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Student Support Forum: '[Long] How to Thread expressions with FindMinimum that Partially Execute?' topicStudent Support Forum > General > Archives > "[Long] How to Thread expressions with FindMinimum that Partially Execute?"

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Author Comment/Response
Tom McKendree
01/29/00 06:16am

Help!!

I have put together an intricate function (costhprtotal) that at the highest level is a summation, and that embeds FindMinimum.

The problem I have is that I need to evaluate this while leaving one element as a variable, and every way I try, the threading on the summation fails. Instead of replacing ''j'' in each summation exression with a value (1, 2, 3, etc.), it just leaves a ''j'', which cannot be evaluated later.

I have successfully done this sort of thing many times, with ordinary arithmetic functions, so I suspect that the problem is weirdness in FindMinimum that leaks out into the evaluation. I have tried removing the ''Hold All'' attribute from FindMinimum, but that does not seem to help.

Any way to make this work would be very appreciated.


Thank you,

Tom McKendree
tmckendree@west.raytheon.com


There are two very long stretches of Mathematica code below. The first is the code I am using, that does not work. The second is a similar batch of code, that does work.







Notebook[{

Cell[CellGroupData[{
Cell[''\<\
Here Are the Equations, Including Example Data Sets, and They Do \
Not Work\
\>'', ''Subtitle''],

Cell[BoxData[
\(\[Copyright]2000, \ Tom\ McKendree, \ All\ Rights\ Reserved\)],
''Input''],

Cell[BoxData[
\(hprunitcost[gm_, ra_, rb_, v_, cinv_, cf_, v0_] :=
cinv\[Times]\(ra + rb\)\/v +
cf\/\(2\ \[Pi]\)\
\[Times]NIntegrate[
\(-1\) +
Re[\[ExponentialE]^
\((\(1\/v0\)
\((\@\(gm\/rb + v\^2 -
2\ \@\(gm\/rb\)\ v\
Sin[Arg[E\^\(\(-I\)\ \[Theta]\)\ ra - rb]]
\) + \@\(gm\/ra + v\^2 -
2\ \@\(gm\/ra\)\ v\
Sin[Arg[\(-ra\) + E\^\(I\ \[Theta]\)\ rb]]\))
\))\)], {\[Theta], 0, 2\ \[Pi]}]\)], ''Input''],

Cell[BoxData[
\(hprcost[gm_, ra_, rb_, cinv_, cf_, v0_, \[Lambda]_] :=
If[ra == rb, 0,
\[Lambda]\[Times]\(FindMinimum[
Abs[hprunitcost[gm, ra, rb, v, cinv, cf, v0]], {v, 1000, 1020}]
\)\[LeftDoubleBracket]1\[RightDoubleBracket]]\)], ''Input''],

Cell[BoxData[
\(hprescapevcost[gm_, ra_, rb_, cinv_, cf_, v0_, \[Lambda]_] :=
If[ra == rb, 0,
\[Lambda]\[Times]hprunitcost[gm, ra, rb, Sqrt[2\[Times]gm/ra], cinv,
cf, v0]]\)], ''Input''],

Cell[BoxData[
\(noinputs[inp_] := Length[inp]; nomarket[m_] := Length[m];
\[Lambda]in[inp_, i_] :=
inp\[LeftDoubleBracket]i, 3, 1, 2\[RightDoubleBracket];
\[Lambda]m[m_, i_] :=
m\[LeftDoubleBracket]i, 3, 2, 2\[RightDoubleBracket];
cf[m_, i_] := m\[LeftDoubleBracket]i, 3, 3, 4\[RightDoubleBracket];
rin[inp_, i_] := inp\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket];
rm[m_, i_] := m\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket];
rff[f_, i_] := f\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket];
c1[inp_, i_] := inp\[LeftDoubleBracket]i, 3, 1, 5\[RightDoubleBracket];
c2[inp_, i_] := inp\[LeftDoubleBracket]i, 3, 1, 6\[RightDoubleBracket];
c3[f_, i_] := f\[LeftDoubleBracket]i, 3, 2, 5\[RightDoubleBracket];
c4[f_, i_] := f\[LeftDoubleBracket]i, 3, 2, 6\[RightDoubleBracket];
c5[m_, i_] := m\[LeftDoubleBracket]i, 3, 2, 5\[RightDoubleBracket];
rlowmin[m_, i_] := m\[LeftDoubleBracket]i, 4, 1\[RightDoubleBracket];
clowmin[m_, i_] := m\[LeftDoubleBracket]i, 4, 2\[RightDoubleBracket];
rhighmin[m_, i_] := m\[LeftDoubleBracket]i, 4, 3\[RightDoubleBracket];
chighmin[m_, i_] := m\[LeftDoubleBracket]i, 4, 4\[RightDoubleBracket];
cr[m_, i_] := m\[LeftDoubleBracket]i, 4, 5\[RightDoubleBracket];
rhpr2escape[m_, i_] := m\[LeftDoubleBracket]i, 5, 3\[RightDoubleBracket];
\)], ''Input''],

