And to think…

Many of you must have watched the news headlines on TV this week; many might have gathered it from the ‘net.

Mumbai—and much of Maharashtra—has gone down under. Under water.

And to think that all this water is now going to go purely to waste, completely unused.

… And that, starting some time right from say February next year, we are once again going to yell desperately about water shortage, about how water-tankers have already begun plying on the “roads” near the drought-hit villages. … May be we will get generous and send not just 4-wheeler tankers but also an entire train to a drought-hit city or two…

Depressing!


OK. Here’s something less depressing. [H/t Jennifer Ouellette (@JenLucPiquant) ]:

“More than 2,000 years ago, people were able to create ice in the desert even with temperatures above freezing!” [^]

The write-up mentions a TED video by Prof. Aaswath Raman. Watched it out of idle interest, checked out his Web site, and found another TED video by him, here [^]. Raman cites statistics that blew me!

They spend “only” $24 billion on supermarket refrigeration (and other food-related cooling), but they already spend $42 billion on data-center cooling!!


But, any way, I did some further “research” and landed at a few links, like the Wiki on Yakhchal [^], on wind-catcher [^], etc.  Prof. Raman’s explanation in terms of the radiative cooling was straight-forwards, but I am not sure I understand the mechanism behind the use of a qanat [^] in Yakhchal/windcatcher cooling. It would be cool to do some CFD simulations though.


Finally, since I am once again out of a job (and out of all my saved money and in fact also into credit-card loans due to some health issue cropping up once again), I was just idly wondering about all this renewable energy business, when something struck me.


The one big downside of windmills is that the electricity they generate fluctuates too much. You can’t rely on it; the availability is neither 24X7 nor uniform. Studies in fact also show that in accommodating the more or less “random” output of windmills into the conventional grid, the price of electricity actually goes up—even if the cost of generation alone at the windmill tower may be lower. Further, battery technology has not improved to such a point that you could store the randomly generated electricity economically.

So, I thought, why not use that randomly fluctuating windmill electricity in just producing the hydrogen gas?

No, I didn’t let out a Eureka. Instead, I let out a Google search. After all, the hydrogen gas could be used in fuel-cells, right? Would the cost of packaging and transportation of hydrogen gas be too much? … A little searching later, I landed at this link: [^]. Ummm… No, no, no…. Why shoot it into the natural gas grid? Why not compress it into cylinders and transport by trains? How does the cost economics work out in that case? Any idea?


Addendum on the same day, but after about a couple of hours:

Yes, I did run into this link: “Hydrogen: Hope or Hype?” [^] (with all the links therein, and then, also this: [^]).

But before running into those links, even as my googling on “hydrogen fuel energy density” still was in progress, I thought of this idea…

Why at all transport the hydrogen fuel from the windmill farm site to elsewhere? Why not simply install a fuel cell electricity generator right at the windmill farm? That is to say, why not use the hydrogen fuel generated via electrolysis as a flywheel of sorts? Get the idea? You introduce a couple of steps in between the windmill’s electricity and the conventional grid. But you also take out the fluctuations, the bad score on the 24X7 availability. And, you don’t have to worry about the transportation costs either.

What do you think?


Addendum on 12th July 2018, 13:27 hrs IST

Further, I also browsed a few links that explore another,  solution: using compressed air: a press report [^], and a technical paper [^]. (PDF of the paper is available, but the paper would be accessible only to mechanical engineers though. Later Update: As to the press report, well, the company it talks about has already merged with another company, and has abandoned the above-ground storage of compressed air [^])

I think that such a design reduces the number of steps of energy conversions. However, that does not necessarily mean that the solution involving hydrogen fuel generation and utilization (both right at the wind-farm) isn’t going to be economical.

Economics determines (or at least must determine) the choice. Enough on this topic for now. Wish I had a student working with me; I could have then written a paper after studying the solution I have proposed above. (The idea is worth a patent too. Too bad I don’t have the money to file one. Depressing, once again!!)


OK. Enough for the time being. I may later on add the songs section if I feel like it. And, iterative modifications will always be done, but will be mostly limited to small editorial changes. Bye for now.

 

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“The spiritual heritage of India”

I wrote a few comments at Prof. Scott Aaronson’s blog, in response to his post of the title: “30 of my favorite books”, here [^].

Let me give you the links to my comments: [^], [^], [^] and [^].


Let me reproduce the last one of my four comments, with just so slight bit of editing. [You know I couldn’t have resisted the opportunity, right?]:

Since I mentioned the “upnishad”s above (i.e. here [ ^]), and as far as this topic is concerned, since the ‘net is so full of the reading material on this topic which isn’t so suitable for this audience, let me leave you with a right kind of a reco.

If it has to be just one short book, then the one which I would pick up is this:

Swami Prabhavananda (with assistance of Frederick Manchester), “The Spiritual Heritage of India,” Doubleday, New York, 1963.

A few notes:

1. The usual qualifications apply. For instance, I of course don’t agree with everything that has been said in the book. And, more. I may not even agree that a summary of something provided here is, in fact, a good summary thereof.

2. I read it very late in life, when I was already past my 50. Wish I had laid my hands on it, say, in my late 20s, early 30s, or so. I simply didn’t happen to know about it, or run into a copy, back then.

3. Just one more thing: a tip on how to read the above book:

First, read the Preface. Go through it real fast. (Reading it faster than you read the newspapers would be perfectly OK—by me).

Then, if you are an American who has at least a smattering of a knowledge about Buddhism, then jump directly on to the chapter on Jainism. (Don’t worry, they both advocate not eating meat!) And, vice-versa!!

If you are not an American, or,  if you have never come across any deeper treatment on any Indian tradition before, then: still jump on to the chapter on Jainism. (It really is a very good summary of this tradition, IMHO.)

Then, browse through some more material.

Then, take a moment and think: if you have appreciated what you’ve just read, think of continuing with the rest of the text.

Else, simple: just call it a book! (It’s very inexpensive.)

 


No need to add anything, but looking at the tone of the comments (referring to the string “Ayn Rand”) that got generated on this above-mentioned thread, I find myself thinking that, may be, given my visitor-ship pattern (there are more Americans hits today to my blog than either Indian or British), I should explain a bit of a word-play which I attempted in that comment (and evidently, very poorly—i.e. non-noticeably). It comes near the end of my above-quoted reply.

“Let’s call it a day” is a very neat British expression. In case you don’t know its meaning, please look it up on the ‘net. Here’s my take on it (without looking it up):

Softly folding away a day, with a shade of an anticipation that a day even better might be about to arrive tomorrow, and so, softly reminding yourself that you better leave the party or the function for now, so as to be able to get ready for yet another party, yet another function, some other day, later…

A sense of something like that, is implied by that expression.

I just attempted a word-play, and so, substituted “book” for the “day”.

Anyway, good night. Do read my last post, the document attached to it, and the links therefrom.

Bye for now.


Oh, yes! There is a song that’s been playing off-and-on at the back of my mind for some time. Let me share it with you.


A Song I Like:

(Hindi) “dil kaa diyaa jala ke gayaa…”
Lyrics: Majrooh Sultaanpuri
Singer: Lata Mangeshkar
Music: Chitragupt

[PS: The order of the listing of the credits, once again, is completely immaterial here.]


Anyway, enjoy the song, and the book mentioned in the quotes (and hopefully, also my past few posts and their attachments)… I should come back soon, with a maths-related puzzle/teaser/question. … Until then, take care and bye!

My small contribution towards the controversies surrounding the important question of “1, 2, 3, …”

As you know, I have been engaged in writing about scalars, vectors, tensors, and CFD.

However, at the same time, while writing my notes, I also happened to think of the “1, 2, 3, …” controversy. Here is my small, personal, contribution to the same.


The physical world evidently consists of a myriad variety of things. Attributes are the metaphysically inseparable aspects that together constitute the identity of a thing. To exist is to exist with all the attributes. But getting to know the identity of a thing does not mean having a knowledge of all of its attributes. The identity of a thing is grasped, or the thing is recognized, on the basis of just a few attributes/characteristics—those which are the defining attributes (including properties, characteristics, actions, etc.), within a given context.

