Skip to content
Permalink
master
Switch branches/tags
Go to file
 
 
Cannot retrieve contributors at this time
(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 12.0' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 158, 7]
NotebookDataLength[ 385370, 8102]
NotebookOptionsPosition[ 359739, 7694]
NotebookOutlinePosition[ 360129, 7711]
CellTagsIndexPosition[ 360086, 7708]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell[CellGroupData[{
Cell["\<\
Random variate generation/Importance sampling using integral transforms\
\>", "Title",
CellChangeTimes->{{3.795032017231084*^9,
3.7950320337266483`*^9}},ExpressionUUID->"376fb3b5-8f42-4328-a329-\
a8c5d740ac91"],
Cell[TextData[{
StyleBox["Eugene d\[CloseCurlyQuote]Eon\nNVIDIA",
FontSlant->"Italic"],
"\n",
StyleBox["Wellington, New Zealand\nApril 5, 2020 (updated May 10, 2020)",
FontSlant->"Italic"]
}], "Text",
CellChangeTimes->{{3.795032037499178*^9, 3.79503204585292*^9}, {
3.795032366450389*^9, 3.79503239318498*^9}, {3.79806199103398*^9,
3.798061992210981*^9}},ExpressionUUID->"58bbfc09-933a-48bd-a1b1-\
c38a7859874a"],
Cell[CellGroupData[{
Cell["Abstract", "Subsection",
CellChangeTimes->{{3.795032054434959*^9,
3.795032056094051*^9}},ExpressionUUID->"93d82261-c3cf-4c8c-b0b4-\
a3501bce6459"],
Cell["\<\
We discuss various analytic and exact methods for deriving \
importance-sampling procedures for 1D distributions f(r) on the half line r \
\[Element] [0,\[Infinity]). Specifically:\
\>", "Text",
CellChangeTimes->{{3.795032060224909*^9, 3.795032168863987*^9}, {
3.795032234103797*^9, 3.7950322599290752`*^9}, {3.79503319548806*^9,
3.795033211407243*^9}, {3.795033287431382*^9, 3.795033287716819*^9}, {
3.79503702889462*^9, 3.795037029780951*^9}, {3.795069655718581*^9,
3.795069661924973*^9}},ExpressionUUID->"89ae6f3b-384b-4f3a-b1d1-\
f7f90a7f14d2"],
Cell[CellGroupData[{
Cell["Finding convolutions using the forward Laplace transform", "Item",
CellChangeTimes->{{3.795032266903244*^9, 3.795032291536994*^9}, {
3.795032422311908*^9,
3.79503242231203*^9}},ExpressionUUID->"bb729839-b5fa-4066-8c5d-\
7894a8b1f9f6"],
Cell["\<\
Sampling as a superposition of exponentials using the inverse Laplace \
transform\
\>", "Item",
CellChangeTimes->{{3.795032266903244*^9, 3.795032291536994*^9}, {
3.7950324232876387`*^9,
3.795032468908576*^9}},ExpressionUUID->"33ede29d-0b28-4468-9708-\
dd6ffa54600c"],
Cell["The Gaussian transform", "Item",
CellChangeTimes->{{3.795032266903244*^9, 3.795032291536994*^9}, {
3.7950324232876387`*^9,
3.7950324721154222`*^9}},ExpressionUUID->"34a93a0c-d624-452f-a1c1-\
d136d9b03990"],
Cell["Generalized Box-Muller projections", "Item",
CellChangeTimes->{{3.795032266903244*^9, 3.795032291536994*^9}, {
3.7950324232876387`*^9,
3.795032449761136*^9}},ExpressionUUID->"401e0e0a-94ab-46c8-91da-\
f330c82c3a15"],
Cell["Sampling using a track-length estimator", "Item",
CellChangeTimes->{{3.795032266903244*^9, 3.795032291536994*^9}, {
3.7950324232876387`*^9,
3.795032461369626*^9}},ExpressionUUID->"2e6d1597-eb36-442b-b20c-\
30312de0ae17"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["1. Introduction", "Section",
CellChangeTimes->{{3.795032523636134*^9, 3.795032524762113*^9}, {
3.795043932720166*^9,
3.795043933001771*^9}},ExpressionUUID->"a731bad4-b6d3-4a2b-8d90-\
3204a8841ef6"],
Cell[TextData[{
"Simulating the linear transport of light or particles often involves \
drawing random variates from some given distribution on the half line, such \
as free-path length sampling in homogeneous classical media where we draw \
samples from the collision-rate density ",
Cell[BoxData[
FormBox[
RowBox[{
SubscriptBox["\[Sigma]", "t"],
SuperscriptBox["e",
RowBox[{
RowBox[{"-",
SubscriptBox["\[Sigma]", "t"]}], " ", "r"}]]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"2f901349-ac93-40c8-864e-2e035c88aba6"],
" using the CDF inversion method, leading to"
}], "Text",
CellChangeTimes->{{3.7950325277954903`*^9, 3.795032599540038*^9}, {
3.795032683438308*^9, 3.7950327020955267`*^9}, {3.7950370788064003`*^9,
3.795037079783433*^9}},ExpressionUUID->"60bd9654-2bd8-4267-9f36-\
52cca3f7f7fd"],
Cell[BoxData[
RowBox[{"r", "=",
RowBox[{
RowBox[{"-",
RowBox[{"Log", "[",
RowBox[{"1", "-",
RowBox[{"RandomReal", "[", "]"}]}], "]"}]}], "/",
RowBox[{
SubscriptBox["\[Sigma]", "t"], "."}]}]}]], "Input",
CellChangeTimes->{{3.795032706361977*^9, 3.7950327241869297`*^9}, {
3.7950327552918787`*^9, 3.795032755438345*^9},
3.795037096563971*^9},ExpressionUUID->"d82f0def-f76a-4f87-b868-\
0ddf2d82b14b"],
Cell["\<\
There are times where we wish to sample other distributions f(r) on the half \
line where the CDF inversion method is not possible analytically. For \
example, in non-classical transport where free-path lengths between collision \
are not exponentially-distributed, we need sampling procedures for a large \
class of distributions that give the chord-lengths in a given class of random \
microstructure.\
\>", "Text",
CellChangeTimes->{{3.795032794890018*^9, 3.7950328942233973`*^9}, {
3.795032947663761*^9, 3.79503294926781*^9}, {3.795033277462796*^9,
3.795033278962041*^9}, {3.795037116314426*^9,
3.7950371565016003`*^9}},ExpressionUUID->"51649930-e67e-45a4-ab2f-\
0256a1f097c7"],
Cell[TextData[{
"In this paper we discuss methods for sampling from a 1D distribution f(r) \
that is normalized on the half line ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"],
RowBox[{
RowBox[{"f", "(", "r", ")"}],
RowBox[{"\[DifferentialD]", "r"}]}]}], " ", "=", " ", "1"}],
TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"0680abaa-17b3-48e0-b20f-e0af4c97eabd"],
". In most cases, more than 1 random number will be required to sample f(r)."
}], "Text",
CellChangeTimes->{{3.795033224669462*^9, 3.795033275568008*^9}, {
3.79503344951571*^9, 3.795033461469551*^9}, {3.795069798823147*^9,
3.795069815107641*^9}},ExpressionUUID->"b3cc3a10-84a9-4782-ab5d-\
1616374421cd"]
}, Open ]],
Cell[CellGroupData[{
Cell["2. Deconvolution", "Section",
CellChangeTimes->{{3.795037479472991*^9, 3.795037481001073*^9}, {
3.795043935153154*^9,
3.795043935404752*^9}},ExpressionUUID->"ac5b0c4d-9830-4f9b-bedf-\
43fd2270949a"],
Cell[TextData[{
"If f(r) can be expressed as a convolution of more than 1 distribution, each \
of which has known sampling procedures, then f(r) can be sampled as a whole \
by sampling each of the distributions in its convolution and summing the \
values ",
Cell[BoxData[
FormBox[
RowBox[{
SubscriptBox["r", "i"], "."}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"2bee8094-bd7c-44f8-9132-104ca2895564"],
" The Laplace transform can be used to factor f(r) into its deconvolved \
components using the convolution property of Laplace transforms: the Laplace \
transform of a convolution is the product of the two Laplace transforms, ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
RowBox[{
SubscriptBox["\[ScriptCapitalL]", "r"], "[",
RowBox[{
RowBox[{
SubscriptBox["f", "1"], "(", "r", ")"}], " ", "\[Star]", " ",
RowBox[{
SubscriptBox["f", "2"], "(", "r", ")"}]}], " ", "]"}],
RowBox[{"(", "s", ")"}]}], "=", " "}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"77660dc1-a8c0-49ca-beeb-645f43a03a3e"],
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SubscriptBox["\[ScriptCapitalL]", "r"], "[",
RowBox[{
SubscriptBox["f", "1"], "(", "r", ")"}], "]"}],
RowBox[{"(", "s", ")"}], " ",
RowBox[{
SubscriptBox["\[ScriptCapitalL]", "r"], "[",
RowBox[{
SubscriptBox["f", "2"], "(", "r", ")"}], "]"}],
RowBox[{"(", "s", ")"}]}], TraditionalForm]],ExpressionUUID->
"72c5d288-3298-478e-a964-4c81effbe1c8"],
"."
}], "Text",
CellChangeTimes->{{3.7950374897871237`*^9, 3.795037564551323*^9}, {
3.795072723014064*^9, 3.795072723694457*^9}, {3.795072786941876*^9,
3.795072816276946*^9}, {3.795072866764824*^9,
3.795072933323966*^9}},ExpressionUUID->"af060f7c-15ad-4cbc-a79e-\
94b1021f451f"],
Cell[CellGroupData[{
Cell["Example 2.1:", "Subsection",
CellChangeTimes->{{3.7950375686958647`*^9, 3.795037569881187*^9}, {
3.795043937689226*^9,
3.795043937854801*^9}},ExpressionUUID->"257b414b-f449-434e-a65f-\
3ee4ce1eea34"],
Cell[TextData[{
"Consider the Erlang-2 / Gamma-2 distribution, f(r) = ",
Cell[BoxData[
FormBox[
RowBox[{
SuperscriptBox["e",
RowBox[{"-", "r"}]],
RowBox[{"r", "."}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"e5606e6a-c19e-42c3-ae33-fb5765698209"]
}], "Text",
CellChangeTimes->{{3.795037572019298*^9,
3.7950376250042963`*^9}},ExpressionUUID->"839b2de3-e1e3-4fbf-a7fd-\
993ab7247c2a"],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{
RowBox[{"Clear", "[", "f", "]"}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"f", "[", "r_", "]"}], ":=",
RowBox[{
RowBox[{"Exp", "[",
RowBox[{"-", "r"}], "]"}], "r"}]}], ";"}], "\[IndentingNewLine]",
RowBox[{" ",
RowBox[{"Integrate", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",",
RowBox[{"{",
RowBox[{"r", ",", "0", ",", "Infinity"}], "}"}]}], "]"}]}]}], "Input",
CellChangeTimes->{{3.795037582582942*^9, 3.795037610496169*^9}},
CellLabel->
"In[1651]:=",ExpressionUUID->"d69642a7-67e4-4215-8222-e7e2c8c14786"],
Cell[BoxData["1"], "Output",
CellChangeTimes->{{3.795037606972011*^9, 3.795037630336089*^9}},
CellLabel->
"Out[1653]=",ExpressionUUID->"84f72d9e-fc81-41b6-a527-0f24eba71af8"]
}, Open ]],
Cell["We can\[CloseCurlyQuote]t analytically invert the CDF of f(r):", "Text",
CellChangeTimes->{{3.795037646399641*^9,
3.7950376538535223`*^9}},ExpressionUUID->"9313aa89-5e54-494f-aa8d-\
10a493da7600"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Integrate", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",",
RowBox[{"{",
RowBox[{"r", ",", "0", ",", "k"}], "}"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"k", ">", "0"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.7950376352525463`*^9, 3.79503764202941*^9}},
CellLabel->
"In[1654]:=",ExpressionUUID->"a2dcd86c-2718-40c8-83f9-e1203808ea03"],
Cell[BoxData[
RowBox[{"1", "-",
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{"-", "k"}]], " ",
RowBox[{"(",
RowBox[{"1", "+", "k"}], ")"}]}]}]], "Output",
CellChangeTimes->{3.7950376427235193`*^9},
CellLabel->
"Out[1654]=",ExpressionUUID->"fdf7b779-e2fc-4275-96f1-4a8d6ca3290f"]
}, Open ]],
Cell["Using the Laplace transform of f(r):", "Text",
CellChangeTimes->{{3.795037660266518*^9,
3.795037666596673*^9}},ExpressionUUID->"8d995cd3-58bb-483a-a844-\
dac6aecc376d"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"LaplaceTransform", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",", "r", ",", "s"}], "]"}]], "Input",
CellChangeTimes->{{3.7950376680839157`*^9, 3.795037670568407*^9}},
CellLabel->
"In[1655]:=",ExpressionUUID->"386b4508-b3b5-4b2e-941f-634dcac8dcd5"],
Cell[BoxData[
FractionBox["1",
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "+", "s"}], ")"}], "2"]]], "Output",
CellChangeTimes->{3.7950376708872013`*^9},
CellLabel->
"Out[1655]=",ExpressionUUID->"100f82a2-a120-4097-b8c9-cc2d3db74d85"]
}, Open ]],
Cell[TextData[{
"we note that f is the convolution of a function with itself, whose laplace \
transform is ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
RowBox[{
FractionBox["1",
RowBox[{"1", "+", "s"}]], ".", " ", "Using"}], " ", "the", " ",
"inverse", " ", "Laplace", " ", "transform"}], ",", " ",
RowBox[{"we", " ", "find"}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"5b5c9a7f-64a4-4bb1-b7fc-7e9d2aa37041"]
}], "Text",
CellChangeTimes->{{3.795037674640583*^9, 3.795037711926779*^9}, {
3.7950698606509733`*^9,
3.7950698608162317`*^9}},ExpressionUUID->"b8d4837c-7835-46ef-a516-\
6d7f30820627"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"InverseLaplaceTransform", "[",
