A cute find closed form of sum problem

19692339_10155494787829320_1775528957_n

A friend pinged me this on Facebook. I decided to look at it to exercise my technical chops. Well, the value of the denominator is given by the hint. In the sum of the first n triangular numbers, k is summed n+1-k times, and the number of ways to split n+1 items in a line and pick one on each side of the split is the same as the number of ways to select 3 items from n+2, with the middle one representing the split point. Finally do a partial fractions to telescope. You’ll get \frac{1/2}{n} - \frac{1}{n+1} + \frac{1/2}{n+2}.

Another characterization of compactness

The canonical definition of compactness of a topological space X is every open cover has finite sub-cover. We can via contraposition translate this to every family of open sets with no finite subfamily that covers X is not a cover. Not a cover via de Morgan’s laws can be characterized equivalently as has complements (which are all closed sets) which have finite intersection. The product is:

A topological space is compact iff for every family of closed sets with the finite intersection property, the intersection of that family is non-empty.

Grassmannian manifold

We all know of real projective space \mathbb{R}P^n. It is in fact a special space of the Grassmannian manifold, which denoted G_{k,n}(\mathbb{R}), is the set of k-dimensional subspaces of \mathbb{R}^n. Such can be represented via the ranges of the k \times n matrices of rank k, k \leq n. On application of that operator we can apply any g \in GL(k, \mathbb{R}) and the range will stay the same. Partitioning by range, we introduce the equivalence relation \sim by \bar{A} \sim A if there exists g \in GL(k, \mathbb{R}) such that \bar{A} = gA. This Grassmannian can be identified with M_{k,n}(\mathbb{R}) / GL(k, \mathbb{R}).

Now we find the charts of it. There must be a minor k \times k with nonzero determinant. We can assume without loss of generality (as swapping columns changes not the range) that the first minor made of the first k columns is one of such, for the convenience of writing A = (A_1, \tilde{A_1}), where the \tilde{A_1} is k \times (n-k). We get

A_1^{-1}A = (I_k, A_1^{-1}\tilde{A_1}).

Thus the degrees of freedom are given by the k \times (n-k) matrix on the right, so k(n-k). If that submatrix is not the same between two full matrices reduced via inverting by minor, they cannot be the same as an application of any non identity element in GL(k, \mathbb{R}) would alter the identity matrix on the left.

I’ll leave it to the reader to run this on the real projective case, where k = 1, n = n+1.

An unpacking of Hurwitz’s theorem in complex analysis

Let’s first state it.

Theorem (Hurwitz’s theorem). Suppose \{f_k(z)\} is a sequence of analytic functions on a domain D that converges normally on D to f(z), and suppose that f(z) has a zero of order N at z_0. Then there exists \rho > 0 such that for k large, f_k(z) has exactly N zeros in the disk \{|z - z_0| < \rho\}, counting multiplicity, and these zeros converge to z_0 as k \to \infty.

As a refresher, normal convergence on D is convergence uniformly on every closed disk contained by it. We know that the argument principle comes in handy for counting zeros within a domain. That means

The number of zeros inside |z - z_0| = \rho, \rho arbitrarily small goes to the number of zeros inside the same circle of f, provided that

\frac{1}{2\pi i}\int_{|z - z_0| = \rho} \frac{f'_k(z)}{f_k(z)}dz \longrightarrow \frac{1}{2\pi i}\int_{|z - z_0| = \rho} \frac{f'(z)}{f(z)}dz.

To show that boils down to a few technicalities. First of all, let \rho > 0 be sufficiently small that the closed disk \{|z - z_0| \leq \rho\} is contained in D, with f(z) \neq 0 inside it everywhere except for z_0. Since f_k(z) converges to f(z) uniformly inside that closed disk, f_k(z) is not zero on its boundary, the domain integrated over, for sufficiently large k. Further, since f_k \to f uniformly, so does f'_k / f_k \to f' / f, so we have condition such that convergence is preserved on application of integral to the elements of the sequence and to its convergent value. With \rho arbitrarily small, the zeros of f_k(z) must accumulate at z_0.

