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n -inner product spaces and projections

Aleksander MisiakAlicja Ryż — 2000

Mathematica Bohemica

This paper is a continuation of investigations of n -inner product spaces given in [, , ] and an extension of results given in [] to arbitrary natural n . It concerns families of projections of a given linear space L onto its n -dimensional subspaces and shows that between these families and n -inner products there exist interesting close relations.

Equivariant maps between certain G -spaces with  G = O ( n - 1 , 1 ) .

Aleksander MisiakEugeniusz Stasiak — 2001

Mathematica Bohemica

In this note, there are determined all biscalars of a system of s n linearly independent contravariant vectors in n -dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation F ( A 1 u , A 2 u , , A s u ) = ( sign ( det A ) ) F ( 1 u , 2 u , , s u ) for an arbitrary pseudo-orthogonal matrix A of index one and the given vectors 1 u , 2 u , , s u .

G -space of isotropic directions and G -spaces of ϕ -scalars with G = O ( n , 1 , )

Aleksander MisiakEugeniusz Stasiak — 2008

Mathematica Bohemica

There exist exactly four homomorphisms ϕ from the pseudo-orthogonal group of index one G = O ( n , 1 , ) into the group of real numbers 0 . Thus we have four G -spaces of ϕ -scalars ( , G , h ϕ ) in the geometry of the group G . The group G operates also on the sphere S n - 2 forming a G -space of isotropic directions ( S n - 2 , G , * ) . In this note, we have solved the functional equation F ( A * q 1 , A * q 2 , , A * q m ) = ϕ ( A ) · F ( q 1 , q 2 , , q m ) for given independent points q 1 , q 2 , , q m S n - 2 with 1 m n and an arbitrary matrix A G considering each of all four homomorphisms. Thereby we have determined all equivariant mappings F : ( S n - 2 ) m . ...

Equivariant mappings from vector product into G -space of vectors and ε -vectors with G = O ( n , 1 , )

Barbara GlancAleksander MisiakZofia Stepień — 2005

Mathematica Bohemica

In this note all vectors and ε -vectors of a system of m 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 1 u , A 2 u , , A m u ) = ( det A ) λ · A · F ( 1 u , 2 u , , m u ) with λ = 0 and λ = 1 , for an arbitrary pseudo-orthogonal matrix A of index one and given vectors 1 u , 2 u , , m u .

Equivariant mappings from vector product into G -spaces of ϕ -scalars with G = O n , 1 ,

Barbara GlancAleksander MisiakMaria Szmuksta-Zawadzka — 2007

Mathematica Bohemica

There are four kinds of scalars in the n -dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of m n linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation F ( A 1 u , A 2 u , , A m u ) = ϕ A · F ( 1 u , 2 u , , m u ) using two homomorphisms ϕ from a group G into the group of real numbers 0 = 0 , · .

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