1,108 research outputs found
Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata
Functions which are equivariant or invariant under the transformations of a
compact linear group acting in an euclidean space , can profitably
be studied as functions defined in the orbit space of the group. The orbit
space is the union of a finite set of strata, which are semialgebraic manifolds
formed by the -orbits with the same orbit-type. In this paper we provide a
simple recipe to obtain rational parametrizations of the strata. Our results
can be easily exploited, in many physical contexts where the study of
equivariant or invariant functions is important, for instance in the
determination of patterns of spontaneous symmetry breaking, in the analysis of
phase spaces and structural phase transitions (Landau theory), in equivariant
bifurcation theory, in crystal field theory and in most areas where use is made
of symmetry adapted functions.
A physically significant example of utilization of the recipe is given,
related to spontaneous polarization in chiral biaxial liquid crystals, where
the advantages with respect to previous heuristic approaches are shown.Comment: Figures generated through texdraw package; revised version appearing
in J. Phys. A: Math. Ge
Action Principle for the Generalized Harmonic Formulation of General Relativity
An action principle for the generalized harmonic formulation of general
relativity is presented. The action is a functional of the spacetime metric and
the gauge source vector. An action principle for the Z4 formulation of general
relativity has been proposed recently by Bona, Bona--Casas and Palenzuela
(BBP). The relationship between the generalized harmonic action and the BBP
action is discussed in detail.Comment: This version is contains more thorough presentations and discussions
of the key results. To be published in PRD. (8 pages, no figures
Cosmology and astrophysics from relaxed galaxy clusters - IV: Robustly calibrating hydrostatic masses with weak lensing
This is the fourth in a series of papers studying the astrophysics and
cosmology of massive, dynamically relaxed galaxy clusters. Here, we use
measurements of weak gravitational lensing from the Weighing the Giants project
to calibrate Chandra X-ray measurements of total mass that rely on the
assumption of hydrostatic equilibrium. This comparison of X-ray and lensing
masses provides a measurement of the combined bias of X-ray hydrostatic masses
due to both astrophysical and instrumental sources. Assuming a fixed cosmology,
and within a characteristic radius (r_2500) determined from the X-ray data, we
measure a lensing to X-ray mass ratio of 0.96 +/- 9% (stat) +/- 9% (sys). We
find no significant trends of this ratio with mass, redshift or the
morphological indicators used to select the sample. In accordance with
predictions from hydro simulations for the most massive, relaxed clusters, our
results disfavor strong, tens-of-percent departures from hydrostatic
equilibrium at these radii. In addition, we find a mean concentration of the
sample measured from lensing data of c_200 = . Anticipated
short-term improvements in lensing systematics, and a modest expansion of the
relaxed lensing sample, can easily increase the measurement precision by
30--50%, leading to similar improvements in cosmological constraints that
employ X-ray hydrostatic mass estimates, such as on Omega_m from the cluster
gas mass fraction.Comment: 13 pages. Submitted to MNRAS. Comments welcom
The oxygen isotopic composition of phytolith assemblages from tropical rainforest soil tops (Queensland, Australia): validation of a new paleoenvironmental tool
Phytoliths are micrometric particles of amorphous silica that form inside or between the cells of higher plant tissues throughout the life of a plant. With plant decay, phytoliths are either incorporated into soils or exported to sediments via regional watersheds. Phytolith morphological assemblages are increasingly used as proxy of grassland diversity and tree cover density in inter-tropical areas. Here, we investigate whether, along altitudinal gradients in northeast Queensland (Australia), changes in the &delta;<sup>18</sup>O signature of soil top phytolith assemblages reflect changes in mean annual temperature (MAT) and in the oxygen isotopic composition of precipitation (&delta;<sup>18</sup>O<sub>precipitation</sub>), as predicted by equilibrium temperature coefficients previously published for silica. Oxygen isotopic analyses were performed on 16 phytolith samples, after controlled isotopic exchange (CIE), using the IR Laser-Heating Fluorination Technique. Long-term mean annual precipitation (MAP) and MAT values at the sampled sites were calculated by the ANUCLIM software. &delta;<sup>18</sup>O<sub>precipitation</sub> estimates were calculated using the Bowen and Wilkinson (2002) model, slightly modified. An empirical temperature-dependant relationship was obtained: &delta;<sup>18</sup>O<sub>wood phytolith-precipitation</sub> (&permil; vs. VSMOW) = −0.4 (±0.2) <i>t</i> (°C) + 46 (±3) (<i>R</i><sup>2</sup> = 0.4, <i>p</i> < 0.05; <i>n</i> = 12). Despite the various unknowns introduced when estimating &delta;<sup>18</sup>O<sub>precipitation</sub> values and the large uncertainties on &delta;<sup>18</sup>O<sub>wood phytolith</sub> values, the temperature coefficient (−0.4 ± 0.2&permil; °C<sup>−1</sup>) is in the range of values previously obtained for natural quartz, fresh and sedimentary diatoms and harvested grass phytoliths (from −0.2 to −0.5&permil; °C<sup>−1</sup>). The consistency supports the reliability of &delta;<sup>18</sup>O<sub>wood phytolith</sub> signatures for recording relative changes in mean annual &delta;<sup>18</sup>O<sub>soil water</sub> values (which are assumed to be equivalent to the weighted annual &delta;<sup>18</sup>O<sub> precipitation</sub> values in rainforests environments) and MAT, provided these changes were several &permil; and/or several °C in magnitude
