Monthly Archives: April 2017

The power of the unseen, the abstract: applications of mathematics

Applications of math are everywhere…anywhere we see, use, test/taste, touch, etc…

I have made a quick compilation of some such examples below:

  1. Crystallography
  2. Coding Theory (Error Correction) (the stuff like Hamming codes, parity check codes; used in 3G, 4G etc.) Used in data storage also. Bar codes, QR codes, etc.
  3. Medicine: MRI, cancer detection, Tomography,etc.
  4. Image processing: JPEG2000; Digital enhancement etc.
  5. Regulating traffic: use of probability theory and queuing theory
  6. Improving performance in sports
  7. Betting and bidding; including spectrum auction using John Nash’s game theory.
  8. Robotics
  9. Space Exploration
  10. Wireless communications including cellular telephony. (You can Google search this; for example, Fourier Series is used in Digital Signal Processing (DSP). Even some concepts of convergence of a series are necessary!) Actually, this is a digital communications systems and each component of this requires heavy use of mathematical machinery: as the information bearing signal is passed from source to sink, it under goes several steps one-by-one: like Source Coding, encryption (like AES, or RSA or ECC), Error Control Coding and Modulation/Transmission via physical channel. On the receiver or sink side, the “opposite” steps are carried out. This is generally taught in Electrical Engineering. You can Google search these things.
  11. DNA Analysis
  12. Exploring oceans (example, with unmanned underwater vehicles)
  13. Packing (physical and electronic)
  14. Aircraft designing
  15. Pattern identification
  16. Weather forecasting.
  17. GPS also uses math. It uses physics also. Perhaps, just to satisfy your curiosity, GPS uses special relativity.
  18. Computer Networks: of course, they use Queuing theory. Long back, the TCP/IP slow start algorithm was designed and developed by van Jacobson.(You can Google search all this — but the stuff is arcande right now due to your current education level.)
  19. Architecture, of course, uses geometry. For example, Golden ratio.
  20. Analyzing fluid flows.
  21. Designing contact lenses for the eyes. Including coloured contact lenses to enhance beauty or for fashion.
  22. Artificial Intelligence and Machine Intelligence.
  23. Internet Security.
  24. Astronomy, of course. Who can ever forget this? Get yourself a nice telescope and get hooked. You can also Stellarium.org freeware to learn to identify stars and planets, and constellations.
  25. Analyzing chaos and fractals: the classic movie “Jurassic Park” was based on fractal geometry. The dino’s were, of course, simulations!
  26. Forensics
  27. Combinatorial optimization; the travelling salesman problem.
  28. Computational Biology

We will try to look at bit deeper into these applications in later blogs. And, yes, before I forget “Ramanujan’s algorithm to compute \pi up to a million digits is used to test the efficacy and efficiency of supercomputers. Of course, there will be other testing procedures also, for testing supercomputers.

There will be several more. Kindly share your views.

-Nalin Pithwa.

Announcement: Scholarships for RMO Training

https://madhavamathcompetition.com/

Mathematics Hothouse.

Somewhere — find out where !!

You have been left in the middle of an island on which there are two villages. In village A, all of the residents, no matter where they are, always tell the truth. In village B, all of the residents, no matter where they are, always tell lies. After walking a few miles from the middle of the island, you find yourself in a village square where there is a resident of one of the villages sitting on some stone steps. What one question would you ask the resident so that you would know which of  the two villages you were in?

PS: Of course, there is no GPS device with you! Or if you have, there is no connectivity!

Have fun!

Nalin Pithwa.

Udaipur boy achieves perfect score in JEE Main

(Nashik’s Vrunda Rathi secures first rank among girl students nationwide)

(Ref: The DNA Newspaper, print edition, Mumbai, April 28, 2017, Friday, Education Section, author: Ankita Bhatkhande; E-mail: ankita.bhatkhande@dnaindia.net)

(Reproduced here for the express purpose of motivating my own students/readers of this blog-NP.)

