Error: export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-linux-gnu/lib; LANG=en_US.UTF-8 /usr/local/lib/fricas/target/x86_64-linux-gnu/bin/fricas -nosman < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7701307392394560546-25px.axm
Killed
Checking for foreign routines
FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu"
spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so"
foreign routines found
openServer result -2
FriCAS Computer Algebra System
Version: FriCAS 1.3.10 built with sbcl 2.2.9.debian
Timestamp: Wed 10 Jan 02:19:45 CET 2024
-----------------------------------------------------------------------------
Issue )copyright to view copyright notices.
Issue )summary for a summary of useful system commands.
Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> (1) -> Scalar := Expression Integer
$$
Expression
\left(
{Integer}
\right)
\leqno(1)
$$
Type: Type
vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x)
Function declaration vect : List(Expression(Integer)) -> Matrix(
Expression(Integer)) has been added to workspace.
Type: Void
vect [a0,a1,a2,a3]
Compiling function vect with type List(Expression(Integer)) ->
Matrix(Expression(Integer))
$$
\left[
\begin{array}{c}
a0 \
a1 \
a2 \
a3
\end{array}
\right]
\leqno(3)
$$
Type: Matrix(Expression(Integer))
(4) -> ID:=diagonalMatrix([1,1,1,1])
$$
\left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(4)
$$
Type: Matrix(Integer)
(5) -> htrigs2exp == rule
cosh(a) == (exp(a)+exp(-a))/2
sinh(a) == (exp(a)-exp(-a))/2
Type: Void
sinhcosh == rule
?cexp(a)+?cexp(-a) == 2ccosh(a)
?cexp(a)-?cexp(-a) == 2csinh(a)
?cexp(a-b)+?cexp(b-a) == 2ccosh(a-b)
?cexp(a-b)-?cexp(b-a) == 2csinh(a-b)
Type: Void
expandhtrigs == rule
cosh(:x+y) == sinh(x)sinh(y)+cosh(x)cosh(y)
sinh(:x+y) == cosh(x)sinh(y)+sinh(x)cosh(y)
cosh(2x) == 2cosh(x)^2-1
sinh(2x) == 2sinh(x)*cosh(x)
Type: Void
expandhtrigs2 == rule
cosh(2x+2y) == 2cosh(x+y)^2-1
sinh(2x+2y) == 2sinh(x+y)cosh(x+y)
cosh(2x-2y) == 2cosh(x-y)^2-1
sinh(2x-2y) == 2sinh(x-y)cosh(x-y)
Type: Void
Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x
Function declaration Simplify : Expression(Integer) -> Expression(
Integer) has been added to workspace.
Type: Void
possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )
Type: Void
is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean
Function declaration is? : Equation(Expression(Integer)) -> Boolean
has been added to workspace.
Type: Void
Is?(eq:Equation(Matrix(Scalar))):Boolean ==
(map(Simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber =
zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean
Function declaration Is? : Equation(Matrix(Expression(Integer))) ->
Boolean has been added to workspace.
