October 2, 2007--Class 11 - Final Project, Intro. to Mathematica,
			Random Walks

Activities:

	Final Project

		It is time to start thinking about your final project.  Some 
		of you may have a problem related to your research that can 
		be used for the project.  If not, there may be some problem
		that interests you that would serve as a project.  If you 
		don't have an idea of your own, you might like to look through
		our text book.  If there is a chapter that we will not cover, 
		you might find some homework problems that can be combined to
		make a project.  The book also has some suggestions for 
		projects.  Some of these, even in chapters will will cover some
		of may make suitable projects.  So start to think about what 
		you want to do, because in a couple of weeks, I will be asking
		everyone to get their project approved by me.

		Here is how the final project will be graded:
        	Statement       10
        	Eq Solved       5
        	Num. Method     5
        	Code            5
        	Results         10
        	Discussion      10
        	Critique        5
        	------------------
        	Total           50

	Introduction to Mathematica

	An eight page handout on Mathematica is available as 
		~sg/mathematica_notes_sg.pdf
	Over the next four classes we will go through the notes
	and the examples it contains.
	
	The material in sections A-E was elementary and known to most
	people.  Here is some elaboration about using mathematica to
	define a random walk.  We did not actually do the statistics of a
	random walk after N steps in class.

	Random walks
	
	We can create a random walk with these two statements

	randomStep[x_] := x + 2 Random[Integer] - 1
	w = NestList[randomStep, 0, 5000];

	We can make 1000 random walks of length 5000 and print the
	ending value from each walk

	v = Table[ w = NestList[randomStep, 0, 5000]; w[[Length[w]]], {i, 1000}] 
	Do[ Write["oct13.dat", v[[i]] ], {i, Length[v]}]

	Note use of Length function to make sure we get the last value of
	the list w, and that all of v is printed without needing to know 
	how long it is.

	We expect that for a random walk of length N the mean value of the 
	final position is 0 and the the mean value of the final position
	squared is N.  We can use Mathematica's built in functions to test.

   	Mathetmatica can create histograms.  These are part of the
	   Graphics add-on package.

	   Needs["Graphics`Graphics`"]
	   Histogram[v]

	   Mathematica can do statistical analysis.  For this, you will need
	   the DescriptiveStatistics sub-package in the Statistics package:

	   Needs["Statistics`DescriptiveStatistics`"]
	   Mean[v]
	   Variance[v]

	We did not actually need to run the Needs command to use Histogram,
	Mean or Variance.