wpmath0000016_0

(+)-alpha-barbatene_synthase.html

  1. \rightleftharpoons

(+)-alpha-pinene_synthase.html

  1. \rightleftharpoons

(+)-alpha-santalene_synthase_((2Z,6Z)-farnesyl_diphosphate_cyclizing).html

  1. \rightleftharpoons

(+)-alpha-terpineol_synthase.html

  1. \rightleftharpoons

(+)-beta-caryophyllene_synthase.html

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(+)-beta-pinene_synthase.html

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(+)-camphene_synthase.html

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(+)-car-3-ene_synthase.html

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(+)-cubenene_synthase.html

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(+)-delta-selinene_synthase.html

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(+)-endo-beta-bergamotene_synthase_((2Z,6Z)-farnesyl_diphosphate_cyclizing).html

  1. \rightleftharpoons

(+)-epi-alpha-bisabolol_synthase.html

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(+)-epicubenol_synthase.html

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(+)-gamma-cadinene_synthase.html

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(+)-germacrene_D_synthase.html

  1. \rightleftharpoons

(+)-sabinene_synthase.html

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(+)-sativene_synthase.html

  1. \rightleftharpoons

(+)-T-muurolol_synthase.html

  1. \rightleftharpoons

(-)-alpha-cuprenene_synthase.html

  1. \rightleftharpoons

(-)-alpha-pinene_synthase.html

  1. \rightleftharpoons

(-)-alpha-terpineol_synthase.html

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(-)-beta-caryophyllene_synthase.html

  1. \rightleftharpoons

(-)-beta-pinene_synthase.html

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(-)-camphene_synthase.html

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(-)-delta-cadinene_synthase.html

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(-)-endo-alpha-bergamotene_synthase_((2Z,6Z)-farnesyl_diphosphate_cyclizing).html

  1. \rightleftharpoons

(-)-gamma-cadinene_synthase_((2Z,6E)-farnesyl_diphosphate_cyclizing).html

  1. \rightleftharpoons

(-)-germacrene_D_synthase.html

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(-)-sabinene_synthase.html

  1. \rightleftharpoons

(1-4)-a-D-glucan_1-a-D-glucosylmutase.html

  1. \rightleftharpoons

(17246)_2000_GL74.html

  1. tan ( θ 2 ) = radius of moon distance from surface of asteroid to center of moon \scriptstyle{\mathrm{tan}\left(\frac{\theta}{2}\right)=\frac{\mathrm{radius~{}% of~{}moon}}{\mathrm{distance~{}from~{}surface~{}of~{}asteroid~{}to~{}center~{}% of~{}moon}}}

(177049)_2003_EE16.html

  1. ( 100 5 ) 24 - 6.5 10000000 (\sqrt[5]{100})^{24-6.5}\approx 10000000

(2,2,3-trimethyl-5-oxocyclopent-3-enyl)acetyl-CoA_synthase.html

  1. \rightleftharpoons

(2R)-sulfolactate_sulfo-lyase.html

  1. \rightleftharpoons

(31345)_1998_PG.html

  1. tan ( θ 2 ) = radius of moon distance from surface of asteroid to center of moon \scriptstyle{\mathrm{tan}\left(\frac{\theta}{2}\right)=\frac{\mathrm{radius~{}% of~{}moon}}{\mathrm{distance~{}from~{}surface~{}of~{}asteroid~{}to~{}center~{}% of~{}moon}}}

(3R,6E)-nerolidol_synthase.html

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(3S,6E)-nerolidol_synthase.html

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(4S)-beta-phellandrene_synthase_(geranyl-diphosphate-cyclizing).html

  1. \rightleftharpoons

(60621)_2000_FE8.html

  1. tan ( θ 2 ) = radius of moon distance from surface of asteroid to center of moon \scriptstyle{\mathrm{tan}\left(\frac{\theta}{2}\right)=\frac{\mathrm{radius~{}% of~{}moon}}{\mathrm{distance~{}from~{}surface~{}of~{}asteroid~{}to~{}center~{}% of~{}moon}}}

(66063)_1998_RO1.html

  1. tan ( θ 2 ) = radius of moon distance from surface of asteroid to center of moon \scriptstyle{\mathrm{tan}\left(\frac{\theta}{2}\right)=\frac{\mathrm{radius~{}% of~{}moon}}{\mathrm{distance~{}from~{}surface~{}of~{}asteroid~{}to~{}center~{}% of~{}moon}}}

(a,b)-decomposability.html

  1. G G
  2. g g
  3. g 4 g\geq 4
  4. g 5 g\geq 5
  5. g 6 g\geq 6
  6. g 8 g\geq 8
  7. G G
  8. G G

(butirosin_acyl-carrier_protein)—L-glutamate_ligase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

(E)-2-epi-beta-caryophyllene_synthase.html

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(E)-beta-ocimene_synthase.html

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(E)-gamma-bisabolene_synthase.html

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(E,E)-germacrene_B_synthase.html

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(S)-beta-bisabolene_synthase.html

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(S)-beta-macrocarpene_synthase.html

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(Z)-gamma-bisabolene_synthase.html

  1. \rightleftharpoons

1,8-cineole_synthase.html

  1. \rightleftharpoons

1-(5-phosphoribosyl)-5-((5-phosphoribosylamino)methylideneamino)imidazole-4-carboxamide_isomerase.html

  1. \rightleftharpoons

10-epi-gamma-eudesmol_synthase.html

  1. \rightleftharpoons

100_prisoners_problem.html

  1. ( 1 7 5 ) ( 2 4 8 ) ( 3 6 ) (1~{}7~{}5)(2~{}4~{}8)(3~{}6)
  2. ( 1 3 7 4 5 8 2 ) ( 6 ) (1~{}3~{}7~{}4~{}5~{}8~{}2)(6)
  3. l l
  4. l l
  5. l > 50 l>50
  6. ( 100 l ) {\textstyle\left({{100}\atop{l}}\right)}
  7. ( l - 1 ) ! (l-1)!
  8. l l
  9. ( 100 - l ) ! (100-l)!
  10. l > 50 l>50
  11. ( 100 l ) ( l - 1 ) ! ( 100 - l ) ! = 100 ! l {\left({{100}\atop{l}}\right)}\cdot(l-1)!\cdot(100-l)!=\frac{100!}{l}
  12. 1 - 1 100 ! ( 100 ! 51 + + 100 ! 100 ) = 1 - ( 1 51 + + 1 100 ) = 1 - ( H 100 - H 50 ) 0.31183 1-\frac{1}{100!}\left(\frac{100!}{51}+\ldots+\frac{100!}{100}\right)=1-\left(% \frac{1}{51}+\ldots+\frac{1}{100}\right)=1-(H_{100}-H_{50})\approx 0.31183
  13. H n H_{n}
  14. n n
  15. 2 n 2n
  16. n n
  17. 1 - ( H 2 n - H n ) = 1 - ( H 2 n - ln 2 n ) + ( H n - ln n ) - ln 2 1-(H_{2n}-H_{n})=1-(H_{2n}-\ln 2n)+(H_{n}-\ln n)-\ln 2
  18. γ \gamma
  19. n n\to\infty
  20. lim n ( H n - ln n ) = γ \lim_{n\to\infty}(H_{n}-\ln n)=\gamma
  21. lim n ( 1 - H 2 n + H n ) = 1 - γ + γ - ln 2 = 1 - ln 2 0.30685 \lim_{n\to\infty}(1-H_{2n}+H_{n})=1-\gamma+\gamma-\ln 2=1-\ln 2\approx 0.30685

16807_(number).html

  1. X k + 1 = 16807 X k mod 2147483647 X_{k+1}=16807\cdot X_{k}~{}~{}\bmod~{}~{}2147483647

16S_rRNA_pseudouridine516_synthase.html

  1. \rightleftharpoons

1999_OJ4.html

  1. tan ( θ 2 ) = radius of moon distance from surface of asteroid to center of moon \scriptstyle{\mathrm{tan}\left(\frac{\theta}{2}\right)=\frac{\mathrm{radius~{}% of~{}moon}}{\mathrm{distance~{}from~{}surface~{}of~{}asteroid~{}to~{}center~{}% of~{}moon}}}

2,3-diketo-5-methylthiopentyl-1-phosphate_enolase.html

  1. \rightleftharpoons

2-(1,2-epoxy-1,2-dihydrophenyl)acetyl-CoA_isomerase.html

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2-deoxy-scyllo-inosose_synthase.html

  1. \rightleftharpoons

2-hydroxychromene-2-carboxylate_isomerase.html

  1. \rightleftharpoons

2-hydroxyhexa-2,4-dienoate_hydratase.html

  1. \rightleftharpoons

2-hydroxymuconate_tautomerase.html

  1. \rightleftharpoons

2-hydroxypropyl-CoM_lyase.html

  1. \rightleftharpoons
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2-methylisoborneol_synthase.html

  1. \rightleftharpoons

2014–15_Blackburn_Rovers_F.C._season.html

  1. 1 2 \tfrac{1}{2}

21S_rRNA_pseudouridine2819_synthase.html

  1. \rightleftharpoons

23S_rRNA_pseudouridine1911::1915::1917_synthase.html

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23S_rRNA_pseudouridine2457_synthase.html

  1. \rightleftharpoons

23S_rRNA_pseudouridine2604_synthase.html

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23S_rRNA_pseudouridine2605_synthase.html

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23S_rRNA_pseudouridine746_synthase.html

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23S_rRNA_pseudouridine955::2504::2580_synthase.html

  1. \rightleftharpoons

2D_Adaptive_Filters.html

  1. P ( z 1 , z 2 ) P(z_{1},z_{2})
  2. H ( z 1 , z 2 ) H(z_{1},z_{2})
  3. e ( n 1 , n 2 ) e(n_{1},n_{2})
  4. d ( n 1 , n 2 ) d(n_{1},n_{2})
  5. y ( n 1 , n 2 ) y(n_{1},n_{2})
  6. P ( z 1 , z 2 ) P(z_{1},z_{2})
  7. H ( z 1 , z 2 ) H(z_{1},z_{2})
  8. H ( z 1 , z 2 ) H(z_{1},z_{2})
  9. y ( n 1 , n 2 ) = m 1 m 2 w ( m 1 , m 2 ) x ( n 1 - m 1 , n 2 - m 2 ) y(n_{1},n_{2})=\sum_{m_{1}}\sum_{m_{2}}w(m_{1},m_{2})x(n_{1}-m_{1},n_{2}-m_{2})
  10. d ( n 1 , n 2 ) d(n_{1},n_{2})
  11. F [ e ( n 1 , n 2 ) ] F[e(n_{1},n_{2})]
  12. e ( n 1 , n 2 ) = d ( n 1 , n 2 ) - y ( n 1 , n 2 ) e(n_{1},n_{2})=d(n_{1},n_{2})-y(n_{1},n_{2})
  13. w j + 1 ( n 1 , n 2 ) = w j ( n 1 , n 2 ) + F [ e ( n 1 , n 2 ) ] w_{j+1}(n_{1},n_{2})=w_{j}(n_{1},n_{2})+F[e(n_{1},n_{2})]
  14. w w
  15. y ( n 1 , n 2 ) = m 1 = 0 N - 1 m 2 = 0 N - 1 w j ( m 1 , m 2 ) x ( n 1 - m 1 , n 2 - m 2 ) y(n_{1},n_{2})=\sum_{m_{1}=0}^{N-1}\sum_{m_{2}=0}^{N-1}w_{j}(m_{1},m_{2})x(n_{% 1}-m_{1},n_{2}-m_{2})
  16. e j e_{j}
  17. e j = d ( n 1 , n 2 ) - m 1 = 0 N - 1 m 2 = 0 N - 1 w j ( m 1 , m 2 ) x ( n 1 - m 1 , n 2 - m 2 ) e_{j}=d(n_{1},n_{2})-\sum_{m_{1}=0}^{N-1}\sum_{m_{2}=0}^{N-1}w_{j}(m_{1},m_{2}% )x(n_{1}-m_{1},n_{2}-m_{2})
  18. w j + 1 ( n 1 , n 2 ) = w j ( n 1 , n 2 ) + 2 μ e j x ( n 1 , n 2 ) w_{j+1}(n_{1},n_{2})=w_{j}(n_{1},n_{2})+2\mu e_{j}x(n_{1},n_{2})
  19. μ \mu
  20. y ( n 1 , n 2 ) = m 1 = 0 M 1 m 2 = 0 M 2 a ( m 1 , m 2 ) x ( n 1 - m 1 , n 2 - m 2 ) - m 1 = 0 L 1 m 2 = 0 L 2 ( m 1 , m 2 ) ( 0 , 0 ) b ( m 1 , m 2 ) y ( n 1 - m 1 , n 2 - m 2 ) y(n_{1},n_{2})=\sum_{m_{1}=0}^{M_{1}}\sum_{m_{2}=0}^{M_{2}}a(m_{1},m_{2})x(n_{% 1}-m_{1},n_{2}-m_{2})-{\sum_{m_{1}=0}^{L_{1}}\sum_{m_{2}=0}^{L_{2}}}_{(m_{1},m% _{2})\neq(0,0)}b(m_{1},m_{2})y(n_{1}-m_{1},n_{2}-m_{2})
  21. y ( n 1 , n 2 ) y(n_{1},n_{2})
  22. x ( n 1 , n 2 ) x(n_{1},n_{2})
  23. b ( m 1 , m 2 ) b(m_{1},m_{2})
  24. a ( m 1 , m 2 ) a(m_{1},m_{2})
  25. e ( n 1 , n 2 ) = d ( n 1 , n 2 ) - y ( n 1 , n 2 ) e(n_{1},n_{2})=d(n_{1},n_{2})-y(n_{1},n_{2})
  26. d ( n 1 , n 2 ) d(n_{1},n_{2})
  27. E { e 2 ( n 1 , n 2 ) } E\{e^{2}(n_{1},n_{2})\}

2D_Filters.html

  1. x ( n 1 , n 2 ) x\left(n_{1},n_{2}\right)
  2. y ( n 1 , n 2 ) y\left(n_{1},n_{2}\right)
  3. y ( n 1 , n 2 ) = l 1 = 0 L 1 - 1 l 2 = 0 L 2 - 1 a ( l 1 , l 2 ) x ( n 1 - l 1 , n 2 - l 2 ) - k 1 = 0 K 1 - 1 k 2 = 0 K 2 - 1 b ( k 1 , k 2 ) y ( n 1 - k 1 , n 2 - k 2 ) y\left(n_{1},n_{2}\right)=\sum_{l_{1}=0}^{L_{1}-1}\sum_{l_{2}=0}^{L_{2}-1}a(l_% {1},l_{2})x(n_{1}-l_{1},n_{2}-l_{2})-\sum_{k_{1}=0}^{K_{1}-1}\sum_{k_{2}=0}^{K% _{2}-1}b(k_{1},k_{2})y(n_{1}-k_{1},n_{2}-k_{2})
  4. δ ( n 1 , n 2 ) \delta(n_{1},n_{2})
  5. h ( n 1 , n 2 ) h\left(n_{1},n_{2}\right)
  6. h ( n 1 , n 2 ) = a ( l 1 , l 2 ) - k 1 = 0 K 1 - 1 k 2 = 0 K 2 - 1 b ( k 1 , k 2 ) h ( n 1 - k 1 , n 2 - k 2 ) h\left(n_{1},n_{2}\right)=a(l_{1},l_{2})-\sum_{k_{1}=0}^{K_{1}-1}\sum_{k_{2}=0% }^{K_{2}-1}b(k_{1},k_{2})h(n_{1}-k_{1},n_{2}-k_{2})
  7. H ( z 1 , z 2 ) H\left(z_{1},z_{2}\right)
  8. H z ( z 1 , z 2 ) = l 1 = 0 L 1 - 1 l 2 = 0 L 2 - 1 a ( l 1 , l 2 ) z 1 - l 1 z 2 - l 2 k 1 = 0 K 1 - 1 k 2 = 0 K 2 - 1 b ( k 1 , k 2 ) z 1 - k 1 z 2 - k 2 = A z ( z 1 , z 2 ) B z ( z 1 , z 2 ) H_{z}(z_{1},z_{2})=\frac{\sum_{l_{1}=0}^{L_{1}-1}\sum_{l_{2}=0}^{L_{2}-1}a(l_{% 1},l_{2})z_{1}^{-l_{1}}z_{2}^{-l_{2}}}{\sum_{k_{1}=0}^{K_{1}-1}\sum_{k_{2}=0}^% {K_{2}-1}b(k_{1},k_{2})z_{1}^{-k_{1}}z_{2}^{-k_{2}}}=\frac{A_{z}(z_{1},z_{2})}% {B_{z}(z_{1},z_{2})}
  9. A ( z 1 , z 2 ) A(z_{1},z_{2})
  10. 1 / B ( z 1 , z 2 ) 1/B(z_{1},z_{2})
  11. H ( z 1 , z 2 ) H(z_{1},z_{2})
  12. H z p ( z 1 , z 2 ) = i = 1 N H z ( i ) ( z 1 , z 2 ) H_{z}^{p}(z_{1},z_{2})=\sum_{i=1}^{N}H_{z}^{(i)}(z_{1},z_{2})
  13. H z ( i ) ( z 1 , z 2 ) = A z ( i ) ( z 1 , z 2 ) B z ( i ) ( z 1 , z 2 ) H_{z}^{(i)}(z_{1},z_{2})=\frac{A_{z}^{(i)}(z_{1},z_{2})}{B_{z}^{(i)}(z_{1},z_{% 2})}
  14. H z p ( z 1 , z 2 ) = A z p ( z 1 , z 2 ) B z p ( z 1 , z 2 ) = j = 1 N i j A z ( j ) ( z 1 , z 2 ) B z ( j ) ( z 1 , z 2 ) i = 1 N B z ( i ) ( z 1 , z 2 ) H_{z}^{p}(z_{1},z_{2})=\frac{A_{z}^{p}(z_{1},z_{2})}{B_{z}^{p}(z_{1},z_{2})}=% \frac{\sum_{j=1}^{N}\prod_{ij}A_{z}^{(j)}(z_{1},z_{2})B_{z}^{(j)}(z_{1},z_{2})% }{\prod_{i=1}^{N}B_{z}^{(i)}(z_{1},z_{2})}

3,4-dihydroxyphenylalanine_reductive_deaminase.html

  1. \rightleftharpoons

3-hydroxybenzoate—CoA_ligase.html

  1. \rightleftharpoons

3-hydroxypropionyl-CoA_synthase.html

  1. \rightleftharpoons

3-subset_meet-in-the-middle_attack.html

  1. 2 112 ( = 2 2 * 56 ) 2^{112}(=2^{2*56})
  2. 2 57 ( = 2 * 2 56 ) 2^{57}(=2*2^{56})
  3. f f
  4. g g
  5. A 0 A_{0}
  6. A 1 A_{1}
  7. f . f.
  8. A 2 A_{2}
  9. g g
  10. A 0 A_{0}
  11. i i
  12. A 1 A_{1}
  13. j j
  14. A 2 A_{2}
  15. i i
  16. j j
  17. 2 75.170 2^{75.170}
  18. 2 80 2^{80}
  19. 2 75.170 2^{75.170}
  20. k 32 , k 39 , k 44 , k 61 , k 66 , k 75 k_{32},k_{39},k_{44},k_{61},k_{66},k_{75}
  21. k 3 , k 20 , k 41 , k 47 , k 63 , k 74 k_{3},k_{20},k_{41},k_{47},k_{63},k_{74}
  22. A 0 A_{0}
  23. A 1 A_{1}
  24. A 2 A_{2}
  25. i i
  26. j j
  27. i i
  28. j j
  29. i i
  30. j j
  31. i i
  32. j j
  33. 2 75.170 2^{75.170}
  34. 2 80 2^{80}
  35. 2 75.044 2^{75.044}
  36. 2 75.584 2^{75.584}
  37. 2 72.9 2^{72.9}

3D_sound_localization.html

  1. E = K + 2 m 1 = 1 M - 1 m 2 = 0 m 1 - 1 R R W P H A T i , j ( τ m 1 - τ m 2 ) E=K+2\sum_{{m}_{1}=1}^{M-1}\sum_{{m}_{2}=0}^{{m}_{1}-1}{{R}^{RWPHAT}}_{i,j}% \left({\tau}_{{m}_{1}}-{\tau}_{{m}_{2}}\right)
  2. R R W P H A T i , j ( τ m 1 - τ m 2 ) {{R}^{RWPHAT}}_{i,j}\left({\tau}_{{m}_{1}}-{\tau}_{{m}_{2}}\right)
  3. R R W P H A T i , j ( τ ) = k = 0 L - 1 ζ i ( k ) X i ( k ) ζ j ( k ) X j * ( k ) | X i ( k ) | | X j ( k ) | e j 2 π k τ / L {{R}^{RWPHAT}}_{i,j}\left(\tau\right)=\sum_{k=0}^{L-1}\frac{{\zeta}_{i}\left(k% \right){X}_{i}\left(k\right){\zeta}_{j}\left(k\right){{X}_{j}}^{*}\left(k% \right)}{\left|{X}_{i}\left(k\right)\right|\left|{X}_{j}\left(k\right)\right|}% {e}^{j2\pi k\tau/L}
  4. ζ n i ( k ) {{\zeta}^{n}}_{i}\left(k\right)
  5. ζ n i ( k ) = ξ n i ( k ) ξ n i ( k ) + 1 {{\zeta}^{n}}_{i}\left(k\right)=\frac{{{\xi}^{n}}_{i}\left(k\right)}{{{\xi}^{n% }}_{i}\left(k\right)+1}
  6. ξ n i ( k ) {{\xi}^{n}}_{i}\left(k\right)
  7. i t h i^{th}
  8. n n
  9. k k
  10. x m n x_{m_{n}}
  11. m t h {m}^{th}
  12. τ m n {\tau}_{{m}_{n}}
  13. P U ( r ) = ± A U 360 × 2 π × r PU\left(r\right)=\frac{\pm AU}{360}\times 2\pi\times r
  14. d i s t ( d i r 1 , d i r 2 ) = ( v 1 × v 2 ) × p 1 p 2 | v 1 × v 2 | dist\left(dir_{1},dir_{2}\right)=\frac{\left(\overrightarrow{v_{1}}\times% \overrightarrow{v_{2}}\right)\times\overrightarrow{p_{1}p_{2}}}{\left|% \overrightarrow{v_{1}}\times\overrightarrow{v_{2}}\right|}
  15. d i r 1 dir_{1}
  16. d i r 2 dir_{2}
  17. v i v_{i}
  18. p i p_{i}
  19. d i s t ( d i r 1 , d i r 2 ) < a b s ( P U 1 ( r 1 ) ) + a b s ( P U 2 ( r 2 ) ) dist(dir_{1},dir_{2})<abs(PU_{1}(r_{1}))+abs(PU_{2}(r_{2}))
  20. P O S s o u r c e = ( P O S 1 × w 1 + P O S 2 × w 2 ) w 1 + w 2 POS_{source}=\frac{\left(POS_{1}\times w_{1}+POS_{2}\times w_{2}\right)}{w_{1}% +w_{2}}
  21. P O S s o u r c e POS_{source}
  22. P O S n POS_{n}
  23. w n w_{n}
  24. w n w_{n}
  25. P U PU
  26. r r
  27. c s p i j ( k ) = I F F T { F F T [ s i ( n ) ] F F T [ s j ( n ) ] * | F F T [ s i ( n ) ] | | F F T [ s j ( n ) ] | } csp_{ij}(k)=IFFT\left\{\frac{FFT[s_{i}(n)]\cdot FFT[s_{j}(n)]^{*}}{\left|FFT[s% _{i}(n)]\right|\cdot\left|FFT[s_{j}(n)]\right|\quad}\right\}\quad
  28. s i ( n ) s_{i}(n)
  29. s j ( n ) s_{j}(n)
  30. i i
  31. j j
  32. τ \tau
  33. τ = a r g max { c s p i j ( k ) } {\tau}=arg\max\{csp_{ij}(k)\}
  34. θ = c o s - 1 v τ d m a x F s {\theta}=cos^{-1}\frac{v\cdot\tau}{d_{max}\cdot F_{s}}
  35. v v
  36. F s F_{s}
  37. d m a x d_{max}

4'-demethylrebeccamycin_synthase.html

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4-hydroxy-tetrahydrodipicolinate_reductase.html

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4-hydroxy-tetrahydrodipicolinate_synthase.html

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4-oxalomesaconate_tautomerase.html

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4-phosphopantoate—beta-alanine_ligase.html

  1. \rightleftharpoons

5-epi-alpha-selinene_synthase.html

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5-epiaristolochene_synthase.html

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5-phosphonooxy-L-lysine_phospho-lyase.html

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6-phospho-3-hexuloisomerase.html

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7-carboxy-7-deazaguanine_synthase.html

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7-cyano-7-deazaguanine_synthase.html

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7-epi-alpha-selinene_synthase.html

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7-epi-sesquithujene_synthase.html

  1. \rightleftharpoons

8192_(number).html

  1. 2 13 2^{13}

9,12-octadecadienoate_8-hydroperoxide_8R-isomerase.html

  1. \rightleftharpoons

9,12-octadecadienoate_8-hydroperoxide_8S-isomerase.html

  1. \rightleftharpoons

A._Dale_Kaiser.html

  1. 10 5 10^{5}

ABJM_superconformal_field_theory.html

  1. A d S 4 × S 7 AdS_{4}\times S^{7}

Abraham_Ziv.html

  1. 2 n - 1 2n-1
  2. / n \mathbb{Z}/n\mathbb{Z}
  3. n n

Academy_Color_Encoding_System.html

  1. [ X Y Z ] = [ 0.9525523959 0.0000000000 0.0000936786 0.3439664498 0.7281660966 - 0.0721325464 0.0000000000 0.0000000000 1.0088251844 ] [ R G B ] \begin{bmatrix}X\\ Y\\ Z\end{bmatrix}=\begin{bmatrix}0.9525523959&0.0000000000&0.0000936786\\ 0.3439664498&0.7281660966&-0.0721325464\\ 0.0000000000&0.0000000000&1.0088251844\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}
  2. [ R G B ] = [ 1.0498110175 0.0000000000 - 0.0000974845 - 0.4959030231 1.3733130458 0.0982400361 0.0000000000 0.0000000000 0.9912520182 ] [ X Y Z ] \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}1.0498110175&0.0000000000&-0.0000974845\\ -0.4959030231&1.3733130458&0.0982400361\\ 0.0000000000&0.0000000000&0.9912520182\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}

Acceleration_(differential_geometry).html

  1. M M
  2. Γ \Gamma
  3. γ : \R M \gamma\colon\R\to M
  4. M M
  5. γ ˙ ( t ) {\dot{\gamma}}(t)
  6. γ \gamma
  7. γ ˙ γ ˙ \nabla_{\dot{\gamma}}{\dot{\gamma}}
  8. \nabla
  9. Γ \Gamma
  10. γ \gamma
  11. γ ˙ γ ˙ = γ ˙ d t . \nabla_{\dot{\gamma}}{\dot{\gamma}}=\frac{\nabla\dot{\gamma}}{dt}.
  12. M M