Cell[CellGroupData[{

Cell[''Data Files, Must Be Entered to Run'', ''Subsubsection''],

Cell[BoxData[
\(costinput = 1; costproduct = 10; costfuel = 1;
interestrate = \((1/10)\)/\((60*60*24*365)\); gm = 1.32712438\ *\ 10\^20;
v0 = 500000; \)], ''Input''],

Cell[BoxData[
\(\(f004 =
\t{{Factory, \ {rf, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, \(-18\), 0, 0, costinput*interestrate, 0}, {product, 9,
0, 0, costproduct*interestrate, costproduct*interestrate}, \n
\t\t\t\t\ {fuel, 0, 0, costfuel, 0, 0}}, {rf, 0, rf, 0, 0}, {
rtbd1, rtbd2, rescape2hprin}}}; \)\)], ''Input''],

Cell[BoxData[
\(\(m004 = {{
Venus, {1.08\[Times]10\^11, 0, 1, \ \ 0, 0, 0, 0, 0,
1}, {{input, 0, 0, 0, 0, 0}, {product, \(-3\), 0, 0,
costproduct*interestrate, costproduct*interestrate}, {fuel, 0,
0, costfuel, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin,
cr}, {rtbd1, rtbd2, 1.37293\[Times]10\^11}}, {
Earth, {1.49\[Times]10\^11, 0, 1, \ \ 0, 0, 0, 0, 0,
1}, {{input, 0, 0, 0, 0, 0}, {product, \(-4\), 0, 0,
costproduct*interestrate, costproduct*interestrate}, {fuel, 0,
0, costfuel, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin,
cr}, {rtbd1, rtbd2, 1.27545\[Times]10\^11}},
\t{Mars, {2.28\[Times]10\^11, 0, 1, \ \ 0, 0, 0, 0, 0,
1}, {{input, 0, 0, 0, 0, 0}, {product, \(-2\), 0, 0,
costproduct*interestrate, costproduct*interestrate}, {fuel, 0,
0, costfuel, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin,
cr}, {rtbd1, rtbd2, 1.09189\[Times]10\^11}}}; \)\)], ''Input''],

Cell[BoxData[
\(\(inp004 = {{Ceres, \ {4.1377*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 6, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin, cr}, {rtbd1,
rtbd2, 3.9711\[Times]10\^11}}, {Mathilde,
\ {3.9558*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 5, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}},
\ {rlowmin, clowmin, rhighmin, chighmin, cr}, {rtbd1, rtbd2,
4.02811\[Times]10\^11}}, {Eros,
\ {2.1807*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 4, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin, cr}, {rtbd1,
rtbd2, 4.30022\[Times]10\^11}}, {Ida,
\ {4.2804*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 3, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}},
\ {rlowmin, clowmin, rhighmin, chighmin, cr}, {rtbd1, rtbd2,
3.9251\[Times]10\^11}}}; \)\)], ''Input'']
}, Open ]],

