during which with Soviet aid, China modernized essentially, developing the industrial foundation that even after the Soviet Union withdrew its support for China, though it had a significant negative effect development, China was able to do okay. In the 60s, the international situation was very unfavorable for China, but by 1970, China was high up enough in terms of capability that America had no choice but to recognize it, seeing that there was no way the old regime in Taiwan could retake the mainland. At that time, China was still extremely poor standard of living wise, but there was already a fair degree of technological sophistication. China was also very lucky not to suffer the demise that the Soviet Union did that is literally impossible to recover from. Why that did not happen, why America did not succeed in 1989 in bringing China down, is a very complex question. The Chinese elite were not as foolish as the Soviet ones. Since then, China has made tremendous progress in terms of developing economics and standard of living and also STEM, and though of course, China is still behind in certain areas, it is only a matter of time as many believe before the gap closes. Throughout the last 50 years, these “experts” have doubted the PRC, but the PRC keeps proving them wrong. Maybe these “experts” should stop deluding themselves on many matters.

It is interesting how many very intelligent people in the West, including the person I mentioned in this very post, believes some rather peculiar notions on China related matters. It still puzzles me where they’re coming from with all that. They can not like China or see China as a threatening competitor (and I won’t be offended by that, as people are entitled to their own view), but they should still try to be objective, as unpleasant as the facts may be for them to bear. Penalizing someone or downgrading someone’s ability or accomplishment out of an antipathy for that person’s background or political/religious beliefs is the act of a little person, an insecure person. Also, when you discriminate against someone and they still beat you, it’ll only make them more formidable and yourself more insecure.

Last but not least, I’ll reiterate again that Anglo culture is still dominant across the globe, as a legacy of British colonialism as well as subsequent American supremacy. With that said, international discourse will necessarily be biased towards the interests of that group, an obvious fact that apparently still needs to be noted, and a rationalist would apply some correction to account for the bias. On the other hand, Chinese language and culture is still alien to most of the world, and a derivative of that is that there is much vital information accessed little outside of China of much more validity than what the Anglo media chooses to promulgate. I know that there are ones keen on using such means to alter political opinion and whatnot, so as to bring down a regime they don’t like, as was done in Ukraine in 2014, but these are rogue tactics that will eventually reflect badly on its instigators. Plus, time and time again, Chinese have proved not foolish enough to fall for these tricks.

## Various thoughts, here and there

Another exhausting week of work is past, and I am presented with another chance to wind down. Last night, through various casual reading, I was reminded of the concept of “effeminacy.” In many contexts, if you’re a man, the worst you can be seen as is “effeminate.” And of course, American culture stereotypes Asian men (specifically East Asian men) as effeminate. On this, there is this and this, among many other similar articles, especially the notorious this, which I stumbled on several years ago. I’ll say that there is as far as I can tell some truth to this from an objective biological point of view, in the likes of higher and softer voices and lower testosterone levels in East Asian men relative to white men, and also in white men relative to black men. On this note, I was also reminded of some comment of Michael O Church on reddit (which I cannot easily find anymore) that used “high IQ androgyny” as a factor to illustrate how super smart people (like +3.5 even +4 sigma g) get smashed in the corporate world, politically. It brought me to wonder if far tail g men really are more effeminate, which is likely to be the case, as there has got to be some biological tradeoff for the substantially larger brain that is the material source of such extreme cognitive ability. The light, nasal (or whatever you call it, for lack of better word I can think of) of voices of various mathematicians echoed back to me one after another, in contrast to the deeper and superficially more assertive and aggressive (and masculine) voices of those dumber business guys (mostly WASPs) in positions of power in the corporate world. Speaking of WASP, I could see also how Jews are perhaps more effeminate as well; I’d seen and heard enough that I feel I could intuitively recognize a Jewish voice as well, with of course there being the style of language in combination with the vocal mannerisms that sounds it more distinctively Jewish. I’ve long noticed that though East Asians reared in US mostly speak without an accent, it is still often easy to tell that it is an East Asian voice; such just goes to show how real race is, how rearing in an alien land and culture changes not the deeply engrained racial characteristics which go beyond physical appearance. It is quite a marvel indeed and a beautiful product of human evolution, a panorama of human biodiversity, that of course varies significantly more among individuals of groups than among the group averages. East Asians may be effeminate in the sense aforementioned, but interestingly, there is this “mad Asian guy” on Steve Hsu’s blog who in comments has written of the East Asian cognitive profile relative to the white one as akin to the male one vis-a-vis the female one, which is the undebatable truth born out by the result of testing that has across generations revealed East Asians to be higher by about if not at least two-thirds standard deviation in math and visual spatial, thereby making white people more effeminate in another sense.