Acetophenone_carboxylase.html

  1. \rightleftharpoons

Achilleol_B_synthase.html

  1. \rightleftharpoons

Acoustic_tweezers.html

  1. F 𝑟𝑎𝑑 = - U \mathbf{\mathit{F^{rad}}}=-\nabla\mathit{U}
  2. F 𝑟𝑎𝑑 = - U \mathbf{\mathit{F^{rad}}}=-\nabla\mathit{U}
  3. U \mathit{U}
  4. U = V 0 ( p i n 2 ¯ 2 ρ f c f 2 f 1 - 3 ρ f v i n 2 ¯ 4 f 2 ) U={V_{0}}({{\overline{p_{in}^{2}}}\over{2{\rho_{f}}c_{f}^{2}}}{f_{1}}-{{3{\rho% _{f}}\overline{v_{in}^{2}}}\over 4}{f_{2}})
  5. V 0 \mathit{V_{\mathit{0}}}
  6. p 𝑖𝑛 \mathit{p_{\mathit{in}}}
  7. v 𝑖𝑛 \mathit{v_{\mathit{in}}}
  8. ρ f \mathit{\rho}_{\mathit{f}}
  9. c f \mathit{c_{\mathit{f}}}
  10. f 1 \mathit{f_{\mathit{1}}}
  11. f 2 \mathit{f_{\mathit{2}}}
  12. < * > ¯ \overline{<*>}
  13. 1 T 0 T ( * ) d t {1\over T}\int\limits_{0}^{T}{(*)}dt
  14. f 1 \mathit{f_{\mathit{1}}}
  15. f 2 \mathit{f_{\mathit{2}}}
  16. f 1 = 1 - ρ f c f 2 ρ p c p 2 {f_{1}}=1-{{{\rho_{f}}c_{f}^{2}}\over{{\rho_{p}}c_{p}^{2}}}
  17. f 2 = 2 ( ρ p - ρ f ) 2 ρ p + ρ f {f_{2}}={{2({\rho_{p}}-{\rho_{f}})}\over{2{\rho_{p}}+{\rho_{f}}}}
  18. ρ p \mathit{\rho}_{\mathit{p}}
  19. c p \mathit{c}_{\mathit{p}}
  20. A i n ( x , t ) = < m t p l > P 0 ρ f ω cos ( k x ) cos ( ω t ) {A_{in}}(x,t)={<mtpl>{{P_{0}}}\over{{\rho_{f}}\omega}}\cos(kx)\cos(\omega t)
  21. p i n ( x , t ) = P 0 cos ( k x ) sin ( ω t ) {p_{in}}(x,t)={P_{0}}\cos(kx)\sin(\omega t)
  22. 𝑣 𝐢𝐧 ( x , t ) = P 0 ρ f c f sin ( 𝑘𝑥 ) cos ( ω t ) ) 𝑒 𝐱 \,\textbf{\,{v}}_{\,\textit{\,{in}}}(\mathit{x},\mathit{t})=\frac{\mathit{P}_{% 0}}{\mathit{\rho}_{\mathit{f}}\mathit{c}_{\mathit{f}}}\sin(\mathit{kx})\cos(% \mathit{\omega t}))\,\textbf{\,{e}}_{\,\textit{\,{x}}}
  23. A 𝑖𝑛 \mathit{A_{\mathit{in}}}
  24. P 0 \mathit{P_{\mathit{0}}}
  25. ω \mathit{\omega}
  26. k \mathit{k}
  27. p i n 2 ¯ = 1 2 P 0 2 cos 2 ( k x ) \overline{p_{in}^{2}}={1\over 2}P_{0}^{2}{\cos^{2}}(kx)
  28. v i n 2 ¯ = P 0 2 2 ρ f 2 c f 2 sin 2 ( k x ) \overline{v_{in}^{2}}={{P_{0}^{2}}\over{2\rho_{f}^{2}c_{f}^{2}}}{\sin^{2}}(kx)
  29. U = V 0 P 0 2 4 ρ f c f 2 [ cos 2 ( k x ) f 1 - 3 2 sin 2 ( k x ) f 2 ] U={V_{0}}{{P_{0}^{2}}\over{4{\rho_{f}}c_{f}^{2}}}[{\cos^{2}}(kx){f_{1}}-{3% \over 2}{\sin^{2}}(kx){f_{2}}]
  30. F x r a d = - x U = V 0 k E a c sin ( 2 k x ) Φ {F_{x}}^{rad}=-{\partial_{x}}U={V_{0}}k{E_{ac}}\sin(2kx)\Phi
  31. E a c = P 0 2 4 ρ f c f 2 {E_{ac}}={{P_{0}^{2}}\over{4{\rho_{f}}c_{f}^{2}}}
  32. Φ = f 1 + 3 2 f 2 = 5 ρ p - 2 ρ f 2 ρ p + ρ f - ρ f c f 2 ρ p c p 2 \Phi={f_{1}}+{3\over 2}{f_{2}}={{5{\rho_{p}}-2{\rho_{f}}}\over{2{\rho_{p}}+{% \rho_{f}}}}-{{{\rho_{f}}c_{f}^{2}}\over{{\rho_{p}}c_{p}^{2}}}
  33. E 𝑎𝑐 \mathit{E_{\mathit{ac}}}
  34. Φ \Phi
  35. sin ( 2 k x ) \sin(\mathit{2kx})
  36. Φ \Phi
  37. F B ( x ) = 4 π a 6 [ ( ρ p - ρ f ) 2 ( 3 cos 2 θ - 1 ) 6 ρ f d 4 v i n 2 ( x ) - ω 2 ρ f 9 d 2 ( 1 ρ p c p 2 - 1 ρ f c f 2 ) 2 p i n 2 ( x ) ] {F_{B}}(x)=4\pi{a^{6}}[{{{{({\rho_{p}}-{\rho_{f}})}^{2}}(3{{\cos}^{2}}\theta-1% )}\over{6{\rho_{f}}{d^{4}}}}{v_{in}}^{2}(x)-{{\omega^{2}{\rho_{f}}}\over{9{d^{% 2}}}}{({1\over{{\rho_{p}}c_{p}^{2}}}-{1\over{{\rho_{f}}c_{f}^{2}}})^{2}}p_{in}% ^{2}(x)]
  38. a \mathit{a}
  39. d \mathit{d}
  40. θ \mathit{\theta}
  41. v 𝑖𝑛 ( x ) \mathit{v}_{\mathit{in}}(\mathit{x})
  42. p 𝑖𝑛 ( x ) \mathit{p}_{\mathit{in}}(\mathit{x})
  43. t ρ = - ( ρ 𝐯 ) \partial_{\mathit{t}}\mathit{\rho}=-\nabla\cdot(\mathit{\rho}\,\textit{\,{v}})
  44. ρ [ t 𝐯 + ( 𝐯 ) 𝐯 ] = - p + μ 2 𝐯 + β μ ( 𝐯 ) ) \mathit{\rho}[\partial_{\mathit{t}}\,\textit{\,{v}}+(\,\textit{\,{v}}\cdot% \nabla)\,\textit{\,{v}}]=-\nabla\mathit{p}+\mathit{\mu}\nabla^{2}\,\textit{\,{% v}}+\mathit{\beta}\mathit{\mu}\nabla(\nabla\cdot\,\textit{\,{v}}))
  45. ρ \mathit{\rho}
  46. 𝐯 \,\textit{\,{v}}
  47. p \mathit{p}
  48. μ \mathit{\mu}
  49. β \mathit{\beta}
  50. p = p 0 + p 1 + p 2 p={p_{0}}+{p_{1}}+{p_{2}}
  51. 𝐯 = 𝟎 + 𝐯 𝟏 + 𝐯 𝟐 \,\textit{\,{v}}=\,\textit{\,{0}}+\,\textit{\,{v}}_{\,\textit{\,{1}}}+\,% \textit{\,{v}}_{\,\textit{\,{2}}}
  52. ρ = ρ 0 + ρ 1 + ρ 2 \mathit{\rho}=\mathit{\rho}_{0}+\mathit{\rho}_{1}+\mathit{\rho}_{2}
  53. p 1 = c f 2 ρ 1 {p_{1}}=c_{f}^{2}{\rho_{1}}
  54. t p 1 = - ρ 0 c f 2 𝐯 𝟏 \partial_{\mathit{t}}\mathit{p}_{1}=-\mathit{\rho}_{0}\mathit{c}_{\mathit{f}}^% {2}\nabla\cdot\,\textit{\,{v}}_{\,\textit{\,{1}}}
  55. ρ 0 t 𝐯 𝟏 = - p 1 + μ 2 𝐯 𝟏 + β μ ( 𝐯 𝟏 ) \mathit{\rho}_{0}\partial_{\mathit{t}}\,\textit{\,{v}}_{\,\textit{\,{1}}}=-% \nabla\mathit{p}_{1}+\mathit{\mu}\nabla^{2}\,\textit{\,{v}}\textit{\,{1}}+% \mathit{\beta}\mathit{\mu}\nabla(\nabla\cdot\,\textit{\,{v}}\textit{\,{1}})
  56. t ρ 2 = - ρ 0 𝐯 𝟐 - ( ρ 1 𝐯 𝟏 ) \partial_{\mathit{t}}\mathit{\rho}_{2}=-\mathit{\rho}_{0}\nabla\cdot\,\textit{% \,{v}}\textit{\,{2}}-\nabla\cdot(\mathit{\rho}_{1}\,\textit{\,{v}}_{\,\textit{% \,{1}}})
  57. ρ 0 t 𝐯 𝟐 = - p 2 + μ 2 𝐯 𝟐 + β μ ( 𝐯 𝟐 ) - ρ 1 t 𝐯 𝟏 - ρ 0 ( 𝐯 𝟏 ) 𝐯 𝟏 \mathit{\rho}_{0}\partial_{\mathit{t}}\,\textit{\,{v}}_{\,\textit{\,{2}}}=-% \nabla\mathit{p}_{2}+\mathit{\mu}\nabla^{2}\,\textit{\,{v}}_{\,\textit{\,{2}}}% +\mathit{\beta}\mathit{\mu}\nabla(\nabla\cdot\,\textit{\,{v}}_{\,\textit{\,{2}% }})-\mathit{\rho}_{1}\partial_{t}\,\textit{\,{v}}_{\,\textit{\,{1}}}-\mathit{% \rho}_{0}(\,\textit{\,{v}}_{\,\textit{\,{1}}}\cdot\nabla)\,\textit{\,{v}}_{\,% \textit{\,{1}}}
  58. 1 c f 2 t 2 p 1 = [ 1 - ( 1 - β ) μ ρ 0 c f 2 t ] 2 p 1 {1\over{c_{f}^{2}}}\partial_{t}^{2}{p_{1}}=[1-{{(1-\beta)\mu}\over{{\rho_{0}}c% _{f}^{2}}}{\partial_{t}}]{\nabla^{2}}{p_{1}}
  59. p 1 \mathit{p}_{1}
  60. 𝐯 𝟏 \,\textit{\,{v}}_{\,\textit{\,{1}}}
  61. [ - ( ρ 1 𝐯 𝟏 ) ) ] [-\nabla\cdot(\mathit{\rho}_{1}\,\textit{\,{v}}_{\,\textit{\,{1}}}))]
  62. [ - ρ 1 t 𝐯 𝟏 - ρ 0 ( 𝐯 𝟏 ) 𝐯 𝟏 ] [-\rho_{1}\partial_{t}\,\textit{\,{v}}_{\,\textit{\,{1}}}-\rho_{0}(\,\textit{% \,{v}}_{\,\textit{\,{1}}}\cdot\nabla)\,\textit{\,{v}}_{\,\textit{\,{1}}}]
  63. 𝐯 𝟐 ¯ \overline{\,\textit{\,{v}}_{\,\textit{\,{2}}}}
  64. t ρ 2 ¯ = 0 \overline{\partial_{\mathit{t}}\mathit{\rho}_{2}}=0
  65. ρ 0 𝐯 𝟐 ¯ = - ( ρ 1 𝐯 𝟏 ) ¯ \mathit{\rho}_{0}\nabla\cdot\overline{\,\textit{\,{v}}_{\,\textit{\,{2}}}}=-% \nabla\cdot\overline{(\mathit{\rho}_{1}\,\textit{\,{v}}_{\,\textit{\,{1}}})}
  66. μ 2 𝐯 𝟐 ¯ + β μ ( 𝐯 𝟐 ¯ ) - p 2 ¯ = ρ 1 t 𝐯 𝟏 ¯ - ρ 0 ( 𝐯 𝟏 ) 𝐯 𝟏 ¯ \mathit{\mu}\nabla^{2}\overline{\,\textit{\,{v}}_{\,\textit{\,{2}}}}+\mathit{% \beta}\mathit{\mu}\nabla(\nabla\cdot\overline{\,\textit{\,{v}}_{\,\textit{\,{2% }}}})-\overline{\nabla\mathit{p}_{2}}=\overline{\mathit{\rho}_{1}\partial_{% \mathit{t}}\,\textit{\,{v}}_{\,\textit{\,{1}}}}-\mathit{\rho}_{0}\overline{(\,% \textit{\,{v}}_{\,\textbf{1}}\cdot\nabla)\,\textit{\,{v}}_{\,\textit{\,{1}}}}
  67. m d 𝐮 𝑑𝑡 = 𝐅 𝐫𝐚𝐝 + 𝐅 𝐝𝐫𝐚𝐠 \mathit{m}\frac{d\,\textit{\,{u}}}{\mathit{dt}}=\,\textit{\,{F}}^{\,\textit{\,% {rad}}}+\,\textit{\,{F}}^{\,\textit{\,{drag}}}
  68. 𝐅 𝐝𝐫𝐚𝐠 = 6 π a μ ( 𝐯 - 𝐮 ) \,\textit{\,{F}}^{\,\textit{\,{drag}}}=6\pi\mathit{a}\mathit{\mu}(\,\textit{\,% {v}}-\,\textit{\,{u}})
  69. 𝐯 \,\textit{\,{v}}
  70. 𝐮 \,\textit{\,{u}}

ACP-SH:acetate_ligase.html

  1. \rightleftharpoons

Action_model_learning.html

  1. E E
  2. e = ( s , a , s ) e=(s,a,s^{\prime})
  3. s , s s,s^{\prime}
  4. t , t t,t^{\prime}
  5. a a
  6. t t
  7. D , P \langle D,P\rangle
  8. D D
  9. P P
  10. D D
  11. P P

Acute_and_obtuse_triangles.html

  1. \underbrace{\qquad\qquad\qquad\qquad\qquad\qquad}
  2. A > B if and only if a > b . A>B\quad\,\text{if and only if}\quad a>b.
  3. c 2 2 < a 2 + b 2 < c 2 , \frac{c^{2}}{2}<a^{2}+b^{2}<c^{2},
  4. a 2 + b 2 > c 2 , b 2 + c 2 > a 2 , c 2 + a 2 > b 2 . a^{2}+b^{2}>c^{2},\quad b^{2}+c^{2}>a^{2},\quad c^{2}+a^{2}>b^{2}.
  5. 1 h c 2 < 1 a 2 + 1 b 2 , \frac{1}{h_{c}^{2}}<\frac{1}{a^{2}}+\frac{1}{b^{2}},
  6. 4 c 2 + 9 a 2 b 2 > 16 m a 2 m b 2 4c^{2}+9a^{2}b^{2}>16m_{a}^{2}m_{b}^{2}
  7. m c > R m_{c}>R
  8. 27 ( b 2 + c 2 - a 2 ) 2 ( c 2 + a 2 - b 2 ) 2 ( a 2 + b 2 - c 2 ) 2 ( 4 A ) 6 , 27(b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2}\leq(4A% )^{6},
  9. cos 2 A + cos 2 B + cos 2 C < 1 , \cos^{2}A+\cos^{2}B+\cos^{2}C<1,
  10. a cos 3 A + b cos 3 B + c cos 3 C a b c 4 R 2 a\cos^{3}A+b\cos^{3}B+c\cos^{3}C\leq\frac{abc}{4R^{2}}
  11. cos 3 A + cos 3 B + cos 3 C + cos A cos B cos C 1 2 . \cos^{3}A+\cos^{3}B+\cos^{3}C+\cos A\cos B\cos C\geq\frac{1}{2}.
  12. sin 2 A + sin 2 B + sin 2 C > 2 , \sin^{2}A+\sin^{2}B+\sin^{2}C>2,
  13. sin A sin B + sin B sin C + sin C sin A ( cos A + cos B + cos C ) 2 . \sin A\cdot\sin B+\sin B\cdot\sin C+\sin C\cdot\sin A\leq(\cos A+\cos B+\cos C% )^{2}.
  14. tan A + tan B + tan C = tan A tan B tan C > 0 \tan A+\tan B+\tan C=\tan A\cdot\tan B\cdot\tan C>0
  15. tan A + tan B + tan C 2 ( sin 2 A + sin 2 B + sin 2 C ) \tan A+\tan B+\tan C\geq 2(\sin 2A+\sin 2B+\sin 2C)
  16. ( tan A + tan B + tan C ) 2 ( sec A + 1 ) 2 + ( sec B + 1 ) 2 + ( sec C + 1 ) 2 . (\tan A+\tan B+\tan C)^{2}\geq(\sec A+1)^{2}+(\sec B+1)^{2}+(\sec C+1)^{2}.
  17. a tan A + b tan B + c tan C 10 R - 2 r . a\tan A+b\tan B+c\tan C\geq 10R-2r.
  18. ( cot A + cot B + cot C ) 2 K r 2 . (\sqrt{\cot A}+\sqrt{\cot B}+\sqrt{\cot C})^{2}\leq\frac{K}{r^{2}}.
  19. R + r < a + b 2 R+r<\frac{a+b}{2}
  20. m a 2 + m b 2 + m c 2 > 6 R 2 m_{a}^{2}+m_{b}^{2}+m_{c}^{2}>6R^{2}
  21. r 2 + r a 2 + r b 2 + r c 2 < 8 R 2 , r^{2}+r_{a}^{2}+r_{b}^{2}+r_{c}^{2}<8R^{2},
  22. s - r > 2 R , s-r>2R,
  23. a b + b c + c a 2 R ( R + r ) + 8 K 3 . ab+bc+ca\geq 2R(R+r)+\frac{8K}{\sqrt{3}}.
  24. O H < R , OH<R,
  25. I H < r 2 , IH<r\sqrt{2},
  26. 1 x a x b 2 2 3 0.94. 1\geq\frac{x_{a}}{x_{b}}\geq\frac{2\sqrt{2}}{3}\approx 0.94.
  27. a p + b q < c r . ap+bq<cr.

Additive_State_Decomposition.html

  1. n = n p + n s and P = P p P s n=n_{p}+n_{s}\,\text{ and }P=P_{p}\oplus P_{s}
  2. n = n p = n s and P = P p + P s n=n_{p}=n_{s}\,\text{ and }P=P_{p}+P_{s}
  3. x ˙ = f ( t , x , u ) , x ( 0 ) = x 0 \dot{x}=f(t,x,u),x(0)=x_{0}
  4. x \R n x\in\R^{n}
  5. x ˙ p = f p ( t , x p , u p ) , \dot{x}_{p}=f_{p}(t,x_{p},u_{p}),
  6. x p ( 0 ) = x p , 0 x_{p}(0)=x_{p,0}
  7. x p \R n . x_{p}\in\R^{n}.
  8. x ˙ - x ˙ p = f ( t , x , u ) - f p ( t , x p , u p ) , x ( 0 ) = x 0 \dot{x}-\dot{x}_{p}=f(t,x,u)-f_{p}(t,x_{p},u_{p}),x(0)=x_{0}
  9. x s \R n x_{s}\in\R^{n}
  10. x s = x - x p , x_{s}=x-x_{p},
  11. u s = u - u p . u_{s}=u-u_{p}.
  12. x ˙ s = f ( t , x p + x s , u p + u s ) \dot{x}_{s}=f(t,x_{p}+x_{s},u_{p}+u_{s})
  13. - f p ( t , x p , u p ) , x s ( 0 ) = x 0 - x p , 0 , -f_{p}(t,x_{p},u_{p}),x_{s}(0)=x_{0}-x_{p,0},
  14. x ( t ) = x p ( t ) + x s ( t ) , x(t)=x_{p}(t)+x_{s}(t),
  15. t 0. t\geq 0.
  16. x ˙ r = f ( t , x r , u r ) , \dot{x}_{r}=f(t,x_{r},u_{r}),
  17. x r ( 0 ) = x r , 0 x_{r}(0)=x_{r,0}
  18. x ˙ e \dot{x}_{e}
  19. = f ( t , x e + x r , u ) - f ( t , x r , u r ) , =f(t,x_{e}+x_{r},u)-f(t,x_{r},u_{r}),
  20. x e ( 0 ) = x 0 - x r , 0 x_{e}(0)=x_{0}-x_{r,0}
  21. x e = x - x r x_{e}=x-x_{r}
  22. e ( t ) = e p ( t ) + e s ( t ) e(t)=e_{p}(t)+e_{s}(t)
  23. e ( t ) e p ( t ) + e s ( t ) \|e(t)\|\leq\|e_{p}(t)\|+\|e_{s}(t)\|
  24. x ˙ = A x + B ( u 1 + u 2 ) \dot{x}=Ax+B(u_{1}+u_{2})
  25. x ( 0 ) = 0 x(0)=0
  26. x ˙ p = A x p + B u 1 , x p ( 0 ) = 0 \dot{x}_{p}=Ax_{p}+Bu_{1},x_{p}(0)=0
  27. x ˙ s = A x s + B u 2 , x s ( 0 ) = 0 \dot{x}_{s}=Ax_{s}+Bu_{2},x_{s}(0)=0

Adenosylcobinamide-phosphate_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

ADP-dependent_NAD(P)H-hydrate_dehydratase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Affine_gauge_theory.html

  1. X X
  2. X = 3 X=\mathbb{R}^{3}
  3. X X
  4. T X TX
  5. n n
  6. X X
  7. A T X ATX
  8. A F X AFX
  9. X X
  10. G A ( n , ) GA(n,\mathbb{R})
  11. T X TX
  12. F X FX
  13. G L ( n , ) GL(n,\mathbb{R})
  14. G A ( n , ) GA(n,\mathbb{R})
  15. G L ( n , ) GL(n,\mathbb{R})
  16. T n T^{n}
  17. F X FX
  18. A F X AFX
  19. T X TX
  20. ( x μ , x ˙ μ ) , x ˙ μ = x μ x ν x ˙ ν , ( 1 ) (x^{\mu},\dot{x}^{\mu}),\qquad\dot{x}^{\prime\mu}=\frac{\partial x^{\prime\mu}% }{\partial x^{\nu}}\dot{x}^{\nu},\qquad\qquad(1)
  21. T X TX
  22. ( x μ , x ~ μ = x ˙ μ + a μ ( x α ) ) , x ~ μ = x μ x ν x ~ ν + b μ ( x α ) . ( 2 ) (x^{\mu},\widetilde{x}^{\mu}=\dot{x}^{\mu}+a^{\mu}(x^{\alpha})),\qquad% \widetilde{x}^{\prime\mu}=\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}% \widetilde{x}^{\nu}+b^{\mu}(x^{\alpha}).\qquad\qquad(2)
  23. A T X ATX
  24. A A
  25. A F X AFX
  26. A T X = T X ATX=TX
  27. A A
  28. A = d x λ [ λ + ( Γ λ ( x α ) μ ν x ˙ ν + σ λ μ ( x α ) ) ˙ μ ] . ( 3 ) A=dx^{\lambda}\otimes[\partial_{\lambda}+(\Gamma_{\lambda}{}^{\mu}{}_{\nu}(x^{% \alpha})\dot{x}^{\nu}+\sigma_{\lambda}^{\mu}(x^{\alpha}))\dot{\partial}_{\mu}]% .\qquad\qquad(3)
  29. Γ = d x λ [ λ + Γ λ ( x α ) μ ν x ˙ ν ˙ μ ] ( 4 ) \Gamma=dx^{\lambda}\otimes[\partial_{\lambda}+\Gamma_{\lambda}{}^{\mu}{}_{\nu}% (x^{\alpha})\dot{x}^{\nu}\dot{\partial}_{\mu}]\qquad\qquad(4)
  30. T X TX
  31. F X FX
  32. Γ \Gamma
  33. T X X TX\to X
  34. A Γ A\Gamma
  35. A T X ATX
  36. Γ \Gamma
  37. A T X = T X ATX=TX
  38. A Γ = d x λ [ λ + ( Γ λ ( x α ) μ ν x ~ ν + s μ ( x α ) ) ~ μ ] , s μ = - Γ λ a ν μ ν + λ a μ , A\Gamma=dx^{\lambda}\otimes[\partial_{\lambda}+(\Gamma_{\lambda}{}^{\mu}{}_{% \nu}(x^{\alpha})\widetilde{x}^{\nu}+s^{\mu}(x^{\alpha}))\widetilde{\partial}_{% \mu}],\qquad s^{\mu}=-\Gamma_{\lambda}{}^{\mu}{}_{\nu}a^{\nu}+\partial_{% \lambda}a^{\mu},
  39. A A
  40. A T X X ATX\to X
  41. A = A Γ + σ ( 5 ) A=A\Gamma+\sigma\qquad\qquad(5)
  42. A Γ A\Gamma
  43. σ = σ λ μ ( x α ) d x λ μ ( 6 ) \sigma=\sigma_{\lambda}^{\mu}(x^{\alpha})dx^{\lambda}\otimes\partial_{\mu}% \qquad\qquad(6)
  44. T X TX
  45. ˙ μ = μ \dot{\partial}_{\mu}=\partial_{\mu}
  46. V A T X = A T X × X T X VATX=ATX\times_{X}TX
  47. V A T X VATX
  48. A T X ATX
  49. A = Γ + σ A=\Gamma+\sigma
  50. Γ \Gamma
  51. σ \sigma
  52. σ \sigma
  53. T X TX
  54. X X
  55. u ( x ) u ( x ) + a ( x ) u(x)\to u(x)+a(x)
  56. u k u^{k}
  57. k = 1 , 2 , 3 k=1,2,3
  58. u k u k + a k ( x ) u^{k}\to u^{k}+a^{k}(x)
  59. X = 3 X=\mathbb{R}^{3}
  60. A = d x i ( i + A i j ( x k ) ~ j ) A=dx^{i}\otimes(\partial_{i}+A^{j}_{i}(x^{k})\widetilde{\partial}_{j})
  61. A l j A^{j}_{l}
  62. D j u i = j u i - A j i D_{j}u^{i}=\partial_{j}u^{i}-A^{i}_{j}
  63. F j i k = j A i k - i A j k F^{k}_{ji}=\partial_{j}A^{k}_{i}-\partial_{i}A^{k}_{j}
  64. L ( σ ) = μ D i u k D i u k + λ 2 ( D i u i ) 2 - ϵ F k F k i j , i j L_{(\sigma)}=\mu D_{i}u^{k}D^{i}u_{k}+\frac{\lambda}{2}(D_{i}u^{i})^{2}-% \epsilon F^{k}{}_{ij}F_{k}{}^{ij},
  65. μ \mu
  66. λ \lambda
  67. u k ( x ) u^{k}(x)
  68. X X
  69. T X TX
  70. X X
  71. T X TX
  72. Γ \Gamma
  73. σ \sigma
  74. σ \sigma
  75. σ \sigma
  76. T X T * X TX\otimes T^{*}X
  77. F X FX
  78. σ \sigma
  79. X X
  80. s : T X λ λ ( θ + σ ) = ( δ λ ν + σ λ ν ) ν T X , s:TX\ni\partial_{\lambda}\to\partial_{\lambda}\rfloor(\theta+\sigma)=(\delta_{% \lambda}^{\nu}+\sigma_{\lambda}^{\nu})\partial_{\nu}\in TX,
  81. θ = d x μ μ \theta=dx^{\mu}\otimes\partial_{\mu}
  82. ( g , Γ ) (g,\Gamma)
  83. g ~ μ ν = s α μ s β ν g α β \widetilde{g}^{\mu\nu}=s^{\mu}_{\alpha}s^{\nu}_{\beta}g^{\alpha\beta}
  84. σ \sigma
  85. L ( σ ) = 1 2 [ a 1 T μ T α ν μ + ν α a 2 T μ ν α T μ ν α + a 3 T μ ν α T ν μ α + a 4 ϵ μ ν α β T γ T β ν α μ γ - μ σ μ σ ν ν + μ λ σ μ σ ν μ ] ν - g L_{(\sigma)}=\frac{1}{2}[a_{1}T^{\mu}{}_{\nu\mu}T_{\alpha}{}^{\nu\alpha}+a_{2}% T_{\mu\nu\alpha}T^{\mu\nu\alpha}+a_{3}T_{\mu\nu\alpha}T^{\nu\mu\alpha}+a_{4}% \epsilon^{\mu\nu\alpha\beta}T^{\gamma}{}_{\mu\gamma}T_{\beta\nu\alpha}-\mu% \sigma^{\mu}{}_{\nu}\sigma^{\nu}{}_{\mu}+\lambda\sigma^{\mu}{}_{\mu}\sigma^{% \nu}{}_{\nu}]\sqrt{-g}
  86. ϵ μ ν α β \epsilon^{\mu\nu\alpha\beta}
  87. T α = ν μ D ν σ α - μ D μ σ α ν T^{\alpha}{}_{\nu\mu}=D_{\nu}\sigma^{\alpha}{}_{\mu}-D_{\mu}\sigma^{\alpha}{}_% {\nu}
  88. Γ \Gamma
  89. σ \sigma

Affine_monoid.html

  1. M M
  2. m 1 , , m n M m_{1},\dots,m_{n}\in M
  3. M = m 1 + + + m n + M=m_{1}\mathbb{Z_{+}}+\dots+m_{n}\mathbb{Z_{+}}
  4. x + y = x + z x+y=x+z
  5. y = z y=z
  6. x , y , z M x,y,z\in M
  7. + +
  8. M M
  9. M M
  10. n x = n y nx=ny
  11. x = y x=y
  12. n n\in\mathbb{N}
  13. x , y M x,y\in M
  14. N N
  15. M M
  16. M M
  17. M M
  18. \mathbb{Z}
  19. \mathbb{Z}
  20. { ( x , y ) × y > 0 } { ( 0 , 0 ) } \{(x,y)\in\mathbb{Z}\times\mathbb{Z}\mid y>0\}\cup\{(0,0)\}
  21. × \mathbb{Z}\times\mathbb{Z}
  22. M M
  23. g p ( M ) gp(M)
  24. M M
  25. g p ( M ) gp(M)
  26. x - y x-y
  27. x - y = u - v x-y=u-v
  28. x + v + z = u + y + z x+v+z=u+y+z
  29. z M z\in M
  30. ( x - y ) + ( u - v ) = ( x + u ) - ( y + v ) (x-y)+(u-v)=(x+u)-(y+v)
  31. M M
  32. g p ( M ) gp(M)
  33. M M
  34. r \mathbb{Z}^{r}
  35. g p ( M ) M gp(M)\cong\mathbb{Z}M
  36. M \mathbb{Z}M
  37. r \mathbb{Z}^{r}
  38. M M
  39. ι : M g p ( M ) \iota:M\to gp(M)
  40. ι ( x ) = x + 0 \iota(x)=x+0
  41. φ : M G \varphi:M\to G
  42. G G
  43. ψ : g p ( M ) G \psi:gp(M)\to G
  44. φ = ψ ι \varphi=\psi\circ\iota
  45. ι \iota
  46. M M
  47. N N
  48. M ^ N = { x N m x M , m } \hat{M}_{N}=\{x\in N\mid mx\in M,m\in\mathbb{N}\}
  49. M M
  50. N N
  51. M = M N ^ M=\hat{M_{N}}
  52. M M
  53. M M
  54. M M
  55. g p ( M ) gp(M)
  56. M M
  57. M M
  58. M M
  59. M M
  60. + M \mathbb{R}_{+}M
  61. M = r + M M=\mathbb{Z}^{r}\cap\mathbb{R}_{+}M
  62. M M
  63. R R
  64. R [ M ] R[M]
  65. R R
  66. M M
  67. f R [ M ] f\in R[M]
  68. f = i = 1 n f i x i f=\sum_{i=1}^{n}f_{i}x_{i}
  69. f i R , x i M f_{i}\in R,x_{i}\in M
  70. n n\in\mathbb{N}
  71. R [ M ] R[M]
  72. M M
  73. R R
  74. C C
  75. n \mathbb{R}^{n}
  76. L L
  77. n \mathbb{Q}^{n}
  78. C L C\cap L
  79. M M
  80. n \mathbb{R}^{n}
  81. + M \mathbb{R}_{+}M
  82. M M
  83. M M
  84. n \mathbb{R}^{n}
  85. C C
  86. g p ( M ) gp(M)
  87. M C M\cap C
  88. P P
  89. n \mathbb{R}^{n}
  90. C C
  91. P P
  92. L L
  93. n \mathbb{Q}^{n}
  94. P L P\cap L
  95. C L C\cap L

Affinity_propagation.html

  1. k k
  2. k k
  3. k k
  4. s s
  5. 𝐑 \mathbf{R}
  6. r ( i , k ) r(i,k)
  7. 𝐀 \mathbf{A}
  8. a ( i , k ) a(i,k)
  9. r ( i , k ) s ( i , k ) - max k k { a ( i , k ) + s ( i , k ) } r(i,k)\leftarrow s(i,k)-\max_{k^{\prime}\neq k}\left\{a(i,k^{\prime})+s(i,k^{% \prime})\right\}
  10. a ( i , k ) min ( 0 , r ( k , k ) + i { i , k } max ( 0 , r ( i , k ) ) ) a(i,k)\leftarrow\min\left(0,r(k,k)+\sum_{i^{\prime}\not\in\{i,k\}}\max(0,r(i^{% \prime},k))\right)
  11. i k i\neq k
  12. a ( k , k ) i k max ( 0 , r ( i , k ) ) a(k,k)\leftarrow\sum_{i^{\prime}\neq k}\max(0,r(i^{\prime},k))
  13. k k
  14. k k

Albert_van_den_Berg_(physicist).html

  1. μ \mu

Aldehyde_ferredoxin_oxidoreductase.html

  1. \rightleftharpoons

Alexandros_Chapsiadis.html

  1. B 1 B_{1}
  2. B 2 B_{2}
  3. E E
  4. L ^ \hat{L}
  5. L 2 L^{2}
  6. N 2 N\geq 2

Alfvén's_Theorem.html

  1. d Φ B d t = S B t . d S + C B . v × d l , {d\Phi_{B}\over dt}=\int_{S}{\partial\vec{B}\over\partial t}.d\vec{S}+\oint_{C% }\vec{B.}\vec{v}\times d\vec{l},
  2. B \vec{B}
  3. v \vec{v}
  4. S \vec{S}
  5. C C
  6. d l d\vec{l}
  7. B t = × ( v × B ) , {\partial\vec{B}\over\partial t}=\vec{\nabla}\times(\vec{v}\times\vec{B}),
  8. d Φ B d t = S × ( v × B ) . d S + C B . v × d l . {d\Phi_{B}\over dt}=\int_{S}\vec{\nabla}\times(\vec{v}\times\vec{B}).d\vec{S}+% \oint_{C}\vec{B.}\vec{v}\times d\vec{l}.
  9. ( A × B ) . C = - B . ( A × C ) (\vec{A}\times\vec{B}).\vec{C}=-\vec{B}.(\vec{A}\times\vec{C})
  10. S B . d S = c o n s t . \int_{S}\vec{B.}d\vec{S}=const.
  11. C C
  12. × ( w × B ) = η 2 B + × ( v × B ) , \vec{\nabla}\times(\vec{w}\times\vec{B})=\eta{\vec{\nabla}}^{2}\vec{B}+\vec{% \nabla}\times(\vec{v}\times\vec{B}),
  13. v v
  14. w w

Algebra_over_an_operad.html

  1. f : O O f:O\to O^{\prime}
  2. n \mathcal{E}_{n}

Algebraic_vector_bundle.html

  1. S U ( 2 ) SU(2)
  2. S 4 S^{4}

Alignment-free_sequence_analysis.html

  1. L ( A , B ) L(A,B)
  2. L ( A , B ) L(A,B)
  3. L ( A , B ) L(A,B)
  4. L ( A , B ) / log ( m ) L(A,B)/\log(m)
  5. d ( A , A ) d(A,A)
  6. d ( A , B ) = [ log ( m ) / L ( A , B ) ] - [ log ( n ) / L ( A , A ] d(A,B)=[\log(m)/L(A,B)]-[\log(n)/L(A,A]
  7. d ( A , B ) d(A,B)
  8. d s ( A , B ) = d s ( B , A ) = ( d ( A , B ) + d ( B , A ) ) / 2 d_{s}(A,B)=d_{s}(B,A)=(d(A,B)+d(B,A))/2
  9. T i j ( K ) = l = 1 K P i j ( l ) . log 2 ( P i j ( l ) P i P j ) T_{ij}(K)=\sum_{l=1}^{K}P_{ij}(l).\log_{2}\left(\frac{P_{ij}(l)}{P_{i}P_{j}}\right)
  10. P i P_{i}
  11. P j P_{j}
  12. P i j ( l ) P_{ij}(l)
  13. I C l = - 2 i P i log 2 P i + i j P i j ( l ) log 2 P i j ( l ) IC_{l}=-2\sum_{i}P_{i}\log_{2}P_{i}+\sum_{ij}P_{ij}(l)\log_{2}P_{ij}(l)
  14. P I C i j ( l ) = ( P i j ( l ) - P i P j ( l ) ) 2 PIC_{ij}(l)=(P_{ij}(l)-P_{i}P_{j}(l))^{2}
  15. V = I C l P I C i j ( l ) V={IC_{l}\over PIC_{ij}(l)}
  16. l \isin { l 0 , l 0 + 1 , , l 0 + n } l\isin\left\{l_{0},l_{0}+1,...,l_{0}+n\right\}

Allocation_(oil_and_gas).html

  1. Q n = Q i * M F * C T L * S F * S W Q_{n}=Q_{i}*MF*CTL*SF*SW
  2. Q i Q_{i}
  3. Q k = C k Q T k = 1 N C k Q_{k}=C_{k}{Q_{T}\over\sum_{k=1}^{N}C_{k}}
  4. Q T Q_{T}
  5. C k C_{k}
  6. Q k Q_{k}
  7. F F = Q n k = 1 N T H k FF={Q_{n}\over\sum_{k=1}^{N}TH_{k}}
  8. T H k TH_{k}
  9. Q n Q_{n}
  10. W A k = T H k * F F WA_{k}=TH_{k}*FF

Almost_holomorphic_modular_form.html

  1. f ( ( a τ + b ) / ( c τ + d ) ) = ( c τ + d ) k f ( τ ) f((a\tau+b)/(c\tau+d))=(c\tau+d)^{k}f(\tau)
  2. E 2 ( τ ) - 3 / π ( τ ) , E 4 ( τ ) , E 6 ( τ ) E_{2}(\tau)-3/\pi\Im(\tau),E_{4}(\tau),E_{6}(\tau)
  3. E 2 ( τ ) , E 4 ( τ ) , E 6 ( τ ) E_{2}(\tau),E_{4}(\tau),E_{6}(\tau)
  4. d E 2 d τ = E 2 2 - E 4 12 d E 4 d τ = E 2 E 4 - E 6 3 d E 6 d τ = E 2 E 6 - E 4 2 2 \begin{aligned}\displaystyle\frac{dE_{2}}{d\tau}&\displaystyle=\frac{E_{2}^{2}% -E_{4}}{12}\\ \displaystyle\frac{dE_{4}}{d\tau}&\displaystyle=\frac{E_{2}E_{4}-E_{6}}{3}\\ \displaystyle\frac{dE_{6}}{d\tau}&\displaystyle=\frac{E_{2}E_{6}-E_{4}^{2}}{2}% \end{aligned}

Almost_linear_hash_function.html

  1. h ( x + y ) = h ( x ) + h ( y ) h(x+y)=h(x)+h(y)
  2. h ( x + y ) = h ( x ) + h ( y ) + 1 h(x+y)=h(x)+h(y)+1
  3. h ( x ) = a x mod m h(x)=ax\bmod m
  4. h ( x ) = ( a x mod M ) ÷ ( M / m ) h(x)=(ax\mod M)\div(M/m)
  5. ÷ \div
  6. h ( x ) = ( x mod 64 ) ÷ 4 h(x)=(x\bmod 64)\div 4
  7. x = 4 ( x ÷ 4 ) + x mod 4 x=4(x\div 4)+x\bmod 4
  8. y = 4 ( y ÷ 4 ) + y mod 4 y=4(y\div 4)+y\bmod 4
  9. x + y = 4 ( x ÷ 4 + y ÷ 4 ) + ( x mod 4 ) + ( y mod 4 ) x+y=4(x\div 4+y\div 4)+(x\bmod 4)+(y\bmod 4)
  10. ( x mod 4 ) + ( y mod 4 ) < 4 (x\bmod 4)+(y\bmod 4)<4
  11. h ( x + y ) = x ÷ 4 + y ÷ 4 = h ( x ) + h ( y ) h(x+y)=x\div 4+y\div 4=h(x)+h(y)
  12. 4 ( x mod 4 ) + ( y mod 4 ) < 8 4\leq(x\bmod 4)+(y\bmod 4)<8
  13. h ( x + y ) = x ÷ 4 + y ÷ 4 + 1 = h ( x ) + h ( y ) + 1 h(x+y)=x\div 4+y\div 4+1=h(x)+h(y)+1

