wpmath0000011_0

(-)-endo-fenchol_synthase.html

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(2,3-dihydroxybenzoyl)adenylate_synthase.html

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(3-methyl-2-oxobutanoate_dehydrogenase_(acetyl-transferring)).html

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(4S)-limonene_synthase.html

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(acetyl-CoA_carboxylase)-phosphatase.html

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(acetyl-CoA_carboxylase)_kinase.html

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(acyl-carrier-protein)_phosphodiesterase.html

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(acyl-carrier-protein)_S-acetyltransferase.html

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(acyl-carrier-protein)_S-malonyltransferase.html

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(carboxyethyl)arginine_beta-lactam-synthase.html

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(citrate_(pro-3S)-lyase)_ligase.html

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(deoxy)adenylate_kinase.html

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(deoxy)nucleoside-phosphate_kinase.html

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(glutamate—ammonia-ligase)_adenylyltransferase.html

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(glycogen-synthase-D)_phosphatase.html

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(hydroxymethylglutaryl-CoA_reductase_(NADPH))-phosphatase.html

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(myelin-proteolipid)_O-palmitoyltransferase.html

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(phosphorylase)_phosphatase.html

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(protein-PII)_uridylyltransferase.html

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(pyruvate_dehydrogenase_(acetyl-transferring))-phosphatase.html

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(pyruvate_kinase)-phosphatase.html

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(R)-2-methylmalate_dehydratase.html

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(R)-3-amino-2-methylpropionate—pyruvate_transaminase.html

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(R)-limonene_synthase.html

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(RNA-polymerase)-subunit_kinase.html

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(S)-2-methylmalate_dehydratase.html

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(S)-3-amino-2-methylpropionate_transaminase.html

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(S)-methylmalonyl-CoA_hydrolase.html

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(S)-N-acetyl-1-phenylethylamine_hydrolase.html

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(S)-norcoclaurine_synthase.html

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(Skp1-protein)-hydroxyproline_N-acetylglucosaminyltransferase.html

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(tyrosine_3-monooxygenase)_kinase.html

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1,2-alpha-L-fucosidase.html

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1,2-diacylglycerol_3-glucosyltransferase.html

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1,3-beta-D-glucan_phosphorylase.html

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1,3-beta-galactosyl-N-acetylhexosamine_phosphorylase.html

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1,3-beta-oligoglucan_phosphorylase.html

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1,4-beta-D-xylan_synthase.html

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1,4-lactonase.html

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1,5-anhydro-D-fructose_dehydratase.html

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1-acylglycerol-3-phosphate_O-acyltransferase.html

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1-acylglycerophosphocholine_O-acyltransferase.html

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1-alkenyl-2-acylglycerol_choline_phosphotransferase.html

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1-alkenylglycerophosphocholine_O-acyltransferase.html

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1-alkenylglycerophosphoethanolamine_O-acyltransferase.html

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1-alkyl-2-acetylglycerol_O-acyltransferase.html

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1-alkyl-2-acetylglycerophosphocholine_esterase.html

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1-alkylglycerophosphocholine_O-acetyltransferase.html

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1-alkylglycerophosphocholine_O-acyltransferase.html

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1-aminocyclopropane-1-carboxylate_deaminase.html

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1-deoxy-D-xylulose-5-phosphate_synthase.html

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1-methyladenosine_nucleosidase.html

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1-phosphatidylinositol-4-phosphate_5-kinase.html

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1-phosphatidylinositol-5-phosphate_4-kinase.html

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1-phosphatidylinositol_4-kinase.html

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1-pyrroline-4-hydroxy-2-carboxylate_deaminase.html

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1.96.html

  1. P ( X > 1.96 ) = 0.025 , \mathrm{P}(X>1.96)=0.025,\,
  2. P ( X < 1.96 ) = 0.975 , \mathrm{P}(X<1.96)=0.975,\,
  3. P ( - 1.96 < X < 1.96 ) = 0.95. \mathrm{P}(-1.96<X<1.96)=0.95.\,
  4. 1 2 π z .025 e - x 2 / 2 d x = 0.025. \frac{1}{\sqrt{2\pi}}\int_{z_{.025}}^{\infty}e^{-x^{2}/2}\,\mathrm{d}x=0.025.

10-deacetylbaccatin_III_10-O-acetyltransferase.html

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10-hydroxytaxane_O-acetyltransferase.html

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11-cis-retinyl-palmitate_hydrolase.html

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13-hydroxydocosanoate_13-beta-glucosyltransferase.html

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13-hydroxylupinine_O-tigloyltransferase.html

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16-alpha-hydroxyprogesterone_dehydratase.html

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16:10.html

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1D-1-guanidino-3-amino-1,3-dideoxy-scyllo-inositol_transaminase.html

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2',3'-cyclic-nucleotide_2'-phosphodiesterase.html

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2'-phosphotransferase.html

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2,1-fructan:2,1-fructan_1-fructosyltransferase.html

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2,3,4,5-tetrahydropyridine-2,6-dicarboxylate_N-succinyltransferase.html

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2,3-diaminopropionate_N-oxalyltransferase.html

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2,3-dihydroxybenzoate—serine_ligase.html

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2,5-diaminovalerate_transaminase.html

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2,5-dioxopiperazine_hydrolase.html

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2-(acetamidomethylene)succinate_hydrolase.html

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2-(hydroxymethyl)-3-(acetamidomethylene)succinate_hydrolase.html

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2-acylglycerol-3-phosphate_O-acyltransferase.html

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2-acylglycerol_O-acyltransferase.html

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2-acylglycerophosphocholine_O-acyltransferase.html

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2-amino-4-hydroxy-6-hydroxymethyldihydropteridine_diphosphokinase.html

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2-aminoadipate_transaminase.html

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2-aminoethylphosphonate—pyruvate_transaminase.html

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2-aminohexanoate_transaminase.html

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2-aminomuconate_deaminase.html

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2-C-methyl-D-erythritol_4-phosphate_cytidylyltransferase.html

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2-Carboxy-D-arabinitol-1-phosphatase.html

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2-coumarate_O-beta-glucosyltransferase.html

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2-dehydro-3-deoxy-L-arabinonate_dehydratase.html

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2-dehydro-3-deoxygalactonokinase.html

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2-dehydro-3-deoxygluconokinase.html

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2-deoxyglucose-6-phosphatase.html

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2-deoxyglucosidase.html

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2-ethylmalate_synthase.html

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2-furoate—CoA_ligase.html

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2-hydroxy-3-oxoadipate_synthase.html

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2-Hydroxyacylsphingosine_1-beta-galactosyltransferase.html

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2-hydroxyglutarate_synthase.html

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2-hydroxyisoflavanone_dehydratase.html

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2-isopropylmalate_synthase.html

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2-methylcitrate_dehydratase.html

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2-methylcitrate_synthase.html

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2-methylisocitrate_dehydratase.html

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2-oxoglutarate_carboxylase.html

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2-oxopent-4-enoate_hydratase.html

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2-phosphosulfolactate_phosphatase.html

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2-pyrone-4,6-dicarboxylate_lactonase.html

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2alpha-hydroxytaxane_2-O-benzoyltransferase.html

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3'(2'),5'-bisphosphate_nucleotidase.html

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3',5'-cyclic-GMP_phosphodiesterase.html

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3'-nucleotidase.html

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3,4-dichloroaniline_N-malonyltransferase.html

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3-alpha,7-alpha-dihydroxy-5-beta-cholestanate—CoA_ligase.html

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3-cyanoalanine_hydratase.html

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3-dehydroquinate_dehydratase.html

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3-dehydroquinate_synthase.html

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3-deoxy-8-phosphooctulonate_synthase.html

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3-deoxy-manno-octulosonate-8-phosphatase.html

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3-deoxy-manno-octulosonate_cytidylyltransferase.html

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3-deoxyoctulosonase.html

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3-ethylmalate_synthase.html

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3-galactosyl-N-acetylglucosaminide_4-alpha-L-fucosyltransferase.html

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3-hydroxybutyryl-CoA_dehydratase.html

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3-hydroxyisobutyryl-CoA_hydrolase.html

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

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

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3-oxoadipate_enol-lactonase.html

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3-oxoadipyl-CoA_thiolase.html

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3-phosphoglycerate_phosphatase.html

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3-phosphoglyceroyl-phosphate—polyphosphate_phosphotransferase.html

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3-phytase.html

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3-propylmalate_synthase.html

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3alpha(S)-strictosidine_beta-glucosidase.html

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3alpha,7alpha,12alpha-trihydroxy-5beta-cholest-24-enoyl-CoA_hydratase.html

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4-(cytidine_5'-diphospho)-2-C-methyl-D-erythritol_kinase.html

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4-acetamidobutyrate_deacetylase.html

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4-acetamidobutyryl-CoA_deacetylase.html

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4-aminobutyrate_transaminase.html

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4-chlorobenzoate—CoA_ligase.html

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4-Coumarate-CoA_ligase.html

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4-galactosyl-N-acetylglucosaminide_3-alpha-L-fucosyltransferase.html

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4-hydroxybenzoate_4-O-beta-D-glucosyltransferase.html

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4-hydroxybenzoate_nonaprenyltransferase.html

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4-hydroxybenzoate—CoA_ligase.html

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4-hydroxybenzoyl-CoA_thioesterase.html

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4-hydroxyglutamate_transaminase.html

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4-methyleneglutamate—ammonia_ligase.html

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4-methyleneglutaminase.html

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4-methyloxaloacetate_esterase.html

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4-nitrophenylphosphatase.html

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4-oxalmesaconate_hydratase.html

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4-phytase.html

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4-pyridoxolactonase.html

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4a-hydroxytetrahydrobiopterin_dehydratase.html

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5'-acylphosphoadenosine_hydrolase.html

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5-(3,4-diacetoxybut-1-ynyl)-2,2'-bithiophene_deacetylase.html

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5-(carboxyamino)imidazole_ribonucleotide_synthase.html

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5-aminopentanamidase.html

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5-aminovalerate_transaminase.html

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5-dehydro-2-deoxygluconokinase.html

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5-dehydro-4-deoxyglucarate_dehydratase.html

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5-formyltetrahydrofolate_cyclo-ligase.html

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5-methyldeoxycytidine-5'-phosphate_kinase.html

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5-oxoprolinase_(ATP-hydrolysing).html

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5-phytase.html

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5alpha-hydroxysteroid_dehydratase.html

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6'-Deoxychalcone_synthase.html

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6-Acetyl-2,3,4,5-tetrahydropyridine.html

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6-acetylglucose_deacetylase.html

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6-aminohexanoate-cyclic-dimer_hydrolase.html

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6-aminohexanoate-dimer_hydrolase.html

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6-carboxyhexanoate—CoA_ligase.html

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6-methylsalicylic-acid_synthase.html

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6-phospho-beta-galactosidase.html

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6-phospho-beta-glucosidase.html

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6-phosphofructo-2-kinase.html

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6-pyruvoyltetrahydropterin_synthase.html

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6G-fructosyltransferase.html

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8-amino-7-oxononanoate_synthase.html

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A-equivalence.html

  1. 𝒜 \mathcal{A}
  2. M M
  3. N N
  4. f , g : ( M , x ) ( N , y ) f,g:(M,x)\to(N,y)
  5. f f
  6. g g
  7. 𝒜 \mathcal{A}
  8. ϕ : ( M , x ) ( M , x ) \phi:(M,x)\to(M,x)
  9. ψ : ( N , y ) ( N , y ) \psi:(N,y)\to(N,y)
  10. ψ f = g ϕ . \psi\circ f=g\circ\phi.
  11. 𝒜 \mathcal{A}
  12. M M
  13. N N
  14. Ω ( M x , N y ) \Omega(M_{x},N_{y})
  15. ( M , x ) ( N , y ) . (M,x)\to(N,y).
  16. diff ( M x ) \mbox{diff}~{}(M_{x})
  17. ( M , x ) ( M , x ) (M,x)\to(M,x)
  18. diff ( N y ) \mbox{diff}~{}(N_{y})
  19. ( N , y ) ( N , y ) . (N,y)\to(N,y).
  20. G := diff ( M x ) × diff ( N y ) G:=\mbox{diff}~{}(M_{x})\times\mbox{diff}~{}(N_{y})
  21. Ω ( M x , N y ) \Omega(M_{x},N_{y})
  22. ( ϕ , ψ ) f = ψ - 1 f ϕ . (\phi,\psi)\cdot f=\psi^{-1}\circ f\circ\phi.
  23. f , g : ( M , x ) ( N , y ) f,g:(M,x)\to(N,y)
  24. 𝒜 \mathcal{A}
  25. g g
  26. f f
  27. g orb ( f ) G g\in\mbox{orb}~{}_{G}(f)
  28. G := diff ( M x ) × diff ( N y ) G:=\mbox{diff}~{}(M_{x})\times\mbox{diff}~{}(N_{y})
  29. Ω ( M x , N y ) \Omega(M_{x},N_{y})
  30. k k
  31. k k
  32. o r b G ( f ) . orb_{G}(f).
  33. f f
  34. ( n , 0 ) ( , 0 ) (\mathbb{R}^{n},0)\to(\mathbb{R},0)
  35. 1 n 3 1\leq n\leq 3
  36. A k A_{k}
  37. k k\in\mathbb{N}
  38. D 4 + k D_{4+k}
  39. k k\in\mathbb{N}
  40. E 6 , E_{6},
  41. E 7 , E_{7},
  42. E 8 . E_{8}.

Abequosyltransferase.html

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Abietadiene_synthase.html

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Absolute_electrode_potential.html

  1. E ( abs ) M = E ( SHE ) M + ( 4.44 ± 0.02 ) V E^{M}_{\rm{(abs)}}=E^{M}_{\rm{(SHE)}}+(4.44\pm 0.02)\ {\mathrm{V}}
  2. E E
  3. 1 2 \frac{1}{2}
  4. 1 2 \frac{1}{2}
  5. E M ( abs ) = ϕ M + Δ S M ψ E^{M}{\rm(abs)}=\phi^{M}+\Delta^{M}_{S}\psi
  6. ϕ M \phi^{M}
  7. Δ S M ψ \Delta^{M}_{S}\psi
  8. E ( H + / H 2 ) ( abs ) = ϕ Hg + Δ S Hg ψ σ = 0 - E σ = 0 Hg ( SHE ) E^{\ominus}{\rm(H^{+}/H_{2})(abs)}=\phi^{\rm{Hg}}+\Delta^{\rm{Hg}}_{S}\psi^{% \ominus}_{\sigma=0}-E^{\rm{Hg}}_{\sigma=0}\rm{(SHE)}
  9. E ( H + / H 2 ) ( abs ) E^{\ominus}{\rm(H^{+}/H_{2})(abs)}
  10. σ = 0 σ=0

Abstract_family_of_acceptors.html

  1. ( Γ , I , f , g ) (\Gamma,I,f,g)
  2. Γ \Gamma
  3. I I
  4. f f
  5. f : Γ * × I Γ * { } f:\Gamma^{*}\times I\rightarrow\Gamma^{*}\cup\{\}
  6. g g
  7. Γ * \Gamma^{*}
  8. Γ * \Gamma^{*}
  9. g ( ϵ ) = { ϵ } g(\epsilon)=\{\epsilon\}
  10. ϵ \epsilon
  11. g ( γ ) g(\gamma)
  12. γ = ϵ \gamma=\epsilon
  13. ϵ \epsilon
  14. γ \gamma
  15. g ( Γ * ) g(\Gamma^{*})
  16. 1 γ 1_{\gamma}
  17. I I
  18. f ( γ , 1 γ ) = γ f(\gamma^{\prime},1_{\gamma})=\gamma^{\prime}
  19. γ \gamma^{\prime}
  20. γ \gamma
  21. g ( γ ) g(\gamma^{\prime})
  22. Γ u \Gamma_{u}
  23. Γ \Gamma
  24. Γ 1 \Gamma_{1}
  25. Γ \Gamma
  26. γ \gamma
  27. Γ 1 * \Gamma_{1}^{*}
  28. f ( γ , u ) f(\gamma,u)\neq
  29. f ( γ , u ) f(\gamma,u)
  30. ( Γ 1 Γ u ) * (\Gamma_{1}\cup\Gamma_{u})^{*}
  31. ( Ω , 𝒟 ) (\Omega,\mathcal{D})
  32. Ω \Omega
  33. K K
  34. Σ \Sigma
  35. Γ \Gamma
  36. I I
  37. f f
  38. g g
  39. Γ \Gamma
  40. I I
  41. f f
  42. g g
  43. K K
  44. Σ \Sigma
  45. 𝒟 \mathcal{D}
  46. D D
  47. K 1 K_{1}
  48. Σ 1 \Sigma_{1}
  49. δ \delta
  50. q 0 q_{0}
  51. F F
  52. K 1 K_{1}
  53. Σ 1 \Sigma_{1}
  54. K K
  55. Σ \Sigma
  56. F F
  57. K 1 K_{1}
  58. q 0 q_{0}
  59. K 1 K_{1}
  60. δ \delta
  61. K 1 × ( Σ 1 { ϵ } ) × g ( Γ * ) K_{1}\times(\Sigma_{1}\cup\{\epsilon\})\times g(\Gamma^{*})
  62. K 1 × I K_{1}\times I
  63. G D = { γ G_{D}=\{\gamma
  64. δ ( q , a , γ ) \delta(q,a,\gamma)
  65. q q
  66. a } a\}
  67. \vdash
  68. K 1 × Σ 1 * × Γ * K_{1}\times\Sigma_{1}^{*}\times\Gamma^{*}
  69. a a
  70. Σ 1 { ϵ } \Sigma_{1}\cup\{\epsilon\}
  71. ( p , a w , γ ) ( p , w , γ ) (p,aw,\gamma)\vdash(p^{\prime},w,\gamma^{\prime})
  72. γ ¯ \overline{\gamma}
  73. u u
  74. γ ¯ \overline{\gamma}
  75. g ( γ ) g(\gamma)
  76. ( p , u ) (p^{\prime},u)
  77. δ ( p , a , γ ¯ ) \delta(p,a,\overline{\gamma})
  78. f ( γ , u ) = γ f(\gamma,u)=\gamma^{\prime}
  79. * \vdash^{*}
  80. \vdash
  81. ( Ω , 𝒟 ) (\Omega,\mathcal{D})
  82. D D
  83. K 1 K_{1}
  84. Σ 1 \Sigma_{1}
  85. δ \delta
  86. q 0 q_{0}
  87. F F
  88. D D
  89. L ( D ) L(D)
  90. { w Σ 1 * | q F . ( q 0 , w , ϵ ) * ( q , ϵ , ϵ ) } \{w\in\Sigma_{1}^{*}|\exists q\in F.(q_{0},w,\epsilon)\vdash^{*}(q,\epsilon,% \epsilon)\}
  91. \mathcal{E}
  92. 𝒟 \mathcal{D}
  93. ( ) = { L ( D ) | D } \mathcal{L}(\mathcal{E})=\{L(D)|D\in\mathcal{E}\}
  94. L f ( D ) L_{f}(D)
  95. { w Σ 1 * | ( q F ) ( γ Γ * ) . ( q 0 , w , ϵ ) * ( q , ϵ , γ ) } \{w\in\Sigma_{1}^{*}|\exists(q\in F)\exists(\gamma\in\Gamma^{*}).(q_{0},w,% \epsilon)\vdash^{*}(q,\epsilon,\gamma)\}
  96. \mathcal{E}
  97. 𝒟 \mathcal{D}
  98. f ( ) = { L f ( D ) | D } \mathcal{L}_{f}(\mathcal{E})=\{L_{f}(D)|D\in\mathcal{E}\}
  99. Γ \Gamma
  100. I I
  101. f f
  102. g g
  103. \vdash
  104. L f ( D ) L_{f}(D)
  105. D D
  106. L ( D ) L(D)
  107. D D
  108. \mathcal{L}
  109. = ( 𝒟 ) \mathcal{L}=\mathcal{L}(\mathcal{D})
  110. ( Ω , 𝒟 ) (\Omega,\mathcal{D})
  111. \mathcal{L}
  112. = f ( 𝒟 ) \mathcal{L}=\mathcal{L}_{f}(\mathcal{D})
  113. ( Ω , 𝒟 ) (\Omega,\mathcal{D})

Accelerated_failure_time_model.html

  1. λ ( t | θ ) = θ λ 0 ( θ t ) \lambda(t|\theta)=\theta\lambda_{0}(\theta t)
  2. θ \theta
  3. θ = exp ( - [ β 1 X 1 + + β p X p ] ) \theta=\exp(-[\beta_{1}X_{1}+\cdots+\beta_{p}X_{p}])
  4. f ( t | θ ) = θ f 0 ( θ t ) f(t|\theta)=\theta f_{0}(\theta t)
  5. S ( t | θ ) = S 0 ( θ t ) S(t|\theta)=S_{0}(\theta t)
  6. T T
  7. T θ T\theta
  8. T 0 T_{0}
  9. l o g ( T ) log(T)
  10. l o g ( T ) = - l o g ( θ ) + l o g ( T θ ) := - l o g ( θ ) + ϵ log(T)=-log(\theta)+log(T\theta):=-log(\theta)+\epsilon
  11. l o g ( T 0 ) log(T_{0})
  12. θ \theta
  13. - l o g ( θ ) -log(\theta)
  14. ϵ \epsilon
  15. ϵ \epsilon
  16. T 0 T_{0}
  17. T i > t i T_{i}>t_{i}
  18. T i = t i T_{i}=t_{i}
  19. T 0 T_{0}
  20. θ \theta
  21. θ = 2 \theta=2
  22. λ ( t | θ ) \lambda(t|\theta)
  23. l o g ( T 0 ) log(T_{0})
  24. S ( t | θ ) = 1 - F ( t | θ ) S(t|\theta)=1-F(t|\theta)

Accessible_category.html

  1. K K
  2. C C
  3. X X
  4. C C
  5. K K
  6. H o m ( X , - ) Hom(X,-)
  7. K K
  8. C C
  9. K K
  10. C C
  11. K K
  12. C C
  13. P P
  14. K K
  15. C C
  16. K K
  17. P P
  18. C C
  19. C C
  20. K K
  21. K K
  22. 0 \aleph_{0}
  23. 0 \aleph_{0}
  24. R R
  25. R R
  26. R R
  27. R R
  28. 1 \aleph_{1}
  29. 1 \aleph_{1}
  30. C C
  31. C C

Acetate_kinase_(diphosphate).html

  1. \rightleftharpoons

Acetate—CoA_ligase_(ADP-forming).html

  1. \rightleftharpoons

Acetoacetate—CoA_ligase.html

  1. \rightleftharpoons

Acetoacetyl-CoA_hydrolase.html

  1. \rightleftharpoons

Acetoin—ribose-5-phosphate_transaldolase.html

  1. \rightleftharpoons

Acetone_carboxylase.html

  1. \rightleftharpoons

Acetoxybutynylbithiophene_deacetylase.html

  1. \rightleftharpoons

Acetyl-CoA_C-acetyltransferase.html

  1. \rightleftharpoons

Acetyl-CoA_C-myristoyltransferase.html

  1. \rightleftharpoons

Acetyl-CoA_hydrolase.html

  1. \rightleftharpoons

Acetylalkylglycerol_acetylhydrolase.html

  1. \rightleftharpoons

Acetylenecarboxylate_hydratase.html

  1. \rightleftharpoons

Acetylesterase.html

  1. \rightleftharpoons

Acetylglutamate_kinase.html

  1. \rightleftharpoons

Acetylornithine_deacetylase.html

  1. \rightleftharpoons

Acetylornithine_transaminase.html

  1. \rightleftharpoons

Acetylputrescine_deacetylase.html

  1. \rightleftharpoons

Acetylsalicylate_deacetylase.html

  1. \rightleftharpoons

Acetylspermidine_deacetylase.html

  1. \rightleftharpoons

Acid—CoA_ligase_(GDP-forming).html

  1. \rightleftharpoons

Acridone_synthase.html

  1. \rightleftharpoons

Actinomycin_lactonase.html

  1. \rightleftharpoons

ACV_synthetase.html

  1. \rightleftharpoons

Acyl-(acyl-carrier-protein)—phospholipid_O-acyltransferase.html

  1. \rightleftharpoons

Acyl-(acyl-carrier-protein)—UDP-N-acetylglucosamine_O-acyltransferase.html

  1. \rightleftharpoons

Acyl-CoA_hydrolase.html

  1. \rightleftharpoons

Acyl-lysine_deacylase.html

  1. \rightleftharpoons

Acyl-phosphate—hexose_phosphotransferase.html

  1. \rightleftharpoons

Acylagmatine_amidase.html

  1. \rightleftharpoons

Acylcarnitine_hydrolase.html

  1. \rightleftharpoons

Acylglycerol_kinase.html

  1. \rightleftharpoons

Acyloxyacyl_hydrolase.html

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Adenine_deaminase.html

  1. \rightleftharpoons

Adenosine-phosphate_deaminase.html

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Adenosine-tetraphosphatase.html

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Adenosine_nucleosidase.html

  1. \rightleftharpoons

Adenosyl-fluoride_synthase.html

  1. \rightleftharpoons

Adenosylcobinamide-phosphate_guanylyltransferase.html

  1. \rightleftharpoons

Adenosylcobinamide_hydrolase.html

  1. \rightleftharpoons

Adenosylcobinamide_kinase.html

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Adenosylcobyric_acid_synthase_(glutamine-hydrolysing).html

  1. \rightleftharpoons

Adenosylhomocysteine_nucleosidase.html

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Adenosylmethionine_cyclotransferase.html

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Adenosylmethionine—8-amino-7-oxononanoate_transaminase.html

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Adenylate_dimethylallyltransferase.html

  1. \rightleftharpoons

Adenylyl-(glutamate—ammonia_ligase)_hydrolase.html

  1. \rightleftharpoons

Adenylyl-sulfate_kinase.html

  1. \rightleftharpoons

Adenylylsulfatase.html

  1. \rightleftharpoons

Adenylylsulfate—ammonia_adenylyltransferase.html

  1. \rightleftharpoons

Adiabatic_quantum_computation.html

  1. H ( t ) H(t)
  2. t t
  3. T = O ( 1 g m i n 2 ) T=O\left(\frac{1}{g_{min}^{2}}\right)
  4. g m i n g_{min}
  5. H ( t ) H(t)
  6. H = i h i Z i + i < j J i j Z i Z i + i < j K i j X i X i H=\sum_{i}h_{i}Z_{i}+\sum_{i<j}J^{ij}Z_{i}Z_{i}+\sum_{i<j}K^{ij}X_{i}X_{i}
  7. Z , X Z,X
  8. σ z , σ x \sigma_{z},\sigma_{x}
  9. C 1 C 2 C M C_{1}\wedge C_{2}\wedge\cdots\wedge C_{M}
  10. C i C_{i}
  11. x j { 0 , 1 } x_{j}\in\{0,1\}
  12. C i C_{i}
  13. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  14. H B H_{B}
  15. H B = H B 1 + H B 2 + + H B M H_{B}=H_{B_{1}}+H_{B_{2}}+\dots+H_{B_{M}}
  16. H B i H_{B_{i}}
  17. C i C_{i}
  18. H B i H_{B_{i}}
  19. H P H_{P}
  20. H P = C H P , C H_{P}=\sum\limits_{C}H_{P,C}
  21. H P , C H_{P,C}
  22. h C ( z 1 C , z 2 C z n C ) = { 0 c l a u s e C s a t i s f i e d 1 c l a u s e C v i o l a t e d h_{C}(z_{1C},z_{2C}\dots z_{nC})=\begin{cases}0&clause\ C\ satisfied\\ 1&clause\ C\ violated\end{cases}
  23. H ( t ) = ( 1 - t / T ) H B + ( t / T ) H P H(t)=(1-t/T)H_{B}+(t/T)H_{P}
  24. s = t / T s=t/T
  25. H ~ ( s ) = ( 1 - s ) H B + s H P \tilde{H}(s)=(1-s)H_{B}+sH_{P}
  26. H B H_{B}
  27. H P H_{P}
  28. z 1 , z 2 , , z n z_{1},z_{2},\dots,z_{n}
  29. ε / g m i n 2 \varepsilon/g_{min}^{2}
  30. g m i n = min 0 s 1 ( E 1 ( s ) - E 0 ( s ) ) g_{min}=\min_{0\leq s\leq 1}(E_{1}(s)-E_{0}(s))

ADP-dependent_medium-chain-acyl-CoA_hydrolase.html

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ADP-dependent_short-chain-acyl-CoA_hydrolase.html

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ADP-phosphoglycerate_phosphatase.html

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ADP-ribosyl-(dinitrogen_reductase)_hydrolase.html

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ADP-specific_glucokinase.html

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ADP-specific_phosphofructokinase.html

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ADP-sugar_diphosphatase.html

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ADP_deaminase.html

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ADP—thymidine_kinase.html

  1. \rightleftharpoons

AD–AS_model.html

  1. Y = Y d ( M P , G , T , Z 1 ) Y=Y^{d}(\tfrac{M}{P},G,T,Z_{1})
  2. Y = Y s ( W / P , P / P e , Z 2 ) Y=Y^{s}(W/P,\ \ P/P^{e},\ \ Z_{2})
  3. Y = Y s ( Z 2 ) Y=Y^{s}(Z_{2})

Aerobactin_synthase.html

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Ag+-exporting_ATPase.html

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Agaritine_gamma-glutamyltransferase.html

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Agmatinase.html

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Agmatine_deiminase.html

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Agmatine_kinase.html

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Agmatine_N4-coumaroyltransferase.html

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Alanine—glyoxylate_transaminase.html

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Alanine—oxo-acid_transaminase.html

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Alanine—oxomalonate_transaminase.html

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Alanine—tRNA_ligase.html

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Alanylphosphatidylglycerol_synthase.html

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Alcohol_O-acetyltransferase.html

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Alcohol_O-cinnamoyltransferase.html

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Aldose-1-phosphate_adenylyltransferase.html

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Aldose-1-phosphate_nucleotidyltransferase.html

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Aldose_beta-D-fructosyltransferase.html

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Algebraic-group_factorisation_algorithm.html

  1. d i r d_{i}r
  2. / N [ t ] \mathbb{Z}/N\mathbb{Z}[\sqrt{t}]
  3. 2 p 2\sqrt{p}

Algebraic_Reconstruction_Technique.html

  1. x x
  2. A A
  3. b b
  4. m × n m\times n
  5. A A
  6. b b
  7. x k + 1 = x k + λ k b i - a i , x k a i 2 a i x^{k+1}=x^{k}+\lambda_{k}\frac{b_{i}-\langle a_{i},x^{k}\rangle}{\lVert a_{i}% \rVert^{2}}a_{i}
  8. i = k mod m + 1 i=k\,\bmod\,m+1
  9. a i a_{i}
  10. A A
  11. b i b_{i}
  12. b b
  13. λ k \lambda_{k}

Alginate_synthase.html

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Aliphatic_nitrilase.html

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Alizarin_2-beta-glucosyltransferase.html

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Alkylacetylglycerophosphatase.html

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Alkylamidase.html

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Alkylglycerol_kinase.html

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Alkylglycerone_kinase.html

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Alkylglycerophosphate_2-O-acetyltransferase.html

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Alkylglycerophosphoethanolamine_phosphodiesterase.html

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All-trans-retinyl-palmitate_hydrolase.html

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Allantoate_deiminase.html

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Allantoicase.html

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Allantoinase.html

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Allophanate_hydrolase.html

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Allose_kinase.html

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Alpha,alpha-phosphotrehalase.html

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Alpha,alpha-trehalase.html

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Alpha,alpha-trehalose-phosphate_synthase_(GDP-forming).html

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Alpha,alpha-trehalose-phosphate_synthase_(UDP-forming).html

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Alpha,alpha-trehalose_phosphorylase.html

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Alpha,alpha-trehalose_phosphorylase_(configuration-retaining).html

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Alpha-1,3-glucan_synthase.html

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Alpha-1,4-glucan-protein_synthase_(ADP-forming).html

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Alpha-amino-acid_esterase.html

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Alpha-factor-transporting_ATPase.html

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Alpha-glucan,_water_dikinase.html

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Alpha-glucuronidase.html

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Alpha-L-fucosidase.html

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Alpha-N-acetylgalactosaminide_alpha-2,6-sialyltransferase.html

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Alpha-N-acetylneuraminate_alpha-2,8-sialyltransferase.html

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Alpha-ribazole_phosphatase.html

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Alpha-tubulin_N-acetyltransferase.html

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Alternating_direction_implicit_method.html

  1. u t = ( 2 u x 2 + 2 u y 2 ) = ( u x x + u y y ) = Δ u {\partial u\over\partial t}=\left({\partial^{2}u\over\partial x^{2}}+{\partial% ^{2}u\over\partial y^{2}}\right)=(u_{xx}+u_{yy})=\Delta u
  2. u i j n + 1 - u i j n Δ t = 1 2 ( δ x 2 + δ y 2 ) ( u i j n + 1 + u i j n ) {u_{ij}^{n+1}-u_{ij}^{n}\over\Delta t}={1\over 2}\left(\delta_{x}^{2}+\delta_{% y}^{2}\right)\left(u_{ij}^{n+1}+u_{ij}^{n}\right)
  3. δ p \delta_{p}
  4. Δ t \Delta t
  5. u i j n + 1 / 2 - u i j n Δ t / 2 = ( δ x 2 u i j n + 1 / 2 + δ y 2 u i j n ) {u_{ij}^{n+1/2}-u_{ij}^{n}\over\Delta t/2}=\left(\delta_{x}^{2}u_{ij}^{n+1/2}+% \delta_{y}^{2}u_{ij}^{n}\right)
  6. u i j n + 1 - u i j n + 1 / 2 Δ t / 2 = ( δ x 2 u i j n + 1 / 2 + δ y 2 u i j n + 1 ) . {u_{ij}^{n+1}-u_{ij}^{n+1/2}\over\Delta t/2}=\left(\delta_{x}^{2}u_{ij}^{n+1/2% }+\delta_{y}^{2}u_{ij}^{n+1}\right).

