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From Poirier's Bohmian Mechanics without Wavefunctions to Hall's Many Interacting Worlds

Ref:

  1. Quantum Phenomena Modeled by Interactions between Many Classical Worlds

    Michael J. W. Hall Dirk-André Deckert and Howard M. Wiseman,

    PHYSICAL REVIEW X 4, 041013 (23 October 2014)

  2. Quantum Mechanics Without Wavefunctions

    Jeremy Schiff and Bill Poirier

    [quant-ph] 11 Jan 2012

    http://arxiv.org/abs/1201.2382v1

  3. Verlet integration

    set

    for n=1,2,... iterate

%sage # Uncomment to use local version of FriCAS #os.environ['PATH'] = '%s/bin:%s'%(os.environ['HOME'],os.environ['PATH']) # Better version of FriCAS/Axiom mode for SageMathCloud from fricas import fricas execfile('fricas_md.py')
%default_mode fricas_md
)clear completely )set output tex on )set output algebra off )set output mathml off )set message type off

All user variables and function definitions have been cleared.

All )browse facility databases have been cleared.

Internally cached functions and constructors have been cleared.

)clear completely is finished.

1-D

Hall's MIW quantum force eqs. (24,25)

sif(n)==1/(subscript('x,[n])-subscript('x,[n-1])) sigma(n)==1/sif(n)^2*(sif(n+1)-2*sif(n)+sif(n-1)) --sigma(n) r(n)==1/4*(sigma(n+1)-sigma(n)) --numer r(n) --factor denom(r(n))
x := operator('x); display(x,w+->subscript(name x,w)) C:List Symbol := [subscript('x,[0])] d := #C J := matrix [[D(x(C(1),t),C(1))]] K := inverse J

$$

x

$$

$$

\left[

{x _{0}}

\right]

$$

$$

1

$$

$$

\left[

\begin{array}{c}

{{x _{{,1}}}

\left(

{{x _{0}}, : t}

\right)}

\end{array}

\right]

$$

$$

\left[

\begin{array}{c}

{1 \over {{x _{{,1}}}

\left(

{{x _{0}}, : t}

\right)}}

\end{array}

\right]

$$

Schiff and Poirier eq. (18)

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]

$$

\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]

$$

In 1-D this is equivalent to eq. (10).

Now replace derivatives with differences

diff1(x,n)==x(n)-x(n-1) diff1(x,n) Req18b:=eval(Req18a,D(x(C(1),t),C(1))=diff1(x,n)); diff2(x,n)==diff1(x,n)-diff1(x,n-1) diff2(x,n) Req18c:=eval(Req18b,D(x(C(1),t),[C(1),C(1)])=diff2(x,n)); diff3(x,n)==diff2(x,n+1)-diff2(x,n) diff3(x,n) Req18d:=eval(Req18c,D(x(C(1),t),[C(1),C(1),C(1)])=diff3(x,n)); diff4(x,n)==diff3(x,n+1)-diff3(x,n) diff4(x,n) Req18e:=eval(Req18d,D(x(C(1),t),[C(1),C(1),C(1),C(1)])=diff4(x,n));

Compiling function diff1 with type (BasicOperator,Variable(n)) -> Expression(Integer)

$$

{x _{n}} -{x _{{n -1}}}

$$

Compiling function diff1 with type (BasicOperator,Polynomial(Integer)) -> Expression(Integer)

Compiling function diff2 with type (BasicOperator,Variable(n)) -> Expression(Integer)

$$

{x _{n}} -{2 \ {x _{{n -1}}}}+{x _{{n -2}}}

$$

Compiling function diff2 with type (BasicOperator,Polynomial(Integer)) -> Expression(Integer)

Compiling function diff3 with type (BasicOperator,Variable(n)) -> Expression(Integer)

$$

{x _{{n+1}}} -{3 \ {x _{n}}}+{3 \ {x _{{n -1}}}} -{x _{{n -2}}}

$$

Compiling function diff3 with type (BasicOperator,Polynomial(Integer)) -> Expression(Integer)

Compiling function diff4 with type (BasicOperator,Variable(n)) -> Expression(Integer)

$$

{x _{{n+2}}} -{4 \ {x _{{n+1}}}}+{6 \ {x _{n}}} -{4 \ {x _{{n -1}}}}+{x _{{n -2}}}

$$

factor numer Req18e.1 factor denom Req18e.1 numer r(n) factor denom r(n)

$$

-{\left( {{\left( {{{x _{n}}} ^{2}} -{2 \ {x _{{n -1}}} \ {x _{n}}}+{{{x _{{n -1}}}} ^{2}}

\right)}

\ {x _{{n+2}}}}+{{\left( -{{12} \ {{{x _{n}}} ^{2}}}+{{\left( {{32} \ {x _{{n -1}}}} -{8 \ {x _{{n -2}}}}

