Here are a few interesting links I browsed recently, listed in no particular order:

“Mathematicians Tame Turbulence in Flattened Fluids” [^].

The operative word here, of course, is: “flattened.” But even then, it’s an interesting read. Another thing: though the essay is pop-sci, the author gives the Navier-Stokes equations, complete with fairly OK explanatory remarks about each term in the equation.

(But I don’t understand why every pop-sci write-up gives the NS equations only in the Lagrangian form, never Eulerian.)

“A Twisted Path to Equation-Free Prediction” [^]. …

“Empirical dynamic modeling.” Hmmm….

“Machine Learning’s `Amazing’ Ability to Predict Chaos” [^].

Click-bait: They use data science ideas to predict chaos!

8 Lyapunov times is impressive. But ignore the other, usual kind of hype: “…the computer tunes its own formulas in response to data until the formulas replicate the system’s dynamics. ” [italics added.]

“Your Simple (Yes, Simple) Guide to Quantum Entanglement” [^].

Click-bait: “Entanglement is often regarded as a uniquely quantum-mechanical phenomenon, but it is not. In fact, it is enlightening, though somewhat unconventional, to consider a simple non-quantum (or “classical”) version of entanglement first. This enables us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.”

Don’t dismiss the description in the essay as being too simplistic; the author is Frank Wilczek.

“A theoretical physics FAQ” [^].

Click-bait: Check your answers with those given by an expert! … Do spend some time here…

Tensor product versus Cartesian product.

If you are engineer and if you get interested in quantum entanglement, beware of the easily confusing terms: The tensor product and the Cartesian product.

The tensor product, you might think, is like the Cartesian product. But it is not. See mathematicians’ explanations. Essentially, the basis sets (and the operations) are different. [^] [^].

But what the mathematicians don’t do is to take some simple but non-trivial examples, and actually work everything out in detail. Instead, they just jump from this definition to that definition. For example, see: “How to conquer tensorphobia” [^] and “Tensorphobia and the outer product”[^]. Read any of these last two articles. Any one is sufficient to give you tensorphobia even if you never had it!

You will never run into a mathematician who explains the difference between the two concepts by first directly giving you a vague feel: by directly giving you a good worked out example in the context of finite sets (including enumeration of all the set elements) that illustrates the key difference, i.e. the addition vs. the multiplication of the unit vectors (aka members of basis sets).

A third-class epistemology when it comes to explaining, mathematicians typically have.

A Song I Like:

(Marathi) “he gard niLe megha…”
Music: Rushiraj
Lyrics: Muralidhar Gode

[As usual, a little streamlining may occur later on.]

# HNY (Marathi). Also, a bit about modern maths.

Happy New (Marathi) Year!

OK.

I will speak in “aaeechee bhaashaa”  (lit.: mother’s language).

“gudhi-paaDawyaachyaa haardik shubhechchhaa.” (lit.: hearty compliments [on the occasion] of “gudhi-paaDawaa” [i.e. the first day of the Marathi new year  [^]].)

I am still writing up my notes on scalars, vectors, tensors, and CFD (cf. my last post). The speed is good. I am making sure that I remain below the RSI [^] detection levels.

BTW, do you know how difficult it can get to explain even the simplest of concepts once mathematicians have had a field day about it? (And especially after Americans have praised them for their efforts?) For instance, even a simple idea like, say, the “dual space”?

Did any one ever give you a hint (or even a hint of a hint) that the idea of “dual space” is nothing but a bloody stupid formalization based on nothing but the idea of taking the transpose of a vector and using it in the dot product? Or the fact that the idea of the transpose of a vector essentially means nothing than more than taking the same old three (or $n$ number of) scalar components, but interpreting them to mean a (directed) planar area instead of an arrow (i.e. a directed line segment)? Or the fact that this entire late 19th–early 20th century intellectual enterprise springs from no grounds more complex than the fact that the equation to the line is linear, and so is the equation to the plane?

[Yes, dear American, it’s the equation not an equation, and the equation is not of a line, but to the line. Ditto, for the case of the plane.]

Oh, but no. You go ask any mathematician worth his salt to explain the idea (say of the dual space), and this modern intellectual idiot would immediately launch himself into blabbering endlessly about “fields” (by which he means something other than what either a farmer or an engineer means; he also knows that he means something else; further, he also knows that not knowing this fact, you are getting confused; but, he doesn’t care to even mention this fact to you let alone explain it (and if you catch him, he ignores you and turns his face towards that other modern intellectual idiot aka the theoretical physicist (who is all ears to the mathematician, BTW))), “space” (ditto), “functionals” (by which term he means two different things even while strictly within the context of his own art: one thing in linear algebra and quite another thing in the calculus of variations), “modules,” (neither a software module nor the lunar one of Apollo 11—and generally speaking, most any modern mathematical idiot would have become far too generally incompetent to be able to design either), “ring” (no, he means neither an engagement nor a bell), “linear forms,” (no, neither Picasso nor sticks), “homomorphism” (no, not not a gay in the course of adding on or shedding body-weight), etc. etc. etc.

What is more, the idiot would even express surprise at the fact that the way he speaks about his work, it makes you feel as if you are far too incompetent to understand his art and will always be. And that’s what he wants, so that his means of livelihood is protected.