Altronate_dehydratase.html

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Amidinoaspartase.html

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Aminoacyl-tRNA_hydrolase.html

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Aminoacylase.html

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Aminodeoxychorismate_synthase.html

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Aminoglycoside_N3'-acetyltransferase.html

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Aminoglycoside_N6'-acetyltransferase.html

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Aminoimidazolase.html

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Aminolevulinate_transaminase.html

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Ammonia_kinase.html

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AMP_nucleosidase.html

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AMP—thymidine_kinase.html

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Amygdalin_beta-glucosidase.html

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Amylosucrase.html

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Analytic_torsion.html

  1. ζ i ( s ) = λ j > 0 λ j - s \zeta_{i}(s)=\sum_{\lambda_{j}>0}\lambda_{j}^{-s}
  2. Δ i = exp ( - ζ i ( 0 ) ) \Delta_{i}=\exp(-\zeta^{\prime}_{i}(0))
  3. T ( M , E ) = exp ( i ( - 1 ) i i ζ i ( 0 ) / 2 ) = i Δ i - ( - 1 ) i i / 2 . T(M,E)=\exp\left(\sum_{i}(-1)^{i}i\zeta^{\prime}_{i}(0)/2\right)=\prod_{i}% \Delta_{i}^{-(-1)^{i}i/2}.
  4. X X
  5. π := π 1 ( X ) \pi:=\pi_{1}(X)
  6. X ~ {\tilde{X}}
  7. U U
  8. π \pi
  9. H n π ( X ; U ) := H n ( U 𝐙 [ π ] C * ( X ~ ) ) = 0 H^{\pi}_{n}(X;U):=H_{n}(U\otimes_{\mathbf{Z}[\pi]}C_{*}({\tilde{X}}))=0
  10. C * ( X ~ ) C_{*}({\tilde{X}})
  11. 𝐑 \mathbf{R}
  12. U U
  13. D * := U 𝐙 [ π ] C * ( X ~ ) D_{*}:=U\otimes_{\mathbf{Z}[\pi]}C_{*}({\tilde{X}})
  14. 𝐑 \mathbf{R}
  15. γ * : D * D * + 1 \gamma_{*}:D_{*}\to D_{*+1}
  16. d n + 1 γ n + γ n - 1 d n = i d D n d_{n+1}\circ\gamma_{n}+\gamma_{n-1}\circ d_{n}=id_{D_{n}}
  17. ( d * + γ * ) o d d : D o d d D e v e n (d_{*}+\gamma_{*})_{odd}:D_{odd}\to D_{even}
  18. D o d d := n o d d D n D_{odd}:=\oplus_{n\,odd}\,D_{n}
  19. D e v e n := n e v e n D n D_{even}:=\oplus_{n\,even}\,D_{n}
  20. ρ ( X ; U ) := | d e t ( A ) | - 1 𝐑 > 0 \rho(X;U):=|\mathop{det}(A)|^{-1}\in\mathbf{R}^{>0}
  21. ( d * + γ * ) o d d (d_{*}+\gamma_{*})_{odd}
  22. ρ ( X ; U ) \rho(X;U)
  23. C * ( X ~ ) C_{*}({\tilde{X}})
  24. U U
  25. γ * \gamma_{*}
  26. M M
  27. ρ : π ( M ) G L ( E ) \rho:\pi(M)\rightarrow GL(E)
  28. M M
  29. μ d e t H * ( M ) \mu\in\mathop{det}H_{*}(M)
  30. τ M ( ρ : μ ) 𝐑 + \tau_{M}(\rho:\mu)\in\mathbf{R}^{+}
  31. τ M ( ρ : μ ) \tau_{M}(\rho:\mu)
  32. M M
  33. ρ \rho
  34. μ \mu
  35. P o P_{o}
  36. P o : det ( H q ( M ) ) ( det ( H n - q ( M ) ) ) - 1 P_{o}:\operatorname{det}(H_{q}(M))\stackrel{\sim}{\longrightarrow}(% \operatorname{det}(H_{n-q}(M)))^{-1}
  37. Δ ( t ) = ± t n Δ ( 1 / t ) . \Delta(t)=\pm t^{n}\Delta(1/t).
  38. ( M , g ) (M,g)
  39. ρ : π ( M ) G L ( E ) \rho:\pi(M)\rightarrow\mathop{GL}(E)
  40. M M
  41. Λ 0 d 0 Λ 1 d 1 d n - 1 Λ n \Lambda^{0}\stackrel{d_{0}}{\longrightarrow}\Lambda^{1}\stackrel{d_{1}}{% \longrightarrow}\cdots\stackrel{d_{n-1}}{\longrightarrow}\Lambda^{n}
  42. d p d_{p}
  43. δ p \delta_{p}
  44. E q E_{q}
  45. Δ p = δ p d p + d p - 1 δ p - 1 . \Delta_{p}=\delta_{p}d_{p}+d_{p-1}\delta_{p-1}.
  46. M = 0 \partial M=0
  47. 0 λ 0 λ 1 . 0\leq\lambda_{0}\leq\lambda_{1}\leq\cdots\rightarrow\infty.
  48. Δ q \Delta_{q}
  49. Λ q ( E ) \Lambda^{q}(E)
  50. ζ q ( s ; ρ ) = λ j > 0 λ j - s = 1 Γ ( s ) 0 t s - 1 T r ( e - t Δ q - P q ) d t , R e ( s ) > n 2 \zeta_{q}(s;\rho)=\sum_{\lambda_{j}>0}\lambda_{j}^{-s}=\frac{1}{\Gamma(s)}\int% ^{\infty}_{0}t^{s-1}\mathop{Tr}(e^{-t\Delta_{q}}-P_{q})dt,\ \ \ \mathop{Re}(s)% >\frac{n}{2}
  51. P P
  52. L 2 Λ ( E ) L^{2}\Lambda(E)
  53. q ( E ) \mathcal{H}^{q}(E)
  54. Δ q \Delta_{q}
  55. ζ q ( s ; ρ ) \zeta_{q}(s;\rho)
  56. s 𝐂 s\in\mathbf{C}
  57. s = 0 s=0
  58. T M ( ρ ; E ) T_{M}(\rho;E)
  59. T M ( ρ ; E ) = exp ( 1 2 q = 0 n ( - l ) q q d d s ζ q ( s ; ρ ) | s = 0 ) . T_{M}(\rho;E)=\exp\biggl(\frac{1}{2}\sum^{n}_{q=0}(-l)^{q}q\frac{d}{ds}\zeta_{% q}(s;\rho)\biggl|_{s=0}\biggr).
  60. T M ( ρ ; E ) = τ M ( ρ ; μ ) T_{M}(\rho;E)=\tau_{M}(\rho;\mu)
  61. ρ \rho

Anthocyanidin_3-O-glucosyltransferase.html

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Anthocyanin_3'-O-beta-glucosyltransferase.html

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Anthocyanin_5-aromatic_acyltransferase.html

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Anthranilate_adenylyltransferase.html

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Anthranilate_N-benzoyltransferase.html

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Anthranilate_N-malonyltransferase.html

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Anthranilate_phosphoribosyltransferase.html

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Anthranilate—CoA_ligase.html

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Applications_of_multiple_coordinate_systems.html

  1. v = i = 1 n x i a i v=\sum_{i=1}^{n}x_{i}a_{i}
  2. v = i = 1 n x i a i v=\sum_{i=1}^{n}x_{i}a_{i}
  3. b 1 = x 11 a 1 + x 12 a 2 + + x 1 n a n b 2 = x 21 a 1 + x 22 a 2 + + x 2 n a n b n = x n 1 a 1 + x n 2 a 2 + + x n n a n \begin{array}[]{lcl}b_{1}=x_{11}a_{1}+x_{12}a_{2}+\cdots+x_{1n}a_{n}\\ b_{2}=x_{21}a_{1}+x_{22}a_{2}+\cdots+x_{2n}a_{n}\\ \cdots\\ b_{n}=x_{n1}a_{1}+x_{n2}a_{2}+\cdots+x_{nn}a_{n}\\ \end{array}
  4. ( P | I ) \begin{pmatrix}P|I\end{pmatrix}
  5. ( I | P - 1 ) \begin{pmatrix}I|P^{-1}\end{pmatrix}
  6. b 1 = x 11 a 1 + x 12 a 2 + + x 1 n a n b_{1}=x_{11}a_{1}+x_{12}a_{2}+\cdots+x_{1n}a_{n}
  7. [ 1 0 0 1 ] \begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  8. [ 1 0 0 1 ] \begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  9. [ 1 0 0 1 ] \begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  10. [ 100 50 ] \begin{bmatrix}100\\ 50\end{bmatrix}
  11. x A = 1 × x B + 0 × y B + 100 = 1 × 4 + 100 = 104 y A = 0 × x B + 1 × y B + 50 = 1 × 10 + 50 = 60 \begin{array}[]{lcl}x_{A}=1\times x_{B}+0\times y_{B}+100&=1\times 4+100&=104% \\ y_{A}=0\times x_{B}+1\times y_{B}+50&=1\times 10+50&=60\end{array}
  12. c 1 = ( 1 c o s ( - 45 ) , 1 s i n ( - 45 ) ) = ( 1 / 2 , - 1 / 2 ) = a 1 / 2 - a 2 / 2 c 2 = ( 1 c o s ( 45 ) , 1 s i n ( 45 ) ) = ( 1 / 2 , 1 / 2 ) = a 1 / 2 + a 2 / 2 \begin{array}[]{lcr}c_{1}=(1cos(-45),1sin(-45))&=(1/\sqrt{2},-1/\sqrt{2})&=a_{% 1}/\sqrt{2}-a_{2}/\sqrt{2}\\ c_{2}=(1cos(45),1sin(45))&=(1/\sqrt{2},1/\sqrt{2})&=a_{1}/\sqrt{2}+a_{2}/\sqrt% {2}\end{array}
  13. [ 1 / 2 - 1 / 2 1 / 2 1 / 2 ] \begin{bmatrix}1/\sqrt{2}&-1/\sqrt{2}\\ 1/\sqrt{2}&1/\sqrt{2}\end{bmatrix}
  14. [ 1 / 2 1 / 2 - 1 / 2 1 / 2 ] \begin{bmatrix}1/\sqrt{2}&1/\sqrt{2}\\ -1/\sqrt{2}&1/\sqrt{2}\end{bmatrix}
  15. [ 1 / 2 1 / 2 - 1 / 2 1 / 2 ] \begin{bmatrix}1/\sqrt{2}&1/\sqrt{2}\\ -1/\sqrt{2}&1/\sqrt{2}\end{bmatrix}
  16. [ 100 50 ] \begin{bmatrix}100\\ 50\end{bmatrix}
  17. [ 1 0 ] \begin{bmatrix}1\\ 0\end{bmatrix}
  18. [ 1 / 2 1 / 2 - 1 / 2 1 / 2 ] [ 1 0 ] + [ 100 50 ] = [ 1 / 2 + 100 - 1 / 2 + 50 ] \begin{bmatrix}1/\sqrt{2}&1/\sqrt{2}\\ -1/\sqrt{2}&1/\sqrt{2}\end{bmatrix}\begin{bmatrix}1\\ 0\end{bmatrix}+\begin{bmatrix}100\\ 50\end{bmatrix}=\begin{bmatrix}1/\sqrt{2}+100\\ -1/\sqrt{2}+50\end{bmatrix}

Apply.html

  1. g : ( X × Y ) Z g:(X\times Y)\to Z
  2. g \isin [ ( X × Y ) Z ] g\isin[(X\times Y)\to Z]
  3. [ A B ] [A\to B]
  4. curry ( g ) : X [ Y Z ] \mbox{curry}~{}(g):X\to[Y\to Z]
  5. Apply : ( [ Y Z ] × Y ) Z \mbox{Apply}~{}:([Y\to Z]\times Y)\to Z
  6. Apply ( f , y ) = f ( y ) \mbox{Apply}(f,y)=f(y)
  7. Apply ( curry ( g ) × id ) Y = g \mbox{Apply}~{}\circ\left(\mbox{curry}~{}(g)\times\mbox{id}~{}_{Y}\right)=g
  8. [ A B ] [A\to B]
  9. [ A B ] [A\to B]
  10. B A B^{A}
  11. λ \lambda
  12. λ g \lambda g
  13. λ \lambda

Arabinonate_dehydratase.html

  1. \rightleftharpoons

Arachidonate—CoA_ligase.html

  1. \rightleftharpoons

Arginine_deiminase.html

  1. \rightleftharpoons

Arginine_kinase.html

  1. \rightleftharpoons

Arginine_N-succinyltransferase.html

  1. \rightleftharpoons

Arginine—pyruvate_transaminase.html

  1. \rightleftharpoons

Arginine—tRNA_ligase.html

  1. \rightleftharpoons

Arginyltransferase.html

  1. \rightleftharpoons

Argument_(complex_analysis).html

  1. φ φ
  2. φ φ
  3. z = x + i y z=x+iy
  4. a r g z argz
  5. φ φ
  6. z z
  7. φ φ
  8. z = r ( cos φ + i sin φ ) z=r(\cos\varphi+i\sin\varphi)
  9. r r
  10. r r
  11. z z
  12. z z
  13. r = x 2 + y 2 . r=\sqrt{x^{2}+y^{2}}.
  14. 2 π
  15. s i n sin
  16. c o s cos
  17. A r g Arg
  18. 1 + i 1+i
  19. π / 4 π/4
  20. φ φ
  21. ( π r a d , π r a d ] (−πrad,πrad]
  22. π −π
  23. π π
  24. π −π
  25. [ 0 , 2 π ) [0,2π)
  26. A r g z Argz
  27. a r g arg
  28. A r g Arg
  29. A r g Arg
  30. arg z = { Arg z + 2 π n | n } . \arg z=\{\operatorname{Arg}z+2\pi n\;|\;n\in\mathbb{Z}\}.
  31. a r g arg
  32. a r g arg
  33. a r g z ( t ) argz(t)
  34. z z
  35. t t
  36. 2 π
  37. z z
  38. z ( t ) z(t)
  39. a r g arg
  40. arg : + { 0 } × ( r , φ ) φ . \begin{aligned}\displaystyle\arg:\mathbb{R}^{+}\smallsetminus\{0\}\times% \mathbb{R}&\displaystyle\to\mathbb{R}\\ \displaystyle(r,\,\varphi)&\displaystyle\mapsto\varphi.\end{aligned}
  41. { 0 } \mathbb{C}\smallsetminus\{0\}
  42. + { 0 } × 𝕊 1 . \mathbb{R}^{+}\smallsetminus\{0\}\times\mathbb{S}^{1}.
  43. A r g Arg
  44. ( π , π ] (−π,π]
  45. Arg : + { 0 } × \displaystyle\operatorname{Arg}:\mathbb{R}^{+}\smallsetminus\{0\}\times\mathbb% {R}
  46. A r g Arg
  47. x + i y x+iy
  48. a t a n 2 ( y , x ) atan2(y,x)
  49. ( π , π ] (−π,π]
  50. a r c t a n ( y / x ) arctan(y/x)
  51. y / x y/x
  52. a r c t a n arctan
  53. x > 0 x>0
  54. π / 2 −π/2
  55. π / 2 π/2
  56. x x
  57. x > 0 x>0
  58. x x
  59. y > 0 y>0
  60. Arg ( x + i y ) = atan2 ( y , x ) = { arctan ( y x ) if x > 0 , arctan ( y x ) + π if x < 0 and y 0 , arctan ( y x ) - π if x < 0 and y < 0 , + π 2 if x = 0 and y > 0 , - π 2 if x = 0 and y < 0 , undefined if x = 0 and y = 0. \operatorname{Arg}(x+iy)=\operatorname{atan2}(y,\,x)=\begin{cases}\arctan(% \frac{y}{x})&\,\text{if }x>0,\\ \arctan(\frac{y}{x})+\pi&\,\text{if }x<0\,\text{ and }y\geq 0,\\ \arctan(\frac{y}{x})-\pi&\,\text{if }x<0\,\text{ and }y<0,\\ +\frac{\pi}{2}&\,\text{if }x=0\,\text{ and }y>0,\\ -\frac{\pi}{2}&\,\text{if }x=0\,\text{ and }y<0,\\ \,\text{undefined}&\,\text{if }x=0\,\text{ and }y=0.\end{cases}
  61. A r g Arg
  62. [ 0 , 2 π ) [0,2π)
  63. 2 π
  64. Arg ( x + i y ) = { 2 arctan ( y x 2 + y 2 + x ) if x > 0 or y 0 , π if x < 0 and y = 0 , undefined if x = 0 and y = 0. \operatorname{Arg}(x+iy)=\begin{cases}2\arctan\biggl(\frac{y}{\sqrt{x^{2}+y^{2% }}+x}\biggr)&\,\text{if }x>0\,\text{ or }y\neq 0,\\ \pi&\,\text{if }x<0\,\text{ and }y=0,\\ \,\text{undefined}&\,\text{if }x=0\,\text{ and }y=0.\end{cases}
  65. x x
  66. A r g Arg
  67. Arg ( x + i y ) = { 2 arctan ( x 2 + y 2 - x y ) if y 0 , 0 if x > 0 and y = 0 , π if x < 0 and y = 0 , undefined if x = 0 and y = 0. \operatorname{Arg}(x+iy)=\begin{cases}2\arctan\biggl(\frac{\sqrt{x^{2}+y^{2}}-% x}{y}\biggr)&\,\text{if }y\neq 0,\\ 0&\,\text{if }x>0\,\text{ and }y=0,\\ \pi&\,\text{if }x<0\,\text{ and }y=0,\\ \,\text{undefined}&\,\text{if }x=0\,\text{ and }y=0.\end{cases}
  68. A r g Arg
  69. z z
  70. z = | z | e i Arg z . z=\left|z\right|e^{i\operatorname{Arg}z}.
  71. z z
  72. z = 0 z=0
  73. A r g ( 0 ) Arg(0)
  74. Arg ( z 1 z 2 ) Arg ( z 1 ) + Arg ( z 2 ) ( mod ( - π , π ] ) , \operatorname{Arg}(z_{1}z_{2})\equiv\operatorname{Arg}(z_{1})+\operatorname{% Arg}(z_{2})\;\;(\mathop{{\rm mod}}(-\pi,\pi]),
  75. Arg ( z 1 z 2 ) Arg ( z 1 ) - Arg ( z 2 ) ( mod ( - π , π ] ) . \operatorname{Arg}\biggl(\frac{z_{1}}{z_{2}}\biggr)\equiv\operatorname{Arg}(z_% {1})-\operatorname{Arg}(z_{2})\;\;(\mathop{{\rm mod}}(-\pi,\pi]).
  76. z 0 z≠0
  77. n n
  78. Arg ( z n ) n Arg ( z ) ( mod ( - π , π ] ) . \operatorname{Arg}\left(z^{n}\right)\equiv n\operatorname{Arg}(z)\;\;(\mathop{% {\rm mod}}(-\pi,\pi]).
  79. Arg ( - 1 - i i ) = Arg ( - 1 - i ) - Arg ( i ) = - 3 π 4 - π 2 = - 5 π 4 = 3 π 4 ( mod ( - π , π ] ) . \operatorname{Arg}\biggl(\frac{-1-i}{i}\biggr)=\operatorname{Arg}(-1-i)-% \operatorname{Arg}(i)=-\frac{3\pi}{4}-\frac{\pi}{2}=-\frac{5\pi}{4}=\frac{3\pi% }{4}\;\;(\mathop{{\rm mod}}(-\pi,\pi]).

Aristolochene_synthase.html

  1. \rightleftharpoons

Aromatic-amino-acid_transaminase.html

  1. \rightleftharpoons

Aromatic-amino-acid—glyoxylate_transaminase.html

  1. \rightleftharpoons

Aromatic-hydroxylamine_O-acetyltransferase.html

  1. \rightleftharpoons

Arsenite-transporting_ATPase.html

  1. \rightleftharpoons

Artstein's_theorem.html

  1. u ( x ) u(x)

Aryl-acylamidase.html

  1. \rightleftharpoons

Arylacetonitrilase.html

  1. \rightleftharpoons

Arylalkyl_acylamidase.html

  1. \rightleftharpoons

Arylamine_glucosyltransferase.html

  1. \rightleftharpoons

Arylamine_N-acetyltransferase.html

  1. \rightleftharpoons

Aryldialkylphosphatase.html

  1. \rightleftharpoons

Arylesterase.html

  1. \rightleftharpoons

Arylformamidase.html

  1. \rightleftharpoons

Asparagine—oxo-acid_transaminase.html

  1. \rightleftharpoons

Asparagine—tRNA_ligase.html

  1. \rightleftharpoons

Asparaginyl-tRNA_synthase_(glutamine-hydrolysing).html

  1. \rightleftharpoons

Aspartate_N-acetyltransferase.html

  1. \rightleftharpoons

Aspartate—ammonia_ligase.html

  1. \rightleftharpoons

Aspartate—ammonia_ligase_(ADP-forming).html

  1. \rightleftharpoons

Aspartate—phenylpyruvate_transaminase.html

  1. \rightleftharpoons

Aspartate—prephenate_aminotransferase.html

  1. \rightleftharpoons

Aspartate—tRNA(Asn)_ligase.html

  1. \rightleftharpoons

Aspartate—tRNA_ligase.html

  1. \rightleftharpoons

Aspartyltransferase.html

  1. \rightleftharpoons

Aspulvinone_dimethylallyltransferase.html

  1. \rightleftharpoons

Asymptotic_computational_complexity.html

  1. O ( n 3 ) . O(n^{3}).

ATP-dependent_NAD(P)H-hydrate_dehydratase.html

  1. \rightleftharpoons

ATP_adenylyltransferase.html

  1. \rightleftharpoons

ATP_citrate_synthase.html

  1. \rightleftharpoons

ATP_deaminase.html

  1. \rightleftharpoons

ATP_diphosphatase.html

  1. \rightleftharpoons

ATP_phosphoribosyltransferase.html

  1. \rightleftharpoons

Atwood_number.html

  1. A = ρ 1 - ρ 2 ρ 1 + ρ 2 \mathrm{A}=\frac{\rho_{1}-\rho_{2}}{\rho_{1}+\rho_{2}}
  2. ρ 1 \rho_{1}
  3. ρ 2 \rho_{2}
  4. A g t 2 \mathrm{A}gt^{2}

Aubin–Lions_lemma.html

  1. W = { u L p ( [ 0 , T ] ; X 0 ) | u ˙ L q ( [ 0 , T ] ; X 1 ) } . W=\{u\in L^{p}([0,T];X_{0})|\dot{u}\in L^{q}([0,T];X_{1})\}.

