5 research outputs found
A graph signal processing solution for defective directed graphs
The main purpose of this thesis is to nd a method that allows to systematically adapt GSP
techniques so they can be used on most non-diagonalizable graph operators.
In Chapter 1 we begin by presenting the framework in which GSP is developed, giving
some basic de nitions in the eld of graph theory and in relation with graph signals. We also
present the concept of a Graph Fourier Tranform (GFT), which will be of great importance
in the proposed solution.
Chapter 2 presents the actual motivation of the research: Why the computation of the
GFT is problematic for some directed graphs, and the speci c cases in which this happen. We
will see that the issue can not be assigned to a very speci c graph topography, and therefore
it is important to develop solutions that can be applied to any directed graph.
In Chapter 3 we introduce our proposed new method, which can be used to form, based on
the spectral decomposition of a matrix obtained through its Schur decomposition, a complete
basis of vectors that can be used as a replacement of the previously mentioned Graph Fourier
Transform. The proposed method, the Graph Schur Transform (GST), aims to o er a valid
operator to perform a spectral decomposition of a graph that can be used even in the case of
defective matrices.
Finally, in Chapter 4 we study the main properties of the proposed method and compare
them with the corresponding properties o ered by the Di usion Wavelets design. In the last
section we prove, for a large set of directed graphs, that the GST provides a valid solution for
the proble
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum