The Volterra Function
The Volterra Function
whose derivative is Riemann Non-integrable
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ClearAll["Global`*"];
The Smith-Volterra-Cantor Set
The Smith-Volterra-Cantor Set
In this section, we construct several iterations of Smith-Volterra-Cantor set by marking its pair of points indicating the interval being removed.
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ClearAll[SVC,SVCG];SVC[0]={0,1};SVC[n_]:=Ifn>0,Sort@FlattenJoinTableIfOddQ[k]&&k<Length[SVC[n-1]],-,+,Nothing,{k,1,Length[SVC[n-1]]},SVC[n-1],SVC[0];
SVC[n-1][[k]]+SVC[n-1][[k+1]]
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SVC[n-1][[k]]+SVC[n-1][[k+1]]
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SVC[4]
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0,,,,,,,,,,,,,,,1
9
128
11
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32
7
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37
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39
128
3
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89
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91
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25
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27
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117
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128
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SVCG=SVC[4];Graphics[Table[If[OddQ[k],Line[{{SVCG[[k]],0},{SVCG[[k+1]],0}}]],{k,1,Length[SVCG]}],ImageSizeLarge,PlotRange{{-0.2,1.2},{-0.2,0.2}}]
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The Volterra Function
The Volterra Function
The Volterra function is constructed recursively, by using sin.
2
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1
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1
.Consider subinterval , find the maximum zero for the derivative of sin in this interval, and make a piecewise defined function such that (x)sin on but (x)sin on . Define the function (x) symmetrically on , set (x)0 elsewhere, and shift the graph of (x) to the right by the amount , such that the .
0,
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8
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[0,]
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supp⊂,
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.Consider subinterval , find the maximum zero for the derivative of sin inside of this interval, and make a piecewise defined function such that (x)sin on but (x)sin on . Define the function (x) symmetrically on , set (x)0 elsewhere, and shift the graph of (x) to the right by the amount and respectively.
0,
1
2·
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2·
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2·
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32
25
32
3
.Do the same thing again and again.
4
.The Volterra function is defined by adding altogether all such functions.
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Chi[a_,b_,x_]:=Piecewise[{{1,a<x<b}},0];V[a_,b_,x_]:=Module{z},z=Maxu/.NSolve2uSin-Cos==0&&u>&&u<,u;SinChi[a,a+z,x]+SinChi[b-z,b,x]+SinChi[a+z,b-z,x];Vd[a_,b_,x_]:=Module{z},z=Maxu/.NSolve2uSin-Cos==0&&u>&&u<,u;2(x-a)Sin-CosChi[a,a+z,x]+2(b-x)Sin-CosChi[b-z,b,x];
1
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100
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100
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x-a
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1
b-x
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Volterra[x_]:=Sum[If[OddQ[k],0,V[SVCG[[k]],SVCG[[k+1]],x]],{k,1,Length[SVCG]-1}];DVolterra[x_]:=Sum[If[OddQ[k],0,Vd[SVCG[[k]],SVCG[[k+1]],x]],{k,1,Length[SVCG]-1}];Plot[Evaluate[Volterra[x]],{x,0,1},PlotRange->All]//AbsoluteTimingPlot[Evaluate[DVolterra[x]],{x,0,1},PlotRange->All]//AbsoluteTiming
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45.7245,
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46.0275,