wpmath0000015_0

(+)-Camphor_6-endo-hydroxylase.html

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(+)-Camphor_6-exo-hydroxylase.html

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(+)-Caryolan-1-ol_synthase.html

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(+)-Larreatricin_hydroxylase.html

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(+)-Menthofuran_synthase.html

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(+)-Pulegone_reductase.html

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(+)-Sabinene_3-hydroxylase.html

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(+)-Thujan-3-ol_dehydrogenase.html

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(-)-Endo-fenchol_dehydrogenase.html

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(-)-Isopiperitenone_reductase.html

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(13E)-labda-7,13-dien-15-ol_synthase.html

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(2,2,3-Trimethyl-5-oxocyclopent-3-enyl)acetyl-CoA_1,5-monooxygenase.html

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(2R)-3-sulfolactate_dehydrogenase_(NADP+).html

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(2Z,6Z)-farnesyl_diphosphate_synthase.html

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(3-methyl-2-oxobutanoate_dehydrogenase_(2-methylpropanoyl-transferring))-phosphatase.html

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(410777)_2009_FD.html

  1. ( 100 5 ) 23 - 6.5 3981071 (\sqrt[5]{100})^{23-6.5}\approx 3981071

(6-4)DNA_photolyase.html

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(a-N-acetylneuraminyl-2,3-b-galactosyl-1,3)-N-acetyl-galactosaminide_6-a-sialyltransferase.html

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(d)CMP_kinase.html

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(Fructose-bisphosphate_aldolase)-lysine_N-methyltransferase.html

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(G,X)-manifold.html

  1. ( G , X ) (G,X)
  2. G G
  3. X X
  4. g G g\in G
  5. x g x x\mapsto gx
  6. X X
  7. M M
  8. ( G , X ) (G,X)
  9. { U α } \{U_{\alpha}\}
  10. M M
  11. { φ α : U α V α } \{\varphi_{\alpha}\colon U_{\alpha}\to V_{\alpha}\}
  12. X X
  13. g G g\in G
  14. g x = ϕ α ϕ β - 1 ( x ) gx=\phi_{\alpha}\circ\phi_{\beta}^{-1}(x)
  15. x V α V β x\in V_{\alpha}\cap V_{\beta}
  16. G G
  17. ϕ α ϕ β - 1 : V α V β V α V β \phi_{\alpha}\circ\phi_{\beta}^{-1}\colon V_{\alpha}\cap V_{\beta}\to V_{% \alpha}\cap V_{\beta}
  18. G G
  19. V α V β V_{\alpha}\cap V_{\beta}

(Histone-H3)-lysine-36_demethylase.html

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(KDO)-lipid_IVA_3-deoxy-D-manno-octulosonic_acid_transferase.html

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(KDO)2-lipid_IVA_(2-8)_3-deoxy-D-manno-octulosonic_acid_transferase.html

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(KDO)3-lipid_IVA_(2-4)_3-deoxy-D-manno-octulosonic_acid_transferase.html

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(Methyl-Co(III)_methanol-specific_corrinoid_protein):coenzyme_M_methyltransferase.html

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(Methyl-Co(III)_methylamine-specific_corrinoid_protein):coenzyme_M_methyltransferase.html

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(Methyl-Co(III)_tetramethylammonium-specific_corrinoid_protein):coenzyme_M_methyltransferase.html

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(N-acetylneuraminyl)-galactosylglucosylceramide_N-acetylgalactosaminyltransferase.html

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(Phosphatase_2A_protein)-leucine-carboxy_methyltransferase.html

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(Pyruvate,_phosphate_dikinase)-phosphate_phosphotransferase.html

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(Pyruvate,_phosphate_dikinase)_kinase.html

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(Pyruvate,_water_dikinase)-phosphate_phosphotransferase.html

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(Pyruvate,_water_dikinase)_kinase.html

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(R)-amidase.html

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(R)-benzylsuccinyl-CoA_dehydrogenase.html

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(R)-citramalate_synthase.html

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(S)-1-phenylethanol_dehydrogenase.html

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(S)-2-hydroxypropylphosphonic_acid_epoxidase.html

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(S)-hydroxynitrile_lyase.html

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(S)-sulfolactate_dehydrogenase.html

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(Z)-3-hexen-1-ol_acetyltransferase.html

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1,2-dihydroxynaphthalene_dioxygenase.html

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1,4-dihydroxy-2-naphthoate_polyprenyltransferase.html

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1,4-dihydroxy-2-naphthoyl-CoA_hydrolase.html

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1,8-Cineole_2-endo-monooxygenase.html

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1,8-Cineole_2-exo-monooxygenase.html

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1-Deoxy-11beta-hydroxypentalenate_dehydrogenase.html

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1-Deoxypentalenic_acid_11beta-hydroxylase.html

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1-hydroxy-2-naphthoate_hydroxylase.html

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1-Hydroxycarotenoid_3,4-desaturase.html

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1-Octanol.html

  1. log ( K s c / w ) = a + b log ( K w / o ) \log(K_{sc/w})=a+b\log(K_{w/o})
  2. K s c / w K_{sc/w}
  3. K w / o K_{w/o}

1-Pyrroline_dehydrogenase.html

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11-cis-retinol_dehydrogenase.html

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15-Cis-phytoene_desaturase.html

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16S_rRNA_(adenine1408-N1)-methyltransferase.html

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16S_rRNA_(adenine1518-N6::adenine1519-N6)-dimethyltransferase.html

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16S_rRNA_(cytidine1402-2'-O)-methyltransferase.html

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16S_rRNA_(cytidine1409-2'-O)-methyltransferase.html

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16S_rRNA_(cytosine1402-N4)-methyltransferase.html

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16S_rRNA_(cytosine1407-C5)-methyltransferase.html

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16S_rRNA_(cytosine967-C5)-methyltransferase.html

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16S_rRNA_(guanine1207-N2)-methyltransferase.html

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16S_rRNA_(guanine1405-N7)-methyltransferase.html

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16S_rRNA_(guanine1516-N2)-methyltransferase.html

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16S_rRNA_(guanine527-N7)-methyltransferase.html

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16S_rRNA_(guanine966-N2)-methyltransferase.html

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16S_rRNA_(uracil1498-N3)-methyltransferase.html

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18S_rRNA_(adenine1779-N6::adenine1780-N6)-dimethyltransferase.html

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1_vs._100_(Hong_Kong_game_show).html

  1. Prize gained = [ HK$200,000 N o . o f o p p o n e n t s ] × No. of wrong answers \textrm{Prize\ gained}=\left[\frac{\textrm{HK\$200,000}}{\textrm{}}{No.\ of\ % opponents}\right]\times\textrm{No.\ of\ wrong\ answers}
  2. [ ] [\ ]

1L-myo-inositol_1-phosphate_cytidylyltransferase.html

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2'-Deamino-2'-hydroxyneamine_transaminase.html

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2'-N-acetylparomamine_deacetylase.html

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2,3-bisphosphoglycerate_3-phosphatase.html

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2,4,7-trihydroxy-1,4-benzoxazin-3-one-glucoside_7-O-methyltransferase.html

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2,4-Dihydroxy-1,4-benzoxazin-3-one-glucoside_dioxygenase.html

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2,4-Dihydroxy-7-methoxy-2H-1,4-benzoxazin-3(4H)-one_2_-D-glucosyltransferase.html

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2,5-diamino-6-(ribosylamino)-4(3H)-pyrimidinone_5'-phosphate_reductase.html

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2,5-diketocamphane_1,2-monooxygenase.html

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2,6-dihydroxypseudooxynicotine_hydrolase.html

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2,7,4'-Trihydroxyisoflavanone_4'-O-methyltransferase.html

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2-amino-3,7-dideoxy-D-threo-hept-6-ulosonate_synthase.html

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2-Amino-4-deoxychorismate_dehydrogenase.html

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2-Amino-4-deoxychorismate_synthase.html

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2-amino-5-formylamino-6-ribosylaminopyrimidin-4(3H)-one_5'-monophosphate_deformylase.html

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2-Deoxy-scyllo-inosamine_dehydrogenase.html

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2-Deoxy-scyllo-inosamine_dehydrogenase_(SAM-dependent).html

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2-Deoxystreptamine_glucosyltransferase.html

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2-Deoxystreptamine_N-acetyl-D-glucosaminyltransferase.html

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2-Formylbenzoate_dehydrogenase.html

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2-haloacid_dehalogenase_(configuration-inverting).html

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2-haloacid_dehalogenase_(configuration-retaining).html

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2-Hydroxy-1,4-benzoxazin-3-one_monooxygenase.html

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2-hydroxy-3-keto-5-methylthiopentenyl-1-phosphate_phosphatase.html

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2-hydroxy-4-carboxymuconate_semialdehyde_hemiacetal_dehydrogenase.html

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2-hydroxy-6-oxo-6-(2-aminophenyl)hexa-2,4-dienoate_hydrolase.html

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2-hydroxy-6-oxonona-2,4-dienedioate_hydrolase.html

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2-hydroxy-dATP_diphosphatase.html

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2-hydroxyethylphosphonate_dioxygenase.html

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2-Hydroxymuconate-6-semialdehyde_dehydrogenase.html

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2-iminoacetate_synthase.html

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2-Ketoarginine_methyltransferase.html

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2-Methoxy-6-polyprenyl-1,4-benzoquinol_methylase.html

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2-methylcitrate_dehydratase_(2-methyl-trans-aconitate_forming).html

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2-nitroimidazole_nitrohydrolase.html

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2-oxo-3-(5-oxofuran-2-ylidene)propanoate_lactonase.html

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2-oxoglutaramate_amidase.html

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2-oxoglutarate::L-arginine_monooxygenase::decarboxylase_(succinate-forming).html

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2-oxoglutarate_dioxygenase_(ethylene-forming).html

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2-Phospho-L-lactate_guanylyltransferase.html

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2-phospho-L-lactate_transferase.html

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2-Polyprenyl-6-hydroxyphenyl_methylase.html

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2012–13_Al_Nassr_FC_season.html

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21S_rRNA_(uridine2791-2'-O)-methyltransferase.html

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23S_rRNA_(adenine1618-N6)-methyltransferase.html

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23S_rRNA_(adenine2085-N6)-dimethyltransferase.html

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23S_rRNA_(adenine2503-C2)-methyltransferase.html

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23S_rRNA_(adenine2503-C2,C8)-dimethyltransferase.html

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23S_rRNA_(adenine2503-C8)-methyltransferase.html

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23S_rRNA_(adenosine1067-2'-O)-methyltransferase.html

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23S_rRNA_(cytidine1920-2'-O)-methyltransferase.html

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23S_rRNA_(cytidine2498-2'-O)-methyltransferase.html

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23S_rRNA_(cytosine1962-C5)-methyltransferase.html

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23S_rRNA_(guanine1835-N2)-methyltransferase.html

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23S_rRNA_(guanine2069-N7)-methyltransferase.html

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23S_rRNA_(guanine2445-N2)-methyltransferase.html

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23S_rRNA_(guanine2535-N1)-methyltransferase.html

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23S_rRNA_(guanine745-N1)-methyltransferase.html

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23S_rRNA_(guanine748-N1)-methyltransferase.html

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23S_rRNA_(guanosine2251-2'-O)-methyltransferase.html

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23S_rRNA_(pseudouridine1915-N3)-methyltransferase.html

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23S_rRNA_(uracil1939-C5)-methyltransferase.html

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23S_rRNA_(uracil747-C5)-methyltransferase.html

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23S_rRNA_(uridine2479-2'-O)-methyltransferase.html

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23S_rRNA_(uridine2552-2'-O)-methyltransferase.html

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24-hydroxycholesterol_7α-hydroxylase.html

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25-hydroxycholesterol_7α-hydroxylase.html

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27S_pre-rRNA_(guanosine2922-2'-O)-methyltransferase.html

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3',5'-cyclic-AMP_phosphodiesterase.html

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3(or_17)a-hydroxysteroid_dehydrogenase.html

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3,4-Dehydroadipyl-CoA_semialdehyde_dehydrogenase_(NADP+).html

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3,4-dihydroxyphenylalanine_oxidative_deaminase.html

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3,5,7-Trioxododecanoyl-CoA_synthase.html

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3-(3-hydroxyphenyl)propanoate_hydroxylase.html

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3-(cis-5,6-dihydroxycyclohexa-1,3-dien-1-yl)propanoate_dehydrogenase.html

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3-dehydroquinate_synthase_II.html

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3-dehydroshikimate_dehydratase.html

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3-Deoxy-D-manno-octulosonic_acid_kinase.html

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3-Epi-6-deoxocathasterone_23-monooxygenase.html

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3-fumarylpyruvate_hydrolase.html

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3-hexulose-6-phosphate_synthase.html

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3-Hydroxy-9,10-secoandrosta-1,3,5(10)-triene-9,17-dione_monooxygenase.html

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3-hydroxy-D-aspartate_aldolase.html

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3-hydroxydecanoyl-(acyl-carrier-protein)_dehydratase.html

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3-hydroxyindolin-2-one_monooxygenase.html

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3-hydroxypropionate_dehydrogenase_(NADP+).html

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3-hydroxypropionyl-CoA_dehydratase.html

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3-Ketosteroid_9alpha-monooxygenase.html

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3-Methyl-2-oxobutanoate_dehydrogenase_(acetyl-transferring)_kinase.html

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3-O-alpha-D-glucosyl-L-rhamnose_phosphorylase.html

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3-O-alpha-D-mannopyranosyl-alpha-D-mannopyranose_xylosylphosphotransferase.html

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3-Oxo-5,6-dehydrosuberyl-CoA_semialdehyde_dehydrogenase.html

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3-Succinoylsemialdehyde-pyridine_dehydrogenase.html

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4,4'-Diapophytoene_desaturase.html

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4,4'-Diapophytoene_synthase.html

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4,5:9,10-diseco-3-hydroxy-5,9,17-trioxoandrosta-1(10),2-diene-4-oate_hydrolase.html

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4-(Gamma-L-glutamylamino)butanoyl-(BtrI_acyl-carrier_protein)_monooxygenase.html

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4-aminobutyrate—pyruvate_transaminase.html

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4-Dimethylallyltryptophan_N-methyltransferase.html

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4-Hydroxy-7-methoxy-3-oxo-3,4-dihydro-2H-1,4-benzoxazin-2-yl_glucoside_beta-D-glucosidase.html

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4-Hydroxybenzoate_geranyltransferase.html

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4-hydroxybutanoyl-CoA_dehydratase.html

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4-Hydroxycoumarin_synthase.html

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4-Methylaminobutanoate_oxidase_(formaldehyde-forming).html

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4-Methylaminobutanoate_oxidase_(methylamine-forming).html

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4-Nitrocatechol_4-monooxygenase.html

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4-Nitrophenol_4-monooxygenase.html

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4-O-beta-D-mannosyl-D-glucose_phosphorylase.html

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4-Sulfomuconolactone_hydrolase.html

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5,6-dimethylbenzimidazole_synthase.html

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5-epiaristolochene_1,3-dihydroxylase.html

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5-Exo-hydroxycamphor_dehydrogenase.html

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5-hydroxyfuranocoumarin_5-O-methyltransferase.html

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5-Methyltetrahydrofolate:corrinoid::iron-sulfur_protein_Co-methyltransferase.html

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5-methyltetrahydrosarcinapterin:corrinoid::iron-sulfur_protein_Co-methyltransferase.html

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5-nitroanthranilic_acid_aminohydrolase.html

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5-nitrosalicylate_dioxygenase.html

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6-carboxytetrahydropterin_synthase.html

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6-Deoxy-5-ketofructose_1-phosphate_synthase.html

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6-Hydroxy-3-succinoylpyridine_3-monooxygenase.html

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6-Hydroxynicotinate_3-monooxygenase.html

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6-Hydroxypseudooxynicotine_dehydrogenase.html

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6-oxocamphor_hydrolase.html

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7,8-didemethyl-8-hydroxy-5-deazariboflavin_synthase.html

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7-Chloro-L-tryptophan_oxidase.html

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7-Dimethylallyltryptophan_synthase.html

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7-Hydroxymethyl_chlorophyll_a_reductase.html

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7-Methylxanthine_demethylase.html

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7-simplex_honeycomb.html

  1. A ~ 7 {\tilde{A}}_{7}
  2. A ~ 7 {\tilde{A}}_{7}
  3. E ~ 7 {\tilde{E}}_{7}
  4. A ~ 7 {\tilde{A}}_{7}
  5. E ~ 7 {\tilde{E}}_{7}
  6. A ~ 7 {\tilde{A}}_{7}
  7. A 7 A_{7}
  8. 7 2 {}^{2}_{7}
  9. 7 4 {}^{4}_{7}
  10. 7 2 {}^{2}_{7}
  11. 7 * {}^{*}_{7}
  12. 7 8 {}^{8}_{7}
  13. A ~ 7 {\tilde{A}}_{7}
  14. C ~ 4 {\tilde{C}}_{4}

8'-apo-beta-carotenoid_14',13'-cleaving_dioxygenase.html

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8-Hydroxy-5-deazaflavin:NADPH_oxidoreductase.html

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8-hydroxyfuranocoumarin_8-O-methyltransferase.html

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8-Hydroxygeraniol_dehydrogenase.html

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8-oxo-dGDP_phosphatase.html

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8-oxo-dGTP_diphosphatase.html

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8-oxoguanine_deaminase.html

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8-simplex_honeycomb.html

  1. A ~ 8 {\tilde{A}}_{8}
  2. A ~ 8 {\tilde{A}}_{8}
  3. E ~ 8 {\tilde{E}}_{8}
  4. A ~ 8 {\tilde{A}}_{8}
  5. E ~ 8 {\tilde{E}}_{8}
  6. A ~ 8 {\tilde{A}}_{8}
  7. A 8 A_{8}
  8. 8 3 {}^{3}_{8}
  9. 8 * {}^{*}_{8}
  10. 8 9 {}^{9}_{8}
  11. A ~ 8 {\tilde{A}}_{8}
  12. C ~ 4 {\tilde{C}}_{4}

801_(number).html

  1. 801 = 17 2 + 8 3 = 26 2 + 5 3 = 1 3 + 2 3 + 4 3 + 6 3 + 8 3 = 2 3 + 4 3 + 9 3 . 801=17^{2}+8^{3}=26^{2}+5^{3}=1^{3}+2^{3}+4^{3}+6^{3}+8^{3}=2^{3}+4^{3}+9^{3}.

9,9'-Dicis-zeta-carotene_desaturase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

9-cis-beta-carotene_9',10'-cleaving_dioxygenase.html

  1. \rightleftharpoons

9-cis-epoxycarotenoid_dioxygenase.html

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  2. \rightleftharpoons
  3. \rightleftharpoons

9beta-pimara-7,15-diene_oxidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

A-1,3-mannosyl-glycoprotein_2-b-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

A-1,3-mannosyl-glycoprotein_4-b-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

A-1,4-glucan-protein_synthase_(UDP-forming).html

  1. \rightleftharpoons

A-1,6-mannosyl-glycoprotein_2-b-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

A-1,6-mannosyl-glycoprotein_6-b-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Abel_equation_of_the_first_kind.html

  1. y = f 3 ( x ) y 3 + f 2 ( x ) y 2 + f 1 ( x ) y + f 0 ( x ) y^{\prime}=f_{3}(x)y^{3}+f_{2}(x)y^{2}+f_{1}(x)y+f_{0}(x)\,
  2. f 3 ( x ) 0 f_{3}(x)\neq 0
  3. f 3 ( x ) = 0 f_{3}(x)=0
  4. f 0 ( x ) = 0 f_{0}(x)=0
  5. f 2 ( x ) = 0 f_{2}(x)=0
  6. f 0 ( x ) = 0 f_{0}(x)=0
  7. f 3 ( x ) = 0 f_{3}(x)=0
  8. y = 1 u y=\dfrac{1}{u}
  9. u u = - f 0 ( x ) u 3 - f 1 ( x ) u 2 - f 2 ( x ) u - f 3 ( x ) . uu^{\prime}=-f_{0}(x)u^{3}-f_{1}(x)u^{2}-f_{2}(x)u-f_{3}(x).\,
  10. ξ \displaystyle\xi
  11. u = u 3 + ϕ ( ξ ) . u^{\prime}=u^{3}+\phi(\xi).\,

Abel–Plana_formula.html

  1. n = 0 f ( n ) = 0 f ( x ) d x + 1 2 f ( 0 ) + i 0 f ( i t ) - f ( - i t ) e 2 π t - 1 d t . \sum_{n=0}^{\infty}f(n)=\int_{0}^{\infty}f(x)\,dx+\frac{1}{2}f(0)+i\int_{0}^{% \infty}\frac{f(it)-f(-it)}{e^{2\pi t}-1}\,dt.
  2. ζ ( s , α ) = n = 0 1 ( n + α ) s = α 1 - s s - 1 + 1 2 α s + 2 0 sin ( s arctan t α ) ( α 2 + t 2 ) s 2 d t e 2 π t - 1 . \zeta(s,\alpha)=\sum_{n=0}^{\infty}\frac{1}{(n+\alpha)^{s}}=\frac{\alpha^{1-s}% }{s-1}+\frac{1}{2\alpha^{s}}+2\int_{0}^{\infty}\frac{\sin\left(s\arctan\frac{t% }{\alpha}\right)}{(\alpha^{2}+t^{2})^{\frac{s}{2}}}\frac{dt}{e^{2\pi t}-1}.
  3. n = 0 ( - 1 ) n f ( n ) = 1 2 f ( 0 ) + i 0 f ( i t ) - f ( - i t ) 2 sinh ( π t ) d t . \sum_{n=0}^{\infty}(-1)^{n}f(n)=\frac{1}{2}f(0)+i\int_{0}^{\infty}\frac{f(it)-% f(-it)}{2\sinh(\pi t)}\,dt.

Aberrations_of_the_eye.html

  1. Z n m Z^{m}_{n}
  2. Z 4 2 Z^{2}_{4}
  3. Z 4 - 2 Z^{-2}_{4}
  4. Z 0 0 Z^{0}_{0}
  5. Z 1 1 Z^{1}_{1}
  6. Z 1 - 1 Z^{-1}_{1}
  7. Z 2 0 Z^{0}_{2}
  8. Z 2 2 Z^{2}_{2}
  9. Z 2 - 2 Z^{-2}_{2}
  10. Z 4 2 Z^{2}_{4}
  11. Z 4 - 2 Z^{-2}_{4}
  12. Z 4 0 Z^{0}_{4}
  13. Z 3 1 Z^{1}_{3}
  14. Z 3 - 1 Z^{-1}_{3}
  15. Z 3 3 Z^{3}_{3}
  16. Z 3 - 3 Z^{-3}_{3}
  17. Z 4 4 Z^{4}_{4}
  18. Z 4 - 4 Z^{-4}_{4}

Abieta-7,13-dien-18-al_dehydrogenase.html

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Abietadiene_hydroxylase.html

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Abietadienol_hydroxylase.html

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Abscisate_beta-glucosyltransferase.html

  1. \rightleftharpoons

Absolute_angular_momentum.html

  1. 𝐋 \mathbf{L}
  2. 𝐫 \mathbf{r}
  3. 𝐩 \mathbf{p}
  4. 𝐋 = 𝐫 × m 𝐯 \mathbf{L}=\mathbf{r}\times m\mathbf{v}
  5. 𝐋 \mathbf{L}
  6. | 𝐋 m | = M = u r cos ( ϕ ) + Ω r 2 cos 2 ( ϕ ) \left|\frac{\mathbf{L}}{m}\right|=M=ur\cos(\phi)+\Omega r^{2}\cos^{2}(\phi)
  7. m m
  8. m s \frac{m}{s}
  9. r a d rad
  10. r a d s \frac{rad}{s}
  11. M u a cos ( ϕ ) + Ω a 2 cos 2 ( ϕ ) M\approx ua\cos(\phi)+\Omega a^{2}\cos^{2}(\phi)
  12. < v a r > a <var>a

Abstract_cell_complex.html

  1. C = ( E , B , d i m ) C=(E,B,dim)
  2. B ( a , b ) B(a,b)
  3. d i m ( a ) < d i m ( b ) dim(a)<dim(b)

Acetoacetyl-CoA_synthase.html

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Acetoin_dehydrogenase.html

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Acetyl-S-ACP:malonate_ACP_transferase.html

  1. \rightleftharpoons

Acetylajmaline_esterase.html

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  2. \rightleftharpoons

Acetylgalactosaminyl-O-glycosyl-glycoprotein_b-1,3-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Acetylgalactosaminyl-O-glycosyl-glycoprotein_b-1,6-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Acireductone_synthase.html

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  2. \rightleftharpoons
  3. \rightleftharpoons

Acoustic_attenuation.html

  1. P ( x + Δ x ) = P ( x ) e - α ( ω ) Δ x , α ( ω ) = α 0 ω η P(x+\Delta x)=P(x)e^{-\alpha(\omega)\Delta x},\alpha(\omega)=\alpha_{0}\omega^% {\eta}
  2. ω \omega
  3. Δ x \Delta x
  4. α ( ω ) \alpha(\omega)
  5. α 0 \alpha_{0}
  6. η \eta
  7. η \eta
  8. η = 2 \eta=2
  9. η \eta
  10. η \eta
  11. η \eta
  12. 2 u - 1 c 0 2 2 u t 2 + τ σ α α t α 2 u - τ ϵ β c 0 2 β + 2 u t β + 2 = 0. {\nabla^{2}u-\dfrac{1}{c_{0}^{2}}\frac{\partial^{2}u}{\partial t^{2}}+\tau_{% \sigma}^{\alpha}\dfrac{\partial^{\alpha}}{\partial t^{\alpha}}\nabla^{2}u-% \dfrac{\tau_{\epsilon}^{\beta}}{c_{0}^{2}}\dfrac{\partial^{\beta+2}u}{\partial t% ^{\beta+2}}=0.}

Acoustic_camera.html

  1. ( θ 0 , ϕ 0 ) \left(\theta_{0},\phi_{0}\right)
  2. ( θ 0 , ϕ 0 ) \left(\theta_{0},\phi_{0}\right)

