New Graph
[2, 3, 2, 3], [4, 4, 1, 1]
π =
[1, 1, 1, 1]
POSSIBLE RANKS
1 x 4
2 x 2
BASE DETERMINANT
117/512, .2285156250
NullSpace of Δ
{1, 2, 3, 4}
Nullspace of A
[{1, 4},{2, 3}]
1
.
Coloring, {}
R:
[2, 3, 2, 3]
B:
[4, 4, 1, 1]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `-1` (` - 1 + τ
` )` ,
1` (` 1 + τ
` )` ,
1` (` 1 + τ
` )` ,
-1` (` - 1 + τ
` )``]`
For τ=1/2, [1, 3, 3, 1]
. FixedPtCheck, [1, 3, 3, 1]
det(A + τ Δ) =
0 Delta Range :
[-y1 - y2 - y3, y1, y2, y3]
[1, 1, 1, 1]
+
-
Δ
See Matrices
$ [
[0, 2, 2, 0]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
] $
$ [
[2, 0, 0, 2]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
] $
$ [
[-1, 1, 1, -1]
,
[0, 0, 0, 0]
,
[0, 0, 0, 0]
] $
[y1, -y1, -y1, y1]
S+
S-
NM
See Matrices
$ [
[1, 0, 1, 0]
,
[1, 1, 0, 0]
,
[0, 0, 1, 1]
,
[0, 1, 0, 1]
] $
$ [
[0, 0, 1, 1]
,
[1, 0, 1, 0]
,
[0, 1, 0, 1]
,
[1, 1, 0, 0]
] $
$ [
[2, 1, 1, 0]
,
[1, 2, 0, 1]
,
[1, 0, 2, 1]
,
[0, 1, 1, 2]
] $
CmmCk
true, true, true
p' =
s 2
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 1 vs 3 |
1 vs 3 |
1 vs 3 |
1 vs 2 |
1 vs 2 |
Omega Rank for R :
cycles:
{{2, 3}}, net cycles:
1
.
order:
2
See Matrix
$ [
[0, 2, 2, 0]
,
[0, 2, 2, 0]
] $
[0, y1, y1, 0]
p =
- s + s 2
Omega Rank for B :
cycles:
{{1, 4}}, net cycles:
1
.
order:
2
See Matrix
$ [
[2, 0, 0, 2]
,
[2, 0, 0, 2]
] $
[y1, 0, 0, y1]
p =
- s + s 2
« NOT SYNC'D »
Nullspace of {Ω&Deltai} :
[0, x1, x2]
For A+2Δ :
[-3 y1 - 3 y2 - y3, y1, y2, y3]
For A-2Δ :
[y1, -3 y1 - y2 - 3 y3, y2, y3]
Range of {ΩΔi}:
[μ1, -μ1, -μ1, μ1]
rank of M is
4
, rank of N is
3
M
N
$ [
[0, 0, 0, 1]
,
[0, 0, 1, 0]
,
[0, 1, 0, 0]
,
[1, 0, 0, 0]
] $
$ [
[0, 1, 1, 2]
,
[1, 0, 2, 1]
,
[1, 2, 0, 1]
,
[2, 1, 1, 0]
] $
Range M, [x4, x1, x2, x3]
τ=
8
, r'=
1/2
Ranges
Action of R on ranges, [[2], [2]]
Action of B on ranges, [[1], [1]]
β({1, 4})
=
1/2
β({2, 3})
=
1/2
ker N, [μ1, -μ1, -μ1, μ1]
Range of
N
[y2, y3, y1, -y2 + y3 + y1]
Partitions
Action of R on partitions, [[2], [2]]
Action of B on partitions, [[1], [1]]
α([{1, 2}, {3, 4}]) = 1/2
α([{1, 3}, {2, 4}]) = 1/2
b1 = {1, 2}
` , ` b2 = {1, 3}
` , ` b3 = {3, 4}
` , ` b4 = {2, 4}
Action of R and B on the blocks of the partitions:
=
[2, 4, 4, 2]
[3, 3, 1, 1]
with invariant measure
[1, 1, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-2 partition graph. `
| Sandwich |
|---|
| Coloring |
{}
|
|---|
| Rank | 2 |
|---|
| R,B |
[2, 3, 2, 3], [4, 4, 1, 1]
|
|---|
| π2 |
[0, 0, 1, 1, 0, 0]
|
|---|
| u2 |
[1, 1, 2, 2, 1, 1]
(dim 1) |
|---|
| wpp |
[2, 2, 2, 2]
|
|---|
2
.
