eon

Posted by tpc at November 7th, 2008

Read something quite cool. Professor Brown from Dalhousie, used fourier transforms to reconstruct a mysterious chord from the Beatles `A Hard Day’s Night’

Posted by tpc at October 12th, 2008

In the midst of recession and oil prices falling from $145 to below $80, the ridiculous power company announced that electricity prices are going up by 21%, based forward oil pricing. The best thing is that our electricity are generated through natural gas, not crude oil!

Even more ridiculous is the following quote taken by the report written by one Liaw Wy-Cin.

The scope for savings is high considering that power consumption patterns show that 40 per cent of households, from one-room flats to landed properties, use more than the monthly average, said Mr Khoo (Chin Hean, EMA chief executive.) EMA is the electricity and gas industry regulator.

Well, if the report was accurate, the statement defies common sense! By the very definition of average, 50% of every household would use more! I’m guessing the 40% comes because they banded the average consumption into ranges. (I made a terrific blunder here, see the comments.)

Another report from Reuters said

People who take long spells of sick leave at least once in three years face a higher risk of early death,…

What a magnificent revelation, almost as strange as rooster crows leading to a higher chance of the sun coming out. Come on, which part of ’sick’ in sick leave do you not understand? People suffering from serious ailment take sick leave and have a high chance of premature death. Common sense tells you that. Again, either it was a worthless paper in the British Medical Journal or it’s the reporters missing the point.

Posted by tpc at September 28th, 2008

Any rigid body displacement where a point is fixed is equivalent to a rotation. I saw this neat proof from Don Koks’ Explorations in Mathematical Physics.

By the hypothesis

Hence is orthogonal and But should vary continuously from the identity transformation, allowing us to conclude So


since we are working in 3D. We are forced to conclude
for some .
Thus has an eigenvector with eigenvalue 1. The transformation is thus the rotation about the direction .

Posted by tpc at September 24th, 2008

While perusing some blogs, while waiting for the rain to stop, I’ve found more evidence to support learning maths than replying on calculators.

A TRAFFIC warden gave parking fines to innocent motorists – because he did not know how to tell the time.

The bungling parking attendant had to use a calculator to work out the expiry time on tickets displayed in motorists’ windscreens.

But with calculators working in decimals rather than minutes and hours, the ticket-happy warden had his book out before realising his maths was letting him down.

One of his victims was IT manager Dave Alsop, who tried his best to show the warden the errors of his arithmetic ways. But to no avail.

The 29-year-old from Torbay had parked in the Terrace car park close to Torquay harbour and paid £1.20p for 75 minutes. His ticket was issued at 2.49pm and would have covered him until 4.04pm.

But when he returned at 3.41pm, he discovered he had been given a £50 parking fine.

He found the traffic warden nearby and asked him why he had been booked when his ticket clearly showed he had time remaining.

The warden disagreed and tried to prove his point using a calculator.

He tapped in 14.49 and added 0.75 to produce a total of 15.24, which he claimed meant Mr Alsop’s ticket expired at 3.24pm – 17 minutes before he returned to his car.

Mr Alsop said: “I tried to explain to the warden but he didn’t have a clue. He thought he was doing things correctly. He just carried on doing other cars parked there.”

The warden insisted he was right and issued fines to two other unsuspecting motorists.

Mr Alsop, who works for Pavey Insurance brokers in Torquay’s Abbey Road, appealed, had his fine waived and received an apology from Torbay Council.

Via -> Natural Blogarithms -> 360 -> God Plays Dice -> Eric Berlin

Posted by tpc at September 23rd, 2008

Yes, it is plural because two such primes were found within two weeks. See the
press release from GIMPS. I guess we can’t really comprehend how big a 10 million digit number is. My ruler tells me that the normal font size of this blog gives 6 characters per cm, hence 600 characters or digits per metre. 10 million / 600, gives about 16.6 km. That’s how long the number would be written out in a line. It would take me almost 2 hours to run 17km. But that’s just writing it out! We haven’t even talked about doing arithmetic on it, let alone checking that it is indeed a prime number!

UCLA who found the larger of the two primes first is getting about $50,000 of the prize money. There goes my dream of making an impact in mathematics and getting some useful cash out of it. And no, I’m realistic enough to not dream about solving Riemann’s hypothesis or the BSD. In my younger days, I dreamt about Goldbach’s conjecture, which is still open but without cash prizes. Even that may be knocked down within the next few years. We do live in exciting times.

Still it is momentous to break the 10 million mark, important enough for me to brush the dust off this blog and write something. Join GIMPS and keep searching. 100 million is not very far away.