Type: Void
(13) -> G:=diagonalMatrix [-1,1,1,1]
$$
\left[
\begin{array}{cccc}
-1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(13)
$$
Type: Matrix(Integer)
(14) -> g(x) == transpose(x)*G
Type: Void
g(vect [a0,a1,a2,a3])
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
(Expression(Integer))
$$
\left[
\begin{array}{cccc}
-a0 & a1 & a2 & a3
\end{array}
\right]
\leqno(15)
$$
Type: Matrix(Expression(Integer))
(16) -> dot(x,y) == (g(x)*y)::Scalar
Type: Void
dot(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
Compiling function dot with type (Matrix(Expression(Integer)),
Matrix(Expression(Integer))) -> Expression(Integer)
$$
{a3 \ b3}+{a2 \ b2}+{a1 \ b1} -{a0 \ b0}
\leqno(17)
$$
Type: Expression(Integer)
(18) -> tensor(x,y) == x*g(y)
Type: Void
tensor(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
Compiling function tensor with type (Matrix(Expression(Integer)),
Matrix(Expression(Integer))) -> Matrix(Expression(Integer))
$$
\left[
\begin{array}{cccc}
-{a0 \ b0} & {a0 \ b1} & {a0 \ b2} & {a0 \ b3} \
-{a1 \ b0} & {a1 \ b1} & {a1 \ b2} & {a1 \ b3} \
-{a2 \ b0} & {a2 \ b1} & {a2 \ b2} & {a2 \ b3} \
-{a3 \ b0} & {a3 \ b1} & {a3 \ b2} & {a3 \ b3}
\end{array}
\right]
\leqno(19)
$$
Type: Matrix(Expression(Integer))
(20) -> P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1,-p2,-p3];
Type: Matrix(Expression(Integer))
dot(P,P)
$$
-1
\leqno(21)
$$
Type: Expression(Integer)
Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1,-q2,-q3];
Type: Matrix(Expression(Integer))
R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1,-r2,-r3];
Type: Matrix(Expression(Integer))
S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]
$$
\left[
\begin{array}{c}
{\frac{1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
-{\frac{s1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
-{\frac{s2}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
-{\frac{s3}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(24)
$$
Type: Matrix(Expression(Integer))
dot(S,S)
$$
-1
\leqno(25)
$$
Type: Expression(Integer)
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3]
$$
\left[
\begin{array}{c}
{\frac{1}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \
-{\frac{t1}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \
-{\frac{t2}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \
-{\frac{t3}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(26)
$$
Type: Matrix(Expression(Integer))
U:=vect [cosh(u),sinh(u),0,0]
$$
\left[
\begin{array}{c}
{\cosh
\left(
{u}
\right)}
\
{\sinh
\left(
{u}
\right)}
\
0 \
0
\end{array}
\right]
\leqno(27)
$$
Type: Matrix(Expression(Integer))
simplify dot(U,U)
$$
-1
\leqno(28)
$$
Type: Expression(Integer)
V:=vect [cosh(v),sinh(v),0,0]
$$
\left[
\begin{array}{c}
{\cosh
\left(
{v}
\right)}
\
{\sinh
\left(
{v}
\right)}
\
0 \
0
\end{array}
\right]
\leqno(29)
$$
Type: Matrix(Expression(Integer))
Simplify dot(U,V)
Compiling body of rule htrigs2exp to compute value of type Ruleset(
Integer,Integer,Expression(Integer))
Compiling body of rule sinhcosh to compute value of type Ruleset(
Integer,Integer,Expression(Integer))
Compiling function Simplify with type Expression(Integer) ->
Expression(Integer)
$$
-{\cosh
\left(
{{v -u}}
\right)}
\leqno(30)
$$
Type: Expression(Integer)
W:=vect [cosh(w),0,sinh(w),0]
$$
\left[
\begin{array}{c}
{\cosh
\left(
