2,159 research outputs found
Agnostic cosmology in the CAMEL framework
Cosmological parameter estimation is traditionally performed in the Bayesian
context. By adopting an "agnostic" statistical point of view, we show the
interest of confronting the Bayesian results to a frequentist approach based on
profile-likelihoods. To this purpose, we have developed the Cosmological
Analysis with a Minuit Exploration of the Likelihood ("CAMEL") software.
Written from scratch in pure C++, emphasis was put in building a clean and
carefully-designed project where new data and/or cosmological computations can
be easily included.
CAMEL incorporates the latest cosmological likelihoods and gives access from
the very same input file to several estimation methods: (i) A high quality
Maximum Likelihood Estimate (a.k.a "best fit") using MINUIT ; (ii) profile
likelihoods, (iii) a new implementation of an Adaptive Metropolis MCMC
algorithm that relieves the burden of reconstructing the proposal distribution.
We present here those various statistical techniques and roll out a full
use-case that can then used as a tutorial. We revisit the CDM
parameters determination with the latest Planck data and give results with both
methodologies. Furthermore, by comparing the Bayesian and frequentist
approaches, we discuss a "likelihood volume effect" that affects the optical
reionization depth when analyzing the high multipoles part of the Planck data.
The software, used in several Planck data analyzes, is available from
http://camel.in2p3.fr. Using it does not require advanced C++ skills.Comment: Typeset in Authorea. Online version available at:
https://www.authorea.com/users/90225/articles/104431/_show_articl
On the minimization of Dirichlet eigenvalues of the Laplace operator
We study the variational problem \inf \{\lambda_k(\Omega): \Omega\
\textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \},
where is the 'th eigenvalue of the Dirichlet Laplacian
acting in , \h(\partial \Omega) is the - dimensional
Hausdorff measure of the boundary of , and is the Lebesgue
measure of . If , and , then there exists a convex
minimiser . If , and if is a minimiser,
then is also a
minimiser, and is connected. Upper bounds are
obtained for the number of components of . It is shown that if
, and then has at most components.
Furthermore is connected in the following cases : (i) (ii) and (iii) and (iv) and
. Finally, upper bounds on the number of components are obtained for
minimisers for other constraints such as the Lebesgue measure and the torsional
rigidity.Comment: 16 page
Relieving tensions related to the lensing of CMB temperature power spectra
The angular power spectra of the cosmic microwave background (CMB)
temperature anisotropies reconstructed from Planck data seem to present too
much gravitational lensing distortion. This is quantified by the control
parameter that should be compatible with unity for a standard cosmology.
With the Class Boltzmann solver and the profile-likelihood method, for this
parameter we measure a 2.6 shift from 1 using the Planck public
likelihoods. We show that, owing to strong correlations with the reionization
optical depth and the primordial perturbation amplitude , a
tension on also appears between the results obtained with
the low () and high () multipoles
likelihoods. With Hillipop, another high- likelihood built from Planck
data, this difference is lowered to . In this case, the value
is still in disagreement with unity by , suggesting a non-trivial
effect of the correlations between cosmological and nuisance parameters. To
better constrain the nuisance foregrounds parameters, we include the very high
measurements of the Atacama Cosmology Telescope (ACT) and South Pole
Telescope (SPT) experiments and obtain . The
Hillipop+ACT+SPT likelihood estimate of the optical depth is
which is now fully compatible with the low
likelihood determination. After showing the robustness of our results with
various combinations, we investigate the reasons for this improvement that
results from a better determination of the whole set of foregrounds parameters.
We finally provide estimates of the CDM parameters with our combined
CMB data likelihood.Comment: accepted by A&
Optimization problem for extremals of the trace inequality in domains with holes
We study the Sobolev trace constant for functions defined in a bounded domain
\O that vanish in the subset We find a formula for the first variation
of the Sobolev trace with respect to hole. As a consequence of this formula, we
prove that when \O is a centered ball, the symmetric hole is critical when we
consider deformation that preserve volume but is not optimal for some case.Comment: 13 page
About the connection between the power spectrum of the Cosmic Microwave Background and the Fourier spectrum of rings on the sky
In this article we present and study a scaling law of the CMB
Fourier spectrum on rings which allows us (i) to combine spectra corresponding
to different colatitude angles (e.g. several detectors at the focal plane of a
telescope), and (ii) to recover the power spectrum once the
coefficients have been measured. This recovery is performed numerically below
the 1% level for colatitudes degrees. In addition, taking
advantage of the smoothness of the and of the , we provide
analytical expressions which allow to recover one of the spectrum at the 1%
level, the other one being known.Comment: 8 pages, 8 figure
Maximizing Neumann fundamental tones of triangles
We prove sharp isoperimetric inequalities for Neumann eigenvalues of the
Laplacian on triangular domains.
The first nonzero Neumann eigenvalue is shown to be maximal for the
equilateral triangle among all triangles of given perimeter, and hence among
all triangles of given area. Similar results are proved for the harmonic and
arithmetic means of the first two nonzero eigenvalues
Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity
We present some open problems and obtain some partial results for spectral
optimization problems involving measure, torsional rigidity and first Dirichlet
eigenvalue.Comment: 18 pages, 4 figure
Approximation of the critical buckling factor for composite panels
This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented
A family of diameter-based eigenvalue bounds for quantum graphs
We establish a sharp lower bound on the first non-trivial eigenvalue of the
Laplacian on a metric graph equipped with natural (i.e., continuity and
Kirchhoff) vertex conditions in terms of the diameter and the total length of
the graph. This extends a result of, and resolves an open problem from, [J. B.
Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17
(2016), 2439--2473, Section 7.2], and also complements an analogous lower bound
for the corresponding eigenvalue of the combinatorial Laplacian on a discrete
graph. We also give a family of corresponding lower bounds for the higher
eigenvalues under the assumption that the total length of the graph is
sufficiently large compared with its diameter. These inequalities are sharp in
the case of trees.Comment: Substantial revision of v1. The main result, originally for the first
eigenvalue, has been generalised to the higher ones. The title has been
changed and the proofs substantially reorganised to reflect the new result,
and a section containing concluding remarks has been adde
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