The complex period constants of center type Lambda_1
Author
Yusen Wu
Title
The complex period constants of center type Lambda_1
Description
Weak Bi-Center and Critical Period Bifurcations of a Z2-Equivariant Quintic System
Category
Academic Articles & Supplements
Keywords
Weak bi-center, critical period bifurcation, period constant, Z2-equivariant quintic system
URL
http://www.notebookarchive.org/2021-08-5ztybxs/
DOI
https://notebookarchive.org/2021-08-5ztybxs
Date Added
2021-08-13
Date Last Modified
2021-08-13
File Size
430.76 kilobytes
Supplements
Rights
Redistribution rights reserved
Download
Open in Wolfram Cloud
In[]:=
ckj=Sum[((k-α+1)-(j-(3-α)+1))c[k-α+1,j-(3-α)+1],{α,0,3}]+Sum[((k-α+1)-(j-(4-α)+1))c[k-α+1,j-(4-α)+1],{α,0,4}]+Sum[((k-α+1)-(j-(5-α)+1))c[k-α+1,j-(5-α)+1],{α,0,5}]+Sum[((k-α+1)-(j-(6-α)+1))c[k-α+1,j-(6-α)+1],{α,0,6}]
a
α,3-α-1
b
3-α,α-1
a
α,4-α-1
b
4-α,α-1
a
α,5-α-1
b
5-α,α-1
a
α,6-α-1
b
6-α,α-1
Out[]=
c[-2+k,1+j]((-2+k)-(1+j))+c[-3+k,1+j]((-3+k)-(1+j))+c[-4+k,1+j]((-4+k)-(1+j))+c[-5+k,1+j]((-5+k)-(1+j))+c[-1+k,j]((-1+k)-j)+c[-2+k,j]((-2+k)-j)+c[-3+k,j]((-3+k)-j)+c[-4+k,j]((-4+k)-j)+c[k,-1+j](k-(-1+j))+c[-1+k,-1+j]((-1+k)-(-1+j))+c[-2+k,-1+j]((-2+k)-(-1+j))+c[-3+k,-1+j]((-3+k)-(-1+j))+c[1+k,-2+j]((1+k)-(-2+j))+c[k,-2+j](k-(-2+j))+c[-1+k,-2+j]((-1+k)-(-2+j))+c[-2+k,-2+j]((-2+k)-(-2+j))+c[1+k,-3+j]((1+k)-(-3+j))+c[k,-3+j](k-(-3+j))+c[-1+k,-3+j]((-1+k)-(-3+j))+c[1+k,-4+j]((1+k)-(-4+j))+c[k,-4+j](k-(-4+j))+c[1+k,-5+j]((1+k)-(-5+j))
a
3,-1
b
0,2
a
4,-1
b
0,3
a
5,-1
b
0,4
a
6,-1
b
0,5
a
2,0
b
1,1
a
3,0
b
1,2
a
4,0
b
1,3
a
5,0
b
1,4
a
1,1
b
2,0
a
2,1
b
2,1
a
3,1
b
2,2
a
4,1
b
2,3
a
0,2
b
3,-1
a
1,2
b
3,0
a
2,2
b
3,1
a
3,2
b
3,2
a
0,3
b
4,-1
a
1,3
b
4,0
a
2,3
b
4,1
a
0,4
b
5,-1
a
1,4
b
5,0
a
0,5
b
6,-1
In[]:=
a
3,-1
b
3,-1
a
4,-1
b
4,-1
a
5,-1
b
5,-1
a
6,-1
b
6,-1
In[]:=
ckj
Out[]=
(1+k)c[1+k,-2+j]+(1+k)c[1+k,-3+j]+(1+k)c[1+k,-4+j]+(1+k)c[1+k,-5+j]-(1+j)c[-2+k,1+j]-(1+j)c[-3+k,1+j]-(1+j)c[-4+k,1+j]-(1+j)c[-5+k,1+j]+c[-1+k,j]((-1+k)-j)+c[-2+k,j]((-2+k)-j)+c[-3+k,j]((-3+k)-j)+c[-4+k,j]((-4+k)-j)+c[k,-1+j](k-(-1+j))+c[-1+k,-1+j]((-1+k)-(-1+j))+c[-2+k,-1+j]((-2+k)-(-1+j))+c[-3+k,-1+j]((-3+k)-(-1+j))+c[k,-2+j](k-(-2+j))+c[-1+k,-2+j]((-1+k)-(-2+j))+c[-2+k,-2+j]((-2+k)-(-2+j))+c[k,-3+j](k-(-3+j))+c[-1+k,-3+j]((-1+k)-(-3+j))+c[k,-4+j](k-(-4+j))
a
0,2
a
0,3
a
0,4
a
0,5
b
0,2
b
0,3
b
0,4
b
0,5
a
2,0
b
1,1
a
3,0
b
1,2
a
4,0