华罗庚

我很佩服华罗庚。他是个传奇。贫苦出生,无金续学,迫而辍,惟有初中文凭。那时中国社会黑暗,洪水没村,瘟神染民,罗庚金坛乡遭于此,其为患儿之一。华佗暂离,疗不到,则华腿永残。在这种艰难的情况下,他并非绝望,反而发愤图强,独学并发表,神使他看中于当时中国数学之要机构人,从而他登上专业数学。

他是一位有勇气的人,气贯长虹,敢于独特。他忘名舍利,纯粹做学,剑桥学位放弃,因为他能力太强了,不需要那张人为纸,也瞧不起那些学术界,尤其是现在,诸多的玩小人技的微人。同时,他的人民意识动人,抗战时,可以留外避暴的他,却回国与庶民同难。后来,在美国学界舒适绝蜚声,忽报人间曾伏虎,再次归祖。

当时的他,是绝有威望的人。想起,我三年级夏天回国,中文全忘,就读小学一二年级课文而补,其中有讲华罗庚的,那也是我第一次有意识的知道这个人的存在。后来,看到他回国时写至在国外中国留学生及工作人公开信,劝大家回国投入刚解放的,百废待兴的祖国的建设。

百度文库有,链接为:https://wenku.baidu.com/view/8167d0d7b9f3f90f76c61b6c.html。链接非永恒,则我再复制,也免读者再载一页:

朋友们:

不一一道别,我先诸位而回去了。我有千言万语,但愧无生花之笔来一一地表达出来。但我敢说,这信中充满着真挚的感情,一字一句都是由衷心吐出来的。

坦白地说,这信中所说的是我这一年来思想战斗的结果。讲到决心归国的理由,有些是独自冷静思索的果实,有些是和朋友们谈话和通信所得的结论。朋友们,如果你们有同样的苦闷,这封信可以做你们决策的参考;如果你们还没有这种感觉,也请细读一遍,由此可以知道这种苦闷的发生,不是偶然的。

让我先从大处说起。现在的世界很明显地分为两个营垒:一个是为大众谋福利的,另一个是专为少数的统治阶级打算利益的,前者是站在正义方面,有真理根据的;后者是充满着矛盾的。一面是与被压迫民族为朋友的,另一面是把所谓“文明”建筑在不幸者身上的。所以凡是世界上的公民都应当有所抉择:为人类的幸福,应当抉择在真理的光明的一面,应当选择在为多数人利益的一面。

朋友们如果细细地想一想,我们身受过移民律的限制,肤色的歧视,哪一件不是替我们规定了一个圈子。当然,有些所谓“杰出”的个人,已经跳出了这圈子,已经得到特别“恩典”,“准许”“归化”了的,但如果扪心一想,我们的同胞们都在被人欺凌,被人歧视,如因个人的被“赏识”,便沾沾沾自喜,这是何种心肝!同时,很老实的说吧,现在他们正想利用这些“人杰”。

也许有人要说,他们的社会有“民主”和“自由”,这是我们所应当爱好的。但我说诸位,不要被“字面”迷惑了,当然被字面迷惑也不是从今日开始。

我们细细想想资本家握有一切的工具——无线电、报纸、杂志、电影,他说一句话的力量当然不是我们一句话所可以比拟的;等于在人家锣鼓喧天的场合下,我们在古琴独奏。固然我们都有“自由”,但我敢断言,在手酸弦断之下,人家再也不会听到你古琴的妙音。在经济不平等的情况下,谈“民主”是自欺欺人;谈“自由” 是自找枷锁。人类的真自由、真民主,仅可能在真平等中得之;没有平等的社会的所谓“自由”、“民主”,仅仅是统治阶级的工具。

