大數學 维基
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(未显示同一用户的10个中间版本)
第38行: 第38行:
 
|2<1>2
 
|2<1>2
 
|\(\omega2+1\)
 
|\(\omega2+1\)
  +
|-
  +
|2<1>3
  +
|\(\omega2+2\)
 
|-
 
|-
 
|3<1>1
 
|3<1>1
第77行: 第80行:
 
|1<2>2
 
|1<2>2
 
|\(\omega^\omega+1\)
 
|\(\omega^\omega+1\)
  +
|-
  +
|1<2>3
  +
|\(\omega^\omega+2\)
 
|-
 
|-
 
|1<2>1<1>1
 
|1<2>1<1>1
第83行: 第89行:
 
|1<2>1<1>2
 
|1<2>1<1>2
 
|\(\omega^\omega+\omega+1\)
 
|\(\omega^\omega+\omega+1\)
  +
|-
  +
|1<2>1<1>3
  +
|\(\omega^\omega+\omega+2\)
 
|-
 
|-
 
|1<2>2<1>1
 
|1<2>2<1>1
 
|\(\omega^\omega+\omega2\)
 
|\(\omega^\omega+\omega2\)
  +
|-
  +
|1<2>3<1>1
  +
|\(\omega^\omega+\omega3\)
 
|-
 
|-
 
|1<2>1<1>1<1>1
 
|1<2>1<1>1<1>1
第95行: 第107行:
 
|2<2>1
 
|2<2>1
 
|\(\omega^\omega2\)
 
|\(\omega^\omega2\)
  +
|-
  +
|3<2>1
  +
|\(\omega^\omega3\)
 
|-
 
|-
 
|1<1>1<2>1
 
|1<1>1<2>1
 
|\(\omega^{\omega+1}\)
 
|\(\omega^{\omega+1}\)
  +
|-
  +
|1<1>1<2>2
  +
|\(\omega^{\omega+1}+1\)
  +
|-
  +
|1<1>1<2>3
  +
|\(\omega^{\omega+1}+2\)
  +
|-
  +
|1<1>1<2>1<1>1
  +
|\(\omega^{\omega+1}+\omega\)
  +
|-
  +
|1<1>1<2>1<1>2
  +
|\(\omega^{\omega+1}+\omega+1\)
  +
|-
  +
|1<1>1<2>1<1>3
  +
|\(\omega^{\omega+1}+\omega+2\)
  +
|-
  +
|1<1>1<2>2<1>1
  +
|\(\omega^{\omega+1}+\omega2\)
  +
|-
  +
|1<1>1<2>3<1>1
  +
|\(\omega^{\omega+1}+\omega3\)
  +
|-
  +
|1<1>1<2>1<1>1<1>1
  +
|\(\omega^{\omega+1}+\omega^2\)
  +
|-
  +
|1<1>1<2>1<1>1<1>1<1>1
  +
|\(\omega^{\omega+1}+\omega^3\)
 
|-
 
|-
 
|1<1>2<2>1
 
|1<1>2<2>1
 
|\(\omega^{\omega+1}+\omega^\omega\)
 
|\(\omega^{\omega+1}+\omega^\omega\)
  +
|-
  +
|1<1>3<2>1
  +
|\(\omega^{\omega+1}+\omega^\omega2\)
 
|-
 
|-
 
|2<1>1<2>1
 
|2<1>1<2>1
第122行: 第167行:
 
|1<1<1>1>1
 
|1<1<1>1>1
 
|\(\omega^{\omega^\omega}\)
 
|\(\omega^{\omega^\omega}\)
  +
|-
  +
|2<1<1>1>1
  +
|\(\omega^{\omega^\omega}2\)
  +
|-
  +
|1<1<1>1>1<1<1>1>1
  +
|\(\omega^{\omega^\omega2}\)
  +
|-
  +
|1<1<1>1>1<1<1>1>1<1<1>1>1
  +
|\(\omega^{\omega^\omega3}\)
 
