## 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.

## A derivation of a Riemann zeta function identity

Yesterday, I saw the following Riemann zeta function identity:

$\displaystyle\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}$.

I took some time to try to derive it myself and to my great pleasure, I succeeded.

Eventually, I realized that it suffices to show that

$\{(dd_a, dd_b, d^2 n) : d_a | n, d_b | n : d, d_a, d_b, n \in \mathbb{Z}\}$

and

$\{(dd_a, dd_b, n) : dd_a d_b | n : d, d_a, d_b, n \in \mathbb{Z}\}$

are equal as multisets. As sets, they are both representations of the set of $3$-tuples of positive integers such that the third is a multiple of the least common multiple of the first two. In the latter one, the frequency of $(a,b,c)$ is the number of $d$ that divides both $a$ and $b$ such that $ab | cd$. In the other one, if we write $(a,b,c)$ as $(d_1 d_2 a', d_1 d_2 b', c)$ where $\mathrm{gcd}(a', b') = 1$, the $ab | cd$ condition equates to $d_1^2 d_2 a'b' | c$, which corresponds to the number of $d_1$ dividing $a$ and $b$ and such that $d_1^2 | c$ and with that, $d_2a', d_2b'$ both dividing $d_1^2 / c$, which is the frequency of $(a,b,c)$ via the former representation.

The coefficients $\{a_n\}$ of the Dirichlet series of the LHS of that identity can be decomposed as follows:

$a_n = \displaystyle\sum_{d^2 | n, d_a | \frac{n}{d^2}, d_b | \frac{n}{d^2}} (dd_a)^a (dd_b)^b$.

The coefficients $\{b_n\}$ of the Dirichlet series of the RHS of that identity are

$b_n = \displaystyle\sum_{dd_a d_b | n} (dd_a)^a (dd_b)^b$.

Observe how both are equivalent in that via the multiset equivalence proved above, $n$ determines the same multiset of $(dd_a, dd_b)$ for both and across that, the values of the same function $(dd_a)^a (dd_b)^b$ are summed. Hence the two series are equal.

## Automorphisms of quaternion group

I learned this morning from Brian Bi that the automorphism group of the quaternion group is in fact $S_4$. Why? The quaternion group is generated by any two of $i,j,k$ all of which have order $4$. $\pm i, \pm j, \pm k$ correspond to the six faces of a cube. Remember that the symmetries orientation preserving of cube form $S_4$ with the objects permuted the space diagonals. Now what do the space diagonals correspond to? Triplet bases $(i,j,k), (-i,j,-k), (j,i,-k), (-j,i,k)$, which correspond to four different corners of the cube, no two of which are joined by a space diagonal. We send both our generators $i,j$ to two of $\pm i, \pm j, \pm k$; there are $6\cdot 4 = 24$ choices. There are by the same logic $24$ triplets $(x,y,z)$ of quaternions such that $xy = z$. We define an equivalence relation with $(x,y,z) \sim (-x,-y,z)$ and $(x,y,z) \sim (y,z,x) \sim (z,x,y)$ that is such that if two elements are in the same equivalence class, then results of the application of any automorphism on those two elements will be as well. Furthermore, no two classes are mapped to the same class. Combined, this shows that every automorphism is a bijection on the equivalence classes.

## A recurrence relation

I noticed that

$(x_1 - x_k)\displaystyle\sum_{i_1+\cdots+i_k=n} x_1^{i_1}\cdots x_k^{i_k} = \displaystyle\sum_{i_1+\cdots+i_{k-1}=n+1} x_1^{i_1}\cdots x_{k-1}^{i_{k-1}} - \displaystyle\sum_{i_2+\cdots+i_k=n+1} x_2^{i_2}\cdots x_k^{i_k}.$

In the difference on the RHS, it is apparent that terms without $x_1$ or $x_k$ will vanish. Thus, all the negative terms which are not cancelled out have a $x_k$ and all such positive terms have a $x_1$. Combinatorially, all terms of degree $n+1$ with $x_k$ can be generated by multiplying $x_k$ on all terms of degree $n$. Analogous holds for the positive terms. The terms with only $x_1$ and $x_k$ are cancelled out with the exception of the $x_1^{n+1} - x_k^{n+1}$ that remains.