Coloring, {2}
R:
[2, 4, 2, 3]
B:
[4, 3, 1, 1]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `2` (` 5 + 2τ + τ 2
` )`` (` - 1 + τ
` )` ,
2` (` 1 + τ
` )`` (` - 5 + τ 2
` )` ,
-2` (` 1 + τ
` )`` (` 5 - 2τ + τ 2
` )` ,
2` (` - 5 - τ - 3τ 2 + τ 3
` )``]`
For τ=1/2, [-25, -57, -51, -49]
. FixedPtCheck, [25, 57, 51, 49]
det(A + τ Δ) =
1` (` τ
` )`` (` 1 + τ
` )`` (` - 1 + τ
` )`
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 3 |
4 vs 4 |
4 vs 4 |
3 vs 3 |
2 vs 3 |
See Matrix for A+τΔ
bi =
$ [
[0, 1/4, 0, 3/4]
,
[0, 0, 3/4, 1/4]
,
[3/4, 1/4, 0, 0]
,
[3/4, 0, 1/4, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[-35/52, 20/13, 32/13, -40/13]
,
[69/52, -6/13, 32/13, -40/13]
,
[21/52, 14/13, -40/13, 24/13]
,
[-3/52, -28/13, -24/13, 56/13]
] $
x
$ [
[3/2, 1/2, 1, 1]
,
[3/2, 5/8, 5/8, 5/4]
,
[45/32, 17/32, 25/32, 41/32]
,
[99/64, 35/64, 23/32, 19/16]
] $
Check x AllOnes:
[1, 1, 1, 1]
Omega Rank for R :
cycles:
{{2, 3, 4}}, net cycles:
1
.
order:
3
[0, y
2, y
3, y
1]
See Matrices
R =
$ [
[0, 1, 0, 0]
,
[0, 0, 0, 1]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
] $
x
$ [
[0, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[3/4, -1/4, -1/4]
,
[-1/4, 3/4, -1/4]
,
[3/4, -1/4, -1/4]
,
[-1/4, -1/4, 3/4]
] $
x
$ [
[0, 2, 1, 1]
,
[0, 1, 1, 2]
,
[0, 1, 2, 1]
] $
See Matrix
$ [
[2, 0, 1, 1]
,
[2, 0, 0, 2]
,
[2, 0, 0, 2]
] $
[y1 + y2, 0, y1, y2]
p =
- s 2 + s 3
» SYNC'D
1/4
,
0.2500000000
3
.
Coloring, {3}
R:
[2, 3, 1, 3]
B:
[4, 4, 2, 1]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `2` (` - 5 - τ - 3τ 2 + τ 3
` )` ,
-2` (` 1 + τ
` )`` (` 5 - 2τ + τ 2
` )` ,
2` (` 1 + τ
` )`` (` - 5 + τ 2
` )` ,
2` (` - 1 + τ
` )`` (` 5 + 2τ + τ 2
` )``]`
For τ=1/2, [-49, -51, -57, -25]
. FixedPtCheck, [49, 51, 57, 25]
det(A + τ Δ) =
1` (` 1 + τ
` )`` (` τ
` )`` (` - 1 + τ
` )`
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 3 |
4 vs 4 |
4 vs 4 |
3 vs 3 |
2 vs 3 |
See Matrix for A+τΔ
bi =
$ [
[0, 1/4, 0, 3/4]
,
[0, 0, 1/4, 3/4]
,
[1/4, 3/4, 0, 0]
,
[3/4, 0, 1/4, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[-3/52, -28/13, -24/13, 56/13]
,
[21/52, 14/13, -40/13, 24/13]
,
[69/52, -6/13, 32/13, -40/13]
,
[-35/52, 20/13, 32/13, -40/13]
] $
x
$ [
[1, 1, 1/2, 3/2]
,
[5/4, 5/8, 5/8, 3/2]
,
[41/32, 25/32, 17/32, 45/32]
,
[19/16, 23/32, 35/64, 99/64]
] $
Check x AllOnes:
[1, 1, 1, 1]
Omega Rank for R :
cycles:
{{1, 2, 3}}, net cycles:
1
.
