Japan

I watched part of this documentary on Japan in WWII. It is a very high quality one, with many personal remembrances of various Japan individuals, from high-ranking politicians and military men to schoolchildren, on their experience spoken in Japanese accented English. (No film experience on my part, but I can most certainly guess that many if not most of those were acted as opposed to real.) I believe it depicted rather realistically the Japanese perspective of the war.

Many individual Chinese openly express detest of Japan for obvious reasons, and in China, it is in some sense taught that Japan is pure evil. Kids will believe that. As one matures, one can of course develop a more realistic and less emotional perspective on the matter. Of course, there are many in China with family members who were killed or suffered tremendously under Japan, which has the most barbaric military culture of any country in the world, and thus, the reaction to Japan is bound to be traumatic and emotional, especially for the older folks.

I actually know little about Japan and am eager to learn more. I’ve never been there, aside from the Narita airport, which doesn’t count. I am increasingly impressed by Japan, by its ability particularly in science and technology. There is the popular stereotype (in China as well) that Japanese are uncreative copycats (they stole Chinese characters) and later they modernized learning from the West, but such is obviously not so given the plethora of original, and in some cases groundbreaking, creations by Japan since they have been an advanced country, which one can put as the 1930s. The masses see directly Japanese cameras and cars, and also their anime, and the elite intelligentsia are well aware of their contributions to pure science (on that, Japan has won sizable chunk of the Nobel prizes since 2000).

Japan modernized very rapidly and successfully with the help of the West. At that time, which was second half of 19th century, it was clear that the West was leagues ahead, having developed modern science, and later modern, industrial technology. By then, not surprisingly, Japan was obsessed with learning from the West (seeing what the defeated and colonized non-Western people of the world, especially China, were suffering), and initially, for good reason, Japanese were not sure they could ever compete with Westerners. As they made progress, doubts on that gradually dispersed, and expectedly, Japan defeated China in 1895, which devastated the Chinese national psyche much more so than did the repeated losses to the Western powers did, on the basis that China historically had always seen Japan as this puny country much as its vassal, which had relied on her as its cultural mother. Needless to say, Japan became the undisputed king of Asia after that, taking over both Taiwan and Korea. Japan experienced a tremendous boost in international status and confidence in itself when it prevailed in the Russo-Japanese War in 1905, against a white, Western power, which was a huge deal at the time when white supremacy was the norm, for good reasons. That also inspired greatly the so called colonized and subjugated colored peoples of the world.

Nonetheless, the Western powers refused to treat Japan as an equal. From my limited knowledge, they made it such that Japan, despite being the victor, did not get very favorable peace terms. The same was in the aftermath of WWI. Japan was obviously resentful that the West treated it as inferior despite all the evidence that Japan was as advanced and competent as the West was, and perhaps more so in some respects. I still recall reading in this silly American grade school textbook: “Japan beat the Europeans at their own game.” So, Japan, very naturally, viewed WWII as a necessity to further prove and assert itself, and justified it on the basis of liberating Asia from Western colonialism and domination. Even though Japan lost that war, it had demonstrated itself more than formidable in cases such as the Battle of Singapore, fighting a fully modern war centered on Navy and Air Force that they lost largely owing to lack of resources and men, such that the West could not but recognize it, despite their being foreign and a latecomer.

Again, post-war, Japan stunned the world with their “economic miracle” that is well represented by the pervasion of its cars and electronics through global (that includes first-class, Western) markets, and it is regarded by many in the West familiar with it, such as Jared Taylor, as one of the best run places on earth. There was some panic in the 1980s in America pertaining to that.

It is apparent that now, Japan, as impressive as it is, has peaked, having endured a so-called Lost Decade and faced competition against its high-tech products from South Korea and now China that cut away some of its market share, further contributing to their decline in economic growth. Still, in absolute terms, it is without doubt that Japan is very high up.

