20 September 2011

Official driving licence photo

(Thanks Lisa for pointing this out. Please feel free to comment.)

A European driving licence photograph is subject to strict regulations. One of them states that headgear is only allowed for confessional reasons, i.e. if it is part of the driver's religion.

Mr Niko Alm applied for licence three years ago demanding that he wear a pasta strainer on his head. He claimed it was part of his religion, pastafarianism, or the Church of the Flying Spaghetti Monster. And he managed to obtain his licence with the photo he needed:

This is a very good practical application of rationality. Indeed, there is absolutely no commonly agreed definition of what a religion is. Informally, something is classified as religion if it has been around for a while, if it has lots of devotees, if it does not go against the political status quo, etc. But none of these things is well-defined, nor has it ever been legalized. So, it is perfectly logical for someone to start a religion.

By acting as Mr Alm, eventually, will force society to properly define "religion". Or try to. And then there will be chaos, for there is no definition that fits all, neither one that will encompass religions to be.

We need examples, and counterexamples, in order to test a theory. Despite the efforts of Tony Blair to unify religions, this cannot happen. For sure, there will be one or more religions left out.

Let us start thinking, then, what on Earth constitutes a religion? Why is religion A better than B?--oops, I'm not allowed to say so! So why are all religions equal?--oops, but then I may have to include loonies like these, who get together and "speak in tongues".

So, congratulations to Mr Alm, for posing the right question. In his words,
"I am ridiculing the authorities," he said. "If anybody is offended there is nothing I can do, but I am offended too, if logic and reason is offended."

17 September 2011

Mathematics screening test for university students

I am writing this in response to my friend Joe who tends to believe that things in US education are bad. In fact, for anyone who thinks that things are bad in the particular place he or she happens to work.

This is a screening test I gave to second year university students (school of mathematical sciences of a UK  university I worked at earlier). The rationale behind a screening test is to alert the students that they should not take a further course without having basic skills acquired in earlier courses and that, if they have not learnt earlier material, they should repeat the courses before proceeding further.

The front page of the linked document contains statistics of students' responses. The remaining pages contain the questions I asked, together with sample responses. I would say that out of 60 students who took the test, there was probably one who could, perhaps, qualify as a university student. The remaining ones had no clue.

Although the document is self-explanatory, here are some of the questions, along with the most funny answers:

Q:  Given two polynomials $p(x) = \sum_{k=0}^n a_k x^k$ and $q(x) = \sum_{k=0}^m b_k x^k$, express the coefficient of the term $x^k$ of the product $r(x) = p(x)q(x)$ in terms of the coefficients $(a_k)$ and $(b_k)$.
A: $a_{k^{1/2}} b_{k^{1/2}}$.

Q: Define the concept of the derivative of a function $f : R \rightarrow R$ at a point $x$.
A: This is the distance of the point $x$ from the origin on a plain  [sic].

Q: Explain what we mean by the integral $\int_0^1 f(x) dx$ of a function $f : [0; 1]  \rightarrow R$. (The answer "area under the curve" is not acceptable.)
A: By integrating this function, we are being asked to calculate an area, and by providing definate [sic] integrals, the question asks us to provide a specific area.

Q: In how many ways can you put 5 indistinguishable balls in 7 distinctly numbered boxes and why?
A: 21/5.

Q: Expand $(a + b)^5$, where $a, b$ are real numbers.
A: $(a + b)^5 = \binom{a}{0} + \binom{a}{1} a^4 b + \binom{a}{2} \frac{a^3 b^2}{2!} + \binom{a}{3} \frac{a^2 b^3}{3!} + \binom{a}{4} \frac{ab^4}{4!} + \binom{a}{5} \frac{b^5}{5!}$.

Q: Compute the (indefinite) integral $\int dx/\sqrt{x}$.
A: $-2u+C$.

The huge problem in education, around the world, is that the meaning of the verb "to learn" is frequently disassociated from the verb "to understand". This is convenient for students. It is also convenient for many teachers who do not want to bother to understand and explain. It is convenient for politicians. It is convenient for administrators. In short, it is convenient for everyone. Except that the result is the production of generations of students who get a degree in, say, mathematics, but know very little mathematics. What is worse, is that they think they know. It is more dangerous to have people who believe they know rather than people who know they do not know (and, therefore, may try to learn whenever necessary). Someone who is convinced of his/her skills will do nothing to improve them.

An example of bad mathematics textbooks for schoolchildren

A friend of mine sent me the following entry from a mathematics textbook for schoolchildren in the US around the age of 12. Here is a question asked, together with the suggested solution:
Question: Tell whether the following events are dependent or independent. If they are independent, find the probability that both events occur.
Event C: Choosing the letter F from a bag containing the alphabet.
Event D: Choosing the letter V from a bag containing the alphabet after already choosing F and not replacing the letter.
Solution: Events C and D are dependent events. Once a letter has been picked from the bag and not replaced, it changes the probability of picking another letter from the bag.
Here is the problem with the way the question is formulated. In defining event D, the writer of this has inserted the description of the sample space. Instead of clearly defining the experiment first and then the events, he/she mixed the two things together in defining the event D. The result is confusion. The correct way of stating the problem is:
We have a bag containing the 26 letters of the alphabet. We pick a letter at random, put it away, and then pick another letter at random and put it away. Define the following events:
Event C: The first letter we pick is F.
Event D: The second letter we pick is V.
Question: Determine whether the two events are independent or not.
The suggested solution is even more confusing. It says that events C and D are dependent (correct, provided you have understood what event D is, i.e., what the author wanted to say but did not say), but the explanation given is not very good: "Once a letter has been picked from the bag and not replaced, it changes the probability of picking another letter from the bag." It changes the probability of what? What does the changing of a probability have to do with the definition of dependence?

