Understanding tensors (of engineering sciences)—part 2: yet another DIY experiment

I continue from my last post.

There is another simple DIY experiment that you can perform at home. The idea of this experiment had occurred quite some time back, but I had completely forgotten it. (I had forgotten it even while delivering lectures for the FEM courses which I taught in 2009 and 2012). Last night, I happened to recall the idea once again, and thought of immediately sharing it with you via this blog post.

A Fun DIY Experiment # 2:

Get that piece of men’s innerware which is known in India as the “banian” or “banyan,” and in American English as the “vest” [^].  (If not sure, check out the “aaraam kaa maamla” ads.) Basically, a banyan (at least these days) is a cotton garment like a T-shirt, but it’s bit smaller in size, and as an inner-ware, it is also meant to be more closely fitting to the body. That also makes it more easily stretchable, and therefore, better suited to our purposes. It’s also very inexpensive.

Start with a new (i.e. unused and never washed (i.e. never stretched/wrinkled)) banyan. The cloth should be easily stretchable. The fabric should be plain and simple, and without any special knitting pattern; e.g. no “self-stripes” etc. Cut it open and lay the cloth flat on a table. Mark a set of regular Cartesian grid-points on it with the help of an ink pen. You can easily make a bigger grid (say of the size 15 cm X 15 cm) at a regular spacing, say of 1 cm.

Lay the cloth flat and unstretched on the glass surface of a computer scanner (or even a Xerox machine), and obtain an image, say PH1. Next, with the help of a friend, stretch the cloth non-uniformly, by pulling unevenly along many directions. Make sure that the stretch is non-uniform but completely planar, and, of course, that there are no wrinkles. Scan it in this stretched state, and thus obtain the second image, PH2.

Advantages of this second experiment are easy to see: (i) As compared to the balloon rubber, is easier to lay the banyan cloth flat and without wrinkles. (ii) It is easier to stretch it in many directions. (iii) It is easier to mark out a regular grid—the regularity of the fabric of the cloth actually helps in ensuring regularity.

Also, even if you manage to get a good piece of a large rubber balloon, it should anyway be easier to obtain the image of a grid on it using an image scanner/Xerox machine rather than using a digital camera—the issues of having to maintain the same zoom and distance don’t arise.

Process the images as mentioned in the previous post, and keep them ready.

In the meanwhile, also consult the references mentioned in the last post, and make sure to go through the following concepts in particular: (i) position vector for a point-particle; (ii) displacement vector for a point-particle; (iii) the position vectors for an infinity of points in a continuum—i.e. the position vector field; (iv) the line segment, i.e., the relative position vector (i.e., the difference between two position vectors); (v) the translation and rotation of a line segment; (vi) the relative displacement vector of a line segment (i.e., the relative displacement vector of a relative position vector!); (vii) the rigid-body translation and rotation vs. the change of size and shape of a continuum body; (viii) the displacement gradient tensor at a point in a continuum; (ix) the rotation tensor at a point in a continuum body vs. its rigid-body rotation as a whole; (viii) the strain tensor at a point in a continuum body; etc. …

We will of course look into all these concepts—in fact, we will calculate the particular values that all these quantities assume in our simple experiment, using the basic data of the two images that our simple experiment generates. That will be our topic in the next post.

But before coming to it, let’s take a pause for a moment to recall what the purpose of this whole exercise is. It is: to know the physical meanings/correspondents of the mathematical concepts; to try and develop a proper hierarchical order the concepts; to develop a physical “feel” for the more abstract concepts involved. And, as far as the last is concerned (developing a physical feel for abstract concepts), there’s no substitute to realizing what the more concrete context of a given more abstract concept is. In understanding the proper context of mathematical concepts, there is no substitute to physical observation. That’s why, no matter how ridiculously simple these experiments might look like, do not skip the step of actually performing one of these two experiments.

And, BTW, in this series, more DIY experiments and fun ideas are going to follow.

More, later.

