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).

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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.

论文革

在网上胡扫碰到此,看了看,想起几年前在某地方看到文革时期,后重庆唱红打黑之熙来同志,反而提倡血统论,老子英雄儿好汉,老子反动儿混蛋那一套,与当时的高干子弟红卫兵联动组织之元一致。不过,看完后,其文将来源写为大纪元,使我对此之可信度有了怀疑。

想起我是八年级左右首次得知文革,那时的我中文不明,毕竟小学就一直在美国上的,所以看的都是英文的那些反共的扯淡。过一年,历史课有讲中国那时,美国老师不用说是讲的完全不符合事实,有夸张,有偏见,以“将农民当rocket scientist“以及类似之之非正当之语形容。反正给的印象就是文革彻底毁灭了中国,是共产党最大的罪之一,是破坏中国文化的。

当时的我,却漏掉了关键,那就是文革是针对当时的当权派,因为毛主席怕他们走资,怕他们欺压百姓,打了江山后建立新的皇朝,以在次循环中国的朝代,至少很多人认为是这样的,当时的毛主席到底在想什么,这很难完全搞清楚。当然,一般人都是会以文革想为灾难,这是默认的看法,我当时也一样,都想得如此简单。

让我惊奇的是,在中国却看到诸多骂邓小平的,甚至说毛泽东错误的没有把他彻底整掉,把邓小平述为中国腐败之父。我没有在中国工作过,也只在那儿上了一点小学,对中国的认识都是从夏天回去,加上网上的阅读,和与在美国的华人的接触。大家都说中国很腐败,但是我对此不知道任何具体,他们的腐败是以什么样的技巧那?我想这,随着自己的不断成熟和社会经验,会慢慢得知的。有人说,因为文化大革命,把那些干部整了一顿,他们才不敢做的太过度,不然中国很可能也会像苏联一样被内部流氓毁灭掉。因为文革,中国没有再次成为人之奴,对自己的制度保留了一定的信心,在国际舆论掌握于美国的情况下,因为我们都知道,你想超越别人,必须开辟自己独特的路,让别人跟随,被动随人,必不为一。科学是这样,政治也是这样。在中国,得知一个用以形容小平同志丧国之具例为运十大飞机的撤销,其由于此已得不少进展的项目是与已否定的四人帮有紧密联系的。要不是这样,中国可能早就造出自己的飞机了,可能九十年代就有了,而非去年,这是晚了二十年啊。有人说,老邓的那批小人为了他们的孩子能够出国,谋取与外企相连的要职,放弃了不少自己研制的项目,觉得也许造船不如买船,买船不如租船,这是多么没有远见的,汉奸式的做法啊!同时,我也看了,祖国伟大的钢琴家殷承宗,绝对的音乐超人,却因为是四人帮的音乐家,文革后与新政府不和,才迫使他跑到美国去。这些给了我一个新的观点。他绝对是有一定的道理。

美国怕毛泽东,毛泽东思想武装的军队,在朝鲜战场上让美国人提心吊胆,美国人不服,封锁中国,七零年,中国两弹一星有了,美国越南无路,不得不投降。在艰难的情况下,有了严峻的,但被外国人极度夸张,误描的饥荒,有了一个接一个的政治动乱,中国的科学技术工作者以他们的天分加刻苦加上一定的”信仰“却把一个一穷二白,百年受列国欺凌的灿烂古文化转至为有了一定工业和科技能力的国家。落后于先进国家,如美国,如苏联,如英法德日,不用说,毕竟起点太低,加上环境也较差。

现在,大部分在美国的人觉得是老邓解开了共产主义的铁链子,拯救了中国,使得中国富强起来。这些大多都是一些不懂科学的人,都是一些弱智,很多走是在我所在的软件的没有真正科学的行业里,他们好多以为自己挣钱多,不得了,典型的Dunning-Kruger。因为我们懂科学的人都知道软件,尤其是business方面的软件是没有技术含量的,与氢弹没法比。但他们对小平还不满,觉得他六四表现太差,最好的时机他们没有抓住,现在中国越来越强大,无可阻止,令一些脑构失常的,阴谋诡计的白宫人及其服从之人无可奈何。他们愿意相信什么,就让他们相信什么吧。

我真的觉得美国的主流文化没有什么好,是低级趣味的文化。上学一直处于此环境中,孩子的我也受了他们一定的影响,此难以避免。不过,随着自己的成熟,我通过网上所获取的信息不断地得知另一个世界的存在,越大越能够找到素质能力高一些的人,从大学到工作,能够接触更多非凡的有趣的人。学习科学,学习中文,学习俄文我越长越孜孜不倦,同时英文和写代码也有大的提高,因为这些毫无疑问是有相连的,相同的一部分。这些给了我一定的精神力量,其愈来愈来超越于某些环境带来的精神压制的力量,让我兴奋不已,让我所有的心上事,精神故障,纸船明烛照天烧!

