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Published in S.P. Bhattacharyya, L.H. Keel, of UNCERTAIN DYNAMIC SYSTEMS, 2020
Edmond A. Jonckheere, Jonathan R. Bar-on
It is argued that the real issue is the topology of the mapping m. To be more specific, assume that D is a two-dimensional manifold, from which it follows that D × Ω is a 3-manifold. On the other hand, the Nyquist template Ν is a 2-manifold. For stability margin calculation, assuming stability for all Δ’s, the critical things occur on the “perimeter” of the Nyquist template, near 0 + j0. The situation would be much simplified if the inverse image of the “perimeter” of Ν would be included on the “walls” of D × Ω. Indeed in this case, for stability margin calculation, it would suffice to restrict the search to the “walls” of D × Ω. The same search restricted to the “walls” occurs in other applications: For open-loop stable systems, stability for all parameters is guaranteed if the perimeter of Ν does not encircle 0 + j0, and to check this last property it would again suffice to check the walls of D × Ω.
A topological characterization of periodic flows
Published in Dynamical Systems, 2023
Let W be the whitehead manifold. It is well known that W is a contractible open 3-manifold not homeomorphic to , but the product is homeomorphic to . For every , let and let be a homeomorphism between and . Then the flow is a periodic flow on which is strongly reversible by , where R is the involution of defined by . Clearly, which is homeomorphic to . Moreover, is a global section for G which is a 4-manifold since W is a 3-manifold. However, is not homeomorphic to since its boundary is not homeomorphic to . Obviously, G cannot be conjugate to a flow of rotations since which is not homeomorphic to .
Computational modeling of consistent observation of asynchronous distributed computation on N–manifold
Published in Cogent Engineering, 2018
The 3-manifold structure illustrates existence of multiple local supremum and infimum points. However, a global supremum and infimum can be computed representing global state of computation. The majority of surface areas are relatively smoother during computation indicating finite bound in computation. In this model of computation, the overall message complexity is minimum in stable network condition because the communication is cyclic unicast and no acknowledgment is transacted between processes. The corresponding embedded lattice chain on 3-manifold representing a consistent distributed computing sequence is presented in Figure 6.