Acoustoelastic_effect.html

  1. λ \lambda
  2. μ \mu
  3. l l
  4. m m
  5. n n
  6. s y m b o l P symbol{P}
  7. s y m b o l P T = s y m b o l N symbol{P}^{T}=symbol{N}
  8. s y m b o l E symbol{E}
  9. s y m b o l P = s y m b o l F W s y m b o l E or P i j = F i k W E k j , i , j = 1 , 2 , 3 , symbol{P}=symbol{F}\cdot\frac{\partial W}{\partial symbol{E}}\qquad\,\text{or}% \qquad P_{ij}=F_{ik}~{}\frac{\partial W}{\partial E_{kj}},\qquad i,j=1,2,3~{},
  10. s y m b o l F symbol{F}
  11. W W
  12. ρ 0 \rho_{0}
  13. W ( s y m b o l E ) W(symbol{E})
  14. s y m b o l E symbol{E}
  15. W C 0 + C i j E i j + 1 2 ! C i j k l E i j E k l + 1 3 ! C i j k l m n E i j E k l E m n + W\approx C_{0}+C_{ij}E_{ij}+\frac{1}{2!}C_{ijkl}E_{ij}E_{kl}+\frac{1}{3!}C_{% ijklmn}E_{ij}E_{kl}E_{mn}+\cdots
  16. W ( E i j = 0 ) = 0 W(E_{ij}=0)=0
  17. W 1 2 ! C i j k l E i j E k l + 1 3 ! C i j k l m n E i j E k l E m n + , W\approx\frac{1}{2!}C_{ijkl}E_{ij}E_{kl}+\frac{1}{3!}C_{ijklmn}E_{ij}E_{kl}E_{% mn}+\cdots,
  18. C i j k l C_{ijkl}
  19. C i j k l m n C_{ijklmn}
  20. E i j = E j i E_{ij}=E_{ji}
  21. W W
  22. C i j k l C_{ijkl}
  23. C i j k l = C j i k l = C i j l k , C_{ijkl}=C_{jikl}=C_{ijlk},
  24. C i j k l = C k l i j , C_{ijkl}=C_{klij},
  25. C i j k l m n C_{ijklmn}
  26. C i j k l = C I J C_{ijkl}=C_{IJ}
  27. C i j k l m n = C I J K C_{ijklmn}=C_{IJK}
  28. F i j = u i X j + δ i j , F_{ij}=\frac{\partial u_{i}}{\partial X_{j}}+\delta_{ij},
  29. u i u_{i}
  30. P P
  31. X i X_{i}
  32. x i x_{i}
  33. E k l E_{kl}
  34. u u
  35. P i j = C i j k l u k X l + 1 2 M i j k l m n u k X l u m X n + 1 3 M i j k l m n p q u k X l u m X n u p X q + , P_{ij}=C_{ijkl}\frac{\partial u_{k}}{\partial X_{l}}+\frac{1}{2}M_{ijklmn}% \frac{\partial u_{k}}{\partial X_{l}}\frac{\partial u_{m}}{\partial X_{n}}+% \frac{1}{3}M_{ijklmnpq}\frac{\partial u_{k}}{\partial X_{l}}\frac{\partial u_{% m}}{\partial X_{n}}\frac{\partial u_{p}}{\partial X_{q}}+\cdots,
  36. M i j k l m n = C i j k l m n + C i j l n δ k m + C j n k l δ i m + C j l m n δ i k , M_{ijklmn}=C_{ijklmn}+C_{ijln}\delta_{km}+C_{jnkl}\delta_{im}+C_{jlmn}\delta_{% ik},
  37. u k / X l \partial u_{k}/\partial X_{l}
  38. P i j = C i j k l u k X l , P_{ij}=C_{ijkl}\frac{\partial u_{k}}{\partial X_{l}},
  39. P i j P_{ij}
  40. u k / X l \partial u_{k}/\partial X_{l}
  41. C i j k l C_{ijkl}
  42. s y m b o l X symbol{X}
  43. s y m b o l x symbol{x}
  44. s y m b o l x symbol{x^{\prime}}
  45. s y m b o l u = s y m b o l u ( 0 ) + s y m b o l u ( 1 ) = s y m b o l x - s y m b o l X , symbol{u}=symbol{u}^{(0)}+symbol{u}^{(1)}=symbol{x}-symbol{X},
  46. s y m b o l u ( 0 ) = s y m b o l x - s y m b o l X , s y m b o l u ( 1 ) = s y m b o l x - s y m b o l x symbol{u}^{(0)}=symbol{x}-symbol{X},\qquad symbol{u}^{(1)}=symbol{x^{\prime}}-% symbol{x}
  47. s y m b o l u ( 1 ) symbol{u}^{(1)}
  48. s y m b o l u ( 0 ) symbol{u}^{(0)}
  49. | s y m b o l u ( 1 ) | | s y m b o l u ( 0 ) | | |symbol{u}^{(1)}|<<|symbol{u}^{(0)}||
  50. Div s y m b o l P T = ρ 0 s y m b o l x ¨ . \operatorname{Div}symbol{P}^{T}=\rho_{0}\ddot{symbol{x}^{\prime}}.
  51. t t
  52. Div \operatorname{Div}
  53. s y m b o l X symbol{X}
  54. ρ 0 s y m b o l x ¨ = 2 t 2 ( s y m b o l u ( 0 ) + s y m b o l u ( 1 ) + s y m b o l X ) = 2 s y m b o l u ( 1 ) t 2 \begin{aligned}\displaystyle\rho_{0}\ddot{symbol{x}^{\prime}}&\displaystyle=% \frac{\partial^{2}}{\partial t^{2}}(symbol{u}^{(0)}+symbol{u}^{(1)}+symbol{X})% \\ &\displaystyle=\frac{\partial^{2}symbol{u}^{(1)}}{\partial t^{2}}\end{aligned}
  55. 2 / t 2 ( s y m b o l u ( 0 ) ) = 2 / t 2 ( s y m b o l X ) = 0 \partial^{2}/\partial t^{2}(symbol{u}^{(0)})=\partial^{2}/\partial t^{2}(% symbol{X})=0
  56. X j X_{j}
  57. x j x_{j}
  58. X j = x j + u k , j ( 0 ) x k + \frac{\partial}{\partial X_{j}}=\frac{\partial}{\partial x_{j}}+u^{(0)}_{k,j}% \frac{\partial}{\partial x_{k}}+\cdots
  59. u k , j ( 0 ) u k ( 0 ) / x j u^{(0)}_{k,j}\equiv\partial u^{(0)}_{k}/\partial x_{j}
  60. P i j X j P i j x j + u p . j ( 0 ) P i j x p \frac{\partial P_{ij}}{\partial X_{j}}\approx\frac{\partial P_{ij}}{\partial x% _{j}}+u_{p.j}^{(0)}\frac{\partial P_{ij}}{\partial x_{p}}
  61. s y m b o l u ( 0 ) symbol{u}^{(0)}
  62. ( D i v ) s y m b o l P ( 0 ) = s y m b o l 0 \operatorname{(}Div)symbol{P}^{(0)}=symbol{0}
  63. u m , n ( 0 ) u_{m,n}^{(0)}
  64. s y m b o l u ( 0 ) symbol{u}^{(0)}
  65. s y m b o l u ( 1 ) ( s y m b o l x , t ) symbol{u}^{(1)}(symbol{x},t)
  66. B i j k l 2 u k ( 1 ) x j x l = ρ 0 2 u i ( 1 ) t 2 , B_{ijkl}\frac{\partial^{2}u_{k}^{(1)}}{\partial x_{j}\partial x_{l}}=\rho_{0}% \frac{\partial^{2}u_{i}^{(1)}}{\partial t^{2}},
  67. B i j k l = C i j k l + δ i k C j l q r u q , r ( 0 ) + δ i k C j l q r u q , r ( 0 ) + C r j k l u i , r ( 0 ) + C i r k l u j , r ( 0 ) + C i j r l u k , r ( 0 ) + C i j k r u l , r ( 0 ) + C i j k l m n u m , n ( 0 ) . B_{ijkl}=C_{ijkl}+\delta_{ik}C_{jlqr}u_{q,r}^{(0)}+\delta_{ik}C_{jlqr}u_{q,r}^% {(0)}+C_{rjkl}u_{i,r}^{(0)}+C_{irkl}u_{j,r}^{(0)}+C_{ijrl}u_{k,r}^{(0)}+C_{% ijkr}u_{l,r}^{(0)}+C_{ijklmn}u_{m,n}^{(0)}.
  68. s y m b o l u ( 1 ) ( s y m b o l x , t ) = s y m b o l m f ( s y m b o l N \cdotsymbol x - c t ) , symbol{u}^{(1)}(symbol{x},t)=symbol{m}\,f(symbol{N}\cdotsymbol{x}-ct),
  69. s y m b o l N symbol{N}
  70. s y m b o l k = k s y m b o l N symbol{k}=ksymbol{N}
  71. s y m b o l m symbol{m}
  72. c c
  73. f f
  74. s y m b o l Q ( s y m b o l N ) s y m b o l m = ρ 0 c 2 s y m b o l m symbol{Q}(symbol{N})symbol{m}=\rho_{0}c^{2}symbol{m}
  75. s y m b o l Q ( s y m b o l N ) symbol{Q}(symbol{N})
  76. s y m b o l N symbol{N}
  77. [ s y m b o l Q ( s y m b o l N ) ] i j = B i j k l N j N l . [symbol{Q}(symbol{N})]_{ij}=B_{ijkl}N_{j}N_{l}.
  78. s y m b o l n symbol{n}
  79. det ( s y m b o l Q ( s y m b o l N ) - ρ 0 c 2 s y m b o l I ) = 0 , \operatorname{det}(symbol{Q}(symbol{N})-\rho_{0}c^{2}symbol{I})=0,
  80. det \operatorname{det}
  81. s y m b o l I symbol{I}
  82. s y m b o l Q ( s y m b o l N ) symbol{Q}(symbol{N})
  83. ρ 0 c 2 \rho_{0}c^{2}
  84. s y m b o l N symbol{N}
  85. ρ 0 c 2 = s y m b o l Q ( s y m b o l N ) s y m b o l m \cdotsymbol m = B i j k l N j N l m i m k , \rho_{0}c^{2}=symbol{Q}(symbol{N})symbol{m}\cdotsymbol{m}=B_{ijkl}N_{j}N_{l}m_% {i}m_{k},
  86. B i j k l N j N l m i m k > 0 B_{ijkl}N_{j}N_{l}m_{i}m_{k}>0
  87. s y m b o l N symbol{N}
  88. s y m b o l m symbol{m}
  89. s y m b o l m = s y m b o l N symbol{m}=symbol{N}
  90. s y m b o l m \cdotsymbol N = 0 symbol{m}\cdotsymbol{N}=0
  91. s y m b o l E symbol{E}
  92. tr s y m b o l E q \operatorname{tr}symbol{E}^{q}
  93. tr \operatorname{tr}
  94. q { 1 , 2 , 3 , } q\in\left\{1,2,3,\dots\right\}
  95. W ( s y m b o l E ) = W ( tr s y m b o l E q ) , k { 1 , 2 , 3 , } W(symbol{E})=W(\operatorname{tr}symbol{E}^{q}),\,k\in\left\{1,2,3,\ldots\right\}
  96. W = λ 2 ( tr s y m b o l E ) 2 + μ tr s y m b o l E 2 + C 3 ( tr s y m b o l E ) 3 + B ( tr s y m b o l E ) tr s y m b o l E 2 + A 3 tr s y m b o l E 3 + , W=\frac{\lambda}{2}(\operatorname{tr}symbol{E})^{2}+\mu\operatorname{tr}symbol% {E}^{2}+\frac{C}{3}(\operatorname{tr}symbol{E})^{3}+B(\operatorname{tr}symbol{% E})\operatorname{tr}symbol{E}^{2}+\frac{A}{3}\operatorname{tr}symbol{E}^{3}+\cdots,
  97. λ , μ , A , B , C \lambda,\mu,A,B,C
  98. λ \lambda
  99. μ \mu
  100. A , B , A,B,
  101. C C
  102. l , m , l,m,
  103. n n
  104. C i j k l = λ δ i j δ k l + 2 μ δ I i j k l , C i j k l m n = 2 C δ i j δ k l δ m n + 2 B ( δ i j I k l m n + δ k l I m n i j + δ m n I i j k l ) + 1 2 A ( δ i k I j k m n + δ i l I j k m n + δ j k I i l m n + δ j l I i k m n ) , \begin{aligned}\displaystyle C_{ijkl}&\displaystyle=\lambda\delta_{ij}\delta_{% kl}+2\mu\delta I_{ijkl},\\ \displaystyle C_{ijklmn}&\displaystyle=2C\delta_{ij}\delta_{kl}\delta_{mn}+2B(% \delta_{ij}I_{klmn}+\delta_{kl}I_{mnij}+\delta_{mn}I_{ijkl})+\frac{1}{2}A(% \delta_{ik}I_{jkmn}+\delta_{il}I_{jkmn}+\delta_{jk}I_{ilmn}+\delta_{jl}I_{ikmn% }),\end{aligned}\!\,
  105. I i j k l = 1 2 ( δ i k δ j l + δ i l δ j k ) I_{ijkl}=\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})
  106. C I J K C_{IJK}
  107. A A
  108. ν 1 = 2 C \nu_{1}=2C
  109. l = B + C l=B+C
  110. α = 1 3 C \alpha=\frac{1}{3}C
  111. l E = 1 3 A + B + 1 3 C l_{E}=\frac{1}{3}A+B+\frac{1}{3}C
  112. C 123 = 2 C C_{123}=2C
  113. C 111 = 2 A + 6 B + 2 C C_{111}=2A+6B+2C
  114. B B
  115. ν 2 = B \nu_{2}=B
  116. m = 1 2 A + B m=\frac{1}{2}A+B
  117. β = B \beta=B
  118. m E = - A - 2 B m_{E}=-A-2B
  119. C 144 = B C_{144}=B
  120. C 112 = 2 B + 2 C C_{112}=2B+2C
  121. C C
  122. ν 3 = 1 4 A \nu_{3}=\frac{1}{4}A
  123. n = A n=A
  124. γ = 1 3 A \gamma=\frac{1}{3}A
  125. n E = A n_{E}=A
  126. C 456 = 1 4 A C_{456}=\frac{1}{4}A
  127. C 166 = 1 2 A + B C_{166}=\frac{1}{2}A+B
  128. λ \lambda
  129. μ \mu
  130. ν 1 \nu_{1}
  131. ν 2 \nu_{2}
  132. ν 3 \nu_{3}
  133. 111 ± 1 111\pm 1
  134. 82.1 ± 0.5 82.1\pm 0.5
  135. - 385 ± 70 -385\pm 70
  136. - 282 ± 30 -282\pm 30
  137. - 177 ± 8 -177\pm 8
  138. 110.5 ± 1 110.5\pm 1
  139. 82.0 ± 0.5 82.0\pm 0.5
  140. - 134 ± 20 -134\pm 20
  141. - 261 ± 20 -261\pm 20
  142. - 167 ± 6 -167\pm 6
  143. 109 ± 1 109\pm 1
  144. 81.9 ± 0.5 81.9\pm 0.5
  145. - 323 ± 50 -323\pm 50
  146. - 265 ± 30 -265\pm 30
  147. - 177 ± 10 -177\pm 10
  148. 109 ± 1 109\pm 1
  149. 81.8 ± 0.5 81.8\pm 0.5
  150. - 175 ± 50 -175\pm 50
  151. - 240 ± 50 -240\pm 50
  152. - 169 ± 15 -169\pm 15
  153. 87 ± 2 87\pm 2
  154. 71.6 ± 3 71.6\pm 3
  155. 34 ± 20 34\pm 20
  156. - 552 ± 80 -552\pm 80
  157. - 100 ± 10 -100\pm 10
  158. λ \lambda
  159. μ \mu
  160. l l
  161. m m
  162. n n
  163. 109.0 ± 1 109.0\pm 1
  164. 81.7 ± 0.2 81.7\pm 0.2
  165. - 56 ± 20 -56\pm 20
  166. - 671 ± 6 -671\pm 6
  167. - 785 ± 7 -785\pm 7
  168. 115.8 ± 2.3 % 115.8\pm 2.3\%
  169. 79.9 ± 2.3 % 79.9\pm 2.3\%
  170. - 248 ± 2.8 % -248\pm 2.8\%
  171. - 623 ± 4.1 % -623\pm 4.1\%
  172. - 714 ± 2.7 % -714\pm 2.7\%
  173. 110.7 ± 2.3 % 110.7\pm 2.3\%
  174. 82.4 ± 2.3 % 82.4\pm 2.3\%
  175. - 302 ± 2.8 % -302\pm 2.8\%
  176. - 616 ± 4.1 % -616\pm 4.1\%
  177. - 724 ± 2.7 % -724\pm 2.7\%
  178. X i [ 0 , L i ] , i = 1 , 2 , 3 X_{i}\in[0,L_{i}],\,i=1,2,3
  179. L i L_{i}
  180. x 1 x_{1}
  181. x 1 = λ 1 X 1 , x 2 = λ 2 X 2 , x 3 = λ 3 X 3 x_{1}=\lambda_{1}X_{1},x_{2}=\lambda_{2}X_{2},x_{3}=\lambda_{3}X_{3}
  182. e i l i / L i - 1 = λ i - 1 e_{i}\equiv l_{i}/L_{i}-1=\lambda_{i}-1
  183. x i x_{i}
  184. l i l_{i}
  185. i i
  186. λ i l i / L i \lambda_{i}\equiv l_{i}/L_{i}
  187. s y m b o l F = s y m b o l R U = s y m b o l V R symbol{F}=symbol{RU}=symbol{VR}
  188. s y m b o l R = s y m b o l I symbol{R}=symbol{I}
  189. λ i \lambda_{i}
  190. e i e_{i}
  191. x 1 x_{1}
  192. P 11 > 0 P_{11}>0
  193. e 1 e_{1}
  194. P 22 = P 33 = 0 P_{22}=P_{33}=0
  195. e 2 e_{2}
  196. e 3 e_{3}
  197. e 2 , e 3 ( - 1 , 0 ] e_{2},e_{3}\in(-1,0]
  198. e 2 = e 3 e_{2}=e_{3}
  199. e 2 = e 3 = - 1 e_{2}=e_{3}=-1
  200. e 2 = e 3 = 0 e_{2}=e_{3}=0
  201. B i j k l N j N l m i m k > 0 B_{ijkl}N_{j}N_{l}m_{i}m_{k}>0
  202. s y m b o l m symbol{m}
  203. s y m b o l N symbol{N}
  204. x 1 x_{1}
  205. x 3 x_{3}
  206. s y m b o l N = [ 0 , 0 , 1 ] symbol{N}=[0,0,1]
  207. { s y m b o l m } = { 𝐦 1 = 𝐱 ^ 1 = [ 1 , 0 , 0 ] to applied tension 𝐦 2 = 𝐱 ^ 2 = [ 0 , 1 , 0 ] to applied tension 𝐦 3 = 𝐱 ^ 3 = [ 0 , 0 , 1 ] to 𝐍 \{symbol{m}\}=\begin{cases}\mathbf{m}_{1}=\mathbf{\hat{x}}_{1}=[1,0,0]&\|\,% \textrm{to}\,\textrm{applied}\,\textrm{tension}\\ \mathbf{m}_{2}=\mathbf{\hat{x}}_{2}=[0,1,0]&\perp\textrm{to}\,\textrm{applied}% \,\textrm{tension}\\ \mathbf{m}_{3}=\mathbf{\hat{x}}_{3}=[0,0,1]&\|\,\textrm{to}\,\mathbf{N}\end{cases}
  208. ρ 0 c 33 2 = B 3333 , ρ 0 c 31 2 = B 1313 , ρ 0 c 32 2 = B 2323 , \rho_{0}c^{2}_{33}=B_{3333},\qquad\rho_{0}c^{2}_{31}=B_{1313},\qquad\rho_{0}c^% {2}_{32}=B_{2323},
  209. i i
  210. c i j c_{ij}
  211. x 3 x_{3}
  212. j j
  213. j = i j=i
  214. i i
  215. j i j\neq i
  216. C i j k l C_{ijkl}
  217. C i j k l m n C_{ijklmn}
  218. λ , μ \lambda,\mu
  219. A , B , C A,B,C
  220. ρ 0 c 33 2 = λ + 2 μ + a 33 e 1 , ρ 0 c 3 k 2 = μ + a 3 k e 1 , k = 1 , 2 \rho_{0}c^{2}_{33}=\lambda+2\mu+a_{33}e_{1},\qquad\rho_{0}c^{2}_{3k}=\mu+a_{3k% }e_{1},\quad k=1,2
  221. a 33 = - 2 λ ( λ + 2 μ ) + λ A + 2 ( λ - μ ) B - 2 μ C λ + μ a_{33}=-\frac{2\lambda(\lambda+2\mu)+\lambda A+2(\lambda-\mu)B-2\mu C}{\lambda% +\mu}
  222. a 31 = ( λ + 2 μ ) ( 4 μ + A ) + 4 μ B 4 ( λ + μ ) a_{31}=\frac{(\lambda+2\mu)(4\mu+A)+4\mu B}{4(\lambda+\mu)}
  223. a 32 = - λ ( 4 μ + A ) - 2 μ B 2 ( λ + μ ) a_{32}=-\frac{\lambda(4\mu+A)-2\mu B}{2(\lambda+\mu)}
  224. c = d t c=\frac{d}{t}
  225. d d
  226. t t
  227. c = f λ c=f\lambda
  228. f f
  229. λ \lambda
  230. n n
  231. s y m b o l m symbol{m}
  232. s y m b o l N symbol{N}

Acrylyl-CoA_reductase_(NADH).html

  1. \rightleftharpoons

Acrylyl-CoA_reductase_(NADPH).html

  1. \rightleftharpoons

Active_redundancy.html

  1. A o N = 0.99 U p T i m e A_{o}^{N}=0.99\ Up\ Time
  2. f a i l e d 90 h o u r / y e a r \approx failed\ 90\ hour/year
  3. A o N + 1 = 1 - ( ( 1 - A o N ) × ( 1 - A o N ) ) = 0.9999 U p T i m e A_{o}^{N+1}=1-\left((1-A_{o}^{N})\times(1-A_{o}^{N})\right)=0.9999\ Up\ Time
  4. f a i l e d 50 m i n u t e s / y e a r \approx failed\ 50\ minutes/year
  5. A o N + 2 = 1 - ( ( 1 - A o N ) × ( 1 - A o N ) × ( 1 - A o N ) ) = 0.999999 U p T i m e A_{o}^{N+2}=1-\left((1-A_{o}^{N})\times(1-A_{o}^{N})\times(1-A_{o}^{N})\right)% =0.999999\ Up\ Time
  6. f a i l e d 30 s e c o n d s / y e a r \approx failed\ 30\ seconds/year

Active_return.html

  1. R p R_{p}
  2. R b R_{b}
  3. R p - R b R_{p}-R_{b}

Acyclic_orientation.html

  1. G G
  2. G G
  3. G G
  4. G G
  5. T G T_{G}
  6. G G
  7. G G
  8. T G ( 2 , 0 ) T_{G}(2,0)
  9. T G ( 0 , 2 ) T_{G}(0,2)
  10. χ G \chi_{G}
  11. | χ G ( - 1 ) | |\chi_{G}(-1)|

Acyl-homoserine-lactone_acylase.html

  1. \rightleftharpoons

Acyl-homoserine-lactone_synthase.html

  1. \rightleftharpoons

Adaptive_coordinate_descent.html

  1. 10 - 10 10^{-10}
  2. x 0 = ( - 3 , - 4 ) x_{0}=(-3,-4)

Adenosylcobinamide-GDP_ribazoletransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Adenozil-chloride_synthase.html

  1. \rightleftharpoons

ADF-GLS_test.html

  1. y t = d t + u t y_{t}=d_{t}+u_{t}\,
  2. u t = ρ u t - 1 + e t u_{t}=\rho u_{t-1}+e_{t}\,
  3. d t d_{t}\,
  4. u t u_{t}\,
  5. y t y_{t}\,
  6. ρ \rho\,
  7. d t d_{t}\,
  8. y t y_{t}\,
  9. ρ = 1 - c T \rho=1-\frac{c}{T}\,
  10. T T\,
  11. 1 - c ¯ T L 1-\frac{\bar{c}}{T}L\,
  12. L L\,
  13. y ¯ t = y t - ( c ¯ / T ) y t - 1 \bar{y}_{t}=y_{t}-(\bar{c}/T)y_{t-1}\,
  14. y ¯ t \bar{y}_{t}\,
  15. y t y_{t}\,
  16. c ¯ \bar{c}\,
  17. ρ = 1 - c / T \rho=1-c/T\,
  18. c = c ¯ c=\bar{c}\,
  19. d t d_{t}\,
  20. c ¯ \bar{c}\,

Admittance_(geophysics).html

  1. m / s 2 Pa \frac{\mathrm{m}/\mathrm{s}^{2}}{\mathrm{Pa}}
  2. m / s 2 N / m 2 \frac{\mathrm{m}/\mathrm{s}^{2}}{\mathrm{N}/\mathrm{m}^{2}}
  3. m 3 N s 2 \frac{\mathrm{m}^{3}}{\mathrm{N}\cdot\mathrm{s}^{2}}
  4. m 2 kg \frac{\mathrm{m}^{2}}{\mathrm{kg}}

ADP-ribosylarginine_hydrolase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Adrenodoxin-NADP+_reductase.html

  1. \rightleftharpoons

Aeolian_sound.html

  1. f Aeolian = α v d f_{\,\text{Aeolian}}\ =\ \frac{\alpha v}{d}

Affine-regular_polygon.html

  1. m m
  2. m m
  3. n n
  4. n > 4 n>4
  5. m m
  6. m m
  7. n n

Affine_bundle.html

  1. π ¯ : Y ¯ X \overline{\pi}:\overline{Y}\to X
  2. F ¯ \overline{F}
  3. π ¯ : Y ¯ X \overline{\pi}:\overline{Y}\to X
  4. π : Y X \pi:Y\to X
  5. F F
  6. F ¯ \overline{F}
  7. Y x Y_{x}
  8. Y Y
  9. Y ¯ x \overline{Y}_{x}
  10. Y ¯ \overline{Y}
  11. Y X Y\to X
  12. ( x μ , y i ) (x^{\mu},y^{i})
  13. y i = A j i ( x ν ) y j + b i ( x ν ) . y^{\prime i}=A^{i}_{j}(x^{\nu})y^{j}+b^{i}(x^{\nu}).
  14. Y × X Y ¯ Y , ( y i , y ¯ i ) y i + y ¯ i , Y\times_{X}\overline{Y}\longrightarrow Y,\qquad(y^{i},\overline{y}^{i})% \longmapsto y^{i}+\overline{y}^{i},
  15. Y × X Y Y ¯ , ( y i , y i ) y i - y i , Y\times_{X}Y\longrightarrow\overline{Y},\qquad(y^{i},y^{\prime i})\longmapsto y% ^{i}-y^{\prime i},
  16. ( y ¯ i ) (\overline{y}^{i})
  17. Y ¯ \overline{Y}
  18. y ¯ i = A j i ( x ν ) y ¯ j \overline{y}^{\prime i}=A^{i}_{j}(x^{\nu})\overline{y}^{j}
  19. π : Y X \pi:Y\to X
  20. π ¯ : Y ¯ X \overline{\pi}:\overline{Y}\to X
  21. s s
  22. Y X Y\to X
  23. Y y y - s ( π ( y ) ) Y ¯ , Y ¯ y ¯ s ( π ( y ) ) + y ¯ Y . Y\ni y\to y-s(\pi(y))\in\overline{Y},\qquad\overline{Y}\ni\overline{y}\to s(% \pi(y))+\overline{y}\in Y.
  24. Y Y
  25. s = 0 s=0
  26. Y Y
  27. T X TX
  28. X X
  29. Y X Y\to X
  30. G A ( m , ) GA(m,\mathbb{R})
  31. V V
  32. m m
  33. G L ( m , ) GL(m,\mathbb{R})
  34. Φ : Y Y \Phi:Y\to Y^{\prime}
  35. Y Y
  36. Φ : Y Y \Phi:Y\to Y^{\prime}
  37. Y Y
  38. Y ¯ \overline{Y}
  39. Y Y^{\prime}
  40. Y ¯ \overline{Y}^{\prime}
  41. Φ ¯ : Y ¯ Y ¯ , y ¯ i = Φ i y j y ¯ j , \overline{\Phi}:\overline{Y}\to\overline{Y}^{\prime},\qquad\overline{y}^{% \prime i}=\frac{\partial\Phi^{i}}{\partial y^{j}}\overline{y}^{j},
  42. Φ \Phi

Affine_term_structure_model.html

  1. r ( t ) r(t)
  2. d r ( t ) = μ ( t , r ( t ) ) d t + σ ( t , r ( t ) ) d W ( t ) dr(t)=\mu(t,r(t))\,dt+\sigma(t,r(t))\,dW(t)
  3. T T
  4. p ( t , T ) p(t,T)
  5. t t
  6. p ( t , T ) = F T ( t , r ( t ) ) p(t,T)=F^{T}(t,r(t))
  7. F F
  8. F T ( t , r ) = e A ( t , T ) - B ( t , T ) r F^{T}(t,r)=e^{A(t,T)-B(t,T)r}
  9. A A
  10. B B
  11. μ \mu
  12. σ \sigma
  13. F F
  14. A t ( t , T ) - ( 1 + B t ( t , T ) ) r - μ ( t , r ) B ( t , T ) + 1 2 σ 2 ( t , r ) B 2 ( t , T ) = 0 A_{t}(t,T)-(1+B_{t}(t,T))r-\mu(t,r)B(t,T)+\frac{1}{2}\sigma^{2}(t,r)B^{2}(t,T)=0
  15. F T ( T , r ) = 1 F^{T}(T,r)=1
  16. A ( T , T ) \displaystyle A(T,T)
  17. μ \mu
  18. σ 2 \sigma^{2}
  19. r r
  20. μ ( t , r ) \displaystyle\mu(t,r)
  21. A t ( t , T ) - β ( t ) B ( t , T ) + 1 2 δ ( t ) B 2 ( t , T ) - [ 1 + B t ( t , T ) + α ( t ) B ( t , T ) - 1 2 γ ( t ) B 2 ( t , T ) ] r = 0 A_{t}(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^{2}(t,T)-\left[1+B_{t}(t,T)+% \alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^{2}(t,T)\right]r=0
  22. r r
  23. t t
  24. T T
  25. r r
  26. 1 + B t ( t , T ) + α ( t ) B ( t , T ) - 1 2 γ ( t ) B 2 ( t , T ) = 0 1+B_{t}(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^{2}(t,T)=0
  27. A t ( t , T ) - β ( t ) B ( t , T ) + 1 2 δ ( t ) B 2 ( t , T ) = 0 A_{t}(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^{2}(t,T)=0
  28. μ \mu
  29. σ 2 \sigma^{2}
  30. r r
  31. A A
  32. B B
  33. 1 + B t ( t , T ) + α ( t ) B ( t , T ) - 1 2 γ ( t ) B 2 ( t , T ) \displaystyle 1+B_{t}(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^{2}(t,T)
  34. d r = ( b - a r ) d t + σ d W dr=(b-ar)\,dt+\sigma\,dW
  35. p ( t , T ) \displaystyle p(t,T)

Aggregative_game.html

  1. S i S_{i}\subseteq\mathbb{R}
  2. S = S 1 × S 2 × × S n S=S_{1}\times S_{2}\times\ldots\times S_{n}
  3. f i : S f_{i}:S\to\mathbb{R}
  4. f ~ i : S i × \tilde{f}_{i}:S_{i}\times\mathbb{R}\to\mathbb{R}
  5. s S s\in S
  6. f i ( s ) = f ~ i ( s i , j = 1 n s j ) f_{i}(s)=\tilde{f}_{i}\left(s_{i},\sum_{j=1}^{n}s_{j}\right)
  7. s j \sum s_{j}
  8. f i ( s ) = s i P ( s j ) - C i ( s i ) f_{i}(s)=s_{i}P\left(\sum s_{j}\right)-C_{i}(s_{i})
  9. P P
  10. C i C_{i}
  11. f i ( s ) = f ~ i ( s i , s j ) f_{i}(s)=\tilde{f}_{i}\left(s_{i},\sum s_{j}\right)
  12. f ~ i ( s i , X ) = s i P ( X ) - C i ( s i ) \tilde{f}_{i}(s_{i},X)=s_{i}P(X)-C_{i}(s_{i})
  13. g : S g:S\to\mathbb{R}
  14. h 0 , h 1 , , h n : h_{0},h_{1},\ldots,h_{n}:\mathbb{R}\to\mathbb{R}
  15. g ( s ) = h 0 ( i h i ( s i ) ) g(s)=h_{0}(\sum_{i}h_{i}(s_{i}))
  16. f ~ i : S i × \tilde{f}_{i}:S_{i}\times\mathbb{R}\to\mathbb{R}
  17. f i ( s ) = f ~ i ( s i , g ( s 1 , , s n ) ) f_{i}(s)=\tilde{f}_{i}(s_{i},g(s_{1},\ldots,s_{n}))
  18. s S s\in S
  19. g ( s 1 , , s n ) = g i g(s_{1},\ldots,s_{n})=\sum g_{i}

Agrawal's_conjecture.html

  1. n n
  2. r r
  3. ( X - 1 ) n X n - 1 ( mod n , X r - 1 ) (X-1)^{n}\equiv X^{n}-1\;\;(\mathop{{\rm mod}}n,X^{r}-1)\,
  4. n n
  5. n 2 1 ( mod r ) n^{2}\equiv 1\;\;(\mathop{{\rm mod}}r)
  6. O ~ ( log 6 n ) \tilde{O}(\log^{6}n)
  7. O ~ ( log 3 n ) ) \tilde{O}(\log^{3}n))
  8. r < 100 r<100
  9. n < 10 10 n<10^{10}
  10. 1 n ε \tfrac{1}{n^{\varepsilon}}
  11. ε > 0 \varepsilon>0
  12. n n
  13. r r
  14. ( X - 1 ) n X n - 1 ( mod n , X r - 1 ) (X-1)^{n}\equiv X^{n}-1\;\;(\mathop{{\rm mod}}n,X^{r}-1)
  15. ( X + 2 ) n X n + 2 ( mod n , X r - 1 ) (X+2)^{n}\equiv X^{n}+2\;\;(\mathop{{\rm mod}}n,X^{r}-1)
  16. n n
  17. n 2 1 ( mod r ) n^{2}\equiv 1\;\;(\mathop{{\rm mod}}r)

Air-gap_flash.html

  1. t d i s c h a r g e L C t_{discharge}\propto{\sqrt{LC}}
  2. E = C V 2 2 E={{CV^{2}}\over 2}

AKNS_system.html

  1. p t = + i p 2 q - i 2 p x x p_{t}=+ip^{2}q-\frac{i}{2}p_{xx}
  2. q t = - i q 2 p + i 2 q x x q_{t}=-iq^{2}p+\frac{i}{2}q_{xx}

Alcohol-forming_fatty_acyl-CoA_reductase.html

  1. \rightleftharpoons

Alcohol_dehydrogenase_(azurin).html

  1. \rightleftharpoons

Alcohol_dehydrogenase_(cytochrome_c).html

  1. \rightleftharpoons

Alcohol_dehydrogenase_(nicotinoprotein).html

  1. \rightleftharpoons

Alcohol_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Alcuin's_sequence.html

  1. x 3 ( 1 - x 2 ) ( 1 - x 3 ) ( 1 - x 4 ) = x 3 + x 5 + x 6 + 2 x 7 + x 8 + 3 x 9 + . \frac{x^{3}}{(1-x^{2})(1-x^{3})(1-x^{4})}=x^{3}+x^{5}+x^{6}+2x^{7}+x^{8}+3x^{9% }+\cdots.