Alpha-amyrin_synthase.html

  1. \rightleftharpoons

Alpha-bisabolene_synthase.html

  1. \rightleftharpoons

Alpha-copaene_synthase.html

  1. \rightleftharpoons

Alpha-eudesmol_synthase.html

  1. \rightleftharpoons

Alpha-farnesene_synthase.html

  1. \rightleftharpoons

Alpha-guaiene_synthase.html

  1. \rightleftharpoons

Alpha-gurjunene_synthase.html

  1. \rightleftharpoons

Alpha-humulene_synthase.html

  1. \rightleftharpoons

Alpha-isocomene_synthase.html

  1. \rightleftharpoons

Alpha-longipinene_synthase.html

  1. \rightleftharpoons

Alpha-muurolene_synthase.html

  1. \rightleftharpoons

Alpha-santalene_synthase.html

  1. \rightleftharpoons

Alpha-seco-amyrin_synthase.html

  1. \rightleftharpoons

Alpha-terpinene_synthase.html

  1. \rightleftharpoons

Alternated_hexagonal_tiling_honeycomb.html

  1. P ¯ 3 {\bar{P}}_{3}
  2. V ¯ 3 {\bar{V}}_{3}
  3. Y ¯ 3 {\bar{Y}}_{3}
  4. Z ¯ 3 {\bar{Z}}_{3}
  5. V P ¯ 3 {\bar{VP}}_{3}
  6. P P ¯ 3 {\bar{PP}}_{3}
  7. P ¯ 3 {\bar{P}}_{3}
  8. P ¯ 3 {\bar{P}}_{3}
  9. P ¯ 3 {\bar{P}}_{3}

Ambisonic_data_exchange_formats.html

  1. Y m ( θ , ϕ ) = N | m | P | m | ( sin ( ϕ ) ) { sin ( | m | θ ) if m < 0 1 if m = 0 cos ( | m | θ ) if m > 0 Y_{\ell}^{m}(\theta,\phi)=N_{\ell}^{|m|}P_{\ell}^{|m|}(\sin(\phi))\cdot\begin{% cases}{\sin(|m|\theta)}&\mbox{if }~{}m<0\\ {1}&\mbox{if }~{}m=0\\ {\cos(|m|\theta)}&\mbox{if }~{}m>0\end{cases}
  2. Y Y
  3. \ell
  4. m m
  5. - m + -\ell\leq m\leq+\ell
  6. N N
  7. P m P_{\ell}^{m}
  8. \ell
  9. m m
  10. θ \theta
  11. ϕ \phi
  12. \ell
  13. m m
  14. S S
  15. ( θ , ϕ ) (\theta,\phi)
  16. B m B_{\ell}^{m}
  17. B m = Y m ( θ , ϕ ) S B_{\ell}^{m}=Y_{\ell}^{m}(\theta,\phi)\cdot S
  18. W X Y Z WXYZ
  19. R S T U V RSTUV
  20. K L M N O P KLMNOP
  21. Y m n σ Y_{mn}^{\sigma}
  22. Y | m | s g n ( m ) Y_{\ell|m|}^{sgn(m)}
  23. A C N = 2 + + m ACN=\ell^{2}+\ell+m
  24. W W
  25. N , m SN3D = 2 - δ m 4 π ( - | m | ) ! ( + | m | ) ! , δ m { 1 if m = 0 0 if m 0 N_{\ell,m}\text{SN3D}=\sqrt{{2-\delta_{m}\over 4\pi}{(\ell-|m|)!\over(\ell+|m|% )!}},\delta_{m}\begin{cases}1&\mbox{if }~{}m=0\\ 0&\mbox{if }~{}m\neq 0\end{cases}
  26. N , m N3D = N , m SN3D 2 + 1 N_{\ell,m}\text{N3D}=N_{\ell,m}\text{SN3D}\sqrt{2\ell+1}
  27. ( - 1 ) m (-1)^{m}
  28. B B
  29. Y Y
  30. B B
  31. Y m ( Y A C N ) Y_{\ell}^{m}(\equiv Y_{ACN})
  32. \ell
  33. m m
  34. W W
  35. 1 1
  36. 1 1
  37. 1 2 1\over{\sqrt{2}}
  38. Y Y
  39. 3 sin ( θ ) cos ( ϕ ) \sqrt{3}\sin(\theta)\cos(\phi)
  40. 1 3 1\over{\sqrt{3}}
  41. 1 3 1\over{\sqrt{3}}
  42. Z Z
  43. 3 sin ( ϕ ) \sqrt{3}\sin(\phi)
  44. 1 3 1\over{\sqrt{3}}
  45. 1 3 1\over{\sqrt{3}}
  46. X X
  47. 3 cos ( θ ) cos ( ϕ ) \sqrt{3}\cos(\theta)\cos(\phi)
  48. 1 3 1\over{\sqrt{3}}
  49. 1 3 1\over{\sqrt{3}}
  50. V V
  51. 15 2 sin ( 2 θ ) cos 2 ( ϕ ) \frac{\sqrt{15}}{2}\sin(2\theta)\cos^{2}(\phi)
  52. 1 5 1\over{\sqrt{5}}
  53. 2 15 2\over\sqrt{15}
  54. T T
  55. 15 2 sin ( θ ) sin ( 2 ϕ ) \frac{\sqrt{15}}{2}\sin(\theta)\sin(2\phi)
  56. 1 5 1\over{\sqrt{5}}
  57. 2 15 2\over\sqrt{15}
  58. R R
  59. 5 2 ( 3 sin 2 ( ϕ ) - 1 ) \frac{\sqrt{5}}{2}(3\sin^{2}(\phi)-1)
  60. 1 5 1\over{\sqrt{5}}
  61. 1 5 1\over{\sqrt{5}}
  62. S S
  63. 15 2 cos ( θ ) sin ( 2 ϕ ) \frac{\sqrt{15}}{2}\cos(\theta)\sin(2\phi)
  64. 1 5 1\over{\sqrt{5}}
  65. 2 15 2\over\sqrt{15}
  66. U U
  67. 15 2 cos ( 2 θ ) cos 2 ( ϕ ) \frac{\sqrt{15}}{2}\cos(2\theta)\cos^{2}(\phi)
  68. 1 5 1\over{\sqrt{5}}
  69. 2 15 2\over\sqrt{15}
  70. Q Q
  71. 35 8 sin ( 3 θ ) cos 3 ( ϕ ) \sqrt{35\over{8}}\sin(3\theta)\cos^{3}(\phi)
  72. 1 7 1\over\sqrt{7}
  73. 8 35 \sqrt{8\over 35}
  74. O O
  75. 105 2 sin ( 2 θ ) sin ( ϕ ) cos 2 ( ϕ ) \frac{\sqrt{105}}{2}\sin(2\theta)\sin(\phi)\cos^{2}(\phi)
  76. 1 7 1\over\sqrt{7}
  77. 3 35 3\over\sqrt{35}
  78. M M
  79. 21 8 sin ( θ ) ( 5 sin 2 ( ϕ ) - 1 ) cos ( ϕ ) \sqrt{21\over{8}}\sin(\theta)(5\sin^{2}(\phi)-1)\cos(\phi)
  80. 1 7 1\over\sqrt{7}
  81. 45 224 \sqrt{45\over 224}
  82. K K
  83. 7 2 sin ( ϕ ) ( 5 sin 2 ( ϕ ) - 3 ) \frac{\sqrt{7}}{2}\sin(\phi)(5\sin^{2}(\phi)-3)
  84. 1 7 1\over\sqrt{7}
  85. 1 7 1\over\sqrt{7}
  86. L L
  87. 21 8 cos ( θ ) ( 5 sin 2 ( ϕ ) - 1 ) cos ( ϕ ) \sqrt{21\over{8}}\cos(\theta)(5\sin^{2}(\phi)-1)\cos(\phi)
  88. 1 7 1\over\sqrt{7}
  89. 45 224 \sqrt{45\over 224}
  90. N N
  91. 105 2 cos ( 2 θ ) sin ( ϕ ) cos 2 ( ϕ ) \frac{\sqrt{105}}{2}\cos(2\theta)\sin(\phi)\cos^{2}(\phi)
  92. 1 7 1\over\sqrt{7}
  93. 3 35 3\over\sqrt{35}
  94. P P
  95. 35 8 cos ( 3 θ ) cos 3 ( ϕ ) \sqrt{35\over{8}}\cos(3\theta)\cos^{3}(\phi)
  96. 1 7 1\over\sqrt{7}
  97. 8 35 \sqrt{8\over 35}
  98. 3 8 35 sin ( 4 θ ) cos 4 ( ϕ ) \frac{3}{8}\sqrt{35}\sin(4\theta)\cos^{4}(\phi)
  99. 1 3 \frac{1}{3}
  100. 3 2 35 2 sin ( 3 θ ) sin ( ϕ ) cos 3 ( ϕ ) \frac{3}{2}\sqrt{\frac{35}{2}}\sin(3\theta)\sin(\phi)\cos^{3}(\phi)
  101. 1 3 \frac{1}{3}
  102. 3 4 5 sin ( 2 θ ) ( 7 sin 2 ( ϕ ) - 1 ) cos 2 ( ϕ ) \frac{3}{4}\sqrt{5}\sin(2\theta)(7\sin^{2}(\phi)-1)\cos^{2}(\phi)
  103. 1 3 \frac{1}{3}
  104. 3 4 5 2 sin ( θ ) sin ( 2 ϕ ) ( 7 sin 2 ( ϕ ) - 3 ) \frac{3}{4}\sqrt{\frac{5}{2}}\sin(\theta)\sin(2\phi)(7\sin^{2}(\phi)-3)
  105. 1 3 \frac{1}{3}
  106. 3 8 ( 35 sin 4 ( ϕ ) - 30 sin 2 ( ϕ ) + 3 ) \frac{3}{8}(35\sin^{4}(\phi)-30\sin^{2}(\phi)+3)
  107. 1 3 \frac{1}{3}
  108. 3 4 5 2 cos ( θ ) sin ( 2 ϕ ) ( 7 sin 2 ( ϕ ) - 3 ) \frac{3}{4}\sqrt{\frac{5}{2}}\cos(\theta)\sin(2\phi)(7\sin^{2}(\phi)-3)
  109. 1 3 \frac{1}{3}
  110. 3 4 5 cos ( 2 θ ) ( 7 sin 2 ( ϕ ) - 1 ) cos 2 ( ϕ ) \frac{3}{4}\sqrt{5}\cos(2\theta)(7\sin^{2}(\phi)-1)\cos^{2}(\phi)
  111. 1 3 \frac{1}{3}
  112. 3 2 35 2 cos ( 3 θ ) sin ( ϕ ) cos 3 ( ϕ ) \frac{3}{2}\sqrt{\frac{35}{2}}\cos(3\theta)\sin(\phi)\cos^{3}(\phi)
  113. 1 3 \frac{1}{3}
  114. 3 8 35 cos ( 4 θ ) cos 4 ( ϕ ) \frac{3}{8}\sqrt{35}\cos(4\theta)\cos^{4}(\phi)
  115. 1 3 \frac{1}{3}
  116. W W
  117. W X WX
  118. W X Y WXY
  119. W X Y Z WXYZ
  120. W X Y U V WXYUV
  121. W X Y Z U V WXYZUV
  122. W X Y U V P Q WXYUVPQ
  123. W X Y Z V U P Q WXYZVUPQ
  124. W X Y Z R S T U V WXYZRSTUV
  125. W X Y Z R S T U V P Q WXYZRSTUVPQ
  126. W X Y Z R S T U V K L M N O P Q WXYZRSTUVKLMNOPQ
  127. u \vec{u}
  128. 2 m + 1 \sqrt{2m+1}

Ambisonic_reproduction_systems.html

  1. r E \vec{r_{E}}
  2. ϕ \phi
  3. 2 + 1 2\ell+1
  4. r V \vec{r_{V}}
  5. r E \vec{r_{E}}

Analysis_of_parallel_algorithms.html

  1. p p
  2. p p
  3. p p
  4. p p
  5. p p
  6. Ω ( n ) Ω(n)
  7. n n
  8. p p
  9. N N
  10. T p T N + T 1 - T N p , T_{p}\leq T_{N}+\frac{T_{1}-T_{N}}{p},
  11. T p = O ( T N + T 1 p ) . T_{p}=O\left(T_{N}+\frac{T_{1}}{p}\right).
  12. T 1 p T p < T 1 p + T \frac{T_{1}}{p}\leq T_{p}<\frac{T_{1}}{p}+T_{\infty}

Analytically_irreducible_ring.html

  1. w = m > 0 a m x m w=\sum_{m>0}a_{m}x^{m}
  2. z 1 = ( y + w ) 2 z_{1}=(y+w)^{2}
  3. z n + 1 = ( z 1 - ( y + 0 < m < n a m x m ) 2 ) / x n z_{n+1}=(z_{1}-(y+\sum_{0<m<n}a_{m}x^{m})^{2})/x^{n}

Analytically_unramified_ring.html

  1. n j n_{j}\to\infty
  2. J j ¯ J n j \overline{J^{j}}\subset J^{n_{j}}

Andreotti–Norguet_formula.html

  1. 0
  2. n > 1 n> 1
  3. n > 1 n> 1
  4. ζ , z [ u S u p , u n ] ζ,z∈ ℂ[u^{\prime}Sup^{\prime},u^{\prime}n^{\prime}]
  5. D [ u S u p , u n ] D⊂ ℂ[u^{\prime}Sup^{\prime},u^{\prime}n^{\prime}]
  6. D ¯ \overline{D}
  7. A ( D ) A(D)
  8. D D
  9. D ∂D
  10. α α
  11. f A ( D ) f∈A(D)
  12. α f = | α | f z 1 α 1 z n α n . \partial^{\alpha}f=\frac{\partial^{|\alpha|}f}{\partial z_{1}^{\alpha_{1}}% \cdots\partial z_{n}^{\alpha_{n}}}.
  13. α α
  14. ω [ u S u b , u 3 b 1 ] ( ζ , z ) ω[u^{\prime}Sub^{\prime},u^{\prime}\u{0}3b1^{\prime}](ζ,z)
  15. ζ ζ
  16. ( n , n 1 ) (n,n− 1)
  17. ω α ( ζ , z ) = ( n - 1 ) ! α 1 ! α n ! ( 2 π i ) n j = 1 n ( - 1 ) j - 1 ( ζ ¯ j - z ¯ j ) d ζ ¯ α + I [ j ] and d ζ ( | z 1 - ζ 1 | 2 ( α 1 + 1 ) + + | z n - ζ n | 2 ( α n + 1 ) ) n , \omega_{\alpha}(\zeta,z)=\frac{(n-1)!\alpha_{1}!\cdots\alpha_{n}!}{(2\pi i)^{n% }}\sum_{j=1}^{n}\frac{(-1)^{j-1}(\bar{\zeta}_{j}-\overline{z}_{j})\,d\bar{% \zeta}^{\alpha+I}[j]\and d\zeta}{\left(|z_{1}-\zeta_{1}|^{2(\alpha_{1}+1)}+% \cdots+|z_{n}-\zeta_{n}|^{2(\alpha_{n}+1)}\right)^{n}},
  18. d ζ ¯ α + I [ j ] = d ζ ¯ 1 α 1 + 1 and and d ζ ¯ j - 1 α j + 1 + 1 and d ζ ¯ j + 1 α j - 1 + 1 and and d ζ ¯ n α n + 1 d\bar{\zeta}^{\alpha+I}[j]=d\bar{\zeta}_{1}^{\alpha_{1}+1}\and\cdots\and d\bar% {\zeta}_{j-1}^{\alpha_{j+1}+1}\and d\bar{\zeta}_{j+1}^{\alpha_{j-1}+1}\and% \cdots\and d\bar{\zeta}_{n}^{\alpha_{n}+1}
  19. f A ( D ) f∈A(D)
  20. z D z∈D
  21. α α
  22. α f ( z ) = D f ( ζ ) ω α ( ζ , z ) . \partial^{\alpha}f(z)=\int_{\partial D}f(\zeta)\omega_{\alpha}(\zeta,z).

Angles_between_flats.html

  1. F F
  2. G G
  3. k k
  4. l l
  5. n n
  6. E n E^{n}
  7. F F
  8. G G
  9. F F
  10. G G
  11. G G
  12. G G
  13. F F
  14. F F
  15. G G
  16. x 1 , , x ρ , x_{1},\dots,x_{\rho},
  17. y 1 , , y σ , y_{1},\dots,y_{\sigma},
  18. z 1 , , z τ , z_{1},\dots,z_{\tau},
  19. u 1 , , u υ , u_{1},\dots,u_{\upsilon},
  20. v 1 , , x α , v_{1},\dots,x_{\alpha},
  21. w 1 , , w α w_{1},\dots,w_{\alpha}
  22. E n E^{n}
  23. F F
  24. G G
  25. x 1 = 0 , , x ρ = 0 , x_{1}=0,\dots,x_{\rho}=0,
  26. u 1 = 0 , , u υ = 0 , u_{1}=0,\dots,u_{\upsilon}=0,
  27. v 1 = 0 , , v α = 0 v_{1}=0,\dots,v_{\alpha}=0
  28. x 1 = 0 , , x ρ = 0 , x_{1}=0,\dots,x_{\rho}=0,
  29. z 1 = 0 , , z τ = 0 , z_{1}=0,\dots,z_{\tau}=0,
  30. v 1 cos θ 1 + w 1 sin θ 1 = 0 , , v α cos θ α + w α sin θ α = 0 v_{1}\cos\theta_{1}+w_{1}\sin\theta_{1}=0,\dots,v_{\alpha}\cos\theta_{\alpha}+% w_{\alpha}\sin\theta_{\alpha}=0
  31. 0 < θ i < π / 2 , i = 1 , , α 0<\theta_{i}<\pi/2,i=1,\dots,\alpha
  32. θ i \theta_{i}
  33. F F
  34. G G
  35. ρ , σ , τ , υ , α \rho,\sigma,\tau,\upsilon,\alpha
  36. ρ + σ + τ + υ + 2 α = n , \rho+\sigma+\tau+\upsilon+2\alpha=n,
  37. σ + τ + α = k , \sigma+\tau+\alpha=k,
  38. σ + υ + α = l . \sigma+\upsilon+\alpha=l.
  39. n , k n,k
  40. l l
  41. α \alpha
  42. θ i \theta_{i}
  43. σ \sigma
  44. y i y_{i}
  45. F F
  46. G G
  47. σ \sigma
  48. F G F\cap G
  49. θ i \theta_{i}
  50. σ \sigma
  51. 0
  52. F G F\cap G
  53. E n E^{n}
  54. n n
  55. n \mathbb{C}^{n}
  56. n \mathbb{C}^{n}
  57. F F
  58. G G
  59. n n
  60. F F
  61. G G
  62. ξ \xi
  63. ξ ^ \hat{\xi}
  64. ξ \xi
  65. y ^ 1 , , y ^ σ , \hat{y}_{1},\dots,\hat{y}_{\sigma},
  66. w ^ 1 , , w ^ α , \hat{w}_{1},\dots,\hat{w}_{\alpha},
  67. z ^ 1 , , z ^ τ \hat{z}_{1},\dots,\hat{z}_{\tau}
  68. F F
  69. y ^ 1 , , y ^ σ , \hat{y}_{1},\dots,\hat{y}_{\sigma},
  70. w ^ 1 , , w ^ α , \hat{w}^{\prime}_{1},\dots,\hat{w}^{\prime}_{\alpha},
  71. u ^ 1 , , u ^ υ \hat{u}_{1},\dots,\hat{u}_{\upsilon}
  72. G G
  73. w ^ i = w ^ i cos θ i - v ^ i sin θ i , i = 1 , , α . \hat{w}^{\prime}_{i}=\hat{w}_{i}\cos\theta_{i}-\hat{v}_{i}\sin\theta_{i},\quad i% =1,\dots,\alpha.
  74. a i , i = 1 , , k a_{i},i=1,\dots,k
  75. F F
  76. b i , i = 1 , , l b_{i},i=1,\dots,l
  77. G G
  78. a i , b j \langle a_{i},b_{j}\rangle
  79. i i
  80. j j
  81. y ^ i , y ^ i = 1 , i = 1 , , σ , \langle\hat{y}_{i},\hat{y}_{i}\rangle=1,\quad i=1,\dots,\sigma,
  82. w ^ i , w ^ i = cos θ i , i = 1 , , α . \langle\hat{w}_{i},\hat{w}^{\prime}_{i}\rangle=\cos\theta_{i},\quad i=1,\dots,\alpha.
  83. a i , b j \langle a_{i},b_{j}\rangle
  84. ( a i , i = 1 , , k ) (a^{\prime}_{i},i=1,\dots,k)
  85. ( b i , i = 1 , , l ) (b^{\prime}_{i},i=1,\dots,l)
  86. F F
  87. G G
  88. ( a i ) (a^{\prime}_{i})
  89. ( a i ) (a_{i})
  90. ( b i ) (b^{\prime}_{i})
  91. ( b i ) (b_{i})
  92. a i , b j \langle a^{\prime}_{i},b^{\prime}_{j}\rangle
  93. a i , b i \langle a_{i},b_{i}\rangle
  94. y ^ i \hat{y}_{i}
  95. w ^ i \hat{w}_{i}
  96. w ^ i \hat{w}^{\prime}_{i}
  97. v ^ i \hat{v}_{i}
  98. w ^ i \hat{w}_{i}
  99. θ i \theta_{i}
  100. w ^ i \hat{w}^{\prime}_{i}
  101. v ^ i \hat{v}_{i}
  102. 1 1
  103. cos 0 \cos\,0
  104. 0
  105. F G F\cap G
  106. 0
  107. cos π / 2 \cos\pi/2
  108. F G F\cap G^{\bot}
  109. F G F^{\bot}\cap G
  110. \bot
  111. 0
  112. π / 2 \pi/2
  113. V V
  114. 𝒰 , 𝒲 \mathcal{U},\mathcal{W}
  115. dim ( 𝒰 ) = k dim ( 𝒲 ) := l \operatorname{dim}(\mathcal{U})=k\leq\operatorname{dim}(\mathcal{W}):=l
  116. k k
  117. 0 θ 1 θ 2 θ k π / 2 0\leq\theta_{1}\leq\theta_{2}\leq\cdots\leq\theta_{k}\leq\pi/2
  118. θ 1 := min { arccos ( | u , w | u w ) | u 𝒰 , w 𝒲 } = ( u 1 , w 1 ) , \theta_{1}:=\min\left\{\arccos\left(\left.\frac{|\langle u,w\rangle|}{\|u\|\|w% \|}\right)\right|u\in\mathcal{U},w\in\mathcal{W}\right\}=\angle(u_{1},w_{1}),
  119. , \langle\cdot,\cdot\rangle
  120. \|\cdot\|
  121. u 1 u_{1}
  122. w 1 w_{1}
  123. θ i := min { arccos ( | u , w | u w ) | u 𝒰 , w 𝒲 , u u j , w w j j { 1 , , i - 1 } } . \theta_{i}:=\min\left\{\left.\arccos\left(\frac{|\langle u,w\rangle|}{\|u\|\|w% \|}\right)\right|u\in\mathcal{U},~{}w\in\mathcal{W},~{}u\perp u_{j},~{}w\perp w% _{j}\quad\forall j\in\{1,\ldots,i-1\}\right\}.
  124. ( θ 1 , , θ k ) (\theta_{1},\ldots,\theta_{k})
  125. 𝒰 \mathcal{U}
  126. 𝒲 \mathcal{W}
  127. 𝒰 \mathcal{U}
  128. 𝒲 \mathcal{W}
  129. θ 1 = θ 2 = 0 \theta_{1}=\theta_{2}=0
  130. θ 1 = 0 \theta_{1}=0
  131. u 1 u_{1}
  132. w 1 w_{1}
  133. 𝒰 𝒲 \mathcal{U}\cap\mathcal{W}
  134. θ 2 > 0 \theta_{2}>0
  135. 𝒰 \mathcal{U}
  136. 𝒲 \mathcal{W}
  137. 𝒰 𝒲 \mathcal{U}\cap\mathcal{W}
  138. θ 2 > 0 \theta_{2}>0
  139. 𝒰 \mathcal{U}
  140. u 1 = ( 1 , 0 , 0 , 0 ) u_{1}=(1,0,0,0)
  141. u 2 = ( 0 , 1 , 0 , 0 ) u_{2}=(0,1,0,0)
  142. 𝒲 \mathcal{W}
  143. w 1 = ( 1 , 0 , 0 , a ) / 1 + a 2 w_{1}=(1,0,0,a)/\sqrt{1+a^{2}}
  144. w 2 = ( 0 , 1 , b , 0 ) / 1 + b 2 w_{2}=(0,1,b,0)/\sqrt{1+b^{2}}
  145. a a
  146. b b
  147. | a | < | b | |a|<|b|
  148. u 1 u_{1}
  149. w 1 w_{1}
  150. θ 1 \theta_{1}
  151. cos ( θ 1 ) = 1 / 1 + a 2 \cos(\theta_{1})=1/\sqrt{1+a^{2}}
  152. u 2 u_{2}
  153. w 2 w_{2}
  154. θ 2 \theta_{2}
  155. cos ( θ 2 ) = 1 / 1 + b 2 \cos(\theta_{2})=1/\sqrt{1+b^{2}}
  156. k k
  157. θ 1 , , θ k \theta_{1},\ldots,\theta_{k}
  158. 2 k 2k
  159. 𝒰 \mathcal{U}
  160. ( e 1 , , e k ) (e_{1},\ldots,e_{k})
  161. ( e 1 , , e n ) (e_{1},\ldots,e_{n})
  162. n 2 k n\geq 2k
  163. 𝒲 \mathcal{W}
  164. ( cos ( θ 1 ) e 1 + sin ( θ 1 ) e k + 1 , , cos ( θ k ) e k + sin ( θ k ) e 2 k ) . (\cos(\theta_{1})e_{1}+\sin(\theta_{1})e_{k+1},\ldots,\cos(\theta_{k})e_{k}+% \sin(\theta_{k})e_{2k}).

Angular_momentum_diagrams_(quantum_mechanics).html

  1. | j , m |j,m\rangle
  2. j , m | \langle j,m|
  3. T 𝐱 ^ T = 𝐱 ^ T\hat{\mathbf{x}}T^{\dagger}=\hat{\mathbf{x}}
  4. T 𝐩 ^ T = - 𝐩 ^ T\hat{\mathbf{p}}T^{\dagger}=-\hat{\mathbf{p}}
  5. T 𝐒 ^ T = - 𝐒 ^ T\hat{\mathbf{S}}T^{\dagger}=-\hat{\mathbf{S}}
  6. T 𝐋 ^ T = - 𝐋 ^ T\hat{\mathbf{L}}T^{\dagger}=-\hat{\mathbf{L}}
  7. T 𝐉 ^ T = - 𝐉 ^ T\hat{\mathbf{J}}T^{\dagger}=-\hat{\mathbf{J}}
  8. | j , m |j,m\rangle
  9. T | j , m | T ( j , m ) = ( - 1 ) j - m | j , - m T\left|j,m\right\rangle\equiv\left|T(j,m)\right\rangle={(-1)}^{j-m}\left|j,-m\right\rangle
  10. j 2 , m 2 | j 1 , m 1 = δ j 1 j 2 δ m 1 m 2 \langle j_{2},m_{2}|j_{1},m_{1}\rangle=\delta_{j_{1}j_{2}}\delta_{m_{1}m_{2}}
  11. m j , m | j , m = 2 j + 1 \sum_{m}\langle j,m|j,m\rangle=2j+1
  12. | j 2 , m 2 j 1 , m 1 | \left|j_{2},m_{2}\right\rangle\left\langle j_{1},m_{1}\right|
  13. m | j , m j , m | \displaystyle\sum_{m}|j,m\rangle\langle j,m|
  14. | j , m |j,m\rangle
  15. | j 1 , m 1 , j 2 , m 2 , j n , m n | j 1 , m 1 | j 2 , m 2 | j n , m n | j 1 , m 1 | j 2 , m 2 | j n , m n \begin{aligned}\displaystyle\left|j_{1},m_{1},j_{2},m_{2},...j_{n},m_{n}\right% \rangle&\displaystyle\equiv\left|j_{1},m_{1}\right\rangle\otimes\left|j_{2},m_% {2}\right\rangle\otimes\cdots\otimes\left|j_{n},m_{n}\right\rangle\\ &\displaystyle\equiv\left|j_{1},m_{1}\right\rangle\left|j_{2},m_{2}\right% \rangle\cdots\left|j_{n},m_{n}\right\rangle\end{aligned}
  16. \circlearrowright
  17. \circlearrowleft
  18. j n , m n , , j 2 , m 2 , j 1 , m 1 | j 1 , m 1 , j 2 , m 2 , j n , m n \displaystyle\left\langle j^{\prime}_{n},m^{\prime}_{n},...,j^{\prime}_{2},m^{% \prime}_{2},j^{\prime}_{1},m^{\prime}_{1}|j_{1},m_{1},j_{2},m_{2},...j_{n},m_{% n}\right\rangle

Annuity.html

  1. a n ¯ | i = 1 - ( 1 + i ) - n i , a_{\overline{n}|i}=\frac{1-\left(1+i\right)^{-n}}{i},
  2. n n
  3. i i
  4. R R
  5. P V ( i , n , R ) = R × a n ¯ | i PV(i,n,R)=R\times a_{\overline{n}|i}
  6. I I
  7. i = I / 12 i=I/12
  8. s n ¯ | i = ( 1 + i ) n - 1 i s_{\overline{n}|i}=\frac{(1+i)^{n}-1}{i}
  9. n n
  10. i i
  11. R R
  12. F V ( i , n , R ) = R × s n ¯ | i FV(i,n,R)=R\times s_{\overline{n}|i}
  13. P V ( 0.12 / 12 , 5 × 12 , $ 100 ) = $ 100 × a 60 ¯ | 0.01 = $ 4 , 495.50 PV(0.12/12,5\times 12,\$100)=\$100\times a_{\overline{60}|0.01}=\$4,495.50
  14. s n ¯ | i = ( 1 + i ) n × a n ¯ | i s_{\overline{n}|i}=(1+i)^{n}\times a_{\overline{n}|i}
  15. 1 a n ¯ | i - 1 s n ¯ | i = i \frac{1}{a_{\overline{n}|i}}-\frac{1}{s_{\overline{n}|i}}=i
  16. a ¨ n | ¯ i = ( 1 + i ) × a n | ¯ i = 1 - ( 1 + i ) - n d \ddot{a}_{\overline{n|}i}=(1+i)\times a_{\overline{n|}i}=\frac{1-\left(1+i% \right)^{-n}}{d}
  17. s ¨ n | ¯ i = ( 1 + i ) × s n | ¯ i = ( 1 + i ) n - 1 d \ddot{s}_{\overline{n|}i}=(1+i)\times s_{\overline{n|}i}=\frac{(1+i)^{n}-1}{d}
  18. n n
  19. i i
  20. d d
  21. d = i / ( i + 1 ) d=i/(i+1)
  22. s ¨ n ¯ | i = ( 1 + i ) n × a ¨ n ¯ | i \ddot{s}_{\overline{n}|i}=(1+i)^{n}\times\ddot{a}_{\overline{n}|i}
  23. 1 a ¨ n ¯ | i - 1 s ¨ n ¯ | i = d \frac{1}{\ddot{a}_{\overline{n}|i}}-\frac{1}{\ddot{s}_{\overline{n}|i}}=d
  24. F V d u e ( 0.09 / 12 , 7 × 12 , $ 100 ) = $ 100 × s ¨ 84 ¯ | 0.0075 = $ 11 , 730.01. FV_{due}(0.09/12,7\times 12,\$100)=\$100\times\ddot{s}_{\overline{84}|0.0075}=% \$11,730.01.
  25. a ¨ n | ¯ i = a n ¯ | i ( 1 + i ) = a n - 1 | ¯ i + 1 \ddot{a}_{\overline{n|}i}=a_{\overline{n}|i}(1+i)=a_{\overline{n-1|}i}+1
  26. s ¨ n | ¯ i = s n ¯ | i ( 1 + i ) = s n + 1 | ¯ i - 1 \ddot{s}_{\overline{n|}i}=s_{\overline{n}|i}(1+i)=s_{\overline{n+1|}i}-1
  27. lim n P V ( i , n , R ) = R i \lim_{n\,\rightarrow\,\infty}\,PV(i,n,R)\,=\,\frac{R}{i}
  28. a ¯ | i = 1 / i ; a ¨ ¯ | i = 1 / d . a_{\overline{\infty}|i}=1/i;\qquad\ddot{a}_{\overline{\infty}|i}=1/d.
  29. i i
  30. d = i / ( 1 + i ) d=i/(1+i)
  31. a n ¯ | i = k = 1 n 1 ( 1 + i ) k = 1 1 + i k = 0 n - 1 ( 1 1 + i ) k = 1 1 + i ( 1 - ( 1 + i ) - n 1 - ( 1 + i ) - 1 ) (Geom. series) = 1 - ( 1 + i ) - n 1 + i - 1 = 1 - ( 1 1 + i ) n i \begin{aligned}\displaystyle a_{\overline{n}|i}&\displaystyle=\sum_{k=1}^{n}% \frac{1}{(1+i)^{k}}=\frac{1}{1+i}\sum_{k=0}^{n-1}\left(\frac{1}{1+i}\right)^{k% }\\ &\displaystyle=\frac{1}{1+i}\left(\frac{1-(1+i)^{-n}}{1-(1+i)^{-1}}\right)% \quad\quad\,\text{(Geom. series)}\\ &\displaystyle=\frac{1-(1+i)^{-n}}{1+i-1}\\ &\displaystyle=\frac{1-\left(\frac{1}{1+i}\right)^{n}}{i}\end{aligned}
  32. s n ¯ | i = 1 + ( 1 + i ) + ( 1 + i ) 2 + + ( 1 + i ) n - 1 = ( 1 + i ) n a n ¯ | i = ( 1 + i ) n - 1 i s_{\overline{n}|i}=1+(1+i)+(1+i)^{2}+\cdots+(1+i)^{n-1}=(1+i)^{n}a_{\overline{% n}|i}=\frac{(1+i)^{n}-1}{i}
  33. R i - ( 1 + i ) n ( R i - P ) \frac{R}{i}-\left(1+i\right)^{n}\left(\frac{R}{i}-P\right)
  34. R / i R/i
  35. R R
  36. R / i - P R/i-P
  37. i i
  38. R [ 1 i - ( i + 1 ) n - N i ] = R × a N - n ¯ | i R\left[\frac{1}{i}-\frac{(i+1)^{n-N}}{i}\right]=R\times a_{\overline{N-n}|i}

