Regularity at space-like and null infinity

We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino-Schoen/Chrusciel-Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ``Morawetz vector field'' to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino-Schoen/Chrusciel-Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows large classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino-Schoen/Chrusciel-Delay space-times

Null controllability of one-dimensional parabolic equations by the flatness approach

We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which providesexplicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u\_t$.The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u\_x)\_x-u\_t=0$ with a(x)\textgreater{}0 for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u\_{xx}+(\mu / |x|^2)u-u\_t=0$with $\mu\ge 1/4$

On the reachable states for the boundary control of the heat equation

We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable controls can be extended to analytic functions onsome square of C, and conversely, that analytic functions defined on a certain disk can be reached by using boundary controlsthat are Gevrey functions of order 2. The method of proof combines the flatness approach with some new Borel interpolation theorem in some Gevrey class witha specified value of the loss in the uniform estimates of the successive derivatives of the interpolating function

A design-for-casting integrated approach based on rapid simulation and modulus criterion

This paper presents a new approach to the design of cast components and their associated tools. The current methodology is analysed through a case study and its main disadvantages underlined. Then, in order to overcome these identified drawbacks, a new approach is proposed. Knowing that this approach is mainly based on a rapid simulation of the process, basics of a simplified physical model of solidification are presented as well as an associated modulus criterion. Finally, technical matters for a software prototype regarding the implementation of this Rapid Simulation Approach (RSA) in a CAD environment are detailed

Null controllability of the 1D heat equation using flatness

We derive in a straightforward way the null controllability of a 1-D heat equation with boundary control. We use the so-called {\em flatness approach}, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control law achieving the exact steering to zero. We also give accurate error estimates when the various series involved are replaced by their partial sums, which is paramount for an actual numerical scheme. Numerical experiments demonstrate the relevance of the approach

Controllability of the 1D Schrodinger equation by the flatness approach

We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control input achieving the exact controllability in the energy space. As an application, we derive an explicit pair of control inputs achieving the exact steering to zero for a simply-supported beam

Gravitational Wave Polarization Modes in f(R)f(R) Theories

Many studies have been carried out in the literature to evaluate the number of polarization modes of gravitational waves in modified theories, in particular in $f(R)$ theories. In the latter ones, besides the usual two transverse-traceless tensor modes present in general relativity, there are two additional scalar ones: a massive longitudinal mode and a massless transverse mode (the so-called breathing mode). This last mode has often been overlooked in the literature, due to the assumption that the application of the Lorenz gauge implies transverse-traceless wave solutions. We however show that this is in general not possible and, in particular, that the traceless condition cannot be imposed due to the fact that we no longer have a Minkowski background metric. Our findings are in agreement with the results found using the Newman-Penrose formalism, and thus clarify the inconsistencies found so far in the literature.Comment: 7 pages; accepted for publication in Phys. Rev.

Generating constrained random graphs using multiple edge switches

The generation of random graphs using edge swaps provides a reliable method to draw uniformly random samples of sets of graphs respecting some simple constraints, e.g. degree distributions. However, in general, it is not necessarily possible to access all graphs obeying some given con- straints through a classical switching procedure calling on pairs of edges. We therefore propose to get round this issue by generalizing this classical approach through the use of higher-order edge switches. This method, which we denote by "k-edge switching", makes it possible to progres- sively improve the covered portion of a set of constrained graphs, thereby providing an increasing, asymptotically certain confidence on the statistical representativeness of the obtained sample.Comment: 15 page

Patterned ferrimagnetic thin films of spinel ferrites obtained directly by laser irradiation

Some spinel ferrites can be oxidized or transformed at moderate temperatures. Such modifications werecarried out on thin films of mixed cobalt copper ferrites and maghemite, by heating small regions with alow-power laser spot applied for about 100 ns. The very simple laser heating process, which can be donedirectly with a conventional photolithographic machine, made it possible to generate two-dimensionalmagnetization heterogeneities in ferrimagnetic films. Such periodic structures could display the specificproperties of magneto-photonic or magnonic crystals