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SandBoxHermitianIsomorphisms

Edit detail for SandBoxHermitianIsomorphisms revision 5 of 7

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Editor: Bill Page Time: 2011/06/26 12:54:18 GMT-7
Note: jacobian

changed:
-p:Complex Fraction Polynomial Integer:=complex(Rp,Ip)
-q:Complex Fraction Polynomial Integer:=complex(Rq,Iq)
-r:Complex Fraction Polynomial Integer:=complex(Rr,Ir)
-t:Complex Fraction Polynomial Integer:=complex(Rt,0)
p:Complex Fraction Polynomial Integer:=complex(ℜp,𝔍p)
q:Complex Fraction Polynomial Integer:=complex(ℜq,𝔍q)
r:Complex Fraction Polynomial Integer:=complex(ℜr,𝔍r)
t:Complex Fraction Polynomial Integer:=complex(ℜt,0)

added:
s0:=solve(imag d,ℜr)
eval(trace(ρ*ρ),s0)

changed:
-s1:=solve(determinant subMatrix(H1,2,5,1,4),Rr)
s1:=solve(determinant subMatrix(H1,2,5,1,4),ℜr)

axiom

p:Complex Fraction Polynomial Integer:=complex(ℜp,𝔍p)

$\label{eq1}� � p +{�� p \ i}$

(1)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q:Complex Fraction Polynomial Integer:=complex(ℜq,𝔍q)

$\label{eq2}� � q +{�� q \ i}$

(2)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

r:Complex Fraction Polynomial Integer:=complex(ℜr,𝔍r)

$\label{eq3}� � r +{�� r \ i}$

(3)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

t:Complex Fraction Polynomial Integer:=complex(ℜt,0)

$\label{eq4}� � t$

(4)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

ρ:=matrix [[t/2+p,q],[r,t/2-p]]

$\label{eq5}\left[ \begin{array}{cc} {{{� � t +{2 \ � � p}}\over 2}+{�� p \ i}}&{� � q +{�� q \ i}} \ {� � r +{�� r \ i}}&{{{� � t -{2 \ � � p}}\over 2}-{�� p \ i}}$

(5)

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

axiom

trace ρ

$\label{eq6}� � t$

(6)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

d:=determinant ρ

$\label{eq7}{{{4 \ �� q \ �� r}+{4 \ {�� p^2}}+{� � t^2}-{4 \ � � q \ � � r}-{4 \ {� � p^2}}}\over 4}+{{\left(-{� � q \ �� r}-{� � r \ �� q}-{2 \ � � p \ �� p}\right)}\ i}$

(7)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

s0:=solve(imag d,ℜr)

$\label{eq8}\left[{� � r ={{-{� � q \ �� r}-{2 \ � � p \ �� p}}\over �� q}}\right]$

(8)

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

axiom

eval(trace(ρ*ρ),s0)

$\label{eq9}{{{\left(-{4 \ {�� q^2}}-{4 \ {� � q^2}}\right)}\ �� r}+{{\left(-{4 \ {�� p^2}}+{� � t^2}+{4 \ {� � p^2}}\right)}\ �� q}-{8 \ � � p \ � � q \ �� p}}\over{2 \ �� q}$

(9)

Type: Fraction(Polynomial(Complex(Integer)))

axiom

test(p^2+r*q=(1/4)*t^2-d)

$\label{eq10} \mbox{\rm true}$

(10)

Type: Boolean

axiom

h:Matrix Complex Polynomial Integer:=matrix [[a,complex(b,c)],[complex(b,-c),e]]

$\label{eq11}\left[ \begin{array}{cc} a &{b +{c \ i}} \ {b -{c \ i}}& e$

(11)

Type: Matrix(Complex(Polynomial(Integer)))

axiom

htranspose(h)==map(x+->conjugate(x),transpose h)

Type: Void

axiom

test(h = htranspose h)

axiom

Compiling function htranspose with type Matrix(Complex(Polynomial(
      Integer))) -> Matrix(Complex(Polynomial(Integer)))

$\label{eq12} \mbox{\rm true}$

(12)

Type: Boolean

axiom

H:=htranspose(ρ)*h-h*ρ

axiom

Compiling function htranspose with type Matrix(Complex(Fraction(
      Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial(
      Integer))))

$\label{eq13}\left[ \begin{array}{cc} {{\left(-{2 \ b \ �� r}-{2 \ a \ �� p}-{2 \ c \ � � r}\right)}\ i}&{{e \ � � r}-{a \ � � q}+{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}} \ {-{e \ � � r}+{a \ � � q}-{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}}&{{\left(-{2 \ b \ �� q}+{2 \ e \ �� p}+{2 \ c \ � � q}\right)}\ i}$

(13)

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

axiom

Hlist:=concat(H::List List ?)