Aldos-2-ulose_dehydratase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Alexander_Arhangelskii.html

  1. | X | 2 χ ( X ) L ( X ) |X|\leq 2^{\chi(X)L(X)}

ALG10_(enzyme_class).html

  1. \rightleftharpoons

ALG6_(enzyme_class).html

  1. \rightleftharpoons

ALG8_(enzyme_class).html

  1. \rightleftharpoons

Aliphatic_(R)-hydroxynitrile_lyase.html

  1. \rightleftharpoons

All-trans-10'-apo-beta-carotenal_13,14-cleaving_dioxygenase.html

  1. \rightleftharpoons

All-trans-8'-apo-beta-carotenal_15,15'-oxygenase.html

  1. \rightleftharpoons

All-trans-decaprenyl-diphosphate_synthase.html

  1. \rightleftharpoons

All-trans-nonaprenyl-diphosphate_synthase_(geranyl-diphosphate_specific).html

  1. \rightleftharpoons

All-trans-nonaprenyl_diphosphate_synthase_(geranylgeranyl-diphosphate_specific).html

  1. \rightleftharpoons

All-trans-octaprenyl-diphosphate_synthase.html

  1. \rightleftharpoons

All-trans-phytoene_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

All-trans-retinyl_ester_13-cis_isomerohydrolase.html

  1. \rightleftharpoons

All-trans-zeta-carotene_desaturase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Allen–Cahn_equation.html

  1. η t = M η [ ϵ η 2 2 η - f ( η ) ] {{\partial\eta}\over{\partial t}}=M_{\eta}[\epsilon^{2}_{\eta}\nabla^{2}\eta-f% ^{\prime}(\eta)]
  2. M η M_{\eta}
  3. f f
  4. η \eta

Almost_commutative_ring.html

  1. gr A = A i / A i - 1 \operatorname{gr}A=\oplus A_{i}/{A_{i-1}}

Almost_ideal_demand_system.html

  1. log ( c ( u , p ) ) = α 0 + k α k log ( p k ) + 1 2 k j γ k j * log ( p k ) log ( p j ) + u β 0 k p k β k \log(c(u,p))=\alpha_{0}+\sum_{k}\alpha_{k}\log(p_{k})+\frac{1}{2}\sum_{k}\sum_% {j}\gamma_{kj}^{*}\log(p_{k})\log(p_{j})+u\beta_{0}\prod_{k}p_{k}^{\beta_{k}}
  2. w i = α i + j γ i j log ( p j ) + β i log { x / P } w_{i}=\alpha_{i}+\sum_{j}\gamma_{ij}\log(p_{j})+\beta_{i}\log\{x/P\}
  3. γ i j = 1 / 2 ( γ i j * + γ j i * ) \gamma_{ij}=1/2(\gamma^{*}_{ij}+\gamma^{*}_{ji})
  4. log ( P ) α 0 + k α k log ( p k ) + 1 2 k j γ k j log ( p k ) log ( p j ) \log(P)\equiv\alpha_{0}+\sum_{k}\alpha_{k}\log(p_{k})+\frac{1}{2}\sum_{k}\sum_% {j}\gamma_{kj}\log(p_{k})\log(p_{j})
  5. α , β , γ \alpha,\beta,\gamma
  6. w i = 1 \sum w_{i}=1

Alpha,alpha-trehalose_synthase.html

  1. \rightleftharpoons

Alpha-1,6-mannosyl-glycoprotein_4-b-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Alpha-D-ribose_1-methylphosphonate_5-triphosphate_diphosphatase.html

  1. \rightleftharpoons

Alpha-D-ribose_1-methylphosphonate_5-triphosphate_synthase.html

  1. \rightleftharpoons

Alpha-humulene_10-hydroxylase.html

  1. \rightleftharpoons

Alpha-pinene_monooxygenase.html

  1. \rightleftharpoons

Alpha_Indexes.html

  1. R S , t + 1 = S t + 1 - S t + D S , t + 1 S t R_{S,t+1}={{S_{t+1}-S_{t}+D_{S,t+1}}\over S_{t}}
  2. S t S_{t}
  3. D s , t D_{s,t}
  4. R M , t + 1 = M t + 1 - M t + D M , t + 1 M t R_{M,t+1}={{M_{t+1}-M_{t}+D_{M,t+1}}\over M_{t}}
  5. I b , t + 1 = I b , t × ( 1 + R S , t + 1 ) ( 1 + R M , t + 1 ) b I_{b,t+1}=I_{b,t}\times{(1+R_{S,t+1})\over(1+R_{M,t+1})^{b}}
  6. σ = σ S 2 + b 2 σ M 2 - 2 b ρ S M σ S σ M \sigma=\sqrt{\sigma_{S}^{2}+b^{2}\sigma_{M}^{2}-2b\rho_{SM}\sigma_{S}\sigma_{M}}
  7. ρ S M \rho_{SM}
  8. σ S \sigma_{S}
  9. σ M \sigma_{M}

Alternant_hydrocarbon.html

  1. E = α + x β E=\alpha+x\beta
  2. E = α - x β E=\alpha-x\beta

Aluminium-ion_battery.html

  1. Al + 7 A l C l 4 - 4 A l 2 Cl 7 - + 3 e - \mathrm{Al}+\mathrm{7AlCl_{4}^{-}}\leftrightarrows\mathrm{4Al_{2}Cl_{7}^{-}}+% \mathrm{3e^{-}}
  2. 2 M n O 2 + Li + + e - LiMn 2 O 4 \mathrm{2MnO_{2}}+\mathrm{Li^{+}}+\mathrm{e^{-}}\leftrightarrows\mathrm{LiMn_{% 2}O_{4}}
  3. Al + 7 A l C l 4 - + 6 M n O 2 + 3 L i + 4 A l 2 Cl 7 - + 3 L i M n 2 O 4 \mathrm{Al}+\mathrm{7AlCl_{4}^{-}}+\mathrm{6MnO_{2}}+\mathrm{3Li^{+}}% \leftrightarrows\mathrm{4Al_{2}Cl_{7}^{-}}+\mathrm{3LiMn_{2}O_{4}}

Ambit_field.html

  1. Y Y
  2. Y Y
  3. χ × \chi\times\mathbb{R}
  4. \mathbb{R}
  5. μ , A t ( x ) , B t ( x ) \mu\in\mathbb{R},A_{t}(x),B_{t}(x)
  6. χ × + , g , q \chi\times\mathbb{R}_{+},g,q
  7. a a
  8. σ 0 \sigma\geq 0
  9. L L
  10. Y Y
  11. Y t = μ + A t ( x ) g ( η , s , x , t ) σ s ( η ) L ( d η , d s ) + B t ( x ) q ( η , s , x , t ) a s ( η ) d η d s Y_{t}=\mu+\int_{A_{t}(x)}g(\eta,s,x,t)\sigma_{s}(\eta)L(d\eta,ds)+\int_{B_{t}(% x)}q(\eta,s,x,t)a_{s}(\eta)\,d\eta\,ds
  12. A t ( x ) A_{t}(x)
  13. B t ( x ) B_{t}(x)
  14. ( t , x ) χ × (t,x)\in\chi\times\mathbb{R}
  15. A t ( x ) A_{t}(x)
  16. B t ( x ) B_{t}(x)
  17. ( t , x ) , Y t ( x ) (t,x),Y_{t}(x)
  18. t t
  19. τ ( θ ) = ( x ( θ ) , t ( θ ) ) χ × \tau(\theta)=(x(\theta),t(\theta))\in\chi\times\mathbb{R}
  20. X θ = Y t ( θ ) ( x ( θ ) ) X_{\theta}=Y_{t(\theta)}(x(\theta))
  21. σ \sigma
  22. σ t 2 ( x ) = C t ( x ) h ( η , s , x , t ) L ~ ( d η , d s ) \sigma^{2}_{t}(x)=\int_{C_{t}(x)}h(\eta,s,x,t)\tilde{L}(d\eta,ds)
  23. L ~ \tilde{L}
  24. A t ( x ) g ( η , s , x , t ) σ s ( η ) L ( d η , d s ) \int_{A_{t}(x)}g(\eta,s,x,t)\sigma_{s}(\eta)L(d\eta,ds)
  25. ( Λ ( A ) : A 𝔹 b ( S ) ) (\Lambda(A):A\in\mathbb{B}_{b}(S))
  26. d \mathbb{R}^{d}
  27. S S
  28. Λ ( A ) \Lambda(A)
  29. A 𝔹 b ( S ) A\in\mathbb{B}_{b}(S)
  30. A 1 , A 2 , , A n 𝔹 b ( S ) A_{1},A_{2},\ldots,A_{n}\in\mathbb{B}_{b}(S)
  31. Λ ( A 1 ) , Λ ( A 2 ) , , Λ ( A n ) \Lambda(A_{1}),\Lambda(A_{2}),\ldots,\Lambda(A_{n})
  32. A 1 , A 2 , 𝔹 b ( S ) A_{1},A_{2},\ldots\in\mathbb{B}_{b}(S)
  33. i = 1 A i 𝔹 b ( S ) \bigcup_{i=1}^{\infty}A_{i}\in\mathbb{B}_{b}(S)
  34. Λ ( i = 1 A i ) = i = 1 Λ ( A i ) \Lambda(\bigcup_{i=1}^{\infty}A_{i})=\sum_{i=1}^{\infty}\Lambda(A_{i})
  35. Y Y
  36. Y t = μ + A t ( x ) g ( η , t - s , x ) σ s ( η ) L ( d η , d s ) + B t ( x ) q ( η , t - s , x ) a s ( η ) d η d s Y_{t}=\mu+\int_{A_{t}(x)}g(\eta,t-s,x)\sigma_{s}(\eta)L(d\eta,ds)+\int_{B_{t}(% x)}q(\eta,t-s,x)a_{s}(\eta)\,d\eta\,ds
  37. A t ( x ) , B t ( x ) A_{t}(x),B_{t}(x)
  38. A t ( x ) = A + ( x , t ) A_{t}(x)=A+(x,t)
  39. A A
  40. B B
  41. g ( η , t , x ) = q ( η , t , x ) = 0 g(\eta,t,x)=q(\eta,t,x)=0
  42. t 0 t\leq 0
  43. σ \sigma
  44. a a
  45. σ \sigma
  46. σ t 2 ( x ) = C t ( x ) h ( η , t - s , x ) L ~ ( d η , d s ) \sigma^{2}_{t}(x)=\int_{C_{t}(x)}h(\eta,t-s,x)\tilde{L}(d\eta,ds)
  47. L ~ \tilde{L}
  48. h h

Aminopyrimidine_aminohydrolase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Ammonia_monooxygenase.html

  1. \rightleftharpoons

Amorpha-4,11-diene_12-monooxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

AmpliPHOX.html

  1. x ¯ \bar{x}
  2. x ¯ \bar{x}
  3. x ¯ \bar{x}
  4. x ¯ \bar{x}

Anatree.html

  1. α 1 \alpha_{1}
  2. α 2 \alpha_{2}
  3. α l \alpha_{l}
  4. n 1 n_{1}
  5. n 2 n_{2}
  6. n l n_{l}
  7. n 1 n_{1}
  8. α 1 \alpha_{1}
  9. n 2 n_{2}
  10. α 2 \alpha_{2}
  11. { A A E E N R T - > { ′′ a n a t r e e ′′ } } \{AAEENRT->\{^{\prime\prime}anatree^{\prime\prime}\}\}
  12. f ( A ) - > 2 , f ( E ) - > 2 , f ( N ) - > 1 , f ( R ) - > 1 , f ( T ) - > 2 f(A)->2,f(E)->2,f(N)->1,f(R)->1,f(T)->2
  13. W n α W_{n}^{\alpha}
  14. n α s n\alpha s
  15. α \alpha
  16. D α = n n | W n α | | W | D_{\alpha}=\sum_{n}n\frac{|W_{n}^{\alpha}|}{|W|}
  17. d o g dog
  18. o g d ogd
  19. 1 1
  20. 1 1
  21. d d
  22. w w
  23. l l
  24. A A
  25. O ( w l ( l o g | A | ) O(wl(log|A|)
  26. O ( w ( | A | l o g l + l l o g | A | ) ) O(w(|A|logl+llog|A|))
  27. O ( w | A | ( l o g | A | + l ) ) O(w|A|(log|A|+l))
  28. O ( | w | ( l + w | A | 2 ) ) O(|w|(l+w|A|^{2}))

Andres_and_Marzo's_delta.html

  1. K Σ i i - n n ( K - 1 ) \frac{K\Sigma_{ii}-n}{n(K-1)}

Andrews_plot.html

  1. x = { x 1 , x 2 , x d } x=\left\{x_{1},x_{2},\ldots x_{d}\right\}
  2. f x ( t ) = x 1 2 + x 2 sin ( t ) + x 3 cos ( t ) + x 4 sin ( 2 t ) + x 5 cos ( 2 t ) + f_{x}(t)=\frac{x_{1}}{\sqrt{2}}+x_{2}\sin(t)+x_{3}\cos(t)+x_{4}\sin(2t)+x_{5}% \cos(2t)+\ldots
  3. - π < t < π -\pi<t<\pi
  4. - π -\pi
  5. π \pi
  6. ( 1 2 , sin ( t ) , cos ( t ) , sin ( 2 t ) , cos ( 2 t ) , ) \left(\frac{1}{\sqrt{2}},\sin(t),\cos(t),\sin(2t),\cos(2t),\ldots\right)

Angelicin_synthase.html

  1. \rightleftharpoons

Anhydro-N-acetylmuramic_acid_kinase.html

  1. \rightleftharpoons

Anthranilate_3-monooxygenase_(FAD).html

  1. \rightleftharpoons

Aperture_value.html

  1. A v A_{v}

Apple_Video.html

  1. color1 = 21 32 * color0 + 11 32 * color3 2 3 * color0 + 1 3 * color3 \mathrm{color1}=\frac{21}{32}*\mathrm{color0}+\frac{11}{32}*\mathrm{color3}% \approx\frac{2}{3}*\mathrm{color0}+\frac{1}{3}*\mathrm{color3}
  2. color2 = 11 32 * color0 + 21 32 * color3 1 3 * color0 + 2 3 * color3 \mathrm{color2}=\frac{11}{32}*\mathrm{color0}+\frac{21}{32}*\mathrm{color3}% \approx\frac{1}{3}*\mathrm{color0}+\frac{2}{3}*\mathrm{color3}

Aqion.html

  1. { i } = K i j = 1 N B { j } ν i , j \{i\}\,=\,K_{i}\,\prod_{j=1}^{N_{B}}\,\{j\}^{\nu_{i,j}}
  2. [ j ] T O T = [ j ] + i = 1 N S ν i , j [ i ] [j]_{TOT}\,=\,[j]+\sum_{i=1}^{N_{S}}\,\nu_{i,j}\,[i]
  3. { i } = γ i [ i ] \{i\}\,=\,\gamma_{i}\,[i]
  4. f j ( c 1 , c 2 , , c N B ) = [ j ] T O T - [ j ] - i = 1 N S ν i , j γ i K i k = 1 N B { k } ν i , k = 0 f_{j}(c_{1},c_{2},...,c_{N_{B}})\,=\,[j]_{TOT}-[j]-\sum^{N_{S}}_{i=1}\frac{\nu% _{i,j}}{\gamma_{i}}\,K_{i}\,\prod^{N_{B}}_{k=1}\{k\}^{\nu_{i,k}}\,=\,0

Aralkylamine_dehydrogenase_(azurin).html

  1. \rightleftharpoons

Archaeosine_synthase.html

  1. \rightleftharpoons

Arching_or_Compressive_Membrane_Action_in_Reinforced_Concrete_Slabs.html

  1. f c f_{\,\text{c}}
  2. f c = 0.8 f cu γ m f_{\,\text{c}}=\frac{0.8f_{\,\text{cu}}}{\gamma_{\,\text{m}}}
  3. ε c \varepsilon_{\,\text{c}}
  4. ε c = ( - 400 + 60 f c - 0.33 f c 2 ) x 10 - 6 \varepsilon_{\,\text{c}}={\left(-400+60f_{\,\text{c}}-0.33f_{\,\text{c}}^{2}% \right)x10^{-6}}
  5. R R
  6. R = ε c L r 2 h 2 R=\frac{\varepsilon_{\,\text{c}}L_{\,\text{r}}^{2}}{h^{2}}
  7. R R
  8. R R
  9. k k
  10. k = 0.0525 ( 4.3 - 16.1 3.3 x 10 - 4 + 0.1243 R ) k={0.0525\left(4.3-16.1\sqrt{3.3x10^{-4}+0.1243R}\right)}
  11. ρ e \rho_{\,\text{e}}
  12. ρ e = k [ f c 240 ] [ h d ] 2 \rho_{\,\text{e}}=k\left[\frac{f_{\,\text{c}}}{240}\right]\left[\frac{h}{d}% \right]^{2}
  13. P ps P_{\,\text{ps}}
  14. P ps = 1.52 ( ϕ + d ) d f c ( 100 ρ e ) 0.25 P_{\,\text{ps}}={1.52\left(\phi+d\right)d\sqrt{f_{\,\text{c}}}\left(100\rho_{% \,\text{e}}\right)^{0.25}}
  15. d d
  16. f cu f_{\,\text{cu}}
  17. h h
  18. L r L_{\,\text{r}}
  19. ϕ \phi
  20. γ m \gamma_{\,\text{m}}

Argumentation_framework.html

  1. A A
  2. A A
  3. R R
  4. S S
  5. S = A , R S=\langle A,R\rangle
  6. A = { a , b , c , d } A=\{a,b,c,d\}
  7. R = { ( a , b ) , ( b , c ) , ( d , c ) } R=\{(a,b),(b,c),(d,c)\}
  8. a , b , c a,b,c
  9. d d
  10. a a
  11. b b
  12. b b
  13. c c
  14. d d
  15. c c
  16. a A a\in A
  17. E A E\subseteq A
  18. E E
  19. a a
  20. b A \forall b\in A
  21. ( b , a ) R , c E (b,a)\in R,\exists c\in E
  22. ( c , b ) R (c,b)\in R
  23. E E
  24. a , b E , ( a , b ) R \forall a,b\in E,(a,b)\not\in R
  25. E E
  26. E E
  27. S = A , R S=\langle A,R\rangle
  28. E E
  29. S S
  30. E E
  31. E E
  32. E E
  33. S S
  34. S S
  35. E E
  36. S S
  37. E E
  38. a A \ E , b S \forall a\in A\backslash E,\exists b\in S
  39. ( b , a ) R (b,a)\in R
  40. E E
  41. S S
  42. S S
  43. a 0 , a 1 , , a n , a_{0},a_{1},\dots,a_{n},\dots
  44. i , ( a i , a i + 1 ) R \forall i,(a_{i},a_{i+1})\in R
  45. E x t σ ( S ) Ext_{\sigma}(S)
  46. σ \sigma
  47. S S
  48. S S
  49. E x t σ ( S ) = { { a , d } } Ext_{\sigma}(S)=\{\{a,d\}\}
  50. a a
  51. d d
  52. ( 𝑎𝑟𝑔𝑢𝑚𝑒𝑛𝑡 , 𝑙𝑎𝑏𝑒𝑙 ) (\mathit{argument},\mathit{label})
  53. L L
  54. S = A , R S=\langle A,R\rangle
  55. a A , L ( a ) = 𝑖𝑛 \forall a\in A,L(a)=\mathit{in}
  56. b A \forall b\in A
  57. ( b , a ) R , L ( b ) = 𝑜𝑢𝑡 (b,a)\in R,L(b)=\mathit{out}
  58. a A , L ( a ) = 𝑜𝑢𝑡 \forall a\in A,L(a)=\mathit{out}
  59. b A \exists b\in A
  60. ( b , a ) R (b,a)\in R
  61. L ( b ) = 𝑖𝑛 L(b)=\mathit{in}
  62. a A , L ( a ) = 𝑢𝑛𝑑𝑒𝑐 \forall a\in A,L(a)=\mathit{undec}
  63. L ( a ) 𝑖𝑛 L(a)\neq\mathit{in}
  64. L ( a ) 𝑜𝑢𝑡 L(a)\neq\mathit{out}
  65. S S
  66. L = { ( a , 𝑖𝑛 ) , ( b , 𝑜𝑢𝑡 ) , ( c , 𝑜𝑢𝑡 ) , ( d , 𝑖𝑛 ) } L=\{(a,\mathit{in}),(b,\mathit{out}),(c,\mathit{out}),(d,\mathit{in})\}
  67. σ \sigma
  68. σ \sigma
  69. a a
  70. b b
  71. a a
  72. b b
  73. σ \sigma
  74. C r σ ( S ) Cr_{\sigma}(S)
  75. σ \sigma
  76. S c σ ( S ) Sc_{\sigma}(S)
  77. σ \sigma
  78. σ \sigma
  79. σ \sigma
  80. 𝐸𝑄 1 \mathit{EQ_{1}}
  81. σ \sigma
  82. S 1 1 S 2 E x t σ ( S 1 ) = E x t σ ( S 2 ) S_{1}\equiv_{1}S_{2}\Leftrightarrow Ext_{\sigma}(S_{1})=Ext_{\sigma}(S_{2})
  83. 𝐸𝑄 2 \mathit{EQ_{2}}
  84. S 1 2 S 2 S c σ ( S 1 ) = S c σ ( S 2 ) S_{1}\equiv_{2}S_{2}\Leftrightarrow Sc_{\sigma}(S_{1})=Sc_{\sigma}(S_{2})
  85. 𝐸𝑄 2 \mathit{EQ_{2}}
  86. S 1 3 S 2 C r σ ( S 1 ) = C r σ ( S 2 ) S_{1}\equiv_{3}S_{2}\Leftrightarrow Cr_{\sigma}(S_{1})=Cr_{\sigma}(S_{2})
  87. S 1 S_{1}
  88. S 2 S_{2}
  89. S 3 S_{3}
  90. S 1 S_{1}
  91. S 3 S_{3}
  92. S 2 S_{2}
  93. S 3 S_{3}
  94. ( Φ , α ) (\Phi,\alpha)
  95. Φ \Phi\nvdash\bot
  96. Φ α \Phi\vdash\alpha
  97. Φ \Phi
  98. Δ \Delta
  99. α \alpha
  100. Δ \Delta
  101. α \alpha
  102. Φ \Phi
  103. Φ \Phi
  104. α \alpha
  105. A × A A\times A
  106. ( Ψ , β ) (\Psi,\beta)
  107. ( Φ , α ) (\Phi,\alpha)
  108. β ¬ ( ϕ 1 ϕ n ) \beta\vdash\neg(\phi_{1}\wedge\dots\wedge\phi_{n})
  109. { ϕ 1 , , ϕ n } Φ \{\phi_{1},\dots,\phi_{n}\}\subseteq\Phi
  110. ( Ψ , β ) (\Psi,\beta)
  111. ( Φ , α ) (\Phi,\alpha)
  112. β = ¬ ( ϕ 1 ϕ n ) \beta=\neg(\phi_{1}\wedge\dots\wedge\phi_{n})
  113. { ϕ 1 , , ϕ n } Φ \{\phi_{1},\dots,\phi_{n}\}\subseteq\Phi
  114. ( Ψ , β ) (\Psi,\beta)
  115. ( Φ , α ) (\Phi,\alpha)
  116. β ¬ α \beta\Leftrightarrow\neg\alpha
  117. V A F = A , R , V , v a l , v a l p r e f s VAF=\langle A,R,V,val,valprefs\rangle
  118. A A
  119. R R
  120. V V
  121. v a l val
  122. A A
  123. V V
  124. v a l p r e f s valprefs
  125. V × V V\times V
  126. a a
  127. b b
  128. a a
  129. b b
  130. ( a , b ) R (a,b)\in R
  131. ( v a l ( b ) , v a l ( a ) ) v a l p r e f s (val(b),val(a))\not\in valprefs
  132. b b
  133. a a

Arithmetic_number.html

  1. 1 + 2 + 3 + 6 4 = 3 , \frac{1+2+3+6}{4}=3,
  2. exp ( - c log log X ) \exp\left({-c\sqrt{\log\log X}}\right)

Arithmetico-geometric_sequence.html

  1. [ a + ( n - 1 ) d ] r n - 1 [a+(n-1)d]r^{n-1}
  2. k = 1 n [ a + ( k - 1 ) d ] r k - 1 = a + [ a + d ] r + [ a + 2 d ] r 2 + + [ a + ( n - 1 ) d ] r n - 1 \sum_{k=1}^{n}\left[a+(k-1)d\right]r^{k-1}=a+[a+d]r+[a+2d]r^{2}+\cdots+[a+(n-1% )d]r^{n-1}
  3. S n = k = 1 n [ a + ( k - 1 ) d ] r k - 1 = a - [ a + ( n - 1 ) d ] r n 1 - r + d r ( 1 - r n - 1 ) ( 1 - r ) 2 . S_{n}=\sum_{k=1}^{n}\left[a+(k-1)d\right]r^{k-1}=\frac{a-[a+(n-1)d]r^{n}}{1-r}% +\frac{dr(1-r^{n-1})}{(1-r)^{2}}.
  4. S n = a + [ a + d ] r + [ a + 2 d ] r 2 + + [ a + ( n - 1 ) d ] r n - 1 S_{n}=a+[a+d]r+[a+2d]r^{2}+\cdots+[a+(n-1)d]r^{n-1}
  5. r S n = a r + [ a + d ] r 2 + [ a + 2 d ] r 3 + + [ a + ( n - 1 ) d ] r n rS_{n}=ar+[a+d]r^{2}+[a+2d]r^{3}+\cdots+[a+(n-1)d]r^{n}
  6. ( 1 - r ) S n = [ a + ( a + d ) r + ( a + 2 d ) r 2 + + [ a + ( n - 1 ) d ] r n - 1 ] - [ a r + ( a + d ) r 2 + ( a + 2 d ) r 3 + + [ a + ( n - 1 ) d ] r n ] = a + d ( r + r 2 + + r n - 1 ) - [ a + ( n - 1 ) d ] r n = a + d r ( 1 - r n - 1 ) 1 - r - [ a + ( n - 1 ) d ] r n \begin{aligned}\displaystyle(1-r)S_{n}&\displaystyle=&\displaystyle\left[a+(a+% d)r+(a+2d)r^{2}+\cdots+[a+(n-1)d]r^{n-1}\right]\\ &&\displaystyle-\left[ar+(a+d)r^{2}+(a+2d)r^{3}+\cdots+[a+(n-1)d]r^{n}\right]% \\ &\displaystyle=&\displaystyle a+d\left(r+r^{2}+\cdots+r^{n-1}\right)-\left[a+(% n-1)d\right]r^{n}\\ &\displaystyle=&\displaystyle a+\frac{dr(1-r^{n-1})}{1-r}-[a+(n-1)d]r^{n}\end{aligned}
  7. lim n S n = a 1 - r + r d ( 1 - r ) 2 \lim_{n\to\infty}S_{n}=\frac{a}{1-r}+\frac{rd}{(1-r)^{2}}

Arnold–Givental_conjecture.html

  1. L L
  2. L L
  3. L L
  4. M M
  5. L L
  6. L L
  7. | L φ H ( L ) | k = 0 n b k ( L ; 𝐙 2 ) \left|L\cap\varphi_{H}(L)\right|\geq\sum_{k=0}^{n}b_{k}\left(L;\mathbf{Z}_{2}\right)

Arsenate-mycothiol_transferase.html

  1. \rightleftharpoons

Arsenate_reductase_(cytochrome_c).html

  1. \rightleftharpoons

Arsenite_methyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Artemisinic_aldehyde_Delta11(13)-reductase.html

  1. \rightleftharpoons

Ascorbate_ferrireductase_(transmembrane).html

  1. \rightleftharpoons

Associate_family.html

  1. x k ( ζ ) = { 0 ζ φ k ( z ) d z } + c k , k = 1 , 2 , 3 x_{k}(\zeta)=\Re\left\{\int_{0}^{\zeta}\varphi_{k}(z)\,dz\right\}+c_{k},\qquad k% =1,2,3
  2. x k ( ζ , θ ) = { e i θ 0 ζ φ k ( z ) d z } + c k , θ [ 0 , 2 π ] x_{k}(\zeta,\theta)=\Re\left\{e^{i\theta}\int_{0}^{\zeta}\varphi_{k}(z)\,dz% \right\}+c_{k},\qquad\theta\in[0,2\pi]

Associated_graded_ring.html

  1. gr I R = n = 0 I n / I n + 1 \operatorname{gr}_{I}R=\oplus_{n=0}^{\infty}I^{n}/I^{n+1}
  2. gr I R \operatorname{gr}_{I}R
  3. gr I M = 0 I n M / I n + 1 M \operatorname{gr}_{I}M=\oplus_{0}^{\infty}I^{n}M/I^{n+1}M
  4. gr I R \operatorname{gr}_{I}R
  5. a I i / I i + 1 a\in I^{i}/I^{i+1}
  6. b I j / I j + 1 b\in I^{j}/I^{j+1}
  7. a I i a^{\prime}\in I^{i}
  8. b I j b^{\prime}\in I^{j}
  9. a b ab
  10. a b a^{\prime}b^{\prime}
  11. I i + j / I i + j + 1 I^{i+j}/I^{i+j+1}
  12. I i + j + 1 I^{i+j+1}
  13. f M f\in M
  14. gr I M \operatorname{gr}_{I}M
  15. in ( f ) \mathrm{in}(f)
  16. I m M / I m + 1 M I^{m}M/I^{m+1}M
  17. f I m M f\in I^{m}M
  18. f I m M f\in I^{m}M
  19. in ( f ) = 0 \mathrm{in}(f)=0
  20. N M N\subset M
  21. in ( N ) \mathrm{in}(N)
  22. gr I M \operatorname{gr}_{I}M
  23. { in ( f ) | f N } \{\mathrm{in}(f)|f\in N\}
  24. gr I M \operatorname{gr}_{I}M
  25. gr I R \operatorname{gr}_{I}R
  26. 𝔤 \mathfrak{g}
  27. gr U \operatorname{gr}U
  28. k [ 𝔤 * ] k[\mathfrak{g}^{*}]
  29. R = I 0 I 1 I 2 R=I_{0}\supset I_{1}\supset I_{2}\supset\cdots
  30. I j I k I j + k I_{j}I_{k}\subset I_{j+k}
  31. gr F R = n = 0 I n / I n + 1 \operatorname{gr}_{F}R=\oplus_{n=0}^{\infty}I_{n}/I_{n+1}

Asymmetric_simple_exclusion_process.html

  1. p , q 0 , p + q = 1 p,q\geqslant 0,\,p+q=1
  2. S = { 0 , 1 } S=\{0,1\}^{\mathbb{Z}}
  3. p p
  4. q q
  5. p - q p-q

Asymptotic_safety_in_quantum_gravity.html

  1. R μ ν - 1 2 g μ ν R + g μ ν Λ = 8 π G T μ ν \textstyle R_{\mu\nu}-{1\over 2}g_{\mu\nu}\,R+g_{\mu\nu}\Lambda=8\pi G\,T_{\mu\nu}
  2. g μ ν g_{\mu\nu}
  3. T μ ν T_{\mu\nu}
  4. - 2 -2
  5. d = 2 + ϵ d=2+\epsilon
  6. 1 16 π G d 2 x g R \textstyle{1\over 16\pi G}\int\mathrm{d}^{2}x\sqrt{g}\,R
  7. G G
  8. ϵ \epsilon
  9. β \beta
  10. ϵ \epsilon
  11. d = 2 + ϵ d=2+\epsilon
  12. d = 4 d=4
  13. ϵ \epsilon
  14. 2 + ϵ 2+\epsilon
  15. f ( R ) f(R)
  16. k k\rightarrow\infty
  17. { g α } \{g_{\alpha}\}
  18. { g α } { g α ( k ) } \{g_{\alpha}\}\equiv\{g_{\alpha}(k)\}
  19. k k
  20. { g α ( k ) } \{g_{\alpha}(k)\}
  21. k k
  22. k 0 k\rightarrow 0
  23. k k\rightarrow\infty
  24. { g α * } \{g_{\alpha}^{*}\}
  25. β γ ( { g α * } ) = 0 \beta_{\gamma}(\{g_{\alpha}^{*}\})=0
  26. γ \gamma
  27. k k\rightarrow\infty
  28. n n
  29. n n
  30. n n
  31. g α * = 0 g_{\alpha}^{*}=0
  32. g α g_{\alpha}
  33. g α * 0 g_{\alpha}^{*}\neq 0
  34. g α g_{\alpha}
  35. k k
  36. Γ k \Gamma_{k}
  37. k k
  38. Φ \Phi
  39. Φ ¯ \bar{\Phi}
  40. Γ k \Gamma_{k}
  41. k k Γ k [ Φ , Φ ¯ ] = 1 2 STr [ ( Γ k ( 2 ) [ Φ , Φ ¯ ] + k [ Φ ¯ ] ) - 1 k k k [ Φ ¯ ] ] . k\partial_{k}\Gamma_{k}\big[\Phi,\bar{\Phi}\big]=\frac{1}{2}\,\mbox{STr}~{}% \Big[\big(\Gamma_{k}^{(2)}\big[\Phi,\bar{\Phi}\big]+\mathcal{R}_{k}[\bar{\Phi}% ]\big)^{-1}k\partial_{k}\mathcal{R}_{k}[\bar{\Phi}]\Big].
  42. Γ k ( 2 ) \Gamma_{k}^{(2)}
  43. Γ k \Gamma_{k}
  44. Φ \Phi
  45. Φ ¯ \bar{\Phi}
  46. k [ Φ ¯ ] \mathcal{R}_{k}[\bar{\Phi}]
  47. k k
  48. p 2 k 2 p^{2}\ll k^{2}
  49. p 2 k 2 p^{2}\gg k^{2}
  50. ( STr ) (\mbox{STr}~{})
  51. p 2 k 2 p^{2}\approx k^{2}
  52. Γ k \Gamma_{k}
  53. Γ k \Gamma_{k}
  54. k k\rightarrow\infty
  55. Γ [ Φ ] = Γ k = 0 [ Φ , Φ ¯ = Φ ] \Gamma[\Phi]=\Gamma_{k=0}\big[\Phi,\bar{\Phi}=\Phi\big]
  56. k 0 k\rightarrow 0
  57. Γ * = Γ k \Gamma_{*}=\Gamma_{k\rightarrow\infty}
  58. { P α [ ] } \{P_{\alpha}[\,\cdot\,]\}
  59. P α P_{\alpha}
  60. Γ k \Gamma_{k}
  61. Γ k [ Φ , Φ ¯ ] = α = 1 g α ( k ) P α [ Φ , Φ ¯ ] . \Gamma_{k}[\Phi,\bar{\Phi}]=\sum\limits_{\alpha=1}^{\infty}g_{\alpha}(k)P_{% \alpha}[\Phi,\bar{\Phi}].
  62. k k g α ( k ) = β α ( g 1 , g 2 , ) k\partial_{k}g_{\alpha}(k)=\beta_{\alpha}(g_{1},g_{2},\cdots)
  63. g α ( k ) g_{\alpha}(k)
  64. Γ k \Gamma_{k}
  65. β \beta
  66. { P α [ ] } \{P_{\alpha}[\,\cdot\,]\}
  67. α = 1 , , N \alpha=1,\cdots,N
  68. Γ k [ Φ , Φ ¯ ] = α = 1 N g α ( k ) P α [ Φ , Φ ¯ ] , \Gamma_{k}[\Phi,\bar{\Phi}]=\sum\limits_{\alpha=1}^{N}g_{\alpha}(k)P_{\alpha}[% \Phi,\bar{\Phi}],
  69. k k g α ( k ) = β α ( g 1 , , g N ) k\partial_{k}g_{\alpha}(k)=\beta_{\alpha}(g_{1},\cdots,g_{N})
  70. β \beta
  71. Γ k \Gamma_{k}
  72. G k G_{k}
  73. Λ k \Lambda_{k}
  74. k k
  75. g μ ν g_{\mu\nu}
  76. g ¯ μ ν \bar{g}_{\mu\nu}
  77. Γ k \Gamma_{k}
  78. d d
  79. Γ k [ g , g ¯ , ξ , ξ ¯ ] = 1 16 π G k d d x g ( - R ( g ) + 2 Λ k ) + Γ k gf [ g , g ¯ ] + Γ k gh [ g , g ¯ , ξ , ξ ¯ ] . \Gamma_{k}[g,\bar{g},\xi,\bar{\xi}]=\frac{1}{16\pi G_{k}}\int\,\text{d}^{d}x\,% \sqrt{g}\,\big(-R(g)+2\Lambda_{k}\big)+\Gamma_{k}\text{gf}[g,\bar{g}]+\Gamma_{% k}\text{gh}[g,\bar{g},\xi,\bar{\xi}].
  80. R ( g ) R(g)
  81. g μ ν g_{\mu\nu}
  82. Γ k gf \Gamma_{k}\text{gf}
  83. Γ k gh \Gamma_{k}\text{gh}
  84. ξ \xi
  85. ξ ¯ \bar{\xi}
  86. β \beta
  87. g k = k d - 2 G k g_{k}=k^{d-2}G_{k}
  88. λ k = k - 2 Λ k \lambda_{k}=k^{-2}\Lambda_{k}
  89. d d
  90. 4 4
  91. d = 4 d=4
  92. g g
  93. λ \lambda
  94. d = 2 + ϵ d=2+\epsilon
  95. d = 2 + ϵ d=2+\epsilon
  96. β \beta
  97. ϵ \epsilon
  98. β \beta
  99. d d
  100. d = 4 d=4
  101. ϵ \epsilon
  102. R 2 R^{2}
  103. R R
  104. f ( R ) f(R)