Human (biological) development is quite fascinating, especially those of outliers, of which I am one to quite a degree, though probably not enough to make me genuinely distinguished (as in 1 in 100,000 or even 1,000,000 in terms of rarity of ability and accomplishment) in any way ever. We all know that top athletes in legit sports like Shaq are physical freaks of nature who deviate dramatically from the norm in their physical development but in one still constrained by the bell curve, physically speaking, but more consequentially, there are geniuses of mind the brains of which are extremely unusual in such a way that concrete output, in terms of quality and profundity, and to a lesser extent quantity is exponentially increasing as one goes further along the tail in our artificially created bell curve. These people, by virtue of their unusual brain structure, are able to perceive the world, engage in a form of consciousness, far beyond, and more objectively correct than, what the normal human experiences. One can only put oneself in another mind rationally; it is impossible to do so for real. As an example to illustrate, it is impossible to feel the way a mentally sick person does when one is healthy and that does not change for one who has recovered from a depressive episode with respect to oneself. Saying this brings to my mind the following quote of Huxley:

Perhaps men of genius are the only true men. In all the history of the race there have been only a few thousand real men. And the rest of us–what are we? Teachable animals. Without the help of the real man, we should have found out almost nothing at all. Almost all the ideas with which we are familiar could never have occurred to minds like ours.

This quote obviously has eugenicist overtones. It hints at a need of measure to protect against degeneration of a human species that for all its wonder and power, relative to real animals, is still mostly degenerate. It also evokes fear, especially in this day and age when genomic prediction and selection is imminent, that some elite will use it to rule an ignorant, subservient masses. We are and already have been in such a world for ages, with the heritability of ability coupled with transmission of money and power in a feudalistic manner, just now in disguise, with additional channels for mobility now for those of ability from lower background. The elite still rules and exploits the masses, just in a way more benign than ages ago with slavery and priestly spiritual opium abound. There is all the evidence that the best we can create as a society is to provide for everyone an environment and position and work appropriate for his ability and temperament taking into account of course the needs of society. There is also the observation that in general, those of higher ability have higher expectations in terms of what they do and also in terms of their material life, just as kids expect less than adults, in terms of their lifestyle. While there are significant differences in ability, and it probably wouldn’t hurt to make people a bit smarter, I would not say there is anything terribly particular one should be, and that everyone should try to find and bring out what is suitable for one’s ability and one’s circumstances, and hopefully something that is never seen before. I’ve lived long enough to be aware that every stage there is a unique personal challenge, and that one’s position relative to others does not necessarily affect one’s happiness much.

History has witnessed clashes, often in the form of war, between races, cultures, beliefs, systems. Groups have, in contempt or hatred of another, sought the other’s destruction, the other’s subjugation, the other’s cultural conversion, particularly in the religious sphere. It is human nature and also human weakness. There is a human inability to respect another far different from oneself for what the other is and a tendency in such cases to be imposing, with this’s being particularly prominent in certain religious cultures. I have had such myself, but they are as far as I can tell largely cleansed away, upon my realization of the differences in humans inherent and not permanently malleable, as evidence by significant changed in people once removed from parental pressure, once they are more free to do as they choose. In this regard, I take a more liberal attitude and come to cherish the diversity in culture, the diversity in styles of thinking, the diversity in talents across the world and across professions. I’ve come to realize over time that to feel contempt for another for his beliefs and tastes is futile, a waste of energy, and in such cases, parting ways, and viewing the difference as a mere reality, one observed and chuckled at, is the best way to go. Moreover, contempt can even metamorphize into some form of appreciation for the richness of our world in the variety it offers, in which one is but one constituent.

## Math sunday

I had a chill day thinking about math today without any pressure whatsoever. First I figured out, calculating inductively, that the order of $GL_n(\mathbb{F}_p)$ is $(p^n - 1)(p^n - p)(p^n - p^2)\cdots (p^n - p^{n-1})$. You calculate the number of $k$-tuples of column vectors linear independent and from there derive $p^k$ as the number of vectors that cannot be appended if linear independence is to be preserved. A Sylow $p$-group of that is the group of upper triangular matrices with ones on the diagonal, which has the order $p^{n(n-1)/2}$ that we want.