Autoacceleration.html

  1. R p = k p [ M ] ( f k d [ I ] k t ) 1 / 2 R_{p}=k_{p}\cdot[M]\left(\frac{f\cdot k_{d}\cdot[I]}{k_{t}}\right)^{1/2}

Autocatalytic_reaction.html

  1. α A + β B σ S + τ T \alpha A+\beta B\rightleftharpoons\sigma S+\tau T
  2. k + { A } α { B } β = k - { S } σ { T } τ k_{+}\left\{A\right\}^{\alpha}\left\{B\right\}^{\beta}=k_{-}\left\{S\right\}^{% \sigma}\left\{T\right\}^{\tau}\,
  3. α \alpha
  4. α \alpha
  5. d d t { A } = - α k + { A } α { B } β + α k - { S } σ { T } τ {d\over dt}\left\{A\right\}=-\alpha k_{+}\left\{A\right\}^{\alpha}\left\{B% \right\}^{\beta}+\alpha k_{-}\left\{S\right\}^{\sigma}\left\{T\right\}^{\tau}\,
  6. d d t { B } = - β k + { A } α { B } β + β k - { S } σ { T } τ {d\over dt}\left\{B\right\}=-\beta k_{+}\left\{A\right\}^{\alpha}\left\{B% \right\}^{\beta}+\beta k_{-}\left\{S\right\}^{\sigma}\left\{T\right\}^{\tau}\,
  7. d d t { S } = σ k + { A } α { B } β - σ k - { S } σ { T } τ {d\over dt}\left\{S\right\}=\sigma k_{+}\left\{A\right\}^{\alpha}\left\{B% \right\}^{\beta}-\sigma k_{-}\left\{S\right\}^{\sigma}\left\{T\right\}^{\tau}\,
  8. d d t { T } = τ k + { A } α { B } β - τ k - { S } σ { T } τ {d\over dt}\left\{T\right\}=\tau k_{+}\left\{A\right\}^{\alpha}\left\{B\right% \}^{\beta}-\tau k_{-}\left\{S\right\}^{\sigma}\left\{T\right\}^{\tau}\,
  9. A + B 2 B A+B\rightleftharpoons 2B
  10. d d t { A } = - k + { A } { B } + k - { B } 2 {d\over dt}\left\{A\right\}=-k_{+}\left\{A\right\}\left\{B\right\}+k_{-}\left% \{B\right\}^{2}\,
  11. d d t { B } = + k + { A } { B } - k - { B } 2 {d\over dt}\left\{B\right\}=+k_{+}\left\{A\right\}\left\{B\right\}-k_{-}\left% \{B\right\}^{2}\,
  12. A + X 2 X A+X\rightarrow 2X
  13. X + Y 2 Y X+Y\rightarrow 2Y
  14. Y E Y\rightarrow E
  15. d d t { X } = k 1 { A } { X } - k 2 { X } { Y } {d\over dt}\left\{X\right\}=k_{1}\left\{A\right\}\left\{X\right\}-k_{2}\left\{% X\right\}\left\{Y\right\}\,
  16. d d t { Y } = k 2 { X } { Y } - k 3 { Y } {d\over dt}\left\{Y\right\}=k_{2}\left\{X\right\}\left\{Y\right\}-k_{3}\left\{% Y\right\}\,
  17. k 1 k_{1}
  18. k 2 k_{2}
  19. k 3 k_{3}
  20. A X A\rightarrow X
  21. 2 X + Y 3 X 2X+Y\rightarrow 3X
  22. B + X Y + D B+X\rightarrow Y+D
  23. X E X\rightarrow E
  24. d d t { X } = { A } + { X } 2 { Y } - { B } { X } - { X } {d\over dt}\left\{X\right\}=\left\{A\right\}+\left\{X\right\}^{2}\left\{Y% \right\}-\left\{B\right\}\left\{X\right\}-\left\{X\right\}\,
  25. d d t { Y } = { B } { X } - { X } 2 { Y } {d\over dt}\left\{Y\right\}=\left\{B\right\}\left\{X\right\}-\left\{X\right\}^% {2}\left\{Y\right\}\,
  26. { Y } = B A \left\{Y\right\}={B\over A}\,
  27. B > 1 + A 2 B>1+A^{2}\,
  28. ( K B r O 3 ) (KBrO_{3})
  29. ( C H 2 ( C O O H ) 2 ) (CH_{2}(COOH)_{2})
  30. ( M n S O 4 ) (MnSO_{4})
  31. ( H 2 S O 4 ) (H_{2}SO_{4})
  32. d d t { X 1 } = { A } + { X 1 } 2 { Y 1 } - { B } { X 1 } - { X 1 } + D x ( X 2 - X 1 ) {d\over dt}\left\{X_{1}\right\}=\left\{A\right\}+\left\{X_{1}\right\}^{2}\left% \{Y_{1}\right\}-\left\{B\right\}\left\{X_{1}\right\}-\left\{X_{1}\right\}+D_{x% }\left(X_{2}-X_{1}\right)\,
  33. d d t { Y 1 } = { B } { X 1 } - { X 1 } 2 { Y 1 } + D y ( Y 2 - Y 1 ) {d\over dt}\left\{Y_{1}\right\}=\left\{B\right\}\left\{X_{1}\right\}-\left\{X_% {1}\right\}^{2}\left\{Y_{1}\right\}+D_{y}\left(Y_{2}-Y_{1}\right)\,
  34. d d t { X 2 } = { A } + { X 2 } 2 { Y 2 } - { B } { X 2 } - { X 2 } + D x ( X 1 - X 2 ) {d\over dt}\left\{X_{2}\right\}=\left\{A\right\}+\left\{X_{2}\right\}^{2}\left% \{Y_{2}\right\}-\left\{B\right\}\left\{X_{2}\right\}-\left\{X_{2}\right\}+D_{x% }\left(X_{1}-X_{2}\right)\,
  35. d d t { Y 2 } = { B } { X 2 } - { X 2 } 2 { Y 2 } + D y ( Y 1 - Y 2 ) {d\over dt}\left\{Y_{2}\right\}=\left\{B\right\}\left\{X_{2}\right\}-\left\{X_% {2}\right\}^{2}\left\{Y_{2}\right\}+D_{y}\left(Y_{1}-Y_{2}\right)\,
  36. glucose + 2 ADP + 2 P + i 2 NAD 2 ( pyruvate ) + 2 ATP + 2 NADH \mbox{glucose}~{}+2\mbox{ADP}~{}+2\mbox{P}~{}_{i}+2\mbox{NAD}~{}\rightarrow 2(% \mbox{pyruvate}~{})+2\mbox{ATP}~{}+2\mbox{NADH}~{}\,

Autonomous_convergence_theorem.html

  1. x x
  2. X n X\subseteq\mathbb{R}^{n}
  3. x ˙ = f ( x ) \dot{x}=f(x)
  4. X X
  5. f f
  6. x ^ X \hat{x}\in X
  7. f ( x ^ ) = 0 f(\hat{x})=0
  8. μ \mu
  9. J ( x ) = D x f J(x)=D_{x}f
  10. μ ( J ( x ) ) < 0 \mu(J(x))<0
  11. x x
  12. x ^ \hat{x}
  13. f f

Autoregressive_conditional_duration.html

  1. τ t ~{}\tau_{t}~{}
  2. τ t = θ t z t ~{}\tau_{t}=\theta_{t}z_{t}~{}
  3. z t z_{t}
  4. E ( z t ) = 1 \operatorname{E}(z_{t})=1
  5. θ t ~{}\theta_{t}~{}
  6. θ t = α 0 + α 1 τ t - 1 + + α q τ t - q + β 1 θ t - 1 + + β p θ t - p = α 0 + i = 1 q α i τ t - i + i = 1 p β i θ t - i \theta_{t}=\alpha_{0}+\alpha_{1}\tau_{t-1}+\cdots+\alpha_{q}\tau_{t-q}+\beta_{% 1}\theta_{t-1}+\cdots+\beta_{p}\theta_{t-p}=\alpha_{0}+\sum_{i=1}^{q}\alpha_{i% }\tau_{t-i}+\sum_{i=1}^{p}\beta_{i}\theta_{t-i}
  7. α 0 > 0 ~{}\alpha_{0}>0~{}
  8. α i 0 \alpha_{i}\geq 0
  9. β i 0 \beta_{i}\geq 0
  10. i > 0 ~{}i>0

Average-case_complexity.html

  1. Pr x R D n [ t A ( x ) t ] p ( n ) t ϵ \Pr_{x\in_{R}D_{n}}[t_{A}(x)\geq t]\leq\frac{p(n)}{t^{\epsilon}}
  2. E x R D n [ t A ( x ) ϵ n ] C E_{x\in_{R}D_{n}}[\frac{t_{A}(x)^{\epsilon}}{n}]\leq C
  3. n c ( t A ( n ) ) ϵ \frac{n^{c}}{(t_{A}(n))^{\epsilon}}
  4. μ ( x ) = y { 0 , 1 } n : y x Pr [ y ] \mu(x)=\sum\limits_{y\in\{0,1\}^{n}:y\leq x}\Pr[y]
  5. x : f ( x ) = y D n ( x ) p ( n ) D m ( n ) ( y ) \sum\limits_{x:f(x)=y}D_{n}(x)\leq p(n)D^{\prime}_{m(n)}(y)

Average_treatment_effect.html

  1. y 0 i y_{0i}
  2. i i
  3. y 1 i y_{1i}
  4. i i
  5. y 0 i y_{0i}
  6. y 1 i y_{1i}
  7. i i
  8. y 1 i - y 0 i = β i y_{1i}-y_{0i}=\beta_{i}
  9. E [ . ] E[.]
  10. E [ y 1 i - y 0 i ] E[y_{1i}-y_{0i}]
  11. y 1 i y_{1i}
  12. y 0 i y_{0i}
  13. y 1 i - y 0 i y_{1i}-y_{0i}
  14. 1 N i = 1 N ( y 1 i - y 0 i ) \frac{1}{N}\cdot\sum_{i=1}^{N}(y_{1i}-y_{0i})
  15. N N
  16. y 1 i y_{1i}
  17. y 0 i y_{0i}
  18. y 1 i y_{1i}
  19. y 0 i y_{0i}
  20. y 0 i y_{0i}
  21. y 1 i y_{1i}
  22. y = B 0 + δ 0 d 2 + B 1 d T + δ 1 d 2 d T , y=B_{0}+\delta_{0}d2+B_{1}dT+\delta_{1}d2\cdot dT,
  23. δ 1 \delta_{1}
  24. δ ^ 1 = ( y ¯ 2 , T - y ¯ 1 , T ) - ( y ¯ 2 , C - y ¯ 1 , C ) , \hat{\delta}_{1}=(\bar{y}_{2,T}-\bar{y}_{1,T})-(\bar{y}_{2,C}-\bar{y}_{1,C}),
  25. δ 1 \delta_{1}

Bag-of-words_model_in_computer_vision.html

  1. V V
  2. w w
  3. w w
  4. w w
  5. v v
  6. w v = 1 w^{v}=1
  7. w u = 0 w^{u}=0
  8. u v u\neq v
  9. 𝐰 \mathbf{w}
  10. 𝐰 = [ w 1 , w 2 , , w N ] \mathbf{w}=[w_{1},w_{2},\cdots,w_{N}]
  11. d j d_{j}
  12. j j
  13. c c
  14. z z
  15. π \pi
  16. c * = arg max c p ( c | 𝐰 ) = arg max c p ( c ) p ( 𝐰 | c ) = arg max c p ( c ) n = 1 N p ( w n | c ) c^{*}=\arg\max_{c}p(c|\mathbf{w})=\arg\max_{c}p(c)p(\mathbf{w}|c)=\arg\max_{c}% p(c)\prod_{n=1}^{N}p(w_{n}|c)
  17. X 2 X^{2}

Balance_puzzle.html

  1. 3 × 3 = 9 \scriptstyle 3\,\times\,3\;=\;9
  2. n \mathbb{R}^{n}
  3. n n
  4. [ e 1 , e 2 ] [\mathrm{e}^{1},\mathrm{e}^{2}]
  5. e 1 \mathrm{e}^{1}
  6. e 2 \mathrm{e}^{2}
  7. n , \mathbb{R}^{n},
  8. e = ( e 1 , , e n ) n \mathrm{e}=(e_{1},\dots,e_{n})\in\mathbb{R}^{n}
  9. E = { e j } n , E=\{\mathrm{e}^{j}\}\subseteq\mathbb{R}^{n},
  10. ( ) * (\cdot)^{*}
  11. ( ) + (\cdot)^{+}
  12. e * = ( s i g n ( e i ) ) i \mathrm{e}^{*}=(sign(e_{i}))_{i}
  13. E * = { ( e j ) * } E^{*}=\{(\mathrm{e}^{j})^{*}\}
  14. e + = ( | s i g n ( e i ) | ) i \mathrm{e}^{+}=(|sign(e_{i})|)_{i}
  15. E + = { ( e j ) + } . E^{+}=\{(\mathrm{e}^{j})^{+}\}.
  16. I n I^{n}
  17. n \mathbb{R}^{n}
  18. n n
  19. I = { - 1 , 0 , 1 } I=\{-1,0,1\}
  20. I t n = { x I n | w ( x ) t } I n I^{n}_{t}=\{\mathrm{x}\in I^{n}|w(\mathrm{x})\leq t\}\subseteq I^{n}
  21. t t
  22. w ( ) w()
  23. 0. \mathrm{0}.
  24. n n
  25. x = ( x 1 , , x n ) I n , \mathrm{x}=(x_{1},\dots,x_{n})\in I^{n},
  26. i i
  27. x i = 0 ; x_{i}=0;
  28. i i
  29. x i = 1 x_{i}=1
  30. x i = - 1 x_{i}=-1
  31. x + \mathrm{x}^{+}
  32. h I n ; \mathrm{h}\in I^{n};
  33. x I n \mathrm{x}\in I^{n}
  34. s ( x ; h ) = s i g n ( [ x ; h ] ) . s(\mathrm{x};\mathrm{h})=sign([\mathrm{x};\mathrm{h}]).
  35. h = ( h 1 , , h n ) \mathrm{h}=(h_{1},\dots,h_{n})
  36. i i
  37. h i 0 h_{i}\neq 0
  38. h i < 0 h_{i}<0
  39. h i > 0. h_{i}>0.
  40. h \mathrm{h}
  41. r ( h ) = [ h , 1 1 ] r(h)=[h,1\dots 1]
  42. s ( x ; h ) s(\mathrm{x};\mathrm{h})
  43. s ( x ; h ) = 0 s(\mathrm{x};\mathrm{h})=0
  44. s ( x ; h ) = - 1 s(\mathrm{x};\mathrm{h})=-1
  45. s ( x ; h ) = 1. s(\mathrm{x};\mathrm{h})=1.
  46. Z I n , Z\subseteq I^{n},
  47. z Z z\in Z
  48. I n I^{n}
  49. [ x ; h ] = 0 [\mathrm{x};\mathrm{h}]=0
  50. W ( s | I ; h ) = { x I | s ( x ; h ) = s } W(s|I;\mathrm{h})=\{\mathrm{x}\in I|s(\mathrm{x};\mathrm{h})=s\}
  51. s I , s\in I,
  52. Z = W ( 0 | Z , h ) + W ( 1 | Z , h ) + W ( - 1 | Z , h ) , Z=W(0|Z,\mathrm{h})+W(1|Z,\mathrm{h})+W(-1|Z,\mathrm{h}),
  53. W ( s | Z , h ) = W ( s | I , h ) Z . W(s|Z,\mathrm{h})=W(s|I,\mathrm{h})\cap Z.
  54. 𝒜 \mathcal{A}
  55. m m
  56. 𝒜 = < A 1 , , A m > , \mathcal{A}=<\mathrm{A}_{1},\dots,\mathrm{A}_{m}>,
  57. A j : I j - 1 I n \mathrm{A}_{j}:I^{j-1}\to I^{n}
  58. h j = A j ( s j - 1 ) ; h j I n , \mathrm{h}^{j}=\mathrm{A}_{j}(s^{j-1});\mathrm{h}^{j}\in I^{n},
  59. j j
  60. j = 1 , 2 , , m , j=1,2,\dots,m,
  61. s j - 1 = ( s 1 , , s j - 1 ) I j - 1 \mathrm{s}^{j-1}=(s_{1},\dots,s_{j-1})\in I^{j-1}
  62. h 1 = A 1 ( ) \mathrm{h}^{1}=\mathrm{A}_{1}()
  63. S ( Z , 𝒜 ) S(Z,\mathcal{A})
  64. ( Z , 𝒜 ) (Z,\mathcal{A})
  65. W ( s | 𝒜 ) I W(s|\mathcal{A})\subseteq I
  66. s s
  67. W ( s | 𝒜 ) = { z I m | s ( z | 𝒜 ) = s } W(s|\mathcal{A})=\{\mathrm{z}\in I^{m}|s(z|\mathcal{A})=s\}
  68. W ( s | Z ; 𝒜 ) = W ( s | 𝒜 ) Z . W(s|Z;\mathcal{A})=W(s|\mathcal{A})\cap Z.
  69. 𝒜 \mathcal{A}
  70. Z Z
  71. | W ( s | Z , 𝒜 | = 1 |W(s|Z,\mathcal{A}|=1
  72. s S ( Z 𝒜 ) ; s\in S(Z\mathcal{A});
  73. Z Z
  74. | W + ( s | Z 𝒜 ) | = 1 |W^{+}(s|Z\mathcal{A})|=1
  75. s S ( Z 𝒜 ) . s\in S(Z\mathcal{A}).
  76. Z Z
  77. Z . Z.
  78. n n
  79. m m
  80. t t

Barber–Layden–Power_effect.html

  1. R E 1 / 5 ρ 0 - 1 / 5 t 2 / 5 R\propto E^{1/5}{{\rho}_{0}}^{-1/5}t^{2/5}

Baskakov_operator.html

  1. [ n ( f ) ] ( x ) = k = 0 ( - 1 ) k x k k ! ϕ n ( k ) ( x ) f ( k n ) [\mathcal{L}_{n}(f)](x)=\sum_{k=0}^{\infty}{(-1)^{k}\frac{x^{k}}{k!}\phi_{n}^{% (k)}(x)f\left(\frac{k}{n}\right)}
  2. x [ 0 , b ) x\in[0,b)\subset\mathbb{R}
  3. b b
  4. \infty
  5. n n\in\mathbb{N}
  6. ( ϕ n ) n (\phi_{n})_{n\in\mathbb{N}}
  7. [ 0 , b ] [0,b]
  8. n , k n,k\in\mathbb{N}
  9. ϕ n 𝒞 [ 0 , b ] \phi_{n}\in\mathcal{C}^{\infty}[0,b]
  10. ϕ n \phi_{n}
  11. [ 0 , b ) [0,b)
  12. ϕ n ( 0 ) = 1 \phi_{n}(0)=1
  13. ϕ n \phi_{n}
  14. ( - 1 ) k ϕ n ( k ) 0 (-1)^{k}\phi_{n}^{(k)}\geq 0
  15. c c
  16. ϕ n ( k + 1 ) = - n ϕ n + c ( k ) \phi_{n}^{(k+1)}=-n\phi_{n+c}^{(k)}
  17. n > max { 0 , - c } n>\max\{0,-c\}

BATON_Overlay.html

  1. O ( log N ) O(\log N)
  2. p p
  3. p - 2 x p-2^{x}
  4. x 0 x\geq 0
  5. p + 2 y p+2^{y}
  6. y 0 y\geq 0

Bayes_linear_statistics.html

  1. B = ( Y 1 , Y 2 ) , D = ( X 1 , X 2 ) . B=(Y_{1},Y_{2}),~{}D=(X_{1},X_{2}).
  2. E ( Y 1 ) = 5 , E ( Y 2 ) = 3 , E ( X 1 ) = 5 , E ( X 2 ) = 3 E(Y_{1})=5,~{}E(Y_{2})=3,~{}E(X_{1})=5,~{}E(X_{2})=3
  3. X 1 X 2 Y 1 Y 2 X 1 1 u γ γ X 2 u 1 γ γ Y 1 γ γ 1 v Y 2 γ γ v 1 . \begin{matrix}&X_{1}&X_{2}&Y_{1}&Y_{2}\\ X_{1}&1&u&\gamma&\gamma\\ X_{2}&u&1&\gamma&\gamma\\ Y_{1}&\gamma&\gamma&1&v\\ Y_{2}&\gamma&\gamma&v&1\\ \end{matrix}.
  4. c 0 + c 1 X 1 + c 2 X 2 c_{0}+c_{1}X_{1}+c_{2}X_{2}
  5. c 0 , c 1 c_{0},c_{1}
  6. c 2 c_{2}
  7. Y 1 , Y 2 Y_{1},Y_{2}
  8. Y 1 Y_{1}
  9. E ( [ Y 1 - c 0 - c 1 X 1 - c 2 X 2 ] 2 ) E([Y_{1}-c_{0}-c_{1}X_{1}-c_{2}X_{2}]^{2})\,
  10. c 0 , c 1 , c 2 c_{0},c_{1},c_{2}\,
  11. Y 1 Y_{1}
  12. E D ( X ) = i = 0 k h i D i . E_{D}(X)=\sum^{k}_{i=0}h_{i}D_{i}.
  13. h 0 , , h k h_{0},\dots,h_{k}
  14. E ( [ X - i = 0 k h i D i ] 2 ) . E\left(\left[X-\sum^{k}_{i=0}h_{i}D_{i}\right]^{2}\right).
  15. E D ( X ) = E ( X ) + C o v ( X , D ) V a r ( D ) - 1 ( D - E ( D ) ) . E_{D}(X)=E(X)+Cov(X,D)Var(D)^{-1}(D-E(D)).\,
  16. X X
  17. D D
  18. V a r D ( X ) = V a r ( X ) - C o v ( X , D ) V a r ( D ) - 1 C o v ( D , X ) . Var_{D}(X)=Var(X)-Cov(X,D)Var(D)^{-1}Cov(D,X).

BCMP_network.html

  1. μ j \scriptstyle{\mu_{j}}
  2. L ( s ) = N ( s ) D ( s ) . L(s)=\frac{N(s)}{D(s)}.
  3. P i j P_{ij}
  4. 1 - j = 1 m P i j 1-\sum_{j=1}^{m}P_{ij}
  5. π ( x 1 , x 2 , , x m ) = C π 1 ( x 1 ) π 2 ( x 2 ) π m ( x m ) , \pi(x_{1},x_{2},\ldots,x_{m})=C\pi_{1}(x_{1})\pi_{2}(x_{2})\cdots\pi_{m}(x_{m}),
  6. π i ( ) \scriptstyle{\pi_{i}(\cdot)}

Benzoate—CoA_ligase.html

  1. \rightleftharpoons

Benzophenone_synthase.html

  1. \rightleftharpoons

Beta-adrenergic-receptor_kinase.html

  1. \rightleftharpoons

Beta-alanine—pyruvate_transaminase.html

  1. \rightleftharpoons

Beta-apiosyl-beta-glucosidase.html

  1. \rightleftharpoons

Beta-aspartyl-N-acetylglucosaminidase.html

  1. \rightleftharpoons

Beta-galactoside_alpha-2,3-sialyltransferase.html

  1. \rightleftharpoons

Beta-galactoside_alpha-2,6-sialyltransferase.html

  1. \rightleftharpoons

Beta-glucan-transporting_ATPase.html

  1. \rightleftharpoons

Beta-glucogallin_O-galloyltransferase.html

  1. \rightleftharpoons

Beta-glucogallin—tetrakisgalloylglucose_O-galloyltransferase.html

  1. \rightleftharpoons

Beta-glucoside_kinase.html

  1. \rightleftharpoons

Beta-ketoacyl-(acyl-carrier-protein)_synthase_III.html

  1. \rightleftharpoons

Beta-ketoacyl-acyl-carrier-protein_synthase_I.html

  1. \rightleftharpoons

Beta-ketoacyl-acyl-carrier-protein_synthase_II.html

  1. \rightleftharpoons

Beta-L-arabinosidase.html

  1. \rightleftharpoons

Beta-primeverosidase.html

  1. \rightleftharpoons

Beta-pyrazolylalanine_synthase.html

  1. \rightleftharpoons

Beta-ureidopropionase.html

  1. \rightleftharpoons

Beurling_algebra.html

  1. f ( x ) = a n e i n x f(x)=\sum a_{n}e^{inx}
  2. c k = sup | n | k | a n | c_{k}=\sup_{|n|\geq k}|a_{n}|
  3. k 0 c k < . \sum_{k\geq 0}c_{k}<\infty.
  4. \mathbb{Z}
  5. w ( m + n ) w ( m ) w ( n ) , w ( 0 ) = 1 w(m+n)\leq w(m)w(n),\quad w(0)=1
  6. A w ( 𝕋 ) = { f : f ( t ) = n a n e i n t , f w = n | a n | w ( n ) < } ( w 1 ( ) ) A_{w}(\mathbb{T})=\{f:f(t)=\sum_{n}a_{n}e^{int},\,\|f\|_{w}=\sum_{n}|a_{n}|w(n% )<\infty\}\,(\sim\ell^{1}_{w}(\mathbb{Z}))

Bilateral_filter.html

  1. I filtered ( x ) = 1 W p x i Ω I ( x i ) f r ( I ( x i ) - I ( x ) ) g s ( x i - x ) , I\text{filtered}(x)=\frac{1}{W_{p}}\sum_{x_{i}\in\Omega}I(x_{i})f_{r}(\|I(x_{i% })-I(x)\|)g_{s}(\|x_{i}-x\|),
  2. W p = x i Ω f r ( I ( x i ) - I ( x ) ) g s ( x i - x ) W_{p}=\sum_{x_{i}\in\Omega}{f_{r}(\|I(x_{i})-I(x)\|)g_{s}(\|x_{i}-x\|)}
  3. I filtered I\text{filtered}
  4. I I
  5. x x
  6. Ω \Omega
  7. x x
  8. f r f_{r}
  9. g s g_{s}
  10. W p W_{p}
  11. ( i , j ) (i,j)
  12. ( k , l ) (k,l)
  13. ( k , l ) (k,l)
  14. ( i , j ) (i,j)
  15. w ( i , j , k , l ) = e ( - ( i - k ) 2 + ( j - l ) 2 2 σ d 2 - I ( i , j ) - I ( k , l ) 2 2 σ r 2 ) w(i,j,k,l)=e^{(-\frac{(i-k)^{2}+(j-l)^{2}}{2\sigma_{d}^{2}}-\frac{\|I(i,j)-I(k% ,l)\|^{2}}{2\sigma_{r}^{2}})}
  16. ( i , j ) (i,j)
  17. ( k , l ) (k,l)
  18. I D ( i , j ) = k , l I ( k , l ) * w ( i , j , k , l ) k , l w ( i , j , k , l ) I_{D}(i,j)=\frac{\sum_{k,l}{I(k,l)*w(i,j,k,l)}}{\sum_{k,l}{w(i,j,k,l)}}
  19. I D I_{D}
  20. ( i , j ) (i,j)

