一周终过

新工作首周刚完,经过了繁忙紧张漫长的五天,对新环境,新代码,新系统的适应,以及新领域知识的吸收,周末终于来临,可趁机缓冲一下。这周收获相当大,感到对新工作所做的事情已得以大致的了解,足以下周完成更细节式的工作。加上,跟其他人也有一定融合之感。工作之余,比如在坐车时,也稍微想了想,学了学数学,有点增加。

昨晚,我跟那位犹裔美国IMO金牌在脸书上讨论犹太人与中国人在最高智力层次相比的问题,想起有两位我所认识的以基督教传统长大的学习理论科学的美国人所我当时难以思议的东亚人智力上强于犹太人的观点。怎么说那,虽然在这前五十年,日本人和华人在理论科学上做出了的不少伟大的贡献,占有美国好研究大学不少教职,加上我这一代的华人在竞赛中出色的表现,可是还是感觉在科学里的绝顶,犹太人更多,以犹太人更具有一定的高瞻远瞩,可促以颠覆性的跨越,苏联那批犹裔数学大师为典型例子。同时,这个人,作为组合数学为学习及研究方向的高材生,又提醒我犹太人在理论计算机以及匈牙利式组合数学所有的牛耳。他说世界上最聪明的人是亚洲人,他的名字是Terry Tao,可是前一百犹太人综合强于前一百亚洲人的综合。对此,我问他:你了解任何Tao所做的工作吗,可肯定他是世界上最聪明的人?他回:我读过Green-Tao定理的证明。我没啥好说的,只言那还算比较前沿的东西,又跟他说我在对一些华罗庚撰的数论引导,虽引导,可以包含一些我现在认为相当深的数论,如Selberg所做的一些。Tao是个神,可是我也有朋友说:我有事想是否Tao未有过以自己不如von Neumann聪明而心里不安,加上数学那么难,连Tao都差点没有通过博士生资格考试。加上,von Neumann精通数门外语,具有即兴无迟钝翻译之能,以及过目不忘的记忆力,而我都看到过有些中国人在网上以将自己视为”primarily an Australian”的Tao对中国文化一无认同和他对中文一无所知表示反感。我在此博客上前所提到那位犹裔数学博士,念到深到Goro Shimura所做的工作,也觉得Tao有点overrated,觉得他的工作没有例如陈省身所做的深远及原创,说Tao至今还没有创造新的领域。关于犹亚之比,我想到的还有环境的因素,在这一点华人还是比较吃亏,由于经济原因,也由于名字及文化陌生原因,老一辈的华人还在为了自己及国家的生存挣扎,没有那么多经历投入科学研究。或许现在歧视对华人,即使在理论科学界,还是相当严重,虽理论科学少有集体性及宣传及政治因素,与比如生物或软件开发不同,可是人都是有偏见的,这包括评审委员会,如我听到的诺贝尔委员会对苏联科学家的工作的贬值。我这一代,华人在那些完全公平没有任何主观因素的竞赛里已经遥遥胜于犹太人,而那些是最好的对纯粹智力顶级的测试。我有时候想:中国人现在最缺的不是科学技术人才,而是反抗歧视,争取话语权的人才。在外国人眼中,中国人经常有性格被动的刻板印象,的确有这一点,但是好多也是不太客观的媒体所造成的。加上,中国人在美国也是少数,又有语言文化障碍,这又是一个视为寻常的Asian penalty.

数学上,我闻到了在\mathbb{F}_p域下的次数整除n的不可约首一多项式的积等于非常干净的x^{p^n} - x。此多项式很容易看到没有平方因式,用典型的此与此导数非共有因子去证。同时,取任意次数dd | n的不可约首一多项式\phi,则\mathbb{F}_p[x] / (\phi)是个p^d元素的域,则所有元素是x^{p^d} - x的根(x也是此域一元),从此可以得到任意多项式(这包括x)代到x^{p^d} - xx里都在模\phi等于零,也就是说他会是\phi的倍数。因d | nx^{p^d} - x | x^{p^n} - x,则\phi | x^{p^n} - x。不难证明\mathrm{gcd}(x^{p^n} - x, x^{p^d} - x) = x^{p^{\mathrm{gcd}(n, d)}} - x. 若d \nmid n,次数d的多项式若要整除x^{p^n} - x,必整除x^{p^{\mathrm{gcd}(n, d)}} - x,可以用归纳法证明此不可能,在\mathrm{gcd}(n, d)< d的情况下。从此,可以得到x^{p^n} - x没有因子次数非整除n。证闭。

 

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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.

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.

