SolvRun

                         [2, 3, 2, 3], [4, 4, 1, 1]
                                 

 


 

$\tilde\pi$ = [1, 1, 1, 1]
$\delta$ = [2, 2, 2, 2]

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}]

STRATIFIED CYCLE COVERS

1

0

v[2] v[3] + v[1] v[4]

v[2] v[4] v[3] + v[1] v[4] v[3] + v[1] v[2] v[4] + v[1] v[2] v[3]

2 v[1] v[2] v[4] v[3]

============================================================================

1, [1, 1, 1, 1]

"Coloring", {}

"R", [2, 3, 2, 3]
"B", [4, 4, 1, 1]

NOT SYNC'D

Kernel has RANK, 2,

N= $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 1 \\ \frac{1}{2} & 0 & 1 & \frac{1}{2} \\ \frac{1}{2} & 1 & 0 & \frac{1}{2} \\ 1 & \frac{1}{2} & \frac{1}{2} & 0 \end{pmatrix} $

"R CYCLES", 1 + v[2] v[3]
"B CYCLES", 1 + v[1] v[4]

"Char Poly V" $ \frac{-1}{5} $ ( $ - 5 + t ^ { 2 } $ )

CHECKS

"OM-R" $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"Kernel Check OMEGA-R", true $ \begin{pmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{pmatrix} $

"RANKING B from RC" 1, "vs", 2

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix} $ 1, "vs", 2

"CHECKING OMEGA-B", true

"Kernel Check OMEGA-B", true $ \begin{pmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{pmatrix} $

"RANKING R from BC" 1, "vs", 2

"RANKING R-NAT from BC" 1, "vs", 2

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \end{pmatrix} $ 1, "vs", 2

NOT SYNC'D

"RANK of R is ", 2
"R ranking is ", 1, "vs", 2
$\text{R}^{\natural}$ = [3, 2, 3, 2]
"RBAR ranking", 1, "vs", 2

"RANK of B is ", 2
"B ranking is ", 1, "vs", 2
$\text{B}^{\natural}$ = [1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2

$\text{R}^{\natural}\text{B}^{\natural}$ = [4, 1, 4, 1]
$\text{B}^{\natural}\text{R}^{\natural}$ = [3, 3, 2, 2]

"RNATBNAT ranking", 1, "vs", 2

"BNATRNAT ranking", 1, "vs", 2

"Centralizer " [4, 3, 2, 1] [1, 2, 3, 4]

"Char Poly Commutator" $ - 2 x ^ { 2 } + x ^ { 4 } $

"Min Poly Commutator" $ - 2 x ^ { 2 } + x ^ { 4 } $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , 1 + v[1] v[4]
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , 1 + v[2] v[3]

$\text{R}^{\natural}\text{B}$ = [1, 4, 1, 4], (v[1] + 1) (v[4] + 1)
$\text{B}^{\natural}\text{R}$ = [2, 2, 3, 3], (v[2] + 1) (v[3] + 1)

"RNATB NAT RANK", 2
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 2, "LOCAL TRACE", 0
"IDEMSOLVRANK", 2, "LOCAL TRACE", 0

"IDEMSOLVER", [1, 4, 1, 4]
"IDEMSOLVER", [2, 2, 3, 3]

"IDEMSOLVABLE?", false

"ABELIAN? " false

"CMM = " $ \begin{pmatrix} 0 & 0 & -1 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 0 \end{pmatrix} $

"ISIDEM?", false

RBSOLVRANK, 2, with index , 2

NOT SOLVABLE

"COLORING IS CC"

============================================================================

2, [1, -1, 1, 1]

"Coloring", {2}

"R", [2, 4, 2, 3]
"B", [4, 3, 1, 1]

SYNC'D

"R CYCLES", 1 + v[2] v[4] v[3]
"B CYCLES", 1 + v[1] v[4]

"Char Poly V" $ \frac{-1}{10} $ ( $ - 10 - 3 t + t ^ { 3 } $ )

CHECKS

"OM-R" $ \begin{pmatrix} 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} \frac{2}{3} & 0 & \frac{1}{3} & 0 \\ \frac{1}{3} & 0 & 0 & \frac{2}{3} \\ \frac{2}{3} & 0 & 0 & \frac{1}{3} \end{pmatrix} $ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{pmatrix} $ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
$\text{R}^{\natural}$ = [3, 2, 3, 4]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
$\text{B}^{\natural}$ = [1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2

$\text{R}^{\natural}\text{B}^{\natural}$ = [4, 1, 4, 4]
$\text{B}^{\natural}\text{R}^{\natural}$ = [3, 3, 4, 4]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" $ x ^ { 4 } $