Similarities and differences are perceptually evident. When two or more concretely real things possess the same attribute, they are directly perceived as being similar. Two mangoes are similar, and so are two bananas. The differences between two or more things of the same kind are the differences in the sizes of those attribute(s) which are in common to them. All mangoes share a great deal of attributes between them, and the differences in the two mangoes are not just the basic fact that they are two separate mangoes, but also that they differ in their respective colors, shapes, sizes, etc.

Sizes or magnitudes (lit.: “bigness”) refer to sizes of things; sizes do not metaphysically exist independent of the things of which they are sizes.

Numbers are the concepts that can be used to measure the sizes of things (and also of their attributes, characteristics, actions, etc.).


It is true that sizes can be grasped and specified without using numbers.

For instance, we can say that this mango is bigger than that. The preceding statement did not involve any number. However, it did involve a comparative statement that ordered two different things in accordance with the sizes of some common attribute possessed by each, e.g., the weight of, or the volume occupied by, each of the two mangoes. In the case of concrete objects such as two mangoes differing in size, the comparative differences in their sizes are grasped via direct perception; one mango is directly seen/felt as being bigger than the other; the mental process involved at this level is direct and automatic.

A certain issue arises when we try to extend the logic to three or more mangoes. To say that the mango A is bigger than the mango B, and that the mango B is bigger than the mango C, is perfectly fine.

However, it is clear from common experience that the size-wise difference between A and B may not exactly be the same as the size-wise difference between B and C. The simple measure: “is bigger than”, thus, is crude.

The idea of numbers is the means through which we try to make the quantitative comparative statements more refined, more precise, more accurately capturing of the metaphysically given sizes.

An important point to note here is that even if you use numbers, a statement involving sizes still remains only a comparative one. Whenever you say that something is bigger or smaller, you are always implicitly adding: as in comparison to something else, i.e., some other thing. Contrary to what a lot of thinkers have presumed, numbers do not provide any more absolute a standard than what is already contained in the comparisons on which a concept of numbers is based.


Fundamentally, an attribute can metaphysically exist only with some definite size (and only as part of the identity of the object which possesses that attribute). Thus, the idea of a size-less attribute is a metaphysical impossibility.

Sizes are a given in the metaphysical reality. Each concretely real object by itself carries all the sizes of all its attributes. An existent or an object, i.e., when an object taken singly, separately, still does possess all its attributes, with all the sizes with which it exists.

However, the idea of measuring a size cannot arise in reference to just a single concrete object. Measurements cannot be conducted on single objects taken out of context, i.e., in complete isolation of everything else that exists.

You need to take at least two objects that differ in sizes (in the same attribute), and it is only then that any quantitative comparison (based on that attribute) becomes possible. And it is only when some comparison is possible that a process for measurements of sizes can at all be conceived of. A process of measurement is a process of comparison.

A number is an end-product of a certain mathematical method that puts a given thing in a size-wise quantitative relationship (or comparison) with other things (of the same kind).


Sizes or magnitudes exist in the raw nature. But numbers do not exist in the raw nature. They are an end-product of certain mathematical processes. A number-producing mathematical process pins down (or defines) some specific sense of what the size of an attribute can at all be taken to mean, in the first place.

Numbers do not exist in the raw nature because the mathematical methods which produce them themselves do not exist in the raw nature.

A method for measuring sizes has to be conceived of (or created or invented) by a mind. The method settles the question of how the metaphysically existing sizes of objects/attributes are to be processed via some kind of a comparison. As such, sure, the method does require a prior grasp of the metaphysical existents, i.e., of the physical reality.

However, the meaning of the method proper itself is not to be located in the metaphysically differing sizes themselves; it is to be located in how those differences in sizes are grasped, processed, and what kind of an end-product is produced by that process.

Thus, a mathematical method is an invention of using the mind in a certain way; it is not a discovery of some metaphysical facts existing independent of the mind grasping (and holding, using, etc.) it.

However, once invented by someone, the mathematical method can be taught to others, and can be used by all those who do know it, but only in within the delimited scope of the method itself, i.e., only in those applications where that particular method can at all be applied.


The simplest kind of numbers are the natural numbers: 1, 2, 3, \dots. As an aside, to remind you, natural numbers do not include the zero; the set of whole numbers does that.

Reaching the idea of the natural numbers involves three steps:

(i) treating a group of some concrete objects of the same kind (e.g. five mangoes) as not only a collection of so many separately existing things, but also as if it were a single, imaginary, composite object, when the constituent objects are seen as a group,

(ii) treating a single concrete object (of the same aforementioned kind, e.g. one mango) not only as a separately existing concrete object, but also as an instance of a group of the aforementioned kind—i.e. a group of the one,

and

(iii) treating the first group (consisting of multiple objects) as if it were obtained by exactly/identically repeating the second group (consisting of a single object).

The interplay between the concrete perception on the one hand and a more abstract, conceptual-level grasp of that perception on the other hand, occurs in each of the first two steps mentioned above. (Ayn Rand: “The ability to regard entities as mental units \dots” [^].)

In contrast, the synthesis of a new mental process that is suitable for making quantitative measurements, which means the issue in the third step, occurs only at an abstract level. There is nothing corresponding to the process of repetition (or for that matter, to any method of quantitative measurements) in the concrete, metaphysically given, reality.

In the third step, the many objects comprising the first group are regarded as if they were exact replicas of the concrete object from the second (singular) group.

This point is important. Primitive humans would use some uniform-looking symbols like dots (.) or circles (\bullet) or sticks (`|‘), to stand for the concrete objects that go in making up either of the aforementioned two groups—the group of the many mangoes vs. the group of the one mango. Using the same symbol for each occurrence of a concrete object underscores the idea that all other facts pertaining to those concrete objects (here, mangoes) are to be summarily disregarded, and that the only important point worth retaining is that a next instance of an exact replica (an instance of an abstract mango, so to speak) has become available.

At this point, we begin representing the group of five mangoes as G_1 = \lbrace\, \bullet\,\bullet\,\bullet\,\bullet\,\bullet\, \rbrace, and the single concretely existing mango as a second abstract group: G_2 = \lbrace\,\bullet\,\rbrace.


Next comes a more clear grasp of the process of repetition. It is seen that the process of repetition can be stopped at discrete stages. For instance:

  1. The process P_1 produces \lbrace\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet once).
  2. The process P_2 produces \lbrace\,\bullet\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet twice)
  3. The process P_3 produces \lbrace\,\bullet\,\bullet\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet thrice)
    etc.

At this point, it is recognized that each output or end-product that a terminated repetition-process produces, is precisely identical to certain abstract group of objects of the first kind.

Thus, each of the P_1 \equiv \lbrace\,\bullet\,\rbrace, or P_2 \equiv \lbrace\,\bullet\,\bullet\,\rbrace, or  P_3 \equiv \lbrace\,\bullet\,\bullet\,\bullet\,\rbrace, $\dots$ is now regarded as if it were a single (composite) object.

Notice how we began by saying that P_1, P_2, P_3 etc. were processes, and then ended up saying that we now see single objects in them.

Thus, the size of each abstract group of many objects (the groups of one, of two, of three, of n objects) gets tied to a particular length of a terminated process, here, of repetitions. As the length of the process varies, so does the size of its output i.e. the abstract composite object.

It is in this way that a process (here, of repetition) becomes capable of measuring the size of the abstract composite object. And it does so in reference to the stage (or the length of repetitions) at which the process was terminated.

It is thus that the repetition process becomes a process of measuring sizes. In other words, it becomes a method of measurement. Qua a method of measurement, the process has been given a name: it is called “counting.”

The end-products of the terminated repetition process, i.e., of the counting process, are the mathematical objects called the natural numbers.


More generally, what we said for the natural numbers also holds true for any other kind of a number. Any kind of a number stands for an end-product that is obtained when a well-defined process of measurement is conducted to completion.

