Expanding on the procedure of expanding: Where is the procedure to do that?

Update on 18th June 2017:

See the update to the last post; I have added three more diagrams depicting the mathematical abstraction of the problem, and also added a sub-question by way of clarifying the problem a bit. Hopefully, the problem is clearer and also its connection to QM a bit more apparent, now.


Here I partly expand on the problem mentioned in my last post [^]. … Believe me, it will take more than one more post to properly expand on it.


The expansion of an expanding function refers to and therefore requires simultaneous expansions of the expansions in both the space and frequency domains.

The said expansions may be infinite [in procedure].


In the application of the calculus of variations to such a problem [i.e. like the one mentioned in the last post], the most important consideration is the very first part:

Among all the kinematically admissible configurations…

[You fill in the rest, please!]


A Song I Like:

[I shall expand on this bit a bit later on. Done, right today, within an hour.]

(Hindi) “goonji see hai, saari feezaa, jaise bajatee ho…”
Music: Shankar Ahasaan Loy
Singers: Sadhana Sargam, Udit Narayan
Lyrics: Javed Akhtar

 

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I’m not…

“I’m not half the man I used to be.”

That’s what they all say when they become, say, mature. These days, the “they” includes me. Yep. That’s right. I have realized that I really am not half the man I used to be. … Let me count the ways…

I no longer write code every day.

I also no longer debug code every day.

I also no longer use C++ as my first programming language.

I also no longer read pop-science books on QM.

…You get the idea. …

But, actually, it’s become worse, much worse. I perhaps am no longer even a quarter of the man I used to be. …

The reason is, not only do I read through (all the pages of) the QM and maths text-books these days, I have in fact also begun going through some maths journals these days… And it doesn’t even stop there… I mean, I also have a recommendation for a maths journal, for you!

OK. Don’t be overly concerned about me. I am quite OK, that way…

About the only maths journal whose many issues I have browsed through (and have recommendation for) is the “SIAM Undergraduate Research Online (SIURO)” [^].

Do check it out. The published papers are available for free. The link to the published papers is right on the home page; it goes to here [^].

Neat… May be Americans have begun becoming smart these days or something…

[I forgot what it was that I was looking for when I stumbled across one of the papers in this journal… If I remember that paper, I will come back and mention it… [I told you, I’m not half the man I used to be…]]


A Song I Like:
(English) “Yesterday”
Band: The Beatles

[Anti-poignancy (or should it be un-poignancy?): The Beatles were in their early twenties when they penned these lyrics.]

[E&OE]

Mathematics—Historic, Contemporary, and Its Relation to Physics

The title of this post does look very ambitious, but in fact the post itself isn’t. I mean, I am not going to even attempt to integrate these diverse threads at all. Instead, I am going to either just jot down a few links, or copy-paste my replies (with a bit editing) that I had made at some other blogs.

 

1. About (not so) ancient mathematics:

1.1 Concerning calculus: It was something of a goose-bumps moment for me to realize that the historic Indians had very definitely gotten to that branch of mathematics which is known as calculus. You have to understand the context behind it.

Some three centuries ago, there were priority battles concerning invention of calculus (started by Newton, and joined by Liebniz and his supporters). Echoes of these arguments could still be heard in popular science writings as recently as when I was a young man, about three decades ago.

Against this backdrop, it was particularly wonderful that an Indian mathematician as early as some eight centuries ago had gotten to the basic idea of calculus.

The issue was highlighted by Prof. Abinandanan at the blog nanpolitan, here [^]. It was based on an article by Prof. Biman Nath that had appeared in the magazine Frontline [^]. My replies can be found at Abi’s post. I am copy-pasting my replies here. I am also taking the opportunity to rectify a mistake—somehow, I thought that Nath’s article appeared in the Hindu newspaper, and not in the Frontline magazine. My comment (now edited just so slightly):

A few comments:

0. Based on my earlier readings of the subject matter (and I have never been too interested in the topic, and so, it was generally pretty much a casual reading), I used to believe that the Indians had not reached that certain abstract point which would allow us to say that they had got to calculus. They had something of a pre-calculus, I thought.

Based (purely) on Prof. Nath’s article, I have now changed my opinion.