Anthony_Hilton.html

  1. n - 1 n-1
  2. n × n n\times n
  3. n 2 - n + 1 n^{2}-n+1
  4. 2 n 2n
  5. k 12 n / 7 k\geq 12n/7

Anti-unification_(computer_science).html

  1. V V
  2. V V
  3. T T
  4. V T V\subseteq T
  5. T T
  6. \equiv
  7. T T
  8. t u t\equiv u
  9. t t
  10. u u
  11. \equiv
  12. \oplus
  13. t u t\equiv u
  14. u u
  15. t t
  16. \oplus
  17. V V
  18. C C
  19. F n F_{n}
  20. n n
  21. n 1 n\geq 1
  22. T T
  23. V T V\subseteq T
  24. C T C\subseteq T
  25. n n
  26. t 1 , , t n t_{1},\ldots,t_{n}
  27. n n
  28. f F n f\in F_{n}
  29. f ( t 1 , , t n ) f(t_{1},\ldots,t_{n})
  30. x V x\in V
  31. 1 C 1\in C
  32. 𝑎𝑑𝑑 F 2 \,\textit{add}\in F_{2}
  33. x T x\in T
  34. 1 T 1\in T
  35. a d d ( x , 1 ) T add(x,1)\in T
  36. x + 1 x+1
  37. + +
  38. σ : V T \sigma:V\longrightarrow T
  39. { x 1 t 1 , , x k t k } \{x_{1}\mapsto t_{1},\ldots,x_{k}\mapsto t_{k}\}
  40. x i x_{i}
  41. t i t_{i}
  42. i = 1 , , k i=1,\ldots,k
  43. t t
  44. t { x 1 t 1 , , x k t k } t\{x_{1}\mapsto t_{1},\ldots,x_{k}\mapsto t_{k}\}
  45. x i x_{i}
  46. t t
  47. t i t_{i}
  48. t σ t\sigma
  49. σ \sigma
  50. t t
  51. t t
  52. { x h ( a , y ) , z b } \{x\mapsto h(a,y),z\mapsto b\}
  53. f ( f(
  54. x x
  55. , a , g ( ,a,g(
  56. z z
  57. ) , y ) ),y)
  58. f ( f(
  59. h ( a , y ) h(a,y)
  60. , a , g ( ,a,g(
  61. b b
  62. ) , y ) ),y)
  63. t t
  64. u u
  65. t σ u t\sigma\equiv u
  66. σ \sigma
  67. t t
  68. u u
  69. u u
  70. t t
  71. x a x\oplus a
  72. a b a\oplus b
  73. \oplus
  74. ( x a ) { x b } = b a a b (x\oplus a)\{x\mapsto b\}=b\oplus a\equiv a\oplus b
  75. \equiv
  76. f ( x 1 , a , g ( z 1 ) , y 1 ) f(x_{1},a,g(z_{1}),y_{1})
  77. f ( x 2 , a , g ( z 2 ) , y 2 ) f(x_{2},a,g(z_{2}),y_{2})
  78. f ( x 1 , a , g ( z 1 ) , y 1 ) { x 1 x 2 , y 2 y 2 , z 1 z 2 } = f ( x 2 , a , g ( z 2 ) , y 2 ) f(x_{1},a,g(z_{1}),y_{1})\{x_{1}\mapsto x_{2},y_{2}\mapsto y_{2},z_{1}\mapsto z% _{2}\}=f(x_{2},a,g(z_{2}),y_{2})
  79. f ( x 2 , a , g ( z 2 ) , y 2 ) { x 1 x 1 , y 2 y 1 , z 2 z 1 } = f ( x 1 , a , g ( z 1 ) , y 1 ) f(x_{2},a,g(z_{2}),y_{2})\{x_{1}\mapsto x_{1},y_{2}\mapsto y_{1},z_{2}\mapsto z% _{1}\}=f(x_{1},a,g(z_{1}),y_{1})
  80. f ( x 1 , a , g ( z 1 ) , y 1 ) f(x_{1},a,g(z_{1}),y_{1})
  81. f ( x 2 , a , g ( x 2 ) , x 2 ) f(x_{2},a,g(x_{2}),x_{2})
  82. { x 1 x 2 , z 1 x 2 , y 1 x 2 } \{x_{1}\mapsto x_{2},z_{1}\mapsto x_{2},y_{1}\mapsto x_{2}\}
  83. σ \sigma
  84. τ \tau
  85. x σ x\sigma
  86. x τ x\tau
  87. x x
  88. { x f ( u ) , y f ( f ( u ) ) } \{x\mapsto f(u),y\mapsto f(f(u))\}
  89. { x z , y f ( z ) } \{x\mapsto z,y\mapsto f(z)\}
  90. f ( u ) f(u)
  91. f ( f ( u ) ) f(f(u))
  92. z z
  93. f ( z ) f(z)
  94. t 1 , t 2 \langle t_{1},t_{2}\rangle
  95. t t
  96. t 1 t_{1}
  97. t 2 t_{2}
  98. t σ 1 t 1 t\sigma_{1}\equiv t_{1}
  99. t σ 2 t 2 t\sigma_{2}\equiv t_{2}
  100. σ 1 , σ 2 \sigma_{1},\sigma_{2}
  101. S S
  102. t S t\in S
  103. S S
  104. T T
  105. V V
  106. C C
  107. F n F_{n}
  108. n n
  109. \equiv
  110. t 1 , t 2 \langle t_{1},t_{2}\rangle
  111. { t } \{t\}
  112. t t
  113. t 1 t_{1}
  114. t 2 t_{2}
  115. t 1 t_{1}
  116. t 2 t_{2}
  117. t t
  118. S 1 S_{1}
  119. S 2 S_{2}
  120. S 1 = { s 1 } S_{1}=\{s_{1}\}
  121. S 2 = { s 2 } S_{2}=\{s_{2}\}
  122. s 1 s_{1}
  123. s 2 s_{2}
  124. ϕ : T × T V \phi:T\times T\longrightarrow V
  125. s , t s,t
  126. ϕ ( s , t ) \phi(s,t)
  127. f ( s 1 , , s n ) f ( t 1 , , t n ) f(s_{1},\dots,s_{n})\sqcup f(t_{1},\ldots,t_{n})
  128. \rightsquigarrow
  129. f ( s 1 t 1 , , s n t n ) f(s_{1}\sqcup t_{1},\ldots,s_{n}\sqcup t_{n})
  130. s t s\sqcup t
  131. \rightsquigarrow
  132. ϕ ( s , t ) \phi(s,t)
  133. ( 0 * 0 ) ( 4 * 4 ) ( 0 4 ) * ( 0 4 ) ϕ ( 0 , 4 ) * ϕ ( 0 , 4 ) x * x (0*0)\sqcup(4*4)\rightsquigarrow(0\sqcup 4)*(0\sqcup 4)\rightsquigarrow\phi(0,% 4)*\phi(0,4)\rightsquigarrow x*x
  134. a ( b f ( x ) ) a ( f ( x ) b ) ( b f ( x ) ) a ( f ( x ) b ) a a\oplus(b\oplus f(x))\equiv a\oplus(f(x)\oplus b)\equiv(b\oplus f(x))\oplus a% \equiv(f(x)\oplus b)\oplus a
  135. V V
  136. T × T T\times T
  137. ϕ \phi
  138. s , t , ϕ ( s , t ) \langle s,t,\phi(s,t)\rangle

Antigen-antibody_interaction.html

  1. [ A b ] + [ A g ] [ A b A g ] [Ab]+[Ag]\leftrightharpoons[AbAg]
  2. K a = k o n k o f f = [ A b A g ] [ A b ] [ A g ] K_{a}=\frac{k_{on}}{k_{off}}=\frac{[AbAg]}{[Ab][Ag]}
  3. K d = k o f f k o n = [ A b ] [ A g ] [ A b A g ] K_{d}=\frac{k_{off}}{k_{on}}=\frac{[Ab][Ag]}{[AbAg]}
  4. K = K a K d = [ A b A g ] [ A b ] [ A g ] K=\frac{K_{a}}{K_{d}}=\frac{[AbAg]}{[Ab][Ag]}
  5. K a = k o n k o f f = [ A b A g ] [ A b ] [ A g ] K_{a}=\frac{k_{on}}{k_{off}}=\frac{[AbAg]}{[Ab][Ag]}

Antiperovskite_(structure).html

  1. ( r a + r x ) 2 ( r b + r x ) \frac{(r_{a}+r_{x})}{\sqrt{2(r_{b}+r_{x})}}

Anton-Schmidt_equation_of_state.html

  1. p ( V ) = - β ( V V 0 ) n ln ( V V 0 ) p(V)=-\beta\left(\frac{V}{V_{0}}\right)^{n}\ln\left(\frac{V}{V_{0}}\right)
  2. p ( V ) p(V)
  3. E E
  4. V V
  5. E ( V ) = - V p ( V ) d V E(V)=-\int_{V}^{\infty}p(V^{\prime})dV^{\prime}
  6. E ( V ) = β V 0 n + 1 ( V V 0 ) n + 1 [ ln ( V V 0 ) - 1 n + 1 ] - E E(V)=\frac{\beta V_{0}}{n+1}\left(\frac{V}{V_{0}}\right)^{n+1}\left[\ln\left(% \frac{V}{V_{0}}\right)-\frac{1}{n+1}\right]-E_{\infty}
  7. β , n \beta,n
  8. V 0 V_{0}
  9. β \beta
  10. K 0 K_{0}
  11. V 0 V_{0}
  12. n n
  13. n = - 1 6 - γ G n=-\frac{1}{6}-\gamma_{G}
  14. E E_{\infty}

Aphidicolan-16beta-ol_synthase.html

  1. \rightleftharpoons

Approximate_max-flow_min-cut_theorem.html

  1. k k
  2. s i s_{\,\text{i}}
  3. t i t_{\,\text{i}}
  4. D i D_{\,\text{i}}
  5. D i D_{\,\text{i}}
  6. i i
  7. s i s_{\,\text{i}}
  8. t i t_{\,\text{i}}
  9. i i
  10. f f
  11. f f
  12. f D i fD_{\,\text{i}}
  13. i i
  14. i i
  15. φ \varphi
  16. v V v\in V
  17. G = ( V , E ) G=(V,E)
  18. u u
  19. v v
  20. u u
  21. v v
  22. u V u\in V
  23. G G
  24. G G
  25. k k
  26. 1 / k 1/k
  27. n n
  28. n n
  29. f f
  30. φ \varphi
  31. f O ( φ log n ) f\leq O(\frac{\varphi}{\log n})
  32. Ω ( φ log n ) f φ \Omega(\frac{\varphi}{\log n})\leq f\leq\varphi
  33. f f
  34. φ \varphi
  35. k k
  36. Ω ( φ log k ) f φ \Omega(\frac{\varphi}{\log k})\leq f\leq\varphi
  37. f f
  38. φ \varphi
  39. n n
  40. Ω ( φ log n ) f φ \Omega(\frac{\varphi}{\log n})\leq f\leq\varphi
  41. f f
  42. φ \varphi
  43. k k
  44. Ω ( φ log k ) f φ \Omega(\frac{\varphi}{\log k})\leq f\leq\varphi
  45. f f
  46. φ \varphi
  47. G = ( V , E ) G=(V,E)
  48. O ( log n ) O(\log n)
  49. O ( log p ) O(\log p)
  50. p p
  51. G = ( V , E ) G=(V,E)
  52. b π ( V ) π ( U ) ( 1 - b ) π ( V ) b\pi(V)\leq\pi(U)\leq(1-b)\pi(V)
  53. π ( U ) \pi(U)
  54. U U
  55. G G
  56. S S
  57. O ( S log ( n / b ) - b ) O(S\log(n/b)-b^{\prime})
  58. O ( log 6 n ) O(\log^{6}n)
  59. n n
  60. G G
  61. K n K_{n}
  62. G G
  63. K n K_{n}
  64. G G
  65. O ( log n ) O(\log n)
  66. G G
  67. O ( ( R log n + n R ) log n R ) O((R\log n+\sqrt{nR})\log\frac{n}{R})
  68. G G
  69. R R
  70. G G
  71. n n
  72. R R

Approximate_tangent_space.html

  1. M n M\subset\mathbb{R}^{n}
  2. m \mathcal{H}^{m}
  3. m M \mathcal{H}^{m}\llcorner M
  4. P n P\subset\mathbb{R}^{n}
  5. M M
  6. x x
  7. T x M = P T_{x}M=P
  8. ( m M ) x , λ m P \left(\mathcal{H}^{m}\llcorner M\right)_{x,\lambda}\rightharpoonup\mathcal{H}^% {m}\llcorner P
  9. λ 0 \lambda\downarrow 0
  10. μ \mu
  11. μ x , λ \mu_{x,\lambda}
  12. μ x , λ ( A ) := λ - n μ ( x + λ A ) , A n \mu_{x,\lambda}(A):=\lambda^{-n}\mu(x+\lambda A),\qquad A\subset\mathbb{R}^{n}
  13. M 1 := { ( x , x 2 ) : x } 2 M_{1}:=\{(x,x^{2}):x\in\mathbb{R}\}\subset\mathbb{R}^{2}
  14. ( 0 , 0 ) M 1 (0,0)\in M_{1}
  15. T ( 0 , 0 ) M 1 = × { 0 } T_{(0,0)}M_{1}=\mathbb{R}\times\{0\}
  16. M 2 := { ( x , x 2 ) : x } { ( x , - x 2 ) : x } 2 2 M_{2}:=\{(x,x^{2}):x\in\mathbb{R}\}\cup\{(x,-x^{2}):x\in\mathbb{R}\}\subset% \mathbb{R}^{2}\subset\mathbb{R}^{2}
  17. M 2 M_{2}
  18. M 2 M_{2}
  19. × { 0 } \mathbb{R}\times\{0\}
  20. μ \mu
  21. n \mathbb{R}^{n}
  22. P n P\subset\mathbb{R}^{n}
  23. μ \mu
  24. x x
  25. θ ( x ) ( 0 , ) \theta(x)\in(0,\infty)
  26. T x μ = P T_{x}\mu=P
  27. θ ( x ) \theta(x)
  28. μ x , λ θ m P \mu_{x,\lambda}\rightharpoonup\theta\;\mathcal{H}^{m}\llcorner P
  29. λ 0 \lambda\downarrow 0
  30. P P
  31. μ := n M \mu:=\mathcal{H}^{n}\llcorner M
  32. M M
  33. μ := 1 M 2 \mu:=\mathcal{H}^{1}\llcorner M_{2}
  34. T ( 0 , 0 ) μ = × { 0 } T_{(0,0)}\mu=\mathbb{R}\times\{0\}
  35. M n M\subset\mathbb{R}^{n}
  36. M M
  37. m \mathcal{H}^{m}
  38. θ : M ( 0 , ) \theta:M\to(0,\infty)
  39. μ ( A ) = A θ ( x ) d m ( x ) \mu(A)=\int_{A}\theta(x)\,d\mathcal{H}^{m}(x)
  40. T x μ T_{x}\mu
  41. m \mathcal{H}^{m}
  42. x M x\in M

Approximation-preserving_reduction.html

  1. A A
  2. B B
  3. x x
  4. A A
  5. O P T ( x ) OPT(x)
  6. c A ( x , y ) c_{A}(x,y)
  7. y y
  8. x x
  9. A A
  10. ( f , g ) (f,g)
  11. f f
  12. x x
  13. A A
  14. x x^{\prime}
  15. B B
  16. g g
  17. y y^{\prime}
  18. B B
  19. y y
  20. A A
  21. g g
  22. R A ( x , y ) = max ( c A ( x , O P T ( x ) ) c A ( x , y ) , c A ( x , y ) c A ( x , O P T ( x ) ) ) R_{A}(x,y)=\max\left(\frac{c_{A}(x,OPT(x))}{c_{A}(x,y)},\frac{c_{A}(x,y)}{c_{A% }(x,OPT(x))}\right)
  23. R A ( x , y ) R B ( x , y ) R_{A}(x,y)\leq R_{B}(x^{\prime},y^{\prime})
  24. x = f ( x ) , y = g ( y ) x^{\prime}=f(x),y=g(y^{\prime})
  25. c A ( x , y ) = c B ( x , y ) c_{A}(x,y)=c_{B}(x^{\prime},y^{\prime})
  26. g g
  27. r r
  28. c c
  29. R B ( x , y ) r R A ( x , y ) c ( r ) R_{B}(x^{\prime},y^{\prime})\leq r\rightarrow R_{A}(x,y)\leq c(r)
  30. R B ( x , y ) c ( r ) R A ( x , y ) r R_{B}(x^{\prime},y^{\prime})\leq c(r)\rightarrow R_{A}(x,y)\leq r
  31. p p
  32. β \beta
  33. p p
  34. β \beta
  35. c B ( O P T B ( x ) ) p ( | x | ) c A ( O P T A ( x ) ) c_{B}(OPT_{B}(x^{\prime}))\leq p(|x|)c_{A}(OPT_{A}(x))
  36. | x | |x|
  37. y y^{\prime}
  38. B B
  39. R A ( x , y ) 1 + β ( R B ( x , y ) - 1 ) R_{A}(x,y)\leq 1+\beta\cdot(R_{B}(x^{\prime},y^{\prime})-1)
  40. c ( r ) = 1 + β ( r - 1 ) c(r)=1+\beta\cdot(r-1)
  41. c ( r ) = 1 + ( r - 1 ) / β c(r)=1+(r-1)/\beta
  42. α \alpha
  43. R B ( x , y ) r R A ( x , y ) 1 + α ( r - 1 ) R_{B}(x^{\prime},y^{\prime})\leq r\rightarrow R_{A}(x,y)\leq 1+\alpha\cdot(r-1)
  44. g g
  45. r r
  46. f , g f,g

Aquifer_properties.html

  1. n = V V V T = V V V S + V V = e 1 + e n=\frac{V_{V}}{V_{T}}=\frac{V_{V}}{V_{S}+V_{V}}=\frac{e}{1+e}
  2. e e
  3. n n

Arabidiol_synthase.html

  1. \rightleftharpoons

Arachidonate_15-lipoxygenase.html

  1. \rightleftharpoons

Arakawa–Kaneko_zeta_function.html

  1. ξ k ( s ) \xi_{k}(s)
  2. ξ k ( s ) = 1 Γ ( s ) 0 + t s - 1 e t - 1 Li k ( 1 - e - t ) d t \xi_{k}(s)=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{t^{s-1}}{e^{t}-1}\mathrm% {Li}_{k}(1-e^{-t})\,dt
  3. Li k ( z ) = n = 1 z n n k . \mathrm{Li}_{k}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{k}}\ .
  4. ( s ) > 0 \Re(s)>0
  5. ξ k ( s ) \xi_{k}(s)
  6. ξ 1 ( s ) = s ζ ( s + 1 ) \xi_{1}(s)=s\zeta(s+1)
  7. ζ \zeta
  8. ξ k ( 1 ) = ζ ( k + 1 ) \xi_{k}(1)=\zeta(k+1)
  9. ζ \zeta
  10. ξ k ( m ) = ζ m * ( k , 1 , , 1 ) \xi_{k}(m)=\zeta_{m}^{*}(k,1,\ldots,1)
  11. ζ n * ( k 1 , , k n - 1 , k n ) = 0 < m 1 < m 2 < < m n 1 m 1 k 1 m n - 1 k n - 1 m n k n . \zeta_{n}^{*}(k_{1},\dots,k_{n-1},k_{n})=\sum_{0<m_{1}<m_{2}<\cdots<m_{n}}% \frac{1}{m_{1}^{k_{1}}\cdots m_{n-1}^{k_{n-1}}m_{n}^{k_{n}}}\ .

Arellano–Bond_estimator.html

  1. N N
  2. T T
  3. y i t = X i t β + h i + u i t y_{it}=X_{it}\mathbf{\beta}+h_{i}+u_{it}
  4. t = 1 , , T t=1,\ldots,T
  5. i = 1 , , N i=1,\ldots,N
  6. y i t y_{it}
  7. i i
  8. t , t,
  9. X i t X_{it}
  10. 1 × k 1\times k
  11. α i \alpha_{i}
  12. u i t u_{it}
  13. X i t X_{it}
  14. h i h_{i}
  15. α i \alpha_{i}
  16. y i t - 1 y_{it-1}
  17. y i t = X i t β + α y i t - 1 + h i + u i t y_{it}=X_{it}\mathbf{\beta}+\alpha y_{it-1}+h_{i}+u_{it}
  18. t = 1 , , T t=1,\ldots,T
  19. i = 1 , , N i=1,\ldots,N
  20. Δ y i t = y i t - y i t - 1 = Δ X i t β + α Δ y i t - 1 + Δ u i t \Delta y_{it}=y_{it}-y_{it-1}=\Delta X_{it}\beta+\alpha\Delta y_{it-1}+\Delta u% _{it}
  21. t = 1 , , T t=1,\ldots,T
  22. i = 1 , , N . i=1,\ldots,N.
  23. Δ y = Δ R π + Δ u . \Delta y=\Delta R\pi+\Delta u.
  24. π EGMM = [ Δ R Z ( Z Ω Z ) - 1 Z Δ R ] - 1 Δ R Z ( Z Ω Z ) - 1 Z y \pi\text{EGMM}=[\Delta R^{\prime}Z(Z^{\prime}\Omega Z)^{-1}Z^{\prime}\Delta R]% ^{-1}\Delta R^{\prime}Z(Z^{\prime}\Omega Z)^{-1}Z^{\prime}y
  25. Z Z
  26. Δ R \Delta R
  27. Ω \Omega
  28. u i t u_{it}

Armadillo_projection.html

  1. x = R ( 1 + cos φ ) sin λ - λ 0 2 x=R\left(1+\cos\varphi\right)\sin\frac{\lambda-\lambda_{0}}{2}
  2. y = R [ 1 + sin 20 - cos 20 2 + sin φ cos 20 - ( 1 + cos φ ) sin 20 cos ( λ - λ 0 2 ) ] y=R\left[\frac{1+\sin 20^{\circ}-\cos 20^{\circ}}{2}+\sin\varphi\cos 20^{\circ% }-\left(1+\cos\varphi\right)\sin 20^{\circ}\cos\left(\frac{\lambda-\lambda_{0}% }{2}\right)\right]
  3. φ s = - tan - 1 ( cos λ - λ 0 2 tan 20 ) \varphi_{s}=-\tan^{-1}\left(\frac{\cos\frac{\lambda-\lambda_{0}}{2}}{\tan 20^{% \circ}}\right)
  4. φ s \varphi_{s}

Arrott_plot.html

  1. M 2 M^{2}
  2. H / M H/M
  3. F ( M ) = - H M + a ( T - T c ) M 2 + b M 4 + F(M)=-HM+a(T-T_{c})M^{2}+bM^{4}+\ldots
  4. M M
  5. H H
  6. T c T_{c}
  7. a , b a,b
  8. M 2 = 1 b H M - a b ϵ M^{2}=\frac{1}{b}\frac{H}{M}-\frac{a}{b}\epsilon
  9. ϵ = T - T c T c \epsilon=\frac{T-T_{c}}{T_{c}}
  10. M 2 M^{2}
  11. H / M H/M

Arthur_Hobbs_(mathematician).html

  1. γ ( G ) = max H G ( | E ( H ) | | V ( H ) | - ω ( H ) ) , \gamma(G)=\max_{H\subseteq G}\left({{|E(H)}|\over{|V(H)|-\omega(H)}}\right),
  2. η ( G ) = min S E ( G ) ( | S | ω ( G - S ) - ω ( G ) ) , \eta(G)=\min_{S\subseteq E(G)}\left({|S|\over{\omega(G-S)-\omega(G)}}\right),

