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…”
Singers: Lata Mangeshkar, Mohammad Rafi
Music: Shankar-Jaikishan
Lyrics: Shailendra


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

 

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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… .]