order:
3
[y
2, y
3, y
1, 0]
See Matrices
R =
$ [
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[1, 0, 0, 0]
,
[0, 0, 1, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 0]
] $
=
$ [
[-1/4, -1/4, 3/4]
,
[3/4, -1/4, -1/4]
,
[-1/4, 3/4, -1/4]
,
[3/4, -1/4, -1/4]
] $
x
$ [
[1, 1, 2, 0]
,
[2, 1, 1, 0]
,
[1, 2, 1, 0]
] $
See Matrix
$ [
[1, 1, 0, 2]
,
[2, 0, 0, 2]
,
[2, 0, 0, 2]
] $
[-y2 + y1, y2, 0, y1]
p =
- s 2 + s 3
» SYNC'D
1/4
,
0.2500000000
4
.
Coloring, {4}
R:
[2, 3, 2, 1]
B:
[4, 4, 1, 3]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `-2` (` - 1 + τ
` )`` (` 5 + 2τ + τ 2
` )` ,
2` (` 5 - 2τ + τ 2
` )`` (` 1 + τ
` )` ,
2` (` 5 - τ + 3τ 2 + τ 3
` )` ,
2` (` - 1 + τ
` )`` (` - 5 + τ 2
` )``]`
For τ=1/2, [25, 51, 43, 19]
. FixedPtCheck, [25, 51, 43, 19]
det(A + τ Δ) =
1` (` - 1 + τ
` )`` (` τ
` )`` (` 1 + τ
` )`
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 3 |
4 vs 4 |
4 vs 4 |
2 vs 3 |
3 vs 3 |
See Matrix for A+τΔ
bi =
$ [
[0, 1/4, 0, 3/4]
,
[0, 0, 1/4, 3/4]
,
[3/4, 1/4, 0, 0]
,
[1/4, 0, 3/4, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[-19/20, -2, -8/15, 56/15]
,
[37/20, 0, -16/15, -8/15]
,
[1/4, 0, 8/3, -8/3]
,
[-3/20, 2, -16/15, -8/15]
] $
x
$ [
[1, 1/2, 1, 3/2]
,
[9/8, 1/2, 5/4, 9/8]
,
[39/32, 19/32, 31/32, 39/32]
,
[33/32, 35/64, 17/16, 87/64]
] $
Check x AllOnes:
[1, 1, 1, 1]
Omega Rank for R :
cycles:
{{2, 3}}, net cycles:
0
.
order:
2
See Matrix
$ [
[1, 2, 1, 0]
,
[0, 2, 2, 0]
,
[0, 2, 2, 0]
] $
[y1 - y2, y1, y2, 0]
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{1, 3, 4}}, net cycles:
1
.
order:
3
[y
3, 0, y
2, y
1]
See Matrices
B =
$ [
[0, 0, 0, 1]
,
[0, 0, 0, 1]
,
[1, 0, 0, 0]
,
[0, 0, 1, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 0, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[3/4, -1/4, -1/4]
,
[3/4, -1/4, -1/4]
,
[-1/4, -1/4, 3/4]
,
[-1/4, 3/4, -1/4]
] $
x
$ [
[1, 0, 1, 2]
,
[1, 0, 2, 1]
,
[2, 0, 1, 1]
] $
5
.
Coloring, {2, 3}
R:
[2, 4, 1, 3]
B:
[4, 3, 2, 1]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `1 ,
1 ,
1 ,
1`]`
For τ=1/2, [1, 1, 1, 1]
. FixedPtCheck, [1, 1, 1, 1]
det(A + τ Δ) =
1` (` τ
` )`` (` 1 + τ 2
` )` Delta Range :
[-y1 - y2 - y3, y1, y2, y3]
[1, 1, 1, 1]
+
-
Δ
See Matrices
$ [
[1, 1, 1, 1]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
] $
$ [
[1, 1, 1, 1]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
] $
$ [
[0, 0, 0, 0]
,
[0, 0, 0, 0]
,
[0, 0, 0, 0]
] $
[0, 0, 0, 0]
S+
S-
NM
See Matrices
$ [
[0, 1, 0, 0]
,
[0, 0, 0, 1]
,
[1, 0, 0, 0]
,
[0, 0, 1, 0]
] $
$ [
[0, 0, 0, 1]
,
[0, 0, 1, 0]
,
[0, 1, 0, 0]
,
[1, 0, 0, 0]
] $
$ [
[3, 2, 2, 2]
,
[2, 3, 2, 2]
,
[2, 2, 3, 2]
,
[2, 2, 2, 3]
] $
CmmCk
true, true, true
p' =
s 2
p' =
s
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 0 vs 3 |
1 vs 4 |
1 vs 4 |
1 vs 4 |
1 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3, 4}}, net cycles:
1
.