The Central Board of Secondary Education (CBSE) released the results of the JEE Main exam on Thursday. The exam was held on April 2 in offline mode and on April 8 and April 9 in the online mode, across 113 cities in the country. Out of the total 11.86 lakh aspirants for Paper I, around 2.20 lakh students managed to qualify for the JEE Advanced exam scheduled for May 21.

The cut-off for the general category is 81, which is lower than the predicted cut-offs (95-105) depending on last year’s cut-off for the general category (100).

For the first time, the All India Ranks (AIRs) were declared along with the JEE Main results. Kalpit Veerwal from Udaipur scored a perfect 360 with AIR 1. Nashik’s Vrunda Rathi stood first among the girls in the exam.

Veerwal, whose father Pushkarlal, works as a compounder has appeared for his Class 12 boards from MDS Public School in Kota. “We are extremely happy with his success, and we are looking forward to his success in the Advanced exams,” said his  mother, Pushpa.

Rathi, a student of Nashik’s Loknete Vyankatrao Hiray Arts, Science and Commerce, and IITian’s Pace, said she is now looking forward to a career in research.

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(with inputs from Simran Motwani) thanks to DNA and team — N.P.

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IITJEE Mains 2017: Inspirational: Hard work and grit pays off

Reference: The DNA newspaper, print edition, Mumbai, Education Section, April 28 2017, Friday:

(Reproduced here on my blog with the express purpose of motivating students and parents alike; readers of my blog)

Dhananjay Subhash Chandra, son of an auto driver in Thane has cleared the JEE Main 2017 exams with a score of 291 out of 360. Attributing the success to his father, Subhash Chandra Tiwari, Dhananjay said that he is now preparing for the JEE Advanced and aspires to get into IIT Bombay. “My father struggled very hard to make ends meet for us. I am happy that I could make it this far and aim to do even better in the Advanced exams,” he said.

Congratulations Dhananjay and to your dad! And, also congratulations to DNA for such a true, motivational story!!

-Nalin Pithwa.

Growing Up Gifted

Source: Growing Up Gifted

A cute illustrative problem on basics of inequalities: IITJEE Foundation Maths

Problem:

Graph the functions f(x)=\frac{3}{x-1} and \frac{2}{x+1} together to identify the values of x for which we can get \frac{3}{x-1} < \frac{2}{x+1}

Also, confirm your findings in the above graph algebraically/analytically.

Also, then find the points of intersection of these graphs using a TI graphing calculator and also algebraically/analytically.

Solution:

Just a suggestion, it helps to have TI graphing calculator, a bit pricey, but …it might help a lot…If you wish, you can grab one from the following:

http://www.amazon.in/TEXAS-GRAPHIC-CALCULATOR-NSPIRE-CX/dp/B00A49F98U/ref=sr_1_fkmr0_1?s=electronics&ie=UTF8&qid=1493101690&sr=1-1-fkmr0&keywords=TI+graphing+calculator+Non+CAS

Of course, there might be excellent freeware software packages that plot graphs, but I am strongly biased: I like the TI graphing calculator toy very much!! I go to the extent of claiming that it is better for a kid to have a nice graphing calculator like a TI than a smartphone! 🙂 🙂 🙂

Later on, I hope to show how a detailed study of graphs helps right from high school level. In fact, an immortal Russian mathematician, I. M. Gel’fand had written a book titled “Functions and Graphs” for high-school children. You can check if it is available in Amazon India or Flipkart or Infibeam, etc.

Oh, one more thing: I love this topic of inequalities because of several reasons. Just now, one of my students, Darpan Gajra asked me certain questions about inequalities when he tried to solve the above problem —- I think his questions were good, and I hope this little explanation that I gave him helps many of other students also.

Explanation:

Consider the given inequality: \frac{3}{x-1} < \frac{2}{x+1}.

You might be tempted to solve this very fast in the following way:

Just cross-multiply: So, we get 3(x+1)<2(x-1), which in turn means 3x+3<2x-2, that is, x<-5. This is the answer by “fluke”!!!

