Cubes

[Last edition: 28/07/2006]

As I promised two weeks ago, I'll talk about cubes today. One of the things Prof. Caticha taught me in his visit is an interesting way of representing cubes in any dimension using matrices. First, let me explain what is a n-dimensional cube, or simply a n-cube. Let us use an iterative method to construct it. Let us set up our notation: the representation of the n-cube will be denoted by the letter


The 0-dimensional cube is just a point and we will represent it by a 1x1 matrix, or as it is more commonly known, by a number: the number 1. We will consider n-cubes as being non-directed graphs and represent them by their connectivity (or incidence) matrix: the (i,j) element of the matrix will be 1 if the i-th vertex of the graph is connected with the j-th vertex and 0 otherwise. This justifies using the number 1 to represent the 0-cube, once it has just 1 point which we will consider as connected to itself.

Now, let us proceed two the 1-cube. The one-dimensional cube is just a line segment joining two points. We will see it as two 0-cubes joined by their vertices and then it will be represented by the 2x2 matrix:


In the end, a n-cube can be described by a block matrix derived from the (n-1)-cube with the form


Where I_{n-1} (in LaTeX notation :) ) is the (n-1) x (n-1) identity matrix and its role is to connect the vertices of the two C_{n-1} cubes to form its edges. An algebraic way to write this is


Where, if A is a 2x2 matrix, then


is the exterior, tensor or Kroenecker product between the two matrices A and B, which gives as a result a block matrix as indicated, and


is a Pauli matrix. Actually, I think there is a slightly more beautiful formula, but I forgot now and I'm waiting for Prof. Caticha to answer me an email I sent him...

Okay, now that I did what I promised, some more random things.

Latest (and some old) interesting news:

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On my desk:

1. "The Bethe lattice spin glass revisited", Marc Mezard, Giorgio Parisi (cond-mat/0009418)
2. "Against 'Realism'" - Travis Norsen (quant-ph/0607057)
   

Picture: A projection in 3 dimensions of a 12-cube taken from the site Equality Set Projection - A new algorithm for the projection of polytopes in halfspace representation

Blogging...

I should have blogged yesterday, but I needed to check a formula which I wish to put here, which I will do in the next week (I hope). I will just talking here today about some random things.

First, today is the 37th anniversary of the first visit of men to another celestial body. Of course I'm talking about the moon, as we didn't manage to visit (at least personally) any other, which I hope to happen still in my lifespan. Indeed, i would like to do it one day (I know, I'm dreaming too much...). You can have more details in the Wikipedia article: Apollo 11.

Another thing is that I found an interesting paper in the arXiv yesterday:

Physical limits on information processing
Stephen D. H. Hsu
hep-th/0607082

It is about bounds on the velocity of information processes in nature. This article is one of many I've been following in the last years which shows a trend to study information as a physical quantity. I'd like to call it Information Physics, or maybe, to be a little more modern, iPhysics.

Anyway, I sent an email to Stephen asking about the definition of information process. He was kind enough to answer me very fast and I will reproduce it here as it may be interesting:

Hi Roberto,

I'm using the Margolus-Levitin (ML) definition, which is evolution
from some initial state i to some orthogonal final state f (e.g.,
| f > = 0 ).That is defined as a single operation.

In classical computation this seems quite reasonable, as "flipping a
bit" presumably means moving some part of the system from, e.g., one
energy eigenstate to another, which would mean a transition between
two orthogonal states.

For quantum computation it is not so clear how to define a discrete
unit of computation. However, I think what ML chose is very
reasonable. If you haven't evolved the system into a different
orthogonal state, it isn't really distinguishable from the initial
state as it still has some overlap with the initial state.

Hope that makes sense.

Cheers,
Steve

Stephen also pointed me to his blog "Information Processing", which is very interesting and I'm adding to my list in the lateral column.

Picture: Astronaut Buzz Aldrin on the moon, from NASA.

Visitor

Professor Nestor Caticha, my former PhD advisor in Brazil, has been visiting Aston University for three weeks and departed in the last Saturday to attend to the MaxEnt 2006 conference in Paris, in which he will present two papers, one of them in collaboration with me: Online Learning in Discrete Hidden Markov Models

We had a good time and, as always, we talked about several different and interesting matters, among them, geometricalgebra, integration and hypercubes. I will not talk here about geometric algebra, once that I have already written a post about that in the past (see Geometric Algebra). I'll talk about integration and let hypercubes to the next week.

You may think 'What could be interesting in integration?'. Well, we didn't talk about ordinary integration, of course. What we did talk about was something that, for a lack of a better term, I will call 'fractional integration'. Consider the function:



D is just a constant. For t going to zero, the gaussian becomes a Dirac delta function and the integration simply gives the function f(x). For t going to infinity, the square root outside the integral cancels the numerator inside it and the exponential becomes 1, giving the ordinary integral of f. So, as the parameter t varies from 0 to infinity, we vary from the ordinary function to the totally integrated function in a continuous way. He said to me that it has applications in renormalization group theory and, consequently, in QFT. He aso told me that his brother, Ariel Caticha, used this kind of integral in his PhD thesis to expand the action in path integrals in QCD. I need to play a little more with this, but seems a very interesting idea that could have more applications.

Well, changing the subject, I was browsing the Wikipedia and found this interesting Raven Paradox with a Bayesian solution. Probably Osame already know it, but It could be interesting for his students. :)

(As you may note, my posts are becoming smaller with time. I apologise, but the quantity of work here in Aston is increasing and my available time does not scale with it. So, I'll keep things this way for a while. Sorry again.)

Picture: Prof. Nestor Caticha in his office at the University of Sao Paulo, Brazil.

Branes, Descartes and Reality

When I was young I tought that science was concerned with the ultimate nature of reality. It was a long road before I realized that reality is a complicated concept that inhabits the realm of philosophy. Sometime ago, when I was reading about braneworlds, I thought about a situation that can illustrate it quite well.

Imagine that the universe contains two branes that, by hypothesis, do not interact in any way. Each brane has a set of physical laws that enables the development of intelligent life. Let us say that we humans live in one of these branes. As there is no interaction (and will never be) with beings in the other brane, their existence is something that science is not concerned with. Indeed, as the hypothesis of their existence is not falsifiable, it is ruled out as science by its the very definition. Of course our science cannot deny our existence, but as the situation of both branes is symmetric, for the scientists of the other brane talking about our existence is also unscientific. If you take by heart Descartes' Cogito ergo sum, then both civilizations exist but only for themselves, the existence of one to the other is not a scientific hypothesis in each brane. And it is not just about the civilizations, but the non-interaction effect rules out the entire brane from the scientific realm of the other.

A friend with whom I talked about this told me that it is similar to the traditional paradox of the tree falling without nobody to watch, but the brane paradox above is more deep, because the tree can in principle leave traces of its fall, but the branes cannot sense the existence of each other by hypothesis!

Well, that is why I started to study a little philosophy in addition to physics. Although it may be not falsifiable, the nature of reality is an interesting question and I remember that, when I was a child, curiosity was the true feeling that led me to choose to be a scientist.

By the way, today is another Holy Day for physics, maybe even holiest than the day of my last post. In this day, in 1687, Philosophiae Naturalis Principia Mathematica (which appeared in the Da Vinci Code depicted as a book about astronomy and gravity :) ) was first published by Isaac Newton. We can say in a certain way that, for physics, this was the book (an the man) that really started it all in the way we know it today.

Picture: http://www.nevis.columbia.edu/~conrad/visuals/hep_images.html