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A Song I Like:

(Hindi) “ek ghar banaoonga, tere ghar ke saamane…”
Singers: Mohammad Rafi and Lata Mangeshkar
Music: S. D. Burman
Lyrics: Hasrat Jaipuri

[E&OE]

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So, it is a QC (at least this week!)

I wanted to write on tensors etc., but a few very fresh inputs concerning the D-Wave device have appeared, all barely within the past 24 hours or less.

First, it was Prof. David Poulin commenting at Prof. Scott Aaronson’s blog once again [^], alerting some new work from Prof. Troyer. Unlike in his last comment (on the same post, when he thought that it was not a QC), Poulin has now come closer towards (or has started) supporting the position that the D-Wave device is a QC:

“…the problem instances that are easy for the D-wave device can sometimes be hard for the SS model. This is interesting new evidence supporting the quantum nature of the D-wave device.”

Next, a very valuable comment by one Bill Kaminsky appeared on Aaronson’s blog, very neatly explaining the Smolin and Smith model [^], and then contrasting it with the new result by Troyer. [Guess this Bill Kaminsky is the same as one William Kaminsky, who, in turn, is a PhD student in QIS at MIT. (… Just a Google search, that’s all!)] … Incidentally, more explanatory material concerning the adiabatic quantum optimization, quantum annealing, and classical annealing, written by Kaminsky, had already been put up last week at Henning Dekant’s blog; see here [^].

Finally, while idly thinking about all these things, even as idly browsing Prof. Poulin’s home page, I just idly happened to hit the “New on quant-ph” link [^] at its bottom, and thereby landed at the arXiv site; and once there, I noticed a new paper by Troyer (and (eight!) pals): [^].

Essentially, what Troyer et al. now say is that the D-Wave device does something that the classical devices apparently don’t, and so, the D-Wave device must be quantum! … If not all the classical devices, then at least the two devices: one, considered by they themselves, and the other, considered by Smolin and Smith. The D-Wave device behaves unlike both.

Further, Troyer et al. offer the following conjecture to account for the difference between the D-Wave chip and the [semi-]classical models:

“…The question of why SQA and semi-classical spin models correlate so differently with the D-Wave device is obviously important and interesting. We note that while SQA captures decoherence in the instantaneous energy eigenbasis of the system, so that each energy eigenstate—in particular the ground state—is itself a coherent superposition of computational basis states, semi-classical spin models assume that each qubit decoheres locally, thus removing all coherence from the ground state. We conjecture that the fact that the D-Wave machine succeeds with high probability on certain instances which the semi-classical models finds hard, can be understood in terms of this difference.”

[emphasis mine]

So, looks like, it is a quantum computer, after all. … At least, for this week!

* * *

Clearly, more studies required. So, here are a few questions to the QC research community:

What needs to be done to study the above conjecture more closely? Would some simple and special-purpose simulations that directly allow for a parametric control of the degrees of decoherence, help at least to illustrate (if not to fully support) the above conjecture? Such simulations could be highly simplified (say involving just a linear graph) but, still, sufficiently complete so as to be able to isolate, study, and possibly help settle, this issue.

How do you square off the quantum-ness of the D-Wave chip, and the “absence” of a speed-up, as discussed on Aaronson’s blog?

What measures would you suggest to capture the “percentage quantum-ness” of a QC? of an adiabatic quantum device such as D-Wave’s?

On these measures, how quantum are the current two D-Wave chips (D-Wave One and Two)? What is your estimate?

* * *

May be, more, later. (Who knows, it might once again collapse back to being a simple classical computer, next Monday!)

 

[May be I will come back (right today) and edit this post a bit, so as to make the write-up a bit more streamlined.]

 

[E&OE]

 

Is it a QC?

This post began its life as a supposedly brief update to my earlier post [^] on D-Wave’s paper, but the text soon grew long enough to become a separate post by itself. So, here we go.

Predictably, a controversy concerning the D-Wave paper (and its coverage in the media) came up soon later, at Prof. Scott Aaronson’s blog [^]. At 300+ comments (as of publishing this post), there is a lot of speculation, skepticism, and hilarity of the usenet/slashdot kind going on over there, apart from also some commentary.

