Diferencia entre revisiones de «Norm of product»
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== sixth order identity == | == sixth order identity == | ||
The non trivial identities for real numbers obtained to sixth order | The non trivial identities for real numbers obtained to sixth order | ||
series expansion of the product identity <math>\prod_{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=\prod_{i=1}^{n}\left(1-a_{i}^{2}\right)\prod_{i=1}^{n}\left(1-b_{i}^{2}\right)</math> | series expansion of the product identity <center><math>\prod_{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=\prod_{i=1}^{n}\left(1-a_{i}^{2}\right)\prod_{i=1}^{n}\left(1-b_{i}^{2}\right)</math></center> | ||
are | are | ||
<math> | <center><math> | ||
\sum_{i<j}^{n}\left[a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)\right]+\sum_{i<j<k}^{n}\left[a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right]=\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right) | \sum_{i<j}^{n}\left[a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)\right]+\sum_{i<j<k}^{n}\left[a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right]=\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right) | ||
</math> | </math></center> | ||
and its counterpart, obtained by interchanging the variables <math>a</math> and <math>b</math>. | and its counterpart, obtained by interchanging the variables <math>a</math> and <math>b</math>. | ||
Expand the product identity in series up to sixth order. The LHS is | Expand the product identity in series up to sixth order. The LHS is | ||
<math> | <math> | ||
\prod_{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=1+\sum_{i=1}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right) | \prod_{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=1+\sum_{i=1}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right) | ||
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+\sum_{i<j<k}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)\left(-a_{j}^{2}-b_{j}^{2}+a_{j}^{2}b_{j}^{2}\right)\left(-a_{k}^{2}-b_{k}^{2}+a_{k}^{2}b_{k}^{2}\right)+\mathcal{O}^{7+}. | +\sum_{i<j<k}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)\left(-a_{j}^{2}-b_{j}^{2}+a_{j}^{2}b_{j}^{2}\right)\left(-a_{k}^{2}-b_{k}^{2}+a_{k}^{2}b_{k}^{2}\right)+\mathcal{O}^{7+}. | ||
</math> | </math> | ||
Consider only the sixth order terms | Consider only the sixth order terms | ||
<math> | <math> | ||
Línea 40: | Línea 40: | ||
-\sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)-\sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right) | -\sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)-\sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right) | ||
</math> | </math> | ||
The RHS of the product identity is similarly expanded in series up | |||
to sixth order | The RHS of the product identity is similarly expanded in series up to sixth order | ||
<math> | <math> | ||
Línea 47: | Línea 47: | ||
\left(1-\sum_{i=1}^{n}b_{i}^{2}+\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}+\mathcal{O}^{7+}\right), | \left(1-\sum_{i=1}^{n}b_{i}^{2}+\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}+\mathcal{O}^{7+}\right), | ||
</math> | </math> | ||
and only sixth order terms retained | |||
and only sixth order terms retained | |||
<math> | <math> | ||
\mathcal{O}^{6}\left(\textrm{RHS}\right)=-\sum_{i<j<k}^{n}a_{i}^{2}a_{j}^{2}a_{k}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}-\left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right)-\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right). | \mathcal{O}^{6}\left(\textrm{RHS}\right)=-\sum_{i<j<k}^{n}a_{i}^{2}a_{j}^{2}a_{k}^{2}-\sum_{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}-\left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right)-\left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right). | ||
</math> | </math> | ||
These two results are equated for equal powers of <math>a^{n}b^{m}</math>. The | These two results are equated for equal powers of <math>a^{n}b^{m}</math>. The | ||
terms <math>a^{6}</math> and <math>b^{6}</math> give trivial identities whereas the terms | terms <math>a^{6}</math> and <math>b^{6}</math> give trivial identities whereas the terms | ||
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order identities | order identities | ||
<math> | <center><math> | ||
\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right) | \sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}b_{i}^{2}\right)\left(\sum_{i<j}^{n}a_{i}^{2}a_{j}^{2}\right) | ||
</math> | </math></center> | ||
and | |||
<math> | <center><math> | ||
\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\left(a_{i}^{2}+a_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right). | \sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\left(a_{i}^{2}+a_{j}^{2}\right)+ \sum_{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)= \left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i<j}^{n}b_{i}^{2}b_{j}^{2}\right). | ||
</math> | </math></center> |
Revisión del 13:41 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.
The fourth order identity gives Lagrange's identity.
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 .
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.