An uncompleted process is just that: a process that is still continuing. The notion of an end-product applies only to a process that has come to an end. Numbers are the end-products of size-measuring processes.

Since an infinite process is not a completed process, infinity is not a number; it is merely a short-hand to denote some aspect of the measurement process other than the use of the process in measuring a size.

The only valid use of infinity is in the context of establishing the limiting values of sequences, i.e., in capturing the essence of the trend in the numbers produced by the nature (or identity) of a given sequence-producing process.

Thus, infinity is a concept that helps pin down the nature of the trend in the numbers belonging to a sequence. On the other hand, a number is a product of a process when it is terminated after a certain, definite, length.

With the concept of infinity, the idea that the process never terminates is not crucial; the crucial thing is that you reach an independence  from the length of a sequence. Let me give you an example.

Consider the sequence for which the n-th term is given by the formula:

S_n = \dfrac{1}{n}.

Thus, the sequence is: 1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dots.

If we take first two terms, we can see that the value has decreased, from 1 to 0.5. If we go from the second to the third term, we can see that the value has decreased even further, to 0.3333. The difference in the decrement has, however, dropped; it has gone from 1 - \dfrac{1}{2} = 0.5 to \dfrac{1}{2} - \dfrac{1}{3} =  0.1666666\dots. Go from the third to the fourth term, and we can see that while the value goes still down, and the decrement itself also has decreased, it has now become 0.08333 . Thus, two trends are unmistakable: (i) the value keeps dropping, but (ii) the decrement also becomes sluggish.  If the values were to drop uniformly, i.e. if the decrement were to stay the same, we would have immediately hit 0, and then gone on to the negative numbers. But the second factor, viz., that the decrement itself is progressively decreasing, seems to play a trick. It seems intent on keeping you afloat, above the 0 value. We can verify this fact. No matter how big n might get, it still is a finite number, and so, its reciprocal is always going to be a finite number, not zero. At the same time, we now have observed that the differences between the subsequent reciprocals has been decreasing. How can we capture this intuition? What we want to say is this: As you go further and further down in the sequence, the value must become smaller and ever smaller. It would never actually become 0. But it will approach 0 (and no number other than 0) better and still better. Take any small but definite positive number, and we can say that our sequence would eventually drop down below the level of that number, in a finite number of steps. We can say this thing for any given definite positive number, no matter how small. So long as it is a definite number, we are going to hit its level in a finite number of steps. But we also know that since n is positive, our sequence is never going to go so far down as to reach into the regime of the negative numbers. In fact, as we just said, let alone the range of the negative numbers, our sequence is not going to hit even 0, in finite number of steps.

To capture all these facts, viz.: (i) We will always go below the level any positive real number R, no matter how small R may be, in a finite number of steps, (ii) the number of steps n required to go below a specified R level would always go on increasing as R becomes smaller, and (iii) we will never reach 0 in any finite number of steps no matter how large n may get, but will always experience decrement with increasing n, we say that:

the limit of the sequence S_n as n approaches infinity is 0.

The word “infinity” in the above description crucially refers to the facts (i) and (ii), which together clearly establish the trend in the values of the sequence S_n. [The fact (iii) is incidental to the idea of “infinity” itself, though it brings out a neat property of limits, viz., the fact that the limit need not always belong to the set of numbers that is the sequence itself. ]


With the development of mathematical knowledge, the idea of numbers does undergo changes. The concept number gets more and more complex/sophisticated, as the process of measurement becomes more and more complex/sophisticated.

We can form the process of addition starting from the process of counting.

The simplest addition is that of adding a unit (or the number 1) to a given number. We can apply the process of addition by 1, to the number 1, and see that the number we thus arrive at is 2. Then we can apply the process of addition by 1, to the number 2, and see that the number we thus arrive at is 3. We can continue to apply the logic further, and thereby see that it is possible to generate any desired natural number.

The so-called natural numbers thus state the sizes of groups of identical objects, as measured via the process of counting. Since natural numbers encapsulate the sizes of such groups, they obviously can be ordered by the sizes they encapsulate. One way to see how the order 1, then 2, then 3, \dots, arises is to observe that in successively applying the process of addition starting from the number 1, it is the number 2 which comes immediately after the number $1$, but before the number 3, etc.

The process of subtraction is formed by inverting the process of addition, i.e., by seeing the logic of addition in a certain, reverse, way.

The process of addition by 1, when repeatedly applied to a given natural number, is capable of generating all the natural numbers greater than the given number. The process of subtraction by 1, when repeatedly applied to a given natural number, is capable of generating all the natural numbers smaller than the given number.

When the process of subtraction by 1 is applied right to the number 1 itself, we reach the idea of the zero. [Dear Indian, now you know that the idea of the number zero was not all that breath-taking, was it?]

In a further development, the idea of the negative numbers is established.


Thus, the concept of numbers develops from the natural numbers (1, 2, 3, \dots) to whole numbers (0, 1, 2, \dots) to integers (\dots, -2, -1, 0, 1, 2, \dots).

At each such a stage, the idea of what a number means—its definition—undergoes a definite change; at any such a stage, there is a well-defined mathematical process, of increasing conceptual complexity, of measuring sizes, whose end-products that idea of numbers represents.


The idea of multiplication follows from that of repeated additions; the idea of division follows from that of the repeated subtractions; the two process are then recognized as the multiplicative inverses of each other. It’s only then that the idea of fractions follows. The distinction between the rational and irrational fractions is then recognized, and then, the concept of numbers gets extended to include the idea of the irrational as well as rational numbers.

A crucial lesson learnt from this entire expansion of knowledge of what it means to be a number, is the recognition of the fact that for any well-defined and completed process of measurement, there must follow a certain number (and only that unique number, obviously!).


Then, in a further, distinct, development, we come to recognize that while some process must exist to produce a number, any well-defined process producing a number would do just as well.

With this realization, we then come to a stage whereby, we can think of conceptually omitting specifying any specific process of measurement.

We thus come to retain only the fact while some process must be specified, any valid process can be, and then, the end-product still would be just a number.

It is with this realization that we come to reach the idea of the real numbers.


The purpose of forming the idea of real numbers is that they allow us to form statements that would hold true for any number qua a number.


The crux of the distinction of the real numbers from any of the preceding notion of numbers (natural, whole, integers) is the following statement, which can be applied to real numbers, and only to real numbers—not to integers.

The statement is this: there is an infinity of real numbers existing between any two distinct real numbers R_1 and R_2, no matter how close they might be to each other.

There is a wealth of information contained in that statement, but if some aspects are to be highlighted and appreciated more than the others, they would be these:

(i) Each of the two numbers R_1 and R_2 are recognized as being an end-product of some or the other well-defined process.

The responsibility of specifying what precise size is meant when you say R_1 or R_2 is left entirely up to you; the definition of real numbers does not take that burden. It only specifies that some well-defined process must exist to produce R_1 as well as R_2, so that what they denote indeed are numbers.

A mathematical process may produce a result that corresponds to a so-called “irrational” number, and yet, it can be a definite process. For instance, you may specify the size-measurement process thus: hold in a compass the distance equal to the diagonal of a right-angled isoscales triangle having the equal sides of 1, and mark this distance out from the origin on the real number-line. This measurement process is well-specified even if \sqrt{2} can be proved to be an irrational number.

(ii) You don’t have to specify any particular measurement process which might produce a number strictly in between R_1 and R_2, to assert that it’s a number. This part is crucial to understand the concept of real numbers.

The real numbers get all their power precisely because their idea brings into the jurisdiction of the concept of numbers not only all those specific definitions of numbers that have been invented thus far, but also all those definitions which ever possibly would be. That’s the crucial part to understand.

