Petri nets have been among the most succinct models that can describe the structure and dynamics of discrete event-driven systems. In this paper, a necessary condition for a 1-safe Petri net generating all the binary n-vectors and the existence of 1-safe Petri nets which generate every binary n-vector as one of their marking vectors exactly once in the smallest possible number of steps have been established.

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In this paper, we consider the risk reserve process with Erlang type claims. We make the model that the claim inter-arrival time has an exponential distribution and the claim size has an Erlang distribution. For this model we obtain a general formula that derives the non-ruin probability in finite time. In order to have an easy calculation for the non-ruin probability we reduce the multi-summation to the singlesummation.

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The purpose of the paper is to consider a necessary and sufficient constraint qualification for local optimality conditions in DC programming problems with convex inequality constraints. Also, we consider necessary and sufficient constraint qualifications for local optimality conditions in fractional programming problems and weakly convex programming problems.

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A store with a bounded tank sells a random resource to customers arriving according to the Poisson process. It reasonably has to be managed in a balance between a fixed ordering cost and a proportional penalty cost. The store should lay down the safety stock level so as to keep these losses as minimal as possible. Then he adopts an ordering policy in which the tank is filled with resource when the stock level falls to the safety stock level. We decide the optimal safety stock level so as to minimize the expected cost per unit time in the infinite periods.

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Surprisingly small varieties of Heyting algebras have very complex categorical structure and contain the largest possible number of subquasivarieties.

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