90,897 research outputs found
Dualities in persistent (co)homology
We consider sequences of absolute and relative homology and cohomology groups
that arise naturally for a filtered cell complex. We establish algebraic
relationships between their persistence modules, and show that they contain
equivalent information. We explain how one can use the existing algorithm for
persistent homology to process any of the four modules, and relate it to a
recently introduced persistent cohomology algorithm. We present experimental
evidence for the practical efficiency of the latter algorithm.Comment: 16 pages, 3 figures, submitted to the Inverse Problems special issue
on Topological Data Analysi
On the Topological Spectra of Composite Molecular Systems
It has been shown that topological spectrum of some large
molecules comprises the complete set of eigenvalues of the topological
matrix of some of their constituting fragments. Several relationships
between the coefficients of the fragmental eigenvectors and
those of the composite system have been derived. Some sufficient
conditions that a system comprise a spectrum of its parts have been
found and their use illustrated
Recursive Algorithms for Distributed Forests of Octrees
The forest-of-octrees approach to parallel adaptive mesh refinement and
coarsening (AMR) has recently been demonstrated in the context of a number of
large-scale PDE-based applications. Although linear octrees, which store only
leaf octants, have an underlying tree structure by definition, it is not often
exploited in previously published mesh-related algorithms. This is because the
branches are not explicitly stored, and because the topological relationships
in meshes, such as the adjacency between cells, introduce dependencies that do
not respect the octree hierarchy. In this work we combine hierarchical and
topological relationships between octree branches to design efficient recursive
algorithms.
We present three important algorithms with recursive implementations. The
first is a parallel search for leaves matching any of a set of multiple search
criteria. The second is a ghost layer construction algorithm that handles
arbitrarily refined octrees that are not covered by previous algorithms, which
require a 2:1 condition between neighboring leaves. The third is a universal
mesh topology iterator. This iterator visits every cell in a domain partition,
as well as every interface (face, edge and corner) between these cells. The
iterator calculates the local topological information for every interface that
it visits, taking into account the nonconforming interfaces that increase the
complexity of describing the local topology. To demonstrate the utility of the
topology iterator, we use it to compute the numbering and encoding of
higher-order nodal basis functions.
We analyze the complexity of the new recursive algorithms theoretically, and
assess their performance, both in terms of single-processor efficiency and in
terms of parallel scalability, demonstrating good weak and strong scaling up to
458k cores of the JUQUEEN supercomputer.Comment: 35 pages, 15 figures, 3 table
Smooth critical points of planar harmonic mappings
In a work in 1992, Lyzzaik studies local properties of light harmonic
mappings. More precisely, he classifies their critical points and accordingly
studies their topological and geometrical behaviours. We will focus our study
on smooth critical points of light harmonic maps. We will establish several
relationships between miscellaneous local invariants, and show how to connect
them to Lyzzaik's models. With a crucial use of Milnor fibration theory, we get
a fundamental and yet quite unexpected relation between three of the numerical
invariants, namely the complex multiplicity, the local order of the map and the
Puiseux pair of the critical value curve. We also derive similar results for a
real and complex analytic planar germ at a regular point of its Jacobian
level-0 curve. Inspired by Whitney's work on cusps and folds, we develop an
iterative algorithm computing the invariants. Examples are presented in order
to compare the harmonic situation to the real analytic one.Comment: 36 pages, 5 figure
Deep Reinforcement Learning-Based Channel Allocation for Wireless LANs with Graph Convolutional Networks
Last year, IEEE 802.11 Extremely High Throughput Study Group (EHT Study
Group) was established to initiate discussions on new IEEE 802.11 features.
Coordinated control methods of the access points (APs) in the wireless local
area networks (WLANs) are discussed in EHT Study Group. The present study
proposes a deep reinforcement learning-based channel allocation scheme using
graph convolutional networks (GCNs). As a deep reinforcement learning method,
we use a well-known method double deep Q-network. In densely deployed WLANs,
the number of the available topologies of APs is extremely high, and thus we
extract the features of the topological structures based on GCNs. We apply GCNs
to a contention graph where APs within their carrier sensing ranges are
connected to extract the features of carrier sensing relationships.
Additionally, to improve the learning speed especially in an early stage of
learning, we employ a game theory-based method to collect the training data
independently of the neural network model. The simulation results indicate that
the proposed method can appropriately control the channels when compared to
extant methods
- …