word algebra

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If $A=\left(\begin{array}{ccc}0&1&0\\ 0&0&1\\ -a_0&-a_1&-a_2\end{array}\right)$ then $\det(Ex-A)=x^3+a_2x^2+a_1x+a_0$, where $E=\left(\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right)$

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For $G=\langle a,b\ |\ a^2=e,\ b^2=e\rangle$, let us write some elements:

word length : words

0 : $e$

1 : $a,b$

2 : $ab,\quad ba$

3 : $aba,\quad bab$

4 : $abab,\quad baba$

5 : $ababa,\quad babab$

6 : $ababab,\quad bababa$

Note that for odd-length-words, like $aba$ we have: $(aba)(aba)=e$,

but for even-length-words, like $abab$, we have: $(abab)(baba)=e$.

So, this group has an infinite many elements of order two and the even-length words
form a subgroup, $H$, isomorphic to the group of the integers, and its index is $[G:H]=2$.

$\S\S$

Now the group $\mathbb{Z}_2*\mathbb{Z}_3$.

Its presentation is $\langle a,b\ |\ a^2=e,\ b^3=e\rangle$

Words like $awa$, where $w$ is a word that begins in $b$ and ends in $b$ also, form a subgroup:

Examples $aba,\ abba,\ ababa,\ ababba,\ abbaba,\ abbabba,...$

$\S\S$

Sample of word algebra of operators

$\partial_1= \left(\begin{array}{c} \frac{\partial x^1}{\partial v^1}\\ \frac{\partial x^2}{\partial v^1}\\ \frac{\partial x^3}{\partial v^1} \end{array}\right)$

and

$J\partial_2= \left(\begin{array}{ccc} \frac{\partial }{\partial x^1}\left(\frac{\partial x^1}{\partial v^2}\right) & \frac{\partial }{\partial x^2}(\frac{\partial x^1}{\partial v^2}) & \frac{\partial }{\partial x^3}(\frac{\partial x^1}{\partial v^2})\\ \frac{\partial }{\partial x^1}(\frac{\partial x^2}{\partial v^2}) & \frac{\partial }{\partial x^2}(\frac{\partial x^2}{\partial v^2}) & \frac{\partial }{\partial x^3}(\frac{\partial x^2}{\partial v^2})\\ \frac{\partial }{\partial x^1}(\frac{\partial x^3}{\partial v^2}) & \frac{\partial }{\partial x^2}(\frac{\partial x^3}{\partial v^2}) & \frac{\partial }{\partial x^3}(\frac{\partial x^3}{\partial v^2}) \end{array}\right)$

so

$D_{\partial_1}{\partial_2}=[J\partial_2]{\partial_1}=\left(\begin{array}{ccc} \frac{\partial }{\partial x^1}(\frac{\partial x^1}{\partial v^2}) & \frac{\partial }{\partial x^2}(\frac{\partial x^1}{\partial v^2}) & \frac{\partial }{\partial x^3}(\frac{\partial x^1}{\partial v^2})\\ \frac{\partial }{\partial x^1}(\frac{\partial x^2}{\partial v^2}) & \frac{\partial }{\partial x^2}(\frac{\partial x^2}{\partial v^2}) & \frac{\partial }{\partial x^3}(\frac{\partial x^2}{\partial v^2})\\ \frac{\partial }{\partial x^1}(\frac{\partial x^3}{\partial v^2}) & \frac{\partial }{\partial x^2}(\frac{\partial x^3}{\partial v^2}) & \frac{\partial }{\partial x^3}(\frac{\partial x^3}{\partial v^2}) \end{array}\right) \left(\begin{array}{c} \frac{\partial x^1}{\partial v^1}\\ \frac{\partial x^2}{\partial v^1}\\ \frac{\partial x^3}{\partial v^1} \end{array}\right)$

$= \left(\begin{array}{c} \frac{\partial }{\partial x^1}(\frac{\partial x^1}{\partial v^2}) \frac{\partial x^1}{\partial v^1}+\frac{\partial }{\partial x^2}(\frac{\partial x^1}{\partial v^2})\frac{\partial x^2}{\partial v^1}+ \frac{\partial }{\partial x^3}(\frac{\partial x^1}{\partial v^2})\frac{\partial x^3}{\partial v^1}\\ \frac{\partial }{\partial x^1}(\frac{\partial x^2}{\partial v^2}) \frac{\partial x^1}{\partial v^1}+\frac{\partial }{\partial x^2}(\frac{\partial x^2}{\partial v^2})\frac{\partial x^2}{\partial v^1}+ \frac{\partial }{\partial x^2}(\frac{\partial x^2}{\partial v^2})\frac{\partial x^3}{\partial v^1}\\ \frac{\partial }{\partial x^1}(\frac{\partial x^3}{\partial v^2}) \frac{\partial x^1}{\partial v^1}+\frac{\partial }{\partial x^2}(\frac{\partial x^3}{\partial v^2})\frac{\partial x^2}{\partial v^1}+ \frac{\partial }{\partial x^3}(\frac{\partial x^3}{\partial v^2})\frac{\partial x^3}{\partial v^1} \end{array}\right)$

$=\left(\begin{array}{c}\frac{\partial^2 x^1}{\partial v^1\partial v^2}\\\frac{\partial^2 x^2}{\partial v^1\partial v^2}\\\frac{\partial^2 x^3}{\partial v^1\partial v^2}\end{array}\right)=\frac{\partial^2\Phi}{\partial v^1\partial v^2}$