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A, f must be an increasing bijection of R≥0 . So define f (ξ) =: sinh g (ξ), ξ ≥ 0, and g must be an increasing bijection of R≥0 as well. 47) implies sinh g (ξ − η) = sinh g (ξ) − g (η) for all ξ ≥ η ≥ 0. Hence g (ξ − η) = g (ξ) − g (η) and we obtain again g (ξ) = lξ for all ξ ≥ 0 with a constant l > 0. 48) for all t ≥ 0. 48) holds true for all t ∈ R with a constant l > 0. a we get ψ (h) = 1 + δh2 . H. a) In the case δ = 0, k · x−y l holds true for all x, y ∈ X and, moreover, (h, t) = lt. 11. A common characterization 31 for all x, y ∈ X and cosh hyp (p, q) = hyp(p, q) ≥ 0, for p, q ∈ X.

Assume now that there is another hyperbolic line g p with l = g ⊥ H. Hence 0 ∈ g because all hyperbolic lines through 0 are of the form Rb. Put g ∩ H =: {r}. e. 1 + p2 = 1 + r2 ( 1 + r2 1 + p2 −rp). But p ∈ Ra, r ∈ H implies pr = 0. e. r = 0, a contradiction. 54 Chapter 2. Euclidean and Hyperbolic Geometry The distance d (p, H) between a point p and a hyperplane H is defined by d (p, r), where r is the point of intersection of H and the line l p orthogonal to H. This applies for (X, eucl) as well as for (X, hyp).

Hence kri = g (ri ) ≤ g (ζ) ≤ g (si ) = ksi and thus g (ζ) = lim kri = kζ. The equation g (ξ + η) = g (ξ) + g (η) is called a Cauchy equation in the theory of functional equations. For the other Cauchy equations see J. Acz´el [1]. 11. A common characterization 25 D. a) To the elements x = y of X there exist ω1 , ω2 ∈ O (X) and λ, t ∈ R with λ > 0 and ω1 Tt ω2 (x) = 0, ω1 Tt ω2 (y) = λe. b) The constant k of statement C is positive. c) d (x, y) = d (y, x) for all x, y ∈ X. d) If x, y ∈ X, then d (x, y) = 0 if, and only if, x = y.

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A Method for Combating Random Geometric Attack on Image Watermarking


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