We all know in WWII, Japan easily took over China’s cities with modern against primitive, and it could not win the war against China mainly due to the vast size, as well as the geographic barriers, of mainland China, coupled with their being outnumbered. It was impossible to Japan to control the smaller, rural areas in China, where there was of course a popular anti-Japan resistance. I find it rather ironic that it was China, as weak and backwards as it was, managed to fight America to a stalemate, winning the North Korean side of the war, only 6 years after the end of WWII, when there were still like a million Japanese soldiers in China. That war though, in stark contrast to Japan vs America in WWII, was mainly a land one, one where numbers and human fighting ability mattered more and military technology less. Owing to that, China faced a very different and much fiercer international discrimination and obstacle than Japan had, but it was able to triumph through it miraculously, and that was a much more of a miracle than the Japanese post-war economic miracle. In 1964, China became the first Asian nuclear power. Though at that time, China was still for the most part behind technologically, it is fair to call that the point when China reclaimed its position as the leader of Asia from Japan. China is obviously much more of a threat to the West given its size, not possessed by Japan, as well as its having had its formative years modernization wise entirely in defiance of the West under an economic embargo, after the US was forced to officially treat it as an equal in the Korean Armistice Agreement. China is much indebted to the Soviet Union, which suffered a very sad, tragic demise and further economic disintegration by taking ridiculous advice of Western leaders eager to ruin it, for the generous aid it provided to China in the 1950s, the decade when the critical foundation of modern China was built. Though there was a Sino-Soviet split, with the two in direct conflict a decade later, the immense contribution of the Soviet Union to China’s current success was a decisive factor and cannot be overstated. I believe that the ties between China and Russia are so strong and friendly today largely due to this, which the Chinese people remember well.

Though primarily an (autistic) math nerd, I do take a casual interest in power politics, as you can tell, and I have developed somewhat of a cynical attitude towards it. It is perhaps deep embedded in our human DNA for powerful groups and tribes to want to rule the world. You can see this with the (rather rogue, and also getting outdated) attitude that the American neocons and British empire nostalgists (for lack of a better word) towards international affairs. They are stupid and let their own exceptionalism delude themselves such that they demand the entirely unreasonable and refuse to give credit, in an utterly egregious way, to those they don’t like. For example, they won’t acknowledge, far from it, that it was mostly the USSR who defeated Hitler, which is obvious. They, being as powerful as they were, could sort of get away it, but now that time is past, with the rise and catching up of the rest of the world, China in particular. We all know that powerful individuals or nations generally don’t get along well and exhibit mutual distrust. It’s not uncommon for the most powerful to use a less threatening competitor against its primary competitor, and such is natural behavior. However, the US and the British do it too nastily without shame, and spread the most ridiculous propaganda that is patently false, not respecting history at all. There is also the entitlement to trample and enslave the weak without any reservation whatsoever that is patently manifested in that elite ruling class today. Take a look at the following picture, of Kate Middleton (with Prince William) in Tuvalu, which I will let speak for itself.

Kate-Middleton-carried-elevated-chair-upon-arrived

We all know that it is a human tendency for the rich and powerful to oppress and exploit the poor and marginalized, as has happened throughout history, just about everywhere. Aspiration for status is in our genes, and any social group operates on a hierarchy in some form or another. There will always be winners and losers, superiors and inferiors. But, this could be done in a more benign way than what is reflected in the above photo, which shows devoid of virtue the Anglo elites in the global “leadership” position they cling onto today, in desperation.

Innate mathematical ability

This morning I had the great pleasure of reading an article on LessWrong on innate ability by Jonah Sinick. Jonah has been one of my greatest influences and inspirations, having interacted with him substantially. He is unusual in one of the best ways possible. I would not be surprised if he goes on to do something extraordinary.

When I catch with Jonah, I like to talk with him about math, mathematicians, and IQ, which happens to be what that article of his on LessWrong is about. 😉 That article resonates with me deeply because I myself had similar experiences as he did. It is hypothesized by me that I was also twice exceptional, albeit in different ways, with its effects compounded by my unusual background, all of which mediocrities within the American public school system are not good at dealing with in an effectual way.

This writing of Jonah has brought forth reflections in my own minds with regard to mathematical ability, development, and style. I’ll say that as a little kid under 6, I was very good at arithmetic and even engaged in it obsessively. However, by age 8, after two years of adjusting to life in America starting off not knowing a word of English, I had forgotten most of that. I was known to be good at math among the normal normal students; of course, that doesn’t mean much. In grade school, I was not terribly interested in math or anything academic; I was more interested in playing and watching sports, particularly basketball and baseball.

I didn’t have any mathematical enrichment outside of school other than this silly after school math olympiad program. Nonetheless, I managed to test into two year accelerated math once I reached junior high, not that it means anything. In junior high, we were doing this stupid “core math” with graphing calculators and “experiments.” I didn’t realize that I was actually a joke at math until I failed miserably at the state mathcounts contest, having not prepared for it, unlike all those other tiger mommed Asian kids, who to me seemed way beyond me at that time. It only occurred to me that I might have some real talent for math when I made the AIME in 10th grade, taking the AMCs for the first time, being one of four in my high school of about 2000 to do so. I thought it was fun solving some of those math contest problems, which were more g-loaded, with an emphasis on the pattern recognition side.