b
1,3
a
5,0
b
1,4
a
1,1
b
2,0
a
2,1
b
2,1
a
3,1
b
2,2
a
4,1
b
2,3
a
1,2
b
3,0
a
2,2
b
3,1
a
3,2
b
3,2
a
1,3
b
4,0
a
2,3
b
4,1
a
1,4
b
5,0
In[]:=
dkj=Sum[((k-α+1)-(j-(3-α)+1))d[k-α+1,j-(3-α)+1],{α,0,3}]+Sum[((k-α+1)-(j-(4-α)+1))d[k-α+1,j-(4-α)+1],{α,0,4}]+Sum[((k-α+1)-(j-(5-α)+1))d[k-α+1,j-(5-α)+1],{α,0,5}]+Sum[((k-α+1)-(j-(6-α)+1))d[k-α+1,j-(6-α)+1],{α,0,6}]
b
α,3-α-1
a
3-α,α-1
b
α,4-α-1
a
4-α,α-1
b
α,5-α-1
a
5-α,α-1
b
α,6-α-1
a
6-α,α-1
Out[]=
-(1+j)d[-2+k,1+j]-(1+j)d[-3+k,1+j]-(1+j)d[-4+k,1+j]-(1+j)d[-5+k,1+j]+(1+k)d[1+k,-2+j]+(1+k)d[1+k,-3+j]+(1+k)d[1+k,-4+j]+(1+k)d[1+k,-5+j]+d[k,-1+j](-(-1+j)+k)+d[k,-2+j](-(-2+j)+k)+d[k,-3+j](-(-3+j)+k)+d[k,-4+j](-(-4+j)+k)+d[-1+k,j](-j+(-1+k))+d[-1+k,-1+j](-(-1+j)+(-1+k))+d[-1+k,-2+j](-(-2+j)+(-1+k))+d[-1+k,-3+j](-(-3+j)+(-1+k))+d[-2+k,j](-j+(-2+k))+d[-2+k,-1+j](-(-1+j)+(-2+k))+d[-2+k,-2+j](-(-2+j)+(-2+k))+d[-3+k,j](-j+(-3+k))+d[-3+k,-1+j](-(-1+j)+(-3+k))+d[-4+k,j](-j+(-4+k))
a
0,2
a
0,3
a
0,4
a
0,5
b
0,2
b
0,3
b
0,4
b
0,5
a
2,0
b
1,1
a
3,0
b
1,2
a
4,0
b
1,3
a
5,0
b
1,4
a
1,1
b
2,0
a
2,1
b
2,1
a
3,1
b
2,2
a
4,1
b
2,3
a
1,2
b
3,0
a
2,2
b
3,1
a
3,2
b
3,2
a
1,3
b
4,0
a
2,3
b
4,1
a
1,4
b
5,0
In[]:=
pj=Sum[((j-α+2)-(j-(3-α)+1))c[j-α+2,j-(3-α)+1],{α,0,3}]+Sum[((j-α+2)-(j-(4-α)+1))c[j-α+2,j-(4-α)+1],{α,0,4}]+Sum[((j-α+2)-(j-(5-α)+1))c[j-α+2,j-(5-α)+1],{α,0,5}]+Sum[((j-α+2)-(j-(6-α)+1))c[j-α+2,j-(6-α)+1],{α,0,6}]
a
α,3-α-1
b
3-α,α-1
a
α,4-α-1
b
4-α,α-1
a
α,5-α-1
b
5-α,α-1
a
α,6-α-1
b
6-α,α-1
Out[]=
(2+j)c[2+j,-2+j]+(2+j)c[2+j,-3+j]+(2+j)c[2+j,-4+j]+(2+j)c[2+j,-5+j]-(1+j)c[-1+j,1+j]-(1+j)c[-2+j,1+j]-(1+j)c[-3+j,1+j]-(1+j)c[-4+j,1+j]+c[j,j](j-j)+c[-1+j,j]((-1+j)-j)+c[-2+j,j]((-2+j)-j)+c[-3+j,j]((-3+j)-j)+c[1+j,-1+j]((1+j)-(-1+j))+c[j,-1+j](j-(-1+j))+c[-1+j,-1+j]((-1+j)-(-1+j))+c[-2+j,-1+j]((-2+j)-(-1+j))+c[1+j,-2+j]((1+j)-(-2+j))+c[j,-2+j](j-(-2+j))+c[-1+j,-2+j]((-1+j)-(-2+j))+c[1+j,-3+j]((1+j)-(-3+j))+c[j,-3+j](j-(-3+j))+c[1+j,-4+j]((1+j)-(-4+j))
a
0,2
a
0,3
a
0,4
a
0,5
b
0,2
b
0,3
b
0,4
b
0,5
a
2,0
b
1,1
a
3,0
b
1,2
a
4,0
b
1,3
a
5,0
b
1,4
a
1,1
b
2,0
a
2,1
b
2,1
a
3,1
b
2,2
a
4,1
b
2,3
a
1,2
b
3,0
a
2,2
b
3,1
a
3,2
b
3,2
a
1,3
b
4,0
a
2,3
b
4,1
a
1,4
b
5,0
In[]:=
qj=Sum[((j-α+2)-(j-(3-α)+1))d[j-α+2,j-(3-α)+1],{α,0,3}]+Sum[((j-α+2)-(j-(4-α)+1))d[j-α+2,j-(4-α)+1],{α,0,4}]+Sum[((j-α+2)-(j-(5-α)+1))d[j-α+2,j-(5-α)+1],{α,0,5}]+Sum[((j-α+2)-(j-(6-α)+1))d[j-α+2,j-(6-α)+1],{α,0,6}]
b
α,3-α-1
a
3-α,α-1