Here are a few points to note:

1. How “jyaa” turned to “sine” makes for a fascinating story. Thanks for its inclusion, Prof. Nath.

2. Aaryabhata didn’t have calculus. Neither did Bramhagupta [my spelling is correct]. But if you wonder why the latter might have laid such an emphasis on the zero about the same time that he tried taking Aaryabhata’s invention further, chances are, there might have been some churning in Bramhagupta’s mind regarding the abstraction of the infinitesimal, though, with the evidence available, he didn’t reach it.

3. Bhaaskara II, if the evidence in the article is correct, clearly did reach calculus. No doubt about it.

He did not only reach a more abstract level, he even finished the concept by giving it a name: “taatkaalik.” Epistemologically speaking, the concept formation was complete.

I wonder why Prof. Nath, writing for the Frontline, didn’t allocate a separate section to Bhaaskara II. The “giant leap” richly deserved it.

And, he even got to the max-min problem by setting the derivative to zero. IMO, this is a second giant leap. Conceptually, it is so distinctive to calculus that even just a fleeting mention of it would be enough to permanently settle the issue.

You can say that Aaryabhata and Bramhagupta had some definite anticipation of calculus. And you can’t possible much more further about Archimedes’ method of exhaustion either. But, as a sum total, I think, they still missed calculus per say.

But with this double whammy (or, more accurately, the one-two punch), Bhaaskara II clearly had got the calculus.

Yes, it would have been nice if he could have left for the posterity a mention of the limit. But writing down the process of reaching the invention has always been so unlike the ancient Indians. Philosophically, the atmosphere would generally be antithetical to such an idea; the scientist, esp. the mathematician, may then be excused.

But then, if mathematicians had already been playing with infinite series with ease, and were already performing the calculus of finite differences in the context of these infinite series, even explicitly composing verses about their results, then they can be excused for not having conceptualized limits.

After all, even Newton initially worked only with the fluxion and Leibniz with the infinitesimal. The modern epsilon-delta definition still was some one–two centuries (in the three–four centuries of modern science) in the coming.

But when you explicitly say “instantaneous,” (i.e. after spelling out the correct thought process leading to it), there is no way one can say that some distance had yet to be travelled to reach calculus. The destination was already there.

And as if to remove any doubt still lingering, when it comes to the min-max condition, no amount of merely geometric thinking would get you there. Reaching of that conclusion means that the train had not already left the first station after entering the calculus territory, but also that it had in fact gone past the second or the third station as well. Complete with an application from astronomy—the first branch of physics.

I would like to know if there are any counter-arguments to the new view I now take of this matter, as spelt out above.

4. Maadhava missed it. The 1/4 vs. 1/6 is not hair-splitting. It is a very direct indication of the fact that either Maadhava did a “typo” (not at all possible, considering that these were verses to be by-hearted by repetition by the student body), or, obviously, he missed the idea of the repeated integration (which in turn requires considering a progressively greater domain even if only infinitesimally). Now this latter idea is at the very basis of the modern Taylor series. If Maadhava were to perform that repeated integration (and he would be a capable mathematical technician to be able to do that should the idea have struck him), then he would surely get 1/6. He would get that number, even if he were not to know anything about the factorial idea. And, if he could not get to 1/6, it’s impossible that he would get the idea of the entire infinite series i.e. the Taylor series, right.

5. Going by the content of the article, Prof. Nath’s conclusion in the last paragraph is, as indicated above, in part, non-sequitur.

6. But yes, I, too, very eagerly look forward to what Prof. Nath has to say subsequently on this and related issues.

But as far as the issues such as the existence of progress only in fits here and there, and indeed the absence of a generally monotonously increasing build-up of knowledge (observe the partial regression in Bramhagupta from Aaryabhat, or in Maadhav from Bhaaskar II), I think that philosophy as the fundamental factor in human condition, is relevant.

7. And, oh, BTW, is “Matteo Ricci” a corrupt form of the original “Mahadeva Rishi” [or “Maadhav Rishi”] or some such a thing? … May Internet battles ensue!

1.2 Concerning “vimaan-shaastra” and estimating \pi: Once again, this was a comment that I made at Abi’s blog, in response to his post on the claims concerning “vimaan-shaastra” and all, here[^]. Go through that post, to know the context in which I wrote the following comment (reproduced here with a bit of copy-editing):

I tend not to out of hand dismiss claims about the ancient Indian tradition. However, this one about the “Vimaan”s and all does seem to exceed even my limits.