Let's see first what the correct solution is:
P(C) clearly equals 1/26.
P(D) also equals 1/26.
The reason is: From the definition of event D, as an event which has nothing to do with whatever we picked at the first pick, and because of symmetry, we can see that P(D)=1/26 as well. What I mean by this is the following: if, say, the letters were arranged in a random order on a line and defined event C to be the leftmost letter, while D the right most one, then it would have been even clearer that P(C)=P(D). This is why the two probabilities are the same.
As for the probability P(C & D) of their intersection we have 
P(C & D) = P(C) P(D | C) = (1/25) x (1/26).

The reason is this:  P(D | C) can be computed on the new sample space containing 25 letters because we are conditioning on having picked the letter F first. Since there are 25 letters remaining, we have P(D | C) = 1/25.
Since P(D & C)  ≠  P(D) P(C), it follows that C and D are dependent events.
A clearly stated problem, and a logical solution is the only way that the kids in the school can learn something which they can subsequently find useful. The kids will remember, even subconsciously, these kind of things and if they have been taught incorrectly, they will have to unlearn everything when they go to the university (provided--this is an assumption--that their university teachers know how to teach or bother to do so).

No wonder why when people ask me what I do for living and I say mathematics they invariable give the same response: "Oh, I was so bad in mathematics at school." "I understand", I reply. "but, most likely, so was your teacher".

P.S. My friend also told me the following idiotic question the kids get in their mathematics class:
Question: What do you call the shape whose area is given by the formula L⋅H?
There is no end to idiocy in this world. 

16 September 2011

Swedish sunset

Nordic darkness will be here soon. Meanwhile, we're getting some nice sunsets. The photo below was taken yesterday in front of  Ångströmlaboratoriet.

15 September 2011

Three Swedish fetishes

Moving from one country to another makes it easy to observe the differences and things which are unique to a particular place. It is much easier for a newcomer to spot them than for people who've lived in the country all their lives. I've noticed at least three things which seem to be uniquely Swedish, in the sense that they exist in abundance and are really loved by people in Sweden. I decided to call them Swedish fetishes because these things are useless, probably addictive, and serve no real purpose.

Fetish no. 1: Candy
If you've ever been to one of the American-style cinemas you cannot fail but notice rows upon rows of stinky, sticky, sickly candy. If your olfactory system works well, then, probably, you'll want to go past them quickly so that you don't have to vomit.
It's really a huge surprise to see that this kind of junk food exists in abundance in Sweden, not only in cinemas, but, really, everywhere: in petrol stations, in supermarkets, in convenience stores, at the university, in the hospital cantine, on the streets (whenever vendors go out), at train stations...
There is no end to the amount of candy that is available in Sweden, everywhere and at all points of time. Sweden is a country where everything closes early. However, candy you can find at almost all times. It is even served during official meetings. You may be out at night searching for milk, which may be hard to find. But candy, you will find, with little difficulty:

Police scooping candy in huge quantities, around 11 pm, on 10 September 2011:
It is rather suprising, but Swedes consume more candy than Americans per person. In fact, it seems that Sweden is the world leader in candy consumption per capita (excluding chocolate). To prove my point further (that Swedes have some kind of peculiar relationship with sweets), here is a recent TV advertisement. In it, we see children with three mouths (!?), evolved this way so that they can consume more candy. Yes, it is quite disturbing image. Enjoy:

Fetish no 2: Tatoo magazines.
"Why are there so many tatoo magazines in Sweden?", I asked a colleague a few months ago. He didn't know. In fact, he hadn't observed it. As I mentioned above, it is easier for a newcomer to spot the differences. To prove my point, I went to the local Pressbyrån and took a picture:
You can perhaps count 27 tatoo magazines in the middle shelf. This number should be compared with the boats magazines (12), the number of newsmagazines (11), etc. Not only is there an amazing large number of tatoo magazines in the shop, but that number is the largest of magazines of all kinds.

Truly peculiar.

Sometime later, I went to another shop in Gothenburg. Same story:
Here you can count about 24 tatoo magazines.
The question, then, is why? Why are there so many customers attracted to tatoo magazines? The obvious answer that people are attracted to the women on the covers and inside is not satisfying. There is something else, and this is something that has to do with Sweden. I don't know what it is.
There are definitely more tatoo magazines in a Swedish magazine shop than in an American shop. Even rednecks don't have the need to consult so many magazines for their body modification.

Fetish no. 3: Old (mostly junk) American cars.
Uppsala is a city with not so wide streets, and a huge number of bicyclists. Once in a while, however, the city is transformed by the peculiar site of platoons of old, mostly ugly, junk, American cars, occupied by a large number of passengers, all of which sit packed together on the front seat, drunk and loud and proudly driving their pile of metal, obviously feeling a sense of achievement, demonstrating their possession to pedestrians who do not have the chance of having a vehicle of this sort in their backyard.

The drivers want to be seen, to be noticed. The other day there was even an American car show in Uppsala. A huge number of cars, probably in the hundreds, had come to Uppsala, parked in a central area. Their owners were dressed like American rednecks and many of them had beer bellies as well! Why would anyone in the world try to behave like a redneck is totally incomprehensible. I hope, however, that the Swedish redneck-wannabes do not have the essential tool of the redneck trade: a gun-rack full of guns. But who knows?

One is inclined to conjecture that Sweden has the largest number of redneck-wannabed than any other country besides redneckland. But I lack statistics, so I will refrain from formulating this kind of conjecture.


What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant