LaTeX Tip: Dynkin Diagrams using Tikz

Hi everyone, I’ve include the code for drawing Dynkin diagrams using Tikz package in LaTeX

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,0) node[anchor=east]  {$A_n$};
    \foreach \x in {0,...,5}
    \draw[xshift=\x cm,thick] (\x cm,0) circle (.3cm);
    \draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
    \foreach \y in {1.15,...,4.15}
    \draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,0) node[anchor=east]  {$B_n$};
    \foreach \x in {0,...,4}
    \draw[xshift=\x cm,thick] (\x cm,0) circle (.3cm);
    \draw[xshift=5 cm,thick,fill=black] (5 cm, 0) circle (.3 cm);
    \draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
    \foreach \y in {1.15,...,3.15}
    \draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
    \draw[thick] (8.3 cm, .1 cm) -- +(1.4 cm,0);
    \draw[thick] (8.3 cm, -.1 cm) -- +(1.4 cm,0);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,0) node[anchor=east]  {$C_n$};
    \foreach \x in {0,...,4}
    \draw[xshift=\x cm,thick,fill=black] (\x cm,0) circle (.3cm);
    \draw[xshift=5 cm,thick] (5 cm, 0) circle (.3 cm);
    \draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
    \foreach \y in {1.15,...,3.15}
    \draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
    \draw[thick] (8.3 cm, .1 cm) -- +(1.4 cm,0);
    \draw[thick] (8.3 cm, -.1 cm) -- +(1.4 cm,0);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,0) node[anchor=east]  {$D_n$};
    \foreach \x in {0,...,4}
    \draw[xshift=\x cm,thick] (\x cm,0) circle (.3cm);
    \draw[xshift=8 cm,thick] (30: 17 mm) circle (.3cm);
    \draw[xshift=8 cm,thick] (-30: 17 mm) circle (.3cm);
    \draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
    \foreach \y in {1.15,...,3.15}
    \draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
    \draw[xshift=8 cm,thick] (30: 3 mm) -- (30: 14 mm);
    \draw[xshift=8 cm,thick] (-30: 3 mm) -- (-30: 14 mm);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,0) node[anchor=east]  {$G_2$};
    \draw[thick] (0 ,0) circle (.3 cm);
    \draw[thick,fill=black] (2 cm,0) circle (.3 cm);
    \draw[thick] (30: 3mm) -- +(1.5 cm, 0);
    \draw[thick] (0: 3 mm) -- +(1.4 cm, 0);
    \draw[thick] (-30: 3 mm) -- +(1.5 cm, 0);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-3,0) node[anchor=east]  {$F_4$};
    \draw[thick] (-2 cm ,0) circle (.3 cm);
    \draw[thick] (0 ,0) circle (.3 cm);
    \draw[thick,fill=black] (2 cm,0) circle (.3 cm);
    \draw[thick,fill=black] (4 cm,0) circle (.3 cm);
    \draw[thick] (15: 3mm) -- +(1.5 cm, 0);
    \draw[xshift=-2 cm,thick] (0: 3 mm) -- +(1.4 cm, 0);
    \draw[thick] (-15: 3 mm) -- +(1.5 cm, 0);
    \draw[xshift=2 cm,thick] (0: 3 mm) -- +(1.4 cm, 0);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,1) node[anchor=east]  {$E_6$};
    \foreach \x in {0,...,4}
    \draw[thick,xshift=\x cm] (\x cm,0) circle (3 mm);
    \foreach \y in {0,...,3}
    \draw[thick,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
    \draw[thick] (4 cm,2 cm) circle (3 mm);
    \draw[thick] (4 cm, 3mm) -- +(0, 1.4 cm);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,1) node[anchor=east]  {$E_7$};
    \foreach \x in {0,...,5}
    \draw[thick,xshift=\x cm] (\x cm,0) circle (3 mm);
    \foreach \y in {0,...,4}
    \draw[thick,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
    \draw[thick] (4 cm,2 cm) circle (3 mm);
    \draw[thick] (4 cm, 3mm) -- +(0, 1.4 cm);
  \end{tikzpicture}
\end{center}

\begin{center}
  \begin{tikzpicture}[scale=.4]
    \draw (-1,1) node[anchor=east]  {$E_8$};
    \foreach \x in {0,...,6}
    \draw[thick,xshift=\x cm] (\x cm,0) circle (3 mm);
    \foreach \y in {0,...,5}
    \draw[thick,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
    \draw[thick] (4 cm,2 cm) circle (3 mm);
    \draw[thick] (4 cm, 3mm) -- +(0, 1.4 cm);
  \end{tikzpicture}
\end{center}

Bye

DOX

Fluid Flows and Infinite-Dimensional Manifolds I

Reblogged from Azimuth:

• 

Or: waves that take the shortest path through infinity

guest post by Tim van Beek

Water waves can do a lot of things that light waves cannot, like “breaking”:

In mathematical models this difference shows up through the kind of partial differential equation (PDE) that models the waves:

• light waves are modelled by linear equations while

• water waves are modelled by nonlinear equations.

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The Education of a Scientist

Reblogged from Azimuth:

Why are scientists like me getting so worked up over Elsevier and other journal publishers? It must seem strange from the outside. This cartoon explains it very clearly. It’s hilarious—except that it’s TRUE!!! This is why we need a revolution.

(It’s true except for one small thing: in math and physics, Elsevier and Springer let us put our papers on our websites and free electronic archives… though not the final version, only the near-final draft.

Read more… 349 more words

Unfortunately it's true

A Noether Theorem for Markov Processes

Reblogged from Azimuth:

I’ll start you off with two puzzles. Their relevance should become clear by the end of this post:

• Puzzle 1. Suppose I have a box of jewels. The average value of a jewel in the box is $10. I randomly pull one out of the box. What’s the probability that its value is at least $100?

• Puzzle 2. Suppose I have a box full of numbers—they can be arbitrary real numbers.

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Noether theorem...

LaTeX Tip: New name after compilation

If you want to compile a file, say Base.tex, and get the PDF with other name, use the -jobname option

$ pdflatex -jobname MyBook Base.tex

In the example you’ll get a PDF file called MyBook.pdf

Enjoy

Table of all possible irreps of Lie groups

Last weekend, I wrote a program in SAGE that list all possible irreps of a Lie group, one specified the group (in Cartan’s classification) and the maximum sum of the Dynkin labels.

Check the notebook in here.

Since I’m not a programmer, please feel free to leave comments, specially if you find ways to optimize the program

Cheers,

DOX

Entropic Forces

Reblogged from Azimuth:

In 2009, Erik Verlinde argued that gravity is an entropic force. This created a big stir—and it helped him win about $6,500,000 in prize money and grants! But what the heck is an ‘entropic force’, anyway?

Entropic forces are nothing unusual: you’ve felt one if you’ve ever stretched a rubber band. Why does a rubber band pull back when you stretch it?

Read more… 1,452 more words

Entropic Forces?