Would it happen to me, too? …Also, other interesting stories / links

1. Would it happen to me, too?

“My Grandfather Thought He Solved a Cosmic Mystery,”

reports Veronique Greenwood for The Atlantic [^] [h/t the CalTech physicist Sean Carroll’s twitter feed]. The story has the subtitle:

“His career as an eminent physicist was derailed by an obsession. Was he a genius or a crackpot?”

If you visit the URL for this story, the actual HTML page which loads into your browser has another title, similar to the one above:

“Science Is Full of Mavericks Like My Grandfather. But Was His Physics Theory Right?”

Hmmm…. I immediately got interested. After all, I do work also on foundations of quantum mechanics. … “Will it happpen to me, too?” I thought.

At this point, you should really go through Greenwood’s article, and continue reading here only after you have finished reading it.


Any one who has worked on any conceptually new approach would find something in Greenwood’s article that resonates with him.

As to me, well, right at the time that attempts were being made to find examiners for my PhD, my guide (and even I) had heard a lot of people say very similar things as Greenwood now reports: “I don’t understand what you are saying, so please excuse me.” This, when I thought that my argument should be accessible even to an undergraduate in engineering!

And now that I continue working on the foundations of QM, having developed a further, completely new (and more comprehensive) approach, naturally, Greenwood’s article got me thinking: “Would it happen to me, too? Once again? What if it does?”


…Naah, it wouldn’t happen to me—that was my conclusion. Not even if I continue talking about, you know, QM!


But why wouldn’t something similar happen to me? Especially given the fact that a good part of it has already happened to me in the past?

The reason, in essence, is simple.

I am not just a physicist—not primarily, anyway. I am primarily an engineer, a computational modeller. That’s why, things are going to work out in a different way for me.

As to my past experience: Well, I still earned my PhD degree. And with it, the most critical part of the battle is already behind me. There is a lot of resistance to your acceptance before you have a PhD. Things do become a lot easier once you have gone successfully past it. That’s another reason why things are going to work out in a different way now. … Let me explain in detail.


I mean to say, suppose that I have a brand-new approach for resolving all the essential quantum mechanical riddles. [I think I actually do!]

Suppose that I try to arrange for a seminar to be delivered by me to a few physics professors and students, say at an IIT, IISER, or so. [I actually did!]

Suppose that they don’t respond very favorably or very enthusiastically. Suppose they are outright skeptical when I say that in principle, it is possible to think of a classical mechanically functioning analog simulator which essentially exhibits all the essential quantum mechanical features. Suppose that they get stuck right at that point—may be because they honestly and sincerely believe that no classical system can ever simulate the very quantum-ness of QM. And so, short of calling me a crack-pot or so, they just directly (almost sternly) issue the warning that there are a lot of arguments against a classical system reproducing the quantum features. [That’s what has actually happened; that’s what one of the physics professors I contacted wrote back to me.]

Suppose, then, that I send an abstract to an international conference or so. [This too has actually happend, too, recently.]

Suppose that, in the near future, the conference organizers too decline my submission. [In actual reality, I still don’t know anything about the status of my submission. It was in my routine searches that I came across this conference, and noticed that I did have about 4–5 hours’ time to meet the abstracts submissions deadline. I managed to submit my abstract within time. But since then, the conference Web site has not got updated. There is no indication from the organizers as to when the acceptance or rejection of the submitted abstracts would be communicated to the authors. An enquiry email I wrote to the organizers has gone unanswered for more than a week by now. Thus, the matter is still open. But, just for the sake of the argument, suppose that they end up rejecting my abstract. Suppose that’s what actually happens.]

So what?

Since I am not a physicist “proper”, it wouldn’t affect me the way it might have, if I were to be one.

… And, that way, I could even say that I am far too smart to let something like that (I mean some deep disappointment or something like that) happen to me! … No, seriously! Let me show you how.

Suppose that the abstract I sent to an upcoming conference was written in theoretical/conceptual terms. [In actual reality, it was.]

Suppose now that it therefore gets rejected.

So what?

I would simply build a computational model based on my ideas. … Here, remember, I have already begun “talking things” about it [^]. No one has come up with a strong objection so far. (May be because they know the sort of a guy I am.)

So, if my proposed abstract gets rejected, what I would do is to simply go ahead and perform a computer simulation of a classical system of this sort (one which, in turn, simulates the QM phenomena). I might even publish a paper or two about it—putting the whole thing in purely classical terms, so that I manage to get it published. (Before doing that, I might even discuss the technical issues involved on blogs, possibly even at iMechanica!)

After such a paper (ostensibly only on the classical mechanics) gets accepted and published, I will simply write a blog post, either here or at iMechanica, noting how that system actually simulates the so-and-so quantum mechanical feature. … Then, I would perform another simulation—say using DFT. (And it is mainly for DFT that I would need help from iMechanicians or so.) After it too gets accepted and published, I will write yet another blog post, explaining how it does show some quantum mechanical-ness. … Who knows such a sequence could continue…

But such a series (of the simulations) wouldn’t be very long, either! The thing is this.

If your idea does indeed simplify certain matters, then you don’t have to argue a lot about it—people can see its truth real fast. Especially if it has to do with “hard” sciences like engineering—even physics!

If your basic idea itself isn’t so good, then, putting it in the engineering terms makes it more likely that even if you fail to get the weakness of your theory, someone else would. All in all, well and good for you.

As to the other possibility, namely, if your idea is good, but, despite putting it in the simpler terms (say in engineering or simulation terms), people still fail to see it, then, well, so long as your job (or money-making potential) itself is not endangered, then I think that it is a good policy to leave the mankind to its own follies. It is not your job to save the world, said Ayn Rand. Here, I believe her. (In fact, I believed in this insight even before I had ever run into Ayn Rand.)


As to the philosophic issues such as those involved in the foundations of QM—well, these are best tackled philosophically, not physics-wise. I wouldn’t use a physics-based argument to take a philosophic argument forward. Neither would I use a philosophical argument to take a physics-argument forward. The concerns and the methods of each are distinctly different, I have come to learn over a period of years.

Yes, you can use a physics situation as being illustrative of a philosophic point. But an illustration is not an argument; it is merely a device to make understanding easier. Similarly, you could try to invoke a philosophic point (say an epistemological point) to win a physics-based argument. But your effort would be futile. Philosophic ideas are so abstract that they can often be made to fit several different, competing, physics-related arguments. I would try to avoid both these errors.

But yes, as a matter of fact, certain issues that can only be described as philosophic ones, do happen to get involved when it comes to the area of the foundations of QM.

