A minimal siphon (or alternatively a structural deadlock) of a Petri net is defined as a minimal set S of places such that existence of any edge from a transition t to a place of S implies that there is an edge from some place of S to t. The subject of the paper is to find a minimal siphon containing a given set of specified places of a general Petri net.

A path diversity is an effective technique to get highly reliable communications in the sensor network. In this paper, the path diversity is examined for a tree network composed of binary symmetric channels (BSC) from the view point of bit error probability (BEP). End-nodes of the network are connected to a fusion center, which sums up the received data. The probability density function (pdf) of decision variable conditioned on a source node data is derived by an iterative algorithm to obtain BEP. Numerical results show that in the case of a majority decision, BEP at the fusion center is almost the same as the BSC crossover probability due to the path diversity effects, even if the number of relay links increases.

A 1.9GHz film bulk acoustic resonator (FBAR)-based low-phase-noise complementary cross-coupled voltage-controlled oscillator (VCO) is presented. The FBAR-VCO is designed and fabricated in 0.18µm CMOS process. The DC latch and the low frequency instability are resolved by employing the NMOS source coupling capacitor and the DC blocked cross-coupled pairs. Since no additional voltage headroom is required, the proposed FBAR-VCO can be operated at a low power supply voltage of 1.1V with a wide voltage swing of 0.9V. An effective phase noise optimization is realized by a reasonable trade-off between the output resistance and the trans-conductance of the cross-coupled pairs. The measured performance shows the proposed FBAR-VCO achieves a phase noise of -148dBc/Hz at 1MHz offset with a figure of merit (FoM) of -211.6dB.

A siphon (or alternatively a structutal deadlock) of a Petri net is defined as a set S of places such that existence of any edge from a transition t to a place of S implies that there is an edge from some place of S to t. A minimal siphon is a siphon such that any proper subset is not a siphon. The results of the paper are as follows. (1) The problem of deciding whether or not a given Petri net has a minimum siphon (i.e., a minimum-cardinality minimal siphon) is NP-complete. (2) A polynomial-time algorithm to find, if any, a minimal siphon or even a maximal calss of mutually disjoint minimal siphons of a general Petri net is proposed.