[15] (a) Show that C(0n|n) c, where c is a constant independent of n. (b) Show that C(1:n|n) c, where = 3.1415

[15]

(a) Show that C(0n|n) ≤

c, where c is a constant independent of n.

(b) Show that C(π1:n|n) ≤

c, where π = 3.1415 ... and c is some constant independent of n.

(c) Show that we can expect C(a1:n|n) ≤ 1 4n, where ai is the ith bit in Shakespeare’s Romeo and Juliet.

(d) What is C(a1:n|n), where ai is the ith bit in the expansion of the fine structure constant a = e2/c, in physics.

Comments. Hint: For Item

(c) use known facts concerning the letter frequencies (entropy) in written English. Source: [T.M. Cover, The Impact of Processing Technique on Communications; J.K. Skwirzynski, ed., Martinus Nijhof, 1985, pp. 23–33].