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Editor: Bill Page
Time: 2008/04/24 07:08:13 GMT-7
Note: Jordan

added:
Ref: http://arxiv.org/abs/0711.3220

Fourvector algebra

Author: Diego Saa

(Submitted on 20 Nov 2007)

Abstract: The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the excellent ability to perform rotations with the use of fourvectors, as well as their use in relativity for producing Lorentz boosts, which are understood as simple rotations. 


added:

Axiom has a domain for NonAssociative Algebra
\begin{axiom}
)show NonAssociativeAlgebra
\end{axiom}

Ref: http://arxiv.org/abs/0711.3220

Fourvector algebra

Author: Diego Saa

(Submitted on 20 Nov 2007)

Abstract: The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the excellent ability to perform rotations with the use of fourvectors, as well as their use in relativity for producing Lorentz boosts, which are understood as simple rotations.

axiom
all:=[a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]
LatexWiki Image(1)
Type: List OrderedVariableList? [a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?
axiom
a:Vector DMP(all,INT):=[a1,a2,a3,a4]
LatexWiki Image(2)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)
axiom
b:Vector DMP(all,INT):=[b1,b2,b3,b4]
LatexWiki Image(3)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)
axiom
c:Vector DMP(all,INT):=[c1,c2,c3,c4]
LatexWiki Image(4)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)
axiom
_*_*(x,y)==concat(x(1) * y(1) + dot(x(2..), y(2..)), x(1) * y(2..) - x(2..) * y(1) + cross(x(2..), y(2..)))
Type: Void
axiom
-- Jordan? a ** (b ** c) + c ** (a ** b) + b ** (c ** a)
axiom
Compiling function ** with type (Vector 
      DistributedMultivariatePolynomial([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,
      c3,c4],Integer),Vector DistributedMultivariatePolynomial([a1,a2,
      a3,a4,b1,b2,b3,b4,c1,c2,c3,c4],Integer)) -> Vector 
      DistributedMultivariatePolynomial([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,
      c3,c4],Integer)
LatexWiki Image(5)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)
axiom
a0:Vector DMP(all,INT):=[0,a2,a3,a4]
LatexWiki Image(6)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)
axiom
b0:Vector DMP(all,INT):=[0,b2,b3,b4]
LatexWiki Image(7)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)
axiom
c0:Vector DMP(all,INT):=[0,c2,c3,c4]
LatexWiki Image(8)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)
axiom
-- Jordan? a0 ** (b0 ** c0) + c0 ** (a0 ** b0) + b0 ** (c0 ** a0)
LatexWiki Image(9)
Type: Vector DistributedMultivariatePolynomial?([a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]?,Integer)

Axiom has a domain for NonAssociative? Algebra

axiom
)show NonAssociativeAlgebra NonAssociativeAlgebra R: CommutativeRing is a category constructor Abbreviation for NonAssociativeAlgebra is NAALG This constructor is exposed in this frame. Issue )edit /usr/local/lib/axiom/target/x86_64-unknown-linux/../../src/algebra/NAALG.spad to see algebra source code for NAALG ------------------------------- Operations -------------------------------- ?*? : (R,%) -> % ?*? : (%,R) -> % ?*? : (%,%) -> % ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % ?-? : (%,%) -> % -? : % -> % ?=? : (%,%) -> Boolean 0 : () -> % antiCommutator : (%,%) -> % associator : (%,%,%) -> % coerce : % -> OutputForm commutator : (%,%) -> % hash : % -> SingleInteger latex : % -> String sample : () -> % zero? : % -> Boolean ?~=? : (%,%) -> Boolean ?*? : (NonNegativeInteger,%) -> % leftPower : (%,PositiveInteger) -> % plenaryPower : (%,PositiveInteger) -> % rightPower : (%,PositiveInteger) -> % subtractIfCan : (%,%) -> Union(%,"failed")