Three Dimensional Patterns Generation Problems

en-tropy such as in3.4.2. Equation (3.4.55) implies h(Ax;2×2×2) > 0, if a diagonal periodic cycle applied, with a maximum eigenvalue in (3.4.55) larger than 1.

This method powerfully yields the positivity of spatial entropy, which is hard in examining the the complexity of patterns generation problems.

The rest of this paper is organized as follows. Section 3.2, we derive a recursive formula to obtain the ordering matrix Ax;2×m2×2 for Σ_2×m2×2 from A_x;2×2×2. Convert the ordering [x] into [ˆx]. Then, construct the similar recur-sive formula for ordering matrix Axˆ;2×m2×m3 from Axˆ;2×m2×2. Section3.3 we derives the recursive formula for the associated higher order transition ma-trices Txˆ;2×m2×m3 from Tx;2×2×2. Section 3.4 derives the connecting operator C_m which can recursively reduce higher elementary patterns to patterns of lower order. Then, the lower-bound of spatial entropy can be found by com-puting the maximum eigenvalues of the diagonal periodic cycles of sequence Sˆx;m3;m1m.

§ 3.2 Three Dimensional Patterns Generation Problems

This section describes three dimensional patterns generation. Let S be a set of p colors, Zm₁×m2×m3 be a fixed finite rectangular sublattice of Z³, where Z³ denotes the integer lattice on R³ and (m₁, m₂, m₃) a 3-tuple of positive integers. Functions U : Z³ → S and U^m1×m2×m3 : Zm1×m2×m3 → S are called global patterns and local patterns on Zm₁×m2×m3 respectively. The set of all patterns U is denoted by ΣP ≡ S^Z3, i.e., Σp is the set of all patterns with p different colors in 3-dimensional lattice. For clarity, we begin by studying two symbols, i.e., S = {0, 1}. There are three coordinates, let x-, y- and z-coordinate represent the 1st-, 2ed- and 3rd-coordinate respectively. There are six orderings [O] ordering could be represented as follows:

[x] : [1] ≻ [2] ≻ [3], [y] : [2] ≻ [1] ≻ [3], [z] : [3] ≻ [1] ≻ [2], [ˆx] : [1] ≻ [3] ≻ [2], [ˆy] : [2] ≻ [3] ≻ [1], [ˆz] : [3] ≻ [2] ≻ [1].

(3.2.1)

On a fixed finite lattice Zm₁×m2×m3, we firstly give an ordering [O] = O^m1×m2×m3

on Zm1×m2×m3 which belongs to any one of above orderings on Zm1×m2×m3

by [O] : [i] ≻ [j] ≻ [k]

O(α¹, α2, α3) = mjmk(αi− 1) + m^k(αj − 1) + α^k. (3.2.2)

84 Pattern Generation Problems

By identifying the pictorial patterns by numbers O(U), it becomes highly ef-fective in proving theorems since computations can now be performed on O(U). For example, the orderings on Z2×2×2could be represented as follows:

[x]-ordering [ˆx]-ordering

3.2. THREE DIMENSIONAL PATTERNS GENERATION PROBLEMS85 where α1 means the α1-th layer in x-coordinate. As denoted by the 1 × m²× m₃ pattern

a_{x;1×m2×m3;iα1} =

uα₁1m3 uα₁2m3 · · · uα₁m₂m₃

... ... . .. ...

uα₁12 uα₁22 · · · uα₁m₂2

u_α111 u_α121 · · · u_α1m21 (3.2.6) .

In particular, when m₂ = 2 and m₃ = 2 as denoted by ax;1×2×2;iα1, where iα1 = 1 + 2³uα111+ 2²uα112+ 2uα121+ uα122

(3.2.7) and

ax;1×2×2;iα1 = uα112 uα122

uα111 uα121

.

A 2 × 2 × 2 pattern U = (u^α1α2α3) can now be obtained by [x]-direct sum of two 1 × 2 × 2 patterns using [x]-ordering, i.e.,

ax;2×2×2;i¹i2 = ax;1×2×2;i¹ ⊕ a^x;1×2×2;i²

= (3.2.8) ,

where iα1 as in (3.2.7) and α1 ∈ {1, 2}. Therefore, the complete set of 2⁸ patterns in Σ_2×2×2 can be listed by a 16 × 16 matrix A^x;2×2×2 = [ax;2×2×2;i1i2] as its entries in

where .