However, as far as I am concerned, the most interesting part in (re) examining the paper and the related claims, was the following doubt which the controversy helped highlight: whether this particular D-Wave device had actually succeeded in exploiting, at least in part, the specifically quantum-mechanical effects, or not; whether there was an engineering success in controlling, at least in part (and to a practically significant extent), the quantum decoherence effects, or not.

The controversy was not entirely unexpected; recall this bit from the first New York Times story [^]:

““There is no sense in which this is the definitive statement about quantum computing,” Ms. McGeoch said. “I’m more interested in how well it works, not whether or not it is quantum.””

Though they called it a “quantum computer” (and I repeated the term), the term obviously was being used in a somewhat loose sense.

And, yes, I will admit it: without going through the paper well, I rather relied on the peer-review process, and so certainly thought, at least at the time of writing my earlier post, that D-Wave had a more impeccable and comprehensive result than what now seems to be the case.

But returning to who is interested in what: Well, as far as I am concerned, the issue of whether they got any speed-up or not, is strictly secondary—it’s “just” a consequence!

(In fact, I even don’t care if a QC research group cannot factor any composite beyond some single digit number, as of today. So long as they demonstrate a practically significant control of decoherence, and some clue about how they expect to scale it up, even their success in factoring only a small number would still make sense to me. Any future value of a QC in cracking open secret codes, or in designing better drugs through quantum chemical modeling, would be “just” a consequence, as far as I am concerned.)

To my mind, the real issue is: whether D-Wave succeeded in building a quantum computer (with some promise of some significant levels of a future scalability), or not.

So, from this angle, the most significant comment at Aaronson’s blog has been this one [^] by Prof. David Poulin, alerting the appearance of a paper by John Smolin and Graeme Smith, both of IBM, at arXiv, yesterday [^]. In case you are wondering whether to give this paper a read or not, let me remind you that IBM is a (corporate-sector) competitor to D-Wave. And, if that isn’t going to help, let me quote a bit from the main text of the paper:

“Since classical simulated annealing is intrinsically random and ‘quantum annealing’ is not…”

[emphasis mine]

and a line from their conclusions section:

“The deterministic nature of quantum annealing leads to rather different behaviors than the random processes of simulated annealing.”

[emphasis mine]

Interesting, no? (LOL!)

Of course, my own interests are in the foundations of QM, in providing a proper conceptual explanation for (and even mathematical expression to) the specifically quantum-mechanical effects/paradoxes/oddities, and not in the details of this or that quantum-mechanical process, whether it has some/a lot of/very great merit in building a scalable QC, or not.

So, I am not going to look too closely into this IBM paper either. Or provide a commentary on the position(s) it takes, its merits, or any polemical value it provides in this controversy (or in any other!). Or, add in any other way, to this D-Wave-related  controversy. … That way, I am not totally averse to controversies, but as far as this one goes, I find that it is a greater fun taking a ring-side view, here.

For another thing, these days, I am also thinking of quite different (and between them, somewhat unrelated) things: diffusion, small dams and water resources engineering/management, and tensors. Expect a post or two on these topics, soon enough.

So, all in all, even if I am having fun watching this controversy develop and grow, I guess I am going to sign off blogging about it. I won’t write any further on this topic, unless, of course something even more funny (or definitive, even if a bit serious) emerges from it.

[E&OE]

The QC pulls ahead of the CC

The first peer-reviewed paper to demonstrate that a quantum computer (QC) outperforms a conventional computer (CC), is here (PDF) [^]. [HT to Henning Dekant [^]].

The New York Times’ story is here [^].

Oh, BTW, this is one of those rare occasions when a peer-reviewed PDF of a scientific paper is being made available from a newspaper’s commercial servers—not from a server at some government-run Important National Lab, or a taxpayer-funded Wonderful State University, or for that matter, even arXiv! An interesting bit by itself, don’t you think?

(And, BBTW, I am old enough that as soon as I read this news, I instinctively slipped into wondering as to the time when the Russians might come forward with some “evidence” to show that they had accomplished the same thing some a few years earlier. … I guess I should go and enquire with the folks at the JNU New Delhi, ISI Kolkata, or IIT Bombay—they should know.)