Artin_transfer_(group_theory).html

  1. G G
  2. H < G H<G
  3. n = ( G : H ) n=(G:H)
  4. H H
  5. G G
  6. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  7. H H
  8. G G
  9. G = ˙ i = 1 n g i H G=\dot{\bigcup}_{i=1}^{n}\,g_{i}H
  10. H H
  11. G G
  12. ( d 1 , , d n ) (d_{1},\ldots,d_{n})
  13. H H
  14. G G
  15. G = ˙ i = 1 n H d i G=\dot{\bigcup}_{i=1}^{n}\,Hd_{i}
  16. H H
  17. G G
  18. 1 i 0 n 1\leq i_{0}\leq n
  19. g i 0 H g_{i_{0}}\in H
  20. d i 0 H d_{i_{0}}\in H
  21. 1 1
  22. G G
  23. H H
  24. G G
  25. ( g 1 - 1 , , g n - 1 ) (g_{1}^{-1},\ldots,g_{n}^{-1})
  26. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  27. H H
  28. G G
  29. G = ˙ i = 1 n g i H G=\dot{\bigcup}_{i=1}^{n}\,g_{i}H
  30. G = G - 1 = ˙ i = 1 n ( g i H ) - 1 = ˙ i = 1 n H - 1 g i - 1 = ˙ i = 1 n H g i - 1 G=G^{-1}=\dot{\bigcup}_{i=1}^{n}\,(g_{i}H)^{-1}=\dot{\bigcup}_{i=1}^{n}\,H^{-1% }g_{i}^{-1}=\dot{\bigcup}_{i=1}^{n}\,Hg_{i}^{-1}
  31. H G H\triangleleft G
  32. G G
  33. H H
  34. G G
  35. x H = H x xH=Hx
  36. x G x\in G
  37. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  38. H < G H<G
  39. n = ( G : H ) n=(G:H)
  40. G G
  41. x G x\in G
  42. π x S n \pi_{x}\in S_{n}
  43. H H
  44. G G
  45. x g i H = g π x ( i ) H xg_{i}H=g_{\pi_{x}(i)}H
  46. x g i g π x ( i ) H xg_{i}\in g_{\pi_{x}(i)}H
  47. h x ( i ) := g π x ( i ) - 1 x g i H h_{x}(i):=g_{\pi_{x}(i)}^{-1}xg_{i}\in H
  48. 1 i n 1\leq i\leq n
  49. ( d 1 , , d n ) (d_{1},\ldots,d_{n})
  50. H H
  51. G G
  52. x G x\in G
  53. ρ x S n \rho_{x}\in S_{n}
  54. H H
  55. G G
  56. H d i x = H d ρ x ( i ) Hd_{i}x=Hd_{\rho_{x}(i)}
  57. d i x H d ρ x ( i ) d_{i}x\in Hd_{\rho_{x}(i)}
  58. η x ( i ) := d i x d ρ x ( i ) - 1 H \eta_{x}(i):=d_{i}xd_{\rho_{x}(i)}^{-1}\in H
  59. 1 i n 1\leq i\leq n
  60. G S n , x π x G\to S_{n},\ x\mapsto\pi_{x}
  61. x ρ x x\mapsto\rho_{x}
  62. G G
  63. S n S_{n}
  64. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  65. ( d 1 , , d n ) (d_{1},\ldots,d_{n})
  66. G H n × S n , x ( h x ( 1 ) , , h x ( n ) ; π x ) G\to H^{n}\times S_{n},\ x\mapsto(h_{x}(1),\ldots,h_{x}(n);\pi_{x})
  67. x ( η x ( 1 ) , , η x ( n ) ; ρ x ) x\mapsto(\eta_{x}(1),\ldots,\eta_{x}(n);\rho_{x})
  68. G G
  69. H n × S n H^{n}\times S_{n}
  70. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  71. ( d 1 , , d n ) (d_{1},\ldots,d_{n})
  72. ( g 1 - 1 , , g n - 1 ) (g_{1}^{-1},\ldots,g_{n}^{-1})
  73. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  74. η x ( i ) = g i - 1 x g ρ x ( i ) \eta_{x}(i)=g_{i}^{-1}xg_{\rho_{x}(i)}
  75. h x ( i ) - 1 = ( g π x ( i ) - 1 x g i ) - 1 = g i - 1 x - 1 g π x ( i ) = g i - 1 x - 1 g ρ x - 1 ( i ) = η x - 1 ( i ) h_{x}(i)^{-1}=(g_{\pi_{x}(i)}^{-1}xg_{i})^{-1}=g_{i}^{-1}x^{-1}g_{\pi_{x}(i)}=% g_{i}^{-1}x^{-1}g_{\rho_{x^{-1}}(i)}=\eta_{x^{-1}}(i)
  76. 1 i n 1\leq i\leq n
  77. x G x\in G
  78. ρ x - 1 = π x \rho_{x^{-1}}=\pi_{x}
  79. η x - 1 ( i ) = h x ( i ) - 1 \eta_{x^{-1}}(i)=h_{x}(i)^{-1}
  80. 1 i n 1\leq i\leq n
  81. G G
  82. H < G H<G
  83. n = ( G : H ) n=(G:H)
  84. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  85. ( d 1 , , d n ) (d_{1},\ldots,d_{n})
  86. H H
  87. G G
  88. T G , H : G H / H T_{G,H}:\ G\to H/H^{\prime}
  89. G G
  90. H H
  91. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  92. ( d 1 , , d n ) (d_{1},\ldots,d_{n})
  93. T G , H ( g ) ( x ) := i = 1 n g π x ( i ) - 1 x g i H T_{G,H}^{(g)}(x):=\prod_{i=1}^{n}\,g_{\pi_{x}(i)}^{-1}xg_{i}\cdot H^{\prime}
  94. T G , H ( x ) = i = 1 n h x ( i ) H T_{G,H}(x)=\prod_{i=1}^{n}\,h_{x}(i)\cdot H^{\prime}
  95. T G , H ( d ) ( x ) := i = 1 n d i x d ρ x ( i ) - 1 H T_{G,H}^{(d)}(x):=\prod_{i=1}^{n}\,d_{i}xd_{\rho_{x}(i)}^{-1}\cdot H^{\prime}
  96. T G , H ( x ) = i = 1 n η x ( i ) H T_{G,H}(x)=\prod_{i=1}^{n}\,\eta_{x}(i)\cdot H^{\prime}
  97. x G x\in G
  98. ( γ 1 , , γ n ) (\gamma_{1},\ldots,\gamma_{n})
  99. H H
  100. G G
  101. G = ˙ i = 1 n γ i H G=\dot{\cup}_{i=1}^{n}\,\gamma_{i}H
  102. σ S n \sigma\in S_{n}
  103. g i H = γ σ ( i ) H g_{i}H=\gamma_{\sigma(i)}H
  104. 1 i n 1\leq i\leq n
  105. h i := g i - 1 γ σ ( i ) H h_{i}:=g_{i}^{-1}\gamma_{\sigma(i)}\in H
  106. γ σ ( i ) = g i h i \gamma_{\sigma(i)}=g_{i}h_{i}
  107. h i H h_{i}\in H
  108. 1 i n 1\leq i\leq n
  109. x G x\in G
  110. λ x S n \lambda_{x}\in S_{n}
  111. γ λ x ( σ ( i ) ) H = x γ σ ( i ) H = x g i h i H = x g i H = g π x ( i ) H = g π x ( i ) h π x ( i ) H = γ σ ( π x ( i ) ) H \gamma_{\lambda_{x}(\sigma(i))}H=x\gamma_{\sigma(i)}H=xg_{i}h_{i}H=xg_{i}H=g_{% \pi_{x}(i)}H=g_{\pi_{x}(i)}h_{\pi_{x}(i)}H=\gamma_{\sigma(\pi_{x}(i))}H
  112. 1 i n 1\leq i\leq n
  113. G G
  114. ( γ 1 , , γ n ) (\gamma_{1},\ldots,\gamma_{n})
  115. λ x σ = σ π x \lambda_{x}\circ\sigma=\sigma\circ\pi_{x}
  116. λ x = σ π x σ - 1 S n \lambda_{x}=\sigma\circ\pi_{x}\circ\sigma^{-1}\in S_{n}
  117. x G x\in G
  118. k x ( i ) := γ λ x ( i ) - 1 x γ i H k_{x}(i):=\gamma_{\lambda_{x}(i)}^{-1}x\gamma_{i}\in H
  119. h x ( i ) := g π x ( i ) - 1 x g i H h_{x}(i):=g_{\pi_{x}(i)}^{-1}xg_{i}\in H
  120. k x ( σ ( i ) ) = γ λ x ( σ ( i ) ) - 1 x γ σ ( i ) = γ σ ( π x ( i ) ) - 1 x g i h i k_{x}(\sigma(i))=\gamma_{\lambda_{x}(\sigma(i))}^{-1}x\gamma_{\sigma(i)}=% \gamma_{\sigma(\pi_{x}(i))}^{-1}xg_{i}h_{i}
  121. = ( g π x ( i ) h π x ( i ) ) - 1 x g i h i = h π x ( i ) - 1 g π x ( i ) - 1 x g i h i = h π x ( i ) - 1 h x ( i ) h i =(g_{\pi_{x}(i)}h_{\pi_{x}(i)})^{-1}xg_{i}h_{i}=h_{\pi_{x}(i)}^{-1}g_{\pi_{x}(% i)}^{-1}xg_{i}h_{i}=h_{\pi_{x}(i)}^{-1}h_{x}(i)h_{i}
  122. 1 i n 1\leq i\leq n
  123. H / H H/H^{\prime}
  124. σ , π x \sigma,\pi_{x}
  125. T G , H ( γ ) ( x ) = i = 1 n k x ( σ ( i ) ) H = i = 1 n h π x ( i ) - 1 h x ( i ) h i H T_{G,H}^{(\gamma)}(x)=\prod_{i=1}^{n}\,k_{x}(\sigma(i))\cdot H^{\prime}=\prod_% {i=1}^{n}\,h_{\pi_{x}(i)}^{-1}h_{x}(i)h_{i}\cdot H^{\prime}
  126. = i = 1 n h x ( i ) i = 1 n h π x ( i ) - 1 i = 1 n h i H = i = 1 n h x ( i ) 1 H = i = 1 n h x ( i ) H = T G , H ( g ) ( x ) =\prod_{i=1}^{n}\,h_{x}(i)\prod_{i=1}^{n}\,h_{\pi_{x}(i)}^{-1}\prod_{i=1}^{n}% \,h_{i}\cdot H^{\prime}=\prod_{i=1}^{n}\,h_{x}(i)\cdot 1\cdot H^{\prime}=\prod% _{i=1}^{n}\,h_{x}(i)\cdot H^{\prime}=T_{G,H}^{(g)}(x)
  127. ( g 1 - 1 , , g n - 1 ) (g_{1}^{-1},\ldots,g_{n}^{-1})
  128. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  129. H / H H/H^{\prime}
  130. T G , H ( g - 1 ) ( x ) = i = 1 n g i - 1 x g ρ x ( i ) H = i = 1 n η x ( i ) H = i = 1 n h x - 1 ( i ) - 1 H = ( i = 1 n h x - 1 ( i ) H ) - 1 T_{G,H}^{(g^{-1})}(x)=\prod_{i=1}^{n}\,g_{i}^{-1}xg_{\rho_{x}(i)}\cdot H^{% \prime}=\prod_{i=1}^{n}\,\eta_{x}(i)\cdot H^{\prime}=\prod_{i=1}^{n}\,h_{x^{-1% }}(i)^{-1}\cdot H^{\prime}=(\prod_{i=1}^{n}\,h_{x^{-1}}(i)\cdot H^{\prime})^{-1}
  131. = ( T G , H ( g ) ( x - 1 ) ) - 1 = T G , H ( g ) ( x ) =(T_{G,H}^{(g)}(x^{-1}))^{-1}=T_{G,H}^{(g)}(x)
  132. x , y G x,y\in G
  133. T G , H ( x ) = i = 1 n g π x ( i ) - 1 x g i H T_{G,H}(x)=\prod_{i=1}^{n}\,g_{\pi_{x}(i)}^{-1}xg_{i}\cdot H^{\prime}
  134. T G , H ( y ) = j = 1 n g π y ( j ) - 1 y g j H T_{G,H}(y)=\prod_{j=1}^{n}\,g_{\pi_{y}(j)}^{-1}yg_{j}\cdot H^{\prime}
  135. H / H H/H^{\prime}
  136. π y \pi_{y}
  137. T G , H ( x ) T G , H ( y ) = i = 1 n g π x ( i ) - 1 x g i H j = 1 n g π y ( j ) - 1 y g j H = j = 1 n g π x ( π y ( j ) ) - 1 x g π y ( j ) H j = 1 n g π y ( j ) - 1 y g j H T_{G,H}(x)\cdot T_{G,H}(y)=\prod_{i=1}^{n}\,g_{\pi_{x}(i)}^{-1}xg_{i}H^{\prime% }\cdot\prod_{j=1}^{n}\,g_{\pi_{y}(j)}^{-1}yg_{j}\cdot H^{\prime}=\prod_{j=1}^{% n}\,g_{\pi_{x}(\pi_{y}(j))}^{-1}xg_{\pi_{y}(j)}H^{\prime}\cdot\prod_{j=1}^{n}% \,g_{\pi_{y}(j)}^{-1}yg_{j}\cdot H^{\prime}
  138. = j = 1 n g π x ( π y ( j ) ) - 1 x g π y ( j ) g π y ( j ) - 1 y g j H = j = 1 n g ( π x π y ) ( j ) ) - 1 x y g j H = T G , H ( x y ) =\prod_{j=1}^{n}\,g_{\pi_{x}(\pi_{y}(j))}^{-1}xg_{\pi_{y}(j)}g_{\pi_{y}(j)}^{-% 1}yg_{j}\cdot H^{\prime}=\prod_{j=1}^{n}\,g_{(\pi_{x}\circ\pi_{y})(j))}^{-1}% xyg_{j}\cdot H^{\prime}=T_{G,H}(xy)
  139. T G , H T_{G,H}
  140. G S n , x π x G\to S_{n},\ x\mapsto\pi_{x}
  141. π x y = π x π y \pi_{xy}=\pi_{x}\circ\pi_{y}
  142. x , y x,y
  143. T G , H ( x ) = i = 1 n h x ( i ) H T_{G,H}(x)=\prod_{i=1}^{n}\,h_{x}(i)\cdot H^{\prime}
  144. T G , H ( y ) = j = 1 n h y ( j ) H T_{G,H}(y)=\prod_{j=1}^{n}\,h_{y}(j)\cdot H^{\prime}
  145. x y xy
  146. T G , H ( x y ) = j = 1 n h x ( π y ( j ) ) h y ( j ) H T_{G,H}(xy)=\prod_{j=1}^{n}\,h_{x}(\pi_{y}(j))\cdot h_{y}(j)\cdot H^{\prime}
  147. x h x x\mapsto h_{x}
  148. H 1 ( G , M ) \mathrm{H}^{1}(G,M)
  149. G G
  150. M M
  151. h x y = h x y h y h_{xy}=h_{x}^{y}\cdot h_{y}
  152. H n × S n H^{n}\times S_{n}
  153. H S n H\wr S_{n}
  154. H H
  155. S n S_{n}
  156. { 1 , , n } \{1,\ldots,n\}
  157. x , y G x,y\in G
  158. ( h x ( 1 ) , , h x ( n ) ; π x ) ( h y ( 1 ) , , h y ( n ) ; π y ) (h_{x}(1),\ldots,h_{x}(n);\pi_{x})\cdot(h_{y}(1),\ldots,h_{y}(n);\pi_{y})
  159. : = ( h x ( π y ( 1 ) ) h y ( 1 ) , , h x ( π y ( n ) ) h y ( n ) ; π x π y ) :=(h_{x}(\pi_{y}(1))\cdot h_{y}(1),\ldots,h_{x}(\pi_{y}(n))\cdot h_{y}(n);\pi_% {x}\circ\pi_{y})
  160. = ( h x y ( 1 ) , , h x y ( n ) ; π x y ) =(h_{xy}(1),\ldots,h_{xy}(n);\pi_{xy})
  161. G H S n , x ( h x ( 1 ) , , h x ( n ) ; π x ) G\to H\wr S_{n},\ x\mapsto(h_{x}(1),\ldots,h_{x}(n);\pi_{x})
  162. G G
  163. K H G K\leq H\leq G
  164. ( G : K ) = ( G : H ) ( H : K ) = n m (G:K)=(G:H)\cdot(H:K)=n\cdot m
  165. T G , K T_{G,K}
  166. T ~ H , K : H / H K / K \tilde{T}_{H,K}:\ H/H^{\prime}\to K/K^{\prime}
  167. T G , H T_{G,H}
  168. T G , K = T ~ H , K T G , H T_{G,K}=\tilde{T}_{H,K}\circ T_{G,H}
  169. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  170. H H
  171. G G
  172. ( h 1 , , h m ) (h_{1},\ldots,h_{m})
  173. K K
  174. H H
  175. G = ˙ i = 1 n g i H G=\dot{\cup}_{i=1}^{n}\,g_{i}H
  176. H = ˙ j = 1 m h j K H=\dot{\cup}_{j=1}^{m}\,h_{j}K
  177. G = ˙ i = 1 n ˙ j = 1 m g i h j K G=\dot{\cup}_{i=1}^{n}\,\dot{\cup}_{j=1}^{m}\,g_{i}h_{j}K
  178. G G
  179. K K
  180. x G x\in G
  181. y H y\in H
  182. π x S n \pi_{x}\in S_{n}
  183. σ y S m \sigma_{y}\in S_{m}
  184. h x ( i ) := g π x ( i ) - 1 x g i H h_{x}(i):=g_{\pi_{x}(i)}^{-1}xg_{i}\in H
  185. 1 i n 1\leq i\leq n
  186. k y ( j ) := h σ y ( j ) - 1 y h j K k_{y}(j):=h_{\sigma_{y}(j)}^{-1}yh_{j}\in K
  187. 1 j m 1\leq j\leq m
  188. T G , H ( x ) = i = 1 n h x ( i ) H T_{G,H}(x)=\prod_{i=1}^{n}\,h_{x}(i)\cdot H^{\prime}
  189. T ~ H , K ( y H ) = T H , K ( y ) = j = 1 m k y ( j ) K \tilde{T}_{H,K}(y\cdot H^{\prime})=T_{H,K}(y)=\prod_{j=1}^{m}\,k_{y}(j)\cdot K% ^{\prime}
  190. 1 i n 1\leq i\leq n
  191. 1 j m 1\leq j\leq m
  192. x g i h j = g π x ( i ) g π x ( i ) - 1 x g i h j = g π x ( i ) h x ( i ) h j = g π x ( i ) h σ y i ( j ) k y i ( j ) xg_{i}h_{j}=g_{\pi_{x}(i)}g_{\pi_{x}(i)}^{-1}xg_{i}h_{j}=g_{\pi_{x}(i)}h_{x}(i% )h_{j}=g_{\pi_{x}(i)}h_{\sigma_{y_{i}}(j)}k_{y_{i}}(j)
  193. h σ y i ( j ) - 1 g π x ( i ) - 1 x g i h j = k y i ( j ) h_{\sigma_{y_{i}}(j)}^{-1}g_{\pi_{x}(i)}^{-1}xg_{i}h_{j}=k_{y_{i}}(j)
  194. y i := h x ( i ) y_{i}:=h_{x}(i)
  195. x x
  196. T G , K T_{G,K}
  197. T G , K ( x ) = i = 1 n j = 1 m k y i ( j ) K = i = 1 n j = 1 m h σ y i ( j ) - 1 g π x ( i ) - 1 x g i h j K T_{G,K}(x)=\prod_{i=1}^{n}\,\prod_{j=1}^{m}\,k_{y_{i}}(j)\cdot K^{\prime}=% \prod_{i=1}^{n}\,\prod_{j=1}^{m}\,h_{\sigma_{y_{i}}(j)}^{-1}g_{\pi_{x}(i)}^{-1% }xg_{i}h_{j}\cdot K^{\prime}
  198. = i = 1 n j = 1 m h σ y i ( j ) - 1 h x ( i ) h j K = i = 1 n j = 1 m h σ y i ( j ) - 1 y i h j K =\prod_{i=1}^{n}\,\prod_{j=1}^{m}\,h_{\sigma_{y_{i}}(j)}^{-1}h_{x}(i)h_{j}% \cdot K^{\prime}=\prod_{i=1}^{n}\,\prod_{j=1}^{m}\,h_{\sigma_{y_{i}}(j)}^{-1}y% _{i}h_{j}\cdot K^{\prime}
  199. = i = 1 n T ~ H , K ( y i H ) = T ~ H , K ( i = 1 n y i H ) = T ~ H , K ( i = 1 n h x ( i ) H ) = T ~ H , K ( T G , H ( x ) ) =\prod_{i=1}^{n}\,\tilde{T}_{H,K}(y_{i}\cdot H^{\prime})=\tilde{T}_{H,K}(\prod% _{i=1}^{n}\,y_{i}\cdot H^{\prime})=\tilde{T}_{H,K}(\prod_{i=1}^{n}\,h_{x}(i)% \cdot H^{\prime})=\tilde{T}_{H,K}(T_{G,H}(x))
  200. G K n m × S n m G\to K^{n\cdot m}\times S_{n\cdot m}
  201. x ( x ( 1 , 1 ) , , x ( n , m ) ; γ x ) x\mapsto(\ell_{x}(1,1),\ldots,\ell_{x}(n,m);\gamma_{x})
  202. x ( i , j ) := ( ( g h ) γ x ( i , j ) ) - 1 x ( g h ) ( i , j ) K \ell_{x}(i,j):=((gh)_{\gamma_{x}(i,j)})^{-1}x(gh)_{(i,j)}\in K
  203. γ x S n m \gamma_{x}\in S_{n\cdot m}
  204. ( g h ) ( i , j ) := g i h j (gh)_{(i,j)}:=g_{i}h_{j}
  205. 1 i n 1\leq i\leq n
  206. 1 j m 1\leq j\leq m
  207. x ( i , j ) = h σ y i ( j ) - 1 g π x ( i ) - 1 x g i h j \ell_{x}(i,j)=h_{\sigma_{y_{i}}(j)}^{-1}g_{\pi_{x}(i)}^{-1}xg_{i}h_{j}
  208. γ x \gamma_{x}
  209. [ 1 , n ] × [ 1 , m ] [1,n]\times[1,m]
  210. γ x ( i , j ) = ( π x ( i ) , σ h x ( i ) ( j ) ) \gamma_{x}(i,j)=(\pi_{x}(i),\sigma_{h_{x}(i)}(j))
  211. σ h x ( i ) S m \sigma_{h_{x}(i)}\in S_{m}
  212. γ x S n m \gamma_{x}\in S_{n\cdot m}
  213. ( π x ; σ h x ( 1 ) , , σ h x ( n ) ) S n × S m n , (\pi_{x};\sigma_{h_{x}(1)},\ldots,\sigma_{h_{x}(n)})\in S_{n}\times S_{m}^{n},
  214. γ x \gamma_{x}
  215. G K n m × S n m G\to K^{n\cdot m}\times S_{n\cdot m}
  216. x ( x ( 1 , 1 ) , , x ( n , m ) ; γ x ) x\mapsto(\ell_{x}(1,1),\ldots,\ell_{x}(n,m);\gamma_{x})
  217. [ 1 , n ] × [ 1 , m ] [1,n]\times[1,m]
  218. n n
  219. S m S n S_{m}\wr S_{n}
  220. S m S_{m}
  221. S n S_{n}
  222. { 1 , , n } \{1,\ldots,n\}
  223. S m n × S n S_{m}^{n}\times S_{n}
  224. γ x γ z \gamma_{x}\cdot\gamma_{z}
  225. = ( σ h x ( 1 ) , , σ h x ( n ) ; π x ) ( σ h z ( 1 ) , , σ h z ( n ) ; π z ) =(\sigma_{h_{x}(1)},\ldots,\sigma_{h_{x}(n)};\pi_{x})\cdot(\sigma_{h_{z}(1)},% \ldots,\sigma_{h_{z}(n)};\pi_{z})
  226. = ( σ h x ( π z ( 1 ) ) σ h z ( 1 ) , , σ h x ( π z ( n ) ) σ h z ( n ) ; π x π z ) =(\sigma_{h_{x}(\pi_{z}(1))}\circ\sigma_{h_{z}(1)},\ldots,\sigma_{h_{x}(\pi_{z% }(n))}\circ\sigma_{h_{z}(n)};\pi_{x}\circ\pi_{z})
  227. = ( σ h x z ( 1 ) , , σ h x z ( n ) ; π x z ) =(\sigma_{h_{xz}(1)},\ldots,\sigma_{h_{xz}(n)};\pi_{xz})
  228. = γ x z =\gamma_{xz}
  229. x , z G x,z\in G
  230. D ( g f ) ( x ) = D ( g ) ( f ( x ) ) D ( f ) ( x ) D(g\circ f)(x)=D(g)(f(x))\circ D(f)(x)
  231. x E x\in E
  232. f : E F f:\,E\to F
  233. g : F G g:\,F\to G
  234. G S m S n , x ( σ h x ( 1 ) , , σ h x ( n ) ; π x ) G\to S_{m}\wr S_{n},\ x\mapsto(\sigma_{h_{x}(1)},\ldots,\sigma_{h_{x}(n)};\pi_% {x})
  235. G G
  236. S m S n S_{m}\wr S_{n}
  237. G G
  238. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  239. H < G H<G
  240. n = ( G : H ) n=(G:H)
  241. G G
  242. x G x\in G
  243. π x S n \pi_{x}\in S_{n}
  244. H H
  245. G G
  246. x g i H = g π x ( i ) H xg_{i}H=g_{\pi_{x}(i)}H
  247. g π x ( i ) - 1 x g i = : h x ( i ) H g_{\pi_{x}(i)}^{-1}xg_{i}=:h_{x}(i)\in H
  248. 1 i n 1\leq i\leq n
  249. π x \pi_{x}
  250. π x = j = 1 t ζ j \pi_{x}=\prod_{j=1}^{t}\,\zeta_{j}
  251. ζ j S n \zeta_{j}\in S_{n}
  252. f j 1 f_{j}\geq 1
  253. ( g j H , g ζ j ( j ) H , g ζ j 2 ( j ) H , , g ζ j f j - 1 ( j ) H ) = ( g j H , x g j H , x 2 g j H , , x f j - 1 g j H ) (g_{j}H,g_{\zeta_{j}(j)}H,g_{\zeta_{j}^{2}(j)}H,\ldots,g_{\zeta_{j}^{f_{j}-1}(% j)}H)=(g_{j}H,xg_{j}H,x^{2}g_{j}H,\ldots,x^{f_{j}-1}g_{j}H)
  254. 1 j t 1\leq j\leq t
  255. j = 1 t f j = n \sum_{j=1}^{t}\,f_{j}=n
  256. x x
  257. T G , H T_{G,H}
  258. T G , H ( x ) = j = 1 t g j - 1 x f j g j H T_{G,H}(x)=\prod_{j=1}^{t}\,g_{j}^{-1}x^{f_{j}}g_{j}\cdot H^{\prime}
  259. H H
  260. G G
  261. γ j , k := x k g j \gamma_{j,k}:=x^{k}g_{j}
  262. 0 k f j - 1 0\leq k\leq f_{j}-1
  263. 1 j t 1\leq j\leq t
  264. G = ˙ j = 1 t ˙ k = 0 f j - 1 x k g j H G=\dot{\cup}_{j=1}^{t}\,\dot{\cup}_{k=0}^{f_{j}-1}\,x^{k}g_{j}H
  265. 1 j t 1\leq j\leq t
  266. 0 k f j - 2 0\leq k\leq f_{j}-2
  267. x γ j , k = x x k g j = x k + 1 g j = γ j , k + 1 = γ j , k + 1 1 x\gamma_{j,k}=xx^{k}g_{j}=x^{k+1}g_{j}=\gamma_{j,k+1}=\gamma_{j,k+1}\cdot 1
  268. h x ( j , k ) = 1 h_{x}(j,k)=1
  269. k = f j - 1 k=f_{j}-1
  270. x γ j , f j - 1 = x x f j - 1 g j = x f j g j g j H x\gamma_{j,f_{j}-1}=xx^{f_{j}-1}g_{j}=x^{f_{j}}g_{j}\in g_{j}H
  271. g j - 1 x f j g j = h x ( j , f j - 1 ) H g_{j}^{-1}x^{f_{j}}g_{j}=h_{x}(j,f_{j}-1)\in H
  272. T G , H ( x ) = j = 1 t k = 0 f j - 1 h x ( j , k ) H = j = 1 t ( k = 0 f j - 2 1 ) h x ( j , f j - 1 ) H = j = 1 t g j - 1 x f j g j H T_{G,H}(x)=\prod_{j=1}^{t}\,\prod_{k=0}^{f_{j}-1}\,h_{x}(j,k)\cdot H^{\prime}=% \prod_{j=1}^{t}\,(\prod_{k=0}^{f_{j}-2}\,1)\cdot h_{x}(j,f_{j}-1)\cdot H^{% \prime}=\prod_{j=1}^{t}\,g_{j}^{-1}x^{f_{j}}g_{j}\cdot H^{\prime}
  273. G = ˙ j = 1 t x g j H G=\dot{\cup}_{j=1}^{t}\,\langle x\rangle g_{j}H
  274. G G
  275. x \langle x\rangle
  276. H H
  277. H G H\triangleleft G
  278. n = ( G : H ) n=(G:H)
  279. G G
  280. x H = H x xH=Hx
  281. x G x\in G
  282. G / H G/H
  283. n n
  284. x G x\in G
  285. f := ord ( x H ) f:=\mathrm{ord}(xH)
  286. x H xH
  287. G / H G/H
  288. x H \langle xH\rangle
  289. f f
  290. G / H G/H
  291. ( g 1 , , g t ) (g_{1},\ldots,g_{t})
  292. x , H \langle x,H\rangle
  293. G G
  294. t = n / f t=n/f
  295. G = ˙ j = 1 t g j x , H G=\dot{\cup}_{j=1}^{t}\,g_{j}\langle x,H\rangle
  296. G = ˙ j = 1 t ˙ k = 0 f - 1 g j x k H G=\dot{\cup}_{j=1}^{t}\,\dot{\cup}_{k=0}^{f-1}\,g_{j}x^{k}H
  297. H H
  298. G G
  299. x x
  300. T G , H T_{G,H}
  301. T G , H ( x ) = j = 1 t g j - 1 x f g j H T_{G,H}(x)=\prod_{j=1}^{t}\,g_{j}^{-1}x^{f}g_{j}\cdot H^{\prime}
  302. f f
  303. j j
  304. x H x\in H
  305. f = 1 f=1
  306. T G , H ( x ) = j = 1 t g j - 1 x g j H = j = 1 t x g j H = x j = 1 t g j H T_{G,H}(x)=\prod_{j=1}^{t}\,g_{j}^{-1}xg_{j}\cdot H^{\prime}=\prod_{j=1}^{t}\,% x^{g_{j}}\cdot H^{\prime}=x^{\sum_{j=1}^{t}\,g_{j}}\cdot H^{\prime}
  307. Tr G ( H ) = j = 1 t g j [ G ] \mathrm{Tr}_{G}(H)=\sum_{j=1}^{t}\,g_{j}\in\mathbb{Z}[G]
  308. H H
  309. G G
  310. x G H x\in G\setminus H
  311. G G
  312. H H
  313. G = x , H G=\langle x,H\rangle
  314. f = n f=n
  315. n n
  316. T G , H ( x ) = j = 1 1 1 - 1 x n 1 H = x n H T_{G,H}(x)=\prod_{j=1}^{1}\,1^{-1}\cdot x^{n}\cdot 1\cdot H^{\prime}=x^{n}% \cdot H^{\prime}
  317. G G
  318. G / G G/G^{\prime}
  319. ( p , p ) (p,p)
  320. G G
  321. p + 1 p+1
  322. H i < G H_{i}<G
  323. ( 1 i p + 1 ) (1\leq i\leq p+1)
  324. ( G : H i ) = p (G:H_{i})=p
  325. 1 i p + 1 1\leq i\leq p+1
  326. T i : G H i / H i T_{i}:\,G\to H_{i}/H_{i}^{\prime}
  327. G G
  328. H i H_{i}
  329. G G
  330. d ( G ) = 2 d(G)=2
  331. G = x , y G=\langle x,y\rangle
  332. x , y x,y
  333. x p , y p G x^{p},y^{p}\in G^{\prime}
  334. H i G H_{i}\triangleleft G
  335. h i h_{i}
  336. G G^{\prime}
  337. t i t_{i}
  338. H i = h i , G H_{i}=\langle h_{i},G^{\prime}\rangle
  339. G = t i , H i G=\langle t_{i},H_{i}\rangle
  340. h 1 = y h_{1}=y
  341. t 1 = x t_{1}=x
  342. h i = x y i - 2 h_{i}=xy^{i-2}
  343. t i = y t_{i}=y
  344. 2 i p + 1 2\leq i\leq p+1
  345. 1 i p + 1 1\leq i\leq p+1
  346. T i ( h i ) = h i Tr G ( H i ) H i = h i 1 + t i + t i 2 + + t i p - 1 H i T_{i}(h_{i})=h_{i}^{\mathrm{Tr}_{G}(H_{i})}\cdot H_{i}^{\prime}=h_{i}^{1+t_{i}% +t_{i}^{2}+\cdots+t_{i}^{p-1}}\cdot H_{i}^{\prime}
  347. h i t i - 1 h i t i t i - 2 h i t i 2 t i - p + 1 h i t i p - 1 H i = ( h i t i - 1 ) p t i p H i h_{i}\cdot t_{i}^{-1}h_{i}t_{i}\cdot t_{i}^{-2}h_{i}t_{i}^{2}\cdots t_{i}^{-p+% 1}h_{i}t_{i}^{p-1}\cdot H_{i}^{\prime}=(h_{i}t_{i}^{-1})^{p}t_{i}^{p}\cdot H_{% i}^{\prime}
  348. T i ( t i ) = t i p H i T_{i}(t_{i})=t_{i}^{p}\cdot H_{i}^{\prime}
  349. f = ord ( h i H i ) = 1 f=\mathrm{ord}(h_{i}H_{i})=1
  350. f = ord ( t i H i ) = p f=\mathrm{ord}(t_{i}H_{i})=p
  351. H i H_{i}^{\prime}
  352. G G^{\prime}
  353. p p
  354. H i H_{i}
  355. H i = [ H i , H i ] = [ G , H i ] = ( G ) h i - 1 H_{i}^{\prime}=[H_{i},H_{i}]=[G^{\prime},H_{i}]=(G^{\prime})^{h_{i}-1}
  356. G G
  357. G = s 1 , , s n G^{\prime}=\langle s_{1},\ldots,s_{n}\rangle
  358. H i = [ s 1 , h i ] , , [ s n , h i ] H_{i}^{\prime}=\langle[s_{1},h_{i}],\ldots,[s_{n},h_{i}]\rangle
  359. G G
  360. G / G G/G^{\prime}
  361. ( p 2 , p ) (p^{2},p)
  362. G G
  363. p + 1 p+1
  364. H i < G H_{i}<G
  365. ( 1 i p + 1 ) (1\leq i\leq p+1)
  366. ( G : H i ) = p (G:H_{i})=p
  367. p + 1 p+1
  368. U i < G U_{i}<G
  369. ( 1 i p + 1 ) (1\leq i\leq p+1)
  370. ( G : U i ) = p 2 (G:U_{i})=p^{2}
  371. 1 i p + 1 1\leq i\leq p+1
  372. T 1 , i : G H i / H i T_{1,i}:\,G\to H_{i}/H_{i}^{\prime}
  373. T 2 , i : G U i / U i T_{2,i}:\,G\to U_{i}/U_{i}^{\prime}
  374. G G
  375. H i H_{i}
  376. U i U_{i}
  377. G G
  378. d ( G ) = 2 d(G)=2
  379. G = x , y G=\langle x,y\rangle
  380. x , y x,y
  381. x p 2 , y p G x^{p^{2}},y^{p}\in G^{\prime}
  382. H i G H_{i}\triangleleft G
  383. ( 1 i p ) (1\leq i\leq p)
  384. h i = x y i - 1 h_{i}=xy^{i-1}
