<<< 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" ] ] ] ] ] ] ] >>> eqn matrix{ ccol{ {roman "Bernoulli Trials"} above {roman "Cauchy-Schwarz Inequality"} above {roman "Cauchy Formula"} above {roman "Cross Product"} above {roman "Vandermonde Determinant"} above {roman "Lorenz Equations"} above {roman "Maxwell's Equations"} above {roman "Einstein Field Equations"} above {roman "Ramanujan Identity"} above {roman "Another Ramanujan identity"} above {roman "Rogers-Ramanujan Identity"} above {roman "Commutative Diagram"} } ccol{ {{P ( E )} = left ( n over k right ) p sub {""} sup k {( 1 - p )} sub {""} sup {n - k}} above {{left ( sum from {k = 1} to n a sub k sup {""} b sub k sup {""} right )} sub {""} sup 2 <= left ( sum from {k = 1} to n a sub k sup 2 right ) left ( sum from {k = 1} to n b sub k sup 2 right )} above {f ( z ) ^ cdot Ind sub gamma sup {""} ( z ) = 1 over {2 pi i} \[u222E] from gamma to {""} {f ( xi )} over {xi - z} ^ d xi} above {V sub 1 sup {""} times V sub 2 sup {""} = left | matrix{ ccol{ i above {{partial X} over {partial u}} above {{partial X} over {partial v}} } ccol{ j above {{partial Y} over {partial u}} above {{partial Y} over {partial v}} } ccol{ k above 0 above 0 } } right |} above {left | matrix{ ccol{ 1 above {v sub 1 sup {""}} above {v sub 1 sup 2} above \[u22EE] above {v sub 1 sup {n - 1}} } ccol{ 1 above {v sub 2 sup {""}} above {v sub 2 sup 2} above \[u22EE] above {v sub 2 sup {n - 1}} } ccol{ cdots above cdots above cdots above \[u22F1] above cdots } ccol{ 1 above {v sub n sup {""}} above {v sub n sup 2} above \[u22EE] above {v sub n sup {n - 1}} } } right | = prod from {1 <= i < j <= n} to {""} ( v sub j sup {""} - v sub i sup {""} )} above {matrix{ ccol{ {{x from {""}} to \[u02D9]} above {{y from {""}} to \[u02D9]} above {{z from {""}} to \[u02D9]} } ccol{ = above = above = } ccol{ {sigma ( y - x )} above {rho x - y - x z} above {- beta z + x y} } }} above {left { matrix{ ccol{ {grad fwd 0 times {B from {""}} to \[u21BC] - ^ 1 over c ^ {partial fwd 0 {E from {""}} to \[u21BC]} over {partial fwd 0 t}} above {grad fwd 0 cdot {E from {""}} to \[u21BC]} above {grad fwd 0 times {E from {""}} to \[u21BC] ^ + ^ 1 over c ^ {partial fwd 0 {B from {""}} to \[u21BC]} over {partial fwd 0 t}} above {grad fwd 0 cdot {B from {""}} to \[u21BC]} } ccol{ = above = above = above = } ccol{ {{4 pi} over c ^ {j from {""}} to \[u21BC]} above {4 pi rho} above {{0 from {""}} to \[u21BC]} above 0 } } right ""} above {R sub {mu nu} sup {""} - 1 over 2 ^ g sub {mu nu} sup {""} ^ R = {8 pi G} over {c sub {""} sup 4} ^ T sub {mu nu} sup {""}} above {1 over {( sqrt {varphi sqrt 5} - varphi ) e sub {""} sup {25 over pi}} = 1 + {e sub {""} sup {- 2 pi}} over {1 + {e sub {""} sup {- 4 pi}} over {1 + {e sub {""} sup {- 6 pi}} over {1 + {e sub {""} sup {- 8 pi}} over {1 + ...}}}}} above {sum from {k = 1} to inf 1 over {2 sub {""} sup {\[u230A] k cdot fwd 0 varphi \[u230B]}} = 1 over {2 sub {""} sup 0 + 1 over {2 sub {""} sup 1 + cdots}}} above {1 + {sum from {k = 1} to inf {q sub {""} sup {k sub {""} sup 2 + k}} over {( 1 - q ) ( 1 - q sub {""} sup 2 ) cdots ( 1 - q sub {""} sup k )}} = {prod from {j = 0} to inf 1 over {( 1 - q sub {""} sup {5 j + 2} ) ( 1 - q sub {""} sup {5 j + 3} )}} , roman "  " roman "  " {f o r} ~ | q | < 1 .} above {matrix{ ccol{ H above \[u2193] above H } ccol{ <- above {fwd 0} above -> } ccol{ K above \[u2191] above K } }} } }