23 July 2010

Harmonic series (further test in LaTeX)

I learned yesterday, through Evolutionblog, that I can write LaTeX as long as I put a little script at the bottom of the posting, which can be found here. This is my attempt to make it work.

Well, since the actual posting on Evolution blog was on harmonic series, let me write Pietro Mengoli's proof of its divergence. Recall that the harmonic series is
S = 1 + \frac{1}{2} + \frac{1}{3} + \cdots.
Let us prove that $S=\infty$. Mengoli did the following. He grouped all terms, except the first one, in triples:
S = 1 + \left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} \right)
+ \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} \right)
+ \left(\frac{1}{8} + \frac{1}{9} + \frac{1}{10} \right) + \cdots
Then he observed that each triple is larger than three times the middle term:
\frac{1}{n-1} + \frac{1}{n} + \frac{1}{n+1} > \frac{3}{n}.
And so he wrote
S > 1 + \frac{3}{3} + \frac{3}{6} + \frac{3}{9} + \cdots = 1 + 3S.
Since no finite positive number can be larger than 3 times itself plus 1, he concluded that $S=\infty$.

To see that the inequality above is true write it as
\frac{1}{n-1} + \frac{1}{n+1} > \frac{2}{n},
which is equivalent to
\frac{2n}{n^2-1} > \frac{2}{n}
which is obviously true.

Another way to see the inequality (and more) is to observe that if $X$ is a positive random variable, which is not a constant, then $E(1/X) > 1/E(X)$. To see this, let $Y$ have the same distribution as $X$ but be independent of it. Since $X^2+Y^2 > 2 XY$ we have $2 < \frac{X}{Y} + \frac{Y}{X}$, and, by taking expectations, $2 < E(X) E(1/Y) + E(Y) E(1/X) = 2 E(X) E(1/X)$, as claimed. Then apply this to a random variable $X$ which takes values $n-1$ or $n$ or $n+1$, each with probability $1/3$. This gives Mengoli's inequality. Mengoli also showed that the alternating harmonic series converges to the natural logarithm of 2:
1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots = \log 2.
Mengoli was born in 1626 in Bologna and died in 1686 in the same town. He also computed the sums
 \sum_{n=1}^\infty \frac{1}{n(n+k)},
for $k=1,2,3,\ldots$ and showed that the result is always a rational number. He naturally wondered what the sum equals to when $k=0$. This was the famous Basel problem, which he posed in 1644. It was shown by Leonhard Euler in 1735 that the sum, for $k=0$, equals $\pi^2/6$. It is not surprising that Mengoli could not find this.

I'm still not happy with this way of writing LaTeX. I can't figure out how to number equations. If only html and LaTeX were fully compatible...

22 July 2010

Successful blogging

I guess this is why my blog is not too successful.

this is a test

$\int_{M} d \omega = \int_{\partial M} \omega$

20 July 2010

New date for the end of the world: 21 May 2011

A certain simpleton, called Harold Camping, has "proved", using "mathematics", that the world will not end in 2012, as the Mayans, allegedly, predicted, and as the recent Hollywood movie proclaimed, but in 21 May 2011.

Here is how he did it:
  • Jesus was crucified on 1 April 33.
  • 2011-33 = 1978 years.
  • 1978 x 365.2422 days (the number of days in each solar year) = 722,449.0716 days. That is, approximately, 722,500 days.
  • But 722,500 = (5 x 10 x 17) x (5 x 10 x 17).
  • And the Bible says (so says the above simpleton):
  • 5 = Atonement
  • 10 = Completeness
  • 17 = Heaven
And then he said:
"I tell ya, I just about fell off my chair when I realized that: 5 times 10 times 17 is telling you a story, it's the story from the time Christ made payment for your sins until you're completely saved."
Why, he is absolutely right! Perfect line of reasoning. No flaws, no nothing. Just the truth, a heavenly one, that is.

We know that this time, Camping is right. Indeed, we can forgive his earlier mistake:
On Sept. 6, 1994, dozens of Camping's believers gathered inside Alameda's Veterans Memorial Building to await the return of Christ, an event Camping had promised for two years. Followers dressed children in their Sunday best and held Bibles open-faced toward heaven. But the world did not end. Camping allowed that he may have made a mathematical error. He spent the next decade running new calculations, as well as overseeing a media company that has grown significantly in size and reach.[source]
And his efforts were redeemed: not only did Camping learn more mathematics (indeed, how could he ever come up with the complicated formulas above), not only did he read the Bible line by line, but he also asked for forgiveness for his previous error and is now, I suppose, preparing for the doomsday. Apparently he has more followers now.
Rick LaCasse, who attended the September 1994 service in Alameda, said that 15 years later, his faith in Camping has only strengthened.
"Evidently, he was wrong," LaCasse allowed, "but this time it is going to happen. There was some doubt last time, but we didn't have any proofs. This time we do."
Yeah! PROOFS! They have proofs. And so we must accept them. They are fullproof foolsproofs proofs.

