Primitive root mod p

Transire suum pectus mundoque potiri

Περί επιστήμης και συνεδρίων 28 June, 2009

Filed under: Science — Nikolas Karalis @ 8:59 pm

Ένα δροσερό απόγευμα του Ιουνίου, σε ένα όμορφο μικρό γαλλικό χωριό των Πυρηναίων. Η πανεπιστημιακή ελίτ δειπνεί, εν μέσω εποικοδομητικών συζητήσεων, σε ένα ακριβό τουριστικό θέρετρο. Τις σαλάτες διαθέχτηκαν τα καλά κρέατα και αυτά τα ακριβά τυριά με  confiture και ένα καλό κρασί και τσιγάρο στο χορτάρι με θέα το καταπράσινο δάσος και τα βοσκοτόπια  στο βάθος του τοπίου. Είναι απλά ένα ακόμα απόγευμα απο τα πολλά που έχω παρεβρεθεί και πιθανόν να ακολουθήσουν. Μόνο τα πρόσωπα και ο χρόνος αλλάζουν. Αλλά το σκηνικό είναι το ίδιο.

Ίσως να αναρωτηθεί κανείς… Το επίπεδο της δουλειάς σου τα αξίζει/δικαιούται αυτά; Σίγουρα όχι είναι η απάντηση. Αλλά και ποιός απο όλους αυτούς τα δικαιούται; Πρόσωπα χαρωπά, ρούχα καθαρά και δάχτυλα που η μόνη ταλαιπωρία που τους επιφύλασε η ζωή είναι τα χαιδολογήματα με το πληκτρολόγιο, μεταξύ επιστολών, εργασιών και παρουσιάσεων.

Τα σταυρωμένα μαχαιροπίρουνα μετά το δείπνο και το  savoir vivre μαζί με τα γεμάτα στομάχια και τις ανούσιες συζητήσεις των παρευρισκομένων περί των τεκταινόμενων στην επιστημονική κοινότητα δεν μπορεί παρά να φέρουν στο νού τη σπατάλη των ήδη πενιχρών κρατικών επιχορηγήσεων για την εκπαίδευση.

Ανάμεσα στις συζητήσεις, ίσως ακουστεί και κάποια δυσαρέσκεια για την άνοδο της (ακρο)δεξιάς στην Ευρώπη μεταξύ των πιο ανοιχτόμυαλων, όπως προστάζουν τα trends της ακαδημαϊκής κοινότητας.
Αλλά οι συζητήσεις αφορούν κατά κύριο λόγο τα διδακτορικά, τις εργασίες, του υπολογιστές και τα προγράμματα και για να σπάει η μονοτονία μερικές διαμάχες για τα πειράματα σε ζώα και άλλα ζητήματα επιστημονικής ηθικής.
Μεγέθη απο τα μικροsecond μέχρι τα TeraHerz, απο τα μικρόμετρα στα Terabytes. Μεγέθη που εύκολα μιλάς για αυτά και δύσκολα τα αντιλαμβάνεται ο νούς σου.

Και μετά αναρωτιέται κανείς πως ο Ramanujan, όπως και πολλοί άλλοι πριν και μετά απο αυτόν, υπέφεραν απο κατάθλιψη και καταστράφηκαν, μέσα στην “κουλτούρα” του δυτικού πολιτισμού, ενώ τους εκμεταλευόταν το ακαδημαϊκό κατεστημένο.

Αλλά όταν το διεθνές σύστημα για την έρευνα έχει στηθεί απο κάποιους μεγάλους πανεπιστημιακούς και το μυστικό της ανόδου στην πανεπιστημιακή κλίμακα κληροδοτείται απο γενιά σε γενιά με συστατικές επιστολές, συνδημοσιεύσεις και προσκλήσεις σε συνέδρια, τίποτα απο αυτά δεν είναι απορίας άξιο.
Απο την αρχαιότητα μέχρι σήμερα, η επιστήμη και η “μόρφωση” ήταν πάντα προνόμιο και χόμπι των πλούσιων και βολεμένων.
Μπορείς να βρείς πολλές δικαιολογίες για το γιατί σπουδάζεις και κάνεις έρευνα και επιστήμη.
Αλλά στο τέλος της μέρας, θα είσαι απλά αυτός που κοιμήθηκε με το στομάχι γεμάτο…

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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.

 

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. 😛

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 😛 ) 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.

