Primitive root mod p

Transire suum pectus mundoque potiri

Consciousness and Applied Mathematics 5 October, 2008

Filed under: Mathematics,Science — Nikolas Karalis @ 11:46 pm

This blog post is about 3 documents I found recently, which are both interesting and correlating.

The first is the 1998 article by Francis Crick and Christof Koch with the title : Consciousness and Neuroscience.

The second is a brochure by SIAM (Society for Industrial and Applied Mathematics) about the possibility of a carrier as a mathematician.

And finally, Wolfgang Klimesch’s article on EEG oscillations in alpha and theta band.

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More adventures from Copenhagen 19 August, 2008

Filed under: /dev/random,Mathematics,Science — Nikolas Karalis @ 11:14 pm

End of the first week here at DTU and the 66th ESGI and we already work for 2 days on the Industrial Problems.

We had to give some assignments on the end of the first part of the DCAMM Ph.D. course, so I gave one in modeling a company and another one about the use of Charpit’s Equations to solve a Geometrical Optics problem.

I’m on the problem proposed by Novo Nordisc, regarding the Protein Structure Modeling and we try to understand how we can predict the 3d structure of a protein based only on it’s amino acids sequence.

I also found an interesting game, called FoldIt, where people try to solve protein folding puzzles, which is difficult, funny and helps the science.

I’m also happy for buying a card game called SET, which I played last year in Vancouver and couldn’t find it in Greece. The day I bought it, I met some great Danish guys in Copenhagen and we played SET while drinking beers on a garbage container, in one of the most touristic places of Copenhagen. πŸ˜›

We also went to a thing called Kro, which means “inn”, and is a popular kind of danish pubs.

There we played a danish game with dices, which is something between Kranen and Liar’s dice (I’m not really sure).

Sunday was dedicated to sightseeing. I went to some touristic locations in Copenhagen, saw the famous Little Mermaid, tried to find the Google offices in downtown Copenhagen (no luck with that), spent some time studying at the beautiful garden of the Museum of Decoration (or something πŸ˜› ) and then I spent the afternoon in Christiania, where I followed the music and ended up at a punk concert (no live bands at that moment) where a strange and original bike-war was taking place. The teams had modified bicycles turning them into some sort of war trucks and they were trying to destroy the opponent’s truck. It was funny and had a lot of people there.

Apart from that I walked to the beautiful deer park near DTU, found the Dyrehavsbakken, the world’s oldest amusement park (1583) and ended up to the beach, where I was able to wet my hands with the waters of Øresund. There I found a party organized by the local Gymnasium’s students which was taking place at a beautiful location. Tonight I did the same thing but with a borrowed bike, which allowed me to cover many more kilometers in a fraction of the time.

In my next post I’m going to discuss some facts about Denmark.

 

Regina Week 2 – IPSW 20 June, 2008

Filed under: /dev/random,Mathematics — Nikolas Karalis @ 7:49 pm

The IPSW week is ending today, so it is time for a new post.

I worked on the problem of modeling cancer cells growth and trying to find the best combination of anti-angiogenic drugs, chemotherapy and radiotherapy in terms of dosage, frequency, scheduling and side effects.

The problem proved to be *H*U*G*E* and extremely difficult (of course! πŸ˜› ). We worked on many different aspects of the problem and we learned a lot in the process.

During the past week, I had the chance to learn many things about Partial Differential Equations (PDEs), since I was trying to solve a 3 coupled PDEs system. I learned how to use Maple’ s PDE features (thanks to Matt) which has A LOT of weaknesses and also the COMSOL Multiphysics program, for modeling of this kind of difficult problems. I also remembered the LaTeX Beamer presentation features.

But the last week was not only about work.

We had the chance to go for kayaking in the beautiful Wascana Center lake, I played a lot of basketball (until I turned my ankle yesterday 😦 ) and some cards and we also watched Iron Man in an almost empty cinema where we bought some HUGE pop corn bags! πŸ˜›

P.S. 1 : Firefox 3 was released this week, so you can see above an interesting screenshot that I was lucky to take.