But, still, I do believe that it can also be very easy to dismiss such claims without giving them due consideration. Yes, so many of them are ridiculous. But not all. Indeed, as a less noted fact, some of the defenders themselves do contradict each other, but never do notice this fact.

Let me give you an example. I am unlike some who would accept a claim only if there is a direct archaeological evidence for it. IMO, theirs is a materialistic position, and materialism is a false premise; it’s the body of the mind-body dichotomy (in Ayn Rand’s sense of the terms). And, so, I am willing to consider the astronomical references contained in the ancient verses as an evidence. So, in that sense, I don’t dismiss a 10,000+ old history of India; I don’t mindlessly accept 600 BC or so as the starting point of civilization and culture, a date so convenient to the missionaries of the Abrahamic traditions. IMO, not every influential commentator to come from the folds of the Western culture can be safely assumed to have attained the levels obtained by the best among the Greek or enlightenment thinkers.

And, so, I am OK if someone shows, based on the astronomical methods, the existence of the Indian culture, say, 5000+ years ago.

Yet, there are two notable facts here. (i) The findings of different proponents of this astronomical method of dating of the past events (say the dates of events mentioned in RaamaayaNa or Mahaabhaarata) don’t always agree with each other. And, more worrisome is the fact that (ii) despite Internet, they never even notice each other, let alone debate the soundness of their own approaches. All that they—and their supporters—do is to pick out Internet (or TED etc.) battles against the materialists.

A far deeper thinking is required to even just approach these (and such) issues. But the proponents don’t show the required maturity.

It is far too easy to jump to conclusions and blindly assert that there were material “Vimaana”s; that “puShpak” etc. were neither a valid description of a spiritual/psychic phenomenon nor a result of a vivid poetic imagination. It is much more difficult, comparatively speaking, to think of a later date insertion into a text. It is most difficult to be judicious in ascertaining which part of which verse of which book, can be reliably taken as of ancient origin, which one is a later-date interpolation or commentary, and which one is a mischievous recent insertion.

Earlier (i.e. decades earlier, while a school-boy or an undergrad in college etc.), I tended to think the very last possibility as not at all possible. Enough people couldn’t possibly have had enough mastery of Sanskrit, practically speaking, to fool enough honest Sanskrit-knowing people, I thought.

Over the decades, guess, I have become wiser. Not only have I understood the possibilities of the human nature better on the up side, but also on the down side. For instance, one of my colleagues, an engineer, an IITian who lived abroad, could himself compose poetry in Sanskrit very easily, I learnt. No, he wouldn’t do a forgery, sure. But could one say the same for every one who had a mastery of Sanskrit, without being too naive?

And, while on this topic, if someone knows the exact reference from which this verse quoted on Ramesh Raskar’s earlier page comes, and drops a line to me, I would be grateful. http://www.cs.unc.edu/~raskar/ . As usual, when I first read it, I was impressed a great deal. Until, of course, other possibilities struck me later. (It took years for me to think of these other possibilities.)

BTW, Abi also had a follow-up post containing further links about this issue of “vimaan-shaastra” [^].

But, in case you missed it, I do want to highlight my question again: Do you know the reference from which this verse quoted by Ramesh Raskar (now a professor at MIT Media Lab) comes? If yes, please do drop me a line.

 

2. An inspiring tale of a contemporary mathematician:

Here is an inspiring story of a Chinese-born mathematician who beat all the odds to achieve absolutely first-rank success.

I can’t resist the temptation to insert my trailer: As a boy, Yitang Zhang could not even attend school because he was forced into manual labor on vegetable-growing farms—he lived in the Communist China. As a young PhD graduate, he could not get a proper academic job in the USA—even if he got his PhD there. He then worked as an accountant of sorts, and still went on to solve one of mathematics’ most difficult problems.

Alec Wilkinson writes insightfully, beautifully, and with an authentic kind of admiration for man the heroic, for The New Yorker, here [^]. (H/T to Prof. Phanish Suryanarayana of GeorgiaTech, who highlighted this article at iMechanica [^].)

 

3. FQXi Essay Contest 2015:

(Hindi) “Picture abhi baaki nahin hai, dost! Picture to khatam ho gai” … Or, welcome back to the “everyday” reality of the modern day—modern day physics, modern day mathematics, and modern day questions concerning the relation between the two.