Cell[CellGroupData[{

Cell[''Continue'', ''Subsubsection''],

Cell[BoxData[
\(costhprtotal[gm_, v0_, inp_, f_, m_] :=
\[Sum]\+\(j = 1\)\%\(Length[inp]\)\((
If[rhpr2escape[inp, j] < rff[f, 1],
hprcost[gm, rff[f, 1], rin[inp, j], c2[inp, j], cf[inp, j], v0,
\[Lambda]in[inp, j]],
hprescapevcost[gm, rff[f, 1], rin[inp, j], c2[inp, j],
cf[inp, j], v0, \[Lambda]in[inp, j]]])\)\ \ +
\[Sum]\+\(j = 1\)\%\(Length[m]\)\((
If[rhpr2escape[m, j] < rff[f, 1],
hprcost[gm, rff[f, 1], rm[m, j], c4[f, 1], cf[f, 1], v0,
\[Lambda]m[m, j]],
hprescapevcost[gm, rff[f, 1], rm[m, j], c4[f, 1], cf[f, 1], v0,
\[Lambda]m[m, j]]])\)\)], ''Input'']
}, Open ]],

Cell[CellGroupData[{

Cell[''Run Cases'', ''Subsubsection''],

Cell[TextData[
''When I then enter the expression below, it chokes. In particular, the \
\''j\''s are not replaced with their threaded values, so the various \
replacement rules (e.g., rin[inp,j], c2[inp,j], \[Lambda]m[m,j]) fail to \
execute, and since the values intended for each \''j\'' are lost, they will not \
execute later. My ideal would be to have every replacement rule that can, \
execute immediately (so that the resulting output expression is shorter), \
while ending up with an expression that, if the unspecified parameter(s) is \
given an explicit value, will evaluate to an explicit final value.''],
''Text''],

Cell[BoxData[
\(costhprtotal[gm, v0, inp004, f004, m004]\)], ''Input''],

Cell[BoxData[
\(\(\n\n\n\n\n\n\n\n\n\n\n\n\)\)], ''Input'']
}, Open ]]
}, Open ]],

Cell[CellGroupData[{

Cell[''\<\
Here Is a Similar Set of Equations, Including Example Data Sets, \
That Do Work\
\>'', ''Subtitle''],

Cell[BoxData[
\(\[Copyright]1999, \ Tom\ McKendree, \ All\ Rights\ Reserved\)],
''Input''],

Cell[CellGroupData[{

Cell[BoxData[
\(costinput = 1; costproduct = 2; costfuel = 1;
interestrate = \((1/10)\)/\((60*60*24*365)\); gm = 1.32712438\ *\ 10\^20;
v0 = 900\)], ''Input''],

Cell[BoxData[
\(900\)], ''Output'']
}, Closed]],

Cell[BoxData[
\(\(f001 =
\t{{Factory, \ {rf, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, \(-18\), 0, 0, costinput*interestrate, 0}, {product, 9,
0, 0, costproduct*interestrate, costproduct*interestrate}, \n
\t\t\t\t\ {fuel, 0, 0, costfuel, 0, 0}}, {rf, 0, rf, 0, 0}}};
\)\)], ''Input''],

Cell[BoxData[
\(\(m001 = {{
Venus, {1.08\[Times]10\^11, 0, 1, \ \ 0, 0, 0, 0, 0,
1}, {{input, 0, 0, 0, 0, 0}, {product, \(-3\), 0, 0,
costproduct*interestrate, costproduct*interestrate}, {fuel, 0,
0, costfuel, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin,
cr}}, {Earth, {1.49\[Times]10\^11, 0, 1, \ \ 0, 0, 0, 0, 0,
1}, {{input, 0, 0, 0, 0, 0}, {product, \(-4\), 0, 0,
costproduct*interestrate, costproduct*interestrate}, {fuel, 0,
0, costfuel, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin,
cr}}, \t{
Mars, {2.28\[Times]10\^11, 0, 1, \ \ 0, 0, 0, 0, 0,
1}, {{input, 0, 0, 0, 0, 0}, {product, \(-2\), 0, 0,
costproduct*interestrate, costproduct*interestrate}, {fuel, 0,
0, costfuel, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin,
cr}}}; \)\)], ''Input''],

Cell[BoxData[
\(\(inp001 = {{Ceres, \ {4.1377*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 6, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin, cr}}, {
Mathilde, \ {3.9558*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 5, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}}, \ {rlowmin, clowmin, rhighmin, chighmin, cr}}, {
Eros, \ {2.1807*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 4, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}}, {rlowmin, clowmin, rhighmin, chighmin, cr}}, {Ida,
\ {4.2804*^11, 0, 1, \ \ 0, 0, 0, 0, 0, 1},
\ {{input, 3, 0, 0, costinput*interestrate,
costinput*interestrate}, {product, 0, 0, 0, 0, 0}, {fuel, 0, 0,
1, 0, 0}}, \ {rlowmin, clowmin, rhighmin, chighmin, cr}}};
\)\)], ''Input''],

