> restart: with(Statistics): with(CurveFitting):
 

Subroutine 

Define the function to plot the area under the curve wich is used in some graphical representations. 

> area_plot := proc(f1,a,b,d,e)
local box,i,n,x1,x2,y1,y2,A,delta:
n:= 50; delta := (b-a)/n; x2 := a;
for i from 1 to n do
x1 := evalf(x2); y1 := evalf(f1(x1));
x2 := evalf(a + i*delta); y2 := evalf(f1(x2));
if( y1 > 0 )
then A||i :=plots[polygonplot]( [[x1,0],[x1,y1],[x2,y2],[x2,0]],color=coral,style=patchnogrid);
else A||i:=plots[polygonplot]( [[x1,0],[x1,y1],[x2,y2],[x2,0]],color=red, style=patchnogrid);
fi; od;
plots[display]([plot(f1(x), x = d..e, color=blue,thickness=2,discont=true), seq(A||i,i=1..n)]);
end:
 

Model prerequisites 

Capability Quotient 

Definition: We define the individual capability quotient (CQ) as the sum of all qualities that make up a leader dived by the avarage sum of such qualities.
Similar to the known intelligence quotient (IQ) we multiply the quotient by 100 to allow a percentage interpretation. Then outstanding leaders posses more than 100% of the average capabilities while people with CQ below 100 are not.
We further assume that, similar to IQ, CQ presents a nomal distribution if the sample is large enough, so we begin our consideration with a mean of 100 and a standard deviation of 15.  
 

> mean := 100; sd := 15;
N := RandomVariable(Normal(mean, sd)):
f1 := t-> PDF(N, t):
K1 := 0: L1 := 200:
A1p := plot(f1(x), x = K1 .. L1, color = blue, thickness = 2, labels = ["CQ", ""]):
(plots[display])(A1p);
 

100 

15 

Plot 

Individually Distorted Perception 

Next, we assume that an individuals are incapable to appreciate capabilities that they do not posses themselves. That leads to the conclusion that individuals fail to appreciate a CQ better than the own one, and this leads to a distorted perception of the normal distribution curve cutting off the portion above the individual CQ.
The coral areas in the below examples demonstrates what an individual believes to be of one own's competency.
 

Example CQ = 80 

> CQ := 80:
A1CQ80p := area_plot(f1, CQ, 200, 0, 200):
plots[display](A1CQ80p);
 

Plot 

Example CQ =120 

> CQ := 120:
A1CQ120p := area_plot(f1, 120, 200, 0, 200):
plots[display](A1CQ120p);
 

Plot 

Idividual Selection 

The model finally assumes that an individual selects the best possible leader from all those people who are perceived of the same - the own - capability. If so the selection is made based on probability. That means the selected individual is the one that is the most likely chosen in the part of the spectrum where no differences can be perceived.
The most likely chosen CQ is calculated by the cetroid of the area.
conjugate(x) = Int(x*f(x), x = a .. b)/Int(f(x), x = a .. b)
 

 

In our case a is the individuals CQ and b is assumed to be the best ever possible CQ (assumed to be 200). 

 

Example CQ = 80 

> x80 := evalf(int(s*f1(s), s = 80 .. L1)/int(f1(s), s = 80 .. L1));
B80p := plot([[x80, 0], [x80, f1(x80)]], linestyle = 3, thickness = 2, color = COLOR(RGB, .3, .2, .1)):
plots[display](A1CQ80p, B80p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
 

102.7070659 

Plot 

Example CQ = 120 

> x120 := evalf(int(s*f1(s), s = 120 .. L1)/int(f1(s), s = 120 .. L1));
B120p := plot([[x120, 0], [x120, f1(x120)]], linestyle = 3, thickness = 2, color = COLOR(RGB, .3, .2, .1)):
plots[display](A1CQ120p, B120p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
 