The crucial part is not the fact that there are an infinity of numbers lying between any two R_1 and R_2. In fact, the existence of an infinity of numbers is damn easy to prove: just take the average of R_1 and R_2 and show that it must fall strictly in between them—in fact, it divides the line-segment from R_1 to R_2 into two equal halves. Then, take each half separately, and take the average of its end-points to hit the middle point of that half. In the first step, you go from one line-segment to two (i.e., you produce one new number that is the average). In the next step, you go from the two segments to the four (i.e. in all, three new numbers). Now, go easy; wash-rinse-repeat! … The number of the numbers lying strictly between R_1 and R_2 increases without bound—i.e., it blows “up to” infinity. [Why not “down to” infinity? Simple: God is up in his heavens, and so, we naturally consider the natural numbers rather than the negative integers, first!]

Since the proof is this simple, obviously, it just cannot be the real meat, it just cannot be the real reason why the idea of real numbers is at all required.

The crucial thing to realize here now is this part: Even if you don’t specify any specific process like hitting the mid-point of the line-segment by taking average, there still would be an infinity of numbers between the end-points.


Another closely related and crucial thing to realize is this part: No matter what measurement (i.e. number-producing) process you conceive of, if it is capable of producing a new number that lies strictly between the two bounds, then the set of real numbers has already included it.

Got it? No? Go read that line again. It’s important.

This idea that

“all possible numbers have already been subsumed in the real numbers set”

has not been proven, nor can it be—not on the basis of any of the previous notions of what it means to be a number. In fact, it cannot be proven on the basis of any well-defined (i.e. specified) notion of what it means to be a number. So long as a number-producing process is specified, it is known, by the very definition of real numbers, that that process would not exhaust all real numbers. Why?

Simple. Because, someone can always spin out yet another specific process that generates a different set of numbers, which all would still belong only to the real number system, and your prior process didn’t cover those numbers.

So, the statement cannot be proven on the basis of any specified system of producing numbers.

Formally, this is precisely what [I think] is the issue at the core of the “continuum hypothesis.”

The continuum hypothesis is just a way of formalizing the mathematician’s confidence that a set of numbers such as real numbers can at all be defined, that a concept that includes all possible numbers does have its uses in theory of measurements.

You can’t use the ideas like some already defined notions of numbers in order to prove the continuum hypothesis, because the hypothesis itself is at the base of what it at all means to be a number, when the term is taken in its broadest possible sense.


But why would mathematicians think of such a notion in the first place?

Primarily, so that those numbers which are defined only as the limits (known or unknown, whether translatable using the already known operations of mathematics or otherwise) of some infinite processes can also be treated as proper numbers.

And hence, dramatically, infinite processes also can be used for measuring sizes of actual, metaphysically definite and mathematically finite, objects.

Huh? Where’s the catch?

The catch is that these infinite processes must have limits (i.e., they must have finite numbers as their output); that’s all! (LOL!).


It is often said that the idea of real numbers is a bridge between algebra and geometry, that it’s the counterpart in algebra of what the geometer means by his continuous curve.

True, but not quite hitting the bull’s eye. Continuity is a notion that geometer himself cannot grasp or state well unless when aided by the ideas of the calculus.

Therefore, a somewhat better statement is this: the idea of the real numbers is a bridge between algebra and calculus.

OK, an improvement, but still, it, too, misses the mark.

The real statement is this:

The idea of real numbers provides the grounds in algebra (and in turn, in the arithmetics) so that the (more abstract) methods such as those of the calculus (or of any future method that can ever get invented for measuring sizes) already become completely well-defined qua producers of numbers.

The function of the real number system is, in a way, to just go nuts, just fill the gaps that are (or even would ever be) left by any possible number system.


In the preceding discussion, we had freely made use of the 1:1 correspondence between the real numbers and the beloved continuous curve of our school-time geometry.

This correspondence was not always as obvious as it is today; in fact, it was a towering achievement of, I guess, Descartes. I mean to say, the algebra-ization of geometry.

In the simplest (1D) case, points on a line can be put in 1:1 correspondence with real numbers, and vice-versa. Thus, for every real number there is one and only one point on the real-number line, and for any point actually (i.e. well-) specified on the real number-line, there is one and only one real number corresponding to it.

But the crucial advancement represented by the idea of real numbers is not that there is this correspondence between numbers (an algebraic concept) and geometry.

The crux is this: you can (or, rather, you are left free to) think of any possible process that ends up cutting a given line segment into two (not necessarily equal) halves, and regardless of the particular nature of that process, indeed, without even having to know anything about its particular nature, we can still make a blanket statement:

if the process terminates and ends up cutting the line segment at a certain geometrical point, then the number which corresponds to that geometrical point is already included in the infinite set of real numbers.


Since the set of real numbers exhausts all possible end-products of all possible infinite limiting processes too, it is fully capable of representing any kind of a continuous change.


We in engineering often model the physical reality using the notion of the continuum.

Inasmuch as it’s a fact that to any arbitrary but finite part of a continuum there does correspond a number, when we have the real number system at hand, we already know that this size is already included in the set of real numbers.

Real numbers are indispensable to us the engineers—theoretically speaking. It gives us the freedom to invent any new mathematical methods for quantitatively dealing with continua, by giving us the confidence that all that they would produce, if valid, is already included in the numbers-set we already use; that our numbers-set will never ever let us down, that it will never ever fall short, that we will never ever fall in between the two stools, so to speak. Yes, we could use even the infinite processes, such as those of the calculus, with confidence, so long as they are limiting.

That’s the [theoretical] confidence which the real number system brings us [the engineers].


A Song I Don’t Like:

[Here is a song I don’t like, didn’t ever like, and what’s more, I am confident, I would never ever like either. No, neither this part of it nor that. I don’t like any part of it, whether the partition is made “integer”-ly, or “real”ly.

Hence my confidence. I just don’t like it.

But a lot of Indian [some would say “retards”] do, I do acknowledge this part. To wit [^].

But to repeat: no, I didn’t, don’t, and wouldn’t ever like it. Neither in its 1st avataar, nor in the 2nd, nor even in an hypothetically \pi-th avataar. Teaser: Can we use a transcendental irrational number to denote the stage of iteration? Are fractional derivatives possible?

OK, coming back to the song itself. Go ahead, listen to it, and you will immediately come to know why I wouldn’t like it.]

(Hindi) “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 \n …” [OK, yes, read the finite sequence before the newline character, using Hindi.]
Credits: [You go hunt for them. I really don’t like it.]


PS: As usual, I may come back and make this post even better. BTW, in the meanwhile, I am thinking of relying on my more junior colleagues to keep me on the track towards delivering on the promised CFD FDP. Bye for now, and take care…

 

I am keeping my New Year’s…

I am keeping my NYR [^], made last year.

How about you?


No, really. I AM keeping my NYR. Here’s how.


December is meant for making resolutions. (It doesn’t matter whether it’s the 1st or the 31st; the month is [the?] December; that’s what matters.)

Done.

January is meant for making a time-table. … But it must be something on which you can execute. I have been actively engaged doing that. … You could see that, couldn’t you? … And, what’s more, you could’ve bet about it at any time in the past, too, couldn’t you?

Since execution can only follow, and not precede, planning, it must be February before execution proper itself can begin. As far as I am concerned, I will make sure of that. [And you know me. You know that I always deliver on all my promises, don’t you?]

March is known for madness. To be avoided, of course.

April is known for foolishness. To be avoided, as far as possible, but, hey, as “friends” of this blog, you know, it’s nothing to be afraid of!

May, in this part of the world, is far too hot for any one to handle it right, OK? The work-efficiency naturally drops down. This fact must be factored into any and every good piece of planning, I say! (Recall the British Governors and Other officers of the Bombay Presidency shifting their offices to Matheran/Mahabaleshwar during summer? [Anyone ever cared to measure the efficiency of this measure on their part? I mean, on work?])

Now, yes, June does bring in the [very welcome] monsoons, finally! But then, monsoon also is universally known to be the most romantic of all seasons. [It leaves a certain something of a feeling which ordinarily would require you to down a Sundowner or so. [I am trying to be honest, here!]… And then, even Kalidas would seem to agree. Remember (Sanskrit) “aashaaDhasya pratham…”? Naturally, the month is not very conducive to work, is it?]


OK.