b
α,4-α-1
a
4-α,α-1
b
α,5-α-1
a
5-α,α-1
b
α,6-α-1
a
6-α,α-1
Out[]=
-(1+j)d[-1+j,1+j]-(1+j)d[-2+j,1+j]-(1+j)d[-3+j,1+j]-(1+j)d[-4+j,1+j]+(2+j)d[2+j,-2+j]+(2+j)d[2+j,-3+j]+(2+j)d[2+j,-4+j]+(2+j)d[2+j,-5+j]+d[1+j,-1+j](-(-1+j)+(1+j))+d[1+j,-2+j](-(-2+j)+(1+j))+d[1+j,-3+j](-(-3+j)+(1+j))+d[1+j,-4+j](-(-4+j)+(1+j))+d[j,j](-j+j)+d[j,-1+j](-(-1+j)+j)+d[j,-2+j](-(-2+j)+j)+d[j,-3+j](-(-3+j)+j)+d[-1+j,j](-j+(-1+j))+d[-1+j,-1+j](-(-1+j)+(-1+j))+d[-1+j,-2+j](-(-2+j)+(-1+j))+d[-2+j,j](-j+(-2+j))+d[-2+j,-1+j](-(-1+j)+(-2+j))+d[-3+j,j](-j+(-3+j))
a
0,2
a
0,3
a
0,4
a
0,5
b
0,2
b
0,3
b
0,4
b
0,5
a
2,0
b
1,1
a
3,0
b
1,2
a
4,0
b
1,3
a
5,0
b
1,4
a
1,1
b
2,0
a
2,1
b
2,1
a
3,1
b
2,2
a
4,1
b
2,3
a
1,2
b
3,0
a
2,2
b
3,1
a
3,2
b
3,2
a
1,3
b
4,0
a
2,3
b
4,1
a
1,4
b
5,0
In[]:=
zT=(-48+32a2+128a6-32a7-56+48a2-16a3+96a6-48a7-16a8-22+24a2-16a3-8a4+32a6-24a7-16a8+8a9-3+4a10+4a2-4a3-4a4+4a6-4a7-4a8+4a9+128z+160wz-64a2wz+64a7wz+24z-48a2z+48a3z+96a6z+48a7z+48a8z-24z+32a3z+32a4z+64a6z+32a8z-32a9z-7z-20a10z+4a2z+4a3z+12a4z+12a6z-4a7z+4a8z-12a9z+208+32a2-128a6-32a7+216w-48a2w-48a3w-96a6w+48a7w-48a8w+60-48a2-48a4+48a7+48a9+2+40a10-8a2+8a3-8a4+8a6+8a7+8a8+8a9+136+48a2+16a3-96a6-48a7+16a8+104w-32a3w+32a4w-64a6w-32a8w-32a9w+18-40a10-8a2-8a3-8a4-8a6+8a7-8a8+8a9+42+24a2+16a3-8a4-32a6-24a7+16a8+8a9+17w+20a10w+4a2w-4a3w+12a4w-12a6w-4a7w-4a8w-12a9w+5-4a10+4a2+4a3-4a4-4a6-4a7+4a8+4a9);
1
128
2
w
2
w
2
w
2
w
3
w
3
w
3
w
3
w
3
w
3
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
5
w
5
w
5
w
5
w
5
w
5
w
5
w
5
w
5
w
2
w
2
w
2
w
2
w
2
w
2
w
3
w
3
w
3
w
3
w
3
w
3
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
w
2
z
2
w
2
z
2
w
2
z
2
w
2
z
2
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
5
z
5
z
5
z
5
z
5
z
5
z
5
z
5
z
5
z
In[]:=
wT=(-128w-208+32a2-128a6+32a7-136+48a2-16a3-96a6+48a7+16a8-42+24a2-16a3-8a4-32a6+24a7+16a8-8a9-5-4a10+4a2-4a3-4a4-4a6+4a7+4a8-4a9-160wz-64a2wz-64a7wz-216z-48a2z+48a3z-96a6z-48a7z-48a8z-104z+32a3z+32a4z-64a6z-32a8z+32a9z-17z+20a10z+4a2z+4a3z+12a4z-12a6z+4a7z-4a8z+12a9z+48+32a2+128a6+32a7-24w-48a2w-48a3w+96a6w-48a7w+48a8w-60-48a2-48a4-48a7-48a9-18-40a10-8a2+8a3-8a4-8a6-8a7-8a8-8a9+56+48a2+16a3+96a6+48a7-16a8+24w-32a3w+32a4w+64a6w+32a8w+32a9w-2+40a10-8a2-8a3-8a4+8a6-8a7+8a8-8a9+22+24a2+16a3-8a4+32a6+24a7-16a8-8a9+7w-20a10w+4a2w-4a3w+12a4w+12a6w+4a7w+4a8w+12a9w+3+4a10+4a2+4a3-4a4+4a6+4a7-4a8-4a9);
1
128
2
w
2
w
2
w
2
w
3
w
3
w
3
w
3