(3.2.9)

It is easy to verify that

x(ax;2×2×2;i1i2) = 2⁴(i1− 1) + i², (3.2.10)

86 Pattern Generation Problems i.e., we are counting local patterns in Σ_2×2×2 by going through each row successively in (3.2.9). Correspondingly, Ax;2×2×2 can be referred to as an ordering matrix for Σ_2×2×2. A 2×2×2 pattern can also be viewed as [x]-direct sum of two 1 × 2 × 2 patterns using [ˆx]-ordering, i.e.,

a_{xˆ;2×2×2; ˆi1}iˆ₂ = a_{x; ˆˆi1} ⊕ ax; ˆˆi₂

(3.2.11) where

iˆα1 = 1 + 2³uα111+ 2²uα121+ 2uα112+ uα122, α1 ∈ {1, 2}, (3.2.12)

such as in (3.2.5). And the ordering matrix Axˆ;2×2×2 can be represented as

where .

(3.2.13)

It could be verified that ˆ

x(a_{x; ˆˆi1}iˆ2) = 2⁴( ˆi₁− 1) + ˆi₂. (3.2.14)

Similarly, a 2 × 2 × 2 pattern can also be viewed as a [y]-direct ([ˆy]-direct) and [z]-direct ([ˆz]-direct) sum of 2 × 1 × 2 and 2 × 2 × 1 pattern, i.e.,

a_y;j1j₂ = ay;j1 ⊕ a^y;j2, a_{y; ˆˆj1}jˆ2 = a_{y; ˆˆj1} ⊕ ay; ˆˆj2, a_z;k1k₂ = az;k1 ⊕ a^z;k2, a_{ˆz; ˆk1}kˆ2 = a_{z; ˆˆk1} ⊕ az; ˆˆk2, where

jα2 = 1 + 2³u1α21+ 2²u1α22+ 2u2α21 + u2α22, α2 ∈ {1, 2}, (3.2.15)

jˆα2 = 1 + 2³u1α21+ 2²u2α21+ 2u1α22 + u2α22, α2 ∈ {1, 2}, (3.2.16)

kα3 = 1 + 2³u11α3 + 2²u12α3 + 2u21α3 + u22α3, α3 ∈ {1, 2}, (3.2.17)

kˆα3 = 1 + 2³u11α3 + 2²u21α3 + 2u12α3 + u22α3, α3 ∈ {1, 2}.

(3.2.18)

3.2. THREE DIMENSIONAL PATTERNS GENERATION PROBLEMS87 A 16 × 16 matrix A^y;2×2×2 = [ay;2×2×2;j1j₂] or Az;2×2×2 = [az;2×2×2;k1k₂] can also be obtained for Σ_2×2×2, i.e., we have Ay;2×2×2 =

where ,

(3.2.19)

or Az;2×2×2

where .

(3.2.20)

The relations between Aω;2×2×2must be explored, where ω ∈ {x, y, z, ˆx, ˆy, ˆz}.

Before explaining the relations we denote column matrix and row matrix. Let A= [aij] be a m²× m² matrix, the column matrix A^(c) of A is defined by

A^(c) = 







A^(c)₁ A^(c)₂ · · · A^(c)m²

A^(c)_m2+1 A^(c)_m2+2 · · · A^(c)2m²

... ... . .. ...

A^(c)_(m2−1)m²+1 A^(c)_(m2−1)m²+2 · · · A^(c)m⁴







, (3.2.21)

A^(c)_alpha = 





a1α a2α · · · a^m2α

a_(m²_+1)α a_(m²_+2)α · · · a^(2m2)α

... ... . .. ...

a_((m²_−1)m²_+1)α a_((m²_−1)m²_+2)α · · · a^m4α





, (3.2.22)

88 Pattern Generation Problems where1 ≤ α ≤ m⁴.

And the row matrix A^(r) of A is defined by A^(r) = 







A^(r)₁ A^(r)₂ · · · A^(r)m²

A^(r)_m2+1 A^(r)_m2+2 · · · A^(r)2m²

... ... . .. ...