RowBox[{
FractionBox["1",
RowBox[{"1", "+", "s"}]], ",", "s", ",", "r"}], "]"}]], "Input",
CellChangeTimes->{{3.7950377132554073`*^9, 3.79503771714287*^9}},
CellLabel->
"In[1656]:=",ExpressionUUID->"dc37631a-ec8f-44c8-ad68-c34258c5e598"],
Cell[BoxData[
SuperscriptBox["\[ExponentialE]",
RowBox[{"-", "r"}]]], "Output",
CellChangeTimes->{3.795037717464224*^9},
CellLabel->
"Out[1656]=",ExpressionUUID->"e69252de-ccb0-4472-893f-a539e13bcf76"]
}, Open ]],
Cell["\<\
We find the exponential, which we know how to sample. Therefore, Erlang-2 \
can be sampled using the sum of 2 calls to sample() procedure for the \
exponential:\
\>", "Text",
CellChangeTimes->{{3.795037719844305*^9, 3.795037745425192*^9}, {
3.7950379024704447`*^9,
3.795037904032846*^9}},ExpressionUUID->"1b37e5f2-1f57-481c-9c50-\
8ca88b42a3bc"],
Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{"sampleExp", "[", "]"}], ":=",
RowBox[{"-",
RowBox[{"Log", "[",
RowBox[{"RandomReal", "[", "]"}], "]"}]}]}], ";"}]], "Input",
CellChangeTimes->{{3.795037926417675*^9, 3.79503793583424*^9}},
CellLabel->
"In[1672]:=",ExpressionUUID->"9593c1ff-6042-4858-9680-d5836f895ab9"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Show", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"Histogram", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"Table", "[",
RowBox[{
RowBox[{
RowBox[{"sampleExp", "[", "]"}], "+",
RowBox[{"sampleExp", "[", "]"}]}], ",",
RowBox[{"{",
RowBox[{"i", ",",
RowBox[{"Range", "[", "100000", "]"}]}], "}"}]}], "]"}], ",", "200",
",", "\"\<PDF\>\""}], "]"}], ",", "\[IndentingNewLine]",
RowBox[{"Plot", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",",
RowBox[{"{",
RowBox[{"r", ",", "0", ",", "7"}], "}"}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}]}],
"\[IndentingNewLine]", "]"}]], "Input",
CellChangeTimes->{{3.7950377472421637`*^9, 3.795037877992696*^9}, {
3.795037934010397*^9, 3.795037947190937*^9}},
CellLabel->
"In[1675]:=",ExpressionUUID->"167785f3-ad1e-42c2-84aa-0ed022709274"],
Cell[BoxData[
GraphicsBox[{{
{RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[
Opacity[0.]], {},
{RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[
Opacity[0.]], RectangleBox[{0., 0}, {0.05, 0.0258},
RoundingRadius->0], RectangleBox[{0.05, 0}, {0.1, 0.0698},
RoundingRadius->0], RectangleBox[{0.1, 0}, {0.15, 0.1088},
RoundingRadius->0], RectangleBox[{0.15, 0}, {0.2, 0.151},
RoundingRadius->0], RectangleBox[{0.2, 0}, {0.25, 0.1818},
RoundingRadius->0], RectangleBox[{0.25, 0}, {0.3, 0.1972},
RoundingRadius->0], RectangleBox[{0.3, 0}, {0.35, 0.236},
RoundingRadius->0], RectangleBox[{0.35, 0}, {0.4, 0.2514},
RoundingRadius->0], RectangleBox[{0.4, 0}, {0.45, 0.2814},
RoundingRadius->0], RectangleBox[{0.45, 0}, {0.5, 0.2798},
RoundingRadius->0], RectangleBox[{0.5, 0}, {0.55, 0.298},
RoundingRadius->0], RectangleBox[{0.55, 0}, {0.6, 0.321},
RoundingRadius->0], RectangleBox[{0.6, 0}, {0.65, 0.3422},
RoundingRadius->0], RectangleBox[{0.65, 0}, {0.7, 0.3408},
RoundingRadius->0], RectangleBox[{0.7, 0}, {0.75, 0.3508},
RoundingRadius->0], RectangleBox[{0.75, 0}, {0.8, 0.3524},
RoundingRadius->0], RectangleBox[{0.8, 0}, {0.85, 0.3706},
RoundingRadius->0], RectangleBox[{0.85, 0}, {0.9, 0.3566},
RoundingRadius->0], RectangleBox[{0.9, 0}, {0.95, 0.3592},
RoundingRadius->0], RectangleBox[{0.95, 0}, {1., 0.3774},
RoundingRadius->0], RectangleBox[{1., 0}, {1.05, 0.3674},
RoundingRadius->0], RectangleBox[{1.05, 0}, {1.1, 0.3636},
RoundingRadius->0], RectangleBox[{1.1, 0}, {1.15, 0.3614},
RoundingRadius->0], RectangleBox[{1.15, 0}, {1.2, 0.3704},
RoundingRadius->0], RectangleBox[{1.2, 0}, {1.25, 0.3654},
RoundingRadius->0], RectangleBox[{1.25, 0}, {1.3, 0.3544},
RoundingRadius->0], RectangleBox[{1.3, 0}, {1.35, 0.3506},
RoundingRadius->0], RectangleBox[{1.35, 0}, {1.4, 0.3438},
RoundingRadius->0], RectangleBox[{1.4, 0}, {1.45, 0.3554},
RoundingRadius->0], RectangleBox[{1.45, 0}, {1.5, 0.3364},
RoundingRadius->0], RectangleBox[{1.5, 0}, {1.55, 0.3352},
RoundingRadius->0], RectangleBox[{1.55, 0}, {1.6, 0.3308},
RoundingRadius->0], RectangleBox[{1.6, 0}, {1.65, 0.3172},
RoundingRadius->0], RectangleBox[{1.65, 0}, {1.7, 0.31},
RoundingRadius->0], RectangleBox[{1.7, 0}, {1.75, 0.3102},
RoundingRadius->0], RectangleBox[{1.75, 0}, {1.8, 0.2992},
RoundingRadius->0], RectangleBox[{1.8, 0}, {1.85, 0.3074},
RoundingRadius->0], RectangleBox[{1.85, 0}, {1.9, 0.2774},
RoundingRadius->0], RectangleBox[{1.9, 0}, {1.95, 0.2846},
RoundingRadius->0], RectangleBox[{1.95, 0}, {2., 0.2754},
RoundingRadius->0], RectangleBox[{2., 0}, {2.05, 0.278},
RoundingRadius->0], RectangleBox[{2.05, 0}, {2.1, 0.2694},
RoundingRadius->0], RectangleBox[{2.1, 0}, {2.15, 0.2458},
RoundingRadius->0], RectangleBox[{2.15, 0}, {2.2, 0.2396},
RoundingRadius->0], RectangleBox[{2.2, 0}, {2.25, 0.2392},
RoundingRadius->0], RectangleBox[{2.25, 0}, {2.3, 0.2256},
RoundingRadius->0], RectangleBox[{2.3, 0}, {2.35, 0.2244},
RoundingRadius->0], RectangleBox[{2.35, 0}, {2.4, 0.2288},
RoundingRadius->0], RectangleBox[{2.4, 0}, {2.45, 0.2244},
RoundingRadius->0], RectangleBox[{2.45, 0}, {2.5, 0.2012},
RoundingRadius->0], RectangleBox[{2.5, 0}, {2.55, 0.2026},
RoundingRadius->0], RectangleBox[{2.55, 0}, {2.6, 0.2016},
RoundingRadius->0], RectangleBox[{2.6, 0}, {2.65, 0.1842},
RoundingRadius->0], RectangleBox[{2.65, 0}, {2.7, 0.193},
RoundingRadius->0], RectangleBox[{2.7, 0}, {2.75, 0.1862},
RoundingRadius->0], RectangleBox[{2.75, 0}, {2.8, 0.173},
RoundingRadius->0], RectangleBox[{2.8, 0}, {2.85, 0.1606},
RoundingRadius->0], RectangleBox[{2.85, 0}, {2.9, 0.16},
RoundingRadius->0], RectangleBox[{2.9, 0}, {2.95, 0.1544},
RoundingRadius->0], RectangleBox[{2.95, 0}, {3., 0.151},
RoundingRadius->0], RectangleBox[{3., 0}, {3.05, 0.1556},
RoundingRadius->0], RectangleBox[{3.05, 0}, {3.1, 0.1376},
RoundingRadius->0], RectangleBox[{3.1, 0}, {3.15, 0.1346},
RoundingRadius->0], RectangleBox[{3.15, 0}, {3.2, 0.1234},
RoundingRadius->0], RectangleBox[{3.2, 0}, {3.25, 0.1226},
RoundingRadius->0], RectangleBox[{3.25, 0}, {3.3, 0.1216},
RoundingRadius->0], RectangleBox[{3.3, 0}, {3.35, 0.1182},
RoundingRadius->0], RectangleBox[{3.35, 0}, {3.4, 0.1128},
RoundingRadius->0], RectangleBox[{3.4, 0}, {3.45, 0.1148},
RoundingRadius->0], RectangleBox[{3.45, 0}, {3.5, 0.1084},
RoundingRadius->0], RectangleBox[{3.5, 0}, {3.55, 0.0962},
RoundingRadius->0], RectangleBox[{3.55, 0}, {3.6, 0.0972},
RoundingRadius->0], RectangleBox[{3.6, 0}, {3.65, 0.0994},
RoundingRadius->0], RectangleBox[{3.65, 0}, {3.7, 0.0984},
RoundingRadius->0], RectangleBox[{3.7, 0}, {3.75, 0.0918},
RoundingRadius->0], RectangleBox[{3.75, 0}, {3.8, 0.0884},
RoundingRadius->0], RectangleBox[{3.8, 0}, {3.85, 0.095},
RoundingRadius->0], RectangleBox[{3.85, 0}, {3.9, 0.0838},
RoundingRadius->0], RectangleBox[{3.9, 0}, {3.95, 0.0692},
RoundingRadius->0], RectangleBox[{3.95, 0}, {4., 0.0842},
RoundingRadius->0], RectangleBox[{4., 0}, {4.05, 0.0668},
RoundingRadius->0], RectangleBox[{4.05, 0}, {4.1, 0.0712},
RoundingRadius->0], RectangleBox[{4.1, 0}, {4.15, 0.067},
RoundingRadius->0], RectangleBox[{4.15, 0}, {4.2, 0.0578},
RoundingRadius->0], RectangleBox[{4.2, 0}, {4.25, 0.0614},
RoundingRadius->0], RectangleBox[{4.25, 0}, {4.3, 0.0646},
RoundingRadius->0], RectangleBox[{4.3, 0}, {4.35, 0.0612},
RoundingRadius->0], RectangleBox[{4.35, 0}, {4.4, 0.0558},
RoundingRadius->0], RectangleBox[{4.4, 0}, {4.45, 0.051},
RoundingRadius->0], RectangleBox[{4.45, 0}, {4.5, 0.0472},
RoundingRadius->0], RectangleBox[{4.5, 0}, {4.55, 0.047},
RoundingRadius->0], RectangleBox[{4.55, 0}, {4.6, 0.0514},
RoundingRadius->0], RectangleBox[{4.6, 0}, {4.65, 0.0442},
RoundingRadius->0], RectangleBox[{4.65, 0}, {4.7, 0.0468},
RoundingRadius->0], RectangleBox[{4.7, 0}, {4.75, 0.0378},
RoundingRadius->0], RectangleBox[{4.75, 0}, {4.8, 0.0414},
RoundingRadius->0], RectangleBox[{4.8, 0}, {4.85, 0.0386},
RoundingRadius->0], RectangleBox[{4.85, 0}, {4.9, 0.0378},
RoundingRadius->0], RectangleBox[{4.9, 0}, {4.95, 0.0378},
RoundingRadius->0], RectangleBox[{4.95, 0}, {5., 0.033},
RoundingRadius->0], RectangleBox[{5., 0}, {5.05, 0.0302},
RoundingRadius->0], RectangleBox[{5.05, 0}, {5.1, 0.0308},
RoundingRadius->0], RectangleBox[{5.1, 0}, {5.15, 0.0352},
RoundingRadius->0], RectangleBox[{5.15, 0}, {5.2, 0.0278},
RoundingRadius->0], RectangleBox[{5.2, 0}, {5.25, 0.0326},
RoundingRadius->0], RectangleBox[{5.25, 0}, {5.3, 0.0238},
RoundingRadius->0], RectangleBox[{5.3, 0}, {5.35, 0.0298},
RoundingRadius->0], RectangleBox[{5.35, 0}, {5.4, 0.0224},
RoundingRadius->0], RectangleBox[{5.4, 0}, {5.45, 0.025},
RoundingRadius->0], RectangleBox[{5.45, 0}, {5.5, 0.0238},
RoundingRadius->0], RectangleBox[{5.5, 0}, {5.55, 0.019},
RoundingRadius->0], RectangleBox[{5.55, 0}, {5.6, 0.0226},
RoundingRadius->0], RectangleBox[{5.6, 0}, {5.65, 0.0198},
RoundingRadius->0], RectangleBox[{5.65, 0}, {5.7, 0.0192},
RoundingRadius->0], RectangleBox[{5.7, 0}, {5.75, 0.0178},
RoundingRadius->0], RectangleBox[{5.75, 0}, {5.8, 0.018},
RoundingRadius->0], RectangleBox[{5.8, 0}, {5.85, 0.0192},
RoundingRadius->0], RectangleBox[{5.85, 0}, {5.9, 0.0202},
RoundingRadius->0], RectangleBox[{5.9, 0}, {5.95, 0.0166},
RoundingRadius->0], RectangleBox[{5.95, 0}, {6., 0.013},
RoundingRadius->0], RectangleBox[{6., 0}, {6.05, 0.0142},
RoundingRadius->0], RectangleBox[{6.05, 0}, {6.1, 0.0156},
RoundingRadius->0], RectangleBox[{6.1, 0}, {6.15, 0.012},
RoundingRadius->0], RectangleBox[{6.15, 0}, {6.2, 0.0136},
RoundingRadius->0], RectangleBox[{6.2, 0}, {6.25, 0.0144},
RoundingRadius->0], RectangleBox[{6.25, 0}, {6.3, 0.014},
RoundingRadius->0], RectangleBox[{6.3, 0}, {6.35, 0.0128},
RoundingRadius->0], RectangleBox[{6.35, 0}, {6.4, 0.0098},
RoundingRadius->0], RectangleBox[{6.4, 0}, {6.45, 0.014},
RoundingRadius->0], RectangleBox[{6.45, 0}, {6.5, 0.0076},
RoundingRadius->0], RectangleBox[{6.5, 0}, {6.55, 0.01},
RoundingRadius->0], RectangleBox[{6.55, 0}, {6.6, 0.0098},
RoundingRadius->0], RectangleBox[{6.6, 0}, {6.65, 0.0084},
RoundingRadius->0], RectangleBox[{6.65, 0}, {6.7, 0.0068},
RoundingRadius->0], RectangleBox[{6.7, 0}, {6.75, 0.0064},
RoundingRadius->0], RectangleBox[{6.75, 0}, {6.8, 0.0058},
RoundingRadius->0], RectangleBox[{6.8, 0}, {6.85, 0.0068},
RoundingRadius->0], RectangleBox[{6.85, 0}, {6.9, 0.004},
RoundingRadius->0], RectangleBox[{6.9, 0}, {6.95, 0.0068},
RoundingRadius->0], RectangleBox[{6.95, 0}, {7., 0.007},
RoundingRadius->0], RectangleBox[{7., 0}, {7.05, 0.0044},
RoundingRadius->0], RectangleBox[{7.05, 0}, {7.1, 0.007},
RoundingRadius->0], RectangleBox[{7.1, 0}, {7.15, 0.0068},
RoundingRadius->0], RectangleBox[{7.15, 0}, {7.2, 0.0048},
RoundingRadius->0], RectangleBox[{7.2, 0}, {7.25, 0.0046},
RoundingRadius->0], RectangleBox[{7.25, 0}, {7.3, 0.0062},
RoundingRadius->0], RectangleBox[{7.3, 0}, {7.35, 0.0036},
RoundingRadius->0], RectangleBox[{7.35, 0}, {7.4, 0.0064},
RoundingRadius->0], RectangleBox[{7.4, 0}, {7.45, 0.0048},
RoundingRadius->0], RectangleBox[{7.45, 0}, {7.5, 0.0032},
RoundingRadius->0], RectangleBox[{7.5, 0}, {7.55, 0.0044},
RoundingRadius->0], RectangleBox[{7.55, 0}, {7.6, 0.0038},
RoundingRadius->0], RectangleBox[{7.6, 0}, {7.65, 0.0038},
RoundingRadius->0], RectangleBox[{7.65, 0}, {7.7, 0.004},
RoundingRadius->0], RectangleBox[{7.7, 0}, {7.75, 0.0048},
RoundingRadius->0], RectangleBox[{7.75, 0}, {7.8, 0.0036},
RoundingRadius->0], RectangleBox[{7.8, 0}, {7.85, 0.0032},
RoundingRadius->0], RectangleBox[{7.85, 0}, {7.9, 0.0028},
RoundingRadius->0], RectangleBox[{7.9, 0}, {7.95, 0.002},