It was after that I started to read up on the history of mathematics and mathematicians. I taught myself some calculus and was fascinated by it, not that I understood it very well. But I could easily sense that this was much more significant than many of those contrived contest problems, and soon, I began to lose interest in the contest stuff. It was also after that that I learned about proving things, which the American public school math doesn’t teach. I finally realized what mathematics is really about.

Like Jonah, I had some difficulties with careless errors and mental organization. I don’t think my raw intellectual horsepower was very high back in high school, but fortunately, it has improved substantially since then that it is for the most part no longer the major impediment.

I took calculus officially in 11th grade, and it was a breeze for me. I could easily compute the areas and volumes and such but the entire time, I felt quite dissatisfied, because I could not actually understand that stuff at a rigorous, theoretical level as I poured through our textbook that went up to vector calculus during lecture, which was rather inane, expected if one considers the mismatch between cognitive threshold relative to the distribution of ability of the students. I knew from reading online the rich world of math far beyond what we were covering, most of which I was not intellectually mature enough to access at that time. However, I vividly remember during summer after 11th grade, while attending a math summer program, I was able to comfortably write out the delta epsilon definition of limit with understanding of why it was reasonably defined that way. Still, I would say I was still quite weak in terms of both my mathematical maturity and overall intellectual ability. There were too many things I wasn’t aware of, including the g factor, that I easily would have been had I been higher in verbal ability, which would have enabled me to read, absorb, and internalize information much more rapidly and broadly. In contrast, Jonah had discovered independently, or so he says, the lack of free will at the age of 7!

I made some incremental advances in my math knowledge from reading and thinking outside of school the next year. As for contest math, I almost made the USAMO. Though I had improved, I was still not terribly quick and careful with solving contest style problems and doing computations. I think close to graduation, I also solved some Putnam problems.

Only in undergrad did I learn real math more seriously, but even there, nothing too advanced. US undergrad is a joke, and I also was one, just to a lesser extent than most of my “peers.” Almost certainly, Jonah, based on he’s told me, had gained much deeper and broader knowledge at the same stage, from the reading works of giants like Euler and Riemann.

I’ve noticed how there are a lot of Chinese-(American) kids really into those high school math contests, and they now also dominate USAMO and Putnam (though careful, as in the latter, there you’ve got some of Chinese internationals drawn from the elite from China). I will say that at the lower levels, many of those kids have some pretty low taste and an inability to think outside the system that would enable them to discover the existence of real math, as opposed to this artificial math game that they enjoy playing or are pressured to doing so for college. Though those contests have a high pattern recognition component to them, there is not really much depth or substantial math knowledge. It is also my belief, with reference to Jonah’s article, that math contests are mostly M loaded while real math is more V loaded. So this behavior is consistent with the lopsidedness in favor of M and perhaps also short term working memory of Chinese students. It has also been Jonah’s belief that controlling for g, these contests select for low taste and value judgement, and I surely identify with that perspective. So maybe college admissions are somewhat fair to assess an Asian penalty?

Of the thesis of Jonah’s article, a representative figure is Terry Tao. There, Jonah also pointed out that Tao’s research in math is more concrete and problem solving oriented by pure math standards, in line with what appears to be the same lopsided (modulo the absolute level, as Terry is a far far outlier) cognitive profile of his based on testing at age 9 and 10. Again, people enjoy what they are best at, and though, Terry Tao is almost certainly at least +4 sigma at verbal, he is far more rare, at least +5 sigma, a real übermensch, in the (in some sense dual) pattern recognition component, which means he leans towards the areas of math more loaded on the latter. I have heard the saying that even other Fields medalists are intimidated by Terry Tao. The breadth and volume and technical power of his work is almost unrivaled and otherworldly. The media makes it seem like Terry is a league above even the other Fields medalists. However, Jonah seems to believe that the deepest and most leading of mathematicians are the ones who are more theory builders, who create through leaps of insight and synthesis new fields and directions that keep mathematicians busy for decades, and even centuries. That would be say Grothendieck or SS Chern, and an ability that is more loaded on verbal ability, crudely speaking. Again, I have felt the same. This might explain why the advantage of Chinese students is not anywhere near as pronounced in math research as in contests, and why some people say that generally speaking, the Chinese mathematicians are more problem solving and technical than theoretical, more analysis than algebra. Likewise, we can predict the opposite for Jews who are skewed in favor of verbal. A corollary of this would be that the Jews produce the deepest thinkers, adjusted somewhat for population, which is almost certainly the case, if you look at the giants of mathematics and theoretical physics.

