(************** Content-type: application/mathematica **************
CreatedBy='Mathematica 4.2'
Mathematica-Compatible Notebook
This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.
To get the notebook into a Mathematica-compatible application, do
one of the following:
* Save the data starting with the line of stars above into a file
with a name ending in .nb, then open the file inside the
application;
* Copy the data starting with the line of stars above to the
clipboard, then use the Paste menu command inside the application.
Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode. Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.
For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
web: http://www.wolfram.com
email: info@wolfram.com
phone: +1-217-398-0700 (U.S.)
Notebook reader applications are available free of charge from
Wolfram Research.
*******************************************************************)
(*CacheID: 232*)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 21135, 594]*)
(*NotebookOutlinePosition[ 21828, 618]*)
(* CellTagsIndexPosition[ 21784, 614]*)
(*WindowFrame->Normal*)
Notebook[{
Cell[TextData[StyleBox["Biophysics 101, Fall 2002\nProblem Set 1, part 3", \
"DisplayFormula"]], "Text",
TextAlignment->Right,
TextJustification->0],
Cell["Warm-up Notebook", "Subtitle",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"This notebook is for beginners that have done the ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" tutorial. Most of it is no harder than a coloring book It starts with \
cell manipulation and formatting, and it works its way up to discrete \
recursive functions that operate on ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" tables and arrays. Many of us are new to ",
StyleBox["Mathematica",
FontSlant->"Italic"],
", so ",
StyleBox["if you discover anything useful, please include them in your \
finished notebook.",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text"],
Cell[CellGroupData[{
Cell[TextData[StyleBox["1: Saving notebooks", "Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell["\<\
Your first challenge is to save a copy of this notebook, and you get to name \
it after you. On the menu bar at the top of the screen its \"File-> Save \
As.... \". If your name is Jack Flapjack then name it Jack_Flapjack.nb. If \
your name is not Jack Flapjack then name it after your own name instead. Put \
your answers in your self-named copy and turn it in to your TF. \
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[{
StyleBox["2: Inserting and deleting ", "Subsection"],
StyleBox["Mathematica", "Subsection",
FontSlant->"Italic"],
StyleBox[" cells", "Subsection"]
}], "Text",
TextAlignment->Center,
TextJustification->0],
Cell["\<\
If you click in the thin horizontal space between cells, a thin black line \
appears. Hit the enter key, and voila, a new cell. Insert a cell between the \
second and third cells in this notebook, and type your name in it. If you are \
feeling stylish, format your name text with the many options available under \
the format menu. On the menu, check out \"Format->Show Toolbar\" Though short \
on features, the formatting toolbar may be worth it's screen real estate. To \
delete a cell, click to highlight it's blue bracket on the right, and press \
the delete key. To copy/paste a cell - highlight the blue bracket of the cell \
you want to copy, ->copy,then make the thin horizontal black \
line appear where you want it to go, and ->paste.\
\>", "Text",
TextAlignment->Left,
TextJustification->0]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[{
StyleBox["3: ", "Subsection"],
StyleBox["Mathematica", "Subsection",
FontSlant->"Italic"],
StyleBox[" cell types", "Subsection"]
}], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"Notice the subtle differences in the cell bracket shapes. They tell you \
what type of cell it is. This type, with the double line at the top, means \
its a non-evaluatable text cell. If you press while the cursor \
is in this type of cell, ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" will beep at you.\n",
StyleBox["(2 pts) Try it on the cell below. Now to make it evaluatable, \
highlight the cell bracket, and then its \"Cell->Cell Properties->Cell \
Evaluatable\"",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[TextData[StyleBox["2+2", "Subsubsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell["\<\
The default cell type, for example the one you just typed your name in, is \
called \"Input Form\", and its cell bracket has a small triangle at the top. \
Change the property of the cell containing your name to non-evaluatable, and \
convert its display type to Text thru the menu via \"Cell->Display \
As->Text\".\
\>", "Text",
TextAlignment->Left,
TextJustification->0],
Cell["\<\
This is an input type cell in text display mode. If you press \
in this cell, you will see what this sentence means to the Mathematica kernel \
(not much).
