171 research outputs found
A Generic Framework for Engineering Graph Canonization Algorithms
The state-of-the-art tools for practical graph canonization are all based on
the individualization-refinement paradigm, and their difference is primarily in
the choice of heuristics they include and in the actual tool implementation. It
is thus not possible to make a direct comparison of how individual algorithmic
ideas affect the performance on different graph classes.
We present an algorithmic software framework that facilitates implementation
of heuristics as independent extensions to a common core algorithm. It
therefore becomes easy to perform a detailed comparison of the performance and
behaviour of different algorithmic ideas. Implementations are provided of a
range of algorithms for tree traversal, target cell selection, and node
invariant, including choices from the literature and new variations. The
framework readily supports extraction and visualization of detailed data from
separate algorithm executions for subsequent analysis and development of new
heuristics.
Using collections of different graph classes we investigate the effect of
varying the selections of heuristics, often revealing exactly which individual
algorithmic choice is responsible for particularly good or bad performance. On
several benchmark collections, including a newly proposed class of difficult
instances, we additionally find that our implementation performs better than
the current state-of-the-art tools
Unfreezing Casimir invariants: singular perturbations giving rise to forbidden instabilities
The infinite-dimensional mechanics of fluids and plasmas can be formulated as
"noncanonical" Hamiltonian systems on a phase space of Eulerian variables.
Singularities of the Poisson bracket operator produce singular Casimir elements
that foliate the phase space, imposing topological constraints on the dynamics.
Here we proffer a physical interpretation of Casimir elements as
\emph{adiabatic invariants} ---upon coarse graining microscopic angle
variables, we obtain a macroscopic hierarchy on which the separated action
variables become adiabatic invariants. On reflection, a Casimir element may be
\emph{unfrozen} by recovering a corresponding angle variable; such an increase
in the number of degrees of freedom is, then, formulated as a \emph{singular
perturbation}. As an example, we propose a canonization of the
resonant-singularity of the Poisson bracket operator of the linearized
magnetohydrodynamics equations, by which the ideal obstacle (resonant Casimir
element) constraining the dynamics is unfrozen, giving rise to a tearing-mode
instability
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
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