我们再来细心分析一下:我们怎样出国的?也许以为当然靠了自己的聪明和努力,才能考试获选出国的,靠了自己的本领和技能,才可能在这儿立足的。因之,也许可以得到一结论:我们在这儿的享受,是我们自己的本领,我们这儿的地位,是我们自己的努力。但据我看来,这是并不尽然的,何以故?谁给我们的特殊学习机会,而使得我们大学毕业?谁给我们所必需的外汇,因之可以出国学习。还不是我们胼手胝足的同胞吗?还不是我们千辛万苦的父母吗?受了同胞们的血汗栽培,成为人材之后,不为他们服务,这如何可以谓之公平?如何可以谓之合理?朋友们,我们不能过河拆桥,我们应当认清:我们既然得到了优越的权利,我们就应当尽我们应尽的义务,尤其是聪明能干的朋友们,我们应当负担起中华人民共和国空前巨大的人民的任务!

现在再让我们看看新生的祖国,怎样在伟大胜利基础上继续迈进!今年元旦新华社的《新年献词》告诉我们说:一九四九年,是中国人民解放战争获得伟大胜利和中华人民共和国宣告诞生的一年。这一年,我们击破了中外反动派 的和平攻势,扫清了中国大陆上的国民党匪帮……,解放了全国百分之九十以上的人口,赢得了战争的基本胜利。这一年,全国民主力量的代表人物举行了人民政治协商会议,通过了国家根本大法共同纲领,成立了中央人民政府。

这个政府不但受到全国人民的普遍拥护,而且受到了全世界反帝国主义阵营的普遍欢迎。苏联和各人民民主国家都迅速和我国建立平等友好的邦交。这一年,我们解放了和管理了全国的大城市和广大乡村,在这些地方迅速地建立了初步的革命秩序,镇压了反革命活动,并初步地发动和组织了劳动群众。在许多城市中已经召集了各界人民代表会议。在许多乡村中,已经肃清了土匪,推行了合理负担政策,展开了减租减息和反恶霸运动。这一年,我们克服了敌人的破坏封锁和严重的旱灾、水灾所加给我们的困难。在财政收支不平衡的条件下,尽可能地进行了恢复生产和交通的工作,并已得到了相当成绩……

中国是在迅速的进步着,一九四九年的胜利,比一年前人们所预料的要大得多,快得多。在一九五〇年,我们有了比一九四九年好得多的条件,因此我们所将要得到的成绩,也会比我们现在所预料的更大些、更快些。当武装的敌人在全中国的土地上被肃清以后,当全中国人民的觉悟性和组织性普遍地提高起来以后,我们的国家就将逐步地脱离长期战争所造成的严重困难,并逐步走上幸福的境地了。

朋友们!“梁园虽好,非久居之乡”,归去来兮!

但也许有朋友说:“我年纪还轻,不妨在此稍待。”但我说:“这也不必。”朋友们,我们都在有为之年,如果我们迟早要回去,何不早回去,把我们的精力都用之于有用之所呢?

总之,为了抉择真理,我们应当回去;为了国家民族,我们应当回去;为了为人民服务,我们也应当回去;就是为了个人出路,也应当早日回去,建立我们工作的基础,为我们伟大祖国的建设和发展而奋斗!

朋友们!语重心长,今年在我们首都北京见面吧!

1950年2月归国途中

最近,在读Erdos的传记时,闻五十年代初,他被美国政府逼问,源于曾跟这位华共匪有通信。华罗庚在中国是家喻户晓,但在美国是鲜为人知的,好多学数学的人也并不知道他。我想这肯定不是偶然的,因为华罗庚成了美国敌对的红色中国的一个象征性人物。

回国后,华好像做了一些多元复变的工作,丘成桐称之为不得了,领先西方至少十年,吾初学者当然无资格评。在他的领导,培养下,出了一批杰出的新一代的数学家,如陈景润,还有什么陆启铿,龚升,等等。文革时,他如诸多中国的知识分子,科学家,遭迫害,受不公正对待,在这种情况下,他转入统筹法,将数学与生产相结合,使祖国实业大大提升,成为了人民的数学家。

大多数那一代的中国科学精英,如他的数学对偶陈省身都出生于富贵书香门第,而他是平民背景,使得更加得以老百姓的认同。加上,他据说人品特好,一点都不是个asshole,这是很难得的。他也不是个纯粹读书的人,对人类社会进步有关心,抗战时就连到一些进步组织。

同时,他文采丰盛,看到他的几首诗,很明显他有丰富的词汇量,更加明示他的超人的脑子。这,当然,是一点不出于预料的。

华罗庚的名字,在数学史,在中国史,永垂不朽!