I’ll conclude with the following remark. I used to revere somewhat those who placed very highly on those contests, until I realized that many of them are actually somewhat weak in terms of deep understanding and thinking at a more theoretical level. Yes, I have met MOSPers who got destroyed by real math and who are not very intellectually versatile, with glaring weaknesses; I was quite surprised initially that even I seemed to be smarter if not a lot than some of them. Once upon a time, I couldn’t understand those who appeared very strong at real math (and often also science and/or engineering and/or humanities) who struggled with more concrete math and/or contest-style problem solving, like Jonah, who has written on LessWrong of his difficulties with accuracy on the trivial math SAT. I’ve met this other guy, who I thought was an idiot for being unable to perform simple computations, who is leagues beyond me in the most abstract of math, who writes prolifically about partially V-loaded areas of math like model theory. Now, the more metacognitive me has awakened to the reality that I may never by deficit of my neurobiology be able to fathom and experience what they’re capable of. After all, there are plenty I am almost certain are and are essentially doomed to be very delusional by nature relative to me, and since I’m at the far tail but not quite so much, there are bound to be people who view me the same. I can only hope that I can become more like them through some combination of exposure and organic neurobiological growth, but I as a realist will not deem that very likely.

More on Asian stereotypes

I just stumbled upon this wonderful essay by Gwydion Madawc Williams on why the Ming voyages led by Zheng He (郑和) led to nothing. The quote of it particularly memorable to me was this:

The separation of craft and education as represented by China’s illiterate shipwrights was indeed a genuine weakness in the Chinese system.  Christian Europe always remembered that St Peter had been a fisherman and St Paul a tent-maker, and it was quite acceptable for learned people to also be involved in manufacturing.  The weakness of Confucianism was not so much that it rated agriculture and craft above merchant trade, but that it insisted on the educated being a learned caste distanced from all of these matters.

Again, it’s the Asian stereotype of being a study hard grind lacking in practical, hands-on skills and “well-roundedness” and “social skills” and all that that admissions officers use to justify denying Asian applicants. I’ll say that from what I know, that is still very limited, Confucianism was very much like that. The quote that epitomized this was: 劳心者治人,劳力者治于人, which translates to roughly “the worker of the mind governs, the physical worker is governed.” The whole imperial examination system essentially created an upper class of bookworms for whom any form of hands on labor was beneath. To be a true 君子, gentlemen, you were supposed to study the classics and write poetry and engage in all that Confucian bull shit. I myself don’t have a very high opinion of Confucianism. It’s too conservative for me, with all the emphasis on ritual and filial piety. It discouraged any form of innovation outside the system, outside what was already there, which is partly why China could not make the giant leaps in science that the West did. I’ve read some of the Analects of Confucius and know some of the quotes, and I don’t think Confucius was a deep philosopher at all; there is little actual substance in what he said. On the other hand, Mo Tzu was a much further reaching, more scientific, and surprisingly modern thinker, and had China followed his path instead of banishing his school of thought into obscurity, the world would be completely different now, with China likely having made many more leaps of progress than it had actually done. I’ll say that the West was able to escape the shackles of Christianity, but China could not by itself escape those of Confucianism, until its dire situation, with reached its nadir in 1900, forced it too.

Apparently, the elite college admissions officers aren’t terribly good at filtering out the real Asian grinds either, as I know one who went to Princeton, who I found ridiculous. He said that all he did in college was study, and even though he majored in math, he hardly knew any. Like, he didn’t know what a topological space is. When I went ice skating with him and some others, he was near the edge the whole time, and he characterized my skating backwards (not well at all) as “scary.” I told him I’m not very athletic and wasn’t even any good, unlike the girl he was dating at that time, who could do spins among other fancy “figure skating” things she was trying out. I did show him the video taken of this 360 somersault I did off a 15 feet cliff in Hawaii, into the water, which was the first time I had done anything like that. He was like: “that’s so scary.” I honestly didn’t know what to say. To justify himself, he was like: “Chinese parents only want their kids to study.” I told him that in China, there are some very athletic people who attend special sports schools. On that, he was like: “but those aren’t normal people.” I also remember when we went camping once, everybody else got drunk, so I got to drive that kid’s BMW back. He had told us that his father does business in Beijing, which might explain why he drives that kind of car. He came to US at age 4. His Chinese is absolutely awful though, and he doesn’t realize it. He will of course say: “I already know enough. Some people can’t even speak it.” 怎么说那,不仅是个书呆子,而且是个书都读不好的书呆子,连这样的sb还都被Princeton录取了。I’ve talked with one of my very smart Asian friends about this, and he was like: “but he’s socially normal, unlike us.” And more recently: “Maybe they do accept Asian grinds, just not the ones with bad social skills.”