{w}
\right)}
\
0 \
{\sinh
\left(
{w}
\right)}
\
0
\end{array}
\right]
\leqno(31)
$$
Type: Matrix(Expression(Integer))
Simplify dot(U,W)
$$
\frac{{-{\cosh
\left(
{{w+u}}
\right)}
-{\cosh
\left(
{{w -u}}
\right)}}}{2}
\leqno(32)
$$
Type: Expression(Integer)
(33) -> vect [1,0,0,0]
$$
\left[
\begin{array}{c}
1 \
0 \
0 \
0
\end{array}
\right]
\leqno(33)
$$
Type: Matrix(Expression(Integer))
dot(%,%)
$$
-1
\leqno(34)
$$
Type: Expression(Integer)
(35) -> ω(P,Q)==-P/dot(P,Q)-Q
Type: Void
ω(P,Q)
Compiling function ω with type (Matrix(Expression(Integer)), Matrix(
Expression(Integer))) -> Matrix(Expression(Integer))
$$
\left[
\begin{array}{c}
{\frac{{{{\left( {p3 \ q3}+{p2 \ q2}+{p1 \ q1}
\right)}
\ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}}+{{\left(
-{{q3} \sp {2}} -{{q2} \sp {2}} -{{q1} \sp {2}}
\right)}
\ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}}}}}{{{{\sqrt
{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp
{2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \
q1}}}} \
{\frac{{{q1 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q1 \ q3}
-{p2 \ q1 \ q2} -{p1 \ {{q1} \sp {2}}} -p1}}{{{{\sqrt {{{{p3} \sp
{2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp
{2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \
{\frac{{{q2 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q2 \ q3}
-{p2 \ {{q2} \sp {2}}} -{p1 \ q1 \ q2} -p2}}{{{{\sqrt {{{{p3} \sp
{2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp
{2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \
{\frac{{{q3 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ {{q3} \sp
{2}}}+{{\left( -{p2 \ q2} -{p1 \ q1}
\right)}
\ q3} -p3}}{{{{\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2
\ q2} -{p1 \ q1}}}}
\end{array}
\right]
\leqno(36)
$$
Type: Matrix(Expression(Integer))
ω(S,T)
$$
\left[
\begin{array}{c}
{\frac{{{{t3} \sp {2}} -{s3 \ t3}+{{t2} \sp {2}} -{s2 \ t2}+{{t1} \sp {2}}
-{s1 \ t1}}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{-{s1 \ {{t3} \sp {2}}}+{s3 \ t1 \ t3} -{s1 \ {{t2} \sp {2}}}+{s2
\ t1 \ t2} -t1+s1}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{-{s2 \ {{t3} \sp {2}}}+{s3 \ t2 \ t3}+{{\left( {s1 \ t1} -1
\right)}
\ t2} -{s2 \ {{t1} \sp {2}}}+s2}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1}
-1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{{{\left( {s2 \ t2}+{s1 \ t1} -1
\right)}
\ t3} -{s3 \ {{t2} \sp {2}}} -{s3 \ {{t1} \sp {2}}}+s3}}{{{\left( {s3 \
t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}}
\end{array}
\right]
\leqno(37)
$$
Type: Matrix(Expression(Integer))
(38) -> map(x+->Simplify x,ω(U,V))
$$
\left[
\begin{array}{c}
{\frac{{-{\cosh
\left(
{{{2 \ v} -u}}
\right)}+{\cosh
\left(
{u}
\right)}}}{{2
\ {\cosh
\left(
{{v -u}}
\right)}}}}
\
{\frac{{-{\sinh
\left(
{{{2 \ v} -u}}
\right)}+{\sinh
\left(
{u}
\right)}}}{{2
\ {\cosh
\left(
{{v -u}}
\right)}}}}
\
0 \
0
\end{array}
\right]
\leqno(38)
$$
Type: Matrix(Expression(Integer))
vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]
$$
\left[
\begin{array}{c}
{\frac{{-{{\cosh
\left(
{{v -u}}
\right)}
\ {\cosh
\left(
{v}
\right)}}+{\cosh
\left(
{u}
\right)}}}{{\cosh
\left(
{{v -u}}
\right)}}}
\
{\frac{{-{{\cosh
\left(
{{v -u}}
\right)}
\ {\sinh
\left(
{v}
\right)}}+{\sinh
\left(
{u}
\right)}}}{{\cosh
\left(
{{v -u}}
\right)}}}
\
0 \
0
\end{array}
\right]
\leqno(39)
