BTW, remember: (i) this blog is in copyright, (ii) your feedback is welcome.
A song I like:
(Hindi) “agar main kahoon”
Lyrics: Javed Akhtar
Singers: Alka Yagnik, Udit Narayan
Multiple header images, and the problem with them:
As noted in my last post, I have made quite a few changes to the layout of this blog, including adding a “Less transient” page [^].
Another important change was that now, there were header images too, at the top.
Actually, initially, there was only one image. For the record, it was this: [^] However, there weren’t enough equations in it. So, I made another image. It was this [^]. But as I had already noted in the last post, this image was already crowded, and even then, it left out some other equations that I wanted to include.
Then, knowing that WordPress allows multiple images that can be shown at random, I created three images, and uploaded them. These are what is being displayed currently.
However, randomizing means that even after re-loading a page a couple of times, there still is a good chance that you will miss some or the other image, out of those three.
A quick question:
Here is the problem statement:
There are three different header images for this blog. The server shows you only one of them during a single visit. Refreshing the page in the browser also counts as a separate visit. In each visit, the server will once again select an image completely at random.
Assume also that the PDF for the random sequence is uniform. That is to say, there is no greater probability for any of the three images during any visit. Cookies, e.g., play no role.
Now, suppose you make only three visits to this blog. For instance, suppose you visit some page on this blog, and then refresh the same page twice in the browser. The problem is to estimate the chances that you will get to see:
Don’t read further until you solve this problem, right now: right on-the-fly and right in your head (i.e. without using paper and pencil).
(Hint [LOL!]: There are three balls of different colors (say Red, Green, and Blue) in a box, and .)
Ummm… Still with me?
OK. That tells me that you are now qualified to read further.
Just in case you were wondering what was there in the “other” header images, here is a little document I am uploading for you. Go, see it (.PDF [^]), but also note the caveat below.
Caveats: It is a work in progress. If you spot a mistake or even just a typo, then please do let me know. Also, don’t rely on this work.
For example, the definition of stress given in the document is what I have not so far read in any book. So, take it with a pinch of the salt—even if I feel confident that it is correct. Similarly, there might be some other changes, especially those related to the definition of the flux and its usage in the generic equation. Also, I am not sure if the product ansatz for the separation of variables technique began with d’Alembert or not. I vaguely remember its invention being attributed to him, but it was a long time ago, and I am no longer sure. May be it was before him. May be it was much later, at the hands of Fourier, or, even still later, by Lame. … Anyway let it be…
…BTW, the equations in the images currently being shown are slightly different—the PDF document is the latest thing there is.
Also, let me have your suggestions for any further inclusions, too, if any. (As to me: Yes, I would like to add a bit on the finite volume method, too.)
As usual, I may change the PDF document at any time in future. However, the document will always carry the date of compilation as the “version number”.
These days, I am also busy converting my already posted CFD snippets [^] into an FVM-based code.
The earlier posted code was done using FDM, not FVM, but it was not my choice—SPPU (Pune University) had thrust it upon me.
Writing an illustrative code for teaching purposes is fairly simple and straight-forward, esp. in Python—and especially if you treat the numpy arrays exactly as if they were Python arrays!! (That is, very inefficiently.) But I also thought of writing some notes on at least some initial parts of FVM (in a PDF document) to go with the code. That’s why, it is going to take a bit of time.
Once all this work is over, I will also try to model the Schrodinger equation using FVM. … Let’s see how it all goes…
…Alright, time to sign off, already! So, OK, take care and bye for now. …
A Song I Like:
(Hindi) “baharon, mera jeevan bhee savaron…”
Singer: Lata Mangeshkar
Lyrics: Kaifi Aazmi
[The obligatory PS: In all probability, I won’t make any changes to the text of this post. However, the linked PDF document is bound to undergo changes, including addition of new material, reorganization, etc. When I do revise that document, I will note the updates in the post, too.]
The changes at this blog:
In case you haven’t noticed it already, notice [what else?] that the layout of this blog has undergone a change. Hopefully for the better!
In particular, I’ve made the following changes:
A note for reading on the mobile:
In case you read this blog on a mobile phone, then to see the “less transient” page, you will have to press the menu button appearing at the top to get to the new page. On a desktop, however, the menu is by default seen as expanded.
The image at the top:
Just for the record, the equations in the top image, as of today (13 August 2018, 11:31 hrs), are the following:
An important note on the definition of stress as given in the header image:
I haven’t yet seen this definition in any solid/fluid/continuum mechanics text. So, please treat it with caution.
Also, please do drop me a line if you find it erroneous, problematic, or simply not general enough.
On the other hand, if you run into this definition anywhere, then please do bring the reference to my attention; thanks in advance. [This definition is a part of my planned paper on stress and strain.]
Some of the equations that got left out:
The equations which I would have liked to have in the header, but which got left out for a lack of space, are the following (in no particular order):
Perhaps also, things like:
On the other hand, some of the equations that are generally of great importance, but which have not come to preoccupy me a lot, are the following:
I must have left out quite a few more in both the lists.
However, I am sure that the three laws of thermodynamics probably would not make it to the header image, despite all their grandeur, their all-encompassing scope.
The reason is this: a computational modeler like me seldom works in a very direct manner with the laws of thermodynamics themselves. These laws do inform his theory; the derivation of the equations he uses indeed are based on them, even if only indirectly. However, the equations he works with happen to be much more detailed (and of far more delimited scope). For instance: the Navier-Stokes system (CFD)—an expression of the first law; the stress-strain fields (FEM)—which makes for merely a part of the internal energy; or the Maxwell system (FDTD)—ditto. Etc.
Further change may be coming:
All in all, I am not quite happy with the top image as it exists right now. … It is too crowded, and speaking from a visual aesthetics point of view, its layout is not well-balanced.
So, on both these counts (too much crowding already, and too many good equations being left out), I am thinking of a further idea: may be I should create a sequence of images, each containing only a few equations, and let the server show you one of them at random. Whaddaya think?
Do check out the “less transient” page:
But yes, if you are interested, check out the “less transient” page too, and let me know if something I wrote in the past should be there or not.
So… does that mean that I’ve gone “mathy”?
Though I exclusively include only equations in the header image—no pictures or visualizations at all, no code, and not much text either—it doesn’t mean that I have gone “mathy”. … Hell, no! Not at all! … Just check out my less transient page [^].
A song I like:
(Hindi) “aankhon aankhon mein hum tum, ho gaye…”
Singers: Kishore Kumar, Asha Bhosale
Lyrics: Anand Bakshi