Now, here, given the nature of philosophy, and of its typical practitioners today (including those physicists who do dabble in philosophy), even if I become satisfied that I have resolved all the essential QM riddles, I still wouldn’t expect these philosophers to accept my ideas—not immediately anyway. In fact, as I anticipate things, philosophers, taken as a group, would never come to accept my position, I think. Such an happenstance is not necessarily to be ascribed to the personal failings of the individual philosophers (even if a lot of them actually do happen to be world-class stupid). That’s just how philosophy (as a discipline of studies) itself is like. A philosophy is a comprehensive view of existence—whether realistic or otherwise. That’s why it’s futile to expect that all of the philosophers would come to agree with you!

But yes, I would expect them to get the essence of my argument. And, many of them would, actually, get my argument, its logic—this part, I am quite sure of. But just the fact that they do understand my argument would not necessarily lead them to accept my positions, especially the idea that all the QM riddles are thereby resolved. That’s what I think.


Similarly, there also are a lot of mathematicians who dabble in the area of foundations of QM. What I said for philosophers also applies more or less equally well to them. They too would get my ideas immediately. But they too wouldn’t, therefore, come to accept my positions. Not immediately anyway. And in all probability, never ever in my lifetime or theirs.


So, there. Since I don’t expect an overwhelming acceptance of my ideas in the first place, there isn’t going to be any great disappointment either. The very expectations do differ.

Further, I must say this: I would never ever be able to rely on a purely abstract argument. That would feel like too dicey or flimsy to me. I would have to offer my arguments in terms of physically existing things, even if of a brand new kind. And, machines built out of them. At least, some working simulations. I would have to have these. I would not be able to rest on an abstract argument alone. To be satisfactory to me, I would have to actually build a machine—a soft machine—that works. And, doing just this part itself is going to be far more than enough to keep me happy. They don’t have to accept the conceptual arguments or the theory that goes with the design of such (soft) machines. It is enough that I play with my toys. And that’s another reason why I am not likely to derive a very deep sense of disenchantment or disappointment.


But if you ask me, the way I really, really like think about it is this:

If they decline my submission to the conference, I will write a paper about it, and send it, may be, to Sean Carroll or Sabine Hosenfelder or so. … The way I imagine things, he is then going to immediately translate my paper into German, add his own name to ensure its timely publication, and … . OK, you get the idea.

[In the interests of making this post completely idiot-proof, let me add: Here, in this sub-section, I was just kidding.]


2. The problem with the Many Worlds:

“Why the Many Worlds interpretation has many problems.”

Philip Ball argues in an article for the Quanta Mag [^] to the effect that many worlds means no world at all.

No, this is not exactly what he says. But what he says is clear enough that it is this conclusion which becomes inescapable.

As to what he actually says: Well, here is a passage, for instance:

“My own view is that the problems with the MWI are overwhelming—not because they show it must be wrong, but because they render it incoherent. It simply cannot be articulated meaningfully.”

In other words, Ball’s actual position is on the epistemic side, not on the ontic. However, his arguments are clear enough (and they often enough touch on issues that are fundamental enough) that the ontological implications of what he actually says, also become inescapable. OK, sometimes, the article unnecessarily takes detours into non-essentials, even into something like polemics. Still, overall, the write up is very good. Recommended very strongly.

Homework for you: If the Many Worlds idea is that bad, then explain why it might be that many otherwise reasonable people (for instance, Sean Carroll) do find the Many Worlds approach attractive. [No cheating. Think on your own and write. But if cheating is what you must do, then check out my past comment at some blog—I no longer remember where I wrote it, but probably it was on Roger Schlafly’s blog. My comment had tackled precisely this latter issue, in essential terms. Hints for your search: My comment had spoken about data structures like call-stacks and trees, and their unfolding.]


3. QM as an embarrassment to science:

“Why quantum mechanics is an “embarrassment” to science”

Brad Plumer in his brief note at the Washington Post [^] provides a link to a video by Sean Carroll.

Carroll is an effective communicator.

[Yes, he is the same one who I imagine is going to translate my article into German and… [Once again, to make this post idiot-proof: I was just kidding.]]


4. Growing younger…

I happened to take up a re-reading of David Ruelle’s book: “Chance and Chaos”. The last time I read it was in the early 1990s.

I felt younger! … May be if something strikes me while I am going through it after a gap of decades, I will come back and note it here.


5. Good introductory resources on nonlinear dynamics, catastrophe theory, and chaos theory:

If you are interested in the area of nonlinear dynamics, catastrophe theory and chaos theory, here are a few great resources:

  • For a long time, the best introduction to the topic was a brief write-up by Prof. Harrison of UToronto; it still remains one of the best [^].
  • Prof. Zeeman’s 1976 article for SciAm on the catastrophe theory is a classic. Prof. Zhigang Suo (of Harvard) has written a blog post of title “Recipe for catastrophe”at iMechanica [^], in which he helpfully provides a copy of Zeeman’s article. I have strongly recommended Zeeman’s write-up before, and I strongly recommend it once again. Go through it even if only to learn how to write for the layman and still not lose precision or quality.
  • As to a more recent introductory expositions, do see Prof. Geoff Boeing’s blog post: “Chaos theory and the logistic map” [^]. Boeing is a professor of urban planning, and not of engineering, physics, CS, or maths. But it is he who gives the clearest idea about the distinction between randomness and chaos that I have ever run into. (However, I only later gathered that he does have a UG in CS, and a PG in Information Management.) Easy to understand. Well ordered. Overall, very highly recommended.

Apart from it all:

Happy Diwali!


A song I like:

(Hindi) “tere humsafar geet hai tere…”
Music: R. D. Burman
Singers: Kishore Kumar, Mukesh, Asha Bhosale
Lyrics: Majrooh Sultanpuri

[Has this song been lifted from some Western song? At least inspired from one?

Here are the reasons for this suspicion: (1) It has a Western-sounding tune. It doesn’t sound Indian. There is no obvious basis either in the “raag-daari,” or in the Indian folk music. (ii) There are (beautiful) changes in the chords here. But there is no concept of chords in the traditional Indian music—basically, there is no concept of harmony in it, only of melody. (iii) Presence of “yoddling” (if that’s the right word for it). That too, by a female singer. That too, in the early 1970’s! Despite all  the “taan”s and “firat”s and all that, this sort of a thing (let’s call it yoddling) has never been a part of the traditional Indian music.

Chances are good that some of the notes were (perhaps very subconsciously) inspired from a Western tune. For instance, I can faintly hear “jingle bells” in the refrain. … But the question is: is there a more direct correspondence to a Western tune, or not.

