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last edited 8 years ago by Bill Page |
Edit detail for SandBoxPoirier revision 1 of 2
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Editor: Bill Page
Time: 2015/07/31 18:17:05 GMT+0 |
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| Note: | ||
changed: - Poirier to Hall 1-D \begin{axiom} x := operator('x) C:List Symbol := [subscript('x,[0])] d := #C J := matrix [[D(x(C(1),t),C(1))]] K := inverse J Req18a:=[reduce(+,[reduce(+,[reduce(+,[reduce(+,[ (1/4)*D( K(k,i)*K(m,j)*D( D( K(l,j),C(k) ),C(l) ),C(m) ) for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d] diff1(n)==subscript('x,[n])-subscript('x,[n-1]) Req18b:=eval(Req18a,D(x(C(1),t),C(1))=diff1(n)) diff2(n)==diff1(n+1)-diff1(n) diff2(n) Req18c:=eval(Req18b,D(x(C(1),t),[C(1),C(1)])=diff2(n)); numer Req18c.1 diff3(n)==diff2(n)-diff2(n-1) diff3(n) Req18d:=eval(Req18c,D(x(C(1),t),[C(1),C(1),C(1)])=diff3(n)); numer Req18d.1 diff4(n)==diff3(n+1)-diff3(n) diff4(n) Req18e:=eval(Req18d,D(x(C(1),t),[C(1),C(1),C(1),C(1)])=diff4(n)); numer Req18e.1 factor % \end{axiom} 2-D \begin{axiom} x := operator('x) y := operator('y) C:List Symbol := [subscript('x,[0]),subscript('y,[0])] d := #C J := matrix [[D(x(C(1),C(2),t),C(1)), D(x(C(1),C(2),t),C(2))], [D(y(C(1),C(2),t),C(1)), D(y(C(1),C(2),t),C(2))]] K := inverse J Req18:=[reduce(+,[reduce(+,[reduce(+,[reduce(+,[ (1/4)*D( K(k,i)*K(m,j)*D( D( K(l,j),C(k) ),C(l) ),C(m) ) for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d]; #Req18 kernels Req18.1 \end{axiom}
Poirier to Hall
1-D
fricas
x := operator('x)
 | (1) |
Type: BasicOperator?
fricas
C:List Symbol := [subscript('x,
![\label{eq2}\left[{x_{0}}\right]](images/5334186771225139228-16.0px.png "\label{eq2}\left[{x_{0}}\right]") | (2) |
Type: List(Symbol)
fricas
d := #C
 | (3) |
Type: PositiveInteger?
fricas
J := matrix [[D(x(C(1),t), C(1))]]
 | (4) |
Type: Matrix(Expression(Integer))
fricas
K := inverse J
 | (5) |
Type: Union(Matrix(Expression(Integer)),
fricas
Req18a:=[reduce(+,[reduce(+, [reduce(+, [reduce(+, [ (1/4)*D( K(k, i)*K(m, j)*D( D( K(l, j), C(k) ), C(l) ), C(m) ) for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d]
![\label{eq6}\left[{-{{{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{2}}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{8 \ {{x_{, 1}}\left({{x_{0}}, \: t}\right)}\ {{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}\ {{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{10}\ {{{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}^{3}}}}\over{4 \ {{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{6}}}\right]](images/4169336034929011785-16.0px.png "\label{eq6}\left[{-{{{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{2}}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{8 \ {{x_{, 1}}\left({{x_{0}}, \: t}\right)}\ {{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}\ {{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{10}\ {{{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}^{3}}}}\over{4 \ {{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{6}}}\right]") | (6) |
Type: List(Expression(Integer))
fricas
diff1(n)==subscript('x,
Type: Void
fricas
Req18b:=eval(Req18a,D(x(C(1), t), C(1))=diff1(n))
fricas
Compiling function diff1 with type Variable(n) -> Polynomial(Integer
)![\label{eq7}\left[{{{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{{\left({8 \ {x_{n}}}-{8 \ {x_{n - 1}}}\right)}\ {{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}\ {{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{10}\ {{{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}^{3}}}}\over{{4 \ {{x_{n}}^{6}}}-{{24}\ {x_{n - 1}}\ {{x_{n}}^{5}}}+{{60}\ {{x_{n - 1}}^{2}}\ {{x_{n}}^{4}}}-{{80}\ {{x_{n - 1}}^{3}}\ {{x_{n}}^{3}}}+{{60}\ {{x_{n - 1}}^{4}}\ {{x_{n}}^{2}}}-{{24}\ {{x_{n - 1}}^{5}}\ {x_{n}}}+{4 \ {{x_{n - 1}}^{6}}}}\right]](images/4637904600355829722-16.0px.png "\label{eq7}\left[{{{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{{\left({8 \ {x_{n}}}-{8 \ {x_{n - 1}}}\right)}\ {{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}\ {{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{10}\ {{{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}^{3}}}}\over{{4 \ {{x_{n}}^{6}}}-{{24}\ {x_{n - 1}}\ {{x_{n}}^{5}}}+{{60}\ {{x_{n - 1}}^{2}}\ {{x_{n}}^{4}}}-{{80}\ {{x_{n - 1}}^{3}}\ {{x_{n}}^{3}}}+{{60}\ {{x_{n - 1}}^{4}}\ {{x_{n}}^{2}}}-{{24}\ {{x_{n - 1}}^{5}}\ {x_{n}}}+{4 \ {{x_{n - 1}}^{6}}}}\right]") | (7) |
