Fooling with mathematicians

28/02/2013

ResearchBlogging.org

I am still working with stochastic processes and, as my readers know, I have proposed a new view of quantum mechanics assuming that at the square root of a Wiener process can be attached a meaning (see here and here). I was able to generate it through a numerical code. A square root of a number can always be taken, irrespective of any deep and beautiful mathematical analysis. The reason is that this is something really new and deserves a different approach much in the same way it happened to the Dirac’s delta that initially met with skepticism from the mathematical community (simply it did not make sense with the knowledge of the time). Here I give you some Matlab code if you want to try by yourselves:

nstep = 500000;
dt = 50;
t=0:dt/nstep:dt;
B = normrnd(0,sqrt(dt/nstep),1,nstep);
dB = cumsum(B);
% Square root of the Brownian motion
dB05=(dB).^(1/2);

Nothing can prevent you from taking the square root of  a number as is a Brownian displacement and so all this has a well sound meaning numerically. The point is just to understand how to give this a full mathematical meaning. The wrong approach in this case is just to throw all away claiming all this does not exist. This is exactly the behavior I met from Didier Piau. Of course, Didier is a good mathematician but simply refuses to accept the possibility that such concepts can have a meaning at all based on what has been so far coded in the area of stochastic processes. This notwithstanding that they can be easily computed on your personal computer at home.

But this saga is not over yet. This time I was trying to compute the cubic root of a Wiener process and I posted this at Mathematics Stackexchange. I put this question with  the simple idea in mind to consider a stochastic process with a random mean and I did not realize that I was provoking a small crisis again. This time the question is the existence of the process {\rm sign}(dW). Didier Piau immediately wrote down that it does not exist. Again I give here the Matlab code that computes it very easily:

nstep = 500000;
dt = 50;
t=0:dt/nstep:dt;
B = normrnd(0,sqrt(dt/nstep),1,nstep);
dB = cumsum(B);
% Sign and absolute value of a Wiener process
dS = sign(dB);
dA = dB./dS;

Didier Piau and a colleague of him just complain on the Matlab way the sign operation is performed. My view is that it is all legal as Matlab takes + or – depending on the sign of the displacement, a thing that can be made by hand and that does not imply anything exotic.  What it is exotic here it the strong opposition this evidence meets notwithstanding is easily understandable by everybody and, of course, easily computable on a tabletop computer. The expected distribution for the signs of Brownian displacements is a Bernoulli with p=1/2. Here is the histogram from the above code

Histogram sign(dW)This has mean 0 and variance 1 as it should for N=\pm 1 and p=\frac{1}{2} but this can be verified after some Montecarlo runs. This is in agreement with what I discussed here at Mathematics Stackexchange as a displacement in a Brownian motion is a physics increment or decrement of the moving particle and has a sign that can be managed statistically. My attempt to compare all this to the case of Dirac’s delta turns out into a complain of overstatement as delta was really useful and my approach is not (but when Dirac put forward his idea this was just airy-fairy for the time). Of course, a reformulation of quantum mechanics would be a rather formidable support to all this but this mathematician does not seem to realize it.

So, in the end, I am somewhat surprised by the behavior of the community against novelties. I can understand skepticism, it belongs to our profession, but for facing new concepts that can be easily checked numerically to exist I would prefer a more constructive behavior trying to understand rather than an immediate dismissal. It appears like history of science never taught anything leaving us with a boring repetition of stereotyped reactions to something that instead would be worthwhile further consideration. Meanwhile, I hope my readers will enjoy playing around with these new computations using some exotic mathematical operations on a stochastic process.

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v2


Lucasian chair again

13/02/2013

On 22 May, Professor Michael Green, Professor Michael Greenthe incumbent Lucasian Professor at Cambridge University, will be 67 and must retire. He succeeded Stephen Hawking that left this chair for the same reason on 2009. Well before Hawking’s retirement, Cambridge University issued an announcement asking for possible candidates and, after the selection ended, Professor Green come out as the chosen one. This time, no announcement is out from Cambridge and so, it is possible that the successor of Professor Green should be already known. I think the news will be released in the next few months. Till now, no rumors spread.