Cell[BoxData[
\(hohmannperiod[gm_, \ ra_, \ rb_]\ :=
If[ra\ == \ rb, 0, \
Pi\ \ Sqrt[\((\((ra\ + \ rb)\)^3)\)/\((8\ \ gm)\)]]\)], ''Input''],

Cell[BoxData[
\(hohmann\[CapitalDelta]v[gm_, \ ra_, \ rb_]\ := \
\@gm\ \((Abs[
\((1\/\@ra - \@\(\(2\[Times]rb\)\/\(ra\[Times]\((ra + rb)\)\)\))
\) - \((
1\/\@rb - \@\(\(2\[Times]ra\)\/\(rb\[Times]\((ra + rb)\)\)\))
\)])\)\)], ''Input''],

Cell[BoxData[
\(hohmannmassratio[gm_, \ v0_, \ ra_, \ rb_]\ := \
Exp[hohmann\[CapitalDelta]v[gm, \ ra, \ rb]/v0]\)], ''Input''],

Cell[BoxData[
\(hohmannfuel[gm_, \ v0_, \ ra_, \ rb_] := \
hohmannmassratio[gm, \ v0, \ ra, \ rb]\ - 1\)], ''Input''],

Cell[BoxData[
\(synodicperiod[gm_, \ ra_, \ rb_]\ \ := \
If[ra\ == \ rb, 0,
Abs[2\ \[Pi]/\((\((1/Sqrt[ra^3\ /gm])\) - \((1/Sqrt[rb^3\ /gm])\))\)]]
\)], ''Input''],

Cell[BoxData[
\(hohmannpipelineinventory[gm_, ra_, rb_, \[Lambda]i_] :=
hohmannperiod[gm, \ ra, \ rb]\ \ *\[Lambda]i\)], ''Input''],

Cell[BoxData[
\(cyclicinventoryoneside[gm_, ra_, rb_, \[Lambda]i_] :=
synodicperiod[gm, \ ra, \ rb]*\[Lambda]i/2\)], ''Input''],

Cell[BoxData[
\(noinputs[inp_] := Length[inp]; nomarket[m_] := Length[m];
\[Lambda]in[inp_, i_] :=
inp\[LeftDoubleBracket]i, 3, 1, 2\[RightDoubleBracket];
\[Lambda]m[m_, i_] :=
m\[LeftDoubleBracket]i, 3, 2, 2\[RightDoubleBracket];
cf[m_, i_] := m\[LeftDoubleBracket]i, 3, 3, 4\[RightDoubleBracket];
rin[inp_, i_] := inp\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket];
rm[m_, i_] := m\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket];
rff[f_, i_] := f\[LeftDoubleBracket]i, 2, 1\[RightDoubleBracket];
c1[inp_, i_] := inp\[LeftDoubleBracket]i, 3, 1, 5\[RightDoubleBracket];
c2[inp_, i_] := inp\[LeftDoubleBracket]i, 3, 1, 6\[RightDoubleBracket];
c3[f_, i_] := f\[LeftDoubleBracket]i, 3, 2, 5\[RightDoubleBracket];
c4[f_, i_] := f\[LeftDoubleBracket]i, 3, 2, 6\[RightDoubleBracket];
c5[m_, i_] := m\[LeftDoubleBracket]i, 3, 2, 5\[RightDoubleBracket]; \)],
''Input''],

