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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.

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(1) -> _*_*(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

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e:Vector INT:=[1,0,0,0]

$\label{eq1}\left[ 1, \: 0, \: 0, \: 0 \right]$

(1)

Type: Vector(Integer)

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i:Vector INT:=[0,1,0,0]

$\label{eq2}\left[ 0, \: 1, \: 0, \: 0 \right]$

(2)

Type: Vector(Integer)

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j:Vector INT:=[0,0,1,0]

$\label{eq3}\left[ 0, \: 0, \: 1, \: 0 \right]$

(3)

Type: Vector(Integer)

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k:Vector INT:=[0,0,0,1]

$\label{eq4}\left[ 0, \: 0, \: 0, \: 1 \right]$

(4)

Type: Vector(Integer)

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test(e**e=e)  and _
test(i**i=e)  and _
test(j**j=e)  and _
test(k**k=e)  and _
test(e**i=i)  and _
test(e**j=j)  and _
test(e**k=k)  and _
test(i**e=-i) and _
test(j**e=-j) and _
test(k**e=-k) and _
test(i**j=k)  and _
test(j**i=-k) and _
test(k**i=j)  and _
test(i**k=-j) and _
test(j**k=i)  and _
test(k**j=-i)

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Compiling function ** with type (Vector(Integer), Vector(Integer))
       -> Vector(Integer)

$\label{eq5} \mbox{\rm true}$

(5)

Type: Boolean

Axiom has a domain for NonAssociative? Algebra

This is documented in the article: Computations in Algebras of Fixed Rank by Johannes Grabmeir and Robert Wisbauer, from the book "Computational Algebra" By Klaus G. Fischer, Philippe Loustaunau, Jay Shapiro.

The algebra above can be given by structural constants.

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)clear all

   All user variables and function definitions have been cleared.
sc:Vector Matrix Fraction Integer := [ _
[[ 1, 0, 0, 0], _
 [ 0, 1, 0, 0], _
 [ 0, 0, 1, 0], _
 [ 0, 0, 0, 1]], _
[[ 0, 1, 0, 0], _
 [-1, 0, 0, 0], _
 [ 0, 0, 0, 1], _
 [ 0, 0,-1, 0]], _
[[ 0, 0, 1, 0], _
 [ 0, 0, 0,-1], _
 [-1, 0, 0, 0], _
 [ 0, 1, 0, 0]], _
[[ 0, 0, 0, 1], _
 [ 0, 0, 1, 0], _
 [ 0,-1, 0, 0], _
 [-1, 0, 0, 0]]];

Type: Vector(Matrix(Fraction(Integer)))

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V:=AlgebraGivenByStructuralConstants(Fraction Integer, 4, [e,i,j,k],sc)

$\label{eq6}\hbox{\axiomType{AlgebraGivenByStructuralConstants}\ } \left({{\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \: 4, \:{\left[ e , \: i , \: j , \: k \right]}, \:{\left[{\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(6)

Type: Type

Multiplication

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a:=basis()$V

$\label{eq7}\left[ e , \: i , \: j , \: k \right]$

(7)

Type: Vector(AlgebraGivenByStructuralConstants?(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]]))

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matrix([[(a.i * a.j) for j in 1..4] for i in 1..4])$OutputForm

$\label{eq8}\left[ \begin{array}{cccc} e & i & j & k \ - i & e & k & - j \ - j & - k & e & i \ - k & j & - i & e$

(8)

Type: OutputForm?

Commutator and Associator

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matrix([[commutator(a.x,a.y) for x in 1..4] for y in 1..4])$OutputForm

$\label{eq9}\left[ \begin{array}{cccc} 0 & -{2 \ i}& -{2 \ j}& -{2 \ k} \ {2 \ i}& 0 & -{2 \ k}&{2 \ j} \ {2 \ j}&{2 \ k}& 0 & -{2 \ i} \ {2 \ k}& -{2 \ j}&{2 \ i}& 0$

(9)

Type: OutputForm?

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[matrix([[associator(a.x,a.y,a.z) for x in 1..4] for y in 1..4])$OutputForm for z in 1..4]

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 0 &{2 \ i}&{2 \ j}&{2 \ k} \ 0 &{2 \ e}& 0 & 0 \ 0 & 0 &{2 \ e}& 0 \ 0 & 0 & 0 &{2 \ e}$

(10)

Type: List(OutputForm?)