A^(r)_(m2−1)m²+1 A^(r)_(m2−1)m²+2 · · · A^(r)m⁴







, (3.2.23)

A^(r)α = 





a_α1 a_α2 · · · a^αm2

a_α(m²₊₁₎ a_α(m²₊₂₎ · · · aα(2m²)

... ... . .. ...

a_α((m²_−1)m²₊₁₎ a_α((m²_−1)m²₊₂₎ · · · a^αm4





, (3.2.24)

where 1 ≤ α ≤ m⁴. Therefore, from some observations, Ax;2×2×2 can be represented by a_y;j1j2 as

A_x;2×2×2 = A^(r)_y;2×2×2. (3.2.25)

The remainder of this subsection is devoted to construct Axˆ;2×m2×m3 from A_x;2×2×2 by the following three steps, where Axˆ;2×m2×m3 represented the or-dering matrix of Σ_2×m2×m3 according to [ˆx]-ordering generated from Σ_2×2×2.

Step I : Use [x]-ordering on Z_1×m2×2 by

2m2-2 ₂

2m2-3 2m2-1

y

(3.2.26)

and introduce ordering matrix Ax;2×m2×2 for Σ_2×m2×2.

Step II : Convert [x]-ordering into [ˆx]-ordering on Z_1×m2×2 by

2 2 2

(3.2.27)

and introduce ordering matrix Axˆ;2×m2×2 for Σ_2×m2×2.

3.2. THREE DIMENSIONAL PATTERNS GENERATION PROBLEMS89 Step III : Define [ˆx]-ordering on Z_1×m2×m3 by

(m3-1)m2+1 (m3-1)m2+2 3 2

(3.2.28) z

and introduce ordering matrix Axˆ;2×m2×m3 for Σ_2×m2×m3. To introduce Ax;2×m²×2, define

ay;2×m2×2;j1j₂...j_2m2 = ay;2×2×2;j1j₂⊕aˆ ^y;2×2×2;j2j₃⊕ · · · ˆˆ ⊕a^y;2×2×2;jm2−1j_m2

= ay;j1 ⊕ a^y;j2⊕ · · · ⊕ a^y;jm2, (3.2.29)

where 1 ≤ j^k ≤ 2⁴ and 1 ≤ k ≤ m2. Herein, a wedge direct sum ˆ⊕ is used for 2 × 2 × 2 patterns whenever they can attached together.

Now, Ax;2×m2×2 can be obtained as follows.

Theorem 3.1. For any m2 ≥ 2, Σ2×m2×2 = {a^y;j1j2...j_m2}, where a^y;j1j2...j_2m2 is given in (3.2.29). Furthermore, the ordering matrix Ax;2×m2×2 = [ay;j1j₂...j_m2] which is a 2^2m2 × 2^2m2 matrix can be decomposed into following matrices

A_x;2×m2×2 = [Ax;2×m2×2;j1]2^m2×2^m2,

where 1 ≤ j1 ≤ 2^2m2. For fixed j₁, j₂, . . . , jk∈ {1, 2, . . . , 2^2m2}, Ax;2×m²×2;j¹j2...jk = [Ax;2×m²×2;j¹j2...jkj_k+1]m2×m²,

where 1 ≤ jk+1 ≤ 2^2m2 and k ∈ {1, 2, · · · , m2−2}. For fixed j1, j₂,· · · , j^m2−1, Ax;2×m²×2;j¹j2...j_m2−1 = [ay;2×m²×2;j¹j2...j_m2−1j_m2]₂m2×2^m2,

where ay;2×m2×2;j1j2...j_m2 is defined in (3.2.29).

Proof. From (3.2.15), uα₁α₂α₃ can be solved in terms of jα₂, i.e., we have u_1α21 = [jα₂ − 1

2³ ], (3.2.30)

u_1α22 = [jα2 − 1 − 2³u1α21

2² ],

(3.2.31)

u2α21 = [jα2 − 1 − 2³u1α21 − 2²u1α22

2 ],

(3.2.32)

u2α22 = jα2 − 1 − 2³u1α21− 2²u1α22 − 2u^2α21, (3.2.33)

90 Pattern Generation Problems where [ ] is the Gauss symbol. From (3.2.30) to (3.2.33), we have the following table.

jα2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

u1α21 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

u_1α22 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

u_2α21 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

u2α22 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

For any m2 ≥ 2, we have

im2;1 = 1 +

m₂

X

α2=1

(2^{2(m2−α2)+1}u1α21+ 2^2(m2−α2)u1α22), (3.2.34)

im₂;2 = 1 +

m2

X

α2=1

(2^{2(m2−α2)+1}u_2α21+ 2^2(m2−α2)u_2α22).

