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

Barbara Glanc; Aleksander Misiak; Maria Szmuksta-Zawadzka

Mathematica Bohemica (2007)

  • Volume: 132, Issue: 3, page 325-332
  • ISSN: 0862-7959

Abstract

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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 , · .

How to cite

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Glanc, Barbara, Misiak, Aleksander, and Szmuksta-Zawadzka, Maria. "Equivariant mappings from vector product into $ G$-spaces of $\varphi $-scalars with $G=O\left( n,1,\mathbb {R}\right) $." Mathematica Bohemica 132.3 (2007): 325-332. <http://eudml.org/doc/250266>.

@article{Glanc2007,
abstract = {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 $m\le n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F( A\underset\{1\}\{\rightarrow \}\{u\},A \underset\{2\}\{\rightarrow \}\{u\},\dots ,A\underset\{m\}\{\rightarrow \}\{u\}) = \varphi \left( A\right) \cdot F( \underset\{1\}\{\rightarrow \}\{u\},\underset\{2\}\{\rightarrow \}\{u\},\dots ,\underset\{m\}\{\rightarrow \}\{u\})$ using two homomorphisms $\varphi $ from a group $G$ into the group of real numbers $\mathbb \{R\}_\{0\}=\left( \mathbb \{R\}\setminus \left\rbrace 0\right\lbrace ,\cdot \right)$.},
author = {Glanc, Barbara, Misiak, Aleksander, Szmuksta-Zawadzka, Maria},
journal = {Mathematica Bohemica},
keywords = {$G$-space; equivariant map; pseudo-Euclidean geometry; -space; equivariant map; pseudo-Euclidean geometry},
language = {eng},
number = {3},
pages = {325-332},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equivariant mappings from vector product into $ G$-spaces of $\varphi $-scalars with $G=O\left( n,1,\mathbb \{R\}\right) $},
url = {http://eudml.org/doc/250266},
volume = {132},
year = {2007},
}

TY - JOUR
AU - Glanc, Barbara
AU - Misiak, Aleksander
AU - Szmuksta-Zawadzka, Maria
TI - Equivariant mappings from vector product into $ G$-spaces of $\varphi $-scalars with $G=O\left( n,1,\mathbb {R}\right) $
JO - Mathematica Bohemica
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 132
IS - 3
SP - 325
EP - 332
AB - 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 $m\le n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F( A\underset{1}{\rightarrow }{u},A \underset{2}{\rightarrow }{u},\dots ,A\underset{m}{\rightarrow }{u}) = \varphi \left( A\right) \cdot F( \underset{1}{\rightarrow }{u},\underset{2}{\rightarrow }{u},\dots ,\underset{m}{\rightarrow }{u})$ using two homomorphisms $\varphi $ from a group $G$ into the group of real numbers $\mathbb {R}_{0}=\left( \mathbb {R}\setminus \left\rbrace 0\right\lbrace ,\cdot \right)$.
LA - eng
KW - $G$-space; equivariant map; pseudo-Euclidean geometry; -space; equivariant map; pseudo-Euclidean geometry
UR - http://eudml.org/doc/250266
ER -

References

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  1. Functionalgleichungen der Theorie der geometrischen Objekte, Panstwowe Wydawnietvo Naukove, Warszawa, 1960. (1960) MR0133763
  2. Sur deux formes équivalents de la notion de ( r , s ) -orientation de la géométrie de Klein, Publ. Math. Debrecen 35 (1988), 43–50. (1988) MR0971951
  3. The homomorphisms of the pseudo-orthogonal group of index one into an abelian group, Demonstratio Math. 22 (1989), 763–771. (1989) MR1041913
  4. Über die Grundlagen der Kleinschen Geometrie, Period. Math. Hungar. 8 (1977), 83–89. (1977) Zbl0335.50001MR0493695
  5. Equivariant maps between certain G -spaces with G = O n - 1 , 1 , Math. Bohem. 126 (2001), 555–560. (2001) MR1970258
  6. O pewnym działaniu grupy pseudoortogonalnej o indeksie jeden O n , 1 , na sferze S n - 2 , Prace Naukowe P.S. 485 (1993). (1993) 
  7. Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1, Publ. Math. Debrecen 57 (2000), 55–69. (2000) Zbl0966.53012MR1771671

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