Bile-acid-CoA_hydrolase.html

  1. \rightleftharpoons

Bile-acid_7alpha-dehydratase.html

  1. \rightleftharpoons

Bile_acid-CoA:amino_acid_N-acyltransferase.html

  1. \rightleftharpoons

Bilinear_time–frequency_distribution.html

  1. P V f ( u , ξ ) = - f ( u + τ 2 ) . f * ( u - τ 2 ) . e - i τ ξ d τ {{P}_{V}}f(u,\xi)=\int_{-\infty}^{\infty}{f(u+\frac{\tau}{2}).}{{f}^{*}}(u-% \frac{\tau}{2}).{{e}^{-i\tau\xi}}d\tau
  2. P V f ( u , ξ ) = 1 2 π - f ^ ( ξ + γ 2 ) . f ^ * ( ξ - γ 2 ) . e i γ u d γ {{P}_{V}}f(u,\xi)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\hat{f}(\xi+\frac{% \gamma}{2}).}{{\hat{f}}^{*}}(\xi-\frac{\gamma}{2}).{{e}^{i\gamma u}}d\gamma
  3. - P V f ( u , ξ ) . d u = | f ^ ( ξ ) | 2 \int\limits_{-\infty}^{\infty}{{{P}_{V}}f(u,\xi)}.du={{\left|\hat{f}(\xi)% \right|}^{2}}
  4. - P V f ( u , ξ ) . d ξ = 2 π | f ( u ) | 2 \int\limits_{-\infty}^{\infty}{{{P}_{V}}f(u,\xi)}.d\xi=2\pi{{\left|f(u)\right|% }^{2}}
  5. 2 π | - f ( t ) g * ( t ) d t | 2 = P V f ( u , ξ ) P V g ( u , ξ ) d u . d ξ 2\pi{{\left|\int\limits_{-\infty}^{\infty}{f(t)g*(t)dt}\right|}^{2}}=\iint{{{P% }_{V}}f(u,\xi)}{{P}_{V}}g(u,\xi)du.d\xi
  6. P V f ( u , ξ ) {{P}_{V}}f(u,\xi)
  7. f ^ {\hat{f}}
  8. P V f ( u , ξ ) {{P}_{V}}f(u,\xi)
  9. f ^ {\hat{f}}
  10. f a ( t ) = a ( t ) e i ϕ ( t ) {{f}_{a}}(t)=a(t){{e}^{i\phi(t)}}
  11. ϕ ( u ) = - ξ P V f a ( u , ξ ) d ξ - P V f a ( u , ξ ) d ξ {{\phi}^{{}^{\prime}}}(u)=\frac{\int\limits_{-\infty}^{\infty}{\xi{{P}_{V}}{{f% }_{a}}(u,\xi)d\xi}}{\int\limits_{-\infty}^{\infty}{{{P}_{V}}{{f}_{a}}(u,\xi)d% \xi}}
  12. f = f 1 + f 2 f={{f}_{1}}+{{f}_{2}}
  13. P V f = P V f 1 + P V f 2 + P V [ f 1 , f 2 ] + P V [ f 2 , f 1 ] {{P}_{V}}f={{P}_{V}}{{f}_{1}}+{{P}_{V}}{{f}_{2}}+{{P}_{V}}[{{f}_{1}},{{f}_{2}}% ]+{{P}_{V}}[{{f}_{2}},{{f}_{1}}]
  14. P V [ h , g ] ( u , ξ ) = - h ( u + τ 2 ) . g * ( u - τ 2 ) . e - i τ ξ d τ {{P}_{V}}[h,g](u,\xi)=\int_{-\infty}^{\infty}{h(u+\frac{\tau}{2}).}{{g}^{*}}(u% -\frac{\tau}{2}).{{e}^{-i\tau\xi}}d\tau
  15. I [ f 1 , f 2 ] = P V [ f 1 , f 2 ] + P V [ f 2 , f 1 ] I[{{f}_{1}},{{f}_{2}}]={{P}_{V}}[{{f}_{1}},{{f}_{2}}]+{{P}_{V}}[{{f}_{2}},{{f}% _{1}}]
  16. ( u , ξ ) (u,\xi)
  17. f a ( t ) {{f}_{a}}(t)
  18. P V f {{P}_{V}}f
  19. θ \theta
  20. P θ f ( u , ξ ) = - - P V f ( u , ξ ) . θ ( u , u , ξ , ξ ) d u . d ξ {{P}_{\theta}}f(u,\xi)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{{P}_{V% }}f({{u}^{{}^{\prime}}},{{\xi}^{{}^{\prime}}})}}.\theta(u,{{u}^{{}^{\prime}}},% \xi,{{\xi}^{{}^{\prime}}})d{{u}^{{}^{\prime}}}.d\xi^{\prime}
  21. θ \theta
  22. ( u , ξ ) (u,\xi)
  23. P θ f ( u , ξ ) 0 ( u , ξ ) 2 {{P}_{\theta}}f(u,\xi)\geq 0\forall(u,\xi)\in{{\mathbb{R}}^{2}}
  24. T f ( γ ) = f , ϕ γ Tf(\gamma)=\left\langle f,{{\phi}_{\gamma}}\right\rangle
  25. { ϕ γ } γ Γ {{\left\{{{\phi}_{\gamma}}\right\}}_{\gamma\in\Gamma}}
  26. ( u , ξ ) (u,\xi)
  27. ϕ γ ( u , ξ ) {{\phi}_{\gamma(u,\xi)}}
  28. ( u , ξ ) (u,\xi)
  29. P T f ( u , ξ ) = | f , ϕ γ ( u , ξ ) | 2 {{P}_{T}}f(u,\xi)={{\left|\left\langle f,{{\phi}_{\gamma(u,\xi)}}\right\rangle% \right|}^{2}}
  30. P T f ( u , ξ ) = 1 2 π - - P V f ( u , ξ ) . P V ϕ γ ( u , ξ ) ( u , ξ ) d u . d ξ {{P}_{T}}f(u,\xi)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty% }{{{P}_{V}}f({{u}^{{}^{\prime}}},{{\xi}^{{}^{\prime}}})}}.{{P}_{V}}{{\phi}_{% \gamma(u,\xi)}}({{u}^{{}^{\prime}}},{{\xi}^{{}^{\prime}}})d{{u}^{{}^{\prime}}}% .d\xi^{\prime}
  31. θ ( u , u , ξ , ξ ) = 1 2 π P V ϕ γ ( u , ξ ) ( u , ξ ) \theta(u,u^{\prime},\xi,\xi^{\prime})=\frac{1}{2\pi}{{P}_{V}}{{\phi}_{\gamma(u% ,\xi)}}({{u}^{{}^{\prime}}},{{\xi}^{{}^{\prime}}})
  32. P V ϕ γ ( u , ξ ) ( u , ξ ) {{P}_{V}}{{\phi}_{\gamma(u,\xi)}}({{u}^{{}^{\prime}}},{{\xi}^{{}^{\prime}}})
  33. ( u , ξ ) (u,\xi)
  34. ϕ γ ( u , ξ ) ( t ) = g ( t - u ) . e i ξ t {{\phi}_{\gamma(u,\xi)}}(t)=g(t-u).{{e}^{i\xi t}}
  35. θ ( u , u , ξ , ξ ) = 1 2 π P V ϕ γ ( u , ξ ) ( u , ξ ) = 1 2 π P V g ( u - u , ξ - ξ ) \theta(u,u^{\prime},\xi,\xi^{\prime})=\frac{1}{2\pi}{{P}_{V}}{{\phi}_{\gamma(u% ,\xi)}}({{u}^{{}^{\prime}}},{{\xi}^{{}^{\prime}}})=\frac{1}{2\pi}{{P}_{V}}g({{% u}^{{}^{\prime}}}-u,{{\xi}^{{}^{\prime}}}-\xi)
  36. P V g {{P}_{V}}g
  37. P V g {{P}_{V}}g
  38. P V f {{P}_{V}}f
  39. P V f {{P}_{V}}f
  40. θ \theta
  41. - P f ( u , ξ ) . d ξ = 2 π | f ( u ) | 2 \int\limits_{-\infty}^{\infty}{Pf(u,\xi)}.d\xi=2\pi{{\left|f(u)\right|}^{2}}
  42. - P f ( u , ξ ) . d u = | f ^ ( ξ ) | 2 \int\limits_{-\infty}^{\infty}{Pf(u,\xi)}.du={{\left|\hat{f}(\xi)\right|}^{2}}
  43. C x ( t , f ) = - - A x ( η , τ ) Φ ( η , τ ) exp ( j 2 π ( η t - τ f ) ) d η d τ , C_{x}(t,f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A_{x}(\eta,\tau)\Phi(% \eta,\tau)\exp(j2\pi(\eta t-\tau f))\,d\eta\,d\tau,
  44. A x ( η , τ ) A_{x}\left(\eta,\tau\right)
  45. Φ ( η , τ ) \Phi\left(\eta,\tau\right)
  46. Φ 1 \Phi\equiv 1
  47. C x ( t , f ) = - - W x ( θ , ν ) Π ( t - θ , f - ν ) d θ d ν , = [ W x Π ] ( t , f ) C_{x}(t,f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_{x}(\theta,\nu)\Pi(% t-\theta,f-\nu)\,d\theta\,d\nu,=[W_{x}\,\ast\,\Pi](t,f)
  48. Π ( t , f ) \Pi(t,f)
  49. Π = δ ( 0 , 0 ) \Pi=\delta_{(0,0)}
  50. Φ = t f - 1 Π \Phi=\mathcal{F}_{t}\mathcal{F}^{-1}_{f}\Pi
  51. Φ ( η , τ ) = - - Π ( t , f ) exp ( - j 2 π ( t η - f τ ) ) d t d f , \Phi(\eta,\tau)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Pi(t,f)\exp(-j2% \pi(t\,\eta-f\,\tau))\,dt\,df,
  52. Π ( t , f ) = - - Φ ( η , τ ) exp ( j 2 π ( η t - τ f ) ) d η d τ , \Pi(t,f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Phi(\eta,\tau)\exp(j2% \pi(\eta\,t-\tau\,f))\,d\eta\,d\tau,
  53. P x ( f ) P_{x}\left(f\right)
  54. R x ( τ ) R_{x}\left(\tau\right)
  55. P x ( f ) = - R x ( τ ) e - j 2 π f τ d τ , P_{x}(f)=\int_{-\infty}^{\infty}R_{x}(\tau)e^{-j2\pi f\tau}\,d\tau,
  56. R x ( τ ) = - x ( t + τ / 2 ) x * ( t - τ / 2 ) d t . R_{x}(\tau)=\int_{-\infty}^{\infty}\left.x(t+\tau/2)x^{*}(t-\tau/2)\right.\,dt.
  57. x ( t ) x\left(t\right)
  58. x ( t ) x\left(t\right)
  59. W x ( t , f ) = - R x ( t , τ ) e - j 2 π f τ d τ , W_{x}(t,f)=\int_{-\infty}^{\infty}R_{x}(t,\tau)e^{-j2\pi f\tau}\,d\tau,
  60. R x ( t , τ ) = x ( t + τ / 2 ) x * ( t - τ / 2 ) . R_{x}\left(t,\tau\right)=x(t+\tau/2)x^{*}(t-\tau/2).
  61. t t
  62. τ \tau
  63. A x ( η , τ ) = - x ( t + τ / 2 ) x * ( t - τ / 2 ) e j 2 π t η d t . A_{x}(\eta,\tau)=\int_{-\infty}^{\infty}x(t+\tau/2)x^{*}(t-\tau/2)e^{j2\pi t% \eta}\,dt.
  64. Φ ( η , τ ) = 1 \Phi\left(\eta,\tau\right)=1
  65. Φ ( η , τ ) 1 \Phi\left(\eta,\tau\right)\neq 1
  66. η , τ \eta,\tau
  67. η , τ \eta,\tau
  68. θ ( u , ξ ) \theta(u,\xi)
  69. θ ^ ( τ , γ ) = - - θ ( u , ξ ) . e - i ( u γ + ξ τ ) d u . d ξ \hat{\theta}(\tau,\gamma)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{% \theta(u,\xi)}}.{{e}^{-i(u\gamma+\xi\tau)}}du.d\xi
  70. P θ {{P}_{\theta}}
  71. - P θ f ( u , ξ ) . d ξ = 2 π | f ( u ) | 2 \int\limits_{-\infty}^{\infty}{{{P}_{\theta}}f(u,\xi)}.d\xi=2\pi{{\left|f(u)% \right|}^{2}}
  72. - P θ f ( u , ξ ) . d u = | f ^ ( ξ ) | 2 \int\limits_{-\infty}^{\infty}{{{P}_{\theta}}f(u,\xi)}.du={{\left|\hat{f}(\xi)% \right|}^{2}}
  73. f L 2 ( ) f\in{{L}^{2}}(\mathbb{R})
  74. ( τ , γ ) 2 , θ ^ ( τ , 0 ) = θ ^ ( 0 , γ ) = 1 \forall(\tau,\gamma)\in{{\mathbb{R}}^{2}},\hat{\theta}(\tau,0)=\hat{\theta}(0,% \gamma)=1
  75. Φ ( η , τ ) = 1 \Phi\left(\eta,\tau\right)=1
  76. W x ( t , f ) = - x ( t + τ / 2 ) x * ( t - τ / 2 ) e - j 2 π f τ d τ . W_{x}(t,f)=\int_{-\infty}^{\infty}x(t+\tau/2)x^{*}(t-\tau/2)e^{-j2\pi f\tau}\,% d\tau.
  77. 1 s f ( t s ) P V f ( u s , s ξ ) \frac{1}{\sqrt{s}}f\left(\frac{t}{s}\right)\leftrightarrow{{P}_{V}}f\left(% \frac{u}{s},s\xi\right)
  78. g ( t ) = 1 s f ( t s ) g(t)=\frac{1}{\sqrt{s}}f\left(\frac{t}{s}\right)
  79. P θ g ( u , ξ ) = P θ f ( u s , s ξ ) {{P}_{\theta}}g(u,\xi)={{P}_{\theta}}f\left(\frac{u}{s},s\xi\right)
  80. s R + , θ ( s u , ξ s ) = θ ( u , ξ ) , \forall s\in{{R}^{+}},\theta\left(su,\frac{\xi}{s}\right)=\theta(u,\xi),
  81. θ ( u , ξ ) = θ ( u ξ , 1 ) = β ( u ξ ) \theta(u,\xi)=\theta(u\xi,1)=\beta(u\xi)
  82. Φ ( η , τ ) = exp [ - α ( η τ ) 2 ] , \Phi\left(\eta,\tau\right)=\exp\left[-\alpha\left(\eta\tau\right)^{2}\right],\,
  83. Φ ( η , τ ) = exp ( - i 2 π η τ 2 ) , \Phi\left(\eta,\tau\right)=\exp\left(-i2\pi\frac{\eta\tau}{2}\right),\,
  84. C x ( t , f ) = x ( t ) x ^ * ( f ) e i 2 π t f C_{x}(t,f)=x(t)\,\hat{x}^{*}(f)\,e^{i2\pi\,tf}
  85. Φ ( η , τ ) = sin ( π η τ ) π η τ exp ( - 2 π α τ 2 ) , \Phi\left(\eta,\tau\right)=\frac{\sin\left(\pi\eta\tau\right)}{\pi\eta\tau}% \exp\left(-2\pi\alpha\tau^{2}\right),
  86. α \alpha
  87. X ( t ) X\left(t\right)
  88. R ( t , s ) = E { X ( t ) X ( s ) } R\left(t,s\right)=E\left\{X\left(t\right)X\left(s\right)\right\}
  89. f L 2 ( ) f\in{{L}^{2}}\left(\mathbb{R}\right)
  90. K f ( t ) = - R ( t , s ) f ( s ) d s Kf\left(t\right)=\int_{-\infty}^{\infty}{R\left(t,s\right)f\left(s\right)ds}
  91. R ( t , s ) R\left(t,s\right)
  92. τ = t - s \tau=t-s
  93. u = t + s 2 u=\frac{t+s}{2}
  94. R ( t , s ) = R ( u + τ 2 , u - τ 2 ) = C ( u , τ ) R\left(t,s\right)=R\left(u+\frac{\tau}{2},u-\frac{\tau}{2}\right)=C\left(u,% \tau\right)
  95. τ = t - s \tau=t-s
  96. K f ( t ) = - C ( t - s ) . f ( s ) d s = C * f ( t ) Kf\left(t\right)=\int_{-\infty}^{\infty}{C\left(t-s\right).f\left(s\right)ds=C% *f\left(t\right)}
  97. e i ω t {{e}^{i\omega t}}
  98. P X ( ω ) = - C ( τ ) e - i ω τ d τ {{P}_{X}}\left(\omega\right)=\int\limits_{-\infty}^{\infty}{C\left(\tau\right)% }{{e}^{-i\omega\tau}}d\tau
  99. P X ( u , ξ ) = - C ( u , τ ) e - i ξ τ d τ = - E { X ( u + τ 2 ) X ( u - τ 2 ) } e - i ξ τ d τ {{P}_{X}}\left(u,\xi\right)=\int\limits_{-\infty}^{\infty}{C\left(u,\tau\right% )}{{e}^{-i\xi\tau}}d\tau=\int\limits_{-\infty}^{\infty}{E\left\{X\left(u+\frac% {\tau}{2}\right)X\left(u-\frac{\tau}{2}\right)\right\}}{{e}^{-i\xi\tau}}d\tau
  100. C ( u , τ ) C\left(u,\tau\right)
  101. τ \tau
  102. P X ( u , ξ ) = E { P V X ( u , ξ ) } {{P}_{X}}\left(u,\xi\right)=E\left\{{{P}_{V}}X\left(u,\xi\right)\right\}
  103. P V X ( u , ξ ) = - { X ( u + τ 2 ) X ( u - τ 2 ) } e - i ξ τ d τ {{P}_{V}}X\left(u,\xi\right)=\int\limits_{-\infty}^{\infty}{\left\{X\left(u+% \frac{\tau}{2}\right)X\left(u-\frac{\tau}{2}\right)\right\}}{{e}^{-i\xi\tau}}d\tau

Bilirubin-glucuronoside_glucuronosyltransferase.html

  1. \rightleftharpoons

Binary_cyclic_group.html

  1. C 2 n C_{2n}
  2. C n C_{n}
  3. C n < SO ( 3 ) C_{n}<\operatorname{SO}(3)
  4. Spin ( 3 ) SO ( 3 ) \operatorname{Spin}(3)\to\operatorname{SO}(3)\,
  5. Spin ( 3 ) Sp ( 1 ) \operatorname{Spin}(3)\cong\operatorname{Sp}(1)

Binder_parameter.html

  1. U L = 1 - s 4 L 3 s 2 L 2 U_{L}=1-\frac{{\langle s^{4}\rangle}_{L}}{3{\langle s^{2}\rangle}^{2}_{L}}
  2. B = 1 2 ( 3 - q 4 ¯ q 2 ¯ 2 ) {B=\frac{1}{2}\left(3-\frac{\overline{\langle q^{4}\rangle}}{\overline{\langle q% ^{2}\rangle}^{2}}\right)}
  3. \langle\cdot\rangle
  4. ¯ \overline{\cdot}
  5. q q
  6. B B
  7. L L
  8. T T c T\approx T_{c}
  9. B ( T , L ) = b ( ϵ L 1 / ν ) B(T,L)=b(\epsilon L^{1/\nu})
  10. ϵ = T - T c T \epsilon=\frac{T-T_{c}}{T}

Biological_neuron_model.html

  1. y i = ϕ ( j w i j x j ) y_{i}=\phi\left(\sum_{j}w_{ij}x_{j}\right)
  2. i i
  3. j j
  4. i i
  5. j j
  6. φ φ
  7. I ( t ) I(t)
  8. I ( t ) = C m d V m ( t ) d t I(t)=C_{\mathrm{m}}\frac{dV_{\mathrm{m}}(t)}{dt}
  9. Q = C V Q=CV
  10. f ( I ) = I C m V th + t ref I \,\!f(I)=\frac{I}{C_{\mathrm{m}}V_{\mathrm{th}}+t_{\mathrm{ref}}I}
  11. I ( t ) - V m ( t ) R m = C m d V m ( t ) d t I(t)-\frac{V_{\mathrm{m}}(t)}{R_{\mathrm{m}}}=C_{\mathrm{m}}\frac{dV_{\mathrm{% m}}(t)}{dt}
  12. f ( I ) = { 0 , I I th [ t ref - R m C m log ( 1 - V th I R m ) ] - 1 , I > I th f(I)=\begin{cases}0,&I\leq I_{\mathrm{th}}\\ {[}t_{\mathrm{ref}}-R_{\mathrm{m}}C_{\mathrm{m}}\log(1-\tfrac{V_{\mathrm{th}}}% {IR_{\mathrm{m}}}){]}^{-1},&I>I_{\mathrm{th}}\end{cases}
  13. d X d t = Δ T exp ( X - X T Δ T ) \frac{dX}{dt}=\Delta_{T}\exp\left(\frac{X-X_{T}}{\Delta_{T}}\right)
  14. X X
  15. X T X_{T}
  16. Δ T \Delta_{T}
  17. X T X_{T}
  18. C m d V ( t ) d t = - i I i ( t , V ) C_{\mathrm{m}}\frac{dV(t)}{dt}=-\sum_{i}I_{i}(t,V)
  19. I ( t , V ) = g ( t , V ) ( V - V eq ) I(t,V)=g(t,V)\cdot(V-V_{\mathrm{eq}})
  20. g ( t , V ) g(t,V)
  21. m m
  22. h h
  23. g ( t , V ) = g ¯ m ( t , V ) p h ( t , V ) q g(t,V)=\bar{g}\cdot m(t,V)^{p}\cdot h(t,V)^{q}
  24. d m ( t , V ) d t = m ( V ) - m ( t , V ) τ m ( V ) = α m ( V ) ( 1 - m ) - β m ( V ) m \frac{dm(t,V)}{dt}=\frac{m_{\infty}(V)-m(t,V)}{\tau_{\mathrm{m}}(V)}=\alpha_{% \mathrm{m}}(V)\cdot(1-m)-\beta_{\mathrm{m}}(V)\cdot m
  25. h h
  26. τ τ
  27. α α
  28. β β
  29. d V d t = V - V 3 - w + I ext τ d w d t = V - a - b w \begin{array}[]{rcl}\dfrac{dV}{dt}&=&V-V^{3}-w+I_{\mathrm{ext}}\\ \\ \tau\dfrac{dw}{dt}&=&V-a-bw\end{array}
  30. w w
  31. a = - 0.7 , b = 0.8 , τ = 1 / 0.08 a=-0.7,b=0.8,τ=1/0.08
  32. C d V d t = - I ion ( V , w ) + I d w d t = ϕ w - w τ w \begin{array}[]{rcl}C\dfrac{dV}{dt}&=&-I_{\mathrm{ion}}(V,w)+I\\ \\ \dfrac{dw}{dt}&=&\phi\cdot\dfrac{w_{\infty}-w}{\tau_{w}}\end{array}
  33. I ion ( V , w ) = g ¯ Ca m ( V - V Ca ) + g ¯ K w ( V - V K ) + g ¯ L ( V - V L ) I_{\mathrm{ion}}(V,w)=\bar{g}_{\mathrm{Ca}}m_{\infty}\cdot(V-V_{\mathrm{Ca}})+% \bar{g}_{\mathrm{K}}w\cdot(V-V_{\mathrm{K}})+\bar{g}_{\mathrm{L}}\cdot(V-V_{% \mathrm{L}})
  34. d x d t = y + 3 x 2 - x 3 - z + I d y d t = 1 - 5 x 2 - y d z d t = r ( 4 ( x + 8 5 ) - z ) \begin{array}[]{rcl}\dfrac{dx}{dt}&=&y+3x^{2}-x^{3}-z+I\\ \\ \dfrac{dy}{dt}&=&1-5x^{2}-y\\ \\ \dfrac{dz}{dt}&=&r\cdot(4(x+\tfrac{8}{5})-z)\end{array}
  35. z z
  36. x x
  37. G i n = G tanh ( L ) + G L 1 + ( G L / G ) tanh ( L ) G_{in}=\frac{G_{\infty}\tanh(L)+G_{L}}{1+(G_{L}/G_{\infty})\tanh(L)}
  38. L L
  39. G D = G m A D tanh ( L D ) / L D \,\!G_{D}=G_{m}A_{D}\tanh(L_{D})/L_{D}
  40. l l
  41. n n
  42. G N = G S + j = 1 n A D j F d g a j G_{N}=G_{S}+\sum_{j=1}^{n}A_{D_{j}}F_{dga_{j}}
  43. l l
  44. d d
  45. i i
  46. G = π d 3 / 2 2 R i R m G_{\infty}=\tfrac{\pi d^{3/2}}{2\sqrt{R_{i}R_{m}}}
  47. B out , i = B in , i + 1 ( d i + 1 / d i ) 3 / 2 R m , i + 1 / R m , i B_{\mathrm{out},i}=\frac{B_{\mathrm{in},i+1}(d_{i+1}/d_{i})^{3/2}}{\sqrt{R_{% \mathrm{m},i+1}/R_{\mathrm{m},i}}}
  48. B in , i = B out , i + tanh X i 1 + B out , i tanh X i B_{\mathrm{in},i}=\frac{B_{\mathrm{out},i}+\tanh X_{i}}{1+B_{\mathrm{out},i}% \tanh X_{i}}
  49. B out , par = B in , dau1 ( d dau1 / d par ) 3 / 2 R m , dau1 / R m , par + B in , dau2 ( d dau2 / d par ) 3 / 2 R m , dau2 / R m , par + B_{\mathrm{out,par}}=\frac{B_{\mathrm{in,dau1}}(d_{\mathrm{dau1}}/d_{\mathrm{% par}})^{3/2}}{\sqrt{R_{\mathrm{m,dau1}}/R_{\mathrm{m,par}}}}+\frac{B_{\mathrm{% in,dau2}}(d_{\mathrm{dau2}}/d_{\mathrm{par}})^{3/2}}{\sqrt{R_{\mathrm{m,dau2}}% /R_{\mathrm{m,par}}}}+\ldots
  50. X i = l i 4 R i d i R m X_{i}=\tfrac{l_{i}\sqrt{4R_{i}}}{\sqrt{d_{i}R_{m}}}
  51. G N = A soma R m , soma + j B in , stem , j G , j G_{N}=\frac{A_{\mathrm{soma}}}{R_{\mathrm{m,soma}}}+\sum_{j}B_{\mathrm{in,stem% },j}G_{\infty,j}
  52. I AMPA ( t , V ) = g ¯ AMPA [ O ] ( V ( t ) - E AMPA ) I_{\mathrm{AMPA}}(t,V)=\bar{g}_{\mathrm{AMPA}}\cdot[O]\cdot(V(t)-E_{\mathrm{% AMPA}})
  53. I NMDA ( t , V ) = g ¯ NMDA B ( V ) [ O ] ( V ( t ) - E NMDA ) I_{\mathrm{NMDA}}(t,V)=\bar{g}_{\mathrm{NMDA}}\cdot B(V)\cdot[O]\cdot(V(t)-E_{% \mathrm{NMDA}})
  54. I GABA A ( t , V ) = g ¯ GABA A ( [ O 1 ] + [ O 2 ] ) ( V ( t ) - E Cl ) I_{\mathrm{GABA_{A}}}(t,V)=\bar{g}_{\mathrm{GABA_{A}}}\cdot([O_{1}]+[O_{2}])% \cdot(V(t)-E_{\mathrm{Cl}})
  55. I GABA B ( t , V ) = g ¯ GABA B [ G ] n [ G ] n + K d ( V ( t ) - E K ) I_{\mathrm{GABA_{B}}}(t,V)=\bar{g}_{\mathrm{GABA_{B}}}\cdot\tfrac{[G]^{n}}{[G]% ^{n}+K_{\mathrm{d}}}\cdot(V(t)-E_{\mathrm{K}})
  56. E E
  57. O O OO
  58. B ( V ) B(V)
  59. G G GG
  60. K < s u b > d K<sub>d

Biotin_carboxylase.html

  1. \rightleftharpoons

Biotin—(acetyl-CoA-carboxylase)_ligase.html

  1. \rightleftharpoons

Biotin—(methylcrotonoyl-CoA-carboxylase)_ligase.html

  1. \rightleftharpoons

Biotin—(methylmalonyl-CoA-carboxytransferase)_ligase.html

  1. \rightleftharpoons

Biotin—(propionyl-CoA-carboxylase_(ATP-hydrolyzing))_ligase.html

  1. \rightleftharpoons

Biotin—CoA_ligase.html

  1. \rightleftharpoons

Biphenyl_synthase.html

  1. \rightleftharpoons

Birkhoff–Grothendieck_theorem.html

  1. 1 \mathbb{CP}^{1}
  2. \mathcal{E}
  3. 1 \mathbb{CP}^{1}
  4. 𝒪 ( a 1 ) 𝒪 ( a n ) . \mathcal{E}\cong\mathcal{O}(a_{1})\oplus\cdots\oplus\mathcal{O}(a_{n}).
  5. k 1 \mathbb{P}^{1}_{k}
  6. k k

Bis(2-ethylhexyl)phthalate_esterase.html

  1. \rightleftharpoons

Bis(5'-adenosyl)-triphosphatase.html

  1. \rightleftharpoons

Bis(5'-nucleosyl)-tetraphosphatase_(asymmetrical).html

  1. \rightleftharpoons

Bis(5'-nucleosyl)-tetraphosphatase_(symmetrical).html

  1. \rightleftharpoons

Bisphosphoglycerate_phosphatase.html

  1. \rightleftharpoons

Biuret_amidohydrolase.html

  1. \rightleftharpoons

Blasticidin-S_deaminase.html

  1. \rightleftharpoons

Blum–Shub–Smale_machine.html

  1. I I
  2. N + 1 N+1
  3. 0 , 1 , , N 0,1,\dots,N
  4. ( k , r , w , x ) (k,r,w,x)
  5. ( 0 , 0 , 0 , x ) (0,0,0,x)
  6. k = N k=N
  7. ( x 0 ) (x_{0})
  8. x 0 := g k ( x ) x_{0}:=g_{k}(x)
  9. g k g_{k}
  10. r := 0 r:=0
  11. r := r + 1 r:=r+1
  12. ( x 0 , l ) (x_{0},l)
  13. x 0 0 x_{0}\geq 0
  14. x r , x w x_{r},x_{w}
  15. x r x_{r}
  16. x w x_{w}

Bonnor–Ebert_mass.html

  1. P 0 P_{0}
  2. M B E = c B E v T 4 P 0 1 2 G 3 2 M_{BE}={c_{BE}v_{T}^{4}\over{P_{0}^{1\over{2}}G^{3\over{2}}}}
  3. v T k T μ v_{T}\equiv\sqrt{kT\over{\mu}}
  4. γ = 1 \gamma=1
  5. μ \mu
  6. c B E c_{BE}
  7. c B E 1.18. c_{BE}\simeq 1.18.

Boosting_methods_for_object_categorization.html

  1. 10 - 5 10^{-5}

Borel_determinacy_theorem.html

  1. I a 1 a 3 a 5 II a 2 a 4 a 6 \begin{matrix}\mathrm{I}&a_{1}&&a_{3}&&a_{5}&&\cdots\\ \mathrm{II}&&a_{2}&&a_{4}&&a_{6}&\cdots\end{matrix}
  2. a 1 , a 2 , a 3 \langle a_{1},a_{2},a_{3}\ldots\rangle
  3. I a 1 = f ( ) a 3 = f ( a 1 , a 2 ) a 5 = f ( a 1 , a 2 , a 3 , a 4 ) II a 2 a 4 a 6 . \begin{matrix}\mathrm{I}&a_{1}=f(\langle\rangle)&&a_{3}=f(\langle a_{1},a_{2}% \rangle)&&a_{5}=f(\langle a_{1},a_{2},a_{3},a_{4}\rangle)&&\cdots\\ \mathrm{II}&&a_{2}&&a_{4}&&a_{6}&\cdots.\end{matrix}
  4. I a 1 a 3 a 5 II a 2 = g ( a 1 ) a 4 = g ( a 1 , a 2 , a 3 ) a 6 = g ( a 1 , a 2 , a 3 , a 4 , a 5 ) . \begin{matrix}\mathrm{I}&a_{1}&&a_{3}&&a_{5}&&\cdots\\ \mathrm{II}&&a_{2}=g(\langle a_{1}\rangle)&&a_{4}=g(\langle a_{1},a_{2},a_{3}% \rangle)&&a_{6}=g(\langle a_{1},a_{2},a_{3},a_{4},a_{5}\rangle)&\cdots.\end{matrix}

Born–Infeld_model.html

  1. = - b 2 - det ( η + F b ) + b 2 \mathcal{L}=-b^{2}\sqrt{-\det\left(\eta+{F\over b}\right)}+b^{2}
  2. = - b 2 1 - E 2 - B 2 b 2 - ( E B ) 2 b 4 + b 2 \mathcal{L}=-b^{2}\sqrt{1-\frac{E^{2}-B^{2}}{b^{2}}-\frac{(\vec{E}\cdot\vec{B}% )^{2}}{b^{4}}}+b^{2}
  3. = - T - det ( η + 2 π α F ) \mathcal{L}=-T\sqrt{-\det\left(\eta+2\pi\alpha^{\prime}F\right)}

Boundedly_generated_group.html

  1. g = s 1 k 1 s m k m , g=s_{1}^{k_{1}}\cdots s_{m}^{k_{m}},
  2. s i S s_{i}\in S
  3. k i k_{i}
  4. F ( g ) F α , z ( g ) = γ α F(g)\equiv F_{\alpha,z}(g)=\int_{\gamma}\,\alpha
  5. F ( g h ) - F ( g ) - F ( h ) = Δ d α , F(gh)-F(g)-F(h)=\int_{\Delta}\,d\alpha,
  6. f α ( g ) = lim n F α , z ( g n ) / n f_{\alpha}(g)=\lim_{n\rightarrow\infty}F_{\alpha,z}(g^{n})/n

Boussinesq_approximation_(water_waves).html

  1. φ = φ b + ( z + h ) [ φ z ] z = - h + 1 2 ( z + h ) 2 [ 2 φ z 2 ] z = - h + 1 6 ( z + h ) 3 [ 3 φ z 3 ] z = - h + 1 24 ( z + h ) 4 [ 4 φ z 4 ] z = - h + , \begin{aligned}\displaystyle\varphi\,=&\displaystyle\varphi_{b}\,+\,(z+h)\,% \left[\frac{\partial\varphi}{\partial z}\right]_{z=-h}\,+\,\frac{1}{2}\,(z+h)^% {2}\,\left[\frac{\partial^{2}\varphi}{\partial z^{2}}\right]_{z=-h}\\ &\displaystyle+\,\frac{1}{6}\,(z+h)^{3}\,\left[\frac{\partial^{3}\varphi}{% \partial z^{3}}\right]_{z=-h}\,+\,\frac{1}{24}\,(z+h)^{4}\,\left[\frac{% \partial^{4}\varphi}{\partial z^{4}}\right]_{z=-h}\,+\,\cdots,\end{aligned}
  2. φ = \displaystyle\varphi\,=
  3. η t + u η x - w = 0 φ t + 1 2 ( u 2 + w 2 ) + g η = 0 , \begin{aligned}\displaystyle\frac{\partial\eta}{\partial t}&\displaystyle+\,u% \,\frac{\partial\eta}{\partial x}\,-\,w\,=\,0\\ \displaystyle\frac{\partial\varphi}{\partial t}&\displaystyle+\,\frac{1}{2}\,% \left(u^{2}+w^{2}\right)\,+\,g\,\eta\,=\,0,\end{aligned}
  4. η t + x [ ( h + η ) u b ] = 1 6 h 3 3 u b x 3 , u b t + u b u b x + g η x = 1 2 h 2 3 u b t x 2 . \begin{aligned}\displaystyle\frac{\partial\eta}{\partial t}&\displaystyle+\,% \frac{\partial}{\partial x}\,\left[\left(h+\eta\right)\,u_{b}\right]\,=\,\frac% {1}{6}\,h^{3}\,\frac{\partial^{3}u_{b}}{\partial x^{3}},\\ \displaystyle\frac{\partial u_{b}}{\partial t}&\displaystyle+\,u_{b}\,\frac{% \partial u_{b}}{\partial x}\,+\,g\,\frac{\partial\eta}{\partial x}\,=\,\frac{1% }{2}\,h^{2}\,\frac{\partial^{3}u_{b}}{\partial t\,\partial x^{2}}.\end{aligned}
  5. 2 η t 2 - g h 2 η x 2 - g h 2 x 2 ( 3 2 η 2 h + 1 3 h 2 2 η x 2 ) = 0. \frac{\partial^{2}\eta}{\partial t^{2}}\,-\,gh\,\frac{\partial^{2}\eta}{% \partial x^{2}}\,-\,gh\,\frac{\partial^{2}}{\partial x^{2}}\left(\frac{3}{2}\,% \frac{\eta^{2}}{h}\,+\,\frac{1}{3}\,h^{2}\,\frac{\partial^{2}\eta}{\partial x^% {2}}\right)\,=\,0.
  6. 2 ψ τ 2 - 2 ψ ξ 2 - 2 ξ 2 ( 3 ψ 2 + 2 ψ ξ 2 ) = 0 , \frac{\partial^{2}\psi}{\partial\tau^{2}}\,-\,\frac{\partial^{2}\psi}{\partial% \xi^{2}}\,-\,\frac{\partial^{2}}{\partial\xi^{2}}\left(\,3\,\psi^{2}\,+\,\frac% {\partial^{2}\psi}{\partial\xi^{2}}\,\right)\,=\,0,
  7. ψ = 1 2 η h \psi\,=\,\frac{1}{2}\,\frac{\eta}{h}
  8. τ = 3 t g h \tau\,=\,\sqrt{3}\,t\,\sqrt{\frac{g}{h}}
  9. ξ = 3 x h \xi\,=\,\sqrt{3}\,\frac{x}{h}
  10. c 2 = g h 1 + 1 6 k 2 h 2 1 + 1 2 k 2 h 2 , c^{2}\,=\;gh\,\frac{1\,+\,\frac{1}{6}\,k^{2}h^{2}}{1\,+\,\frac{1}{2}\,k^{2}h^{% 2}},
  11. c 2 = g h ( 1 - 1 3 k 2 h 2 ) . c^{2}\,=\,gh\,\left(1\,-\,\frac{1}{3}\,k^{2}h^{2}\right).

