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2013 Abel Prize

Posted on 23/5/2013 by tpc

A well written note by Frenkel that attempts to explain Deligne’s work that won him this year’s Abel Prize.

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Multitasking with studying

Posted on 11/5/2013 by tpc

Another report on how students don’t learn as well when they attempt to multi-task.

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With the recent korean wave, we get to see lots of Korean pop, tv, movie stars in the media. But I found that I can never tell them apart. Especially, the ladies, tall, thin, beautiful but too homogenous. (Of course, korea is well known to be a cosmetic surgery heaven, but let’s not go there.)

Here’s finally some no-nonsense maths to prove the point. A great post analyzing the 2013 contestants of the Miss Korea competiton. In a nutshell, there are 6 eigenfaces – great word – and all the rest of the faces can be reconstructed by some linear combinations of these 6.

Posted in Applications, Fun Stuff, Linear Algebra | Leave a comment

How Not To Excel

Posted on 27/4/2013 by tpc

Apparently a couple of influential papers — by two world famous economists Reinhart and Rogoff from Harvard — that were quoted by politicians to shape policies contain serious errors that invalidated their claims. They made the common mistake of not selecting all the entries in the column when averaging, and reported a negative value when in fact the average was positive.

More details can be found here. There was no byline for the post, but the blog is by David Bailey and Jon Borwein, respected mathematicians.

So should we stop using excel because it has tremendous potential for disasters such as this? Yes, if you ascribe to the “everyone is an idiot” philosophy, and then you make everyone do all the calculations and data entry manually thereby increasing the chance of more errors.

Let’s end with a joke. “Why should doctors, nurses, economists, judges, auditors, accountants learn mathematics?”

“To prevent idiots from getting into these professions.”

Posted in General, Technology | Leave a comment

According to Socrates

Posted on 20/4/2013 by tpc

… it is no true wisdom that you offer your disciples, but only its semblance, for by telling them of many things without teaching them you will make them seem to know much, while for the most part they know nothing …

This quote caught my attention for it seems to make a case of “telling” vs “teaching”. Of course it is somewhat taken out of context. Socrates was actually commenting on the technological advancement (back then) known to us as writing or books. And many people nowadays use the passage as a critique of the internet. A more complete transcript may be found here.

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Can computers write proofs

Posted on 8/4/2013 by tpc

that human being can understand? By that I mean not those formal logic nonsense. Tim Gowers is doing an interesting experiment to get readers to judge proofs of exercises in metric space theory. The three proofs are supposed to be written by an undergraduate, a phd student and a computer. I only looked at the first problem which is quite straightforward: prove that the intersection of two open metric spaces is open.

Honestly, I can’t tell which one is the computer. The main problem is one does not know how the program was written. Here’s my analysis of the solutions. The first two proofs are very similar in the sense that they both use open balls. (The third did not.) The writing is also much clearer with good choice of notations. One thing that is clear is that the first two writers already knew what is the key to the proof – take the minimum of the the radii of the two open balls. The third proof, on the other hand, seemed to be using the various conditions to force the choice. Perhaps that makes it the computer, assuming that the programmer do not actually instruct the computer to reorganize and rewrite the proof. If the first two were humans, I still cannot tell which is which. The second proof looks to be more carefully written, but that does not actually imply the level of mathematical sophistication. Or does it?

Posted in Geometry/Topology, Problems, Technology | Leave a comment

Whither the golden ratio

Posted on 20/2/2013 by tpc

According to this report from the Guardian, someone claims that the golden ratio pleases the eye because of scanning ratios, i.e. how fast the eye scans things horizontally versus vertically. The actual paper is supposed to be free access (with a little digging) but you need to sign up for an account, and presumably some amount of spam mail.

I used to give talks about the golden ratio, mainly because I see it appearing everywhere. Over the years, I started to remove connections that seemed spurious, or added disclaimers, like for example the dimensions of the parthenon. Generally, I’m not as excited as before. Of course, the mathematics remain beautiful, perhaps I’m just getting a little old and a little jaded.