In other words, they still don’t get it—the relation between mathematics and physics. That’s why FQXi [^] has got an essay contest about it. They even call it “mysterious.” More details here [^]. (H/T to Roger Schlafly [^].)

Though this last link looks like a Web page of some government lab (American government, not Indian), do check out the second section on that same page: “II Evaluation Criteria.” The main problem description appears in this section. Let me quote the main problem description right in this post:

The theme for this Essay Contest is: “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”.

In many ways, physics has developed hand-in-hand with mathematics. It seems almost impossible to imagine physics without a mathematical framework; at the same time, questions in physics have inspired so many discoveries in mathematics. But does physics simply wear mathematics like a costume, or is math a fundamental part of physical reality?

Why does mathematics seem so “unreasonably” effective in fundamental physics, especially compared to math’s impact in other scientific disciplines? Or does it? How deeply does mathematics inform physics, and physics mathematics? What are the tensions between them — the subtleties, ambiguities, hidden assumptions, or even contradictions and paradoxes at the intersection of formal mathematics and the physics of the real world?

This essay contest will probe the mysterious relationship between physics and mathematics.

Further, this section actually carries a bunch of thought-provocative questions to get you going in your essay writing. … And, yes, the important dates are here [^].

Now, my answers to a few questions about the contest:

Is this issue interesting enough? Yes.

Will I write an essay? No.

Why? Because I haven’t yet put my thoughts in a sufficiently coherent form.

However, I notice that the contest announcement itself includes so many questions that are worth attempting. And so, I will think of jotting down my answers to these questions, even if in a bit of a hurry.

However, I will neither further forge the answers together in a single coherent essay, nor will I participate in the contest.

And even if I were to participate… Well, let me put it this way. Going by Max Tegmark’s and others’ inclinations, I (sort of) “know” that anyone with my kind of answers would stand a very slim chance of actually landing the prize. … That’s another important reason for me not even to try.

But, yes, at least this time round, many of the detailed questions themselves are both valid and interesting. And so, it should be worth your while addressing them (or at least knowing what you think of them for your answers). …

As far as I am concerned, the only issue is time. … Given my habits, writing about such things—the deep and philosophical, and therefore fascinating things, the things that are interesting by themselves—have a way of totally getting out of control. That is, even if you know you aren’t going to interact with anyone else. And, mandatory interaction, incidentally, is another FQXi requirement that discourages me from participating.

So, as the bottom-line: no definitive promises, but let me see if I can write a post or a document by just straight-forwardly jotting down my answers to those detailed questions, without bothering to explain myself much, and without bothering to tie my answers together into a coherent whole.

Ok. Enough is enough. Bye for now.

[May be I will come back and add the “A Song I Like” section or so. Not sure. May be I will; may be I won’t. Bye.]

[E&OE]

 

The Other Clay Maths Problem

[Major updates to this post are now all complete. 2014.08.31 12:10 PM.]

Everybody knows about The Clay Maths Problem. There are claims, and then there are counter-claims. … First, there are some claims regarding The Clay Maths Problem. Then there are some claims going counter to them… And then, there also are claims about the claims about The Clay Maths Problem. Here is an example of a claim of the third kind.

If what Prof. Scott Aaronson often writes on his blog [^] about this issue is to be taken even semi-seriously, then he routinely receives something like [the particular estimates being mine] 2^{n^n} emails per week, all seeking his opinion about what he thinks of the 2^n arXiv article submissions per week claiming to have proved the P-vs-NP problem one way or the other, where n is a very, very large number; who knows, it might even be approaching \infty.

[Since this is a problem from theoretical computer science, for all estimates, the base has to be `2′; any thing else would be unacceptable. Aaronson is usually silent on precisely where the partition lies: whether the number of claims proving P = NP is statistically equal to those proving P != NP. However, he seems to hint that the two are equal. If so, then the CS-favorite number 2 would slip in once again, now as a divisor.]

The P-vs-NP problem is, thus, THE well-known Clay Maths Problem. Everyone knows about it.

Few people also know that there also is/was one more Clay Maths problem. … For example, they know that some decidedly crazy guy fooled them all—first, the mathematicians, and then, also himself. They—the mathematicians—accepted his solution, but he declined to accept the award, even the $1 million prize money that goes with it [^]. [Since the definition of a proper solution for the award is acceptance by mathematicians, it is easily conceivable that someone manages to fool them for two years and collects the prize.]

