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This paper is a continuation of investigations of -inner product spaces given in [, , ] and an extension of results given in [] to arbitrary natural . It concerns families of projections of a given linear space onto its -dimensional subspaces and shows that between these families and -inner products there exist interesting close relations.
In this note, there are determined all biscalars of a system of linearly independent contravariant vectors in -dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation for an arbitrary pseudo-orthogonal matrix of index one and the given vectors .
There exist exactly four homomorphisms from the pseudo-orthogonal group of index one into the group of real numbers Thus we have four -spaces of -scalars in the geometry of the group The group operates also on the sphere forming a -space of isotropic directions In this note, we have solved the functional equation for given independent points with and an arbitrary matrix considering each of all four homomorphisms. Thereby we have determined all equivariant mappings
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In this note all vectors and -vectors of a system of linearly independent contravariant vectors in the -dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation with and , for an arbitrary pseudo-orthogonal matrix of index one and given vectors
There are four kinds of scalars in the -dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation using two homomorphisms from a group into the group of real numbers .
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