Cell[BoxData[
\(costhohmanntotal[gm_, v0_, inp_, f_, m_] := \n\t
\[Sum]\+\(i = 1\)\%\(noinputs[inp]\)\((
\((\[Lambda]in[inp, i]*cf[inp, i]*
hohmannfuel[gm, \ v0, rin[inp, i], rff[f, 1]]\ + \
c2[inp, i]*
hohmannpipelineinventory[gm, rin[inp, i], rff[f, 1],
\[Lambda]in[inp, i]]\ +
\((2*c1[inp, i])\)*
cyclicinventoryoneside[gm, rin[inp, i], rff[f, 1],
\[Lambda]in[inp, i]])\))\) +
\[Sum]\+\(i = 1\)\%\(nomarket[m]\)\((
\((\(-\[Lambda]m[m, i]\)*cf[f, 1]*
hohmannfuel[gm, \ v0, rff[f, 1], rm[m, i]]\ \ + \
c4[f, 1]*
hohmannpipelineinventory[gm, rff[f, 1], rm[m, i],
\(-\[Lambda]m[m, i]\)]\ + \
\((c3[f, 1] + c5[m, i])\)*
cyclicinventoryoneside[gm, rff[f, 1], rm[m, i],
\(-\[Lambda]m[m, i]\)])\))\)\)], ''Input''],

Cell[''\<\
When I now enter the expression below, I get a reduced expression, \
with only the variable \''rf\'' remaining to be defined to get a numberical \
value. The resulting expression can be plotted, FindMinimum can be run on \
it, etc. I need the same sort of flexibility out of the evaluated expression \
in the subtitled code above.\
\>'', ''Text''],