Relatively fewer people still know that as many as seven such million dollar problems were announced by the Clay Institute at the turn of the millennium.

As to the other problems, still fewer people ever bother to get past talking about the Yang-Mills problem. Even when it comes to this problem, as usual, they do not entertain any hope about seeing its resolution in the near future, where the “near” is left unquantified. But they all agree that it is a problem from mathematics—not physics.

What theoreticians agree on is always more interesting than what they disagree on. And, guess it was Ayn Rand who said it: also more dangerous.

Then, there are even fewer people who at all know anything about the Navier-Stokes problem—the mathematical version of it.

And, from my Web searches yesterday, there are very, very, very few people, at most a handful, who do something serious about it. This post is about them.

* * * * *  * * * * *   * * * * *

The first gentleman who purportedly continues to remain concerned about this Navier-Stokes problem is: Prof. Charles Fefferman. He should have no choice in the matter, I suppose, because it was he who wrote the official problem statement for Mr. Clay in the first place. Prof. Fefferman is an American mathematician [^].

The second man person to stay worried about this problem for as long a period as one entire month, was one maths professor from Lehigh, one Ms. Penny Smith [^]. She—an American—soon later retracted her solution [^]. Judging by the dates of the v1 and v5 versions on abstract page of her arXiv paper, the retraction took about half a month. (The number two, again!) Most of the ‘net discussions regarding her solution seem to have since then undergone a process of typographical mistake-making [^], or of plain vaporization [^]. Peter Woit, an American professor of Mathematics Physics Mathematics at Columbia, often better known on the ‘net for his proclivity to arbitrarily delete others’ replies on his blog even after having first having allowed their publication there, however, tenaciously holds on to some of that discussion [^]. (Here, I read only the first and the last still-published comments.)

The third person to stay sufficiently bothered about this problem so as to go to the extent of  writing a significant paper on it, seems to have been Prof. Mukhtarbay Otelbayev of Kazakhstan [^].

The first guy to venture discussing Otelbayev’s solution on the Mathematics StackExchange forum, chose to do so anonymously [^]. There must be something about the character of this problem that makes even people from Berlin, Germany, behave this way—writing anonymously—even on the Mathematics part of the StackExchange forum. Indeed, the question to strike this “Unknown” Berlin-based guy wasn’t the correctness or otherwise of Otelbayev’s solution; it was: Did Otelbayev solve the same problem as was posed by the Clay Institute? [^].

There were other follow-up discussions, but soon enough [within a month, of course] the author admitted that there was a mistake in his proof. The ‘net discussion on his proposal is still available [^]. [Professor Otelbayev is not an American.]

To my utter and great surprise, I also found (during an Internet search right this week) that in the meanwhile, there also has been none other than Terry Tao himself jumping into the frey issue. (No one calls him Professor Terence Tao. That’s exactly like how very, very few people, if any one, anywhere, ever, calls Prof. Aaronson by his last name and/or profession.)

My surprise was not entirely baseless. Terry Tao is a Fields medallist [^]. It was plain inconceivable that someone who already is a Fields medallist, would directly take on a(ny) million [American-] dollar problem.

That, indeed, turns out to have been the actual case. Terry Tao didn’t directly tackle the Clay Maths problem itself. See the Simons Foundation’s original coverage here [^], or the San Francisco-based Scientific American’s copy-paste job, here [^]. What Terry instead did is to pose a similar, and related, problem, and then solved it [^].

The “Unknown” Berlin-er mentioned above, was absolutely on the right track. It’s one thing to pose a problem. It’s another thing to pose it well.

That’s the light in which you might want to examine both the Clay Institute’s formulation of the problem and Tao’s recent efforts concerning it.  Realize, Terry didn’t solve the original problem. He solved another, well-posed, problem. And, as to the well-posed-ness of the original Clay Maths problem, there has been another notable effort.

Challenging the well posed-ness of the Clay Maths NS problem seems to have been the track adopted by Prof. Claes Johnson [^] for quite some time by now—several years or so.

Johnson is not a pure mathematician, but a fluid dynamicist. In fact, he is a computational fluid dynamicist, who has actually worked on some practical fluid dynamical problems [^][^]. He seems to be an interesting fellow. Despite having an h-Index of 52 [^], he has written against the climate-warming alarmism [(.pptx) ^][^]. Also see his response after having been selected for the Prandtl medal [^].

