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Mathematics Math21b Spring 2009

Linear Algebra and Differential Equations

Exhibit: Linear algebra: zipf law

Course Head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Here is a nice data fitting problem which empirically leads to Zipfs law.
If we read in the data of the population of cities:
A=ReadList["data.txt","Number"]; B=Partition[A,2]; n=Length[B]; B1=Table[{Log[k],Log[B[[k,1]]]},{k,27}]; Fit[B1,{1,x},x]and do a linear fit, we measure a power law. The linear fit is 17-0.77 k as mentioned in the New York Times article New York Times: Math and the city. Note that the law only applies to the top. If we take all data, we get 17.5-0.95 k: A=ReadList["data.txt","Number"]; B=Partition[A,2]; n=Length[B]; B1=Table[{Log[k],Log[B[[k,1]]]},{k,100}]; Fit[B1,{1,x},x]and a quadratic fit is probably better A=ReadList["data.txt","Number"]; B=Partition[A,2]; n=Length[B]; B1=Table[{Log[k],Log[B[[k,1]]]},{k,n}]; Fit[B1,{1,x,x^2},x] |

Please send questions and comments to math21b@fas.harvard.edu

Math21b (Exam Group 1)| Oliver Knill | Spring 2009 |
Department of Mathematics |
Faculty of Art and Sciences |
Harvard University