And, if it was not lifted or inspired from a Western song, then it’s nothing but a work of an absolute genius. RD anyway was one—whether this particular song was inspired from some other song, or not.

But yes, I liked this song a great deal as a school-boy. It happened to strike me once again only recently (within the last couple of weeks or so). I found that I still love it just as much, if not more.]


[As usual, may be I will come back tomorrow or so, and edit/streamline this post a bit. One update done on 2018.11.04 08:26 IST. A second update done on 2018.11.04 21:01 IST. I will now leave this post in whatever shape it is in. Got to move on to trying out a few things in Python and all. Will keep you informed, probably after Diwali. In the meanwhile, take care and bye for now…]

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Just a blog-filler

Am travelling…

… May be I should’ve had a bit to report to you by now, but then, no, no such beans—well, at least not by now.

Anyway, as I said, I am travelling. And, also have not been keeping exactly very well. (But this was the case even before this short travel began.)

Also, even after returning back home, there would be a stack of other things that have to be taken care of (unrelated to ANNs, QM, etc.). So, won’t be blogging for a while. May be should come back in first week November.

In the meanwhile, you enjoy this song. But if I’ve run it already, then sure do let me know; I will play another one in its place immediately.

…Take care, and bye for now.


A song I like:

(Hindi) “aanchal mein kyaa jee”
Singers: Kishore Kumar, Asha Bhosale
Music: S. D. Burman
Lyrics: Majrooh Sultanpuri

[Two notes: (i) The video is recommended, in particular, this one [^]. Don’t miss the beginning (which was the reason it was this particular video which was taken up for recommendation). (ii) Here, it’s the girl who gives the initial cat-call; the guy follows. (iii) Guess they were already married by the time this song was shot.]

[PS: (iii) As usual, a further extra: Have loved this song for decades, but today happened to be the very first time in my life that I watched the visual of this song. (No, I had somehow had missed the movie back then when I used to watch them.)]

 

 

Back on the ‘net!

Hushshshsh… Finally I am back on the ‘net (I mean to say, in a real way—not via smartphone). But it’s not after having the broken laptop repaired. On the contrary, it seems as if it’s no longer viable to revive my broken laptop.

So, right now, I am writing this post using an even older laptop that I had. I dug it up from the cupboard, and revived it.

Actually, you can’t call it a laptop; even the manufacturer called it a notebook (the 2008 model Compaq Presario C700, 32-bit Intel Core Duo @ 1.83 GHz, 1 GB RAM). I used to use about a decade ago. All my PhD time programs and data were sitting on it. (The thesis had been written and submitted even earlier, even before I bought this notebook; it was written on a desktop I had back then. However, after submitting thesis, it took them 2 years to arrange for the defence. That’s how, it was on this machine that I had prepared my final defence slides.)

It was a dual boot machine, with one partition running XP, and the other, Vista. I had forgotten the password of the Vista, but fortunately, not of the XP.

The trouble with XP was that both the Mozilla and Internet Explorer installed on it had already become absolutely obsolete. Also Java. (The support for these software had vanished by 2014, I now gathered.) The browsers were so old that they couldn’t handle even simplest of today’s https requests—the SSL layer itself must have been too old. So, I couldn’t have used that OS even for just surfing on the ‘net.

So, what I did over the past couple of days was to first take a backup of my PhD-time data. Then, I proceeded to reformat the whole hard disk, and installed Lubuntu 18.04.1 on it.

Yes, Lubuntu does manage to run even on a 32-bit 1 GB machine. However, even with this comparatively light-weight OS, there is enormous disk-thrashing, particularly if I try to use a programming IDE. (After bootup, the OS by itself eats up something like 600–700 MB of RAM.) So far, I’ve tried installing and using PyCharm, Spyder and VSCode. They all do run, but very, very slowly. Sometimes, you have to even wait for a minute or so just for a context-switch between two processes (say, the browser and the IDE).

So, looks like despite all my valiant tries, this machine isn’t going to be useful for my ANN studies; it would be usable only for browsing. … May be I should borrow money and buy a new laptop….

But one way or the other, this decade-old machine still is better, much much better, than my new (late-2017 times) smart-phone. … As I said recently, the smart-phone is a bad idea. …

… Anyway, now that I am on the ‘net (can use a real keyboard), I should be back pretty soon, say tomorrow or the day after, with something which is much more exciting. That’s a promise. So, bye for now, but stay tuned.


A song I like:
(Marathi) “ekaach yaa janmi jaNu…”
Music: Sudhir Phadke
Lyrics: Sudhir Moghe
Singer: Asha Bhosale

[Usually, when you say “song,” what you usually mean is the basic tune. OK, sometimes, first the words and then the tune. However, this song is odd.

The real beauty of this song lies not in any one of its elements but in the way it unfolds—the way the music composer leads you through an interplay between various musical phrases and the lyrical ones. Especially noteworthy are the violin pieces which transition you from the stanza to the refrain and back. Also noteworthy is the interplay between the Western and the Indian instruments…

So, what rules here is not just the tune, not just the words, not just the orchestration, and not just the rendering in the voice by the singer. Instead, it is the skillfully arranged interplay between all these elements which truly gives the defining character to this song and makes it so beautiful.

So, it’s the music composer who really stands out here, even if the entire team is outstanding… Or so I believe. … Anyway, bye for now. ]

[May be a little streamlining, later on.]

The quantum mechanical features of my laptop…

My laptop has developed certain quantum mechanical features after its recent repairs [^]. In particular, if I press the “power on” button, it does not always get “measured” into the “power-on” state.

That’s right. In starting the machine, it is not possible to predict when the power-on button may work, when it may lead to an actual boot-up. Sometimes it does, sometimes it doesn’t.

For instance, the last time I shut it down was on the last night, just before dinner. Then, after dinner, when I tried to restart it, the quantum mechanical features kicked in and the associated randomness was such that it simply refused the request. Ditto, this morning. Ditto, early afternoon today. But now (at around 18:00 hrs on 09 October), it somehow got up and going!


Fortunately, I have taken backup of crucial data (though not all). So, I can afford to look at it with a sense of humour.

But still, if I don’t come back for a somewhat longer period of time than is usual (about 8–10 days), then know that, in all probability, I was just waiting helplessly in getting this thing repaired, once again. (I plan to take it to the repairsman tomorrow morning.) …

…The real bad part isn’t this forced break in browsing or blogging. The real bad part is: my inability to continue with my ANN studies. It’s not possible to maintain any tempo in studies in this now-on-now-off sort of a manner—i.e., when the latter is not chosen by you.