 

Greek Summer Schools and Seminars 4 May, 2008

Filed under: Science — Nikolas Karalis @ 8:31 pm
Last Edited : 26 June 2008

Here is a list of the summer schools organized in Greece this summer and a few interesting seminars in the next months. As you can see, this is a (scientifically) hot summer in Greece! 😉

Summer Schools

Economics & Econometrics Advanced Summer School (Rethymno, Crete)

EKEFE Demokritos Summer School (Athens)

Mathematical Modeling (Athens)

Non Linear Science & Complexity (Athens)

5th Mathematics Summer School (Heraklion, Creta)

20th Advanced Physics School (Heraklion, Creta)

9th Astronomy Summer School (Volos)

Onassis Foundation Science Lectures (Heraklion, Creta)

Looks in Modern Physics — for high school students

Seminars and Congresses

Stoa tou Bibliou – Free University

12th Panhellenic Conference in Mathematical Analysis (University of Athens)

8th Panhellenic Conference in Algebra, Number Theory and Application (NTUA)

4th International Summer School on Emerging Technologies in Biomedicine (Patra)

8th International Conference on Bioinformatics and Bioengineering (Athens)

18th European Conference in Artificial Intelligence (Patra)

 

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.

                  

 

Millenium Problems Bet 26 January, 2008

Filed under: Mathematics,Science — Nikolas Karalis @ 1:46 am

Like the Hawking-Thorne bet, we have been talking with my good friend and fellow mathematician Primelude about the Clay Millennium Problems and if and how they are going to be solved. For a few of them we agree and for others not. So we decided to bet on that. We will post our predictions, and wait if and how some of these are going to be solved. The bet is a one year subscription to a (scientific? 😛 ) magazine of the winners choice.

So here are my predictions…

It will be proved to be True, using tools from the Etale cohomology.

Probably False. Counterexample will be found.

Probably will be proved to be False. But even if it turns out to be True,
computationally (–> practically), P will still be not equal to NP.

************************

Grigori Perelman

************************

Smooth solutions always exist for both Navier-Stokes Equations and Euler Equations.

True. I can even predict who is going to prove it, if it happens in the next 30 years.

No! I’m not telling you! 😉

My guess, it is going to happen by finding attributes of the distribution of the primes.

Com’ on! I almost said who! 😛

Btw, i think that if one of the 7 problems is unsolvable,then this is the one.




There exists a theory and it can explain the mass gap… True, true true! 🙂


And now we sit and wait! Or try to win the bet (together with the 1.000.000$) 😛


P.S.
On Goldbach conjecture :

I believe it is true, but it will take us at least 100 more years before it is being solved.

 

Synæsthesia & Algebra 18 January, 2008

Filed under: Mathematics,Science — Nikolas Karalis @ 5:53 pm

Synæsthesia is the neurological condition where the stimulation of a cognitive pathway leads to the involuntary stimulation of another cognitive pathway. In simple English, it mean that when for example you see a number, your brain reacts as if you have seen a color. This also happens for shapes, letters, sounds etc.

People under the influence of psychedelic drugs can sometimes feel it (the popular “hear the colors and see the sounds” effect) or even people after a stroke.

The most common explanations are two.

1) The cross-activation of adjacent regions of the brain which are involved in the color and numbers/letters recognition.

2) The reduction in the amount of inhibition along feedback pathways (can explain the LSD/mescaline effects).

But what do all of these have to do with Algebra?

Well, last night while studying Algebra (from the wonderful textbook A First Course in Abstract AlgebraJ.Fraleigh, which i highly recommend) i stopped on a phrase saying that the Dihedral D4 symmetry group is simple and beautiful.

And this is true. D4 is beautiful. But how can you see that in this Cayley table?

o I T R R2 R3 RT R2T R3T
I I T R R2 R3 RT R2T R3T
T T I R3T R2T RT R3 R2 R
R R RT R2 R3 I R2T R3T T
R2 R2 R2T R3 I R R3T T RT
R3 R3 R3T I R R2 T RT R2T
RT RT R T R3T R2T I R3 R2
R2T R2T R2 RT T R3T R I R3
R3T R3T R3 R2T RT T R2 R I

I believe that the following representation is much better for seeing the underlying beauty and symmetry.

So, i decided to create a web application for doing exactly this. You provide the Cayley table of the group, and it returns you the color representation of it.

You can find it here : http://papaya.hep.ntua.gr/projects/algebra/

For any suggestions/problems, fell free to contact me.