Type: List(Expression(Integer))
fricas
diff2(n)==diff1(n+1)-diff1(n)
Type: Void
fricas
diff2(n)
fricas
Compiling function diff1 with type Polynomial(Integer) -> Polynomial
(Integer)fricas
Compiling function diff2 with type Variable(n) -> Polynomial(Integer
) | (8) |
Type: Polynomial(Integer)
fricas
Req18c:=eval(Req18b,D(x(C(1), t), [C(1), C(1)])=diff2(n));
Type: List(Expression(Integer))
fricas
numer Req18c.1
![\label{eq9}\begin{array}{@{}l}
\displaystyle
{{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{16}\ {{x_{n}}^{2}}}+{{\left({8 \ {x_{n + 1}}}+{{24}\ {x_{n - 1}}}\right)}\ {x_{n}}}-
\
\
\displaystyle
{8 \ {x_{n - 1}}\ {x_{n + 1}}}-{8 \ {{x_{n - 1}}^{2}}}](images/1109480983925935953-16.0px.png "\label{eq9}\begin{array}{@{}l}
\displaystyle
{{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{16}\ {{x_{n}}^{2}}}+{{\left({8 \ {x_{n + 1}}}+{{24}\ {x_{n - 1}}}\right)}\ {x_{n}}}-
\
\
\displaystyle
{8 \ {x_{n - 1}}\ {x_{n + 1}}}-{8 \ {{x_{n - 1}}^{2}}}") | (9) |
Type: SparseMultivariatePolynomial?(Integer,
fricas
diff3(n)==diff2(n)-diff2(n-1)
Type: Void
fricas
diff3(n)
fricas
Compiling function diff2 with type Polynomial(Integer) -> Polynomial
(Integer)fricas
Compiling function diff3 with type Variable(n) -> Polynomial(Integer
) | (10) |
Type: Polynomial(Integer)
fricas
Req18d:=eval(Req18c,D(x(C(1), t), [C(1), C(1), C(1)])=diff3(n));
Type: List(Expression(Integer))
fricas
numer Req18d.1
![\label{eq11}\begin{array}{@{}l}
\displaystyle
{{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{{128}\ {{x_{n}}^{3}}}+
\
\
\displaystyle
{{\left(-{{160}\ {x_{n + 1}}}-{{240}\ {x_{n - 1}}}+{{16}\ {x_{n - 2}}}\right)}\ {{x_{n}}^{2}}}+
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{68}\ {{x_{n + 1}}^{2}}}+{{\left({{192}\ {x_{n - 1}}}-{8 \ {x_{n - 2}}}\right)}\ {x_{n + 1}}}+
\
\
\displaystyle
{{156}\ {{x_{n - 1}}^{2}}}-{{24}\ {x_{n - 2}}\ {x_{n - 1}}}](images/8301132264775566650-16.0px.png "\label{eq11}\begin{array}{@{}l}
\displaystyle
{{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{{128}\ {{x_{n}}^{3}}}+
\
\
\displaystyle
{{\left(-{{160}\ {x_{n + 1}}}-{{240}\ {x_{n - 1}}}+{{16}\ {x_{n - 2}}}\right)}\ {{x_{n}}^{2}}}+
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{68}\ {{x_{n + 1}}^{2}}}+{{\left({{192}\ {x_{n - 1}}}-{8 \ {x_{n - 2}}}\right)}\ {x_{n + 1}}}+
\
\
\displaystyle
{{156}\ {{x_{n - 1}}^{2}}}-{{24}\ {x_{n - 2}}\ {x_{n - 1}}}") | (11) |
Type: SparseMultivariatePolynomial?(Integer,
fricas
diff4(n)==diff3(n+1)-diff3(n)
Type: Void
fricas
diff4(n)
fricas
Compiling function diff3 with type Polynomial(Integer) -> Polynomial
(Integer)fricas
Compiling function diff4 with type Variable(n) -> Polynomial(Integer
) | (12) |
Type: Polynomial(Integer)
fricas
Req18e:=eval(Req18d,D(x(C(1), t), [C(1), C(1), C(1), C(1)])=diff4(n));
Type: List(Expression(Integer))
fricas
numer Req18e.1
![\label{eq13}\begin{array}{@{}l}
\displaystyle
{{122}\ {{x_{n}}^{3}}}+{{\left({
\begin{array}{@{}l}
\displaystyle
-{x_{n + 2}}-{{156}\ {x_{n + 1}}}-{{224}\ {x_{n - 1}}}+
\
\
\displaystyle
{{15}\ {x_{n - 2}}}](images/7995758085648227231-16.0px.png "\label{eq13}\begin{array}{@{}l}
\displaystyle
{{122}\ {{x_{n}}^{3}}}+{{\left({
\begin{array}{@{}l}
\displaystyle
-{x_{n + 2}}-{{156}\ {x_{n + 1}}}-{{224}\ {x_{n - 1}}}+
\
\
\displaystyle
{{15}\ {x_{n - 2}}}") | (13) |
Type: SparseMultivariatePolynomial?(Integer,
fricas
factor %
![\label{eq14}\begin{array}{@{}l}
\displaystyle
{{122}\ {{x_{n}}^{3}}}+{{\left({
\begin{array}{@{}l}
\displaystyle
-{x_{n + 2}}-{{156}\ {x_{n + 1}}}-{{224}\ {x_{n - 1}}}+
\
\
\displaystyle
{{15}\ {x_{n - 2}}}](images/1638684053588715180-16.0px.png "\label{eq14}\begin{array}{@{}l}
\displaystyle
{{122}\ {{x_{n}}^{3}}}+{{\left({
\begin{array}{@{}l}
\displaystyle
-{x_{n + 2}}-{{156}\ {x_{n + 1}}}-{{224}\ {x_{n - 1}}}+
\
\
\displaystyle
{{15}\ {x_{n - 2}}}") | (14) |
Type: Factored(SparseMultivariatePolynomial?(Integer,
2-D
fricas
x := operator('x)
 | (15) |
Type: BasicOperator?
fricas
y := operator('y)
 | (16) |
Type: BasicOperator?