Principal values (of integrals)

I’ve been looking through my Gamelin’s Complex Analysis quite a bit lately. I’ve solved some exercises, which I’ve written up in private. I was just going over the section on principal values, which had a very neat calculation. I’ll give a sketch of that one here.

Take an integral \int_a^b f(x)dx such that on some x_0 \in (a,b) there is a singularity, such as \int_{-1}^1 \frac{1}{x}dx. The principal value of that is defined as

PV \int_a^b f(x)dx = \lim_{\epsilon \to 0}\left(\int_a^{x_0 - \epsilon} + \int_{x_0 + \epsilon}^b\right)f(x)dx.

The example the book presented was

PV\int_{-\infty}^{\infty} \frac{1}{x^3 - 1} = -\frac{\pi}{\sqrt{3}}.

Its calculation invokes both the residue theorem and the fractional residue theorem. Our integrand, complexly viewed, has a singularity at e^{2\pi i / 3}, with residue \frac{1}{3z^2}|_{z = e^{2\pi i / 3}} = \frac{e^{2\pi i / 3}}{3}, which one can arrive at with so called Rule 4 in the book, or more from first principles, l’Hopital’s rule. That is the residue to calculate if we had the half-disk in the half plane, arbitrarily large. However, with our pole at 1 we must indent it there. The integral along the arc obviously vanishes. The infinitesimal arc spawned by the indentation, the integral along which, can be calculated by the fractional residue theorem, with any -\pi, the minus accounting for the clockwise direction. This time the residue is at 1, with \frac{1}{3z^2}|_{z = 1} = \frac{1}{3}. So that integral, no matter how small \epsilon is, is -\frac{\pi}{3}i. 2\pi i times the first residue we calculated minus that, which is extra with respect to the integral, the principal value thereof, that we wish to calculate, yields -\frac{\pi}{\sqrt{3}} for the desired answer.

Let’s generalize. Complex analysis provides the machinery to compute integrals not to be integrated easily by real means, or something like that. Canonical is having the value on an arc go to naught as the arc becomes arbitrarily large, and equating the integral with a constant times the sum of the residues inside. We’ve done that here. Well, it turns out that if the integral has an integrand that explodes somewhere on the domain of integration, we can make a dent there, and minus out the integral along its corresponding arc.

A possible switch in focus from math to natural science

I find myself becoming more keen on natural reality over the last week or so, though my time has still been mostly concentrated on mathematics. It is possible that I am actually more suited to natural science than to mathematics, who knows. To estimate the expected extent of that finer, I’m going to go learn some natural science, like what else would I do.

I want to first talk about my experience with science in college, high school, middle school, and perhaps even earlier. In elementary school, in sixth grade, we had this science fair. My partner and I chose to do ours on wastewater treatment plants. There were some people who did solar power and even one who did population growth the previous year (social science is science I guess). I learned absolutely nothing from that; it’s like, how many kids at that age can actually learn science that isn’t bull shit?

In 7th grade, we had for our science course life science. It was mostly taking notes on various types of life, from fish to reptiles to plants. That’s when I first learned of the Linnaeus classification system. We didn’t do experiments really. Tests were mostly regurgitation of notes. There was nothing quantitative.

In 8th grade, it was earth science. The teacher was so dumb that in math class, this kid was like: how can you like Mrs.    ? She’s as dumb as a rock! To that, the math teacher, who later realized by me was a complete moron who didn’t even know what math was, was not terribly accepting, I’ll put it that way. We studied volcanos and earthquakes, watched documentaries on those types of things, and played around a bit with Bunsen burners and random equipment typical in chemistry laboratory, the names of which I know not. For names, I guess use this as a reference? I didn’t like that class and didn’t do well in it at all. My ADHD or what not was particularly severe in it.