Anyway, coming back to this exciting bit of news itself: at least at the time of going to wordpress, far too many American blogs on quantum computing still were completely silent. Especially those being maintained by the American academics. Several days over, and still not even a cursory acknowledgment!

Yet, this bit of news is not a hype; the advancement is for real.  Check out the following links (many of which were mentioned in Henning’s post, anyway): New Scientist [^], MIT’s Technology Review [^], IEEE [^], and even Nature [^].

So, an exciting news item, this one surely is. But what is comprehensively missing is one thing: that American (Hindi word) “taDkaa.”

The MIT Technology Review story, for instance, has this as the subtitle of its online story:

“Tests suggest that a CIA-backed quantum computing technology can be very powerful for some kinds of problems.”

Very careful.

“A” quantum computing technology—not the first to get a definite practical success.  “CIA-backed”—which means, this hint: the CIA has the money to pour into some potentially wasteful projects, and also have the means to choke out any adverse news reports if they fail, unlike the real innovative, open, democratic institutions like certain US universities. And, only “some” kind of problems would become solvable—it’s certain that with more research at MIT and Berkeley, the hardware is bound to get intelligent, but don’t expect it to be omniscient, that’s all. (Parenthetically: the company is Canadian.)

Sooooooo careful.

So, all in all, what I am missing out on is that American “taDkaa.” Even if Lockheed Martin, an American firm, already has gone ahead with the plans to use it [^], and an American by name Bo Ewald has become involved with the DWave [^]. [Full disclosure: I worked with e-Stamp roughly around the same time that Bo Ewald did. [Hi Bo!]]

The major reason I want to see some real American “masaalaa” and “taDkaa” on and around this topic, and if not that, then at least some ordinary hype on it, is: so that people get mysterious about this whole thing. Remember, the field of quantum computing carries two highfalutin words: “quantum” and “computing.” Even if the second word has lost a bit of a shine (Steve Jobs is no longer around, Chairman Bill is no longer the Chairman, and even the DC threatens Google only once in a while—there is no real DoJ action), it still carries a lot of aura. And, till date, they have managed to keep the first word, neatly wrapped up in a thick, impenetrable kind of an aura of a mystery.

When you combine the two together, there should be a multiplicative/exponential kind of a synergy. “Quantum Computing,” you know, should sound big. BIG. VERY BIG.

It, then, would be such a fun to step in on to the scene, and begin explaining how quantum computing is such a simple thing, after all! … How it is not all that big a mystery; how it really works. Explaining quantum computing on the basis of [clears the throat] my novel approach, would be fun, provided there is a preceding American “taDkaa” to it. In sufficient quantities. Together with “masaalaa.” To make it all mysterious in the first place.

There is no fun carrying just a pin around, no matter how sharp it may be. It’s no fun if you do have the pin, but there is no balloon in the first place—or, as in this case, there is that balloon, but still, no one is willing to inflate it.

* * *

Congratulations to the engineers and physicists at the D-Wave, anyway!

 

[E&OE]

Where the Mind Stops—Not!

The way people use language, changes.

In the mid- and late-1990s, when the Internet was new, when blogs had yet to become widespread, when people would often use their own Web sites (or the feedback forms and “guestbooks” at others’ Web sites) to express their own personal thoughts, opinions and feelings—in short, when it still was Web 1.0—one would often run into expressions of the title sort. For example: XYZ is a very great course—NOT! XYZ university has a very great student housing—NOT! XYZ is a very cute product—NOT! … You get the idea—you really do! (NO not!!)… That’s the sense in which the title of this post is to be taken.

For quite some time, I had been thinking of a problem, a deceptively simple problem, from engineering sciences and mechanics. Actually, it’s not a problem, but a way of modeling problems.

Consider a body or a physical object, say a piece of chalk. Break it into two pieces. Easy to do so physically? … Fine. Now, consider how you would represent this scenario mathematically. That is the problem under consideration. … Let me explain further.