Yes, I do like browsing, but once I get into the mood of studying a new topic (and, BTW, just reading through pop-sci articles does not count as studies) and especially if the studies also involve programming, then having these forced breaks is really bad. …

Anyway, bye for now, and take care.


PS: I added that note on browsing and then it struck me. Check out a few resources while I am gone and following up with the laptop repairs (and no links because right while writing this postscript, the machine crashed, and so I am somehow completing it using smartphone—I hate this stuff, I mean typing using at most two fingers, modtly just one):

  1. As to Frauchiger and Renner’s controversial much-discussed result, Chris Lee’s account at ArsTechnica is the simplest to follow. Go through it before any other sources/commentaries, whether to the version published recently in Nature Comm. or the earlier ones, since 2016.
  2. Carver Mead’s interview in the American Spectator makes for an interesting read even after almost two decades.
  3. Vinod Khosla’s prediction in 2017 that AI will make radiologists obsolete in 5 years’ time. One year is down already. And that way, the first time he made remarks to that sort of an effect were some 6+ years ago, in 2012!
  4. As to AI’s actual status today, see the Quanta Magazine article: “Machine learning confronts the elephant in the room” by Kevin Hartnett. Both funny and illuminating (esp. if you have some idea about how ML works).
  5. And, finally, a pretty interesting coverage of something about which I didn’t have any idea beforehand whatsoever: “New AI strategy mimics how brains learn to smell” by Jordana Cepelwicz in Quanta Mag.

Ok. Bye, really, for now. See you after the laptop begins working.


A Song I Like:
Indian, instrumental: Theme song of “Malgudi Days”
Music: L. Vaidyanathan

 

 

Suspension of blogging

Earlier, within a day of my posting the last blog entry here, i.e. right by 26th September morning, my laptop developed a problem, which led to a series of problems, which meant that, for a while, I could not at all blog or even surf on the ‘net effectively.

The smartphone screen is too small for me to do any serious browsing very effectively, let alone doing any blogging / writing / coding.


Never did buy into that idiotic Steve Jobs’ ridiculous claims anyway; bought my smartphone only because it’s good for things like storing phone numbers and listening to songs—and, yes, also for browsing a bit on google maps, and for taking snaps once in a while. But that’s about it. Nothing more than that. In particular, no social media, no banking, no e-payments, no emails, no real browsing. And, as to that prized (actually wretched) thin-ness and/or the delicate-ness of this goddamn thing. It is annoying. Just hold the damn thing in your palm, and it seems as if it itself auto-punches a few buttons and proceeds to close all the windows you had kept active. Or, worse: it launches a new window all by itself, forcing you to take a hike into an ad-link or sundry news item.

A good 1 inch thick and sturdy instrument with goodly big buttons would have been a better design choice, far better—not those bloody thin slivers on the sides which pass for buttons.


Anyway, the troubles with the laptop were these:

(i) In 2014, the screen panel of the laptop had cracked near a corner a bit, and then, subsequently, over a period of years or so, the front and the back covering parts of the screen panel had come to split apart, though only just slightly, only partially. I had shown the problem to the authorized dealer. He had advised me to do nothing about it. (If the problem were to be worse, he would have advised me to replace the screen, he had said. This was about 2 years ago.)

(ii) Then, slowly, friction began developing in only one of the hinges of the screen panel, the hinge near the same (cracked) corner. Finally, came this day when this partial splitting suddenly led to the panel-studs breaking apart (with a clean, brittle fracture). How did it happen? Because—I figured out only after the fact—the friction in the hinges together with the partial split up meant that an interior part of the screen panel was getting excessively bent, near the broken panel corner. This excessive bending was putting enormous bending moment on the studs holding the two parts of the partially split up panel near the hinges. (The overall frame of the screen panel was effectively acting as a large lever arm serving to bend the small plastic studs.)

(iii) In getting the above-mentioned problem fixed (by 29th September), some short-circuiting also occurred, with the result that now the graphics chip conked up. (No, the authorized dealer didn’t accept the machine. He advised replacement of both the mother-board and the screen. So, I did a google search and went through two private repairs-men, one of them being much better than the other. He fixed it right.)

Fixing the graphics problem took time because a replacement chip was not readily available in the local market, and there was a national holiday in between (on 2nd October) which kept the concerned courier services closed on that day.

(iv) Then, after replacing the graphics chip, once the screen finally started working, now it was the turn of the USB ports to begin malfunctioning. I got the delivery of my laptop last evening, and noticed it only after coming home.

This is a problem which has not yet been fixed. Getting it fixed is important because only 1 out of the 3 USB ports is currently functioning, and if it too is gone, backups will become impossible. I am not willing to lose my data once again.


The problem with the machine meant that my studies (and programming) of ANNs too got interrupted.

They still remain interrupted.

I guess the remaining problem (regarding the malfunctioning USB ports) is relatively a minor issue.

What I mean to say is that I could have resumed my regular sort of blogging.

However, last night, at around 00:40 hrs IST on 05 October 2018 there was a psychic attack on me which woke me up from sleep. (Also note the update in my last post). In view of this attack, I have finally decided to say it clear and loud, (perhaps once again):

“To hell with you, LA!”


If you wonder why I was so confident about “LA,” check out the visits pattern for the earlier part of the day yesterday, and juxtapose them with the usual patterns of visits here, overall.

In case you don’t know, all local newspapers of all California towns have been full of advertisements for psychic “consultants” providing their “services” for a fee—which would be almost nothing when measured in US dollars.


I have had enough of these bitches and bastards. That’s why, I am temporarily suspending my blog. When the psychic attacks come to a definitive stop, I will resume my own blogging, as also my commenting on other blogs, and posting any research notes etc.


And, yes, one more point: No, don’t believe what Ayn Rand Institute tells you. Psychic attacks are for real (though they are much, much rarer, and they indeed are effected in far more controlled ways, than what folklore or your average street-side vendor of the “services” says.)


No songs section this time round, for obvious reasons.


PS: BTW, no, I still haven’t seen my approach to QM mentioned in any of the papers / books, or in any discussions of any papers anywhere (including some widely followed blogs / twitter feeds), as yet. Apparently, my judgment that my approach is indeed new, continues to hold.

 

 

Hushshshsh…

The title word of this post is Marathi for “phew!”—not for “hush hush.” (But, to me, the Marthi word is more expressive. [BTW, the “hu” is to be pronounced as “hoo”.])