Augmented-fourths_tuning.html

  1. n n
  2. 12 - n 12-n

Aureusidin_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Autowave.html

  1. f ( u ) \vec{f}(\vec{u})
  2. u ˙ = f ( u ) \dot{\vec{u}}=\vec{f}(\vec{u})
  3. f f
  4. N N
  5. u u
  6. f ( u ) f(u)

AW*-algebra.html

  1. A A
  2. p A p\in A
  3. p * = p = p 2 p^{*}=p=p^{2}
  4. A A
  5. S A S\subseteq A
  6. Ann R ( S ) = { a A s S , a s = 0 } \mathrm{Ann}_{R}(S)=\{a\in A\mid\forall s\in S,as=0\}\,
  7. p p
  8. A A
  9. q q
  10. S A p , q Proj ( A ) : Ann R ( S ) = A p , Ann L ( S ) = q A \forall S\subseteq A\,\exists p,q\in\mathrm{Proj}(A)\colon\mathrm{Ann}_{R}(S)=% Ap,\quad\mathrm{Ann}_{L}(S)=qA
  11. A A
  12. X X\to\mathbb{C}
  13. X X
  14. A A
  15. X X

Axial_fan_design.html

  1. C 1 = C 2 = C C_{\rm 1}=C_{\rm 2}=C
  2. P 1 P 2 P_{\rm 1}\neq P_{\rm 2}
  3. A = π D 2 4 A=\frac{\pi D^{2}}{4}
  4. m ˙ = ρ A C \dot{m}={\rho AC}
  5. F x = m ˙ ( C s - C u ) = ρ A C ( C s - C u ) F_{\rm x}={\dot{m}}{(C_{\rm s}-C_{\rm u})}={\rho AC}{(C_{\rm s}-C_{\rm u})}
  6. P a + 1 2 ρ C u 2 \displaystyle P_{a}+\frac{1}{2}{\rho C_{u}^{2}}
  7. P 2 - P 1 = 1 2 ρ ( C s 2 - C u 2 ) P_{2}-P_{1}=\frac{1}{2}\rho(C_{s}^{2}-C_{u}^{2})
  8. F x = A ( P 2 - P 1 ) = 1 2 ρ A ( C s 2 - C u 2 ) F_{x}=A(P_{2}-P_{1})=\frac{1}{2}\rho A\left(C_{s}^{2}-C_{u}^{2}\right)
  9. C = C s + C u 2 C=\frac{C_{s}+C_{u}}{2}
  10. C = ( 1 + a ) C u C=(1+a)C_{u}
  11. a = C C u - 1 a=\frac{C}{C_{u}}-1
  12. C s = ( 1 + 2 a ) C u C_{s}=(1+2a)C_{u}
  13. Δ h o = Δ h o d - Δ h o u = ( h d + 1 2 C s 2 ) - ( h u + 1 2 C u 2 ) = 1 2 ( C s 2 - C u 2 ) \Delta h_{o}=\Delta h_{o}d-\Delta h_{o}u=\left(h_{d}+\frac{1}{2}C_{s}^{2}% \right)-\left(h_{u}+\frac{1}{2}C_{u}^{2}\right)=\frac{1}{2}\left(C_{s}^{2}-C_{% u}^{2}\right)
  14. P i = m ˙ Δ h o P_{i}=\dot{m}{\Delta h_{o}}
  15. m ˙ = ρ A C \dot{m}=\rho AC
  16. P = F x C u P=F_{x}C_{u}
  17. η p = Actual Power ( P ) Ideal Power ( P i ) = F x C u 1 2 ρ A C ( C s 2 - C u 2 ) = C u C = 1 1 + a \eta_{p}=\frac{\,\text{Actual Power}(P)}{\,\text{Ideal Power}(P_{i})}=\frac{F_% {x}C_{u}}{\frac{1}{2}\rho AC\left(C_{s}^{2}-C_{u}^{2}\right)}=\frac{C_{u}}{C}=% \frac{1}{1+a}
  18. C π D 2 4 = C s π D s 2 4 D s 2 = C C s D 2 \begin{aligned}\displaystyle C\frac{\pi D^{2}}{4}&\displaystyle=C_{s}\frac{\pi D% _{s}^{2}}{4}\\ \displaystyle\Rightarrow D_{s}^{2}&\displaystyle=\frac{C}{C_{s}}D^{2}\end{aligned}
  19. C s = 1 + 2 a 1 + a C C_{s}=\frac{1+2a}{1+a}C
  20. D s 2 = ( 1 + a 1 + 2 a ) D 2 D_{s}^{2}=\left(\frac{1+a}{1+2a}\right)D^{2}
  21. Δ F x = Δ L sin ( β ) - Δ D cos ( β ) \Delta F_{x}=\Delta L\sin(\beta)-\Delta D\cos(\beta)
  22. Δ F y = Δ L cos ( β ) - Δ D sin ( β ) \Delta F_{y}=\Delta L\cos(\beta)-\Delta D\sin(\beta)
  23. Lift ( Δ L ) = 1 2 C L ρ w 2 ( l d r ) \mathrm{Lift}(\Delta L)=\frac{1}{2}C_{L}\rho w^{2}(ldr)
  24. Drag ( Δ L ) = 1 2 C D ρ w 2 ( l d r ) \mathrm{Drag}(\Delta L)=\frac{1}{2}C_{D}\rho w^{2}(ldr)
  25. tan ( ϕ ) = Δ D Δ L = C D C L \tan(\phi)=\frac{\Delta D}{\Delta L}=\frac{C_{D}}{C_{L}}
  26. Δ F x = Δ L ( cos ϕ - Δ D Δ L sin ϕ ) = Δ L ( cos ϕ - tan ϕ sin ϕ ) = 1 2 C L ρ w 2 l d r sin ( β - ϕ ) cos ϕ \Delta F_{x}=\Delta L(\cos\phi-\frac{\Delta D}{\Delta L}\sin\phi)=\Delta L(% \cos\phi-\tan\phi\sin\phi)=\frac{1}{2}C_{L}\rho w^{2}ldr\frac{\sin(\beta-\phi)% }{\cos\phi}
  27. s = 2 π r z s=\frac{2\pi r}{z}
  28. Δ p ( 2 π r d r ) = z Δ F x \Delta p(2\pi rdr)=z\Delta F_{x}
  29. Δ p = 1 2 C L ρ w 2 ( l s ) sin ( β - ϕ ) cos ϕ = 1 2 C D ρ w 2 ( l s ) sin ( β - ϕ ) sin ϕ \Rightarrow\Delta p=\frac{1}{2}C_{L}\rho w^{2}(\frac{l}{s})\frac{\sin(\beta-% \phi)}{\cos\phi}=\frac{1}{2}C_{D}\rho w^{2}(\frac{l}{s})\frac{\sin(\beta-\phi)% }{\sin\phi}
  30. Δ F y = 1 2 C L ρ w 2 l d r cos ( β - ϕ ) cos ϕ \Delta F_{y}=\frac{1}{2}C_{L}\rho w^{2}ldr\frac{\cos(\beta-\phi)}{\cos\phi}
  31. ( Torque ) Δ Q = r Δ F y (\mathrm{Torque})\Delta Q=r\Delta F_{y}

Azimi_Q_models.html

  1. α ( w ) = a 1 | w | 1 - γ ( 1.1 ) \alpha(w)=a_{1}|w|^{1-\gamma}\quad(1.1)
  2. 1 c ( w ) = 1 c + a 1 | w | - γ + c o t ( π γ 2 ) ( 1.2 ) \frac{1}{c(w)}=\frac{1}{c_{\infty}}+a_{1}|w|^{-\gamma}+cot(\frac{\pi\gamma}{2}% )\quad(1.2)
  3. α ( w ) = a 2 | w | 1 + a 3 | w | ( 2.1 ) \alpha(w)=\frac{a_{2}|w|}{1+a_{3}|w|}\quad(2.1)
  4. 1 c ( w ) = 1 c - 2 a 2 l n ( a 3 w ) π ( 1 - a 3 2 w 2 ) ( 1.2 ) \frac{1}{c(w)}=\frac{1}{c_{\infty}}-\frac{2a_{2}ln(a_{3}w)}{\pi(1-a_{3}^{2}w^{% 2})}\quad(1.2)

B-1,3-galactosyl-O-glycosyl-glycoprotein_b-1,3-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

B-1,3-galactosyl-O-glycosyl-glycoprotein_b-1,6-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

B-1,4-mannosyl-glycoprotein_4-b-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Background_subtraction.html

  1. P [ F ( t ) ] = P [ I ( t ) ] - P [ B ] P[F(t)]=P[I(t)]-P[B]\,
  2. | P [ F ( t ) ] - P [ F ( t + 1 ) ] | > Threshold |P[F(t)]-P[F(t+1)]|>\mathrm{Threshold}\,
  3. B ( x , y ) = 1 N i = 1 N V ( x , y , t - i ) B(x,y)={1\over N}\sum_{i=1}^{N}V(x,y,t-i)
  4. | V ( x , y , t ) - B ( x , y ) | > Th |V(x,y,t)-B(x,y)|>\mathrm{Th}\,
  5. n n
  6. t t
  7. μ t \mu_{t}
  8. σ t 2 \sigma^{2}_{t}
  9. μ 0 = I 0 \mu_{0}=I_{0}
  10. σ 0 2 = < \sigma^{2}_{0}=<
  11. Align g t ; &gt;
  12. I t I_{t}
  13. t t
  14. t t
  15. μ t = ρ I t + ( 1 - ρ ) μ t - 1 \mu_{t}=\rho I_{t}+(1-\rho)\mu_{t-1}
  16. σ t 2 = d 2 ρ + ( 1 - ρ ) σ t - 1 2 \sigma^{2}_{t}=d^{2}\rho+(1-\rho)\sigma^{2}_{t-1}
  17. d = | ( I t - μ t ) | d=|(I_{t}-\mu_{t})|
  18. ρ \rho
  19. ρ = 0.01 \rho=0.01
  20. d d
  21. | ( I t - μ t ) | σ t > k 𝐹𝑜𝑟𝑒𝑔𝑟𝑜𝑢𝑛𝑑 \frac{|(I_{t}-\mu_{t})|}{\sigma_{t}}>k\longrightarrow\mathit{Foreground}
  22. | ( I t - μ t ) | σ t k 𝐵𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑 \frac{|(I_{t}-\mu_{t})|}{\sigma_{t}}\leq k\longrightarrow\mathit{Background}
  23. k k
  24. k = 2.5 k=2.5
  25. k k
  26. k k
  27. μ t = M μ t - 1 + ( 1 - M ) ( I t ρ + ( 1 - ρ ) μ t - 1 ) \mu_{t}=M\mu_{t-1}+(1-M)(I_{t}\rho+(1-\rho)\mu_{t-1})
  28. M = 1 M=1
  29. I t I_{t}
  30. M = 0 M=0
  31. M = 1 M=1
  32. x 0 , y 0 x_{0},y_{0}
  33. X 1 , , X t = { V ( x 0 , y 0 , i ) : 1 i t } X_{1},\ldots,X_{t}=\{V(x_{0},y_{0},i):1\leqslant i\leqslant t\}\,
  34. P ( X t ) = i = 1 K ω i , t N ( X t μ i , t , Σ i , t ) P(X_{t})=\sum_{i=1}^{K}\omega_{i,t}N\left(X_{t}\mid\mu_{i,t},\Sigma_{i,t}\right)
  35. N ( X t μ i t , Σ i , t ) = 1 ( 2 π ) D / 2 1 | Σ i , t | 1 / 2 exp ( - 1 2 ( X t - μ i , t ) T Σ i , t - 1 ( X t - μ i , t ) ) N\left(X_{t}\mid\mu_{it},\Sigma_{i,t}\right)=\dfrac{1}{(2\pi)^{D/2}}{1\over|% \Sigma_{i,t}|^{1/2}}\exp\left(-{1\over 2}(X_{t}-\mu_{i,t})^{T}\Sigma_{i,t}^{-1% }\left(X_{t}-\mu_{i,t}\right)\right)
  36. P ( X t ) = i = 1 K ω i , t η ( X t μ i , t , Σ i , t ) P(X_{t})=\sum_{i=1}^{K}\omega_{i,t}\eta\left(X_{t}\,\mu_{i,t},\Sigma_{i,t}\right)
  37. η ( X t μ i , t , Σ i , t ) = 1 ( 2 / p i ) n / 2 Σ i , t 0.5 exp ( - 1 2 ( X t - μ i , t ) Σ i , t ( X t - μ i , t ) ) \eta\left(X_{t}\,\mu_{i,t},\Sigma_{i,t}\right)=\dfrac{1}{(2/pi)^{n/2}\Sigma_{i% ,t}^{0.5}}\exp\left(-{1\over 2}(X_{t}-\mu_{i,t})\Sigma_{i,t}\left(X_{t}-\mu_{i% ,t}\right)\right)
  38. B = a r g m i n ( Σ i - 1 B ω i , t > T ) B=argmin\left(\Sigma_{i-1}^{B}\omega_{i,t}>T\right)
  39. t + 1 t+1
  40. ( ( X t + 1 - μ t + 1 ) T Σ i - 1 b ( X t + 1 - μ t + 1 ) ) 0.5 < k * σ i , t \left(\left(X_{t+1}-\mu_{t+1}\right)^{T}\Sigma_{i-1}^{b}\left(X_{t+1}-\mu_{t+1% }\right)\right)^{0.5}<k*\sigma_{i,t}
  41. 2.5 2.5
  42. σ i , t + 1 = ( 1 - ρ ) σ i , t 2 + ρ ( X x + 1 - μ x + 1 ) ( X x + 1 - μ x + 1 ) T \sigma_{i,t+1}=\left(1-\rho\right)\sigma_{i,t}^{2}+\rho\left(X_{x+1}-\mu_{x+1}% \right)\left(X_{x+1}-\mu_{x+1}\right)^{T}
  43. σ i , t + 1 = ( 1 - α ) ω i , t + α P ( k X t , ϕ ) \sigma_{i,t+1}=\left(1-\alpha\right)\omega_{i,t}+\alpha P\left(k\mid\ X_{t},% \phi\right)
  44. P ( k X t , ϕ ) P\left(k\mid\ X_{t},\phi\right)
  45. M ( k , t ) M_{(}k,t)
  46. M ( k , t ) = 1 ( m a t c h ) , M ( k , t ) = 0 ( o t h e r w i s e ) M_{(}k,t)=1\left(match\right),M_{(}k,t)=0\left(otherwise\right)
  47. K K
  48. K K
  49. k i . t = l o w P r i o r W e i g h t k_{i.t}=lowPriorWeight
  50. μ i , t + 1 = X t + 1 \mu_{i,t+1}=X_{t+1}
  51. k i . t + 1 = L a r g e I n i t i a l W e i g h t k_{i.t+1}=LargeInitialWeight

Bahcall–Wolf_cusp.html

  1. ρ ( r ) r - 7 / 4 . \rho(r)\propto r^{-7/4}.
  2. E = s y m b o l v 2 / 2 - G M / r E=symbol{v}^{2}/2-GM/r
  3. N ( E ) d E = N 0 | E | - 9 / 4 d E , N(E)\,dE=N_{0}|E|^{-9/4}dE,
  4. \propto
  5. \propto
  6. \lesssim

Baicalein_7-O-glucuronosyltransferase.html

  1. \rightleftharpoons

Bandwidth-sharing_game.html

  1. n n
  2. i i
  3. U i ( x ) U_{i}(x)
  4. x x
  5. i i
  6. w i w_{i}
  7. x x
  8. U i ( x ) - w i U_{i}(x)-w_{i}
  9. B B
  10. U i ( x ) U_{i}(x)
  11. U i ( x ) 0 U_{i}(x)\geq 0
  12. U i ( x ) U_{i}(x)
  13. U ( x ) U(x)
  14. p p
  15. a r g m a x x U i ( x ) - p x argmax_{x}U_{i}(x)-px
  16. U i ( x ) = p U_{i}^{^{\prime}}(x)=p
  17. i i
  18. w i w_{i}
  19. x i x_{i}
  20. x i = ( w i j w j ) * ( B ) x_{i}=(\frac{w_{i}}{\sum_{j}w_{j}})*(B)
  21. p = j w j B p=\frac{\sum_{j}w_{j}}{B}
  22. x 1 , , x n x_{1},...,x_{n}
  23. a r g m a x x i U i ( x i ) - w i argmax_{x_{i}}U_{i}(x_{i})-w_{i}
  24. a r g m a x w i U i ( w i j w j * B ) - w i \implies argmax_{w_{i}}U_{i}(\frac{w_{i}}{\sum_{j}w_{j}}*B)-w_{i}
  25. U i ( w i j w j * B ) ( 1 j w j * B - w i ( j w j ) 2 * B ) - 1 = 0 \implies U^{^{\prime}}_{i}(\frac{w_{i}}{\sum_{j}w_{j}}*B)(\frac{1}{\sum_{j}w_{% j}}*B-\frac{w_{i}}{(\sum_{j}w_{j})^{2}}*B)-1=0
  26. U i ( x i ) ( 1 p - 1 p * x i B ) - 1 = 0 \implies U^{^{\prime}}_{i}(x_{i})(\frac{1}{p}-\frac{1}{p}*\frac{x_{i}}{B})-1=0
  27. U i ( x i ) ( 1 - x i B ) = p \implies U^{^{\prime}}_{i}(x_{i})(1-\frac{x_{i}}{B})=p
  28. x i B \frac{x_{i}}{B}

Bargmann–Wigner_equations.html

  1. j j
  2. j = 1 , 2 , 3... j=1,2,3...
  3. j = 1 / 2 , 3 / 2 , 5 / 2... j={1}/{2},{3}/{2},{5}/{2}...
  4. s s
  5. j j
  6. ( - γ μ P ^ μ + m c ) Ψ = 0 , (-\gamma^{\mu}\hat{P}_{\mu}+mc)\Psi=0\,,
  7. Ψ = Ψ ( 𝐫 , t ) Ψ=Ψ(\mathbf{r},t)
  8. 𝐫 \mathbf{r}
  9. t t
  10. α α
  11. P ^ μ = i μ \hat{P}_{\mu}=i\hbar\partial_{\mu}
  12. m c mc
  13. - γ μ P ^ μ + m c = - γ 0 E ^ c - s y m b o l γ ( - 𝐩 ^ ) + m c = - ( I 2 0 0 - I 2 ) E ^ c + ( 0 s y m b o l σ 𝐩 ^ - s y m b o l σ 𝐩 ^ 0 ) + ( I 2 0 0 I 2 ) m c = ( - E ^ / c + m c 0 p ^ z p ^ x - i p ^ y 0 - E ^ / c + m c p ^ x + p ^ y - p ^ z - p ^ z - ( p ^ x - i p ^ y ) E ^ / c + m c 0 - ( p ^ x + i p ^ y ) p ^ z 0 E ^ / c + m c ) \begin{aligned}\displaystyle-\gamma^{\mu}\hat{P}_{\mu}+mc&\displaystyle=-% \gamma^{0}\frac{\hat{E}}{c}-symbol{\gamma}\cdot(-\hat{\mathbf{p}})+mc\\ &\displaystyle=-\begin{pmatrix}I_{2}&0\\ 0&-I_{2}\\ \end{pmatrix}\frac{\hat{E}}{c}+\begin{pmatrix}0&symbol{\sigma}\cdot\hat{% \mathbf{p}}\\ -symbol{\sigma}\cdot\hat{\mathbf{p}}&0\\ \end{pmatrix}+\begin{pmatrix}I_{2}&0\\ 0&I_{2}\\ \end{pmatrix}mc\\ &\displaystyle=\begin{pmatrix}-\hat{E}/c+mc&0&\hat{p}_{z}&\hat{p}_{x}-i\hat{p}% _{y}\\ 0&-\hat{E}/c+mc&\hat{p}_{x}+\hat{p}_{y}&-\hat{p}_{z}\\ -\hat{p}_{z}&-(\hat{p}_{x}-i\hat{p}_{y})&\hat{E}/c+mc&0\\ -(\hat{p}_{x}+i\hat{p}_{y})&\hat{p}_{z}&0&\hat{E}/c+mc\\ \end{pmatrix}\end{aligned}
  14. ( - E ^ + m c ) ψ 1 , 2 = ( - s y m b o l σ 𝐩 ^ ) ψ 3 , 4 (-\hat{E}+mc)\psi_{1,2}=(-symbol{\sigma}\cdot\hat{\mathbf{p}})\psi_{3,4}
  15. ( E ^ + m c ) ψ 3 , 4 = ( s y m b o l σ 𝐩 ^ ) ψ 1 , 2 (\hat{E}+mc)\psi_{3,4}=(symbol{\sigma}\cdot\hat{\mathbf{p}})\psi_{1,2}
  16. Ψ = ( ψ 1 , 2 ψ 3 , 4 ) ψ 1 , 2 = ( ψ 1 ψ 2 ) ψ 3 , 4 = ( ψ 3 ψ 4 ) . \Psi=\begin{pmatrix}\psi_{1,2}\\ \psi_{3,4}\\ \end{pmatrix}\,\quad\psi_{1,2}=\begin{pmatrix}\psi_{1}\\ \psi_{2}\\ \end{pmatrix}\,\quad\psi_{3,4}=\begin{pmatrix}\psi_{3}\\ \psi_{4}\\ \end{pmatrix}\,.
  17. e e
  18. j j
  19. 2 j 2j
  20. ( - γ μ P ^ μ + m c ) α 1 α 1 ψ α 1 α 2 α 3 α 2 j = 0 ( - γ μ P ^ μ + m c ) α 2 α 2 ψ α 1 α 2 α 3 α 2 j = 0 ( - γ μ P ^ μ + m c ) α 2 j α 2 j ψ α 1 α 2 α 3 α 2 j = 0 \begin{aligned}&\displaystyle(-\gamma^{\mu}\hat{P}_{\mu}+mc)_{\alpha_{1}\alpha% _{1}^{\prime}}\psi_{\alpha^{\prime}_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{2j}}=% 0\\ &\displaystyle(-\gamma^{\mu}\hat{P}_{\mu}+mc)_{\alpha_{2}\alpha_{2}^{\prime}}% \psi_{\alpha_{1}\alpha^{\prime}_{2}\alpha_{3}\cdots\alpha_{2j}}=0\\ &\displaystyle\qquad\vdots\\ &\displaystyle(-\gamma^{\mu}\hat{P}_{\mu}+mc)_{\alpha_{2j}\alpha^{\prime}_{2j}% }\psi_{\alpha_{1}\alpha_{2}\alpha_{3}\cdots\alpha^{\prime}_{2j}}=0\\ \end{aligned}
  21. r = 1 , 2 , 2 j r=1,2,...2j
  22. Ψ = Ψ ( 𝐫 , t ) Ψ=Ψ(\mathbf{r},t)
  23. ψ α 1 α 2 α 3 α 2 j ( 𝐫 , t ) \psi_{\alpha_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{2j}}(\mathbf{r},t)
  24. Ψ Ψ
  25. 2 ( 2 j + 1 ) 2(2j+1)
  26. n = 2 j n=2j
  27. n n
  28. j j
  29. Ψ Ψ
  30. Ψ Ψ
  31. E 2 = ( p c ) 2 + ( m c 2 ) 2 E^{2}=(pc)^{2}+(mc^{2})^{2}
  32. Ψ = ( ψ 1 α 2 α 3 α 2 j ψ 2 α 2 α 3 α 2 j ψ 3 α 2 α 3 α 2 j ψ 4 α 2 α 3 α 2 j ) \Psi=\begin{pmatrix}\psi_{1\alpha_{2}\alpha_{3}\cdots\alpha_{2j}}\\ \psi_{2\alpha_{2}\alpha_{3}\cdots\alpha_{2j}}\\ \psi_{3\alpha_{2}\alpha_{3}\cdots\alpha_{2j}}\\ \psi_{4\alpha_{2}\alpha_{3}\cdots\alpha_{2j}}\\ \end{pmatrix}
  33. 22 j 22j
  34. 2 j 2j
  35. 2 j + 1 2j+1
  36. r r
  37. γ r μ = I 4 I 4 r - 1 matrices γ μ I 4 \gamma_{r}^{\mu}=\underbrace{I_{4}\otimes I_{4}\otimes\cdots}_{r-1\,\,\text{% matrices}}\gamma^{\mu}\cdots\otimes I_{4}
  38. r = 1 , 2...2 j r=1,2...2j
  39. ( γ r μ P ^ μ - m c ) Ψ = 0 (\gamma_{r}^{\mu}\hat{P}_{\mu}-mc)\Psi=0
  40. r r
  41. ( 1 2 j r = 1 2 j γ r μ P ^ μ - m c ) Ψ = 0 \left(\frac{1}{2j}\sum_{r=1}^{2j}\gamma_{r}^{\mu}\hat{P}_{\mu}-mc\right)\Psi=0
  42. 1 / 2 j 1/2j
  43. ± 1 , ± i ±1,±i
  44. 2 j 2j
  45. r r
  46. r + 1 r+1
  47. ( γ r μ - γ r + 1 μ ) P ^ μ ψ = 0 (\gamma_{r}^{\mu}-\gamma_{r+1}^{\mu})\hat{P}_{\mu}\psi=0
  48. r = 1 , 2...2 j 1 r=1,2...2j−1
  49. 2 ( 2 j + 1 ) × 2 ( 2 j + 1 ) 2(2j+1)×2(2j+1)
  50. γ μ 1 μ 2 μ 2 j \gamma^{\mu_{1}\mu_{2}\cdots\mu_{2j}}
  51. [ ( i ) 2 j γ μ 1 μ 2 μ 2 j μ 1 μ 2 μ 2 j + ( m c ) 2 j ] Ψ = 0 [(i\hbar)^{2j}\gamma^{\mu_{1}\mu_{2}\cdots\mu_{2j}}\partial_{\mu_{1}}\partial_% {\mu_{2}}\cdots\partial_{\mu_{2j}}+(mc)^{2j}]\Psi=0
  52. ( a c b d ) ( ψ 1 ψ 2 ) = ( χ 1 χ 2 ) \begin{pmatrix}a&c\\ b&d\\ \end{pmatrix}\begin{pmatrix}\psi_{1}\\ \psi_{2}\\ \end{pmatrix}=\begin{pmatrix}\chi_{1}\\ \chi_{2}\\ \end{pmatrix}
  53. a ψ 1 + c ψ 2 \displaystyle a\psi_{1}+c\psi_{2}
  54. 2 j 2j
  55. ( a ψ 1 + c ψ 2 ) j + m ( b ψ 1 + d ψ 2 ) j - m = χ 1 j + m χ 2 j - m , (a\psi_{1}+c\psi_{2})^{j+m}(b\psi_{1}+d\psi_{2})^{j-m}=\chi_{1}^{j+m}\chi_{2}^% {j-m}\,,
  56. m = j , j + 1 , j 1 , j m=−j,−j+1,...j−1,j
  57. A A
  58. m m
  59. ( s y m b o l σ 𝐩 ^ ) [ 2 j ] = ( i | 𝐩 ^ | ) 2 j e - i π 𝐉 ( j ) 𝐧 (symbol{\sigma}\cdot\hat{\mathbf{p}})^{[2j]}=(i|\hat{\mathbf{p}}|)^{2j}e^{-i% \pi\mathbf{J}^{(j)}\cdot\mathbf{n}}
  60. m , m = j , j + 1... j m,m′=−j,−j+1...j
  61. m m mm′
  62. ( 2 j + 1 ) × ( 2 j + 1 ) (2j+1)×(2j+1)
  63. ( s y m b o l σ 𝐩 ^ ) m m [ 2 j ] = ( - 1 ) m - m r = - ( - 1 ) r p - j ( - p ^ z ) j - m - r p ^ z j + m - r ( - p + ) m - m + r r ! ( j - m - r ) ! ( j + m - r ) ! ( m - m + r ) ! ( j + m ) ! ( j - m ) ! ( j + m ) ! ( j - m ) ! {(symbol{\sigma}\cdot\hat{\mathbf{p}})^{[2j]}_{mm^{\prime}}=(-1)^{m^{\prime}-m% }\sum_{r=-\infty}^{\infty}\frac{(-1)^{r}p_{-}^{j}(-\hat{p}_{z})^{j-m^{\prime}-% r}\hat{p}_{z}^{j+m-r}(-p_{+})^{m^{\prime}-m+r}}{r!(j-m^{\prime}-r)!(j+m-r)!(m^% {\prime}-m+r)!}\sqrt{(j+m)!(j-m)!(j+m^{\prime})!(j-m^{\prime})!}}
  64. 𝐧 = 𝐩 / | 𝐩 | \mathbf{n}=\mathbf{p}/|\mathbf{p}|
  65. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  66. x Λ x x→Λx
  67. j j
  68. σ σ
  69. D D
  70. ψ ( x ) D ( Λ ) ψ ( Λ - 1 x ) \psi(x)\rightarrow D(\Lambda)\psi(\Lambda^{-1}x)
  71. D ( Λ ) D(Λ)
  72. ψ ψ
  73. σ σ
  74. j j
  75. σ σ
  76. σ σ
  77. j j
  78. ( A , B ) (A,B)
  79. ( 1 2 , 1 2 ) (\frac{1}{2},\frac{1}{2})
  80. ( 1 2 , 0 ) ( 0 , 1 2 ) (\frac{1}{2},0)⊕(0,\frac{1}{2})
  81. ( A , B ) (A,B)
  82. j = A + B , A + B - 1 , , | A - B | , j=A+B,A+B-1,...,|A-B|,
  83. D BW = r = 1 2 j [ D r ( 1 / 2 , 0 ) D r ( 0 , 1 / 2 ) ] . D^{\mathrm{BW}}=\bigotimes_{r=1}^{2j}\left[D_{r}^{(1/2,0)}\oplus D_{r}^{(0,1/2% )}\right]\,.
  84. j j
  85. ( A , B ) (A,B)
  86. j j
  87. D JW = D ( j , 0 ) D ( 0 , j ) . D^{\mathrm{JW}}=D^{(j,0)}\oplus D^{(0,j)}\,.
  88. j j
  89. j j
  90. j j
  91. [ γ i , γ j ] + = 2 η i j [\gamma^{i},\gamma^{j}]_{+}=2\eta^{ij}
  92. i , j = 1 , 2 , 3 i,j=1,2,3
  93. [ γ μ , γ ν ] + = 2 g μ ν [\gamma^{\mu},\gamma^{\nu}]_{+}=2g^{\mu\nu}
  94. μ , ν = 0 , 1 , 2 , 3 μ,ν=0,1,2,3
  95. 𝒟 μ = μ + Ω μ \mathcal{D}_{\mu}=\partial_{\mu}+\Omega_{\mu}
  96. Ω Ω
  97. ω ω
  98. Ω μ = 1 4 μ ω i j ( γ i γ j - γ j γ i ) \Omega_{\mu}=\frac{1}{4}\partial_{\mu}\omega^{ij}(\gamma_{i}\gamma_{j}-\gamma_% {j}\gamma_{i})
  99. ψ ψ
  100. 𝒟 μ ψ D ( Λ ) 𝒟 μ ψ \mathcal{D}_{\mu}\psi\rightarrow D(\Lambda)\mathcal{D}_{\mu}\psi
  101. ( - i γ μ 𝒟 μ + m c ) α 1 α 1 ψ α 1 α 2 α 3 α 2 j = 0 \displaystyle(-i\hbar\gamma^{\mu}\mathcal{D}_{\mu}+mc)_{\alpha_{1}\alpha_{1}^{% \prime}}\psi_{\alpha^{\prime}_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{2j}}=0
  102. μ = ( / t , ) ∂_{μ}=(∂/∂t,∇)
  103. μ = ( i / t , ) ∂_{μ}=(−i∂/∂t,∇)
  104. x x
  105. y y
  106. p < s u b > ± = p 1 ± i p 2 = p x ± i p y p<sub>±=p_{1}±ip_{2}=p_{x}±ip_{y}

Barsoum_elements.html

  1. 1 / [ u r a d i c a l , u r ] 1/[u^{\prime}radical^{\prime},u^{\prime}r^{\prime}]
  2. 1 / [ u r a d i c a l , u r ] 1/[u^{\prime}radical^{\prime},u^{\prime}r^{\prime}]
  3. K 1 = 2 μ 2 π ( 4 V B - V C ) ( k + 1 ) L K_{1}=\frac{2\mu\sqrt{2\pi}(4V_{B}-V_{C})}{(k+1)\sqrt{L}}

Bartlett's_theorem.html

  1. μ ( t ) = - t A ( s ) p ( s , t ) d t . \mu(t)=\int_{-\infty}^{t}A(s)p(s,t)\,\mathrm{d}t.