order:
4
See Matrix
$ [
[1, 1, 1, 1]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
] $
[y1, y1, y1, y1]
p' =
- 1 + s 2
p' =
- 1 + s 3
p' =
- 1 + s
Omega Rank for B :
cycles:
{{1, 4}, {2, 3}}, net cycles:
2
.
order:
2
See Matrix
$ [
[1, 1, 1, 1]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
,
[1, 1, 1, 1]
] $
[y1, y1, y1, y1]
p' =
- 1 + s
p' =
- 1 + s 2
p' =
- 1 + s 3
« NOT SYNC'D »
Nullspace of {Ω&Deltai} :
[x1, x2, x3]
For A+2Δ :
[-y1 - y2 - y3, y1, y2, y3]
For A-2Δ :
[-y1 - y2 - y3, y1, y3, y2]
Range of {ΩΔi}:
[0, 0, 0, 0]
rank of M is
4
, rank of N is
4
M
N
$ [
[0, 1, 1, 1]
,
[1, 0, 1, 1]
,
[1, 1, 0, 1]
,
[1, 1, 1, 0]
] $
$ [
[0, 1, 1, 1]
,
[1, 0, 1, 1]
,
[1, 1, 0, 1]
,
[1, 1, 1, 0]
] $
Range M, [x4, x3, x2, x1]
τ=
4
, r'=
3/4
Ranges
Action of R on ranges, [[1]]
Action of B on ranges, [[1]]
β({1, 2, 3, 4})
=
1/1
ker N, [0, 0, 0, 0]
Range of
N
[y1, y2, y3, y4]
Partitions
α([{1}, {2}, {3}, {4}]) = 1/1
b1 = {1}
` , ` b2 = {2}
` , ` b3 = {3}
` , ` b4 = {4}
Action of R and B on the blocks of the partitions:
=
[3, 1, 4, 2]
[4, 3, 2, 1]
with invariant measure
[1, 1, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-4 partition graph. `
| Right Group |
|---|
| Coloring |
{2, 3}
|
|---|
| Rank | 4 |
|---|
| R,B |
[2, 4, 1, 3], [4, 3, 2, 1]
|
|---|
| π2 |
[1, 1, 1, 1, 1, 1]
|
|---|
| u2 |
[1, 1, 1, 1, 1, 1]
(dim 2) |
|---|
| wpp |
[1, 1, 1, 1]
|
|---|
| π4 |
[1]
|
|---|
| u4 |
[1]
|
|---|
6
.
Coloring, {2, 4}
R:
[2, 4, 2, 1]
B:
[4, 3, 1, 3]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `1` (` 5 - τ + 3τ 2 + τ 3
` )` ,
1` (` 1 + τ
` )`` (` 5 - 2τ + τ 2
` )` ,
-1` (` - 1 + τ
` )`` (` 5 + 2τ + τ 2
` )` ,
-1` (` - 5 - τ - 3τ 2 + τ 3
` )``]`
For τ=1/2, [43, 51, 25, 49]
. FixedPtCheck, [43, 51, 25, 49]
det(A + τ Δ) =
0
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 3 |
3 vs 3 |
3 vs 3 |
3 vs 3 |
3 vs 3 |
See Matrix for A+τΔ
bi =
$ [
[0, 1/4, 0, 3/4]
,
[0, 0, 3/4, 1/4]
,
[3/4, 1/4, 0, 0]
,
[1/4, 0, 3/4, 0]
] $
x
$ [
[91/100, 27/100, 3/100, -9/100]
,
[27/100, 19/100, -9/100, 27/100]
,
[3/100, -9/100, 99/100, 3/100]
,
[-9/100, 27/100, 3/100, 91/100]
] $
=
$ [
[-3/4, -1, 2]
,
[5/4, -3/5, -2/5]
,
[-3/4, 7/5, -2/5]
,
[5/4, 1/5, -6/5]
] $
x
$ [
[1, 1/2, 3/2, 1]
,
[11/8, 5/8, 9/8, 7/8]
,
[17/16, 5/8, 9/8, 19/16]
] $
Check x AllOnes:
[1, 1, 1, 1]
Omega Rank for R :
cycles:
{{1, 2, 4}}, net cycles:
1
.