RoundingRadius->0], RectangleBox[{7.95, 0}, {8., 0.0032},
RoundingRadius->0], RectangleBox[{8., 0}, {8.05, 0.0022},
RoundingRadius->0], RectangleBox[{8.05, 0}, {8.1, 0.0026},
RoundingRadius->0], RectangleBox[{8.1, 0}, {8.15, 0.003},
RoundingRadius->0], RectangleBox[{8.15, 0}, {8.2, 0.0018},
RoundingRadius->0], RectangleBox[{8.2, 0}, {8.25, 0.0014},
RoundingRadius->0], RectangleBox[{8.25, 0}, {8.3, 0.002},
RoundingRadius->0], RectangleBox[{8.3, 0}, {8.35, 0.0018},
RoundingRadius->0], RectangleBox[{8.35, 0}, {8.4, 0.0022},
RoundingRadius->0], RectangleBox[{8.4, 0}, {8.45, 0.0018},
RoundingRadius->0], RectangleBox[{8.45, 0}, {8.5, 0.0016},
RoundingRadius->0], RectangleBox[{8.5, 0}, {8.55, 0.002},
RoundingRadius->0], RectangleBox[{8.55, 0}, {8.6, 0.0016},
RoundingRadius->0], RectangleBox[{8.6, 0}, {8.65, 0.001},
RoundingRadius->0], RectangleBox[{8.65, 0}, {8.7, 0.001},
RoundingRadius->0], RectangleBox[{8.7, 0}, {8.75, 0.0018},
RoundingRadius->0], RectangleBox[{8.75, 0}, {8.8, 0.001},
RoundingRadius->0], RectangleBox[{8.8, 0}, {8.85, 0.0016},
RoundingRadius->0], RectangleBox[{8.85, 0}, {8.9, 0.0006},
RoundingRadius->0], RectangleBox[{8.9, 0}, {8.95, 0.0014},
RoundingRadius->0], RectangleBox[{8.95, 0}, {9., 0.0016},
RoundingRadius->0], RectangleBox[{9., 0}, {9.05, 0.0006},
RoundingRadius->0], RectangleBox[{9.05, 0}, {9.1, 0.001},
RoundingRadius->0], RectangleBox[{9.1, 0}, {9.15, 0.001},
RoundingRadius->0], RectangleBox[{9.15, 0}, {9.2, 0.0016},
RoundingRadius->0], RectangleBox[{9.2, 0}, {9.25, 0.0012},
RoundingRadius->0], RectangleBox[{9.25, 0}, {9.3, 0.0004},
RoundingRadius->0], RectangleBox[{9.3, 0}, {9.35, 0.0004},
RoundingRadius->0], RectangleBox[{9.35, 0}, {9.4, 0.0004},
RoundingRadius->0], RectangleBox[{9.4, 0}, {9.45, 0.0006},
RoundingRadius->0], RectangleBox[{9.45, 0}, {9.5, 0.0002},
RoundingRadius->0], RectangleBox[{9.5, 0}, {9.55, 0.0008},
RoundingRadius->0], RectangleBox[{9.55, 0}, {9.6, 0.0012},
RoundingRadius->0], RectangleBox[{9.6, 0}, {9.65, 0.001},
RoundingRadius->0], RectangleBox[{9.65, 0}, {9.7, 0.0014},
RoundingRadius->0], RectangleBox[{9.7, 0}, {9.75, 0.0006},
RoundingRadius->0], RectangleBox[{9.75, 0}, {9.8, 0.0012},
RoundingRadius->0], RectangleBox[{9.8, 0}, {9.85, 0.0014},
RoundingRadius->0], RectangleBox[{9.85, 0}, {9.9, 0.0012},
RoundingRadius->0], RectangleBox[{9.9, 0}, {9.95, 0.0006},
RoundingRadius->0], RectangleBox[{9.95, 0}, {10., 0.0006},
RoundingRadius->0], RectangleBox[{10., 0}, {10.05, 0.0004},
RoundingRadius->0], RectangleBox[{10.05, 0}, {10.1, 0.0008},
RoundingRadius->0], RectangleBox[{10.1, 0}, {10.15, 0.0008},
RoundingRadius->0], RectangleBox[{10.15, 0}, {10.2, 0.0006},
RoundingRadius->0], RectangleBox[{10.2, 0}, {10.25, 0.0006},
RoundingRadius->0], RectangleBox[{10.3, 0}, {10.35, 0.0002},
RoundingRadius->0], RectangleBox[{10.4, 0}, {10.45, 0.0002},
RoundingRadius->0], RectangleBox[{10.55, 0}, {10.6, 0.0006},
RoundingRadius->0], RectangleBox[{10.65, 0}, {10.7, 0.0002},
RoundingRadius->0], RectangleBox[{10.75, 0}, {10.8, 0.0002},
RoundingRadius->0], RectangleBox[{10.8, 0}, {10.85, 0.0006},
RoundingRadius->0], RectangleBox[{10.85, 0}, {10.9, 0.0006},
RoundingRadius->0], RectangleBox[{10.9, 0}, {10.95, 0.0002},
RoundingRadius->0], RectangleBox[{10.95, 0}, {11., 0.0002},
RoundingRadius->0], RectangleBox[{11., 0}, {11.05, 0.0002},
RoundingRadius->0], RectangleBox[{11.05, 0}, {11.1, 0.0002},
RoundingRadius->0], RectangleBox[{11.2, 0}, {11.25, 0.0002},
RoundingRadius->0], RectangleBox[{11.25, 0}, {11.3, 0.0002},
RoundingRadius->0], RectangleBox[{11.3, 0}, {11.35, 0.0004},
RoundingRadius->0], RectangleBox[{11.35, 0}, {11.4, 0.0004},
RoundingRadius->0], RectangleBox[{11.4, 0}, {11.45, 0.0002},
RoundingRadius->0], RectangleBox[{11.5, 0}, {11.55, 0.0006},
RoundingRadius->0], RectangleBox[{11.55, 0}, {11.6, 0.0002},
RoundingRadius->0], RectangleBox[{11.7, 0}, {11.75, 0.0002},
RoundingRadius->0], RectangleBox[{11.8, 0}, {11.85, 0.0002},
RoundingRadius->0], RectangleBox[{12.05, 0}, {12.1, 0.0002},
RoundingRadius->0], RectangleBox[{12.15, 0}, {12.2, 0.0004},
RoundingRadius->0], RectangleBox[{12.45, 0}, {12.5, 0.0002},
RoundingRadius->0], RectangleBox[{12.5, 0}, {12.55, 0.0002},
RoundingRadius->0], RectangleBox[{13.6, 0}, {13.65, 0.0002},
RoundingRadius->
0]}, {}, {}}, {{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}}}, {{{}, {},
TagBox[
{RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6],
Opacity[1.], LineBox[CompressedData["
1:eJwV1Xk4VN8fB3BJ30iIbJmylBZLm+yp9yUqJSkla2mzpFGiyBaKRIstvgip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"]]},
Annotation[#, "Charting`Private`Tag$5035213#1"]& ]}, {}}},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->{True, True},
AxesLabel->{None, None},
AxesOrigin->{-0.136, 0},
FrameLabel->{{None, None}, {None, None}},
FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
GridLines->{None, None},
GridLinesStyle->Directive[
GrayLevel[0.5, 0.4]],
PlotRange->NCache[{{0,
Rational[34, 5]}, {All, All}}, {{0, 6.8}, {All, All}}],
PlotRangePadding->{{
Scaled[0.02],
Scaled[0.02]}, {
Scaled[0.02],
Scaled[0.05]}},
Ticks->{Automatic, Automatic}]], "Output",
CellChangeTimes->{{3.795037784230567*^9, 3.795037878460627*^9},
3.7950379480378113`*^9},
CellLabel->
"Out[1675]=",ExpressionUUID->"0a9d1ea3-f2e9-4fab-9e1c-16f849c22653"]
}, Open ]],
Cell[TextData[{
"In general the Gamma distribution f(r) = ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
RowBox[{
SuperscriptBox["e",
RowBox[{"-", "r"}]],
RowBox[{
SuperscriptBox["r",
RowBox[{"a", "-", "1"}]], "/",
RowBox[{"Gamma", "[", "a", "]"}]}], " ", "can", " ", "be", " ",
"sampled", " ", "as", " ", "the", " ", "sum", " ", "of", " ", "a", " ",
"exponential", " ", "random", " ",
RowBox[{"variates", ".", " ", "In"}], " ", "the", " ", "case", " ",
"of", " ", "non"}], "-",
RowBox[{"integer", " ", "a"}]}], ",", " ",
RowBox[{
RowBox[{"Marsaglia", "'"}], "s", " ", "method", " ", "can", " ", "be",
" ",
RowBox[{"used", "."}]}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"1f6cf433-8d33-41d9-8d60-a6d598582103"]
}], "Text",
CellChangeTimes->{{3.7950698956955214`*^9,
3.795069947315837*^9}},ExpressionUUID->"1b123bb1-e637-4189-a104-\
1d738e0526cc"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["3. Discrete Decomposition", "Section",
CellChangeTimes->{{3.7950334000773497`*^9, 3.7950334223425694`*^9}, {
3.7950439433658733`*^9,
3.795043943718939*^9}},ExpressionUUID->"e39533d0-1de1-486d-809d-\
b072d3e13e8d"],
Cell[TextData[{
"Sometimes f(r) can be written as the sum of several non-negative \
distributions f(r) = ",
Cell[BoxData[
FormBox[
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"i", "=", "1"}], "n"],
RowBox[{
SubscriptBox["w", "i"],
RowBox[{
SubscriptBox["f", "i"], "(", "r", ")"}]}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"aa4113eb-2f15-4703-b6f9-912587fa7cc7"],
", each of which is normalized ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"],
RowBox[{
RowBox[{
SubscriptBox["f", "i"], "(", "r", ")"}],
RowBox[{"\[DifferentialD]", "r"}]}]}], " ", "=", " ", "1"}],
TraditionalForm]],ExpressionUUID->"5e4cb6a2-425f-46d9-88fc-db31dc1b942c"],
" and an analytic sampling procedure is known for each ",
Cell[BoxData[
FormBox[
RowBox[{
SubscriptBox["f", "i"], "(", "r", ")"}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"5c5f6d89-4813-4abb-bfa9-efdd365321f9"],
". In this case, one random number can be used to first select one of the \
",
Cell[BoxData[
FormBox[
RowBox[{
SubscriptBox["f", "i"], " ", "distributions", " ", "using", " ", "a", " ",
"CDF", " ", "built", " ", "from", " ", "the", " ", "weights", " ",
SubscriptBox["w", "i"]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"80969bd5-29e0-4c7a-b4a4-00cf2ad150f4"],
", and then a second independent random number can be used to sample ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SubscriptBox["f", "i"], ".", " ", "The"}], " ", "resulting", " ",
"distance", " ", "r", " ", "is", " ", "then", " ", "distributed", " ",
"by", " ",
RowBox[{
RowBox[{"f", "(", "r", ")"}], "."}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"912b0ee8-dd33-47f5-8e50-6b00df54f915"]
}], "Text",
CellChangeTimes->{{3.7950334056681547`*^9, 3.795033443357107*^9}, {
3.795033483765088*^9, 3.7950335614624653`*^9}, {3.7950335986085377`*^9,
3.79503368480424*^9}, {3.795037215602681*^9, 3.7950373366224833`*^9}, {
3.79506998786063*^9,
3.795069999892305*^9}},ExpressionUUID->"1a636909-f3bd-4c3b-8484-\
f1ff9b67edb2"]
}, Open ]],
Cell[CellGroupData[{
Cell["4. Continuous superpositions", "Section",
CellChangeTimes->{{3.795034008461308*^9, 3.7950340116105337`*^9},
3.795034048726885*^9, {3.7950439458375387`*^9,
3.7950439461707487`*^9}},ExpressionUUID->"0e4a8200-95b9-4677-bd65-\
28a8e83da6e2"],
Cell[TextData[{
"A continuous generalization of the previous discrete decomposition is when \
f(r) can be expressed as a continuous superposition of a family of \
distributions g(r,s), where an exact analytic sampling procedure for g(r,s) \
is known: f(r) = ",
Cell[BoxData[
FormBox[
RowBox[{
SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"],
RowBox[{
RowBox[{"g", "(",
RowBox[{"r", ",", "s"}], ")"}],
RowBox[{"w", "(", "s", ")"}],
RowBox[{"\[DifferentialD]", "s"}]}]}], TraditionalForm]],ExpressionUUID->
"7cd2e163-c656-42be-b9c7-cabbe6a31667"],
", where s is a parameter for distribution g (such as the mean of g) and \
w(s) \[GreaterEqual] 0, for s \[GreaterEqual] 0. If a sampling procedure is \
also known for w(s), then we can first sample s from w, and then sample \
g(r,s) with the sampled s parameter value. See also [Devroye 2006 - Section \
1.2]."
}], "Text",
CellChangeTimes->{{3.795034050707748*^9, 3.795034230598435*^9}, {
3.7950342682292337`*^9, 3.795034290923036*^9}, {3.795037378687886*^9,
3.795037385555725*^9}, {3.79806196781586*^9,
3.798061968161107*^9}},ExpressionUUID->"a521c78c-5256-47ff-b702-\
e850a116a650"],
Cell[CellGroupData[{
Cell[TextData[{
"Example 4.1 - ",
Cell[BoxData[
FormBox[
SubscriptBox["BesselK", "0"], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"8abb1571-2471-4bff-a1a9-7f1875feacb7"]
}], "Subsection",
CellChangeTimes->{{3.7950342939097652`*^9, 3.7950342947347927`*^9}, {
3.795038337954341*^9, 3.795038341195614*^9}, {3.795043948543998*^9,
3.795043948828019*^9}},ExpressionUUID->"adb24fc9-8bf6-4c56-8a2a-\
87327c7cd64d"],
Cell[TextData[{
"Consider the distribution ",
Cell[BoxData[
FormBox[
RowBox[{
FractionBox["2", "\[Pi]"],
RowBox[{
SubscriptBox["K", "0"], "(", "r", ")"}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"c05c3d02-3184-43a1-875c-6a4a58b19cf4"],
":"
}], "Text",
CellChangeTimes->{{3.795038343491076*^9, 3.795038349931592*^9}, {
3.795041413985859*^9,
3.795041424087284*^9}},ExpressionUUID->"002aafe1-1999-428f-892a-\
b0f680b5645d"],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{
RowBox[{"Clear", "[", "f", "]"}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"f", "[", "r_", "]"}], ":=",
RowBox[{
FractionBox["2", "Pi"],
RowBox[{"BesselK", "[",
RowBox[{"0", ",", "r"}], "]"}]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{"Integrate", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",",
RowBox[{"{",
RowBox[{"r", ",", "0", ",", "Infinity"}], "}"}]}], "]"}]}], "Input",
CellChangeTimes->{{3.7950383519798307`*^9, 3.795038366846962*^9}},
CellLabel->
"In[1779]:=",ExpressionUUID->"c0fe22a9-70ae-4ea8-8312-69a0ac00f359"],
Cell[BoxData["1"], "Output",
CellChangeTimes->{3.795038370711545*^9, 3.795053555443206*^9},
CellLabel->
"Out[1781]=",ExpressionUUID->"9e1f0e7f-2ab8-4183-a6ad-568cf1db3d2a"]
}, Open ]],