w
3
w
3
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
5
w
5
w
5
w
5
w
5
w
5
w
5
w
5
w
5
w
2
w
2
w
2
w
2
w
2
w
2
w
3
w
3
w
3
w
3
w
3
w
3
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
4
w
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
z
2
w
2
z
2
w
2
z
2
w
2
z
2
w
2
z
2
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
w
2
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
2
w
3
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
4
z
5
z
5
z
5
z
5
z
5
z
5
z
5
z
5
z
5
z
In[]:=
zT=Expand[zT]
Out[]=
-+a2+a6--+a2-+a6--a8-+a2--a4+a6--a8+-+a10+a2--a4+a6--a8++z+-a2wz++-a2z+a3z+a6z+a7z+a8z-+a3z+a4z+a6z+a8z-a9z--a10z+a2z+a3z+a4z+a6z-a7z+a8z-a9z++a2-a6-+-a2w-a3w-a6w+a7w-a8w+-a2-a4+a7+a9++a10-a2+a3-a4+a6+a7+a8+a9++a2+-a6-+a8+-a3w+a4w-a6w-a8w-a9w+-a10-a2-a3-a4-a6+a7-a8+a9++a2+-a4-a6-+a8+++a10w+a2w-a3w+a4w-a6w-a7w-a8w-a9w+-a10+a2+-a4-a6-+a8+
3
2
w
8
1
4
2
w
2
w
a7
2
w
4
7
3
w
16
3
8
3
w
a3
3
w
8
3
4
3
w
3a7
3
w
8
1
8
3
w
11
4
w
64
3
16
4
w
a3
4
w
8
1
16
4
w
1
4
4
w
3a7
4
w
16
1
8
4
w
a9
4
w
16
3
5
w
128
1
32
5
w
1
32
5
w
a3
5
w
32
1
32
5
w
1
32
5
w
a7
5
w
32
1
32
5
w
a9
5
w
32
5wz
4
1
2
a7wz
2
3z
2
w
16
3
8
2
w
3
8
2
w
3
4
2
w
3
8
2
w
3
8
2
w
3z
3
w
16
1
4
3
w
1
4
3
w
1
2
3
w
1
4
3
w
1
4
3
w
7z
4
w
128
5
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
3
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
13
2
z
8
1
4
2
z
2
z
a7
2
z
4
27w
2
z
16
3
8
2
z
3
8
2
z
3
4
2
z
3
8
2
z
3
8
2
z
15
2
w
2
z
32
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
3
w
2
z
64
5
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
17
3
z
16
3
8
3
z
a3
3
z
8
3
4
3
z
3a7
3
z
8
1
8
3
z
13w
3
z
16
1
4
3
z
1
4
3
z
1
2
3
z
1
4
3
z
1
4
3
z
9
2
w
3
z
64
5
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
21
4
z
64
3
16
4
z
a3
4
z
8
1
16
4
z
1
4
4
z
3a7
4
z
16
1
8
4
z
a9
4
z
16
17w
4
z
128
5
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
3
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
5
5
z
128
1
32
5
z
1
32
5
z
a3
5
z
32
1
32
5
z
1
32
5
z
a7
5
z
32
1
32
5
z
a9
5
z
32
In[]:=
wT=Expand[wT]
Out[]=
-w-+a2-a6+-+a2--a6++a8-+a2--a4-a6++a8---a10+a2--a4-a6++a8---a2wz---a2z+a3z-a6z-a7z-a8z-+a3z+a4z-a6z-a8z+a9z-+a10z+a2z+a3z+a4z-a6z+a7z-a8z+a9z++a2+a6+--a2w-a3w+a6w-a7w+a8w--a2-a4-a7-a9--a10-a2+a3-a4-a6-a7-a8-a9++a2++a6+-a8+-a3w+a4w+a6w+a8w+a9w-+a10-a2-a3-a4+a6-a7+a8-a9++a2+-a4+a6+-a8-+-a10w+a2w-a3w+a4w+a6w+a7w+a8w+a9w++a10+a2+-a4+a6+-a8-