\
\>", "Input",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[
FormBox[
StyleBox[\(This\ is\ a\ traditional\ form\ cell . \ \ It\ has\ the\ \
coolest\ bracket\ shape, \
but\ it' s\ a\ relic\ of\ old\ Mathematica\ versions\ that'
s\ \ still\ \ compatable\ with\ version\ \ \(\(4.2\)\(.\)\)\),
"Text"], TraditionalForm]], "Input"]
}, Open ]],
Cell[TextData[StyleBox["4: Cell grouping", "Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell["\<\
Notice how most of these cells are nested, but the two cells in this section \
are not. To group them, first go on the menu to \"Cell->Cell Grouping\" and \
make sure \"Manual Grouping\" is checked. Then select both brackets by \
dragging the mouse along the right margin, and its \"Cell->Cell \
Grouping->Group Cells\". Double click on the outermost bracket to make the \
nest expand and collapse.\
\>", "Text"],
Cell[CellGroupData[{
Cell[TextData[StyleBox["5: Named lists", "Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"In the next cell there are some named lists and a name based list \
operation. ",
StyleBox["(2 pts)",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]],
" ",
StyleBox["Evaluate the expressions by placing the cursor to the right of \
the last line, and . To get the total oil consumption for all \
five years, replace the \"*\" with a \".\" and evaluate again.",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text"],
Cell[BoxData[{
\(census\ = \ {7, \ 12, \ 11, \ 17, \ 28}\), "\[IndentingNewLine]",
\(perCapitaOil\ = \ {300, \ 445, \ 425, \ 625, \
850}\), "\[IndentingNewLine]",
\(census\ *\ perCapitaOil\)}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["6: Calculated lists using the Table method", \
"Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"The first example creates a length b array of a. ",
StyleBox["Evaluate the expression",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]],
"."
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(a = 3\), "\[IndentingNewLine]",
\(b = 4\), "\[IndentingNewLine]",
\(Table[a, \ {b}]\)}], "Input"],
Cell["Evaluate these expressions.", "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(Prime[5]\), "\[IndentingNewLine]",
\(Table[Prime[n], \ {n, \ 1, \ 10}]\), "\[IndentingNewLine]",
\(Random[Real, \ {0, 1}]\)}], "Input"],
Cell[TextData[StyleBox["(2 pts) Now insert a cell below this one that creates \
a list of 20 random numbers. Choose your own range (in the above case the \
range is between 0 and 1), and your number type choices are (Integer, Real, \
or Complex).",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]], "Text",
TextAlignment->Left,
TextJustification->0],
Cell["\<\
Evaluate these different ways to count to ten.\
\>", "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(Table[x, \ {x, \ 10}]\), "\[IndentingNewLine]",
\(Table[x, \ {x, \ 1, \ 10}]\), "\[IndentingNewLine]",
\(Table[x/2, \ {x, 2, 20, 2}]\), "\[IndentingNewLine]",
\(Range[10]\)}], "Input"],
Cell[TextData[StyleBox["(2 pts) Inset a formula in a cell at the bottom of \
this group that counts backwards from ten.",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]], "Text",
TextAlignment->Left,
TextJustification->0]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox[" 7: List generation using functions with the Array \
method", "Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"This example introduces functions and arrays. Notice the underscore (x_) \
after the variable definition in the function. Arg 1 in the Array \
specification is the creation method that takes an integer, arg 2 is the \
array length, arg 3 is the starting point. ",
StyleBox["(2 pts)",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]],
" ",
StyleBox["Evaluate these expressions. ",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{\(f[x_]\ := \ x^2;\), "\[IndentingNewLine]", \(qT = \
Array[f, \ 20, \ \(-12\)]\), "\[IndentingNewLine]",
"qT", "\[IndentingNewLine]", \(ListPlot[qT];\), "\[IndentingNewLine]",
RowBox[{"ListPlot", "[",
RowBox[{"qT", ",", " ",
RowBox[{
StyleBox["PlotJoined",
"MR"], " ", "\[Rule]", " ",
StyleBox["True",
"MR"]}]}], "]"}]}], "Input"],