This recurrence appears in calculation of the determinant of the Vandermonde matrix.

## 两首诗

### Чанша

В день осенний, холодный
Я стою над рекой многоводной,
Над текущим на север Сянцзяном.
Вижу горы и рощи в наряде багряном,
Изумрудные воды прозрачной реки,
По которой рыбачьи снуют челноки.
Вижу: сокол взмывает стрелой к небосводу,
Рыба в мелкой воде промелькнула, как тень.
Всё живое стремится сейчас на свободу
В этот ясный, подёрнутый инеем день.
Увидав многоцветный простор пред собою,
Что теряется где-то во мгле,
Задаёшься вопросом: кто правит судьбою
Всех живых на бескрайной земле?
Мне припомнились дни отдалённой весны,
Те друзья, с кем учился я в школе…
Все мы были в то время бодры и сильны
И мечтали о будущей воле.
По-студенчески, с жаром мы споры вели
О вселенной, о судьбах родимой земли
И стихами во время досуга
Вдохновляли на подвиг друг друга.
В откровенных беседах своих молодёжь
Не щадила тогдашних надменных вельмож.
Наши лодки неслись всем ветрам вопреки,
Но в пути задержали нас волны реки…

## Follow-up on the chosen people

Readers of my blog might recall my post titled The Chosen People. I’ve had the privilege to have had some fruitful and stimulating discussions with a friend of mine who was raised Christian on this matter. Such reminds me of Bobby Fischer, and I had just found my way to this, which has a transcript of some of Fischer’s antisemitic remarks. I was also reminded of sayings of a CCTV reporter along the same lines. When I was a child, I enjoyed watching the kids show Arthur. Particularly memorable was this rich girl Muffy Crosswire. I remember particularly vividly this clip, where when that girl’s psychopath auto CEO daddy turned on the speaker that exposed the dealings within the “strategy room,” it was like: so what if the engine falls out, once they’re off the lot, it’s their problem! Hmm, maybe that’s why American cars couldn’t compete with Japanese ones, because there were such people in executive management at Ford and General Motors? Also, there was the special Christmas episode of Authur, from which I learned that Muffy is the Gentile while her bestie Francine, the poor daughter of garbage man, is the Jew. Given how so a high percentage of the Russian oligarch robber barons are Jewish as are so many of the financiers and media moguls in America, aren’t the roles a bit reversed? Now, given that, would Marc Brown have had difficulties in his career had the portrayal been more in phase with the statistical reality? On this group that is the subject matter of the blog post, I have also heard people say in private things in the likes of: it’s always they win others lose, that’s why nobody likes them. And they’re basically doing to the Palestinians what the Nazis did to them!

The Jews are a very unique group. They seem to take almost exclusively jobs in the intellectual class. You see very few in the low end menial jobs. Now that is a natural consequence of their high average IQ over 115ish. This naturally breeds resentment. People love to say how now Asians are the new Jews, but keep in mind there most Chinese do menial labor for living, especially in China. Even in the US, many of them work at restaurants and laundromats and such and they are very socially disjoint with the Chinese intellectual class, the ones who work for Microsoft or profess at universities. Also, there are almost no Chinese in positions of power in the US right now. We all know that ruling classes of virtually all cultures are scumbags in one way or another. However, I would say that some are more benign than others. Remember that during the Cultural Revolution in China, children of the communist elite were actually forced to go work in the countryside to get a taste of what the majority of working class people go through on a daily basis, and many of the current leaders of China are in that category. Steve Hsu has pointed out a possible lower prevalence of psychopaths among those of East Asian descent as a partial explanation to the dearth of East Asians in “leadership” positions in America. There is also the (very politically incorrect theory) that the Holocaust wiped out the weaker and nicer portion of the Jewish population that further contributes to the phenomenon here described. Now we all know that people who are too smart for their own good are a root of resentment but those people are quite benign and contribute much to our culture and are responsible for the radical scientific and technological advances which enrich (hopefully) people’s lives. However, there is more cause for hate when you have a group of parasites who do no menial labor and hoard the wealth created by the blood and sweat of working masses for themselves and even feel entitled to do so.

I have not much more to say at this moment on this matter. You, the reader, be the judge.