Branched-chain-fatty-acid_kinase.html

  1. \rightleftharpoons

Bromoxynil_nitrilase.html

  1. \rightleftharpoons

Brown–Forsythe_test.html

  1. z i j = | y i j - y ~ j | z_{ij}=\left|y_{ij}-\tilde{y}_{j}\right|
  2. y ~ j \tilde{y}_{j}
  3. F = ( N - p ) ( p - 1 ) j = 1 p n j ( z ~ j - z ~ ) 2 j = 1 p i = 1 n j ( z i j - z ~ j ) 2 F=\frac{(N-p)}{(p-1)}\frac{\sum_{j=1}^{p}n_{j}(\tilde{z}_{\cdot j}-\tilde{z}_{% \cdot\cdot})^{2}}{\sum_{j=1}^{p}\sum_{i=1}^{n_{j}}(z_{ij}-\tilde{z}_{\cdot j})% ^{2}}
  4. z ~ j \tilde{z}_{\cdot j}
  5. z i j z_{ij}
  6. z ~ \tilde{z}_{\cdot\cdot}
  7. z i j z_{ij}

Burton_Wold_Wind_Farm.html

  1. 43 , 416 MWh ( 366 days ) × ( 24 hours/day ) × ( 20 MW ) = 0.2471 25 % \frac{43,416\mbox{MWh}~{}}{(366\mbox{days}~{})\times(24\mbox{hours/day}~{})% \times(20\mbox{MW}~{})}=0.2471\approx{25\%}

Butyrate_kinase.html

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

Butyrate—CoA_ligase.html

  1. \rightleftharpoons

Bühlmann_model.html

  1. X ¯ i = 1 m j = 1 m X i j \scriptstyle\bar{X}_{i}=\frac{1}{m}\sum_{j=1}^{m}X_{ij}
  2. Θ i \Theta_{i}
  3. m ( ϑ ) = E [ X i j | Θ i = ϑ ] m(\vartheta)=\operatorname{E}\left[X_{ij}|\Theta_{i}=\vartheta\right]
  4. Π = E ( m ( ϑ ) | X i 1 , X i 2 , X i m ) \Pi=\operatorname{E}(m(\vartheta)|X_{i1},X_{i2},...X_{im})
  5. μ = ( m ( ϑ ) ) \mu=\operatorname{(}m(\vartheta))
  6. s 2 ( ϑ ) = Var [ X i j | Θ i = ϑ ] s^{2}(\vartheta)=\operatorname{Var}\left[X_{ij}|\Theta_{i}=\vartheta\right]
  7. σ 2 = E [ s 2 ( ϑ ) ] \sigma^{2}=\operatorname{E}\left[s^{2}(\vartheta)\right]
  8. v 2 = Var [ m ( ϑ ) ] v^{2}=\operatorname{Var}\left[m(\vartheta)\right]
  9. m ( ϑ ) m(\vartheta)
  10. s 2 ( ϑ ) s^{2}(\vartheta)
  11. ϑ \vartheta
  12. arg min a i 0 , a i 1 , , a i m E [ ( a i 0 + j = 1 m a i j X i j - Π ) 2 ] \underset{a_{i0},a_{i1},...,a_{im}}{\operatorname{arg\,min}}\operatorname{E}% \left[\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right)^{2}\right]
  13. a i 0 + j = 1 m a i j X i j a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}
  14. Π \Pi
  15. Z X ¯ i + ( 1 - Z ) μ Z\bar{X}_{i}+(1-Z)\mu
  16. Z = 1 1 + σ 2 v 2 m Z=\frac{1}{1+\frac{\sigma^{2}}{v^{2}m}}
  17. f = E [ ( a i 0 + j = 1 m a i j X i j - m ( ϑ ) ) 2 ] m i n f=\operatorname{E}\left[\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)% \right)^{2}\right]\rightarrow min
  18. E [ ( a i 0 + j = 1 m a i j X i j - m ( ϑ ) ) 2 ] \operatorname{E}\left[\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)% \right)^{2}\right]
  19. = E [ ( a i 0 + j = 1 m a i j X i j - Π ) 2 ] + E [ ( m ( ϑ ) - Π ) 2 ] + 2 E [ ( a i 0 + j = 1 m a i j X i j - Π ) ( m ( ϑ ) - Π ) ] =\operatorname{E}\left[\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right)^{2}% \right]+\operatorname{E}\left[\left(m(\vartheta)-\Pi\right)^{2}\right]+2E\left% [\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right)\left(m(\vartheta)-\Pi% \right)\right]
  20. = E [ ( a i 0 + j = 1 m a i j X i j - Π ) 2 ] + E [ ( m ( ϑ ) - Π ) 2 ] =\operatorname{E}\left[\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right)^{2}% \right]+\operatorname{E}\left[\left(m(\vartheta)-\Pi\right)^{2}\right]
  21. E [ ( a i 0 + j = 1 m a i j X i j - Π ) ( m ( ϑ ) - Π ) ] \operatorname{E}\left[\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right)\left(% m(\vartheta)-\Pi\right)\right]
  22. E Θ { E X [ ( a i 0 + j = 1 m a i j X i j - Π ) ( m ( ϑ ) - Π ) | X i 1 , X i 2 , X i m ] } \operatorname{E}_{\Theta}\left\{E_{X}\left[\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_% {ij}-\Pi\right)\left(m(\vartheta)-\Pi\right)|X_{i1},X_{i2},X_{im}\right]\right\}
  23. = ( a i 0 + j = 1 m a i j X i j - Π ) E Θ { E X [ ( m ( ϑ ) - Π ) | X i 1 , X i 2 , X i m ] } = 0 =\left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right)\operatorname{E}_{\Theta}% \left\{\operatorname{E}_{X}\left[\left(m(\vartheta)-\Pi\right)|X_{i1},X_{i2},X% _{im}\right]\right\}=0
  24. Π = E ( m ( ϑ ) | X i 1 , X i 2 , X i m ) \Pi=\operatorname{E}(m(\vartheta)|X_{i1},X_{i2},...X_{im})
  25. 1 2 f a 01 = \frac{1}{2}\frac{\partial f}{\partial a_{01}}=
  26. E [ a i 0 + j = 1 m a i j X i j - m ( ϑ ) ] = a i 0 + j = 1 m a i j E ( X i j ) - E ( m ( ϑ ) ) = a i 0 - ( j = 1 m a i j - 1 ) μ \operatorname{E}\left[a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right]=a_% {i0}+\sum_{j=1}^{m}a_{ij}\operatorname{E}(X_{ij})-\operatorname{E}(m(\vartheta% ))=a_{i0}-\left(\sum_{j=1}^{m}a_{ij}-1\right)\mu
  27. a i 0 = ( j = 1 m a i j - 1 ) μ a_{i0}=\left(\sum_{j=1}^{m}a_{ij}-1\right)\mu
  28. k 0 k\neq 0
  29. 1 2 f a i k = E [ X i k ( a i 0 + j = 1 m a i j X i j - m ( ϑ ) ) ] \frac{1}{2}\frac{\partial f}{\partial a_{ik}}=\operatorname{E}\left[X_{ik}% \left(a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right)\right]
  30. = E [ X i k ] a i 0 + j = 1 , j k m a i j E [ X i k X i j ] + a i k E [ X i k 2 ] - E [ X i k m ( ϑ ) ] = 0 =\operatorname{E}\left[X_{ik}\right]a_{i0}+\sum_{j=1,j\neq k}^{m}a_{ij}% \operatorname{E}[X_{ik}X_{ij}]+a_{ik}\operatorname{E}[X^{2}_{ik}]-% \operatorname{E}[X_{ik}m(\vartheta)]=0
  31. E [ X i j X i k ] = E [ E [ X i j X i k | ϑ ] ] = E [ c o v ( X i j X i k | ϑ ) + E ( X i j | ϑ ) E ( X i k | ϑ ) ] = E [ ( m ( ϑ ) ) 2 ] = v 2 + μ 2 \operatorname{E}[X_{ij}X_{ik}]=\operatorname{E}[\operatorname{E}[X_{ij}X_{ik}|% \vartheta]]=\operatorname{E}[cov(X_{ij}X_{ik}|\vartheta)+\operatorname{E}(X_{% ij}|\vartheta)\operatorname{E}(X_{ik}|\vartheta)]=\operatorname{E}[(m(% \vartheta))^{2}]=v^{2}+\mu^{2}
  32. E [ X i k 2 ] = E [ E [ X i k 2 | ϑ ] ] = E [ s 2 ( ϑ ) + ( m ( ϑ ) ) 2 ] = σ 2 + v 2 + μ 2 \operatorname{E}[X^{2}_{ik}]=\operatorname{E}[\operatorname{E}[X^{2}_{ik}|% \vartheta]]=\operatorname{E}[s^{2}(\vartheta)+(m(\vartheta))^{2}]=\sigma^{2}+v% ^{2}+\mu^{2}
  33. E [ X i k m ( ϑ ) ] = E [ E [ X i k m ( ϑ ) | Θ i ] = E [ ( m ( ϑ ) ) 2 ] = v 2 + μ 2 \operatorname{E}[X_{ik}m(\vartheta)]=\operatorname{E}[\operatorname{E}[X_{ik}m% (\vartheta)|\Theta_{i}]=\operatorname{E}[(m(\vartheta))^{2}]=v^{2}+\mu^{2}
  34. 1 2 f a i k \frac{1}{2}\frac{\partial f}{\partial a_{ik}}
  35. = ( 1 - j = 1 m a i j ) μ 2 + j = 1 , j k m a i j ( v 2 + μ 2 ) + a i k ( σ 2 + v 2 + μ 2 ) - ( v 2 + μ 2 ) = a i k σ 2 - ( 1 - j = 1 m a i j ) v 2 = 0 =\left(1-\sum_{j=1}^{m}a_{ij}\right)\mu^{2}+\sum_{j=1,j\neq k}^{m}a_{ij}(v^{2}% +\mu^{2})+a_{ik}(\sigma^{2}+v^{2}+\mu^{2})-(v^{2}+\mu^{2})=a_{ik}\sigma^{2}-% \left(1-\sum_{j=1}^{m}a_{ij}\right)v^{2}=0
  36. σ 2 a i k = v 2 ( 1 - j = 1 m a i j ) \sigma^{2}a_{ik}=v^{2}\left(1-\sum_{j=1}^{m}a_{ij}\right)
  37. a i k a_{ik}
  38. a i 1 = a i 2 = = a i m = v 2 σ 2 + m v 2 a_{i1}=a_{i2}=...=a_{im}=\frac{v^{2}}{\sigma^{2}+mv^{2}}
  39. a i 0 a_{i0}
  40. a i 0 = ( 1 - m a i k ) μ = ( 1 - m v 2 σ 2 + m v 2 ) μ a_{i0}=(1-ma_{ik})\mu=\left(1-\frac{mv^{2}}{\sigma^{2}+mv^{2}}\right)\mu
  41. a i 0 + j = 1 m a i j X i j = m v 2 σ 2 + m v 2 X i ¯ + ( 1 - m v 2 σ 2 + m v 2 ) μ = Z X i ¯ + ( 1 - Z ) μ a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}=\frac{mv^{2}}{\sigma^{2}+mv^{2}}\bar{X_{i}}+% \left(1-\frac{mv^{2}}{\sigma^{2}+mv^{2}}\right)\mu=Z\bar{X_{i}}+(1-Z)\mu

C-minimal_theory.html

  1. x y z [ C ( x ; y z ) C ( x ; z y ) ] , \forall xyz\,[C(x;yz)\rightarrow C(x;zy)],
  2. x y z [ C ( x ; y z ) ¬ C ( y ; x z ) ] , \forall xyz\,[C(x;yz)\rightarrow\neg C(y;xz)],
  3. x y z w [ C ( x ; y z ) ( C ( w ; y z ) C ( x ; w z ) ) ] , \forall xyzw\,[C(x;yz)\rightarrow(C(w;yz)\vee C(x;wz))],
  4. x y [ x y z y C ( x ; y z ) ] . \forall xy\,[x\neq y\rightarrow\exists z\neq y\,C(x;yz)].
  5. C ( a ; b c ) | b - c | p < | a - c | p C(a;bc)\iff|b-c|_{p}<|a-c|_{p}

Cadmium-transporting_ATPase.html

  1. \rightleftharpoons

Caldesmon-phosphatase.html

  1. \rightleftharpoons

Caltech_101.html

  1. N train 30 \mathrm{N}_{\mathrm{train}}\leq 30

Canopy_clustering_algorithm.html

  1. T 1 T_{1}
  2. T 2 T_{2}
  3. T 1 > T 2 T_{1}>T_{2}
  4. T 1 T_{1}
  5. T 2 T_{2}

Capacity_of_a_set.html

  1. C ( Σ , S ) = - 1 ( n - 2 ) σ n S u ν d σ , C(\Sigma,S)=-\frac{1}{(n-2)\sigma_{n}}\int_{S^{\prime}}\frac{\partial u}{% \partial\nu}\,\mathrm{d}\sigma^{\prime},
  2. u ν ( x ) = u ( x ) ν ( x ) \frac{\partial u}{\partial\nu}(x)=\nabla u(x)\cdot\nu(x)
  3. C ( Σ , S ) = 1 ( n - 2 ) σ n D | u | 2 d x . C(\Sigma,S)=\frac{1}{(n-2)\sigma_{n}}\int_{D}|\nabla u|^{2}\mathrm{d}x.
  4. I [ v ] = 1 ( n - 2 ) σ n D | v | 2 d x I[v]=\frac{1}{(n-2)\sigma_{n}}\int_{D}|\nabla v|^{2}\mathrm{d}x
  5. C ( K ) = n K | u | 2 d x . C(K)=\int_{\mathbb{R}^{n}\setminus K}|\nabla u|^{2}\mathrm{d}x.
  6. C ( K ) = S u ν d σ . C(K)=\int_{S}\frac{\partial u}{\partial\nu}\,\mathrm{d}\sigma.
  7. C ( K ) = lim r C ( Σ , S r ) . C(K)=\lim_{r\to\infty}C(\Sigma,S_{r}).
  8. ( A u ) = 0 \nabla\cdot(A\nabla u)=0
  9. I [ u ] = D ( u ) T A ( u ) d x I[u]=\int_{D}(\nabla u)^{T}A(\nabla u)\,\mathrm{d}x

Capsular-polysaccharide-transporting_ATPase.html

  1. \rightleftharpoons

Carbamate_kinase.html

  1. \rightleftharpoons

Carboxylesterase.html

  1. \rightleftharpoons

Carboxymethylenebutenolidase.html

  1. \rightleftharpoons

Carboxymethylhydantoinase.html

  1. \rightleftharpoons

Carboxymethyloxysuccinate_lyase.html

  1. \rightleftharpoons

Carboxyvinyl-carboxyphosphonate_phosphorylmutase.html

  1. \rightleftharpoons

Carleman_matrix.html

  1. f ( x ) f(x)
  2. M [ f ] j k = 1 k ! [ d k d x k ( f ( x ) ) j ] x = 0 , M[f]_{jk}=\frac{1}{k!}\left[\frac{d^{k}}{dx^{k}}(f(x))^{j}\right]_{x=0}~{},
  3. ( f ( x ) ) j = k = 0 M [ f ] j k x k . (f(x))^{j}=\sum_{k=0}^{\infty}M[f]_{jk}x^{k}.
  4. f ( x ) f(x)
  5. f ( x ) = k = 0 M [ f ] 1 , k x k . f(x)=\sum_{k=0}^{\infty}M[f]_{1,k}x^{k}.~{}
  6. M [ f ] M[f]
  7. [ 1 , x , x 2 , x 3 , ] τ \left[1,x,x^{2},x^{3},...\right]^{\tau}
  8. M [ f ] M[f]
  9. f ( x ) f(x)
  10. f ( x ) 2 = k = 0 M [ f ] 2 , k x k , f(x)^{2}=\sum_{k=0}^{\infty}M[f]_{2,k}x^{k}~{},
  11. f ( x ) f(x)
  12. M [ f ] M[f]
  13. f ( x ) 0 = 1 = k = 0 M [ f ] 0 , k x k = 1 + k = 1 0 * x k . f(x)^{0}=1=\sum_{k=0}^{\infty}M[f]_{0,k}x^{k}=1+\sum_{k=1}^{\infty}0*x^{k}~{}.
  14. M [ f ] M[f]
  15. [ 1 , x , x 2 , ] τ \left[1,x,x^{2},...\right]^{\tau}
  16. [ 1 , f ( x ) , f ( x ) 2 , ] τ \left[1,f(x),f(x)^{2},...\right]^{\tau}
  17. M [ f ] * [ 1 , x , x 2 , x 3 , ] τ = [ 1 , f ( x ) , ( f ( x ) ) 2 , ( f ( x ) ) 3 , ] τ . M[f]*\left[1,x,x^{2},x^{3},...\right]^{\tau}=\left[1,f(x),(f(x))^{2},(f(x))^{3% },...\right]^{\tau}.
  18. f ( x ) f(x)
  19. B [ f ] j k = 1 j ! [ d j d x j ( f ( x ) ) k ] x = 0 , B[f]_{jk}=\frac{1}{j!}\left[\frac{d^{j}}{dx^{j}}(f(x))^{k}\right]_{x=0}~{},
  20. ( f ( x ) ) k = j = 0 B [ f ] j k x j , (f(x))^{k}=\sum_{j=0}^{\infty}B[f]_{jk}x^{j}~{},
  21. f ( x ) = a 1 x + k = 2 a k x k f(x)=a_{1}x+\sum_{k=2}^{\infty}a_{k}x^{k}
  22. M [ f ] x 0 = M x [ x - x 0 ] M [ f ] M x [ x + x 0 ] M[f]_{x_{0}}=M_{x}[x-x_{0}]M[f]M_{x}[x+x_{0}]
  23. M [ f ] x 0 = M [ g ] M[f]_{x_{0}}=M[g]
  24. g ( x ) = f ( x + x 0 ) - x 0 g(x)=f(x+x_{0})-x_{0}
  25. ( M [ f ] x 0 ) n = M x [ x - x 0 ] M [ f ] n M x [ x + x 0 ] (M[f]_{x_{0}})^{n}=M_{x}[x-x_{0}]M[f]^{n}M_{x}[x+x_{0}]
  26. M [ f g ] = M [ f ] M [ g ] , M[f\circ g]=M[f]M[g]~{},
  27. B [ f g ] = B [ g ] B [ f ] , B[f\circ g]=B[g]B[f]~{},
  28. f ( x ) f(x)
  29. f ( x ) f(x)
  30. f g f\circ g
  31. f ( g ( x ) ) f(g(x))
  32. M [ f n ] = M [ f ] n \,M[f^{n}]=M[f]^{n}
  33. f n \,f^{n}
  34. M [ f - 1 ] = M [ f ] - 1 \,M[f^{-1}]=M[f]^{-1}
  35. f - 1 \,f^{-1}
  36. M [ a ] = ( 1 0 0 a 0 0 a 2 0 0 ) M[a]=\left(\begin{array}[]{cccc}1&0&0&\cdots\\ a&0&0&\cdots\\ a^{2}&0&0&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}\right)
  37. M x [ x ] = ( 1 0 0 0 1 0 0 0 1 ) M_{x}[x]=\left(\begin{array}[]{cccc}1&0&0&\cdots\\ 0&1&0&\cdots\\ 0&0&1&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}\right)
  38. M x [ a + x ] = ( 1 0 0 a 1 0 a 2 2 a 1 ) M_{x}[a+x]=\left(\begin{array}[]{cccc}1&0&0&\cdots\\ a&1&0&\cdots\\ a^{2}&2a&1&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}\right)
  39. M x [ c x ] = ( 1 0 0 0 c 0 0 0 c 2 ) M_{x}[cx]=\left(\begin{array}[]{cccc}1&0&0&\cdots\\ 0&c&0&\cdots\\ 0&0&c^{2}&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}\right)
  40. M x [ a + c x ] = ( 1 0 0 a c 0 a 2 2 a c c 2 ) M_{x}[a+cx]=\left(\begin{array}[]{cccc}1&0&0&\cdots\\ a&c&0&\cdots\\ a^{2}&2ac&c^{2}&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}\right)
  41. f ( x ) = k = 1 f k x k f(x)=\sum_{k=1}^{\infty}f_{k}x^{k}
  42. M [ f ] = ( 1 0 0 0 f 1 f 2 0 0 f 1 2 ) M[f]=\left(\begin{array}[]{cccc}1&0&0&\cdots\\ 0&f_{1}&f_{2}&\cdots\\ 0&0&f_{1}^{2}&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}\right)
  43. f ( x ) = k = 0 f k x k f(x)=\sum_{k=0}^{\infty}f_{k}x^{k}
  44. M [ f ] = ( 1 0 0 f 0 f 1 f 2 f 0 2 2 f 0 f 1 f 1 2 + 2 f 0 f 2 ) M[f]=\left(\begin{array}[]{cccc}1&0&0&\cdots\\ f_{0}&f_{1}&f_{2}&\cdots\\ f_{0}^{2}&2f_{0}f_{1}&f_{1}^{2}+2f_{0}f_{2}&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{array}\right)

CaRMetal.html

  1. The radius of the circle would be 4 π 1.13 \mbox{The radius of the circle would be }~{}\sqrt{\frac{4}{\pi}}\simeq 1.13

Carnitinamidase.html

  1. \rightleftharpoons

Carnitine_dehydratase.html

  1. \rightleftharpoons

Carnitine_O-acetyltransferase.html

  1. \rightleftharpoons

Carnitine_O-octanoyltransferase.html

  1. \rightleftharpoons

Carnosine_synthase.html

  1. \rightleftharpoons

Carvill_Hurricane_Index.html

  1. C H I = ( V V 0 ) 3 + 3 2 ( R R 0 ) ( V V 0 ) 2 CHI=\left(\frac{V}{V_{0}}\right)^{3}+\frac{3}{2}\left(\frac{R}{R_{0}}\right)% \left(\frac{V}{V_{0}}\right)^{2}

Casbene_synthase.html

  1. \rightleftharpoons

Casey's_theorem.html

  1. t 12 t 34 + t 41 t 23 - t 13 t 24 = 0 t_{12}\cdot t_{34}+t_{41}\cdot t_{23}-t_{13}\cdot t_{24}=0
  2. O \,O
  3. R \,R
  4. O 1 , O 2 , O 3 , O 4 \,O_{1},O_{2},O_{3},O_{4}
  5. O \,O
  6. t i j \,t_{ij}
  7. O i , O j \,O_{i},O_{j}
  8. t 12 t 34 + t 41 t 23 = t 13 t 24 . \,t_{12}\cdot t_{34}+t_{41}\cdot t_{23}=t_{13}\cdot t_{24}.
  9. O i \,O_{i}
  10. R i \,R_{i}
  11. O \,O
  12. K i \,K_{i}
  13. O , O i \,O,O_{i}
  14. t i j 2 = O i O j ¯ 2 - ( R i - R j ) 2 . \,t_{ij}^{2}=\overline{O_{i}O_{j}}^{2}-(R_{i}-R_{j})^{2}.
  15. K i , K j \,K_{i},K_{j}
  16. O i O O j \,O_{i}OO_{j}
  17. O i O j ¯ 2 = O O i ¯ 2 + O O j ¯ 2 - 2 O O i ¯ O O j ¯ cos O i O O j \overline{O_{i}O_{j}}^{2}=\overline{OO_{i}}^{2}+\overline{OO_{j}}^{2}-2% \overline{OO_{i}}\cdot\overline{OO_{j}}\cdot\cos\angle O_{i}OO_{j}
  18. O , O i \,O,O_{i}
  19. O O i ¯ = R - R i , O i O O j = K i O K j \overline{OO_{i}}=R-R_{i},\,\angle O_{i}OO_{j}=\angle K_{i}OK_{j}
  20. C \,C
  21. O \,O
  22. K i C K j \,K_{i}CK_{j}
  23. K i K j ¯ = 2 R sin K i C K j = 2 R sin K i O K j 2 \overline{K_{i}K_{j}}=2R\cdot\sin\angle K_{i}CK_{j}=2R\cdot\sin\frac{\angle K_% {i}OK_{j}}{2}
  24. cos K i O K j = 1 - 2 sin 2 K i O K j 2 = 1 - 2 ( K i K j ¯ 2 R ) 2 = 1 - K i K j ¯ 2 2 R 2 \cos\angle K_{i}OK_{j}=1-2\sin^{2}\frac{\angle K_{i}OK_{j}}{2}=1-2\cdot\left(% \frac{\overline{K_{i}K_{j}}}{2R}\right)^{2}=1-\frac{\overline{K_{i}K_{j}}^{2}}% {2R^{2}}
  25. O i O j ¯ 2 = ( R - R i ) 2 + ( R - R j ) 2 - 2 ( R - R i ) ( R - R j ) ( 1 - K i K j ¯ 2 2 R 2 ) \overline{O_{i}O_{j}}^{2}=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})\left% (1-\frac{\overline{K_{i}K_{j}}^{2}}{2R^{2}}\right)
  26. O i O j ¯ 2 = ( R - R i ) 2 + ( R - R j ) 2 - 2 ( R - R i ) ( R - R j ) + ( R - R i ) ( R - R j ) K i K j ¯ 2 R 2 \overline{O_{i}O_{j}}^{2}=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})+(R-R% _{i})(R-R_{j})\cdot\frac{\overline{K_{i}K_{j}}^{2}}{R^{2}}
  27. O i O j ¯ 2 = ( ( R - R i ) - ( R - R j ) ) 2 + ( R - R i ) ( R - R j ) K i K j ¯ 2 R 2 \overline{O_{i}O_{j}}^{2}=((R-R_{i})-(R-R_{j}))^{2}+(R-R_{i})(R-R_{j})\cdot% \frac{\overline{K_{i}K_{j}}^{2}}{R^{2}}
  28. t i j = O i O j ¯ 2 - ( R i - R j ) 2 = R - R i R - R j K i K j ¯ R t_{ij}=\sqrt{\overline{O_{i}O_{j}}^{2}-(R_{i}-R_{j})^{2}}=\frac{\sqrt{R-R_{i}}% \cdot\sqrt{R-R_{j}}\cdot\overline{K_{i}K_{j}}}{R}
  29. K 1 K 2 K 3 K 4 \,K_{1}K_{2}K_{3}K_{4}
  30. t 12 t 34 + t 14 t 23 = 1 R 2 R - R 1 R - R 2 R - R 3 R - R 4 ( K 1 K 2 ¯ K 3 K 4 ¯ + K 1 K 4 ¯ K 2 K 3 ¯ ) t_{12}t_{34}+t_{14}t_{23}=\frac{1}{R^{2}}\cdot\sqrt{R-R_{1}}\sqrt{R-R_{2}}% \sqrt{R-R_{3}}\sqrt{R-R_{4}}\left(\overline{K_{1}K_{2}}\cdot\overline{K_{3}K_{% 4}}+\overline{K_{1}K_{4}}\cdot\overline{K_{2}K_{3}}\right)
  31. = 1 R 2 R - R 1 R - R 2 R - R 3 R - R 4 ( K 1 K 3 ¯ K 2 K 4 ¯ ) = t 13 t 24 =\frac{1}{R^{2}}\cdot\sqrt{R-R_{1}}\sqrt{R-R_{2}}\sqrt{R-R_{3}}\sqrt{R-R_{4}}% \left(\overline{K_{1}K_{3}}\cdot\overline{K_{2}K_{4}}\right)=t_{13}t_{24}
  32. O i , O j \,O_{i},O_{j}
  33. O \,O
  34. t i j \,t_{ij}
  35. O i , O j \,O_{i},O_{j}
  36. O \,O
  37. t i j \,t_{ij}

Cass_criterion.html

  1. p t p_{t}
  2. t t
  3. t = 0 1 p t < \sum_{t=0}^{\infty}\frac{1}{\|p_{t}\|}<\infty

