427,606 research outputs found
Symmetry within Solutions
We define the concept of an internal symmetry. This is a symmety within a
solution of a constraint satisfaction problem. We compare this to solution
symmetry, which is a mapping between different solutions of the same problem.
We argue that we may be able to exploit both types of symmetry when finding
solutions. We illustrate the potential of exploiting internal symmetries on two
benchmark domains: Van der Waerden numbers and graceful graphs. By identifying
internal symmetries we are able to extend the state of the art in both cases.Comment: AAAI 2010, Proceedings of Twenty-Fourth AAAI Conference on Artificial
Intelligenc
Revealing the state space of turbulent pipe flow by symmetry reduction
Symmetry reduction by the method of slices is applied to pipe flow in order
to quotient the stream-wise translation and azimuthal rotation symmetries of
turbulent flow states. Within the symmetry-reduced state space, all travelling
wave solutions reduce to equilibria, and all relative periodic orbits reduce to
periodic orbits. Projections of these solutions and their unstable manifolds
from their -dimensional symmetry-reduced state space onto suitably
chosen 2- or 3-dimensional subspaces reveal their interrelations and the role
they play in organising turbulence in wall-bounded shear flows. Visualisations
of the flow within the slice and its linearisation at equilibria enable us to
trace out the unstable manifolds, determine close recurrences, identify
connections between different travelling wave solutions, and find, for the
first time for pipe flows, relative periodic orbits that are embedded within
the chaotic attractor, which capture turbulent dynamics at transitional
Reynolds numbers.Comment: 24 pages, 12 figure
Chiral symmetry breaking in dimensionally regularized nonperturbative quenched QED
In this paper we study dynamical chiral symmetry breaking in dimensionally
regularized quenched QED within the context of Dyson-Schwinger equations. In D
< 4 dimensions the theory has solutions which exhibit chiral symmetry breaking
for all values of the coupling. To begin with, we study this phenomenon both
numerically and, with some approximations, analytically within the rainbow
approximation in the Landau gauge. In particular, we discuss how to extract the
critical coupling alpha_c = pi/3 relevant in four dimensions from the D
dimensional theory. We further present analytic results for the chirally
symmetric solution obtained with the Curtis-Pennington vertex as well as
numerical results for solutions exhibiting chiral symmetry breaking. For these
we demonstrate that, using dimensional regularization, the extraction of the
critical coupling relevant for this vertex is feasible. Initial results for
this critical coupling are in agreement with cut-off based work within the
currently achievable numerical precision.Comment: 24 pages, including 5 figures; submitted to Phys. Rev.
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