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Edit detail for SandBoxMIWHall revision 11 of 14

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Editor: Bill Page Time: 2015/07/31 17:21:16 GMT+0
Note:

changed:
-From BillPage Fri Jul 31 17:16:56 +0000 2015
-From: Bill Page
-Date: Fri, 31 Jul 2015 17:16:56 +0000
-Subject: 
-Message-ID: <20150731171656+0000@axiom-wiki.newsynthesis.org>
-
-1-D

Poirier

  1-D

changed:
-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))]]
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))]]

changed:
-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:=[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];

Many Interacting Worlds - Hall, et al.

fricas

P:=operator('P);

Type: BasicOperator?

fricas

Q:=operator('Q);

Type: BasicOperator?

fricas

g:=operator('g);

Type: BasicOperator?

fricas

U:=operator('U);

Type: BasicOperator?

fricas

r:=operator('r);

Type: BasicOperator?

fricas

ℏ:=h;

Type: Variable(h)

Hall equations 15,19,20

fricas

eq15:= r(q) = -D(U(q),q)

$\label{eq1}{r \left({q}\right)}= -{{U_{\ }^{,}}\left({q}\right)}$

(1)

Type: Equation(Expression(Integer))

fricas

eq19:= U(q) = 1/(2*m)*g(q)^2

$\label{eq2}{U \left({q}\right)}={{{g \left({q}\right)}^{2}}\over{2 \ m}}$

(2)

Type: Equation(Expression(Integer))

fricas

eq20:= g(q) = ℏ/2 * 1/P(q)*D(P(q),q)

$\label{eq3}{g \left({q}\right)}={{h \ {{P_{\ }^{,}}\left({q}\right)}}\over{2 \ {P \left({q}\right)}}}$

(3)

Type: Equation(Expression(Integer))

fricas

eval(eq19,eq20)

$\label{eq4}{U \left({q}\right)}={{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{2}}}\over{8 \ m \ {{P \left({q}\right)}^{2}}}}$

(4)

Type: Equation(Expression(Integer))

fricas

RHall:=eval(eq15,D(lhs %,q)=D(rhs %,q))

$\label{eq5}{r \left({q}\right)}={{-{{{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{,}}\left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}+{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{3}}}}\over{4 \ m \ {{P \left({q}\right)}^{3}}}}$

(5)

Type: Equation(Expression(Integer))

Hall equations 6, 7, A3

fricas

eq6:= r(q) = -D(Q(q),q)

$\label{eq6}{r \left({q}\right)}= -{{Q_{\ }^{,}}\left({q}\right)}$

(6)

Type: Equation(Expression(Integer))

fricas

eq7:= Q(q) = 1/sqrt(P(q))*-ℏ^2/2/m*D(sqrt(P(q)),[q,q])

$\label{eq7}{Q \left({q}\right)}={{{2 \ {{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}-{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{2}}}}\over{8 \ m \ {{P \left({q}\right)}^{2}}}}$

(7)

Type: Equation(Expression(Integer))

fricas

Rforce:=eval(eq6,D(lhs eq7,q)=D(rhs eq7,q))

$\label{eq8}\begin{array}{@{}l} \displaystyle {r \left({q}\right)}={{\left( \begin{array}{@{}l} \displaystyle -{{{h}^{2}}\ {{P \left({q}\right)}^{2}}\ {{P_{\ }^{, , ,}}\left({q}\right)}}+ \ \ \displaystyle {2 \ {{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{,}}\left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}-{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{3}}}$

(8)

Type: Equation(Expression(Integer))

Gaussian

fricas

PdfNorm(x)==1/2*sqrt(2)*exp(-1/2*x^2)/sqrt(%pi)

Type: Void

fricas

eval(RHall,[P(q)=PdfNorm(q), _
            D(P(q),q)=D(PdfNorm(q),q), _
            D(P(q),[q,q])=D(PdfNorm(q),[q,q])])

fricas

Compiling function PdfNorm with type Variable(q) -> Expression(
      Integer)

$\label{eq9}{r \left({q}\right)}= -{{{{h}^{2}}\ q}\over{4 \ m}}$

(9)

Type: Equation(Expression(Integer))

fricas

eval(Rforce, [P(q)=PdfNorm(q), _
              D(P(q),q)=D(PdfNorm(q),q), _
              D(P(q),[q,q])=D(PdfNorm(q),[q,q]), _
              D(P(q),[q,q,q])=D(PdfNorm(q),[q,q,q])])

$\label{eq10}{r \left({q}\right)}= -{{{{h}^{2}}\ q}\over{4 \ m}}$

(10)

Type: Equation(Expression(Integer))

Cauchy

fricas

PdfCauchy(x)==1/%pi/(1+x^2)

Type: Void

fricas

eval(RHall,[P(q)=PdfCauchy(q), _
            D(P(q),q)=D(PdfCauchy(q),q), _
            D(P(q),[q,q])=D(PdfCauchy(q),[q,q])])

fricas

Compiling function PdfCauchy with type Variable(q) -> Expression(
      Integer)

