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Separability in Persistent Petri Nets

Best E., Darondeau P.

Fundamenta Informaticae

2011

Vol. 113, nr 3/4

179-203

Separability in Petri nets means the property for a net k źN with an initial marking k źM to behave in the same way as k parallel instances of the same net N with an initial markingM, thus divided by k. We prove the separability of plain, bounded, reversible and persistent Petri nets, a class of nets that extends the well-known live and bounded marked graphs. We establish first a weak form of separability, already known to hold for marked graphs, in which every firing sequence of k ź N is simulated by a firing sequence of k parallel instances of N with an identical firing count. We establish on top of this a strong form of separability, in which every firing sequence of k ź N is simulated by an identical firing sequence of k parallel instances of N.

On some properties of Musielak-Orlicz sequence spaces

Shragin I.V.

Commentationes Mathematicae

2008

Vol. 48, [Z] 2

147-153

We consider a nontrivial vector space X and a semimodularM : X→[0,∞] with property: [...] . The function M generates in X a metric d with [formula] At the same time M generates a metric in Musielak-Orlicz sequence space lM, namely [formula] It is proved that the space (lM, p) is complete if and only if the space (X, d) is complete. [formula] Several examples are considered.