19 July 2010

"a Nadder!"'s Friday Links

The Friday Links from the blog of a Nadder are not to be missed, He does a great job and I'd like to encourage him by reposting here:



Last night I made the mistake of going to watch Inception, a new film by Christopher Nolan. The verdict:
Complete, utter trash.
It starts as a boring story. It gets more boring by trying to be complicated, it develops into a boring scenario, when it becomes so boring that you have to walk out.

So I did: After an hour or so, I couldn't take it any more and walked out feeling that I had (i) wasted money, (ii) wasted time, and (iii) become irritated by my stupid choice to go see an idiotic film.

But I should have known better. I should have read the New Yorker's review of the film:
Christopher Nolan, the British-born director of “Memento” and of the two most recent Batman movies, appears to believe that if he can do certain things in cinema—especially very complicated things—then he has to do them. But why? To what end? His new movie, “Inception,” is an astonishment, an engineering feat, and, finally, a folly.
He has spent 10 years contemplating the movie and finally came up with total trash.
He has been contemplating the movie for ten years, and as movie technology changed he must have realized that he could do more and more complex things. He wound up overcooking the idea.
If only I had spent 5 minutes looking at the review I would have realized the unfathomable stupidity of the film whose main idea is that we are watching people dreaming about dreaming:
Nolan gives us dreams within dreams (people dream that they’re dreaming); he also stages action within different levels of dreaming—deep, deeper, and deepest, with matching physical movements played out at each level—all of it cut together with trombone-heavy music by Hans Zimmer, which pounds us into near-deafness, if not quite submission.
I would have known that Nolan makes films at the level of Big Brother, for audiences who love watching films in order to kill 2 hours of their time:
Dreams, of course, are a fertile subject for moviemakers. Buñuel created dream sequences in the teasing masterpieces “Belle de Jour” (1967) and “The Discreet Charm of the Bourgeoisie” (1972), but he was not making a hundred-and-sixty-million-dollar thriller. He hardly needed to bother with car chases and gun battles; he was free to give his work the peculiar malign intensity of actual dreams. Buñuel was a surrealist— Nolan is a literal-minded man.
If only I had realized that Nolan was the one who took a wonderful Norwegian film, Insomnia, a 1997 film by Erik Skjoldbjærg, and made it into a Holywood blockbuster in 2002, a very poor immitation of the 1997 original, I would have known that Nolan's films are to be avoided at all costs. But I hadn't relized that Nolan is the same joke of a director who spoiled the Norwegian film.

David Denby, in his review for the New Yorker, concludes thus:
In any case, I would like to plant in Christopher Nolan’s head the thought that he might consider working more simply next time. His way of dodging powerful emotion is beginning to look like a grand-scale version of a puzzle-maker’s obsession with mazes and tropes.
I absolutely agree. However, I'm afraid that as long as there are Big Brother watchers, Nolan's films will keep generating money for him and his sponsors.

17 July 2010

Mao feng tea

I am, at the moment, enjoying a cup of mao feng tea (), one of the top varieties of Chinese green teas,

 grown in Mount Huang (Huangshan) in the Anhui province of China.

I discovered the mao feng tea thanks to my ex-colleague Yuanhua Feng (who managed to escape from Heriot-Watt university some time ago, and is now professor in the University of Paderborn, Germany) who had given me a box of it a couple of years ago.

The shape of the processed leaves resembles the peak of a mountain. An ancient Chinese legend about the tea goes as follows:
It is said that young scholar and a beautiful woman who worked in a tea plantation were madly in love. One day a local landowner saw the woman and wanting her for his own, seized her and forced her to become her concubine. The woman escaped only to find that the landowner had killed the young scholar. When she found this out, she immediately went to his grave which was located high on a mountainside. She wept uncontrollably until she became the rain, and the young scholar became a tea tree. It is said this is why the area where Huangshan Mao Feng Tea grows is cloudy and humid all year.
As for the meaning of the name "mao feng", and the qualities of the tea, we note that
"Mao" means fluffy, and "Feng" means mountain peaks. Although it is not a scented tea, it has many properties similar to scented tea. The tea liquor has an apricot flavor, and a fragrance like magnolias. No apricot trees or magnolias grow in the area, so it is unknown how Huangshan Mao Feng gets its unique flavors. There is however wild peach trees that grow in close proximity to the tea plants, and that may be the cause. Huangshan Mao Feng Tea drinkers say that the first brewing is fragrant, the second brewing is sweet, and the third brewing is strong. The tea tastes clean and refreshing, and lasts a long time on the tongue.
I agree with the last two sentences.


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