The reason for this somewhat accentuated and prolonged exhalation is this: I am done with the version “0.1-beta” of my note on flux (see my last post). I need a break.

As of now, this note is about 27 pages, and with figures and some further (< 5%) additions, the final number of pages for the version 0.1 should easily go into the early 30s. … To be readable, it will have to brought down to about 15 pages or fewer, including diagrams. Preferably, < 10 pages. Some other day. For now, I find that I have grown plain sick and tired of working on that topic. I need to get away from it all. I am sure that I will return to it later on—may be after a month or so. But for the time being, I simply need a break—from it. I’ve had enough of this flux and vectors and tensors thing. … Which brings me to the next topic of this post.

There are also other announcements.


I think that I have also had enough of QM.

QM was interesting, very interesting, to me. It remained that way for some four decades. But now that I have cracked it (to my satisfaction), my interest in the topic has begun dwindling down very rapidly.

Sure I will conduct a few simulations, deliver the proposed seminar(s) I had mentioned in the past, and also write a paper or two about it. But I anticipate that I won’t go much farther than what I have already understood. The topic, now, simply has ceased to remain all that interesting.

But, yes, I have addressed all the essential QM riddles. That’s for certain.


And then, I was taking a stock of my current situation, and here are a few things that stood out:

  • I am not getting an academic job in Pune (because of the stupid / evil SPPU rules), and frankly, the time when a (full) Professor’s job could have meant something to me is already over. If I were to get such a job well in time—which means years ago—then I could have done some engineering research (especially in CFD), guided a few students (in general in computational science and engineering), taught courses, developed notes, etc. But after having lost a decade or so due to that stupid and/or evil Metallurgy-vs-Mechanical Branch Jumping issue, I don’t have the time to pursue all that sort of a thing, any more.
  • You would know this: All my savings are over; I am already in debts.
  • I do not wish to get into a typical IT job. It could be well paying, but it involves absolutely no creativity and originality—I mean creativity involving theoretical aspects. Deep down in my heart, I remain a “theoretician”—and a programmer. But not a manager. There is some scope for creativity in the Indian IT industry, but at my “seniority,” it is mostly limited, in one way or the other, only to “herd-management” (to use an expression I have often heard from my friends in the industry). And, I am least bothered about that. So, to say that by entering the typical Indian IT job, my best skills (even “gifts”) would go under-utilized, would be an understatement.
  • For someone like me, there is no more scope, in Pune, in the CFD field either. Consultants and others are already well established. I could keep my eyes and ears open. But it looks dicey to rely on this option. The best period for launching industrial careers in CFD here was, say, up to the early naughties. … I still could continue with some research in CFD. But guess it no longer is a viable career option for me. Not in Pune.
  • Etc.

However, all is not gloomy. Not at all. Au contraire.

I am excited that I am now entering a new field.

I will not ask you to take a guess. This career route for people with my background and skills is so well-established by now, that there aren’t any more surprises left in it. Even an ANN would be able to guess it right.

Yes, that’s right. From now on, I am going to pursue Data Science.


This field—Data Science—has a lot of attractive features, as far as I am concerned. The way I see it, the following two stand out:

  1. There is a very wide variety of application contexts; and
  2. There is a fairly wide range of mathematical skills that you have to bring to bear on these problems.

Notice, the emphasis is on the width, not on the depth.

The above-mentioned two features, in turn, lead to or help explain many other features, like:

  1. A certain open ended-ness of solutions—pretty much like what you have in engineering research and design. In particular, one size doesn’t fit all.
  2. A relatively higher premium on the individual thinking skills—unlike what your run-of-the-mill BE in CS does, these days [^].

Yes, Data Science, as a field, will come to mature, too. The first sign that it is reaching the first stage of maturity would be an appearance of a book like “Design Patterns.”

However, even this first stage is, I anticipate, distant in future. All in all, I anticipate that the field will not come to mature before some 7–10 years pass by. And that’s long enough a time for me to find some steady career option in the meanwhile.


There are also some other plus points that this field holds from my point of view.

I have coded extensively—more than 1 lakh (100,000) lines of C++ code in all, before I came to stop using C++, which happened especially after I entered academia. I am already well familiar with Python and its eco-system, though am not an expert when it comes to writing the fastest possible numerical code in Python.

I have handled a great variety of maths. The list of equations mentioned in my recent post [^] is not even nearly exhaustive. (For instance, it does not even mention whole topics like probability and statistics, stereology, and many such matters.) When it comes to Data Science, a prior experience with a wide variety of maths is a plus point.

I have not directly worked with topics like artificial neural networks, deep learning, the more advanced regression analysis, etc.

However, realize that for someone like me, i.e., someone who taught FEM, and had thought of accelerating solvers via stochastic means, the topic of constrained optimization would not be an entirely unknown animal. Some acquaintance has already been made with the conjugate gradient (though I can’t claim mastery of it). Martingales theory—the basic idea—is not a complete unknown. (I had mentioned a basic comparison of my approach vis-a-vis the simplest or the most basic martingales theory, in my PhD thesis.)

Other minor points are these. This field also (i) involves visualization abilities, (ii) encourages good model building at the right level of abstraction, and (iii) places premium on presentation. I am not sure if I am good on the third count, but I sure am confident that I do pretty well on the first two. The roots of all my new research ideas, in fact, can be traced back to having to understand physical principles in 3+1 D settings.


Conclusion 1: I should be very much comfortable with Data Science. (Not sure if Data Science itself (i.e., Data Scientists themselves) would be comfortable with me or not. But that’s something I could deal later on.)

Conclusion 2: Expect blogging here going towards Data Science in the future.


A Song I Like:

(Marathi) “uff, teri adaa…”
Music: Shankar-Ahsaan-Loy
Singer: Shankar Mahadevan
Lyrics: Javed Akhtar

[By any chance, was this tune at least inspired (if not plagiarized) from some Western song? Or is it through and through original? …In any case, I like it a lot. I find it wonderful. It’s upbeat, but not at all banging on the ears. (For contrast, attend any Ganapati procession, whether on the first day, the last day, or any other day in between. You will have ample opportunities to know what having your ears banged out to deafness means. Nay, these opportunities will be thrust upon you, whether you like it or not. It’s our “culture.”)]

 

 

Where are those other equations?

Multiple header images, and the problem with them:

As noted in my last post, I have made quite a few changes to the layout of this blog, including adding a “Less transient” page [^].

Another important change was that now, there were header images too, at the top.

Actually, initially, there was only one image. For the record, it was this: [^] However, there weren’t enough equations in it. So, I made another image. It was this [^]. But as I had already noted in the last post, this image was already crowded, and even then, it left out some other equations that I wanted to include.