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for x in 1..4 repeat
  for y in 1..4 repeat
    for z in 1..4 repeat
      if associator(a.x,a.y,a.z) ~= 0$V then
        output([[a.x,a.y,a.z],"=",associator(a.x,a.y,a.z)])

   [[i, e, e], "=", 2 i]
   [[i, e, i], "=", - 2 e]
   [[i, e, j], "=", - 2 k]
   [[i, e, k], "=", 2 j]
   [[i, i, e], "=", 2 e]
   [[i, i, i], "=", 2 i]
   [[i, i, j], "=", 2 j]
   [[i, i, k], "=", 2 k]
   [[j, e, e], "=", 2 j]
   [[j, e, i], "=", 2 k]
   [[j, e, j], "=", - 2 e]
   [[j, e, k], "=", - 2 i]
   [[j, j, e], "=", 2 e]
   [[j, j, i], "=", 2 i]
   [[j, j, j], "=", 2 j]
   [[j, j, k], "=", 2 k]
   [[k, e, e], "=", 2 k]
   [[k, e, i], "=", - 2 j]
   [[k, e, j], "=", 2 i]
   [[k, e, k], "=", - 2 e]
   [[k, k, e], "=", 2 e]
   [[k, k, i], "=", 2 i]
   [[k, k, j], "=", 2 j]
   [[k, k, k], "=", 2 k]

Type: Void

Volume form?

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a.2 * (a.3 * a.4) = (a.2 * a.3) * a.4

$\label{eq11}e = e$

(11)

Type: Equation(AlgebraGivenByStructuralConstants?(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]]))

Check standard properties

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leftUnit()$V

$\label{eq12}e$

(12)

Type: Union(AlgebraGivenByStructuralConstants?(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]]),...)

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rightUnit()$V

$\label{eq13}\verb#"failed"#$

(13)

Type: Union("failed",...)

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alternative?()$V

$\label{eq14} \mbox{\rm false}$

(14)

Type: Boolean

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leftAlternative?()$V

$\label{eq15} \mbox{\rm false}$

(15)

Type: Boolean

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rightAlternative?()$V

$\label{eq16} \mbox{\rm false}$

(16)

Type: Boolean

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associative?()$V

$\label{eq17} \mbox{\rm false}$

(17)

Type: Boolean

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antiAssociative?()$V

$\label{eq18} \mbox{\rm false}$

(18)

Type: Boolean

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--powerAssociative?()$V
commutative?()$V

$\label{eq19} \mbox{\rm false}$

(19)

Type: Boolean

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antiCommutative?()$V

$\label{eq20} \mbox{\rm false}$

(20)

Type: Boolean

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jordanAlgebra?()$V

$\label{eq21} \mbox{\rm false}$

(21)

Type: Boolean

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jordanAdmissible?()$V

$\label{eq22} \mbox{\rm false}$

(22)

Type: Boolean

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noncommutativeJordanAlgebra?()$V

$\label{eq23} \mbox{\rm false}$

(23)

Type: Boolean

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lieAlgebra?()$V

$\label{eq24} \mbox{\rm false}$

(24)

Type: Boolean

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lieAdmissible?()$V

$\label{eq25} \mbox{\rm false}$

(25)

Type: Boolean

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jacobiIdentity?()$V

$\label{eq26} \mbox{\rm false}$

(26)

Type: Boolean

Commuting elements

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V has FramedNonAssociativeAlgebra(Fraction Integer)

$\label{eq27} \mbox{\rm true}$

(27)

Type: Boolean

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basisOfCommutingElements()$AlgebraPackage(Fraction Integer,V)

$\label{eq28}\left[ \right]$

(28)

Type: List(AlgebraGivenByStructuralConstants?(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]]))

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basisOfCenter()$AlgebraPackage(Fraction Integer,V)

$\label{eq29}\left[ \right]$

(29)

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basisOfCentroid()$AlgebraPackage(Fraction Integer,V)

$\label{eq30}\left[ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(30)

Type: List(Matrix(Fraction(Integer)))

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basisOfNucleus()$AlgebraPackage(Fraction Integer,V)

$\label{eq31}\left[ \right]$

(31)

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basisOfLeftNucloid()$AlgebraPackage(Fraction Integer,V)

$\label{eq32}\left[ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(32)

Type: List(Matrix(Fraction(Integer)))

Symbolic computations

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G:=GenericNonAssociativeAlgebra(Fraction Integer, 4, [e,i,j,k],sc)

$\label{eq33}\hbox{\axiomType{GenericNonAssociativeAlgebra}\ } \left({{\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \: 4, \:{\left[ e , \: i , \: j , \: k \right]}, \:{\left[{\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(33)

Type: Type

Look for Idempotents

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conditionsForIdempotents()$G

$\label{eq34}\left[{{{\%x 4}^{2}}+{{\%x 3}^{2}}+{{\%x 2}^{2}}+{{\%x 1}^{2}}- \%x 1}, \: - \%x 2, \: - \%x 3, \: - \%x 4 \right]$

(34)

Type: List(Polynomial(Fraction(Integer)))

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gb:=groebnerFactorize %

$\label{eq35}\left[{\left[ \%x 4, \: \%x 3, \: \%x 2, \: \%x 1 \right]}, \:{\left[ \%x 4, \: \%x 3, \: \%x 2, \:{\%x 1 - 1}\right]}\right]$

(35)

Type: List(List(Polynomial(Fraction(Integer))))

