Diferencia entre revisiones de «Norm of product»
(Página creada con ' == sixth order identity == Sixth and higher order terms produce other series identities. The non trivial identities for real numbers obtained to sixth order series expansion …') |
|||
(No se muestran 7 ediciones intermedias del mismo usuario) | |||
Línea 1: | Línea 1: | ||
==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 <math>\mathbb{R}^{n}</math> | |||
are possible <ref>P. Fjelstad and S. G. Gal., ''n-dimensional hyperbolic complex numbers'', Adv. Appl. Clifford Alg., 8(1), 1998, p. 47–68</ref>, <ref>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</ref>. | |||
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 <ref> 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.</ref>. | |||
The product identity used as a starting point here, is a consequence | |||
of the <math>\left\Vert \mathbf{ab}\right\Vert =\left\Vert \mathbf{a}\right\Vert \left\Vert \mathbf{b}\right\Vert </math> | |||
equality for scator algebras. However, care should be taken to avoid the divisors of zero. | |||
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>. | |||
== sixth order identity == | == sixth order identity == | ||
The non trivial identities for real numbers obtained to sixth order 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> | |||
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> | |||
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>. To prove it, 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) | ||
Línea 20: | Línea 32: | ||
+\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 25: | Línea 38: | ||
-\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 32: | Línea 45: | ||
\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 | ||
Línea 41: | Línea 57: | ||
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 | |||
<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). | |||
</math></center> | |||
---- | |||
<references/> | |||
[[categoría:matematicas]] | |||
Revisión actual - 13:57 27 oct 2018
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]