  385. H i = h i , G H_{i}=\langle h_{i},G^{\prime}\rangle
  386. H i / G H_{i}/G^{\prime}
  387. p 2 p^{2}
  388. H p + 1 H_{p+1}
  389. H p + 1 / G H_{p+1}/G^{\prime}
  390. ( p , p ) (p,p)
  391. h p + 1 = y h_{p+1}=y
  392. h 0 = x p h_{0}=x^{p}
  393. H p + 1 = h p + 1 , h 0 , G H_{p+1}=\langle h_{p+1},h_{0},G^{\prime}\rangle
  394. t i t_{i}
  395. G = t i , H i G=\langle t_{i},H_{i}\rangle
  396. 1 i p + 1 1\leq i\leq p+1
  397. t i = y t_{i}=y
  398. 1 i p 1\leq i\leq p
  399. t p + 1 = x t_{p+1}=x
  400. 1 i p + 1 1\leq i\leq p+1
  401. T 1 , i ( h i ) = h i Tr G ( H i ) H i = h i 1 + t i + t i 2 + + t i p - 1 H i T_{1,i}(h_{i})=h_{i}^{\mathrm{Tr}_{G}(H_{i})}\cdot H_{i}^{\prime}=h_{i}^{1+t_{% i}+t_{i}^{2}+\ldots+t_{i}^{p-1}}\cdot H_{i}^{\prime}
  402. ( h i t i - 1 ) p t i p H i (h_{i}t_{i}^{-1})^{p}t_{i}^{p}\cdot H_{i}^{\prime}
  403. T 1 , i ( t i ) = t i p H i T_{1,i}(t_{i})=t_{i}^{p}\cdot H_{i}^{\prime}
  404. f = ord ( h i H i ) = 1 f=\mathrm{ord}(h_{i}H_{i})=1
  405. f = ord ( t i H i ) = p f=\mathrm{ord}(t_{i}H_{i})=p
  406. U i G U_{i}\triangleleft G
  407. ( 1 i p + 1 ) (1\leq i\leq p+1)
  408. u 1 = y u_{1}=y
  409. u i = x p y i - 1 u_{i}=x^{p}y^{i-1}
  410. 2 i p 2\leq i\leq p
  411. u p + 1 = x p u_{p+1}=x^{p}
  412. U i = u i , G U_{i}=\langle u_{i},G^{\prime}\rangle
  413. U p + 1 = x p , G = G p G U_{p+1}=\langle x^{p},G^{\prime}\rangle=G^{p}\cdot G^{\prime}
  414. t i , w i t_{i},w_{i}
  415. G = t i , w i , U i G=\langle t_{i},w_{i},U_{i}\rangle
  416. t i = x , w i = x p t_{i}=x,w_{i}=x^{p}
  417. 1 i p 1\leq i\leq p
  418. t p + 1 = x , w p + 1 = y t_{p+1}=x,w_{p+1}=y
  419. f = ord ( u i U i ) = 1 f=\mathrm{ord}(u_{i}U_{i})=1
  420. f = ord ( t i U i ) = p 2 f=\mathrm{ord}(t_{i}U_{i})=p^{2}
  421. f = ord ( w i U i ) = p f=\mathrm{ord}(w_{i}U_{i})=p
  422. 1 i p + 1 1\leq i\leq p+1
  423. f = ord ( t p + 1 U p + 1 ) = p f=\mathrm{ord}(t_{p+1}U_{p+1})=p
  424. T 2 , i ( u i ) = u i Tr G ( U i ) U i = u i j = 0 p - 1 k = 0 p - 1 w i j t i k U i = j = 0 p - 1 k = 0 p - 1 ( w i j t i k ) - 1 u i w i j t i k U i T_{2,i}(u_{i})=u_{i}^{\mathrm{Tr}_{G}(U_{i})}\cdot U_{i}^{\prime}=u_{i}^{\sum_% {j=0}^{p-1}\,\sum_{k=0}^{p-1}\,w_{i}^{j}t_{i}^{k}}\cdot U_{i}^{\prime}=\prod_{% j=0}^{p-1}\,\prod_{k=0}^{p-1}\,(w_{i}^{j}t_{i}^{k})^{-1}u_{i}w_{i}^{j}t_{i}^{k% }\cdot U_{i}^{\prime}
  425. T 2 , i ( t i ) = t i p 2 U i T_{2,i}(t_{i})=t_{i}^{p^{2}}\cdot U_{i}^{\prime}
  426. T 2 , p + 1 ( t p + 1 ) = ( t p + 1 p ) 1 + w p + 1 + w p + 1 2 + + w p + 1 p - 1 U p + 1 T_{2,p+1}(t_{p+1})=(t_{p+1}^{p})^{1+w_{p+1}+w_{p+1}^{2}+\ldots+w_{p+1}^{p-1}}% \cdot U_{p+1}^{\prime}
  427. T 2 , i ( w i ) = ( w i p ) 1 + t i + t i 2 + + t i p - 1 U i T_{2,i}(w_{i})=(w_{i}^{p})^{1+t_{i}+t_{i}^{2}+\ldots+t_{i}^{p-1}}\cdot U_{i}^{\prime}
  428. 1 i p + 1 1\leq i\leq p+1
  429. H i H_{i}^{\prime}
  430. U i U_{i}^{\prime}
  431. G G
  432. G / G G/G^{\prime}
  433. ( H i ) i I (H_{i})_{i\in I}
  434. H i G H_{i}\triangleleft G
  435. G G^{\prime}
  436. I I
  437. i I i\in I
  438. T i := T G , H i T_{i}:=T_{G,H_{i}}
  439. G G
  440. H i / H i H_{i}/H_{i}^{\prime}
  441. ϰ H ( G ) = ( ker ( T i ) ) i I \varkappa_{H}(G)=(\ker(T_{i}))_{i\in I}
  442. G G
  443. ( H i ) i I (H_{i})_{i\in I}
  444. τ H ( G ) = ( H i / H i ) i I \tau_{H}(G)=(H_{i}/H_{i}^{\prime})_{i\in I}
  445. G G
  446. ( H i ) i I (H_{i})_{i\in I}
  447. G G
  448. G / G G/G^{\prime}
  449. ( p , p ) (p,p)
  450. G G
  451. p + 1 p+1
  452. H i < G H_{i}<G
  453. ( 1 i p + 1 ) (1\leq i\leq p+1)
  454. ( G : H i ) = p (G:H_{i})=p
  455. 1 i p + 1 1\leq i\leq p+1
  456. T i : G H i / H i T_{i}:\,G\to H_{i}/H_{i}^{\prime}
  457. G G
  458. H i H_{i}
  459. ϰ H ( G ) = ( ker ( T i ) ) 1 i p + 1 \varkappa_{H}(G)=(\ker(T_{i}))_{1\leq i\leq p+1}
  460. G G
  461. H 1 , , H p + 1 H_{1},\ldots,H_{p+1}
  462. ( ϰ ( i ) ) 1 i p + 1 (\varkappa(i))_{1\leq i\leq p+1}
  463. ϰ ( i ) = { 0 if ker ( T i ) = G , j if ker ( T i ) = H j for some 1 j p + 1. \varkappa(i)=\begin{cases}0&\,\text{if }\ker(T_{i})=G,\\ j&\,\text{if }\ker(T_{i})=H_{j}\,\text{ for some }1\leq j\leq p+1.\end{cases}
  464. ker ( T i ) \ker(T_{i})
  465. G G^{\prime}
  466. G G
  467. H i / H i H_{i}/H_{i}^{\prime}
  468. ker ( T i ) = G \ker(T_{i})=G^{\prime}
  469. K i = H π ( i ) K_{i}=H_{\pi(i)}
  470. V i = T π ( i ) V_{i}=T_{\pi(i)}
  471. π S p + 1 \pi\in S_{p+1}
  472. λ K ( G ) = ( ker ( V i ) ) 1 i p + 1 \lambda_{K}(G)=(\ker(V_{i}))_{1\leq i\leq p+1}
  473. K 1 , , K p + 1 K_{1},\ldots,K_{p+1}
  474. ( λ ( i ) ) 1 i p + 1 (\lambda(i))_{1\leq i\leq p+1}
  475. λ ( i ) = { 0 if ker ( V i ) = G , j if ker ( V i ) = K j for some 1 j p + 1. \lambda(i)=\begin{cases}0&\,\text{ if }\ker(V_{i})=G,\\ j&\,\text{ if }\ker(V_{i})=K_{j}\,\text{ for some }1\leq j\leq p+1.\end{cases}
  476. λ K ( G ) ϰ H ( G ) \lambda_{K}(G)\sim\varkappa_{H}(G)
  477. K λ ( i ) = ker ( V i ) = ker ( T π ( i ) ) = H ϰ ( π ( i ) ) = K π ~ - 1 ( ϰ ( π ( i ) ) ) K_{\lambda(i)}=\ker(V_{i})=\ker(T_{\pi(i)})=H_{\varkappa(\pi(i))}=K_{\tilde{% \pi}^{-1}(\varkappa(\pi(i)))}
  478. λ \lambda
  479. ϰ \varkappa
  480. λ = π ~ - 1 ϰ π \lambda=\tilde{\pi}^{-1}\circ\varkappa\circ\pi
  481. λ \lambda
  482. ϰ S p + 1 \varkappa^{S_{p+1}}
  483. ϰ \varkappa
  484. ( π , μ ) π ~ - 1 μ π (\pi,\mu)\mapsto\tilde{\pi}^{-1}\circ\mu\circ\pi
  485. S p + 1 S_{p+1}
  486. { 1 , , p + 1 } \{1,\ldots,p+1\}
  487. { 0 , , p + 1 } \{0,\ldots,p+1\}
  488. π ~ S p + 2 \tilde{\pi}\in S_{p+2}
  489. π S p + 1 \pi\in S_{p+1}
  490. π ~ ( 0 ) = 0 \tilde{\pi}(0)=0
  491. H 0 = G H_{0}=G
  492. K 0 = G K_{0}=G
  493. ϰ ( G ) = ϰ S p + 1 \varkappa(G)=\varkappa^{S_{p+1}}
  494. ϰ \varkappa
  495. G G
  496. # 0 ( G ) := # { 1 i p + 1 ϰ ( i ) = 0 } \#\mathcal{H}_{0}(G):=\#\{1\leq i\leq p+1\mid\varkappa(i)=0\}
  497. ker ( T i ) = G \ker(T_{i})=G
  498. G G
  499. p p
  500. 0 n p + 1 0\leq n\leq p+1
  501. G G
  502. G / G G/G^{\prime}
  503. ( p , p ) (p,p)
  504. # 0 ( G ) = n \#\mathcal{H}_{0}(G)=n
  505. p = 2 p=2
  506. 2 2
  507. G G
  508. G / G ( 2 , 2 ) G/G^{\prime}\simeq(2,2)
  509. # 0 ( G ) 2 \#\mathcal{H}_{0}(G)\geq 2
  510. 2 2
  511. G = C 2 × C 2 G=C_{2}\times C_{2}
  512. # 0 ( G ) = 3 \#\mathcal{H}_{0}(G)=3
  513. # 0 ( G ) \#\mathcal{H}_{0}(G)
  514. p = 3 p=3
  515. # 0 ( G ) = 0 \#\mathcal{H}_{0}(G)=0
  516. G = 27 , 4 G=\langle 27,4\rangle
  517. 9 9
  518. ϰ = ( 1111 ) \varkappa=(1111)
  519. # 0 ( G ) = 1 \#\mathcal{H}_{0}(G)=1
  520. G { 243 , 6 , 243 , 8 } G\in\{\langle 243,6\rangle,\langle 243,8\rangle\}
  521. ϰ { ( 0122 ) , ( 2034 ) } \varkappa\in\{(0122),(2034)\}
  522. # 0 ( G ) = 2 \#\mathcal{H}_{0}(G)=2
  523. G = 243 , 3 G=\langle 243,3\rangle
  524. ϰ = ( 0043 ) \varkappa=(0043)
  525. # 0 ( G ) = 3 \#\mathcal{H}_{0}(G)=3
  526. G = 81 , 7 G=\langle 81,7\rangle
  527. ϰ = ( 2000 ) \varkappa=(2000)
  528. # 0 ( G ) = 4 \#\mathcal{H}_{0}(G)=4
  529. G = 27 , 3 G=\langle 27,3\rangle
  530. 3 3
  531. ϰ = ( 0000 ) \varkappa=(0000)
  532. G G
  533. G / G G/G^{\prime}
  534. ( p 2 , p ) (p^{2},p)
  535. G G
  536. p + 1 p+1
  537. H i < G H_{i}<G
  538. ( 1 i p + 1 ) (1\leq i\leq p+1)
  539. ( G : H i ) = p (G:H_{i})=p
  540. p + 1 p+1
  541. U i < G U_{i}<G
  542. ( 1 i p + 1 ) (1\leq i\leq p+1)
  543. ( G : U i ) = p 2 (G:U_{i})=p^{2}
  544. H p + 1 = j = 1 p + 1 U j H_{p+1}=\prod_{j=1}^{p+1}\,U_{j}
  545. p 2 p^{2}
  546. U p + 1 = j = 1 p + 1 H j U_{p+1}=\cap_{j=1}^{p+1}\,H_{j}
  547. p 2 p^{2}
  548. Φ ( G ) \Phi(G)
  549. G G
  550. 1 i p + 1 1\leq i\leq p+1
  551. T 1 , i : G H i / H i T_{1,i}:\,G\to H_{i}/H_{i}^{\prime}
  552. G G
  553. H i H_{i}
  554. ϰ 1 , H , U ( G ) = ( ker ( T 1 , i ) ) 1 i p + 1 \varkappa_{1,H,U}(G)=(\ker(T_{1,i}))_{1\leq i\leq p+1}
  555. G G
  556. H 1 , , H p + 1 H_{1},\ldots,H_{p+1}
  557. U 1 , , U p + 1 U_{1},\ldots,U_{p+1}
  558. ( ϰ 1 ( i ) ) 1 i p + 1 (\varkappa_{1}(i))_{1\leq i\leq p+1}
  559. ϰ 1 ( i ) = { 0 if ker ( T 1 , i ) = H p + 1 , j if ker ( T 1 , i ) = U j for some 1 j p + 1. \varkappa_{1}(i)=\begin{cases}0&\,\text{ if }\ker(T_{1,i})=H_{p+1},\\ j&\,\text{ if }\ker(T_{1,i})=U_{j}\,\text{ for some }1\leq j\leq p+1.\end{cases}
  560. p p
  561. G G^{\prime}
  562. H j H_{j}
  563. 1 j p 1\leq j\leq p
  564. H j / G H_{j}/G^{\prime}
  565. p 2 p^{2}
  566. H p + 1 / G H_{p+1}/G^{\prime}
  567. ( p , p ) (p,p)
  568. 1 i p + 1 1\leq i\leq p+1
  569. T 2 , i : G U i / U i T_{2,i}:\,G\to U_{i}/U_{i}^{\prime}
  570. G G
  571. U i U_{i}
  572. ϰ 2 , U , H ( G ) = ( ker ( T 2 , i ) ) 1 i p + 1 \varkappa_{2,U,H}(G)=(\ker(T_{2,i}))_{1\leq i\leq p+1}
  573. G G
  574. U 1 , , U p + 1 U_{1},\ldots,U_{p+1}
  575. H 1 , , H p + 1 H_{1},\ldots,H_{p+1}
  576. ( ϰ 2 ( i ) ) 1 i p + 1 (\varkappa_{2}(i))_{1\leq i\leq p+1}
  577. ϰ 2 ( i ) = { 0 if ker ( T 2 , i ) = G , j if ker ( T 2 , i ) = H j for some 1 j p + 1. \varkappa_{2}(i)=\begin{cases}0&\,\text{ if }\ker(T_{2,i})=G,\\ j&\,\text{ if }\ker(T_{2,i})=H_{j}\,\text{ for some }1\leq j\leq p+1.\end{cases}
  578. ϰ H , U ( G ) = ( ϰ 1 , H , U ( G ) ; ϰ 2 , U , H ( G ) ) \varkappa_{H,U}(G)=(\varkappa_{1,H,U}(G);\varkappa_{2,U,H}(G))
  579. G G
  580. H 1 , , H p + 1 H_{1},\ldots,H_{p+1}
  581. U 1 , , U p + 1 U_{1},\ldots,U_{p+1}
  582. H p + 1 H_{p+1}
  583. U p + 1 = Φ ( G ) U_{p+1}=\Phi(G)
  584. G G
  585. K i = H τ ( i ) K_{i}=H_{\tau(i)}
  586. ( 1 i p ) (1\leq i\leq p)
  587. V 1 , i = T 1 , τ ( i ) V_{1,i}=T_{1,\tau(i)}
  588. τ S p \tau\in S_{p}
  589. W i = U σ ( i ) W_{i}=U_{\sigma(i)}
  590. ( 1 i p ) (1\leq i\leq p)
  591. p 2 p^{2}
  592. V 2 , i = T 2 , σ ( i ) V_{2,i}=T_{2,\sigma(i)}
  593. σ S p \sigma\in S_{p}
  594. λ 1 , K , W ( G ) = ( ker ( V 1 , i ) ) 1 i p + 1 \lambda_{1,K,W}(G)=(\ker(V_{1,i}))_{1\leq i\leq p+1}
  595. K 1 , , K p + 1 K_{1},\ldots,K_{p+1}
  596. W 1 , , W p + 1 W_{1},\ldots,W_{p+1}
  597. ( λ 1 ( i ) ) 1 i p + 1 (\lambda_{1}(i))_{1\leq i\leq p+1}
  598. λ 1 ( i ) = { 0 if ker ( V 1 , i ) = K p + 1 , j if ker ( V 1 , i ) = W j for some 1 j p + 1 , \lambda_{1}(i)=\begin{cases}0&\,\text{ if }\ker(V_{1,i})=K_{p+1},\\ j&\,\text{ if }\ker(V_{1,i})=W_{j}\,\text{ for some }1\leq j\leq p+1,\end{cases}
  599. λ 2 , W , K ( G ) = ( ker ( V 2 , i ) ) 1 i p + 1 \lambda_{2,W,K}(G)=(\ker(V_{2,i}))_{1\leq i\leq p+1}
  600. W 1 , , W p + 1 W_{1},\ldots,W_{p+1}
  601. K 1 , , K p + 1 K_{1},\ldots,K_{p+1}
  602. ( λ 2 ( i ) ) 1 i p + 1 (\lambda_{2}(i))_{1\leq i\leq p+1}
  603. λ 2 ( i ) = { 0 if ker ( V 2 , i ) = G , j if ker ( V 2 , i ) = K j for some 1 j p + 1. \lambda_{2}(i)=\begin{cases}0&\,\text{ if }\ker(V_{2,i})=G,\\ j&\,\text{ if }\ker(V_{2,i})=K_{j}\,\text{ for some }1\leq j\leq p+1.\end{cases}
  604. λ 1 , K , W ( G ) ϰ 1 , H , U ( G ) \lambda_{1,K,W}(G)\sim\varkappa_{1,H,U}(G)
  605. λ 2 , W , K ( G ) ϰ 2 , U , H ( G ) \lambda_{2,W,K}(G)\sim\varkappa_{2,U,H}(G)
  606. W λ 1 ( i ) = ker ( V 1 , i ) = ker ( T 1 , τ ^ ( i ) ) = U ϰ 1 ( τ ^ ( i ) ) = W σ ~ - 1 ( ϰ 1 ( τ ^ ( i ) ) ) W_{\lambda_{1}(i)}=\ker(V_{1,i})=\ker(T_{1,\hat{\tau}(i)})=U_{\varkappa_{1}(% \hat{\tau}(i))}=W_{\tilde{\sigma}^{-1}(\varkappa_{1}(\hat{\tau}(i)))}
  607. K λ 2 ( i ) = ker ( V 2 , i ) = ker ( T 2 , σ ^ ( i ) ) = H ϰ 2 ( σ ^ ( i ) ) = K τ ~ - 1 ( ϰ 2 ( σ ^ ( i ) ) ) K_{\lambda_{2}(i)}=\ker(V_{2,i})=\ker(T_{2,\hat{\sigma}(i)})=H_{\varkappa_{2}(% \hat{\sigma}(i))}=K_{\tilde{\tau}^{-1}(\varkappa_{2}(\hat{\sigma}(i)))}
  608. λ 1 \lambda_{1}
  609. ϰ 1 \varkappa_{1}
  610. λ 2 \lambda_{2}
  611. ϰ 2 \varkappa_{2}
  612. λ 1 = σ ~ - 1 ϰ 1 τ ^ \lambda_{1}=\tilde{\sigma}^{-1}\circ\varkappa_{1}\circ\hat{\tau}
  613. λ 2 = τ ~ - 1 ϰ 2 σ ^ \lambda_{2}=\tilde{\tau}^{-1}\circ\varkappa_{2}\circ\hat{\sigma}
  614. λ = ( λ 1 , λ 2 ) \lambda=(\lambda_{1},\lambda_{2})
  615. ϰ S p × S p \varkappa^{S_{p}\times S_{p}}
  616. ϰ = ( ϰ 1 , ϰ 2 ) \varkappa=(\varkappa_{1},\varkappa_{2})
  617. ( ( σ , τ ) , ( μ 1 , μ 2 ) ) ( σ ~ - 1 μ 1 τ ^ , τ ~ - 1 μ 2 σ ^ ) ((\sigma,\tau),(\mu_{1},\mu_{2}))\mapsto(\tilde{\sigma}^{-1}\circ\mu_{1}\circ% \hat{\tau},\tilde{\tau}^{-1}\circ\mu_{2}\circ\hat{\sigma})
  618. S p × S p S_{p}\times S_{p}
  619. { 1 , , p + 1 } \{1,\ldots,p+1\}
  620. { 0 , , p + 1 } \{0,\ldots,p+1\}
  621. π ^ S p + 1 \hat{\pi}\in S_{p+1}
  622. π ~ S p + 2 \tilde{\pi}\in S_{p+2}
  623. π S p \pi\in S_{p}
  624. π ^ ( p + 1 ) = π ~ ( p + 1 ) = p + 1 \hat{\pi}(p+1)=\tilde{\pi}(p+1)=p+1
  625. π ~ ( 0 ) = 0 \tilde{\pi}(0)=0
  626. H 0 = K 0 = G H_{0}=K_{0}=G
  627. K p + 1 = H p + 1 K_{p+1}=H_{p+1}
  628. U 0 = W 0 = H p + 1 U_{0}=W_{0}=H_{p+1}
  629. W p + 1 = U p + 1 = Φ ( G ) W_{p+1}=U_{p+1}=\Phi(G)
  630. ϰ ( G ) = ϰ S p × S p \varkappa(G)=\varkappa^{S_{p}\times S_{p}}
  631. ϰ = ( ϰ 1 , ϰ 2 ) \varkappa=(\varkappa_{1},\varkappa_{2})
  632. G G
  633. T 2 , i : G U i / U i T_{2,i}:\,G\to U_{i}/U_{i}^{\prime}
  634. G G
  635. U i U_{i}
  636. ( G : U i ) = p 2 (G:U_{i})=p^{2}
  637. 1 i p + 1 1\leq i\leq p+1
  638. T 2 , i = T ~ H j , U i T 1 , j T_{2,i}=\tilde{T}_{H_{j},U_{i}}\circ T_{1,j}
  639. T ~ H j , U i : H j / H j U i / U i \tilde{T}_{H_{j},U_{i}}:\,H_{j}/H_{j}^{\prime}\to U_{i}/U_{i}^{\prime}
  640. H j H_{j}
  641. U i U_{i}
  642. T 1 , j : G H j / H j T_{1,j}:\,G\to H_{j}/H_{j}^{\prime}
  643. G G
  644. H j H_{j}
  645. U i < H j < G U_{i}<H_{j}<G
  646. ( G : H j ) = p (G:H_{j})=p
  647. 1 j p + 1 1\leq j\leq p+1
  648. U 1 , , U p U_{1},\ldots,U_{p}
  649. H p + 1 H_{p+1}
  650. U p + 1 = Φ ( G ) U_{p+1}=\Phi(G)
  651. H 1 , , H p + 1 H_{1},\ldots,H_{p+1}
  652. ϰ 2 ( G ) \varkappa_{2}(G)
  653. ker ( T 2 , i ) = ker ( T ~ H j , U i T 1 , j ) ker ( T 1 , j ) \ker(T_{2,i})=\ker(\tilde{T}_{H_{j},U_{i}}\circ T_{1,j})\supset\ker(T_{1,j})
  654. ker ( T 2 , i ) ker ( T 1 , p + 1 ) \ker(T_{2,i})\supset\ker(T_{1,p+1})
  655. 1 i p 1\leq i\leq p
  656. ker ( T 2 , p + 1 ) j = 1 p + 1 ker ( T 1 , j ) \ker(T_{2,p+1})\supset\langle\cup_{j=1}^{p+1}\,\ker(T_{1,j})\rangle
  657. G = x , y G=\langle x,y\rangle
  658. x p G x^{p}\notin G^{\prime}
  659. y p G y^{p}\in G^{\prime}
  660. x y k - 1 xy^{k-1}
  661. 1 k p 1\leq k\leq p
  662. p 2 p^{2}
  663. G G^{\prime}
  664. ker ( T 2 , i ) \ker(T_{2,i})
  665. p p
  666. x p x^{p}
  667. ker ( T 1 , j ) \ker(T_{1,j})
  668. U i < H j < G U_{i}<H_{j}<G
  669. x y k - 1 ker ( T 2 , i ) xy^{k-1}\in\ker(T_{2,i})
  670. 1 i , k p 1\leq i,k\leq p
  671. ϰ 1 ( p + 1 ) = p + 1 \varkappa_{1}(p+1)=p+1
  672. x y k - 1 ker ( T 2 , p + 1 ) xy^{k-1}\in\ker(T_{2,p+1})
  673. 1 k p 1\leq k\leq p
  674. ϰ 1 = ( ( p + 1 ) p + 1 ) \varkappa_{1}=((p+1)^{p+1})
  675. ϰ 1 ( j ) = p + 1 \varkappa_{1}(j)=p+1
  676. 1 j p + 1 1\leq j\leq p+1
  677. π ( G ) \pi(G)
  678. G / N G/N
  679. G G
  680. N G N\triangleleft G
  681. φ \varphi
  682. G G
  683. G ~ \tilde{G}
  684. ker ( φ ) \ker(\varphi)
  685. N G N\triangleleft G
  686. φ : G A \varphi:\ G\to A
  687. G G
  688. A A
  689. φ ~ : G / G A \tilde{\varphi}:\ G/G^{\prime}\to A
  690. φ = φ ~ ω \varphi=\tilde{\varphi}\circ\omega
  691. ω : G G / G \omega:\ G\to G/G^{\prime}
  692. φ ~ \tilde{\varphi}
  693. ker ( φ ~ ) = ker ( φ ) / G \ker(\tilde{\varphi})=\ker(\varphi)/G^{\prime}
  694. φ ~ \tilde{\varphi}
  695. φ = φ ~ ω \varphi=\tilde{\varphi}\circ\omega
  696. φ ~ \tilde{\varphi}
  697. φ ~ ( x G ) = φ ~ ( ω ( x ) ) = ( φ ~ ω ) ( x ) = φ ( x ) \tilde{\varphi}(xG^{\prime})=\tilde{\varphi}(\omega(x))=(\tilde{\varphi}\circ% \omega)(x)=\varphi(x)
  698. x G x\in G
  699. φ ~ ( x G y G ) = φ ~ ( ( x y ) G ) = φ ( x y ) = φ ( x ) φ ( y ) = φ ~ ( x G ) φ ~ ( x G ) \tilde{\varphi}(xG^{\prime}\cdot yG^{\prime})=\tilde{\varphi}((xy)G^{\prime})=% \varphi(xy)=\varphi(x)\cdot\varphi(y)=\tilde{\varphi}(xG^{\prime})\cdot\tilde{% \varphi}(xG^{\prime})
  700. x , y G x,y\in G
  701. φ ~ \tilde{\varphi}
  702. x , y G x,y\in G
  703. φ ( [ x , y ] ) = [ φ ( x ) , φ ( y ) ] = 1 \varphi([x,y])=[\varphi(x),\varphi(y)]=1
  704. A A
  705. G G^{\prime}
  706. G G
  707. ker ( φ ) \ker(\varphi)
  708. φ ~ \tilde{\varphi}
  709. x G = y G xG^{\prime}=yG^{\prime}
  710. \Rightarrow
  711. x - 1 y G ker ( φ ) x^{-1}y\in G^{\prime}\leq\ker(\varphi)
  712. \Rightarrow
  713. φ ~ ( x G ) - 1 φ ~ ( y G ) = φ ~ ( x - 1 y G ) = φ ( x - 1 y ) = 1 \tilde{\varphi}(xG^{\prime})^{-1}\cdot\tilde{\varphi}(yG^{\prime})=\tilde{% \varphi}(x^{-1}yG^{\prime})=\varphi(x^{-1}y)=1
  714. \Rightarrow
  715. φ ~ ( x G ) = φ ~ ( y G ) \tilde{\varphi}(xG^{\prime})=\tilde{\varphi}(yG^{\prime})
  716. G G
  717. G ~ \tilde{G}
  718. G ~ = φ ( G ) \tilde{G}=\varphi(G)
  719. G G
  720. φ : G G ~ \varphi:\ G\to\tilde{G}
  721. H ~ = φ ( H ) \tilde{H}=\varphi(H)
  722. H G H\leq G
  723. H ~ \tilde{H}
  724. H H
  725. H ~ = φ ( H ) \tilde{H}^{\prime}=\varphi(H^{\prime})
  726. ker ( φ ) H \ker(\varphi)\leq H
  727. H ~ H / ker ( φ ) \tilde{H}\simeq H/\ker(\varphi)
  728. φ \varphi
  729. φ ~ : H / H H ~ / H ~ \tilde{\varphi}:\ H/H^{\prime}\to\tilde{H}/\tilde{H}^{\prime}
  730. H ~ / H ~ \tilde{H}/\tilde{H}^{\prime}
  731. H / H H/H^{\prime}
  732. H / H H/H^{\prime}
  733. ker ( φ ) H \ker(\varphi)\leq H^{\prime}
  734. H ~ H / ker ( φ ) \tilde{H}^{\prime}\simeq H^{\prime}/\ker(\varphi)
  735. φ ~ \tilde{\varphi}
  736. H ~ / H ~ H / H \tilde{H}/\tilde{H}^{\prime}\simeq H/H^{\prime}
  737. φ ( H ) = φ ( [ H , H ] ) = φ ( [ u , v ] u , v H ) \varphi(H^{\prime})=\varphi([H,H])=\varphi(\langle[u,v]\mid u,v\in H\rangle)
  738. = [ φ ( u ) , φ ( v ) ] u , v H = [ φ ( H ) , φ ( H ) ] = φ ( H ) = H ~ =\langle[\varphi(u),\varphi(v)]\mid u,v\in H\rangle=[\varphi(H),\varphi(H)]=% \varphi(H)^{\prime}=\tilde{H}^{\prime}
  739. ker ( φ ) H \ker(\varphi)\leq H
  740. φ \varphi
  741. φ | H : H H ~ \varphi|_{H}:\ H\to\tilde{H}
  742. H ~ = φ ( H ) H / ker ( φ ) \tilde{H}=\varphi(H)\simeq H/\ker(\varphi)
  743. ( ω H ~ φ | H ) : H H ~ / H ~ (\omega_{\tilde{H}}\circ\varphi|_{H}):\ H\to\tilde{H}/\tilde{H}^{\prime}
  744. H H
  745. H ~ / H ~ \tilde{H}/\tilde{H}^{\prime}
  746. H / H H/H^{\prime}
  747. φ ~ : H / H H ~ / H ~ \tilde{\varphi}:\ H/H^{\prime}\to\tilde{H}/\tilde{H}^{\prime}
  748. φ ~ ω H = ω H ~ φ | H \tilde{\varphi}\circ\omega_{H}=\omega_{\tilde{H}}\circ\varphi|_{H}
  749. H ~ / H ~ ( H / H ) / ker ( φ ~ ) \tilde{H}/\tilde{H}^{\prime}\simeq(H/H^{\prime})/\ker(\tilde{\varphi})
  750. φ ~ \tilde{\varphi}
  751. ker ( φ ~ ) = ker ( ω H ~ φ | H ) / H = ( H ker ( φ ) ) / H \ker(\tilde{\varphi})=\ker(\omega_{\tilde{H}}\circ\varphi|_{H})/H^{\prime}=(H^% {\prime}\cdot\ker(\varphi))/H^{\prime}
  752. ker ( φ ) H \ker(\varphi)\leq H^{\prime}
  753. H ~ = φ ( H ) H / ker ( φ ) \tilde{H}^{\prime}=\varphi(H^{\prime})\simeq H^{\prime}/\ker(\varphi)
  754. φ ~ \tilde{\varphi}
  755. ker ( φ ~ ) = H / H = 1 \ker(\tilde{\varphi})=H^{\prime}/H^{\prime}=1
  756. H ~ / H ~ H / H \tilde{H}/\tilde{H}^{\prime}\preceq H/H^{\prime}
  757. H ~ / H ~ ( H / H ) / ker ( φ ~ ) \tilde{H}/\tilde{H}^{\prime}\simeq(H/H^{\prime})/\ker(\tilde{\varphi})
  758. H ~ / H ~ = H / H \tilde{H}/\tilde{H}^{\prime}=H/H^{\prime}
  759. H ~ / H ~ H / H \tilde{H}/\tilde{H}^{\prime}\simeq H/H^{\prime}
  760. G G
  761. G ~ \tilde{G}
  762. G ~ = φ ( G ) \tilde{G}=\varphi(G)
  763. G G
  764. φ : G G ~ \varphi:\ G\to\tilde{G}
  765. H ~ = φ ( H ) \tilde{H}=\varphi(H)
  766. H G H\leq G
  767. n = ( G : H ) n=(G:H)
  768. T G , H T_{G,H}
  769. G G
  770. H / H H/H^{\prime}
  771. T G ~ , H ~ T_{\tilde{G},\tilde{H}}
  772. G ~ \tilde{G}
  773. H ~ / H ~ \tilde{H}/\tilde{H}^{\prime}
  774. ker ( φ ) H \ker(\varphi)\leq H
  775. ( φ ( g 1 ) , , φ ( g n ) ) (\varphi(g_{1}),\ldots,\varphi(g_{n}))
  776. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  777. H H
  778. G G
  779. H ~ \tilde{H}
  780. G ~ \tilde{G}
  781. φ ( ker ( T G , H ) ) ker ( T G ~ , H ~ ) \varphi(\ker(T_{G,H}))\leq\ker(T_{\tilde{G},\tilde{H}})
  782. ker ( φ ) H \ker(\varphi)\leq H^{\prime}
  783. φ ( ker ( T G , H ) ) = ker ( T G ~ , H ~ ) \varphi(\ker(T_{G,H}))=\ker(T_{\tilde{G},\tilde{H}})
  784. ( g 1 , , g n ) (g_{1},\ldots,g_{n})
  785. H H
  786. G G
  787. G = ˙ i = 1 n g i H G=\dot{\cup}_{i=1}^{n}\,g_{i}H
  788. φ ( G ) = ˙ i = 1 n φ ( g i ) φ ( H ) \varphi(G)=\dot{\cup}_{i=1}^{n}\,\varphi(g_{i})\varphi(H)
  789. 1 j , k n 1\leq j,k\leq n
  790. φ ( g j ) φ ( H ) = φ ( g k ) φ ( H ) \varphi(g_{j})\varphi(H)=\varphi(g_{k})\varphi(H)
  791. \Leftrightarrow
  792. φ ( H ) = φ ( g j ) - 1 φ ( g k ) φ ( H ) = φ ( g j - 1 g k ) φ ( H ) \varphi(H)=\varphi(g_{j})^{-1}\varphi(g_{k})\varphi(H)=\varphi(g_{j}^{-1}g_{k}% )\varphi(H)
  793. \Leftrightarrow
  794. φ ( g j - 1 g k ) = φ ( h ) \varphi(g_{j}^{-1}g_{k})=\varphi(h)
  795. h H h\in H
  796. \Leftrightarrow
  797. φ ( h - 1 g j - 1 g k ) = 1 \varphi(h^{-1}g_{j}^{-1}g_{k})=1
  798. \Leftrightarrow
  799. h - 1 g j - 1 g k = : k ker ( φ ) h^{-1}g_{j}^{-1}g_{k}=:k\in\ker(\varphi)