From what I’ve seen, there are plenty of super conformist Asian grinds like him, but there are also many who aren’t, who are actually smart and interesting, like myself (or at least I hope). I think what he said about Chinese parents is somewhat true actually; after all, I saw many growing up. They do see academics as a way to get ahead more so than others, largely because in China, to get out of your rural village and/or not be stuck with a working class job, you had to do sufficiently well on the gaokao to get into a good major at a good university. It’s funny that I’ve actually seen a ton of ignorant, narrow-minded, and risk-averse uncool tiger Chinese parents. And I have also seen some extremely impressive ones, not just academically. There is again quite a wide range and variety.

There is a phenomenon I’ve witnessed, which is that if a person is extremely strong at X and merely above average at Y, then that person will seem weak at Y, even compared to another person about as good at Y but less lopsided. It seems a natural human cognitive bias to think this way. This is in fact applied rather perversely to Asians in stereotyping. For example, Asian students are perceived as weak at language and humanities because they are generally stronger at STEM. We all know that in fact math IQ and verbal IQ (which we can use crudely as proxies for STEM ability and humanities ability respectively) are highly correlated, which makes it highly unlikely that a STEM star is actually legitimately weak at humanities. He might not be interested in reading novels and such but that’s rather different. There is also that humanities is more cultural exposure loaded with a much higher subjective element to it, with much less of a uniform metric. It actually seems to me based on personal experience that is by no means representative that in terms of precise use of language and the learning of foreign languages, mathematicians and theoretical physicists are at or near the top in terms of ability. On this, I will give an opposing perspective that I identify with somewhat, which is that even if you’re very strong at Y, having an X that you are significantly more talented at is a weakness for Y, because engaging in Y deprives the joy derived from engaging in the X, which often leads to loss of interest over time. Maybe this is why employers shy from hiring people who they deem “overqualified?” On this, I have thought of how possibly the lopsided cognitive profile in East Asians (with what is likely at least 2/3 SD differential between math/visuo-spatial and verbal, normalizing on white European scores) predisposed the thinking of the elite (assuming that lopsidedness is preserved at the far tail) as well as the development of that society at large in certain ways, some of which may have been not the most conducive for, say, the development of theoretical science. This is of course very speculative, and I would actually hypothesize that the far tail cognitive elite among East Asians is more balanced in terms of the math/visuo-spatial and verbal split, given the great extent to which the imperial examination system, which tested almost exclusively literary things, selected for V at the tail instead of for M.

On the aforementioned bias, I’ll give another illustrative example. I once said to this friend of mine, a math PhD student, not Asian, how there’s the impression that people who are weaker academically tend to be better at certain practical things, like starting restaurants and businesses. We sure all know there are plenty who weren’t good at school but were very shrewd and successful at business, at practical things. That guy responded with reference to Berkson’s paradox. He said something like: “That’s because you are unlikely to see those who are bad at both. They tend to be in prison or in the lower classes.” I could only agree.

I’ll conclude with another more dramatic example. I used to, when I knew nothing about the subject, think that people who were really at math were weirdos and socially awkward. For one, there was this kid in my high school who was way better than me at math at the time, who was incredibly autistic. Also, summer after 10th grade, I saw Beautiful Mind, which depicts the mathematician as mentally crazy. Now I would bet the incidence of schizophrenia among the mathematically gifted is lower than it is in the whole population. It just happens that certain combinations of extreme traits are vastly more noticeable or exposed by the media to the public (a mathematician or physicist may think of this as weighing those with such combinations with a delta function, or something along that direction at least). I wasn’t quite aware of that at that time though. Only later, after meeting more math people did I realize that math people are not actually that socially out of it in general, far from it, at least once they’re past a certain age, by which they will have had the chance to interact with more people like them and form their own peer group.

It is my hope that people can be more cognizant of these biases described in this blog post.

 

Hahn-Banach theorem

I’m pleased to say that I find the derivation of the Hahn-Banach theorem pretty straightforward by now. Let me first state it, for the real case.

Hahn-Banach theorem: Let V be a real vector space. Let p: V \to \mathbb{R} be sublinear. If f : U \to \mathbb{R} be a linear functional on the subspace U \subset V with f(x) \leq p(x) for x \in U, then there exists a linear extension of f to all of V (call it g) such that f(x) \leq g(x) for x \in V with f(x) = g(x) for x \in U and g(x) \leq p(x) for all x \in V.

To show this, start by taking any x_0 \in V \setminus U. We wish to assign some \alpha to x_0 that keeps p as the dominating function in the vector space U + \mathbb{R}x_0. For this to happen, applying the linearity of f and the domination constraint, we can derive

\frac{f(y) - p(y - \lambda x_0)}{\lambda} \leq \alpha \leq \frac{p(y+\lambda x_0) - f(y)}{\lambda}, \quad y \in U, \lambda > 0.