In fact, the problem requires a detailed solution as follows:

Firstly, let us see what is wrong with the above fast solution:

Consider any inequality a<b. Now, we know that only if x>0, then ax<bx. But, if x<0, then ax>bx. Now, in the present question, if you simply cross-multiply, you are not clearing assuming whether (x-1)>0 or (x-1)<0; (x+1)>0 or (x+1)<0.

Also, one simple yet, I call it a golden rule is: even though it is not mentioned explicitly in the question, when you solve the question, immediately write the big restrictions: x \neq 1, x \neq -1 as these values make the denominators of the two fractions zero. Develop this good habit.

So, now, coming to the solution:

\frac{3}{x-1} < \frac{2}{x+1}, where x \neq \pm 1

\frac{3}{x-1} - \frac{2}{x+1}<0

\frac{3x+3-2x+2}{x^{2}-1}<0

\frac{x+5}{x^{2}-1}<0

Case I: (x^{2}-1)>0: Multiplying both sides of the above fraction inequality by (x^{2}-1) gives us:

x+5<0, that is, x<-5, but also (x^{2}-1)>0 is a restriction.  So, it means that x<-5 AND (x-1)(x+1)>0.

Subcase Ia: (x+1)>0 and (x-1)>0, which gives, x>-1 and x>1, which together imply that x>1, but we need x<-5 also as a restriction/assumption. So, in this subcase Ia, solution set is empty.

Subcase Ib: x+1<0 and x-1<0, which in turn imply, x<-1 and x<1, that is, x<-1, but we need x<-5 also. Hence, in subcase Ib: x<-5 and x<-1, we get x<-5 as the solution set.

Case II: (x^{2}-1)<0: Multiplying both sides of \frac{x+5}{x^{2}-1}<0 by (x^{2}-1), we get:

(x+5)>0, hence, x>-5. But, we also need the restriction/assumption: (x^{2}-1)<0, which implies that (x+1)(x-1)<0.

Subcase IIa: (x+1)>0 and (x-1)<0, that is, -1<x<1

So, in subcase IIa, we have (x>-5) \bigcap (-1<x<1), that is, (-1<x<1).

Subcase IIb: (x+1)<0, and (x-1)>0, that is, x<-1 and x>1. But, this itself is an empty set. Hence, in subcase IIb, the solution set is empty.

Now, the final solution is \{ case I\} OR \{ case II\}, that is, \{ case I\} \bigcup \{ case II\}, that is, \{ x<-5\}\bigcup {-1<x<1}

Homework:

a) Graph the function f(x)=\frac{x}{2} and g(x)=1+\frac{4}{x} together to identify the values of x for which \frac{x}{2}>1+\frac{4}{x}

(b) Confirm your findings in (a) algebraically.

More later,

Nalin Pithwa.

Use of mathematics by a financial analyst

“maths: make your career count”

AMSI (Australian Mathematical Sciences Institute) and ICE-EM have done a commendable job …to motivate all high students interested in various careers to excel in maths also:

Have a look at the following, for example:

http://mathscareers.org.au/index.php?option=com_content&view=article&id=19&Itemid=19

Cheers to maths!

Nalin Pithwa.

Can anyone have fun with infinite series?

Below is list of finitely many puzzles on infinite series to keep you a bit busy !! 🙂 Note that these puzzles do have an academic flavour, especially concepts of convergence and divergence of an infinite series.

Puzzle 1: A grandmother’s vrat (fast) requires her to keep odd number of lamps of finite capacity lit in a temple at any time during 6pm to 6am the next morning. Each oil-filled lamp lasts 1 hour and it burns oil at a constant rate. She is not allowed to light any lamp after 6pm but she can light any number of lamps before 6pm and transfer oil from some to the others throughout the night while keeping odd number of lamps lit all the time. How many fully-filled oil lamps does she need to complete her vrat?