"Min Poly Commutator" $ x ^ { 3 } $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , v[4] + 1
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , v[4] + 1

$\text{R}^{\natural}\text{B}$ = [1, 3, 1, 1], v[1] + 1
$\text{B}^{\natural}\text{R}$ = [2, 2, 3, 3], (v[2] + 1) (v[3] + 1)

"RNATB NAT RANK", 1
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [4, 4, 4, 4]
"IDEMSOLVER", [4, 4, 4, 4]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $ \begin{pmatrix} 0 & 0 & -1 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} $

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

3, [1, 1, -1, 1]

"Coloring", {3}

"R", [2, 3, 1, 3]
"B", [4, 4, 2, 1]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[3]
"B CYCLES", 1 + v[1] v[4]

"Char Poly V" $ \frac{-1}{10} $ ( $ - 10 - 3 t + t ^ { 3 } $ )

CHECKS

"OM-R" $ \begin{pmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} 0 & \frac{1}{3} & 0 & \frac{2}{3} \\ \frac{2}{3} & 0 & 0 & \frac{1}{3} \\ \frac{1}{3} & 0 & 0 & \frac{2}{3} \end{pmatrix} $ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 \end{pmatrix} $ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
$\text{R}^{\natural}$ = [1, 2, 3, 2]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
$\text{B}^{\natural}$ = [1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2

$\text{R}^{\natural}\text{B}^{\natural}$ = [1, 1, 4, 1]
$\text{B}^{\natural}\text{R}^{\natural}$ = [1, 1, 2, 2]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" $ x ^ { 4 } $

"Min Poly Commutator" $ x ^ { 3 } $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , v[1] + 1
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , v[1] + 1

$\text{R}^{\natural}\text{B}$ = [4, 4, 2, 4], v[4] + 1
$\text{B}^{\natural}\text{R}$ = [2, 2, 3, 3], (v[2] + 1) (v[3] + 1)

"RNATB NAT RANK", 1
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [1, 1, 1, 1]
"IDEMSOLVER", [1, 1, 1, 1]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 0 \end{pmatrix} $

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

4, [1, 1, 1, -1]

"Coloring", {4}

"R", [2, 3, 2, 1]
"B", [4, 4, 1, 3]

SYNC'D

"R CYCLES", 1 + v[2] v[3]
"B CYCLES", 1 + v[1] v[4] v[3]

"Char Poly V" $ \frac{1}{10} $ ( $ 10 - 3 t + t ^ { 3 } $ )

CHECKS

"OM-R" $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 3

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 \end{pmatrix} $ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 2

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} \frac{1}{3} & \frac{2}{3} & 0 & 0 \\ 0 & \frac{1}{3} & \frac{2}{3} & 0 \\ 0 & \frac{2}{3} & \frac{1}{3} & 0 \end{pmatrix} $ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 2, "vs", 3
$\text{R}^{\natural}$ = [3, 2, 3, 2]
"RBAR ranking", 1, "vs", 2

"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
$\text{B}^{\natural}$ = [1, 1, 3, 4]
"BBAR ranking", 3, "vs", 3

$\text{R}^{\natural}\text{B}^{\natural}$ = [3, 1, 3, 1]
$\text{B}^{\natural}\text{R}^{\natural}$ = [3, 3, 3, 2]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" $ x ^ { 4 } $

"Min Poly Commutator" $ x ^ { 3 } $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , v[3] + 1
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , v[3] + 1

$\text{R}^{\natural}\text{B}$ = [1, 4, 1, 4], (v[1] + 1) (v[4] + 1)
$\text{B}^{\natural}\text{R}$ = [2, 2, 2, 1], v[2] + 1

"RNATB NAT RANK", 2
"BNATR NAT RANK", 1

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [3, 3, 3, 3]
"IDEMSOLVER", [3, 3, 3, 3]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \end{pmatrix} $

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

5, [1, -1, -1, 1]

"Coloring", {2, 3}

"R", [2, 4, 1, 3]
"B", [4, 3, 2, 1]

NOT SYNC'D

Kernel has RANK, 4,

N= $ \begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{pmatrix} $

"R CYCLES", 1 + v[1] v[2] v[4] v[3]
"B CYCLES", (1 + v[2] v[3]) (1 + v[1] v[4])

"Char Poly V" $ \frac{1}{5} $ ( $ 5 - 2 t + t ^ { 2 } $ ) ( $ 1 + t $ )

"CYCLE CHECK B"

CHECKS

"OM-R" $ \begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"Kernel Check OMEGA-R", true $ \begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix} $