  800. ker ( φ ) H \ker(\varphi)\leq H
  801. g j - 1 g k = h k H g_{j}^{-1}g_{k}=hk\in H
  802. j = k j=k
  803. φ ~ : H / H H ~ / H ~ \tilde{\varphi}:\ H/H^{\prime}\to\tilde{H}/\tilde{H}^{\prime}
  804. x G x\in G
  805. φ ~ ( T G , H ( x ) ) = φ ~ ( i = 1 n g π x ( i ) - 1 x g i H ) = i = 1 n φ ( g π x ( i ) ) - 1 φ ( x ) φ ( g i ) ) φ ( H ) = \tilde{\varphi}(T_{G,H}(x))=\tilde{\varphi}(\prod_{i=1}^{n}\,g_{\pi_{x}(i)}^{-% 1}xg_{i}\cdot H^{\prime})=\prod_{i=1}^{n}\,\varphi(g_{\pi_{x}(i)})^{-1}\varphi% (x)\varphi(g_{i}))\cdot\varphi(H^{\prime})=
  806. φ ( H ) = φ ( H ) = H ~ \varphi(H^{\prime})=\varphi(H)^{\prime}=\tilde{H}^{\prime}
  807. T G ~ , H ~ ( φ ( x ) ) T_{\tilde{G},\tilde{H}}(\varphi(x))
  808. ( φ ( g 1 ) , , φ ( g n ) ) (\varphi(g_{1}),\ldots,\varphi(g_{n}))
  809. H ~ \tilde{H}
  810. G ~ \tilde{G}
  811. ker ( φ ) H \ker(\varphi)\leq H
  812. φ ~ T G , H = T G ~ , H ~ φ \tilde{\varphi}\circ T_{G,H}=T_{\tilde{G},\tilde{H}}\circ\varphi
  813. φ ( ker ( T G , H ) ) ker ( T G ~ , H ~ ) \varphi(\ker(T_{G,H}))\leq\ker(T_{\tilde{G},\tilde{H}})
  814. ker ( φ ) H \ker(\varphi)\leq H
  815. ker ( φ ) H \ker(\varphi)\leq H^{\prime}
  816. φ ~ \tilde{\varphi}
  817. T G , H = φ ~ - 1 T G ~ , H ~ φ T_{G,H}=\tilde{\varphi}^{-1}\circ T_{\tilde{G},\tilde{H}}\circ\varphi
  818. φ ( ker ( T G , H ) ) = ker ( T G ~ , H ~ ) \varphi(\ker(T_{G,H}))=\ker(T_{\tilde{G},\tilde{H}})
  819. ker ( T G , H ) ker ( T G ~ , H ~ ) \ker(T_{G,H})\preceq\ker(T_{\tilde{G},\tilde{H}})
  820. φ ( ker ( T G , H ) ) ker ( T G ~ , H ~ ) \varphi(\ker(T_{G,H}))\leq\ker(T_{\tilde{G},\tilde{H}})
  821. ker ( T G , H ) = ker ( T G ~ , H ~ ) \ker(T_{G,H})=\ker(T_{\tilde{G},\tilde{H}})
  822. φ ( ker ( T G , H ) ) = ker ( T G ~ , H ~ ) \varphi(\ker(T_{G,H}))=\ker(T_{\tilde{G},\tilde{H}})
  823. G G
  824. G ~ \tilde{G}
  825. G ~ = φ ( G ) \tilde{G}=\varphi(G)
  826. G G
  827. φ : G G ~ \varphi:\ G\to\tilde{G}
  828. G / G G ~ / G ~ G/G^{\prime}\simeq\tilde{G}/\tilde{G}^{\prime}
  829. ( H i ) i I (H_{i})_{i\in I}
  830. H i G H_{i}\triangleleft G
  831. G G^{\prime}
  832. I I
  833. H i ~ = φ ( H i ) \tilde{H_{i}}=\varphi(H_{i})
  834. H i H_{i}
  835. φ \varphi
  836. i I i\in I
  837. i I i\in I
  838. T i := T G , H i T_{i}:=T_{G,H_{i}}
  839. G G
  840. H i / H i H_{i}/H_{i}^{\prime}
  841. T ~ i := T G ~ , H ~ i \tilde{T}_{i}:=T_{\tilde{G},\tilde{H}_{i}}
  842. G ~ \tilde{G}
  843. H ~ i / H ~ i \tilde{H}_{i}/\tilde{H}_{i}^{\prime}
  844. J I J\subseteq I
  845. I I
  846. ϰ H ( G ) = ( ker ( T j ) ) j J \varkappa_{H}(G)=(\ker(T_{j}))_{j\in J}
  847. G G
  848. ( H j ) j J (H_{j})_{j\in J}
  849. τ H ( G ) = ( H j / H j ) j J \tau_{H}(G)=(H_{j}/H_{j}^{\prime})_{j\in J}
  850. G G
  851. ( H j ) j J (H_{j})_{j\in J}
  852. ker ( φ ) j J H j \ker(\varphi)\leq\cap_{j\in J}\,H_{j}
  853. τ H ~ ( G ~ ) τ H ( G ) \tau_{\tilde{H}}(\tilde{G})\preceq\tau_{H}(G)
  854. H ~ j / H ~ j H j / H j \tilde{H}_{j}/\tilde{H}_{j}^{\prime}\preceq H_{j}/H_{j}^{\prime}
  855. j J j\in J
  856. ϰ H ( G ) ϰ H ~ ( G ~ ) \varkappa_{H}(G)\preceq\varkappa_{\tilde{H}}(\tilde{G})
  857. ker ( T j ) ker ( T ~ j ) \ker(T_{j})\preceq\ker(\tilde{T}_{j})
  858. j J j\in J
  859. ker ( φ ) j J H j \ker(\varphi)\leq\cap_{j\in J}\,H_{j}^{\prime}
  860. τ H ~ ( G ~ ) = τ H ( G ) \tau_{\tilde{H}}(\tilde{G})=\tau_{H}(G)
  861. H ~ j / H ~ j = H j / H j \tilde{H}_{j}/\tilde{H}_{j}^{\prime}=H_{j}/H_{j}^{\prime}
  862. j J j\in J
  863. ϰ H ( G ) = ϰ H ~ ( G ~ ) \varkappa_{H}(G)=\varkappa_{\tilde{H}}(\tilde{G})
  864. ker ( T j ) = ker ( T ~ j ) \ker(T_{j})=\ker(\tilde{T}_{j})
  865. j J j\in J
  866. G G
  867. G ~ \tilde{G}
  868. G ~ = φ ( G ) G / ker ( φ ) \tilde{G}=\varphi(G)\simeq G/\ker(\varphi)
  869. G G
  870. φ : G G ~ \varphi:\ G\to\tilde{G}
  871. σ Aut ( G ) \sigma\in\mathrm{Aut}(G)
  872. G G
  873. σ ( ker ( φ ) ) ker ( φ ) \sigma(\ker(\varphi))\leq\ker(\varphi)
  874. σ ~ : G ~ G ~ \tilde{\sigma}:\,\tilde{G}\to\tilde{G}
  875. φ σ = σ ~ φ \varphi\circ\sigma=\tilde{\sigma}\circ\varphi
  876. σ ( ker ( φ ) ) = ker ( φ ) \sigma(\ker(\varphi))=\ker(\varphi)
  877. σ ~ Aut ( G ~ ) \tilde{\sigma}\in\mathrm{Aut}(\tilde{G})
  878. G ~ = φ ( G ) G / ker ( φ ) \tilde{G}=\varphi(G)\simeq G/\ker(\varphi)
  879. σ ~ ( g ker ( φ ) ) = ^ σ ~ ( φ ( g ) ) = φ ( σ ( g ) ) = ^ σ ( g ) ker ( φ ) \tilde{\sigma}(g\ker(\varphi))\hat{=}\tilde{\sigma}(\varphi(g))=\varphi(\sigma% (g))\hat{=}\sigma(g)\ker(\varphi)
  880. g G g\in G
  881. σ ~ \tilde{\sigma}
  882. σ ( ker ( φ ) ) ker ( φ ) \sigma(\ker(\varphi))\leq\ker(\varphi)
  883. g ker ( φ ) = h ker ( φ ) g\ker(\varphi)=h\ker(\varphi)
  884. \Rightarrow
  885. h - 1 g ker ( φ ) h^{-1}g\in\ker(\varphi)
  886. \Rightarrow
  887. σ ( h - 1 g ) ker ( φ ) \sigma(h^{-1}g)\in\ker(\varphi)
  888. \Rightarrow
  889. σ ( g ) ker ( φ ) = σ ( h ) ker ( φ ) \sigma(g)\ker(\varphi)=\sigma(h)\ker(\varphi)
  890. σ ( ker ( φ ) ) = ker ( φ ) \sigma(\ker(\varphi))=\ker(\varphi)
  891. σ ~ \tilde{\sigma}
  892. σ ~ ( g ker ( φ ) ) = ^ σ ( g ) ker ( φ ) = ker ( φ ) \tilde{\sigma}(g\ker(\varphi))\hat{=}\sigma(g)\ker(\varphi)=\ker(\varphi)
  893. \Rightarrow
  894. σ ( g ) ker ( φ ) \sigma(g)\in\ker(\varphi)
  895. \Rightarrow
  896. g = σ - 1 ( σ ( g ) ) ker ( φ ) g=\sigma^{-1}(\sigma(g))\in\ker(\varphi)
  897. σ - 1 ( ker ( φ ) ) ker ( φ ) \sigma^{-1}(\ker(\varphi))\leq\ker(\varphi)
  898. G G
  899. G / G G/G^{\prime}
  900. ω : G G / G \omega:\,G\to G/G^{\prime}
  901. σ ¯ Aut ( G / G ) \bar{\sigma}\in\mathrm{Aut}(G/G^{\prime})
  902. ω σ = σ ¯ ω \omega\circ\sigma=\bar{\sigma}\circ\omega
  903. σ ¯ ( g G ) = σ ¯ ( ω ( g ) ) = ω ( σ ( g ) ) = σ ( g ) G \bar{\sigma}(gG^{\prime})=\bar{\sigma}(\omega(g))=\omega(\sigma(g))=\sigma(g)G% ^{\prime}
  904. g G g\in G
  905. σ ¯ \bar{\sigma}
  906. σ ( g ) G = σ ¯ ( g G ) = G \sigma(g)G^{\prime}=\bar{\sigma}(gG^{\prime})=G^{\prime}
  907. \Rightarrow
  908. σ ( g ) G \sigma(g)\in G^{\prime}
  909. \Rightarrow
  910. g = σ - 1 ( σ ( g ) ) G g=\sigma^{-1}(\sigma(g))\in G^{\prime}
  911. G G^{\prime}
  912. G G
  913. G G
  914. σ Aut ( G ) \sigma\in\mathrm{Aut}(G)
  915. G / G G/G^{\prime}
  916. σ ( g ) G = σ ¯ ( g G ) = g - 1 G \sigma(g)G^{\prime}=\bar{\sigma}(gG^{\prime})=g^{-1}G^{\prime}
  917. σ ( g ) g - 1 ( mod G ) \sigma(g)\equiv g^{-1}\;\;(\mathop{{\rm mod}}G^{\prime})
  918. g G g\in G
  919. G G
  920. σ \sigma
  921. σ ( ker ( φ ) ) = ker ( φ ) \sigma(\ker(\varphi))=\ker(\varphi)
  922. G ~ \tilde{G}
  923. σ \sigma
  924. σ ~ \tilde{\sigma}
  925. φ \varphi
  926. σ ( g ) G = σ ¯ ( g G ) = g - 1 G \sigma(g)G^{\prime}=\bar{\sigma}(gG^{\prime})=g^{-1}G^{\prime}
  927. g G g\in G
  928. σ ~ ( x ) G ~ = σ ~ ( φ ( g ) ) G ~ = φ ( σ ( g ) ) φ ( G ) = φ ( g - 1 ) φ ( G ) = φ ( g ) - 1 G ~ = x - 1 G ~ \tilde{\sigma}(x)\tilde{G}^{\prime}=\tilde{\sigma}(\varphi(g))\tilde{G}^{% \prime}=\varphi(\sigma(g))\varphi(G^{\prime})=\varphi(g^{-1})\varphi(G^{\prime% })=\varphi(g)^{-1}\tilde{G}^{\prime}=x^{-1}\tilde{G}^{\prime}
  929. x = φ ( g ) φ ( G ) = G ~ x=\varphi(g)\in\varphi(G)=\tilde{G}
  930. π ( G ) \pi(G)
  931. G G
  932. π ( G ) = G / N \pi(G)=G/N
  933. G G
  934. N = γ c ( G ) G N=\gamma_{c}(G)\triangleleft G
  935. G G
  936. c c
  937. G G
  938. π \pi
  939. G G
  940. π ( G ) = G / γ c ( G ) \pi(G)=G/\gamma_{c}(G)
  941. ker ( π ) = γ c ( G ) \ker(\pi)=\gamma_{c}(G)
  942. p p
  943. G G
  944. c = cl ( G ) 2 c=\mathrm{cl}(G)\geq 2
  945. G G
  946. π ( G ) \pi(G)
  947. τ ( π ( G ) ) τ ( G ) \tau(\pi(G))\preceq\tau(G)
  948. ϰ ( G ) ϰ ( π ( G ) ) \varkappa(G)\preceq\varkappa(\pi(G))
  949. G H G G^{\prime}\leq H\leq G
  950. ker ( π ) = γ c ( G ) γ 2 ( G ) = G H \ker(\pi)=\gamma_{c}(G)\leq\gamma_{2}(G)=G^{\prime}\leq H
  951. c 2 c\geq 2
  952. G G
  953. G / G G/G^{\prime}
  954. 2 2
  955. ( p , p ) (p,p)
  956. G G
  957. cc ( G ) = 1 \mathrm{cc}(G)=1
  958. c = cl ( G ) 3 c=\mathrm{cl}(G)\geq 3
  959. p p
  960. τ ( G ) \tau(G)
  961. ϰ ( G ) \varkappa(G)
  962. π ( G ) \pi(G)
  963. p 3 p\geq 3
  964. τ ( G ) i = ( p , p ) \tau(G)_{i}=(p,p)
  965. ϰ ( G ) i = 0 \varkappa(G)_{i}=0
  966. 2 i p + 1 2\leq i\leq p+1
  967. c 3 c\geq 3
  968. ker ( π ) = γ c ( G ) γ 3 ( G ) = H i \ker(\pi)=\gamma_{c}(G)\leq\gamma_{3}(G)=H_{i}^{\prime}
  969. p p
  970. H 2 , , H p + 1 H_{2},\ldots,H_{p+1}
  971. G G
  972. c 3 c\geq 3
  973. p 3 p\geq 3
  974. p p
  975. G = G 0 3 ( 0 , 1 ) G=G^{3}_{0}(0,1)
  976. p 3 p^{3}
  977. p 2 p^{2}
  978. c = 2 c=2
  979. p p
  980. ϰ = ( 1 p + 1 ) \varkappa=(1^{p+1})
  981. ϰ = ( 0 p + 1 ) \varkappa=(0^{p+1})
  982. π ( G ) \pi(G)
  983. p p
  984. ( p , p ) (p,p)
  985. p = 2 p=2
  986. 2 2
  987. 1 1
  988. c = 2 c=2
  989. G = G 0 3 ( 0 , 1 ) G=G^{3}_{0}(0,1)
  990. ϰ = ( 123 ) \varkappa=(123)
  991. G = G 0 3 ( 0 , 0 ) G=G^{3}_{0}(0,0)
  992. ϰ = ( 023 ) \varkappa=(023)
  993. π ( G ) = C 2 × C 2 \pi(G)=C_{2}\times C_{2}
  994. ϰ = ( 000 ) \varkappa=(000)
  995. G G
  996. cc ( G ) = 1 \mathrm{cc}(G)=1
  997. c = m - 1 = cl ( G ) 4 c=m-1=\mathrm{cl}(G)\geq 4
  998. m 5 m\geq 5
  999. p + 1 p+1
  1000. τ ( G ) \tau(G)
  1001. ϰ ( G ) \varkappa(G)
  1002. π ( G ) \pi(G)
  1003. k = k ( G ) 1 k=k(G)\geq 1
  1004. k 1 k\geq 1
  1005. p 3 p\geq 3
  1006. ϰ ( G ) i = 0 \varkappa(G)_{i}=0
  1007. 1 i p + 1 1\leq i\leq p+1
  1008. m 5 m\geq 5
  1009. k 1 k\geq 1
  1010. ker ( π ) = γ m - 1 ( G ) γ m - k ( G ) H i \ker(\pi)=\gamma_{m-1}(G)\leq\gamma_{m-k}(G)\leq H_{i}^{\prime}
  1011. p + 1 p+1
  1012. H 1 , , H p + 1 H_{1},\ldots,H_{p+1}
  1013. G G
  1014. k 1 k\geq 1
  1015. c 4 c\geq 4
  1016. G = G 0 c + 1 ( 0 , 1 ) G=G^{c+1}_{0}(0,1)
  1017. ϰ = ( 10 p ) \varkappa=(10^{p})
  1018. G = G 0 c + 1 ( 1 , 0 ) G=G^{c+1}_{0}(1,0)
  1019. ϰ = ( 20 p ) \varkappa=(20^{p})
  1020. k = 0 k=0
  1021. ϰ = ( 0 p + 1 ) \varkappa=(0^{p+1})
  1022. π ( G ) = G 0 c ( 0 , 0 ) \pi(G)=G^{c}_{0}(0,0)
  1023. p = 3 p=3
  1024. G G
  1025. G / G ( 3 , 3 ) G/G^{\prime}\simeq(3,3)
  1026. cc ( G ) 2 \mathrm{cc}(G)\geq 2
  1027. c = cl ( G ) 4 c=\mathrm{cl}(G)\geq 4
  1028. τ ( G ) \tau(G)
  1029. ϰ ( G ) \varkappa(G)
  1030. π ( G ) \pi(G)
  1031. c 4 c\geq 4
  1032. ker ( π ) = γ c ( G ) γ 4 ( G ) H i \ker(\pi)=\gamma_{c}(G)\leq\gamma_{4}(G)\leq H_{i}^{\prime}
  1033. H 3 , H 4 H_{3},H_{4}
  1034. G G
  1035. c 4 c\geq 4
  1036. 3 3
  1037. c = 3 c=3
  1038. G { 243 , 3 , 243 , 6 , 243 , 8 } G\in\{\langle 243,3\rangle,\langle 243,6\rangle,\langle 243,8\rangle\}
  1039. ϰ { ( 0043 ) , ( 0122 ) , ( 2034 ) } \varkappa\in\{(0043),(0122),(2034)\}
  1040. ϰ = ( 0000 ) \varkappa=(0000)
  1041. π ( G ) = G 0 3 ( 0 , 0 ) \pi(G)=G^{3}_{0}(0,0)
  1042. p = 3 p=3
  1043. G G
  1044. G / G ( 3 , 3 ) G/G^{\prime}\simeq(3,3)
  1045. cc ( G ) 2 \mathrm{cc}(G)\geq 2
  1046. c = cl ( G ) 4 c=\mathrm{cl}(G)\geq 4
  1047. ζ 1 ( G ) \zeta_{1}(G)
  1048. τ ( G ) \tau(G)
  1049. ϰ ( G ) \varkappa(G)
  1050. π ( G ) \pi(G)
  1051. ker ( π ) = γ c ( G ) = ζ 1 ( G ) H i \ker(\pi)=\gamma_{c}(G)=\zeta_{1}(G)\leq H_{i}^{\prime}
  1052. H 1 , , H 4 H_{1},\ldots,H_{4}
  1053. G G
  1054. γ c ( G ) = ζ 1 ( G ) \gamma_{c}(G)=\zeta_{1}(G)
  1055. γ c ( G ) \gamma_{c}(G)
  1056. H 2 H_{2}^{\prime}
  1057. γ c ( G ) < ζ 1 ( G ) \gamma_{c}(G)<\zeta_{1}(G)
  1058. H 1 H_{1}^{\prime}
  1059. p = 3 p=3
  1060. 3 3
  1061. 2 2
  1062. ( 7 ) \mathcal{B}(7)
  1063. ( 5 ) \mathcal{B}(5)
  1064. ( 6 ) \mathcal{B}(6)
  1065. ( 7 ) \mathcal{B}(7)
  1066. ( 8 ) \mathcal{B}(8)
  1067. ( j ) ( 7 ) \mathcal{B}(j)\simeq\mathcal{B}(7)
  1068. j 9 j\geq 9
  1069. ( j ) ( 8 ) \mathcal{B}(j)\simeq\mathcal{B}(8)
  1070. j 10 j\geq 10
  1071. 3 3
  1072. R = 243 , 6 R=\langle 243,6\rangle
  1073. | R | = 3 5 = 243 |R|=3^{5}=243
  1074. 6 6
  1075. 𝒯 ( R ) \mathcal{T}(R)
  1076. 2 2
  1077. 𝒯 2 ( R ) \mathcal{T}^{2}(R)
  1078. ϰ \varkappa
  1079. τ ( 1 ) \tau(1)
  1080. 1 1
  1081. 2 2
  1082. ( 3 , 3 ) (3,3)
  1083. τ = [ A ( 3 , c ) , ( 3 , 3 , 3 ) , ( 9 , 3 ) , ( 9 , 3 ) ] \tau=[A(3,c),(3,3,3),(9,3),(9,3)]
  1084. τ ( 1 ) = A ( 3 , c ) \tau(1)=A(3,c)
  1085. 3 3
  1086. 3 c 3^{c}
  1087. τ ( 2 ) = ( 3 , 3 , 3 ) = ^ ( 1 3 ) \tau(2)=(3,3,3)\hat{=}(1^{3})
  1088. τ ( 3 ) = τ ( 4 ) = ( 9 , 3 ) = ^ ( 21 ) \tau(3)=\tau(4)=(9,3)\hat{=}(21)
  1089. 3 3
  1090. 2 2
  1091. 3 n 3^{n}
  1092. c = n - 2 c=n-2
  1093. σ \sigma
  1094. σ \sigma
  1095. ϰ = ( 0122 ) \varkappa=(0122)
  1096. 𝒯 2 ( R ) \mathcal{T}^{2}_{\ast}(R)
  1097. ϰ = ( 2122 ) \varkappa=(2122)
  1098. σ \sigma
  1099. ϰ = ( 1122 ) \varkappa=(1122)
  1100. τ ( 1 ) = ( 43 ) \tau(1)=(43)
  1101. G = G 3 ( K ) = Gal ( F 3 ( K ) | K ) G=\mathrm{G}_{3}^{\infty}(K)=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)|K)
  1102. 3 3
  1103. 3 3
  1104. F 3 ( K ) \mathrm{F}_{3}^{\infty}(K)
  1105. K = ( - 9748 ) K=\mathbb{Q}(\sqrt{-9748})
  1106. Q = G / G ′′ = G 3 2 ( K ) = Gal ( F 3 2 ( K ) | K ) Q=G/G^{\prime\prime}=\mathrm{G}_{3}^{2}(K)=\mathrm{Gal}(\mathrm{F}_{3}^{2}(K)|K)
  1107. G G
  1108. 3 3
  1109. F 3 2 ( K ) \mathrm{F}_{3}^{2}(K)
  1110. K K
  1111. 78 78
  1112. 3 3
  1113. G = G 3 ( K ) G=\mathrm{G}_{3}^{\infty}(K)
  1114. 3 3
  1115. G 3 3 ( K ) = Gal ( F 3 3 ( K ) | K ) \mathrm{G}_{3}^{3}(K)=\mathrm{Gal}(\mathrm{F}_{3}^{3}(K)|K)
  1116. dl ( G ) = 3 \mathrm{dl}(G)=3
  1117. 3 3
  1118. K K
  1119. 3 3
  1120. F 3 3 ( K ) \mathrm{F}_{3}^{3}(K)
  1121. K K
  1122. ϰ \varkappa
  1123. τ \tau
  1124. 1 1
  1125. 9 9
  1126. 9 , 2 \langle 9,2\rangle
  1127. ( 0000 ) (0000)
  1128. [ ( 1 ) ( 1 ) ( 1 ) ( 1 ) ] [(1)(1)(1)(1)]
  1129. 3 3
  1130. 3 3
  1131. 3 / 2 ; 3 / 3 ; 1 / 1 3/2;3/3;1/1
  1132. 2 2
  1133. 27 27
  1134. 27 , 3 \langle 27,3\rangle
  1135. ( 0000 ) (0000)
  1136. [ ( 1 2 ) ( 1 2 ) ( 1 2 ) ( 1 2 ) ] [(1^{2})(1^{2})(1^{2})(1^{2})]
  1137. 2 2
  1138. 4 4
  1139. 4 / 1 ; 7 / 5 4/1;7/5
  1140. 3 3
  1141. 243 243
  1142. 243 , 8 \langle 243,8\rangle
  1143. ( 2034 ) (2034)
  1144. [ ( 21 ) ( 21 ) ( 21 ) ( 21 ) ] [(21)(21)(21)(21)]
  1145. 1 1
  1146. 3 3
  1147. 4 / 4 4/4
  1148. 4 4
  1149. 729 729
  1150. 729 , 54 \langle 729,54\rangle
  1151. ( 2034 ) (2034)
  1152. [ ( 21 ) ( 2 2 ) ( 21 ) ( 21 ) ] [(21)(2^{2})(21)(21)]
  1153. 2 2
  1154. 4 4
  1155. 8 / 3 ; 6 / 3 8/3;6/3
  1156. 5 5
  1157. 2187 2187
  1158. 2187 , 302 \langle 2187,302\rangle
  1159. ( 2334 ) (2334)
  1160. [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] [(21)(32)(21)(21)]
  1161. 0
  1162. 3 3
  1163. 0 / 0 0/0
  1164. 5 5
  1165. 2187 2187
  1166. 2187 , 306 \langle 2187,306\rangle
  1167. ( 2434 ) (2434)
  1168. [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] [(21)(32)(21)(21)]
  1169. 0
  1170. 3 3
  1171. 0 / 0 0/0
  1172. 5 5
  1173. 2187 2187
  1174. 2187 , 303 \langle 2187,303\rangle
  1175. ( 2034 ) (2034)
  1176. [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] [(21)(32)(21)(21)]
  1177. 1 1
  1178. 4 4
  1179. 5 / 2 5/2
  1180. 5 5
  1181. 6561 6561
  1182. 729 , 54 - # 2 ; 2 \langle 729,54\rangle-\#2;2
  1183. ( 2334 ) (2334)
  1184. [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] [(21)(32)(21)(21)]
  1185. 0
  1186. 2 2
  1187. 0 / 0 0/0
  1188. 5 5
  1189. 6561 6561
  1190. 729 , 54 - # 2 ; 6 \langle 729,54\rangle-\#2;6
  1191. ( 2434 ) (2434)
  1192. [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] [(21)(32)(21)(21)]
  1193. 0
  1194. 2 2
  1195. 0 / 0 0/0
  1196. 5 5
  1197. 6561 6561
  1198. 729 , 54 - # 2 ; 3 \langle 729,54\rangle-\#2;3
  1199. ( 2034 ) (2034)
  1200. [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] [(21)(32)(21)(21)]
  1201. 1 1
  1202. 3 3
  1203. 4 / 4 4/4
  1204. 6 6
  1205. 6561 6561
  1206. 2187 , 303 - # 1 ; 1 \langle 2187,303\rangle-\#1;1
  1207. ( 2034 ) (2034)
  1208. [ ( 21 ) ( 3 2 ) ( 21 ) ( 21 ) ] [(21)(3^{2})(21)(21)]
  1209. 1 1
  1210. 4 4
  1211. 7 / 3 7/3
  1212. 6 6
  1213. 19683 19683
  1214. 729 , 54 - # 2 ; 3 - # 1 ; 1 \langle 729,54\rangle-\#2;3-\#1;1
  1215. ( 2034 ) (2034)
  1216. [ ( 21 ) ( 3 2 ) ( 21 ) ( 21 ) ] [(21)(3^{2})(21)(21)]
  1217. 2 2
  1218. 4 4
  1219. 8 / 3 ; 6 / 3 8/3;6/3
  1220. d d
  1221. p = 3 p=3
  1222. d = r 3 ( K ) = d ( Cl 3 ( K ) ) d=r_{3}(K)=d(\mathrm{Cl}_{3}(K))
  1223. 3 3
  1224. K K
  1225. 3 3
  1226. Cl 3 ( K ) ( 1 2 ) \mathrm{Cl}_{3}(K)\simeq(1^{2})
  1227. K K
  1228. 3 3
  1229. 9 , 2 \langle 9,2\rangle
  1230. 𝒯 ( 9 , 2 ) \mathcal{T}(\langle 9,2\rangle)
  1231. 3 3
  1232. G G
  1233. K K
  1234. ϰ { ( 2334 ) , ( 2434 ) } \varkappa\in\{(2334),(2434)\}
  1235. τ = [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] \tau=[(21)(32)(21)(21)]
  1236. G G
  1237. σ \sigma
  1238. K K
  1239. 9 , 2 \langle 9,2\rangle
  1240. 27 , 3 \langle 27,3\rangle
  1241. ( 1 2 ) (1^{2})
  1242. 9 , 2 \langle 9,2\rangle
  1243. 1 1
  1244. G / γ 2 ( G ) G/\gamma_{2}(G)
  1245. G G
  1246. 27 , 3 \langle 27,3\rangle
  1247. 2 2
  1248. G / γ 3 ( G ) G/\gamma_{3}(G)
  1249. G G
  1250. 𝒯 ( 27 , 3 ) = 𝒯 1 ( 27 , 3 ) ˙ 𝒯 2 ( 27 , 3 ) \mathcal{T}(\langle 27,3\rangle)=\mathcal{T}^{1}(\langle 27,3\rangle)\dot{\cup% }\mathcal{T}^{2}(\langle 27,3\rangle)
  1251. 𝒯 1 ( 27 , 3 ) \mathcal{T}^{1}(\langle 27,3\rangle)
  1252. ϰ = ( 000 ) \varkappa=(\ast 000)
  1253. 3 3
  1254. 243 , 8 \langle 243,8\rangle
  1255. 3 3
  1256. G / γ 4 ( G ) G/\gamma_{4}(G)
  1257. G G
  1258. σ \sigma
  1259. 729 , 54 \langle 729,54\rangle
  1260. 243 , 8 \langle 243,8\rangle
  1261. 4 4
  1262. G / γ 5 ( G ) G/\gamma_{5}(G)
  1263. G G
  1264. 𝒯 ( 729 , 54 ) = 𝒯 2 ( 729 , 54 ) ˙ 𝒯 3 ( 729 , 54 ) \mathcal{T}(\langle 729,54\rangle)=\mathcal{T}^{2}(\langle 729,54\rangle)\dot{% \cup}\mathcal{T}^{3}(\langle 729,54\rangle)
  1265. 𝒢 ( 3 , 2 ) \mathcal{G}(3,2)
  1266. 𝒢 ( 3 , 3 ) \mathcal{G}(3,3)
  1267. Q = G / G ′′ Q=G/G^{\prime\prime}
  1268. G G
  1269. Q { 2187 , 302 , 2187 , 306 } Q\in\{\langle 2187,302\rangle,\langle 2187,306\rangle\}
  1270. r = 3 > 2 = d r=3>2=d
  1271. 3 3
  1272. G G
  1273. σ \sigma
  1274. 729 , 54 - # 2 ; 2 \langle 729,54\rangle-\#2;2
  1275. 729 , 54 - # 2 ; 6 \langle 729,54\rangle-\#2;6
  1276. r = 2 = d r=2=d
  1277. 2187 , 303 - # 1 ; 1 𝒢 ( 3 , 2 ) \langle 2187,303\rangle-\#1;1\in\mathcal{G}(3,2)
  1278. 729 , 54 - # 2 ; 3 - # 1 ; 1 𝒢 ( 3 , 3 ) \langle 729,54\rangle-\#2;3-\#1;1\in\mathcal{G}(3,3)
  1279. τ = [ ( 21 ) ( 3 2 ) ( 21 ) ( 21 ) ] > [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] \tau=[(21)(3^{2})(21)(21)]>[(21)(32)(21)(21)]
  1280. [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] [(21)(32)(21)(21)]
  1281. P c = G / γ c + 1 ( G ) P_{c}=G/\gamma_{c+1}(G)
  1282. 3 3
  1283. G = G 3 ( K ) G=\mathrm{G}_{3}^{\infty}(K)
  1284. K = ( - 9748 ) K=\mathbb{Q}(\sqrt{-9748})
  1285. c = cl ( P c ) c=\mathrm{cl}(P_{c})
  1286. ν = ν ( P c ) \nu=\nu(P_{c})
  1287. μ = μ ( P c ) \mu=\mu(P_{c})
  1288. 3 3
  1289. c c
  1290. G c , n ( z , w ) = x , y , s 2 , t 3 , s 3 , , s c x 3 = s c w , y 3 = s 3 2 s 4 s c z , s j 3 = s j + 2 2 s j + 3 for 2 j c - 3 , s c - 2 3 = s c 2 , t 3 3 = 1 , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j - 1 , x ] for 3 j c , \begin{aligned}\displaystyle G^{c,n}(z,w)=&\displaystyle\langle x,y,s_{2},t_{3% },s_{3},\ldots,s_{c}\mid\\ &\displaystyle x^{3}=s_{c}^{w},\ y^{3}=s_{3}^{2}s_{4}s_{c}^{z},\ s_{j}^{3}=s_{% j+2}^{2}s_{j+3}\,\text{ for }2\leq j\leq c-3,\ s_{c-2}^{3}=s_{c}^{2},\ t_{3}^{% 3}=1,\\ &\displaystyle s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]\,\text{ for }% 3\leq j\leq c\rangle,\end{aligned}
  1291. c 5 c\geq 5
  1292. 3 n 3^{n}
  1293. n = c + 2 n=c+2
  1294. 0 w 1 0\leq w\leq 1
  1295. - 1 z 1 -1\leq z\leq 1
  1296. G = G c , n ( z , w ) G=G^{c,n}(z,w)
  1297. c c
  1298. w , z w,z
  1299. τ = [ A ( 3 , c ) , ( 3 , 3 , 3 ) , ( 9 , 3 ) , ( 9 , 3 ) ] \tau=[A(3,c),(3,3,3),(9,3),(9,3)]
  1300. G = G c , n ( z , w ) G=G^{c,n}(z,w)
  1301. c c
  1302. w , z w,z
  1303. ϰ = ( 0122 ) \varkappa=(0122)
  1304. w = z = 0 w=z=0
  1305. ϰ = ( 2122 ) \varkappa=(2122)
  1306. w = 0 , z = ± 1 w=0,z=\pm 1
  1307. ϰ = ( 1122 ) \varkappa=(1122)
  1308. w = 1 , z = 0 w=1,z=0
  1309. ϰ { ( 4122 ) , ( 3122 ) } \varkappa\in\{(4122),(3122)\}
  1310. w = 1 , z = ± 1 w=1,z=\pm 1
  1311. z z
  1312. [ a , x ] = 1 [a,x]=1
  1313. a { s c , t 3 } a\in\{s_{c},t_{3}\}
  1314. [ a , y ] = 1 [a,y]=1
  1315. a { s 3 , , s c , t 3 } a\in\{s_{3},\ldots,s_{c},t_{3}\}
  1316. ζ 1 ( G ) = s c , t 3 \zeta_{1}(G)=\langle s_{c},t_{3}\rangle
  1317. [ a , x y ] = [ a , y ] [ a , x ] [ [ a , x ] , y ] [a,xy]=[a,y]\cdot[a,x]\cdot[[a,x],y]
  1318. [ a , y 2 ] = [ a , y ] 1 + y [a,y^{2}]=[a,y]^{1+y}
  1319. [ s 2 , x y ] = s 3 t 3 [s_{2},xy]=s_{3}t_{3}
  1320. [ s 2 , x y 2 ] = s 3 t 3 2 [s_{2},xy^{2}]=s_{3}t_{3}^{2}
  1321. [ s j , x y ] = [ s j , x y 2 ] = [ s j , x ] = s j + 1 [s_{j},xy]=[s_{j},xy^{2}]=[s_{j},x]=s_{j+1}
  1322. j 3 j\geq 3
  1323. G G
  1324. H 1 = y , G H_{1}=\langle y,G^{\prime}\rangle