Puzzle 2: Two number theorists, bored in a chemistry lab, played a game with a large flask containing 2 liters of a colourful chemical solution and an ultra-accurate pipette. The game was that they would take turns to recall a prime number p such that p+2 is also a prime number. Then, the first number theorist would pipette out \frac{1}{p} litres of chemical and the second \frac{1}{(p+2)} litres. How many times do they have to play this game to empty the flask completely?

Puzzle 3: How farthest from the edge of a table can a deck of playing cards be stably overhung if the cards are stacked on top of one another? And, how many of them will be overhanging completely away from the edge of the table?

Puzzle 4: Imagine a tank that can be filled with infinite taps and can be emptied with infinite drains. The taps, turned on alone, can fill the empty tank to its full capacity in 1 hour, 3 hours, 5 hours, 7 hours and so on. Likewise, the drains opened alone, can drain a full tank in 2 hours, 4 hours, 6 hours, and so on. Assume that the taps and drains are sequentially arranged in the ascending order of their filling and emptying durations.

Now, starting with an empty tank, plumber A alternately turns on a tap for 1 hour and opens the drain for 1 hour, all operations done one at a time in a sequence. His sequence, by using t_{i} for i^{th} tap and d_{j} for j^{th} drain, can be written as follows: \{ t_{1}, d_{1}, t_{2}, d_{2}, \ldots\}_{A}.

When he finishes his operation, mathematically, after using all the infinite taps and drains, he notes that the tank is filled to a certain fraction, say, n_{A}<1.

Then, plumber B turns one tap on for 1 hour and then opens two drains for 1 hour each and repeats his sequence: \{ (t_{1},d_{1},d_{2}), (t_{2},d_{3},d_{4}), (t_{3},d_{4},d_{5}) \ldots \}_{B}.

At the end of his (B’s) operation, he finds that the tank is filled to a fraction that is exactly half of what plumber A had filled, that is, 0.5n_{A}.

How is this possible even though both have turned on all taps for 1 hour and opened all drains for 1 hour, although in different sequences?

I hope u do have fun!!

-Nalin Pithwa.

Logicalympics — 100 meters!!!

Just as you go to the gym daily and increase your physical stamina, so also, you should go to the “mental gym” of solving hard math or logical puzzles daily to increase your mental stamina. You should start with a laser-like focus (or, concentrate like Shiva’s third eye, as is famous in Hindu mythology/scriptures!!) for 15-30 min daily and sustain that pace for a month at least. Give yourself a chance. Start with the following:

The logicalympics take place every year in a very quiet setting so that the competitors can concentrate on their events — not so much the events themselves, but the results. At the logicalympics every event ends in a tie so that no one goes home disappointed 🙂 There were five entries in the room, so they held five races in order that each competitor could win, and so that each competitor could also take his/her turn in 2nd, 3rd, 4th, and 5th place. The final results showed that each competitor had duly taken taken their turn in finishing in each of the five positions. Given the following information, what were the results of each of the five races?

The five competitors were A, B, C, D and E. C didn’t win the fourth race. In the first race A finished before C who in turn finished after B. A finished in a better position in the fourth race than in the second race. E didn’t win the second race. E finished two places  behind C in the first race. D lost the fourth race. A finished ahead of B in the fourth race, but B finished before A and C in the third race. A had already finished before C in the second race who in turn finished after B again. B was not first in the first race and D was not last. D finished in a better position in the second race than in the first race and finished before B. A wasn’t second in the second race and also finished before B.

So, is your brain racing now to finish this puzzle?

Cheers,

Nalin Pithwa.

PS: Many of the puzzles on my blog(s) are from famous literature/books/sources, but I would not like to reveal them as I feel that students gain the most when they really try these questions on their own rather than quickly give up and ask for help or look up solutions. Students have finally to stand on their own feet! (I do not claim creating these questions or puzzles; I am only a math tutor and sometimes, a tutor on the web.) I feel that even a “wrong” attempt is a “partial” attempt; if u can see where your own reasoning has failed, that is also partial success!