fricas
C:List Symbol := [subscript('x,
![\label{eq17}\left[{x_{0}}, \:{y_{0}}\right]](images/6749273628223114448-16.0px.png "\label{eq17}\left[{x_{0}}, \:{y_{0}}\right]") | (17) |
Type: List(Symbol)
fricas
d := #C
 | (18) |
Type: PositiveInteger?
fricas
J := matrix [[D(x(C(1),C(2), t), C(1)), D(x(C(1), C(2), t), C(2))], [D(y(C(1), C(2), t), C(1)), D(y(C(1), C(2), t), C(2))]]
![\label{eq19}\left[
\begin{array}{cc}
{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}
\
{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}](images/5770634314802573724-16.0px.png "\label{eq19}\left[
\begin{array}{cc}
{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}
\
{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}") | (19) |
Type: Matrix(Expression(Integer))
fricas
K := inverse J
![\label{eq20}\left[
\begin{array}{cc}
-{{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}&{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}
\
{{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}& -{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}](images/3029883782386138039-16.0px.png "\label{eq20}\left[
\begin{array}{cc}
-{{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}&{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}
\
{{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}& -{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}") | (20) |
Type: Union(Matrix(Expression(Integer)),
fricas
Req18:=[reduce(+,[reduce(+, [reduce(+, [reduce(+, [ (1/4)*D( K(k, i)*K(m, j)*D( D( K(l, j), C(k) ), C(l) ), C(m) ) for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d];
Type: List(Expression(Integer))
fricas
#Req18
 | (21) |
Type: PositiveInteger?
fricas
kernels Req18.1
![\label{eq22}\begin{array}{@{}l}
\displaystyle
\left[{{y_{{{{, 1}{, 1}}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 1}{, 1}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 1}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 2}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 1}}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{{{, 1}{, 1}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{{{, 2}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 1}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 1}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 2}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 1}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{{, 1}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 2}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{{, 1}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{, 1}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{, 2}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{, 1}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{, 1}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{, 2}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\right]](images/4314305287642463990-16.0px.png "\label{eq22}\begin{array}{@{}l}
\displaystyle
\left[{{y_{{{{, 1}{, 1}}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 1}{, 1}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 1}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 2}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 1}}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{{{, 1}{, 1}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{{{, 2}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 1}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 1}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 2}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 1}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{{, 1}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 2}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{{, 1}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{, 1}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{, 2}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{x_{{, 1}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{, 1}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{, 2}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right.
\
\
\displaystyle
\left.{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\right]") | (22) |
Type: List(Kernel(Expression(Integer)))