一位哈弗毕业的,发表超弦的,科学通才的,也是无出于意料,犹裔的(现已归以),在脸书上有过几次闲聊。此中,他有提到他的在文化上科学与马克思相连的观点。当然,这不是绝对的且远之。好多伟大的科学家,如Edward Teller,都是坚决反共的。他也说过美国在二战前没有什么科学,加上美国文化好的一面都是从欧洲带来的。他也在网上多次提到过西方物理学家剽窃苏联的行为,尤其在七十年代,此例子众多,这我没法评价,鄙人莫谙物理,基本此盲也,可是我相信是这样的,凭受制于某某利益集团之美国媒体及教育的极其偏见及歪曲现实。我个人也有类似的感觉,虽然马克思的著作我从未认真读过,目前难以解之,仅感觉而已,也许因为我对于美国教育及体制及文化某些地方的叛逆,也许因为在美国的好多科学家工程师,包括拔尖的,都非美美国人。此人也说过,美国是建立于使对方失地,白人优越主义的新国家,未有不已之外入,则一切良素成烟。有一定道理,这样的国家,何可文化不稍偏夷焉!

话多余了!何终之?以自近有之观察,此为倾向于马克思之“思人”,根据自己接触到的有限的,文采期望值高于他人的。孔庆东比袁腾飞,毛泽东比邓小平,列宁比赫鲁晓夫,等等。还有鲁迅那。这是不是隐示马克思真的是代表人类社会下一步进化,或更准确,人,作为智力生物,的下一步进化?我相信未来会给以明确的答案!

 

Programming types

Programming, the intense hacker side of it, attracts a certain breed of person. In short, I would put it as that it attracts those who are higher in autism than in g, though of course one needs to be reasonably high in both, especially the verbal side of g, as its activity is largely one of reading (of logs and documentation) and writing (of code (and its supporting documentation), the quality of which has good variable names as a major component). I do feel at times that programmers, even elite ones, are lacking in scientific taste. Many of them are mathematically null. They thrive on and even love the detailed minutiae involved in the work, such as encodings (like UTF, ASCII, that type of thing), the ins and outs of Unix, and arcane facts of various languages. I had to encounter in my work today parsing of CSV files, and it turned out that the CSV reader was not reading under the correct encoding. I ended up diffing my output with the output generated via a means more or less guaranteed to work to aid such’s diagnosis. I’m not bad at this type of thing any longer, having trained myself or more like grown to be able to patiently resolve such problems in a systematic, foolproof fashion.

Does that mean I enjoy this type of thing? No, not at all, though I find it tolerable, more or less. Too autistic for me. It does not have the depth that mathematics has. It has not the beauty of poetry or of music. It has not the wittiness of words or the expressiveness of (human) language. Nor does it have the significance on the world that politics has. There are more meaningful to be doing than programming, though needless to say there is much demand for it as the world now runs on computer programs, which are written mostly by politically incompetent and often socially awkward who answer to morons with MBAs.

I’ve come to notice that programmers tend to be very narrow. They only know programming. There are of course exceptions. Mathematicians and to a greater extent physicists are more broad, and more deep. It makes them very boring to talk with. The people who are more well rounded who are in programming are often, from my observation, in it for the easy money, which is of course paltry relative to what the parasites of our society suck in, but nonetheless a very good sum by the standards of ordinary folk.

There is of course another world of programming, that of the incompetents, who often know only Java and barely know any computer science even. They’re far from the functional programmers who I work with. This industry is so in need of grunt labor that those people manage to find their way into six figure salaries. Yes, this includes places like Google and Facebook. There are Google engineers who don’t know what the difference between stack memory and heap memory is and who think C++ pointers are scary, who make 200k a year or almost. I won’t talk more about them. Waste of breath.