Then, knowing that WordPress allows multiple images that can be shown at random, I created three images, and uploaded them. These are what is being displayed currently.

However, randomizing means that even after re-loading a page a couple of times, there still is a good chance that you will miss some or the other image, out of those three.

Ummm… OK.


A quick question:

Here is the problem statement:

There are three different header images for this blog. The server shows you only one of them during a single visit. Refreshing the page in the browser also counts as a separate visit. In each visit, the server will once again select an image completely at random.

Assume also that the PDF for the random sequence is uniform. That is to say, there is no greater probability for any of the three images during any visit. Cookies, e.g., play no role.

Now, suppose you make only three visits to this blog. For instance, suppose you visit some page on this blog, and then refresh the same page twice in the browser. The problem is to estimate the chances that you will get to see:

  • all of the three different images, but in only three visits
  • one and the same image, each time, during exactly three visits
  • exactly two different images, during exactly three visits

Don’t read further until you solve this problem, right now: right on-the-fly and right in your head (i.e. without using paper and pencil).

(Hint [LOL!]: There are three balls of different colors (say Red, Green, and Blue) in a box, and \dots.)


 

…No, really!

 


Ummm… Still with me?

OK. That tells me that you are now qualified to read further.

Just in case you were wondering what was there in the “other” header images, here is a little document I am uploading for you. Go, see it (.PDF [^]), but also note the caveat below.

Caveats: It is a work in progress. If you spot a mistake or even just a typo, then please do let me know. Also, don’t rely on this work.

For example, the definition of stress given in the document is what I have not so far read in any book. So, take it with a pinch of the salt—even if I feel confident that it is correct. Similarly, there might be some other changes, especially those related to the definition of the flux and its usage in the generic equation. Also, I am not sure if the product ansatz for the separation of variables technique began with d’Alembert or not. I vaguely remember its invention being attributed to him, but it was a long time ago, and I am no longer sure. May be it was before him. May be it was much later, at the hands of Fourier, or, even still later, by Lame. … Anyway let it be…

…BTW, the equations in the images currently being shown are slightly different—the PDF document is the latest thing there is.

Also, let me have your suggestions for any further inclusions, too, if any. (As to me: Yes, I would like to add a bit on the finite volume method, too.)

As usual, I may change the PDF document at any time in future. However, the document will always carry the date of compilation as the “version number”.


General update:

These days, I am also busy converting my already posted CFD snippets [^] into an FVM-based code.

The earlier posted code was done using FDM, not FVM, but it was not my choice—SPPU (Pune University) had thrust it upon me.

Writing an illustrative code for teaching purposes is fairly simple and straight-forward, esp. in Python—and especially if you treat the numpy arrays exactly as if they were Python arrays!! (That is, very inefficiently.) But I also thought of writing some notes on at least some initial parts of FVM (in a PDF document) to go with the code. That’s why, it is going to take a bit of time.

Once all this work is over, I will also try to model the Schrodinger equation using FVM. … Let’s see how it all goes…

…Alright, time to sign off, already! So, OK, take care and bye for now. …

 


A Song I Like:
(Hindi) “baharon, mera jeevan bhee savaron…”
Music: Khayyam
Singer: Lata Mangeshkar
Lyrics: Kaifi Aazmi


[The obligatory PS: In all probability, I won’t make any changes to the text of this post. However, the linked PDF document is bound to undergo changes, including addition of new material, reorganization, etc. When I do revise that document, I will note the updates in the post, too.]

 

Changes at this blog…

The changes at this blog:

In case you haven’t noticed it already, notice [what else?] that the layout of this blog has undergone a change. Hopefully for the better!

In particular, I’ve made the following changes:

  1. This blog is now concerned not only with the more transient writings of mine, but also with the less transient ones! … Accordingly, I have made a new page which holds links to my less transient writings, too, whether the write-ups were published here or elsewhere. See that page here [^].
  2. The tagline too now reflects the change in the purpose of this blog.
  3. I have added a header image, too. As of now, it holds some of the equations that have come to grab my attention for a long while. This may change in future. (See the separate section below.)
  4. A more minor change is the one made to the font.

A note for reading on the mobile:

In case you read this blog on a mobile phone, then to see the “less transient” page, you will have to press the menu button appearing at the top to get to the new page. On a desktop, however, the menu is by default seen as expanded.


The image at the top:

Just for the record, the equations in the top image, as of today (13 August 2018, 11:31 hrs), are the following:

  • The inner product and the outer product of two vectors, expressed using the more familiar notation of matrices.
  • Definitions of the grad of scalars and vectors, and the div of vectors and tensors.
  • The Taylor series expansion
  • The Fourier series expansion
  • The generic conservation equation for a scalar quantity, in the Eulerian form
  • The conservation equation for momentum, in the Eulerian form. (NB: The source term is in terms of \Phi i.e. the conserved quantity itself, whereas the rest of the terms have the mass-specific term \phi in them. This is correct.)
  • Definition of stress. (See the note for this equation below.)
  • Definitions of the displacement gradient tensor, the strain tensor, and the rotation tensor.
  • Cauchy’s formula (the relation between stress and the net force)
  • The Planck-Einstein relations
  • The most general form of the Schrodinger equation
  • The time-dependent Schrodinger equation in 1D
  • The inner product defined over a Hilbert space, and expansion of a function in terms of its basis set defined in a Hilbert space

An important note on the definition of stress as given in the header image:

I haven’t yet seen this definition in any solid/fluid/continuum mechanics text. So, please treat it with caution.

Also, please do drop me a line if you find it erroneous, problematic, or simply not general enough.

On the other hand, if you run into this definition anywhere, then please do bring the reference to my attention; thanks in advance. [This definition is a part of my planned paper on stress and strain.]


Some of the equations that got left out:

The equations which I would have liked to have in the header, but which got left out for a lack of space, are the following (in no particular order):

  • Newton’s second law defining force
  • Definitions of action (as momentum-dot-displacement and energy-times-time); action as an integral; action as a functional
  • The general equation for the methods of the weighted residuals, and the particular equations for the commonly used test functions (i.e., the Galerkin, the pseudospectral, the least-squares, the method of moments, and the collocation)
  • The Euler identity

Perhaps also, things like:

  • The wavefunction normalization principle, and the Born equation for finding probabilities
  • Structure of probability: simultaneous vs. subsequent events
  • The wave, diffusion and potential equations (juxtaposed with the Schrodinger equation)

On the other hand, some of the equations that are generally of great importance, but which have not come to preoccupy me a lot, are the following:

  • The Euler-Lagrange equations for classical mechanics
  • The Maxwell equations of electrodynamics, supplemented with the “fifth” (i.e. the Lorentz) force equation
  • Boltzmann’s equation, and other equations from statistical mechanics

I must have left out quite a few more in both the lists.