|-
 
|-
 
|1<1<1>2>1
 
|1<1<1>2>1
第153行: 第207行:
 
|1<1/1>2
 
|1<1/1>2
 
|\(\varepsilon_0+1\)
 
|\(\varepsilon_0+1\)
  +
|-
  +
|1<1/1>3
  +
|\(\varepsilon_0+2\)
 
|-
 
|-
 
|1<1/1>1<1>1
 
|1<1/1>1<1>1
 
|\(\varepsilon_0+\omega\)
 
|\(\varepsilon_0+\omega\)
  +
|-
  +
|1<1/1>1<1>2
  +
|\(\varepsilon_0+\omega+1\)
  +
|-
  +
|1<1/1>1<1>3
  +
|\(\varepsilon_0+\omega+2\)
  +
|-
  +
|1<1/1>2<1>1
  +
|\(\varepsilon_0+\omega2\)
  +
|-
  +
|1<1/1>3<1>1
  +
|\(\varepsilon_0+\omega3\)
  +
|-
  +
|1<1/1>1<1>1<1>1
  +
|\(\varepsilon_0+\omega^2\)
  +
|-
  +
|1<1/1>1<1>1<1>1<1>1
  +
|\(\varepsilon_0+\omega^3\)
  +
|-
  +
|1<1/1>1<2>1
  +
|\(\varepsilon_0+\omega^\omega\)
 
|-
 
|-
 
|1<1/1>1<1<1/1>1>1
 
|1<1/1>1<1<1/1>1>1
第219行: 第297行:
 
|1<2/1>1
 
|1<2/1>1
 
|\(BHO\)
 
|\(BHO\)
  +
|-
  +
|1<2/1>1<1<2/1>1>1
  +
|\(\psi(\psi_1(0))2\)
  +
|-
  +
|1<2/1>1<1<2/1>2>1
  +
|\(\psi(\psi_1(0))^\omega\)
  +
|-
  +
|1<2/1>1<1<2/1>1<1<2/1>1>1>1
  +
|\(\psi(\psi_1(0))^{\psi(\psi_1(0))}\)
  +
|-
  +
|1<2/1>1<1/1>1
  +
|\(\psi(\psi_1(0)+1)\)
  +
|-
  +
|1<2/1>1<1/1>1<1/1>1
  +
|\(\psi(\psi_1(0)+\Omega)\)
  +
|-
  +
|1<2/1>1<1/2>1
  +
|\(\psi(\psi_1(0)+\Omega^\omega)\)
  +
|-
  +
|1<2/1>1<1/1<2/1>1>1
  +
|\(\psi(\psi_1(0)2)\)
  +
|-
  +
|1<2/1>1<1/1<2/1>2>1
  +
|\(\psi(\psi_1(0)^\omega)\)
  +
|-
  +
|1<2/1>1<1/1<2/1>1<1<2/1>1>1>1
  +
|\(\psi(\psi_1(0)^{\psi(\psi_1(0))})\)
  +
|-
  +
|1<2/1>1<1/1<2/1>1<1/1>1>1
  +
|\(\psi(\psi_1(0)^\Omega)\)
  +
|-
  +
|1<2/1>1<1/1<2/1>1<1/1<2/1>1>1>1
  +
|\(\psi(\psi_1(0)^{\psi_1(0)})\)
  +
|-
  +
|2<2/1>1
  +
|\(\psi(\psi_1(1))\)
  +
|-
  +
|3<2/1>1
  +
|\(\psi(\psi_1(2))\)
  +
|-
  +
|1<1<2/1>1>1<2/1>1
  +
|\(\psi(\psi_1(\psi(\psi_1(0))))\)
  +
|-
  +
|1<1/1>1<2/1>1
  +
|\(\psi(\psi_1(\Omega))\)
  +
|-
  +
|1<1/1<2/1>1>1<2/1>1
  +
|\(\psi(\psi_1(\psi_1(0)))\)
  +
|-
  +
|1<1/1<1/1<2/1>1>1<2/1>1>1<2/1>1
  +
|\(\psi(\psi_1(\psi_1(\psi_1(0))))\)
 
|-
 
|-
 
|1<2/1>1<2/1>1
 
|1<2/1>1<2/1>1
 
|\(\psi(\Omega_2)\)
 
|\(\psi(\Omega_2)\)
  +
|-
  +
|1<2/2>1
  +
|\(\psi(\Omega_2^\omega)\)
  +
|-
  +
|1<2/1<1/1>1>1
  +
|\(\psi(\Omega_2^\Omega)\)
  +
|-
  +
|1<2/1<2/1>1>1
  +
|\(\psi(\Omega_2^{\Omega_2})\)
  +
|-
  +
|1<2/1<2/1<2/1>1>1>1
  +
|<nowiki>\(\psi(\Omega_2^{\Omega_2^{\Omega_2}})\)</nowiki>
 
|-
 
|-
 
|1<3/1>1
 
|1<3/1>1
第256行: 第397行:
   
 
这里特别注意:1<1/1>2=1<1<1/1>1>2,而2<1/1>1=1<2<1/1>1>1,其中2所在的位置并不相同!
 
这里特别注意:1<1/1>2=1<1<1/1>1>2,而2<1/1>1=1<2<1/1>1>1,其中2所在的位置并不相同!
  +
  +
<br />
 
{| class="fandom-table"
 
{| class="fandom-table"
  +
|+反射模式分析
|+MOTAN增长率分析(\(\psi(\psi_I(0))\)以上)
 
  +
|1<1>1
!MOTAN
 
  +
|1
!FGH
 
 
|-
 
|-
|2<1/1/1>1
+
|1<1>1<1>1
  +
|1-1
|\(\psi(\psi_I(1))\)
 
 
|-
 
|-
|1<1<1/1/1>1/1>1<1/1/1>1
+
|1<1>1<1>1<1>1
  +
|1-1-1
|\(\psi(\psi_I(\psi_I(0)))\)
 
 
|-
 
|-
|1<1/1/1>1<1/1/1>1
+
|1<1/1>1
  +
|2
|\(\psi(I)\)
 
 
|-
 
|-
|1<1/1/2>1
+
|1<1<1>1/1>1
  +
|1-2
|\(\psi(I^\omega)\)
 
 
|-
 
|-
|1<1/2/1>1
+
|1<1/1/1>1
  +
|2 1-2
|\(\psi(\psi_{I+1}(0))\)
 
|-
 
|1<2/1/1>1
 
|\(\psi(\psi_{I_2}(0))\)
 
 
|-
 
|-
 
|1<1/1/1/1>1
 
|1<1/1/1/1>1
  +
|2 1-(2 1-2)
|\(\psi(\psi_{I(1,0)}(0))\)
 
|-
 
|1<1{2}1>1
 
|\(\psi(\psi_{I(\omega,0)}(0))\)
 
 
|-
 
|-
 
|1<1{1{1}1}1>1
 
|1<1{1{1}1}1>1
  +
|2-2
|\(\psi(\psi_{I(1,0,0)}(0))\)=\(\psi(\psi_{\chi(M)}(0))\)
 
|-
 
|2<1{1{1}1}1>1
 
|\(\psi(M)\)=\(\psi(\psi_{I(2,0,0)}(0))\)
 
|-
 
|1<1{1{1}1}1>1<1{1{1}1}1>1
 
|\(\psi(M^2)\)=\(\psi(\psi_{I(1,0,0,0)}(0))\)
 
|-
 
|1<1{1{1}1}2>1
 
|\(\psi(M^\omega)\)
 
|-
 
|1<1{1{1}1}1<1{1<1{1{1}1}1>1}1>1>1
 
|\(\psi(M^\chi(M))\)
 
|-
 
|1<1{1{1}1}1<1{1{1}1}1>1>1
 
|\(\psi(M^M)\)
 
