8,662 research outputs found
A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain
The 3D incompressible Euler equation is an important research topic in the
mathematical study of fluid dynamics. Not only is the global regularity for
smooth initial data an open issue, but the behaviour may also depend on the
presence or absence of boundaries.
For a good understanding, it is crucial to carry out, besides mathematical
studies, high-accuracy and well-resolved numerical exploration. Such studies
can be very demanding in computational resources, but recently it has been
shown that very substantial gains can be achieved first, by using Cauchy's
Lagrangian formulation of the Euler equations and second, by taking advantages
of analyticity results of the Lagrangian trajectories for flows whose initial
vorticity is H\"older-continuous. The latter has been known for about twenty
years (Serfati, 1995), but the combination of the two, which makes use of
recursion relations among time-Taylor coefficients to obtain constructively the
time-Taylor series of the Lagrangian map, has been achieved only recently
(Frisch and Zheligovsky, 2014; Podvigina {\em et al.}, 2016 and references
therein).
Here we extend this methodology to incompressible Euler flow in an
impermeable bounded domain whose boundary may be either analytic or have a
regularity between indefinite differentiability and analyticity.
Non-constructive regularity results for these cases have already been obtained
by Glass {\em et al.} (2012). Using the invariance of the boundary under the
Lagrangian flow, we establish novel recursion relations that include
contributions from the boundary. This leads to a constructive proof of
time-analyticity of the Lagrangian trajectories with analytic boundaries, which
can then be used subsequently for the design of a very high-order
Cauchy--Lagrangian method.Comment: 18 pages, no figure
Functional Multi-Layer Perceptron: a Nonlinear Tool for Functional Data Analysis
In this paper, we study a natural extension of Multi-Layer Perceptrons (MLP)
to functional inputs. We show that fundamental results for classical MLP can be
extended to functional MLP. We obtain universal approximation results that show
the expressive power of functional MLP is comparable to that of numerical MLP.
We obtain consistency results which imply that the estimation of optimal
parameters for functional MLP is statistically well defined. We finally show on
simulated and real world data that the proposed model performs in a very
satisfactory way.Comment: http://www.sciencedirect.com/science/journal/0893608
On the nonexistence of quasi-Einstein metrics
We study complete Riemannian manifolds satisfying the equation by studying the associated PDE for . By developing a gradient estimate for , we show
there are no nonconstant solutions. We then apply this to show that there are
no nontrivial Ricci flat warped products with fibers which have nonpositive
Einstein constant. We also show that for nontrivial steady gradient Ricci
solitons, the quantity is a positive constant.Comment: Final version: Improved exposition of Section 2, corrected minor
typo
Stable variable selection for right censored data: comparison of methods
The instability in the selection of models is a major concern with data sets
containing a large number of covariates. This paper deals with variable
selection methodology in the case of high-dimensional problems where the
response variable can be right censored. We focuse on new stable variable
selection methods based on bootstrap for two methodologies: the Cox
proportional hazard model and survival trees. As far as the Cox model is
concerned, we investigate the bootstrapping applied to two variable selection
techniques: the stepwise algorithm based on the AIC criterion and the
L1-penalization of Lasso. Regarding survival trees, we review two
methodologies: the bootstrap node-level stabilization and random survival
forests. We apply these different approaches to two real data sets. We compare
the methods on the prediction error rate based on the Harrell concordance index
and the relevance of the interpretation of the corresponding selected models.
The aim is to find a compromise between a good prediction performance and ease
to interpretation for clinicians. Results suggest that in the case of a small
number of individuals, a bootstrapping adapted to L1-penalization in the Cox
model or a bootstrap node-level stabilization in survival trees give a good
alternative to the random survival forest methodology, known to give the
smallest prediction error rate but difficult to interprete by
non-statisticians. In a clinical perspective, the complementarity between the
methods based on the Cox model and those based on survival trees would permit
to built reliable models easy to interprete by the clinician.Comment: nombre de pages : 29 nombre de tableaux : 2 nombre de figures :
A remark on Einstein warped products
We prove triviality results for Einstein warped products with non-compact
bases. These extend previous work by D.-S. Kim and Y.-H. Kim. The proof, from
the viewpoint of "quasi-Einstein manifolds" introduced by J. Case, Y.-S. Shu
and G. Wei, rely on maximum principles at infinity and Liouville-type theorems.Comment: 12 pages. Corrected typos. Final version: to appear on Pacific J.
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