This is [just] January, and my time-table is all done up and ready. Or, at least, it’s [at least] half-way through. …

I will really, really begin work in the second half of the year.

Bye until then.


A Song I Don’t Ever Recall Liking Back Then [When Things Mattered Far More Routinely in Far More Respects than They Do Today]

[Not too sure I like it today either. But there were certain happy isolated instances related to my more recent career which are associated with it. I had registered, but hadn’t known this fact, until recently.

But then, recently, I happened suddenly to “re-hear” the phrase (Hindi) “yeh kaunsaa…”, complete with the piece of the “sax” which follows it…

Then, the world had become [in a [comparatively] recent past] a slightly better place to live in.

So, I’d decided, not quite fully certain but still being inclined to this possibility, that I might actually like this song. … But I still don’t fully, you know… But I still do fully want to run it, you know…

Anyway, just listen to it…]

(Hindi) “chocolate, lime juice, ice-cream…” [No, it really is a Hindi song. Just listen to it further…]
Singer: Lata Mangeshkar [A peculiarity of this song is that precisely when [an aged] Lata sounds [a bit] heavy [of course due to the age not to mention the pressures of the day-to-day work and every one’s normal inability to hit the sweet spot every time!], the directors of the movie and the music together focus your attention on a rather cheerfully smiling and dancing Madhuri. [No, never been one of my favorite actresses, but then, that’s an entirely different story altogether.]]
Music: Ramlaxman
Lyrics: Dev Kohli [?]

[PS: And, coming to the video of this song, did you notice that the hero drives a Maruti Gypsy?

I mean, ask any NRI in USA, and they he will tell you that it was because this was an early 90’s movie; the fruits of the [half-/quarter-/oct-something-/etc.] economic liberalization had still not been had by the general public; the liberalization they [I mean these NRIs] had brought about.

If these [I mean the economic freedoms] were to be brought about , they could easily point out, with good amount of references to Hindi movies of the recent years, that the presence on Indian roads of the [government-subsidized-diesel-driven] SUVs could easily have been seen in the same movie!!!

Hmmm…  Point[s] taken.]


How about your NYR?


[A bit of an editing is still due, I am sure… TBD, when I get the time to do so…]

Yes I know it!

Note: A long update was posted on 12th December 2017, 11:35 IST.


This post is spurred by my browsing of certain twitter feeds of certain pop-sci. writers.

The URL being highlighted—and it would be, say, “negligible,” but for the reputation of the Web domain name on which it appears—is this: [^].


I want to remind you that I know the answers to all the essential quantum mysteries.

Not only that, I also want to remind you that I can discuss about them, in person.

It’s just that my circumstances—past, and present (though I don’t know about future)—which compel me to say, definitely, that I am not available for writing it down for you (i.e. for the layman) whether here or elsewhere, as of now. Neither am I available for discussions on Skype, or via video conferencing, or with whatever “remoting” mode you have in mind. Uh… Yes… WhatsApp? Include it, too. Or something—anything—like that. Whether such requests come from some millionaire Indian in USA (and there are tons of them out there), or otherwise. Nope. A flat no is the answer for all such requests. They are out of question, bounds… At least for now.

… Things may change in future, but at least for the time being, the discussions would have to be with those who already have studied (the non-relativistic) quantum physics as it is taught in universities, up to graduate (PhD) level.

And, you have to have discussions in person. That’s the firm condition being set (for the gain of their knowledge 🙂 ).


Just wanted to remind you, that’s all!


Update on 12th December 2017, 11:35 AM IST:

I have moved the update to a new post.

 


A Song I Like:

(Western, Instrumental) “Berlin Melody”
Credits: Billy Vaughn

[The same 45 RPM thingie [as in here [^], and here [^]] . … I was always unsure whether I liked this one better or the “Come September” one. … Guess, after the n-th thought, that it was this one. There is an odd-even thing about it. For odd ‘n” I think this one is better. For even ‘n’, I think the “Come September” is better.

… And then, there also are a few more musical goodies which came my way during that vacation, and I will make sure that they find their way to you too….

Actually, it’s not the simple odd-even thing. The maths here is more complicated than just the binary logic. It’s an n-ary logic. And, I am “equally” divided among them all. (4+ decades later, I still remain divided.)… (But perhaps the “best” of them was a Marathi one, though it clearly showed a best sort of a learning coming from also the Western music. I will share it the next time.)]


[As usual, may be, another revision [?]… Is it due? Yes, one was due. Have edited streamlined the main post, and then, also added a long update on 12th December 2017, as noted above.]

 

 

An INALFO in my backyard

This being a Diwali day, I got up a bit earlier than usual this morning, and as I lazily shuffled out of my bed, as a routine first thing I do, I stumbled my way out to the smallish balcony/porch of our ground-floor flat [i.e. apartment].

While still being in that state of being half-hazy and half awake, in the dim light of the early morning, I spotted an Unidentified Object lying on the ground just 3–4 feet away from me, in the (very smallish) backyard of our house.

It was some 2–3 feet in size; the color was a pink.

Initially, I thought that it could be some piece of a baby clothing that got dropped from one of the flats above ours. Then I thought that it could be a smallish mosquito-net they use for infants. But then, soon enough, I realized that no one lives in the flat directly above ours, and with all the flower-trees and shrubs they have planted around in this backyard, the probability of a piece of clothing—a fairly heavy object—merely getting drifted away in the wind and landing precisely into that very smallish patch of the ground which lies in front of our flat, was very low. After all, there are a couple of 5–7 feet tall trees here, in our patch of the backyard. And, so, the question of how that piece of clothing got landed here was not a very easy one to figure out. … I yawned, decided to think about it later, and turned around to go inside, pick up my tooth-brush.

Some one or one-and-a-half hour later or so, I was in the balcony once again, and checked it out once again. … Whatever it was, I had to go out, pick it up, and may be alert my neighbours later in the day, I thought. The Sun had already been up, and the sky had got brightened up quite well by now.

More important, I had by now become fully awake.

In fact, a few minutes earlier, while sipping up my cup of coffee, while sitting in that same balcony, even “looking at” the same object, I had already thought of some small thing about QM, and so, I had by now picked up this book on QM by Eisberg and Resnick. I was trying to locate the thing I wanted to check out, its presentation in this book.

Yet, at the same time, the Unidentified Object lying out there wouldn’t let me go through the book right. That’s how, I had now decided to check that thing out there, first.

I had to get up from my chair and check it out. So, I leaned out of the balcony a bit, and had a good look at it. Under the brightened sky, and with me not just looking at it but also with my mental focus on it, I could now immediately recognize it for it was.

It was a very specific kind of a Flying Object.

Thus, the object had undergone a direct transformation from being (i) an Unidentified Object, to (i) a specific kind of a Flying Object. [Too bad, there never was an intermediate state of its formally being an Unidentified Flying Object.]

I anyway decided to take a snap of this INALFO (Identified, Not Any Longer Flying, Object). Here it is:

It sure was a Diwali-time balloon (with the hot air generated by a burning candle) of the kind they send it up in the sky at night.

OK. The UFO part of this post is now over.


I then slipped on my chappals and stepped out of our flat. As I picked this balloon up, I realized that something had been printed on the other side of it.

It turned out to be a message of love: there was a screen-printed outline of a rose flower, and also the words appearing in the capital letters: “I LOVE.” … The screen-printer had done a poor job of printing, and so, the “YOU” part had got only partially printed; it was almost completely unreadable.

It was funny, I thought. Right on the morning of the most important day of Diwali, I had happened to have received, literally, a message of love from the heavens. … There was this bit of that “family” / “Greeting Cards” / “Gift from the Heavens” / “TV serials” / “Hindi TV News Channels” / “Indians” sort of a drama built into it—the kind of a drama that everyone seems to so much love or at least appreciate, and, simultaneously, none has any actual use for.