$\label{eq14}\left[{{\left(-{2 \ b \ �� r}-{2 \ a \ �� p}-{2 \ c \ � � r}\right)}\ i}, \:{{e \ � � r}-{a \ � � q}+{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}}, \:{-{e \ � � r}+{a \ � � q}-{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}}, \:{{\left(-{2 \ b \ �� q}+{2 \ e \ �� p}+{2 \ c \ � � q}\right)}\ i}\right]$

(14)

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

axiom

Hreal:=removeDuplicates(select(x+->(x~=0),concat(map(x+->real x, Hlist),map(x+->imag x, Hlist))))

$\label{eq15}\left[{{e \ � � r}-{a \ � � q}+{2 \ b \ � � p}}, \:{-{e \ � � r}+{a \ � � q}-{2 \ b \ � � p}}, \:{-{2 \ b \ �� r}-{2 \ a \ �� p}-{2 \ c \ � � r}}, \:{-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}}, \:{-{2 \ b \ �� q}+{2 \ e \ �� p}+{2 \ c \ � � q}}\right]$

(15)

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

axiom

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial 
H1:=jacobian(Hreal,[a,b,c,e]::List Symbol)

$\label{eq16}\left[ \begin{array}{cccc} - � � q &{2 \ � � p}& 0 & � � r \ � � q & -{2 \ � � p}& 0 & - � � r \ -{2 \ �� p}& -{2 \ �� r}& -{2 \ � � r}& 0 \ - �� q & 0 &{2 \ � � p}& - �� r \ 0 & -{2 \ �� q}&{2 \ � � q}&{2 \ �� p}$

(16)

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

axiom

s1:=solve(determinant subMatrix(H1,2,5,1,4),ℜr)

$\label{eq17}\left[{� � r ={{-{� � q \ �� r}-{2 \ � � p \ �� p}}\over �� q}}\right]$

(17)

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

axiom

H2:=map(x+->eval(x,s1),H1)

$\label{eq18}\left[ \begin{array}{cccc} - � � q &{2 \ � � p}& 0 &{{-{� � q \ �� r}-{2 \ � � p \ �� p}}\over �� q} \ � � q & -{2 \ � � p}& 0 &{{{� � q \ �� r}+{2 \ � � p \ �� p}}\over �� q} \ -{2 \ �� p}& -{2 \ �� r}&{{{2 \ � � q \ �� r}+{4 \ � � p \ �� p}}\over �� q}& 0 \ - �� q & 0 &{2 \ � � p}& - �� r \ 0 & -{2 \ �� q}&{2 \ � � q}&{2 \ �� p}$

(18)

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

axiom

N:=nullSpace(H2)

$\label{eq19}\left[{\left[{{2 \ � � p}\over �� q}, \:{� � q \over �� q}, \: 1, \: 0 \right]}, \:{\left[ -{�� r \over �� q}, \:{�� p \over �� q}, \: 0, \: 1 \right]}\right]$

(19)

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

axiom

H2*(c*N(1)+e*N(2))

$\label{eq20}\left[ 0, \: 0, \: 0, \: 0, \: 0 \right]$

(20)

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

axiom

s2:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)

$\label{eq21}\left[{a ={{-{e \ �� r}+{2 \ c \ � � p}}\over �� q}}, \:{b ={{{e \ �� p}+{c \ � � q}}\over �� q}}, \:{c = c}, \:{e = e}\right]$

(21)

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

axiom

h1:=map(x+->eval(x,s2),h)

$\label{eq22}\left[ \begin{array}{cc} {{-{e \ �� r}+{2 \ c \ � � p}}\over �� q}&{{{i \ c \ �� q}+{e \ �� p}+{c \ � � q}}\over �� q} \ {{-{i \ c \ �� q}+{e \ �� p}+{c \ � � q}}\over �� q}& e$

(22)

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

axiom

map(x+->eval(x,s1),htranspose(ρ)*h1-h1*ρ)

$\label{eq23}\left[ \begin{array}{cc} 0 & 0 \ 0 & 0$

(23)

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