In 9th grade, it was “physical science.” We did some problems in Newtonian mechanics, very simple ones, that’s when I first learned of Newtons, Joules, work, energy, those types of things. I actually found that pretty interesting. There was this project for making an elastic powered, or rubber band powered car to be more explicit. Really, there wasn’t much point in that other than as a way to pass time for kids. Having worked as a software engineer, I can guess that there are very systematic ways of designing and building that stuff. Of course, us kids just tinkered around in a way wherein we didn’t know at all what we were doing. I do remember there was a time when we were playing with this thing, called a crucible I think, that we were not supposed to touch, as doing so would smear black onto our hands, which I nonetheless still did, receiving, consequently, reprimand from the teacher.

In 10th grade, it was chemistry and then biology for the second half of the year. This was now at my grades 10-12 high school. The class was rumored to be impossibly hard, and the teacher was said to be a very demanding guy. There was, unlike the year previous, basically zero hands on. The chemistry part was very quantitative, I remember stoichiometry was a big part. There was nothing really hard about it; the students were simply too dumb to even perform very mechanical calculations. Kids would say: “it’s a lot of math.” At that time, I didn’t know the difference between math and science, and the other kids knew even less. Math is founded on the axiomatic system pioneered by the Greeks, about proving things in an a priori way, while science is about modeling natural phenomena and testing those models. Math in science is just a tool and not the focus. I recall we started off learning about uncertainties in measurement. There’s really nothing especially hard about that stuff, with a very systematic way of going about it, but the atmosphere and the way it was lectured about made it seem like such a grand thing to us. The second half the year as I said was biology. I wasn’t terribly engaged in that. I didn’t like the memorization involved. I liked math more. I made the AIME that year, taking the AMCs for the first time, and was one of four kids out of almost 2000 in our high school to do so, so that brought me to conclude that maybe I actually had some talent for math and science. I knew that physics was the hardest and brainiest of the sciences, with all that fancy math and Einstein, so I was rather keen to learn that. I checked out some physics books from the library I think, and the first thing I learned about was if I remember correctly centripetal acceleration, which confused me quite a lot at that time.

11th and 12th grade was physics with the same teacher. The class was rather dumbed down; it had to, especially on the math end, problem solving wise, since this is an American high school after all. There was quite a focus on phenomena, as opposed to formalism. I didn’t really like that much. I was more comfortable with formalism, with math being my relative forte at that time. We did some experiments, but I wasn’t good at them at all. I remember on the first day, when looking at some uniformly dense rod vertically situated, it occurred neither to me nor my partner to record its position at its center of mass. I didn’t really understand what I was doing throughout the whole time. The other kids, most of them, were worse. There were some who were confused about the difference between energy and power, the latter of which is the derivative of the former of course, after two years of it! I remember the whole time many kids would go: wow! physics! That kind of perspective, later understood by me, makes it almost impossible for one to really learn it. With just about everything, there is a right way of going about it. Discover it (mentally, with the aid of books, lectures, various resources) and you’ll do great. Be in awe of it, and you’ll never get it. The former is in line with the philosophy that you should focus solely on what is true, objectively, and not imagine anything that doesn’t aid in your convergence to the truth, and reminds me of the quote of Einstein that one should make something as simple as it can be but no simpler. Simplicity is gold in science and just about everything. Ability to recognize the redundant and superfluous and to generalize is the essence of intellectual ability, or to put it in more extreme terms, genius. The culture in American high school is the antithesis of that. Kids are always talking about how hard things, especially math loaded subjects, are, when they’re making it hard for themselves by imagining in their minds what is complete bogus from a scientific point of view. To digress, this holds as well as for subjects like history. Focus only and solely on the what are the facts and the truth they bare out. Don’t let political biases and personal wants and wishes interfere in any way. This is to my remembrance advice intellectual Bertrand Russell gave to posterity nearing his death. American history classes are particular awful at this. American teaching of history is very much founded on ignorance and American exceptionalism and a misportrayal of cultures or political systems it, or more like, its blood sucking elite, regards as evil for the simple reason that they are seen as threatening towards their interests. Math and science under the American public school system was pretty dismal. History (or social studies, as they call it) was perhaps more so, in a way more laughable and contemptible.