Base_change_map.html

  1. g * ( R r f * ) R r f * ( g * ) g^{*}(R^{r}f_{*}\mathcal{F})\to R^{r}f^{\prime}_{*}(g^{\prime*}\mathcal{F})
  2. f : X S , f : X S , g : X X , g : S S f:X\to S,f^{\prime}:X^{\prime}\to S^{\prime},g^{\prime}:X^{\prime}\to X,g:S^{% \prime}\to S
  3. \mathcal{F}
  4. X × S T T X\times_{S}T\to T
  5. T S T\to S
  6. s S s\in S
  7. ( R r f * ) s = lim H r ( U , ) = H r ( X s , ) , X s = f - 1 ( s ) (R^{r}f_{*}\mathcal{F})_{s}=\underrightarrow{\lim}H^{r}(U,\mathcal{F})=H^{r}(X% _{s},\mathcal{F}),\quad X_{s}=f^{-1}(s)
  8. s = g ( t ) s=g(t)
  9. g * ( R r f * ) t = H r ( X s , ) = H r ( X t , g * ) = R r f * ( g * ) t . g^{*}(R^{r}f_{*}\mathcal{F})_{t}=H^{r}(X_{s},\mathcal{F})=H^{r}(X^{\prime}_{t}% ,g^{\prime*}\mathcal{F})=R^{r}f^{\prime}_{*}(g^{\prime*}\mathcal{F})_{t}.
  10. g * g^{\prime*}
  11. g * g^{\prime}_{*}
  12. id g * g * \operatorname{id}\to g^{\prime}_{*}\circ g^{\prime*}
  13. R r f * R r f * g * g * . R^{r}f_{*}\to R^{r}f_{*}\circ g^{\prime}_{*}\circ g^{\prime*}.
  14. R r f * g * g * R r ( f g ) * g * = R r ( g f ) * g * g * R r f * g * . R^{r}f_{*}\circ g^{\prime}_{*}\circ g^{\prime*}\to R^{r}(f\circ g^{\prime})_{*% }\circ g^{\prime*}=R^{r}(g\circ f^{\prime})_{*}\circ g^{\prime*}\to g_{*}\circ R% ^{r}f^{\prime}_{*}\circ g^{\prime*}.
  15. R r f * g * R r f * g * . R^{r}f_{*}\to g_{*}\circ R^{r}f^{\prime}_{*}\circ g^{\prime*}.

Basic_solution_(linear_programming).html

  1. P P
  2. 𝐱 * n \mathbf{x}^{*}\in\mathcal{R}^{n}
  3. 𝐱 * \mathbf{x}^{*}
  4. P P
  5. 𝐱 * \mathbf{x}^{*}
  6. n n
  7. n n
  8. 𝐱 \mathbf{x}
  9. P P
  10. P P

Basset–Boussinesq–Oseen_equation.html

  1. d p d_{p}
  2. s y m b o l x = s y m b o l X p ( t ) symbol{x}=symbol{X}_{p}(t)
  3. ρ p \rho_{p}
  4. s y m b o l U p = d s y m b o l X p / d t symbol{U}_{p}=\,\text{d}symbol{X}_{p}/\,\text{d}t
  5. ρ f \rho_{f}
  6. μ \mu
  7. s y m b o l U f : symbol{U}_{f}:
  8. π 6 ρ p d p 3 d s y m b o l U p d t = 3 π μ d p ( s y m b o l U f - s y m b o l U p ) term 1 - π 6 d p 3 s y m b o l p term 2 + π 12 ρ f d p 3 d d t ( s y m b o l U f - s y m b o l U p ) term 3 + 3 2 d p 2 π ρ f μ t 0 t 1 t - τ d d τ ( s y m b o l U f - s y m b o l U p ) d τ term 4 + k s y m b o l F k term 5 . \begin{aligned}\displaystyle\frac{\pi}{6}\rho_{p}d_{p}^{3}\frac{\,\text{d}% symbol{U}_{p}}{\,\text{d}t}&\displaystyle=\underbrace{3\pi\mu d_{p}\left(% symbol{U}_{f}-symbol{U}_{p}\right)}_{\,\text{term 1}}-\underbrace{\frac{\pi}{6% }d_{p}^{3}symbol{\nabla}p}_{\,\text{term 2}}+\underbrace{\frac{\pi}{12}\rho_{f% }d_{p}^{3}\,\frac{\,\text{d}}{\,\text{d}t}\left(symbol{U}_{f}-symbol{U}_{p}% \right)}_{\,\text{term 3}}\\ &\displaystyle+\underbrace{\frac{3}{2}d_{p}^{2}\sqrt{\pi\rho_{f}\mu}\int_{t_{{% }_{0}}}^{t}\frac{1}{\sqrt{t-\tau}}\,\frac{\,\text{d}}{\,\text{d}\tau}\left(% symbol{U}_{f}-symbol{U}_{p}\right)\,\,\text{d}\tau}_{\,\text{term 4}}+% \underbrace{\sum_{k}symbol{F}_{k}}_{\,\text{term 5}}.\end{aligned}
  9. s y m b o l symbol{\nabla}
  10. R e : R_{e}:
  11. R e = max { | s y m b o l U p - s y m b o l U f | } d p μ / ρ f R_{e}=\frac{\max\left\{\left|symbol{U}_{p}-symbol{U}_{f}\right|\right\}\,d_{p}% }{\mu/\rho_{f}}
  12. R e < 1 R_{e}<1
  13. - s y m b o l p = ρ f D s y m b o l u f D t - μ s y m b o l s y m b o l s y m b o l u f , -symbol{\nabla}p=\rho_{f}\frac{\,\text{D}symbol{u}_{f}}{\,\text{D}t}-\mu symbol% {\nabla}\!\cdot\!symbol{\nabla}symbol{u}_{f},
  14. D s y m b o l u f / D t \,\text{D}symbol{u}_{f}/\,\text{D}t
  15. s y m b o l u f . symbol{u}_{f}.
  16. s y m b o l u f ( s y m b o l x , t ) symbol{u}_{f}(symbol{x},t)
  17. s y m b o l U f symbol{U}_{f}
  18. s y m b o l U f ( t ) = s y m b o l u f ( s y m b o l X p ( t ) , t ) . symbol{U}_{f}(t)=symbol{u}_{f}(symbol{X}_{p}(t),t).

Baxter_permutation.html

  1. σ S n \sigma\in S_{n}
  2. a n = k = 1 n ( n + 1 k - 1 ) ( n + 1 k ) ( n + 1 k + 1 ) ( n + 1 1 ) ( n + 1 2 ) . a_{n}\,=\,\sum_{k=1}^{n}\frac{{\left({{n+1}\atop{k-1}}\right)}{\left({{n+1}% \atop{k}}\right)}{\left({{n+1}\atop{k+1}}\right)}}{{\left({{n+1}\atop{1}}% \right)}{\left({{n+1}\atop{2}}\right)}}.
  3. ( n + 1 k - 1 ) ( n + 1 k ) ( n + 1 k + 1 ) ( n + 1 1 ) ( n + 1 2 ) \frac{{\left({{n+1}\atop{k-1}}\right)}{\left({{n+1}\atop{k}}\right)}{\left({{n% +1}\atop{k+1}}\right)}}{{\left({{n+1}\atop{1}}\right)}{\left({{n+1}\atop{2}}% \right)}}

Bayes_error_rate.html

  1. p = C i C max x H i P ( x | C i ) p ( C i ) d x , p=\sum_{C_{i}\neq C\text{max}}\textstyle\int\limits_{x\in H_{i}}P(x|C_{i})p(C_% {i})\,dx,

Bayesian_inference_in_marketing.html

  1. P ( A B ) = P ( A | B ) P ( B ) = P ( B | A ) P ( A ) P(AB)=P(A|B)P(B)=P(B|A)P(A)
  2. A A
  3. B B
  4. ( H ) (H)
  5. ( D ) (D)
  6. P ( D | H ) P(D|H)
  7. ( D ) (D)
  8. ( H ) (H)
  9. P ( H ) P(H)
  10. P ( D ) P(D)
  11. P ( D | H ) P ( H ) P(D|H)P(H)
  12. P ( H | D ) P(H|D)
  13. P ( H | D ) = P ( D | H ) P ( H ) P ( D ) P(H|D)=\frac{P(D|H)P(H)}{P(D)}

Bayesian_inference_in_motor_learning.html

  1. P ( A | B ) = P ( B | A ) P ( A ) P ( B ) . P(A|B)=\frac{P(B|A)\,P(A)}{P(B)}.\,
  2. P ( A | B ) P(A|B)
  3. P ( B | A ) P(B|A)
  4. P ( A ) P(A)
  5. P ( B ) P(B)
  6. E = W p S + W s P W p + W s . E=\frac{W_{p}S+W_{s}P}{W_{p}+W_{s}}.
  7. E E
  8. S S
  9. P P
  10. W p Wp
  11. W s Ws
  12. S S
  13. P P

Bayesian_interpretation_of_kernel_regularization.html

  1. 𝐱 \mathbf{x}^{\prime}
  2. f ^ ( 𝐱 ) \hat{f}(\mathbf{x}^{\prime})
  3. S S
  4. n n
  5. S = ( 𝐗 , 𝐘 ) = ( 𝐱 1 , y 1 ) , , ( 𝐱 n , y n ) S=(\mathbf{X},\mathbf{Y})=(\mathbf{x}_{1},y_{1}),\ldots,(\mathbf{x}_{n},y_{n})
  6. k ( , ) k(\cdot,\cdot)
  7. f ^ ( 𝐱 ) = 𝐤 ( 𝐊 + λ n 𝐈 ) - 1 𝐘 , \hat{f}(\mathbf{x}^{\prime})=\mathbf{k}^{\top}(\mathbf{K}+\lambda n\mathbf{I})% ^{-1}\mathbf{Y},
  8. 𝐊 k ( 𝐗 , 𝐗 ) \mathbf{K}\equiv k(\mathbf{X},\mathbf{X})
  9. 𝐊 i j = k ( 𝐱 i , 𝐱 j ) \mathbf{K}_{ij}=k(\mathbf{x}_{i},\mathbf{x}_{j})
  10. 𝐤 = [ k ( 𝐱 1 , 𝐱 ) , , k ( 𝐱 n , 𝐱 ) ] \mathbf{k}=[k(\mathbf{x}_{1},\mathbf{x}^{\prime}),\ldots,k(\mathbf{x}_{n},% \mathbf{x}^{\prime})]^{\top}
  11. 𝐘 = [ y 1 , , y n ] \mathbf{Y}=[y_{1},\ldots,y_{n}]^{\top}
  12. \mathcal{F}
  13. k \mathcal{H}_{k}
  14. k \mathcal{H}_{k}
  15. k : 𝒳 × 𝒳 k:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}
  16. k ( 𝐱 , ) k(\mathbf{x},\cdot)
  17. k \mathcal{H}_{k}
  18. 𝐱 𝒳 \mathbf{x}\in\mathcal{X}
  19. f ( 𝐱 ) = f , k ( 𝐱 , ) k , f k , f(\mathbf{x})=\langle f,k(\mathbf{x},\cdot)\rangle_{k},\quad\forall\ f\in% \mathcal{H}_{k},
  20. , k \langle\cdot,\cdot\rangle_{k}
  21. k \mathcal{H}_{k}
  22. f ( 𝐱 ) = i k ( 𝐱 i , 𝐱 ) c i f(\mathbf{x})=\sum_{i}k(\mathbf{x}_{i},\mathbf{x})c_{i}
  23. f k 2 = i , j k ( 𝐱 i , 𝐱 j ) c i c j \|f\|_{k}^{2}=\sum_{i,j}k(\mathbf{x}_{i},\mathbf{x}_{j})c_{i}c_{j}
  24. 1 n i = 1 n ( f ( 𝐱 i ) - y i ) 2 + λ f k 2 , \frac{1}{n}\sum_{i=1}^{n}(f(\mathbf{x}_{i})-y_{i})^{2}+\lambda\|f\|_{k}^{2},
  25. f k f\in\mathcal{H}_{k}
  26. k \|\cdot\|_{k}
  27. k \mathcal{H}_{k}
  28. f ( 𝐱 i ) f(\mathbf{x}_{i})
  29. y i y_{i}
  30. f ( 𝐱 i ) f(\mathbf{x}_{i})
  31. y i y_{i}
  32. λ \lambda
  33. λ \lambda
  34. λ \lambda
  35. f ^ ( 𝐱 ) = i = 1 n c i k ( 𝐱 i , 𝐱 ) = 𝐤 𝐜 , \hat{f}(\mathbf{x}^{\prime})=\sum_{i=1}^{n}c_{i}k(\mathbf{x}_{i},\mathbf{x}^{% \prime})=\mathbf{k}^{\top}\mathbf{c},
  36. 𝐜 n \mathbf{c}\in\mathbb{R}^{n}
  37. 𝐜 = [ c 1 , , c n ] \mathbf{c}=[c_{1},\ldots,c_{n}]^{\top}
  38. f ( ) f(\cdot)
  39. f k 2 = f , f k , = i = 1 N c i k ( 𝐱 i , ) , j = 1 N c j k ( 𝐱 j , ) k , = i = 1 N j = 1 N c i c j k ( 𝐱 i , ) , k ( 𝐱 j , ) k , = i = 1 N j = 1 N c i c j k ( 𝐱 i , 𝐱 j ) , = 𝐜 𝐊𝐜 . \begin{aligned}\displaystyle\|f\|_{k}^{2}&\displaystyle=\langle f,f\rangle_{k}% ,\\ &\displaystyle=\left\langle\sum_{i=1}^{N}c_{i}k(\mathbf{x}_{i},\cdot),\sum_{j=% 1}^{N}c_{j}k(\mathbf{x}_{j},\cdot)\right\rangle_{k},\\ &\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}c_{i}c_{j}\langle k(\mathbf{x}_{i},% \cdot),k(\mathbf{x}_{j},\cdot)\rangle_{k},\\ &\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}c_{i}c_{j}k(\mathbf{x}_{i},\mathbf{% x}_{j}),\\ &\displaystyle=\mathbf{c}^{\top}\mathbf{K}\mathbf{c}.\end{aligned}
  40. 1 n 𝐲 - 𝐊𝐜 2 + λ 𝐜 𝐊𝐜 . \frac{1}{n}\|\mathbf{y}-\mathbf{K}\mathbf{c}\|^{2}+\lambda\mathbf{c}^{\top}% \mathbf{K}\mathbf{c}.
  41. 𝐜 \mathbf{c}
  42. 𝐜 \mathbf{c}
  43. - 1 n 𝐊 ( 𝐘 - 𝐊𝐜 ) + λ 𝐊𝐜 = 0 , ( 𝐊 + λ n 𝐈 ) 𝐜 = 𝐘 , 𝐜 = ( 𝐊 + λ n 𝐈 ) - 1 𝐘 . \begin{aligned}\displaystyle-\frac{1}{n}\mathbf{K}(\mathbf{Y}-\mathbf{K}% \mathbf{c})+\lambda\mathbf{K}\mathbf{c}&\displaystyle=0,\\ \displaystyle(\mathbf{K}+\lambda n\mathbf{I})\mathbf{c}&\displaystyle=\mathbf{% Y},\\ \displaystyle\mathbf{c}&\displaystyle=(\mathbf{K}+\lambda n\mathbf{I})^{-1}% \mathbf{Y}.\end{aligned}
  44. f ^ ( 𝐱 ) = 𝐤 ( 𝐊 + λ n 𝐈 ) - 1 𝐘 . \hat{f}(\mathbf{x}^{\prime})=\mathbf{k}^{\top}(\mathbf{K}+\lambda n\mathbf{I})% ^{-1}\mathbf{Y}.
  45. f f
  46. m m
  47. k k
  48. f 𝒢 𝒫 ( m , k ) . f\sim\mathcal{GP}(m,k).
  49. 𝐗 = { 𝐱 i } i = 1 n \mathbf{X}=\{\mathbf{x}_{i}\}_{i=1}^{n}
  50. f ( 𝐗 ) = [ f ( 𝐱 1 ) , , f ( 𝐱 n ) ] f(\mathbf{X})=[f(\mathbf{x}_{1}),\ldots,f(\mathbf{x}_{n})]^{\top}
  51. f ( 𝐗 ) 𝒩 ( 𝐦 , 𝐊 ) , f(\mathbf{X})\sim\mathcal{N}(\mathbf{m},\mathbf{K}),
  52. 𝐦 = m ( 𝐗 ) = [ m ( 𝐱 1 ) , , m ( 𝐱 N ) ] \mathbf{m}=m(\mathbf{X})=[m(\mathbf{x}_{1}),\ldots,m(\mathbf{x}_{N})]^{\top}
  53. 𝐊 = k ( 𝐗 , 𝐗 ) \mathbf{K}=k(\mathbf{X},\mathbf{X})
  54. p ( y | f , 𝐱 , σ 2 ) = 𝒩 ( f ( 𝐱 ) , σ 2 ) . p(y|f,\mathbf{x},\sigma^{2})=\mathcal{N}(f(\mathbf{x}),\sigma^{2}).
  55. σ 2 \sigma^{2}
  56. 𝐗 \mathbf{X}
  57. σ 2 \sigma^{2}
  58. 𝐱 \mathbf{x}^{\prime}
  59. S = { 𝐗 , 𝐘 } S=\{\mathbf{X},\mathbf{Y}\}
  60. p ( f ( 𝐱 ) | S , 𝐱 , s y m b o l ϕ ) = 𝒩 ( m ( 𝐱 ) , σ 2 ( 𝐱 ) ) , p(f(\mathbf{x}^{\prime})|S,\mathbf{x}^{\prime},symbol{\phi})=\mathcal{N}(m(% \mathbf{x}^{\prime}),\sigma^{2}(\mathbf{x}^{\prime})),
  61. s y m b o l ϕ symbol{\phi}
  62. σ 2 \sigma^{2}
  63. k k
  64. m ( 𝐱 ) = 𝐤 ( 𝐊 + σ 2 𝐈 ) - 1 𝐘 , σ 2 ( 𝐱 ) = k ( 𝐱 , 𝐱 ) - 𝐤 ( 𝐊 + σ 2 𝐈 ) - 1 𝐤 . \begin{aligned}\displaystyle m(\mathbf{x}^{\prime})&\displaystyle=\mathbf{k}^{% \top}(\mathbf{K}+\sigma^{2}\mathbf{I})^{-1}\mathbf{Y},\\ \displaystyle\sigma^{2}(\mathbf{x}^{\prime})&\displaystyle=k(\mathbf{x}^{% \prime},\mathbf{x}^{\prime})-\mathbf{k}^{\top}(\mathbf{K}+\sigma^{2}\mathbf{I}% )^{-1}\mathbf{k}.\end{aligned}
  65. Φ : 𝒳 p \Phi:\mathcal{X}\rightarrow\mathbb{R}^{p}
  66. k ( 𝐱 , 𝐱 ) = i = 1 p Φ i ( 𝐱 ) Φ i ( 𝐱 ) . k(\mathbf{x},\mathbf{x}^{\prime})=\sum_{i=1}^{p}\Phi^{i}(\mathbf{x})\Phi^{i}(% \mathbf{x}^{\prime}).
  67. 𝐊 \mathbf{K}
  68. f 𝐰 ( 𝐱 ) = i = 1 p 𝐰 i Φ i ( 𝐱 ) = 𝐰 , Φ ( 𝐱 ) , f_{\mathbf{w}}(\mathbf{x})=\sum_{i=1}^{p}\mathbf{w}^{i}\Phi^{i}(\mathbf{x})=% \langle\mathbf{w},\Phi(\mathbf{x})\rangle,
  69. f 𝐰 k = 𝐰 . \|f_{\mathbf{w}}\|_{k}=\|\mathbf{w}\|.
  70. 𝐰 = [ w 1 , , w p ] \mathbf{w}=[w^{1},\ldots,w^{p}]^{\top}
  71. 𝐰 𝒩 ( 0 , 𝐈 ) exp ( - 𝐰 2 ) . \mathbf{w}\sim\mathcal{N}(0,\mathbf{I})\propto\exp(-\|\mathbf{w}\|^{2}).
  72. P ( 𝐘 | 𝐗 , f ) = 𝒩 ( f ( 𝐗 ) , σ 2 𝐈 ) exp ( - 1 σ 2 f 𝐰 ( 𝐗 ) - 𝐘 2 ) , P(\mathbf{Y}|\mathbf{X},f)=\mathcal{N}(f(\mathbf{X}),\sigma^{2}\mathbf{I})% \propto\exp\left(-\frac{1}{\sigma^{2}}\|f_{\mathbf{w}}(\mathbf{X})-\mathbf{Y}% \|^{2}\right),
  73. f 𝐰 ( 𝐗 ) = ( 𝐰 , Φ ( 𝐱 1 ) , , 𝐰 , Φ ( 𝐱 n ) f_{\mathbf{w}}(\mathbf{X})=(\langle\mathbf{w},\Phi(\mathbf{x}_{1})\rangle,% \ldots,\langle\mathbf{w},\Phi(\mathbf{x}_{n}\rangle)
  74. P ( f | 𝐗 , 𝐘 ) exp ( - 1 σ 2 f 𝐰 ( 𝐗 ) - 𝐘 n 2 + 𝐰 2 ) P(f|\mathbf{X},\mathbf{Y})\propto\exp\left(-\frac{1}{\sigma^{2}}\|f_{\mathbf{w% }}(\mathbf{X})-\mathbf{Y}\|_{n}^{2}+\|\mathbf{w}\|^{2}\right)
  75. f ( 𝐱 ) f(\mathbf{x})
  76. y y
  77. f f
  78. y y

Beck–Fiala_theorem.html

  1. [ n ] = { 1 , , n } [n]=\{1,\ldots,n\}
  2. S 1 , S 2 , , S m [ n ] S_{1},S_{2},\ldots,S_{m}\subseteq[n]
  3. i [ n ] i\in[n]
  4. | { j ; i S j } | t , |\{j;i\in S_{j}\}|\leq t,
  5. x : [ n ] { - 1 , + 1 } x:[n]\rightarrow\{-1,+1\}
  6. x ( S j ) 2 t - 1 , j where x ( S j ) := i S j x i . x(S_{j})\leq 2t-1,\forall j\,\text{ where }x(S_{j}):=\sum_{i\in S_{j}}x_{i}.
  7. x i = 0 x_{i}=0
  8. | S j | > t |S_{j}|>t
  9. t t
  10. n n
  11. i S j x i = 0 \sum_{i\in S_{j}}x_{i}=0
  12. 𝐑 n \mathbf{R}^{n}
  13. + 1 , - 1 +1,-1
  14. t t
  15. + 1 , - 1 +1,-1
  16. t t
  17. x ( S j ) x(S_{j})
  18. 2 t - 1 2t-1

Behrens–Fisher_distribution.html

  1. T 2 cos θ - T 1 sin θ T_{2}\cos\theta-T_{1}\sin\theta\,
  2. X 1 , 1 , , X 1 , n 1 \displaystyle X_{1,1},\ldots,X_{1,n_{1}}
  3. X ¯ 1 \displaystyle\bar{X}_{1}
  4. S pooled 2 = k = 1 n 1 ( X 1 , k - X ¯ 1 ) 2 + k = 1 n 2 ( X 2 , k - X ¯ 2 ) 2 n 1 + n 2 - 2 = ( n 1 - 1 ) S 1 2 + ( n 2 - 1 ) S 2 2 n 1 + n 2 - 2 S_{\mathrm{pooled}}^{2}=\frac{\sum_{k=1}^{n_{1}}(X_{1,k}-\bar{X}_{1})^{2}+\sum% _{k=1}^{n_{2}}(X_{2,k}-\bar{X}_{2})^{2}}{n_{1}+n_{2}-2}=\frac{(n_{1}-1)S_{1}^{% 2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}
  5. ( μ 2 - μ 1 ) - ( X ¯ 2 - X ¯ 1 ) S pooled 2 n 1 + S pooled 2 n 2 \frac{(\mu_{2}-\mu_{1})-(\bar{X}_{2}-\bar{X}_{1})}{\displaystyle\sqrt{\frac{S^% {2}_{\mathrm{pooled}}}{n_{1}}+\frac{S^{2}_{\mathrm{pooled}}}{n_{2}}}}
  6. X ¯ 2 - X 1 ¯ ± A S pooled 1 n 1 + 1 n 2 , \bar{X}_{2}-\bar{X_{1}}\pm A\cdot S_{\mathrm{pooled}}\sqrt{\frac{1}{n_{1}}+% \frac{1}{n_{2}}},
  7. ( μ 2 - μ 1 ) - ( X ¯ 2 - X ¯ 1 ) S 1 2 n 1 + S 2 2 n 2 . \frac{(\mu_{2}-\mu_{1})-(\bar{X}_{2}-\bar{X}_{1})}{\displaystyle\sqrt{\frac{S^% {2}_{1}}{n_{1}}+\frac{S^{2}_{2}}{n_{2}}}}.
  8. T 2 cos θ - T 1 sin θ , T_{2}\cos\theta-T_{1}\sin\theta,\,
  9. T i = μ i - X ¯ i S i / n i for i = 1 , 2 T_{i}=\frac{\mu_{i}-\bar{X}_{i}}{S_{i}/\sqrt{n_{i}}}\,\text{ for }i=1,2\,
  10. tan θ = S 1 / n 1 S 2 / n 2 \tan\theta=\frac{S_{1}/\sqrt{n_{1}}}{S_{2}/\sqrt{n_{2}}}
  11. ( μ 2 - μ 1 ) - ( X ¯ 2 - X ¯ 1 ) S 1 2 n 1 + S 2 2 n 2 \displaystyle\frac{(\mu_{2}-\mu_{1})-(\bar{X}_{2}-\bar{X}_{1})}{\displaystyle% \sqrt{\frac{S^{2}_{1}}{n_{1}}+\frac{S^{2}_{2}}{n_{2}}}}
  12. X ¯ 2 - X ¯ 1 ± A S 1 2 n 1 + S 2 2 n 2 \bar{X}_{2}-\bar{X}_{1}\pm A\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{% 2}}}

Belinski–Zakharov_transform.html

  1. d s 2 = f ( - d ( x 0 ) 2 + d ( x 1 ) 2 ) + g a b d x a d x b ds^{2}=f(-d(x^{0})^{2}+d(x^{1})^{2})+g_{ab}\,dx^{a}\,dx^{b}
  2. a , b = 2 , 3 a,b=2,3
  3. f f
  4. g = g a b g=g_{ab}
  5. x 0 x^{0}
  6. x 1 x^{1}
  7. R μ ν = 0 R_{\mu\nu}=0
  8. g = g a b g=g_{ab}
  9. f f
  10. ζ = x 0 + x 1 , η = x 0 - x 1 \zeta=x^{0}+x^{1},\eta=x^{0}-x^{1}
  11. g g
  12. ( α g , ζ g - 1 ) , η + ( α g , η g - 1 ) , ζ = 0 (\alpha g_{,\zeta}g^{-1})_{,\eta}+(\alpha g_{,\eta}g^{-1})_{,\zeta}=0
  13. α \alpha
  14. g g
  15. det g = α 2 \det g=\alpha^{2}
  16. ( ln f ) , ζ = ( ln α ) , ζ ζ ( ln α ) , ζ + α 4 α , ζ tr ( g , ζ g - 1 g , ζ g - 1 ) (\ln f)_{,\zeta}=\frac{(\ln\alpha)_{,\zeta\zeta}}{(\ln\alpha)_{,\zeta}}+\frac{% \alpha}{4\alpha_{,\zeta}}\operatorname{tr}(g_{,\zeta}g^{-1}g_{,\zeta}g^{-1})
  17. ( ln f ) , η = ( ln α ) , η η ( ln α ) , η + α 4 α , η tr ( g , η g - 1 g , η g - 1 ) (\ln f)_{,\eta}=\frac{(\ln\alpha)_{,\eta\eta}}{(\ln\alpha)_{,\eta}}+\frac{% \alpha}{4\alpha_{,\eta}}\operatorname{tr}(g_{,\eta}g^{-1}g_{,\eta}g^{-1})
  18. g g
  19. α \alpha
  20. α , ζ η = 0 \alpha_{,\zeta\eta}=0
  21. D 1 , D 2 D_{1},D_{2}
  22. D 1 = ζ + 2 α , ζ λ λ - α λ D_{1}=\partial_{\zeta}+\frac{2\alpha_{,\zeta}\lambda}{\lambda-\alpha}\partial_% {\lambda}
  23. D 2 = η - 2 α , η λ λ + α λ D_{2}=\partial_{\eta}-\frac{2\alpha_{,\eta}\lambda}{\lambda+\alpha}\partial_{\lambda}
  24. λ \lambda
  25. α \alpha
  26. [ D 1 , D 2 ] = 0 \left[D_{1},D_{2}\right]=0
  27. ψ = ψ ( ζ , η , λ ) \psi=\psi(\zeta,\eta,\lambda)
  28. D 1 ψ = A λ - α ψ D_{1}\psi=\frac{A}{\lambda-\alpha}\psi
  29. D 2 ψ = B λ + α ψ D_{2}\psi=\frac{B}{\lambda+\alpha}\psi
  30. D 2 D_{2}
  31. D 1 D_{1}
  32. D 1 D_{1}
  33. D 2 D_{2}
  34. g g
  35. g g
  36. g g
  37. ψ \psi
  38. λ 0 \lambda\rightarrow 0
  39. ψ - 1 \psi^{-1}
  40. ψ , ζ ψ - 1 = g , ζ g - 1 \psi_{,\zeta}\psi^{-1}=g_{,\zeta}g^{-1}
  41. ψ , η ψ - 1 = g , η g - 1 \psi_{,\eta}\psi^{-1}=g_{,\eta}g^{-1}
  42. g g
  43. g ( ζ , η ) = ψ ( ζ , η , 0 ) g(\zeta,\eta)=\psi(\zeta,\eta,0)

BELLA_(laser).html

  1. 42.2 J 40 f s = 42.2 J 40 * 10 - 15 s 1 * 10 15 J / s = 1 P W {42.2J\over 40fs}={42.2J\over 40*10^{-15}s}\approx 1*10^{15}J/s=1PW