Then, to let you gauge the size of the balloon, I placed my copy of the QM book over it (which I happened to be still carrying in my hand, absentmindedly, even in the act of stepping out). I then took another snap, which is here:

OK. Now… Taking a second look at this photograph…. I mean… if you take just this one photograph, and try to decipher the message contained in this photograph by reading it very straight-forwardly…

… Well, yes, I do LOVE QUANTUM PHYSICS, don’t I? Hadn’t the message arrived from the heavens only so I come to “learn” and “realize” and “appreciate” this part about myself? What do you think?


OK. The Mars-Man [^] [^] part of this post, too, is over.

But, have a Happy Diwali anyway!

And, as to me, well, let me now get back to the business of the accreditation-related work (which I’ve had to take home this Diwali season). Bye for now.

Incidentals—Part 1

Yes, I have been working hard, very hard, and have been managing a responsibility, a very difficult and demanding and emotionally very draining a responsibility in a singular capacity, and yes, I have been having problems with people—their irrationalities. And, the irrational scripts they follow. [Mind you, the reference is to scripts and not to scriptures.]


But, just check this out this one, for instance (and, the people I have in mind in the above section wouldn’t do that, I am sure; they never do pursue links from such posts of mine, especially if they are just Indians—they are just self-confident, that’s all): [^].

But, leaving them aside—and I find it very, very easy do that at least in a moment like this—here is a suggestion: For tomorrow and the day after, and may be for a week or so, watch out the physics (esp. astronomy-related) Twitter-feeds, news-feeds, even blogs [complete with words like “kicking” people and all [Indians with ability to speak in English regard it as “unparliamentary,” together with words like “bloody”].


For obvious reasons, no “A Song I Like” section for this posts The news I am reporting about is exciting enough, all by itself.

Best,

–Ajit

Update on 21:35 IST the same day:

A couple of related posts are these: [^][^]

A Song I Like:

(Hindi) “meghaa chhaaye aadhee raat…”
Lyrics: Neeraj
Music: S. D. Burman
Singer: Lata Mangeshkar

 

 

Fluxes, scalars, vectors, tensors…. and, running in circles about them!

0. This post is written for those who know something about Thermal Engineering (i.e., fluid dynamics, heat transfer, and transport phenomena) say up to the UG level at least. [A knowledge of Design Engineering, in particular, the tensors as they appear in solid mechanics, would be helpful to have but not necessary. After all, contrary to what many UGC and AICTE-approved (Full) Professors of Mechanical Engineering teaching ME (Mech – Design Engineering) courses in SPPU and other Indian universities believe, tensors not only appear also in fluid mechanics, but, in fact, the fluids phenomena make it (only so slightly) easier to understand this concept. [But all these cartoons characters, even if they don’t know even this plain and simple a fact, can always be fully relied (by anyone) about raising objections about my Metallurgy background, when it comes to my own approval, at any time! [Indians!!]]]

In this post, I write a bit about the following question:

Why is the flux \vec{J} of a scalar \phi a vector quantity, and not a mere number (which is aka a “scalar,” in certain contexts)? Why is it not a tensor—whatever the hell the term means, physically?

And, what is the best way to define a flux vector anyway?


1.

One easy answer is that if the flux is a vector, then we can establish a flux-gradient relationship. Such relationships happen to appear as statements of physical laws in all the disciplines wherever the idea of a continuum was found useful. So the scope of the applicability of the flux-gradient relationships is very vast.

The reason to define the flux as a vector, then, becomes: because the gradient of a scalar field is a vector field, that’s why.

But this answer only tells us about one of the end-purposes of the concept, viz., how it can be used. And then the answer provided is: for the formulation of a physical law. But this answer tells us nothing by way of the very meaning of the concept of flux itself.


2.

Another easy answer is that if it is a vector quantity, then it simplifies the maths involved. Instead of remembering having to take the right \theta and then multiplying the relevant scalar quantity by the \cos of this \theta, we can more succinctly write:

q = \vec{J} \cdot \vec{S} (Eq. 1)

where q is the quantity of \phi, an intensive scalar property of the fluid flowing across a given finite surface, \vec{S}, and \vec{J} is the flux of \Phi, the extensive quantity corresponding to the intensive quantity \phi.

However, apart from being a mere convenience of notation—a useful shorthand—this answer once again touches only on the end-purpose, viz., the fact that the idea of flux can be used to calculate the amount q of the transported property \Phi.

There also is another problem with this, second, answer.

Notice that in Eq. 1, \vec{J} has not been defined independently of the “dotting” operation.

If you have an equation in which the very quantity to be defined itself has an operator acting on it on one side of an equation, and then, if a suitable anti- or inverse-operator is available, then you can apply the inverse operator on both sides of the equation, and thereby “free-up” the quantity to be defined itself. This way, the quantity to be defined becomes available all by itself, and so, its definition in terms of certain hierarchically preceding other quantities also becomes straight-forward.

OK, the description looks more complex than it is, so let me illustrate it with a concrete example.

Suppose you want to define some vector \vec{T}, but the only basic equation available to you is:

\vec{R} = \int \text{d} x \vec{T}, (Eq. 2)

assuming that \vec{T} is a function of position x.

In Eq. 2, first, the integral operator must operate on \vec{T}(x) so as to produce some other quantity, here, \vec{R}. Thus, Eq. 2 can be taken as a definition for \vec{R}, but not for \vec{T}.

However, fortunately, a suitable inverse operator is available here; the inverse of integration is differentiation. So, what we do is to apply this inverse operator on both sides. On the right hand-side, it acts to let \vec{T} be free of any operator, to give you:

\dfrac{\text{d}\vec{R}}{\text{d}x} = \vec{T} (Eq. 3)

It is the Eq. 3 which can now be used as a definition of \vec{T}.

In principle, you don’t have to go to Eq. 3. In principle, you could perhaps venture to use a bit of notation abuse (the way the good folks in the calculus of variations and integral transforms always did), and say that the Eq. 2 itself is fully acceptable as a definition of \vec{T}. IMO, despite the appeal to “principles”, it still is an abuse of notation. However, I can see that the argument does have at least some point about it.

But the real trouble with using Eq. 1 (reproduced below)

q = \vec{J} \cdot \vec{S} (Eq. 1)

as a definition for \vec{J} is that no suitable inverse operator exists when it comes to the dot operator.


3.

Let’s try another way to attempt defining the flux vector, and see what it leads to. This approach goes via the following equation:

\vec{J} \equiv \dfrac{q}{|\vec{S}|} \hat{n} (Eq. 4)

where \hat{n} is the unit normal to the surface \vec{S}, defined thus:

\hat{n} \equiv \dfrac{\vec{S}}{|\vec{S}|} (Eq. 5)

Then, as the crucial next step, we introduce one more equation for q, one that is independent of \vec{J}. For phenomena involving fluid flows, this extra equation is quite simple to find:

q = \phi \rho \dfrac{\Omega_{\text{traced}}}{\Delta t} (Eq. 6)

where \phi is the mass-density of \Phi (the scalar field whose flux we want to define), \rho is the volume-density of mass itself, and \Omega_{\text{traced}} is the volume that is imaginarily traced by that specific portion of fluid which has imaginarily flowed across the surface \vec{S} in an arbitrary but small interval of time \Delta t. Notice that \Phi is the extensive scalar property being transported via the fluid flow across the given surface, whereas \phi is the corresponding intensive quantity.

