Abstract State Machines 2003: Advances in Theory and by Antonia Bertolino (auth.), Egon Börger, Angelo Gargantini,

By Antonia Bertolino (auth.), Egon Börger, Angelo Gargantini, Elvinia Riccobene (eds.)

This e-book constitutes the refereed court cases of the tenth foreign Workshop on summary kingdom Machines, ASM 2003, held in Taormina, Italy in March 2003.

The sixteen revised complete papers provided including eight invited papers and 12 abstracts have been rigorously reviewed and chosen for inclusion within the e-book. The papers mirror the cutting-edge of the summary country computing device process for the layout and research of advanced software/hardware structures. in addition to theoretical effects and methodological growth, software in numerous fields are studied to boot.

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Extra info for Abstract State Machines 2003: Advances in Theory and Practice 10th International Workshop, ASM 2003 Taormina, Italy, March 3–7, 2003 Proceedings

Example text

5, the existence of γ∗ will imply, as required, that H 1 (Γ) = H 1 γ∗ [0, ] ≤ H 1 [0, ] = . 28 Hausdorff measures To construct γ∗ (which is just the parametrization by arc length of γ, defined without using derivatives), we define v : [0, a] → [0, ] by v(t) = (γ, [0, t]), t ∈ [0, a]. Then v(0) = 0, v(a) = and v is strictly increasing, that is, v(t) < v(s) if t < s, as γ is injective. In particular, v is invertible, with a strictly increasing inverse w : [0, ] → [0, a]. Let then γ∗ : [0, ] → Rn be defined by γ∗ (s) = γ(w(s)), s ∈ [0, ].

B, c]) = sup h=1 It is easily seen that (γ; [0, a]) is independent of the parametrization γ of Γ. Therefore, the length of Γ is defined as length(Γ) = (γ; [0, a]) . Whether length(Γ) is finite or not, the following theorem holds true. 8 If Γ is a curve, then H 1 (Γ) = length(Γ). 7 if Γ is a segment. We now consider a parametrization γ : [0, a] → Rn of Γ and set = (γ; [0, a]) = length(Γ). We divide the proof into three steps, and notice that (i) (γ; [b, c]) ≥ |γ(b) − γ(c)|, whenever 0 ≤ b ≤ c ≤ a; (ii) (γ; [b, c]) = (γ; [b, d]) + (γ; [d, c]) whenever 0 ≤ b ≤ d ≤ c ≤ a.

7 if Γ is a segment. We now consider a parametrization γ : [0, a] → Rn of Γ and set = (γ; [0, a]) = length(Γ). We divide the proof into three steps, and notice that (i) (γ; [b, c]) ≥ |γ(b) − γ(c)|, whenever 0 ≤ b ≤ c ≤ a; (ii) (γ; [b, c]) = (γ; [b, d]) + (γ; [d, c]) whenever 0 ≤ b ≤ d ≤ c ≤ a. Step one: We show that H 1 (Γ) ≥ |γ(a) − γ(0)|. 5 we have H 1 (p(Γ)) ≤ H 1 (Γ). At the same time, p(Γ) must contain the segment [γ(0)γ(a)]: otherwise, Γ = γ([0, a]) would be disconnected, against the continuity of γ.

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