There is no Simons Foundation coverage on Johnson’s work. Naturally, any coverage by the Scientific American is plain inconceivable—especially if Johnson is going to write about his positions in springs and summers.

Johnson raises the issue of whether the Clay Maths formulation is well-posed or not. In simple words, can it at all be solved (by any one, ever) or not—whether the problem is in principle amenable to a solution or not. In case you don’t know, the “well-posed-ness” is a technical concept from mathematics [^].

Yes, that way, the issue of whether a problem is well-posed or not, does mean something like: “do you yourself know what you are asking, or not,” but the sense in which Hadamard meant it was certainly a bit more refined.

The distinction of the well-posed vs. ill-posed applies specifically to the solution of differential equations, and it means something like the following:

If you are going to throw a ball so as to hit a distant target (i.e., technically, a two-point BV problem for the second-order differential equation i.e. Newton’s 2nd law), you have the following choice: within appropriate limits, you can select the initial angle for the parabolic trajectory of the ball, in which case you have no choice about its initial speed—the horizontal distance to cover, together with the initial angle, would fix the value of the initial speed with which the ball must be thrown if it is to hit the target. (It would fix how tall the parabola should be, given the initial slope and a fixed horizontal distance.) Alternatively, you can choose the initial speed for the ball, in which case you have no longer have a choice about the initial angle. If hitting the target is your objective, you cannot arbitrarily specify both the auxiliary conditions: the initial angle and the initial speed, at the initial point. The nature of the differential equation is such that specifying both the auxiliary conditions at the same time at the same point renders this differential equation problem ill-posed. That, probably, is the simplest conceivable example of what it means for a problem to be ill-posed or well-posed.

As Hadamard pointed out, a differential equation problem, to be well-posed, must fulfill three conditions: (a) a solution must exist, (b) the solution must be unique, (c) the solution must change continuously with data (i.e., auxiliary conditions, i.e., the boundary and initial conditions).

Sometimes, the solution exists, but is not unique. For example, the diffusion equation problem is well-posed in the forward time direction but not in the reverse, in general. The diffusion process tends to smoothen out any initial sharpness. For example, if you place a drop of ink in the shape of a square on a blotting paper, it soon spreads and becomes a big, thin blot, growing ever rounder and rounder in shape as time passes by. Therefore, the information about the initial shape of the blot gets smeared all over the domain in such a way that starting from this later, bigger and roundish shape, and then going back in time following the diffusion equation, you cannot uniquely recover the initial shape of the blot. Whether the initial ink blot is square or hexagon, they both become round during diffusion. The resulting round shape doesn’t hold enough clue as to the number and locations of the sharp corners in the initial shape. In other words, the information about the initial sharpness is immediately lost during the diffusion process. And so, you can’t uniquely say whether it was a square or a hexagon: both (and infinity of other shapes) are possibilities. [As an aside, I do have some objections to this logic of the diffusion equation, but more on it, in a separate blog post, some time later.] So, the forward diffusion problem is well-posed, but the reverse one is not—no unique solution exists for the latter.

There also are other considerations for well-posed and ill-posed problems, which are more complicated. They refer to the continuous dependence of the solution to the auxiliary data. The auxiliary data, for the time-marching problems like diffusion and fluid flow, crucially means: the initial data.

Thus, the additional relevant consideration concerning the NS system has to do with the smoothness or otherwise of the initial velocity field. Johnson and a colleague rightly point in a paper [(.PDF) ^], that:

“If a vanishingly small perturbation can have a major effect on a solution, then the solution (or problem) is illposed [in the Hadamard sense], and in this case the solution may not carry any meaningful information and thus may be meaningless from both mathematical and applications points of view.

In this perspective it is remarkable that the issue of wellposedness does not appear in the formulation of the Millenium Problem. The purpose of this note is to seek an explanation of this fact, which threatens to make the problem formulation itself illposed in the sense that a resolution is either trivial or impossible.”

I agree. Do see the paper in original.

While Johnson and colleague’s technical paper may be out of the reach of many people—and in any case, at many places in the later half, it certainly is beyond my reach—the first half of the paper as well as Johnson’s blog entries are simple and clear enough to be understood even by any graduate engineer. See the series of his blog posts on this topic, here [^]. For ease of reference, the Clay Maths official problem description is here [(.PDF) ^].