126.9720228 

Plot 

Typesetting:-delayDotProduct(We*now*may*construct*a*function*that*represents*the^2*selected*CQ^2*depending*on*individual, f2(t)) := proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s...
Typesetting:-delayDotProduct(We*now*may*construct*a*function*that*represents*the^2*selected*CQ^2*depending*on*individual, f2(t)) := proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s...
Typesetting:-delayDotProduct(We*now*may*construct*a*function*that*represents*the^2*selected*CQ^2*depending*on*individual, f2(t)) := proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s...
Typesetting:-delayDotProduct(We*now*may*construct*a*function*that*represents*the^2*selected*CQ^2*depending*on*individual, f2(t)) := proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s...
Typesetting:-delayDotProduct(We*now*may*construct*a*function*that*represents*the^2*selected*CQ^2*depending*on*individual, f2(t)) := proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s...
Typesetting:-delayDotProduct(We*now*may*construct*a*function*that*represents*the^2*selected*CQ^2*depending*on*individual, f2(t)) := proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s...
Typesetting:-delayDotProduct(We*now*may*construct*a*function*that*represents*the^2*selected*CQ^2*depending*on*individual, f2(t)) := proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s...If we assume that all individuals of a certain CQ select the same new CQ, so the probability of the new CQ is the same as the probability of the old one. This procedure projects the distribution curve into a new curve after the first selection process.
 

> f2:=t->(int(s*f1(s),s=t..L1))/(int(f1(s),s=t..L1));
B1p := plot([f2(t), f1(t), t = 0 .. L1], color = red, thickness = 2):
plots[display](A1p, B1p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
 

proc (t) options operator, arrow; int(s*f1(s), s = t .. L1)/int(f1(s), s = t .. L1) end proc 

Plot 

Systematic of Manipulations 

Shifting the Sample 

>
 

The original sample can be modified in order to change the outcome of an election. There are two ways to accomplish this. Both differ slightly by their mathematical model, but lead to the same result. (1) One can chose a subsample with an other dstribution function, or (2) from the original sample, some votes are eliminated. As the first ase would not differ mathemathecally from the above consideration we will focus now on the second case, the emilination of a subsample.
 

Let be f[all](x) the distribution function of CQ in the whole sample (A) and `in`(f[b](x)*the*distribution*function*of*CQ, Typesetting:-delayDotProduct(a*subsample*B, Now)*we*are*serching*the*distribution*function*of)
`in`(f[b](x)*the*distribution*function*of*CQ, Typesetting:-delayDotProduct(a*subsample*B, Now)*we*are*serching*the*distribution*function*of)
`in`(f[b](x)*the*distribution*function*of*CQ, Typesetting:-delayDotProduct(a*subsample*B, Now)*we*are*serching*the*distribution*function*of)
 

 

Now we are searching for the distribution function f[c](x) in the subsample A∖B :={x|(x∈A)∧(x∉B)} that is defined by  

f[c](x) = (f[all](x)-p*f[c](x))/(1-p) 

 

with p the probability that an element belongs to subsample B. 

 

 

Example: Eliminating more capable 

> M1:=RandomVariable(Normal(110,12));
f_B110:=t->PDF(M1,t);
p:=0.4;
f_C110:=x->(f1(x)-p*f_B110(x))/(1-p);
A1p110:=plot(f_B110(x),x=0..L1, color = red, thickness = 2);
A2p110:=plot(f_C110(x),x=0..L1, color = green, thickness = 2);
plots[display](A1p, A1p110, A2p110, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
 

_R0 

proc (t) options operator, arrow; (Statistics:-PDF)(M1, t) end proc 

.4 

proc (x) options operator, arrow; (f1(x)-p*f_B110(x))/(1-p) end proc 

A1p110 := INTERFACE_PLOT(CURVES([[0., 0.188507533948787292e-19], [4.35943083333333358, 0.493086853955544876e-18], [8.15254829166666718, 0.758111835351086710e-17], [12.4183055833333338, 0.1454070265252... 

A2p110 := INTERFACE_PLOT(CURVES([[0., 0.990100019817618311e-11], [4.35943083333333358, 0.658861377227747974e-10], [8.15254829166666718, 0.319990942003489860e-9], [12.4183055833333338, 0.17531758251757... 