Back to CUDA

11/02/2013

It is about two years ago when I wrote my last post about CUDA technology by NVIDIA (see here). At that time I added two new graphic cards to my PC, being on the verge to reach 3 Tflops in single precision for lattice computations.  CUDA LogoIndeed, I have had an unlucky turn of events and these cards went back to the seller as they were not working properly and I was completely refunded. Meantime, also the motherboard failed and the hardware was largely changed  and so, I have been for a lot of time without the opportunity to work with CUDA and performing intensive computations as I planned. As it is well-known, one can find a lot of software exploiting this excellent technology provided by NVIDIA and, during these years, it has been spreading largely, both in academia and industry, making life of researchers a lot easier. Personally, I am using it also at my workplace and it is really exciting to have such a computational capability at your hand at a really affordable price.

Nvidia TeslaNow, I am newly able to equip my personal computer at home with a powerful Tesla card. Some of these cards are currently dismissed as they are at the end of activity, due to upgrades of more modern ones, and so can be found at a really small price in bid sites like ebay. So, I bought a Tesla M1060 for about 200 euros.Tesla M1060 As the name says, this card has not been conceived for a personal computer but rather for servers produced by some OEMs. This can also be realized when we look at the card and see a passive cooler. This means that the card should have a proper physical dimension to enter into a server while the active dissipation through fans should be eventually provided by the server itself. Indeed, I added an 80mm Enermax fan to my chassis (also Enermax Enlobal)  to be granted that the motherboard temperature does not reach too high values. My motherboard is an ASUS P8P67 Deluxe. This is  a very good card, as usual for ASUS, providing three PCIe 2.0 slots and, in principle, one can add up to three video cards together. But if you have a couple of NVIDIA cards in SLI configuration, the slots work at x8. A single video card will work at x16.  Of course, if you plan to work with these configurations, you will need a proper PSU. I have a Cooler Master Silent Pro Gold 1000 W and I am well beyond my needs. This is what remains from my preceding configuration and is performing really well. I have also changed my CPU being this now an Intel i3-2125 with two cores at 3.30 GHz and 3Mb Cache. Finally, I added  16 Gb of Corsair Vengeance DDR3 RAM.

The installation of the card went really smooth and I have got it up and running in a few minutes on Windows 8 Pro 64 Bit,  after the installation of the proper drivers. I checked with Matlab 2011b and PGI compilers with CUDA Toolkit 5.0 properly installed. All worked fine. I would like to spend a few words about PGI compilers that are realized by The Portland Group. PGI Workstation compilersI have got a trial license at home and tested them while at my workplace we have a fully working license. These compilers make the realization of accelerated CUDA code absolutely easy. All you need is to insert into your C or Fortran code some preprocessing directives. I have executed some performance tests and the gain is really impressive without ever writing a single line of CUDA code. These compilers can be easily introduced into Matlab to yield mex-files or S-functions even if they are not yet supported by Mathworks (they should!) and also this I have verified without too much difficulty both for C and Fortran.

Finally, I would like to give you an idea on the way I will use CUDA technology for my aims. What I am doing right now is porting some good code for the scalar field and I would like to use it in the limit of large self-interaction to derive the spectrum of the theory. It is well-known that if you take the limit of the self-interaction going to infinity you recover the Ising model. But I would like to see what happens with intermediate but large values as I was not able to get any hint from literature on this, notwithstanding this is the workhorse for any people doing lattice computations. What seems to matter today is to show triviality at four dimensions, a well-acquired evidence. As soon as the accelerate code will run properly, I plan to share it here as it is very easy to get good code to do lattice QCD but it is very difficult to get good code for scalar field theory as well. Stay tuned!


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