(3.2.35)

From above formulae, we have

im2+1;1 = 2²(im2;1− 1) + 2u^1(m2+1)1+ u1(m2+1)2+ 1, im2+1;2 = 2²(im2;2− 1) + 2u^2(m2+1)1 + u2(m2+1)2+ 1.

Now, by induction on m2 the theorem follows from last two formulae and the above table. The proof is complete.

Remark 3.2. By the similar method, the following relations ca be derived but the detailed proof is omitted here for brevity.

A_ˆx;2×2×m³ = [az;2×2×m³;k1k2...k_m3−1k_m3]₂m3×2^m3

(3.2.36)

A_y;m1×2×2 = [ax;m1×2×2;i1i₂...i_m1−1i_m1]2^m1×2^m1

(3.2.37)

A_yˆ;2×2×m3 = [aˆz;2×2×m3;k1k2...k_m3−1k_m3]2^m3×2^m3

(3.2.38)

Az;m1×2×2 = [aˆx;m1×2×2;i1i2...i_m1−1i_m1]2^m1×2^m1

(3.2.39)

Azˆ;2×m2×2 = [ayˆ;2×m2×2;j1j2...j_m2−1j_m2]2^m2×2^m2

(3.2.40)

Next, [x]-ordering is converted into [ˆx]-ordering for Z_1×m2×2. Since Z_1×m2×2 = {(1, α², α3) : 1 ≤ α² ≤ m²,1 ≤ α³ ≤ 2}, the position (α², α3) is the α-th in (3.2.26), where

α= 2(α2− 1) + α³. (3.2.41)

In (3.2.27), the position of (1, α2, α₃) is the ˆα-th, where ˆ

α= m2(α3− 1) + α². (3.2.42)

3.2. THREE DIMENSIONAL PATTERNS GENERATION PROBLEMS91 It is easy to verify

ˆ

α= m2α+ (1 − 2m²)[α− 1

2 ] + (1 − m²), (3.2.43)

or

ˆ

α= k if α= 2k − 1, and

ˆ

α= m2 + k if α = 2k, 1 ≤ k ≤ m².

Now, the ordering [ˆx] in (3.2.27) on Z_1×m2×2 can be extended to Z_1×m2×m3 by (3.2.28). For a fixed m2, [ˆx]-ordering on Z_1×m2×m3 is clearly one di-mensional; it grows in z-direction. With ordering (3.2.28) on Z_1×m2×m3, for U = (uα1α2α3) ∈ Σ2×m2×m3, denoted by

ˆi_α1 = 1 +

m2

X

α2=1 m3

X

α3=1

uα1α2α32^{m2(m3−α3)+(m2−α2)}, (3.2.44)

where α1 = 1, 2. Then, we obtain ˆ

x(U) = 2^m2m3( ˆi₁− 1) + ˆi₂. (3.2.45)

Now, let a_{x; ˆˆi1}iˆ2 = U = (uα1α2α3), then we have new ordering matrix Axˆ;2×m2×2 = [a_{xˆ;2×m2×2; ˆi1}iˆ2] for Σ_2×m2×2. The relationship between Ax;2×m2×2and Axˆ;2×m2×2

is established before constructing A_ˆx;2×m²×m³ from A_ˆx;2×m²×2 for m₃ ≥ 3.

We firstly established a conversion sequence of orderings from (3.2.26) to (3.2.27). Where Pk denotes the permutation of N2m = {1, 2, · · · , 2m²} such that Pk(k + 1) = k,Pk(k) = k + 1 and the other numbers are fixed. We also denote Pk the permutation on Z_1×m2×2 such that it exchanges k and k+1 and maintains the other positions fixed, i.e,

· k+ 1 · ·

· · k ·

Pk

−→ · k · ·

· · k+ 1 ·

(3.2.46)

Obviously (3.2.26) can be converted into (3.2.27) in many ways by using sequence of Pk. Here, we present a systematic approach.

Lemma 3.3. For m2 ≥ 2, (3.2.26) can be converted into (3.2.27) by the following sequences of ^m2(m₂²⁻¹⁾ permutations successively

(P2P₄· · · P^2m2−2)(P3P₅· · · P^2m2−3) · · · (PkPk+2· · · P^2m2−k) · · · (P^m2−1Pm2+1)Pm2, (3.2.47)

2 ≤ k ≤ m².