Now express \Omega_{\text{traced}} in terms of the imagined maximum normal distance from the plane \vec{S} up to which the forward moving front is found extended after \Delta t. Thus,

\Omega_{\text{traced}} = \xi |\vec{S}| (Eq. 7)

where \xi is the traced distance (measured in a direction normal to \vec{S}). Now, using the geometric property for the area of parallelograms, we have that:

\xi = \delta \cos\theta (Eq. 8)

where \delta is the traced distance in the direction of the flow, and \theta is the angle between the unit normal to the plane \hat{n} and the flow velocity vector \vec{U}. Using vector notation, Eq. 8 can be expressed as:

\xi = \vec{\delta} \cdot \hat{n} (Eq. 9)

Now, by definition of \vec{U}:

\vec{\delta} = \vec{U} \Delta t, (Eq. 10)

Substituting Eq. 10 into Eq. 9, we get:

\xi = \vec{U} \Delta t \cdot \hat{n} (Eq. 11)

Substituting Eq. 11 into Eq. 7, we get:

\Omega_{\text{traced}} = \vec{U} \Delta t \cdot \hat{n} |\vec{S}| (Eq. 12)

Substituting Eq. 12 into Eq. 6, we get:

q = \phi \rho \dfrac{\vec{U} \Delta t \cdot \hat{n} |\vec{S}|}{\Delta t} (Eq. 13)

Cancelling out the \Delta t, Eq. 13 becomes:

q = \phi \rho \vec{U} \cdot \hat{n} |\vec{S}| (Eq. 14)

Having got an expression for q that is independent of \vec{J}, we can now use it in order to define \vec{J}. Thus, substituting Eq. 14 into Eq. 4:

\vec{J} \equiv \dfrac{q}{|\vec{S}|} \hat{n} = \dfrac{\phi \rho \vec{U} \cdot \hat{n} |\vec{S}|}{|\vec{S}|} \hat{n} (Eq. 16)

Cancelling out the two |\vec{S}|s (because it’s a scalar—you can always divide any term by a scalar (or even  by a complex number) but not by a vector), we finally get:

\vec{J} \equiv \phi \rho \vec{U} \cdot \hat{n} \hat{n} (Eq. 17)


4. Comments on Eq. 17

In Eq. 17, there is this curious sequence: \hat{n} \hat{n}.

It’s a sequence of two vectors, but the vectors apparently are not connected by any of the operators that are taught in the Engineering Maths courses on vector algebra and calculus—there is neither the dot (\cdot) operator nor the cross \times operator appearing in between the two \hat{n}s.

But, for the time being, let’s not get too much perturbed by the weird-looking sequence. For the time being, you can mentally insert parentheses like these:

\vec{J} \equiv \left[ \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \right) \right] \hat{n} (Eq. 18)

and see that each of the two terms within the parentheses is a vector, and that these two vectors are connected by a dot operator so that the terms within the square brackets all evaluate to a scalar. According to Eq. 18, the scalar magnitude of the flux vector is:

|\vec{J}| = \left( \phi \rho \vec{U}\right) \cdot \left( \hat{n} \right) (Eq. 19)

and its direction is given by: \hat{n} (the second one, i.e., the one which appears in Eq. 18 but not in Eq. 19).


5.

We explained away our difficulty about Eq. 17 by inserting parentheses at suitable places. But this procedure of inserting mere parentheses looks, by itself, conceptually very attractive, doesn’t it?

If by not changing any of the quantities or the order in which they appear, and if by just inserting parentheses, an equation somehow begins to make perfect sense (i.e., if it seems to acquire a good physical meaning), then we have to wonder:

Since it is possible to insert parentheses in Eq. 17 in some other way, in some other places—to group the quantities in some other way—what physical meaning would such an alternative grouping have?

That’s a delectable possibility, potentially opening new vistas of physico-mathematical reasonings for us. So, let’s pursue it a bit.

What if the parentheses were to be inserted the following way?:

\vec{J} \equiv \left( \hat{n} \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 20)

On the right hand-side, the terms in the second set of parentheses evaluate to a vector, as usual. However, the terms in the first set of parentheses are special.

The fact of the matter is, there is an implicit operator connecting the two vectors, and if it is made explicit, Eq. 20 would rather be written as:

\vec{J} \equiv \left( \hat{n} \otimes \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 21)

The \otimes operator, as it so happens, is a binary operator that operates on two vectors (which in general need not necessarily be one and the same vector as is the case here, and whose order with respect to the operator does matter). It produces a new mathematical object called the tensor.

The general form of Eq. 21 is like the following:

\vec{V} = \vec{\vec{T}} \cdot \vec{U} (Eq. 22)

where we have put two arrows on the top of the tensor, to bring out the idea that it has something to do with two vectors (in a certain order). Eq. 22 may be read as the following: Begin with an input vector \vec{U}. When it is multiplied by the tensor \vec{\vec{T}}, we get another vector, the output vector: \vec{V}. The tensor quantity \vec{\vec{T}} is thus a mapping between an arbitrary input vector and its uniquely corresponding output vector. It also may be thought of as a unary operator which accepts a vector on its right hand-side as an input, and transforms it into the corresponding output vector.


6. “Where am I?…”

Now is the time to take a pause and ponder about a few things. Let me begin doing that, by raising a few questions for you:

Q. 6.1:

What kind of a bargain have we ended up with? We wanted to show how the flux of a scalar field \Phi must be a vector. However, in the process, we seem to have adopted an approach which says that the only way the flux—a vector—can at all be defined is in reference to a tensor—a more advanced concept.

Instead of simplifying things, we seem to have ended up complicating the matters. … Have we? really? …Can we keep the physical essentials of the approach all the same and yet, in our definition of the flux vector, don’t have to make a reference to the tensor concept? exactly how?

(Hint: Look at the above development very carefully once again!)

Q. 6.2:

In Eq. 20, we put the parentheses in this way:

\vec{J} \equiv \left( \hat{n} \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 20, reproduced)

What would happen if we were to group the same quantities, but alter the order of the operands for the dot operator?  After all, the dot product is commutative, right? So, we could have easily written Eq. 20 rather as:

\vec{J} \equiv \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \hat{n} \right) (Eq. 21)

What could be the reason why in writing Eq. 20, we might have made the choice we did?

Q. 6.3:

We wanted to define the flux vector for all fluid-mechanical flow phenomena. But in Eq. 21, reproduced below, what we ended up having was the following:

\vec{J} \equiv \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \otimes \hat{n} \right) (Eq. 21, reproduced)

Now, from our knowledge of fluid dynamics, we know that Eq. 21 seemingly stands only for one kind of a flux, namely, the convective flux. But what about the diffusive flux? (To know the difference between the two, consult any good book/course-notes on CFD using FVM, e.g. Jayathi Murthy’s notes at Purdue, or Versteeg and Malasekara’s text.)

Q. 6.4:

Try to pursue this line of thought a bit:

Start with Eq. 1 again:

q = \vec{J} \cdot \vec{S} (Eq. 1, reproduced)

Express \vec{S} as a product of its magnitude and direction:

q = \vec{J} \cdot |\vec{S}| \hat{n} (Eq. 23)

Divide both sides of Eq. 23 by |\vec{S}|:

\dfrac{q}{|\vec{S}|} = \vec{J} \cdot \hat{n} (Eq. 24)

“Multiply” both sides of Eq. 24 by \hat{n}:

\dfrac{q} {|\vec{S}|} \hat{n} = \vec{J} \cdot \hat{n} \hat{n} (Eq. 25)

We seem to have ended up with a tensor once again! (and more rapidly than in the development in section 4. above).

Now, looking at what kind of a change the left hand-side of Eq. 24 undergoes when we “multiply” it by a vector (which is: \hat{n}), can you guess something about what the “multiplication” on the right hand-side by \hat{n} might mean? Here is a hint:

To multiply a scalar by a vector is meaningless, really speaking. First, you need to have a vector space, and then, you are allowed to take any arbitrary vector from that space, and scale it up (without changing its direction) by multiplying it with a number that acts as a scalar. The result at least looks the same as “multiplying” a scalar by a vector.

What then might be happening on the right hand side?

Q.6.5:

Recall your knowledge (i) that vectors can be expressed as single-column or single-row matrices, and (ii) how matrices can be algebraically manipulated, esp. the rules for their multiplications.

Try to put the above developments using an explicit matrix notation.

In particular, pay particular attention to the matrix-algebraic notation for the dot product between a row- or column-vector and a square matrix, and the effect it has on your answer to question Q.6.2. above. [Hint: Try to use the transpose operator if you reach what looks like a dead-end.]

Q.6.6.

Suppose I introduce the following definitions: All single-column matrices are “primary” vectors (whatever the hell it may mean), and all single-row matrices are “dual” vectors (once again, whatever the hell it may mean).