order:
3
[y
1, y
2, 0, y
3]
See Matrices
R =
$ [
[0, 1, 0, 0]
,
[0, 0, 0, 1]
,
[0, 1, 0, 0]
,
[1, 0, 0, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 0, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[3/4, -1/4, -1/4]
,
[-1/4, 3/4, -1/4]
,
[3/4, -1/4, -1/4]
,
[-1/4, -1/4, 3/4]
] $
x
$ [
[1, 2, 0, 1]
,
[1, 1, 0, 2]
,
[2, 1, 0, 1]
] $
[y
3, 0, y
1, y
2]
See Matrices
B =
$ [
[0, 0, 0, 1]
,
[0, 0, 1, 0]
,
[1, 0, 0, 0]
,
[0, 0, 1, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 0, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[-1/4, -1/4, 3/4]
,
[3/4, -1/4, -1/4]
,
[-1/4, 3/4, -1/4]
,
[3/4, -1/4, -1/4]
] $
x
$ [
[1, 0, 2, 1]
,
[2, 0, 1, 1]
,
[1, 0, 1, 2]
] $
7
.
Coloring, {3, 4}
R:
[2, 3, 1, 1]
B:
[4, 4, 2, 3]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `1` (` 1 + τ
` )`` (` 5 - 2τ + τ 2
` )` ,
-1` (` - 5 - τ - 3τ 2 + τ 3
` )` ,
1` (` 5 - τ + 3τ 2 + τ 3
` )` ,
-1` (` - 1 + τ
` )`` (` 5 + 2τ + τ 2
` )``]`
For τ=1/2, [51, 49, 43, 25]
. FixedPtCheck, [51, 49, 43, 25]
det(A + τ Δ) =
0
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 3 |
3 vs 3 |
3 vs 3 |
3 vs 3 |
3 vs 3 |
See Matrix for A+τΔ
bi =
$ [
[0, 1/4, 0, 3/4]
,
[0, 0, 1/4, 3/4]
,
[1/4, 3/4, 0, 0]
,
[1/4, 0, 3/4, 0]
] $
x
$ [
[19/100, 27/100, 27/100, -9/100]
,
[27/100, 91/100, -9/100, 3/100]
,
[27/100, -9/100, 91/100, 3/100]
,
[-9/100, 3/100, 3/100, 99/100]
] $
=
$ [
[5/4, -3/5, -2/5]
,
[5/4, 1/5, -6/5]
,
[-3/4, -1, 2]
,
[-3/4, 7/5, -2/5]
] $
x
$ [
[1/2, 1, 1, 3/2]
,
[5/8, 7/8, 11/8, 9/8]
,
[5/8, 19/16, 17/16, 9/8]
] $
Check x AllOnes:
[1, 1, 1, 1]
Omega Rank for R :
cycles:
{{1, 2, 3}}, net cycles:
1
.
order:
3
[y
2, y
3, y
1, 0]
See Matrices
R =
$ [
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[1, 0, 0, 0]
,
[1, 0, 0, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 0]
] $
=
$ [
[-1/4, 3/4, -1/4]
,
[-1/4, -1/4, 3/4]
,
[3/4, -1/4, -1/4]
,
[3/4, -1/4, -1/4]
] $
x
$ [
[2, 1, 1, 0]
,
[1, 2, 1, 0]
,
[1, 1, 2, 0]
] $
[0, y
1, y
2, y
3]
See Matrices
B =
$ [
[0, 0, 0, 1]
,
[0, 0, 0, 1]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
] $
x
$ [
[0, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[3/4, -1/4, -1/4]
,
[3/4, -1/4, -1/4]
,
[-1/4, -1/4, 3/4]
,
[-1/4, 3/4, -1/4]
] $
x
$ [
[0, 1, 1, 2]
,
[0, 1, 2, 1]
,
[0, 2, 1, 1]
] $
8
.
