# [Article] Further Developments in Anisotropic Plasticity by C. F. Shih, D. Lee

By C. F. Shih, D. Lee

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15. iions of 8. 9) . IIP,)= ln4 . 9) . m i=l (1. ,,;;·. · INFORMATION MEASURES IN SIMPLE COOING PROBLEMS 28 Here Es and var9 denote expectation resp. variance in the case when x• has distribution P;. (1(8) was introduced by Fisher (1925) as a measure of the information contained in one observation from Ps for estimating that the maximum likelihood estimator of asymptotic variance -1 n/(80) if 9=80. The 8 8. His motivation was from a sample of size n has assertion of (b) is known as the Cramer-Rao inequality, cf.

Forb" �(b+i5')1XI. -o. -oo as k-+oo. 11 (Delta-Convention) To every set X resp. 9). Typical sequences are understood with these o;s. } are considered as fixed, and in all assertions, dependence on them will be suppressed. , we shall write T1PJ, T1w1(x), etc. } will also be needed. 10. 12 There exists a sequence e•..... o depending only on lXI and IYI (cf. E; 1-ek for every X E x•. 0 REMARK More explicitly, lXI , , P"(T (P J) 6; 1 - 4ki52 , for every i5 > 0. E; 1 - 4k02 35 §2. TYPES AND TYPICAL SEQUENCES Proof It suffices to prove the inequalities of the remark.

If P"(x)�2-k(H(P)+01. 7. , any finite field) and consider x• as a vector space over this field. A linear source code is a pair of mappi ngs f: X1-+X" a nd Q1: X"-+X• such that f is a linear mapping (QI is arbitrary). Show that for a DMS with gen eric distribution P the r e exist linear source codes (b). Hint . H(P) n . with --+-- and e(f, Q�)-+0. l k logiXI A linear mapping f is defined by a k x n matrix over the given field. Cho ose li near mappings at random, by independent and equiprobab le choice of the matrix elements.