Given these definitions, you can see that any primary vector can be turned into its corresponding dual vector simply by applying the transpose operator to it. Taking the logic to full generality, the entirety of a given primary vector-space can then be transformed into a certain corresponding vector space, called the dual space.

Now, using these definitions, and in reference to the definition of the flux vector via a tensor (Eq. 21), but with the equation now re-cast into the language of matrices, try to identify the physical meaning the concept of “dual” space. [If you fail to, I will sure provide a hint.]

As a part of this exercise, you will also be able to figure out which of the two \hat{n}s forms the “primary” vector space and which \hat{n} forms the dual space, if the tensor product \hat{n}\otimes\hat{n} itself appears (i) before the dot operator or (ii) after the dot operator, in the definition of the flux vector. Knowing the physical meaning for the concept of the dual space of a given vector space, you can then see what the physical meaning of the tensor product of the unit normal vectors (\hat{n}s) is, here.

Over to you. [And also to the UGC/AICTE-Approved Full Professors of Mechanical Engineering in SPPU and in other similar Indian universities. [Indians!!]]

A Song I Like:

[TBD, after I make sure all LaTeX entries have come out right, which may very well be tomorrow or the day after…]

Introducing the world at large to a new concept, viz., “Blog-Filling”—Part 1

I hereby introduce to the world at large, awaiting for it with a withheld breath, a new concept, viz. (which is read as “namely” and not “that is,” though the difference has been lost on the English Newspaper Editors of my current town, apparently, long ago; apparently, out of not only a very poor sense of English, but of equally poor sense of supervision descending here from the likes of Delhi and Mumbai—the two highly despicable towns of India).


The concept itself pertains to the idea of having to fill some column-centimeters (or, column-inches in that deprecated country, viz., USA), with whatever it is that you have to fill with.


The world (including the said USA) has been waiting precisely for such a new concept, and I am particularly glad at having not only announcing it, but also having had developed the requisite skills.


The concept in question may most aptly be named: “Blog-Filling.” Translated into a noun, it reads: a “blog filler.”

This post now is [in case you didn’t already guess] is The Blog Filler. [Guess I might have already announced its arrival, given my psycho-epistemological habits i.e. second natures.]


Ummm… In case you still are found wondering, may I repeat, this post really is a blog-filler.


OK. Honest. I will deliver on the promised count. So, here we go: I mean on the RD+Gulzar+Lata song I had had [and may be, also have had/had had/had have/etc.] promised…


A Song I Like:

(Hindi) “silli hawaa chhoo gayee, sillaa badan chhill_ gayaa”
Credits: Are you so dumb as not to be able to guess even these?
OK. I will tell you what? I will note these down, right here:
Lyrics: Gulzaar
Music: R. D. Burman
Singer: Lata Mangeshkar


A “Philanthropic” Assertion:

Even if you are so dumb, and, as usual, richer-than-me, or an Approved SPPU Mechanical Engineering Faculty (or of Any Other Indian University/AICTE/UGC), as not having been able to even guess it, or, in summary, if you are an American Citizen:

Don’t worry, even if you have not been able to guess it. … It was just a small simple game…

…Continuing on the same lines [which lines, people like me don’t need]: now, take care, and best, and good-bye; I mean it; etc.


Bye for now. Don’t bother me too much.

 

Busy, busy, busy… And will be. (Aka: Random Notings in the Passing)

Have been very busy. [What’s new about that? Read on…]


First, there is that [usual] “busy-ness” on the day job.


Then, Mary Hesse (cf. my last post) does not cover tensor fields.

A tensor is a very neat mathematical structure. Essentially, you get it by taking a Cartesian product of the basis vectors of (a) space(s). A tensor field is a tensor-valued function of, say, the physical (“ambient”) space, itself a vector space and also a vector field.

Yes, that reads like the beginning paragraph of a Wiki article on a mathematical topic. Yes, you got into circles. Mathematicians always do that—esp. to you. … Well, they also try doing that, on me. But, usually, they don’t succeed. … But, yes, it does keep me busy. [Now you know why I’ve been so busy.]


Now, a few other, mostly random, notings in the passing…


As every year, the noise pollution of the Ganapati festival this year, too, has been nothing short of maddening. But this year, it has not been completely maddening. Not at least to me. The reason is, I am out of Pune. [And what a relief it is!]


OK, time to take some cognizance of the usual noises on the QM front. The only way to do that is to pick up the very best among them. … I will do that for you.

The reference is to Roger Schlafly’s latest post: “Looking for new quantum axioms”, here [^]. He in turn makes a reference to a Quanta Mag article [^] by Philip Ball, who in turn makes a reference to the usual kind of QM noises. For the last, I shall not provide you with references. … Then, in his above-cited post, Schlafly also makes a reference to the Czech physicist Lubos Motl’s blog post, here [^].

Schlafly notes that Motl “…adequately trashes it as an anti-quantum crackpot article,” and that he “will not attempt to outdo his [i.e. Motl’s] rant.” Schlafly even states that he agrees with him Motl.

Trailer: I don’t; not completely anyway.

Immediately later, however, Schlafly says quite a remarkable thing, something that is interesting in its own regard:

Instead, I focus on one fallacy at the heart of modern theoretical physics. Under this fallacy, [1] the ideal theory is one that is logically derived from postulates, and [2] where one can have a metaphysical belief in those postulates independent of messy experiments.” [Numbering of the clauses is mine.]

Hmmm…

Yes, [1] is right on, but not [2]. Both the postulates and the belief in them here are of physics; experiments—i.e. [controlled] observations of physical reality—play not just a crucial part; they play the “game-starting” part. Wish Schlafly had noted the distinction between the two clauses.

All in all, I think that, on this issue of Foundations of QM, we all seem to be not talking to each other—we seem to be just looking past each other, so to say. That’s the major reason why the field has been flourishing so damn well. Yet, all in all, I think, Schlafly and Motl are more right about it all than are Ball or the folks he quotes.

But apart from it all, let me say that Schlafly and Motl have been advocating the view that Dirac–von Neumann axioms [^] provide the best possible theoretical organization for the theory of the quantum mechanical phenomena.

I disagree.

My position is that the Dirac-von Neumann axioms have not been done with due care to the scope (and applicability) of all the individual concepts subsuming the different aspects of the quantum physical phenomena. Like all QM physicists of the past century (and continuing with those in this century as well, except for, as far as I know, me!), they confuse on one crucial issue. And that issue is at the heart and the base of the measurement/collapse postulate. Understand that one critical issue well, and the measurement/collapse postulate itself collapses in no time. I can name it—that one critical issue. In fact, it’s just one concept. Just one concept that is already well-known to science, but none thinks of it in the context of Foundations of QM. Not in the right way, anyway. [Meet me in person to learn what it is.]


OK, another thing.

I haven’t yet finished Hesse’s book. [Did you honestly expect me to do that so fast?] That, plus the fact that in my day-job, we would be working even harder, working extra hours (plus may be work on week-ends, as well).

In fact, I have already frozen all my research schedule and put it in the deep freeze section. (Not even on the back-burner, I mean.)

So, allow me to go off the blog once again for yet another 3–4 weeks or so. [And I will do that anyway, even if you don’t allow.]


A Song I Like:

[The value of this song to me is mostly nostalgic; it has some very fond memories of my childhood associated with it. As an added bonus, Shammi Kapoor looks slim(mer than his usual self) in this video, the so-called Part 2 of the song, here [^]—and thereby causes a relatively lesser irritation to the eye. [Yes, sometimes, I do refer to videos too, even in this section.]]

(Hindi) “madahosh hawaa matawaali fizaa”
Lyrics: Farooq Qaisar
Singer: Mohammed Rafi
Music: Shankar-Jaikishan

[BTW, did you guess the RD+Gulzar+Lata song I had alluded to, last time? … May be I will write a post just to note that song. Guess it might make for a  good “blog-filler” sometime during the upcoming several weeks, when I will once again be generally off the blog. … OK, take care, and bye for now….]