I’ll end with something not so spurious, according to mashable twitter was designed with the golden ratio. But how many people actually uses the twitter client?

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Speed Index

Posted on 16/2/2013 by tpc

A clearly written page explaining how one can measure the speed in which a website loads by integrating the area under the curve. link

Posted in Applications, Calculus/Analysis, Web | Leave a comment

3D printing

Posted on 12/2/2013 by tpc

seems to be the rage these days, and the natural question is that if there is anything that you cannot print. The short answer is NO. Thanks to Fubini’s Theorem.

Posted in Applications, Calculus/Analysis | Leave a comment

Mathematical Sense Making

Posted on 8/2/2013 by tpc

I like the following problem.
Match the Graph
It was shown by Alan Schoenfeld in his talk yesterday in Singapore. Among the many things he touched upon, he introduced the Mathematics Assessment Project which has a package of lessons for grades 6 till high school. The above problem came from the lesson on Distance Time Graphs.

Schoenfeld also gave a list of key questions to ask for Mathematics Classes:
1) Was there honest-to-goodness maths in what students and teacher did?
2) Did students engage in “productive struggle,” or was the math dumbed down to the point where they didn’t?
3) Who has the opportunity to engage? A select few, or everyone?
4) Who had a voice? Did students get to say things, develop ownership?
5) Did instruction find out what students know and build on it?

Certainly a tall order!

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Fake curves

Posted on 2/1/2013 by tpc

Interesting post on how a supposed trig function is actually not trig. I have to admit I used to do things like this and use the arc tool (on powerpoint) to create curves that only look vaguely like the actual one. Come to think of it, I sort of still do it because if you want a rational function that has a max turning point at 2, a vertical asymptote at 3 and approaches y=x as x gets large, it’s actually pretty time consuming to try and get the actual equation.

Posted in Calculus/Analysis, Teaching, Technology, Trigonometry | Leave a comment

125th Anniversary of Ramanujan’s Birth

Posted on 22/12/2012 by tpc

It was a privilege to be in Delhi, India to celebrate the 125th Anniversary of the birth of Ramanujan. It was a good conference with good talks by many speakers. Personally for me I didn’t get any new ideas or projects to work on but that is fine. I did get to see the Taj Mahal though. Anyway, the following is the doodle by google in honour of Ramanujan.
Ramanujan's Google Doodle

I have to say although [tex]\pi[/tex] was there to 21 decimal places, as well as a magic square and an Indian (far too thin) drawing in the sand, I thought there could have been more Ramanujan-esque mathematics like 1729, 691, a q-series or at least the partition function.

They did include the following
[tex] \sqrt{x} + y = 7 , x + \sqrt{y} = 11 [/tex] which supposedly had a connection to Ramanujan. But seriously, it is quite trivial to get the integral solutions, even for a mere mortal like yours truly.

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Polya and Ramanujan

Posted on 17/12/2012 by tpc

Taken from the introduction to Bruce Berndt’s Ramanujan’s Notebooks Part I.

In notes left by B. M. Wilson, he tells us how George Polya was captivated by Ramanujan’s formulas. One day in 1925 while Polya was visiting Oxford, he borrowed from Hardy his copy of Ramanujan’s notebooks. A couple of days later, Polya returned them in almost a state of panic explaining that however long he kept them, he would have to keep attempting to verify the formulae therein and never again would have time to establish another original result of his own.

Posted in Quotes/People | 1 Comment

Social interactions and the development of control strategies

Posted on 14/12/2012 by tpc

Is a sub section in Schoenfeld’s Mathematical Problem Solving. The argument is that when two students work together, this interaction can spur cognitive development resulting in approaches to the problem being solved that are qualitatively different than those taken by the students on their own.

But if Vygotsky is not one’s cup of tea, how about this problem also from the book.
Ten people are seated around a table. The average income of these 10 people is $10000. Each person’s income is the average of the people sitting immediately to his left and right. What is the possible range of incomes for each person?