Cell[CellGroupData[{

Cell[BoxData[
\(costhohmanntotal[gm, v0, inp001, f001, m001]\)], ''Input''],

Cell[BoxData[
\(3\ \((
\(-1\) +
E^\((1.28000982828016883`*^7\
Abs[\(-3.04290309725092278`*^-6\) + 1\/\@rf -
464758.001544890131`\
\@\(1\/\(rf\
\((1.08000000000000007`*^11 + rf)\)\)\) +
4.30331482911935214`*^-6\
\@\(rf\/\(1.08000000000000007`*^11 + rf\)\)])\))\) +
4\ \((\(-1\) +
E^\((1.28000982828016883`*^7\
Abs[\(-2.59063880075419916`*^-6\) + 1\/\@rf -
545893.762558247175`\
\@\(1\/\(rf\
\((1.49000000000000003`*^11 + rf)\)\)\) +
3.66371652723655838`*^-6\
\@\(rf\/\(1.49000000000000003`*^11 + rf\)\)])\))\) +
4\ \((\(-1\) +
E^\((1.28000982828016883`*^7\
Abs[2.14142090587653921`*^-6 - 1\/\@rf +
660408.96420324282`\
\@\(1\/\(rf\
\((2.18069999999999986`*^11 + rf)\)\)\) -
3.02842648783988099`*^-6\
\@\(rf\/\(2.18069999999999986`*^11 + rf\)\)])\))\) +
2\ \((\(-1\) +
E^\((1.28000982828016883`*^7\
Abs[\(-2.09426954145847776`*^-6\) + 1\/\@rf -
675277.720645365153`\
\@\(1\/\(rf\
\((2.27999999999999936`*^11 + rf)\)\)\) +
2.96174438879546197`*^-6\
\@\(rf\/\(2.27999999999999936`*^11 + rf\)\)])\))\) +
5\ \((\(-1\) +
E^\((1.28000982828016883`*^7\
Abs[1.58994769317124955`*^-6 - 1\/\@rf +
889471.753345770999`\
\@\(1\/\(rf\
\((3.95579999999999998`*^11 + rf)\)\)\) -
2.24852559114659689`*^-6\
\@\(rf\/\(3.95579999999999998`*^11 + rf\)\)])\))\) +
6\ \((
\(-1\) +
E^\((1.28000982828016883`*^7\
Abs[1.55460657555417069`*^-6 - 1\/\@rf +
909692.255655724402`\
\@\(1\/\(rf\
\((4.13770000000000059`*^11 + rf)\)\)\) -
2.19854570330310217`*^-6\
\@\(rf\/\(4.13770000000000059`*^11 + rf\)\)])\))\) +
3\ \((\(-1\) +
E^\((1.28000982828016883`*^7\
Abs[1.5284731789777382`*^-6 - 1\/\@rf +
925245.913257659324`\
\@\(1\/\(rf\ \((4.2804000000000002`*^11 + rf)\)\)\) -
2.16158749943383599`*^-6\
\@\(rf\/\(4.2804000000000002`*^11 + rf\)\)])\))\) +
If[2.18069999999999986`*^11 == rf, 0,
Abs[\(2\ \[Pi]\)\/\(1
\/\@\(2.18069999999999986`*^11\^3\/1.3271243800000001`*^20
\) - 1\/\@\(rf\^3\/1.3271243800000001`*^20\)\)]]
\/78840000 +
If[2.18069999999999986`*^11 == rf, 0,
\[Pi]\ \@\(\((2.18069999999999986`*^11 + rf)\)\^3\/\(8\
1.3271243800000001`*^20\)\)]\/78840000 +
If[3.95579999999999998`*^11 == rf, 0,
Abs[\(2\ \[Pi]\)\/\(1
\/\@\(3.95579999999999998`*^11\^3\/1.3271243800000001`*^20
\) - 1\/\@\(rf\^3\/1.3271243800000001`*^20\)\)]]
\/63072000 +
If[3.95579999999999998`*^11 == rf, 0,
\[Pi]\ \@\(\((3.95579999999999998`*^11 + rf)\)\^3\/\(8\
1.3271243800000001`*^20\)\)]\/63072000 +
If[4.13770000000000059`*^11 == rf, 0,
Abs[\(2\ \[Pi]\)\/\(1
\/\@\(4.13770000000000059`*^11\^3\/1.3271243800000001`*^20
\) - 1\/\@\(rf\^3\/1.3271243800000001`*^20\)\)]]
\/52560000 +
If[4.13770000000000059`*^11 == rf, 0,
\[Pi]\ \@\(\((4.13770000000000059`*^11 + rf)\)\^3\/\(8\
1.3271243800000001`*^20\)\)]\/52560000 +
If[4.2804000000000002`*^11 == rf, 0,
Abs[\(2\ \[Pi]\)\/\(1
\/\@\(4.2804000000000002`*^11\^3\/1.3271243800000001`*^20
\) - 1\/\@\(rf\^3\/1.3271243800000001`*^20\)\)]]
\/105120000 +
If[4.2804000000000002`*^11 == rf, 0,
\[Pi]\ \@\(\((4.2804000000000002`*^11 + rf)\)\^3\/\(8\
1.3271243800000001`*^20\)\)]\/105120000 +
If[rf == 1.08000000000000007`*^11, 0,
Abs[\(2\ \[Pi]\)\/\(1\/\@\(rf\^3\/1.3271243800000001`*^20\) -
1\/\@\(1.08000000000000007`*^11\^3\/1.3271243800000001`*^20
\)\)]]\/52560000 +
If[rf == 1.08000000000000007`*^11, 0,
\[Pi]\ \@\(\((rf + 1.08000000000000007`*^11)\)\^3\/\(8\
1.3271243800000001`*^20\)\)]\/52560000 +
If[rf == 1.49000000000000003`*^11, 0,
Abs[\(2\ \[Pi]\)\/\(1\/\@\(rf\^3\/1.3271243800000001`*^20\) -
1\/\@\(1.49000000000000003`*^11\^3\/1.3271243800000001`*^20
\)\)]]\/39420000 +
If[rf == 1.49000000000000003`*^11, 0,
\[Pi]\ \@\(\((rf + 1.49000000000000003`*^11)\)\^3\/\(8\
1.3271243800000001`*^20\)\)]\/39420000 +
If[rf == 2.27999999999999936`*^11, 0,
Abs[\(2\ \[Pi]\)\/\(1\/\@\(rf\^3\/1.3271243800000001`*^20\) -
1\/\@\(2.27999999999999936`*^11\^3\/1.3271243800000001`*^20
\)\)]]\/78840000 +
If[rf == 2.27999999999999936`*^11, 0,
\[Pi]\ \@\(\((rf + 2.27999999999999936`*^11)\)\^3\/\(8\
1.3271243800000001`*^20\)\)]\/78840000\)], ''Output'']
}, Closed]]
}, Open ]]
},
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MacintoshSystemPageSetup->''\<\
00@0001804P000000]/2@?oWonh2o`9B7`<5:0?l0040004/0B`000002nL9H04/
02d5X5k/02H40@4101P00BL?00400@0000000000000000010000000000000000
00000000000000000000004000910000\>''
]


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