Cell["We can write f(r) as the superposition of exponentials:", "Text",
CellChangeTimes->{{3.795038378991269*^9,
3.795038388431787*^9}},ExpressionUUID->"e18b68be-fcc6-452e-be13-\
53b6135f2d4d"],
Cell[BoxData[
RowBox[{
RowBox[{"w", "[", "s_", "]"}], ":=",
RowBox[{
FractionBox["2",
RowBox[{"\[Pi]", " ", "s",
SqrtBox[
RowBox[{
RowBox[{"-", "1"}], "+",
SuperscriptBox["s", "2"]}]]}]],
RowBox[{"HeavisideTheta", "[",
RowBox[{"s", "-", "1"}], "]"}]}]}]], "Input",
CellChangeTimes->{{3.795038400229918*^9, 3.795038405216826*^9}},
CellLabel->
"In[1097]:=",ExpressionUUID->"41f82aae-4bce-4b3c-a45d-32ed3ea4128d"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Integrate", "[",
RowBox[{
RowBox[{
RowBox[{"w", "[", "s", "]"}], " ", "s", " ",
RowBox[{"Exp", "[",
RowBox[{
RowBox[{"-", "s"}], " ", "r"}], "]"}]}], ",",
RowBox[{"{",
RowBox[{"s", ",", "0", ",", "Infinity"}], "}"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"r", ">", "0"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.795038408538952*^9, 3.795038421796344*^9}},
CellLabel->
"In[1680]:=",ExpressionUUID->"b5a0c33b-2bc1-4a0c-ae6d-66db2857c8e1"],
Cell[BoxData[
FractionBox[
RowBox[{"2", " ",
RowBox[{"BesselK", "[",
RowBox[{"0", ",", "r"}], "]"}]}], "\[Pi]"]], "Output",
CellChangeTimes->{3.795038422807893*^9},
CellLabel->
"Out[1680]=",ExpressionUUID->"deb7a6b7-0318-47cb-a42b-bf211f51e30c"]
}, Open ]],
Cell["\<\
We check to see we can sample inverse mfp s from w(s). The CDF of the weight \
function w(s) is\
\>", "Text",
CellChangeTimes->{{3.79503844209874*^9, 3.795038458284916*^9}, {
3.7950438430333347`*^9, 3.795043845713229*^9}, {3.795053503225481*^9,
3.795053507347065*^9}},ExpressionUUID->"4ecaf526-2944-4211-a152-\
ec22046eb861"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Integrate", "[",
RowBox[{
RowBox[{"w", "[", "s", "]"}], ",",
RowBox[{"{",
RowBox[{"s", ",", "1", ",", "S"}], "}"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"S", ">", "1"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.7950384609025087`*^9, 3.795038486075193*^9}, {
3.795043808704808*^9, 3.795043810751683*^9}, {3.79692996698703*^9,
3.796929994933724*^9}},
CellLabel->
"In[1101]:=",ExpressionUUID->"c9f85219-5994-471f-9a8c-aba011dc84d0"],
Cell[BoxData[
FractionBox[
RowBox[{"2", " ",
RowBox[{"ArcSec", "[", "S", "]"}]}], "\[Pi]"]], "Output",
CellChangeTimes->{{3.795038467551611*^9, 3.795038487711966*^9},
3.795043812668079*^9, {3.796929969597281*^9, 3.796929996864685*^9}},
CellLabel->
"Out[1101]=",ExpressionUUID->"636739dd-fff3-4c25-8bb4-26b860320899"]
}, Open ]],
Cell["\<\
Here we integrate starting from s = 1, since the Heaviside function in w[s] \
makes the distribution 0 below s = 1. We can invert the result,\
\>", "Text",
CellChangeTimes->{{3.795043849154255*^9, 3.795043851019261*^9}, {
3.7969299014305153`*^9,
3.7969299471463547`*^9}},ExpressionUUID->"4ac2d3ce-a6cb-4d0a-b36f-\
4ee775a7c3a8"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Solve", "[",
RowBox[{
RowBox[{"\[Xi]", "==",
FractionBox[
RowBox[{"2", " ",
RowBox[{"ArcSec", "[", "S", "]"}]}], "\[Pi]"]}], ",", "S"}],
"]"}]], "Input",
CellChangeTimes->{{3.795038475825841*^9, 3.795038537862911*^9}, {
3.795043813416388*^9, 3.7950438151691923`*^9}},
CellLabel->
"In[1694]:=",ExpressionUUID->"5bbae4dd-6253-4a2d-96d0-5767f78a2919"],
Cell[BoxData[
RowBox[{"{",
RowBox[{"{",
RowBox[{"S", "\[Rule]",
RowBox[{"ConditionalExpression", "[",
RowBox[{
RowBox[{"Sec", "[",
FractionBox[
RowBox[{"\[Pi]", " ", "\[Xi]"}], "2"], "]"}], ",",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"\[Xi]", "\[NotEqual]", "1"}], "&&",
RowBox[{"0", "<",
RowBox[{"\[Pi]", " ",
RowBox[{"Re", "[", "\[Xi]", "]"}]}], "<",
RowBox[{"2", " ", "\[Pi]"}]}]}], ")"}], "||",
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"\[Pi]", " ",
RowBox[{"Re", "[", "\[Xi]", "]"}]}], "\[Equal]", "0"}], "&&",
RowBox[{
RowBox[{"\[Pi]", " ",
RowBox[{"Im", "[", "\[Xi]", "]"}]}], "\[GreaterEqual]", "0"}]}],
")"}], "||",
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"\[Pi]", " ",
RowBox[{"Re", "[", "\[Xi]", "]"}]}], "\[Equal]",
RowBox[{"2", " ", "\[Pi]"}]}], "&&",
RowBox[{
RowBox[{"\[Pi]", " ",
RowBox[{"Im", "[", "\[Xi]", "]"}]}], "\[LessEqual]", "0"}]}],
")"}]}]}], "]"}]}], "}"}], "}"}]], "Output",
CellChangeTimes->{{3.79503849628082*^9, 3.795038538145218*^9},
3.79504381538186*^9},
CellLabel->
"Out[1694]=",ExpressionUUID->"8ded88af-5d47-4a17-af21-bb57c42e0161"]
}, Open ]],
Cell[TextData[{
"So we can sample inverse mfp s using ",
Cell[BoxData[
RowBox[{"Sec", "[",
FractionBox[
RowBox[{"\[Pi]", " ",
SubscriptBox["\[Xi]", "1"]}], "2"], "]"}]],
CellChangeTimes->{{3.79503849628082*^9, 3.795038538145218*^9}},
ExpressionUUID->"4dc627b9-e83c-4146-8c2e-3e4b9bc2de9d"],
" where ",
Cell[BoxData[
FormBox[
SubscriptBox["\[Xi]", "1"], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"110cee6d-5ec3-4838-bfe8-31e7b4be6c5a"],
" is a uniform random real in [0,1], and then sample the exponential using: \
",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{"-", "log"}],
RowBox[{
RowBox[{"(",
SubscriptBox["\[Xi]", "2"], ")"}], "/",
RowBox[{"s", ":"}]}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"a0fe0dec-6a56-4d82-bca2-103145098645"]
}], "Text",
CellChangeTimes->{{3.79503858605945*^9,
3.795038641295437*^9}},ExpressionUUID->"419d102e-5a9e-4fc6-9bec-\
a0e7f112b92d"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Show", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"Histogram", "[",
RowBox[{
RowBox[{"Table", "[",
RowBox[{
RowBox[{"(",
RowBox[{"-",
FractionBox[
RowBox[{"Log", "[",
RowBox[{"RandomReal", "[", "]"}], "]"}],
RowBox[{"Sec", "[",
FractionBox[
RowBox[{"\[Pi]", " ",
RowBox[{"RandomReal", "[", "]"}]}], "2"], "]"}]]}], ")"}], ",",
RowBox[{"{",
RowBox[{"i", ",",
RowBox[{"Range", "[", "50000", "]"}]}], "}"}]}], "]"}], ",", "450",
",", "\"\<PDF\>\""}], "]"}], ",", "\[IndentingNewLine]",
RowBox[{"Plot", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",",
RowBox[{"{",
RowBox[{"r", ",", "0", ",", "10"}], "}"}], ",",
RowBox[{"PlotRange", "\[Rule]",
RowBox[{"{",
RowBox[{"0", ",", "3"}], "}"}]}]}], "]"}]}], "\[IndentingNewLine]",
"]"}]], "Input",
CellChangeTimes->{{3.758565174797154*^9, 3.7585651978577213`*^9}, {
3.795027089648484*^9, 3.795027112413389*^9}, {3.7950385590280857`*^9,
3.795038566210376*^9}, {3.795053539688067*^9, 3.795053545542327*^9}},
CellLabel->
"In[1782]:=",ExpressionUUID->"5a8534e6-4090-4598-9a89-12acc96b2d87"],
Cell[BoxData[
GraphicsBox[{{
{RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[
Opacity[0.]], {},
{RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[
Opacity[0.]], RectangleBox[{0., 0}, {0.02, 3.218},
RoundingRadius->0], RectangleBox[{0.02, 0}, {0.04, 2.34},
RoundingRadius->0], RectangleBox[{0.04, 0}, {0.06, 1.988},
RoundingRadius->0], RectangleBox[{0.06, 0}, {0.08, 1.733},
RoundingRadius->0], RectangleBox[{0.08, 0}, {0.1, 1.614},
RoundingRadius->0], RectangleBox[{0.1, 0}, {0.12, 1.483},
RoundingRadius->0], RectangleBox[{0.12, 0}, {0.14, 1.298},
RoundingRadius->0], RectangleBox[{0.14, 0}, {0.16, 1.36},
RoundingRadius->0], RectangleBox[{0.16, 0}, {0.18, 1.23},
RoundingRadius->0], RectangleBox[{0.18, 0}, {0.2, 1.072},
RoundingRadius->0], RectangleBox[{0.2, 0}, {0.22, 1.14},
RoundingRadius->0], RectangleBox[{0.22, 0}, {0.24, 1.076},
RoundingRadius->0], RectangleBox[{0.24, 0}, {0.26, 1.045},
RoundingRadius->0], RectangleBox[{0.26, 0}, {0.28, 0.945},
RoundingRadius->0], RectangleBox[{0.28, 0}, {0.3, 0.884},
RoundingRadius->0], RectangleBox[{0.3, 0}, {0.32, 0.843},
RoundingRadius->0], RectangleBox[{0.32, 0}, {0.34, 0.796},
RoundingRadius->0], RectangleBox[{0.34, 0}, {0.36, 0.828},
RoundingRadius->0], RectangleBox[{0.36, 0}, {0.38, 0.758},
RoundingRadius->0], RectangleBox[{0.38, 0}, {0.4, 0.718},
RoundingRadius->0], RectangleBox[{0.4, 0}, {0.42, 0.712},
RoundingRadius->0], RectangleBox[{0.42, 0}, {0.44, 0.666},
RoundingRadius->0], RectangleBox[{0.44, 0}, {0.46, 0.652},
RoundingRadius->0], RectangleBox[{0.46, 0}, {0.48, 0.693},
RoundingRadius->0], RectangleBox[{0.48, 0}, {0.5, 0.607},
RoundingRadius->0], RectangleBox[{0.5, 0}, {0.52, 0.554},
RoundingRadius->0], RectangleBox[{0.52, 0}, {0.54, 0.59},
RoundingRadius->0], RectangleBox[{0.54, 0}, {0.56, 0.524},
RoundingRadius->0], RectangleBox[{0.56, 0}, {0.58, 0.497},
RoundingRadius->0], RectangleBox[{0.58, 0}, {0.6, 0.508},
RoundingRadius->0], RectangleBox[{0.6, 0}, {0.62, 0.474},
RoundingRadius->0], RectangleBox[{0.62, 0}, {0.64, 0.5},
RoundingRadius->0], RectangleBox[{0.64, 0}, {0.66, 0.445},
RoundingRadius->0], RectangleBox[{0.66, 0}, {0.68, 0.462},
RoundingRadius->0], RectangleBox[{0.68, 0}, {0.7, 0.44},
RoundingRadius->0], RectangleBox[{0.7, 0}, {0.72, 0.419},
RoundingRadius->0], RectangleBox[{0.72, 0}, {0.74, 0.394},
RoundingRadius->0], RectangleBox[{0.74, 0}, {0.76, 0.41},
RoundingRadius->0], RectangleBox[{0.76, 0}, {0.78, 0.379},
RoundingRadius->0], RectangleBox[{0.78, 0}, {0.8, 0.372},
RoundingRadius->0], RectangleBox[{0.8, 0}, {0.82, 0.364},
RoundingRadius->0], RectangleBox[{0.82, 0}, {0.84, 0.308},
RoundingRadius->0], RectangleBox[{0.84, 0}, {0.86, 0.333},
RoundingRadius->0], RectangleBox[{0.86, 0}, {0.88, 0.337},
RoundingRadius->0], RectangleBox[{0.88, 0}, {0.9, 0.303},
RoundingRadius->0], RectangleBox[{0.9, 0}, {0.92, 0.3},
RoundingRadius->0], RectangleBox[{0.92, 0}, {0.94, 0.304},
RoundingRadius->0], RectangleBox[{0.94, 0}, {0.96, 0.292},
RoundingRadius->0], RectangleBox[{0.96, 0}, {0.98, 0.276},
RoundingRadius->0], RectangleBox[{0.98, 0}, {1., 0.281},
RoundingRadius->0], RectangleBox[{1., 0}, {1.02, 0.261},
RoundingRadius->0], RectangleBox[{1.02, 0}, {1.04, 0.256},
RoundingRadius->0], RectangleBox[{1.04, 0}, {1.06, 0.253},
RoundingRadius->0], RectangleBox[{1.06, 0}, {1.08, 0.234},
RoundingRadius->0], RectangleBox[{1.08, 0}, {1.1, 0.241},
RoundingRadius->0], RectangleBox[{1.1, 0}, {1.12, 0.222},
RoundingRadius->0], RectangleBox[{1.12, 0}, {1.14, 0.218},
RoundingRadius->0], RectangleBox[{1.14, 0}, {1.16, 0.222},
RoundingRadius->0], RectangleBox[{1.16, 0}, {1.18, 0.213},
RoundingRadius->0], RectangleBox[{1.18, 0}, {1.2, 0.208},
RoundingRadius->0], RectangleBox[{1.2, 0}, {1.22, 0.204},
RoundingRadius->0], RectangleBox[{1.22, 0}, {1.24, 0.186},
RoundingRadius->0], RectangleBox[{1.24, 0}, {1.26, 0.19},
RoundingRadius->0], RectangleBox[{1.26, 0}, {1.28, 0.22},
RoundingRadius->0], RectangleBox[{1.28, 0}, {1.3, 0.173},
RoundingRadius->0], RectangleBox[{1.3, 0}, {1.32, 0.152},
RoundingRadius->0], RectangleBox[{1.32, 0}, {1.34, 0.183},
RoundingRadius->0], RectangleBox[{1.34, 0}, {1.36, 0.152},
RoundingRadius->0], RectangleBox[{1.36, 0}, {1.38, 0.145},
RoundingRadius->0], RectangleBox[{1.38, 0}, {1.4, 0.14},
RoundingRadius->0], RectangleBox[{1.4, 0}, {1.42, 0.151},
RoundingRadius->0], RectangleBox[{1.42, 0}, {1.44, 0.147},
RoundingRadius->0], RectangleBox[{1.44, 0}, {1.46, 0.147},
RoundingRadius->0], RectangleBox[{1.46, 0}, {1.48, 0.152},
RoundingRadius->0], RectangleBox[{1.48, 0}, {1.5, 0.136},
RoundingRadius->0], RectangleBox[{1.5, 0}, {1.52, 0.118},
RoundingRadius->0], RectangleBox[{1.52, 0}, {1.54, 0.134},
RoundingRadius->0], RectangleBox[{1.54, 0}, {1.56, 0.125},
RoundingRadius->0], RectangleBox[{1.56, 0}, {1.58, 0.127},
RoundingRadius->0], RectangleBox[{1.58, 0}, {1.6, 0.14},
RoundingRadius->0], RectangleBox[{1.6, 0}, {1.62, 0.097},
RoundingRadius->0], RectangleBox[{1.62, 0}, {1.64, 0.095},
RoundingRadius->0], RectangleBox[{1.64, 0}, {1.66, 0.105},
RoundingRadius->0], RectangleBox[{1.66, 0}, {1.68, 0.111},
RoundingRadius->0], RectangleBox[{1.68, 0}, {1.7, 0.114},