I also find the proof of the first Sylow theorem much easier to understand now, the inspiration of it. I had always remembered that the Sylow $p$-group we are looking for can be the stabilizer subgroup of some set of $p^k$ elements of the group where $p^k$ divides the order of the group. By the pigeonhole principle, there can be no more than $p^k$ elements in it. The part to prove that kept boggling my mind was the reverse inequality via orbits. It turns out that that can be viewed in a way that makes its logic feel much more natural than it did before, when like many a proof not understood, seems to spring out of the blue.

We wish to show that the number of times, letting $p^r$ be the largest $p$th power dividing $n$, that the order of some orbit is divided by $p$ is no more than $r-k$. To do that it suffices to show that the sum of the orders of the orbits, $\binom{n}{p^k}$ is divided by $p$ no more than that many times. To show that is very mechanical. Write out as $m\displaystyle\prod_{j = 1}^{p^k-1} \frac{p^k m - j}{p^k - j}$ and divide out each element of the product on both the numerator and denominator by $p$ to the number of times $j$ divides it. With this, the denominator of the product is not a multiple of $p$, which means the number of times $p$ divides the sum of the orders of the orbits is the number of times it divides $m$, which is $r-k$.

Following this, Brian Bi told me about this problem, starred in Artin, which means it was considered by the author to be difficult, that he was stuck on. To my great surprise, I managed to solve it under half an hour. The problem is:

Let $H$ be a proper subgroup of a finite group $G$. Prove that the conjugate subgroups of $H$ don’t cover $G$.

For this, I remembered the relation $|G| = |N(H)||Cl(H)|$, where $Cl(H)$ denotes the number of conjugate subgroups of $H$, which is a special case of the orbit-stabilizer theorem, as conjugation is a group action after all. With this, given that $|N(H)| \geq |H|$ and that conjugate subgroups share the identity, the union of them has less than $|G|$ elements.

I remember Jonah Sinick’s once saying that finite group theory is one of the most g-loaded parts of math. I’m not sure what his rationale is for that exactly. I’ll say that I have a taste for finite group theory though I can’t say I’m a freak at it, unlike Aschbacher, but I guess I’m not bad at it either. Sure, it requires some form of pattern recognition and abstraction visualization that is not so loaded on the prior knowledge front. Brian Bi keeps telling me about how hard finite group theory is, relative to the continuous version of group theory, the Lie groups, which I know next to nothing about at present.

Oleg Olegovich, who told me today that he had proved “some generalization of something to semi-simple groups,” but needs a bit more to earn the label of Permanent Head Damage, suggested upon my asking him what he considers as good mathematics that I look into Arnold’s classic on classical mechanics, which was first to come to mind on his response of “stuff that is geometric and springs out of classical mechanics.” I found a PDF of it online and browsed through it but did not feel it was that tasteful, perhaps because I’m been a bit immersed lately in the number theoretic and abstract algebraic side of math that intersects not with physics, though I had before an inclination towards more physicsy math. I thought of possibly learning PDEs and some physics as a byproduct of it, but I’m also worried about lack of focus. Maybe eventually I can do that casually without having to try too hard as I have done lately for number theory. At least, I have not the right combination of brainpower and interest sufficient for that in my current state of mind.

## The Asian penalty

We all know that elite schools in the US discriminate against Asian applicants, essentially imposing a penalty for being Asian. And they have been rather pathetically pretending that such is not the case in spite of all the statistical evidence to the contrary. On this, people have said things like: where is affirmative action for Asians in the NBA/NFL? Well, today one of my colleagues who is a keen baseball fan, and probably also an NBA one, was talking about how there is even an Asian penalty in the NBA. Like, Asians are typically under-drafted, which means their number or rank in the draft is under commensurate with their actual ability and value at basketball. He says it’s due to the negative perception being Asian is for basketball in terms of how good one is yada yada yada. I haven’t paid attention to basketball for a long time, but I do remember the Linsanity several years ago, and when I was a kid, one who was a keen NBA fan, all the talk about Yao Ming. Last I checked Asians account for 0.2% of the NBA players, which means just a few names. Well, there is the height disadvantage after all. This was actually somewhat surprising to me, perhaps influenced by the fact that Yao Ming seemed to be overvalued due to the money he would bring from all the Chinese fans. So not only is there not affirmative action for Asians in the NBA, there is the same discrimination, the same penalty, the same stereotypes against Asians as in college admissions. Now that really sucks!