|-
 
|1<1{1{1}1}1{1}1>1
 
|\(\psi(\psi_{\chi(\Omega_{M+1})}(0))\)
 
|-
 
|1<1{1{1}1}1{1}1{1}1>1
 
|\(\psi(\psi_{\chi(I_{M+1})}(0))\)
 
|-
 
|1<1{1{1}1}1{1}1{1}1{1}1>1
 
|\(\psi(\psi_{\chi(I(1,M+1))}(0))\)
 
|-
 
|1<1{1{1}1}1{2}1>1
 
|\(\psi(\psi_{\chi(I(\omega,M+1))}(0))\)
 
|-
 
|1<2{1{1}1}1>1
 
|\(\psi(\psi_{\chi(M_2)}(0))\)
 
|-
 
|1<1<1{1{1}1}1>1{1{1}1}1>1
 
|\(\psi(\psi_{\chi(M_M)}(0))\)
 
 
|-
 
|-
 
|1<1{1}1{1{1}1}1>1
 
|1<1{1}1{1{1}1}1>1
  +
|2 1-2-2
|\(\psi(\psi_{\chi(M[(1,0)])}(0))\)
 
 
|-
 
|-
|1<1{1}2{1{1}1}1>1
+
|1<1{1}1{1}1{1{1}1}1>1
  +
|2 1-(2 1-2-2)
|\(\psi(\psi_{\chi(M_{M[(1,0)]+1})}(0))\)
 
 
|-
 
|-
|1<2{1}1{1{1}1}1>1
+
|1<1{1{1}1}1{1{1}1}1>1
  +
|2-2 1-2-2
|\(\psi(\psi_{\chi(M[(1,1)])}(0))\)
 
 
|-
 
|-
|1<1{1}1{1}1{1{1}1}1>1
+
|1<1{2{1}1}1>1
  +
|2-2-2
|\(\psi(\psi_{\chi(M[(2,0)])}(0))\)
 
 
|-
 
|-
|1<1{2}1{1{1}1}1>1
+
|1<1{3{1}1}1>1
  +
|2-2-2-2
|\(\psi(\psi_{\chi(M[(\omega,0)])}(0))\)
 
 
|-
 
|-
|1<1{1{1}1}1{1{1}1}1>1
+
|1<1{1{1}1{1}1}1>1
  +
|3
|\(\psi(\psi_{\chi(M[1,0])}(0))\)
 
 
|-
 
|-
|1<1{1{1}1}1{1{1}1}1{1{1}1}1>1
+
|1<1{1}1{1{1}1{1}1}1>1
  +
|2 1-3
|\(\psi(\psi_{\chi(M[2,0])}(0))\)
 
 
|-
 
|-
|1<1{1{1}2}1>1
+
|1<1{1}1{1}1{1{1}1{1}1}1>1
  +
|2 1-(2 1-3)
|\(\psi(\psi_{\chi(M[\omega,0])}(0))\)
 
 
|-
 
|-
|1<1{2{1}1}1>1
+
|1<1{1{1}1}1{1{1}1{1}1}1>1
  +
|2-2 1-3
|\(\psi(\psi_{\chi(M[1,0,0])}(0))\)=\(\psi(\psi_{\chi(N)}(0))\)
 
|}
 
<br />
 
{| class="fandom-table"
 
|+MOTAN表达式对应的反射序数
 
!MOTAN
 
!反射序数
 
 
|-
 
|-
  +
|1<1{1}1{1{1}1}1{1{1}1{1}1}1>1
|1<1>1
 
  +
|2 1-(2-2 1-3)
|\(\Pi_1 反射\)
 
 
|-
 
|-
|1<1>1<1>1
+
|1<1{1{1}1}1{1{1}1}1{1{1}1{1}1}1>1
  +
|2-2 1-(2-2 1-3)
|\(\Pi_1 反射 onto \Pi_1 反射\)
 
 
|-
 
|-
|1<1/1>1
+
|1<1{2{1}1}1{1{1}1{1}1}1>1
  +
|2-2-2 1-3
|\(\Pi_2 反射\)
 
