601 research outputs found
Simplicial Homology for Future Cellular Networks
Simplicial homology is a tool that provides a mathematical way to compute the
connectivity and the coverage of a cellular network without any node location
information. In this article, we use simplicial homology in order to not only
compute the topology of a cellular network, but also to discover the clusters
of nodes still with no location information. We propose three algorithms for
the management of future cellular networks. The first one is a frequency
auto-planning algorithm for the self-configuration of future cellular networks.
It aims at minimizing the number of planned frequencies while maximizing the
usage of each one. Then, our energy conservation algorithm falls into the
self-optimization feature of future cellular networks. It optimizes the energy
consumption of the cellular network during off-peak hours while taking into
account both coverage and user traffic. Finally, we present and discuss the
performance of a disaster recovery algorithm using determinantal point
processes to patch coverage holes
Chain Homotopies for Object Topological Representations
This paper presents a set of tools to compute topological information of
simplicial complexes, tools that are applicable to extract topological
information from digital pictures. A simplicial complex is encoded in a
(non-unique) algebraic-topological format called AM-model. An AM-model for a
given object K is determined by a concrete chain homotopy and it provides, in
particular, integer (co)homology generators of K and representative (co)cycles
of these generators. An algorithm for computing an AM-model and the
cohomological invariant HB1 (derived from the rank of the cohomology ring) with
integer coefficients for a finite simplicial complex in any dimension is
designed here. A concept of generators which are "nicely" representative cycles
is also presented. Moreover, we extend the definition of AM-models to 3D binary
digital images and we design algorithms to update the AM-model information
after voxel set operations (union, intersection, difference and inverse)
The triangulation of manifolds
A mostly expository account of old questions about the relationship between
polyhedra and topological manifolds. Topics are old topological results, new
gauge theory results (with speculations about next directions), and history of
the questions.Comment: 26 pages, 2 figures. version 2: spellings corrected, analytic
speculations in 4.8.2 sharpene
Simplification Techniques for Maps in Simplicial Topology
This paper offers an algorithmic solution to the problem of obtaining
"economical" formulae for some maps in Simplicial Topology, having, in
principle, a high computational cost in their evaluation. In particular, maps
of this kind are used for defining cohomology operations at the cochain level.
As an example, we obtain explicit combinatorial descriptions of Steenrod k-th
powers exclusively in terms of face operators
- …