This reduces to

\sup_{y \in U} p(y+x_0) - f(y) \leq \inf_{y \in U} f(y) - p(y-x_0).

Such can be proven via

f(y_1) + f(y_2) = f(y_1 + y_2) \leq p(y_1 + y_2) \leq p(y_1 - x_0) +p(y_2 + x_0), \quad y_1, y_2 \in U.

Now take the space of linear functionals defined on some specific subspace dominated by p. Denote an element of it as (f, U). We introduce a partial order wherein (f, U) \leq (f', U') iff f(x) = f'(x) for x \in U and U \subset U'. We can apply Zorn’s lemma on this, as we can take the union to derive an upper bound for any chain. Any maximal element is necessarily (g, V) as if the domain is not the entire vector space, we can by above construct a larger element.

 

Riesz-Thorin interpolation theorem

I had, a while ago, the great pleasure of going through the proof of the Riesz-Thorin interpolation theorem. I believe I understand the general strategy of the proof, though for sure, I glossed over some details. It is my hope that in writing this, I can fill in the holes for myself at the more microscopic level.

Let us begin with a statement of the theorem.

Riesz-Thorin Interpolation Theorem. Suppose that (X,\mathcal{M}, \mu) and (Y, \mathcal{N}, \nu) are measure spaces and p_0, p_1, q_0, q_1 \in [1, \infty]. If q_0 = q_1 = \infty, suppose also that \mu is semifinite. For 0 < t < 1, define p_t and q_t by

\frac{1}{p_t} = \frac{1-t}{p_0} + \frac{t}{p_1}, \qquad  \frac{1}{q_t} = \frac{1-t}{q_0} + \frac{t}{q_1}.

If T is a linear map from L^{p_0}(\mu) + L^{p_1}(\mu) into L^{q_0}(\nu) + L^{q_1}(\nu) such that \left\|Tf\right\|_{q_0} \leq M_0 \left\|f\right\|_{p_0} for f \in L^{p_0}(\mu) and \left\|Tf\right\|_{q_1} \leq M_1 \left\|f\right\|_{p_1} for f \in L^{p_1}(\mu), then \left\|Tf\right\|_{q_t} \leq M_0^{1-t}M_1^t \left\|f\right\|_{p_t} for f \in L^{p_t}(\mu), 0 < t < 1.

We begin by noticing that in the special case where p = p_0 = p_1,

\left\|Tf\right\|_{q_t} \leq \left\|Tf\right\|_{q_0}^{1-t} \left\|Tf\right\|_{q_1}^t \leq M_0^{1-t}M_1^t \left\|f\right\|_p,

wherein the first inequality is a consequence of Holder’s inequality. Thus we may assume that p_0 \neq p_1 and in particular that p_t < \infty.

Observe that the space of all simple functions on X that vanish outside sets of finite measure has in its completion L_p(\mu) for p < \infty and the analogous holds for Y. To show this, take any f \in L^p(\mu) and any sequence of simple f_n that converges to f almost everywhere, which must be such that f_n \in L^p(\mu), from which follows that they are non-zero on a finite measure. Denote the respective spaces of such simple functions with \Sigma_X and \Sigma_Y.

To show that \left\|Tf\right\|_{q_t} \leq M_0^{1-t}M_1^t \left\|f\right\|_{p_t} for all f \in \Sigma_X, we use the fact that

\left\|Tf\right\|_{q_t} = \sup \left\{\left|\int (Tf)g d\nu \right| : g \in \Sigma_Y, \left\|g\right\|_{q_t'} = 1\right\},

where q_t' is the conjugate exponent to q_t. We can rescale f such that \left\|f\right\|_{p_t} = 1.

From this it suffices to show that across all f \in \Sigma_X, g \in \Sigma_Y with \left\|f\right\|_{p_t} = 1 and \left\|g\right\|_{q_t'} = 1, |\int (Tf)g d\nu| \leq M_0^{1-t}M_1^t.

For this, we use the three lines lemma, the inequality of which has the same value on its RHS.

Three Lines Lemma. Let \phi be a bounded continuous function on the strip 0 \leq \mathrm{Re} z \leq 1 that is holomorphic on the interior of the strip. If |\phi(z)| \leq M_0 for \mathrm{Re} z = 0 and |\phi(z)| \leq M_1 for \mathrm{Re} z = 1, then |\phi(z)| \leq M_0^{1-t} M_1^t for \mathrm{Re} z = t, 0 < t < 1.