RoundingRadius->0], RectangleBox[{1.7, 0}, {1.72, 0.104},
RoundingRadius->0], RectangleBox[{1.72, 0}, {1.74, 0.088},
RoundingRadius->0], RectangleBox[{1.74, 0}, {1.76, 0.097},
RoundingRadius->0], RectangleBox[{1.76, 0}, {1.78, 0.087},
RoundingRadius->0], RectangleBox[{1.78, 0}, {1.8, 0.085},
RoundingRadius->0], RectangleBox[{1.8, 0}, {1.82, 0.082},
RoundingRadius->0], RectangleBox[{1.82, 0}, {1.84, 0.088},
RoundingRadius->0], RectangleBox[{1.84, 0}, {1.86, 0.088},
RoundingRadius->0], RectangleBox[{1.86, 0}, {1.88, 0.087},
RoundingRadius->0], RectangleBox[{1.88, 0}, {1.9, 0.077},
RoundingRadius->0], RectangleBox[{1.9, 0}, {1.92, 0.07},
RoundingRadius->0], RectangleBox[{1.92, 0}, {1.94, 0.077},
RoundingRadius->0], RectangleBox[{1.94, 0}, {1.96, 0.078},
RoundingRadius->0], RectangleBox[{1.96, 0}, {1.98, 0.075},
RoundingRadius->0], RectangleBox[{1.98, 0}, {2., 0.071},
RoundingRadius->0], RectangleBox[{2., 0}, {2.02, 0.067},
RoundingRadius->0], RectangleBox[{2.02, 0}, {2.04, 0.071},
RoundingRadius->0], RectangleBox[{2.04, 0}, {2.06, 0.063},
RoundingRadius->0], RectangleBox[{2.06, 0}, {2.08, 0.049},
RoundingRadius->0], RectangleBox[{2.08, 0}, {2.1, 0.056},
RoundingRadius->0], RectangleBox[{2.1, 0}, {2.12, 0.072},
RoundingRadius->0], RectangleBox[{2.12, 0}, {2.14, 0.062},
RoundingRadius->0], RectangleBox[{2.14, 0}, {2.16, 0.069},
RoundingRadius->0], RectangleBox[{2.16, 0}, {2.18, 0.056},
RoundingRadius->0], RectangleBox[{2.18, 0}, {2.2, 0.065},
RoundingRadius->0], RectangleBox[{2.2, 0}, {2.22, 0.046},
RoundingRadius->0], RectangleBox[{2.22, 0}, {2.24, 0.054},
RoundingRadius->0], RectangleBox[{2.24, 0}, {2.26, 0.059},
RoundingRadius->0], RectangleBox[{2.26, 0}, {2.28, 0.053},
RoundingRadius->0], RectangleBox[{2.28, 0}, {2.3, 0.044},
RoundingRadius->0], RectangleBox[{2.3, 0}, {2.32, 0.046},
RoundingRadius->0], RectangleBox[{2.32, 0}, {2.34, 0.04},
RoundingRadius->0], RectangleBox[{2.34, 0}, {2.36, 0.039},
RoundingRadius->0], RectangleBox[{2.36, 0}, {2.38, 0.041},
RoundingRadius->0], RectangleBox[{2.38, 0}, {2.4, 0.044},
RoundingRadius->0], RectangleBox[{2.4, 0}, {2.42, 0.057},
RoundingRadius->0], RectangleBox[{2.42, 0}, {2.44, 0.037},
RoundingRadius->0], RectangleBox[{2.44, 0}, {2.46, 0.038},
RoundingRadius->0], RectangleBox[{2.46, 0}, {2.48, 0.04},
RoundingRadius->0], RectangleBox[{2.48, 0}, {2.5, 0.039},
RoundingRadius->0], RectangleBox[{2.5, 0}, {2.52, 0.04},
RoundingRadius->0], RectangleBox[{2.52, 0}, {2.54, 0.039},
RoundingRadius->0], RectangleBox[{2.54, 0}, {2.56, 0.043},
RoundingRadius->0], RectangleBox[{2.56, 0}, {2.58, 0.029},
RoundingRadius->0], RectangleBox[{2.58, 0}, {2.6, 0.043},
RoundingRadius->0], RectangleBox[{2.6, 0}, {2.62, 0.036},
RoundingRadius->0], RectangleBox[{2.62, 0}, {2.64, 0.035},
RoundingRadius->0], RectangleBox[{2.64, 0}, {2.66, 0.029},
RoundingRadius->0], RectangleBox[{2.66, 0}, {2.68, 0.031},
RoundingRadius->0], RectangleBox[{2.68, 0}, {2.7, 0.035},
RoundingRadius->0], RectangleBox[{2.7, 0}, {2.72, 0.036},
RoundingRadius->0], RectangleBox[{2.72, 0}, {2.74, 0.031},
RoundingRadius->0], RectangleBox[{2.74, 0}, {2.76, 0.033},
RoundingRadius->0], RectangleBox[{2.76, 0}, {2.78, 0.03},
RoundingRadius->0], RectangleBox[{2.78, 0}, {2.8, 0.03},
RoundingRadius->0], RectangleBox[{2.8, 0}, {2.82, 0.033},
RoundingRadius->0], RectangleBox[{2.82, 0}, {2.84, 0.032},
RoundingRadius->0], RectangleBox[{2.84, 0}, {2.86, 0.024},
RoundingRadius->0], RectangleBox[{2.86, 0}, {2.88, 0.021},
RoundingRadius->0], RectangleBox[{2.88, 0}, {2.9, 0.024},
RoundingRadius->0], RectangleBox[{2.9, 0}, {2.92, 0.028},
RoundingRadius->0], RectangleBox[{2.92, 0}, {2.94, 0.026},
RoundingRadius->0], RectangleBox[{2.94, 0}, {2.96, 0.022},
RoundingRadius->0], RectangleBox[{2.96, 0}, {2.98, 0.012},
RoundingRadius->0], RectangleBox[{2.98, 0}, {3., 0.024},
RoundingRadius->0], RectangleBox[{3., 0}, {3.02, 0.027},
RoundingRadius->0], RectangleBox[{3.02, 0}, {3.04, 0.02},
RoundingRadius->0], RectangleBox[{3.04, 0}, {3.06, 0.01},
RoundingRadius->0], RectangleBox[{3.06, 0}, {3.08, 0.017},
RoundingRadius->0], RectangleBox[{3.08, 0}, {3.1, 0.024},
RoundingRadius->0], RectangleBox[{3.1, 0}, {3.12, 0.017},
RoundingRadius->0], RectangleBox[{3.12, 0}, {3.14, 0.013},
RoundingRadius->0], RectangleBox[{3.14, 0}, {3.16, 0.022},
RoundingRadius->0], RectangleBox[{3.16, 0}, {3.18, 0.013},
RoundingRadius->0], RectangleBox[{3.18, 0}, {3.2, 0.016},
RoundingRadius->0], RectangleBox[{3.2, 0}, {3.22, 0.018},
RoundingRadius->0], RectangleBox[{3.22, 0}, {3.24, 0.012},
RoundingRadius->0], RectangleBox[{3.24, 0}, {3.26, 0.029},
RoundingRadius->0], RectangleBox[{3.26, 0}, {3.28, 0.008},
RoundingRadius->0], RectangleBox[{3.28, 0}, {3.3, 0.019},
RoundingRadius->0], RectangleBox[{3.3, 0}, {3.32, 0.019},
RoundingRadius->0], RectangleBox[{3.32, 0}, {3.34, 0.019},
RoundingRadius->0], RectangleBox[{3.34, 0}, {3.36, 0.017},
RoundingRadius->0], RectangleBox[{3.36, 0}, {3.38, 0.015},
RoundingRadius->0], RectangleBox[{3.38, 0}, {3.4, 0.014},
RoundingRadius->0], RectangleBox[{3.4, 0}, {3.42, 0.009},
RoundingRadius->0], RectangleBox[{3.42, 0}, {3.44, 0.006},
RoundingRadius->0], RectangleBox[{3.44, 0}, {3.46, 0.009},
RoundingRadius->0], RectangleBox[{3.46, 0}, {3.48, 0.011},
RoundingRadius->0], RectangleBox[{3.48, 0}, {3.5, 0.021},
RoundingRadius->0], RectangleBox[{3.5, 0}, {3.52, 0.004},
RoundingRadius->0], RectangleBox[{3.52, 0}, {3.54, 0.008},
RoundingRadius->0], RectangleBox[{3.54, 0}, {3.56, 0.014},
RoundingRadius->0], RectangleBox[{3.56, 0}, {3.58, 0.014},
RoundingRadius->0], RectangleBox[{3.58, 0}, {3.6, 0.013},
RoundingRadius->0], RectangleBox[{3.6, 0}, {3.62, 0.006},
RoundingRadius->0], RectangleBox[{3.62, 0}, {3.64, 0.013},
RoundingRadius->0], RectangleBox[{3.64, 0}, {3.66, 0.01},
RoundingRadius->0], RectangleBox[{3.66, 0}, {3.68, 0.009},
RoundingRadius->0], RectangleBox[{3.68, 0}, {3.7, 0.007},
RoundingRadius->0], RectangleBox[{3.7, 0}, {3.72, 0.01},
RoundingRadius->0], RectangleBox[{3.72, 0}, {3.74, 0.006},
RoundingRadius->0], RectangleBox[{3.74, 0}, {3.76, 0.012},
RoundingRadius->0], RectangleBox[{3.76, 0}, {3.78, 0.009},
RoundingRadius->0], RectangleBox[{3.78, 0}, {3.8, 0.01},
RoundingRadius->0], RectangleBox[{3.8, 0}, {3.82, 0.01},
RoundingRadius->0], RectangleBox[{3.82, 0}, {3.84, 0.006},
RoundingRadius->0], RectangleBox[{3.84, 0}, {3.86, 0.006},
RoundingRadius->0], RectangleBox[{3.86, 0}, {3.88, 0.008},
RoundingRadius->0], RectangleBox[{3.88, 0}, {3.9, 0.007},
RoundingRadius->0], RectangleBox[{3.9, 0}, {3.92, 0.003},
RoundingRadius->0], RectangleBox[{3.92, 0}, {3.94, 0.008},
RoundingRadius->0], RectangleBox[{3.94, 0}, {3.96, 0.008},
RoundingRadius->0], RectangleBox[{3.96, 0}, {3.98, 0.006},
RoundingRadius->0], RectangleBox[{3.98, 0}, {4., 0.006},
RoundingRadius->0], RectangleBox[{4., 0}, {4.02, 0.011},
RoundingRadius->0], RectangleBox[{4.02, 0}, {4.04, 0.008},
RoundingRadius->0], RectangleBox[{4.04, 0}, {4.06, 0.006},
RoundingRadius->0], RectangleBox[{4.06, 0}, {4.08, 0.004},
RoundingRadius->0], RectangleBox[{4.08, 0}, {4.1, 0.007},
RoundingRadius->0], RectangleBox[{4.1, 0}, {4.12, 0.008},
RoundingRadius->0], RectangleBox[{4.12, 0}, {4.14, 0.006},
RoundingRadius->0], RectangleBox[{4.14, 0}, {4.16, 0.009},
RoundingRadius->0], RectangleBox[{4.16, 0}, {4.18, 0.008},
RoundingRadius->0], RectangleBox[{4.18, 0}, {4.2, 0.005},
RoundingRadius->0], RectangleBox[{4.2, 0}, {4.22, 0.011},
RoundingRadius->0], RectangleBox[{4.22, 0}, {4.24, 0.007},
RoundingRadius->0], RectangleBox[{4.24, 0}, {4.26, 0.005},
RoundingRadius->0], RectangleBox[{4.26, 0}, {4.28, 0.004},
RoundingRadius->0], RectangleBox[{4.28, 0}, {4.3, 0.005},
RoundingRadius->0], RectangleBox[{4.3, 0}, {4.32, 0.003},
RoundingRadius->0], RectangleBox[{4.32, 0}, {4.34, 0.006},
RoundingRadius->0], RectangleBox[{4.34, 0}, {4.36, 0.002},
RoundingRadius->0], RectangleBox[{4.36, 0}, {4.38, 0.004},
RoundingRadius->0], RectangleBox[{4.38, 0}, {4.4, 0.001},
RoundingRadius->0], RectangleBox[{4.4, 0}, {4.42, 0.002},
RoundingRadius->0], RectangleBox[{4.42, 0}, {4.44, 0.006},
RoundingRadius->0], RectangleBox[{4.44, 0}, {4.46, 0.001},
RoundingRadius->0], RectangleBox[{4.46, 0}, {4.48, 0.005},
RoundingRadius->0], RectangleBox[{4.48, 0}, {4.5, 0.003},
RoundingRadius->0], RectangleBox[{4.5, 0}, {4.52, 0.003},
RoundingRadius->0], RectangleBox[{4.52, 0}, {4.54, 0.003},
RoundingRadius->0], RectangleBox[{4.54, 0}, {4.56, 0.004},
RoundingRadius->0], RectangleBox[{4.56, 0}, {4.58, 0.004},
RoundingRadius->0], RectangleBox[{4.58, 0}, {4.6, 0.005},
RoundingRadius->0], RectangleBox[{4.6, 0}, {4.62, 0.003},
RoundingRadius->0], RectangleBox[{4.62, 0}, {4.64, 0.003},
RoundingRadius->0], RectangleBox[{4.64, 0}, {4.66, 0.003},
RoundingRadius->0], RectangleBox[{4.66, 0}, {4.68, 0.003},
RoundingRadius->0], RectangleBox[{4.68, 0}, {4.7, 0.004},
RoundingRadius->0], RectangleBox[{4.7, 0}, {4.72, 0.003},
RoundingRadius->0], RectangleBox[{4.72, 0}, {4.74, 0.001},
RoundingRadius->0], RectangleBox[{4.74, 0}, {4.76, 0.004},
RoundingRadius->0], RectangleBox[{4.76, 0}, {4.78, 0.002},
RoundingRadius->0], RectangleBox[{4.78, 0}, {4.8, 0.005},
RoundingRadius->0], RectangleBox[{4.8, 0}, {4.82, 0.005},
RoundingRadius->0], RectangleBox[{4.82, 0}, {4.84, 0.002},
RoundingRadius->0], RectangleBox[{4.84, 0}, {4.86, 0.001},
RoundingRadius->0], RectangleBox[{4.86, 0}, {4.88, 0.002},
RoundingRadius->0], RectangleBox[{4.88, 0}, {4.9, 0.002},
RoundingRadius->0], RectangleBox[{4.9, 0}, {4.92, 0.003},
RoundingRadius->0], RectangleBox[{4.92, 0}, {4.94, 0.001},
RoundingRadius->0], RectangleBox[{4.94, 0}, {4.96, 0.004},
RoundingRadius->0], RectangleBox[{4.96, 0}, {4.98, 0.002},
RoundingRadius->0], RectangleBox[{4.98, 0}, {5., 0.007},
RoundingRadius->0], RectangleBox[{5., 0}, {5.02, 0.002},
RoundingRadius->0], RectangleBox[{5.02, 0}, {5.04, 0.001},
RoundingRadius->0], RectangleBox[{5.04, 0}, {5.06, 0.002},
RoundingRadius->0], RectangleBox[{5.06, 0}, {5.08, 0.001},
RoundingRadius->0], RectangleBox[{5.08, 0}, {5.1, 0.001},
RoundingRadius->0], RectangleBox[{5.1, 0}, {5.12, 0.001},
RoundingRadius->0], RectangleBox[{5.14, 0}, {5.16, 0.001},
RoundingRadius->0], RectangleBox[{5.18, 0}, {5.2, 0.003},
RoundingRadius->0], RectangleBox[{5.2, 0}, {5.22, 0.004},
RoundingRadius->0], RectangleBox[{5.22, 0}, {5.24, 0.002},
RoundingRadius->0], RectangleBox[{5.26, 0}, {5.28, 0.003},
RoundingRadius->0], RectangleBox[{5.28, 0}, {5.3, 0.002},
RoundingRadius->0], RectangleBox[{5.32, 0}, {5.34, 0.002},
RoundingRadius->0], RectangleBox[{5.34, 0}, {5.36, 0.001},
RoundingRadius->0], RectangleBox[{5.36, 0}, {5.38, 0.004},
RoundingRadius->0], RectangleBox[{5.38, 0}, {5.4, 0.002},
RoundingRadius->0], RectangleBox[{5.42, 0}, {5.44, 0.003},
RoundingRadius->0], RectangleBox[{5.44, 0}, {5.46, 0.001},
RoundingRadius->0], RectangleBox[{5.48, 0}, {5.5, 0.002},
RoundingRadius->0], RectangleBox[{5.5, 0}, {5.52, 0.001},
RoundingRadius->0], RectangleBox[{5.54, 0}, {5.56, 0.002},
RoundingRadius->0], RectangleBox[{5.56, 0}, {5.58, 0.002},
RoundingRadius->0], RectangleBox[{5.58, 0}, {5.6, 0.001},
RoundingRadius->0], RectangleBox[{5.6, 0}, {5.62, 0.003},
RoundingRadius->0], RectangleBox[{5.62, 0}, {5.64, 0.001},
RoundingRadius->0], RectangleBox[{5.64, 0}, {5.66, 0.003},
RoundingRadius->0], RectangleBox[{5.66, 0}, {5.68, 0.001},
RoundingRadius->0], RectangleBox[{5.74, 0}, {5.76, 0.001},
RoundingRadius->0], RectangleBox[{5.76, 0}, {5.78, 0.002},
RoundingRadius->0], RectangleBox[{5.78, 0}, {5.8, 0.001},
RoundingRadius->0], RectangleBox[{5.8, 0}, {5.82, 0.002},
RoundingRadius->0], RectangleBox[{5.82, 0}, {5.84, 0.001},
