2,308 research outputs found
A geometric approach to scalar field theories on the supersphere
Following a strictly geometric approach we construct globally supersymmetric
scalar field theories on the supersphere, defined as the quotient space
. We analyze the superspace geometry of the
supersphere, in particular deriving the invariant vielbein and spin connection
from a generalization of the left-invariant Maurer-Cartan form for Lie groups.
Using this information we proceed to construct a superscalar field action on
, which can be decomposed in terms of the component fields, yielding a
supersymmetric action on the ordinary two-sphere. We are able to derive
Lagrange equations and Noether's theorem for the superscalar field itself.Comment: 38 pages, 1 figur
Curvature blow up in Bianchi VIII and IX vacuum spacetimes
The maximal globally hyperbolic development of non-Taub-NUT Bianchi IX vacuum
initial data and of non-NUT Bianchi VIII vacuum initial data is C2
inextendible. Furthermore, a curvature invariant is unbounded in the incomplete
directions of inextendible causal geodesics.Comment: 20 pages, no figures. Submitted to Classical and Quantum Gravit
Asymptotic self-similarity breaking at late times in cosmology
We study the late time evolution of a class of exact anisotropic cosmological
solutions of Einstein's equations, namely spatially homogeneous cosmologies of
Bianchi type VII with a perfect fluid source. We show that, in contrast to
models of Bianchi type VII which are asymptotically self-similar at late
times, Bianchi VII models undergo a complicated type of self-similarity
breaking. This symmetry breaking affects the late time isotropization that
occurs in these models in a significant way: if the equation of state parameter
satisfies the models isotropize as regards the shear
but not as regards the Weyl curvature. Indeed these models exhibit a new
dynamical feature that we refer to as Weyl curvature dominance: the Weyl
curvature dominates the dynamics at late times. By viewing the evolution from a
dynamical systems perspective we show that, despite the special nature of the
class of models under consideration, this behaviour has implications for more
general models.Comment: 29 page
When is it Better to Compare than to Score?
When eliciting judgements from humans for an unknown quantity, one often has
the choice of making direct-scoring (cardinal) or comparative (ordinal)
measurements. In this paper we study the relative merits of either choice,
providing empirical and theoretical guidelines for the selection of a
measurement scheme. We provide empirical evidence based on experiments on
Amazon Mechanical Turk that in a variety of tasks, (pairwise-comparative)
ordinal measurements have lower per sample noise and are typically faster to
elicit than cardinal ones. Ordinal measurements however typically provide less
information. We then consider the popular Thurstone and Bradley-Terry-Luce
(BTL) models for ordinal measurements and characterize the minimax error rates
for estimating the unknown quantity. We compare these minimax error rates to
those under cardinal measurement models and quantify for what noise levels
ordinal measurements are better. Finally, we revisit the data collected from
our experiments and show that fitting these models confirms this prediction:
for tasks where the noise in ordinal measurements is sufficiently low, the
ordinal approach results in smaller errors in the estimation
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