Plot 

> f2_B110:=t->(int(s*f_C110(s),s=t..L1))/(int(f_C110(s),s=t..L1));
B1pr := plot([f2(t), f1(t), t = 0 .. L1], color = blue, thickness = 2):
B110p := plot([f2_B110(t), f_C110(t), t = 0 .. L1], color = red, thickness = 2):
plots[display](B1pr, B110p, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
 

proc (t) options operator, arrow; int(s*f_C110(s), s = t .. L1)/int(f_C110(s), s = t .. L1) end proc 

Plot 


 

Example: Eliminating less capable 

> M2:=RandomVariable(Normal(90,12));
f_B90:=t->PDF(M2,t);
p:=0.4;
f_C90:=x->(f1(x)-p*f_B90(x))/(1-p);
A1p90:=plot(f_B90(x),x=0..L1, color = red, thickness = 2);
A2p90:=plot(f_C90(x),x=0..L1, color = green, thickness = 2);
plots[display](A1p, A1p90, A2p90, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
 

_R1 

proc (t) options operator, arrow; (Statistics:-PDF)(M2, t) end proc 

.4 

proc (x) options operator, arrow; (f1(x)-p*f_B90(x))/(1-p) end proc 

A1p90 := INTERFACE_PLOT(CURVES([[0., 0.202860044419084165e-13], [4.35943083333333358, 0.289625051095725898e-12], [8.15254829166666718, 0.262936056616892199e-11], [12.4183055833333338, 0.27886719055774... 

A2p90 := INTERFACE_PLOT(CURVES([[0., 0.988747620777612908e-11], [4.35943083333333358, 0.656930546840172588e-10], [8.15254829166666718, 0.318238040012684902e-9], [12.4183055833333338, 0.173458477606845... 

Plot 

> f2_B90:=t->(int(s*f_C90(s),s=t..L1))/(int(f_C90(s),s=t..L1));
B90p := plot([f2_B90(t), f_C90(t), t = 0 .. L1], color = red, thickness = 2):
plots[display](B1pr, B90p, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
 

proc (t) options operator, arrow; int(s*f_C90(s), s = t .. L1)/int(f_C90(s), s = t .. L1) end proc 

Plot 

Status of Holiness 

'Holy' Candidate in Less Capable CQ=80 

> a1 := plottools[arrow]([x80,.004],[80,.004],.0001,.0005,.003,color=black):
a2 := plottools[arrow]([x80,.006],[200,.006],.0001,.0005,.003,color=black):
plots[display](A1CQ80p, B80p, a1, a2, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
 

Plot 

'Holy' Candidate in More Capable CQ=120 

> a3 := plottools[arrow]([x120,.002],[120,.002],.0001,.0005,.003,color=black):
a4 := plottools[arrow]([x120,.003],[200,.003],.0001,.0005,.003,color=black):
plots[display](A1CQ120p, B120p, a3, a4, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
 

Plot 

Likelihood to Improve 

>
 

Step-by-step Improvements 

Test a Proper Interpolation 

Prerequisit: find a proper Interpolation 

> l:=7:
f_fill:=j->(j)*25:
X:=Vector(l,f_fill):
Y1:=<seq(evalf(f1(X[i])),i=1..l)>:
X1:=<seq(evalf(f2(X[i])),i=1..l)>:
fh:=Spline(X1,Y1,t):
f3:=unapply(%,t);
 

proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
 

Function f3(x) now defines the new distribution function after the first selection.
The following graph demonstrates that the interpolation matches the data for the range considered.
 

> C1p:=ScatterPlot(X1,Y1):
E1p:=plot(f3(x),x=99..160,color=green, thickness = 2):
plots[display](C1p,B1p,E1p);
 

Plot 

> Num_Samp:=7:
f_fill:=j->(j)*25:
X:=Vector(l,f_fill):
Y1:=<seq(evalf(f1(X[i])),i=1..Num_Samp)>:
X1:=<seq(evalf(f2(X[i])),i=1..Num_Samp)>:
fh:=Spline(X1,Y1,t):
f3:=unapply(%,t);
 

proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
proc (t) options operator, arrow; piecewise(t < 100.0231441, -.4443248035+0.444324803499999977e-2*t-0.132731099700000009e-2*(t-100.0000223)^3, t < 101.5670468, -.4441118894+0.444111922299999986e-2*t-0...
 