RoundingRadius->0], RectangleBox[{5.88, 0}, {5.9, 0.002},
RoundingRadius->0], RectangleBox[{5.9, 0}, {5.92, 0.002},
RoundingRadius->0], RectangleBox[{5.92, 0}, {5.94, 0.001},
RoundingRadius->0], RectangleBox[{5.96, 0}, {5.98, 0.001},
RoundingRadius->0], RectangleBox[{6.04, 0}, {6.06, 0.001},
RoundingRadius->0], RectangleBox[{6.06, 0}, {6.08, 0.003},
RoundingRadius->0], RectangleBox[{6.1, 0}, {6.12, 0.001},
RoundingRadius->0], RectangleBox[{6.2, 0}, {6.22, 0.001},
RoundingRadius->0], RectangleBox[{6.28, 0}, {6.3, 0.001},
RoundingRadius->0], RectangleBox[{6.3, 0}, {6.32, 0.001},
RoundingRadius->0], RectangleBox[{6.32, 0}, {6.34, 0.001},
RoundingRadius->0], RectangleBox[{6.46, 0}, {6.48, 0.001},
RoundingRadius->0], RectangleBox[{6.54, 0}, {6.56, 0.001},
RoundingRadius->0], RectangleBox[{6.56, 0}, {6.58, 0.001},
RoundingRadius->0], RectangleBox[{6.6, 0}, {6.62, 0.001},
RoundingRadius->0], RectangleBox[{6.7, 0}, {6.72, 0.001},
RoundingRadius->0], RectangleBox[{6.74, 0}, {6.76, 0.001},
RoundingRadius->0], RectangleBox[{6.76, 0}, {6.78, 0.001},
RoundingRadius->0], RectangleBox[{6.8, 0}, {6.82, 0.001},
RoundingRadius->0], RectangleBox[{6.96, 0}, {6.98, 0.001},
RoundingRadius->0], RectangleBox[{7.02, 0}, {7.04, 0.001},
RoundingRadius->0], RectangleBox[{7.08, 0}, {7.1, 0.001},
RoundingRadius->0], RectangleBox[{7.16, 0}, {7.18, 0.001},
RoundingRadius->0], RectangleBox[{7.22, 0}, {7.24, 0.001},
RoundingRadius->0], RectangleBox[{7.28, 0}, {7.3, 0.001},
RoundingRadius->0], RectangleBox[{7.36, 0}, {7.38, 0.001},
RoundingRadius->0], RectangleBox[{7.4, 0}, {7.42, 0.001},
RoundingRadius->0], RectangleBox[{7.44, 0}, {7.46, 0.001},
RoundingRadius->0], RectangleBox[{7.86, 0}, {7.88, 0.001},
RoundingRadius->0], RectangleBox[{7.9, 0}, {7.92, 0.001},
RoundingRadius->0], RectangleBox[{7.94, 0}, {7.96, 0.001},
RoundingRadius->0], RectangleBox[{8., 0}, {8.02, 0.001},
RoundingRadius->0], RectangleBox[{8.28, 0}, {8.3, 0.001},
RoundingRadius->0], RectangleBox[{8.48, 0}, {8.5, 0.002},
RoundingRadius->0], RectangleBox[{8.5, 0}, {8.52, 0.001},
RoundingRadius->0], RectangleBox[{9.18, 0}, {9.2, 0.001},
RoundingRadius->0], RectangleBox[{9.42, 0}, {9.44, 0.001},
RoundingRadius->0], RectangleBox[{9.98, 0}, {10., 0.001},
RoundingRadius->
0]}, {}, {}}, {{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}}}, {{{}, {},
TagBox[
{RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6],
Opacity[1.], LineBox[CompressedData["
1:eJwdVnc8F/7zt/d42/ttvGVHJBLe95JRREWoUEbJKlGUjA8hmaHsyp5Jtqzs
kS07KXvvvfPt97t/7nGPu8fjnnf3uOcdv8VjPUsiAgKCKUICgv/TtYjIdHvY
Fwj+XyjQx5oQXlK+AJgef/SIm5AcSazef3NiFQp3WD9S6v4hRb/p4v8eDb6F
XrVrbg1rJEgwnVLu4DAS7jx/TLUnTIIeKjs/2uWNhdUHH6mYXIlRYd9Y6pbq
BzC9FBJQMUaEDuy0R9atEkB421D7/U0ilEg4ErCrngRSPDjruF+ESK0j/NTO
YDIc5uVwO9gSormYS7VbNqlAMbqiZ0ZEiILvH5tsHqZBqS9HL0MyAeo9tI5Y
480C60ckJzrzJ/CsGXtmNf8T1DyOYt68cgKc7/rallU/g7COBtOnr3/BQgwR
L1rlwvnX8tqB6cdAdOjateWeB7KXWlPFRI6hb8yAZ0s9Hz5hyfu+5R6By2fq
so3BAvipSK5W0HkIWm9nyDYSC0FHuidAzvIQuF1q9ddtimDAXuLjyskB1Kg+
X1s9LAYyX/e8anQA5L8mhJd5yyDK7ezy/tweDNV8c16aKwOtkibu2dg9+JQe
U7+YXw4ZWIdo0NmDq0+umi6oVsKVycj+9PJdiKIsj5q1qgYzulx35pwdKHqu
9eQpew2wlH5+QWm/Ax8W5Dk23WtgzrmdNvzMDth3M1ptqNcCeMfJ0JVvA+OH
70Trg3XAnLjI+XdkCw7pSrIclOrh3csOoYrMLZh6mXJ9LbEe1lg3Od85b0GJ
1X/xqzYNIMtWxmTNuAVGZ2UvrBw2wsH7IUzUzU1Iak1wXORtARNxgs+0lBuw
FVMsNhrcAl+tsoikRtdBw6ptsme/Bcxt+NXoStZhiXjXoLy3FXIF816uWK7D
eeWrFwJet4PH0cOTDx1r0J17RCS82gVEfu19Vx1XgREo79JKdoNO5T5h1plV
UBSKuPD6fjcUMsaOr6+twJutrE3XH90w/im4Qd5pBWTD+y0tsn8AUVRr0p7v
Mni2SmhJm/bC2/SVRPq2RcjM/yqUHdkLQXba4ctRi9ATc5H4VHsvxGrZxT++
twiCVrcq2RX64PT4f1Z6fxegldhXkoCpH3xK3i6dV1wAVuVfjF1NAyDwp2qY
vGsOkOCD1cvHA8DtS/unOmUObKjX2+rODsK5ZtFvqy5zUPmT7FVJ4iDo+za/
/iI4BxbPZHY/ug7Bm5fUXx+/nIUvuQG/HkoOQ9ftRSPLqzOwwLgQrGU+DGNi
2x38EjMg/EwLLxIxDOnb1OSSVDOQqESdNLk/DPzk1kLSzdPwtiX4gVHjLxip
ndmrU5uGZ5Oh65dMfsPi+u8fkjpTUKCxlnwq9DeMj2CCrKWmYDXruj5x3W+4
Y3CWaoJhCqwdGL5WCf+BhDMtpOGDk2B0/Nb93OYfGJILfPXdchISqNt8GNTH
4POomddO8ATgld1MvO6Ogce93sjuJxMwYi9+bvX5GExwRLf+uTUB7D1BM+2f
xmBRWZkj8tQEhMdoa/pjxuHVneuRt2rHwedUJ93Jr3E4v4NN/kY4DlbwI3bp
ySR4DnPlYNRHgSB5p/xX0CSkvNTKCeUehRgS7pHW1EnQU7hDb7D1B763PMBm
DUxCNYZUOi/1D4joHyVbXpiCpkGCvxEUf2DOWvjzH8JpeLHRig0YGQHbtx5V
3WEz4Fp+dJM2fxiItpNHq7NmwD5+ZFk0bBjibn4nyK2bgdWLHSWuj4ehlZtJ
NWRrBhgz+Bi//puLeEZms+atWeiVcwl9k/sTlip6u+t456CoZp3RqHwI7KdF
J4u+zIP60TWvL+sDMHb3J0Vtyzzk2dR1K/cNwI0hf8mOqXnwvOxz/qhkAM63
zb6Y5lwA5fiTEhKPASDOT8Owvl74xw+h4QrUAxDjzo9/broIcZzE7qNS/VDL
xBGrgFmGucDlla/RvSAb8r1KXXwZvoS7mLzw7IV0MpcpXY1lsG64IWBq1QvB
+wNStu7LYEwtGxcl3wuGoxFNcXPLwBNsFJfzswcWPmG2DmtWoNMNiaoI9gAV
v6/w6Pk10JDsmtf6t0fflcKl/lNfA+3RdOUnVd3gdytenkdvDc7G3Q0vyO4G
orDSS8Z2a/BBdDvUw68bDv4uWQ1+XIOAll2L88rdsDhikNlFtA7kqStpSne7
oD1GWLSmfR1eriq6X77UAYFFstKmP9ehkf5KOxLugMvdKgp/p9eh7sPJoDl5
BzSQm2gqnazD6UMNfp7v7VD5LMymVHoDPnDFhjleaYcc/f2svKgNEDHllzS5
3QZvMG1iSWabsLiBRylhLVAmz1N81X4T8s59Y9t50gJTdx/DkdsmVHno33Uw
aAHFHCb9m9GbQB8tcGGYswXmNE08aDs3gRIvoeee8R1UfZa7Xihuwbms6p6G
5mbY26Z31mPdBt+mCwv9Ak2A47E4OcFtQ0WLhWEqZRNcVSsKyJHehpTHg7zv
1hoh7e2teAqdbfip50ryo6oR9KSSmqt8tgHDLVb13aQRcqxlOMXXt+F97I+H
/8U3gMUv/Wri9h141P/0ynf5eqB6P/7I8ecO7Eb6ceQI1EOBkT336MwOxJWf
1GbT1gPx8OsX5YS7sHyugnZnsg5Sh8rPOsrvApVCVWh5eB3M9PNl/kneBbrS
dsqp9Vqw7V4KK3uxB2uS++Ft9TXAGPYChF/vgep/g0aZ+TVQfo1sJSJiD05h
BTvjEmqAuov3ikPuHmADPJVq3Grgc4ceqfD0HjRkPfhLea4GVlpLX0Rc3wcP
n77OE8NqcGz0tXgsfAAKCxHvbK9+A+yUBtnO2QNoPWVLpSj9DdqIKD+5/7t7
1cohuuLM3+AUClkPuH0AMtERBbbDlfCzPNIrNegA9MiNX5hZV4JKXnr8z9UD
WGnmSCELrACG99+HVUsP4eMZ2mWlyTKoKgv8r7XhEBIauTarWsrAbkhbQPfH
IRB0TCCTvDJoZPlhc3fhEO7o/paa8yiDF2E/91y4j8B/O6JNhbMMxl8tsH7x
OgLbB98wYzdLId+B5ga75jF4rMn6b0yXQHsc04scw2NY1TjHVdNRAjMNnAkX
7x/Dk/wYraziEuDkEF186HkMBmzl6x2vSsC3Vt2ntvgYXBlE35kIl4Aho2eh
rcBf4EvgueX4uBgOCteZvh3+BdOw1LgIxiJg/rOnoEd5An8eR8lGHheCJAWB
2SzrCRzfGjLPmSuEeyZ0nxlkTuBCGROhYHUhtJOIqz2wPgFhSRJh4UeFEG9w
34m+/wTG5DBKDzsKIMnNbuHMTQJ0p/jWtkx8Ppypodx7fYcAjRcICwmF5EMN
SSbp6D0CFMPrvSXmng9jIVN8IQ4ESLZkisPGKB+wiXduzgcQoE0VgzvFbPkQ
13CtIamSANlShba4R+bBW1rZeEYBQoQlPo7zv5ALAno92TYihGhQeSEimDsX
8qMcymokCVFD8U/h3L9foJs3p89ekRCxaL4Mu1b/BehkhKjb9AnRa/XYyn2d
LxBoyO7i85oQfS1t9BuwzgHvhCPdrSVCxJaxbdVTmA3Xz5Q6im8SInW9g0ur
sdmArX0abrFPiJR9VOtPeWVD+cRCdzcpEcoibOUZ1c6GjVM/r+VgidDUwpy1
1+wnsPhcrPPgOhFi0kjdtOT/BCpl9ppDhUTIqSlQmCIjE+i0xGzoy4kQJjv9
3amwTBgZnvbXqCFCdvDX6vaLTHA5Mvle1E6ElFOqFoi1MyEPrlwKnyZCOonC
3803MoCvSVhdi40Y1VXxlUmoZgBB7xiq/PfXrpSShN/bSgPf+aok15fESK54
UsP2TxpQEnwkUvAnRt++5nP5f08DxtO3G4ojiRHZ18Iyio9pcOr1j0u5ecSo
sj16k/NSGmgp1l1NniFGBm8bRR8lpMK7lBQTfz0S5K29fZnHLAXYy19+07hN
gthdWuMGtVPgQ7cpltSMBLmePUucrpAC6cdcYy8fkaAy8htV/zGlQOnNiHtu
/iTIJbzVked7MoxQv7K1ryJBDmrGjUeyySDk9MBFX4wUDaRLSuRxJ8HpC6M3
Es6QIkdPb81pmiSQJbgltSBHigL/80sSPU6Ei8GaM56qpGjZVrx+9Hci3E2V
0M82IUWGT0O5CBITIapvQ4oolBQFMJ/tJBZOBNJznrO5m6TILN6K18wzHmgO
d+sO9klRidwMsem9eGCsdYhXJyBDLd6YEpdL8cCnY2Hwi4YMBYuWsR7Rx4OS
pXo9mRAZivRlHYTkj+AUSZ1w5xYZKlg4I5Da9gGmtqMNqb+RoWjdLHEJ6feQ
rHL26e96MqSoAmLKrO/BNKQzNLeVDAkxCszaHMbBsCBpy40hMkRgJoZjbYqD
HzeeKH7YJEMjhd6mJSZxUJWvzXdajBwp/Tlz7uhNLEQ/Ily4Gk2O9JNr4wbI
YkC/7AMZfzw52hOTfyG8Gg2MpOdxm6nkiJGFPzxsMBrefLA3iS4gRzxzTKTZ
mdHg2/arc7SDHFmU64v9vRINDqIlhQ4kFKhoCmTIo6NAa9rW460jBZreob+9
pxwJGh64z2ouFKjU7LLjS9FIuMgyMrzzHwVyUcvZxLJEwgV1nfPGwRSo2i98
PGopAsTSpDZxGRSIs4mWweVDBFBbblkXj1AgwyBBv+WTd9A+5aE/dIkSSTeH
5LL1v4WrU2HiWCwV6pXNqNIoCoN28ak+E0EqRLOVwX0vLQyuPJX/770YFfrE
gpMIiQqDy0R/utnlqdCF68diVC/C4CKf+DOm61QoLuiuYjo+DM6ZNNZQeFOh
6059/SUxocDdt2+wOU2Favp1QhonQiCOS+dEeokKndILyVb4HgIc9xIzHTao
0KvoQ+amnBBg3dA4XP5LhagcBh4KvwgBDCYicY6NGpV/KcuhZAgBYm3Jxd+a
1Chj+VTWefVgWKi38PqeQ42K+svq06sCgTd8yPZNETV6j1FU208NBP27Vw30
K6hRgtICv2VQIFTtKYiNfadGIqJVwf63AuHtaYa+3Ulq9PBhr87hZgAoRlcL
C3PSoNt4pO4qFQDBttxdr/xoUOtWSqt06WuQZBjgUTWlRcn3fFtcqF/BTMiM
d6QlLdoS8dqt2POFj1S7s7N2tGjmR6cqZsYXaEjYC4JdaJGjZM7OYo0vLOzc
1hgIp0XyHXPE9s99IW3kt71Nwz9/CdHUm1kf4M6crg4VpUOlpTJ8Q73e0Ce4
IzghRYcEVyhn+Oq9ISiJLFBWjg41m/wcdynwhoM4Yf2fF+lQwIH0RcNwbxgK
sZkTMKFDtzXfsTde84Z3T5cxJW/o0IBm++iDrpdABdvmI5t0SDQxqNz4pxe4
qIhQlBzQocaslGPpFi+YUTX+EkpIj64rpfOxl3lB3eXag4v09CiDlruJKdYL
XPVC3mWJ0aMLz292N932gkXLU43PzOkRqVmmFnelJ7QHG4gwdNEjTqZzvvaT
HnAh1L9zoZ8etb1OFB5s8oDM8AqnhhF6xFPE0mD8yQN8o/hrny/QIyI36+el
jh6A7d6RWCXCIB0DmbJKQg8Y+FnXaIDFoKJHvyOGhNzh0rLRroABBjGdjhtM
9nWFeGafXs9bGHTlfdRAm50rbClm544YY5DCp/T3NDdcITHw0CrKAoNSch9z
9vO7woHwhyFKBwzS1Rgz86x5AZ8tfpetBWIQRbAR5hTRC8AMmbpX1WBQ2zX7
DI6o5zBUd5/QSIIB8WWNZTAfOcGayjCnhxQDCjzi9uibdwLy2muyiTIM6LLO
gHfqoBPIVV+wmjnPgNx8BqacCp0gogLT8USdATEfFdj02znBtaLKmKC7DIiz
bWHIf/QpNKUxn/kWxoBI0zY4b/Y8gWL/hrt82wzoZrX9g6wJB8jUamN/vceA
Xuo3rXX2OEAcTU/P8iEDmjbpSSGudwCvsFGNCiJGpJhfcDkuxQG0Yw4kDTGM
iPm037qrpQNMZJw5CRJnRNb/TWMPBh8DXfOHhF1zRnTNhmRUldweHpA6j3V2
MqIVe9tntMW2wMnbMnj4gxFRPZa6VvnRFjrP83SJ9DMi2TD2EU8/W5B72PjN
+xcjutGr1Kx2yxbIelney80xogm7uaH6IxtISywxiCdkQozWxr1+mjYwpbjX
+kiWCRnQvp9NXreCe0/cimnimNDG71q//1wsQU7mT7HoRyZUqpBaKPnAEqg2
UIlGIhO62bX9beWGJeQ5kn31SmdCAwkDMvFSlnDkEF66VcCEFh5Ns5XN3oeo
xxkVI21MyOs6a1uP0X1oedhb+/mYCUmYh54/r30PpKzEO7XNmZEVk/A3ktvm
8MeG93HOfWZ0IlRaQqppDsEPmTB01szo2wFjJouCOcw7Hup22TMjUyZuBQsO
c0hxb+vXdWdGu/4q/p+HzYDtrd1vw2hmxFxOnb5vagYnldlLZh3MSCfAbe30
wl3oYpSgdlZgQQPVt8/yhxgD/tGj+fdKLIj2aF3V09kYvjR/aa4DFnR1zmR8
/Y4xvHGX9sVosCBen+MXzFLGoD0jd5ytx4LOi6iK6vQYQWuZyuqEHQsSaf69
6M5tBI1mN3uvx7Mgfs2nSSbVt6Dii/d7CWJWlJN0FOSjaAhk2gac0WSsaCbt
scqckCHozgvHElKxotJD22tmjIYwi+uIGsCwItJXsuKB8wbAFMv21gvLit4S
RIzXxRiAnc9n/z4FVnQtO96z9EAfuG4NOLs7sKJji8+VCu03wI1Q7Hr7b1aU
FrbDu5WkCzNq1ENC46yIxlLRcitcF677L5m+nGJF7Va+xsTeuiBIn/tYbpEV
6So9ENSx0IV2btnQxL1/+c1Ys9/gdAF7Ht/pxMSGYu/H937LvA519nraPJps
iH5dvHLC9RpQjbheti9iQzWecmPbnNog9zyLNvQrG1qoHRBWItYGc8ahntxy
NrR9NSYvavEKlF4+d2e9hg3Jn7MG/8or8KB41dGpgw1pO+3ZGptegdo399+7
zrCh0zO16wWZWv/47+qKHzs76iP9HpSvqQnT6fwRCe7saLzIU+5JrQYs+Y7v
E3uxI1rjjvufPmvAhkWSqbUPO9pfX63ZjtYAAl5+celAdlTRT1LRaK8BnNF8
dXXR7Mjfz7jhMo8GXH/NuzZdwI6mTDV62jzUodKKR1tinh0t9fs+vKqtBhEi
HCRlBhzo+J2X7zeui2ByVyhj8hYHslvcjDSkuQiCEWe16Ew40O2AnImjIxUo
OtEJu2fBgZzMt04c/qhA34A3N91jDqTvTku1k6QCTK+WZO/5c6AADjlPG3EV
eDtebUlbwYE+F1qQZV1CEBb3oNmcjxM5KU8q+2oow/Nn3k/dcZyI8c1pFhFJ
ZbirF88bLcSJChRqLMZYlEGCauB5uwQnwowqc72ZVoKWF+oi8gqciA5rhB15
pQREtwUDaPQ4UbFEtM/ZVkVwZpvQ+urDidqZaZ4Zm18Ao4g7nTSznCh5+cM4
Z5M8cBdScdktcKLyI0Oy3Dx5GP3x1aplmRP59nxJNXwvD/fpGQn9tjjRKfSZ
bMxBHuwDm86eEHIhO948+5/c8uDtJRW7zs2Fvh7ccLvpIgdZDwnvD+hxoUun
Ws5RKp+DPbW0/YQqLpSj5nZUvi8D7hUJPg21XIhx+MrXznkZIJCJo51v4ELX
80oWV3/KADlvKL9MGxeiLtR5Z1QuA6z7LpoNg1xoWer6xy43GTibox07t8aF
5uqsKyQJZcCeZeu8NI4b1WWk0juxSMPk9EWXutfciJo1uT6pXhIuJnQmiwVx
Iz+iE+rCDElIvGXU8fYNN+Ji0J/+HSQJd9scBe5FciMNduclT31JGM5LbCdO
4UZqklqVGzOnocftL596FTciCrET9sSchjqG8pbmLW7U6befH+ksDsnK0lwd
5jwo0SHdE+MjAnVXqXwO7/Og+PfRSrr2IjBhOrkgas2DAtmfZqfcFgGcd2S5
nz0PKvtMbuFxRgRSm/ZvIzceVERhEUQ0Kgxp1+piCiN4UMbNmGoNEIZM8xus
cc08qB6N9F6nEYKcV84MVqexiL6emNOyHwdLSne8X53BIrH0VsOdBhxIbKlt
ppzFoqZjK63IIhx8smDuH1PAIsr8E+GjdzjIhMIYIw0s6oQKt4c3cJC6v8Z7
1RSL5ktWP3b0CUDco4eS58KxyCNt20dtnB9eG9y7QryNRQyd03alWD5IvjZi
YrmHRQnBiqUqGD6o1DSwbz7EIt6PKyxDhHywoXwpPIiIFyUp52bfbOUFEyHx
QSYML8qVo7owcosXZHY3LATFedGNY4fTGa5Y+BPj7apuzou2+VqT0rq54dxI
UtbrDl4UvcsTFlbLARhJ673cbl7EXGzWQ/aZA5Y8JS8N9fIiSpfVc6FRHJCK
q5gSGeZFRXHOuAE7DmC26+NtneFFnfUpSxfYOODAnfgKGQEfojMSIrvhyA4/
qeVqL3PwoeK6lgeGMmwQI/o+p+PKP/ui+NO4KWbY2tRI89HhQ/MiCT8KfjDD
taqNDwrX+FBH5PiDkSpmILuhFZymx4c2Iwt3LGKZwdl939bjNh8SOD+7pXiN
GfS6boqctuJDldoZ0/RVTEDjzJwS7M2HllYitAcyGMGzJiTmStk/v9Kjw6pM
DCz8ronYL+dDcx/1onZjMWB4uBmWUcmHXmOGzJWDMCAhZxRAXMOHuq6aGa89
wsBQtpBrRRMfshWlCt04iwGZ6Gpj8T4+xJp3OanClx5mHq1jqVf5EE1BnTzh
D1rQ4TJIaxXkRwcTYcnKuVQQMyVZYyzEj+qPG85JxVLBVA7FryVhflTw4Ah7
1ocK3FElhl6cH+G8eEocb1LBZ0uc+w3pf/EWUdrfCaiAKm9db0SZH03NlK12
GFJCs9obghVDfiSS46bVR08BKvbNxgwB/Ojy6mpMVQEpHAzNazQE8iMDw7BS
vQRSKFKlkXkezI9mg7f19oJIQYhDl+J3KD/6pdFY+tySFGgahosyo/jRj8HE
MHUOUhjiWKFFqfzoS5PiOL0vCTg0MtfYV/MjAvFps21LYkjgthBs3+ZH90T8
