19,432 research outputs found
Learning the Irreducible Representations of Commutative Lie Groups
We present a new probabilistic model of compact commutative Lie groups that
produces invariant-equivariant and disentangled representations of data. To
define the notion of disentangling, we borrow a fundamental principle from
physics that is used to derive the elementary particles of a system from its
symmetries. Our model employs a newfound Bayesian conjugacy relation that
enables fully tractable probabilistic inference over compact commutative Lie
groups -- a class that includes the groups that describe the rotation and
cyclic translation of images. We train the model on pairs of transformed image
patches, and show that the learned invariant representation is highly effective
for classification
Bistable flows in precessing spheroids
Precession driven flows are found in any rotating container filled with
liquid, when the rotation axis itself rotates about a secondary axis that is
fixed in an inertial frame of reference. Because of its relevance for planetary
fluid layers, many works consider spheroidal containers, where the uniform
vorticity component of the bulk flow is reliably given by the well-known
equations obtained by Busse in 1968. So far however, no analytical result on
the solutions is available. Moreover, the cases where multiple flows can
coexist have not been investigated in details since their discovery by Noir et
al. (2003). In this work, we aim at deriving analytical results on the
solutions, aiming in particular at, first estimating the ranges of parameters
where multiple solutions exist, and second studying quantitatively their
stability. Using the models recently proposed by Noir \& C{\'e}bron (2013),
which are more generic in the inviscid limit than the equations of Busse, we
analytically describe these solutions, their conditions of existence, and their
stability in a systematic manner. We then successfully compare these analytical
results with the theory of Busse (1968). Dynamical model equations are finally
proposed to investigate the stability of the solutions, which allows to
describe the bifurcation of the unstable flow solution. We also report for the
first time the possibility that time-dependent multiple flows can coexist in
precessing triaxial ellipsoids. Numerical integrations of the algebraic and
differential equations have been efficiently performed with the dedicated
script FLIPPER (supplementary material)
Construction of a giant vortex state in a trapped Fermi system
A superfluid atomic Fermi system may support a giant vortex if the trapping
potential is anharmonic. In such a potential, the single-particle spectrum has
a positive curvature as a function of angular momentum. A tractable model is
put up in which the lowest and next lowest Landau levels are occupied.
Different parameter regimes are identified and characterized. Due to the
dependence of the interaction on angular momentum quantum number, the Cooper
pairing is at its strongest not only close to the Fermi level, but also close
to the energy minimum. It is shown that the gas is superfluid in the interior
of the toroidal density distribution and normal in the outer regions.
Furthermore, the pairing may give rise to a localized density depression in
configuration space.Comment: 12 pages, 14 figure file
Functional summary statistics for point processes on the sphere with an application to determinantal point processes
We study point processes on , the -dimensional unit sphere
, considering both the isotropic and the anisotropic case, and
focusing mostly on the spherical case . The first part studies reduced
Palm distributions and functional summary statistics, including nearest
neighbour functions, empty space functions, and Ripley's and inhomogeneous
-functions. The second part partly discusses the appealing properties of
determinantal point process (DPP) models on the sphere and partly considers the
application of functional summary statistics to DPPs. In fact DPPs exhibit
repulsiveness, but we also use them together with certain dependent thinnings
when constructing point process models on the sphere with aggregation on the
large scale and regularity on the small scale. We conclude with a discussion on
future work on statistics for spatial point processes on the sphere
Structural parameterizations for boxicity
The boxicity of a graph is the least integer such that has an
intersection model of axis-aligned -dimensional boxes. Boxicity, the problem
of deciding whether a given graph has boxicity at most , is NP-complete
for every fixed . We show that boxicity is fixed-parameter tractable
when parameterized by the cluster vertex deletion number of the input graph.
This generalizes the result of Adiga et al., that boxicity is fixed-parameter
tractable in the vertex cover number.
Moreover, we show that boxicity admits an additive -approximation when
parameterized by the pathwidth of the input graph.
Finally, we provide evidence in favor of a conjecture of Adiga et al. that
boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page
Geoids in General Relativity: Geoid Quasilocal Frames
We develop, in the context of general relativity, the notion of a geoid -- a
surface of constant "gravitational potential". In particular, we show how this
idea naturally emerges as a specific choice of a previously proposed, more
general and operationally useful construction called a quasilocal frame -- that
is, a choice of a two-parameter family of timelike worldlines comprising the
worldtube boundary of the history of a finite spatial volume. We study the
geometric properties of these geoid quasilocal frames, and construct solutions
for them in some simple spacetimes. We then compare these results -- focusing
on the computationally tractable scenario of a non-rotating body with a
quadrupole perturbation -- against their counterparts in Newtonian gravity (the
setting for current applications of the geoid), and we compute
general-relativistic corrections to some measurable geometric quantities.Comment: 24 pages, 8 figures; v2: reference added; v3: introduction clarified,
reference adde
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