(No jokes. Just search for any of the quoted terms on the Wiki/Google. Or, actually talk to an actual mathematician about it. Just ask him this one question: Essentially speaking, is there something more to the idea of a dual space than transposing—going from an arrow to a plane?)

So, it’s not just that no one has written about these ideas before. The trouble is that they have, including the extent to which they have and the way they did.

And therefore, writing about the same ideas but in plain(er) language (but sufficiently accurately) gets tough, extraordinarily tough.

But I am trying. … Don’t keep too high a set of hopes… but well, at least, I am trying…

BTW, talking of fields and all, here are a few interesting stories (starting from today’s ToI, and after a bit of a Google search)[^][^] [^][^].

A Song I Like:

(Marathi) “maajhyaa re preeti phulaa”

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# In maths, the boundary is…

In maths, the boundary is a verb, not a noun.

It’s an active something, that, through certain agencies (whose influence, in the usual maths, is wholly captured via differential equations) actually goes on to act [directly or indirectly] over the entirety of a [spatial] region.

Mathematicians have come to forget about this simple physical fact, but by the basic rules of knowledge, that’s how it is.

They love to portray the BV (boundary-value) problems in terms of some dead thing sitting at the boundary, esp. for the Dirichlet variety of problems (esp. for the case when the field variable is zero out there) but that’s not what the basic nature of the abstraction is actually like. You couldn’t possibly build the very abstraction of a boundary unless if first pre-supposed that what it in maths represented was an active [read: physically active] something!

Keep that in mind; keep on reminding yourself at least $10^n$ times every day, where $n$ is an integer $\ge 1$.

A Song I Like:

[Unlike most other songs, this was an “average” one  in my [self-]esteemed teenage opinion, formed after listening to it on a poor-reception-area radio in an odd town at some odd times. … It changed for forever to a “surprisingly wonderful one” the moment I saw the movie in my SE (second year engineering) while at COEP. … And, haven’t yet gotten out of that impression yet… .]

(Hindi) “main chali main chali, peechhe peeche jahaan…”
Music: Shankar-Jaikishan
Lyrics: Shailendra

[May be an editing pass would be due tomorrow or so?]

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# Is something like a re-discovery of the same thing by the same person possible?

Yes, we continue to remain very busy.

However, in spite of all that busy-ness, in whatever spare time I have [in the evenings, sometimes at nights, why, even on early mornings [which is quite unlike me, come to think of it!]], I cannot help but “think” in a bit “relaxed” [actually, abstract] manner [and by “thinking,” I mean: musing, surmising, etc.] about… about what else but: QM!

So, I’ve been doing that. Sort of like, relaxed distant wonderings about QM…

Idle musings like that are very helpful. But they also carry a certain danger: it is easy to begin to believe your own story, even if the story itself is not being borne by well-established equations (i.e. by physic-al evidence).

But keeping that part aside, and thus coming to the title question: Is it possible that the same person makes the same discovery twice?

It may be difficult to believe so, but I… I seemed to have managed to have pulled precisely such a trick.

Of course, the “discovery” in question is, relatively speaking, only a part of of the whole story, and not the whole story itself. Still, I do think that I had discovered a certain important part of a conclusion about QM a while ago, and then, later on, had completely forgotten about it, and then, in a slow, patient process, I seem now to have worked inch-by-inch to reach precisely the same old conclusion.

In short, I have re-discovered my own (unpublished) conclusion. The original discovery was may be in the first half of this calendar year. (I might have even made a hand-written note about it, I need to look up my hand-written notes.)

Now, about the conclusion itself. … I don’t know how to put it best, but I seem to have reached the conclusion that the postulates of quantum mechanics [^], say as stated by Dirac and von Neumann [^], have been conceptualized inconsistently.

Please note the issue and the statement I am making, carefully. As you know, more than 9 interpretations of QM [^][^][^] have been acknowledged right in the mainstream studies of QM [read: University courses] themselves. Yet, none of these interpretations, as far as I know, goes on to actually challenge the quantum mechanical formalism itself. They all do accept the postulates just as presented (say by Dirac and von Neumann, the two “mathematicians” among the physicists).

Coming to me, my positions: I, too, used to say exactly the same thing. I used to say that I agree with the quantum postulates themselves. My position was that the conceptual aspects of the theory—at least all of them— are missing, and so, these need to be supplied, and if the need be, these also need to be expanded.

But, as far as the postulates themselves go, mine used to be the same position as that in the mainstream.

Until this morning.

Then, this morning, I came to realize that I have “re-discovered,” (i.e. independently discovered for the second time), that I actually should not be buying into the quantum postulates just as stated; that I should be saying that there are theoretical/conceptual errors/misconceptions/misrepresentations woven-in right in the very process of formalization which produced these postulates.

Since I think that I should be saying so, consider that, with this blog post, I have said so.

Just one more thing: the above doesn’t mean that I don’t accept Schrodinger’s equation. I do. In fact, I now seem to embrace Schrodinger’s equation with even more enthusiasm than I have ever done before. I think it’s a very ingenious and a very beautiful equation.

A Song I Like:

(Hindi) “tum jo hue mere humsafar”
Music: O. P. Nayyar
Singers: Geeta Dutt and Mohammad Rafi
Lyrics: Majrooh Sultanpuri

Update on 2017.10.14 23:57 IST: Streamlined a bit, as usual.

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