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Bernoulli Trials P ( E ) Probability of event E: Get exactly k heads in n coin flips. = ( n k ) Number of ways to get exactly k heads in n coin flips p Probability of getting heads in one flip k Number of heads ( 1 - p ) Probability of getting tails in one flip n - k Number of tails 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 + 1 2 1 + 1 2 2 + 1 2 3 + 1 2 5 + Rogers-Ramanujan Identity 1 + k = 1 q k 2 + k ( 1 - q ) ( 1 - q 2 ) ( 1 - q k ) q 2 ( 1 - q ) + q 6 ( 1 - q ) ( 1 - q 2 ) + = j = 0 1 ( 1 - q 5 j + 2 ) ( 1 - q 5 j + 3 ) 1 ( 1 - q 2 ) ( 1 - q 3 ) × 1 ( 1 - q 7 ) ( 1 - q 8 ) × ,       f o r | q | < 1 . Commutative Diagram H K H K