<<< native [ EArray [ AlignCenter , AlignCenter ] [ [ [ EText TextNormal "Bernoulli Trials" ] , [ EGrouped [ EGrouped [ EIdentifier "P" , ESymbol Open "(" , EIdentifier "E" , ESymbol Close ")" ] , ESymbol Rel "=" , EDelimited "(" ")" [ Right (EFraction NormalFrac (EIdentifier "n") (EIdentifier "k")) ] , ESubsup (EIdentifier "p") (EGrouped []) (EIdentifier "k") , ESubsup (EGrouped [ ESymbol Open "(" , ENumber "1" , ESymbol Bin "-" , EIdentifier "p" , ESymbol Close ")" ]) (EGrouped []) (EGrouped [ EIdentifier "n" , ESymbol Bin "-" , EIdentifier "k" ]) ] ] ] , [ [ EText TextNormal "Cauchy-Schwarz Inequality" ] , [ EGrouped [ ESubsup (EDelimited "(" ")" [ Right (EUnderover False (ESymbol Op "\8721") (EGrouped [ EIdentifier "k" , ESymbol Rel "=" , ENumber "1" ]) (EIdentifier "n")) , Right (ESubsup (EIdentifier "a") (EIdentifier "k") (EGrouped [])) , Right (ESubsup (EIdentifier "b") (EIdentifier "k") (EGrouped [])) ]) (EGrouped []) (ENumber "2") , ESymbol Rel "\8804" , EDelimited "(" ")" [ Right (EUnderover False (ESymbol Op "\8721") (EGrouped [ EIdentifier "k" , ESymbol Rel "=" , ENumber "1" ]) (EIdentifier "n")) , Right (ESubsup (EIdentifier "a") (EIdentifier "k") (ENumber "2")) ] , EDelimited "(" ")" [ Right (EUnderover False (ESymbol Op "\8721") (EGrouped [ EIdentifier "k" , ESymbol Rel "=" , ENumber "1" ]) (EIdentifier "n")) , Right (ESubsup (EIdentifier "b") (EIdentifier "k") (ENumber "2")) ] ] ] ] , [ [ EText TextNormal "Cauchy Formula" ] , [ EGrouped [ EIdentifier "f" , ESymbol Open "(" , EIdentifier "z" , ESymbol Close ")" , ESpace (1 % 6) , ESymbol Bin "\183" , ESubsup (EIdentifier "Ind") (EIdentifier "\947") (EGrouped []) , ESymbol Open "(" , EIdentifier "z" , ESymbol Close ")" , ESymbol Rel "=" , EFraction NormalFrac (ENumber "1") (EGrouped [ ENumber "2" , EIdentifier "\960" , EIdentifier "i" ]) , EUnderover False (ESymbol Op "\8750") (EIdentifier "\947") (EGrouped []) , EFraction NormalFrac (EGrouped [ EIdentifier "f" , ESymbol Open "(" , EIdentifier "\958" , ESymbol Close ")" ]) (EGrouped [ EIdentifier "\958" , ESymbol Bin "-" , EIdentifier "z" ]) , ESpace (1 % 6) , EIdentifier "d" , EIdentifier "\958" ] ] ] , [ [ EText TextNormal "Cross Product" ] , [ EGrouped [ ESubsup (EIdentifier "V") (ENumber "1") (EGrouped []) , ESymbol Bin "\215" , ESubsup (EIdentifier "V") (ENumber "2") (EGrouped []) , ESymbol Rel "=" , EDelimited "|" "|" [ Right (EArray [ AlignCenter , AlignCenter , AlignCenter ] [ [ [ EIdentifier "i" ] , [ EIdentifier "j" ] , [ EIdentifier "k" ] ] , [ [ EFraction NormalFrac (EGrouped [ ESymbol Ord "\8706" , EIdentifier "X" ]) (EGrouped [ ESymbol Ord "\8706" , EIdentifier "u" ]) ] , [ EFraction NormalFrac (EGrouped [ ESymbol Ord "\8706" , EIdentifier "Y" ]) (EGrouped [ ESymbol Ord "\8706" , EIdentifier "u" ]) ] , [ ENumber "0" ] ] , [ [ EFraction NormalFrac (EGrouped [ ESymbol Ord "\8706" , EIdentifier "X" ]) (EGrouped [ ESymbol Ord "\8706" , EIdentifier "v" ]) ] , [ EFraction NormalFrac (EGrouped [ ESymbol Ord "\8706" , EIdentifier "Y" ]) (EGrouped [ ESymbol Ord "\8706" , EIdentifier "v" ]) ] , [ ENumber "0" ] ] ]) ] ] ] ] , [ [ EText TextNormal "Vandermonde Determinant" ] , [ EGrouped [ EDelimited "|" "|" [ Right (EArray [ AlignCenter , AlignCenter , AlignCenter , AlignCenter ] [ [ [ ENumber "1" ] , [ ENumber "1" ] , [ ESymbol Ord "\8943" ] , [ ENumber "1" ] ] , [ [ ESubsup (EIdentifier "v") (ENumber "1") (EGrouped []) ] , [ ESubsup (EIdentifier "v") (ENumber "2") (EGrouped []) ] , [ ESymbol Ord "\8943" ] , [ ESubsup (EIdentifier "v") (EIdentifier "n") (EGrouped []) ] ] , [ [ ESubsup (EIdentifier "v") (ENumber "1") (ENumber "2") ] , [ ESubsup (EIdentifier "v") (ENumber "2") (ENumber "2") ] , [ ESymbol Ord "\8943" ] , [ ESubsup (EIdentifier "v") (EIdentifier "n") (ENumber "2") ] ] , [ [ ESymbol Rel "\8942" ] , [ ESymbol Rel "\8942" ] , [ ESymbol Rel "\8945" ] , [ ESymbol Rel "\8942" ] ] , [ [ ESubsup (EIdentifier "v") (ENumber "1") (EGrouped [ EIdentifier "n" , ESymbol Bin "-" , ENumber "1" ]) ] , [ ESubsup (EIdentifier "v") (ENumber "2") (EGrouped [ EIdentifier "n" , ESymbol Bin "-" , ENumber "1" ]) ] , [ ESymbol Ord "\8943" ] , [ ESubsup (EIdentifier "v") (EIdentifier "n") (EGrouped [ EIdentifier "n" , ESymbol Bin "-" , ENumber "1" ]) ] ] ]) ] , ESymbol Rel "=" , EUnderover False (ESymbol Op "\8719") (EGrouped [ ENumber "1" , ESymbol Rel "\8804" , EIdentifier "i" , ESymbol Rel "<" , EIdentifier "j" , ESymbol Rel "\8804" , EIdentifier "n" ]) (EGrouped []) , ESymbol Open "(" , ESubsup (EIdentifier "v") (EIdentifier "j") (EGrouped []) , ESymbol Bin "-" , ESubsup (EIdentifier "v") (EIdentifier "i") (EGrouped []) , ESymbol Close ")" ] ] ] , [ [ EText TextNormal "Lorenz Equations" ] , [ EArray [ AlignCenter , AlignCenter , AlignCenter ] [ [ [ EUnderover False (EIdentifier "x") (EGrouped []) (ESymbol Accent "\729") ] , [ ESymbol Rel "=" ] , [ EGrouped [ EIdentifier "\963" , ESymbol Open "(" , EIdentifier "y" , ESymbol Bin "-" , EIdentifier "x" , ESymbol Close ")" ] ] ] , [ [ EUnderover False (EIdentifier "y") (EGrouped []) (ESymbol Accent "\729") ] , [ ESymbol Rel "=" ] , [ EGrouped [ EIdentifier "\961" , EIdentifier "x" , ESymbol Bin "-" , EIdentifier "y" , ESymbol Bin "-" , EIdentifier "x" , EIdentifier "z" ] ] ] , [ [ EUnderover False (EIdentifier "z") (EGrouped []) (ESymbol Accent "\729") ] , [ ESymbol Rel "=" ] , [ EGrouped [ ESymbol Bin "-" , EIdentifier "\946" , EIdentifier "z" , ESymbol Bin "+" , EIdentifier "x" , EIdentifier "y" ] ] ] ] ] ] , [ [ EText TextNormal "Maxwell's Equations" ] , [ EDelimited "{" "" [ Right (EArray [ AlignCenter , AlignCenter , AlignCenter ] [ [ [ EGrouped [ ESymbol Ord "\8711" , ESpace (0 % 1) , ESymbol Bin "\215" , EUnderover False (EIdentifier "B") (EGrouped []) (ESymbol Accent "\8636") , ESymbol Bin "-" , ESpace (1 % 6) , EFraction NormalFrac (ENumber "1") (EIdentifier "c") , ESpace (1 % 6) , EFraction NormalFrac (EGrouped [ ESymbol Ord "\8706" , ESpace (0 % 1) , EUnderover False (EIdentifier "E") (EGrouped []) (ESymbol Accent "\8636") ]) (EGrouped [ ESymbol Ord "\8706" , ESpace (0 % 1) , EIdentifier "t" ]) ] ] , [ ESymbol Rel "=" ] , [ EGrouped [ EFraction NormalFrac (EGrouped [ ENumber "4" , EIdentifier "\960" ]) (EIdentifier "c") , ESpace (1 % 6) , EUnderover False (EIdentifier "j") (EGrouped []) (ESymbol Accent "\8636") ] ] ] , [ [ EGrouped [ ESymbol Ord "\8711" , ESpace (0 % 1) , ESymbol Bin "\183" , EUnderover False (EIdentifier "E") (EGrouped []) (ESymbol Accent "\8636") ] ] , [ ESymbol Rel "=" ] , [ EGrouped [ ENumber "4" , EIdentifier "\960" , EIdentifier "\961" ] ] ] , [ [ EGrouped [ ESymbol Ord "\8711" , ESpace (0 % 1) , ESymbol Bin "\215" , EUnderover False (EIdentifier "E") (EGrouped []) (ESymbol Accent "\8636") , ESpace (1 % 6) , ESymbol Bin "+" , ESpace (1 % 6) , EFraction NormalFrac (ENumber "1") (EIdentifier "c") , ESpace (1 % 6) , EFraction NormalFrac (EGrouped [ ESymbol Ord "\8706" , ESpace (0 % 1) , EUnderover False (EIdentifier "B") (EGrouped []) (ESymbol Accent "\8636") ]) (EGrouped [ ESymbol Ord "\8706" , ESpace (0 % 1) , EIdentifier "t" ]) ] ] , [ ESymbol Rel "=" ] , [ EUnderover False (ENumber "0") (EGrouped []) (ESymbol Accent "\8636") ] ] , [ [ EGrouped [ ESymbol Ord "\8711" , ESpace (0 % 1) , ESymbol Bin "\183" , EUnderover False (EIdentifier "B") (EGrouped []) (ESymbol Accent "\8636") ] ] , [ ESymbol Rel "=" ] , [ ENumber "0" ] ] ]) ] ] ] , [ [ EText TextNormal "Einstein Field Equations" ] , [ EGrouped [ ESubsup (EIdentifier "R") (EGrouped [ EIdentifier "\956" , EIdentifier "\957" ]) (EGrouped []) , ESymbol Bin "-" , EFraction NormalFrac (ENumber "1") (ENumber "2") , ESpace (1 % 6) , ESubsup (EIdentifier "g") (EGrouped [ EIdentifier "\956" , EIdentifier "\957" ]) (EGrouped []) , ESpace (1 % 6) , EIdentifier "R" , ESymbol Rel "=" , EFraction NormalFrac (EGrouped [ ENumber "8" , EIdentifier "\960" , EIdentifier "G" ]) (ESubsup (EIdentifier "c") (EGrouped []) (ENumber "4")) , ESpace (1 % 6) , ESubsup (EIdentifier "T") (EGrouped [ EIdentifier "\956" , EIdentifier "\957" ]) (EGrouped []) ] ] ] , [ [ EText TextNormal "Ramanujan Identity" ] , [ EGrouped [ EFraction NormalFrac (ENumber "1") (EGrouped [ ESymbol Open "(" , ESqrt (EGrouped [ EIdentifier "\966" , ESqrt (ENumber "5") ]) , ESymbol Bin "-" , EIdentifier "\966" , ESymbol Close ")" , ESubsup (EIdentifier "e") (EGrouped []) (EFraction NormalFrac (ENumber "25") (EIdentifier "\960")) ]) , ESymbol Rel "=" , ENumber "1" , ESymbol Bin "+" , EFraction NormalFrac (ESubsup (EIdentifier "e") (EGrouped []) (EGrouped [ ESymbol Bin "-" , ENumber "2" , EIdentifier "\960" ])) (EGrouped [ ENumber "1" , ESymbol Bin "+" , EFraction NormalFrac (ESubsup (EIdentifier "e") (EGrouped []) (EGrouped [ ESymbol Bin "-" , ENumber "4" , EIdentifier "\960" ])) (EGrouped [ ENumber "1" , ESymbol Bin "+" , EFraction NormalFrac (ESubsup (EIdentifier "e") (EGrouped []) (EGrouped [ ESymbol Bin "-" , ENumber "6" , EIdentifier "\960" ])) (EGrouped [ ENumber "1" , ESymbol Bin "+" , EFraction NormalFrac (ESubsup (EIdentifier "e") (EGrouped []) (EGrouped [ ESymbol Bin "-" , ENumber "8" , EIdentifier "\960" ])) (EGrouped [ ENumber "1" , ESymbol Bin "+" , ESymbol Ord "\8230" ]) ]) ]) ]) ] ] ] , [ [ EText TextNormal "Another Ramanujan identity" ] , [ EGrouped [ EUnderover False (ESymbol Op "\8721") (EGrouped [ EIdentifier "k" , ESymbol Rel "=" , ENumber "1" ]) (EIdentifier "\8734") , EFraction NormalFrac (ENumber "1") (ESubsup (ENumber "2") (EGrouped []) (EGrouped [ ESymbol Open "\8970" , EIdentifier "k" , ESymbol Bin "\183" , ESpace (0 % 1) , EIdentifier "\966" , ESymbol Close "\8971" ])) , ESymbol Rel "=" , EFraction NormalFrac (ENumber "1") (EGrouped [ ESubsup (ENumber "2") (EGrouped []) (ENumber "0") , ESymbol Bin "+" , EFraction NormalFrac (ENumber "1") (EGrouped [ ESubsup (ENumber "2") (EGrouped []) (ENumber "1") , ESymbol Bin "+" , ESymbol Ord "\8943" ]) ]) ] ] ] , [ [ EText TextNormal "Rogers-Ramanujan Identity" ] , [ EGrouped [ ENumber "1" , ESymbol Bin "+" , EGrouped [ EUnderover False (ESymbol Op "\8721") (EGrouped [ EIdentifier "k" , ESymbol Rel "=" , ENumber "1" ]) (EIdentifier "\8734") , EFraction NormalFrac (ESubsup (EIdentifier "q") (EGrouped []) (EGrouped [ ESubsup (EIdentifier "k") (EGrouped []) (ENumber "2") , ESymbol Bin "+" , EIdentifier "k" ])) (EGrouped [ ESymbol Open "(" , ENumber "1" , ESymbol Bin "-" , EIdentifier "q" , ESymbol Close ")" , ESymbol Open "(" , ENumber "1" , ESymbol Bin "-" , ESubsup (EIdentifier "q") (EGrouped []) (ENumber "2") , ESymbol Close ")" , ESymbol Ord "\8943" , ESymbol Open "(" , ENumber "1" , ESymbol Bin "-" , ESubsup (EIdentifier "q") (EGrouped []) (EIdentifier "k") , ESymbol Close ")" ]) ] , ESymbol Rel "=" , EGrouped [ EUnderover False (ESymbol Op "\8719") (EGrouped [ EIdentifier "j" , ESymbol Rel "=" , ENumber "0" ]) (EIdentifier "\8734") , EFraction NormalFrac (ENumber "1") (EGrouped [ ESymbol Open "(" , ENumber "1" , ESymbol Bin "-" , ESubsup (EIdentifier "q") (EGrouped []) (EGrouped [ ENumber "5" , EIdentifier "j" , ESymbol Bin "+" , ENumber "2" ]) , ESymbol Close ")" , ESymbol Open "(" , ENumber "1" , ESymbol Bin "-" , ESubsup (EIdentifier "q") (EGrouped []) (EGrouped [ ENumber "5" , EIdentifier "j" , ESymbol Bin "+" , ENumber "3" ]) , ESymbol Close ")" ]) ] , ESymbol Pun "," , EText TextNormal "\8287\8202" , EText TextNormal "\8287\8202" , EGrouped [ EIdentifier "f" , EIdentifier "o" , EIdentifier "r" ] , ESpace (2 % 9) , ESymbol Op "|" , EIdentifier "q" , ESymbol Op "|" , ESymbol Rel "<" , ENumber "1" , EIdentifier "." ] ] ] , [ [ EText TextNormal "Commutative Diagram" ] , [ EArray [ AlignCenter , AlignCenter , AlignCenter ] [ [ [ EIdentifier "H" ] , [ ESymbol Accent "\8592" ] , [ EIdentifier "K" ] ] , [ [ ESymbol Rel "\8595" ] , [ ESpace (0 % 1) ] , [ ESymbol Rel "\8593" ] ] , [ [ EIdentifier "H" ] , [ ESymbol Accent "\8594" ] , [ EIdentifier "K" ] ] ] ] ] ] ] >>> omml Bernoulli Trials P ( E ) = n k p ​ k ( 1 - p ) ​ n - k Cauchy-Schwarz Inequality k = 1 n a k ​ b k ​ ​ 2 ≤ k = 1 n a k 2 k = 1 n b k 2 Cauchy Formula f ( z )   · Ind γ ​ ( z ) = 1 2 π i γ ​ f ( ξ ) ξ - z   d ξ Cross Product V 1 ​ × V 2 ​ = i j k ∂ X ∂ u ∂ Y ∂ u 0 ∂ X ∂ v ∂ Y ∂ v 0 Vandermonde Determinant 1 1 ⋯ 1 v 1 ​ v 2 ​ ⋯ v n ​ v 1 2 v 2 2 ⋯ v n 2 ⋮ ⋮ ⋱ ⋮ v 1 n - 1 v 2 n - 1 ⋯ v n n - 1 = 1 ≤ i < j ≤ n ​ ( v j ​ - v i ​ ) Lorenz Equations x ​ = σ ( y - x ) y ​ = ρ x - y - x z z ​ = - β z + x y Maxwell's Equations ∇ ​ × B ​ -   1 c   ∂ ​ E ​ ∂ ​ t = 4 π c   j ​ ∇ ​ · E ​ = 4 π ρ ∇ ​ × E ​   +   1 c   ∂ ​ B ​ ∂ ​ t = 0 ​ ∇ ​ · B ​ = 0 Einstein Field Equations R μ ν ​ - 1 2   g μ ν ​   R = 8 π G c ​ 4   T μ ν ​ Ramanujan Identity 1 ( φ 5 - φ ) e ​ 25 π = 1 + e ​ - 2 π 1 + e ​ - 4 π 1 + e ​ - 6 π 1 + e ​ - 8 π 1 + … Another Ramanujan identity k = 1 ∞ 1 2 ​ ⌊ k · ​ φ ⌋ = 1 2 ​ 0 + 1 2 ​ 1 + ⋯ Rogers-Ramanujan Identity 1 + k = 1 ∞ q ​ k ​ 2 + k ( 1 - q ) ( 1 - q ​ 2 ) ⋯ ( 1 - q ​ k ) = j = 0 ∞ 1 ( 1 - q ​ 5 j + 2 ) ( 1 - q ​ 5 j + 3 ) ,       f o r   | q | < 1 . Commutative Diagram H ← K ↓ ​ ↑ H → K