This is proven via application of the maximum modulus principle on \phi_{\epsilon}(z) = \phi(z)M_0^{z-1} M_1^{-z} \mathrm{exp}^{\epsilon z(z-1)} for \epsilon > 0. The \mathrm{exp}^{\epsilon z(z-1)} serves of function of |\phi_{\epsilon}(z)| \to 0 as |\mathrm{Im} z| \to \infty for any \epsilon > 0.

We observe that if we construct f_z such that f_t = f for some 0 < \mathrm{Re} t < 1. To do this, we can express for convenience f = \sum_1^m |c_j|e^{i\theta_j} \chi_{E_j} and g = \sum_1^n |d_k|e^{i\theta_k} \chi_{F_k} where the c_j‘s and d_k‘s are nonzero and the E_j‘s and F_k‘s are disjoint in X and Y and take each |c_j| to \alpha(z) / \alpha(t) power for such a fixed t for some \alpha with \alpha(t) > 0. We let t \in (0, 1) be the value corresponding to the interpolated p_t. With this, we have

f_z = \displaystyle\sum_1^m |c_j|^{\alpha(z)/\alpha(t)}e^{i\theta_j}\chi_{E_j}.

Needless to say, we can do similarly for g, with \beta(t) < 1,

g_z = \displaystyle\sum_1^n |d_k|^{(1-\beta(z))/(1-\beta(t))}e^{i\psi_k}\chi_{F_k}.

Together these turn the LHS of the inequality we desire to prove to a complex function that is

\phi(z) = \int (Tf_z)g_z d\nu.

To use the three lines lemma, we must satisfy

|\phi(is)| \leq \left\|Tf_{is}\right\|_{q_0}\left\|g_{is}\right\|_{q_0'} \leq M_0 \left\|f_{is}\right\|_{p_0}\left\|g_{is}\right\|_{q_0'} \leq M_0 \left\|f\right\|_{p_t}\left\|g\right\|_{q_t'} = M_0.

It is not hard to make it such that \left\|f_{is}\right\|_{p_0} = 1 = \left\|g_{is}\right\|_{q_0'}. A sufficient condition for that would be integrands associated with norms are equal to |f|^{p_t/p_0} and |g|^{q_t'/q_0'} respectively, which equates to \mathrm{Re} \alpha(is) = 1 / p_0 and \mathrm{Re} (1-\beta(is)) = 1 / q_0'. Similarly, we find that \mathrm{Re} \alpha(1+is) = 1 / p_1 and \mathrm{Re} (1-\beta(1+is)) = 1 / q_1'. From this, we can solve that

\alpha(z) = (1-z)p_0^{-1}, \qquad \beta(z) = (1-z)q_0^{-1} + zq_1^{-1}.

With these functions inducing a \phi(z) that satisfies the hypothesis of the three lines lemma, our interpolation theorem is shown for such simple functions, from which extend our result to all f \in L^{p_t}(\mu).

To extend this to all of L^p, it suffices that Tf_n \to Tf a.e. for some sequence of measurable simple functions f_n with |f_n| \leq |f| and f_n \to f pointwise. Why? With this, we can invoke Fatou’s lemma (and also that \left\|f_n\right\|_p \to \left\|f\right\|_p by dominated convergence theorem) to obtained the desired result, which is

\left\|Tf\right\|_q \leq \lim\inf \left\|Tf_n\right\|_q \leq \lim\inf M_0^{1-t} M_1^t\left\|Tf_n\right\|_p \leq M_0^{1-t} M_1^t \left\|f\right\|_p.

Recall that convergence in measure is a means to derive a subsequence that converges a.e. So it is enough to show that \displaystyle\lim_{n \to \infty} \mu(\left\|Tf_n - Tf\right\| > \epsilon) = 0 for all \epsilon > 0. This can be done by upper bounding with something that goes to zero. By Chebyshev’s inequality, we have

\mu(\left\|Tf_n - Tf\right\| > \epsilon) \leq \frac{\left\|Tf_n - Tf\right\|_p^p}{\epsilon^p}.

However, recall that in our hypotheses we have constant upper bounds on T in the p_0 and p_1 norms respectively assuming that f is in L^{p_0} and L^{p_1}, which we can make use of.  So apply Chebyshev on any one of q_0 (let’s use this) and q_1, upper bound its upper bound with M_0 or M_1 times \left\|f_n - f\right\|_{p_0}, which must go to zero by pointwise convergence.

Hilbert basis theorem

I remember learning this theorem early 2015, but I could not remember its proof at all. Today, I relearned it. It employed a beautiful induction argument to transfer the Noetherianness (in the form of finite generation) from R to R[x] via the leading coefficient.

Hilbert Basis TheoremIf R is a Noetherian ring, then so is R[x].