92 Pattern Generation Problems Proof. When m2 = 2 and 3, verifying that (3.2.47) can convert (3.2.26) into (3.2.27) is relatively easy.

When m2 ≥ 4, and for any 2 ≤ k ≤ m², applying

(P2P4· · · P^2m2−2)(P3P5· · · P^2m2−3) · · · (P^kPk+2· · · P^2m2−k) (3.2.48)

to (3.2.26), then there are two intermediate cases:

(i) when 2 ≤ k ≤ [^m₂²], then we have

1 2 · · · k k+ 2 k+ 4 · · · k+ 2ℓ · · · · · · ^2m2− 3k + 1 · · · ^2m2− k − 22m2− k k+ 1 k+ 3 · · · 3k − 1 · · · · · · · · · 3k − 1 + 2ℓ · · · ^2m2− k − 12m2− k + 1 · · · 2m2− 1 2m2

(3.2.49)

where 0 ≤ ℓ ≤ m2− 2k.

(ii) when [^m₂²] + 1 ≤ k ≤ m²− 1, then we have

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

1 2 k− 1 k k+ 2 2m2− k

k+ 1 ^{2m2− k − 12m2− k + 12m2− k + 2} 2m2− 1 2m2

(3.2.50)

When k = m2 in (3.2.50), we have (3.2.27). We prove (3.2.49) and (3.2.50) by mathematical induction on k. When k=2, it is relatively easy to verify that (3.2.26) is converted into

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

1 2 4 2m2− 12m2− 2

3 5 2m2− 32m2− 1 2m2

by P2P4· · · P^2m2−2, i.e., (3.2.49) holds for k=2. Next, assume that (3.2.49) holds for k ≤ [^m2]. Then, by applying Pk+1P_k+2· · · P^2m2−k−1to (3.2.49), it can be verified that (3.2.49) holds for k + 1 when k + 1 ≤ [^m₂²] or becomes (3.2.50) when k + 1 ≥ [^m₂²]. When k ≥ [^m₂²] + 1, we apply Pk+1Pk+3· · · P^2m2−k−1 to (3.2.50). It can also be verified that (3.2.50) holds for k+1. Finally, we conclude that (4.27) holds for k = m2. The proof is thus complete.

By using Lemma 3.3, Ax;2×m2×2 can be converted into Ax_ˆ;2×m2×2 by the following construction. Let

P =







1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1





, (3.2.51)

3.2. THREE DIMENSIONAL PATTERNS GENERATION PROBLEMS93 and for 2 ≤ j ≤ 2m²− 2, as denoted by

P_x;2m2;j = I₂^j−1 ⊗ P ⊗ I2^2m2−j−1, (3.2.52)

where Ik is the k × k identity matrix. Furthermore, let P_x;2×m2×2 = (P2m2;2P_2m2;4· · · P^2m2;2m2−2)

· · · (P2m2;k· · · P2m2;2m2−k) · · · (P2m2;m2), (3.2.53)

2 ≤ k ≤ m². Then, we have the following theorem.

Theorem 3.4. For any m2 ≥ 2,

A_xˆ;2×m2×2 = P^t_x;2×m2×2A_x;2×m2×2P_x;2×m2×2. (3.2.54)

Proof. From (3.2.41), in Z_1×m2×2 the position (α2, α₃) is the α-th in (3.2.26), where α = 2(α2− 1) + α³. Define

ℓα = 1 + 2u1α2α3 + u2α2α3, (3.2.55)

1 ≤ ℓ^α ≤ 4 and 1 ≤ α ≤ 2m². For U = (uα1α2α3) ∈ Σ2×m2×2, from Theorem 3.1 it can be denoted by ay;2×m2×2;j1j₂...j_m2 and by (3.2.15) for fixed 1 ≤ α² ≤ m2 we have

jα2 = 1 + 2³u1α21+ 2²u1α22+ 2u2α21+ u2α22

= 2²(ℓ_2α2−1) + ℓ_2α2 + 1,

where 1 ≤ j^α2≤16. Hence the relation between ay;j_α2 and wy;ℓ_2α2−1ℓ_2α2 is







ay;1 ay;2 ay;3 ay;4

a_y;5 a_y;6 a_y;7 a_y;8 a_y;9 a_y;10 a_y;11 a_y;12 ay;13 ay;14 ay;15 ay;16





 =







w11 w12 w21 w22

w₁₃ w₁₄ w₂₃ w₂₄ w₃₁ w₃₂ w₄₁ w₄₂ w33 w34 w43 w44





.

