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Notebook[{
Cell[TextData[StyleBox["Biophysics 101, Fall 2002\nProblem Set 1, part 3", \
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Cell["Warm-up Notebook", "Subtitle",
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Cell[TextData[{
  "This notebook is for beginners that have done the ",
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  " 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 ",
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    FontSlant->"Italic"],
  " tables and arrays. Many of us are new to ",
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  ", so ",
  StyleBox["if you discover anything useful, please include them in your \
finished notebook.",
    FontWeight->"Bold",
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Cell[TextData[StyleBox["1:  Saving notebooks", "Subsection"]], "Text",
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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"]
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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, <right-click>->copy,then  make the thin horizontal black \
line appear where you want it to go, and <right-click>->paste.\
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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 <shift-enter> while the cursor \
is in this type of cell, ",
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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",
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Cell["\<\
This is an input type cell in text display mode. If you press <shift-enter> \
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make sure \"Manual Grouping\" is checked. Then select both brackets by \
dragging the mouse along the right margin, and its \"Cell->Cell \
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Evaluate these different ways to count to ten.\
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forget to re-read this note!",
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Cell[TextData[{
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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) = \
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Cell[TextData[StyleBox["11: Modeling population growth", "Subsection"]], \
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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",
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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 ",
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\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? ",
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    \(s\  = \ 5\), "\[IndentingNewLine]", 
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                pop[t - 1]/k)\)\);\)\), "\[IndentingNewLine]", 
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Cell[TextData[StyleBox["13: Building recursive lists with NestList", \
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Cell[TextData[{
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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."
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Cell[TextData[{
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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.",
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  TextJustification->0],

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        x\  + \ Random[Real, \ {0, 1}]\  - 
          0.5\ ;\)\), "\[IndentingNewLine]", 
    \(\(hist1\  = \ NestList[browny, \ 0, \ 10];\)\), "\[IndentingNewLine]", 
    \(\(hist2\  = \ NestList[browny, \ 0, \ 10];\)\), "\[IndentingNewLine]", 
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*)



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End of Mathematica Notebook file.
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