Posted by tpc at June 3rd, 2008

Most people tend to confuse randomness with uniformly distributed. They believed that if you toss a coin 100 times you would get 50 heads and 50 tails. Lots of people have discussed this, gave examples of how they detected pseudo randomness. I would like to add my very own example.

I have 1600 exam scripts ordered by student matriculation numbers, divided into 16 piles. A total of 11 students asked to checked their scripts. One would think the 11 would come from different piles right? Here’s the actual stats: number of scripts (from pile number)

2 (1), 1 (4), 1 (6), 2 (7), 2 (9), 3 (14).

Posted by tpc at May 25th, 2008

The sad truth is that many of the students in the university are not here for an education. They are here to socialise, play freesbies in the field, run hall activities and yes get a degree. While some still attend classes, many subscribe to the mantra “studying worked solutions will help them pass exams”. To a certain extent, it becomes a self-fulfilling prophecy because the university do not want to fail too many students, and given that the students did not work hard to learn, the only way to let them pass is to set exams which resembles tutorial questions and provide clear written solutions to tutorial questions.

Now in my view, studying solutions to sample problems is really not a pedagogical sin. Well at least not in mathematics. One very important aspect of learning mathematics is mimicking carefully selected examples. But that is only the start. A student need to move on to the next stage of actually solving problems on their own. Many never do.

Because I do not want to adopt the high handed approach of failing 70% of the class, I intend to assign homework for my courses next semester. Simply put, it is to force students to do work which they are supposed to do on their own but never get round to it. Homework is not a popular activity, although the dept has recently made it mandatory for certain courses - a good sign. It requires a lot of extra effort in assigning problems, writing clear solutions, collecting, collating marks and worst of all, marking. Fortunately, we have graders to help us do the latter.

But collecting homework brings with it a new problem. Plagiarism. Some pretend it doesn’t happen, which is the easy way out since if we catch students we also need to go through the difficult process of discipline. Yet, it becomes a farce if the homeworks are merely copied, which renders everything completely meaningless. I especially pity the graders. (Although the graders might actually like to grade 100 copies of the same assignment.)

The university has licensed a system called turnitin, and I attended a talk by a chemistry lecturer who shared his experience. I really appreciated that. The surprising(?) result is that he had a chart that says almost 50% of the lab reports had been plagiarised. Now, the 50% can be interpreted in many ways, but it does sound high. The twist to the story is that the students knew they had to submit their report to plagiarism checking. So they could in theory, lift a passage from wikipedia, changed the adjectives, rearrange the sentences and escape detection. I do not believe the engine is that smart yet. But it turned out that those reports submitted early was not flagged, but those that were submitted later were flagged. The revelation is that many copied from past year reports from seniors which were not in the database of turnitin but were flagged because their classmates who submitted earlier had the same answers.

But with mathematics, I don’t think turnitin will work. For example, in those chemistry lab reports, an example was that when there were exponents like “x 10-5″ , it got flagged. Plus, if I wanted typed reports, I’d get ms word documents! And most important of all, I wouldn’t be surprised if original solutions to mathematical problems look identical.

I’m still thinking of a good way to implement this. I might try turnitin if I decide to ask the students to write some essays on mathematics history, if that day ever comes.

Posted by tpc at May 6th, 2008

I’ve been looking at logic puzzles as a break from all the frenzied examination related activities. An internet search will inevitably throw up the old nugget known as Einstein’s IQ quiz or puzzle. The wiki entry has a variant called the Zebra puzzle and at least a proper reference.

(more…)

Posted by tpc at April 30th, 2008

I’ve finally wiped the dust off my copy of Ahlfors and started reading from page 1. Previously, I’ve only looked at the chapter on Elliptic functions as a reference. I’ve skimmed through four chapters and decided that it is probably not very suitable for the complex analysis course later in the year. It’s a pity because the international version of Ahlfors is readily available locally and costs less than US$20.

I’ve been looking at several nice books, Stein and Shakachi, Gamelin and Needham.

All seem pretty promising, especially Needham’s Visual Complex Analysis. I have read many good reviews of the book, and now that I’m starting to look at the contents seriously, I agree it’s very good, and unconventional.

Posted by tpc at April 30th, 2008

I have no idea who is Heaviside until I started to teach this course which included Laplace transforms. That got me really interested and I checked out P. Nahin’s biography from the library. Perusing the borrowing slip, the book was last borrowed in Sep 2006, prior to that Mar 1990. 16 long years.