13
2
w
8
1
4
2
w
2
w
a7
2
w
4
17
3
w
16
3
8
3
w
a3
3
w
8
3
4
3
w
3a7
3
w
8
1
8
3
w
21
4
w
64
3
16
4
w
a3
4
w
8
1
16
4
w
1
4
4
w
3a7
4
w
16
1
8
4
w
a9
4
w
16
5
5
w
128
1
32
5
w
1
32
5
w
a3
5
w
32
1
32
5
w
1
32
5
w
a7
5
w
32
1
32
5
w
a9
5
w
32
5wz
4
1
2
a7wz
2
27z
2
w
16
3
8
2
w
3
8
2
w
3
4
2
w
3
8
2
w
3
8
2
w
13z
3
w
16
1
4
3
w
1
4
3
w
1
2
3
w
1
4
3
w
1
4
3
w
17z
4
w
128
5
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
3
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
3
2
z
8
1
4
2
z
2
z
a7
2
z
4
3w
2
z
16
3
8
2
z
3
8
2
z
3
4
2
z
3
8
2
z
3
8
2
z
15
2
w
2
z
32
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
9
3
w
2
z
64
5
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
7
3
z
16
3
8
3
z
a3
3
z
8
3
4
3
z
3a7
3
z
8
1
8
3
z
3w
3
z
16
1
4
3
z
1
4
3
z
1
2
3
z
1
4
3
z
1
4
3
z
2
w
3
z
64
5
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
11
4
z
64
3
16
4
z
a3
4
z
8
1
16
4
z
1
4
4
z
3a7
4
z
16
1
8
4
z
a9
4
z
16
7w
4
z
128
5
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
3
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
3
5
z
128
1
32
5
z
1
32
5
z
a3
5
z
32
1
32
5
z
1
32
5
z
a7
5
z
32
1
32
5
z
a9
5
z
32
In[]:=
Z2=Select[zT,Total[Exponent[#,{z,w}]]2&]
Out[]=
-+a2+a6-+-a2wz+++a2-a6-
3
2
w
8
1
4
2
w
2
w
a7
2
w
4
5wz
4
1
2
a7wz
2
13
2
z
8
1
4
2
z
2
z
a7
2
z
4
In[]:=
Z3=Select[zT,Total[Exponent[#,{z,w}]]3&]
Out[]=
-+a2-+a6--a8+-a2z+a3z+a6z+a7z+a8z+-a2w-a3w-a6w+a7w-a8w++a2+-a6-+a8
7
3
w
16
3
8
3
w
a3
3
w
8
3
4
3
w
3a7
3
w
8
1
8
3
w
3z
2
w
16
3
8
2
w
3
8
2
w
3
4
2
w
3
8
2
w
3
8
2
w
27w
2
z
16
3
8
2
z
3
8
2
z
3
4
2
z
3
8
2
z
3
8
2
z
17
3
z
16
3
8
3
z
a3
3
z
8
3
4
3
z
3a7
3
z
8
1
8
3
z
In[]:=
Z4=Select[zT,Total[Exponent[#,{z,w}]]4&]
Out[]=
-+a2--a4+a6--a8+-+a3z+a4z+a6z+a8z-a9z+-a2-a4+a7+a9+-a3w+a4w-a6w-a8w-a9w++a2+-a4-a6-+a8+
11
4
w
64
3
16
4
w
a3
4
w
8
1
16
4
w
1
4
4
w
3a7
4
w
16
1
8
4
w
a9
4
w
16
3z
3
w
16
1
4
3
w
1
4
3
w
1
2
3
w
1
4
3
w
1
4
3
w
15
2
w
2
z
32
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
13w
3
z
16
1
4
3
z
1
4
3
z
1
2
3
z
1
4
3
z
1
4
3
z
21
4
z
64
3
16
4
z
a3
4
z
8
1
16
4
z
1
4
4
z
3a7
4
z
16
1
8
4
z
a9
4
z
16
In[]:=
Z5=Select[zT,Total[Exponent[#,{z,w}]]5&]
Out[]=
-+a10+a2--a4+a6--a8+--a10z+a2z+a3z+a4z+a6z-a7z+a8z-a9z++a10-a2+a3-a4+a6+a7+a8+a9+-a10-a2-a3-a4-a6+a7-a8+a9++a10w+a2w-a3w+a4w-a6w-a7w-a8w-a9w+-a10+a2+-a4-a6-+a8+
3
5
w
128
1
32
5
w
1
32
5
w