The problem would be a mere idle curiosity but for the fact that it has huge economic consequences. I shall illustrate it with just two examples.

Example 1: Consider hot molten metal being poured in sand molds, during casting. Though “thick,” the liquid metal does not necessarily flow very smoothly as it runs everywhere inside the mold cavity. It brushes against mold-walls, splashes, and forms droplets. These flying droplets are more effective than the main body of the molten metal in abrading (“scrubbing”) the mold-walls, and thereby dislodging sand particles off the mold walls. Further, the droplets themselves both oxidize fast, and cool down fast. Both the oxidized and solidified droplets, and the sand particles abraded or dislodged by the droplets, fall into the cooling liquid metal. Due to oxidized layer the solidified droplets (or due to the high melting point of silicates, the sand particles) do not easily remelt once they fall into the main molten metal. The particles remain separate, and thus get embedded into the casting, leading to defective castings. (Second-phase particles like oxidized droplets and sand particles adversely affect the mechanical load-carrying capacity of the casting, and also lead to easier corrosion.) We need the flow here to be smooth, not so much because laminar flow by itself is a wonderful to have (and mathematically easier to handle). We need it to remain smooth mainly in order to prevent splashing and to reduce wall-abrasion. The splashing part involves separation of a contiguous volume of liquid into several bodies (the main body of liquid, and all the splashed droplets). If we can accurately, i.e. mathematically, model how droplets separate out from a liquid, we would be better equipped to handle the task of designing the flow inside a mold cavity.

Example 2: Way back in mid-1980s, when I was doing my MTech at IIT Madras, I had already run into some report which had said that the economic losses due to unintended catastrophic fractures occurring in the US alone were estimated to be some $5 billion annually. … I quote the figure purely from my not-so-reliable memory. However, even today, I do think that the quoted figure seems reasonable. Just consider just one category of fractures: the loss of buildings and human life due to fractures occurring during earthquakes. Fracture mechanics has been an important field of research for more than half a century by now. The process of fracture, if allowed to continue unchecked, results in a component or an object fragmenting into many pieces.

It might surprise many of you (in fact, almost anyone who has not studied fluid mechanics or fracture mechanics) that there simply does not exist any good way to mathematically represent this crucial aspect of droplets formation or fracture: namely, the fact of one body becoming several bodies. More accurately, no one so far (at least to my knowledge) has ever proposed a neat mathematical way to represent such a simple physical fact. Not in any way that could even potentially prove useful in building a better mechanics of fluids or fracture.

Not very surprising. After all, right since Newton’s time, the ruling paradigm of building mathematical models has been: differential equations. Differential equations necessarily assume the existence of a continuum. The region over which a given differential equation is to be integrated, may itself contain holes. Now, sometimes, the existence of holes in a region of space by itself leads to some troubles in some areas of mechanics; e.g., consider how the compatibility criteria of elasticity lose simplicity once you let a body carry holes. Yet, these difficulties are nothing once you theoretically allow the original single body to split apart into two or more fragments. The main difficulty is the following:

A differential equation is nothing but an equation defined in terms of differentials. (That is some insight!) In the sense of its usage in physics/engineering, a differential equation is an equation defined over a differential element. A differential element (or an infinitesimal) is a mathematical abstraction. It begins with a mathematically demarcated finite piece of a continuum, and systematically takes its size towards zero. A “demarcated finite piece” here essentially means that it has boundaries. For example, for a 1D continuum, there would be two separate points serving as the end-points of the finite piece. Such a piece is, then, subjected to the mathematical limiting process, so as to yield a differential element. To be useful, the differential equation has to be integrated over the entire region, taking into consideration the boundary and initial values. (The region must be primarily finite, and it usually is so. However, sometimes, through certain secondary mathematical considerations and tricks involving certain specific kinds of boundary conditions, we can let the region to be indefinitely large in extent as well.)