Bennett's_laws.html

  1. \geqslant
  2. \geqslant
  3. \geqslant
  4. \geqslant
  5. \geqslant

Benzil_reductase_((R)-benzoin_forming).html

  1. \rightleftharpoons

Benzil_reductase_((S)-benzoin_forming).html

  1. \rightleftharpoons

Benzoyl-CoA-dihydrodiol_lyase.html

  1. \rightleftharpoons

Benzoyl-CoA_2,3-dioxygenase.html

  1. \rightleftharpoons

Benzyl_alcohol_O-benzoyltransferase.html

  1. \rightleftharpoons

Bernal_chart.html

  1. ζ \zeta
  2. ξ \xi
  3. ϕ \phi
  4. ζ \zeta
  5. ξ \xi
  6. ζ \zeta
  7. ζ \zeta
  8. ζ \zeta
  9. ζ = 0 \zeta=0
  10. ζ = ζ 1 \zeta=\zeta_{1}
  11. ζ = ζ 2 \zeta=\zeta_{2}
  12. ξ \xi
  13. H R = t a n ( μ ) \frac{H}{R}=tan(\mu)
  14. μ \mu
  15. ζ \zeta
  16. ξ \xi
  17. ζ \zeta
  18. ξ \xi

Beta-amyrin_11-oxidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Beta-amyrin_24-hydroxylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Beta-apo-4'-carotenal_oxygenase.html

  1. \rightleftharpoons

Beta-carotene_15,15'-dioxygenase.html

  1. \rightleftharpoons

Beta-carotene_3-hydroxylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Beta-D-galactosyl-(1-4)-L-rhamnose_phosphorylase.html

  1. \rightleftharpoons

Beta-D-glucopyranosyl_abscisate_beta-glucosidase.html

  1. \rightleftharpoons

Beta-galactosyl-N-acetylglucosaminylgalactosylglucosyl-ceramide_beta-1,3-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Beta-ketodecanoyl-(acyl-carrier-protein)_synthase.html

  1. \rightleftharpoons

Beta-mannosylphosphodecaprenol—mannooligosaccharide_6-mannosyltransferase.html

  1. \rightleftharpoons

Beta_function_(accelerator_physics).html

  1. σ ( s ) = ϵ β ( s ) \sigma(s)=\sqrt{\epsilon\cdot\beta(s)}
  2. s s
  3. σ ( s ) \sigma(s)
  4. ϵ \epsilon
  5. β ( z ) = β * + z 2 β * \beta(z)=\beta^{*}+\dfrac{z^{2}}{\beta^{*}}

Better-quasi-ordering.html

  1. x * {{}_{*}}x
  2. x x
  3. [ ω ] < ω [\omega]^{<\omega}
  4. ω \omega
  5. \triangleleft
  6. [ ω ] < ω [\omega]^{<\omega}
  7. s t s\triangleleft t
  8. u u
  9. s s
  10. u u
  11. t = u * t={}_{*}u
  12. \triangleleft
  13. [ ω ] < ω [\omega]^{<\omega}
  14. B \bigcup B
  15. Q Q
  16. Q Q
  17. Q Q
  18. Q Q
  19. f : B Q f\colon B\to Q
  20. f ( s ) Q f ( t ) f(s)\not\leq_{Q}f(t)
  21. s , t B s,t\in B
  22. s t s\triangleleft t
  23. f f
  24. Q Q
  25. Q Q
  26. \subset
  27. Q Q
  28. Q Q
  29. Q Q
  30. Q Q
  31. [ ω ] ω Q [\omega]^{\omega}\to Q
  32. [ ω ] ω [\omega]^{\omega}
  33. ω \omega
  34. Q Q
  35. Q Q
  36. Q Q
  37. [ A ] ω Q [A]^{\omega}\to Q
  38. A A
  39. ω \omega
  40. Q Q
  41. f f
  42. f ( X ) Q f ( X * ) f(X)\not\leq_{Q}f({{}_{*}}X)
  43. X [ A ] ω X\in[A]^{\omega}
  44. f f
  45. Q Q
  46. Q Q
  47. ( Q , Q ) (Q,\leq_{Q})
  48. \leq^{\prime}
  49. Q Q
  50. Q Q
  51. q r q Q r q\leq^{\prime}r\to q\leq_{Q}r
  52. Q Q
  53. f : [ A ] ω Q f\colon[A]^{\omega}\to Q
  54. g : [ B ] ω Q g\colon[B]^{\omega}\to Q
  55. g * f if B A and g ( X ) f ( X ) for every X [ B ] ω g\leq^{*}f\,\text{ if }B\subseteq A\,\text{ and }g(X)\leq^{\prime}f(X)\,\text{% for every }X\in[B]^{\omega}
  56. g < * f if B A and g ( X ) < f ( X ) for every X [ B ] ω g<^{*}f\,\text{ if }B\subseteq A\,\text{ and }g(X)<^{\prime}f(X)\,\text{ for % every }X\in[B]^{\omega}
  57. Q Q
  58. g g
  59. \leq^{\prime}
  60. Q Q
  61. f f
  62. f < * g f<^{*}g
  63. * \leq^{*}
  64. < <^{\prime}
  65. \leq^{\prime}
  66. Q Q
  67. < * <^{*}
  68. * \leq^{*}
  69. Q Q
  70. f f
  71. Q Q
  72. Q Q
  73. g g
  74. g * f g\leq^{*}f

Biflaviolin_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Bile-acid_7alpha-dehydroxylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Binary_Goppa_code.html

  1. g ( x ) g(x)
  2. t t
  3. G F ( 2 m ) GF(2^{m})
  4. L L
  5. n n
  6. G F ( 2 m ) GF(2^{m})
  7. i , j { 0 , , n - 1 } : L i G F ( 2 m ) and L i L j and g ( L i ) 0 \forall i,j\in\{0,\ldots,n-1\}:L_{i}\in GF(2^{m})\and L_{i}\neq L_{j}\and g(L_% {i})\neq 0
  8. { 0 , 1 } n \{0,1\}^{n}
  9. Γ ( g , L ) = { c { 0 , 1 } n | i = 0 n - 1 c i x - L i 0 mod g ( x ) } \Gamma(g,L)=\left\{c\in\{0,1\}^{n}|\sum_{i=0}^{n-1}\frac{c_{i}}{x-L_{i}}\equiv 0% \mod g(x)\right\}
  10. ( g , L ) (g,L)
  11. 2 t + 1 2t+1
  12. t = ( 2 t + 1 ) - 1 2 t=\left\lfloor\frac{(2t+1)-1}{2}\right\rfloor
  13. n - m t n-mt
  14. n n
  15. H H
  16. H = V D = ( 1 1 1 1 L 0 1 L 1 1 L 2 1 L n - 1 1 L 0 2 L 1 2 L 2 2 L n - 1 2 L 0 t L 1 t L 2 t L n - 1 t ) ( 1 g ( L 0 ) 1 g ( L 1 ) 1 g ( L 2 ) 1 g ( L n - 1 ) ) H=VD=\begin{pmatrix}1&1&1&\cdots&1\\ L_{0}^{1}&L_{1}^{1}&L_{2}^{1}&\cdots&L_{n-1}^{1}\\ L_{0}^{2}&L_{1}^{2}&L_{2}^{2}&\cdots&L_{n-1}^{2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ L_{0}^{t}&L_{1}^{t}&L_{2}^{t}&\cdots&L_{n-1}^{t}\end{pmatrix}\begin{pmatrix}% \frac{1}{g(L_{0})}&&&&\\ &\frac{1}{g(L_{1})}&&&\\ &&\frac{1}{g(L_{2})}&&\\ &&&\ddots&\\ &&&&\frac{1}{g(L_{n-1})}\end{pmatrix}
  17. V V
  18. D D
  19. t / 2 t/2
  20. t t
  21. n n
  22. G F ( 2 m ) GF(2^{m})
  23. m t mt
  24. n n
  25. G F ( 2 m ) GF(2^{m})
  26. m m
  27. t t
  28. c = ( c 0 , , c n - 1 ) c=(c_{0},\dots,c_{n-1})
  29. s ( x ) c i = 1 1 x - L i mod g ( x ) s(x)\equiv\sum_{c_{i}=1}\frac{1}{x-L_{i}}\mod g(x)
  30. s ( x ) s(x)
  31. v ( x ) s ( x ) - 1 - x mod g ( x ) v(x)\equiv\sqrt{s(x)^{-1}-x}\mod g(x)
  32. s ( x ) 0 s(x)\equiv 0
  33. v ( x ) v(x)
  34. a ( x ) a(x)
  35. b ( x ) b(x)
  36. a ( x ) b ( x ) v ( x ) mod g ( x ) a(x)\equiv b(x)\cdot v(x)\mod g(x)
  37. deg ( a ) t / 2 \deg(a)\leq\lfloor t/2\rfloor
  38. deg ( b ) ( t - 1 ) / 2 \deg(b)\leq\lfloor(t-1)/2\rfloor
  39. σ ( x ) = a ( x ) 2 + x b ( x ) 2 \sigma(x)=a(x)^{2}+x\cdot b(x)^{2}
  40. e = ( e 0 , e 1 , , e n - 1 ) e=(e_{0},e_{1},\dots,e_{n-1})
  41. σ ( x ) = e i = 1 ( x - L i ) \sigma(x)=\prod_{e_{i}=1}(x-L_{i})
  42. σ ( x ) \sigma(x)
  43. deg ( g ) \deg(g)
  44. deg ( g ) / 2 \deg(g)/2

Binary_matroid.html

  1. M M
  2. 𝒮 \mathcal{S}
  3. 𝒮 \mathcal{S}
  4. C , D C,D
  5. C C
  6. M M
  7. D D
  8. M M
  9. | C D | |C\cap D|
  10. B , C B,C
  11. B B
  12. M M
  13. C C
  14. M M
  15. C C
  16. B B
  17. C B C\setminus B
  18. M M
  19. U 4 2 U{}^{2}_{4}
  20. K 5 K_{5}
  21. K 3 , 3 K_{3,3}
  22. M M
  23. M M

Binning_(Metagenomics).html

  1. 4 4 = 256 4^{4}=256

Bioconcentration.html

  1. B C F = C o n c e n t r a t i o n B i o t a C o n c e n t r a t i o n W a t e r BCF=\frac{Concentration_{Biota}}{Concentration_{Water}}
  2. l o g B C F = m l o g K O W + b logBCF=mlogK_{OW}+b
  3. K O W = C o n c e n t r a t i o n o c t a n o l C o n c e n t r a t i o n w a t e r = C O C W K_{OW}=\frac{Concentration_{octanol}}{Concentration_{water}}=\frac{C_{O}}{C_{W}}
  4. Z F i s h = P F i s h × B C F H Z_{Fish}=\frac{P_{Fish}\times{BCF}}{H}
  5. l o g B C F = 0.76 l o g K o w - 0.23 logBCF=0.76logKow-0.23
  6. l o g B C F = l o g K o w - 1.32 logBCF=logKow-1.32
  7. l o g B C F = 2.791 - 0.564 l o g S ( S = w a t e r s o l u b i l i t y ) logBCF=2.791-0.564logS(S=watersolubility)
  8. l o g B C F = 3.41 - 0.508 l o g S logBCF=3.41-0.508logS
  9. l o g B C F = 1.119 l o g K o c - 1.579 logBCF=1.119logKoc-1.579
  10. t e S S = 0.00654 K O W + 55.31 t_{eSS}=0.00654\cdot K_{OW}+55.31
  11. d C B d t = ( k 1 C W D ) - ( k 2 + k E + k M + k G ) C B \frac{dC_{B}}{dt}=(k_{1}C_{WD})-(k_{2}+k_{E}+k_{M}+k_{G})C_{B}

Biomaterial_Surface_Modifications.html

  1. γ S V - γ S L = γ L V c o s θ \gamma_{SV}-\gamma_{SL}=\gamma_{LV}cos\theta
  2. γ S V \gamma_{SV}
  3. γ S L \gamma_{SL}
  4. γ L V \gamma_{LV}
  5. S = γ S A + ( γ C A - γ S C ) S=\gamma_{SA}+(\gamma_{CA}-\gamma_{SC})
  6. γ S A \gamma_{SA}
  7. γ C A \gamma_{CA}
  8. γ S C \gamma_{SC}

Biotin-dependent_malonate_decarboxylase.html

  1. \rightleftharpoons

Biotin-independent_malonate_decarboxylase.html

  1. \rightleftharpoons

Biotransducer.html

  1. Δ V = ω P A r Δ T ( 1 + ω 2 τ E 2 ) - 1 / 2 \Delta V=\omega PAr\Delta T\left(1+\omega^{2}\tau_{E}^{2}\right)^{-1/2}
  2. τ E = r C \tau_{E}=rC

Bipartite_matroid.html

  1. U n r U{}^{r}_{n}
  2. r r
  3. r + 1 r+1
  4. U 6 4 U{}^{4}_{6}
  5. U 6 2 U{}^{2}_{6}
  6. U 6 3 U{}^{3}_{6}

Biregular_graph.html

  1. G = ( U , V , E ) G=(U,V,E)
  2. U U
  3. x x
  4. V V
  5. y y
  6. ( x , y ) (x,y)
  7. K a , b K_{a,b}
  8. ( b , a ) (b,a)
  9. ( x , y ) (x,y)
  10. G = ( U , V , E ) G=(U,V,E)
  11. x | U | = y | V | x|U|=y|V|
  12. U U
  13. x | U | x|U|
  14. V V
  15. y | V | y|V|

Birge–Sponer_method.html

  1. Δ G v + 1 2 = G ( v + 1 ) - G ( v ) \Delta G_{v+\frac{1}{2}}=G(v+1)-G(v)
  2. v m a x v_{max}
  3. D 0 = v = 0 v m a x Δ G v + 1 2 D_{0}=\sum_{v=0}^{v_{max}}\Delta G_{v+\frac{1}{2}}
  4. D 0 D_{0}
  5. Δ G v + 1 2 \Delta G_{v+\frac{1}{2}}
  6. v + 1 / 2 v+1/2
  7. v m a x v_{max}
  8. v v

Birks'_Law.html

  1. d L d x = L 0 d E d x 1 + k B d E d x . \frac{dL}{dx}=L_{0}\frac{\frac{dE}{dx}}{1+k_{B}\frac{dE}{dx}}.
  2. × 10 - 2 \times 10^{-2}

Bisection_(software_engineering).html

  1. O ( log N ) O(\log N)
  2. N N

Bivector_(complex).html

  1. x = x 1 + h x 2 , y = y 1 + h y 2 , z = z 1 + h z 2 , h 2 = - 1 = i 2 = j 2 = k 2 . x=x_{1}+hx_{2},\ y=y_{1}+hy_{2},\ z=z_{1}+hz_{2},\quad h^{2}=-1=i^{2}=j^{2}=k^% {2}.
  2. ( x 1 i + y 1 j + z 1 k ) + h ( x 2 i + y 2 j + z 2 k ) (x_{1}i+y_{1}j+z_{1}k)+h(x_{2}i+y_{2}j+z_{2}k)
  3. r 1 = x 1 i + y 1 j + z 1 k , r 2 = x 2 i + y 2 j + z 2 k ) r_{1}=x_{1}i+y_{1}j+z_{1}k,\quad r_{2}=x_{2}i+y_{2}j+z_{2}k)
  4. q = x i + y j + z k = r 1 + h r 2 . q=xi+yj+zk=r_{1}+hr_{2}.
  5. r 1 2 = - 1 = r 2 2 r_{1}^{2}=-1=r_{2}^{2}
  6. ( h v w + h x - w + h x - h v ) \begin{pmatrix}hv&w+hx\\ -w+hx&-hv\end{pmatrix}

Björling_problem.html

  1. c ( s ) c(s)
  2. c ( s ) 0 c^{\prime}(s)\neq 0
  3. n ( s ) n(s)
  4. || n ( t ) || = 1 ||n(t)||=1
  5. c ( t ) n ( t ) = 0 c^{\prime}(t)\cdot n(t)=0
  6. X ( u , v ) = ( c ( w ) - i w 0 w n ( w ) c ( w ) d w ) X(u,v)=\Re\left(c(w)-i\int_{w_{0}}^{w}n(w)\wedge c^{\prime}(w)\,dw\right)
  7. w = u + i v Ω w=u+iv\in\Omega
  8. u 0 I u_{0}\in I
  9. I Ω I\subset\Omega
  10. c ( s ) c(s)
  11. n ( s ) n(s)

Black's_approximation.html

  1. S 0 = $ 40 X = $ 40 σ = 30 % p . a . r = 10 % p . a . T = 6 m o n t h s = .5 y e a r s D = $ 0.70 \begin{aligned}\displaystyle S_{0}&\displaystyle=\$40\\ \displaystyle X&\displaystyle=\$40\\ \displaystyle\sigma&\displaystyle=30\%\;p.a.\\ \displaystyle r&\displaystyle=10\%\;p.a.\\ \displaystyle T&\displaystyle=6\;months=.5\;years\\ \displaystyle D&\displaystyle=\$0.70\\ \end{aligned}
  2. P V \displaystyle PV
  3. P V PV
  4. D 1 , 2 D_{1,2}
  5. r r
  6. Δ t 1 , 2 \Delta t_{1,2}
  7. m m
  8. Δ t \Delta t
  9. m m
  10. Δ t m \frac{\Delta t}{m}
  11. e e
  12. 0.7 e - ( .1 ) ( 3 12 ) + 0.7 e - ( .1 ) ( 5 12 ) = 1.3541 \displaystyle 0.7e^{-(.1)(\frac{3}{12})}+0.7e^{-(.1)(\frac{5}{12})}=1.3541
  13. S 0 S_{0}
  14. S 0 S_{0}^{\prime}
  15. S 0 = 40 - 1.3541 = 38.6459 S_{0}^{\prime}=40-1.3541=38.6459
  16. C \displaystyle C
  17. N ( ) N(\cdot)
  18. T T
  19. S 0 S_{0}
  20. X X
  21. r r
  22. σ \sigma
  23. d 1 \displaystyle d_{1}
  24. P V = D 1 e - ( r ) ( Δ t 1 m ) \begin{aligned}\displaystyle PV&\displaystyle=D_{1}e^{-(r)(\frac{\Delta t_{1}}% {m})}\end{aligned}
  25. T \displaystyle T
  26. P V \displaystyle PV
  27. $ 3.54 > $ 3.50 \$3.54>\$3.50
  28. $ 3.54 \$3.54

Blade_element_momentum_theory.html

  1. 1 2 ρ v 2 + P + ρ g h = const . \frac{1}{2}\rho v^{2}+P+\rho gh=\mathrm{const.}
  2. ρ \rho
  3. v v
  4. P P
  5. g g
  6. h h
  7. 1 2 ρ v 2 + P = const . \frac{1}{2}\rho v^{2}+P=\mathrm{const.}
  8. v 1 v_{1}
  9. P 1 P_{1}
  10. v 2 v_{2}
  11. P 2 P_{2}
  12. 1 2 ρ v 1 2 + P 1 = 1 2 ρ v 2 2 + P 2 \frac{1}{2}\rho v_{1}^{2}+P_{1}=\frac{1}{2}\rho v_{2}^{2}+P_{2}
  13. v v_{\infty}
  14. m ˙ \dot{m}
  15. A A
  16. m ˙ = ρ A v \dot{m}=\rho Av
  17. ρ \rho
  18. v v
  19. P P_{\infty}
  20. 1 2 ρ v 2 + P = 1 2 ρ ( v ( 1 - a ) ) 2 + P D + \frac{1}{2}\rho v_{\infty}^{2}+P_{\infty}=\frac{1}{2}\rho\left(v_{\infty}(1-a)% \right)^{2}+P_{D+}
  21. v ( 1 - a ) v_{\infty}(1-a)
  22. a a
  23. P D + P_{D+}
  24. v ( 1 - a ) v_{\infty}(1-a)
  25. P D - P_{D-}
  26. P P P\rightarrow P_{\infty}
  27. 1 2 ρ ( v ( 1 - a ) ) 2 + P D - = 1 2 ρ v w 2 + P \frac{1}{2}\rho\left(v_{\infty}(1-a)\right)^{2}+P_{D-}=\frac{1}{2}\rho v_{w}^{% 2}+P_{\infty}
  28. P D + - P D - = 1 2 ρ ( v 2 - v w 2 ) P_{D+}-P_{D-}=\frac{1}{2}\rho(v_{\infty}^{2}-v_{w}^{2})
  29. F = Δ P A F=\Delta PA
  30. 1 2 ρ ( v 2 - v w 2 ) A D \frac{1}{2}\rho(v_{\infty}^{2}-v_{w}^{2})A_{D}
  31. A D A_{D}
  32. F = d p d t = m ˙ ( v - v w ) = ρ A D v D ( v - v w ) = ρ A D ( 1 - a ) v ( v - v w ) F=\frac{\mathrm{d}p}{\mathrm{d}t}=\dot{m}(v_{\infty}-v_{w})=\rho A_{D}v_{D}(v_% {\infty}-v_{w})=\rho A_{D}(1-a)v_{\infty}(v_{\infty}-v_{w})
  33. v w = ( 1 - 2 a ) v v_{w}=(1-2a)v_{\infty}
  34. Power e x t = F v D = 2 a ( 1 - a ) 2 v 3 ρ A D \mathrm{Power}_{ext}=Fv_{D}=2a(1-a)^{2}v_{\infty}^{3}\rho A_{D}
  35. Power = 1 2 ρ A v 3 \mathrm{Power}=\frac{1}{2}\rho Av^{3}
  36. ρ \rho
  37. v v
  38. A A
  39. C p C_{p}
  40. Power e x t \mathrm{Power}_{ext}
  41. Power e x t = Power × C p \mathrm{Power}_{ext}=\mathrm{Power}\times C_{p}
  42. C p C_{p}
  43. Power e x t \mathrm{Power}_{ext}
  44. Power e x t \mathrm{Power}_{ext}
  45. a a
  46. dPower e x t d a = 2 v 3 ρ A D × ( ( 1 - a ) 2 - 2 a ( 1 - a ) ) \frac{\mathrm{d}\mathrm{Power}_{ext}}{\mathrm{d}a}=2v_{\infty}^{3}\rho A_{D}% \times\left((1-a)^{2}-2a(1-a)\right)
  47. a a
  48. C P m a x = 16 / 27 C_{P~{}max}=16/27
  49. a a
  50. a a^{\prime}
  51. 1 2 ρ v 1 2 + P 1 = 1 2 ρ v 2 2 + P 2 \frac{1}{2}\rho v_{1}^{2}+P_{1}=\frac{1}{2}\rho v_{2}^{2}+P_{2}
  52. ( r , θ , z ) (r,~{}\theta,~{}z)
  53. v 2 = v r 2 + v θ 2 + v z 2 v^{2}=v_{r}^{2}+v_{\theta}^{2}+v_{z}^{2}
  54. 𝐅 = F r 𝐫 ^ + F θ θ ^ + F z 𝐳 ^ \mathbf{F}=F_{r}\hat{\mathbf{r}}+F_{\theta}\hat{\theta}+F_{z}\hat{\mathbf{z}}
  55. P + 1 2 ρ v u 2 = P D + + 1 2 ρ v D 2 P_{\infty}+\frac{1}{2}\rho v_{u}^{2}=P_{D+}+\frac{1}{2}\rho v_{D}^{2}
  56. v u v_{u}
  57. v D v_{D}
  58. P + 1 2 ρ v 2 = P D + + 1 2 ρ ( v ( 1 - a ) ) 2 P_{\infty}+\frac{1}{2}\rho v_{\infty}^{2}=P_{D+}+\frac{1}{2}\rho(v_{\infty}(1-% a))^{2}
  59. v v_{\infty}
  60. v ( 1 - a ) v_{\infty}(1-a)
  61. v r 0 v_{r}\neq 0
  62. P D - + 1 2 ρ v D 2 = P + 1 2 ρ v w 2 P_{D-}+\frac{1}{2}\rho v_{D}^{2}=P_{\infty}+\frac{1}{2}\rho v_{w}^{2}
  63. v D v_{D}
  64. v D 2 = v D , r 2 + v D , θ 2 + v D , z 2 v_{D}^{2}=v_{D,~{}r}^{2}+v_{D,~{}\theta}^{2}+v_{D,~{}z}^{2}
  65. v D , z = ( 1 - a ) v v_{D,~{}z}=(1-a)v_{\infty}
  66. v θ v_{\theta}
  67. P D - + 1 2 ρ v D , z 2 = P + 1 2 ρ v w , z 2 = P D - + 1 2 ρ ( v ( 1 - a ) ) 2 P_{D-}+\frac{1}{2}\rho v_{D,~{}z}^{2}=P_{\infty}+\frac{1}{2}\rho v_{w,~{}z}^{2% }=P_{D-}+\frac{1}{2}\rho(v_{\infty}(1-a))^{2}
  68. v w , z = ( 1 - 2 a ) v v_{w,z}=(1-2a)v_{\infty}
  69. Power = 2 a ( 1 - a ) 2 v 3 ρ A D \mathrm{Power}=2a(1-a)^{2}v_{\infty}^{3}\rho A_{D}
  70. δ 𝐐 = 2 π r δ r × ρ U ( 1 - a ) × 2 a r ω \delta\mathbf{Q}=2\pi r\delta r\times\rho U_{\infty}(1-a)\times 2a^{\prime}r\omega
  71. v ( 1 - a ) v_{\infty}(1-a)
  72. a ω r a^{\prime}\omega r
  73. ω r \omega r
  74. 𝐯 = ω r ( 1 + a ) θ ^ + v ( 1 - a ) 𝐳 ^ \mathbf{v}=\omega r(1+a^{\prime})\hat{\mathbf{\theta}}+v_{\infty}(1-a)\hat{% \mathbf{z}}
  75. | 𝐯 | 2 = ( ω r ( 1 + a ) ) 2 + ( v ( 1 - a ) ) 2 = W 2 |\mathbf{v}|^{2}=(\omega r(1+a^{\prime}))^{2}+(v_{\infty}(1-a))^{2}=W^{2}
  76. ϕ \phi
  77. sin ϕ = v ( 1 - a ) W \sin\phi=\frac{v_{\infty}(1-a)}{W}
  78. β \beta
  79. α \alpha
  80. α = ϕ - β \alpha=\phi-\beta
  81. c L c_{L}
  82. c D c_{D}
  83. δ r \delta r
  84. c c
  85. δ L = 1 2 ρ N W 2 c × c L ( α ) δ r \delta L=\frac{1}{2}\rho NW^{2}c\times c_{L}(\alpha)\delta r
  86. c L c_{L}
  87. N N
  88. c c
  89. δ D = 1 2 ρ N W 2 c × c D ( α ) δ r \delta D=\frac{1}{2}\rho NW^{2}c\times c_{D}(\alpha)\delta r
  90. 𝐳 ^ \hat{\mathbf{z}}
  91. θ ^ \hat{\theta}
  92. δ F θ = δ L sin ϕ - δ D cos ϕ \delta F_{\theta}=\delta L\sin\phi-\delta D\cos\phi
  93. δ F z = δ L cos ϕ + δ D sin ϕ \delta F_{z}=\delta L\cos\phi+\delta D\sin\phi
  94. F θ F_{\theta}
  95. F z F_{z}
  96. v θ = 2 a ω r v_{\theta}=2a^{\prime}\omega r
  97. | δ 𝐐 | = ρ ( 2 π r δ r ) U ( 1 - a ) × ( 2 Ω a r 2 ) |\mathbf{\delta{Q}}|=\rho(2\pi r\delta r)U_{\infty}(1-a)\times(2\Omega a^{% \prime}r^{2})
  98. 1 2 ρ W 2 N c ( c l sin ϕ - c d cos ϕ ) r δ r = ρ ( 2 π r δ r ) U ( 1 - a ) × ( 2 Ω a r 2 ) \frac{1}{2}\rho W^{2}Nc(c_{l}\sin\phi-c_{d}\cos\phi)r\delta r=\rho(2\pi r% \delta r)U_{\infty}(1-a)\times(2\Omega a^{\prime}r^{2})
  99. 1 2 W 2 N c ( c l sin ϕ - c d cos ϕ ) = 4 π U ( 1 - a ) × Ω a r 2 \frac{1}{2}W^{2}Nc(c_{l}\sin\phi-c_{d}\cos\phi)=4\pi U_{\infty}(1-a)\times% \Omega a^{\prime}r^{2}
  100. δ F z \delta F_{z}
  101. δ F z = ρ ( 2 π r δ r ) U ( 1 - a ) × ( v - v w ) \delta F_{z}=\rho(2\pi r\delta r)U_{\infty}(1-a)\times(v_{\infty}-v_{w})
  102. δ F z = ρ ( 4 π r δ r ) U 2 a ( 1 - a ) \delta F_{z}=\rho(4\pi r\delta r)U^{2}_{\infty}a(1-a)
  103. 1 2 W 2 N c ( c l cos ϕ + c d sin ϕ ) = ρ ( 4 π r δ r ) U 2 a ( 1 - a ) \frac{1}{2}W^{2}Nc(c_{l}\cos\phi+c_{d}\sin\phi)=\rho(4\pi r\delta r)U^{2}_{% \infty}a(1-a)
  104. C y = c l sin ϕ - c d cos ϕ C_{y}=c_{l}\sin\phi-c_{d}\cos\phi
  105. C x = c l cos ϕ + c d sin ϕ C_{x}=c_{l}\cos\phi+c_{d}\sin\phi
  106. 1 2 W 2 N c C y = 4 π U ( 1 - a ) × Ω a r 2 \frac{1}{2}W^{2}NcC_{y}=4\pi U_{\infty}(1-a)\times\Omega a^{\prime}r^{2}
  107. 1 2 ρ W 2 N c C x = 4 π ρ [ ( a Ω r ) 2 + U 2 a ( 1 - a ) ] r \frac{1}{2}\rho W^{2}NcC_{x}=4\pi\rho\left[(a^{\prime}\Omega r)^{2}+U^{2}_{% \infty}a(1-a)\right]r
  108. sin ϕ = U W ( 1 - a ) sin 2 ϕ = ( U W ( 1 - a ) ) 2 \sin\phi=\frac{U_{\infty}}{W}(1-a)\rightarrow\sin^{2}\phi=\left(\frac{U_{% \infty}}{W}(1-a)\right)^{2}
  109. a 1 - a = C x σ r 4 sin 2 ϕ \frac{a}{1-a}=\frac{C_{x}\sigma_{r}}{4\sin^{2}\phi}
  110. a a^{\prime}

Blahut–Arimoto_algorithm.html

  1. X X
  2. p ( x ) p(x)
  3. p ( x ^ | x ) p(\hat{x}|x)
  4. X ^ \hat{X}
  5. d ( x , x ^ ) \langle d(x,\hat{x})\rangle
  6. X X
  7. X ^ \hat{X}
  8. p t + 1 ( x ^ ) = x p ( x ) p t ( x ^ | x ) p_{t+1}(\hat{x})=\sum_{x}p(x)p_{t}(\hat{x}|x)
  9. p t + 1 ( x ^ | x ) = p t ( x ^ ) exp ( - β d ( x , x ^ ) ) x ^ p t ( x ^ ) exp ( - β d ( x , x ^ ) ) p_{t+1}(\hat{x}|x)=\frac{p_{t}(\hat{x})\exp(-\beta d(x,\hat{x}))}{\sum_{\hat{x% }}p_{t}(\hat{x})\exp(-\beta d(x,\hat{x}))}
  10. β \beta
  11. β \beta

Blocked_rotor_test.html

  1. s = 1 s=1
  2. 1 / 4 1/4
  3. I S I_{S}
  4. V S V_{S}
  5. I S N I_{SN}
  6. V V
  7. I S N = I S × V V S I_{SN}=I_{S}\times\frac{V}{V_{S}}
  8. W S W_{S}
  9. V S L V_{SL}
  10. I S L I_{SL}
  11. c o s ϕ S cos\phi_{S}
  12. c o s ϕ S = W S 3 V S L I S L cos\phi_{S}=\frac{W_{S}}{{\sqrt{3}}{V_{SL}}{I_{SL}}}
  13. Z 01 Z_{01}
  14. X 01 X_{01}
  15. Z 01 = s h o r t c i r c u i t v o l t a g e p e r p h a s e short circuit current = V S I S Z_{01}=\frac{\text{}}{shortcircuitvoltageperphase}\text{short circuit current}% =\frac{V_{S}}{I_{S}}
  16. W c u W_{cu}
  17. W c W_{c}
  18. W c u = W S - W c W_{cu}=W_{S}-W_{c}
  19. W c u = 3 × I S 2 R 01 W_{cu}={3}\times{{I_{S}}^{2}{R_{01}}}
  20. R 01 = W c u 3 I S 2 R_{01}=\frac{W_{cu}}{3{I_{S}}^{2}}
  21. X 01 = Z 01 2 - R 01 2 X_{01}=\sqrt{{Z_{01}}^{2}-{R_{01}^{2}}}

