real scators

Submitted by mfg on Tue, 05/03/2016 - 21:12
English

The hyperbolic scator set is represented with a "+" in the following way, $\mathbb{E}_{+}^{1+n}$. These numbers have an algebra of their own. A relevant feature of this algebra is that the product is not distributive over addition. In the additive representation a scator is written as

\begin{equation}
\overset{o}\varphi=f_{0} + \sum_{j=1}^{n}{f_{j}\hat{\mathbf{e}}_{j} } ,
\end{equation}with $f_{0}, f_{j} \in \mathbb{R}$ for all $j:[1,n]$ y $\hat{\mathbf{e}}_{j} \notin \mathbb{R}$. Los escatores se componen de una parte escalar, y $n$ partes directoras. Una particularidad de los escatores reales, similar a los números hiperbólicos, radica en que el producto $\hat{\mathbf{e}}_{j} \hat{\mathbf{e}}_{l}=\delta_{j,l}$, a diferencia de los números escatores imaginarios en los cuales el signo de la delta es negativo.

real scator sum

Let two scators be $\overset{o}\alpha=a_{0} + \sum_{j=1}^{n}{a_{j}\hat{\mathbf{e}}_{j}}$ and $\overset{o}\beta=b_{0} + \sum_{j=1}^{n}{b_{j}\hat{\mathbf{e}}_{j}}$. Their sum is performed component by component,
\begin{equation}
\overset{o}\gamma = g_{0} + \sum_{j=1}^{n}{g_{j}\hat{\mathbf{e}}_{j}} = \overset{o}\alpha + \overset{o}\beta = (a_{0} + b_{0}) + \sum_{j=1}^{n}({a_{j}+ b_{j})\hat{\mathbf{e}}_{j}}.
\end{equation}We can see that the scalar part of the sum is the sum of the scalar components of scators  $\overset{o}\alpha$ y $ \overset{o}\beta$. In a similar fashion, the director part of the sum is equal to the sum of the director coefficients with equal unit director component $\hat{\mathbf{e}}_{j}$.

 

The hyperbolic scator set is represented with a "+" in the following way, $\mathbb{E}_{+}^{1+n}$. These numbers have an algebra of their own. A relevant feature of this algebra is that the product is not distributive over addition. In the additive representation a scator is written as

\begin{equation}
\overset{o}\varphi=f_{0} + \sum_{j=1}^{n}{f_{j}\hat{\mathbf{e}}_{j} } ,
\end{equation}with $f_{0}, f_{j} \in \mathbb{R}$ for all $j:[1,n]$ y $\hat{\mathbf{e}}_{j} \notin \mathbb{R}$. Los escatores se componen de una parte escalar, y $n$ partes directoras. Una particularidad de los escatores reales, similar a los números hiperbólicos, radica en que el producto $\hat{\mathbf{e}}_{j} \hat{\mathbf{e}}_{l}=\delta_{j,l}$, a diferencia de los números escatores imaginarios en los cuales el signo de la delta es negativo.

real scator product

The product definition between two scators can be divided into three cases:

Case 1

If $\overset{o}\alpha$ and $\overset{o}\beta$ two scators such that $a_{0}b_{0}\neq 0$ then, the product between them, i.e. $\overset{o}\gamma=\overset{o}\alpha\overset{o}\beta$ is defined as follows:\begin{equation}\overset { o }\gamma   ={ a }_{ 0 }{ b }_{ 0 }\prod _{ k=1 }^{ n }{ \left( 1+\frac { { a }_{ k }{ b }_{ k } }{ { a }_{ 0 }{ b }_{ 0 } }  \right) +{ a }_{ 0 }{ b }_{ 0 }\sum _{ j=1 }^{ n }{ \left[ \prod _{ k\neq j }^{ n }{ \left( 1+\frac { { a }_{ k }{ b }_{ k } }{ { a }_{ 0 }{ b }_{ 0 } }  \right) \left( \frac { { a }_{ j } }{ { a }_{ 0 } } +\frac { { b }_{ j } }{ { b }_{ 0 } }  \right)  }  \right]  }  } { \hat {\mathbf{ e }  }_{ j }}.\end{equation}

Case 2

If one of the scators involved in the product, only has one direction component non zero  $a_{0}=0, b_{0}\neq 0$, i.e. $\overset{o}\alpha={a}_{l}  { \hat {\mathbf{  e }  }_{ l }}$ and the remanning scator such that $\overset{o}\beta=b_{0} + \sum_{j=1}^{n}{b_{j}}{\hat{\mathbf{e}}_{j}}$, then, the product beween them is defined as follows :\begin{equation}\overset { o }{ \gamma  } =\left( { a }_{ l }{ b }_{ l } \right) +\left( { a }_{ l }{ b }_{ 0 } \right) { \hat { \mathbf{ e }  }_{ l }}+\sum _{ j\neq l }^{ n }{ \left( \frac { { a }_{ l }{ b }_{ l }{ b }_{ j } }{ { b }_{ 0 } }  \right)  } { \hat {\mathbf{ e }  }_{ j }}.\end{equation}

Case 3

Finally, when both scators, have scalar component equal zero, $a_{0}=0, b_{0}= 0$ namely, $\overset{o}\alpha={a}_{l}  { \hat {\mathbf{  e }  }_{ l }}$ and $\overset{o}\beta={b_{j}\hat{\mathbf{e}}_{j}}$. Then the product between these scators is defined as follows: \begin{equation}\overset { o }{ \gamma  } =({a}_{l}  { \hat {\mathbf{  e }  }_{ l }})({b_{j}\hat{\mathbf{e}}_{j}})={a}_{l} b_{j} \delta_{lj},\end{equation}

It is possible to obtain the following case from the previous cases:

Two scators such that, both have scalar component and only have one direction component non zero, namely:$\overset{o}\alpha={a}_{0}+{a}_{l}  { \hat {\mathbf{  e }  }_{ l }}$ and $\overset{o}\beta={b}_{0}+{b}_{j}  { \hat {\mathbf{e}  }_{ j }}$ then, the product between them is defined as follows:\begin{equation}\overset { o }{ \gamma  } ={ a }_{ 0 }{ b }_{ 0 }+{ a }_{ l }{ b }_{ j }{ \delta  }_{ lj }+{ b }_{ 0 }{ a }_{ l }{ \hat { \mathbf{ e }  }_{ l }}+{ a }_{ 0 }{ b }_{ j }{ \hat {\mathbf{  e }  }_{ j }}.\end{equation}

If the scator  $\overset{o}\alpha$  has a scalar component non zero and all his direction components are equals zero, $\overset{o}\alpha={a}_{0}$, from the first case you get

\begin{equation}{a}_{0}\overset{o}\beta={a}_{0}b_{0} + \sum_{j=1}^{n}{a}_{0}{b_{j}\hat{\mathbf{e}}_{j}}\end{equation}

then  $\overset{o}\alpha={a}_{0}$ is a multiple for the scator  $\overset{o}\beta$. Hence the first component  is called the scalar component.