I hardly took science in college, being a math and CS major. I did take two quarters of physics and it was awful. Talking with some actual physics PhD students and physics PhDs gave me a more accurate idea of what physics really was, though I was still pretty clueless. It was evident to me at that time that physics, and probably also chemistry, was far more demanding in terms of cognitive ability as many of the CS majors, who could write code not badly, struggled with even very simple physics. Being in college, I had a closer look at the world of real science, of scientists, in America, which is very foreign. It dawned on me that science, as exciting as it sounds, is in America done mostly by underpaid ubermensch immigrant men, who are of a completely different breed both intellectually and culturally from most of the people I had encountered at that time. Yes, by then I had found my way to this essay by Greenspun. I’ll leave its interpretation up to the reader. 😉

You can probably guess that I think American science education is a complete joke, which is the truth. I felt like I only began to really learn things once I got out of the American school system, although for sure, the transition between high school and college in terms of content and depth and rapidity of learning was quite substantial. However, the transition from undergrad to out into the bigger world, where I could consider myself psychologically as more in the ranks of everyone, regardless of age or national origin, than in the ranks of clueless American undergrads at a mediocre program, was probably just as substantial in the same respect, albeit in a very different way.

Now let’s context switch to some actual science (that’s not pure math or artificial in any way).

*************************************************

A capacitor is made by taking some negative charge off a positive plate and transferring it to the negative plate. This obviously requires work. If the final voltage is \Delta V, then the average during the charging process is half of that. With the change in potential energy \Delta U_E as change times voltage (remember, voltage is potential energy per unit charge), we can write \Delta U_E = \frac{1}{2}Q\Delta V.

How to maintain charge separation? Insert an insulator (or dielectric) between the plates. Curiously, a dielectric always increases the capacitance (Q / \Delta V) of a capacitor. Its existence, via the charge on the plates, makes for a electrically polarized medium, which induces an electric field in the reverse direction that is in addition to the one induced by the capacitors alone. As you see, the negative charges in the dielectric lean towards the positive plate and same holds if you permute negative and positive. So if the plates, by themselves give rise to \mathbf{E}, the addition of the dielectric gives rise to some \mathbf{E_i} in the opposite direction. Call \kappa the coefficient of the reduction in the magnitude of the electric field with

E_{\mathrm{with\;dielectric}} = E_{\mathrm{without\;dielectric}} - E_i = \frac{E} {\kappa}.

We put that coefficient in the denominator so that

C_{\mathrm{with\;dielectric}} = \kappa C_{\mathrm{without\;dielectric}}.

To be more explicit, so that \kappa is proportional capacitance wise, which is reasonable since capacitance is what is more central to the current context. This kappa value is called dielectric constant, varying from material to material, under the constraint that it is always greater than 1.

Now one might ask if the capacitor is charging when the dielectric is inserted. If it isn’t, the voltage across will experience a sudden decrease, with the charge stored constant, and if it is, voltage will experience the same, but the charge on the plates will keep going up, as the voltage will too at a rate proportional to that of the increase of the charge, with the constant of proportionality the increased C. Needless to say, on taking derivative, a linear relation is preserved with the same coefficient of linearity.

The presence of a dielectric presents a potential problem, namely that if the voltage is too high, the electrons in the dielectric material can be ripped out of their atoms and propelled towards the positive plate. Obviously, this discharges the capacitor, as negative and positive meet to neutralize. It is said that this typically burns a hole through the dielectric. This phenomenon is called dielectric breakdown.