a3
5
w
32
1
32
5
w
1
32
5
w
a7
5
w
32
1
32
5
w
a9
5
w
32
7z
4
w
128
5
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
3
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
3
w
2
z
64
5
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
9
2
w
3
z
64
5
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
17w
4
z
128
5
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
3
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
5
5
z
128
1
32
5
z
1
32
5
z
a3
5
z
32
1
32
5
z
1
32
5
z
a7
5
z
32
1
32
5
z
a9
5
z
32
In[]:=
Factor[z+Z2+Z3+Z4+Z5-zT]
Out[]=
0
In[]:=
W2=Select[-wT,Total[Exponent[#,{z,w}]]2&]
Out[]=
13
2
w
8
1
4
2
w
2
w
a7
2
w
4
5wz
4
1
2
a7wz
2
3
2
z
8
1
4
2
z
2
z
a7
2
z
4
In[]:=
W3=Select[-wT,Total[Exponent[#,{z,w}]]3&]
Out[]=
17
3
w
16
3
8
3
w
a3
3
w
8
3
4
3
w
3a7
3
w
8
1
8
3
w
27z
2
w
16
3
8
2
w
3
8
2
w
3
4
2
w
3
8
2
w
3
8
2
w
3w
2
z
16
3
8
2
z
3
8
2
z
3
4
2
z
3
8
2
z
3
8
2
z
7
3
z
16
3
8
3
z
a3
3
z
8
3
4
3
z
3a7
3
z
8
1
8
3
z
In[]:=
W4=Select[-wT,Total[Exponent[#,{z,w}]]4&]
Out[]=
21
4
w
64
3
16
4
w
a3
4
w
8
1
16
4
w
1
4
4
w
3a7
4
w
16
1
8
4
w
a9
4
w
16
13z
3
w
16
1
4
3
w
1
4
3
w
1
2
3
w
1
4
3
w
1
4
3
w
15
2
w
2
z
32
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
3
8
2
w
2
z
3w
3
z
16
1
4
3
z
1
4
3
z
1
2
3
z
1
4
3
z
1
4
3
z
11
4
z
64
3
16
4
z
a3
4
z
8
1
16
4
z
1
4
4
z
3a7
4
z
16
1
8
4
z
a9
4
z
16
In[]:=
W5=Select[-wT,Total[Exponent[#,{z,w}]]5&]
Out[]=
5
5
w
128
1
32
5
w
1
32
5
w
a3
5
w
32
1
32
5
w
1
32
5
w
a7
5
w
32
1
32
5
w
a9
5
w
32
17z
4
w
128
5
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
3
32
4
w
1
32
4
w
1
32
4
w
3
32
4
w
9
3
w
2
z
64
5
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
1
16
3
w
2
z
2
w
3
z
64
5
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
1
16
2
w
3
z
7w
4
z
128
5
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
3
32
4
z
1
32
4
z
1
32
4
z
3
32
4
z
3
5
z
128
1
32
5
z
1
32
5
z
a3
5
z
32
1
32
5
z
1
32
5
z
a7
5
z
32
1
32
5
z
a9
5
z
32
In[]:=
Factor[w+W2+W3+W4+W5+wT]
Out[]=
0
In[]:=
S=Factor[CoefficientRules[Z2,{z,w}]]
Out[]=
{2,0}(13+2a2-8a6-2a7),{1,1}(5-2a2+2a7),{0,2}(3+2a2+8a6+2a7)
1
8
1
4
1
8
In[]:=
{a20,a11,a02}={S[[1,2]],S[[2,2]],S[[3,2]]}
Out[]=
(13+2a2-8a6-2a7),(5-2a2+2a7),(3+2a2+8a6+2a7)
1
8
1
4
1
8
In[]:=
S=Factor[CoefficientRules[Z3,{z,w}]]
Out[]=
{3,0}(17+6a2+2a3-12a6-6a7+2a8),{2,1}(9-2a2-2a3-4a6+2a7-2a8),{1,2}(1-2a2+2a3+4a6+2a7+2a8),{0,3}(7+6a2+2a3+12a6+6a7-2a8)
