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Excerpt from idea of Differential Equations, Vol. 6: half IV.-Partial Differential Equations
213, 214. Riemann's use of the adjoint equation to build the Cauchy essential of the unique equation: with.
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Additional info for Theory of differential equations Volume VI (Part 4): Partial Differential Equations
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Fig. 1. Example of time synchronisation for GT nets – We define a time-elapsing transition , for : P/E → D, as follows, where [p]E denotes the E-equivalence class of p: M[ M iff ∀p ∈ P. M (p) = M (p) <+ ([p]E ). We write M −→ M iff either M [t M or M [ M for some t or . In particular we can consider the following two classes of DTAPNs. The first one requires an absolute synchronisation and was studied in the past, while the other one is a new model — completely asynchronous. – Global timed-arc Petri nets (GT nets): E = P × P .
Information and Computation, 57:125–147, 1983. M. Hanisch. Analysis of place/transition nets with timed-arcs and its application to batch process control. In Application and Theory of Petri Nets, volume 691 of LNCS, pages 282–299, 1993. P. Jancar. Decidability of a temporal logic problem for Petri nets. Theoretical Computer Science, 74(1):71–93, 1990. P. Jancar. Bisimulation equivalence is decidable for one-counter processes. In Automata, Languages and Programming, 24th International Colloquium (ICALP’97), volume 1256 of LNCS, pages 549–559.
A termination predicate ↓ ⊆ N defines in what states a marked net can terminate successfully. A marked net that cannot perform any actions or terminate successfully is in a deadlock. We need two auxiliary definitions: (1) a relation expressing that a marked net can evolve via zero or more τ actions into another marked net; (2) a predicate expressing that a marked net can terminate via zero or more τ actions. 7. The relation =⇒ ⊆ N × N is defined as the smallest relation satisfying, for any p, p , p ∈ N , p =⇒ p and (p =⇒ p ∧ p [τ p ) ⇒ p =⇒ p .