An Algorithm for the Set of All Generators of an Arbitrary Firing Count Vector in Petri Nets

In this paper, an effective method to obtain all nonnegative integer minimal support vectors (U4,V4) at level 4 for an arbitrary homogeneous / inhomogeneous solution of a matrix equation A x = b (A∈Zmxn, b∈Zmxl) starting from nonnegative rational number minimal support vectors (U3,V3) at level 3 is proposed. Although V4 has been derived from all minimal vectors (U5,V5) of level 5 which are obtained starting from vectors at levell, 2, or 3, so far, this proposed method for (U4,V4) gives us a big shortcut comparing with them. However, it is pointed out that obtaining (U5,V5) of level 5 from (U4,V4) of level 4 is difficult

Phase Noise of Multiphase CMOS LC Oscillators Coupled by Mutual Inductors

There have beem many studies on the analysis and design of low-noise oscillators. Recently, much attention has been paid to the noise reduction technique using coupled oscillators. When oscillators are coupled, the coupling method is very important and affects various factors, for example, the level of noise. In this study, we analyze the phase noise of multiphase CMOS LC oscillators coupled by mutual inductors by using the impulse sensitivity function. From the simulation results, using mutual inductors as coupling elements can reduce the phase noise. Also, for oscillators coupled by mutual inductors, we show that setting a coupling coefficient of around 0.2 can realize the lowest phase noise near the oscillation frequency

An Application of Groebner Bases to Behavioral Analyses for Petri Nets by Means of Computer Algebra Systems

One of methods to solve integer programming problems consists in the method which uses groebner bases. An arbitrary solution for a state equation Ax=b(A∈Zmxn,b∈Zmx1) of Petri nets means a firing count vector. Then finding a nonnegative integer solution x∈Znx1+ for Ax=b in Petri nets is one of integer programming problems. In this paper, the method to obtain generators of solutions in Petri nets by using groebner bases is proposed and investigated. moreover, Petri nets have an ill property that the number of minimal support T-invariants increases in exponential when places and transitions are increased. Then, the number of groebner bases and calculation time of groebner bases are measured by using a symbolic computation system or a computer algebra system; Maple 7