China 
Africa 
Explore chapters and articles related to this topic
Topology
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Steve Huntsman, Jimmy Palladino, Michael Robinson
While chain complexes distill abstract simplicial complexes into the realm of algebra, they are still rather complicated. Moreover, the simplicial chain complex contains combinatorial, non-topological information. Homology is a convenient, linear algebraic summary for a chain complex that still preserves the structure of chain maps. Additionally, the homology of the simplicial chain complex is a topological invariant. Although we restrict our considerations to fields for practical purposes, homology is readily defined over rings, with the integers ℤ serving as the case through which all others factor via the universal coefficient theorem (which, incidentally, gave rise to the topics of category theory and homological algebra).
Computational modeling of consistent observation of asynchronous distributed computation on N–manifold
Published in Cogent Engineering, 2018
The formal modeling of computing systems enables to gain an analytical insight to the complex systems, which facilitates robust designs. For example, a formal specification of dependability of computation in large-scale pervasive systems is constructed (Ayara & Najjar, 2008). The stable predicate detection in distributed computation having infinite run is formulated by restricting the run within finite graph model (Garg et al., 2014). However, constraining the infinite computation on finite structure may invite oscillatory execution pattern incorporating indeterminate convergence. A comparatively new approach to model distributed computing systems involves elements of algebraic topology, lattice theory, and homological algebra offering new insights (Bagchi, 2018; Bauer et al., 2014; Conde & Rajsbaum, 2012; Herlihy & Rajsbaum, 1999). This paper proposes a formal model of consistent observation of a distributed computation on N-manifold. The proposed model does not impose any condition of finiteness of computation. The variable filter function can control the granularity of observation of states in a sequence of computation while maintaining consistency conditions. The main contributions of this paper are as follows.Construction of a formal model of observation of asynchronous distributed computing on N-manifold structureIntegrating filtering method to generate consistent observation of distributed computationIncorporation of variability of filter to prepare consistent observation of computation with varying granularityEvaluation of resulting manifold structure of distributed computation in 3-D and lattice chain embedding on it