Posted in Learning, Problems | Leave a comment

Collective Nouns for Mathematicians

Posted on 7/12/2012 by tpc

A creative colleague got me started and I just can’t stop. Of course the most apt term would be “a set of mathematicians”. But how about others, according to their fields? Ignoring the obvious alliterative ones like “a set of set theorists”, here’s the list so far…

A clique of graph theorists
A group of algebraists
A circle of geometers
A triangle of geometers (if there are exactly three, and you now know what to do for the general case)
A union of set theorists
A collection of set theorists
A divide of number theorists
A fraction of number theorists
A composite of number theorists
A (number) field of number theorists
A partition of number theorists/combinatorists
A series of analysts
A convergent of analysts
A residue of complex analysts
A branch of complex analysts
A neighbourhood of analysts/topologists
A (Riemann) sum of analysts
A basis of linear algebraists
A sample of statisticians
A distribution of statisticians
A permutation of combinatorists

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Is 0 even?

Posted on 5/12/2012 by tpc

I first saw this from BBC news. The aftermath of hurricane Sandy resulted in a shortage of fuel and New York City had to implement an odd-even system. The following is taken from their press release:

1) Vehicles with license plates ending in an even number or the number “0” can make purchases of motor fuel on even numbered days.
2) Vehicles with license plates ending in an odd number can make purchases of motor fuel on odd numbered days.
3) Vehicles with licenses plates ending in a letter or other character can make purchases on odd numbered days.
4) Commercial vehicles, emergency vehicles, buses and paratransit vehicles, Medical Doctor (MD) plates and vehicles licensed by the Taxi and Limousine Commission are exempt.

Of course 0 is even and rule 1) can be made more concise. But the point is that not many people know of this, and so putting that down resolves ambiguity. Another interesting thing to note is that splitting it odd and even seems fair because assuming licence plates numbers are uniformly distributed, 50% can make purchases on any day. But rule 3) sort of muddle things up (at least for me) because I have no idea what proportion of plates in NYC ends with a character.

Posted in Number Theory, Probability, Teaching | Leave a comment

Geometry Puzzle

Posted on 5/12/2012 by tpc

A neat little puzzle from John Mason.

How are the blue and red area related?
Continue reading →

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Paraphrasing Russell

Posted on 28/11/2012 by tpc

William Mueller in his 2001 article in the American Mathematical Monthly suggested that we could take Bertrand Russell’s quote* on philosophy and substitute it with pedagogy. So here goes

Pedagogy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation…

*B. Russell, The Problems of Philosophy, 1912.

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4 clicks to Mathematics

Posted on 22/11/2012 by tpc

I learn this game from watching Tim Chartier’s video. (The original game was 5 clicks to Jesus but in this age of religious sensitivity … )

So this is the game, go to Wikipedia and click random article on the left column. Starting from this random article, try to click your way to the article Mathematics and see how many clicks it takes.
Here goes,
Start: Ayumi Tanimoto
1) Japan
2) Programme for International Student Assessment
3) Trends in International Mathematics and Science Study
4) Mathematics

4 clicks on the first try! Actually in the PISA page there are lots of mention of the word mathematics but all were not hyperlinked. (So it should really be 3 clicks.)

Second try
Start: Martin Fillo
1) Czech Republic national football team
2) Czech Republic
3) Kurt Godel
4) Mathematics

4 again.

Last try for the day
Start: List of Galatasaray S.K. records and statistics
1) Harry Kewell
2) United Kingdom
3) Alan Turing
4) Mathematics

Another 4.

PS: One thing I found was that some articles are really too long to read.

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Why are Finland’s Schools Successful?

Posted on 12/11/2012 by tpc

is an insightful article by LynNell Hancock published in the Smithsonian magazine in Sept 2011. It was recently republished in a local newspaper (strangely without reference to the original publication) and circulated by a colleague.

I particularly love this para:

Finland’s schools are publicly funded. The people in the government agencies running them, from national officials to local authorities, are educators, not business people, military leaders or career politicians.

Hmm, let’s have a quick recall of the background of the top man in charge of education locally for the past decade.
1) Formerly from the Monetary Authority 2) Also formerly from the Monetary Authority 3) Surgeon 4) From the Navy. All career politicians.

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