 By S.S.Dragomir

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Geographic Information Science: Mastering the Legal Issues (2005)(2nd ed.)(en)(474s)

Cho (University of Canberra) explores either how the legislation impacts using geographic details and the way geographic info has formed the legislation and coverage in numerous nations. He identifies the standards influencing public use and pricing rules, describes using alternate criteria to proportion and commercialize geographic info, and descriptions a world framework for the improvement of entry regulations.

Additional info for A Survey on Cauchy-Buniakowsky-Schwartz Type Discrete Inequalities

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23) that  1 2 n 1 2 n pk a2k  pk b2k k=1 1 2 n qk a2k − k=1 1 2 n qk b2k k=1 2  k=1 2 n ≥ (pk − qk ) ak bk . 22). The other inequalities are obvious and we omit the details. 11]. CHAPTER 2. REFINEMENTS OF THE (CBS) −INEQUALITY 46 ¯ be sequences of real numbers and ¯ Corollary 74 Let ¯ a, b s = (s1 , . . , sn ) be such that 0 ≤ sk ≤ 1 for any k ∈ {1, . . , n} . 25)  k=1 2 (1 − sk ) ak bk + k=1 s k ak b k k=1 2 n ≥ sk b2k n ≥ 1 2 n n b2k ≥  a2k k=1 n ak b k . k=1 ¯ and ¯ Remark 75 Assume that ¯ a, b s are as in Corollary 74.

N} . 6 A Refinement for Non-Constant Sequences The following result was proved in [3, Theorem 1]. 6. A REFINEMENT FOR NON-CONSTANT SEQUENCES 51 (i) ai = aj and bi = bj for i = j, i, j ∈ N; (ii) pi > 0 for all i ∈ N. 45) 2 pj . Proof. We shall follow the proof in . Let J be a part of H. Define the mapping fJ : R → R given by  pi a2i  fJ (t) = i∈H i∈H\J  pi (bi + t)2  pi b2i + i∈J 2  − i∈H\J p i ai b i + pi ai (bi + t) . i∈J Then by the (CBS) −inequality we have that fJ (t) ≥ 0 for all t ∈ R.

REFINEMENTS OF THE (CBS) −INEQUALITY Proof. We will follow the proof in . 32). 6]. ¯ = (b1 , . . , bn ) and ¯ Theorem 77 Let ¯ a = (a1 , . . , an ), b c = (c1 , . . , cn ) be sequences of real numbers such that (i) |bk | + |ck | = 0 (k ∈ {1, . . , n}) (ii) |ak | ≤ 2|bk ck | |bk |+|ck | for any k ∈ {1, . . , n} . Then one has the inequality n |ak | ≤ 2 k=1 n k=1 |bk | n k=1 (|bk | n k=1 |ck | . 33) Proof. We will follow the proof in . By (ii) we observe that   2 |bk | 2 |bk ck | for any k ∈ {1, .