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Proof lnexy = xy = lnex lney = ln(ex ·ey) Since lnx is onetoone, then exy = ex ·ey 1 = e0 = ex(−x) = ex ·e−x ⇒ e−x = 1 ex ex−y = ex(−y) = ex ·e−y = ex · 1 ey ex ey • For r = m ∈ N, emx = e z }m { x···x = z }m { ex ···ex = (ex)m • For r = 1 n, n ∈ N and n 6= 0, ex = e n n x = e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ N.

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Hence f is concave Convex functions 3–5 Extendedvalue extension extendedvalue extension f˜of f is geometric mean f(x) = (Qn k=1 xk) 1/n on Rn is concave (similar proof as for logsumexp) Convex functions 3–10 Epigraph and sublevel set. C is a fixed number?. º ¥ ß ¼ Ý « ¤ 75 F P'Ç W _ > E £'z í#Õ '¼ b4 · þ b ú ã ^ } g _4 · þ è0Ç _ X 8 Z ç ô º v>0> ¥.

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Suppose you are given the two functions f (x) = 2x 3 and g(x) = –x 2 5Composition means that you can plug g(x) into f (x)This is written as "(f o g)(x)", which is pronounced as "fcomposeg of x"And "( f o g)(x)" means "f (g(x))"That is, you plug something in for x, then you plug that value into g, simplify, and then plug the result into f. Values The Qfunction is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and MathematicaSome values of the Qfunction are given below for reference. A Porphyrinic Zirconium Metal–Organic Framework for Oxygen Reduction Reaction Tailoring the Spacing between ActiveSites through ChainBased Inorganic Building Units.

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