Have been reading QM texts and taking down my notes. Also jotting down my thoughts as they occur during these studies. There have been many threads of such thoughts, but nothing is even remotely near the completion stage. …
QM is hard—it pulls together an incredible variety of mathematical methods and tools, and each is a gem that you must spend some time on. So, the task keeps on growing…
It would have been nice to at least indicate what my thoughts are like, but it just so happens that they are rather like the margin notes. I could easily talk about them with any one (suitable) in a personal discussion, but a blog isn’t a good medium for sharing these—the ratio of the thought to the required context is too small here, and so, if these thoughts are to be converted into blog posts, too much time would be spent in just building the context.
That explains why, though I do have some free time at hand these days, I haven’t felt like blogging over the past two weeks.
But, guess I have to keep the momentum of blogging going too, and so, I will write a bit about a wonderful book on mathematics that I ran into only in the last month.
The book I am talking about packs together a lot of seeming contradictions. It is freely downloadable. It is short (130 pages). Its exercises are simple—so simple that they can’t but bolster your confidence. Yet, the text is written in such a way that it manages to pique your curiosity, so you feel like pursuing the topics covered in more depth later on. The book was written for biologists. But it has introductions to some topics that engineers wouldn’t have either understood so well, or the topics they wouldn’t have been exposed to, during their undergraduate education. By the latter, I mean topics like non-linearity and chaos. So, clearly, this book is useful also to engineers.
The book in question is:
Kirsten ten Tusscher and Alexander Panfilov (2011) “Mathematics for Biologists,” Utrecht University. [^]
A little bit more about the utility of the book to engineers, from my personal experience and viewpoint….
I have taught (simple) introductory courses on finite element method (FEM) to undergraduate students, postgraduate students, and also to practicing engineers. So let me say something with this background in mind.
If you have done a course on FEM, you know that you don’t waste your time trying to by-heart a seemingly countless number of integration formulae, as in your calculus-related mathematics courses.
Instead, you choose to play God—a rather lazy version of a God—and thereby directly assert right at the beginning of solving a(ny) problem that the unknown solution carries the form of only a polynomial. … I haven’t yet run into any book that pursues anything other than polynomials in more than one chapter, usually the introductory one. So, practically speaking, it’s always the polynomials.
How can you assert that it can only be a polynomial? Because, you are playing a God, that’s why. A rather lazy God.
In fact, it’s a polynomial of only a few initial terms, e.g., this one: .
In other words, even if the physical situation to model is such that the actual solution would consist of, say, the first half of the sine curve (the one that runs between and ), you still, in effect, proceed to ban the sine wave out of your universe:
“What, sine curves? Not in my universe. In my universe, sine curves are banned. They are not allowed to exist. Only the polynomials may.” That’s what you effectively say.
“But if polynomial is not the true solution, wouldn’t your `solution’ be in error?,” someone may raise this question.
When it comes to FEM, you have a ready answer: “Yes, but I can make the error as small as my computational power permits me to.”
The ability to reduce the error as small as desirable (or practically possible) is what makes numerical techniques (like FEM) “good.” (It is also what permits us to arbitrarily ban sines out of our universe; the cost to be paid is: high computational power. In short, we can play God because the machine helps us.) Let me give a concrete example of that—of the the idea of continuously reducing errors.
Remember Internet 1.0? Remember how long it took for a .JPG file to download? [Speculations on the nature of the downloaded files is left as an exercise for the reader.] It could take tens of minutes for a single image. (At least in India, it easily could.) What did your browser (say the Netscape Gold, or IE 1.0) do in the meanwhile?
The browser would initially show you a big empty rectangle in place where image was supposed to appear. Some time later, this rectangle would get filled with some big colored square boxes. As the download further progressed, once in a while, the boxes would abruptly become smaller, and thus, the nature of the picture would become somewhat easier to make out. The process of the refinement went on with the amount of the data downloaded, and a greater amount of the actually downloaded data meant: smaller colored boxes, more number of them. That is, a more refined picture. After some 30 minutes, you would have that half-MB of a file completely downloaded on to your local machine (to be shared with your friends, using 1.44 MB floppy disks (assuming these wouldn’t develop bad sectors)).
So, that was the idea: a very coarse beginning but a series of repeated refinements.
The idea works also in the non-image processing contexts. You can begin with a coarse solution and then refine it better and still better. FEM uses precisely such a principle.
So, in FEM, you can always start with a polynomial even if the true (unknown) solution happens to be a sine curve.