Category_of_topological_vector_spaces.html

  1. 𝐓𝐕𝐞𝐜𝐭 K \,\textbf{TVect}_{K}
  2. V V
  3. ( ( V i , τ i ) , f i ) i I ((V_{i},\tau_{i}),f_{i})_{i\in I}
  4. ( V i , τ i ) (V_{i},\tau_{i})
  5. f i : V V i f_{i}:V\to V_{i}
  6. τ \tau
  7. V V
  8. g : Z V g:Z\to V
  9. ( Z , σ ) (Z,\sigma)
  10. g : ( Z , σ ) ( V , τ ) g:(Z,\sigma)\to(V,\tau)
  11. \iff
  12. i I : f i g : ( Z , σ ) ( V i , τ i ) \forall i\in I:f_{i}\circ g:(Z,\sigma)\to(V_{i},\tau_{i})
  13. ( V , τ ) (V,\tau)
  14. V V
  15. ( τ i , f i ) i I (\tau_{i},f_{i})_{i\in I}
  16. 𝐓𝐕𝐞𝐜𝐭 K \,\textbf{TVect}_{K}
  17. 𝐕𝐞𝐜𝐭 K 𝐒𝐞𝐭 𝐓𝐕𝐞𝐜𝐭 K 𝐓𝐨𝐩 \begin{array}[]{ccc}\,\textbf{Vect}_{K}&\rightarrow&\,\textbf{Set}\\ \uparrow&&\uparrow\\ \,\textbf{TVect}_{K}&\rightarrow&\,\textbf{Top}\end{array}
  18. 𝐕𝐞𝐜𝐭 K \,\textbf{Vect}_{K}
  19. 𝐓𝐕𝐞𝐜𝐭 K \,\textbf{TVect}_{K}
  20. 𝐕𝐞𝐜𝐭 K \,\textbf{Vect}_{K}
  21. 𝐓𝐕𝐞𝐜𝐭 K \,\textbf{TVect}_{K}

Cauchy_number.html

  1. Ca = ρ u 2 K \mathrm{Ca}=\frac{\rho u^{2}}{K}
  2. ρ \rho
  3. K s = γ p K_{s}=\gamma p
  4. γ \gamma
  5. K s = γ p = γ ρ R T = ρ a 2 K_{s}=\gamma p=\gamma\rho RT=\,\rho a^{2}
  6. a = γ R T a=\sqrt{\gamma RT}
  7. Ca = u 2 a 2 = M 2 \mathrm{Ca}=\frac{u^{2}}{a^{2}}=\mathrm{M}^{2}

Causality_conditions.html

  1. ( M , g ) (M,g)
  2. p q p\ll q
  3. p q p\prec q
  4. I + ( x ) \,I^{+}(x)
  5. I - ( x ) \,I^{-}(x)
  6. J + ( x ) \,J^{+}(x)
  7. J - ( x ) \,J^{-}(x)
  8. p M p\in M
  9. p ≪̸ p p\not\ll p
  10. p ≪̸ p p\not\ll p
  11. p M p\in M
  12. p q p\prec q
  13. q p q\prec p
  14. p = q p=q
  15. p , q M p,q\in M
  16. I - ( p ) = I - ( q ) p = q I^{-}(p)=I^{-}(q)\implies p=q
  17. U U
  18. p M p\in M
  19. V U , p V V\subset U,p\in V
  20. p p
  21. V V
  22. p , q M p,q\in M
  23. I + ( p ) = I + ( q ) p = q I^{+}(p)=I^{+}(q)\implies p=q
  24. U U
  25. p M p\in M
  26. V U , p V V\subset U,p\in V
  27. p p
  28. V V
  29. p M p\in M
  30. U U
  31. p p
  32. U U
  33. U U
  34. p M p\in M
  35. V U , p V V\subset U,p\in V
  36. V V
  37. M M
  38. U U
  39. M M
  40. t t
  41. M M
  42. a t \nabla^{a}t
  43. M \,M
  44. J + ( x ) J - ( y ) J^{+}(x)\cap J^{-}(y)
  45. x , y M x,y\in M
  46. M M
  47. M M
  48. × S \mathbb{R}\times\!\,S
  49. S S
  50. \mathbb{R}

CCA_tRNA_nucleotidyltransferase.html

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

Cd2+-exporting_ATPase.html

  1. \rightleftharpoons

CD48.html

  1. K D K_{D}
  2. K D K_{D}

CDP-acylglycerol_O-arachidonoyltransferase.html

  1. \rightleftharpoons

CDP-diacylglycerol_diphosphatase.html

  1. \rightleftharpoons

CDP-diacylglycerol—glycerol-3-phosphate_3-phosphatidyltransferase.html

  1. \rightleftharpoons

CDP-diacylglycerol—inositol_3-phosphatidyltransferase.html

  1. \rightleftharpoons

CDP-diacylglycerol—serine_O-phosphatidyltransferase.html

  1. \rightleftharpoons

CDP-glucose_4,6-dehydratase.html

  1. \rightleftharpoons

CDP-glycerol_diphosphatase.html

  1. \rightleftharpoons

CDP-glycerol_glycerophosphotransferase.html

  1. \rightleftharpoons

CDP-ribitol_ribitolphosphotransferase.html

  1. \rightleftharpoons

Cellobiose_phosphorylase.html

  1. \rightleftharpoons

Cellodextrin_phosphorylase.html

  1. \rightleftharpoons

Cellulose_synthase_(GDP-forming).html

  1. \rightleftharpoons

Cellulose_synthase_(UDP-forming).html

  1. \rightleftharpoons

Central_series.html

  1. { 1 } = A 0 A 1 A n = G \{1\}=A_{0}\triangleleft A_{1}\triangleleft\dots\triangleleft A_{n}=G
  2. 1 = Z 0 Z 1 Z i , 1=Z_{0}\triangleleft Z_{1}\triangleleft\cdots\triangleleft Z_{i}\triangleleft\cdots,
  3. Z i + 1 = { x G y G : [ x , y ] Z i } Z_{i+1}=\{x\in G\mid\forall y\in G:[x,y]\in Z_{i}\}
  4. Z λ ( G ) = α < λ Z α ( G ) . Z_{\lambda}(G)=\bigcup_{\alpha<\lambda}Z_{\alpha}(G).

Cephalosporin-C_deacetylase.html

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Cephalosporin-C_transaminase.html

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Ceramide_cholinephosphotransferase.html

  1. \rightleftharpoons

Ceramide_kinase.html

  1. \rightleftharpoons

Ceramography.html

  1. I t = I 0 e - α x I_{t}=I_{0}e^{-\alpha x}
  2. R = I r I i = ( η 1 - η 2 ) 2 + k 2 ( η 1 + η 2 ) 2 + k 2 R=\frac{I_{r}}{I_{i}}=\frac{(\eta_{1}-\eta_{2})^{2}+k^{2}}{(\eta_{1}+\eta_{2})% ^{2}+k^{2}}
  3. H K = 14229 P d 2 HK=14229\frac{P}{d^{2}}
  4. H K = 139.54 P d 2 HK=139.54\frac{P}{d^{2}}
  5. K i c l = 0.016 E H P ( c 0 ) 1.5 K_{icl}=0.016\sqrt{\frac{E}{H}}\frac{P}{(c_{0})^{1.5}}
  6. K i s b = 0.59 ( E H ) 1 / 8 [ σ ( P 1 / 3 ) ] 3 / 4 K_{isb}=0.59\left(\frac{E}{H}\right)^{1/8}[\sigma(P^{1/3})]^{3/4}

Certificate_(complexity).html

  1. n n
  2. f f

Cetraxate_benzylesterase.html

  1. \rightleftharpoons

Channel-conductance-controlling_ATPase.html

  1. \rightleftharpoons

Character_variety.html

  1. G G
  2. π \pi
  3. G G
  4. π \pi
  5. = Hom ( π , G ) . \mathfrak{R}=\operatorname{Hom}(\pi,G).
  6. G G
  7. \mathfrak{R}
  8. \mathbb{C}
  9. G G
  10. [ Hom ( π , G ) ] G . \mathbb{C}[\operatorname{Hom}(\pi,G)]^{G}.
  11. π \pi
  12. G = SL ( 2 , ) G=\mathrm{SL}(2,\mathbb{C})
  13. π \pi
  14. 3 \mathbb{C}^{3}
  15. [ x , y , z ] \mathbb{C}[x,y,z]
  16. G = SU ( 2 ) G=\mathrm{SU}(2)
  17. G = SL ( 2 , ) G=\mathrm{SL}(2,\mathbb{C})
  18. π \pi
  19. 7 \mathbb{C}^{7}
  20. a 2 + b 2 + c 2 + d 2 + x 2 + y 2 + z 2 - ( a b + c d ) x - ( a d + b c ) y - ( a c + b d ) z + a b c d + x y z - 4 = 0. a^{2}+b^{2}+c^{2}+d^{2}+x^{2}+y^{2}+z^{2}-(ab+cd)x-(ad+bc)y-(ac+bd)z+abcd+xyz-% 4=0.
  21. G = SL ( n , ) G=\mathrm{SL}(n,\mathbb{C})
  22. G = SL ( n , ) G=\mathrm{SL}(n,\mathbb{C})
  23. = Hom ( π , H ) \mathfrak{R}=\operatorname{Hom}(\pi,H)
  24. G H . G\not=H.
  25. G = SO ( 2 ) G=\mathrm{SO}(2)
  26. G G
  27. S 1 × S 1 . S^{1}\times S^{1}.
  28. SO ( 2 ) \mathrm{SO}(2)
  29. M M
  30. π 1 ( M ) \pi_{1}(M)
  31. G G

Charge_ordering.html

  1. δ d \delta_{d}
  2. H = - t i σ ( [ 1 + ( - 1 ) i δ d ] c i , σ c i + 1 , σ + h . c ) + U i n i , n i , + V i n i , n i + 1 H=-t\sum_{i}\sum_{\sigma}\left(\left[1+\left(-1\right)^{i}\delta_{d}\right]c^{% \dagger}_{i,\sigma}c_{i+1,\sigma}+h.c\right)+U\sum_{i}n_{i,\uparrow}n_{i,% \downarrow}+V\sum_{i}n_{i},n_{i+1}
  3. c i , σ c^{\dagger}_{i,\sigma}
  4. c i + 1 , σ c_{i+1,\sigma}
  5. σ = , \sigma=\uparrow,\downarrow
  6. i i
  7. i + 1 i+1
  8. n i , , n_{i,\downarrow,\uparrow}
  9. δ d \delta_{d}

Charging_argument.html

  1. h ( J ) = { the interval in EFT(I) with the same job class as J, if one exists the interval with the earliest finishing time amongst all intervals in EFT(I) intersecting J, otherwise h(J)=\begin{cases}\mbox{the interval in EFT(I) with the same job class as J, % if one exists}\\ \mbox{the interval with the earliest finishing time amongst all intervals in % EFT(I) intersecting J, otherwise}\end{cases}

Chenodeoxycholoyltaurine_hydrolase.html

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Chernoff's_distribution.html

  1. Z = argmax s 𝐑 ( W ( s ) - s 2 ) , Z=\underset{s\in\mathbf{R}}{\operatorname{argmax}}\ (W(s)-s^{2}),
  2. V ( a , c ) = argmax s 𝐑 ( W ( s ) - c ( s - a ) 2 ) , V(a,c)=\underset{s\in\mathbf{R}}{\operatorname{argmax}}\ (W(s)-c(s-a)^{2}),
  3. f c ( t ) = 1 2 g c ( t ) g c ( - t ) f_{c}(t)=\frac{1}{2}g_{c}(t)g_{c}(-t)
  4. g ^ c ( s ) = ( 2 / c ) 1 / 3 Ai ( i ( 2 c 2 ) - 1 / 3 s ) , s 𝐑 \hat{g}_{c}(s)=\frac{(2/c)^{1/3}}{\operatorname{Ai}(i(2c^{2})^{-1/3}s)},\ \ \ % s\in\mathbf{R}
  5. f Z ( z ) 1 2 4 4 / 3 | z | Ai ( a ~ 1 ) exp ( - 2 3 | z | 3 + 2 1 / 3 a ~ 1 | z | ) as z f_{Z}(z)\sim\frac{1}{2}\frac{4^{4/3}|z|}{\operatorname{Ai}^{\prime}(\tilde{a}_% {1})}\exp\left(-\frac{2}{3}|z|^{3}+2^{1/3}\tilde{a}_{1}|z|\right)\,\text{ as }% z\rightarrow\infty
  6. a ~ 1 - 2.3381 \tilde{a}_{1}\approx-2.3381
  7. Ai ( a ~ 1 ) 0.7022 \operatorname{Ai}^{\prime}(\tilde{a}_{1})\approx 0.7022

Chitin_deacetylase.html

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Chitin_synthase.html

  1. \rightleftharpoons

Chitobiosyldiphosphodolichol_beta-mannosyltransferase.html

  1. \rightleftharpoons

Chlorogenate_hydrolase.html

  1. \rightleftharpoons

Chlorogenate—glucarate_O-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Chlorophyll_synthase.html

  1. \rightleftharpoons

Chloroplast_protein-transporting_ATPase.html

  1. \rightleftharpoons

Choi–Williams_distribution_function.html

  1. η , τ \eta,\tau
  2. C x ( t , f ) = - - A x ( η , τ ) Φ ( η , τ ) exp ( j 2 π ( η t - τ f ) ) d η d τ , C_{x}(t,f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A_{x}(\eta,\tau)\Phi(% \eta,\tau)\exp(j2\pi(\eta t-\tau f))\,d\eta\,d\tau,
  3. A x ( η , τ ) = - x ( t + τ / 2 ) x * ( t - τ / 2 ) e - j 2 π t η d t , A_{x}(\eta,\tau)=\int_{-\infty}^{\infty}x(t+\tau/2)x^{*}(t-\tau/2)e^{-j2\pi t% \eta}\,dt,
  4. Φ ( η , τ ) = exp [ - α ( η τ ) 2 ] . \Phi\left(\eta,\tau\right)=\exp\left[-\alpha\left(\eta\tau\right)^{2}\right].
  5. η , τ \eta,\tau
  6. α \alpha
  7. η \eta
  8. τ \tau

Choline-phosphate_cytidylyltransferase.html

  1. \rightleftharpoons

Choline-sulfatase.html

  1. \rightleftharpoons

Choline_kinase.html

  1. \rightleftharpoons

Choloyl-CoA_hydrolase.html

  1. \rightleftharpoons

Choloylglycine_hydrolase.html

  1. \rightleftharpoons

Chondro-4-sulfatase.html

  1. \rightleftharpoons

Chondro-6-sulfatase.html

  1. \rightleftharpoons

Chorismate_synthase.html

  1. \rightleftharpoons

Church's_thesis_(constructive_mathematics).html

  1. ( x \exist y ϕ ( x , y ) ) ( \exist e x \exist y , u T ( e , x , y , u ) ϕ ( x , y ) ) (\forall x\;\exist y\;\phi(x,y))\to(\exist e\;\forall x\;\exist y,u\;{T}(e,x,y% ,u)\wedge\phi(x,y))
  2. ( f ) ( e ) ( n ) ( u ) [ 𝐓 ( e , n , f ( n ) , u ) ] (\forall f)(\exists e)(\forall n)(\exists u)[\mathbf{T}(e,n,f(n),u)]
  3. 𝖱𝖢𝖠 0 \mathsf{RCA}_{0}
  4. 𝖱𝖢𝖠 0 \mathsf{RCA}_{0}
  5. ( x ψ ( x ) \exist y ϕ ( x , y ) ) \exist f ( x ψ ( x ) \exist y , u T ( f , x , y , u ) ϕ ( x , y ) ) (\forall x\;\psi(x)\to\exist y\;\phi(x,y))\to\exist f(\forall x\;\psi(x)\to% \exist y,u\;{T}(f,x,y,u)\wedge\phi(x,y))
  6. ψ \psi
  7. E C T 0 ECT_{0}
  8. ψ \psi
  9. Δ 0 0 \Delta^{0}_{0}
  10. H A + E C T 0 ( ϕ ( \exist n n ϕ ) ) HA+ECT_{0}\vdash(\phi\leftrightarrow(\exist n\;n\Vdash\phi))
  11. ( H A + E C T 0 ϕ ) ( H A \exist n ( n ϕ ) ) (HA+ECT_{0}\vdash\phi)\leftrightarrow(HA\vdash\exist n\;(n\Vdash\phi))
  12. n ϕ n\Vdash\phi
  13. n realises ϕ n\,\text{ realises }\phi
  14. H A HA
  15. E C T 0 ECT_{0}
  16. ϕ \phi
  17. H A HA
  18. E C T 0 ECT_{0}
  19. ϕ \phi
  20. H A HA
  21. E C T 0 ECT_{0}
  22. ( H A + E C T 0 + M ϕ ) ( \exist n P A ( n ¯ ϕ ) ) (HA+ECT_{0}+M\vdash\phi)\leftrightarrow(\exist n\;PA\vdash(\bar{n}\Vdash\phi))
  23. ϕ \phi
  24. H A HA
  25. E C T 0 ECT_{0}
  26. M M
  27. ϕ \phi
  28. P A PA
  29. P A PA

CIE_1960_color_space.html

  1. ( ′′ R ′′ ′′ G ′′ ′′ B ′′ ) = ( 3.1956 2.4478 - 0.1434 - 2.5455 7.0492 0.9963 0.0000 0.0000 1.0000 ) ( X Y Z ) \begin{pmatrix}^{\prime\prime}R^{\prime\prime}\\ ^{\prime\prime}G^{\prime\prime}\\ ^{\prime\prime}B^{\prime\prime}\end{pmatrix}=\begin{pmatrix}3.1956&2.4478&-0.1% 434\\ -2.5455&7.0492&0.9963\\ 0.0000&0.0000&1.0000\end{pmatrix}\begin{pmatrix}X\\ Y\\ Z\end{pmatrix}
  2. u = 0.4661 x + 0.1593 y y - 0.15735 x + 0.2424 u=\frac{0.4661x+0.1593y}{y-0.15735x+0.2424}
  3. v = 0.6581 y y - 0.15735 x + 0.2424 v=\frac{0.6581y}{y-0.15735x+0.2424}
  4. u = 5.5932 x + 1.9116 y 12 y - 1.882 x + 2.9088 u=\frac{5.5932x+1.9116y}{12y-1.882x+2.9088}
  5. v = 7.8972 y 12 y - 1.882 x + 2.9088 v=\frac{7.8972y}{12y-1.882x+2.9088}
  6. u = 4 x 12 y - 2 x + 3 u=\frac{4x}{12y-2x+3}
  7. v = 6 y 12 y - 2 x + 3 v=\frac{6y}{12y-2x+3}
  8. U = 2 3 X U=\textstyle{\frac{2}{3}}X
  9. V = Y V=Y\,
  10. W = 1 2 ( - X + 3 Y + Z ) W=\textstyle{\frac{1}{2}}(-X+3Y+Z)
  11. X = 3 2 U X=\textstyle{\frac{3}{2}}U
  12. Y = V Y=V
  13. Z = 3 2 U - 3 V + 2 W Z=\textstyle{\frac{3}{2}}U-3V+2W
  14. u = U U + V + W = 4 X X + 15 Y + 3 Z u=\frac{U}{U+V+W}=\frac{4X}{X+15Y+3Z}
  15. v = V U + V + W = 6 Y X + 15 Y + 3 Z v=\frac{V}{U+V+W}=\frac{6Y}{X+15Y+3Z}
  16. x = 3 u 2 u - 8 v + 4 x=\frac{3u}{2u-8v+4}
  17. y = 2 v 2 u - 8 v + 4 y=\frac{2v}{2u-8v+4}
  18. u = u u^{\prime}=u\,
  19. v = 3 2 v v^{\prime}=\textstyle{\frac{3}{2}}v\,

CIE_1964_color_space.html

  1. U * = 13 W * ( u - u 0 ) , V * = 13 W * ( v - v 0 ) , W * = 25 Y 1 / 3 - 17 U^{*}=13W^{*}(u-u_{0}),\quad V^{*}=13W^{*}(v-v_{0}),\quad W^{*}=25Y^{1/3}-17
  2. Δ U 2 + Δ V 2 = 13 \sqrt{\Delta U^{2}+\Delta V^{2}}=13
  3. Δ E C I E U V W = ( Δ U * ) 2 + ( Δ V * ) 2 + ( Δ W * ) 2 \Delta E_{CIEUVW}=\sqrt{(\Delta U^{*})^{2}+(\Delta V^{*})^{2}+(\Delta W^{*})^{% 2}}

Cinnamate_beta-D-glucosyltransferase.html

  1. \rightleftharpoons

Cis-p-coumarate_glucosyltransferase.html

  1. \rightleftharpoons

Cis-zeatin_O-beta-D-glucosyltransferase.html

  1. \rightleftharpoons

Citrate_(Re)-synthase.html

  1. \rightleftharpoons

Citrate_dehydratase.html

  1. \rightleftharpoons

Citrate_lyase_deacetylase.html

  1. \rightleftharpoons

Citrate—CoA_ligase.html

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Citrullinase.html

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Cl-transporting_ATPase.html

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Classical_Hamiltonian_quaternions.html

  1. q = 𝐒 ( q ) + 𝐕 ( q ) q=\mathbf{S}(q)+\mathbf{V}(q)\,
  2. 𝐔 q \mathbf{U}q
  3. q = 𝐓 q 𝐔 q q=\mathbf{T}q\mathbf{U}q
  4. S ( q ) = 0 S(q)=0
  5. Q = x i + y j + z k Q=xi+yj+zk\,
  6. O A : O B OA:OB\,
  7. α ÷ β \alpha\div\beta
  8. α β \frac{\alpha}{\beta}
  9. α β = q \frac{\alpha}{\beta}=q
  10. q × β = α . {q}\times{\beta}=\alpha.
  11. α β = α × 1 β \frac{\alpha}{\beta}=\,{\alpha}\times\frac{1}{\beta}
  12. α β = q \frac{\alpha}{\beta}=q
  13. α β - 1 = q \alpha\beta^{-1}=q\,
  14. α β . β = α β - 1 . β = α \frac{\alpha}{\beta}.\beta=\alpha\beta^{-1}.\beta=\alpha
  15. × \times
  16. ÷ \div
  17. β ÷ α × α = β \beta\div\alpha\times\alpha=\beta
  18. q × α ÷ α = q q\times\alpha\div\alpha=q
  19. γ = ( γ ÷ β ) × ( β ÷ α ) × α \gamma=(\gamma\div\beta)\times(\beta\div\alpha)\times\alpha
  20. γ ÷ α = ( γ ÷ β ) × ( β ÷ α ) \gamma\div\alpha=(\gamma\div\beta)\times(\beta\div\alpha)
  21. i j = k ij=k\,
  22. k j = i \frac{k}{j}=i
  23. j k = i jk=i\,
  24. i k = j \frac{i}{k}=j
  25. k i = j ki=j\,
  26. j i = k \frac{j}{i}=k
  27. j i = - k ji=-k\,
  28. - k i = j \frac{-k}{i}=j
  29. k j = - i kj=-i\,
  30. - i j = k \frac{-i}{j}=k
  31. i k = - j ik=-j\,
  32. - j k = i \frac{-j}{k}=i
  33. i ( - j ) = - k i(-j)=-k\,
  34. - k - j = i \frac{-k}{-j}=i
  35. i ( - k ) = j i(-k)=j\,
  36. j - k = i \frac{j}{-k}=i
  37. k ( - i ) = - j k(-i)=-j\,
  38. - j - i = k \frac{-j}{-i}=k
  39. k ( - j ) = i k(-j)=i\,
  40. i - j = k \frac{i}{-j}=k
  41. j ( - k ) = - i j(-k)=-i\,
  42. - i - k = j \frac{-i}{-k}=j
  43. j ( - i ) = k j(-i)=k\,
  44. k - i = j \frac{k}{-i}=j
  45. 1 i = i - 1 = - i \frac{1}{i}=i^{-1}=-i\,
  46. 1 a a = ( - a ) a = 1 = a ( - a ) = a 1 a . \frac{1}{a}a=(-a)a=1=a(-a)=a\frac{1}{a}.
  47. i k i i\frac{k}{i}
  48. k i i . \frac{k}{i}i.
  49. k i = k 1 i = k i - 1 = k ( - i ) = - ( k i ) = - ( j ) = - j \frac{k}{i}=k\frac{1}{i}=ki^{-1}=k(-i)=-(ki)=-(j)=-j
  50. i k i = i ( - j ) = - k i\frac{k}{i}=i(-j)=-k
  51. k i i = ( - j ) i = - ( j i ) = - ( - k ) = k \frac{k}{i}i=(-j)i=-(ji)=-(-k)=k
  52. α = a i \alpha=ai\,
  53. β = b i \beta=bi\,
  54. α ÷ β = α β = a i b i = a b \alpha\div\beta=\frac{\alpha}{\beta}=\frac{ai}{bi}=\frac{a}{b}
  55. q = α β q=\frac{\alpha}{\beta}
  56. = T α T β ( cos ϕ + ϵ sin ϕ ) =\frac{T\alpha}{T\beta}(\cos\phi+\epsilon\sin\phi)
  57. q = ( a i + b j + c k ) × ( e i + f j + g k ) q=(ai+bj+ck)\times(ei+fj+gk)\,
  58. q = a e ( i × i ) + a f ( i × j ) + a g ( i × k ) + b e ( j × i ) + b f ( j × j ) + b g ( j × k ) + c e ( k × i ) + c f ( k × j ) + c g ( k × k ) q=ae({i}\times{i})+af({i}\times{j})+ag({i}\times{k})+be({j}\times{i})+bf({j}% \times{j})+bg({j}\times{k})+ce({k}\times{i})+cf({k}\times{j})+cg({k}\times{k})
  59. q = a e ( - 1 ) + a f ( + k ) + a g ( - j ) + b e ( - k ) + b f ( - 1 ) + b g ( + i ) + c e ( + j ) + c f ( - i ) + c g ( - 1 ) q=ae(-1)+af(+k)+ag(-j)+be(-k)+bf(-1)+bg(+i)+ce(+j)+cf(-i)+cg(-1)\,
  60. q = - a e - b f - c g + ( b g - c f ) i + ( c e - a g ) j + ( a f - b e ) k q=-ae-bf-cg+(bg-cf)i+(ce-ag)j+(af-be)k\,
  61. w = - a e - b f - c g w=-ae-bf-cg\,
  62. x = ( b g - c f ) x=(bg-cf)\,
  63. y = ( c e - a g ) y=(ce-ag)\,
  64. z = ( a f - b e ) z=(af-be)\,
  65. q = w + x i + y j + z k q=w+xi+yj+zk\,
  66. α = 𝐕 p \alpha=\mathbf{V}p
  67. β = 𝐕 q \beta=\mathbf{V}q
  68. r = α β ; r=\,\alpha\beta;
  69. r = 𝐒 r + 𝐕 r r=\mathbf{S}r+\mathbf{V}r
  70. q = 𝐒 q + 𝐕 q q=\,\mathbf{S}q+\mathbf{V}q
  71. q = 𝐒 q + 𝐕 q q=\,\mathbf{S}q+\mathbf{V}q
  72. 𝐊 q = 𝐒 q - 𝐕 q \mathbf{K}q=\mathbf{S}\,q-\mathbf{V}q
  73. r = 𝐊 q r=\,\mathbf{K}q
  74. 𝐓 ( 5 ) = 5 \mathbf{T}(5)=5\,
  75. 𝐓 ( - 5 ) = 5 \mathbf{T}(-5)=5\,
  76. α = x i + y j + z k \alpha=xi+yj+zk\,
  77. 𝐓 α = x 2 + y 2 + z 2 \mathbf{T}\alpha=\sqrt{x^{2}+y^{2}+z^{2}}
  78. 𝐓𝐔 α = 1 \mathbf{TU}\alpha=1
  79. q = α β . q=\frac{\alpha}{\beta}.
  80. 𝐓 q = 𝐓 α 𝐓 β . \mathbf{T}q=\frac{\mathbf{T}\alpha}{\mathbf{T}\beta}.
  81. 𝐓 q = w 2 + x 2 + y 2 + z 2 \mathbf{T}q=\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}\,
  82. 𝐓 q = q K q \mathbf{T}q=\sqrt{qKq}\,
  83. ( 𝐓 q ) 2 = 𝐓 ( q 2 ) = 𝐓 q 2 (\mathbf{T}q)^{2}=\mathbf{T}(q^{2})=\mathbf{T}q^{2}
  84. 𝐓𝐊 q = 𝐓 q \mathbf{TK}q=\mathbf{T}q
  85. u = A x . q u=Ax.q\,
  86. θ = q \theta=\angle q
  87. q = α β q=\frac{\alpha}{\beta}
  88. 1 q = q - 1 = β α \frac{1}{q}=q^{-1}=\frac{\beta}{\alpha}
  89. q × α × 1 q {q}\times{\alpha}\times\frac{1}{q}
  90. 1 ( 𝐔 q ) = 𝐒 . 𝐔 q - 𝐕 . 𝐔 q = 𝐊 . 𝐔 q \frac{1}{(\mathbf{U}q)}=\mathbf{S.U}q-\mathbf{V.U}q=\mathbf{K.U}q
  91. 𝐍 q = q 𝐊 q = ( 𝐓 q ) 2 \mathbf{N}q=\,q\mathbf{K}q=\,(\mathbf{T}q)^{2}
  92. 𝐍𝐔 q = 𝐔 q . 𝐊𝐔 q = 1 \mathbf{NU}q=\mathbf{U}q.\mathbf{KU}q=1\,
  93. q 2 = - 1 q^{2}=-1\,
  94. q + q - 1 q+q^{\prime}\sqrt{-1}

Classical_modal_logic.html

  1. A ¬ ¬ A \Diamond A\equiv\lnot\Box\lnot A
  2. A B A B . A\equiv B\vdash\Box A\equiv\Box B.
  3. A ¬ ¬ A \Box A\equiv\lnot\Diamond\lnot A
  4. A B A B . A\equiv B\vdash\Diamond A\equiv\Diamond B.