\right)}

\ {x _{n}}} -{{20} \ {{{x _{{n -1}}}} ^{2}}}+{8 \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {x _{{n+1}}}}+{{40} \ {{{x _{n}}} ^{3}}}+{{\left( -{{172} \ {x _{{n -1}}}}+{{63} \ {x _{{n -2}}}}

\right)}

\ {{{x _{n}}} ^{2}}}+{{\left( {{254} \ {{{x _{{n -1}}}} ^{2}}} -{{194} \ {x _{{n -2}}} \ {x _{{n -1}}}}+{{38} \ {{{x _{{n -2}}}} ^{2}}}

\right)}

\ {x _{n}}} -{{132} \ {{{x _{{n -1}}}} ^{3}}}+{{161} \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}} -{{68} \ {{{x _{{n -2}}}} ^{2}} \ {x _{{n -1}}}}+{{10} \ {{{x _{{n -2}}}} ^{3}}}

\right)}

$$

$$

4 \ {{{\left( {x _{n}} -{x _{{n -1}}}

\right)}}

^{6}}

$$

Compiling function sif with type Polynomial(Integer) -> Fraction(Polynomial(Integer))

Compiling function sigma with type Polynomial(Integer) -> Fraction(Polynomial(Integer))

Compiling function sif with type Variable(n) -> Fraction(Polynomial(Integer))

Compiling function sigma with type Variable(n) -> Fraction(Polynomial(Integer))

Compiling function r with type Variable(n) -> Fraction(Polynomial(Integer))

$$

{{\left( -{x _{{n+2}}}+{x _{{n+1}}}+{x _{{n -1}}} -{x _{{n -2}}}

\right)}

\ {{{x _{n}}} ^{4}}}+{{\left( {{\left( {x _{{n+1}}}+{9 \ {x _{{n -1}}}} -{6 \ {x _{{n -2}}}}

\right)}

\ {x _{{n+2}}}} -{{{x _{{n+1}}}} ^{2}}+{{\left( -{{12} \ {x _{{n -1}}}}+{9 \ {x _{{n -2}}}}

\right)}

\ {x _{{n+1}}}} -{{{x _{{n -1}}}} ^{2}}+{{x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {{{x _{n}}} ^{3}}}+{{\left( {{\left( {{\left( -{{12} \ {x _{{n -1}}}}+{9 \ {x _{{n -2}}}}

\right)}

\ {x _{{n+1}}}} -{{12} \ {{{x _{{n -1}}}} ^{2}}}+{9 \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {x _{{n+2}}}}+{{\left( {{15} \ {x _{{n -1}}}} -{{12} \ {x _{{n -2}}}}

\right)}

\ {{{x _{{n+1}}}} ^{2}}}+{{\left( {{15} \ {{{x _{{n -1}}}} ^{2}}} -{{12} \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {x _{{n+1}}}}

\right)}

\ {{{x _{n}}} ^{2}}}+{{\left( {{\left( {{\left( {5 \ {x _{{n -1}}}} -{5 \ {x _{{n -2}}}}

\right)}

\ {{{x _{{n+1}}}} ^{2}}}+{{\left( {{11} \ {{{x _{{n -1}}}} ^{2}}} -{8 \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {x _{{n+1}}}}+{6 \ {{{x _{{n -1}}}} ^{3}}} -{5 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}}

\right)}

\ {x _{{n+2}}}}+{{\left( -{6 \ {x _{{n -1}}}}+{6 \ {x _{{n -2}}}}

\right)}

\ {{{x _{{n+1}}}} ^{3}}}+{{\left( -{{14} \ {{{x _{{n -1}}}} ^{2}}}+{{11} \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {{{x _{{n+1}}}} ^{2}}}+{{\left( -{6 \ {{{x _{{n -1}}}} ^{3}}}+{5 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}}

\right)}

\ {x _{{n+1}}}}

\right)}

\ {x _{n}}}+{{\left( {{\left( -{x _{{n -1}}}+{x _{{n -2}}}

\right)}

\ {{{x _{{n+1}}}} ^{3}}}+{{\left( -{2 \ {{{x _{{n -1}}}} ^{2}}}+{2 \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {{{x _{{n+1}}}} ^{2}}}+{{\left( -{3 \ {{{x _{{n -1}}}} ^{3}}}+{2 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}}

\right)}

\ {x _{{n+1}}}} -{{{x _{{n -1}}}} ^{4}}+{{x _{{n -2}}} \ {{{x _{{n -1}}}} ^{3}}}

\right)}

\ {x _{{n+2}}}}+{{\left( {x _{{n -1}}} -{x _{{n -2}}}

\right)}

\ {{{x _{{n+1}}}} ^{4}}}+{{\left( {3 \ {{{x _{{n -1}}}} ^{2}}} -{3 \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {{{x _{{n+1}}}} ^{3}}}+{{\left( {3 \ {{{x _{{n -1}}}} ^{3}}} -{2 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}}

