In[]:=
deploy
Sat 24 Jul 2021 11:41:09
math.sePOST
In[]:=
variance[s_,n_]:=​​evals=Table
2s
1.-
1
i
1
i
,{i,1,n}/.Indeterminate1;​​diff[a_,b_]:=
2
(a-b)
;​​sum=Outer[diff,evals,evals]//Flatten//Total;​​
1
2
n
(n+2)
sum​​;​​(*Assuming{n>1,s>1},Integrate
2
2s
1-
1
i
1
i
-
2s
1-
1
j
1
j
,{i,1,n},{j,1,n}*)​​​​approxVariance[s_,n_]:=
1
2
n
(n+2)
*
2
2+4s
(-1+n)
-1-4s
n
1+4s
-2
2
HarmonicNumber[2s]
-
8
2
s
2
HypergeometricPFQ{1,1,1-2s},{2,2},
1
n

2
n
-
8sHypergeometricPFQ{1,1,1-2s},{2,2},
1
n
Log[n]
n
-2
2
Log[n]
+HarmonicNumber[2s]
8sHypergeometricPFQ{1,1,1-2s},{2,2},
1
n

n
+4Log[n];​​​​var2[n_]:=
2
Pi
3
2
n
;​​LogPlot{variance[0,n],variance[1,n],variance[2,n],var2[n]},{n,1,100},PlotRange{
-6
10
,1},PlotLegends"s=0","s=1","s=2",
2
π
3
2
n
,AxesLabel{"n","f"},ImageSizeLarge,PlotStyle{Automatic,Automatic,Automatic,Dashed},PlotLabel"f(s,n)"
Power
:Indeterminate expression
0
0.
encountered.
Power
:Indeterminate expression
0
0.
encountered.
Out[]=
s=0
s=1
s=2
2
π
3
2
n
In[]:=
x0s=CirclePoints[11];​​makeTraj[x0_,numSteps_]:=(​​n=2;​​hs=Table[1/i,{i,1,n}];​​ones=ConstantArray[1,n];​​xs=NestList[(ones-1.4hs)*#&,x0,numSteps];​​trajectory=Graphics[Line[xs]];​​contour=ContourPlot[Total[{x,y}*hs*{x,y}],{x,-1,1},{y,-1,1},ContourShadingNone,ContourStyleBlue];​​Show[contour,trajectory,Graphics[Point[x0s]],Graphics[Point[xs]]]​​)​​frames=Flatten[Table[makeTraj[x0,s],{x0,x0s},{s,1,5}]];
In[]:=
anim=ListAnimate[frames,AnimationRate2,AnimationRepetitions∞]
Out[]=
In[]:=
SetDirectory[NotebookDirectory[]];​​SetDirectory["export"];​​Export["anim.gif",anim,"AnimationRepetitions"∞]
OptionValue
:Unknown option AnimationRepetitions for System`Convert`CommonGraphicsDump`ManipulateToImageList.
OptionValue
:Unknown option AnimationRepetitions for System`Convert`CommonGraphicsDump`ManipulateToImageList.
Out[]=
anim.gif