"RANKING B from RC" 1, "vs", 4

"RANKING B-NAT from RC" 1, "vs", 4

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix} $ 1, "vs", 4

"CHECKING OMEGA-B", true

"Kernel Check OMEGA-B", false $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix} $

"RANKING R from BC" 2, "vs", 4

"RANKING R-NAT from BC" 2, "vs", 4

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix} $ 2, "vs", 4

NOT SYNC'D

"RANK of R is ", 4
"R ranking is ", 1, "vs", 4
$\text{R}^{\natural}$ = [1, 2, 3, 4]
"RBAR ranking", 1, "vs", 4

"RANK of B is ", 4
"B ranking is ", 1, "vs", 4
$\text{B}^{\natural}$ = [1, 2, 3, 4]
"BBAR ranking", 1, "vs", 4

$\text{R}^{\natural}\text{B}^{\natural}$ = [1, 2, 3, 4]
$\text{B}^{\natural}\text{R}^{\natural}$ = [1, 2, 3, 4]

"RNATBNAT ranking", 1, "vs", 4

"BNATRNAT ranking", 1, "vs", 4

"Centralizer " [1, 2, 3, 4] [4, 3, 2, 1] [3, 1, 4, 2] [2, 4, 1, 3]

"Char Poly Commutator" $ x ^ { 4 } $

"Min Poly Commutator" $ x $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , (v[1] + 1) (v[2] + 1) (v[3] + 1) (v[4] + 1)
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , (v[1] + 1) (v[2] + 1) (v[3] + 1) (v[4] + 1)

$\text{R}^{\natural}\text{B}$ = [4, 3, 2, 1], (1 + v[2] v[3]) (1 + v[1] v[4])
$\text{B}^{\natural}\text{R}$ = [2, 4, 1, 3], 1 + v[1] v[2] v[4] v[3]

"RNATB NAT RANK", 4
"BNATR NAT RANK", 4

"IDEMSOLVRANK", 4, "LOCAL TRACE", 4
"IDEMSOLVRANK", 4, "LOCAL TRACE", 4

"IDEMSOLVER", [1, 2, 3, 4]
"IDEMSOLVER", [1, 2, 3, 4]

"IDEMSOLVABLE?", true

"ABELIAN? " true

"CMM = " $ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} $

"ISIDEM?", true

RBSOLVRANK, 4, with index , 4

NOT SOLVABLE

============================================================================

6, [1, -1, 1, -1]

"Coloring", {2, 4}

"R", [2, 4, 2, 1]
"B", [4, 3, 1, 3]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[4]
"B CYCLES", 1 + v[1] v[4] v[3]

"Char Poly V" $ \frac{1}{5} $ ( $ 5 + t ^ { 2 } $ )

CHECKS

"OM-R" $ \begin{pmatrix} \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 3

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} 0 & 0 & \frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & 0 & \frac{1}{3} & 0 \\ \frac{1}{3} & 0 & 0 & \frac{2}{3} \end{pmatrix} $ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} \frac{1}{3} & \frac{2}{3} & 0 & 0 \\ 0 & \frac{1}{3} & 0 & \frac{2}{3} \\ \frac{2}{3} & 0 & 0 & \frac{1}{3} \end{pmatrix} $ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
$\text{R}^{\natural}$ = [1, 2, 1, 4]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
$\text{B}^{\natural}$ = [1, 4, 3, 4]
"BBAR ranking", 3, "vs", 3

$\text{R}^{\natural}\text{B}^{\natural}$ = [1, 4, 1, 4]
$\text{B}^{\natural}\text{R}^{\natural}$ = [1, 4, 1, 4]

"RNATBNAT ranking", 1, "vs", 2

"BNATRNAT ranking", 1, "vs", 2

"Centralizer "

"Char Poly Commutator" $ x ^ { 4 } $

"Min Poly Commutator" $ x $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , (v[1] + 1) (v[4] + 1)
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , (v[1] + 1) (v[4] + 1)

$\text{R}^{\natural}\text{B}$ = [4, 3, 4, 3], 1 + v[4] v[3]
$\text{B}^{\natural}\text{R}$ = [2, 1, 2, 1], 1 + v[1] v[2]

"RNATB NAT RANK", 2
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2

"IDEMSOLVER", [1, 4, 1, 4]
"IDEMSOLVER", [1, 4, 1, 4]

"IDEMSOLVABLE?", true

"ABELIAN? " true

"CMM = " $ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} $

"ISIDEM?", true

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 2, with index , 1

NOT SOLVABLE

============================================================================

7, [1, 1, -1, -1]

"Coloring", {3, 4}

"R", [2, 3, 1, 1]
"B", [4, 4, 2, 3]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[3]
"B CYCLES", 1 + v[2] v[4] v[3]

