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Couldn’t stop giggling seeing this. Nice composition here!

Some folks seriously believed Chechans are from Czechoslovakia! Swaziland neighbors Switzerland? If you thought that was funny, but not as much as Chechans and Czechoslovakia:-)

I was playing a few tricks on a few centrality measures on a few graphs. SO, thought of penning down a quick piece of notes on the notion of centrality (A quick introduction can be found here along with Wikipiedia page on centrality)

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether, say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

I recall hearing this interesting quote. If my memory is correct, I first heard it from Martin Vetterli, who either mentioned this during a talk/class or it was there in the footnote of his forthcoming book. It sounded funny that time, but I didn’t really know to whom this quote is originally attributed to. To my surprise, this has its origin dates to the story man Hamming. Well, why Hamming? Emre Telatar did tell me couple of funny stories about Hamming besides the one famous scream on the computers which eventually led to the discovery of error correcting codes! By the way, Emre is a walking encyclopedia when it comes to stories. He is a treasure in many ways!

Ok, coming back to where we started. Springer has published this nice conversation piece online for free. The title is “Mathematics and Design: Yes, But Will it Fly?”. It is not really a book, but a very interesting conversation discussing the above mentioned quote by Richard Hamming. The preface of the discussion itself couldn’t be more apt, which reads:

“Martin Davis and Matt Insall discuss a quote by Richard W. Hamming about the physical effect of Lebesgue and Riemann integrals and whether it made a difference whether one or the other was used, for example, in the design of an airplane. The gist of Hamming’s quote was that the fine points of mathematical analysis are not relevant to engineering considerations.”

A very fascinating read indeed. An even more fascinating, formal defense on Lebesgue’s right on why aeroplane fly is here. Well, the answer to aeroplane question: here is what Andrew Lewis has to say, “In the event that the reader is consulting this paper in a panic just prior to boarding an airplane, let us answer the question posed in the title of the paper. The answer is, “The question is meaningless as the distinctions between the Riemann and Lebesgue integrals do not, and should not be thought to, contribute to such worldly matters as aircraft design.” However, the salient point is that this is not a valid criticism of the Lebesgue integral.

A few years ago, during undergrad days, myself and  friend Ramani during our lazy 75 paise mini canteen tea outing, were discussing a small riddle. It was motivated from a real world experience from our computer center in NIT Calicut (REC Calicut). In REC those days, we students almost exclusively used rubber slippers (Yes, those Paraqon brand which used to cost 20 rupees or so), usually called by the name ‘chappels’. With that, we were not only comfortable while walking and running around, but we’re equally at ease playing cricket and badminton with the very same foot support; and many other things too, including jogging. Those thin hard rubber slippers used to last an year or more without giving much trouble, other than perhaps an occasional tearing of the rubber tie. In all, we were at peace with that.

But there was an issue, not exclusively for this brand, but for chappals in general (shoes were a luxury of sort in the campus;atleast it wasnt very common). Not for everyone though! If and only if you were fancied of visiting the computer center! Well, computer center wasn’t all that fanciful then, since we were provided with only graphics less Unix terminals (no colour monitors!). You might wonder, huh! what age am I talking about? Besides, Internet and Emails were only taking shape then. Chats and browsing were not quite there yet;Unless you felt a touch inferior to the computer wizkid around, that was not a compelling centre de visite. As, ‘would be‘ electronics and communication engineers we had that occasional inferiority complex!. Computer center was air conditioned and was strictly slippers free. We were expected to keep our valuable slippers outside (no clock room luxury! well that was not a necessity either) before entering to that cooler room, filled with monochromatic terminals. Since most of the chappals dropped outside were alike (in size and also sometimes color) there was a good chance that at the time return, we ended up with a different pair of slippers (Some folks found happy for themselves by a visit to the computer center, just for a pair change, often to an improved lot!).  Sometimes, we ended up having differently colored ones, say left foot white and right foot blue. That wasn’t a problem socially either, as long as you stayed within the campus. It was socially accepted within the walls!

Anyway, coming back to the riddle we were busy conjecturing on. We wanted to automate a clock room. The idea then would be to just deposit the chappals there at random. The clock room work automatically. Upon asking (at the time of return, say) it will select a pair at random and give it to you. Sorry, you cant have a choice. Just accept and hope for the best. We asked the questions:

1) What is the probability that everyone gets their own chappals

2) What is the probability that none of them get their submitted pairs

Assume n number of  people (and hence n pairs). We can assume that, a pair is a single entity (say both left and right slippers are tied and submitted as one) . This simplified the problem to n people n slipper scenario. A simplistic model assumeed that all n people submit their slippers at the same time. We wanted to build that great randomized clocker machine! And we wanted that to work for any n, which means, the algorithm had to be implementable and to work well in expectation!