Inter-species variation in colour perception
Inter-species variation in colour perception poses a serious problem for the view that colours are mind-independent properties. Given that colour perception varies so drastically across species, which species perceives colours as they really are? In this paper, I argue that all do. Specifically, I argue that members of different species perceive properties that are determinates of different, mutually compatible, determinables. This is an instance of a general selectionist strategy for dealing with cases of perceptual variation. According to selectionist views, objects simultaneously instantiate a plurality of colours, all of them genuinely mind-independent, and subjects select from amongst this plurality which colours they perceive. I contrast selectionist views with relationalist views that deny the mind-independence of colour, and consider some general objections to this strategy
The structure of the Kac-Wang-Yan algebra
The Lie algebra of regular differential operators on the circle
has a universal central extension . The invariant subalgebra
under an involution preserving the principal gradation
was introduced by Kac, Wang, and Yan. The vacuum -module
with central charge , and its irreducible quotient
, possess vertex algebra structures, and has a
nontrivial structure if and only if . We show that
for each integer , and are
-algebras of types and
, respectively. These results are formal
consequences of Weyl's first and second fundamental theorems of invariant
theory for the orthogonal group and the symplectic group
, respectively. Based on Sergeev's theorems on the invariant
theory of we conjecture that is of
type , and we prove this for . As an
application, we show that invariant subalgebras of -systems and
free fermion algebras under arbitrary reductive group actions are strongly
finitely generated.Comment: Final versio
Revisiting Weyl's calculation of the gravitational pull in Bach's two-body solution
When the mass of one of the two bodies tends to zero, Weyl's definition of
the gravitational force in an axially symmetric, static two-body solution can
be given an invariant formulation in terms of a force four-vector. The norm of
this force is calculated for Bach's two-body solution, that is known to be in
one-to-one correspondence with Schwarzschild's original solution when one of
the two masses l, l' is made to vanish. In the limit when, say, l' goes to
zero, the norm of the force divided by l' and calculated at the position of the
vanishing mass is found to coincide with the norm of the acceleration of a test
body kept at rest in Schwarzschild's field. Both norms happen thus to grow
without limit when the test body (respectively the vanishing mass l') is kept
at rest in a position closer and closer to Schwarzschild's two-surface.Comment: 11 pages, 2 figures. Text to appear in Classical and Quantum Gravit
The double torus as a 2D cosmos: groups, geometry and closed geodesics
The double torus provides a relativistic model for a closed 2D cosmos with
topology of genus 2 and constant negative curvature. Its unfolding into an
octagon extends to an octagonal tessellation of its universal covering, the
hyperbolic space H^2. The tessellation is analysed with tools from hyperbolic
crystallography. Actions on H^2 of groups/subgroups are identified for SU(1,
1), for a hyperbolic Coxeter group acting also on SU(1, 1), and for the
homotopy group \Phi_2 whose extension is normal in the Coxeter group. Closed
geodesics arise from links on H^2 between octagon centres. The direction and
length of the shortest closed geodesics is computed.Comment: Latex, 27 pages, 5 figures (late submission to arxiv.org
A Note on Infinities in Eternal Inflation
In some well-known scenarios of open-universe eternal inflation, developed by
Vilenkin and co-workers, a large number of universes nucleate and thermalize
within the eternally inflating mega-universe. According to the proposal, each
universe nucleates at a point, and therefore the boundary of the nucleated
universe is a space-like surface nearly coincident with the future light cone
emanating from the point of nucleation, all points of which have the same
proper-time. This leads the authors to conclude that at the proper-time t =
t_{nuc} at which any such nucleation occurs, an infinite open universe comes
into existence. We point out that this is due entirely to the supposition of
the nucleation occurring at a single point, which in light of quantum cosmology
seems difficult to support. Even an infinitesimal space-like length at the
moment of nucleation gives a rather different result -- the boundary of the
nucleating universe evolves in proper-time and becomes infinite only in an
infinite time. The alleged infinity is never attained at any finite time.Comment: 13 pages and 6 figure
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