Yes, using a polynomial where a sine curve solution is expected, does mean that there is an error in the solution—the curve for a polynomial isn’t exactly the same as the sine curve. (Try i.e. for the first half of .) That’s because the two aren’t one and the same function. The difference in the two functions is what we call “error.” But since as a God you have banned sine curves from your universe and allowed only the polynomials, the only thing you can do to save your Godliness is to reduce the error.
At this point, you have two choices, concerning what kind of mathematics to use for reducing errors.
(i) You can increase the solution accuracy by adding more, higher order, terms to the same polynomial. What I mean here is, a single polynomial continues to run across the entire domain all by itself, but the number of terms it carries goes on increasing [^]. Thus, for instance, you initially may use the quadratic polynomial: (it has three terms) [^]. You then keep on adding terms for refinement: (cubic polynomial, four terms) [^];, (quartic, five terms) [^], etc. That’s one way of refining the solution.
(ii) The second way is: you keep the degree of the polynomial the same (say, you use only a cubic polynomial), but you indirectly compensate for the lack of the higher order terms by increasing the number of boxes used for filling the domain. Thus, a lower-order or cruder polynomial is used for interpolations within a single box, but there are a large number of such boxes side by side to cover the entire domain. (The polynomials are arranged in such a way that at the interface between the two adjacent sub-domains, their curves touch each other, so that the solution for the entire domain remains continuous.)
In other words, the two methods of refinement are: (i) using a progressively longer but single polynomial over the entire domain, or (ii) using shorter and simpler polynomials within more and more color-boxes that cover the entire domain.
The FEM course, at this point, gets a bit more clerical—there always is a “routine” to any job. It also gets a little more difficult, for the students, from this point on. The reason is: they have forgotten some very basic maths by the time they begin to study FEM.
For instance, suppose I tell them this: Since the shape of the first half of the sine curve (i.e. the one between to ) is like a bell or a dome—i.e. it has just a single hump—so, obviously, the simplest polynomial which reproduces this broad feature (of the first half of the sine curve) would have to be a quadratic, because a quadratic can be made to carry a bend, whereas a straight line cannot ever be made to do so. On the other hand, if we had to approximately model the full sine curve (i.e. the one between to ), then, since it carries two bends—a hump in the first half and a cup in the second—therefore, the simplest polynomial that still shows this feature of a hump and a cup, will have to be a cubic.
Typically, people—even graduate engineers with fairly good proficiency in solving differential equations—tend to go (at least temporarily) blank at this point. (Yes, some of the best of them also are merely straining themselves, in my actual observation—including those at COEP.) They are unable to connect the logic of the number of the inflection points on a curve and the degree of the approximating polynomial. They can easily rattle off the relation between the number of points through which a polynomial passes and its degree, but not the number of inflection points and the degree.
It happens. Either they never looked into things like that in depth because their JEE/XII exams didn’t need them, or, in the rush of five courses a semester of new maths/application every semester, they hadn’t had the time necessary to integrate ideas. (Or, they have been slacking. Possible.)
There is no easy solution. They are supposed to have read such topics, but they don’t show any evidence of a fast enough recall. The mathematics texts they use (in engineering) don’t cover such basic topics, because these books take them for granted (e.g. check out Kreyszig, Greenberg, Barrett or Arfken). The texts which the students cursorily used in their XI/XII standards are long gone off their hands (and their heads), possibly because, I said, neither the XII boards nor the JEE asked them to do curve-tracing. Among the mathematics texts now nearby them, Wartikar (or other local books) do have this topic covered (at least cursorily), but the illustrations and the print quality in them is so bad that only a mathematician could pick up a book like that for the second time in life. [Try to believe me when I say that I say this in admiration of mathematicians.]
Curve-tracing isn’t the only elementary or basic topic on which my students habitually go blank. There also are many other similar topics.
They go blank also if I ask them what it is that they visualize for the matrix eigenvalue problem. They never have visualized anything related to matrices. (A rectangle of numbers doesn’t count as a visualization.) The very idea that an eigenvalue problem can be visualized, itself is new to them. (They, by habit, never check the Wiki for any of the topics currently ongoing in the class.) Similarly, they also go blank even if I just utter the words: “a set of coupled differential equations.” They remain “blanked out” until I drop the hint like, say, several mass-spring systems connected together via some pins. At this point the bulb lights up, but only momentarily, and then, an even thicker darkness descends on them. But even that momentary flicker is an encouraging sign, to me. That’s because, nothing at all happens if I mention “chaos”—their eyes remain glassy. Or, the precise difference between complex numbers and vectors. The eyes now begin to show a generally tired version of a confident kind of a carelessness.
They need a book. A helpfully written book. A short book. An easy to read book. The book should treat also the basic topics, but rapidly.