[In this update of my blog post, I have edited a lot around here—from the meaning of the well-posed-ness, to what constitutes Johnson’s position. Some of my writing in the very first version of this post simply was some draft in (very rapid) progress, and so didn’t summarize Johnson’s position well. But then, I had noted that I was going to come back and edit this whole post. In particular, in the following couple of updates, I have deleted the line to the effect that Johnson meant that a blow-up won’t occur in the NS, as if this were to be his final, unqualified position, which, of course, it is not. ]

So,  in summary, Johnson has been repeatedly pointing out some important considerations regarding this problem, for a long time. A summary post could be this one, his latest [^].

And then, tarries along Terry Tao, correctly poses and seemingly correctly solves a problem that is teasingly near the original Clay Maths problem. He shows that a blow-up does occur—but only in his system, not necessarily in the NS system.

Terry, obviously, is interested in only teasing his reader, but not yet quite willing to jump into the… [Ahem.] … You see, otherwise, he could have easily dressed up the same result in different terms [say, in fuller clothes] without making any reference to the Clay Institute or its problem. But he does.

So, that’s what Terry Tao does. He wants to be seen as both addressing and not addressing the Clay Maths Problem. [He is an American.]

And, of course, even though Terry Tao responds at his blog to many, many people, he curiously doesn’t at all respond to a well-established European professor with definitely impressive credentials, like Johnson [^]. [Interestingly, though, Tao also does not delete Johnson’s replies once he publishes them. [Tao also is an Australian—he is a dual citizen.]] But the lack of response in such a context takes the matter closer to “tantalizing” than plain “teasing.” At least to someone like me, it does.

The issue is not whether the particular arguments that Prof. Johnson forwards are in themselves general or powerful enough to settle by themselves the Clay Maths Problem, or not.

The real issue is: the broader, valid and extremely relevant point regarding well-posed-ness which he repeatedly raises. Professor Tao should have responded to that. … You simply can’t go on beating [dancing] around the bush [pole], you know! [Professor Tao works in California, USA.]

* * * * *  * * * * *   * * * * *

Ok, enough of “cross-referencing.”

But if you ask me what my take on this whole issue is, let me stick my neck out and say (at least) this much:

I think that:

(i) Prof. Johnson’s broader point, regarding the well-posed-ness is absolutely relevant, and unless it is addressed in a forthright manner by the Clay Institute, this Millenium problem essentially remains muddy, and

(ii)  Prof. Tao’s line of thought—the idea of fluid logic gates and machines to be built out of the ideal fluid, and the fluid computer, etc., etc., etc.—would eventually be regarded as not at all relevant to the very core of resolving the NS issue. Tao at best clarifies some abstract part of the nature of the problem, and this part, IMO, isn’t going to be relevant—not to the NS problem itself. His best contribution concerns the energy cascade across the scales, but he offers absolutely no insight as to how that cascade might respond in the actual NS problem settings.

Let me put the second point in a round-about and enormously hand-waving kind of way: If Tao’s current work is at all relevant to the NS problem, then it means that the mathematical community’s standards are sufficiently lax that a solution to the P-vs-NP problem could also be had without the mathematical community first coming to agree with some new and explicit clarity about some of the particular details about how the continuum hypothesis is to be interpreted in that context. But the second is not going to happen, simply because it is THE Clay Maths Problem. Ergo, Tao’s current work… QED.

On the other hand, to repeat, the points that Johnson raises are relevant, and they will stay relevant.

And, of course, all that is quite apart from another, “related” issue: even if there is or isn’t a blow-up in the NS system (and even if the Clay prize gets awarded for “proving” it either way (i.e. for someone getting the mathematicians to agree with him for two consecutive years)), the real issue would still remain: whether the Navier-Stokes system happens to be a good model for real fluids or not.

Just for the starters, as every one knows, at appropriate Knudsen numbers, the no-slip condition is no good. If you are going to complicate the auxiliary conditions to such an extent that the complexity of their effects exceeds that of the basic governing differential equation, then, you sure gotta ask of what good use your basic differential equation is, in the first place. In physics and engineering, we adopt the differential equation paradigm only because it has an epistemological value: it helps reduce cognitive load. If the bells and whistles are going to weigh a ton, how do you expect a cart powered by a toy-spring (or a 12 V 0.4 A Watt stepper motor) to get off to any start?