bSduEsKFs+KSFrv8SHT5vqm1JiH0aW7J7+3xIwyOx/XgAiFQPPe7InjEj07y
+fYuYgnB4cenJ+5EAsgUa75/b5oAkN9mjQS9AOK1slH77UwAE6u+d0JEBFD2
Ze3MbvwJXrA+I+qqsQDa4r2Bsk8d4Tsl9lJHTAQQVbADURbdEd4l6nKh7V0B
1Duzk122e4hvs5nv8jMXQIUv254xtR7in2DEKautBFAZmYZCxuNDfPWdL25S
TgJImEqY8mLVAd5or8gMEyKAsAcDp0as9/Fhp+vEeqoEkFWXupvv2g4+87wx
pXeNAJLVUbwoM7KDr1bdmpWuE0D5VgJ8G807+JXbQmnhjQKIXpt6LCNhB3/F
L5BXt10A6Xuafpm/toMnG9Vj7v4pgOyGCNNKC7fxbmGTxx1bAkjp0GWEzX8L
/2CT5EeLGA7R3XxXlm62gb/7Sa3TRAKHYkSJnr7Q3cAbmvu2rZ7Goa3fLA23
Lm7gNbqIm1ikcch2RZJYWXADL5RNVGEuj0NsoqI/4ubW8dMWBGn7qjhUbD+Q
mfx0HX+/59BF9A4OTWEsFj+8XcNb5G/wBYTiUFqUivOlqWX8Q+8bR1rhOHTn
didB7fdlvPONokGadzjUuChPpJ+zjPffdn4TFoVDwfwkFE3PlvFfFPYPoz/i
kJw2+ZtBqmX8fu3JQHo2Dp0j84IP8kv4sB6akIZmHPrTNutMl7KAj0t5ZO3X
gkPp0ZxegYEL+BSnTtXLbf/q1Vo7y/NkAV/CGnbQ1olDsfIman4qC/gRIybr
3n4c+vLSqvr++DxeeJJDdWIShxy9n7CpnJrHnylyxaZO45BB7F1yLN08/sKr
X/uWs//wJmNuY3bn8NrCH/PmF3AogNuqR7ZlDv/Ejg+7vo5DODGdbVf7Oby7
0sv9gk0cUnQ2aCO4NYd/RTvR57T9r/5o1QfvVebwMbkpQXt7ODRWU/GRjmUO
n+xFYlV+8K/f27NMs39n8dm6lhfdj/7hUcAT9c3N4osEmnjwf3HokX6lQX/P
LL5qU2j/5ASH1rscDxcqZ/H/A1ENBg0=
"]]},
Annotation[#, "Charting`Private`Tag$6201388#1"]& ]}, {}}},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->{True, True},
AxesLabel->{None, None},
AxesOrigin->{-0.054000000000000006`, 0},
FrameLabel->{{None, None}, {None, None}},
FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
GridLines->{None, None},
GridLinesStyle->Directive[
GrayLevel[0.5, 0.4]],
PlotRange->NCache[{{0,
Rational[27, 10]}, {All, All}}, {{0, 2.7}, {All, All}}],
PlotRangePadding->{{
Scaled[0.02],
Scaled[0.02]}, {
Scaled[0.02],
Scaled[0.05]}},
Ticks->{Automatic, Automatic}]], "Output",
CellChangeTimes->{{3.758565177064314*^9, 3.758565201507423*^9}, {
3.7950270968726797`*^9, 3.7950271129644203`*^9}, {3.7950385610943413`*^9,
3.795038566835258*^9}, {3.795053547307488*^9, 3.79505355803351*^9}},
CellLabel->
"Out[1782]=",ExpressionUUID->"bd4ad9e0-255a-44f5-b022-8e6e297be157"]
}, Open ]]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["5. Finding continuous superpositions of a given f(r)", "Section",
CellChangeTimes->{{3.79503431268128*^9, 3.79503431575909*^9}, {
3.7950439583720427`*^9, 3.79504395881502*^9}, {3.7950535853763733`*^9,
3.795053587689913*^9}},ExpressionUUID->"6c014522-e1a0-409a-9da1-\
00164d5740d1"],
Cell[CellGroupData[{
Cell[TextData[{
"When the inverse Laplace transform ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SubsuperscriptBox["\[ScriptCapitalL]", "r",
RowBox[{"-", "1"}]], "[",
RowBox[{"f", "(", "r", ")"}], "]"}],
RowBox[{"(", "s", ")"}]}], TraditionalForm]],
CellChangeTimes->{3.79503435556837*^9},ExpressionUUID->
"5f1afaca-6dbb-42a7-8822-cb36ed25748b"],
" is known (superposition of exponentials)"
}], "Subsection",
CellChangeTimes->{{3.7950343194514437`*^9, 3.795034360498578*^9}, {
3.795034398277055*^9,
3.795034404678817*^9}},ExpressionUUID->"9179e88b-58b3-48fc-a192-\
4d52ec601909"],
Cell[TextData[{
"When the inverse Laplace transform ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SubsuperscriptBox["\[ScriptCapitalL]", "r",
RowBox[{"-", "1"}]], "[",
RowBox[{"f", "(", "r", ")"}], "]"}],
RowBox[{"(", "s", ")"}]}], TraditionalForm]],
CellChangeTimes->{3.79503435556837*^9},ExpressionUUID->
"8b309a48-77df-458a-8632-6df198e17f0d"],
" of f(r) is known, then a weight function for a superposition of \
exponentials s ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SuperscriptBox["e",
RowBox[{
RowBox[{"-", "s"}], " ", "r"}]], " ", "is", " ",
RowBox[{"w", "(", "s", ")"}]}], " ", "=", " "}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"c7452c0c-e548-4e17-9872-7785aeadcec7"],
Cell[BoxData[
FormBox[
FractionBox[
RowBox[{
RowBox[{
SubsuperscriptBox["\[ScriptCapitalL]", "r",
RowBox[{"-", "1"}]], "[",
RowBox[{"f", "(", "r", ")"}], "]"}],
RowBox[{"(", "s", ")"}]}], "s"], TraditionalForm]],
CellChangeTimes->{3.79503435556837*^9},ExpressionUUID->
"e9ad6b0b-d264-49d9-b374-b19868e42aac"],
". If w(s) can be sampled, then we have a new 2-random-number sampling \
procedure for f(r)."
}], "Text",
CellChangeTimes->{{3.795035231716831*^9, 3.7950352509160337`*^9}, {
3.795043879058278*^9, 3.7950439091234713`*^9}, {3.795053600497447*^9,
3.7950536217096567`*^9}},ExpressionUUID->"27da7768-f303-4542-8099-\
0e62b76fb772"],
Cell[CellGroupData[{
Cell["Example 5.1:", "Subsubsection",
CellChangeTimes->{{3.7950439207937803`*^9, 3.7950439220731897`*^9}, {
3.7950439611910963`*^9,
3.79504396138199*^9}},ExpressionUUID->"84522e16-fe39-460f-bbd4-\
ee8efbda9abb"],
Cell["Consider the distribution f(r) given by:", "Text",
CellChangeTimes->{{3.795044514734785*^9, 3.79504451537891*^9}, {
3.795044593002843*^9,
3.7950446014719152`*^9}},ExpressionUUID->"ed8a83e9-1245-45c3-b86b-\
607316b72aa4"],
Cell[BoxData[
RowBox[{
RowBox[{"f", "[", "r_", "]"}], ":=",
RowBox[{
SqrtBox[
FractionBox["2", "\[Pi]"]], "-",
RowBox[{
SuperscriptBox["\[ExponentialE]",
FractionBox[
SuperscriptBox["r", "2"], "2"]], " ", "r", " ",
RowBox[{"Erfc", "[",
FractionBox["r",
SqrtBox["2"]], "]"}]}]}]}]], "Input",
CellChangeTimes->{{3.795044516890581*^9, 3.795044518561034*^9}},
CellLabel->
"In[1726]:=",ExpressionUUID->"8643fc8a-1146-4901-9361-72b52900a600"],
Cell["We check the inverse Laplace:", "Text",
CellChangeTimes->{{3.7950445238025703`*^9,
3.7950445299201097`*^9}},ExpressionUUID->"f1c22aad-eb46-4c6b-87a4-\
9d84d83a65db"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
FractionBox["1", "s"],
RowBox[{"FullSimplify", "[",
RowBox[{
RowBox[{"InverseLaplaceTransform", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",", "r", ",", "s"}], "]"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"s", ">", "0"}]}]}], "]"}]}]], "Input",
CellChangeTimes->{{3.795044531497806*^9, 3.795044543470605*^9}},
CellLabel->
"In[1727]:=",ExpressionUUID->"4e3b0010-31f2-47fc-a42d-3dc72e40344a"],
Cell[BoxData[
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{"-",
FractionBox[
SuperscriptBox["s", "2"], "2"]}]], " ",
SqrtBox[
FractionBox["2", "\[Pi]"]]}]], "Output",
CellChangeTimes->{3.795044545505468*^9},
CellLabel->
"Out[1727]=",ExpressionUUID->"4ffd8cba-e971-4303-b279-c84aa5a9d2bb"]
}, Open ]],
Cell["\<\
This is a weighting function w(s), which is simply a Gaussian/Normal \
distribution, which is easily sampled.\
\>", "Text",
CellChangeTimes->{{3.7950445556309853`*^9,
3.795044573904298*^9}},ExpressionUUID->"e219c20f-cfc7-4629-92c4-\
f1fb33311337"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Show", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"Histogram", "[", "\[IndentingNewLine]",
RowBox[{
RowBox[{"Table", "[",
RowBox[{
FractionBox[
RowBox[{"-",
RowBox[{"Log", "[",
RowBox[{"RandomReal", "[", "]"}], "]"}]}],
RowBox[{" ",
RowBox[{"Abs", "[",
RowBox[{"RandomVariate", "[",
RowBox[{"NormalDistribution", "[", "]"}], "]"}], "]"}]}]], ",",
RowBox[{"{",
RowBox[{"i", ",",
RowBox[{"Range", "[", "30000", "]"}]}], "}"}]}], "]"}], ",",
RowBox[{"{",
RowBox[{"0", ",", "10", ",", "0.03"}], "}"}], ",", "\"\<PDF\>\""}],
"]"}], ",", "\[IndentingNewLine]",
RowBox[{"Plot", "[",
RowBox[{
RowBox[{"f", "[", "r", "]"}], ",",
RowBox[{"{",
RowBox[{"r", ",", "0", ",", "10"}], "}"}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}]}],
"\[IndentingNewLine]", "]"}]], "Input",
CellChangeTimes->{{3.795044929095886*^9, 3.7950449348473053`*^9}, {
3.795045041681753*^9, 3.795045062339698*^9}},
CellLabel->
"In[1750]:=",ExpressionUUID->"adb35330-59ce-4301-af2f-75a82c7f098c"],
Cell[BoxData[
GraphicsBox[{{
{RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[
Opacity[0.]], {},
{RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[
Opacity[0.]], RectangleBox[{0., 0}, {0.03, 0.8428897008284195},
RoundingRadius->0],
RectangleBox[{0.03, 0}, {0.06, 0.8260077897960907},
RoundingRadius->0], RectangleBox[{0.06, 0}, {0.09, 0.844095551616443},
RoundingRadius->0],
RectangleBox[{0.09, 0}, {0.12, 0.7391865330583994},
RoundingRadius->0],
RectangleBox[{0.12, 0}, {0.15, 0.7572742948787518},
RoundingRadius->0],
RectangleBox[{0.15, 0}, {0.18, 0.7319514283302585},
RoundingRadius->0],
RectangleBox[{0.18, 0}, {0.21, 0.6547769778967552},
RoundingRadius->0],
RectangleBox[{0.21, 0}, {0.24, 0.6427184700165202},
RoundingRadius->0],
RectangleBox[{0.24, 0}, {0.27, 0.6173956034680264},
RoundingRadius->0], RectangleBox[{0.27, 0}, {0.3, 0.6378950668644268},
RoundingRadius->0],
RectangleBox[{0.3, 0}, {0.32999999999999996, 0.6065429463758161},
RoundingRadius->0],
RectangleBox[{0.32999999999999996, 0}, {0.36, 0.6029253940117445},
RoundingRadius->0],
RectangleBox[{0.36, 0}, {0.39, 0.5655440195830164},
RoundingRadius->0],
RectangleBox[{0.39, 0}, {0.42, 0.5185158388501012},
RoundingRadius->0],
RectangleBox[{0.42, 0}, {0.44999999999999996, 0.49439882308963135},
RoundingRadius->0],
RectangleBox[{0.44999999999999996, 0}, {0.48, 0.5088690325459123},
RoundingRadius->0],
RectangleBox[{0.48, 0}, {0.51, 0.5088690325459123},
RoundingRadius->0],
RectangleBox[{0.51, 0}, {0.54, 0.48354616599741906},
RoundingRadius->0],
RectangleBox[{0.54, 0}, {0.57, 0.4582232994489275},
RoundingRadius->0],
RectangleBox[{0.57, 0}, {0.6, 0.38948980453158694},
RoundingRadius->0], RectangleBox[{0.6, 0}, {0.63, 0.4148126710800802},
RoundingRadius->0],
RectangleBox[{0.63, 0}, {0.6599999999999999, 0.42566532817229313},
RoundingRadius->0],
RectangleBox[{0.6599999999999999, 0}, {0.69, 0.3810488490154225},
RoundingRadius->0],
RectangleBox[{0.69, 0}, {0.72, 0.3955190584717044},
RoundingRadius->0],
RectangleBox[{0.72, 0}, {0.75, 0.38828395374356345},
RoundingRadius->0],
RectangleBox[{0.75, 0}, {0.78, 0.3460791761627413},
RoundingRadius->0],
RectangleBox[{0.78, 0}, {0.8099999999999999, 0.361755236407048},
RoundingRadius->0],
RectangleBox[{0.8099999999999999, 0}, {0.84, 0.3364323698585534},
RoundingRadius->0],
RectangleBox[{0.84, 0}, {0.87, 0.3472850269507648},
RoundingRadius->0],
RectangleBox[{0.87, 0}, {0.8999999999999999, 0.33402066828250765},
RoundingRadius->0],
RectangleBox[{0.8999999999999999, 0}, {0.9299999999999999, 0.3340206682825064},
RoundingRadius->0],
RectangleBox[{0.9299999999999999, 0}, {0.96, 0.33040311591843596},
RoundingRadius->0],
RectangleBox[{0.96, 0}, {0.99, 0.30266854779389574},
RoundingRadius->0],
RectangleBox[{0.99, 0}, {1.02, 0.30266854779389574},
RoundingRadius->0],
RectangleBox[{1.02, 0}, {1.05, 0.26649302415319104},
RoundingRadius->0],
RectangleBox[{1.05, 0}, {1.08, 0.2773456812454025},
RoundingRadius->0],
RectangleBox[{1.08, 0}, {1.1099999999999999, 0.26649302415319304},
RoundingRadius->0],
RectangleBox[{1.1099999999999999, 0}, {1.14, 0.24478770996876825},
RoundingRadius->0], RectangleBox[{1.14, 0}, {1.17, 0.278551532033426},
RoundingRadius->0],
RectangleBox[{1.17, 0}, {1.2, 0.24719941154481523},
RoundingRadius->0], RectangleBox[{1.2, 0}, {1.23, 0.268904725729238},
RoundingRadius->0],
RectangleBox[{1.23, 0}, {1.26, 0.25443451627295616},
RoundingRadius->0],
RectangleBox[{1.26, 0}, {1.29, 0.23755260524062732},
RoundingRadius->0],
RectangleBox[{1.29, 0}, {1.3199999999999998, 0.22187654499632362},
RoundingRadius->0],
RectangleBox[{1.3199999999999998, 0}, {1.3499999999999999, 0.22669994814841593},
RoundingRadius->0],
RectangleBox[{1.3499999999999999, 0}, {1.38, 0.19414197687178172},
RoundingRadius->0],
RectangleBox[{1.38, 0}, {1.41, 0.20861218632806358},
RoundingRadius->0],
RectangleBox[{1.41, 0}, {1.44, 0.21343558948015756},
RoundingRadius->0],
RectangleBox[{1.44, 0}, {1.47, 0.20378878317596963},
RoundingRadius->0],
RectangleBox[{1.47, 0}, {1.5, 0.20137708159992265},
RoundingRadius->0], RectangleBox[{1.5, 0}, {1.53, 0.1736425134753824},
RoundingRadius->0],
RectangleBox[{1.53, 0}, {1.56, 0.17002496111131193},
