word algebra


at sight of short range of time, let us talk!

… …

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)

… …

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}

 

schreierx03

 

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