Or, for another matter, if what happens at the small, local, scale is extraordinarily different from what happens at the large, global, scale, or if pathological connections exists between the two scales, then, you have gotta ask some time: Why elevate some scale-based parameters that only give kicks to the mathematicians but does not make life any simpler to anyone else? Why stick to this continuum-based description for all the scales, in the first place? After all, even simplest phenomena like droplet formation or coalescence are in any case beyond the reach of this “basic,” “fundamental,” etc. etc. Navier-Stokes formalism anyway. Why elevate a theory of such obvious flaws to such a high pedestal? Why keep such a narrow mind that it can’t even deal with some of the simplest phenomena of nature? And, if you must do that, then why not take just one more step and declare another million dollar prize for some latest parlour game? What ultimately does distinguish mathematics as we know it, from the parlour games?

But, of course, we need not go all that far, really speaking. In many ways, the NS system is of enormous practical importance. All that we need to do is to bring some astute observations regarding differential equations, into the problem formulation.

Absent that one, no one has proved either a blow-up, or its absence, in the NS system, thus far. Not even Terry Tao. And, he must know that that’s because of the ill-posed-ness of the problem formulation itself. That’s why he must be choosing to remain silent to Johnson’s query, here [^]. Remember, he both works and does not work on the Clay Maths problem?

The current situation, in many, many ways, is something like this:

Suppose that mathematicians are busy building castles out of thin air (say, debating endlessly about whether the d’Almbert paradox is mathematically consistent or not). Suppose that the working epistemology of the culture has reached such a low level that no one can tell if the theories of thermodynamics and EM are consistent within themselves, let alone with each other. So, a rich guy steps in and declares a big mathematical prize for someone proving whether the Rayleigh-Jean blow-up really follows from the Maxwell system or not. Predictably, there is a flurry of activity, and thus there follow a few mathematicians who can’t get it right. And then, there steps in a young, brilliant mathematician, and declares that he can prove that a blow-up cannot occur but that his proof is limited only to a subcategory of the classical EM fields: the ones that are averaged in some sense. And then comes along some industrial physicist (i.e. actually a theoretical engineer, who has got a job declaring himself an applied mathematician). The industrial physicist points out that the mathematician’s argument can also be taken to imply the exactly opposite conclusion: namely, that these average fields must necessarily lead to the ultraviolet catastrophe. The mathematician chooses not to respond. The audience claps for the mathematician, and falls dead silent for the industrial physicist.

And, no one thinks of instituting a new prize that would reward such efforts as of building a new theory of physics which shows how to prevent the blow-up, even if doing so would involve breaking away from the clutches of some deeply held assumptions about the physical nature of reality, even if such a break-away is only a desperate last measure.

Sure, the situation is not exactly analogous. But if you go through the history of QM, and see how no physicist (or mathematician) ever left others’ valid queries unanswered—if you see how, on the contrary, they rapidly and openly communicated if not collaborated with each other—and then, if you see the kind of hype and blogging practices currently going on in our times, you will begin to see some pattern, and if not that, then at least some semblance, somewhere.

And, you will dearly feel something like—what? wistfulness?—about the good old times now so distant from ours: the times when the rational culture of science had already taken strong roots and it still was mostly a free, application-driven enterprise (the studies of cavity radiation, leading to the first physics Nobel, were sponsored precisely so as to help produce brighter bulbs more cheaply, for better business profits); the times when the state control of science was barely in its nascent stages or altogether absent. (Check out the history of Income Tax on Google, for instance. And, remember, that one—the Income Tax—is only for the starters, as far as the means of the state control of science goes.)

Guess I have made the most important two points which I had.

[Guess my major editing is over, except perhaps for a typo here and there. This long week-end for the Ganesh Chaturthee ends today, and from tomorrow, I will be back to my heavy class-room teaching duties—i.e. away from blogging. (In fact, factoring in the preparations for lectures, I would be into my heavy teaching duty starting right this afternoon.) So, bye for now.]

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A Song I Like:
(Hindi) “purvaiyya leke chali…”
Music: Ravindra Jain
Singers: Lata Mangeshkar and Shailendra Singh
Lyrics: ? (Ravindra Jain?)

[E&OE]