"Char Poly V" $ \frac{1}{5} $ ( $ 5 + t ^ { 2 } $ )

CHECKS

"OM-R" $ \begin{pmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 3

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} 0 & \frac{1}{3} & 0 & \frac{2}{3} \\ 0 & 0 & \frac{2}{3} & \frac{1}{3} \\ 0 & \frac{2}{3} & \frac{1}{3} & 0 \end{pmatrix} $ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} \frac{2}{3} & 0 & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{2}{3} & 0 & 0 \\ 0 & \frac{1}{3} & \frac{2}{3} & 0 \end{pmatrix} $ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
$\text{R}^{\natural}$ = [1, 2, 3, 3]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
$\text{B}^{\natural}$ = [2, 2, 3, 4]
"BBAR ranking", 3, "vs", 3

$\text{R}^{\natural}\text{B}^{\natural}$ = [2, 2, 3, 3]
$\text{B}^{\natural}\text{R}^{\natural}$ = [2, 2, 3, 3]

"RNATBNAT ranking", 1, "vs", 2

"BNATRNAT ranking", 1, "vs", 2

"Centralizer "

"Char Poly Commutator" $ x ^ { 4 } $

"Min Poly Commutator" $ x $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , (v[2] + 1) (v[3] + 1)
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , (v[2] + 1) (v[3] + 1)

$\text{R}^{\natural}\text{B}$ = [4, 4, 2, 2], 1 + v[2] v[4]
$\text{B}^{\natural}\text{R}$ = [3, 3, 1, 1], 1 + v[1] v[3]

"RNATB NAT RANK", 2
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2

"IDEMSOLVER", [2, 2, 3, 3]
"IDEMSOLVER", [2, 2, 3, 3]

"IDEMSOLVABLE?", true

"ABELIAN? " true

"CMM = " $ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} $

"ISIDEM?", true

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 2, with index , 1

NOT SOLVABLE

============================================================================

8, [1, -1, -1, -1]

"Coloring", {2, 3, 4}

"R", [2, 4, 1, 1]
"B", [4, 3, 2, 3]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[4]
"B CYCLES", 1 + v[2] v[3]

"Char Poly V" $ \frac{-1}{10} $ ( $ - 10 - 3 t + t ^ { 3 } $ )

CHECKS

"OM-R" $ \begin{pmatrix} \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix} $ "CESARO CHECK", true

"OM-B" $ \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \end{pmatrix} $ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $ \begin{pmatrix} 0 & 0 & \frac{2}{3} & \frac{1}{3} \\ 0 & \frac{2}{3} & \frac{1}{3} & 0 \\ 0 & \frac{1}{3} & \frac{2}{3} & 0 \end{pmatrix} $ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $ \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \end{pmatrix} $ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
$\text{R}^{\natural}$ = [1, 2, 4, 4]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
$\text{B}^{\natural}$ = [3, 2, 3, 2]
"BBAR ranking", 1, "vs", 2

$\text{R}^{\natural}\text{B}^{\natural}$ = [3, 2, 2, 2]
$\text{B}^{\natural}\text{R}^{\natural}$ = [4, 2, 4, 2]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" $ x ^ { 4 } $

"Min Poly Commutator" $ x ^ { 3 } $

$\text{R}^{\natural}\text{B}^{\natural}$ CYCLES , v[2] + 1
$\text{B}^{\natural}\text{R}^{\natural}$ CYCLES , v[2] + 1

$\text{R}^{\natural}\text{B}$ = [4, 3, 3, 3], v[3] + 1
$\text{B}^{\natural}\text{R}$ = [1, 4, 1, 4], (v[1] + 1) (v[4] + 1)

"RNATB NAT RANK", 1
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [2, 2, 2, 2]
"IDEMSOLVER", [2, 2, 2, 2]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $ \begin{pmatrix} 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix} $

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

"SANDWICH SUMMARY", 0

"RG SUMMARY", 0

"SOLVSUMMARY", 4
2 . {2}, rank: 1/1
3 . {3}, rank: 1/1
4 . {4}, rank: 1/1
8 . {2, 3, 4}, rank: 1/1

"IDEMSOLVSUMMARY", 7
2 . {2}, rank: 1/1
3 . {3}, rank: 1/1
4 . {4}, rank: 1/1
5 . {2, 3}, rank: 4/1
6 . {2, 4}, rank: 2/1
7 . {3, 4}, rank: 2/1
8 . {2, 3, 4}, rank: 1/1

"SUMMARY: NON-SYNCED CC", 1
{[1, {}, sw, 2]}