P.S. 2 : Google Code Jam 2008 registration already began. First Prize $10,000 plus a free trip to the Google Headquarters in Mountain View, California for the finals! πŸ™‚

 

Regina Week 1 – GIMMC 16 June, 2008

Filed under: /dev/random,Mathematics — Nikolas Karalis @ 2:07 am

The first week of the GIMMC is over and there is one more week left before I go back to Greece and the exams.
The past week I have been working on the problem of snow removal for the city of Regina.
Our mentor was Dr. Ed Doolittle and we were able to categorize the problem as a Capacitated Arc Routing Problem (CARP). I performed a rough Cost Estimation to show that it probably isn’t worth of optimizing, but people should really use a relatively efficient way of solving it. You can find our presentation here.

Some highlights of the past week :

  • We have been walking A LOT! Downtown Regina is about 4 km from the University, and since the transit system here is not very good it is easier to walk this distance.
  • Regina has about 180.000 people and is so spread out, that almost nobody walks, even downtown.
  • I’ve been 2 times at the Casino Regina, however I didn’t play. Is is interesting watching all these people being there all day long and (mostly) loosing money.
  • We organized a poker tournament one night (guess who won! πŸ˜› ) and a Mafia game another night.
  • I visited some Dollarama and A Buck Or Two shops. They have a huge collection of useless (and sometimes usefull) things in ridiculously low prices (1$ πŸ˜› )

My personal suggestions for Regina :

  1. The Cathedral Village Freehouse – Nice bar/restaurant, with live music.
  2. Bonanza – An All-You-Can-Eat restaurant with nice steaks and good prices.
  3. O’Hanlon’s Pub – The place to be when everything else is empty.
  4. Trifon’s Pizza – The bar/restaurant near the university. Nice meals, very cheap beer, nice atmosphere.
  5. Vintage Vinyl – Great Discs and musical T-Shirts Collection.
  6. Wascana Centre – It rocks! Beautiful lake…
  7. The Old Warehouse District – Most of the bars are in this area.
  8. Lazy OwlThe UofR’s student pub.

P.S. For the last 2 days the Internet connection at the university has some problems, so I can’t upload photos now. But in the next few days, they will be posted on Facebook, so just be patient.

 

Industrial math @ Regina 1 June, 2008

Filed under: Mathematics — Nikolas Karalis @ 5:58 pm

In 5 days from now I will be on a plane going 9000 klm across the world.

I’m going to Regina the capital of the Saskatchewan province of Canada where I will attend the 2008 PIMS Industrial Problem Solving Workshop and Graduate Industrial Mathematics Modelling Camp.

It will take place at University of Regina. It will be a good opportunity to work on some interesting problems and get to learn another city in Canada after Vancouver.

The thing is that the chaos in this wannabe civilized country called Greece has won once again.

I had to reject an offer to participate in a summer program on Mathematical Biology at the Mathematical Biosciences Institute in Ohio, USA since the exams period in our university is in a quantum state.

You never know if and when it is going to happen.

I will write soon with some photos from my first days in Regina.

 

New semester! 30 March, 2008

Filed under: /dev/random,Mathematics — Nikolas Karalis @ 8:51 pm

Exams are over, and the 8th semester already began.

It has been a long time since my last post…Β  In the meantime a lot of strange things happened. A general strike, a few riots etc.

I’ve been enjoying the lack of electricity for a few hours a day, which provides a peaceful and dark environment for thinking and resting your mind..

I don’t have much to say this time, but here is a gift…

The other day I needed to convert an image to pdf, so I created a simple command line utility to do so…

You can find it here. I hope you will find it useful.

So, here are a few links from OCW for some of the courses I’ll be taking this semester in case anyone takes the same courses…

Physics II (YES! Walter Lewin again! :D)

OCW 8.02

Partial Differential Equations

OCW 18.152

Signal Processing

OCW 6.011

Modeling

18.361J

Optimization

15.094J

6.253

Optimal Control

16.323

Logic

24.241

Theory of Probability

18.175Β 

 

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.

Theorem.
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.

QED