Function f3(x) now defines the new distribution function after the first selection.
The following graph demonstrates that the interpolation matches the data for the range considered.
 

> C1p:=ScatterPlot(X1,Y1):
E1p:=plot(f3(x),x=99..160,color=green, thickness = 2):
plots[display](C1p,B1p,E1p);
 

Plot 

Two Step Improvement 

> f4:=t->(int(s*f3(s),s=t..150))/(int(f3(s),s=t..150));
B1pp := plot([f2(t), f1(t), t = 0 .. L1], color = pink, thickness = 2):
F1p:=plot([f4(t),f3(t),t=100..138],color=red, thickness = 2);
plots[display](A1p, B1pp,F1p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
 

proc (t) options operator, arrow; int(s*f3(s), s = t .. 150)/int(f3(s), s = t .. 150) end proc 

F1p := INTERFACE_PLOT(CURVES([[115.956814347544096, 0.147193042265122318e-16], [115.970259716102362, 0.182347092437270251e-2], [115.999253108288982, 0.360354386367030960e-2], [116.038855057812668, 0.5... 

Plot 

>
 

Forecast of Further Improvements 

> Y2:=<seq(evalf(f3(X1[i])),i=1..Num_Samp)>;
X2:=<seq(evalf(f4(X1[i])),i=1..Num_Samp)>;
ff := PolynomialInterpolation ([[0,100],[1,X1[4]],[2,X2[4]]], t);
f8:= unapply(%,t);
S1p := ScatterPlot([0,1,2],[100,X1[4],X2[4]]);
S2p := plot(f8(x),x=0..4,color=green);
plots[display](S1p,S2p);
 

Y2 := Vector[column](%id = 146925476) 

X2 := Vector[column](%id = 149846472) 

-1.672294350*t^2+13.64056275*t+100 

proc (t) options operator, arrow; -1.672294350*t^2+13.64056275*t+100 end proc 

S1p := INTERFACE_PLOT(POINTS([0., 100.], [1., 111.968268399999999], [2., 120.591948099999996])) 

S2p := INTERFACE_PLOT(CURVES([[0., 100.], [0.871886166666666629e-1, 101.176589257768299], [.163050965833333339, 102.179647953029431], [.248366111666666666, 103.284696841059770], [.334246781666666659, ... 

Plot 

>
 

>
 

>
 

Print Output 

> plotsetup(jpeg, plotoutput = `fig0.jpg`, plotoptions = "height=600,width=960"):
plots[display](A1p);
plotsetup(jpeg, plotoutput = `fig1.jpg`, plotoptions = "height=600,width=960"):
plots[display](A1CQ80p, B80p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
plotsetup(jpeg, plotoutput = `fig2.jpg`, plotoptions = "height=600,width=960"):
plots[display](A1CQ120p, B120p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
plotsetup(jpeg, plotoutput = `fig3.jpg`, plotoptions = "height=600,width=960"):
plots[display](A1p, B1p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
plotsetup(jpeg, plotoutput = `fig4.jpg`, plotoptions = "height=600,width=960"):
plots[display](A1p, A1p110, A2p110, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
plotsetup(jpeg, plotoutput = `fig5.jpg`, plotoptions = "height=600,width=960"):
plots[display](B1pr, B110p, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
plotsetup(jpeg, plotoutput = `fig6.jpg`, plotoptions = "height=600,width=960"):
plots[display](A1p, A1p90, A2p90, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
plotsetup(jpeg, plotoutput = `fig7.jpg`, plotoptions = "height=600,width=960"):
plots[display](B1pr, B90p, view = [0 .. 200, 0 .. 0.35e-1], labels = ["CQ", ""]);
plotsetup(jpeg, plotoutput = `fig8.jpg`, plotoptions = "height=600,width=960"):
plots[display](A1p, B1pp, F1p, view = [0 .. 200, 0 .. 0.3e-1], labels = ["CQ", ""]);
plotsetup(inline, plotoptions = "height=600,width=960"):
 

>