Primitive root mod p

Transire suum pectus mundoque potiri

Infinite products 28 February, 2008

Filed under: Mathematics — Nikolas Karalis @ 12:23 am

Background song : Molotov – Frijolero

Since when this blog migrated from my own server to the WordPress one, I always disliked some of the restrictions this implied. One of the most important ones was the lack of LaTeX.
But as you may have already guessed from the previous sentence I have great news.
Wordpress has a new feature, so we can write normal LaTeX code in our posts and comments.

So, I inaugurate this new feature, by extending a result I found at The Everything Seminar, a blog I found recently and since then reading it.

At the post Convergence of Infinite, we see a simple but strong convergence test.

Let a_n be a sequence of positive numbers. Then the infinite product prod_{n=1}^{infty} {(1+a_n)} converges if and only if the series sum_{n=1}^{infty} a_n converges.

It is easy to see that we can use this result also for a sequence of negative numbers, when -1 leqslant a_n leqslant 0 .

Using this we can derive the classic result of the divergence of the harmonic series.

We can also show that the alternating harmonic series converges.

Alternating Harmonic Series

sum_{k=1}^{infty} frac{(-1)^{k+1}}{k}=1-frac{1}{2}+frac{1}{3}-frac{1}{4}+...

We take prod_{n=1}^{infty} {(1+a_n)} = (1+1)(1-frac{1}{2})(1+frac{1}{3})(1-frac{1}{4}) = 2 times frac{1}{2} times frac{4}{3} times frac{5}{4} times ... = 1

So, the alternating harmonic series converges.



La lingvo de la mondo 26 February, 2008

Filed under: /dev/random — Nikolas Karalis @ 11:30 pm

Ni estas mia unua teksto en Esperanto lingvo. (This is my first post in Esperanto language).

After a successful first half of the exams period, I’m here writing this post about my new “discovery”.

Last weekend I accidentally heard of Esperanto, so I thought to give it a try. And here I am, having taken 3 “courses” from a computer program I downloaded.

Esperanto is a constructed language, meaning it was designed on purpose by L. L. Zamenhof , a Polish ophthalmologist in 1887.

The ideal behind it is that a language should be easy and flexible, to be used as a universal second language to foster peace and international understanding.

My opinion so far is positive. The language is relatively easy, well structured and the pronunciation of the words sounds well. I would say it has the structure of German, the orthography and endings of Slavic languages, common roots of the words come from Latin (and a bit ancient Greek) and in the first look it reminds you of Spanish.

They say (and I kinda hope it’s true) that it can provide good ground knowledge to support the easier learning of other languages and of course, as every language there is a whole culture generated by and around the language.

For example, you can take a look at a ska group called Esperanto Desperado which has a few nice Esperanto songs.


I provide a few links here with references and courses.

Esperanto – English translator

lernu! (online courses and material)

MIT Societo por Esperanto (it has a few interesting links)


Cognitive processes 21 February, 2008

Filed under: Science — Nikolas Karalis @ 11:59 am

Now that the snows have gone and the seismology project is over, I can concentrate on my exams… 😛

However, the other night I tried to sleep (after drinking a late coffee). I couldn’t, so while thinking about some numbers, i realized that i can’t think of a numbers without “spelling” it in my mind. For example, it is impossible to think the shape or the idea of 3 without saying “three”. Some times, i can even feel my tongue trying to move when thinking of numbers. Of course, the same thing happens while writing this post, but i guess this could be expected.

From my point of view, this means that (at least for my brain) there is a “connection” between the cognitive processes of speaking and thinking about numbers and words. I’m not sure whether this is a form of synaesthesia or an inability to process numbers in a more abstract level.

I tried to find papers and references on this, but i didn’t have much luck. An interesting paper (on a slightly different subject) i found is : S.Dehaene, L.Cohen Cultural recycling of cortical maps, Neuron. 2007 Oct 25;56 (2):384-98

Any feedback on this would be highly appreciated.

Please write about any similar experiences you had and even try this mental experiment and share your results!

P.S. Two photos I shot yesterday… Our university in white.



Seismology, Snow & Procrastination 18 February, 2008

Filed under: /dev/random,Programming — Nikolas Karalis @ 1:29 am

I haven’t updated my blog for a while. No specific reason… I was just procrastinating since exams were coming and I didn’t have much to say.

The latest news from Greece are the earthquakes and the snow (and of course my exams 😛 ).

The last few days, 2 big earthquakes (6.2 R and 6.9 R) shaked Southern Greece and last night, it began snowing in Athens!

This happens once every one or two years, so it is big news

So last night, a small earthquake reminded me of the passionate discussions about the earthquakes and since i didn’t have much to do because of the snow and my constant last month procrastination, i decided to build a database of the Greek Earthquakes of the last few years.

So, a little bit of coding and a few hours later, I proudly (:P) present my Greek Seismological Database (click here to visit).

EDIT : A new addition : It is automatically updated when a new earthquake happens…

I end this post, with a few photos from Athens in white.