Engineers, that way, are well-exposed to the art of juggling through mathematics. However, even if the pace is somewhat rapid, the book should be a little more than Schaum’s series books (or a compilation of formulae)—there should be some interesting bits of conceptual explanations (or hints), some non-routine kinds of applications mentioned, too.
Further, the book should be small enough that it fits the time that the students have available. (My earlier relevant post on this problem is here [^].)
Finally, the book should not list some challenging—actually, depression-inducing—exercises at the end of a chapter; the book should not turn away the reader from mathematics on that count. (Also, it should preferably have some colorful diagrams.) And, the price should be reasonable.
There are two books that deliever in an excellent manner on all these counts. Both happen to have been originally written for biologists, not engineers.
The first book is: Edward Batschelet (1979) “Introduction to Mathematics for Life Scientists, 3/e” Springer. (Published in 1979, the only feature it misses on is: color diagrams.)
The second book is the one I am talking about, in this post.
On this blog, I had touched upon the first book a while ago, here [^].
The book now under discussion complements the first. It dwells more on the recent topics which were not covered in the first. (The third edition of the first book was in 1979.) It also is just so slightly at a more “advanced” level, though, IMO, 100% understandable to any UG student of engineering.
So, I am happy to strongly recommend the second book as well. Go ahead, check it out for yourself; the .PDF is free to download anyway!
A Song I Like:
(Western “Classical”) “Va, pensiero” (from “Nabucco”)
Original Composer: Giuseppe Verdi
Original Lyrics: Temistocle Solera
[The version which I listened to for the first time in my life was the one by James Last and his orchestra. I still continue liking that version for its own sake even today. The reason is, while the “purely” “classical” (or the classically oriented) performances naturally carry depth, they also tend to sometimes become a shade too pensive or sombre—sometimes even bringing a touch of ghoulishness into rendering.
On the other hand, the pop-instrumental versions (like those by James Last) are more lyrical—they are even lilting in a way. But this form—popular instrumental—itself is such that a performance can’t help but skim over the more serious portions. … The emotional experience of actually finding oneself in the midst of a physical enslavement—the gravitas of that situation—is made light, a bit too light by this form. And therefore, that yearning for the freedom, that soaring affirmation of freedom as a golden value by itself, also becomes that much less moving or stirring. …
As to me, there are times when I want to listen to only the rhythmic affirmation of the positive that such a piece can bring. I want to focus on the transformation that a mind undergoes in the act of even just contemplating the state of freedom. I want to directly sense that heavenly lightness of being which even just a mental contemplation of freedom is able to bring. Why, I want to even just directly sense the fleetingly light experience as was once expressed in (Marathi) “swatanatre, bhagawatee, chaanDaNee cham-cham lakhalakhashee,” or in [continuing with the same Marathi song] “gaalaawarachyaa kusumi kinvaa kusumaanchyaa gaali.” There is this kind of a lightness present also here in Verdi’s “va pensiero,” and there are times when I want to have this part stressed in my experience of the music.
And, of course, there also are other times (in my case these are somewhat more rare) when I must listen to a good “classical” rendering, for a deeper experience of all the aspects of the original music, with all its subtlety and seriousness. I thus have listened to several “classical” renderings of this piece by now, though I haven’t so far had an opportunity to sit through the entire opera. [BTW, I keep putting the scare-quotes around the word classical, because this piece is from the 19th century, and thus, it is, technically, from the Romantic era, not Classical.]
Just one more point. While the words for this song are great, I happen almost never to listen to (or look for) the words here. The music here is just too powerful for the words to matter much one way or the other, even if the words themselves happen to be as good as they are (I mean the English words [^]; I don’t know Italian). In fact, the specific historical context that the opera involves itself means almost nothing to me; only the themes in the abstract do: The interwoven themes of exilement and patriotism. Or, just plain of immigration and nostalgia. But, inescapably, above all, of enslavement and freedom. The overall theme here is complex but universal, and that’s why the specific concretes cease to matter. The words do express the theme well, but compared to the music, they, too, cease to matter…
The music… It’s just too subtle and yet too powerful, it’s too exceptional.
… This piece is supposed to be Verdi’s achievement of a lifetime. I haven’t heard a lot of Verdi, but I find it easy to believe the critic here. Seemingly very simple, it carries quite a few complex layers. It touches on many seemingly familiar musical phrases, but it still remains distinctly innovative, somehow. The music here has drama, and so, its progression does require just a bit of an emotional stamina, but it is not as much as what, say, Beethoven demands of you. The theme here is comparatively on the brighter side, and so, listening to it doesn’t exhaust you….
All in all, it’s a great piece of music! Hope you like it, too.]