\right)}

\ {{{x _{{n+1}}}} ^{2}}}+{{\left( {{{x _{{n -1}}}} ^{4}} -{{x _{{n -2}}} \ {{{x _{{n -1}}}} ^{3}}}

\right)}

\ {x _{{n+1}}}}

$$

$$

4 \ {\left( {x _{{n -1}}} -{x _{{n -2}}}

\right)}

\ {\left( {x _{{n+2}}} -{x _{{n+1}}}

\right)}

\ {\left( {x _{n}} -{x _{{n+1}}}

\right)}

\ {\left( {x _{n}} -{x _{{n -1}}}

\right)}

$$

2-D

1 particle moving in 2 dimensions (x,y) in many worlds, each indexed by 2 uniform parameters (n,m)

-- 1-D test case for comparison --vars:List Symbol := ['x]; d := 1; --X:=[x] --C:=[subscript('x,[0])] --ind:List Expression Integer:=[n]
-- 2-D vars:List Symbol := ['x,'y]; d := #vars; X := map(operator,vars) -- uniformizing trajectory parameters C:List Symbol := map(x+->subscript(x,[0]),vars) -- world indices ind:List Expression Integer:=[n,m] map(x+->display(x,w+->subscript(name x,w)),X); -- position of particle in world (n,m) P:=map(x+->kernel(x,ind)$Expression Integer,X)

$$

\left[

x, : y

\right]

$$

$$

\left[

{x _{0}}, : {y _{0}}

\right]

$$

$$

\left[

n, : m

\right]

$$

$$

\left[

{x _{n, : m}}, : {y _{n, : m}}

\right]

$$

-- Ensemble of Bohmian trajectories E:=map(x+->x concat(C,t),X) -- Jacobian: How trajectories change with trajectory parameters (function of initial position). J:=matrix map(e+->map(c+->D(e,c),C),E) K := inverse J

$$

\left[

{x

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {y

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)}

\right]

$$

$$

\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)}

\end{array}

\right]

$$

$$

\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)}}}}

\end{array}

\right]

$$

eq18a:=[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]; --in the 1-D case --test(eq18a=Req18a) #eq18a sort kernels eq18a.1 # %

$$

2

$$

$$

\left[

{{x _{{,2}}}

\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)},

: {{x _{{{,2}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{,1}{,1}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{,1}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{,2}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{,1}{,1}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{,1}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{,2}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{,1}{,1}}{,1}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{,1}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{,1}{,1}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{,2}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{,1}{,1}}{,1}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{,1}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{,1}{,1}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{{,2}{,2}}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{{,1}{,1}}{,1}}{,1}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{{,1}{,2}}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{{,1}{,1}}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{x _{{{{{,1}{,1}}{,1}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{{,2}{,2}}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{{,1}{,1}}{,1}}{,1}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{{,1}{,2}}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{{,1}{,1}}{,2}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)},

: {{y _{{{{{,1}{,1}}{,1}}{,2}}}}

\left(

{{x _{0}}, : {y _{0}}, : t}

\right)}

\right]

$$

$$

28

$$

Note that there are only 28 unique derivatives since:

test(D(E(1),[C(1),C(2)])=D(E(1),[C(2),C(1)]))

$$

true

$$

etc.

Difference operators

D_-(x,i)==x-eval(x,ind(i)=ind(i)-1) D_+(x,i)==eval(x,ind(i)=ind(i)+1)-x d1:=D_-(kernel(X(1),ind)$Expression Integer,1) d2:=D_-(d1,1) d2':=D_-(d1,2) d3:=D_+(d2,1) d3':=D_+(d2',1) D_+(d3,1)

Compiled code for D- has been cleared.

1 old definition(s) deleted for function or rule D-

Compiled code for D+ has been cleared.

1 old definition(s) deleted for function or rule D+

Compiling function D- with type (Expression(Integer),PositiveInteger) -> Expression(Integer)

$$

{x _{n, : m}} -{x _{{n -1}, : m}}

$$

$$

{x _{n, : m}} -{2 \ {x _{{n -1}, : m}}}+{x _{{n -2}, : m}}

$$

$$

{x _{n, : m}} -{x _{n, : {m -1}}} -{x _{{n -1}, : m}}+{x _{{n -1}, : {m -1}}}

$$

Compiling function D+ with type (Expression(Integer),PositiveInteger) -> Expression(Integer)

$$

{x _{{n+1}, : m}} -{3 \ {x _{n, : m}}}+{3 \ {x _{{n -1}, : m}}} -{x _{{n -2}, : m}}