湾区游

我们在群里所讨论的好多与种族,文化,智商,和学科相关,还记得我曾经对数学尖子开玩笑,美国IMO选手,非亚裔,必犹裔,而他似乎未感到我的玩笑口气,回说他那界有一两位非犹太白人。他还强调自己很美国人,不是那种在以色列待过的。高中时,他选了中文为他的外语,以童话为他爱听的一首中文歌。学术上,给的感觉是专注,单一的纯数学本科生,非常肯定他会走学术道路。他的具体数学兴趣及倾向为组合数学和理论计算机和解析数论。那天晚上,问他知道那几个二次互反律(law of quadratic reciprocity)的证明,他回答一个引用Zolotarev’s lemma的,并且发了个链接。当时,我只知道高斯和(Gauss sum)的那个,而细节已经记不清楚了。我花了一小时细读那个证明,领悟后感觉漂亮无比,引用的工具极其简单。从来没有想到可以将这数论皇后的定理视为,表达为置换的奇偶的积,毕竟Legendre symbol给的是1或-1,与sgn一样,好妙啊。回顾透明Legendre symbol给的基本是在给循环阿贝尔群(\mathbb{Z}/{p\mathbb{Z}})^{\times}的元素奇偶,二次剩余和偶置换在他们对应母群都是指数为2的子群。这又让我想到高斯对于正十七边形可作图的证明也是引用指数2的子群,其由Galois theory对应于度数为2的域扩张。过两天,我又学到了Gauss lemma,就是\left(\frac{a}{p}\right) = (-1)^n, n\{a, 2a, \ldots, \frac{p-1}{2}a\}大于p/2的元素的数量。证明思路很直接,将\{a, 2a, \ldots, \frac{p-1}{2}a\}的大于p/2的元素负掉,可和其他元素从新凑成\{1, 2, \ldots, \frac{p-1}{2}\}. Eisenstein对该定理的证明,我以前知道其存在但没看懂的,引用类似于Gauss lemma的引理,思路及证明策略抓住,这次却清晰了然。二次互反律的美妙我之前无法欣赏,记得对此定理有过一种稀奇古怪,难以思议之感,是没有抓住并且悟觉它的美妙的对称结构。想起在Eisenstein的证明中,画了一个pq的格子,将-1的次数示为格子的下左象限所有的点数,此为(p-1)(q-1)/4的来源。

Continue reading “湾区游”

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

昨天,我学会背了两首中文诗,一首是杜甫的《闻官军收河南河北》,另一首是毛泽东的《沁园春·长沙》

聞官軍收河南河北

劍外忽傳收薊北,初聞涕淚滿衣裳。
卻看妻子愁何在,漫卷詩書喜欲狂。
白日放歌須縱酒,青春作伴好還鄉。
即從巴峽穿巫峽,便下襄陽向洛陽。

杜甫这首诗具体哪一点感动了我,此我难以解释,缺乏文学描述所需要的词汇,同时也还未形成任何诗人的口味。加上,我对当时的中国的历史和文化也几乎一无所知。这还是我学会的第一首杜甫的诗。相反,他的对偶李白的诗我会好几首,如《蜀道难》和《将进酒》。

与杜甫的不同,我对毛泽东的《沁园春·长沙》的能容可有更深刻的理解,由于自己对二十年代的中国的政治形势有过一定的阅读。

沁園春‧長沙
獨立寒秋,湘江北去,橘子洲頭。看萬山紅遍,層林盡染;漫江碧透,百舸爭流。鷹擊長空,魚翔淺底,萬類霜天競自由。悵寥廓,問蒼茫大地,誰主沉浮?
攜來百侶曾游,憶往昔崢嶸歲月稠。恰同學少年,風華正茂;書生意氣,揮斥方遒。指點江山,激揚文字,糞土當年萬戶侯。曾記否,到中流擊水,浪遏飛舟!

这首诗的形势明显跟《沁园春·雪》相同,前段绘画祖国的美丽江山,后段启发式的鼓励壮怀激情的爱国主义革命家,开辟新天地,粉碎军阀混乱之黑暗。也可以说,《念奴娇·昆仑》也有接近或类似的形状。同时也发觉到,原来孔庆东出版的那本描述及讽刺韩国的书的名字却引用的这首诗。

这几月,我在学习俄语,网上找到了毛泽东十八首诗的俄文翻译, 此中有《长沙》的。

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…

阅此非太陌生,令余稍欣慰,表示己有进步,语言能力还不差,当然自己在这方面绝对没什么不得了。此俄语翻译,我还传给了我的几位苏联同志看了看。说起翻译,我昨天还把一段中文翻译成了俄文,至之至大学时给了我,在一个风气腐朽,无知无趣,在我另一位朋友形容为“如家具”的本科生漫天遍野的校园上,不少思想丰富及精神隐蔽的一位绝顶聪明又非昏迷于垃圾美国文化的童年来美的俄罗斯同学。我选择自学俄文大大出于本人本性对此语言及其文化发生的兴趣,但同时,他的鼓励及具体帮助也一直有了一定启发性的作用,有人可分享自己心灵所产生的美感,绝对有一定浪推飞舟的作用。

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.