# Analyse fonctionnelle elementaire by Willem M.

By Willem M.

By Willem M.

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Recent Developments in Complex Analysis and Computer Algebra: This conference was supported by the National Science Foundation through Grant INT-9603029 and the Japan Society for the Promotion of Science through Grant MTCS-134

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Extra resources for Analyse fonctionnelle elementaire

Example text

The module Add1+x k (x, u, t) depicted in Fig. 5 adds 1+fracp(x) to the cost, leaving the conﬁguration unchanged (up to shifts). Lemma 3. Let δ < 12 . Assume module Add1+x k (x, u, t) is entered with valuation ν ∈ codek (n, m) for some < 12 . Controller has a strategy that ensures either reaching a target location with cost −∞ or location 2 with valuation ν satisfying ν = fracp(ν) + k where kx = 1, and kα = kα + 3 for all α ∈ {x , y, y , z, z }, while the cost increases by 1 + fracp(x). We deﬁne similarly a module Add2−x k (x, u, t) that adds 2 − fracp(x) to the cost variable.

Thus, the cost variable grows with derivative S( ) at location , and the sojourn time is either the delay suggested by Perturbator if it is shorter, or the 36 P. Bouyer, N. Markey, and O. Sankur delay suggested by Controller and perturbed by Perturbator otherwise. Notice that, in this semantics the guard of an edge that is taken need not be satisﬁed after a perturbation (hence the term excess). We also consider another natural semantics for perturbation, which we call the δ-conservative perturbation semantics and denote Gδc (A).

B3. The guard g of every transition δ = (q, a, g, r, q ) ∈ Δ is just one region. All the clock values satisfying g are time-reachable from rq . B4. For every transition δ its guard g has no constraints of the form x = c. By the fundamental property of region abstraction, any path of the underlying graph of a region-split TA is realizable as a run that follows the same edges. Proposition 6 ([4]). Given a BDTA accepting a language L, a ﬂeshy regionsplit TA accepting a language L ⊂ L with Vn (L ) = Vn (L) and H(L ) = H(L) can be constructed.