 
|-
 
|-
|1<1<1>1/1>1
+
|1<1{1{1}1{1}1}1{1{1}1{1}1}1>1
  +
|3 1-3
|\(\Pi_1 反射 onto \Pi_2 反射\)
 
 
|-
 
|-
  +
|1<1{1{1}1{1}1}1{1{1}1{1}1}1{1{1}1{1}1}1>1
|1<1/1/1>1
 
  +
|3 1-(3 1-3)
|\(\Pi_2 反射 且 \Pi_1 反射 onto \Pi_2 反射\)
 
 
|-
 
|-
|1<1<1>1/1/1>1
+
|1<1{1{1}2{1}1}1>1
  +
|2-3
|\(\Pi_1 反射 onto (\Pi_2 反射 且 \Pi_1 反射 onto \Pi_2 反射)\)
 
 
|-
 
|-
|1<1/1/1/1>1
+
|1<1{1{1}3{1}1}1>1
  +
|2-2-3
|\(\Pi_2 反射 且 \Pi_1 反射 onto (\Pi_2 反射 且 \Pi_1 反射 onto \Pi_2 反射)\)
 
 
|-
 
|-
|1<1{1{1}1}1>1
+
|1<1{2{1}1{1}1}1>1
  +
|3 2-3
|\(\Pi_2 反射 onto \Pi_2 反射\)
 
 
|-
 
|-
|1<1{2{1}1}1>1
+
|1<1{3{1}1{1}1}1>1
  +
|3 2-(3 2-3)
|\(\Pi_2 反射 onto \Pi_2 反射 onto \Pi_2 反射\)
 
 
|-
 
|-
|1<1{1{1}1{1}1}1>1
+
|1<1{1{1}1{1}1{1}1}1>1
  +
|3-3
|\(\Pi_3 反射\)
 
 
|-
 
|-
|1<1{1{1}1{1}1{1}1}1>1
+
|1<1{1{1}1{1}1{1}1{1}1}1>1
  +
|3-3-3
|\(\Pi_3 反射 onto \Pi_3 反射\)
 
 
|-
 
|-
 
|1<1{1{1{1}1}1}1>1
 
|1<1{1{1{1}1}1}1>1
  +
|4
|\(\Pi_4 反射\)
 
 
|}
 
|}
 
[[Category:博客帖子]]
 
[[Category:博客帖子]]

2021年11月5日 (五) 02:37的最新版本

当n为自然数时,MOTAN的b[n]p跟BEAF的b{n}p是完全相同的,b[n]p=b\(\uparrow^n\)p

3[1<1>2]2=3[1<1>1]3=3[3]3

3[1<1>2]3=3[1<1>1]3[1<1>1]3=3[1<1>1](3[3]3)=3[3[3]3](3[3]3)>3[3[3]3]3>3[4]3=3\(\uparrow\uparrow\uparrow\uparrow\)3=G(1)

3[1<1>2]4>3[1<1>1](G(1))>3[G(1)]3=G(2)

......

3[1<1>2]66>G(64)