1
16
3
16
3
16
1
16
In[]:=
{a30,a21,a12,a03}={S[[1,2]],S[[2,2]],S[[3,2]],S[[4,2]]}
Out[]=
(17+6a2+2a3-12a6-6a7+2a8),(9-2a2-2a3-4a6+2a7-2a8),(1-2a2+2a3+4a6+2a7+2a8),(7+6a2+2a3+12a6+6a7-2a8)
1
16
3
16
3
16
1
16
In[]:=
S=Factor[CoefficientRules[Z4,{z,w}]]
Out[]=
{4,0}(21+12a2+8a3-4a4-16a6-12a7+8a8+4a9),{3,1}(13-4a3+4a4-8a6-4a8-4a9),{2,2}(5-4a2-4a4+4a7+4a9),{1,3}(-3+4a3+4a4+8a6+4a8-4a9),{0,4}(-11+12a2-8a3-4a4+16a6-12a7-8a8+4a9)
1
64
1
16
3
32
1
16
1
64
In[]:=
{a40,a31,a22,a13,a04}={S[[1,2]],S[[2,2]],S[[3,2]],S[[4,2]],S[[5,2]]}
Out[]=
(21+12a2+8a3-4a4-16a6-12a7+8a8+4a9),(13-4a3+4a4-8a6-4a8-4a9),(5-4a2-4a4+4a7+4a9),(-3+4a3+4a4+8a6+4a8-4a9),(-11+12a2-8a3-4a4+16a6-12a7-8a8+4a9)
1
64
1
16
3
32
1
16
1
64
In[]:=
S=Factor[CoefficientRules[Z5,{z,w}]]
Out[]=
{5,0}(5-4a10+4a2+4a3-4a4-4a6-4a7+4a8+4a9),{4,1}(17+20a10+4a2-4a3+12a4-12a6-4a7-4a8-12a9),{3,2}(9-20a10-4a2-4a3-4a4-4a6+4a7-4a8+4a9),{2,3}(1+20a10-4a2+4a3-4a4+4a6+4a7+4a8+4a9),{1,4}(-7-20a10+4a2+4a3+12a4+12a6-4a7+4a8-12a9),{0,5}(-3+4a10+4a2-4a3-4a4+4a6-4a7-4a8+4a9)
1
128
1
128
1
64
1
64
1
128
1
128
In[]:=
{a50,a41,a32,a23,a14,a05}={S[[1,2]],S[[2,2]],S[[3,2]],S[[4,2]],S[[5,2]],S[[6,2]]}
Out[]=
(5-4a10+4a2+4a3-4a4-4a6-4a7+4a8+4a9),(17+20a10+4a2-4a3+12a4-12a6-4a7-4a8-12a9),(9-20a10-4a2-4a3-4a4-4a6+4a7-4a8+4a9),(1+20a10-4a2+4a3-4a4+4a6+4a7+4a8+4a9),(-7-20a10+4a2+4a3+12a4+12a6-4a7+4a8-12a9),(-3+4a10+4a2-4a3-4a4+4a6-4a7-4a8+4a9)
1
128
1
128
1
64
1
64
1
128
1
128
In[]:=
S=Factor[CoefficientRules[W2,{w,z}]]
Out[]=
{2,0}(13-2a2+8a6-2a7),{1,1}(5+2a2+2a7),{0,2}-(-3+2a2+8a6-2a7)
1
8
1
4
1
8
In[]:=
{b20,b11,b02}={S[[1,2]],S[[2,2]],S[[3,2]]}
Out[]=
(13-2a2+8a6-2a7),(5+2a2+2a7),-(-3+2a2+8a6-2a7)
1
8
1
4
1
8
In[]:=
S=Factor[CoefficientRules[W3,{w,z}]]
Out[]=
{3,0}(17-6a2+2a3+12a6-6a7-2a8),{2,1}(9+2a2-2a3+4a6+2a7+2a8),{1,2}(1+2a2+2a3-4a6+2a7-2a8),{0,3}-(-7+6a2-2a3+12a6-6a7-2a8)
1
16
3
16
3
16
1
16
In[]:=
{b30,b21,b12,b03}={S[[1,2]],S[[2,2]],S[[3,2]],S[[4,2]]}
Out[]=
(17-6a2+2a3+12a6-6a7-2a8),(9+2a2-2a3+4a6+2a7+2a8),(1+2a2+2a3-4a6+2a7-2a8),-(-7+6a2-2a3+12a6-6a7-2a8)
1
16
3
16
3
16
1
16
In[]:=
S=Factor[CoefficientRules[W4,{w,z}]]
Out[]=
{4,0}(21-12a2+8a3+4a4+16a6-12a7-8a8+4a9),{3,1}(13-4a3-4a4+8a6+4a8-4a9),{2,2}(5+4a2+4a4+4a7+4a9),{1,3}(-3+4a3-4a4-8a6-4a8-4a9),{0,4}(-11-12a2-8a3+4a4-16a6-12a7+8a8+4a9)
1
64
1
16
3
32
1
16
1
64
In[]:=
{b40,b31,b22,b13,b04}={S[[1,2]],S[[2,2]],S[[3,2]],S[[4,2]],S[[5,2]]}
