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Alternatives: | (mml file) (full) (simple) (plain) (form) (slideshow)
File: | TortureTests/Complexity/complex1
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Author: | Design Science, Inc. (D. Doyle, E. Cannon, E. Tabacman, R.Miner)
Description: | short complexity test
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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