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.

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.

## 4 comments:

Ahahahahah. Essa eu nao esperava! Que tal chamar t de "parametro q" e integral fracionaria de "integração generalizada"?

Afinal, nao é fracionaria, nao tem fracao nenhuma ai...

Alamino, qualquer representação da função delta tipo delta(q) =lim_q F(q) (q-> infty) poderia ser usada para definir a integração generalizada?

Quero dizer, pegamos o F(q) e injetamos na integral, de modo a definir a q-integral. Minha pergunta é: que condições F(q) deve satisfazer? Bom, já percebi que, para q finito, a q-integral nao é unica, depende de F(q). Mas F(q) precisa satisfazer condições (por exemplo tail exponencial?). Ou não? Posso ter uma F(q) com tail lei de potencia?

:) Acho que integracao generalizada e' melhor, mas estou sentindo uma pontinha de Tsallis nessa terminologia. Hehe... Vou pensar um pouco sobre o assunto. O funcional F_q[f], onde f e' a funcao a ser integrada, tender a delta para q-> infinito define a "nao-integracao" da funcao, o parametro q precisa ser colocado na formula de maneira que para q tendendo a algum outro valor a funcao q*F_q[f] tenda `a integral simples de f. Nao tenho certeza, mas a principio acho que so a existencia dos dois limites basta. Eu sei, a questao e', quais as condicoes para a existencia dos limites. Nao sei. Preciso pensar um pouco mais sobre o assunto.

Fiz um post sobre isso no SEMCIENCIA. Com desculpas a todos, ok?

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