Out[]=
(21-12a2+8a3+4a4+16a6-12a7-8a8+4a9),(13-4a3-4a4+8a6+4a8-4a9),(5+4a2+4a4+4a7+4a9),(-3+4a3-4a4-8a6-4a8-4a9),(-11-12a2-8a3+4a4-16a6-12a7+8a8+4a9)
1
64
1
16
3
32
1
16
1
64
In[]:=
S=Factor[CoefficientRules[W5,{w,z}]]
Out[]=
{5,0}(5+4a10-4a2+4a3+4a4+4a6-4a7-4a8+4a9),{4,1}(17-20a10-4a2-4a3-12a4+12a6-4a7+4a8-12a9),{3,2}(9+20a10+4a2-4a3+4a4+4a6+4a7+4a8+4a9),{2,3}(1-20a10+4a2+4a3+4a4-4a6+4a7-4a8+4a9),{1,4}(-7+20a10-4a2+4a3-12a4-12a6-4a7-4a8-12a9),{0,5}(-3-4a10-4a2-4a3+4a4-4a6-4a7+4a8+4a9)
1
128
1
128
1
64
1
64
1
128
1
128
In[]:=
{b50,b41,b32,b23,b14,b05}={S[[1,2]],S[[2,2]],S[[3,2]],S[[4,2]],S[[5,2]],S[[6,2]]}
Out[]=
(5+4a10-4a2+4a3+4a4+4a6-4a7-4a8+4a9),(17-20a10-4a2-4a3-12a4+12a6-4a7+4a8-12a9),(9+20a10+4a2-4a3+4a4+4a6+4a7+4a8+4a9),(1-20a10+4a2+4a3+4a4-4a6+4a7-4a8+4a9),(-7+20a10-4a2+4a3-12a4-12a6-4a7-4a8-12a9),(-3-4a10-4a2-4a3+4a4-4a6-4a7+4a8+4a9)
1
128
1
128
1
64
1
64
1
128
1
128
In[]:=
con={a20,b20,a11,b11,a02,b02,a30,b30,a21,b21,a12,b12,a03,b03,a40,a31,a22,a13,a04,b40,b31,b22,b13,b04,a50,a41,a32,a23,a14,a05,b50,b41,b32,b23,b14,b05};
a
2,0
b
2,0
a
1,1
b
1,1
a
0,2
b
0,2
a
3,0
b
3,0
a
2,1
b
2,1
a
1,2
b
1,2
a
0,3
b
0,3
a
4,0
a
3,1
a
2,2
a
1,3
a
0,4
b
4,0
b
3,1
b
2,2
b
1,3
b
0,4
a
5,0
a
4,1
a
3,2
a
2,3
a
1,4
a
0,5
b
5,0
b
4,1
b
3,2
b
2,3
b
1,4
b
0,5
In[]:=
ckj=ckj//.con
Out[]=
In[]:=
dkj=dkj//.con
Out[]=
In[]:=
pj=pj//.con;
In[]:=
qj=qj//.con;
In[]:=
tj=pj+qj
Out[]=
In[]:=
c[1,0]=1;c[0,1]=0;d[1,0]=1;d[0,1]=0;c[k_,j_]:=0/;(k<0||j<0||(j>0&&k==j+1));d[k_,j_]:=0/;(k<0||j<0||(j>0&&k==j+1));
In[]:=
In[]:=
In[]:=
In[]:=
a6=0;
In[]:=
a8=1/3(a2-2a2*a7);
In[]:=
a9=1/2(a3-a3*a7);
In[]:=
a10=1/5(3a4-2a4*a7);
In[]:=
t1=Factor[t[1]]
Out[]=
1
12
2
a2
2
a7
In[]:=
Solve[t10,a3]
Out[]=
a3-(24+5+18a7+2)
2
9
2
a2
2
a7
In[]:=
Factort1//.a3-(24+5+18a7+2)
2
9
2
a2
2
a7
Out[]=
0
In[]:=
t2=Factor[t[2]]
Out[]=
1
384
2
a2
4
a2
2
a2
2
a3
2
a2
2
a7
2
a2
2
a7
2
a7
3
a7
4
a7
In[]:=
t21=Factort2//.a3-(24+5+18a7+2)
2
9
2
a2
2
a7
Out[]=
1
108
2
a2
4
a2
2
a2
2
a7
2
a2
2
a7
3
a7
4
a7
In[]:=
k[2,1]=Factor[(t[2]-t21)/t[1]]
Out[]=
1
288
2
a2
2
a7
In[]:=
Factor[t[2]-k[2,1]t1]
Out[]=
1
108
2
a2
4
a2
2
a2
2
a7
2
a2
2
a7
3
a7
4
a7
In[]:=
t2=t21;
Cite this as: Yusen Wu, "The complex period constants of center type Lambda_1" from the Notebook Archive (2021), https://notebookarchive.org/2021-08-5ztybxs
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