IJPAM: Volume 79, No. 3 (2012)

GEOMETRICAL WAVE EQUATION AND
THE CAUCHY-LIKE THEOREM FOR OCTONIONS

M.F. Borges$^1$, J.A. Marão$^2$
$^1$UNESP - São Paulo State University
S.J. Rio Preto Campus
15054-000, São José do Rio Preto, BRAZIL
$^2$Department of Mathematics
UFMA - Federal University of Maranhão
65085-580, Maranhão, BRAZIL


Abstract. Riemann surfaces, cohomology and homology groups, Cartan's spinors and triality, octonionic projective geometry, are all well supported by Complex Structures [#!1!#], [#!2!#], [#!3!#], [#!4!#]. Furthermore, in Theoretical Physics, mainly in General Relativity, Supersymmetry and Particle Physics, Complex Theory Plays a Key Role [#!5!#], [#!6!#], [#!7!#], [#!8!#]. In this context it is expected that generalizations of concepts and main results from the Classical Complex Theory, like conformal and quasiconformal mappings [#!9!#], [#!10!#] in both quaternionic and octonionic algebra, may be useful for other fields of research, as for graphical computing enviromment [#!11!#]. In this Note, following recent works by the autors [#!12!#], [#!13!#], the Cauchy Theorem will be extended for Octonions in an analogous way that it has recentely been made for quaternions [#!14!#]. Finally, will be given an octonionic treatment of the wave equation, which means a wave produced by a hyper-string with initial conditions similar to the one-dimensional case.

Received: May 7, 2012

AMS Subject Classification: 30G99, 30E99

Key Words and Phrases: Cauchy integral, hypercomplex, quaternions

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 79
Issue: 3