AOP

Paper-1
TOC-1

Degeneracy

GTODE

Table-1: Uniform Image-Scaling Of a=Q1 Under f=4S3012, a^offf=Q_F

No

A

B

A^{¾ f}

B^{¾ f}

A a B

Aa^offfB

 (Aa B) ^{¾ f}

A^{¾ f} a^offf B^{¾ f}

1

0

0

2

2

0

0

2

2

2

0

1

2

1

0

1

2

2

3

0

2

2

0

0

2

2

2

4

0

3

2

3

0

0

2

2

5

1

0

1

2

0

1

2

2

6

1

1

1

1

1

1

1

1

7

1

2

1

0

1

2

1

1

8

1

3

1

3

1

1

1

1

9

2

0

0

2

0

2

2

2

10

2

1

0

1

1

2

1

1

11

2

2

0

0

2

2

0

0

12

2

3

0

3

2

2

0

0

13

3

0

3

2

0

0

2

2

14

3

1

3

1

1

1

1

1

15

3

2

3

0

2

2

0

0

16

3

3

3

3

3

3

3

3

Table-2: Degenerate Equations Of  Example-2.5.3

#

“offf”

Uniform Degeneracy Of 
A a₁ (B a_OC)=(A a₁B) a_O(A a₁C)
Under offf Operator

#

“offf”

Uniform Degeneracy Of 
A a₁ (B a_OC)=(A a₁B) a_O(A a₁C)
Under offf Operator

1

4S0123

A a_O (B a₁C)=(A a_O B) a₁(A a_OC)

13

4S2013

A a_L (B a_DC)=(A a_LB) a_D(A a_LC)

2

4S0132

A a_I  (B a₂ C)=(A a_I B) a₂  (A a_IC)

14

4S2031

A a_B (B a_EC)=(A a_BB) a_E(A a_B C)

3

4S0213

A a_M (B a₃ C)=(A a_MB) a₃(A a_MC)

15

4S2103

A a_J (B a_FC)=(A a_JB) a_F(A a_J C)

4

4S0231

A a_C (B a₄C)=(A aC_B) a₄(A a_CC)

16

4S2130

A a₅ (B a_GC)=(A a₅B) a_G(A a₅ C)

5

4S0312

A a_G (B a₅ C)=(A a_G B) a₅(A a_GC)

17

4S2301

A a₈ (B a_HC)=(A a₈B) a_H(A a₈ C)

6

4S0321

A a_A (B a₆ C)=(A a_A B) a₆(A a_A C)

18

4S2310

A a₂ (B a_IC)=(A a₂B) a_I(A a₂ C)

7

4S1023

A a_N (B a₇ C)=(A a_N B) a₇(A a_N C)

19

4S3012

A a_F (B a_JC)=(A a_FB) a_J(A a_F C)

8

4S1032

A a_H (B a₈ C)=(A a_H B) a₈(A a_H C)

20

4S3021

A a₉ (B a_KC)=(A a₉B) a_K(A a₉ C)

9

4S1203

A a_K (B a₉ C)=(A a_K B) a₉(A a_K C)

21

4S3102

A a_D (B a_LC)=(A a_DB) a_L(A a_D C)

10

4S1230

A a₆ (B a_A C)=(A a₆ B) a_A(A a₆ C)

22

4S3120

A a₃ (B a_MC)=(A a₃B) a_M(A a₃ C)

11

4S1302

A a_E (B a_B C)=(A a_E B) a_B(A a_E C)

23

4S3201

A a₇ (B a_NC)=(A a₇B) a_N(A a₇ C)

12

4S1320

A a₄ (B a_C C)=(A a₄ B) a_C(A a₄ C)

24

4S3210

A a₁ (B a_OC)=(A a₁B) a_O(A a₁ C)

Table-3: Transferring The Function Table Of Example-3.4.2 Into Orthogonal Theorem-I&II

Function Table

Term

Vectors Table

Representations Table

#

u

v

f(u,v)

Type

Vector

Components

Terms Of

Terms Of

m

X_m2

X_m1

F(X_m)

ü´þ

X_m

X_m2

X_m1

Orthogonal-I

Orthogonal-II

1

0

0

0

ü

X₁=00

X₁₂=0

X₁₁=0

( 0au^{¾  D012} av^{¾  D012}) a*

(u^{¾  D010} av^{¾  D010})a*

2

0

1

1

´

X₂=01

X₂₂=0

X₂₁=1

trivial

trivial

3

0

2

2

þ

X₃=02

X₃₂=0

X₃₁=2

( u^{¾  D012} av^{¾  D212})a*

(u^{¾  D012} av^{¾  D212})a*

4

1

0

1

´

X₄=10

X₄₂=1

X₄₁=0

trivial

trivial

5

1

1

1

´

X₅=11

X₅₂=1

X₅₁=1

trivial

trivial

6

1

2

1

´

X₆=12

X₆₂=1

X₆₁=2

trivial

trivial

7

2

0

2

þ

X₇=20

X₇₂=2

X₇₁=0

( u^{¾  D212} av^{¾  D012})a*

( u^{¾  D212} av ^{¾  D012})a*

8

2

1

1

´

X₈=21

X₈₂=2

X₈₁=1

trivial

trivial

9

2

2

2

þ

X₉=22

X₉₂=2

X₉₁=2

( u^{¾  D212} av^{¾  D212})

(u^{¾  D212} av ^{¾  D212})

Orthogonal-I Rep: f(u,v)= (0au^{¾  D012} av^{¾  D012})a*(u^{¾  D012} av^{¾  D212})a*( u^{¾  D212} av^{¾  D012})a*(u^{¾  D212} av^{¾  D212})

´Trivial term, ü none-trivial term, þ  NMRV-term,  NMRV=2, MRV=1,  a=T₃ , STAS=(T₃,T₅)

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