However, I am sure that the three laws of thermodynamics probably would not make it to the header image, despite all their grandeur, their all-encompassing scope.

The reason is this: a computational modeler like me seldom works in a very direct manner with the laws of thermodynamics themselves. These laws do inform his theory; the derivation of the equations he uses indeed are based on them, even if only indirectly. However, the equations he works with happen to be much more detailed (and of far more delimited scope). For instance: the Navier-Stokes system (CFD)—an expression of the first law; the stress-strain fields (FEM)—which makes for merely a part of the internal energy; or the Maxwell system (FDTD)—ditto. Etc.


Further change may be coming:

All in all, I am not quite happy with the top image as it exists right now. … It is too crowded, and speaking from a visual aesthetics point of view, its layout is not well-balanced.

So, on both these counts (too much crowding already, and too many good equations being left out), I am thinking of a further idea: may be I should create a sequence of images, each containing only a few equations, and let the server show you one of them at random. Whaddaya think?


Do check out the “less transient” page:

But yes, if you are interested, check out the “less transient” page too, and let me know if something I wrote in the past should be there or not.


So… does that mean that I’ve gone “mathy”?

Though I exclusively include only equations in the header image—no pictures or visualizations at all, no code, and not much text either—it doesn’t mean that I have gone “mathy”. … Hell, no! Not at all! … Just check out my less transient page [^].


A song I like:

(Hindi) “aankhon aankhon mein hum tum, ho gaye…”
Music: Kalyanji-Anandji
Singers: Kishore Kumar, Asha Bhosale
Lyrics: Anand Bakshi


 

Blog-Filling—Part 3

Note: A long Update was added on 23 November 2017, at the end of the post.


Today I got just a little bit of respite from what has been a very tight schedule, which has been running into my weekends, too.

But at least for today, I do have a bit of a respite. So, I could at least think of posting something.

But for precisely the same reason, I don’t have any blogging material ready in the mind. So, I will just note something interesting that passed by me recently:

  1. Catastrophe Theory: Check out Prof. Zhigang Suo’s recent blog post at iMechanica on catastrophe theory, here [^]; it’s marked by Suo’s trademark simplicity. He also helpfully provides a copy of Zeeman’s 1976 SciAm article, too. Regular readers of this blog will know that I am a big fan of the catastrophe theory; see, for instance, my last post mentioning the topic, here [^].
  2. Computational Science and Engineering, and Python: If you are into computational science and engineering (which is The Proper And The Only Proper long-form of “CSE”), and wish to have fun with Python, then check out Prof. Hans Petter Langtangen’s excellent books, all under Open Source. Especially recommended is his “Finite Difference Computing with PDEs—A Modern Software Approach” [^]. What impressed me immediately was the way the author begins this book with the wave equation, and not with the diffusion or potential equation as is the routine practice in the FDM (or CSE) books. He also provides the detailed mathematical reason for his unusual choice of ordering the material, but apart from his reason(s), let me add in a comment here: wave \Rightarrow diffusion \Rightarrow potential (Poisson-Laplace) precisely was the historical order in which the maths of PDEs (by which I mean both the formulations of the equations and the techniques for their solutions) got developed—even though the modern trend is to reverse this order in the name of “simplicity.” The book comes with Python scripts; you don’t have to copy-paste code from the PDF (and then keep correcting the errors of characters or indentations). And, the book covers nonlinearity too.
  3. Good Notes/Teachings/Explanations of UG Quantum Physics: I ran across Dan Schroeder’s “Entanglement isn’t just for spin.” Very true. And it needed to be said [^]. BTW, if you want a more gentle introduction to the UG-level QM than is presented in Allan Adam (et al)’s MIT OCW 8.04–8.06 [^], then make sure to check out Schroeder’s course at Weber [^] too. … Personally, though, I keep on fantasizing about going through all the videos of Adam’s course and taking out notes and posting them at my Web site. [… sigh]
  4. The Supposed Spirituality of the “Quantum Information” Stored in the “Protein-Based Micro-Tubules”: OTOH, if you are more into philosophy of quantum mechanics, then do check out Roger Schlafly’s latest post, not to mention my comment on it, here [^].

The point no. 4. above was added in lieu of the usual “A Song I Like” section. The reason is, though I could squeeze in the time to write this post, I still remain far too rushed to think of a song—and to think/check if I have already run it here or not. But I will try add one later on, either to this post, or, if there is a big delay, then as the next “blog filler” post, the next time round.

[Update on 23 Nov. 2017 09:25 AM IST: Added the Song I Like section; see below]

OK, that’s it! … Will catch you at some indefinite time in future here, bye for now and take care…


A Song I Like:

(Western, Instrumental) “Theme from ‘Come September'”
Credits: Bobby Darin (?) [+ Billy Vaughn (?)]

[I grew up in what were absolutely rural areas in Maharashtra, India. All my initial years till my 9th standard were limited, at its upper end in the continuum of urbanity, to Shirpur, which still is only a taluka place. And, back then, it was a decidedly far more of a backward + adivasi region. The population of the main town itself hadn’t reached more than 15,000 or so by the time I left it in my X standard; the town didn’t have a single traffic light; most of the houses including the one we lived in) were load-bearing structures, not RCC; all the roads in the town were of single lanes; etc.

Even that being the case, I happened to listen to this song—a Western song—right when I was in Shirpur, in my 2nd/3rd standard. I first heard the song at my Mama’s place (an engineer, he was back then posted in the “big city” of the nearby Jalgaon, a district place).

As to this song, as soon as I listened to it, I was “into it.” I remained so for all the days of that vacation at Mama’s place. Yes, it was a 45 RPM record, and the permission to put the record on the player and even to play it, entirely on my own, was hard won after a determined and tedious effort to show all the elders that I was able to put the pin on to the record very carefully. And, every one in the house was an elder to me: my siblings, cousins, uncle, his wife, not to mention my parents (who were the last ones to be satisfied). But once the recognition arrived, I used it to the hilt; I must have ended up playing this record for at least 5 times for every remaining day of the vacation back then.

As far as I am concerned, I am entirely positive that appreciation for a certain style or kind of music isn’t determined by your environment or the specific culture in which you grow up.