A result of Cantorian pathologies

We are asked to find a function on [0,1] such that f(0) = f(1) = 1 that has positive derivative almost everywhere on that domain. It occurred to me to use the Cantor set, which is obtained by partitioning remaining intervals into thirds and removing the interior of the second one. So first (1/3, 2/3) then (1/3^2, 2/3^2) and (7/3^2, 8/3^2), and so on. Each time we remove two-thirds of the remaining and summing the geometric series yields a measure of 1 for all that is removed. Another Cantorian construct arrived at from the Cantor set is the Cantor function, or Cantor staircase, called such as it resembles a staircase. That is, it turns out exactly what we need. It is a function with derivative zero almost everywhere, with non zero derivative points as jump points at the Cantor set. It is that discontinuity that facilitates the going from 0 to 1 along an interval with zero derivative almost everywhere. A transformation of that with a function with positive derivative is a step to deliver us what we want. That would be x minus the Cantor function. This has derivative 1 almost everywhere. However, we need to keep its range inside [0,1], which its outputs at [0,1/2] seems to not satisfy entirely. We are done though if we prove the other half to be non-negative, because then we can stretch the other half horizontally by a factor of two. It is needless to say that between 1/2 and 2/3 such is the case. Past 2/3, we have a downshift of 2 in base three to a 1 in base two at the first place, meaning a decrease of at least 1/6 when summing the net change at all digits which decrease in value. Digits increase in value on a change from 2 to 1 in every place past the first, or on a 1 digit in the original in its change to base two, which can occur only once, with all the following of that changed to zero, an increase of 1/2^n - 1/3^n, where n is the index of the place. In the former case, the largest possible increase is 1/2^2 + 1/2^3 + \cdots = 1/2 minus 2/3^2 + 2/3^3 + \cdots = 1/3, which is 1/6. In the latter case, 1/2^n - 1/3^n is exceeded by 1/2^{n-1} - 1/(2 \cdot 3^{n-1}), the total decrease from digit 2 to digit 1 from the nth place on. Thus, the minimum total increase from the increases exceeds the maximum total decrease from the decreases, which completes our proof of non-negativity for x \geq 1/2.

Galois group of x^10+x^5+1

This was a problem from an old qualifying exam, that I solved today, with a few pointers. First of all, is it reducible? It actually is. Note that x^{15} - 1 = (x^5-1)(x^{10}+x^5+1) = (x^3-1)(x^{12} + x^9 + x^6 + x^3 + 1). 1 + x + x^2, as a prime element of \mathbb{Q}[x] that divides not x^5-1 must divide the polynomial, the Galois group of which we are looking for. The other factor of it corresponds to the multiplicative group of \mathbb{F}_{15}, which has 8 elements. Seeing that it has 3 elements of order 2 and 4 elements of order 4 and is abelian, it must be C_2 \times C_4. Thus, the answer is C_2 \times C_2 \times C_4.

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}).

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。

Some speculation on the possible detailed linkage of brain structure to cognitive function

I recall seeing, probably first on Steve Hsu’s blog, that brain structure is highly heritable, as high as g is, surprise surprise. In fact, g, currently measured non-physically through cognitive tests, is, to my intuition, unguided by detailed relevant knowledge of the area, in its scientific essence a function of brain structure that could be computed through data obtained from a direct neurobiological measurement (through MRI and what not).

On this, I have hypothesized a future where we will be able to predict with reasonable accuracy, to a fine grain of detail, behavioral traits as complex as one’s personality or one’s cognitive strengths and weaknesses (like what subjects one will find more intuitive), which have a high heritable component, and not only that, the course of their development across age. A clone of me who lived in that world could be told at age n or so you will become like X, feel subjectively like Y, be at X \pm \sigma across age group at Z, etc, at age n+k, k > 0, k \in \mathbb{N}. Of course, if say n = 6, I would only respond with a sense of awe and incredulity, but across time, I would attain ever deeper and more realistic understanding of what was going on, as can be said for virtually every subject matter for virtually every human that has ever lived on this planet. Perhaps this is pushing it a bit too far. Learning to this degree of precision may be too much; there would not be enough data, even if we sequence every human on the planet and run on a computer cluster orders of magnitude larger and more powerful than, say, Google’s jobs which continuously improve the models. However, I do believe we will get somewhere. We will reach the point where we can deliver with reasonable level of confidence personalized medicine and education and such for every individual in a way that based on rigorous biostatistical data as opposed to on blind human intuition, which of course still works well, most of the time, as is done now. What other comes to mind is, as you can probably guess, the possibility of evolving a population vastly different from our own, perhaps one with ability far beyond that of homo sapiens (say, a population wherein everyone is smarter than John von Neumann (which I highly doubt, as that would be rather non conducive to the well functioning of the world)). As for that, as for what should be done, I have little to say other than that we’ll have to decide for ourselves. I do think it would be not a bad thing to use this newly found knowledge to eliminate severe dysfunction and suffering on this planet, which could be done through the likes of preventing the birth of, say, those prone towards severe depression or towards becoming serial killers.

We’ll have to see what happens. Maybe I will be able to see something exciting, possibly even to the level just described, in my lifetime. I am still young, after all.