13,485 research outputs found
Persistence of invariant manifolds for nonlinear PDEs
We prove that under certain stability and smoothing properties of the
semi-groups generated by the partial differential equations that we consider,
manifolds left invariant by these flows persist under perturbation. In
particular, we extend well known finite-dimensional results to the setting of
an infinite-dimensional Hilbert manifold with a semi-group that leaves a
submanifold invariant. We then study the persistence of global unstable
manifolds of hyperbolic fixed-points, and as an application consider the
two-dimensional Navier-Stokes equation under a fully discrete approximation.
Finally, we apply our theory to the persistence of inertial manifolds for those
PDEs which possess them. teComment: LaTeX2E, 32 pages, to appear in Studies in Applied Mathematic
On the use of Gaia magnitudes and new tables of bolometric corrections
The availability of reliable bolometric corrections and reddening estimates,
rather than the quality of parallaxes will be one of the main limiting factors
in determining the luminosities of a large fraction of Gaia stars. With this
goal in mind, we provide Gaia G, BP and RP synthetic photometry for the entire
MARCS grid, and test the performance of our synthetic colours and bolometric
corrections against space-borne absolute spectrophotometry. We find indication
of a magnitude-dependent offset in Gaia DR2 G magnitudes, which must be taken
into account in high accuracy investigations. Our interpolation routines are
easily used to derive bolometric corrections at desired stellar parameters, and
to explore the dependence of Gaia photometry on Teff, log(g), [Fe/H],
alpha-enhancement and E(B-V). Gaia colours for the Sun and Vega, and
Teff-dependent extinction coefficients, are also provided.Comment: MNRAS Letter. Solar colours: BP-G = 0.33, G-RP = 0.49, BP-RP = 0.82.
Mean extinction coefficients at turn-off: R_G = 2.740 , R_BP = 3.374, R_RP =
2.035. Interpolation routines available at
https://github.com/casaluca/bolometric-correction
Empathic Neural Responses Predict Group Allegiance.
Watching another person in pain activates brain areas involved in the sensation of our own pain. Importantly, this neural mirroring is not constant; rather, it is modulated by our beliefs about their intentions, circumstances, and group allegiances. We investigated if the neural empathic response is modulated by minimally-differentiating information (e.g., a simple text label indicating another's religious belief), and if neural activity changes predict ingroups and outgroups across independent paradigms. We found that the empathic response was larger when participants viewed a painful event occurring to a hand labeled with their own religion (ingroup) than to a hand labeled with a different religion (outgroup). Counterintuitively, the magnitude of this bias correlated positively with the magnitude of participants' self-reported empathy. A multivariate classifier, using mean activity in empathy-related brain regions as features, discriminated ingroup from outgroup with 72% accuracy; the classifier's confidence correlated with belief certainty. This classifier generalized successfully to validation experiments in which the ingroup condition was based on an arbitrary group assignment. Empathy networks thus allow for the classification of long-held, newly-modified and arbitrarily-formed ingroups and outgroups. This is the first report of a single machine learning model on neural activation that generalizes to multiple representations of ingroup and outgroup. The current findings may prove useful as an objective diagnostic tool to measure the magnitude of one's group affiliations, and the effectiveness of interventions to reduce ingroup biases
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