Now to something else that saddens me greatly that is a consequence of the current discriminatory policies against Asians in college admissions. Some Asian-Americans are afraid to check Asian and even afraid to engage in activities/pursuits they have gift and passion for, or at least some intrinsic interest in, under the fear that those are too Asian. Some are even afraid to show their Asian heritage and even reject their roots, which is quite sad, as you are who you are, in terms of your cultural background and denying it mostly makes you look quite pathetic. At least based on what I experienced growing up Asian in the states, many if not most Asian kids, even smart ones, try to distance themselves from their parents’ culture and are reluctant to learn or speak their parents’ native language, under social pressures osmosed in them by the whole American public school experience. Chinese culture is a pretty fucking cool and rich culture, with a beautiful language of artistic virtue that comes with a rich history. It is a pity that it is so misunderstood and that the American education system pressures against it in those from that cultural background. This is anti-intellectual in fact too, ironic as it is instigated partly by elite educational institutions, as reading multiple languages makes one’s mental world and whole spiritual existence a hell of a lot more interesting, an inevitable product of access of more diverse information.

To sum it up, it looks like all across the board America treats Asians as second class citizens. Yes, Asians are mostly new immigrants, but this is in fact overstated. As early as in the 19th century, Chinese in America were made to do much of the most dangerous work building the Transcontinental Railroad only to suffer the Chinese Exclusion Act. In the 20th century especially later on, Chinese as well as Indians, in addition to Japanese, Koreans, and Vietnamese, have created tremendous wealth for America, largely in science and engineering fields. Asians in America have mostly been busy working, busy creating, and some in hard menial labor in wretched conditions, with such being a major contributor to Asians, as a group, being seen as passive and apolitical, creating a self-pertuating stereotype. On this, Steve Hsu has expressed on his blog how slow the Asian community has been to organize against the double standards imposed on them in college applications, relative to the what Jewish community had done when the same had been unjustly instituted against them. Given the voluminous extent to which Asians as a group have contributed to America in terms of innovation and value creation, Asians have every right to demand that they are fairly considered for all positions, and leadership positions in particular, based on merit, which is not happening right now. On this more Asian-Americans ought to muster the courage to speak up for what is right, as Steve Hsu, Yukong Zhao, and Jian Li, among many others, have done amidst resistance.

## Asymptotic formula for square free integers

\begin{aligned} \displaystyle\sum_{n \leq x} \displaystyle\sum_{d^2 | n} \mu(d) & = \displaystyle\sum_{d \leq \sqrt{x}} \mu(d)\left\lfloor \frac{x}{d^2} \right\rfloor \\ & = x\displaystyle\sum_{d \leq \sqrt{x}} \frac{\mu(d)}{d^2} + O(\sqrt{x}) \\ & = x \frac{6}{\pi^2} + O(x\displaystyle\sum_{d > \sqrt{x}} \frac{1}{d^2} + \sqrt{x}) \\ & = x \frac{6}{\pi^2} + O((1 + \sqrt{x}) + \sqrt{x}) \\ & = x \frac{6}{\pi^2} + O(\sqrt{x}) \end{aligned}

## Composition series

My friend after some time in industry is back in school, currently taking graduate algebra. I was today looking at one of his homework and in particular, I thought about and worked out one of the problems, which is to prove the uniqueness part of the Jordan-Hölder theorem. Formally, if $G$ is a finite group and

$1 = N_0 \trianglelefteq N_1 \trianglelefteq \cdots \trianglelefteq N_r = G$ and $1 = N_0' \trianglelefteq N_1' \trianglelefteq \cdots \trianglelefteq N_s' = G$

are composition series of $G$, then $r = s$ and there exists $\sigma \in S_r$ and isomorphisms $N_{i+1} / N_i \cong N_{\sigma(i)+1} / N_{\sigma(i)}$.

Suppose WLOG that $s \geq r$ and as a base case $s = 2$. Then clearly, $s = r$ and if $N_1 \neq N_1'$, $N_1 \cap N_1' = 1$. $N_1 N_1' = G$ must hold as it is normal in $G$. Now, remember there is a theorem which states that if $H, K$ are normal subgroups of $G = HK$ with $H \cap K = 1$, then $G \cong H \times K$. (This follows from $(hkh^{-1})k^{-1} = h(kh^{-1}k^{-1})$, which shows the commutator to be the identity). Thus there are no other normal proper subgroups other than $H$ and $K$.

For the inductive step, take $H = N_{r-1} \cap N_{s-1}'$. By the second isomorphism theorem, $N_{r-1} / H \cong G / N_{s-1}'$. Take any composition series for $H$ to construct another for $G$ via $N_{r-1}$. This shows on application of the inductive hypothesis that $r = s$. One can do the same for $N_{s-1}'$. With both our composition series linked to two intermediary ones that differ only between $G$ and the common $H$ with factors swapped in between those two, our induction proof completes.