Meanwhile, in the preface a quote attributed to Lazarus Long:

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.

Posted by tpc at April 19th, 2008

Came across this curious identity while preparing some trigo notes.

According to wolfram, it is called Morrie’s Law by Feynman after his childhood friend who showed it to him. Wow, all I need now is some future nobel laureate with a big mouth (no disrespect intended) , so that some cute little identity might be named after me.

It’s easy to prove Morrie, just multiply to the numerator and denominator for each of the 3 angles. Use the double angle formula for sine and then cancel away with

The general formula, easily proven with induction is the following:

Posted by tpc at March 16th, 2008

supposedly used by Russian Peasants as claimed by A. Posamentier and I. Lehmann in chapter 6 of their book The (Fabulous) Fibonacci Numbers.

Suppose you want to multiply 23 to 41. What you do is to write the numbers in two columns. In the first column you successively half (round down) while in the second column you double. By the time you reach 1 in the first column, look at those odd numbers in the first column and add the corresponding numbers of the second to get the answer.

23* x 41
11* x 82
5* x 164
2 x 328
1* x 656

The odd numbers are marked by *, hence the answer is
41 + 82 + 164 + 656 = 943.

The interesting question is why does this work? It boils down to binary numbers.

Posted by tpc at January 14th, 2008

See this youtube video. Saw it via Natural Blogarithms.

Posted by tpc at December 7th, 2007

If you’ve watched Disney’s High School Musical, you might remember a scene where the female lead corrected the teacher “shouldn’t the second equation read sixteen over pi?”

What was written on the board looked vaguely familiar, and so it got me trawling over the world wide web looking for details to no avail. I later found out from one of the world’s renown expert on that indeed the equation is one of three series that appeared in Ramanujan’s work “Modular equations and approximations to ” Naturally, I went back to the web and this time hit the jackpot. Two screencaps:


Ramanujan’s series

Now if you want to watch the video, here’s the link.
It happens in the first minute. So you don’t have to wait too long.

Posted by tpc at November 17th, 2007

Was happily reading the Saturday morning papers when I came across the following advertisement by a local defence agency.

It looked easy enough and got me off my butt and to my computer to try and break it. The first guess is of course plain old substitution cipher and in this passage of 36 numbers, you can already make good guesses with frequency analysis.

But I’m a recreational code-breaker and so it took me quite some time to set everything up using a spreadsheet. Halfway through I had a great idea about the most frequent number in the code: 32. But I did not pursue it. After an hour or so of fiddling, I found out that idea was right and how to do this. It’s very easy if you see how.

Posted by tpc at November 8th, 2007

Really cool video on mobius transformation.
http://www.ima.umn.edu/~arnold/moebius/

Posted by tpc at August 4th, 2007

I read in the local newspaper that 54.2% of bioengineering graduates last year found jobs not related to their field of study. Ha, great fodder for me to attack the crazy obsession with life sciences. But since we live in a google age, I thought I better check the data. Turned out that there are only less than 50 per cohort so that 54.2%. So there are two ways to interpret this.

1) only about 25 people found non bio related jobs. This is a small number and could reflect just diverse interest or better opportunities in business and finance. (These days every other study want to go into finance.)

2) then again for such an exclusive club. 50 out of the 1300 engineering graduates, you would think that these students are the creme de la creme, specially honed with tender loving care into bright future stalwarts of the life science industry. Surely, the industry could squeeze out 50 jobs?

Posted by tpc at July 31st, 2007

Cryptography article on Nova

Posted by tpc at June 19th, 2007

a personal perspective by Terence Tao. This is a new edition of a book which was written by Tao more than 15 years ago, which means when he was only 15! It’s a thin little book that takes a leisurely look at solving some competition type problems. The coverage is not huge, but the author take pains to go through in great detail various strategies one can adopt in solving problems. Quite a nice book but very pricey for 102 pages.

I found exercise 2.1 quite fun.

In a parlour game, the ‘magician’ asks one of the participants to think of a three-digit number . Then the magician asks the participant to add the five numbers and , and reveal their sum. Suppose the sum was 3194. What was ?

My solution is this. If we add all the six permutations, we know that the sum equals

.
So we just need to know the multiples, . Take the smallest multiple larger than the given number, and check by subtracting the difference and summing the digits. You do not have to do it with more than 5 different multiples.

.
But , so incorrect.
now .
And and we found our number.

Posted by tpc at June 10th, 2007

Found a very nice post by Alane Tentoni on the topic, together with a link to a great story by Asimov about a future where people cannot do maths without a computer.

via Carnival of Math IX