MOTAN增长率分析(\(0\) ~ \(\varepsilon_0\))
MOTAN FGH
1 \(2\)
2 \(3\)
3 \(4\)
1<1>1 \(\omega\)
1<1>2 \(\omega+1\)
1<1>3 \(\omega+2\)
2<1>1 \(\omega2\)
2<1>2 \(\omega2+1\)
2<1>3 \(\omega2+2\)
3<1>1 \(\omega3\)
1<1>1<1>1 \(\omega^2\)
1<1>1<1>2 \(\omega^2+1\)
1<1>1<1>3 \(\omega^2+2\)
1<1>2<1>1 \(\omega^2+\omega\)
1<1>2<1>2 \(\omega^2+\omega+1\)
1<1>2<1>3 \(\omega^2+\omega+2\)
1<1>3<1>1 \(\omega^2+\omega2\)
2<1>1<1>1 \(\omega^22\)
3<1>1<1>1 \(\omega^23\)
1<1>1<1>1<1>1 \(\omega^3\)
1<2>1 \(\omega^\omega\)
1<2>2 \(\omega^\omega+1\)
1<2>3 \(\omega^\omega+2\)
1<2>1<1>1 \(\omega^\omega+\omega\)
1<2>1<1>2 \(\omega^\omega+\omega+1\)
1<2>1<1>3 \(\omega^\omega+\omega+2\)
1<2>2<1>1 \(\omega^\omega+\omega2\)
1<2>3<1>1 \(\omega^\omega+\omega3\)
1<2>1<1>1<1>1 \(\omega^\omega+\omega^2\)
1<2>1<1>1<1>1<1>1 \(\omega^\omega+\omega^3\)
2<2>1 \(\omega^\omega2\)
3<2>1 \(\omega^\omega3\)
1<1>1<2>1 \(\omega^{\omega+1}\)
1<1>1<2>2 \(\omega^{\omega+1}+1\)
1<1>1<2>3 \(\omega^{\omega+1}+2\)
1<1>1<2>1<1>1 \(\omega^{\omega+1}+\omega\)
1<1>1<2>1<1>2 \(\omega^{\omega+1}+\omega+1\)
1<1>1<2>1<1>3 \(\omega^{\omega+1}+\omega+2\)
1<1>1<2>2<1>1 \(\omega^{\omega+1}+\omega2\)
1<1>1<2>3<1>1 \(\omega^{\omega+1}+\omega3\)
1<1>1<2>1<1>1<1>1 \(\omega^{\omega+1}+\omega^2\)
1<1>1<2>1<1>1<1>1<1>1 \(\omega^{\omega+1}+\omega^3\)
1<1>2<2>1 \(\omega^{\omega+1}+\omega^\omega\)
1<1>3<2>1 \(\omega^{\omega+1}+\omega^\omega2\)
2<1>1<2>1 \(\omega^{\omega+1}2\)
1<1>1<1>1<2>1 \(\omega^{\omega+2}\)
1<1>1<1>1<1>1<2>1 \(\omega^{\omega+3}\)
1<2>1<2>1 \(\omega^{\omega2}\)
1<2>1<2>1<2>1 \(\omega^{\omega3}\)
1<3>1 \(\omega^{\omega^2}\)
1<1<1>1>1 \(\omega^{\omega^\omega}\)
2<1<1>1>1 \(\omega^{\omega^\omega}2\)
1<1<1>1>1<1<1>1>1 \(\omega^{\omega^\omega2}\)
1<1<1>1>1<1<1>1>1<1<1>1>1 \(\omega^{\omega^\omega3}\)
1<1<1>2>1 \(\omega^{\omega^{\omega+1}}\)
1<2<1>1>1 \(\omega^{\omega^{\omega2}}\)
1<1<1>1<1>1>1 \(\omega^{\omega^{\omega^2}}\)
1<1<1>1<1>1<1>1>1 \(\omega^{\omega^{\omega^3}}\)
1<1<2>1>1 \(\omega^{\omega^{\omega^\omega}}\)
1<1<1<1>1>1>1 \(\omega^{\omega^{\omega^{\omega^\omega}}}\)
1<1/1>1 \(\varepsilon_0\)