$$
Type: Matrix(Expression(Integer))
Is?(% = ω(U,V))
Compiling function Is? with type Equation(Matrix(Expression(Integer)
)) -> Boolean
$$
true
\leqno(40)
$$
Type: Boolean
map(x+->Simplify x,ω(U,W))
$$
\left[
\begin{array}{c}
{\frac{{-{\cosh
\left(
{{2 \ w}}
\right)}+1}}{{2
\ {\cosh
\left(
{w}
\right)}}}}
\
{\frac{{2 \ {\sinh
\left(
{u}
\right)}}}{{{\cosh
\left(
{{w+u}}
\right)}+{\cosh
\left(
{{w -u}}
\right)}}}}
\
-{\sinh
\left(
{w}
\right)}
\
0
\end{array}
\right]
\leqno(41)
$$
Type: Matrix(Expression(Integer))
(42) -> dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0]))
$$
\frac{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}}}{{{{p3} \sp {2}}+{{p2}
\sp {2}}+{{p1} \sp {2}}+1}}
\leqno(42)
$$
Type: Expression(Integer)
(43) -> dot(P,ω(Q,P))
$$
0
\leqno(43)
$$
Type: Expression(Integer)
possible dot(ω(Q,P),ω(Q,P))::EXPR Float
Compiling function possible with type Expression(Integer) ->
Expression(Integer)
$$
0.9999995659\_3941083651
\leqno(44)
$$
Type: Expression(Float)
dot(Q,ω(P,Q))
$$
0
\leqno(45)
$$
Type: Expression(Integer)
possible dot(ω(P,Q),ω(P,Q))::EXPR Float
$$
0.9999994628\_9493005142
\leqno(46)
$$
Type: Expression(Float)
(47) -> ω(vect [u0,u1,u2,u3],vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
{\frac{u1}{u0}} \
{\frac{u2}{u0}} \
{\frac{u3}{u0}}
\end{array}
\right]
\leqno(47)
$$
Type: Matrix(Expression(Integer))
ω(R,vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
-{\frac{r1}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
-{\frac{r2}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
-{\frac{r3}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(48)
$$
Type: Matrix(Expression(Integer))
ω(S,vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
-s1 \
-s2 \
-s3
\end{array}
\right]
\leqno(49)
$$
Type: Matrix(Expression(Integer))
map(Simplify, ω(U,vect [1,0,0,0]))
$$
\left[
\begin{array}{c}
0 \
{\frac{{\sinh
\left(
{u}
\right)}}{{\cosh
\left(
{u}
\right)}}}
\
0 \
0
\end{array}
\right]
\leqno(50)
$$
Type: Matrix(Expression(Integer))
(51) -> ω(vect [1,0,0,0],S)
$$
\left[
\begin{array}{c}
{\frac{{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}}}{{\sqrt {{-{{s3} \sp
{2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s2}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s3}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(51)
$$
Type: Matrix(Expression(Integer))
ω(vect [1,0,0,0],R)
$$
\left[
\begin{array}{c}
{\frac{{-{{r3} \sp {2}} -{{r2} \sp {2}} -{{r1} \sp {2}}}}{{\sqrt {{{{r3} \sp
{2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
r1 \
r2 \
r3
\end{array}
\right]
\leqno(52)
$$
Type: Matrix(Expression(Integer))
is?(dot(ω(P,Q),ω(P,Q))=dot(ω(Q,P),ω(Q,P)))
Compiling function is? with type Equation(Expression(Integer)) ->
Boolean
$$
true
\leqno(53)
$$
Type: Boolean
(54) -> L(P,Q) == ID + tensor(P+Q,P+Q)/(1-dot(P,Q)) - 2*tensor(P,Q)
Type: Void
Is?(L(P,P) = ID)
Compiling function L with type (Matrix(Expression(Integer)), Matrix(
Expression(Integer))) -> Matrix(Expression(Integer))
$$
true
\leqno(55)
$$
Type: Boolean
Is?(L(P,Q)*L(Q,P) = ID)
$$
true
\leqno(56)
$$
Type: Boolean
Is?(L(P,Q)*Q=P)
$$
true
\leqno(57)
$$
Type: Boolean
Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))
$$
true
\leqno(58)
$$
Type: Boolean
(59) -> B(P,Q,X) == ID -
( tensor(
2X, dot(P-Q,P-Q)X - 2dot(X,P)(P-Q)
) +
tensor( _