  1325. H 2 = x , G H_{2}=\langle x,G^{\prime}\rangle
  1326. H 3 = x y , G H_{3}=\langle xy,G^{\prime}\rangle
  1327. H 4 = x y 2 , G H_{4}=\langle xy^{2},G^{\prime}\rangle
  1328. H i = ( G ) h i - 1 H_{i}^{\prime}=(G^{\prime})^{h_{i}-1}
  1329. H i = h i , G H_{i}=\langle h_{i},G^{\prime}\rangle
  1330. G = s 2 , t 3 , s 3 , , s c G^{\prime}=\langle s_{2},t_{3},s_{3},\ldots,s_{c}\rangle
  1331. H 1 = s 2 y - 1 = t 3 H_{1}^{\prime}=\langle s_{2}^{y-1}\rangle=\langle t_{3}\rangle
  1332. H 2 = s 2 x - 1 , , s c - 1 x - 1 = s 3 , , s c H_{2}^{\prime}=\langle s_{2}^{x-1},\ldots,s_{c-1}^{x-1}\rangle=\langle s_{3},% \ldots,s_{c}\rangle
  1333. H 3 = s 2 x y - 1 , , s c - 1 x y - 1 = s 3 t 3 , s 4 , , s c H_{3}^{\prime}=\langle s_{2}^{xy-1},\ldots,s_{c-1}^{xy-1}\rangle=\langle s_{3}% t_{3},s_{4},\ldots,s_{c}\rangle
  1334. H 4 = s 2 x y 2 - 1 , , s c - 1 x y 2 - 1 = s 3 t 3 2 , s 4 , , s c H_{4}^{\prime}=\langle s_{2}^{xy^{2}-1},\ldots,s_{c-1}^{xy^{2}-1}\rangle=% \langle s_{3}t_{3}^{2},s_{4},\ldots,s_{c}\rangle
  1335. H 1 H_{1}
  1336. H 1 = t 3 H_{1}^{\prime}=\langle t_{3}\rangle
  1337. ζ 1 ( G ) = s c , t 3 \zeta_{1}(G)=\langle s_{c},t_{3}\rangle
  1338. H 1 / H 1 = y , s 2 , , s c H 1 / H 1 A ( 3 , c ) H_{1}/H_{1}^{\prime}=\langle y,s_{2},\ldots,s_{c}\rangle H_{1}^{\prime}/H_{1}^% {\prime}\simeq A(3,c)
  1339. c c
  1340. ord ( y ) = ord ( s 2 ) = 3 m \mathrm{ord}(y)=\mathrm{ord}(s_{2})=3^{m}
  1341. c = 2 m c=2m
  1342. ord ( y ) = 3 m + 1 , ord ( s 2 ) = 3 m \mathrm{ord}(y)=3^{m+1},\mathrm{ord}(s_{2})=3^{m}
  1343. c = 2 m + 1 c=2m+1
  1344. H 2 / H 2 = x , s 2 , t 3 H 2 / H 2 ( 3 , 3 , 3 ) H_{2}/H_{2}^{\prime}=\langle x,s_{2},t_{3}\rangle H_{2}^{\prime}/H_{2}^{\prime% }\simeq(3,3,3)
  1345. H 3 / H 3 = x y , s 2 , t 3 H 3 / H 3 ( 9 , 3 ) H_{3}/H_{3}^{\prime}=\langle xy,s_{2},t_{3}\rangle H_{3}^{\prime}/H_{3}^{% \prime}\simeq(9,3)
  1346. H 4 / H 4 = x y 2 , s 2 , t 3 H 4 / H 4 ( 9 , 3 ) H_{4}/H_{4}^{\prime}=\langle xy^{2},s_{2},t_{3}\rangle H_{4}^{\prime}/H_{4}^{% \prime}\simeq(9,3)
  1347. ord ( s 2 ) = ord ( t 3 ) = 3 \mathrm{ord}(s_{2})=\mathrm{ord}(t_{3})=3
  1348. ord ( x ) = 3 \mathrm{ord}(x)=3
  1349. H 2 H_{2}
  1350. ord ( x y ) = ord ( x y 2 ) = 9 \mathrm{ord}(xy)=\mathrm{ord}(xy^{2})=9
  1351. H 3 H_{3}
  1352. H 4 H_{4}
  1353. T i : G H i / H i T_{i}:\,G\to H_{i}/H_{i}^{\prime}
  1354. T ~ i : G / G H i / H i \tilde{T}_{i}:\,G/G^{\prime}\to H_{i}/H_{i}^{\prime}
  1355. T ~ i ( g G ) \tilde{T}_{i}(gG^{\prime})
  1356. g G G / G gG^{\prime}\in G/G^{\prime}
  1357. g x j y ( mod G ) g\equiv x^{j}y^{\ell}\;\;(\mathop{{\rm mod}}G^{\prime})
  1358. - 1 j , 1 -1\leq j,\ell\leq 1
  1359. x H 1 x\notin H_{1}
  1360. \Rightarrow
  1361. T ~ 1 ( x G ) = x 3 H 1 = s c w H 1 \tilde{T}_{1}(xG^{\prime})=x^{3}H_{1}^{\prime}=s_{c}^{w}H_{1}^{\prime}
  1362. y H 2 y\notin H_{2}
  1363. \Rightarrow
  1364. T ~ 2 ( y G ) = y 3 H 2 = s 3 2 s 4 s c z H 2 = 1 H 2 \tilde{T}_{2}(yG^{\prime})=y^{3}H_{2}^{\prime}=s_{3}^{2}s_{4}s_{c}^{z}H_{2}^{% \prime}=1\cdot H_{2}^{\prime}
  1365. x , y H 3 , H 4 x,y\notin H_{3},H_{4}
  1366. \Rightarrow
  1367. T ~ i ( x G ) = x 3 H i = s c w H i = 1 H i \tilde{T}_{i}(xG^{\prime})=x^{3}H_{i}^{\prime}=s_{c}^{w}H_{i}^{\prime}=1\cdot H% _{i}^{\prime}
  1368. T ~ i ( y G ) = y 3 H i = s 3 2 s 4 s c z H i = s 3 2 H i \tilde{T}_{i}(yG^{\prime})=y^{3}H_{i}^{\prime}=s_{3}^{2}s_{4}s_{c}^{z}H_{i}^{% \prime}=s_{3}^{2}H_{i}^{\prime}
  1369. 3 i 4 3\leq i\leq 4
  1370. X 2 + X + 1 = ( X - 1 ) 2 + 3 ( X - 1 ) + 3 X^{2}+X+1=(X-1)^{2}+3(X-1)+3
  1371. y H 1 y\in H_{1}
  1372. \Rightarrow
  1373. T ~ 1 ( y G ) = y 1 + x + x 2 H 1 = y 3 + 3 ( x - 1 ) + ( x - 1 ) 2 H 1 = y 3 [ y , x ] 3 [ [ y , x ] , x ] H 1 \tilde{T}_{1}(yG^{\prime})=y^{1+x+x^{2}}H_{1}^{\prime}=y^{3+3(x-1)+(x-1)^{2}}H% _{1}^{\prime}=y^{3}\cdot[y,x]^{3}\cdot[[y,x],x]H_{1}^{\prime}
  1374. = s 3 2 s 4 s c z s 2 3 s 3 H 1 = s 2 3 s 3 3 s 4 s c z H 1 = s c z H 1 =s_{3}^{2}s_{4}s_{c}^{z}s_{2}^{3}s_{3}H_{1}^{\prime}=s_{2}^{3}s_{3}^{3}s_{4}s_% {c}^{z}H_{1}^{\prime}=s_{c}^{z}H_{1}^{\prime}
  1375. x H 2 x\in H_{2}
  1376. \Rightarrow
  1377. T ~ 2 ( x G ) = x 1 + y + y 2 H 2 = x 3 + 3 ( y - 1 ) + ( y - 1 ) 2 H 2 = x 3 [ x , y ] 3 [ [ x , y ] , y ] H 2 = s c w s 2 - 3 t 3 - 1 H 2 = t 3 - 1 H 2 \tilde{T}_{2}(xG^{\prime})=x^{1+y+y^{2}}H_{2}^{\prime}=x^{3+3(y-1)+(y-1)^{2}}H% _{2}^{\prime}=x^{3}\cdot[x,y]^{3}\cdot[[x,y],y]H_{2}^{\prime}=s_{c}^{w}s_{2}^{% -3}t_{3}^{-1}H_{2}^{\prime}=t_{3}^{-1}H_{2}^{\prime}
  1378. T ~ i ( g G ) = T ~ i ( x G ) j T ~ i ( y G ) \tilde{T}_{i}(gG^{\prime})=\tilde{T}_{i}(xG^{\prime})^{j}\tilde{T}_{i}(yG^{% \prime})^{\ell}
  1379. T ~ 1 ( g G ) = s c w j + z H 1 \tilde{T}_{1}(gG^{\prime})=s_{c}^{wj+z\ell}H_{1}^{\prime}
  1380. T ~ 2 ( g G ) = t 3 - j H 2 \tilde{T}_{2}(gG^{\prime})=t_{3}^{-j}H_{2}^{\prime}
  1381. T ~ i ( g G ) = s 3 2 H i \tilde{T}_{i}(gG^{\prime})=s_{3}^{2\ell}H_{i}^{\prime}
  1382. 3 i 4 3\leq i\leq 4
  1383. s c w j + z H 1 = H 1 s_{c}^{wj+z\ell}H_{1}^{\prime}=H_{1}^{\prime}
  1384. \Rightarrow
  1385. j , j,\ell
  1386. w = z = 0 w=z=0
  1387. = 0 \ell=0
  1388. j j
  1389. w = 0 , z = ± 1 w=0,z=\pm 1
  1390. j = 0 j=0
  1391. \ell
  1392. w = 1 , z = 0 w=1,z=0
  1393. j = j=\mp\ell
  1394. w = 1 , z = ± 1 w=1,z=\pm 1
  1395. t 3 - j H 2 = H 2 t_{3}^{-j}H_{2}^{\prime}=H_{2}^{\prime}
  1396. \Rightarrow
  1397. j = 0 j=0
  1398. \ell
  1399. s 3 2 H i = H i s_{3}^{2\ell}H_{i}^{\prime}=H_{i}^{\prime}
  1400. \Rightarrow
  1401. = 0 \ell=0
  1402. j j
  1403. 3 i 4 3\leq i\leq 4
  1404. 1 i 4 1\leq i\leq 4
  1405. j = 0 j=0
  1406. \ell
  1407. \Leftrightarrow
  1408. ker ( T i ) = y , G = H 1 \ker(T_{i})=\langle y,G^{\prime}\rangle=H_{1}
  1409. \Leftrightarrow
  1410. ϰ ( i ) = 1 \varkappa(i)=1
  1411. = 0 \ell=0
  1412. j j
  1413. \Leftrightarrow
  1414. ker ( T i ) = x , G = H 2 \ker(T_{i})=\langle x,G^{\prime}\rangle=H_{2}
  1415. \Leftrightarrow
  1416. ϰ ( i ) = 2 \varkappa(i)=2
  1417. j = j=\ell
  1418. \Leftrightarrow
  1419. ker ( T i ) = x y , G = H 3 \ker(T_{i})=\langle xy,G^{\prime}\rangle=H_{3}
  1420. \Leftrightarrow
  1421. ϰ ( i ) = 3 \varkappa(i)=3
  1422. j = - j=-\ell
  1423. \Leftrightarrow
  1424. ker ( T i ) = x y - 1 , G = H 4 \ker(T_{i})=\langle xy^{-1},G^{\prime}\rangle=H_{4}
  1425. \Leftrightarrow
  1426. ϰ ( i ) = 4 \varkappa(i)=4
  1427. j , j,\ell
  1428. \Leftrightarrow
  1429. ker ( T i ) = x , y , G = G \ker(T_{i})=\langle x,y,G^{\prime}\rangle=G
  1430. \Leftrightarrow
  1431. ϰ ( i ) = 0 \varkappa(i)=0
  1432. w , z w,z
  1433. p p
  1434. p { 2 , 3 , 5 } p\in\{2,3,5\}
  1435. r 1 r\geq 1
  1436. σ \sigma
  1437. p { 2 , 3 , 5 } p\in\{2,3,5\}
  1438. 𝒯 1 ( 4 , 2 ) \mathcal{T}^{1}(\langle 4,2\rangle)
  1439. p = 2 p=2
  1440. 𝒯 1 ( 9 , 2 ) \mathcal{T}^{1}(\langle 9,2\rangle)
  1441. p = 3 p=3
  1442. 𝒯 1 ( 25 , 2 ) \mathcal{T}^{1}(\langle 25,2\rangle)
  1443. p = 5 p=5
  1444. 5 5
  1445. 2 2
  1446. 1 1
  1447. G c , n ( z , w ) = x , y , s 2 , , s c x 2 = s c w , y 2 = s c z , s j 2 = s j + 1 s j + 2 for 2 j c - 2 , s c - 1 2 = s c , s 2 = [ y , x ] , s j = [ s j - 1 , x ] = [ s j - 1 , y ] for 3 j c , \begin{aligned}\displaystyle G^{c,n}(z,w)=&\displaystyle\langle x,y,s_{2},% \ldots,s_{c}\mid\\ &\displaystyle x^{2}=s_{c}^{w},\ y^{2}=s_{c}^{z},\ s_{j}^{2}=s_{j+1}s_{j+2}\,% \text{ for }2\leq j\leq c-2,\ s_{c-1}^{2}=s_{c},\\ &\displaystyle s_{2}=[y,x],\ s_{j}=[s_{j-1},x]=[s_{j-1},y]\,\text{ for }3\leq j% \leq c\rangle,\end{aligned}
  1448. c 3 c\geq 3
  1449. 2 n 2^{n}
  1450. n = c + 1 n=c+1
  1451. w , z w,z
  1452. 1 1
  1453. 1 1
  1454. 1 1
  1455. 3 3
  1456. τ = [ ( 1 2 ) , ( 1 2 ) , A ( 2 , c ) ] \tau=[(1^{2}),(1^{2}),A(2,c)]
  1457. c c
  1458. A ( 2 , c ) A(2,c)
  1459. ϰ = ( 210 ) \varkappa=(210)
  1460. w = z = 0 w=z=0
  1461. ϰ = ( 213 ) \varkappa=(213)
  1462. w = z = 1 w=z=1
  1463. ϰ = ( 211 ) \varkappa=(211)
  1464. w = 1 , z = 0 w=1,z=0
  1465. τ = [ ( 1 ) , ( 1 ) , ( 1 ) ] \tau=[(1),(1),(1)]
  1466. ϰ = ( 000 ) \varkappa=(000)
  1467. τ = [ ( 2 ) , ( 2 ) , ( 2 ) ] \tau=[(2),(2),(2)]
  1468. ϰ = ( 123 ) \varkappa=(123)
  1469. 3 3
  1470. 1 1
  1471. G a c , n ( z , w ) = x , y , s 2 , t 3 , s 3 , , s c x 3 = s c w , y 3 = s 3 2 s 4 s c z , t 3 = s c a , s j 3 = s j + 2 2 s j + 3 for 2 j c - 3 , s c - 2 3 = s c 2 , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j - 1 , x ] for 3 j c , \begin{aligned}\displaystyle G^{c,n}_{a}(z,w)=&\displaystyle\langle x,y,s_{2},% t_{3},s_{3},\ldots,s_{c}\mid\\ &\displaystyle x^{3}=s_{c}^{w},\ y^{3}=s_{3}^{2}s_{4}s_{c}^{z},\ t_{3}=s_{c}^{% a},\ s_{j}^{3}=s_{j+2}^{2}s_{j+3}\,\text{ for }2\leq j\leq c-3,\ s_{c-2}^{3}=s% _{c}^{2},\\ &\displaystyle s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]\,\text{ for }% 3\leq j\leq c\rangle,\end{aligned}
  1472. c 5 c\geq 5
  1473. 3 n 3^{n}
  1474. n = c + 1 n=c+1
  1475. a , w , z a,w,z
  1476. 2 2
  1477. 2 2
  1478. 1 1
  1479. 7 7
  1480. τ = [ A ( 3 , c - a ) , ( 1 2 ) , ( 1 2 ) , ( 1 2 ) ] \tau=[A(3,c-a),(1^{2}),(1^{2}),(1^{2})]
  1481. c c
  1482. a a
  1483. ϰ = ( 0000 ) \varkappa=(0000)
  1484. a = w = z = 0 a=w=z=0
  1485. ϰ = ( 1000 ) \varkappa=(1000)
  1486. a = 0 , w = 1 , z = 0 a=0,w=1,z=0
  1487. ϰ = ( 2000 ) \varkappa=(2000)
  1488. a = w = 0 , z = ± 1 a=w=0,z=\pm 1
  1489. ϰ = ( 0000 ) \varkappa=(0000)
  1490. a = 1 , w { - 1 , 0 , 1 } , z = 0 a=1,w\in\{-1,0,1\},z=0
  1491. τ = [ ( 1 ) , ( 1 ) , ( 1 ) , ( 1 ) ] \tau=[(1),(1),(1),(1)]
  1492. 9 9
  1493. τ = [ ( 1 2 ) , ( 2 ) , ( 2 ) , ( 2 ) ] \tau=[(1^{2}),(2),(2),(2)]
  1494. ϰ = ( 1111 ) \varkappa=(1111)
  1495. 3 3
  1496. A 9 A_{9}
  1497. τ = [ ( 1 3 ) , ( 1 2 ) , ( 1 2 ) , ( 1 2 ) ] \tau=[(1^{3}),(1^{2}),(1^{2}),(1^{2})]
  1498. σ \sigma
  1499. 5 5
  1500. 1 1
  1501. G a c , n ( z , w ) = x , y , s 2 , t 3 , s 3 , , s c x 5 = s c w , y 5 = s c z , t 3 = s c a , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j - 1 , x ] for 3 j c , \begin{aligned}\displaystyle G^{c,n}_{a}(z,w)=&\displaystyle\langle x,y,s_{2},% t_{3},s_{3},\ldots,s_{c}\mid\\ &\displaystyle x^{5}=s_{c}^{w},\ y^{5}=s_{c}^{z},\ t_{3}=s_{c}^{a},\\ &\displaystyle s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]\,\text{ for }% 3\leq j\leq c\rangle,\end{aligned}
  1502. c 3 c\geq 3
  1503. 5 n 5^{n}
  1504. n = c + 1 n=c+1
  1505. a , w , z a,w,z
  1506. 3 3
  1507. 4 4
  1508. 3 3
  1509. 67 67
  1510. τ = [ A ( 5 , c - k ) , ( 1 2 ) 5 ] \tau=[A(5,c-k),(1^{2})^{5}]
  1511. c c
  1512. k k
  1513. ϰ = ( 0 6 ) \varkappa=(0^{6})
  1514. a = w = z = 0 a=w=z=0
  1515. ϰ = ( 10 5 ) \varkappa=(10^{5})
  1516. a = 0 , w = 1 , z = 0 a=0,w=1,z=0
  1517. ϰ = ( 20 5 ) \varkappa=(20^{5})
  1518. a = w = 0 , z 0 a=w=0,z\neq 0
  1519. ϰ = ( 0 6 ) \varkappa=(0^{6})
  1520. a 0 a\neq 0
  1521. τ = [ ( 1 ) 6 ] \tau=[(1)^{6}]
  1522. 25 25
  1523. τ = [ ( 1 2 ) , ( 2 ) 5 ] \tau=[(1^{2}),(2)^{5}]
  1524. ϰ = ( 1 6 ) \varkappa=(1^{6})
  1525. 15625 , 631 \langle 15625,631\rangle
  1526. τ = [ ( 1 5 ) , ( 1 2 ) 5 ] \tau=[(1^{5}),(1^{2})^{5}]
  1527. σ \sigma
  1528. 𝒯 2 ( 243 , 6 ) \mathcal{T}^{2}(\langle 243,6\rangle)
  1529. 𝒯 2 ( 243 , 8 ) \mathcal{T}^{2}(\langle 243,8\rangle)
  1530. 𝒯 2 ( 729 , 40 ) \mathcal{T}^{2}(\langle 729,40\rangle)
  1531. p = 3 p=3
  1532. 𝒯 2 ( 243 , 6 ) \mathcal{T}^{2}(\langle 243,6\rangle)
  1533. 3 3
  1534. 2 2
  1535. G c , n ( z , w ) = x , y , s 2 , t 3 , s 3 , , s c x 3 = s c w , y 3 = s 3 2 s 4 s c z , s j 3 = s j + 2 2 s j + 3 for 2 j c - 3 , s c - 2 3 = s c 2 , t 3 3 = 1 , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j - 1 , x ] for 3 j c , \begin{aligned}\displaystyle G^{c,n}(z,w)=&\displaystyle\langle x,y,s_{2},t_{3% },s_{3},\ldots,s_{c}\mid\\ &\displaystyle x^{3}=s_{c}^{w},\ y^{3}=s_{3}^{2}s_{4}s_{c}^{z},\ s_{j}^{3}=s_{% j+2}^{2}s_{j+3}\,\text{ for }2\leq j\leq c-3,\ s_{c-2}^{3}=s_{c}^{2},\ t_{3}^{% 3}=1,\\ &\displaystyle s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]\,\text{ for }% 3\leq j\leq c\rangle,\end{aligned}
  1536. c 5 c\geq 5
  1537. 3 n 3^{n}
  1538. n = c + 2 n=c+2
  1539. w , z w,z
  1540. 2 2
  1541. 2 2
  1542. 3 3
  1543. 18 18
  1544. τ = [ A ( 3 , c ) , ( 1 3 ) , ( 21 ) , ( 21 ) ] \tau=[A(3,c),(1^{3}),(21),(21)]
  1545. c c
  1546. ϰ = ( 0122 ) \varkappa=(0122)
  1547. w = z = 0 w=z=0
  1548. ϰ = ( 2122 ) \varkappa=(2122)
  1549. w = 0 , z = ± 1 w=0,z=\pm 1
  1550. ϰ = ( 1122 ) \varkappa=(1122)
  1551. w = 1 , z = 0 w=1,z=0
  1552. ϰ = ( 3122 ) \varkappa=(3122)
  1553. w = 1 , z = ± 1 w=1,z=\pm 1
  1554. σ \sigma
  1555. 𝒯 2 ( 243 , 8 ) \mathcal{T}^{2}(\langle 243,8\rangle)
  1556. 3 3
  1557. 2 2
  1558. G c , n ( z , w ) = x , y , t 2 , s 3 , t 3 , , t c y 3 = s 3 t c w , x 3 = t 3 t 4 2 t 5 t c z , t j 3 = t j + 2 2 t j + 3 for 2 j c - 3 , t c - 2 3 = t c 2 , s 3 3 = 1 , t 2 = [ y , x ] , s 3 = [ t 2 , x ] , t j = [ t j - 1 , y ] for 3 j c , \begin{aligned}\displaystyle G^{c,n}(z,w)=&\displaystyle\langle x,y,t_{2},s_{3% },t_{3},\ldots,t_{c}\mid\\ &\displaystyle y^{3}=s_{3}t_{c}^{w},\ x^{3}=t_{3}t_{4}^{2}t_{5}t_{c}^{z},\ t_{% j}^{3}=t_{j+2}^{2}t_{j+3}\,\text{ for }2\leq j\leq c-3,\ t_{c-2}^{3}=t_{c}^{2}% ,\ s_{3}^{3}=1,\\ &\displaystyle t_{2}=[y,x],\ s_{3}=[t_{2},x],\ t_{j}=[t_{j-1},y]\,\text{ for }% 3\leq j\leq c\rangle,\end{aligned}
  1559. c 6 c\geq 6
  1560. 3 n 3^{n}
  1561. n = c + 2 n=c+2
  1562. w , z w,z
  1563. 2 2
  1564. 2 2
  1565. 3 3
  1566. 16 16
  1567. τ = [ ( 21 ) , A ( 3 , c ) , ( 21 ) , ( 21 ) ] \tau=[(21),A(3,c),(21),(21)]
  1568. c c
  1569. ϰ = ( 2034 ) \varkappa=(2034)
  1570. w = z = 0 w=z=0
  1571. ϰ = ( 2134 ) \varkappa=(2134)
  1572. w = 0 , z = ± 1 w=0,z=\pm 1
  1573. ϰ = ( 2234 ) \varkappa=(2234)
  1574. w = 1 , z = 0 w=1,z=0
  1575. ϰ = ( 2334 ) \varkappa=(2334)
  1576. w = 1 , z = ± 1 w=1,z=\pm 1
  1577. σ \sigma
  1578. 𝒯 2 ( 16 , 3 ) \mathcal{T}^{2}(\langle 16,3\rangle)
  1579. 𝒯 2 ( 16 , 4 ) \mathcal{T}^{2}(\langle 16,4\rangle)
  1580. p = 2 p=2
  1581. 𝒯 2 ( 243 , 15 ) \mathcal{T}^{2}(\langle 243,15\rangle)
  1582. 𝒯 2 ( 243 , 17 ) \mathcal{T}^{2}(\langle 243,17\rangle)
  1583. p = 3 p=3
  1584. 𝒯 2 ( 16 , 11 ) \mathcal{T}^{2}(\langle 16,11\rangle)
  1585. p = 2 p=2
  1586. 𝒯 2 ( 81 , 12 ) \mathcal{T}^{2}(\langle 81,12\rangle)
  1587. p = 3 p=3
  1588. 𝒯 3 ( 729 , 13 ) \mathcal{T}^{3}(\langle 729,13\rangle)
  1589. 𝒯 3 ( 729 , 18 ) \mathcal{T}^{3}(\langle 729,18\rangle)
  1590. 𝒯 3 ( 729 , 21 ) \mathcal{T}^{3}(\langle 729,21\rangle)
  1591. p = 3 p=3
  1592. 𝒯 3 ( 32 , 35 ) \mathcal{T}^{3}(\langle 32,35\rangle)
  1593. 𝒯 3 ( 64 , 181 ) \mathcal{T}^{3}(\langle 64,181\rangle)
  1594. p = 2 p=2
  1595. 𝒯 3 ( 243 , 38 ) \mathcal{T}^{3}(\langle 243,38\rangle)
  1596. 𝒯 3 ( 243 , 41 ) \mathcal{T}^{3}(\langle 243,41\rangle)
  1597. p = 3 p=3
  1598. G ( K ) G(K)
  1599. K K
  1600. G ( K ) G(K)
  1601. G ( K ) G(K)
  1602. p p
  1603. F p 2 ( K ) F^{2}_{p}(K)
  1604. K K
  1605. K K
  1606. p p
  1607. G p 2 ( K ) = Gal ( F p 2 ( K ) | K ) G^{2}_{p}(K)=\mathrm{Gal}(F^{2}_{p}(K)|K)
  1608. K K
  1609. 2 2
  1610. K K
  1611. G p ( K ) = Gal ( F p ( K ) | K ) G^{\infty}_{p}(K)=\mathrm{Gal}(F^{\infty}_{p}(K)|K)
  1612. F p ( K ) F^{\infty}_{p}(K)
  1613. K K
  1614. K K
  1615. ( r 1 , r 2 ) (r_{1},r_{2})
  1616. | d | |d|
  1617. d = d ( K ) d=d(K)
  1618. 𝒯 \mathcal{T}
  1619. 𝒢 0 ( p , r ) \mathcal{G}_{0}(p,r)
  1620. 𝒢 ( p , r ) \mathcal{G}(p,r)
  1621. G p 2 ( K ) G^{2}_{p}(K)
  1622. K K
  1623. V 𝒯 V\in\mathcal{T}
  1624. V 𝒢 0 ( p , r ) V\in\mathcal{G}_{0}(p,r)
  1625. K K
  1626. V = G p 2 ( K ) V=G^{2}_{p}(K)
  1627. p = 3 p=3
  1628. K ( d ) = ( d ) K(d)=\mathbb{Q}(\sqrt{d})
  1629. ( 0 , 1 ) (0,1)
  1630. 3 3
  1631. ( 3 , 3 ) (3,3)
  1632. 3 3
  1633. G 3 2 ( K ) G^{2}_{3}(K)
  1634. - 10 6 < d < 0 -10^{6}<d<0
  1635. - 10 8 < d < 0 -10^{8}<d<0
  1636. 𝒯 2 ( 243 , 6 ) \mathcal{T}^{2}(\langle 243,6\rangle)
  1637. 𝒯 2 ( 243 , 8 ) \mathcal{T}^{2}(\langle 243,8\rangle)
  1638. V V
  1639. min { | d | V = G 3 2 ( K ( d ) ) } \min\{|d|\mid V=G^{2}_{3}(K(d))\}
  1640. V V
  1641. 3 3
  1642. G 3 2 ( K ( d ) ) G^{2}_{3}(K(d))
  1643. 3 3
  1644. n \uparrow^{n}
  1645. ϰ = ( 3122 ) \varkappa=(3122)
  1646. ϰ = ( 1122 ) \varkappa=(1122)
  1647. ϰ = ( 2122 ) \varkappa=(2122)
  1648. ϰ = ( 2334 ) \varkappa=(2334)
  1649. ϰ = ( 2234 ) \varkappa=(2234)
  1650. ϰ = ( 2134 ) \varkappa=(2134)
  1651. 0 \uparrow^{0}
  1652. 16 627 16\,627
  1653. 15 544 15\,544
  1654. 21 668 21\,668
  1655. 9 748 9\,748
  1656. 34 867 34\,867
  1657. 17 131 17\,131
  1658. 1 \uparrow^{1}
  1659. 262 744 262\,744
  1660. 268 040 268\,040
  1661. 446 788 446\,788
  1662. 297 079 297\,079
  1663. 370 740 370\,740
  1664. 819 743 819\,743
  1665. 2 \uparrow^{2}
  1666. 4 776 071 4\,776\,071
  1667. 1 062 708 1\,062\,708
  1668. 3 843 907 3\,843\,907
  1669. 1 088 808 1\,088\,808
  1670. 4 087 295 4\,087\,295
  1671. 2 244 399 2\,244\,399
  1672. 3 \uparrow^{3}
  1673. 40 059 363 40\,059\,363
  1674. 27 629 107 27\,629\,107
  1675. 52 505 588 52\,505\,588
  1676. 11 091 140 11\,091\,140
  1677. 19 027 947 19\,027\,947
  1678. 30 224 744 30\,224\,744
  1679. 4 \uparrow^{4}
  1680. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  1681. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  1682. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  1683. 94 880 548 94\,880\,548
  1684. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  1685. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  1686. 3 3
  1687. G 3 2 ( K ( d ) ) G^{2}_{3}(K(d))
  1688. 3 3
  1689. ( 6 ) \mathcal{B}(6)
  1690. ( j ) \mathcal{B}(j)
  1691. j 8 j\geq 8
  1692. ϰ = ( 3122 ) \varkappa=(3122)
  1693. ϰ = ( 1122 ) \varkappa=(1122)
  1694. ϰ = ( 2122 ) \varkappa=(2122)
  1695. 𝒯 2 ( 243 , 6 ) \mathcal{T}^{2}(\langle 243,6\rangle)
  1696. ϰ = ( 2334 ) \varkappa=(2334)
  1697. ϰ = ( 2234 ) \varkappa=(2234)
  1698. ϰ = ( 2134 ) \varkappa=(2134)
  1699. 𝒯 2 ( 243 , 8 ) \mathcal{T}^{2}(\langle 243,8\rangle)
  1700. ϰ = ( 2334 ) \varkappa=(2334)
  1701. 3 3
  1702. | d | |d|
  1703. # \#
  1704. ϰ = ( 3144 ) \varkappa=(3144)
  1705. τ = [ ( 21 ) ( 1 3 ) ( 21 ) ( 21 ) ] \tau=[(21)(1^{3})(21)(21)]
  1706. G = 243 , 5 G=\langle 243,5\rangle
  1707. ϰ = ( 1133 ) \varkappa=(1133)
  1708. τ = [ ( 21 ) ( 1 3 ) ( 21 ) ( 1 3 ) ] \tau=[(21)(1^{3})(21)(1^{3})]
  1709. G = 243 , 7 G=\langle 243,7\rangle
  1710. ϰ = ( 4111 ) \varkappa=(4111)
  1711. τ = [ ( 1 3 ) ( 1 3 ) ( 1 3 ) ( 21 ) ] \tau=[(1^{3})(1^{3})(1^{3})(21)]
  1712. G = 729 , 45 G=\langle 729,45\rangle
  1713. ϰ = ( 2143 ) \varkappa=(2143)
  1714. τ = [ ( 21 ) ( 21 ) ( 21 ) ( 21 ) ] \tau=[(21)(21)(21)(21)]
  1715. G = 729 , 57 G=\langle 729,57\rangle
  1716. b = 10 6 b=10^{6}
  1717. 2 020 2\,020
  1718. 667 ( 33.0 % ) 667\ (33.0\%)
  1719. 269 ( 13.3 % ) 269\ (13.3\%)
  1720. 297 ( 14.7 % ) 297\ (14.7\%)
  1721. 94 ( 4.7 % ) 94\ (4.7\%)
  1722. b = 10 7 b=10^{7}
  1723. 24 476 24\,476
  1724. 7 622 ( 31.14 % ) 7\,622\ (31.14\%)
  1725. 3 625 ( 14.81 % ) 3\,625\ (14.81\%)
  1726. 3 619 ( 14.79 % ) 3\,619\ (14.79\%)
  1727. 1 019 ( 4.163 % ) 1\,019\ (4.163\%)
  1728. b = 10 8 b=10^{8}
  1729. 276 375 276\,375
  1730. 83 353 ( 30.159 % ) 83\,353\ (30.159\%)
  1731. 41 398 ( 14.979 % ) 41\,398\ (14.979\%)
  1732. 40 968 ( 14.823 % ) 40\,968\ (14.823\%)
  1733. 10 426 ( 3.7724 % ) 10\,426\ (3.7724\%)
  1734. 𝒢 0 ( 3 , 2 ) \mathcal{G}_{0}(3,2)
  1735. 𝒢 ( 3 , 2 ) \mathcal{G}(3,2)
  1736. # { | d | < b V = G 3 2 ( K ( d ) ) } \#\{|d|<b\mid V=G^{2}_{3}(K(d))\}
  1737. V V
  1738. 3 3
  1739. G 3 2 ( K ( d ) ) G^{2}_{3}(K(d))
  1740. b b
  1741. b = 10 8 b=10^{8}
  1742. 276 375 276\,375
  1743. 3 3
  1744. ( 3 , 3 ) (3,3)
  1745. - b < d < 0 -b<d<0
  1746. 𝒢 0 ( 3 , 2 ) \mathcal{G}_{0}(3,2)
  1747. 𝒢 ( 3 , 2 ) \mathcal{G}(3,2)
  1748. 3 3
  1749. G 3 2 ( K ( d ) ) G^{2}_{3}(K(d))
  1750. 243 , 5 \langle 243,5\rangle
  1751. 243 , 7 \langle 243,7\rangle
  1752. 729 , 45 \langle 729,45\rangle
  1753. 729 , 57 \langle 729,57\rangle
  1754. p { 3 , 5 , 7 } p\in\{3,5,7\}
  1755. K ( d ) = ( d ) K(d)=\mathbb{Q}(\sqrt{d})
  1756. ( 0 , 1 ) (0,1)
  1757. ( p , p ) (p,p)
  1758. p 5 p^{5}
  1759. 𝒢 0 ( p , 2 ) \mathcal{G}_{0}(p,2)
  1760. 𝒢 ( p , 2 ) \mathcal{G}(p,2)
  1761. Φ 6 \Phi_{6}
  1762. p 6 p^{6}
  1763. p > 3 p>3
  1764. Φ 6 \Phi_{6}
  1765. p + 7 p+7
  1766. 3 3
  1767. Φ 6 \Phi_{6}
  1768. 3 3
  1769. p = 3 p=3
  1770. 4 4
  1771. p > 3 p>3
  1772. Φ 6 \Phi_{6}
  1773. σ \sigma
  1774. 2 2
  1775. p = 3 p=3
  1776. p + 1 p+1
  1777. p > 3 p>3
  1778. 𝒢 0 ( p , 2 ) \mathcal{G}_{0}(p,2)
  1779. 2 2
  1780. p 3 p\geq 3
  1781. p > 3 p>3
  1782. σ \sigma
  1783. 0
  1784. 2 2
  1785. 𝒢 0 ( p , 2 ) \mathcal{G}_{0}(p,2)
  1786. min { | d | V = G p 2 ( K ( d ) ) } \min\{|d|\mid V=G^{2}_{p}(K(d))\}
  1787. V 𝒢 0 ( p , 2 ) V\in\mathcal{G}_{0}(p,2)
  1788. p = 3 p=3
  1789. p = 5 p=5
  1790. p = 7 p=7