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| equality for scator algebras. However, care should be taken to avoid the divisors of zero. | | equality for scator algebras. However, care should be taken to avoid the divisors of zero. |
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| The fourth order identity gives [[Lagrange identity|Lagrange's identity]]. | | The fourth order identity gives [[Lagrange identity|Lagrange's identity]].The sixth order identities are derived here. An extended version of these results are available in an open source journal <ref> M. Fernández-Guasti. Lagrange's identity obtained from product identity, Int. Math. Forum, 70(52):2555-2559, 2012. [http://www.m-hikari.com/imf/imf-2012/49-52-2012/fernandezguastiIMF49-52-2012.pdf]</ref>. |
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| == sixth order identity == | | == sixth order identity == |
Revisión del 14:00 19 ago 2012
Norm of a product
Normed division algebras require that the norm of the product is equal
to the product of the norms. Lagrange's identity exhibits this equality.
Due to Hurwitz theorem, it admits this interpretation only for algebras
isomorphic to the real numbers, complex numbers, quaternions and octonions. If divisors of zero
are allowed, many other algebraic structures in
are possible [1], [2].
One approach has been presented in the context of a deformed Lorentz metric. This latter proposal
is based on a transformation stemming from the product operation and
magnitude definition in hyperbolic scator algebra [3].
The product identity used as a starting point here, is a consequence
of the
equality for scator algebras. However, care should be taken to avoid the divisors of zero.
The fourth order identity gives Lagrange's identity.The sixth order identities are derived here. An extended version of these results are available in an open source journal [4].
sixth order identity
The non trivial identities for real numbers obtained to sixth order series expansion of the product identity
are
and its counterpart, obtained by interchanging the variables and . To prove it, expand the product identity in series up to sixth order. The LHS is
Consider only the sixth order terms
The RHS of the product identity is similarly expanded in series up to sixth order
and only sixth order terms retained
These two results are equated for equal powers of . The
terms and give trivial identities whereas the terms
involving and give the non trivial sixth
order identities
and
- ↑ P. Fjelstad and S. G. Gal., n-dimensional hyperbolic complex numbers, Adv. Appl. Clifford Alg., 8(1), 1998, p. 47–68
- ↑ F. Catoni, R. Cannata, E. Nichelatti, and P. Zampetti, Commmutative hypercomplex numbers and functions of hypercomplex variable: a matrix study, Adv. Appl. Clifford Alg., 15(2), 2005, pag.183–212
- ↑ M. Fernández-Guasti, Alternative realization for the composition of relativistic velocities, Optics and Photonics 2011, vol. 8121 of The nature of light: What are photons? IV, pp. 812108–1–11. SPIE, 2011.
- ↑ M. Fernández-Guasti. Lagrange's identity obtained from product identity, Int. Math. Forum, 70(52):2555-2559, 2012. [1]