Therefore, the pattern in ordering matrix Ax;2×m2×2 can be represented by ay;2×m2×2;j1j₂...j_m2 = ay;j1 ⊕ a^y;j2 ⊕ · · · ⊕ a^y;jm2

= wy;ℓ1ℓ₂ ⊕ w^y;ℓ3ℓ₄ ⊕ · · · ⊕ w^y;ℓ_2m2−1^ℓ_2m2

≡ w^y;ℓ1ℓ2...ℓ_2m2. It is easy to verify that for any 1 ≤ k ≤ 2m²− 1,

P_2m^t _2;kA_x;2×m2×2P2m2;k

= P_2m^t _2;k[w_y;ℓ1ℓ2...ℓkℓ_k+1...ℓ_2m2]P_2m2;k

= [w_y;ℓ1ℓ₂...ℓ_k+1ℓk...ℓ_2m2],

i.e., P2m2;k exchanges ℓk and ℓk+1 in Ax;2×m2×2. Therefore, from (3.2.53) and Lemma 3.3, (3.2.54) follows.

94 Pattern Generation Problems Now, in Theorem 3.4, as denoted by

A_xˆ;2×m2×2 = [a_{xˆ;2×m2×2; ˆi1}iˆ₂], (3.2.56)

1 ≤ ˆi¹,ˆi₂ ≤ 2m². And by Remark 3.2, Aˆx;2×m2×2 could be represented by az;2×m²×2;k¹k2, where 1 ≤ k1, k₂ ≤ 2^2m2. The [ˆx]-expression

A_xˆ;2×m2×2 = A^(r)_z;2×m2×2 (3.2.57)

for Σ_2×m2×2 enable us to construct Aˆx;2×m2×m3 for Σ_2×m2×m3. Indeed, for fixed m₂ ≥ 2 and m3 ≥ 2, let

a_{xˆ;2×m2×m3; ˆi1}iˆ2 = az;2×m2×m3;k1k2...k_m3

= az;2×m2×2;k1k₂⊕aˆ ^z;2×m2×2;k2k₃⊕ · · · ˆˆ ⊕a^z;2×m2×2;km3−1k_m3. (3.2.58)

Therefore, by a similar argument as in proving Theorem 3.1 we have the following theorem for Axˆ;2×m2×m3. The detailed proof is omitted here for brevity.

Theorem 3.5. By fixing m₂ ≥ 2 and for any m3 ≥ 2, the ordering matrix A_{ˆx;2×m2×m3} with respect to [ˆx]-ordering can be expressed as

A_{xˆ;2×m2×m3} = [Axˆ;2×m2×m3;k1]2^m2×2^m2, (3.2.59)

where 1 ≤ k¹ ≤ 2^2m2. For fixed 1 ≤ k¹, k₂,· · · , k^l ≤ 2^2m2, Ax_ˆ;2×m²×m³;k1k2···k^l = [Ax_ˆ;2×m²×m³;k1k2···k^lk_l+1]₂m2×2^m2

(3.2.60)

where 1 ≤ k^l+1 ≤ 2^2m2 and 1 ≤ l ≤ m³− 2. For fixed k¹, k2,· · · , k^m3−1, A_{ˆx;2×m2×m3;k1k2···km3−1} = [az;2×m2×m3;k1k2...k_m3],

(3.2.61)

where az;2×m2×m3;k1k₂...k_m3 is given by (3.2.58).

Remark 3.6. Similarly, the following relations can be derived but the de-tailed proof is omitted here for brevity.

A_x;2×m2×m3 = [ay;2×m2×m3;j1j2...j_m2]2^m2m3×2^m2m3

Ay;mˆ 1×2×m3 = [az;mˆ 1×2×m3;k1k2...k_m3]2^m1m3×2^m1m3

Ay;m1×2×m3 = [ax;m1×2×m3;i1i2...i_m1]2^m1m3×2^m1m3

A_{z;mˆ 1×m2×2} = [a_{y;mˆ 1×m2×2;j1}j2...j_m2]₂m1m2×2^m1m2

A_z;m1×m2×2 = [a_{x;mˆ 1×m2×2;i1}i2...i_m1]₂m1m2×2^m1m2

在文檔中 Pattern Generation Problems (頁 87-99)