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Keywords:
$G$-space; equivariant map; pseudo-Euclidean geometry; functional equation
$G$-space; equivariant map; pseudo-Euclidean geometry; functional equation
Summary:
In this note all vectors and $\varepsilon $-vectors of a system of $m\le n$ linearly independent contravariant vectors in the $n$-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation $F( A{\underset{1}{\rightarrow }u}, A{\underset{2}{\rightarrow }u},\dots ,A{\underset{m}{\rightarrow }u}) =( \det A)^{\lambda }\cdot A\cdot F( {\underset{1}{\rightarrow }u},{\underset{2}{\rightarrow }u},\dots , {\underset{m}{\rightarrow }u})$ with $\lambda =0$ and $\lambda =1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $ {\underset{1}{\rightarrow }u},{\underset{2}{\rightarrow }u},\dots ,{\underset{m}{\rightarrow }u}.$
References:
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