We had thought and pondered about it for a while, then. In the end, we had found that the first one is easy, but the second one a little harder to generalize for beyond n=10 or something.  As busy undergrads, we left the problem after an hour of discussion, probably until we had finished sipping the tea. Aside, we were busy with many other extra curricular activities including a 3 hour daily cricket match at the lush green international hostel ground. The megadeth team, as we proudly grouped ourselves, the electronics and communication batch hardly missed those cricket matches. We were electronics engineers and had taken pride in ourselves by not really bothered to ask any fellow discrete math or combinatorics folks! That perhaps helped in some sense.  Ramani found management more interesting than those technical details of counting. I am sure he took the right career. Anyway…too much digressing already!

Now, it turns out that, the very same problem is akin to a well known problem in combinatorics. It is called the Hatcheck lady problem. It is fairly easy to solve it using the inclusion exclusion principle. The proof outline is shown below. As I type, memory fetches that discussion,  sitting leg-folded on the cement bench at the REC mini-canteen, perhaps an occasional cool breeze around too. 

The inclusion exclusion principle is the following:

\lvert \bigcup_{i=1}^{n} A_{i} \rvert=\displaystyle\sum_{i=1}^{n}{\lvert A_{i}\rvert}-\displaystyle\sum_{1\le i_{1}<i_{2}\le n}^{n}{\lvert A_{i1}\cap A_{i2} \rvert}+\displaystyle\sum_{1\le i_{1}<i_{2}\le n}^{n}{\lvert A_{i1}\cap A_{i2}\cap A_{i3} \rvert}

                 +\displaystyle\sum_{1\le i_{1}<i_{2}\le n}^{n}{\lvert A_{i1}\cap A_{i2}\cap A_{i3} \rvert}+\ldots+

                 +(-1)^{n-1}{\lvert A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n} \rvert}

The Hatchek lady problem can be stated with a similar story as the random clocker machine. (From Harris, Mossinghoff, Hirst’s book on Combinatorics and Graph Theory)

A lazy professor gives a quiz to a class of n students, then collects the papers, shuffles them, and redistribute them randomly to the class for grading. The professor would prefer that no student receives his or her own paper to grade. What is the probability that this occurs? This indeed is an equivalent statement of the well known Hatcheck lady problem (I guess the exact name come from a hatcheck lady who collects hats and absentmindedly return them)

For Hatcheck lady problem, the probability P(n)=\frac{D(n)}{n!}.

D(n)=n!-\lvert A_{1}\cup A_{2}\ldots\cup A_{n}\rvert=n!-\frac{n!}{1!}+\frac{n!}{2!}-\ldots+(-1)^{n}\frac{n!}{n!}

= n!-\displaystyle\sum_{k=1}^{n}{(-1)^{k-1}\binom{n}{k}(n-k)!}=n!-\displaystyle\sum_{k=1}^{n}{(-1)^{k-1}\frac{n!}{k!}}

P(n)= 1-\displaystyle\sum_{k=1}^{n}{(-1)^{k-1}\frac{1}{k!}}

When n gets larger and larger it converges asymptotically to a constant!

\displaystyle\lim_{n\to\infty} P(n)=\displaystyle\lim_{n\to\infty}{\displaystyle \sum_{k=1}^{n}{\frac{1}{k!}}}=\frac{1}{e}

Here is a list of some of my favourite Indian commercial ads.  Thanks to youtube, I get to see them again! The adhisive brand “Fevicol”has produced some inredible and funny ads.  Most of their ads stuck on to the viewers mind.  Among the other funny ads, I liked camlin erasurs and marker ones.  I better dont say too much here. As they say, the fun in an ad is best viewed and chilled out! Have a look and enjoy the fun and appreciate the creativity of these fabulous ad makers.

The old Ericsson mobile phone ad (I guess 1996). The concept of a  “small” phone back in 1996 perhaps is too outdated for today.  All boils down to Moore’s law!

Rimii Sen and Aamir Khan did a nice job here in this Bengali accented conversation.  The coke ad is one of the better ads from the cool drink folks.

Fevicol: Simply superb ad from the popular adhesive brand.

Naukri.com’s famous Hari Sadoo funny ad:

How about this one. To be this is too good an ad from Camlin.

The Peugeot ad used to appear in Channel 4 in UK. It was an incredible ad. I am glad that this is there for viewing in youtube. Superb one.  

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