Coloring, {2, 3, 4}
R:
[2, 4, 1, 1]
B:
[4, 3, 2, 3]
` See graph `
` See pair graph `
Ω for A+τΔ :
`[ `2` (` 1 + τ
` )`` (` - 5 + τ 2
` )` ,
2` (` - 5 - τ - 3τ 2 + τ 3
` )` ,
2` (` 5 + 2τ + τ 2
` )`` (` - 1 + τ
` )` ,
-2` (` 1 + τ
` )`` (` 5 - 2τ + τ 2
` )``]`
For τ=1/2, [-57, -49, -25, -51]
. FixedPtCheck, [57, 49, 25, 51]
det(A + τ Δ) =
1` (` 1 + τ
` )`` (` τ
` )`` (` - 1 + τ
` )`
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 3 |
4 vs 4 |
4 vs 4 |
3 vs 3 |
2 vs 3 |
See Matrix for A+τΔ
bi =
$ [
[0, 1/4, 0, 3/4]
,
[0, 0, 3/4, 1/4]
,
[1/4, 3/4, 0, 0]
,
[1/4, 0, 3/4, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 1, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[69/52, -6/13, 32/13, -40/13]
,
[-3/52, -28/13, -24/13, 56/13]
,
[-35/52, 20/13, 32/13, -40/13]
,
[21/52, 14/13, -40/13, 24/13]
] $
x
$ [
[1/2, 1, 3/2, 1]
,
[5/8, 5/4, 3/2, 5/8]
,
[17/32, 41/32, 45/32, 25/32]
,
[35/64, 19/16, 99/64, 23/32]
] $
Check x AllOnes:
[1, 1, 1, 1]
Omega Rank for R :
cycles:
{{1, 2, 4}}, net cycles:
1
.
order:
3
[y
1, y
2, 0, y
3]
See Matrices
R =
$ [
[0, 1, 0, 0]
,
[0, 0, 0, 1]
,
[1, 0, 0, 0]
,
[1, 0, 0, 0]
] $
x
$ [
[1, 0, 0, 0]
,
[0, 1, 0, 0]
,
[0, 0, 0, 0]
,
[0, 0, 0, 1]
] $
=
$ [
[-1/4, 3/4, -1/4]
,
[-1/4, -1/4, 3/4]
,
[3/4, -1/4, -1/4]
,
[3/4, -1/4, -1/4]
] $
x
$ [
[2, 1, 0, 1]
,
[1, 2, 0, 1]
,
[1, 1, 0, 2]
] $
See Matrix
$ [
[0, 1, 2, 1]
,
[0, 2, 2, 0]
,
[0, 2, 2, 0]
] $
[0, y1 - y2, y1, y2]
p =
s 2 - s 3
» SYNC'D
1/4
,
0.2500000000
| SUMMARY |
|---|
| Graph Type |
| CC |
|---|
| ν(A) |
|
1
|
|---|
| ν(Δ) |
|
1
|
|---|
| π |
|
[1, 1, 1, 1]
|
|---|
| Dbly Stoch |
| true |
|---|
| SANDWICH |
| Total
1
|
|---|
| No . | Coloring | Rank |
|---|
| 1 |
{}
|
2
|
|---|
| RT GROUPS |
| Total
1
|
|---|
| No . | Coloring | Rank | Solv |
|---|
| 1 |
{2, 3}
|
4
|
["group", Not Solvable]
|
|---|
| CC Colorings |
| Total
1
|
|---|
| No . | Coloring | Sandwich,Rank |
|---|
| 1 |
{}
|
true, 2
|
|---|
| Δ-RANK'D | SC'D !RK'D |
τ-RANK'D | R/B RANK'D | NOT SYNC'D |
Total Runs | 2n-1 |
|---|
| 6 |
0 |
6 , 6 |
5 , 3 |
2 |
8 |
8 |