RoundingRadius->0],
RectangleBox[{1.56, 0}, {1.5899999999999999, 0.20861218632806514},
RoundingRadius->0],
RectangleBox[{1.5899999999999999, 0}, {1.6199999999999999, 0.2074063355400401},
RoundingRadius->0],
RectangleBox[{1.6199999999999999, 0}, {1.65, 0.1519371992909596},
RoundingRadius->0],
RectangleBox[{1.65, 0}, {1.68, 0.18328931977957033},
RoundingRadius->0],
RectangleBox[{1.68, 0}, {1.71, 0.19414197687178172},
RoundingRadius->0],
RectangleBox[{1.71, 0}, {1.74, 0.16520155795921798},
RoundingRadius->0],
RectangleBox[{1.74, 0}, {1.77, 0.14711379613886566},
RoundingRadius->0],
RectangleBox[{1.77, 0}, {1.7999999999999998, 0.1676132595352662},
RoundingRadius->0],
RectangleBox[{1.7999999999999998, 0}, {1.8299999999999998, 0.14952549771491264},
RoundingRadius->0],
RectangleBox[{1.8299999999999998, 0}, {1.8599999999999999, 0.16640740874724147},
RoundingRadius->0],
RectangleBox[{1.8599999999999999, 0}, {1.89, 0.1760542150514294},
RoundingRadius->0],
RectangleBox[{1.89, 0}, {1.92, 0.11696752643827843},
RoundingRadius->0],
RectangleBox[{1.92, 0}, {1.95, 0.14831964692688915},
RoundingRadius->0],
RectangleBox[{1.95, 0}, {1.98, 0.14590794535084217},
RoundingRadius->0],
RectangleBox[{1.98, 0}, {2.01, 0.14108454219874925},
RoundingRadius->0],
RectangleBox[{2.01, 0}, {2.04, 0.12179092959037147},
RoundingRadius->0],
RectangleBox[{2.04, 0}, {2.07, 0.1133499740742088},
RoundingRadius->0],
RectangleBox[{2.07, 0}, {2.1, 0.14711379613886455},
RoundingRadius->0],
RectangleBox[{2.1, 0}, {2.13, 0.13505528825863175},
RoundingRadius->0],
RectangleBox[{2.13, 0}, {2.16, 0.12179092959037147},
RoundingRadius->0],
RectangleBox[{2.16, 0}, {2.19, 0.11817337722630279},
RoundingRadius->0],
RectangleBox[{2.19, 0}, {2.2199999999999998, 0.10008561540595032},
RoundingRadius->0],
RectangleBox[{2.2199999999999998, 0}, {2.25, 0.1193792280143245},
RoundingRadius->0],
RectangleBox[{2.25, 0}, {2.28, 0.1157616756502558},
RoundingRadius->0],
RectangleBox[{2.28, 0}, {2.31, 0.1024973169819958},
RoundingRadius->0],
RectangleBox[{2.31, 0}, {2.34, 0.1193792280143263},
RoundingRadius->0],
RectangleBox[{2.34, 0}, {2.37, 0.10370316777001927},
RoundingRadius->0],
RectangleBox[{2.37, 0}, {2.4, 0.12420263116642027},
RoundingRadius->0],
RectangleBox[{2.4, 0}, {2.4299999999999997, 0.09526221225385634},
RoundingRadius->0],
RectangleBox[{2.4299999999999997, 0}, {2.46, 0.09767391382990187},
RoundingRadius->0],
RectangleBox[{2.46, 0}, {2.4899999999999998, 0.10370316777002081},
RoundingRadius->0],
RectangleBox[{2.4899999999999998, 0}, {2.52, 0.11455582486223059},
RoundingRadius->0],
RectangleBox[{2.52, 0}, {2.55, 0.09285051067780933},
RoundingRadius->0],
RectangleBox[{2.55, 0}, {2.58, 0.08802710752571404},
RoundingRadius->0],
RectangleBox[{2.58, 0}, {2.61, 0.08320370437362136},
RoundingRadius->0],
RectangleBox[{2.61, 0}, {2.6399999999999997, 0.08561540594966835},
RoundingRadius->0],
RectangleBox[{2.6399999999999997, 0}, {2.67, 0.10129146619397231},
RoundingRadius->0],
RectangleBox[{2.67, 0}, {2.6999999999999997, 0.10129146619397382},
RoundingRadius->0],
RectangleBox[{2.6999999999999997, 0}, {2.73, 0.09405636146583143},
RoundingRadius->0],
RectangleBox[{2.73, 0}, {2.76, 0.09043880910176234},
RoundingRadius->0],
RectangleBox[{2.76, 0}, {2.79, 0.07838030122152619},
RoundingRadius->0],
RectangleBox[{2.79, 0}, {2.82, 0.07958615200955085},
RoundingRadius->0],
RectangleBox[{2.82, 0}, {2.85, 0.07235104728140879},
RoundingRadius->0],
RectangleBox[{2.85, 0}, {2.88, 0.09043880910176234},
RoundingRadius->0],
RectangleBox[{2.88, 0}, {2.9099999999999997, 0.07717445043350386},
RoundingRadius->0],
RectangleBox[{2.9099999999999997, 0}, {2.94, 0.07355689806943227},
RoundingRadius->0],
RectangleBox[{2.94, 0}, {2.9699999999999998, 0.07717445043350386},
RoundingRadius->0],
RectangleBox[{2.9699999999999998, 0}, {3., 0.06511594255326791},
RoundingRadius->0], RectangleBox[{3., 0}, {3.03, 0.07596859964548036},
RoundingRadius->0],
RectangleBox[{3.03, 0}, {3.06, 0.08320370437362011},
RoundingRadius->0],
RectangleBox[{3.06, 0}, {3.09, 0.07596859964548036},
RoundingRadius->0],
RectangleBox[{3.09, 0}, {3.12, 0.07838030122152619},
RoundingRadius->0],
RectangleBox[{3.12, 0}, {3.15, 0.05305743467303391},
RoundingRadius->0],
RectangleBox[{3.15, 0}, {3.1799999999999997, 0.06391009176524538},
RoundingRadius->0],
RectangleBox[{3.1799999999999997, 0}, {3.21, 0.06391009176524444},
RoundingRadius->0],
RectangleBox[{3.21, 0}, {3.2399999999999998, 0.06029253940117489},
RoundingRadius->0],
RectangleBox[{3.2399999999999998, 0}, {3.27, 0.059086688613150515},
RoundingRadius->0],
RectangleBox[{3.27, 0}, {3.3, 0.06752764412931588},
RoundingRadius->0],
RectangleBox[{3.3, 0}, {3.33, 0.057880837825127034},
RoundingRadius->0],
RectangleBox[{3.33, 0}, {3.36, 0.059086688613151396},
RoundingRadius->0],
RectangleBox[{3.36, 0}, {3.3899999999999997, 0.05305743467303391},
RoundingRadius->0],
RectangleBox[{3.3899999999999997, 0}, {3.42, 0.06752764412931488},
RoundingRadius->0],
RectangleBox[{3.42, 0}, {3.4499999999999997, 0.06873349491733938},
RoundingRadius->0],
RectangleBox[{3.4499999999999997, 0}, {3.48, 0.05667498703710355},
RoundingRadius->0],
RectangleBox[{3.48, 0}, {3.51, 0.06993934570536288},
RoundingRadius->0],
RectangleBox[{3.51, 0}, {3.54, 0.059086688613150515},
RoundingRadius->0],
RectangleBox[{3.54, 0}, {3.57, 0.055469136249080904},
RoundingRadius->0],
RectangleBox[{3.57, 0}, {3.5999999999999996, 0.0566749870371044},
RoundingRadius->0],
RectangleBox[{3.5999999999999996, 0}, {3.63, 0.057880837825127034},
RoundingRadius->0],
RectangleBox[{3.63, 0}, {3.6599999999999997, 0.0542632854610574},
RoundingRadius->0],
RectangleBox[{3.6599999999999997, 0}, {3.69, 0.044616479156868755},
RoundingRadius->0],
RectangleBox[{3.69, 0}, {3.7199999999999998, 0.049439882308963415},
RoundingRadius->0],
RectangleBox[{3.7199999999999998, 0}, {3.75, 0.051851583885009635},
RoundingRadius->0],
RectangleBox[{3.75, 0}, {3.78, 0.043410628368845926},
RoundingRadius->0],
RectangleBox[{3.78, 0}, {3.81, 0.04943988230896267},
RoundingRadius->0],
RectangleBox[{3.81, 0}, {3.84, 0.06632179334129239},
RoundingRadius->0],
RectangleBox[{3.84, 0}, {3.8699999999999997, 0.04099892679279893},
RoundingRadius->0],
RectangleBox[{3.8699999999999997, 0}, {3.9, 0.057880837825127034},
RoundingRadius->0],
RectangleBox[{3.9, 0}, {3.9299999999999997, 0.049439882308963415},
RoundingRadius->0],
RectangleBox[{3.9299999999999997, 0}, {3.96, 0.060292539401173996},
RoundingRadius->0],
RectangleBox[{3.96, 0}, {3.9899999999999998, 0.045822329944892916},
RoundingRadius->0],
RectangleBox[{3.9899999999999998, 0}, {4.02, 0.03858722521675193},
RoundingRadius->0],
RectangleBox[{4.02, 0}, {4.05, 0.03617552364070439},
RoundingRadius->0],
RectangleBox[{4.05, 0}, {4.08, 0.04943988230896267},
RoundingRadius->0],
RectangleBox[{4.08, 0}, {4.109999999999999, 0.04702818073291711},
RoundingRadius->0],
RectangleBox[{4.109999999999999, 0}, {4.14, 0.04099892679279832},
RoundingRadius->0],
RectangleBox[{4.14, 0}, {4.17, 0.04943988230896267},
RoundingRadius->0],
RectangleBox[{4.17, 0}, {4.2, 0.037381374428727875},
RoundingRadius->0],
RectangleBox[{4.2, 0}, {4.2299999999999995, 0.04823403152094063},
RoundingRadius->0],
RectangleBox[{4.2299999999999995, 0}, {4.26, 0.026528717336516558},
RoundingRadius->0],
RectangleBox[{4.26, 0}, {4.29, 0.037381374428727875},
RoundingRadius->0],
RectangleBox[{4.29, 0}, {4.32, 0.03376382206465744},
RoundingRadius->0],
RectangleBox[{4.32, 0}, {4.35, 0.04823403152094063},
RoundingRadius->0],
RectangleBox[{4.35, 0}, {4.38, 0.043410628368845273},
RoundingRadius->0],
RectangleBox[{4.38, 0}, {4.41, 0.053057434673033116},
RoundingRadius->0],
RectangleBox[{4.41, 0}, {4.4399999999999995, 0.04220477758082305},
RoundingRadius->0],
RectangleBox[{4.4399999999999995, 0}, {4.47, 0.04220477758082179},
RoundingRadius->0],
RectangleBox[{4.47, 0}, {4.5, 0.03617552364070439},
RoundingRadius->0],
RectangleBox[{4.5, 0}, {4.53, 0.03496967285268092},
RoundingRadius->0],
RectangleBox[{4.53, 0}, {4.56, 0.021705314184423282},
RoundingRadius->0],
RectangleBox[{4.56, 0}, {4.59, 0.038587225216751356},
RoundingRadius->0],
RectangleBox[{4.59, 0}, {4.62, 0.024117015760469596},
RoundingRadius->0],
RectangleBox[{4.62, 0}, {4.6499999999999995, 0.03496967285268195},
RoundingRadius->0],
RectangleBox[{4.6499999999999995, 0}, {4.68, 0.03255797127663396},
RoundingRadius->0],
RectangleBox[{4.68, 0}, {4.71, 0.030146269700586998},
RoundingRadius->0],
RectangleBox[{4.71, 0}, {4.74, 0.038587225216751356},
RoundingRadius->0],
RectangleBox[{4.74, 0}, {4.77, 0.026528717336517346},
RoundingRadius->0],
RectangleBox[{4.77, 0}, {4.8, 0.038587225216751356},
RoundingRadius->0],
RectangleBox[{4.8, 0}, {4.83, 0.03376382206465744},
RoundingRadius->0],
RectangleBox[{4.83, 0}, {4.859999999999999, 0.031352120488611405},
RoundingRadius->0],
RectangleBox[{4.859999999999999, 0}, {4.89, 0.03617552364070439},
RoundingRadius->0],
RectangleBox[{4.89, 0}, {4.92, 0.03376382206465744},
RoundingRadius->0],
RectangleBox[{4.92, 0}, {4.95, 0.030146269700586998},
RoundingRadius->0],
RectangleBox[{4.95, 0}, {4.9799999999999995, 0.01929361260837625},
RoundingRadius->0],
RectangleBox[{4.9799999999999995, 0}, {5.01, 0.022911164972446118},
RoundingRadius->0],
RectangleBox[{5.01, 0}, {5.04, 0.03376382206465744},
RoundingRadius->0],
RectangleBox[{5.04, 0}, {5.069999999999999, 0.037381374428728985},
RoundingRadius->0],
RectangleBox[{5.069999999999999, 0}, {5.1, 0.025322866548493077},
RoundingRadius->0],
RectangleBox[{5.1, 0}, {5.13, 0.024117015760469596},
RoundingRadius->0],
RectangleBox[{5.13, 0}, {5.16, 0.027734568124540036},
RoundingRadius->0],
RectangleBox[{5.16, 0}, {5.1899999999999995, 0.0385872252167525},
RoundingRadius->0],
RectangleBox[{5.1899999999999995, 0}, {5.22, 0.030146269700586998},
RoundingRadius->0],
RectangleBox[{5.22, 0}, {5.25, 0.03255797127663396},
RoundingRadius->0],
RectangleBox[{5.25, 0}, {5.279999999999999, 0.03496967285268195},
RoundingRadius->0],
RectangleBox[{5.279999999999999, 0}, {5.31, 0.025322866548493077},
RoundingRadius->0],
RectangleBox[{5.31, 0}, {5.34, 0.021705314184422637},
RoundingRadius->0],
RectangleBox[{5.34, 0}, {5.37, 0.030146269700586998},
RoundingRadius->0],
RectangleBox[{5.37, 0}, {5.3999999999999995, 0.031352120488611405},
RoundingRadius->0],
RectangleBox[{5.3999999999999995, 0}, {5.43, 0.030146269700586998},
RoundingRadius->0],
RectangleBox[{5.43, 0}, {5.46, 0.027734568124540036},
RoundingRadius->0],
RectangleBox[{5.46, 0}, {5.49, 0.022911164972446118},
RoundingRadius->0],
RectangleBox[{5.49, 0}, {5.52, 0.018087761820352734},
RoundingRadius->0],
RectangleBox[{5.52, 0}, {5.55, 0.024117015760469596},
RoundingRadius->0],
RectangleBox[{5.55, 0}, {5.58, 0.028940418912563517},
RoundingRadius->0],
RectangleBox[{5.58, 0}, {5.609999999999999, 0.021705314184423282},
RoundingRadius->0],
RectangleBox[{5.609999999999999, 0}, {5.64, 0.027734568124540036},
RoundingRadius->0],
RectangleBox[{5.64, 0}, {5.67, 0.026528717336516558},
RoundingRadius->0],
RectangleBox[{5.67, 0}, {5.7, 0.044616479156868755},
RoundingRadius->0],
RectangleBox[{5.7, 0}, {5.7299999999999995, 0.02773456812454086},
RoundingRadius->0],
RectangleBox[{5.7299999999999995, 0}, {5.76, 0.030146269700586998},
RoundingRadius->0],
RectangleBox[{5.76, 0}, {5.79, 0.01688191103232872},
RoundingRadius->0],
RectangleBox[{5.79, 0}, {5.819999999999999, 0.01929361260837625},
RoundingRadius->0],
RectangleBox[{5.819999999999999, 0}, {5.85, 0.021705314184422637},
RoundingRadius->0],
RectangleBox[{5.85, 0}, {5.88, 0.022911164972446118},