$$

$$

{x _{{n+1}, : m}} -{x _{{n+1}, : {m -1}}} -{2 \ {x _{n, : m}}}+{2 \ {x _{n, : {m -1}}}}+{x _{{n -1}, : m}} -{x _{{n -1}, : {m -1}}}

$$

$$

{x _{{n+2}, : m}} -{4 \ {x _{{n+1}, : m}}}+{6 \ {x _{n, : m}}} -{4 \ {x _{{n -1}, : m}}}+{x _{{n -2}, : m}}

$$

eq18b := eval(eq18a, _ concat [map(x+->D(x,C(i)),p) for i in 1..d], _ concat [map(x+->D_-(kernel(x,ind)$Expression Integer,i),X) for i in 1..d]); kernels eq18b

$$

\left[

{{x _{{{{{,1}{,1}}{,1}}{,1}}}}

\left(

{{x _{0}}, : t}

\right)},

: {{x _{{{{,1}{,1}}{,1}}}}

\left(

{{x _{0}}, : t}

\right)},

: {{x _{{{,1}{,1}}}}

\left(

{{x _{0}}, : t}

\right)},

: {x _{n}}, : {x _{{n -1}}}

\right]

$$

eq18c := eval(eq18b, _ concat concat [[map(x+->D(x,[C(i),C(j)]),p) for j in i..d] for i in 1..d], _ concat concat [[map(x+->D_-(D_-(kernel(x,ind)$Expression Integer,i),j),X) for j in i..d] for i in 1..d]); kernels eq18c

$$

\left[

{{x _{{{{{,1}{,1}}{,1}}{,1}}}}

\left(

{{x _{0}}, : t}

\right)},

: {{x _{{{{,1}{,1}}{,1}}}}

\left(

{{x _{0}}, : t}

\right)},

: {x _{n}}, : {x _{{n -1}}}, : {x _{{n -2}}}

\right]

$$

eq18d := eval(eq18c, _ concat concat concat [[[map(x+->D(x,[C(i),C(j),C(k)]),p) for k in j..d] for j in i..d] for i in 1..d], _ concat concat concat [[[map(x+->D_+(D_-(D_-(kernel(x,ind)$Expression Integer,i),j),k),X) for k in j..d] for j in i..d] for i in 1..d]); kernels eq18d

$$

\left[

{{x _{{{{{,1}{,1}}{,1}}{,1}}}}

\left(

{{x _{0}}, : t}

\right)},

: {x _{{n+1}}}, : {x _{n}}, : {x _{{n -1}}}, : {x _{{n -2}}}

\right]

$$

eq18e := eval(eq18d, _ concat concat concat concat [[[[map(x+->D(x,[C(i),C(j),C(k),C(l)]),p) for l in k..d] for k in j..d] for j in i..d] for i in 1..d], _ concat concat concat concat [[[[map(x+->D_+(D_+(D_-(D_-(kernel(x,ind)$Expression Integer,i),j),k),l),X) for l in k..d] for k in j..d] for j in i..d] for i in 1..d]); sort kernels eq18e

$$

\left[

{x _{{n -2}}}, : {x _{{n -1}}}, : {x _{n}}, : {x _{{n+1}}}, : {x _{{n+2}}}

\right]

$$

# %

$$

5

$$

numer eq18e.1 factor denom eq18e.1

$$

{{\left( -{{{x _{n}}} ^{2}}+{2 \ {x _{{n -1}}} \ {x _{n}}} -{{{x _{{n -1}}}} ^{2}}

\right)}

\ {x _{{n+2}}}}+{{\left( {{12} \ {{{x _{n}}} ^{2}}}+{{\left( -{{32} \ {x _{{n -1}}}}+{8 \ {x _{{n -2}}}}

\right)}

\ {x _{n}}}+{{20} \ {{{x _{{n -1}}}} ^{2}}} -{8 \ {x _{{n -2}}} \ {x _{{n -1}}}}

\right)}

\ {x _{{n+1}}}} -{{40} \ {{{x _{n}}} ^{3}}}+{{\left( {{172} \ {x _{{n -1}}}} -{{63} \ {x _{{n -2}}}}

\right)}

\ {{{x _{n}}} ^{2}}}+{{\left( -{{254} \ {{{x _{{n -1}}}} ^{2}}}+{{194} \ {x _{{n -2}}} \ {x _{{n -1}}}} -{{38} \ {{{x _{{n -2}}}} ^{2}}}

\right)}

\ {x _{n}}}+{{132} \ {{{x _{{n -1}}}} ^{3}}} -{{161} \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}}+{{68} \ {{{x _{{n -2}}}} ^{2}} \ {x _{{n -1}}}} -{{10} \ {{{x _{{n -2}}}} ^{3}}}

$$

$$

4 \ {{{\left( {x _{n}} -{x _{{n -1}}}

\right)}}

^{6}}

$$

test(eq18e.1=Req18e.1)

$$

true

$$