Cell[TextData[{
StyleBox["Note 1) ",
FontWeight->"Bold",
FontColor->RGBColor[0.682353, 0.32549, 0.317647]],
StyleBox["Notice how the arrays are printed out when you evaluate the cell. \
This can be annoying when you want to look at a complicated graph but don't \
want to see all of it's underlying points printed out. Put a \";\" at the end \
of the expressions that you don't want to display the results of.\n\nNote 2) \
It's considered good ",
FontWeight->"Bold",
FontColor->RGBColor[0.682353, 0.32549, 0.317647],
FontVariations->{"CompatibilityType"->0}],
StyleBox["Mathematica",
FontWeight->"Bold",
FontSlant->"Italic",
FontColor->RGBColor[0.682353, 0.32549, 0.317647],
FontVariations->{"CompatibilityType"->0}],
StyleBox[" practice to begin your custom variable names with a lower case \
letter, which will prevent name collisions with the ",
FontWeight->"Bold",
FontColor->RGBColor[0.682353, 0.32549, 0.317647],
FontVariations->{"CompatibilityType"->0}],
StyleBox["Mathematicas ",
FontWeight->"Bold",
FontSlant->"Italic",
FontColor->RGBColor[0.682353, 0.32549, 0.317647],
FontVariations->{"CompatibilityType"->0}],
StyleBox["names, all of which begin with uppercase letters.\n\nNote 3) \
Don't forget to put the colon in front of the equals sign \":=\" when you \
define a function. This can be very frustrating, because leaving it off won't \
cause an error. So if your function is giving you unexpected results, don't \
forget to re-read this note!",
FontWeight->"Bold",
FontColor->RGBColor[0.682353, 0.32549, 0.317647],
FontVariations->{"CompatibilityType"->0}]
}], "Text",
TextAlignment->Left,
TextJustification->0]
}, Open ]],
Cell[TextData[StyleBox["8: Using lists to graph a built in function", \
"Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[CellGroupData[{
Cell[TextData[{
"You can compose your own custom functions with ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" built in functions. Remember N from the tutorial? -",
StyleBox[" evaluate these expressions.",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(\(k\ = \ 3;\)\), "\[IndentingNewLine]",
\(\(f[x_]\ = k*Tanh[x];\)\), "\[IndentingNewLine]",
\(\(f[3];\)\), "\[IndentingNewLine]",
\(N[f[3]]\)}], "Input"],
Cell[TextData[StyleBox["(2 pts) Make a plot of Tanh[x], with the y values \
between -0.9 and 0.9. Bonus: Make it exactly between -0.9 and 0.9 (hint: look \
in Help->Master Index->Built-in Functions->Mathematical Functions).",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]], "Text",
TextAlignment->Left,
TextJustification->0]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["9: The Table method", "Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"Similar to the Array method, these two have a lot of overlapping \
functionality. ",
StyleBox["Evaluate ",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]],
"and notice we didn't have to redefine f, because this variable has \
notebook level scope. ",
StyleBox["(2 pts)",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]],
" ",
StyleBox["Create a second list populated with decimal approximations of f.",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(\(myTableGeneratedList\ =
Table[f[x], \ {x, 2, 20, 2}];\)\), "\[IndentingNewLine]",
\(myTableGeneratedList\)}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["10: Recursive functions", "Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"Recursive functions take whole numbers as inputs, while their output \
variables can be as simple as true/false, whole numbers, or as complicated as \
you can imagine. A recursive function, rF, is made out of a regular function, \
f, and a seed value, s. s is usually the same type of thing as the output of \
f. \nrF(0) = s\nrF(1) = f(s)\nrF(2) = f(f(s)), or generally speaking, rF(n) = \
f(rF(n-1)).\n",
StyleBox["(2 pts)",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]],
" ",
StyleBox["In the cell below is the definition of a ",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]],
StyleBox["Mathematica",
FontWeight->"Bold",
FontSlant->"Italic",
FontColor->RGBColor[0, 0, 1]],