As far as songs like these are concerned, today I am able to discern that what I had immediately though indirectly grasped, even as a 6–7 year old child, was what I today would describe as a certain kind of an “epistemological cleanliness.” There was a clear adherence to certain definitive, delimited kind of specifics, whether in terms of tones or rhythm. Now, it sure did help that this tune was happy. But frankly, I am certain, I would’ve liked a “clean” song like this one—one with very definite “separations”/”delineations” in its phrases, in its parts—even if the song itself weren’t to be so directly evocative of such frankly happy a mood. Indian music, in contrast, tends to keep “continuity” for its own sake, even when it’s not called for, and the certain downside of that style is that it leads to a badly mixed up “curry” of indefinitely stretched out weilings, even noise, very proudly passing as “music”. (In evidence: pick up any traditional “royal palace”/”kothaa” music.) … Yes, of course, there is a symmetrical downside to the specific “separated” style carried by the Western music too; the specific style of noise it can easily slip into is a disjointed kind of a noise. (In evidence, I offer 90% of Western classical music, and 99.99% of Western popular “music”. As to which 90%, well, we have to meet in person, and listen to select pieces of music on the fly.)

Anyway, coming back to the present song, today I searched for the original soundtrack of “Come September”, and got, say, this one [^]. However, I am not too sure that the version I heard back then was this one. Chances are much brighter that the version I first listened to was Billy Vaughn’s, as in here [^].

… A wonderful tune, and, as an added bonus, it never does fail to take me back to my “salad days.” …

… Oh yes, as another fond memory: that vacation also was the very first time that I came to wear a T-shirt; my Mama had gifted it to me in that vacation. The actual choice to buy a T-shirt rather than a shirt (+shorts, of course) was that of my cousin sister (who unfortunately is no more). But I distinctly remember she being surprised to learn that I was in no mood to have a T-shirt when I didn’t know what the word meant… I also distinctly remember her assuring me using sweet tones that a T-shirt would look good on me! … You see, in rural India, at least back then, T-shirts weren’t heard of; for years later on, may be until I went to Nasik in my 10th standard, it would be the only T-shirt I had ever worn. … But, anyway, as far as T-shirts go… well, as you know, I was into software engineering, and so….

Bye [really] for now and take care…]

 

Recovering-ed/Recovered-ing

The general impression among philosophers of science and physicists alike is that maths is simple.

According to this viewpoint, maths may be—nay, even  must be—beautiful. But for all its complexity, speaking in the cultured tones, it is condemned to stay simple. The subtle shades of the evanescent feelings and emotions, say as captured by a piece of poetry or a work of fine art, they say, is not accessible to the hard, cold, “objective,” world of even of science in general, let alone the “world” of mathematics.

And, yes, as a matter of a plain truth, blogs must still be written, for the most part, using plain languages, for instance, in English! Not in mathematics.


Now, as far as I am concerned, I do seem to have sometime in the past much appreciated what folks such as these mean by those words.

But a subtle change took a root in my mind over the course of the last, ummmm…, 6–7 days, whose final culmination is what this post is all going to be about.


I mean to say, over the course of the past week or so, I seemed to be steadily recovering from my RSI (duly reported earlier on this very blog; see my last post).

Yesterday, the situation was that I seemed to have “fully” recovered from it.

And yet, as I was at it—I mean: at my poor keyboard—once again, I developed, you know, … a feeling. A feeling, now, near the base of my right-hand thumb. A feeling of a bit of a pain.

Now, given the really, really smart person that I am, I exactly knew what to do next: I stopped doing work, and ordered for me, through official channels [if you must be ever so curious], a new, more ergonomic, and < Rs. 500 keyboard. And then, I rested upon my newfound hint of an oncoming pain. [“Prevention is better than cure.”]

Then, sometime this late afternoon, as I was toying with the idea of slipping myself out of this sense of a highly diluted but nevertheless all-pervading boredom, I noticed that I cannot express myself at all. I mean to say: in plain English.

The “real truth” of the matter is this:

I think I have recovered—at least with all of today’s (and past few days’) boredom.

The thing also is: I think I have not recovered—not at least with that slight-ish pain, now appearing at the basal region of my right thumb.

Now, see, this is a situation that is so well-captured by maths in the following manner [but before going over that, may I remind you, for the nth time, that the proper spelling of the proper short-form of “mathematics” does naturally carry an ‘s’ at its end]:

Let w be defined as the wellness index. Then, states of well-/ill-ness can be easily expressed according to the following scheme:

  • Illness \Leftrightarrow -1.0 \leq w \leq 0.0
  • Recovered \Leftrightarrow w = 1.0
  • Recovering \Leftrightarrow 0.0 < w < 1.0

Simple enough a scheme, right?

So, now, the only question is: what English phrase do you use for the case which is captured by the expression: 0.999\dots \leq w < 1.0? especially if it also includes a time-evolution? a progress with the passage of time?

If you try to put it in English, referring to the above-mentioned points, there is no word in the English language (or any other “natural” language) to express this thought, this aspect of the actual reality (i.e. the condition of my typing hands). After all, “Recovered” does mean 1.0 but this number is not acceptable because of the use of the strong “<” sign.

As to “Recovering,” the 0.0 < w < 1.0 range, in this case, turns out to be of a rather Very Large Scale. In fact, as compared to the expression: “0.999\dots \leq w < 1.0“, it actually refers to an infinitely large Scale.

So, how do you express yourself in English, as far as that quoted expression, viz., “0.999\dots \leq w < 1.0,” goes?

After taking into account the time-evolution part of it, you would very naturally say something like: “recovering-ed” or “recovered-ing”. … You choose between the two.

Precisely both precisely are kind of usages that the Wren and Martin of my childhood times wouldn’t permit me, or any other child. (It’s not that I took the pair very seriously even back then, but the point is: I’ve come to know what painful book to quote when.)

And yet, the title usage is amply justified. As so well illustrated by the already established correspondence of maths and English.


And so, as I once again get back to typing [a lot]—but not on this (or any other) blog—what do you do in the meanwhile?

You listen to this song which I like…


A Song I Like:

(Marathi) “naval vartale ge maaye, ujaLalaa prakaashu…”
Lyrics: G. D. Madgulkar [Yes, that’s right, the words didn’t come to you from “sant dnyaaneshwara.”  [Yes, you further are wrong, “dnyaaneshwara” is never pronounced as “dnyaaneshwaraa,” let alone a “dnyaaneshwaraaaaaa.”]]
Singer: Asha Bhosale
Music: C. Ramchandra


[PS: May be I will streamline this post just a bit later tomorrow or the day after or so… .]