Blocking_set.html

  1. H H
  2. τ ( H ) \tau(H)
  3. q + q + 1 | B | q 2 - q . q+\sqrt{q}+1\leq|B|\leq q^{2}-\sqrt{q}.
  4. n + n + 1 n+\sqrt{n}+1
  5. n n + 1 n\sqrt{n}+1
  6. 9 \leq 9
  7. H = ( X , E ) H=(X,E)
  8. X X
  9. E E
  10. X X
  11. H H
  12. S S
  13. X X
  14. H H
  15. T T
  16. X X
  17. H H
  18. { C , D } \{C,D\}
  19. X X
  20. C C
  21. D D
  22. C C
  23. D D
  24. b n + q . b\geq n+q.
  25. τ ( H ) \tau(H)

Blood_group_B_linear_chain_alpha-1,3-galactosidase.html

  1. \rightleftharpoons

Boggio's_formula.html

  1. ( - Δ ) m u ( x ) = f ( x ) (-\Delta)^{m}u(x)=f(x)
  2. ( - Δ ) (-\Delta)
  3. ( - Δ ) m G ( x , y ) = δ ( x - y ) (-\Delta)^{m}G(x,y)=\delta(x-y)
  4. δ \delta
  5. G m , n ( x , y ) = C m , n | x - y | 2 m - n 1 | | x | y - x | x | | | x - y | ( v 2 - 1 ) m - 1 v 1 - n d v G_{m,n}(x,y)=C_{m,n}|x-y|^{2m-n}\int_{1}^{\frac{\left||x|y-\frac{x}{|x|}\right% |}{|x-y|}}(v^{2}-1)^{m-1}v^{1-n}dv
  6. C m , n C_{m,n}
  7. C m , n = 1 n e n 4 m - 1 ( ( m - 1 ) ! ) 2 , C_{m,n}=\frac{1}{ne_{n}4^{m-1}((m-1)!)^{2}},
  8. e n = π n 2 Γ ( 1 + n 2 ) e_{n}=\frac{\pi^{\frac{n}{2}}}{\Gamma(1+\frac{n}{2})}

Bony–Brezis_theorem.html

  1. | X ( a ) - X ( b ) | C | a - b | . \displaystyle{|X(a)-X(b)|\leq C|a-b|.}
  2. D ( x + h ) = D ( x ) + min z 2 h ( x - z ) + o ( | h | ) , \displaystyle{D(x+h)=D(x)+\min_{z}2h\cdot(x-z)+o(|h|),}
  3. f ε ( h ) = min z 2 h ( x - z ) , \displaystyle{f_{\varepsilon}(h)=\min_{z}2h\cdot(x-z),}
  4. f 0 ( h ) f ε ( h ) f 0 ( h ) - C ( ε ) | h | , \displaystyle{f_{0}(h)\geq f_{\varepsilon}(h)\geq f_{0}(h)-C(\varepsilon)|h|,}
  5. D ( x + h ) | x + h - z | 2 | z - x | 2 + 2 h ( x - z ) + | h | 2 = D ( x ) + f 0 ( h ) + | h | 2 \displaystyle{D(x+h)\leq|x+h-z|^{2}\leq|z-x|^{2}+2h\cdot(x-z)+|h|^{2}=D(x)+f_{% 0}(h)+|h|^{2}}
  6. D ( x + h ) D ( x ) + f ε ( h ) + | h | 2 . \displaystyle{D(x+h)\geq D(x)+f_{\varepsilon}(h)+|h|^{2}.}
  7. lim δ 0 D ( c ( t + δ ) ) - D ( c ( t ) ) δ = 2 min z X ( c ( t ) ) ( c ( t ) - z ) , \displaystyle{\lim{\delta\downarrow 0}{D(c(t+\delta))-D(c(t))\over\delta}=2% \min_{z}X(c(t))\cdot(c(t)-z),}
  8. 2 X ( c ( t ) ) ( c ( t ) - z ) = 2 X ( z ) ( c ( t ) - z ) - 2 ( X ( z ) - X ( c ( t ) ) ) ( c ( t ) - z ) . \displaystyle{2X(c(t))\cdot(c(t)-z)=2X(z)\cdot(c(t)-z)-2(X(z)-X(c(t)))\cdot(c(% t)-z).}
  9. e - 2 C t D ( c ( t ) ) \displaystyle{e^{-2Ct}D(c(t))}

Boolean_hierarchy.html

  1. B H 2 k = c o N P B H 2 k - 1 BH_{2k}=coNP\wedge BH_{2k-1}
  2. B H 2 k + 1 = N P B H 2 k BH_{2k+1}=NP\vee BH_{2k}
  3. B H 2 k = i = 1 k D P BH_{2k}=\bigvee_{i=1}^{k}DP
  4. B H 2 k + 1 = N P i = 1 k D P BH_{2k+1}=NP\vee\bigvee_{i=1}^{k}DP
  5. B H k = D P B H k - 2 BH_{k}=DP\vee BH_{k-2}

Boom_method.html

  1. = 0.5 1 10 6 100 1 c m = 0.5 × 10 - 4 c m =0.5\frac{1}{10^{6}}\frac{100}{1}cm=0.5\times 10^{-4}cm
  2. V = 2 × 2 3 π r 3 = 4 3 π r 3 V=2\times\frac{2}{3}\pi r^{3}=\frac{4}{3}\pi{r}^{3}

Borel–de_Siebenthal_theory.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 = 𝔷 𝔤 1 𝔤 m , \displaystyle{\mathfrak{g}=\mathfrak{z}\oplus\mathfrak{g}_{1}\oplus\cdots% \oplus\mathfrak{g}_{m},}
  3. 𝔷 \mathfrak{z}
  4. 𝔤 i \mathfrak{g}_{i}
  5. 𝔱 \mathfrak{t}
  6. 𝔱 = 𝔷 𝔱 1 𝔱 m , \displaystyle{\mathfrak{t}=\mathfrak{z}\oplus\mathfrak{t}_{1}\oplus\cdots% \oplus\mathfrak{t}_{m},}
  7. 𝔱 i \mathfrak{t}_{i}
  8. 𝔤 i \mathfrak{g}_{i}
  9. 𝔥 \mathfrak{h}
  10. 𝔥 \mathfrak{h}
  11. 𝔱 \mathfrak{t}
  12. 𝔤 i \mathfrak{g}_{i}
  13. 𝔥 i = 𝔥 𝔤 i , \displaystyle{\mathfrak{h}_{i}=\mathfrak{h}\cap\mathfrak{g}_{i},}
  14. 𝔥 = 𝔷 𝔥 1 𝔥 m . \displaystyle{\mathfrak{h}=\mathfrak{z}\oplus\mathfrak{h}_{1}\oplus\cdots% \oplus\mathfrak{h}_{m}.}
  15. 𝔥 i \mathfrak{h}_{i}
  16. 𝔤 i \mathfrak{g}_{i}
  17. 𝔤 \mathfrak{g}
  18. 𝔱 \mathfrak{t}
  19. 𝔤 𝐂 \mathfrak{g}_{\mathbf{C}}
  20. α 0 = m 1 α 1 + + m n α n \displaystyle{\alpha_{0}=m_{1}\alpha_{1}+\cdots+m_{n}\alpha_{n}}
  21. A = { T 𝔱 : α 1 ( T ) 0 , , α n ( T ) 0 , α 0 ( T ) 1 } . \displaystyle{A=\{T\in\mathfrak{t}:\,\alpha_{1}(T)\geq 0,\dots,\alpha_{n}(T)% \geq 0,\alpha_{0}(T)\leq 1\}.}
  22. 𝔤 \mathfrak{g}
  23. v 0 = 0 , v i = m i - 1 X i , \displaystyle{v_{0}=0,\,\,v_{i}=m_{i}^{-1}X_{i},}
  24. 𝔤 \mathfrak{g}
  25. 𝔥 \mathfrak{h}
  26. 𝔥 \mathfrak{h}
  27. 𝔱 𝐂 α Δ 1 𝔤 α \displaystyle{\mathfrak{t}_{\mathbf{C}}\oplus\bigoplus_{\alpha\in\Delta_{1}}% \mathfrak{g}_{\alpha}}
  28. 𝔤 𝐂 \mathfrak{g}_{\mathbf{C}}
  29. 𝔱 𝐂 \mathfrak{t}_{\mathbf{C}}
  30. 𝔤 \mathfrak{g}
  31. 𝔤 = 𝔨 𝔭 , \displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p},}
  32. 𝔨 \mathfrak{k}
  33. 𝔭 \mathfrak{p}
  34. 𝔨 \mathfrak{k}
  35. 𝔤 \mathfrak{g}
  36. 𝔤 \mathfrak{g}
  37. 𝔤 \mathfrak{g}
  38. 𝔨 \mathfrak{k}
  39. 𝔭 \mathfrak{p}
  40. 𝔤 \mathfrak{g}
  41. 𝔨 \mathfrak{k}
  42. 𝔭 \mathfrak{p}
  43. 𝔨 \mathfrak{k}
  44. 𝔥 \mathfrak{h}
  45. 𝔭 1 \mathfrak{p}_{1}
  46. 𝔭 \mathfrak{p}
  47. 𝔥 = 𝔨 𝔭 1 , 𝔭 1 = 𝔥 𝔭 . \displaystyle{\mathfrak{h}=\mathfrak{k}\oplus\mathfrak{p}_{1},\,\,\,\ % \mathfrak{p}_{1}=\mathfrak{h}\cap\mathfrak{p}.}
  48. 𝔤 \mathfrak{g}
  49. 𝔤 \mathfrak{g}
  50. 𝔤 = i = 1 N 𝔤 i , \displaystyle{\mathfrak{g}=\oplus_{i=1}^{N}\mathfrak{g}_{i},}
  51. 𝔤 1 \mathfrak{g}_{1}
  52. 𝔨 \mathfrak{k}
  53. 𝔭 \mathfrak{p}
  54. 𝔤 1 \mathfrak{g}_{1}
  55. G / K = G 1 / K 1 × × G s / K s , \displaystyle{G/K=G_{1}/K_{1}\times\cdots\times G_{s}/K_{s},}
  56. Γ i = Z ( G i ) / Z ( G i ) K i \displaystyle{\Gamma_{i}=Z(G_{i})/Z(G_{i})\cap K_{i}}
  57. Δ = Δ 1 × × Δ s \displaystyle{\Delta=\Delta_{1}\times\cdots\times\Delta_{s}}
  58. 𝔤 \mathfrak{g}
  59. 𝔤 \mathfrak{g}
  60. 𝔭 \mathfrak{p}
  61. 𝔨 \mathfrak{k}
  62. 𝔭 \mathfrak{p}
  63. 𝔤 \mathfrak{g}
  64. ( [ [ A , B ] , C ] , D ) = ( [ A , B ] , [ C , D ] ) = ( [ [ C , D ] , B ] , A ) . \displaystyle{([[A,B],C],D)=([A,B],[C,D])=([[C,D],B],A).}
  65. [ J A , J B ] = [ A , B ] . \displaystyle{[JA,JB]=[A,B].}
  66. 𝔤 \mathfrak{g}
  67. 𝔨 \mathfrak{k}
  68. 𝔤 \mathfrak{g}
  69. 𝔤 \mathfrak{g}
  70. δ ( X ) = [ T + A , X ] , \displaystyle{\delta(X)=[T+A,X],}
  71. 𝔨 \mathfrak{k}
  72. 𝔭 \mathfrak{p}
  73. 𝔨 \mathfrak{k}
  74. 𝔨 \mathfrak{k}

Born_equation.html

  1. Δ G = - N A z 2 e 2 8 π ϵ 0 r 0 ( 1 - 1 ϵ r ) \Delta G=-\frac{N_{A}z^{2}e^{2}}{8\pi\epsilon_{0}r_{0}}\left(1-\frac{1}{% \epsilon_{r}}\right)
  2. × 10 19 \times 10^{−}19

Born_reciprocity.html

  1. 𝐗 = X μ := ( X 0 , X 1 , X 2 , X 3 ) = ( c t , x , y , z ) \mathbf{X}=X^{\mu}:=\left(X^{0},X^{1},X^{2},X^{3}\right)=\left(ct,x,y,z\right)
  2. 𝐏 = P ν := ( P 0 , P 1 , P 2 , P 3 ) = ( E , p x , p y , p z ) \mathbf{P}=P_{\nu}:=\left(P_{0},P_{1},P_{2},P_{3}\right)=\left(E,p_{x},p_{y},p% _{z}\right)
  3. x ˙ i = H / p i \dot{x}_{i}=\partial H/\partial p_{i}
  4. p ˙ i = - H / x i \dot{p}_{i}=-\partial H/\partial x_{i}
  5. x k x k + p k p k x_{k}x^{k}+p_{k}p^{k}

Botryococcene_C-methyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons
  5. \rightleftharpoons

Botryococcene_synthase.html

  1. \rightleftharpoons

Botryococcus_squalene_synthase.html

  1. \rightleftharpoons

Bott–Samelson_variety.html

  1. P 1 × B P 2 × B × × B P n / B P_{1}\times_{B}P_{2}\times_{B}\times\cdots\times_{B}P_{n}/B

Boundary_conditions_in_CFD.html

  1. y + > 11.63 y^{+}>11.63\,
  2. y + < 11.63 y^{+}<11.63\,
  3. U N I , J = U N I - 1 , J M i n M o u t U_{NI,J}=U_{NI-1,J}\frac{M_{in}}{M_{out}}\,

Boundary_knot_method.html

  1. L u = f ( x , y ) , ( x , y ) Ω Lu=f\left(x,y\right),\ \ \left(x,y\right)\in\Omega
  2. u = g ( x , y ) , ( x , y ) Ω D u=g\left(x,y\right),\ \ \left(x,y\right)\in\partial\Omega_{D}
  3. u n = h ( x , y ) , h ( x , y ) Ω N \frac{\partial u}{\partial n}=h\left(x,y\right),\ \ h\left(x,y\right)\in% \partial\Omega_{N}
  4. L L
  5. Ω \Omega
  6. Ω D \partial\Omega_{D}
  7. Ω N \partial\Omega_{N}
  8. Ω D Ω N = Ω \partial\Omega_{D}\cup\partial\Omega_{N}=\partial\Omega
  9. Ω D Ω N = \partial\Omega_{D}\cap\partial\Omega_{N}=\varnothing
  10. L L
  11. u * ( x , y ) = i = 1 N α i ϕ ( r i ) u^{*}\left(x,y\right)=\sum\limits_{i=1}^{N}\alpha_{i}\phi\left(r_{i}\right)
  12. r i = ( x , y ) - ( x i , y i ) 2 r_{i}=\left\|\left(x,y\right)-\left(x_{i},y_{i}\right)\right\|_{2}
  13. ϕ ( ) \phi\left(\cdot\right)
  14. L ϕ = 0 L\phi=0
  15. g ( x k , y k ) = i = 1 N α i ϕ ( r i ) , k = 1 , , m 1 h ( x k , y k ) = i = 1 N α i ϕ ( r i ) n , k = m 1 + 1 , , m \begin{aligned}&\displaystyle g\left(x_{k},y_{k}\right)=\sum\limits_{i=1}^{N}% \alpha_{i}\phi\left(r_{i}\right),\qquad k=1,\ldots,m_{1}\\ &\displaystyle h\left(x_{k},y_{k}\right)=\sum\limits_{i=1}^{N}\alpha_{i}\frac{% \partial\phi\left(r_{i}\right)}{\partial n},\qquad k=m_{1}+1,\ldots,m\\ \end{aligned}
  16. ( x k , y k ) | k = 1 m 1 \left(x_{k},y_{k}\right)|_{k=1}^{m_{1}}
  17. ( x k , y k ) | k = m 1 + 1 m \left(x_{k},y_{k}\right)|_{k=m_{1}+1}^{m}
  18. α i \alpha_{i}

Bounding_point.html

  1. A X A\subset X
  2. X X
  3. x X x\in X
  4. A A
  5. A A

Bour's_minimal_surface.html

  1. x ( r , θ ) = r cos ( θ ) - ( 1 / 2 ) r 2 cos ( 2 θ ) x(r,\theta)=r\cos(\theta)-(1/2)r^{2}\cos(2\theta)
  2. y ( r , θ ) = - r sin ( θ ) ( r cos ( θ ) + 1 ) y(r,\theta)=-r\sin(\theta)(r\cos(\theta)+1)
  3. z ( r , θ ) = ( 4 / 3 ) r 3 / 2 cos ( 3 θ / 2 ) . z(r,\theta)=(4/3)r^{3/2}\cos(3\theta/2).
  4. f ( z ) = 1 , g ( z ) = z f(z)=1,g(z)=\sqrt{z}

Box_spline.html

  1. d \mathbb{R}^{d}\to\mathbb{R}
  2. ξ d \xi\in\mathbb{R}^{d}
  3. 𝚵 := [ ξ 1 ξ N ] \mathbf{\Xi}:=\left[\xi_{1}\dots\xi_{N}\right]
  4. N = d N=d
  5. 𝚵 \mathbf{\Xi}
  6. M 𝚵 ( 𝐱 ) := 1 det Ξ χ 𝚵 ( 𝐱 ) = { 1 det Ξ 𝐱 = n = 1 d t n ξ n for some 0 t n < 1 0 otherwise . M_{\mathbf{\Xi}}(\mathbf{x}):=\frac{1}{\mid{\det{\Xi}}\mid}\chi_{\mathbf{\Xi}}% (\mathbf{x})=\begin{cases}\frac{1}{\mid{\det{\Xi}}\mid}&\mathbf{x}=\sum_{n=1}^% {d}{t_{n}\xi_{n}}\,\text{ for some }0\leq t_{n}<1\\ 0&\,\text{otherwise}\end{cases}.
  7. ξ \xi
  8. 𝚵 \mathbf{\Xi}
  9. N > d N>d
  10. M 𝚵 ξ ( 𝐱 ) = 0 1 M 𝚵 ( 𝐱 - t ξ ) d t M_{\mathbf{\Xi}\cup\xi}(\mathbf{x})=\int_{0}^{1}{M_{\mathbf{\Xi}}(\mathbf{x}-t% \xi)\,{\rm d}t}
  11. M 𝚵 M_{\mathbf{\Xi}}
  12. N \mathbb{R}^{N}
  13. d \mathbb{R}^{d}
  14. ξ 𝚵 \xi\in\mathbf{\Xi}
  15. N \mathbb{R}^{N}
  16. d \mathbb{R}^{d}
  17. t ξ t\xi
  18. 0 t < 1 0\leq t<1
  19. M 𝚵 = M ξ 1 M ξ 2 M ξ N . M_{\mathbf{\Xi}}=M_{\xi_{1}}\ast M_{\xi_{2}}\dots\ast M_{\xi_{N}}.
  20. κ \kappa
  21. Ξ \Xi
  22. d \mathbb{R}^{d}
  23. κ - 2 \kappa-2
  24. M 𝚵 C κ - 2 ( d ) M_{\mathbf{\Xi}}\in C^{\kappa-2}(\mathbb{R}^{d})
  25. N d N\geq d
  26. Ξ \Xi
  27. d \mathbb{R}^{d}
  28. d \mathbb{R}^{d}
  29. ξ 𝚵 {\xi}\in\mathbf{\Xi}
  30. 𝐜 Ξ := 1 2 n = 1 N ξ n . \mathbf{c}_{\Xi}:=\frac{1}{2}\sum_{n=1}^{N}\xi_{n}.
  31. d d
  32. M ^ Ξ ( ω ) = exp ( - j 𝐜 Ξ ω ) n = 1 N sinc ( ξ n ω ) . \hat{M}_{\Xi}(\omega)=\exp{(-j\mathbf{c}_{\Xi}\cdot\omega)}\prod_{n=1}^{N}{{% \rm sinc}(\xi_{n}\cdot\omega)}.

Bracket_ring.html

  1. ( n d ) {\left({{n}\atop{d}}\right)}

Brahmagupta's_interpolation_formula.html

  1. f ( x ) f(x)
  2. f ( a ) f(a)
  3. x r < a < x r + 1 x_{r}<a<x_{r+1}
  4. h h
  5. D r = f r + 1 - f r D_{r}=f_{r+1}-f_{r}
  6. f ( a ) f(a)
  7. f ( a ) = f r + t D r f(a)=f_{r}+tD_{r}
  8. t = ( a - x r ) / h t=(a-x_{r})/h
  9. f ( a ) f(a)
  10. D r D_{r}
  11. D r D_{r}
  12. D r + 1 D_{r+1}
  13. a - x r a-x_{r}
  14. f r + 1 - f r f_{r+1}-f_{r}
  15. h h
  16. = D r + D r + 1 2 ± t | D r - D r + 1 | 2 . \displaystyle=\frac{D_{r}+D_{r+1}}{2}\pm t\frac{|D_{r}-D_{r+1}|}{2}.
  17. = D r + D r + 1 2 + t D r + 1 - D r 2 . \displaystyle=\frac{D_{r}+D_{r+1}}{2}+t\frac{D_{r+1}-D_{r}}{2}.
  18. f ( a ) \displaystyle f(a)

Brain_connectivity_estimators.html

  1. X ( t ) = j = 1 p A ( j ) X ( t - j ) + E ( t ) X(t)=\sum_{j=1}^{p}A(j)X(t-j)+E(t)
  2. E ( f ) = A ( f ) X ( f ) X ( f ) = A - 1 ( f ) E ( f ) = H ( f ) E ( f ) \begin{array}[]{l}E(f)=A(f)X(f)\\ X(f)=A^{-1}(f)E(f)=H(f)E(f)\end{array}
  3. GCI i j ( t ) = ln ( V i , n ( t ) V i , n - 1 ( t ) ) \mathrm{GCI}_{i\rightarrow j}(t)=\ln\left(\frac{V_{i,n}(t)}{V_{i,n-1}(t)}\right)
  4. DTF j i 2 ( f ) = | H i j ( f ) | 2 m = 1 k | H i m ( f ) | 2 \mathrm{DTF}^{2}_{j\rightarrow i}(f)=\frac{\left|H_{ij}(f)\right|^{2}}{\sum_{m% =1}^{k}\left|H_{im}(f)\right|^{2}}
  5. NDTF j i 2 ( f ) = | H i j ( f ) | 2 \mathrm{NDTF}^{2}_{j\rightarrow i}(f)=\left|H_{ij}(f)\right|^{2}
  6. dDTF j i 2 ( f ) = F i j 2 ( f ) C i j 2 ( f ) F i j 2 ( f ) = | H i j ( f ) | 2 f m = 1 k | H i m ( f ) | 2 \begin{array}[]{l}\mathrm{dDTF}^{2}_{j\rightarrow i}(f)=F_{ij}^{2}(f)C_{ij}^{2% }(f)\\ F^{2}_{ij}(f)=\displaystyle\frac{\left|H_{ij}(f)\right|^{2}}{\sum_{f}\sum_{m=1% }^{k}\left|H_{im}(f)\right|^{2}}\end{array}
  7. P i j ( f ) = A i j ( f ) 𝐚 j * ( f ) 𝐚 j ( f ) P_{ij}(f)=\frac{A_{ij}(f)}{\sqrt{\mathbf{a}^{*}_{j}(f)\mathbf{a}_{j}(f)}}
  8. R ~ i j ( s ) = 1 N T r = 1 N T R i j ( r ) ( s ) = 1 N T r = 1 N T 1 N S t = 1 N S X i ( r ) ( t ) X j ( r ) ( t + s ) \tilde{R}_{ij}(s)=\frac{1}{N_{T}}\sum_{r=1}^{N_{T}}R_{ij}^{(r)}(s)=\frac{1}{N_% {T}}\sum_{r=1}^{N_{T}}\frac{1}{N_{S}}\sum_{t=1}^{N_{S}}X_{i}^{(r)}(t)X_{j}^{(r% )}(t+s)

Bramble_(graph_theory).html

  1. Ω ( k 1 / 2 / log 2 k ) \Omega(k^{1/2}/\log^{2}k)
  2. Ω ( k 1 / 2 / log 3 k ) \Omega(k^{1/2}/\log^{3}k)

Brewer_sum.html

  1. Λ n ( a ) = x mod p ( D n + 1 ( x , a ) p ) \Lambda_{n}(a)=\sum_{x\bmod p}{\left({{D_{n+1}(x,a)}\atop{p}}\right)}
  2. D 0 ( x , a ) = 2 , D 1 ( x , a ) = x , D n + 1 ( x , a ) = x D n ( x , a ) - a D n - 1 ( x , a ) D_{0}(x,a)=2,\quad D_{1}(x,a)=x,\quad D_{n+1}(x,a)=xD_{n}(x,a)-aD_{n-1}(x,a)

Bring's_curve.html

  1. v + w + x + y + z = v 2 + w 2 + x 2 + y 2 + z 2 = v 3 + w 3 + x 3 + y 3 + z 3 = 0. v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0.

Bromide_peroxidase.html

  1. \rightleftharpoons

Brownian_meander.html

  1. W + = { W t + , t [ 0 , 1 ] } W^{+}=\{W_{t}^{+},t\in[0,1]\}
  2. W = { W t , t 0 } W=\{W_{t},t\geq 0\}
  3. τ := sup { t [ 0 , 1 ] : W t = 0 } \tau:=\sup\{t\in[0,1]:W_{t}=0\}
  4. W W
  5. { 0 } \{0\}
  6. W t + := 1 1 - τ | W τ + t ( 1 - τ ) | , t [ 0 , 1 ] . W^{+}_{t}:=\frac{1}{\sqrt{1-\tau}}|W_{\tau+t(1-\tau)}|,\quad t\in[0,1].
  7. p ( s , x , t , y ) d y := P ( W t + d y W s + = x ) p(s,x,t,y)\,dy:=P(W^{+}_{t}\in dy\mid W^{+}_{s}=x)
  8. 0 < s < t 1 0<s<t\leq 1
  9. x , y > 0 x,y>0
  10. ϕ t ( x ) := exp { - x 2 / ( 2 t ) } 2 π t and Φ t ( x , y ) := x y ϕ t ( w ) d w , \phi_{t}(x):=\frac{\exp\{-x^{2}/(2t)\}}{\sqrt{2\pi t}}\quad\,\text{and}\quad% \Phi_{t}(x,y):=\int^{y}_{x}\phi_{t}(w)\,dw,
  11. p ( s , x , t , y ) d y : = P ( W t + d y W s + = x ) = ( ϕ t - s ( y - x ) - ϕ t - s ( y + x ) ) Φ 1 - t ( 0 , y ) Φ 1 - s ( 0 , x ) d y \begin{aligned}\displaystyle p(s,x,t,y)\,dy&\displaystyle:=P(W^{+}_{t}\in dy% \mid W^{+}_{s}=x)\\ &\displaystyle=\bigl(\phi_{t-s}(y-x)-\phi_{t-s}(y+x)\bigl)\frac{\Phi_{1-t}(0,y% )}{\Phi_{1-s}(0,x)}\,dy\end{aligned}
  12. p ( 0 , 0 , t , y ) d y := P ( W t + d y ) = 2 2 π y t ϕ t ( y ) Φ 1 - t ( 0 , y ) d y . p(0,0,t,y)\,dy:=P(W^{+}_{t}\in dy)=2\sqrt{2\pi}\frac{y}{t}\phi_{t}(y)\Phi_{1-t% }(0,y)\,dy.
  13. P ( W 1 + d y ) = y exp { - y 2 / 2 } d y , y > 0 , P(W^{+}_{1}\in dy)=y\exp\{-y^{2}/2\}\,dy,\quad y>0,
  14. W 1 + W^{+}_{1}
  15. 2 𝐞 \sqrt{2\mathbf{e}}
  16. 𝐞 \mathbf{e}

Bunce–Deddens_algebra.html

  1. B ( { n k } ) = lim M n k ( C ( 𝕋 ) ) β k M n k + 1 ( C ( 𝕋 ) ) . B(\{n_{k}\})=\underrightarrow{\lim}\cdots\rightarrow M_{n_{k}}(C(\mathbb{T}))% \;\stackrel{\beta_{k}}{\rightarrow}\;M_{n_{k+1}}(C(\mathbb{T}))\rightarrow\cdots.
  2. β k : M n k ( C ( 𝕋 ) ) M n k + 1 ( C ( 𝕋 ) ) . \beta_{k}:M_{n_{k}}(C(\mathbb{T}))\;\rightarrow\;M_{n_{k+1}}(C(\mathbb{T})).
  3. T = [ 0 T z 1 2 I 1 2 I 0 ] , T=\begin{bmatrix}0&&\cdots&T_{z}\\ \frac{1}{2}I&\ddots&\ddots&\\ &\ddots&\ddots&\vdots\\ &&\frac{1}{2}I&0\end{bmatrix},
  4. 0 𝒦 C * ( T z ) C ( 𝕋 ) 0 , 0\rightarrow\mathcal{K}\;{\rightarrow}\;C^{*}(T_{z})\;{\rightarrow}\;C(\mathbb% {T})\rightarrow 0,
  5. 0 M n ( 𝒦 ) i M n ( C * ( T z ) ) j M n ( C ( 𝕋 ) ) 0 , 0\rightarrow M_{n}(\mathcal{K})\;\stackrel{i}{\hookrightarrow}\;M_{n}(C^{*}(T_% {z}))\;\stackrel{j}{\rightarrow}\;M_{n}(C(\mathbb{T}))\rightarrow 0,
  6. T ~ = [ 0 z 1 2 1 2 0 ] , \tilde{T}=\begin{bmatrix}0&&\cdots&z\\ \frac{1}{2}&\ddots&\ddots&\\ &\ddots&\ddots&\vdots\\ &&\frac{1}{2}&0\end{bmatrix},
  7. T [ 0 T z 1 2 I 0 1 2 I 0 I 0 1 2 I 1 2 I 0 ] . T\mapsto\begin{bmatrix}0&&&&&&&T_{z}\\ \frac{1}{2}I&\ddots&&&&&&0\\ &\ddots&\ddots&&&&&\vdots\\ &&\frac{1}{2}I&0&&&&\\ &&&I&0&&&\\ &&&&\frac{1}{2}I&\ddots&&\\ &&&&&\ddots&\ddots&\vdots\\ &&&&&&\frac{1}{2}I&0\end{bmatrix}.
  8. β k ( T ~ ) = [ 0 z 1 2 0 1 2 0 1 0 1 2 1 2 0 ] . \beta_{k}(\tilde{T})=\begin{bmatrix}0&&&&&&&z\\ \frac{1}{2}&\ddots&&&&&&0\\ &\ddots&\ddots&&&&&\vdots\\ &&\frac{1}{2}&0&&&&\\ &&&1&0&&&\\ &&&&\frac{1}{2}&\ddots&&\\ &&&&&\ddots&\ddots&\vdots\\ &&&&&&\frac{1}{2}&0\end{bmatrix}.
  9. β k ( E i j ) = E i j I 2 \beta_{k}(E_{ij})=E_{ij}\otimes I_{2}
  10. β k ( z E 11 ) = E 11 \Zeta 2 , \beta_{k}(zE_{11})=E_{11}\otimes\Zeta_{2},
  11. \Zeta 2 = [ 0 z 1 0 ] M 2 ( C ( 𝕋 ) ) . \Zeta_{2}=\begin{bmatrix}0&z\\ 1&0\end{bmatrix}\in M_{2}(C(\mathbb{T})).
  12. β k ( f i j ( z ) ) = f i j ( \Zeta 2 ) . \beta_{k}(f_{ij}(z))=f_{ij}(\Zeta_{2}).\;
  13. \mapsto
  14. U t π ( a ) U t * = π ( σ ( t ) ( a ) ) , U_{t}\pi(a)U_{t}^{*}=\pi(\sigma(t)(a)),
  15. A σ G , A\rtimes_{\sigma}G,
  16. A σ G . A\rtimes_{\sigma}G.
  17. X = { 0 , 1 } , X=\prod\{0,1\},
  18. α : X X \alpha:X\rightarrow X
  19. α ( x ) = x + ( , 0 , 0 , 1 ) \alpha(x)=x+(\cdots,0,0,1)

