We prove the following property for safe marked graphs, safe conflict-free Petri nets, and live and safe extended free-choice Petri nets. We prove the following three results. If the Petri net is a marked graph, then the length of the shortest path is at most (|T |−1)·|T |/2. If the Petri net is conflict free, then the length of the shortest path is at most (|T |+1)·|T |/2. If the Petri net is ...

where g : R × R × R → R is a C function. Once we specify a parameter path u(t), t ≥ t1, in the parameter space we can solve (1.1) uniquely for x(t), t ≥ t1, with the given initial condition x(t1) = x0. Alternatively if we specify a parameter path u(t), t ≤ t1, in the parameter space we can solve (1.1) uniquely for x(t), t ≤ t1, with the given terminal condition x(t1) = x0. Suppose now that for ...

Given a tree T with edge lengths and edge weights, and a value B, the length-constrained heaviest path problem is to find a path in T with maximum path weight whose path length is at most B. We present a linear time algorithm for the problem when the edge lengths are uniform, i.e., all one. This algorithm with slight modification can be used to find the heaviest path of length exactly B in T in...

This global viewpoint is in contrast to the calculus of variations approach where we typically look at local variations of trajectories in the vicinity of a given initial path x(·) often starting from a given state x(0) = x0. The DP principle is based on the idea that once an optimal path x∗(t) for t ∈ [t0, tf ] is computed then any segment of this path is optimal, i.e. x∗(t) for t ∈ [t1, tf ] ...

Proterozoic rocks of the southwestern United States exhibit a variety of tectonometamorphic histories that can all be explained by a characteristic looping style of pressure-temperature (P-T) path. The path is fundamentally clockwise. However, because the rocks were cooled isobarically at 3 kbar after tectonism, the cooling path crosses the prograde path, leading to a geometry with a counterclo...

Given a tree T with edge lengths and edge weights, and a value B, the length-constrained heaviest path problem is to find a path in T with maximum path weight whose path length is at most B. We present a linear time algorithm for the problem when the edge lengths are uniform, i.e., all one. This algorithm with slight modification can be used to find the heaviest path of length exactly B in T in...

Quantum mechanical phase space path integrals are re-examined with regard to the physical interpretation of the phase space variables involved. It is demonstrated that the traditional phase space path integral implies a meaning for the variables involved that is manifestly inconsistent. On the other hand, a phase space path integral based on coherent states entails variables that exhibit a self...