CMP-N-acylneuraminate_phosphodiesterase.html

  1. \rightleftharpoons

Coase_conjecture.html

  1. X X
  2. Y Y
  3. x x
  4. y y
  5. x < y < 2 x x<y<2x
  6. p r i c e = y price=y
  7. p r i c e = x price=x
  8. Y Y
  9. x x
  10. Y Y
  11. p r i c e = d x + ( 1 - d ) y price=dx+(1-d)y
  12. d d
  13. d x + ( 1 - d ) y < y dx+(1-d)y<y
  14. Y Y
  15. n n
  16. y y
  17. ( 1 - d n ) y (1-d^{n})y
  18. n n

Cobalt_chelatase.html

  1. \rightleftharpoons

Coherence_condition.html

  1. α A , B , C \alpha_{A,B,C}
  2. α A , B , C : ( A B ) C A ( B C ) \alpha_{A,B,C}\colon(A\otimes B)\otimes C\rightarrow A\otimes(B\otimes C)
  3. A , B , C A,B,C
  4. α A , B , C \alpha_{A,B,C}
  5. ( ( A N A N - 1 ) A N - 2 ) A 1 ) ( A N ( A N - 1 ( A 2 A 1 ) ) . (\cdots(A_{N}\otimes A_{N-1})\otimes A_{N-2})\otimes\cdots\otimes A_{1})% \rightarrow(A_{N}\otimes(A_{N-1}\otimes\cdots\otimes(A_{2}\otimes A_{1})\cdots).
  6. α A , B , C \alpha_{A,B,C}
  7. A , B , C , D A,B,C,D

Coherence_theorem.html

  1. α A , B , C : ( A B ) C A ( B C ) \alpha_{A,B,C}\colon(A\otimes B)\otimes C\rightarrow A\otimes(B\otimes C)
  2. A , B , C A,B,C
  3. A , B , C , D A,B,C,D
  4. ( ( ( A N A N - 1 ) ) A 2 ) A 1 ) ((\cdots(A_{N}\otimes A_{N-1})\otimes\cdots)\otimes A_{2})\otimes A_{1})
  5. ( A N ( A N - 1 ( ( A 2 A 1 ) ) ) (A_{N}\otimes(A_{N-1}\otimes(\cdots\otimes(A_{2}\otimes A_{1})\cdots))
  6. α A , B , C \alpha_{A,B,C}

Coherent_topology.html

  1. i α : C α X α A . i_{\alpha}:C_{\alpha}\to X\qquad\alpha\in A.
  2. { X α , α A } \{X_{\alpha},\alpha\in A\}
  3. X s e t = α A X α X^{set}=\bigcup_{\alpha\in A}X_{\alpha}
  4. i α : X α X s e t i_{\alpha}:X_{\alpha}\to X^{set}
  5. ( x , α ) ( y , β ) x = y (x,\alpha)\sim(y,\beta)\Leftrightarrow x=y
  6. X α A X α / . X\cong\coprod_{\alpha\in A}X_{\alpha}/\sim.
  7. α β \alpha\leq\beta
  8. X α X β X_{\alpha}\subset X_{\beta}
  9. lim X α \underrightarrow{\lim}X_{\alpha}
  10. lim X α \underrightarrow{\lim}X_{\alpha}
  11. f | C α : C α Y f|_{C_{\alpha}}:C_{\alpha}\to Y\,
  12. f α : f - 1 ( D α ) D α f_{\alpha}:f^{-1}(D_{\alpha})\to D_{\alpha}\,

Color_difference.html

  1. Δ E * ΔE*
  2. d E * dE*
  3. d E dE
  4. Δ E ΔE
  5. 2.3 Δ E 2.3ΔE
  6. Δ E ΔE
  7. ( L 1 * , a 1 * , b 1 * ) ({L^{*}_{1}},{a^{*}_{1}},{b^{*}_{1}})
  8. ( L 2 * , a 2 * , b 2 * ) ({L^{*}_{2}},{a^{*}_{2}},{b^{*}_{2}})
  9. Δ E a b * = ( L 2 * - L 1 * ) 2 + ( a 2 * - a 1 * ) 2 + ( b 2 * - b 1 * ) 2 \Delta E_{ab}^{*}=\sqrt{(L^{*}_{2}-L^{*}_{1})^{2}+(a^{*}_{2}-a^{*}_{1})^{2}+(b% ^{*}_{2}-b^{*}_{1})^{2}}
  10. Δ E a b * 2.3 \Delta E_{ab}^{*}\approx 2.3
  11. ( L 1 * , a 1 * , b 1 * ) (L^{*}_{1},a^{*}_{1},b^{*}_{1})
  12. ( L 2 * , a 2 * , b 2 * ) (L^{*}_{2},a^{*}_{2},b^{*}_{2})
  13. Δ E 94 * = ( Δ L * k L S L ) 2 + ( Δ C a b * k C S C ) 2 + ( Δ H a b * k H S H ) 2 \Delta E_{94}^{*}=\sqrt{\left(\frac{\Delta L^{*}}{k_{L}S_{L}}\right)^{2}+\left% (\frac{\Delta C^{*}_{ab}}{k_{C}S_{C}}\right)^{2}+\left(\frac{\Delta H^{*}_{ab}% }{k_{H}S_{H}}\right)^{2}}
  14. Δ L * = L 1 * - L 2 * \Delta L^{*}=L^{*}_{1}-L^{*}_{2}
  15. C 1 * = a 1 * 2 + b 1 * 2 C^{*}_{1}=\sqrt{{a^{*}_{1}}^{2}+{b^{*}_{1}}^{2}}
  16. C 2 * = a 2 * 2 + b 2 * 2 C^{*}_{2}=\sqrt{{a^{*}_{2}}^{2}+{b^{*}_{2}}^{2}}
  17. Δ C a b * = C 1 * - C 2 * \Delta C^{*}_{ab}=C^{*}_{1}-C^{*}_{2}
  18. Δ H a b * = Δ E a b * 2 - Δ L * 2 - Δ C a b * 2 = Δ a * 2 + Δ b * 2 - Δ C a b * 2 \Delta H^{*}_{ab}=\sqrt{{\Delta E^{*}_{ab}}^{2}-{\Delta L^{*}}^{2}-{\Delta C^{% *}_{ab}}^{2}}=\sqrt{{\Delta a^{*}}^{2}+{\Delta b^{*}}^{2}-{\Delta C^{*}_{ab}}^% {2}}
  19. Δ a * = a 1 * - a 2 * \Delta a^{*}=a^{*}_{1}-a^{*}_{2}
  20. Δ b * = b 1 * - b 2 * \Delta b^{*}=b^{*}_{1}-b^{*}_{2}
  21. S L = 1 S_{L}=1
  22. S C = 1 + K 1 C 1 * S_{C}=1+K_{1}C^{*}_{1}
  23. S H = 1 + K 2 C 1 * S_{H}=1+K_{2}C^{*}_{1}
  24. k L k_{L}
  25. K 1 K_{1}
  26. K 2 K_{2}
  27. Δ H a b * \Delta H^{*}_{ab}
  28. Δ E 00 * = ( Δ L k L S L ) 2 + ( Δ C k C S C ) 2 + ( Δ H k H S H ) 2 + R T Δ C k C S C Δ H k H S H \Delta E_{00}^{*}=\sqrt{\left(\frac{\Delta L^{\prime}}{k_{L}S_{L}}\right)^{2}+% \left(\frac{\Delta C^{\prime}}{k_{C}S_{C}}\right)^{2}+\left(\frac{\Delta H^{% \prime}}{k_{H}S_{H}}\right)^{2}+R_{T}\frac{\Delta C^{\prime}}{k_{C}S_{C}}\frac% {\Delta H^{\prime}}{k_{H}S_{H}}}
  29. Δ L = L 2 * - L 1 * \Delta L^{\prime}=L^{*}_{2}-L^{*}_{1}
  30. L ¯ = L 1 * + L 2 * 2 C ¯ = C 1 * + C 2 * 2 \bar{L}=\frac{L^{*}_{1}+L^{*}_{2}}{2}\quad\bar{C}=\frac{C^{*}_{1}+C^{*}_{2}}{2}
  31. a 1 = a 1 * + a 1 * 2 ( 1 - C ¯ 7 C ¯ 7 + 25 7 ) a 2 = a 2 * + a 2 * 2 ( 1 - C ¯ 7 C ¯ 7 + 25 7 ) a_{1}^{\prime}=a_{1}^{*}+\frac{a_{1}^{*}}{2}\left(1-\sqrt{\frac{\bar{C}^{7}}{% \bar{C}^{7}+25^{7}}}\right)\quad a_{2}^{\prime}=a_{2}^{*}+\frac{a_{2}^{*}}{2}% \left(1-\sqrt{\frac{\bar{C}^{7}}{\bar{C}^{7}+25^{7}}}\right)
  32. C ¯ = C 1 + C 2 2 and Δ C = C 2 - C 1 where C 1 = a 1 2 + b 1 * 2 C 2 = a 2 2 + b 2 * 2 \bar{C}^{\prime}=\frac{C_{1}^{\prime}+C_{2}^{\prime}}{2}\mbox{ and }~{}\Delta{% C^{\prime}}=C^{\prime}_{2}-C^{\prime}_{1}\quad\mbox{where }~{}C_{1}^{\prime}=% \sqrt{a_{1}^{{}^{\prime 2}}+b_{1}^{*^{2}}}\quad C_{2}^{\prime}=\sqrt{a_{2}^{{}% ^{\prime 2}}+b_{2}^{*^{2}}}\quad
  33. h 1 = atan2 ( b 1 * , a 1 ) mod 360 , h 2 = atan2 ( b 2 * , a 2 ) mod 360 h_{1}^{\prime}=\,\text{atan2}(b_{1}^{*},a_{1}^{\prime})\mod 360^{\circ},\quad h% _{2}^{\prime}=\,\text{atan2}(b_{2}^{*},a_{2}^{\prime})\mod 360^{\circ}
  34. Δ h = { h 2 - h 1 | h 1 - h 2 | 180 h 2 - h 1 + 360 | h 1 - h 2 | > 180 , h 2 h 1 h 2 - h 1 - 360 | h 1 - h 2 | > 180 , h 2 > h 1 \Delta h^{\prime}=\begin{cases}h_{2}^{\prime}-h_{1}^{\prime}&\left|h_{1}^{% \prime}-h_{2}^{\prime}\right|\leq 180^{\circ}\\ h_{2}^{\prime}-h_{1}^{\prime}+360^{\circ}&\left|h_{1}^{\prime}-h_{2}^{\prime}% \right|>180^{\circ},h_{2}^{\prime}\leq h_{1}^{\prime}\\ h_{2}^{\prime}-h_{1}^{\prime}-360^{\circ}&\left|h_{1}^{\prime}-h_{2}^{\prime}% \right|>180^{\circ},h_{2}^{\prime}>h_{1}^{\prime}\end{cases}
  35. Δ H = 2 C 1 C 2 sin ( Δ h / 2 ) , H ¯ = { ( h 1 + h 2 + 360 ) / 2 | h 1 - h 2 | > 180 ( h 1 + h 2 ) / 2 | h 1 - h 2 | 180 \Delta H^{\prime}=2\sqrt{C_{1}^{\prime}C_{2}^{\prime}}\sin(\Delta h^{\prime}/2% ),\quad\bar{H}^{\prime}=\begin{cases}(h_{1}^{\prime}+h_{2}^{\prime}+360^{\circ% })/2&\left|h_{1}^{\prime}-h_{2}^{\prime}\right|>180^{\circ}\\ (h_{1}^{\prime}+h_{2}^{\prime})/2&\left|h_{1}^{\prime}-h_{2}^{\prime}\right|% \leq 180^{\circ}\end{cases}
  36. T = 1 - 0.17 cos ( H ¯ - 30 ) + 0.24 cos ( 2 H ¯ ) + 0.32 cos ( 3 H ¯ + 6 ) - 0.20 cos ( 4 H ¯ - 63 ) T=1-0.17\cos(\bar{H}^{\prime}-30^{\circ})+0.24\cos(2\bar{H}^{\prime})+0.32\cos% (3\bar{H}^{\prime}+6^{\circ})-0.20\cos(4\bar{H}^{\prime}-63^{\circ})
  37. S L = 1 + 0.015 ( L ¯ - 50 ) 2 20 + ( L ¯ - 50 ) 2 S C = 1 + 0.045 C ¯ S H = 1 + 0.015 C ¯ T S_{L}=1+\frac{0.015\left(\bar{L}-50\right)^{2}}{\sqrt{20+{\left(\bar{L}-50% \right)}^{2}}}\quad S_{C}=1+0.045\bar{C}^{\prime}\quad S_{H}=1+0.015\bar{C}^{% \prime}T
  38. R T = - 2 C ¯ 7 C ¯ 7 + 25 7 sin [ 60 exp ( - [ H ¯ - 275 25 ] 2 ) ] R_{T}=-2\sqrt{\frac{\bar{C}^{\prime 7}}{\bar{C}^{\prime 7}+25^{7}}}\sin\left[6% 0^{\circ}\cdot\exp\left(-\left[\frac{\bar{H}^{\prime}-275^{\circ}}{25^{\circ}}% \right]^{2}\right)\right]
  39. ( L 2 * , C 2 * , h 2 ) (L^{*}_{2},C^{*}_{2},h_{2})
  40. ( L 1 * , C 1 * , h 1 ) (L^{*}_{1},C^{*}_{1},h_{1})
  41. Δ E C M C * = ( L 2 * - L 1 * l S L ) 2 + ( C 2 * - C 1 * c S C ) 2 + ( Δ H a b * S H ) 2 \Delta E^{*}_{CMC}=\sqrt{\left(\frac{L^{*}_{2}-L^{*}_{1}}{lS_{L}}\right)^{2}+% \left(\frac{C^{*}_{2}-C^{*}_{1}}{cS_{C}}\right)^{2}+\left(\frac{\Delta H^{*}_{% ab}}{S_{H}}\right)^{2}}
  42. S L = { 0.511 L 1 * < 16 0.040975 L 1 * 1 + 0.01765 L 1 * L 1 * 16 S C = 0.0638 C 1 * 1 + 0.0131 C 1 * + 0.638 S H = S C ( F T + 1 - F ) S_{L}=\begin{cases}0.511&L^{*}_{1}<16\\ \frac{0.040975L^{*}_{1}}{1+0.01765L^{*}_{1}}&L^{*}_{1}\geq 16\end{cases}\quad S% _{C}=\frac{0.0638C^{*}_{1}}{1+0.0131C^{*}_{1}}+0.638\quad S_{H}=S_{C}(FT+1-F)
  43. F = C 1 * 4 C 1 * 4 + 1900 T = { 0.56 + | 0.2 cos ( h 1 + 168 ) | 164 h 1 345 0.36 + | 0.4 cos ( h 1 + 35 ) | otherwise F=\sqrt{\frac{C^{*^{4}}_{1}}{C^{*^{4}}_{1}+1900}}\quad T=\begin{cases}0.56+|0.% 2\cos(h_{1}+168^{\circ})|&164^{\circ}\leq h_{1}\leq 345^{\circ}\\ 0.36+|0.4\cos(h_{1}+35^{\circ})|&\mbox{otherwise}\end{cases}

Color_triangle.html

  1. 0.37 V + 0.27 U + 0.36 E G = 0.28 S W + 0.72 B K 0.37V+0.27U+0.36EG=0.28SW+0.72BK
  2. 0.33 P C + 0.55 U + 0.12 E G = 0.37 S W + 0.63 B K 0.33PC+0.55U+0.12EG=0.37SW+0.63BK

Color–flavor_locking.html

  1. ψ i α \psi^{\alpha}_{i}
  2. α \alpha
  3. i i
  4. ψ i α C γ 5 ψ j β δ i α δ j β - δ j α δ i β = ϵ α β A ϵ i j A \langle\psi^{\alpha}_{i}C\gamma_{5}\psi^{\beta}_{j}\rangle\propto\delta^{% \alpha}_{i}\delta^{\beta}_{j}-\delta^{\alpha}_{j}\delta^{\beta}_{i}=\epsilon^{% \alpha\beta A}\epsilon_{ijA}

Common-mode_signal.html

  1. U c m = U 1 + U 2 2 U_{cm}=\frac{U_{1}+U_{2}}{2}

Complex_base_systems.html

  1. D D
  2. \C \subset\C
  3. | | |\cdot|
  4. X D X\in D
  5. X = ± ν x ν ρ ν X=\pm\sum_{\nu}x_{\nu}\rho^{\nu}
  6. ρ \rho
  7. D \in D
  8. | ρ | > 1 |\rho|>1
  9. ν \nu
  10. \Z \in\Z
  11. x ν x_{\nu}
  12. Z D Z\subset D
  13. | x ν | < | ρ | . |x_{\nu}|<|\rho|.
  14. R := | Z | R:=|Z|
  15. ρ , Z \left\langle\rho,Z\right\rangle
  16. ρ \rho
  17. Z Z
  18. R R
  19. Z R := { 0 , 1 , 2 , , R - 1 } Z_{R}:=\{0,1,2,\ldots,{R-1}\}
  20. D D
  21. \Z \Z
  22. \Z [ i ] \Z[\mathrm{i}]
  23. \Z [ - 1 + i 7 2 ] \Z[\tfrac{-1+\mathrm{i}\sqrt{7}}{2}]
  24. K := 𝖰𝗎𝗈𝗍 ( D ) K:=\mathsf{Quot}(D)
  25. | | |\cdot|
  26. K := \R K:=\R
  27. K := \C K:=\C
  28. X X
  29. | | |\cdot|
  30. ν - \nu\to-\infty
  31. Z Z
  32. R = | ρ | R=|\rho|
  33. R = | ρ | 2 R=|\rho|^{2}
  34. 10 , Z 10 \left\langle 10,Z_{10}\right\rangle
  35. 2 , Z 2 \left\langle 2,Z_{2}\right\rangle
  36. - 2 , Z 2 \left\langle-2,Z_{2}\right\rangle
  37. 3 , { - 1 , 0 , 1 } \left\langle 3,\{-1,0,1\}\right\rangle
  38. \Z \Z
  39. \R \R
  40. i \mathrm{i}
  41. R , Z R \left\langle\sqrt{R},Z_{R}\right\rangle
  42. ± i 2 , Z 2 \left\langle\pm\mathrm{i}\sqrt{2},Z_{2}\right\rangle
  43. ± 2 i , Z 4 \left\langle\pm 2\mathrm{i},Z_{4}\right\rangle
  44. 2 e ± π 2 i = ± i 2 , Z 2 \left\langle\sqrt{2}e^{\pm\tfrac{\pi}{2}\mathrm{i}}=\pm\mathrm{i}\sqrt{2},Z_{2% }\right\rangle
  45. 2 e ± 3 π 4 i = - 1 ± i , Z 2 \left\langle\sqrt{2}e^{\pm\tfrac{3\pi}{4}\mathrm{i}}=-1\pm\mathrm{i},Z_{2}\right\rangle
  46. R e i φ , Z R \left\langle\sqrt{R}e^{\mathrm{i}}\varphi,Z_{R}\right\rangle
  47. φ = ± arccos ( - β / ( 2 R ) ) \varphi=\pm\arccos{(-\beta/(2\sqrt{R}))}
  48. β < min ( R , 2 R ) \beta<\min(R,2\sqrt{R})
  49. β \beta
  50. R R
  51. β = 1 \beta=1
  52. R = 2 R=2
  53. - 1 + i 7 2 , Z 2 \left\langle\tfrac{-1+\mathrm{i}\sqrt{7}}{2},Z_{2}\right\rangle
  54. 2 e π 3 i , A 4 := { 0 , 1 , e 2 π 3 i , e - 2 π 3 i } \left\langle 2e^{\tfrac{\pi}{3}\mathrm{i}},A_{4}:=\left\{0,1,e^{\tfrac{2\pi}{3% }\mathrm{i}},e^{-\tfrac{2\pi}{3}\mathrm{i}}\right\}\right\rangle
  55. - R , A R 2 \left\langle-R,A_{R}^{2}\right\rangle
  56. A R 2 A_{R}^{2}
  57. r ν = α ν 1 + α ν 2 i r_{\nu}=\alpha_{\nu}^{1}+\alpha_{\nu}^{2}\mathrm{i}
  58. α ν Z R \alpha_{\nu}\in Z_{R}
  59. - 2 , { 0 , 1 , i , 1 + i } \left\langle-2,\{0,1,\mathrm{i},1+\mathrm{i}\}\right\rangle
  60. ρ = ρ 2 , Z 2 \left\langle\rho=\rho_{2},Z_{2}\right\rangle
  61. \rho_2=\begin{cases}
  62. Z 2 = { 0 , 1 } Z_{2}=\{0,1\}
  63. ρ , Z 2 \langle\rho,Z_{2}\rangle
  64. i \mathrm{i}
  65. i \mathrm{i}
  66. - 1 -1
  67. 2 2
  68. - 2 -2
  69. i \mathrm{i}
  70. 2 2
  71. - 1 -1
  72. 10 10
  73. - 10 -10
  74. i \mathrm{i}
  75. 0. 1 ¯ = 1. 0 ¯ = 1 0.\overline{1}=1.\overline{0}=1
  76. - 2 -2
  77. 11 11
  78. 110 110
  79. 10 10
  80. i \mathrm{i}
  81. 0. 01 ¯ = 1. 10 ¯ = 1 3 0.\overline{01}=1.\overline{10}=\tfrac{1}{3}
  82. i 2 \textstyle\mathrm{i}\sqrt{2}
  83. 101 101
  84. 10100 10100
  85. 100 100
  86. 10.101010100010... 10.101010100010...
  87. 0. 0011 ¯ = 11. 1100 ¯ = 1 3 + 1 3 i 2 0.\overline{0011}=11.\overline{1100}=\tfrac{1}{3}+\tfrac{1}{3}\mathrm{i}\sqrt{2}
  88. - 1 + i -1+\mathrm{i}
  89. 11101 11101
  90. 1100 1100
  91. 11100 11100
  92. 11 11
  93. 0. 010 ¯ = 11. 001 ¯ = 1110. 100 ¯ = 1 5 + 3 5 i 0.\overline{010}=11.\overline{001}=1110.\overline{100}=\tfrac{1}{5}+\tfrac{3}{% 5}\mathrm{i}
  94. - 1 + i 7 2 \tfrac{-1+\mathrm{i}\sqrt{7}}{2}
  95. 111 111
  96. 1010 1010
  97. 110 110
  98. 11.110001100111... 11.110001100111...
  99. 1. 011 ¯ = 11. 101 ¯ = 11100. 110 ¯ = 3 + i 7 4 1.\overline{011}=11.\overline{101}=11100.\overline{110}=\tfrac{3+\mathrm{i}% \sqrt{7}}{4}
  100. ρ 2 \rho_{2}
  101. 101 101
  102. 10100 10100
  103. 100 100
  104. 10 10
  105. 0. 0011 ¯ = 11. 1100 ¯ = 1 3 + 1 3 i 0.\overline{0011}=11.\overline{1100}=\tfrac{1}{3}+\tfrac{1}{3}\mathrm{i}

Complex_network_zeta_function.html

  1. r i j \textstyle r_{ij}
  2. i \textstyle i
  3. j \textstyle j
  4. r i j \textstyle r_{ij}
  5. \textstyle\infty
  6. i \textstyle i
  7. j \textstyle j
  8. S ( r ) \textstyle S(r)
  9. r \textstyle r
  10. ζ G ( α ) \textstyle\zeta_{G}(\alpha)
  11. ζ G ( α ) := 1 N i j i r i j - α , \zeta_{G}(\alpha):=\frac{1}{N}\sum_{i}\sum_{j\neq i}r^{-\alpha}_{ij},
  12. N \textstyle N
  13. α \textstyle\alpha
  14. ζ G ( 0 ) \textstyle\zeta_{G}(0)
  15. N - 1 \textstyle N-1
  16. N \textstyle N\rightarrow\infty
  17. α \textstyle\alpha
  18. ζ G ( α ) \textstyle\zeta_{G}(\alpha)
  19. < k > \textstyle<k>
  20. α \textstyle\alpha\rightarrow\infty
  21. k = lim α ζ G ( α ) . \langle k\rangle=\lim_{\alpha\rightarrow\infty}\zeta_{G}(\alpha).
  22. ζ G ( α ) = r S ( r ) / r α . \zeta_{G}(\alpha)=\sum_{r}S(r)/r^{\alpha}.
  23. ζ G ( α ) \textstyle\zeta_{G}(\alpha)
  24. α \textstyle\alpha
  25. ζ G ( α 1 ) > ζ G ( α 2 ) \textstyle\zeta_{G}(\alpha_{1})>\zeta_{G}(\alpha_{2})
  26. α 1 < α 2 \textstyle\alpha_{1}<\alpha_{2}
  27. α \textstyle\alpha
  28. α t r a n s i t i o n \textstyle\alpha_{transition}
  29. S ( r ) \textstyle S(r)
  30. N \textstyle N\rightarrow\infty
  31. 𝐙 d \textstyle\mathbf{Z}^{d}
  32. L 1 \textstyle L^{1}
  33. n 1 = n 1 + + n d , \|\vec{n}\|_{1}=\|n_{1}\|+\cdots+\|n_{d}\|,
  34. α = d \textstyle\alpha=d
  35. N \textstyle N
  36. \textstyle\infty
  37. S 1 ( r ) \textstyle S_{1}(r)
  38. r \textstyle r
  39. ζ G ( α ) \textstyle\zeta_{G}(\alpha)
  40. 2 ζ ( α ) \textstyle 2\zeta(\alpha)
  41. ζ ( α ) \textstyle\zeta(\alpha)
  42. S d + 1 ( r ) = 2 + S d ( r ) + 2 i = 1 r - 1 S d ( i ) . S_{d+1}(r)=2+S_{d}(r)+2\sum^{r-1}_{i=1}S_{d}(i).
  43. S d ( r ) = i = 0 d - 1 ( - 1 ) i 2 d - i ( d i ) ( d + r - i - 1 d - i - 1 ) . S_{d}(r)=\sum^{d-1}_{i=0}(-1)^{i}2^{d-i}{d\choose i}{d+r-i-1\choose d-i-1}.
  44. k \textstyle k
  45. d \textstyle d
  46. i = 1 r i k = r k + 1 ( k + 1 ) + r k 2 + j = 1 ( k + 1 ) / 2 ( - 1 ) j + 1 2 ζ ( 2 j ) k ! r k + 1 - 2 j ( 2 π ) 2 j ( k + 1 - 2 j ) ! . \sum^{r}_{i=1}i^{k}=\frac{r^{k+1}}{(k+1)}+\frac{r^{k}}{2}+\sum^{\lfloor(k+1)/2% \rfloor}_{j=1}\frac{(-1)^{j+1}2\zeta(2j)k!r^{k+1-2j}}{(2\pi)^{2j}(k+1-2j)!}.
  47. k \textstyle k
  48. k = 1 n ( n + 1 k ) i = 1 r i k = ( r + 1 ) ( ( r + 1 ) n - 1 ) . \sum^{n}_{k=1}\bigl(\begin{smallmatrix}n+1\ k\end{smallmatrix}\bigr)\sum^{r}_{% i=1}i^{k}=(r+1)((r+1)^{n}-1).
  49. S d ( r ) O ( 2 d r d - 1 / Γ ( d ) ) \textstyle S_{d}(r)\rightarrow O(2^{d}r^{d-1}/\Gamma(d))
  50. r \textstyle r\rightarrow\infty
  51. d = 1 \textstyle d=1
  52. ζ G ( α ) = 2 ζ ( α ) \textstyle\zeta_{G}(\alpha)=2\zeta(\alpha)
  53. d = 2 \textstyle d=2
  54. ζ G ( α ) = 4 ζ ( α - 1 ) \textstyle\zeta_{G}(\alpha)=4\zeta(\alpha-1)
  55. d = 3 \textstyle d=3
  56. ζ G ( α ) = 4 ζ ( α - 2 ) + 2 ζ ( α \textstyle\zeta_{G}(\alpha)=4\zeta(\alpha-2)+2\zeta(\alpha
  57. d = 4 \textstyle d=4
  58. ζ G ( α ) = 8 3 ζ ( α - 3 ) + 16 3 ζ ( α - 1 ) \textstyle\zeta_{G}(\alpha)=\frac{8}{3}\zeta(\alpha-3)+\frac{16}{3}\zeta(% \alpha-1)
  59. r \textstyle r\rightarrow\infty
  60. ζ G ( α ) = 2 d ζ ( α - d + 1 ) / Γ ( d ) \textstyle\zeta_{G}(\alpha)=2^{d}\zeta(\alpha-d+1)/\Gamma(d)
  61. α \alpha
  62. N \textstyle N
  63. p \textstyle p
  64. N \textstyle N\rightarrow\infty
  65. A \textstyle A
  66. B \textstyle B
  67. C \textstyle C
  68. A \textstyle A
  69. B \textstyle B
  70. C \textstyle C
  71. A \textstyle A
  72. B \textstyle B
  73. ( 1 - p 2 ) \textstyle(1-p^{2})
  74. N - 2 \textstyle N-2
  75. 2 \textstyle 2
  76. A \textstyle A
  77. B \textstyle B
  78. ( 1 - p 2 ) N - 2 \textstyle(1-p^{2})^{N-2}
  79. 2 \textstyle 2
  80. p ( N - 1 ) \textstyle p(N-1)
  81. S ( 1 ) \textstyle S(1)
  82. p ( N - 1 ) \textstyle p(N-1)
  83. S ( 2 ) \textstyle S(2)
  84. ( N - 1 ) ( 1 - p ) \textstyle(N-1)(1-p)
  85. ζ G ( α ) = p ( N - 1 ) + ( N - 1 ) ( 1 - p ) 2 - α . \zeta_{G}(\alpha)=p(N-1)+(N-1)(1-p)2^{-\alpha}.