Burst_error-correcting_code.html

  1. l \textstyle l
  2. C \textstyle C
  3. Y = C + E \textstyle Y=C+E
  4. E \textstyle E
  5. l \textstyle l
  6. E \textstyle E
  7. l \textstyle l
  8. E = ( 0 𝟏𝟎𝟎𝟎𝟎𝟏𝟏 0 ) \textstyle E=(0\,\textbf{1000011}0)
  9. l = 7 \textstyle l=7
  10. l \textstyle l
  11. E \textstyle E
  12. l \textstyle l
  13. l \textstyle l
  14. E = ( 010000110 ) \textstyle E=(010000110)
  15. l = 5 \textstyle l=5
  16. 6 \textstyle 6
  17. 1 \textstyle 1
  18. 0 \textstyle 0
  19. 0 \textstyle 0
  20. ( P , L ) \textstyle(P,L)
  21. P \textstyle P
  22. L \textstyle L
  23. E = ( 010000110 ) \textstyle E=(010000110)
  24. D = ( 1000011 , 1 ) \textstyle D=(1000011,1)
  25. D = ( 11001 , 6 ) \textstyle D^{\prime}=(11001,6)
  26. E \textstyle E
  27. w \textstyle w
  28. E \textstyle E
  29. w \textstyle w
  30. E \textstyle E
  31. E \textstyle E
  32. n \textstyle n
  33. ( P 1 , L 1 ) \textstyle(P_{1},L_{1})
  34. ( P 2 , L 2 ) \textstyle(P_{2},L_{2})
  35. length ( P 1 ) + length ( P 2 ) n + 1 \textstyle\mathrm{length}(P_{1})+\mathrm{length}(P_{2})\leq n+1
  36. length ( y ) \textstyle\mathrm{length}(y)
  37. y \textstyle y
  38. w \textstyle w
  39. E \textstyle E
  40. E \textstyle E
  41. w \textstyle w
  42. w = 0 \textstyle w=0
  43. w = 1 \textstyle w=1
  44. w 2 \textstyle w\geq 2
  45. E \textstyle E
  46. E \textstyle E
  47. E = ( 010000110 ) \textstyle E=(010000110)
  48. w \textstyle w
  49. n - w \textstyle n-w
  50. ( n - length ( P 1 ) ) + ( n - length ( P 2 ) ) \textstyle(n-\mathrm{length}(P_{1}))+(n-\mathrm{length}(P_{2}))
  51. E \textstyle E
  52. length ( P 1 ) + length ( P 2 ) n + 1 \textstyle\mathrm{length}(P_{1})+\mathrm{length}(P_{2})\leq n+1
  53. n - 1 \textstyle\geq n-1
  54. w 2 \textstyle w\geq 2
  55. ( n + 1 ) / 2 \textstyle(n+1)/2
  56. q \textstyle q
  57. 𝔽 q \textstyle\mathbb{F}_{q}
  58. 𝔽 q \textstyle\mathbb{F}_{q}
  59. n - 1 \textstyle\leq n-1
  60. g ( x ) \textstyle g(x)
  61. r \textstyle r
  62. n - 1 \textstyle\leq n-1
  63. g ( x ) \textstyle g(x)
  64. g ( x ) \textstyle g(x)
  65. n - 1 - r \textstyle\leq n-1-r
  66. q n - r \textstyle q^{n-r}
  67. k = n - r \textstyle k=n-r
  68. l = n - k = r \textstyle l=n-k=r
  69. ( n , k ) \textstyle(n,k)
  70. l n - k \textstyle l\leq n-k
  71. r \textstyle r
  72. r \textstyle\leq r
  73. r \textstyle\leq r
  74. g ( x ) \textstyle g(x)
  75. g ( x ) \textstyle g(x)
  76. r \textstyle\leq r
  77. g ( x ) \textstyle g(x)
  78. x i b ( x ) \textstyle x^{i}b(x)
  79. b ( x ) \textstyle b(x)
  80. < r \textstyle<r
  81. b ( x ) \textstyle b(x)
  82. g ( x ) \textstyle g(x)
  83. r \textstyle r
  84. g ( x ) \textstyle g(x)
  85. x \textstyle x
  86. 0 \textstyle 0
  87. x i \textstyle x^{i}
  88. g ( x ) \textstyle g(x)
  89. n - 1 \textstyle\leq n-1
  90. g ( x ) \textstyle g(x)
  91. g ( x ) \textstyle g(x)
  92. g ( x ) \textstyle g(x)
  93. l n - k = r \textstyle l\leq n-k=r
  94. l > r \textstyle l>r
  95. > r \textstyle>r
  96. g ( x ) \textstyle g(x)
  97. 2 l - 2 \textstyle 2^{l-2}
  98. l \textstyle l
  99. 2 l - 2 - r \textstyle 2^{l-2-r}
  100. g ( x ) \textstyle g(x)
  101. 2 - r \textstyle 2^{-r}
  102. l \textstyle l
  103. C \textstyle C
  104. l \textstyle l
  105. l \textstyle l
  106. C \textstyle C
  107. 𝐞 1 \textstyle\mathbf{e}_{1}
  108. 𝐞 2 \textstyle\mathbf{e}_{2}
  109. l \textstyle l
  110. C \textstyle C
  111. 𝐞 1 \textstyle\mathbf{e}_{1}
  112. 𝐞 2 \textstyle\mathbf{e}_{2}
  113. 𝐜 \textstyle\mathbf{c}
  114. ( 𝐜 = 𝐞 1 - 𝐞 2 ) \textstyle(\mathbf{c}=\mathbf{e}_{1}-\mathbf{e}_{2})
  115. 𝐜 \textstyle\mathbf{c}
  116. 𝐞 1 \textstyle\mathbf{e}_{1}
  117. 𝟎 \textstyle\mathbf{0}
  118. 𝐜 \textstyle\mathbf{c}
  119. 𝐞 1 \textstyle\mathbf{e}_{1}
  120. 𝐞 2 \textstyle\mathbf{e}_{2}
  121. C \textstyle C
  122. l \textstyle l
  123. l \textstyle l
  124. C \textstyle C
  125. C \textstyle C
  126. [ n , k ] \textstyle[n,k]
  127. l \textstyle l
  128. 2 l \textstyle 2l
  129. c \textstyle c
  130. 2 l \textstyle 2l
  131. c \textstyle c
  132. ( 0 , 1 , u , v , 1 , 0 ) \textstyle(0,1,u,v,1,0)
  133. u u
  134. v v
  135. l \textstyle\leq l
  136. w = ( 0 , 1 , u , 0 , 0 , 0 ) \textstyle w=(0,1,u,0,0,0)
  137. c \textstyle c
  138. - \textstyle-
  139. w = ( 0 , 0 , 0 , v , 1 , 0 ) \textstyle w=(0,0,0,v,1,0)
  140. l \textstyle\leq l
  141. 2 l \textstyle 2l
  142. ( n , k ) \textstyle(n,k)
  143. l \textstyle\leq l
  144. n \textstyle n
  145. k \textstyle k
  146. l \textstyle l
  147. l \textstyle l
  148. ( n , k ) \textstyle(n,k)
  149. ( n , k ) \textstyle(n,k)
  150. l n - k \textstyle l\leq n-k
  151. l \textstyle\leq l
  152. l \textstyle\leq l
  153. 𝐜 1 \textstyle\mathbf{c}_{1}
  154. 𝐜 2 \textstyle\mathbf{c}_{2}
  155. 𝐛 \textstyle\mathbf{b}
  156. l \textstyle\leq l
  157. 𝐜 1 \textstyle\mathbf{c}_{1}
  158. 𝐜 1 \textstyle\mathbf{c}_{1}
  159. 𝐜 2 \textstyle\mathbf{c}_{2}
  160. 𝐛 \textstyle\mathbf{b}
  161. l \textstyle l
  162. 𝐜 1 \textstyle\mathbf{c}_{1}
  163. 𝐛 \textstyle\mathbf{b}
  164. l \textstyle l
  165. 𝐜 1 \textstyle\mathbf{c}_{1}
  166. 𝐛 \textstyle\mathbf{b}
  167. n - l \textstyle n-l
  168. l \textstyle l
  169. l \textstyle l
  170. q k \textstyle q^{k}
  171. q k q n - l \textstyle q^{k}\leq q^{n-l}
  172. q \textstyle q
  173. l n - k \textstyle l\leq n-k
  174. n \textstyle n
  175. k \textstyle k
  176. l \textstyle l
  177. ( n , k ) \textstyle(n,k)
  178. ( n , k ) \textstyle(n,k)
  179. l n - k - log q ( n - l ) + 2 \textstyle l\leq n-k-\log_{q}(n-l)+2
  180. l \textstyle\leq l
  181. l \textstyle\leq l
  182. 𝐜 1 \textstyle\mathbf{c}_{1}
  183. 𝐜 2 \textstyle\mathbf{c}_{2}
  184. 𝐛 1 \textstyle\mathbf{b}_{1}
  185. 𝐛 2 \textstyle\mathbf{b}_{2}
  186. l \textstyle\leq l
  187. 𝐜 1 \textstyle\mathbf{c}_{1}
  188. 𝐛 1 \textstyle\mathbf{b}_{1}
  189. 𝐜 2 \textstyle\mathbf{c}_{2}
  190. - 𝐛 2 \textstyle-\mathbf{b}_{2}
  191. 𝐜 1 \textstyle\mathbf{c}_{1}
  192. 𝐜 2 \textstyle\mathbf{c}_{2}
  193. l \textstyle l
  194. 𝐜 1 \textstyle\mathbf{c}_{1}
  195. l \textstyle l
  196. 𝐜 \textstyle\mathbf{c}
  197. B ( 𝐜 ) \textstyle B(\mathbf{c})
  198. 𝐜 \textstyle\mathbf{c}
  199. l \textstyle\leq l
  200. B ( 𝐜 ) \textstyle B(\mathbf{c})
  201. 𝐜 \textstyle\mathbf{c}
  202. 𝐜 i \textstyle\mathbf{c}_{i}
  203. 𝐜 j \textstyle\mathbf{c}_{j}
  204. B ( 𝐜 i ) \textstyle B(\mathbf{c}_{i})
  205. B ( 𝐜 j ) \textstyle B(\mathbf{c}_{j})
  206. q k \textstyle q^{k}
  207. q k | B ( 𝐜 ) | q n \textstyle q^{k}|B(\mathbf{c})|\leq q^{n}
  208. ( n - l ) q l - 2 | B ( 𝐜 ) | \textstyle(n-l)q^{l-2}\leq|B(\mathbf{c})|
  209. q \textstyle q
  210. l \textstyle l
  211. [ n , k ] \textstyle[n,k]
  212. 2 l n - k 2l\leq n-k
  213. l \textstyle l
  214. 2 l \textstyle 2l
  215. 2 l \textstyle 2l
  216. l \textstyle l
  217. l \textstyle l
  218. l \textstyle l
  219. 2 l \textstyle 2l
  220. 2 l \textstyle 2l
  221. q 2 l q^{2l}
  222. 2 l \textstyle 2l
  223. l \textstyle\leq l
  224. 1 l ( n + 1 ) / 2 \textstyle 1\leq l\leq(n+1)/2
  225. n 2 l - 1 + 1 \textstyle n2^{l-1}+1
  226. n \textstyle n
  227. l \textstyle\leq l
  228. ( n + 1 ) / 2 \textstyle\leq(n+1)/2
  229. n \textstyle n
  230. 1 \textstyle 1
  231. l \textstyle l
  232. 1 \textstyle 1
  233. l \textstyle l
  234. 2 l - 1 \textstyle 2^{l-1}
  235. n 2 l - 1 \textstyle n2^{l-1}
  236. l \textstyle\leq l
  237. n 2 l - 1 + 1 \textstyle n2^{l-1}+1
  238. l \textstyle\leq l
  239. 1 l ( n + 1 ) / 2 \textstyle 1\leq l\leq(n+1)/2
  240. l \textstyle l
  241. 2 n / ( n 2 l - 1 + 1 ) \textstyle 2^{n}/(n2^{l-1}+1)
  242. l ( n + 1 ) / 2 \textstyle l\leq(n+1)/2
  243. n 2 l - 1 + 1 \textstyle n2^{l-1}+1
  244. l \textstyle\leq l
  245. M \textstyle M
  246. M n 2 l - 1 \textstyle Mn2^{l-1}
  247. l \textstyle\leq l
  248. M \textstyle M
  249. < 1 \textstyle<1
  250. M ( 2 l - 1 + 1 ) 2 n \textstyle M(2^{l-1}+1)\leq 2^{n}
  251. M ( 2 n ) / ( n 2 l - 1 + 1 ) M\leq(2^{n})/(n2^{l-1}+1)
  252. 1 b ( n + 1 ) / 2 \textstyle 1\leq b\leq(n+1)/2
  253. [ n , k ] l \textstyle[n,k]l
  254. n 2 r - b + 1 - 1 , \textstyle n\leq 2^{r-b+1}-1,
  255. r = n - k \textstyle r=n-k
  256. r log 2 ( n + 1 ) + ( b - 1 ) . \textstyle r\geq\lceil\log_{2}(n+1)\rceil+(b-1).
  257. [ n , k ] \textstyle[n,k]
  258. 2 k \textstyle 2^{k}
  259. 2 k 2 n ( n 2 b - 1 + 1 ) \textstyle 2^{k}\leq\frac{2^{n}}{(n2^{b-1}+1)}
  260. n \textstyle n
  261. n 2 r - b + 1 - 2 - b + 1 \textstyle n\leq 2^{r-b+1}-2^{-b+1}
  262. n \textstyle n
  263. n 2 r - b + 1 - 1 \textstyle n\leq 2^{r-b+1}-1
  264. r \textstyle r
  265. l \textstyle l
  266. l \textstyle l
  267. p ( x ) \textstyle p(x)
  268. m \textstyle m
  269. 𝔽 2 \textstyle\mathbb{F}_{2}
  270. p \textstyle p
  271. p ( x ) \textstyle p(x)
  272. p ( x ) \textstyle p(x)
  273. r \textstyle r
  274. p ( x ) ( x r - 1 ) \textstyle p(x)\mid(x^{r}-1)
  275. l \textstyle l
  276. l m \textstyle l\leq m
  277. 2 l - 1 \textstyle 2l-1
  278. p \textstyle p
  279. p \textstyle p
  280. p ( x ) \textstyle p(x)
  281. l \textstyle l
  282. G \textstyle G
  283. g ( x ) = ( x 2 l - 1 + 1 ) p ( x ) \textstyle g(x)=(x^{2l-1}+1)p(x)
  284. p ( x ) \textstyle p(x)
  285. ( x 2 l - 1 + 1 ) \textstyle(x^{2l-1}+1)
  286. d ( x ) = GCD ( p ( x ) , ( x 2 l - 1 + 1 ) ) d(x)=\mathrm{GCD}(p(x),(x^{2l-1}+1))
  287. p ( x ) \textstyle p(x)
  288. deg ( d ( x ) ) \textstyle\deg(d(x))
  289. 0 \textstyle 0
  290. deg ( p ( x ) ) \textstyle\deg(p(x))
  291. deg ( d ( x ) ) \textstyle\deg(d(x))
  292. p ( x ) = c d ( x ) \textstyle p(x)=c\;d(x)
  293. c \textstyle c
  294. ( 1 / c ) p ( x ) \textstyle(1/c)p(x)
  295. ( x 2 l - 1 + 1 ) \textstyle(x^{2l-1}+1)
  296. d ( x ) \textstyle d(x)
  297. ( x 2 l - 1 + 1 ) \textstyle(x^{2l-1}+1)
  298. p ( x ) \textstyle p(x)
  299. ( x 2 l - 1 + 1 ) \textstyle(x^{2l-1}+1)
  300. deg ( d ( x ) ) \textstyle\deg(d(x))
  301. 0 \textstyle 0
  302. p ( x ) \textstyle p(x)
  303. ( x 2 l - 1 + 1 ) \textstyle(x^{2l-1}+1)
  304. p ( x ) \textstyle p(x)
  305. p \textstyle p
  306. p ( x ) \textstyle p(x)
  307. x k - 1 \textstyle x^{k}-1
  308. p k \textstyle p\mid k
  309. p k \textstyle p\mid k
  310. x k - 1 = ( x p - 1 ) ( 1 + x p + x 2 p + + x k / p ) \textstyle x^{k}-1=(x^{p}-1)(1+x^{p}+x^{2p}+\ldots+x^{k/p})
  311. p ( x ) \textstyle p(x)
  312. x k - 1 \textstyle x^{k}-1
  313. p ( x ) \textstyle p(x)
  314. x k - 1 \textstyle x^{k}-1
  315. k p \textstyle k\geq p
  316. k \textstyle k
  317. p \textstyle p
  318. k \textstyle k
  319. k = p \textstyle k=p
  320. k \textstyle k
  321. p \textstyle p
  322. p ( x ) \textstyle p(x)
  323. p \textstyle p
  324. x p - 1 = ( x - 1 ) ( 1 + x + + x p - 1 ) \textstyle x^{p}-1=(x-1)(1+x+\ldots+x^{p-1})
  325. x k - 1 = ( x - 1 ) ( 1 + x + + x k - 1 ) \textstyle x^{k}-1=(x-1)(1+x+\ldots+x^{k-1})
  326. p ( x ) \textstyle p(x)
  327. ( 1 + x + + x p - 1 ) \textstyle(1+x+\ldots+x^{p-1})
  328. ( 1 + x + + x k - 1 ) \textstyle(1+x+\ldots+x^{k-1})
  329. x p ( 1 + x + + x p - k - 1 ) \textstyle x^{p}(1+x+\ldots+x^{p-k-1})
  330. p ( x ) \textstyle p(x)
  331. ( 1 + x + + x p - k - 1 ) \textstyle(1+x+\ldots+x^{p-k-1})
  332. x k - p - 1 = ( x - 1 ) ( 1 + x + + x p - k - 1 ) \textstyle x^{k-p}-1=(x-1)(1+x+\ldots+x^{p-k-1})
  333. p k - p \textstyle p\mid k-p
  334. p k \textstyle p\mid k
  335. p ( x ) = x p - 1 \textstyle p(x)=x^{p}-1
  336. p \textstyle p
  337. p ( x ) \textstyle p(x)
  338. x k - 1 \textstyle x^{k}-1
  339. p k \textstyle p\mid k
  340. l \textstyle l
  341. l \textstyle l
  342. x i a ( x ) \textstyle x^{i}a(x)
  343. x j b ( x ) \textstyle x^{j}b(x)
  344. l 1 - 1 \textstyle l_{1}-1
  345. l 2 - 1 \textstyle l_{2}-1
  346. l 1 \textstyle l_{1}
  347. l 2 \textstyle l_{2}
  348. l 1 l \textstyle l_{1}\leq l
  349. l 2 l \textstyle l_{2}\leq l
  350. i \textstyle i
  351. j \textstyle j
  352. x i a ( x ) \textstyle x^{i}a(x)
  353. x j b ( x ) \textstyle x^{j}b(x)
  354. v ( x ) = x i a ( x ) + x j b ( x ) \textstyle v(x)=x^{i}a(x)+x^{j}b(x)
  355. i j \textstyle i\leq j
  356. j - i \textstyle j-i
  357. 2 l - 1 \textstyle 2l-1
  358. j - i = g ( 2 l - 1 ) + r \textstyle j-i=g(2l-1)+r
  359. g \textstyle g
  360. r \textstyle r
  361. 0 r < 2 l - 1 \textstyle 0\leq r<\textstyle 2l-1
  362. v ( x ) \textstyle v(x)
  363. v ( x ) = x i a ( x ) + x i + g ( 2 l - 1 ) + r \textstyle v(x)=x^{i}a(x)+x^{i+g(2l-1)+r}
  364. = x i a ( x ) + x i + g ( 2 l - 1 ) + r + 2 x i + r b ( x ) \textstyle=x^{i}a(x)+x^{i+g(2l-1)+r}+2x^{i+r}b(x)
  365. = x i ( a ( x ) + x b b ( x ) ) + x i + r b ( x ) ( x g ( 2 l - 1 ) + 1 ) \textstyle=x^{i}(a(x)+x^{b}b(x))+x^{i+r}b(x)(x^{g(2l-1)}+1)
  366. 2 x i + r b ( x ) \textstyle 2x^{i+r}b(x)
  367. 𝔽 2 \textstyle\mathbb{F}_{2}
  368. v ( x ) \textstyle v(x)
  369. g ( x ) \textstyle g(x)
  370. g ( x ) \textstyle g(x)
  371. v ( x ) \textstyle v(x)
  372. x 2 l - 1 + 1 \textstyle x^{2l-1}+1
  373. v ( x ) \textstyle v(x)
  374. x g ( 2 l - 1 ) + 1 \textstyle x^{g(2l-1)}+1
  375. x 2 l - 1 + 1 \textstyle x^{2l-1}+1
  376. a ( x ) + x b b ( x ) \textstyle a(x)+x^{b}b(x)
  377. x 2 l - 1 + 1 \textstyle x^{2l-1}+1
  378. 0 \textstyle 0
  379. a ( x ) + x b b ( x ) = d ( x ) ( x 2 l - 1 + 1 ) \textstyle a(x)+x^{b}b(x)=d(x)(x^{2l-1}+1)
  380. d ( x ) \textstyle d(x)
  381. δ = deg ( d ( x ) ) \textstyle\delta=\deg(d(x))
  382. deg ( d ( x ) ( x 2 l - 1 + 1 ) ) \textstyle\deg(d(x)(x^{2l-1}+1))
  383. δ + 2 l - 1 \textstyle\delta+2l-1
  384. deg ( a ( x ) ) = l 1 - 1 \textstyle\deg(a(x))=l_{1}-1
  385. < 2 l - 1 \textstyle<2l-1
  386. a ( x ) + x b b ( x ) \textstyle a(x)+x^{b}b(x)
  387. x b b ( x ) \textstyle x^{b}b(x)
  388. b + l 2 - 1 \textstyle b+l_{2}-1
  389. b + l 2 - 1 = 2 l - 1 + δ \textstyle b+l_{2}-1=2l-1+\delta
  390. l 1 l \textstyle l_{1}\leq l
  391. l 2 l \textstyle l_{2}\leq l
  392. l 2 \textstyle l_{2}
  393. b l 1 + δ \textstyle b\geq l_{1}+\delta
  394. b > l 1 - 1 \textstyle b>l_{1}-1
  395. b > δ \textstyle b>\textstyle\delta
  396. a ( x ) + x b b ( x ) \textstyle a(x)+x^{b}b(x)
  397. 1 + a 1 x + a 2 x 2 + + x l 1 - 1 + x b ( 1 + b 1 x + b 2 x 2 + + x l 2 - 1 ) . \textstyle 1+a_{1}x+a_{2}x^{2}+\ldots+x^{l_{1}-1}+x^{b}(1+b_{1}x+b_{2}x^{2}+% \ldots+x^{l_{2}-1}).
  398. x b \textstyle x^{b}
  399. δ < b < 2 l - 1 \textstyle\delta<b<2l-1
  400. d ( x ) ( x 2 l - 1 + 1 ) \textstyle d(x)(x^{2l-1}+1)
  401. x b \textstyle x^{b}
  402. d ( x ) = 0 \textstyle d(x)=0
  403. a ( x ) + x b b ( x ) = 0 \textstyle a(x)+x^{b}b(x)=0
  404. b = 0 \textstyle b=0
  405. a ( x ) = b ( x ) \textstyle a(x)=b(x)
  406. j - i \textstyle j-i
  407. g ( 2 l - 1 ) \textstyle g(2l-1)
  408. b = 0 \textstyle b=0
  409. j - i = g ( 2 l - 1 ) \textstyle j-i=g(2l-1)
  410. v ( x ) \textstyle v(x)
  411. v ( x ) = x i b ( x ) ( x j - 1 + 1 ) . \textstyle v(x)=x^{i}b(x)(x^{j-1}+1).
  412. b ( x ) \textstyle b(x)
  413. l 2 - 1 < l \textstyle l_{2}-1<l
  414. deg ( b ( x ) ) < deg ( p ( x ) ) = m \textstyle\deg(b(x))<\deg(p(x))=m
  415. p ( x ) \textstyle p(x)
  416. b ( x ) \textstyle b(x)
  417. p ( x ) \textstyle p(x)
  418. v ( x ) \textstyle v(x)
  419. x j - 1 + 1 \textstyle x^{j-1}+1
  420. p ( x ) \textstyle p(x)
  421. x 2 l - 1 + 1 \textstyle x^{2l-1}+1
  422. j - i \textstyle j-i
  423. p \textstyle p
  424. 2 l - 1 \textstyle 2l-1
  425. n = LCM ( 2 l - 1 , p ) \textstyle n=\mathrm{LCM}(2l-1,p)
  426. j - i \textstyle j-i
  427. n \textstyle n
  428. n \textstyle n
  429. v ( x ) \textstyle v(x)
  430. x i a ( x ) \textstyle x^{i}a(x)
  431. x j b ( x ) \textstyle x^{j}b(x)
  432. 5 \textstyle 5
  433. p ( x ) \textstyle p(x)
  434. l \textstyle l
  435. 2 l - 1 \textstyle 2l-1
  436. p ( x ) \textstyle p(x)
  437. p ( x ) = 1 + x 2 + x 5 \textstyle p(x)=1+x^{2}+x^{5}
  438. l = 5 \textstyle l=5
  439. p ( x ) \textstyle p(x)
  440. 2 5 - 1 = 31 \textstyle 2^{5}-1=31
  441. 2 l - 1 = 9 \textstyle 2l-1=9
  442. 31 \textstyle 31
  443. g ( x ) = ( x 9 + 1 ) ( 1 + x 2 + x 5 ) = 1 + x 2 + x 5 + x 9 + x 11 + x 14 \textstyle g(x)=(x^{9}+1)(1+x^{2}+x^{5})=1+x^{2}+x^{5}+x^{9}+x^{11}+x^{14}
  444. p \textstyle p
  445. 2 l - 1 \textstyle 2l-1
  446. n = LCM ( 9 , 31 ) = 279 \textstyle n=\mathrm{LCM}(9,31)=279
  447. 5 \textstyle 5
  448. 𝔽 2 m \textstyle\mathbb{F}_{2^{m}}
  449. m \textstyle m
  450. C \textstyle C
  451. [ n , k ] \textstyle[n,k]
  452. 𝔽 2 m \textstyle\mathbb{F}_{2^{m}}
  453. C \textstyle C
  454. [ m n , m k ] 2 \textstyle[mn,mk]_{2}
  455. 𝔽 2 \textstyle\mathbb{F}_{2}
  456. m \textstyle m
  457. m \textstyle m
  458. m \textstyle m
  459. ( m + 1 ) \textstyle(m+1)
  460. 2 \textstyle 2
  461. 2 m + 1 \textstyle 2m+1
  462. 3 \textstyle 3
  463. t m + 1 \textstyle tm+1
  464. t + 1 \textstyle t+1
  465. t \textstyle t
  466. ( t - 1 ) m + 1 \textstyle(t-1)m+1
  467. t \textstyle t
  468. 𝔽 2 m \textstyle\mathbb{F}_{2^{m}}
  469. t 1 + ( l + m - 2 ) / m \frac{t}{1+\lfloor(l+m-2)/m\rfloor}
  470. l \textstyle l
  471. t \textstyle t
  472. G \textstyle G
  473. [ 255 , 223 , 33 ] \textstyle[255,223,33]
  474. 𝔽 2 8 \textstyle\mathbb{F}_{2^{8}}
  475. 33 / 2 = 16 \textstyle\lfloor 33/2\rfloor=16
  476. G \textstyle G^{\prime}
  477. G \textstyle G
  478. log 2 ( 255 ) = 8 \textstyle\lceil\log_{2}(255)\rceil=8
  479. [ 2040 , 1784 , 33 ] 2 \textstyle[2040,1784,33]_{2}
  480. l = 121 \textstyle l=121
  481. l \textstyle l
  482. λ \textstyle\lambda
  483. λ l \textstyle\lambda l
  484. ( n , k ) \textstyle(n,k)
  485. l \textstyle\leq l
  486. ( λ n , λ k ) \textstyle(\lambda n,\lambda k)
  487. λ l \textstyle\leq\lambda l
  488. λ \textstyle\lambda
  489. λ k \textstyle\lambda k
  490. λ k \textstyle\lambda k
  491. λ × k \textstyle\lambda\times k
  492. ( n , k ) \textstyle(n,k)
  493. λ × n \textstyle\lambda\times n
  494. h \textstyle h
  495. h λ \textstyle\frac{h}{\lambda}
  496. h λ \textstyle\lfloor\frac{h}{\lambda}\rfloor
  497. h λ \textstyle\lceil\frac{h}{\lambda}\rceil
  498. h λ l \textstyle h\leq\lambda l
  499. h λ l \textstyle\frac{h}{\lambda}\leq l
  500. ( n , k ) \textstyle(n,k)
  501. ( λ n , λ k ) \textstyle(\lambda n,\lambda k)
  502. h \textstyle h
  503. h > λ l \textstyle h>\lambda l
  504. h λ \textstyle\frac{h}{\lambda}
  505. ( n , k ) \textstyle(n,k)
  506. ( λ n , λ k ) \textstyle(\lambda n,\lambda k)
  507. λ l \textstyle\lambda l
  508. ( n , k ) \textstyle(n,k)
  509. 2 λ l λ n - λ k = 2 l n - k \frac{2\lambda l}{\lambda n-\lambda k}\quad=\quad\frac{2l}{n-k}
  510. M × N \textstyle M\times N
  511. N \textstyle N
  512. M × N \textstyle M\times N
  513. l \textstyle l
  514. l M {\dfrac{l}{M}}
  515. M \textstyle M
  516. t \textstyle t
  517. M t \textstyle Mt
  518. M t + 1 \textstyle Mt+1
  519. γ \gamma
  520. γ \gamma
  521. γ = M t + 1 M N t N \gamma={\dfrac{Mt+1}{MN}}\approx{\dfrac{t}{N}}
  522. n \textstyle n
  523. d \textstyle d
  524. n d \textstyle nd
  525. n \textstyle n
  526. l \textstyle l
  527. n d \textstyle nd
  528. l n d + 1 {\dfrac{l}{nd+1}}
  529. t \textstyle t
  530. ( n d + 1 ) ( t - 1 ) \textstyle(nd+1)(t-1)
  531. ( n d + 1 ) ( t - 1 ) + 1 \textstyle(nd+1)(t-1)+1
  532. γ \gamma
  533. ( 0 + 1 + 2 + 3 + + ( n - 1 ) ) d = n d ( n - 1 ) 2 (0+1+2+3+\cdots+(n-1))d=\dfrac{n\cdot d\cdot(n-1)}{2}
  534. γ \gamma
  535. γ = ( n d + 1 ) ( t - 1 ) + 1 n d ( n - 1 ) / 2 \gamma={\dfrac{(nd+1)\cdot(t-1)+1}{nd\cdot(n-1)/2}}
  536. 𝔽 2 16 \textstyle\mathbb{F}_{2}^{16}
  537. 𝔽 2 8 \textstyle\mathbb{F}_{2}^{8}
  538. L 1 R 1 L 2 R 2 L 6 R 6 \textstyle L_{1}R_{1}L_{2}R_{2}\ldots L_{6}R_{6}
  539. L i \textstyle L_{i}
  540. R i \textstyle R_{i}
  541. i t h \textstyle i^{th}
  542. L 1 L 3 L 5 R 1 R 3 R 5 L 2 L 4 L 6 R 2 R 4 R 6 \textstyle L_{1}L_{3}L_{5}R_{1}R_{3}R_{5}L_{2}L_{4}L_{6}R_{2}R_{4}R_{6}
  543. L i , R i \textstyle L_{i},R_{i}
  544. i t h \textstyle i^{th}
  545. 𝔽 256 \textstyle\mathbb{F}_{256}
  546. P 1 P 2 \textstyle P_{1}P_{2}
  547. L 1 L 3 L 5 R 1 R 3 R 5 P 1 P 2 L 2 L 4 L 6 R 2 R 4 R 6 \textstyle L_{1}L_{3}L_{5}R_{1}R_{3}R_{5}P_{1}P_{2}L_{2}L_{4}L_{6}R_{2}R_{4}R_% {6}
  548. R = 24 / 32 \textstyle R=24/32
  549. e \textstyle e
  550. f \textstyle f
  551. 2 e + f < 5 \textstyle 2e+f<5
  552. = 10 - 4 =\textstyle 10^{-4}
  553. 10 - 3 \textstyle 10^{-3}
  554. 10 - 3 \textstyle 10^{-3}
  555. 10 - 4 \textstyle 10^{-4}