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associatorDependence()$G

$\label{eq36}\left[ \right]$

(36)

Type: List(Vector(Fraction(Polynomial(Fraction(Integer)))))

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q:=leftRankPolynomial()$G

$\label{eq37}{{?}^{3}}-{2 \ \%x 1 \ {{?}^{2}}}+{{\left({{\%x 4}^{2}}+{{\%x 3}^{2}}+{{\%x 2}^{2}}+{{\%x 1}^{2}}\right)}\ ?}$

(37)

Type: SparseUnivariatePolynomial?(Fraction(Polynomial(Fraction(Integer))))

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map(factor,coefficients q)

$\label{eq38}\left[ 1, \: -{2 \ \%x 1}, \:{{{\%x 4}^{2}}+{{\%x 3}^{2}}+{{\%x 2}^{2}}+{{\%x 1}^{2}}}\right]$

(38)

Type: List(Factored(Fraction(Polynomial(Fraction(Integer)))))

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rightUnit()$G

$\label{eq39}\verb#"failed"#$

(39)

Type: Union("failed",...)

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p1:=generic([x1,y1,z1,w1])$G

$\label{eq40}{w 1 \ k}+{z 1 \ j}+{y 1 \ i}+{x 1 \ e}$

(40)

Type: GenericNonAssociativeAlgebra?(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]])

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p2:=generic([x2,y2,z2,w2])$G

$\label{eq41}{w 2 \ k}+{z 2 \ j}+{y 2 \ i}+{x 2 \ e}$

(41)

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p3:=generic([x3,y3,z3,w3])$G

$\label{eq42}{w 3 \ k}+{z 3 \ j}+{y 3 \ i}+{x 3 \ e}$

(42)

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leftRecip(p1)$G

$\label{eq43}\begin{array}{@{}l} \displaystyle {{\frac{w 1}{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ k}+ \ \ \displaystyle {{\frac{z 1}{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ j}+ \ \ \displaystyle {{\frac{y 1}{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ i}+ \ \ \displaystyle {{\frac{x 1}{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ e}$

(43)

Type: Union(GenericNonAssociativeAlgebra?(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]]),...)

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rightRecip(p1)$G

$\label{eq44}\verb#"failed"#$

(44)

Type: Union("failed",...)

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leftRegularRepresentation(p1)

$\label{eq45}\left[ \begin{array}{cccc} x 1 & y 1 & z 1 & w 1 \ - y 1 & x 1 & - w 1 & z 1 \ - z 1 & w 1 & x 1 & - y 1 \ - w 1 & - z 1 & y 1 & x 1$

(45)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

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rightRegularRepresentation(p1)

$\label{eq46}\left[ \begin{array}{cccc} x 1 & y 1 & z 1 & w 1 \ y 1 & - x 1 & w 1 & - z 1 \ z 1 & - w 1 & - x 1 & y 1 \ w 1 & z 1 & - y 1 & - x 1$

(46)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

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associator(p1,p2,p3)$G

$\label{eq47}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ x 2 \ y 1 \ z 3}+{2 \ w 3 \ z 1 \ z 2}+{2 \ x 2 \ y 3 \ z 1}+ \ \ \displaystyle {2 \ w 3 \ y 1 \ y 2}+{2 \ w 1 \ x 2 \ x 3}+{2 \ w 1 \ w 2 \ w 3}$

(47)

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associator(p1,p1,p2)

$\label{eq48}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ x 1 \ y 1 \ z 2}+{2 \ w 2 \ {{z 1}^{2}}}+{2 \ x 1 \ y 2 \ z 1}+{2 \ w 2 \ {{y 1}^{2}}}+ \ \ \displaystyle {2 \ w 1 \ x 1 \ x 2}+{2 \ {{w 1}^{2}}\ w 2}$

(48)

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associator(p1,p2,p2)

$\label{eq49}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({2 \ w 2 \ z 1}-{2 \ x 2 \ y 1}\right)}\ z 2}+{2 \ x 2 \ y 2 \ z 1}+{2 \ w 2 \ y 1 \ y 2}+ \ \ \displaystyle {2 \ w 1 \ {{x 2}^{2}}}+{2 \ w 1 \ {{w 2}^{2}}}$

(49)

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p1*p1

$\label{eq50}{\left({{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}\right)}\ e$

(50)

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p1*p2 + p2*p1

$\label{eq51}{\left({2 \ z 1 \ z 2}+{2 \ y 1 \ y 2}+{2 \ x 1 \ x 2}+{2 \ w 1 \ w 2}\right)}\ e$

(51)

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p1*(p1*p1)+(p1*p1)*p1

$\label{eq52}{\left({2 \ x 1 \ {{z 1}^{2}}}+{2 \ x 1 \ {{y 1}^{2}}}+{2 \ {{x 1}^{3}}}+{2 \ {{w 1}^{2}}\ x 1}\right)}\ e$

(52)