36,602 research outputs found
Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory
In this paper we present a general framework for solving the stationary
nonlinear Schr\"odinger equation (NLSE) on a network of one-dimensional wires
modelled by a metric graph with suitable matching conditions at the vertices. A
formal solution is given that expresses the wave function and its derivative at
one end of an edge (wire) nonlinearly in terms of the values at the other end.
For the cubic NLSE this nonlinear transfer operation can be expressed
explicitly in terms of Jacobi elliptic functions. Its application reduces the
problem of solving the corresponding set of coupled ordinary nonlinear
differential equations to a finite set of nonlinear algebraic equations. For
sufficiently small amplitudes we use canonical perturbation theory which makes
it possible to extract the leading nonlinear corrections over large distances.Comment: 26 page
Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames
We consider the problem of designing spectral graph filters for the
construction of dictionaries of atoms that can be used to efficiently represent
signals residing on weighted graphs. While the filters used in previous
spectral graph wavelet constructions are only adapted to the length of the
spectrum, the filters proposed in this paper are adapted to the distribution of
graph Laplacian eigenvalues, and therefore lead to atoms with better
discriminatory power. Our approach is to first characterize a family of systems
of uniformly translated kernels in the graph spectral domain that give rise to
tight frames of atoms generated via generalized translation on the graph. We
then warp the uniform translates with a function that approximates the
cumulative spectral density function of the graph Laplacian eigenvalues. We use
this approach to construct computationally efficient, spectrum-adapted, tight
vertex-frequency and graph wavelet frames. We give numerous examples of the
resulting spectrum-adapted graph filters, and also present an illustrative
example of vertex-frequency analysis using the proposed construction
The Graph Curvature Calculator and the curvatures of cubic graphs
We classify all cubic graphs with either non-negative Ollivier-Ricci
curvature or non-negative Bakry-\'Emery curvature everywhere. We show in both
curvature notions that the non-negatively curved graphs are the prism graphs
and the M\"obius ladders. We also highlight an online tool for calculating the
curvature of graphs under several variants of these curvature notions that we
use in the classification. As a consequence of the classification result we
show, that non-negatively curved cubic expanders do not exist
V.M. Miklyukov: from dimension 8 to nonassociative algebras
In this short survey we give a background and explain some recent
developments in algebraic minimal cones and nonassociative algebras. A good
deal of this paper is recollections of my collaboration with my teacher, PhD
supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that
motivated the present research. This paper is dedicated to his memory.Comment: 19 page
Strong-coupling scales and the graph structure of multi-gravity theories
In this paper we consider how the strong-coupling scale, or perturbative
cutoff, in a multi-gravity theory depends upon the presence and structure of
interactions between the different fields. This can elegantly be rephrased in
terms of the size and structure of the `theory graph' which depicts the
interactions in a given theory. We show that the question can be answered in
terms of the properties of various graph-theoretical matrices, affording an
efficient way to estimate and place bounds on the strong-coupling scale of a
given theory. In light of this we also consider the problem of relating a given
theory graph to a discretised higher dimensional theory, a la dimensional
deconstruction.Comment: 23 pages, 7 figures; v2: additional references included, and minor
typos corrected; version published in JHE
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