MOTAN增长率分析(\(\varepsilon_0\) ~ \(\psi(\psi_I(0))\))
MOTAN FGH
1<1/1>2 \(\varepsilon_0+1\)
1<1/1>3 \(\varepsilon_0+2\)
1<1/1>1<1>1 \(\varepsilon_0+\omega\)
1<1/1>1<1>2 \(\varepsilon_0+\omega+1\)
1<1/1>1<1>3 \(\varepsilon_0+\omega+2\)
1<1/1>2<1>1 \(\varepsilon_0+\omega2\)
1<1/1>3<1>1 \(\varepsilon_0+\omega3\)
1<1/1>1<1>1<1>1 \(\varepsilon_0+\omega^2\)
1<1/1>1<1>1<1>1<1>1 \(\varepsilon_0+\omega^3\)
1<1/1>1<2>1 \(\varepsilon_0+\omega^\omega\)
1<1/1>1<1<1/1>1>1 \(\varepsilon_02\)
1<1/1>2<1<1/1>1>1 \(\varepsilon_03\)
1<1/1>1<1<1/1>1>1<1<1/1>1>1 \(\varepsilon_0^2\)
1<1/1>1<1<1/1>2>1 \(\varepsilon_0^\omega\)
1<1/1>1<1<1/1>1<1<1/1>1>1>1 \(\varepsilon_0^{\varepsilon_0}\)
1<1/1>1<1<1/1>1<1<1/1>1<1<1/1>1>1>1>1 \(\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}\)
2<1/1>1 \(\varepsilon_1\)
3<1/1>1 \(\varepsilon_2\)
1<1>1<1/1>1 \(\varepsilon_{\omega}\)
1<1<1/1>1>1<1/1>1 \(\varepsilon_{\varepsilon_0}\)
1<1<1<1/1>1>1<1/1>1>1<1/1>1 \(\varepsilon_{\varepsilon_{\varepsilon_0}}\)
1<1/1>1<1/1>1 \(\zeta_0\)
1<1/1>2<1/1>1 \(\varepsilon_{\zeta_0+1}\)
2<1/1>1<1/1>1 \(\zeta_1\)
1<1/1>1<1/1>1<1/1>1 \(\eta_0\)
1<1/2>1 \(\varphi(\omega,0)\)
1<1/1<1<1/1>1>1>1 \(\varphi(\varphi(\omega,0),0)\)
1<1/1<1/1>1>1 \(\Gamma_0\)
1<1/1<1>1<1/1>1>1 \(SVO\)
1<1/1<1/1>1<1/1>1>1 \(LVO\)
1<2/1>1 \(BHO\)
1<2/1>1<1<2/1>1>1 \(\psi(\psi_1(0))2\)
1<2/1>1<1<2/1>2>1 \(\psi(\psi_1(0))^\omega\)
1<2/1>1<1<2/1>1<1<2/1>1>1>1 \(\psi(\psi_1(0))^{\psi(\psi_1(0))}\)
1<2/1>1<1/1>1 \(\psi(\psi_1(0)+1)\)
1<2/1>1<1/1>1<1/1>1 \(\psi(\psi_1(0)+\Omega)\)
1<2/1>1<1/2>1 \(\psi(\psi_1(0)+\Omega^\omega)\)
1<2/1>1<1/1<2/1>1>1 \(\psi(\psi_1(0)2)\)
1<2/1>1<1/1<2/1>2>1 \(\psi(\psi_1(0)^\omega)\)
1<2/1>1<1/1<2/1>1<1<2/1>1>1>1 \(\psi(\psi_1(0)^{\psi(\psi_1(0))})\)
1<2/1>1<1/1<2/1>1<1/1>1>1 \(\psi(\psi_1(0)^\Omega)\)
1<2/1>1<1/1<2/1>1<1/1<2/1>1>1>1 \(\psi(\psi_1(0)^{\psi_1(0)})\)
2<2/1>1 \(\psi(\psi_1(1))\)
3<2/1>1 \(\psi(\psi_1(2))\)
1<1<2/1>1>1<2/1>1 \(\psi(\psi_1(\psi(\psi_1(0))))\)
1<1/1>1<2/1>1 \(\psi(\psi_1(\Omega))\)
1<1/1<2/1>1>1<2/1>1 \(\psi(\psi_1(\psi_1(0)))\)
1<1/1<1/1<2/1>1>1<2/1>1>1<2/1>1 \(\psi(\psi_1(\psi_1(\psi_1(0))))\)
1<2/1>1<2/1>1 \(\psi(\Omega_2)\)
1<2/2>1 \(\psi(\Omega_2^\omega)\)
1<2/1<1/1>1>1 \(\psi(\Omega_2^\Omega)\)
1<2/1<2/1>1>1 \(\psi(\Omega_2^{\Omega_2})\)
1<2/1<2/1<2/1>1>1>1 \(\psi(\Omega_2^{\Omega_2^{\Omega_2}})\)
1<3/1>1 \(\psi(\psi_2(0))\)
1<3/1>1<3/1>1 \(\psi(\Omega_3)\)
1<1<1>1/1>1 \(\psi(\Omega_{\omega})\)
1<1<1>2/1>1 \(\psi(\psi_{\omega}(0))\)
1<1<1/1>1/1>1 \(\psi(\Omega_{\Omega})\)
1<1<1<1/1>1/1>1/1>1 \(\psi(\Omega_{\Omega_{\Omega}})\)
1<1/1/1>1 \(\psi(\psi_I(0))\)