$\label{eq11}{r \left({q}\right)}={{{{{h}^{2}}\ {{q}^{3}}}-{{{h}^{2}}\ q}}\over{{m \ {{q}^{6}}}+{3 \ m \ {{q}^{4}}}+{3 \ m \ {{q}^{2}}}+ m}}$

(11)

Type: Equation(Expression(Integer))

fricas

eval(Rforce,[P(q)=PdfCauchy(q), _
             D(P(q),q)=D(PdfCauchy(q),q), _
             D(P(q),[q,q])=D(PdfCauchy(q),[q,q]), _
             D(P(q),[q,q,q])=D(PdfCauchy(q),[q,q,q])])

$\label{eq12}{r \left({q}\right)}={{{2 \ {{h}^{2}}\ {{q}^{3}}}-{4 \ {{h}^{2}}\ q}}\over{{m \ {{q}^{6}}}+{3 \ m \ {{q}^{4}}}+{3 \ m \ {{q}^{2}}}+ m}}$

(12)

Type: Equation(Expression(Integer))

... --Bill Page, Tue, 21 Jul 2015 20:17:15 +0000 reply

fricas

J(i,j)==matrix([[x[i,j]-x[i-1,j],y[i,j]-y[i-1,j]],[x[i,j]-x[j,j-1],y[i,j]-y[j,j-1]]])

Type: Void

fricas

J(i,j)

fricas

Compiling function J with type (Variable(i),Variable(j)) -> Matrix(
      Polynomial(Integer))

$\label{eq13}\left[ \begin{array}{cc} {{x_{i , \: j}}-{x_{{i - 1}, \: j}}}&{{y_{i , \: j}}-{y_{{i - 1}, \: j}}} \ {-{x_{j , \:{j - 1}}}+{x_{i , \: j}}}&{-{y_{j , \:{j - 1}}}+{y_{i , \: j}}}$

(13)

Type: Matrix(Polynomial(Integer))

fricas

determinant(J(i,j))

$\label{eq14}\begin{array}{@{}l} \displaystyle {{\left(-{x_{i , \: j}}+{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left({x_{j , \:{j - 1}}}-{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+ \ \ \displaystyle {{\left(-{x_{j , \:{j - 1}}}+{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}$

(14)

Type: Polynomial(Integer)

fricas

K(i,j)==inverse(J(i,j))

Type: Void

fricas

K(i,j)

fricas

Compiling function K with type (Variable(i),Variable(j)) -> Union(
      Matrix(Fraction(Polynomial(Integer))),"failed")

$\label{eq15}\left[ \begin{array}{cc} {{{y_{j , \:{j - 1}}}-{y_{i , \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}}&{{{y_{i , \: j}}-{y_{{i - 1}, \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}} \ {{-{x_{j , \:{j - 1}}}+{x_{i , \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}}&{{-{x_{i , \: j}}+{x_{{i - 1}, \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}}$

(15)

Type: Union(Matrix(Fraction(Polynomial(Integer))),...)

Poirier

1-D

fricas

x := operator('x)

   Compiled code for J has been cleared.
   Compiled code for K has been cleared.

$\label{eq16}x$

(16)

Type: BasicOperator?

fricas

C:List Symbol := [subscript('x,[0])]

$\label{eq17}\left[{x_{0}}\right]$

(17)

Type: List(Symbol)

fricas

d := #C

$\label{eq18}1$

(18)

Type: PositiveInteger?

fricas

J := matrix [[D(x(C(1),t),C(1))]]

   Function definition for J is being overwritten.

$\label{eq19}\left[ \begin{array}{c} {{x_{, 1}}\left({{x_{0}}, \: t}\right)}$

(19)

Type: Matrix(Expression(Integer))

fricas

K := inverse J

   Function definition for K is being overwritten.

$\label{eq20}\left[ \begin{array}{c} {1 \over{{x_{, 1}}\left({{x_{0}}, \: t}\right)}}$

(20)

Type: Union(Matrix(Expression(Integer)),...)

fricas

[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]

$\label{eq21}\left[{-{{{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{2}}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{8 \ {{x_{, 1}}\left({{x_{0}}, \: t}\right)}\ {{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}\ {{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{10}\ {{{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}^{3}}}}\over{4 \ {{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{6}}}\right]$

(21)

Type: List(Expression(Integer))

2-D

fricas

x := operator('x)

$\label{eq22}x$

(22)

Type: BasicOperator?

fricas

y := operator('y)

$\label{eq23}y$

(23)

Type: BasicOperator?

fricas

C:List Symbol := [subscript('x,[0]),subscript('y,[0])]

$\label{eq24}\left[{x_{0}}, \:{y_{0}}\right]$

(24)

Type: List(Symbol)

fricas

d := #C

$\label{eq25}2$

(25)

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))]]

$\label{eq26}\left[ \begin{array}{cc} {{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)} \ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}$

(26)

Type: Matrix(Expression(Integer))

fricas

K := inverse J

$\label{eq27}\left[ \begin{array}{cc} -{{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}&{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}} \ {{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}& -{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\over{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}$

(27)

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

$\label{eq28}2$

(28)

Type: PositiveInteger?

fricas

kernels Req18.1

$\label{eq29}\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]$

(29)

Type: List(Kernel(Expression(Integer)))