StyleBox[" function that doubles at every step. Insert some expressions \
that evaluate it at a few more steps. What is the function \"f\" in this \
case?",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(\(rF[t_] := \ \(rF[t]\ = \ rF[t - 1]*2\);\)\), "\[IndentingNewLine]",
\(\(rF[0]\ = \ 7;\)\), "\[IndentingNewLine]",
\(rF[3]\)}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["11: Modeling population growth", "Subsection"]], \
"Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"Recursive functions are perfect for modeling systems that change in a step \
by step way at regular time points. Dynamic systems such as populations of \
organisms don't change their numbers on a fixed schedule, but recursive \
functions can be a good rough model, since the population at year n has a big \
influence on the population at year n+1. \n",
StyleBox["(2 pts) Use the methods in steps 6-9 to create a graph of F \
defined above. Can you think of a real biological system that F is a good \
model for?",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["12: A more complex model", "Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"The logistic model is presented below. It is considered to be the simplest \
dynamical system that exhibits chaotic behavior. It is implemented here with \
three parameters: r, k, and g. Set these three numbers to different values, \
evaluate, and observe how the graph changes. In ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" it is necessary to clear a function before you change the parameters that \
are part of it's definition, which explains the first command: Clear[pop]. \
Note: use values for r, 0 < r < 4, or you will get overflow errors. Also, \
when setting r to a whole number, such as 2, put a decimal point after the \
number: write \"r = 2.\" instead of \"r=2\" . The latter will cause ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to hang. \n",
StyleBox["(12 pts) Match the four parameters to the following definitions:\
\n1) The population at time zero.\n2) The number of generations.\n3) The \
capacity of the enviroment to sustain the population.\n4) The fertility of \
the organisims.\nCreate three copies of this cell. In one cell choose \
parameters that produce a graph that shows unchecked exponential growth, \
similar to the graph you created in (11). \nIn the second cell, set \
parameters that produce a graph that shows periodic behavior (i.e. the \
population changes from generation to generation in a regular way.\nIn the \
thrid cell, set parameters that result in chaotic behavior. Which parameter \
controls the chaotic/non-chaotic behavior of the system? ",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(Clear\ [pop]\), "\[IndentingNewLine]",
\(r\ = \ 3.77\), "\[IndentingNewLine]",
\(k\ = \ 100\), "\[IndentingNewLine]",
\(g\ = \ 200\), "\[IndentingNewLine]",
\(s\ = \ 5\), "\[IndentingNewLine]",
\(\(pop[t_]\ := \ \(pop[t]\ = \
r\ pop[t - 1] \((1 - \
pop[t - 1]/k)\)\);\)\), "\[IndentingNewLine]",
\(\(pop[0]\ = \ s;\)\), "\[IndentingNewLine]",
\(\(dt\ = Table[pop[x], \ {x, 0, g, 1}];\)\), "\[IndentingNewLine]",
\(ListPlot[dt, \ PlotJoined\ \[Rule] True, \
PlotRange \[Rule] {0, \ 3 k/2}]\)}], "Input"]
}, Open ]],
Cell[TextData[StyleBox["Extra Sections", "Subtitle"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[StyleBox["13: Building recursive lists with NestList", \
"Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"The ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" NestList method provides another way to model recursiveness. The NestList \
second arg is the inital value of the recursive variable, and the third arg \
is the number of recursive iterations. Notice how NestList can be used with \
multivariate functions, provided the variable to be treated recursively is \
listed last."
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(\(Clear[g1];\)\), "\[IndentingNewLine]",
\(\(g1[z_]\ := \ N[z^2];\)\), "\[IndentingNewLine]",
\(NestList[g1, \ 3, \ 4]\), "\[IndentingNewLine]",
\(\(Clear[g2];\)\), "\[IndentingNewLine]",
\(\(\(\(g2[x_]\)[y_]\)[z_]\ := \ x*y*z;\)\), "\[IndentingNewLine]",