Componendo_and_dividendo.html

  1. If a b = c d , then: \,\text{If }\frac{a}{b}=\frac{c}{d}\,\text{, then: }
  2. b a = d c \frac{b}{a}=\frac{d}{c}
  3. a c = b d \frac{a}{c}=\frac{b}{d}
  4. a + b b = c + d d \frac{a+b}{b}=\frac{c+d}{d}
  5. a - b b = c - d d \frac{a-b}{b}=\frac{c-d}{d}
  6. a + b a - b = c + d c - d \frac{a+b}{a-b}=\frac{c+d}{c-d}
  7. a - b a + b = c - d c + d \frac{a-b}{a+b}=\frac{c-d}{c+d}
  8. x 0 + + x n y 0 + + y n \frac{x_{0}+\cdots+x_{n}}{y_{0}+\cdots+y_{n}}
  9. x 0 x_{0}
  10. y 0 y_{0}
  11. x 1 x 0 , , x n x 0 , y 1 y 0 , , y n y 0 \frac{x_{1}}{x_{0}},\ldots,\frac{x_{n}}{x_{0}},\frac{y_{1}}{y_{0}},\ldots,% \frac{y_{n}}{y_{0}}
  12. x 0 y 0 \frac{x_{0}}{y_{0}}
  13. x 0 + + x n y 0 + + y n = x 0 y 0 ( 1 + x 1 x 0 + + x n x 0 1 + y 1 y 0 + + y n y 0 ) \frac{x_{0}+\cdots+x_{n}}{y_{0}+\cdots+y_{n}}=\frac{x_{0}}{y_{0}}\left(\frac{1% +\frac{x_{1}}{x_{0}}+\cdots+\frac{x_{n}}{x_{0}}}{1+\frac{y_{1}}{y_{0}}+\cdots+% \frac{y_{n}}{y_{0}}}\right)
  14. x + y x - y = x + y x + ( - y ) \frac{x+y}{x-y}=\frac{x+y}{x+(-y)}
  15. 3 + x 3 - x = 2 \frac{\sqrt{3}+x}{\sqrt{3}-x}=2
  16. ( 3 + x ) + ( 3 - x ) ( 3 + x ) - ( 3 - x ) = 2 + 1 2 - 1 \frac{(\sqrt{3}+x)+(\sqrt{3}-x)}{(\sqrt{3}+x)-(\sqrt{3}-x)}=\frac{2+1}{2-1}
  17. = > 2 3 2 x = 3 1 =>\frac{2\sqrt{3}}{2x}=\frac{3}{1}
  18. = > 3 x = 3 =>\frac{\sqrt{3}}{x}=3
  19. = > x = 1 3 =>x=\frac{1}{\sqrt{3}}

Composition_ring.html

  1. : R × R R \circ:R\times R\rightarrow R
  2. f , g , h R f,g,h\in R
  3. ( f + g ) h = ( f h ) + ( g h ) (f+g)\circ h=(f\circ h)+(g\circ h)
  4. ( f g ) h = ( f h ) ( g h ) (f\cdot g)\circ h=(f\circ h)\cdot(g\circ h)
  5. ( f g ) h = f ( g h ) . (f\circ g)\circ h=f\circ(g\circ h).
  6. f g = g f f\circ g=g\circ f
  7. f ( g + h ) f\circ(g+h)
  8. f g f\circ g
  9. f h f\circ h
  10. f g = 0 f\circ g=0
  11. f g = f f\circ g=f
  12. f g = f g f\circ g=fg
  13. X g = g X\circ g=g
  14. f g f\circ g
  15. g 2 n g_{2}^{n}
  16. f 1 f 2 g = f 1 g f 2 g . \frac{f_{1}}{f_{2}}\circ g=\frac{f_{1}\circ g}{f_{2}\circ g}.
  17. f 2 g f_{2}\circ g
  18. \circ
  19. [ x ] {\mathbb{Z}}[x]
  20. F : [ x ] [ x ] F:{\mathbb{Z}}[x]\rightarrow{\mathbb{Z}}[x]
  21. [ x ] {\mathbb{Z}}[x]
  22. F F
  23. x x
  24. f = F ( x ) f=F(x)
  25. f f
  26. [ x ] {\mathbb{Z}}[x]
  27. f [ x ] f\in{\mathbb{Z}}[x]
  28. : [ x ] × [ x ] [ x ] \circ:{\mathbb{Z}}[x]\times{\mathbb{Z}}[x]\rightarrow{\mathbb{Z}}[x]
  29. [ x ] {\mathbb{Z}}[x]
  30. ( x 2 + 3 x + 5 ) ( x - 2 ) = ( x - 2 ) 2 + 3 ( x - 2 ) + 5 = x 2 - x + 3. (x^{2}+3x+5)\circ(x-2)=(x-2)^{2}+3(x-2)+5=x^{2}-x+3.
  31. \mathbb{Z}
  32. \mathbb{Z}\to\mathbb{Z}

Compound_of_two_snub_cubes.html

  1. ξ 3 + ξ 2 + ξ = 1 , \xi^{3}+\xi^{2}+\xi=1,\,
  2. ξ = 1 3 ( 17 + 3 33 3 - - 17 + 3 33 3 - 1 ) \xi=\frac{1}{3}\left(\sqrt[3]{17+3\sqrt{33}}-\sqrt[3]{-17+3\sqrt{33}}-1\right)

Computation_of_cyclic_redundancy_checks.html

  1. x 8 + x 2 + x + 1 x^{8}+x^{2}+x+1
  2. x x
  3. M ( x ) M(x)
  4. x 6 + x 4 + x 2 + x + 1 x^{6}+x^{4}+x^{2}+x+1
  5. x 7 + x 6 + x 5 + x 3 + x x^{7}+x^{6}+x^{5}+x^{3}+x
  6. x 8 x^{8}
  7. x 8 M ( x ) x^{8}M(x)
  8. G ( x ) G(x)
  9. x 7 + x 5 + x x^{7}+x^{5}+x
  10. x 7 + x 4 + x 3 x^{7}+x^{4}+x^{3}
  11. n n
  12. x x
  13. x x
  14. x 0 x^{0}
  15. x n x^{n}
  16. x n - 1 x^{n-1}
  17. x x
  18. x n x^{n}
  19. x x
  20. n n
  21. R ( x ) R(x)
  22. x n - 1 x^{n-1}
  23. x 0 x^{0}
  24. x x
  25. M ( x ) M(x)
  26. x x
  27. x x
  28. x 16 + x 12 + x 5 + 1 x^{16}+x^{12}+x^{5}+1

Conductance_(graph).html

  1. ( S , S ¯ ) (S,\bar{S})
  2. φ ( S ) = i S , j S ¯ a i j min ( a ( S ) , a ( S ¯ ) ) \varphi(S)=\frac{\sum_{i\in S,j\in\bar{S}}a_{ij}}{\min(a(S),a(\bar{S}))}
  3. a i j a_{ij}
  4. a ( S ) = i S j V a i j a(S)=\sum_{i\in S}\sum_{j\in V}a_{ij}
  5. ϕ ( G ) = min S V φ ( S ) . \phi(G)=\min_{S\subseteq V}\varphi(S).
  6. ϕ ( G ) := min S V ; 0 a ( S ) a ( V ) / 2 i S , j S ¯ a i j a ( S ) . \phi(G):=\min_{S\subseteq V;0\leq a(S)\leq a(V)/2}\frac{\sum_{i\in S,j\in\bar{% S}}a_{ij}}{a(S)}.\,
  7. S S
  8. S S
  9. S S
  10. Φ S \Phi_{S}
  11. Φ S \Phi_{S}
  12. S S

Cone-shape_distribution_function.html

  1. t , τ t,\tau
  2. C x ( t , f ) = - - A x ( η , τ ) Φ ( η , τ ) exp ( j 2 π ( η t - τ f ) ) d η d τ , C_{x}(t,f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A_{x}(\eta,\tau)\Phi(% \eta,\tau)\exp(j2\pi(\eta t-\tau f))\,d\eta\,d\tau,
  3. A x ( η , τ ) = - x ( t + τ / 2 ) x * ( t - τ / 2 ) e - j 2 π t η d t , A_{x}(\eta,\tau)=\int_{-\infty}^{\infty}x(t+\tau/2)x^{*}(t-\tau/2)e^{-j2\pi t% \eta}\,dt,
  4. Φ ( η , τ ) = sin ( π η τ ) π η τ exp ( - 2 π α τ 2 ) . \Phi\left(\eta,\tau\right)=\frac{\sin\left(\pi\eta\tau\right)}{\pi\eta\tau}% \exp\left(-2\pi\alpha\tau^{2}\right).
  5. t , τ t,\tau
  6. ϕ ( t , τ ) = { 1 τ exp ( - 2 π α τ 2 ) , | τ | 2 | t | , 0 , otherwise . \phi\left(t,\tau\right)=\begin{cases}\frac{1}{\tau}\exp\left(-2\pi\alpha\tau^{% 2}\right),&|\tau|\geq 2|t|,\\ 0,&\mbox{otherwise}~{}.\end{cases}
  7. t , τ t,\tau
  8. η , τ \eta,\tau
  9. α \alpha
  10. τ \tau
  11. η , τ \eta,\tau
  12. η \eta

Cone_of_curves.html

  1. X X
  2. X X
  3. X X
  4. X X
  5. C = a i C i C=\sum a_{i}C_{i}
  6. C i C_{i}
  7. a i a_{i}\in\mathbb{R}
  8. C C
  9. C C^{\prime}
  10. C D = C D C\cdot D=C^{\prime}\cdot D
  11. D D
  12. X X
  13. N 1 ( X ) N_{1}(X)
  14. X X
  15. N E ( X ) = { a i [ C i ] , 0 a i } NE(X)=\left\{\sum a_{i}[C_{i}],\ 0\leq a_{i}\in\mathbb{R}\right\}
  16. C i C_{i}
  17. X X
  18. [ C i ] [C_{i}]
  19. N 1 ( X ) N_{1}(X)
  20. N E ( X ) NE(X)
  21. D D
  22. X X
  23. D x > 0 D\cdot x>0
  24. x x
  25. N E ( X ) ¯ \overline{NE(X)}
  26. N E ( X ) NE(X)
  27. X X
  28. X X^{\prime}
  29. X X
  30. K X K_{X^{\prime}}
  31. X X
  32. X X^{\prime}
  33. K x K_{x}
  34. N E ( X ) NE(X)
  35. X X
  36. C i C_{i}
  37. X X
  38. 0 < - K X C i dim X + 1 0<-K_{X}\cdot C_{i}\leq\operatorname{dim}X+1
  39. N E ( X ) ¯ = N E ( X ) ¯ K X 0 + i 𝐑 0 [ C i ] . \overline{NE(X)}=\overline{NE(X)}_{K_{X}\geq 0}+\sum_{i}\mathbf{R}_{\geq 0}[C_% {i}].
  40. ϵ \epsilon
  41. H H
  42. N E ( X ) ¯ = N E ( X ) ¯ K X + ϵ H 0 + 𝐑 0 [ C i ] , \overline{NE(X)}=\overline{NE(X)}_{K_{X}+\epsilon H\geq 0}+\sum\mathbf{R}_{% \geq 0}[C_{i}],
  43. N 1 ( X ) N_{1}(X)
  44. K X K_{X}
  45. X X
  46. { C : K X C = 0 } \{C:K_{X}\cdot C=0\}
  47. X X
  48. F N E ( X ) ¯ F\subset\overline{NE(X)}
  49. K X K_{X}
  50. cont F : X Z \operatorname{cont}_{F}:X\rightarrow Z
  51. ( cont F ) * 𝒪 X = 𝒪 Z (\operatorname{cont}_{F})_{*}\mathcal{O}_{X}=\mathcal{O}_{Z}
  52. C C
  53. X X
  54. cont F \operatorname{cont}_{F}
  55. [ C ] F [C]\in F

Configuration_entropy.html

  1. S = k B ln W , S=k_{B}\,\ln W,
  2. S = - k B n = 1 W P n ln P n , S=-k_{B}\,\sum_{n=1}^{W}P_{n}\ln P_{n},

Conformal_dimension.html

  1. 𝒢 \mathcal{G}
  2. Cdim X = inf Y 𝒢 dim H Y \mathrm{Cdim}X=\inf_{Y\in\mathcal{G}}\dim_{H}Y
  3. dim T X Cdim X dim H X \dim_{T}X\leq\mathrm{Cdim}X\leq\dim_{H}X
  4. 𝐑 N \mathbf{R}^{N}

Coniferin_beta-glucosidase.html

  1. \rightleftharpoons

Coniferyl-alcohol_glucosyltransferase.html

  1. \rightleftharpoons

Constellation_model.html

  1. X o = ( x 11 , x 12 , , x 1 N 1 x 21 , x 22 , , x 2 N 2 x T 1 , x T 2 , , x T N T ) X^{o}=\begin{pmatrix}x_{11},x_{12},{\cdots},x_{1N_{1}}\\ x_{21},x_{22},{\cdots},x_{2N_{2}}\\ \vdots\\ x_{T1},x_{T2},{\cdots},x_{TN_{T}}\end{pmatrix}
  2. N i N_{i}\,
  3. i { 1 , , T } i\in\{1,\dots,T\}
  4. x m x^{m}\,
  5. F F\,
  6. F = T F=T\,
  7. F > T F>T\,
  8. h h\,
  9. h i = j h_{i}=j\,
  10. x i j x_{ij}\,
  11. X o X^{o}\,
  12. p ( X o , x m , h ) p(X^{o},x^{m},h)\,
  13. b b\,
  14. n n\,
  15. b b\,
  16. b i = 1 b_{i}=1\,
  17. h i > 0 h_{i}>0\,
  18. b i = 0 b_{i}=0\,
  19. n n\,
  20. n i n_{i}\,
  21. i t h i^{th}
  22. X o X^{o}\,
  23. b b\,
  24. n n\,
  25. h h\,
  26. X o X^{o}\,
  27. p ( X o , x m , h ) = p ( X o , x m , h , n , b ) p(X^{o},x^{m},h)=p(X^{o},x^{m},h,n,b)\,
  28. p ( X o , x m , h , n , b ) = p ( X o , x m | h , n , b ) p ( h | n , b ) p ( n ) p ( b ) p(X^{o},x^{m},h,n,b)=p(X^{o},x^{m}|h,n,b)p(h|n,b)p(n)p(b)\,
  29. p ( n ) = i = 1 T 1 n i ! ( M i ) n i e - M i p(n)=\prod_{i=1}^{T}\frac{1}{n_{i}!}(M_{i})^{n_{i}}e^{-M_{i}}
  30. M i M_{i}\,
  31. i i\,
  32. F F\,
  33. p ( b ) p(b)\,
  34. 2 F 2^{F}\,
  35. F F\,
  36. F F\,
  37. p ( h | n , b ) p(h|n,b)\,
  38. p ( h | n , b ) = { 1 f = 1 F N f b f , if h H ( b , n ) 0 , for other h p(h|n,b)=\begin{cases}\frac{1}{\textstyle\prod_{f=1}^{F}N_{f}^{b_{f}}},&\mbox{% if }~{}h\in H(b,n)\\ 0,&\mbox{for other }~{}h\end{cases}
  39. H ( b , n ) H(b,n)\,
  40. b b\,
  41. n n\,
  42. N f N_{f}\,
  43. f f\,
  44. f = 1 F N f b f \textstyle\prod_{f=1}^{F}N_{f}^{b_{f}}
  45. p ( X o , x m | h , n ) = p f g ( z ) p b g ( x b g ) p(X^{o},x^{m}|h,n)=p_{fg}(z)p_{bg}(x_{bg})\,
  46. z = ( x o x m ) z=(x^{o}x^{m})\,
  47. x b g x_{bg}\,
  48. p f g ( z ) p_{fg}(z)\,
  49. μ \mu\,
  50. Σ \Sigma\,
  51. C 1 C_{1}\,
  52. C 0 C_{0}\,
  53. X o X^{o}\,
  54. p ( C 1 | X o ) p ( C 0 | X o ) h p ( X o , h | C 1 ) p ( X o , h 0 | C 0 ) \frac{p(C_{1}|X^{o})}{p(C_{0}|X^{o})}\propto\frac{\sum_{h}p(X^{o},h|C_{1})}{p(% X^{o},h_{0}|C_{0})}
  55. h 0 h_{0}\,
  56. Θ = { μ , Σ , p ( b ) , M } \Theta=\{\mu,\Sigma,p(b),M\}\,
  57. μ \mu\,
  58. Σ \Sigma\,
  59. p f g ( z ) p_{fg}(z)\,
  60. p ( b ) p(b)\,
  61. M M\,
  62. L ( X o | Θ ) = i = 1 I log h i p ( X i o , x i m , h i | Θ ) d x i m L(X^{o}|\Theta)=\sum_{i=1}^{I}\log\sum_{h_{i}}\int p(X_{i}^{o},x_{i}^{m},h_{i}% |\Theta)dx_{i}^{m}
  63. Θ \Theta\,
  64. Q ( Θ ~ | Θ ) = i = 1 I E [ log p ( X i o , x i m , h i | Θ ~ ) ] Q(\tilde{\Theta}|\Theta)=\sum_{i=1}^{I}E[\log p(X_{i}^{o},x_{i}^{m},h_{i}|% \tilde{\Theta})]
  65. μ ~ = 1 I i = 1 I E [ z i ] \tilde{\mu}=\frac{1}{I}\sum_{i=1}^{I}E[z_{i}]
  66. Σ ~ = 1 I i = 1 I E [ z i z i T ] - μ ~ μ ~ T \tilde{\Sigma}=\frac{1}{I}\sum_{i=1}^{I}E[z_{i}z_{i}^{T}]-\tilde{\mu}\tilde{% \mu}^{T}
  67. p ~ ( b ¯ ) = 1 I i = 1 I E [ δ b , b ¯ ] \tilde{p}(\bar{b})=\frac{1}{I}\sum_{i=1}^{I}E[\delta_{b,\bar{b}}]
  68. M ~ = 1 I i = 1 I E [ n i ] \tilde{M}=\frac{1}{I}\sum_{i=1}^{I}E[n_{i}]
  69. E [ z ] E[z]\,
  70. E [ z z T ] E[zz^{T}]\,
  71. E [ δ b , b ¯ ] E[\delta_{b,\bar{b}}]\,
  72. E [ n ] E[n]\,
  73. p ( h i , x i m | X i o , Θ ) = p ( h i , x i m , X i o | Θ ) h i H b p ( h i , x i m , X i o | Θ ) d x i m p(h_{i},x_{i}^{m}|X_{i}^{o},\Theta)=\frac{p(h_{i},x_{i}^{m},X_{i}^{o}|\Theta)}% {\textstyle\sum_{h_{i}\in H_{b}}\int p(h_{i},x_{i}^{m},X_{i}^{o}|\Theta)dx_{i}% ^{m}}
  74. X X\,
  75. S S\,
  76. A A\,
  77. Θ \Theta\,
  78. R = p ( Object | X , S , A ) p ( No object | X , S , A ) R=\frac{p(\mbox{Object}~{}|X,S,A)}{p(\mbox{No object}~{}|X,S,A)}
  79. = p ( X , S , A | Object ) p ( Object ) p ( X , S , A | No object ) p ( No object ) =\frac{p(X,S,A|\mbox{Object}~{})p(\mbox{Object}~{})}{p(X,S,A|\mbox{No object}~% {})p(\mbox{No object}~{})}
  80. p ( X , S , A | Θ ) p ( Object ) p ( X , S , A | Θ b g ) p ( No object ) \approx\frac{p(X,S,A|\Theta)p(\mbox{Object}~{})}{p(X,S,A|\Theta_{bg})p(\mbox{% No object}~{})}
  81. Θ b g \Theta_{bg}
  82. T T\,
  83. p ( X , S , A | Θ ) = h H p ( X , S , A , h | Θ ) = p(X,S,A|\Theta)=\sum_{h\in H}p(X,S,A,h|\Theta)=
  84. h H p ( A | X , S , h , Θ ) Appearance p ( X | S , h , Θ ) Shape p ( S | h , Θ ) Rel. Scale p ( h | Θ ) Other \sum_{h\in H}\underbrace{p(A|X,S,h,\Theta)}_{\mbox{Appearance}~{}}\underbrace{% p(X|S,h,\Theta)}_{\mbox{Shape}~{}}\underbrace{p(S|h,\Theta)}_{\mbox{Rel. Scale% }~{}}\underbrace{p(h|\Theta)}_{\mbox{Other}~{}}
  85. p p\,
  86. Θ p a p p = { c p , V p } \Theta_{p}^{app}=\{c_{p},V_{p}\}
  87. Θ b g a p p = { c b g , V b g } \Theta_{bg}^{app}=\{c_{bg},V_{bg}\}
  88. p ( A | X , S , h , Θ ) = p ( A | h , Θ ) p(A|X,S,h,\Theta)=p(A|h,\Theta)\,
  89. p ( A | X , S , h , Θ ) p ( A | X , S , h , Θ b g ) = p ( A | h , Θ ) p ( A | h , Θ b g ) \frac{p(A|X,S,h,\Theta)}{p(A|X,S,h,\Theta_{bg})}=\frac{p(A|h,\Theta)}{p(A|h,% \Theta_{bg})}
  90. = p = 1 P ( G ( A ( h p ) | c p , V p ) G ( A ( h p ) | c b g , V b g ) ) b p =\prod_{p=1}^{P}\left(\frac{G(A(h_{p})|c_{p},V_{p})}{G(A(h_{p})|c_{bg},V_{bg})% }\right)^{b_{p}}
  91. h h\,
  92. b b\,
  93. Θ shape = { μ , Σ } \Theta^{\mbox{shape}~{}}=\{\mu,\Sigma\}\,
  94. Θ b g \Theta_{bg}\,
  95. α \alpha\,
  96. f f\,
  97. p ( X | S , h , Θ ) p ( X | S , h , Θ b g ) = G ( X ( h ) | μ , Σ ) α f \frac{p(X|S,h,\Theta)}{p(X|S,h,\Theta_{bg})}=G(X(h)|\mu,\Sigma)\alpha^{f}
  98. p p\,
  99. Θ scale = { t p , U p } \Theta^{\mbox{scale}~{}}=\{t_{p},U_{p}\}\,
  100. Θ b g \Theta_{bg}\,
  101. r r\,
  102. p ( S | h , Θ ) p ( S | h , Θ b g ) = p = 1 P G ( S ( h p ) | t p , U p ) d p r f \frac{p(S|h,\Theta)}{p(S|h,\Theta_{bg})}=\prod_{p=1}^{P}G(S(h_{p})|t_{p},U_{p}% )^{d_{p}}r^{f}
  103. p ( h | Θ ) p ( h | Θ b g ) = p Poiss ( n | M ) p Poiss ( N | M ) 1 C r n ( N , f ) p ( b | Θ ) \frac{p(h|\Theta)}{p(h|\Theta_{bg})}=\frac{p_{\mbox{Poiss}~{}}(n|M)}{p_{\mbox{% Poiss}~{}}(N|M)}\frac{1}{{}^{n}C_{r}(N,f)}p(b|\Theta)
  104. Θ = { μ , Σ , c , V , M , p ( b | Θ ) , t , U } \Theta=\{\mu,\Sigma,c,V,M,p(b|\Theta),t,U\}\,
  105. N N\,
  106. P P\,
  107. H H\,
  108. O ( N P ) O(N^{P})\,
  109. P 6 P\leq 6
  110. N N\,
  111. O ( N 2 P ) O(N^{2}P)\,
  112. O ( N P ) O(N^{P})\,

Contact_analysis_(cryptanalysis).html

  1. P ( X i = a ) P(X_{i}=a)
  2. P ( X i = a X i + 1 = b ) P(X_{i}=a\cap X_{i+1}=b)
  3. P ( X i = b X i - 1 = a ) P(X_{i}=b\mid X_{i-1}=a)
  4. P ( X i = c X i - 2 = a X i - 1 = b ) P(X_{i}=c\mid X_{i-2}=a\cap X_{i-1}=b)
  5. P ( X i \sub S X i - 1 \sub T X i + 1 \sub T ) P(X_{i}\sub S\mid X_{i-1}\sub T\cap X_{i+1}\sub T)
  6. S S
  7. T T

Continuous-time_quantum_walk.html

  1. × \times
  2. A u , v = { 1 if {u,v} ϵ E 0 otherwise A_{u,v}=\left\{\begin{matrix}1&\,\text{if }\textrm{\{u,v\}}\,\text{ }\epsilon% \,\text{ }E\\ 0&\,\text{otherwise}\end{matrix}\right.
  3. × \times
  4. U ( t ) = e - i t L \,U(t)=e^{-itL}
  5. i i
  6. t ϵ t\,\,\epsilon\,\text{ }\mathbb{R}
  7. u u
  8. v v
  9. t t
  10. | v | U ( t ) | u | 2 \left|\langle v|U(t)|u\rangle\right|^{2}
  11. | ψ 0 |\psi_{0}\rangle
  12. t t
  13. | ψ t = U ( t ) | ψ 0 |\psi_{t}\rangle=U(t)|\psi_{0}\rangle
  14. v v
  15. | v | U ( t ) | ψ 0 | 2 \left|\langle v|U(t)|\psi_{0}\rangle\right|^{2}
  16. v v
  17. t t
  18. u u
  19. d d
  20. N N
  21. P N ( d ) = 1 2 N ( N d + N 2 ) P_{N}(d)=\frac{1}{2^{N}}\begin{pmatrix}N\\ \frac{d+N}{2}\end{pmatrix}
  22. N \sqrt{N}
  23. | |\downarrow\rangle
  24. | |\uparrow\rangle
  25. | 0 |0\rangle
  26. | 1 |1\rangle
  27. H | = 1 2 | + 1 2 | H|\uparrow\rangle=\frac{1}{\sqrt{2}}|\uparrow\rangle+\frac{1}{\sqrt{2}}|\downarrow\rangle
  28. H = 1 2 ( 1 1 1 - 1 ) H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}
  29. U = e i p ^ σ z H U=e^{i\hat{p}\sigma_{z}}H
  30. σ z \sigma_{z}
  31. p ^ \hat{p}
  32. 1 2 | + 1 2 | \frac{1}{\sqrt{2}}|\uparrow\rangle+\frac{1}{\sqrt{2}}|\downarrow\rangle
  33. 3 5 N \frac{3}{5}N
  34. G 4 G_{4}
  35. G n G_{n}
  36. 2 - n 2^{-n}
  37. ( 2 n + 1 ) - 1 {(2n+1)}^{-1}
  38. O ( N ) O(\sqrt{N})

Contour_set.html

  1. X X
  2. X 2 \succcurlyeq~{}\subseteq~{}X^{2}
  3. x x
  4. X X
  5. x X x\in X
  6. x x
  7. y y
  8. x x
  9. { y ϶ y x } \left\{y~{}\backepsilon~{}y\succcurlyeq x\right\}
  10. x x
  11. y y
  12. x x
  13. { y ϶ x y } \left\{y~{}\backepsilon~{}x\succcurlyeq y\right\}
  14. x x
  15. y y
  16. x x
  17. x x
  18. { y ϶ ( y x ) ¬ ( x y ) } \left\{y~{}\backepsilon~{}(y\succcurlyeq x)\land\lnot(x\succcurlyeq y)\right\}
  19. x x
  20. y y
  21. x x
  22. x x
  23. { y ϶ ( x y ) ¬ ( y x ) } \left\{y~{}\backepsilon~{}(x\succcurlyeq y)\land\lnot(y\succcurlyeq x)\right\}
  24. = { ( a , b ) ϶ ( a b ) ¬ ( b a ) } \succ~{}=~{}\left\{\left(a,b\right)~{}\backepsilon~{}\left(a\succcurlyeq b% \right)\land\lnot(b\succcurlyeq a)\right\}
  25. a a
  26. b b
  27. b b
  28. a a
  29. x x
  30. { y ϶ y x } \left\{y~{}\backepsilon~{}y\succ x\right\}
  31. x x
  32. { y ϶ x y } \left\{y~{}\backepsilon~{}x\succ y\right\}
  33. f ( ) f()
  34. \triangleright
  35. ( a b ) [ f ( a ) f ( b ) ] (a\succcurlyeq b)~{}\Leftarrow~{}[f(a)\triangleright f(b)]
  36. x x
  37. \geq
  38. x x
  39. x x
  40. x x
  41. x x
  42. x x
  43. x x
  44. x x
  45. x x
  46. ( a b ) [ f ( a ) f ( b ) ] (a\succcurlyeq b)~{}\Leftarrow~{}[f(a)\geq f(b)]
  47. x x
  48. y y
  49. f ( y ) f ( x ) f(y)\geq f(x)
  50. x x
  51. y y
  52. f ( y ) > f ( x ) f(y)>f(x)
  53. x x
  54. y y
  55. f ( x ) f ( y ) f(x)\geq f(y)
  56. x x
  57. y y
  58. f ( x ) > f ( y ) f(x)>f(y)
  59. ( a b ) [ f ( a ) f ( b ) ] (a\succcurlyeq b)~{}\Leftarrow~{}[f(a)\leq f(b)]
  60. f ( ) f()
  61. ( a b ) [ f ( a ) f ( b ) ] (a\succcurlyeq b)~{}\Leftarrow~{}[f(a)\geq f(b)]
  62. f ( ) f()
  63. [ ( a 1 , a 2 , ) ( b 1 , b 2 , ) ] [ f ( a 1 , a 2 , ) f ( b 1 , b 2 , ) ] [(a_{1},a_{2},\ldots)\succcurlyeq(b_{1},b_{2},\ldots)]~{}\Leftarrow~{}[f(a_{1}% ,a_{2},\ldots)\geq f(b_{1},b_{2},\ldots)]
  64. X X
  65. \succ
  66. \succcurlyeq
  67. x x
  68. x x
  69. x x
  70. x x
  71. x x
  72. x x
  73. x x
  74. x x
  75. u ( ) u()
  76. x x
  77. y y
  78. u ( y ) u ( x ) u(y)\geq u(x)
  79. x x
  80. y y
  81. u ( y ) > u ( x ) u(y)>u(x)
  82. x x
  83. y y
  84. u ( x ) u ( y ) u(x)\geq u(y)
  85. x x
  86. y y
  87. u ( x ) > u ( y ) u(x)>u(y)
  88. \succcurlyeq
  89. X X
  90. X 2 \ { y ϶ y x } = { y ϶ x y } X^{2}\backslash\left\{y~{}\backepsilon~{}y\succcurlyeq x\right\}=\left\{y~{}% \backepsilon~{}x\succ y\right\}
  91. X 2 \ { y ϶ x y } = { y ϶ y x } X^{2}\backslash\left\{y~{}\backepsilon~{}x\succ y\right\}=\left\{y~{}% \backepsilon~{}y\succcurlyeq x\right\}
  92. X 2 \ { y ϶ y x } = { y ϶ x y } X^{2}\backslash\left\{y~{}\backepsilon~{}y\succ x\right\}=\left\{y~{}% \backepsilon~{}x\succcurlyeq y\right\}
  93. X 2 \ { y ϶ x y } = { y ϶ y x } X^{2}\backslash\left\{y~{}\backepsilon~{}x\succcurlyeq y\right\}=\left\{y~{}% \backepsilon~{}y\succ x\right\}