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## Tareas para mañana 21 de noviembre

Posets

1. probar el programa [test code]
2. meter ruido a la reconstrucción
2.1 pensar en cómo meter el ruido [add noise]
2.2 reconstrucción usando cortes alfa [alfa reconst.]

Cosets

1. Estudiar el documento
2. trabajar desde donde se dejo y resolver las dudas planteadas
3. volver a estudiar la forma de los grupos para los estados usados. Buscar si caen en alguno de los casos sencillos.

Burocracia
1. Hacer la aplicación
2. Escribir a Hdg sobre la respuesta

Otros.
1. pagina de sonidos. no importa que no sea unica.
2. hacer mejor el review de pdfpc

Categories

## Fuzzy entropy pt1

La entropia fuzzy cuantifica el grado de fuzyness de una lista de pertenencia. De esta forma, su valor es cero para un conjunto crisp y uno para el estado $$1/2\_ con pertenencia a todo. De esto ultimo se puede generar dudas respecto a su utilidad ya que podemos tener un conjunto muy desordenado —en rankings— que no necesariamente tenga la mayor entropía fuzzy. Es facil notar que para el poset más borroso $\tilde P = \begin{pmatrix} 1 & 1/2 & 1/2 \\ 1/2 & 1 & 1/2 \\ 1/2 & 1/2 & 1 \end{pmatrix}$ el valor maximo es \(\frac{n-1}{n+1}$$.

[todo] Hay que hacer una prueba y verificar que un poset con varios 1/2 tenga mayor entropia que uno que sea construido usando una lista “desordenada”.

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## Savings due to zeros pt2

Consult pt1 here.

After a bit of thought, I came to the conclusion that I needed to study the $$\gamma$$ present in the computed group functions (GF). From them, observer if there is any pattern showing how can we exploit the number of zeroes to reduce the computational complexity.

As the main image of this post reveals, is not difficult to see that from the sum of GFs a lot of them are zero. For instance, in the $$SU4/SU3$$ case, in the original —using the original scattering matrix— we use 15 non zero terms. However, using the coset matrix, only 4 are required.

The situation improve in the $$SU5/SU4$$ case.

On another theme, also note the fact that we only require two different double-coset representatives.

So, task for the next time I work on this

1. Study the possible form of the double coset representatives.
2. Create a new Julia method that avoids computing orthogonal irrep elements if the monomial is zero. Then, benchmark it.
3. Count the total number of orthogonal entries saved.
4. Try to find the form of the $$\gamma$$s. That is, the double coset representatives in the case of using a coset matrix.

On the first point. My guess is that the correct non $$e$$ double coset representative is of the form $$(1, n-1, n-2, \dots, 2)$$.

Update. The case that I was studying is not the most interesting. I was seeing the case of multiple photons in a channel. However, the savings, though not as great, they are still present. I added two new tasks.

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## Savings due to zeros pt1

Consult pt2 here.

In this post I will provide a journal about the savings due to the presence of zeros in the submatrix needed to compute the sum rules.

Day 0

Problems found regarding the savings. The $${\rm det}$$ and a transformation are not always valid. Sometimes, the submatrix is not Hessenberg.

Day 1

$T_{U, V}^{\lambda}= \frac{1}{\sqrt{\Theta_{i}^{f}}} \frac{1}{\sqrt{\Theta_{j}^{g}}} \sum_{\cup_\gamma S_{\alpha} \gamma S_{\beta}=S_{n}}\left(\sum_{\sigma \in S_{\alpha} \gamma S_{\beta}} \omega_{i, j}^{\lambda}(\sigma)\right) X_{f\circ\gamma, g}.$

According to the previous equation, if the submatrix present some zeros then at least one $$X_{f\circ\gamma, g}$$ is zero. This in turn leads to avoiding compute some orthogonal irrep (OI) entries.

This is the unique saving I see at the present.

Thanks to Antti Koskinen Post

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## Ideas para el blog

Me parece interesante hacer una bitácora de las distintas cosas que voy aprendiendo:

1. Física
2. Inversiones
3. Posets
4. Escritura
5. Programación

En cuanto un conjunto de elementos ya tenga mucho contenido tal vez hacer una entrada en la wiki.

Categories

## Tasks for tomorrow

I plan to do the following tasks

Think about one application of fuzzy Bayes Theorem
Prepare what I am going to discuss with hdg
Prepare presentation for Tuesday
Write draft and, hopefully, email to cp on bureaucracy.
Write a blog entry about scattershot boson sampling

I really wish to make a good use of this blog.

Edit 1. Put every new function —related to fuzzy entropy— in a new file in the Julia package.