\(NestList[\(g2[b]\)[c], \ a, \ \ 5]\)}], "Input"],
Cell[TextData[StyleBox["14: Sensitivity analysis with list differences", \
"Subsection"]], "Text",
TextAlignment->Center,
TextJustification->0],
Cell[TextData[{
"The code in the cell below offers a technique on how to compare two \
alternative histories. ",
StyleBox["Apply this technique to the Logistic map. Suggested steps: 1) \
Define a non-recursive version of the pop[] function above, with all of the \
appropriate parameters. 2) Use NestList to populate two lists, that are each \
seeded with nearly identical starting populations. 3) Plot the history \
difference. Compare results of the history differences in the chaotic and \
non-chaotic regimes.",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]
}], "Text",
TextAlignment->Left,
TextJustification->0],
Cell[BoxData[{
\(\(Clear[browny];\)\), "\[IndentingNewLine]",
\(\(browny[x_]\ := \
x\ + \ Random[Real, \ {0, 1}]\ -
0.5\ ;\)\), "\[IndentingNewLine]",
\(\(hist1\ = \ NestList[browny, \ 0, \ 10];\)\), "\[IndentingNewLine]",
\(\(hist2\ = \ NestList[browny, \ 0, \ 10];\)\), "\[IndentingNewLine]",
\(ListPlot[hist2 - hist1, \ PlotJoined\ \[Rule] True]\)}], "Input"]
},
FrontEndVersion->"4.2 for Microsoft Windows",
ScreenRectangle->{{0, 1280}, {0, 941}},
WindowToolbars->"EditBar",
CellGrouping->Manual,
WindowSize->{563, 859},
WindowMargins->{{82, Automatic}, {Automatic, 0}}
]
(*******************************************************************
Cached data follows. If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of the file. The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[1754, 51, 152, 3, 52, "Text"],
Cell[1909, 56, 85, 2, 64, "Subtitle"],
Cell[1997, 60, 640, 17, 90, "Text"],
Cell[CellGroupData[{
Cell[2662, 81, 119, 2, 35, "Text"],
Cell[2784, 85, 399, 6, 90, "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[3220, 96, 233, 7, 36, "Text"],
Cell[3456, 105, 842, 13, 166, "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[4335, 123, 215, 7, 36, "Text"],
Cell[4553, 132, 636, 15, 109, "Text"],
Cell[5192, 149, 105, 2, 33, "Text"],
Cell[5300, 153, 387, 8, 71, "Text"],
Cell[5690, 163, 242, 7, 84, "Input"],
Cell[5935, 172, 317, 6, 65, "Input"]
}, Open ]],
Cell[6267, 181, 115, 2, 35, "Text"],
Cell[6385, 185, 422, 7, 90, "Text"],
Cell[CellGroupData[{
Cell[6832, 196, 113, 2, 35, "Text"],
Cell[6948, 200, 479, 12, 71, "Text"],
Cell[7430, 214, 227, 4, 70, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[7694, 223, 143, 3, 35, "Text"],
Cell[7840, 228, 232, 8, 33, "Text"],
Cell[8075, 238, 128, 3, 70, "Input"],
Cell[8206, 243, 90, 2, 33, "Text"],
Cell[8299, 247, 166, 3, 70, "Input"],
Cell[8468, 252, 359, 7, 71, "Text"],
Cell[8830, 261, 117, 4, 33, "Text"],
Cell[8950, 267, 220, 4, 90, "Input"],
Cell[9173, 273, 232, 5, 33, "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[9442, 283, 158, 3, 35, "Text"],
Cell[9603, 288, 530, 14, 71, "Text"],
Cell[10136, 304, 422, 9, 110, "Input"],
Cell[10561, 315, 1718, 37, 223, "Text"]
}, Open ]],
Cell[12294, 355, 144, 3, 35, "Text"],
Cell[CellGroupData[{
Cell[12463, 362, 338, 10, 52, "Text"],
Cell[12804, 374, 191, 4, 90, "Input"],
Cell[12998, 380, 336, 6, 71, "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[13371, 391, 118, 2, 35, "Text"],
Cell[13492, 395, 572, 18, 71, "Text"],
Cell[14067, 415, 156, 3, 50, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[14260, 423, 122, 2, 35, "Text"],
Cell[14385, 427, 1015, 25, 185, "Text"],
Cell[15403, 454, 170, 3, 70, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[15610, 462, 131, 3, 35, "Text"],
Cell[15744, 467, 653, 13, 128, "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[16434, 485, 123, 2, 35, "Text"],
Cell[16560, 489, 1659, 29, 356, "Text"],
Cell[18222, 520, 608, 12, 190, "Input"]
}, Open ]],
Cell[18845, 535, 111, 2, 45, "Text"],
Cell[18959, 539, 143, 3, 35, "Text"],
Cell[19105, 544, 463, 11, 90, "Text"],
Cell[19571, 557, 354, 6, 130, "Input"],
Cell[19928, 565, 147, 3, 35, "Text"],
Cell[20078, 570, 638, 13, 109, "Text"],
Cell[20719, 585, 412, 7, 110, "Input"]
}
]
*)
(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)