As you may have noticed

I have not been writing here for a year or so. Not that I have been playing cards and not that I have not done any mathematics. But time, my friends, time... is not on my side.

Well, I will postpone this a while and then restart again... hopefully not too long from now. Thanks for dropping by.

The approach

To calculate the probabilities of different hands in poker are quite complicated. What makes it complicated are the straights. If poker only concisted of color and numbers, I believe all calculations would be quite straightforward, but the possibilities of straights makes it almost impossible, at least if you want to calculate the probabilities of a winning hand when there are more than 2 players.

Consider the number of combination there is for a Texas Hold'em table of 10. For starters there are 2598960 different table card combinations. But with the rest of the cards, taking in to consideration the place on the table which has some importance of the game, there are 2.31945E+28 different ways of distribute those cards, that is 6,02815E+34 all in all. Even not taking into account the places the number of combinations are 1,6612E+28, a number too big even for the most powerful computer to evaluate by raw counting. There are of course symmetries, the most obvious being the color symmetries, but dividing by 24 doesn't help much either (and 24 is an upper limit for the color symmetries. For some games that figure could be as low as 6).

The most natural approach is therefore simulation, at least when to evaluate a more complex situation. Now poker is also betting, and to put that into calculations, needs even more approximations of the situation, such as which hands hole-cards are playable, and when may a hand be considered dead and so on. There are quite a few situations to evaluate with simulations.

AA is the best starting hand i Texas Hold'em, but going all in in a family pot will only win 31% of the games of 10 players. You need to have better odds than that, if you want to win.

Mathematics of poker

So now it is official. I have become an amateur poker-player. Sorry to dissapoint you who believed that I would involve in something more serious than that. Or maybe it is serious. Poker consists of two quite different variables. One - it is purely mathematical, although the calculations, the probability calculations are both abstract, difficult and rather complicated. Two is the betting procedures, which actually also is mathematical, but mainly psychological. Taste the last word in that sentance: Psycho - the mind, psycho-logical - logics of the mind.

My plan is to mix mathematics with poker. Be my guest to read on.

And so on...

I have started a new project about mathematics. Maybe I will write some more about it, later on, but since it involves consultant work, I cannot write about this work. But maybe some theories around it...

Restarting...

I found a few old Excel-sheets with calculations, when I was cleaning up files on my computer today. Have done a lot of fun stuff on Excel in the past. I found a sheet where I calculated the correct SO3-element on a threedimensinal transformation. Don't remember why I did that at the time, but I have been fascinated by doing correct - photographical 3D imaging, starting from scratch, whithout any help from a CAD-program.

Another one of those sheets where calculations of cubic equations, using one of the most beautiful substitutions I have ever discovered: the Vieta substitution. It is a pease of art.

Didn't find any of my fractal-calculations, but that might been a while since I played with those. My favorite, the Newton-fractal of the equation x^3-1=0, using different coulors for the different solutions is also one of my historical achievements.

One fun thing I found was also an equation for a Klein-bottle. I had a suggestion of how to construct it, but it took some serious calculations on Excelsheets, before I was convinced that my formulae was correct.

So where did all this interest start? I am not sure, but there was quite a few question marks left, once I had left my engineering studies. One was to understand the relation - the homomorphism between SO3 and SU2. Actually, when I remembered the presentation at some obscure course in quantum physics, I remembered it the wrong way. It is nor SO3 that is homomorph to SU2, even though the matrix is smaller (9 compared with 8), it is the other way around. It is not that difficult to realise why. S03 may be seen as a 3-dimensional ball, with radius of pi (half-a-turn). At the edge it is connected (topologically) whith the opposite side. SU2 on the other hand is the three-dimensional surface of a 4-dimensional ball, which is two balls (a positive and a negative 3-dimensional ball connected at their 2-dimensional shell.

SO3 is for those who are not familiar with terminology, the orthonormal matrix, with determinant of 1. SU2 is the komplex 2-dimensional matrix with the determinant of 1. When I investigated them, I even determined specific homomorpisms, how they could be translated into one-another. The subject still fascinates me. I have not yet been able to determine relations between higher orders of SO and SU, where the numbers of unknown are equal.

SO2 and SU1 are isomorphic, but what about SU4 and S06, or SU11 and SO16? I am convinced that at least someone have investigated for real... but maybe it is not that is too much technique and too little theory. SU23 and SO33 *s*

No, I have just taken a pause

I have used computertime to play computer-chess instead of studying algebra. It is fun as well, but I will soon return to my books. Or I already have.

Sweet topology; trying to memorise expressions like homogenious, homeomorphic, regular, and T0, T1, T2 and T3. f is continuous iff the inverse of any open set is an open set. If f is continuous the image of a compact set is compact. Simple, elementary.

Perhaps my aim

is and has been a little too high. Come to think of it maybe Langs algebra is just a little bit too much to study by your own. The thing is - I would if I wanted to, get the hang of the whole book, but maybe it is time to revaluate the plan. I have done some advancement. A few days ago I was stuck in an expression, and it really seemed that I was stuck for good. It was a different stop than before. I had no clues to how to solve it. All of a sudden the book pointed to a concept that definitely wasn't to be found previously in the book. The other problems had just been semantic, but this time it was theoretic.

It said bluntly that sqr(2) was in the kernel of (x^2-2). Well, what was a kernel of a polynomial? Nothing before had pointed out that a polynomial could have a kernel - it was a mystery hidden in an enigma... But what I did realise after thinking just a little bit more on the subject was that, if x=sqr(2), (x^2-2) could be considerred as an element of some kind of ideal, since (x^2-2) multiplied with any polynomial yields the result 0, for sqr(2). That is probably the reason, surely it is close enough to the real answer, and it was good enough for me. But the way it was throughn out, as it was basic knowledge, that is what made me concerned.

However, one of my goals is to get to know some more on Galois theory, and the beginning in Lang of that chapter is really a complete no-no. So I danced around on the internet. Especially one link was intresting. I have come across John Baez before on the net. A nice, happy guy, with good pedagogical skills. The page of today is however http://math.ucr.edu/home/baez/twf_ascii/week201.