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S798 ̑ ̓ ́A ^ Ɋ֘A \ s799 C X g ̃R s C g @ ۈ Õ w @. P V ® s M O p K { y C x ¦ ï å ï ¼ í » ¶ w s T w ® » w q Ð ¯ g æ w Æ ;. 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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Course Title IT ICT401;. This may have been an attempt to write a paper with a longer title than the paper itself In fact, this "paper" is a discussion of our examining a particular problem that having tools like function graphers available might make possible different approaches. 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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And since the P^(j) n all have the same distribution, taking expectations yields, E D(P^ n1kQ) E D(P^ nkQ) as claimed 5 Estimating the entropy The fact that EH(P^ n) H(P) follows from the concavity of the entropy (using Jensen’s inequality), upon noting that E(P^. A a Ma a , I a a B T a Ja M , G a a M E , I c. í S ^ 2(#Ø'¼ b0Û o 2 b v 8'¼ r S " 'g/ ;.

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