P-Q, 2dot(X,X)(P-Q)+4dot(X,Q)X
)
) / ( _
dot(X,X)dot(P-Q,P-Q)+4dot(X,P)*dot(X,Q) _
)
Type: Void
Is?(B(P,P,R) = ID)
Compiling function B with type (Matrix(Expression(Integer)), Matrix(
Expression(Integer)), Matrix(Expression(Integer))) -> Matrix(
Expression(Integer))
$$
true
\leqno(60)
$$
Type: Boolean
Is?(B(P,Q,R)*B(Q,P,R) = ID)
$$
true
\leqno(61)
$$
Type: Boolean
Is?(B(P,Q,R)*P=Q)
$$
true
\leqno(62)
$$
Type: Boolean
--Is?(B(P,Q,R)*ω(P,Q) = -ω(Q,P))
Is?(L(P,Q)=B(Q,P,qQ+pP))
$$
true
\leqno(63)
$$
Type: Boolean
(64) -> map(x+->simplify expandhtrigs2 Simplify x, L(U,V))
Compiling body of rule expandhtrigs2 to compute value of type
Ruleset(Integer,Integer,Expression(Integer))
$$
\left[
\begin{array}{cccc}
{\cosh
\left(
{{v -u}}
\right)}
& -{\sinh
\left(
{{v -u}}
\right)}
& 0 & 0 \
-{\sinh
\left(
{{v -u}}
\right)}
& {\cosh
\left(
{{v -u}}
\right)}
& 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(64)
$$
Type: Matrix(Expression(Integer))
map(x+->simplify expandhtrigs expandhtrigs2 Simplify x, L(U,W))
Compiling body of rule expandhtrigs to compute value of type Ruleset
(Integer,Integer,Expression(Integer))
$$
\left[
\begin{array}{cccc}
{\frac{{{{\left( {2 \ {{{\cosh
\left(
{u}
\right)}}
\sp {2}}} -1
\right)}
\ {{{\cosh
\left(
{w}
\right)}}
\sp {2}}}+{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}
-{{{\cosh
\left(
{u}
\right)}}
\sp {2}}+1}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\left( {\cosh
\left(
{w}
\right)}+{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{u}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\left( {{\left( -{2 \ {{{\cosh
\left(
{u}
\right)}}
\sp {2}}}+1
\right)}
\ {\cosh
\left(
{w}
\right)}}
-{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& 0 \
{\frac{{{\left( {2 \ {\cosh
\left(
{u}
\right)}
\ {{{\cosh
\left(
{w}
\right)}}
\sp {2}}}+{\cosh
\left(
{w}
\right)}
-{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{u}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+{{{\cosh
\left(
{u}
\right)}}
\sp {2}}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\left( -{2 \ {\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}
-1
\right)}
\ {\sinh
\left(
{u}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& 0 \
{\frac{{{\left( -{\cosh
\left(
{w}
\right)}
-{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\sinh
\left(
{u}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{{{\cosh
\left(
{w}
\right)}}
\sp {2}}+{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(65)
$$
Type: Matrix(Expression(Integer))
(66) -> Is?(L(R,P)*L(P,Q) = L(R,Q))
$$
false
\leqno(66)
$$
Type: Boolean
RQ:=aR+bQ;
Type: Matrix(Expression(Integer))
rq:=solve(dot(RQ,RQ)=-1,b); #rq
$$
2
\leqno(68)
$$
Type: PositiveInteger
RQ1:=eval(RQ,rq.1);
Type: Matrix(Expression(Integer))
dot(RQ1,RQ1)
$$
-1
\leqno(70)
$$
Type: Expression(Integer)
Is?(L(R,RQ1)*L(RQ1,Q) = L(R,Q))
$$
true
\leqno(71)
$$
Type: Boolean
RQ2:=eval(RQ,rq.2);
Type: Matrix(Expression(Integer))
Is?(RQ1=RQ2)
$$
false
\leqno(73)
$$
Type: Boolean
dot(RQ2,RQ2)
$$
-1
\leqno(74)
$$
Type: Expression(Integer)
Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q))
$$
true
\leqno(75)
$$
Type: Boolean
(76) -> LRPQ := L(R,P)*L(P,Q);
Type: Matrix(Expression(Integer))
Is?(LRPQQ = L(R,Q)Q)
$$
true
\leqno(77)
$$
Type: Boolean
is?(dot(LRPQ*ω(S,Q),LRPQ*ω(S,Q))=dot(L(R,Q)*ω(S,Q),L(R,Q)*ω(S,Q)))