1<1/1>1是1<1<1/1>1>1的省略写法,省略了最外层的1<>1,它们都是\(\varepsilon_0\)。

1<1/1>2跟1<1<1/1>1>2表示同一序数,都是\(\varepsilon_0+1\)。

1<1/1>1<1<1/1>1>1和2<1<1/1>1>1表示同一序数,都是\(\varepsilon_02\)。

1<1/1>1<1>1<1<1/1>1>1和1<1>1<1<1/1>1>1表示同一序数,都是\(\varepsilon_0\omega\),此时第1个式子开头的1<1/1>1变成了并不影响表达式增长率的“前缀”。

1<1<1/1>2>1和1<1/1>1<1<1/1>2>1表示的都是\(\varepsilon_0^\omega\)

2<1/1>1和1<2<1/1>1>1表示的是同一序数,都是\(\varepsilon_1\)。

这里特别注意:1<1/1>2=1<1<1/1>1>2,而2<1/1>1=1<2<1/1>1>1,其中2所在的位置并不相同!


反射模式分析
1<1>1 1
1<1>1<1>1 1-1
1<1>1<1>1<1>1 1-1-1
1<1/1>1 2
1<1<1>1/1>1 1-2
1<1/1/1>1 2 1-2
1<1/1/1/1>1 2 1-(2 1-2)
1<1{1{1}1}1>1 2-2
1<1{1}1{1{1}1}1>1 2 1-2-2
1<1{1}1{1}1{1{1}1}1>1 2 1-(2 1-2-2)
1<1{1{1}1}1{1{1}1}1>1 2-2 1-2-2
1<1{2{1}1}1>1 2-2-2
1<1{3{1}1}1>1 2-2-2-2
1<1{1{1}1{1}1}1>1 3
1<1{1}1{1{1}1{1}1}1>1 2 1-3
1<1{1}1{1}1{1{1}1{1}1}1>1 2 1-(2 1-3)
1<1{1{1}1}1{1{1}1{1}1}1>1 2-2 1-3
1<1{1}1{1{1}1}1{1{1}1{1}1}1>1 2 1-(2-2 1-3)
1<1{1{1}1}1{1{1}1}1{1{1}1{1}1}1>1 2-2 1-(2-2 1-3)
1<1{2{1}1}1{1{1}1{1}1}1>1 2-2-2 1-3
1<1{1{1}1{1}1}1{1{1}1{1}1}1>1 3 1-3
1<1{1{1}1{1}1}1{1{1}1{1}1}1{1{1}1{1}1}1>1 3 1-(3 1-3)
1<1{1{1}2{1}1}1>1 2-3
1<1{1{1}3{1}1}1>1 2-2-3
1<1{2{1}1{1}1}1>1 3 2-3
1<1{3{1}1{1}1}1>1 3 2-(3 2-3)
1<1{1{1}1{1}1{1}1}1>1 3-3
1<1{1{1}1{1}1{1}1{1}1}1>1 3-3-3
1<1{1{1{1}1}1}1>1 4