Since the basic definition of the differential element itself refers to a continuum, i.e. to a continuous region of space, this entire paradigm requires that cuts or holes not existing initially in the region cannot at all be later introduced. A hole is, as I said above, mostly acceptable in mathematical physics. However, the hole cannot grow so as to actually severe a single contiguous region of space into two (or more) separate regions. A cut cannot be allowed to run all the way through. The reason is: (i) either the differential element spanning the two sides of the cut must be taken out of the model—which cannot be done under the differential equations paradigm, (ii) or the entire model must be rejected as being invalid.

Thus, no cut—no boundary—can be introduced within a differential element. A differential element may be taken to end on a boundary, in a sense. However, it can never be cut apart. (This, incidentally, is the reason why people fall silent when you ask them the question of one of my previous posts: can an infinitesimal carry parts?)

You can look at it as a simple logical consistency requirement. If you model anything with differential elements (i.e. using the differential equations paradigm), then, by the logic of the way this kind of mathematics has been built and works, you are not allowed to introduce a cut into a continuum and make fragments out of it, later on.  In case you are wondering about a logically symmetrical scenario: no, you can also not join two continua into one—the differential equation paradigm does not allow you to do that either. And, no, topology does not lead to any actual progress with this problem either. Topology only helps define some aspects of the problem in mathematically precise terms. But it does not even address the problem I am mentioning here.

Such a nature of continuum modeling is indeed was what I had once hinted at, in one of my comments at iMechanica [^]. I had said (and none contradicted me at that forum for it) that:

As an aside, I think in classical mathematics there is no solution to this issue, and there cannot be—you simply cannot model a situation like “one thing becomes two things” or “two infinitesimally close points become separated by a finite distance” within any continuum theory at all…

In other words, this is a situation where, if one wishes to think about it in mathematical terms, one’s mind stops.

Or does it?

Today, I happened to idly go over these thoughts once again. And then, a dim possibility of appending a NOT appeared.

The reason I say it’s a dim possibility is because: (i) I haven’t yet carefully thought it through; (ii) and so, I am not sure if it really does not carry philosophic inconsistencies (philosophy, here, is to be rather taken in the sense of philosophy of science, of physics and mathematics); (iii) I already know enough to know that this possibility would not in any way help at least that basic fracture mechanical problem which I have mentioned above; and (iv) I think an application simpler than the basic problem of fracture mechanics, should be possible—with some careful provisos in place. May be, just may be. (The reason I am being so tentative is that the idea struck me only this afternoon.)

I still need to go over the matter, and so, I will not provide any more details about that dim possibility, right here, right today. However, I think I have already provided a sufficiently detailed description of the problem (and the supposed difficulty about it) that, probably, anyone else (trained in basic engineering/physics and mathematics) could easily get it.

So, in the meanwhile, if you can think of any solution—or even a solution approach—that could take care of this problem, drop me a line or add a comment.  … If you are looking for a succinct statement of the problem out of this (as usual) verbose blog-post, then take the above-mentioned quote from my iMechanica comment, as the problem statement. … For years (two+ decades) I thought no solution/approach to that problem was possible, and even at iMechanica, it didn’t elicit any response indicating otherwise. … But, now, I think there could perhaps be a way out—if I am consistent by basic philosophic considerations, that is. It’s a simple thing, really speaking, a very obvious one too, and not at all a big deal… However, the point is, now the (or my) mind no longer comes to a complete halt when it comes to that problem…

Enough for the time being. I will consider posting about this issue at iMechanica after a little while. … And, BTW, if you are in a mode to think very deeply about it, also see something somewhat related to this problem, viz., the 2011 FQXi Essay Contest (and what its winners had to say about that problem): [^]. Though related, the two questions are a bit different. For the purpose of this post, the main problem is the one I mentioned above. Think about it, and have fun! And if you have something to say about it, do drop me a line! Bye for now!!

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A Song I Like
(Hindi) “nahin nahin koee tumasaa hanseen…”
Singers: Kishore Kumar, Asha Bhosale
Music: Rajesh Roshan
Lyrics: Anand Bakshi

[PS: Perhaps, a revision to fix simple errors, and possibly to add a bit of content here and there, is still due.]
[E&OE]