Proof: Take some ideal J in R. Notice that if we partition J by degree, we get from the leading coefficients appearing in each an ascending chain (that has to become constant eventually, say at k). Take finite sets A_n \subset J for m \leq n \leq k, where m is the smallest possible non-zero degree such that the I_ns for the leading coefficient ideals are generated. With this we can for any polynomial p construct a finite combination within A = \displaystyle\cup_{n=m}^k A_n that equates to p leading coefficient wise, and thereby subtraction reduces to a lower degree. Such naturally lends itself induction, with m as the base case. For m any lower degree polynomial is the zero polynomial. Now assume, as the inductive hypothesis that A acts as a finite generating set all polynomials with degree at most n. If n+1 \leq k, we can cancel out the leading coefficient using our generating set, and then use the inductive hypothesis. If n+1 > k, we can by our inductive hypothesis generate with A a degree n polynomial with same leading coefficient (and thereby a degree n+1 one multiplying by x) and from that apply our inductive hypothesis again, this time on our difference.

四海翻腾云水怒,五洲震荡风雷激

昨天,我的俄罗斯(或取苏联代之)товарищ,那位本科认识,才华超众,现已明格罗滕迪克(Grothendieck)的工作的,脸书上发给我一张图片:ChinaWonTime

后在得知此来自Time Magazine的刚发布的一篇文章,想这是Time Magazine啊,美国很有威望的畅销杂志。在那儿,闻到中国竟然已在非洲东北岸的吉布提(Djibouti)建立了头一个海外的军事基地。但对我印象最深的文章最后段,为:

The China striding into that spotlight is not guaranteed to win the future. In this fragmenting world, no one government will have the international influence required to continue to set the political and economic rules that govern the global system. But if you had to bet on one country that is best positioned today to extend its influence with partners and rivals alike, you wouldn’t be wise to back the U.S. The smart money would probably be on China.

这是美国,对,美国(!!!)的主流媒体,竟然会这么说,令我吃惊,因为不用说,我么都习惯了美国媒体带有政治偏见的那一套的同时说中国怎么怎么不行,问题障碍如此之严重。看来真的,中国成为世界绝对一把手,在2017年末的这一天,不仅非遥远无可想象,而是所料当中了。同时,在历史一直低估怀疑中华人民共和国的情况下,还真能想象到续几十年,遥遥领先,使外界望尘莫及。我也有想过中国人过去在物质条件极其劣势,外来的歧视极其恶略的情形之下,都有了不少成就,有些让外眼彻底震惊,有了充分的财力支撑能够多么可怕。中国人天分很高,又有数量,又是在内分歧相对少的一个民族,一个文化,领导人又不是傻子,可建立多么效率超高,思想丰富,创新异彩的社会和国家啊!中国用胚胎筛选和基因工程把人类进化到新的,高于人的物种都是有可能的,在这一点,中国文化的态度比较进步,与被过时的某些关于纳粹主义的观念的遗产以及愚昧的基督教基要主义所障碍的西方相比,加上具有庞大的基因组学设施,华大基因(BGI)为首,和这种大科学系统工程所需要的充分的经费和组织力量。

这些使得我回想到我近几年学到的一首给我留下深刻的印象的一首诗的一句,为此博客文章的主题。“四海翻腾云水怒,五洲震荡风雷激”,这是多么激情豪迈,万能无阻的诗句形象啊!而这不就是当日的中国与世界吗?此诗《满江红·和郭沫若同志》创于1963年一月,那时中印战争刚结束,中国已在国际形势对其非常不利的情况下走出三年困难时期,毛主席号召中国和全世界革命进步人民“扫除一切害人虫”,奋勇前进,创造人类新纪元。可是这只取得了最多一半的成功。相反,今天,继承走向人类逃出剥削精神麻木的伟大事业的中国,与往年贫穷落后隔离不同,通过几十年的发展和积累,随着不断地开放,已经具有浓厚的经济资源和先进的科技经验,纵横全球,力量无穷。还想起,诗的上段,”蚂蚁缘槐夸大国,蚍蜉撼树谈何易“,不是用以讽刺性的藐视当时被干脆收拾的自作聪明,不自量力的印度和他的支持者吗?同样,在今天的世界,此妙语所指的对象就是以小人无比,阴谋诡计的方式,企图阻止中国崛起的,文化低贱,贪得无厌,妄想霸权世界的美国保守派及其走狗。这些人的声音和实力和他们所有的威信在不断减小,总会有一天,人类的进步将把他们带到边缘,化成历史的破产。

无话多言。中国赢了。历史新篇章即将来临。太平世界,寰球同此凉热!