Angular Symmetry Breaking Induced by Electromagnetic Field

It is well known that velocities does not commute in presence of an electromagnetic field. This property implies that angular algebra symmetries, such as the sO(3) and Lorentz algebra symmetries, are broken. To restore these angular symmetries we show the necessity of adding the Poincare momentum M to the simple angular momentum L. These restorations performed succesively in a flat space and in a curved space lead in each cases to the generation of a Dirac magnetic monopole. In the particular case of the Lorentz algebra we consider an application of our theory to the gravitoelectromagnetism. In this last case we establish a qualitative relation giving the mass spectrum for dyons.Comment: 19 page

Inverse spectral positivity for surfaces

Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $\Delta + aK + W$, where $\Delta$ is the non-negative Laplacian, $K$ the Gaussian curvature, $W$ a locally integrable function, and $a$ a positive real number. Assuming that the positive part of $W$ is integrable, we address the question "What conclusions on $(M,g)$ and $W$ can one draw from the fact that the operator $\Delta + aK + W$ is non-negative ?" As a consequence of our main result, we get a new proof of Huber's theorem and Cohn-Vossen's inequality, and we improve earlier results in the particular cases in which $W$ is non-positive and $a = 1/4$ or $a \in (0,1/4)$

The weak Pleijel theorem with geometric control

Let $\Omega\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $\lambda\_j(\Omega), j\geq 1$, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues $\lambda\_j(\Omega)$, for which there exists an associated eigenfunction with precisely $j$ nodal domains (Courant-sharp eigenvalues), is finite. The purpose of this note is to determine an upper bound for Courant-sharp eigenvalues, expressed in terms of simple geometric invariants of $\Omega$. We will see that this is connected with one of the favorite problems considered by Y. Safarov.Comment: Revised Oct. 12, 2016. To appear in Journal of Spectral Theory 6 (2016

A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a natural continuation of our critical reading of A. Stern's results for Dirichlet eigenfunctions in the square, see arXiv:14026054.Comment: Accepted for publication in "Monatshefte f{\"u}r Mathematik

Coupling from the past times with ambiguities and perturbations of interacting particle systems

We discuss coupling from the past techniques (CFTP) for perturbations of interacting particle systems on the d-dimensional integer lattice, with a finite set of states, within the framework of the graphical construction of the dynamics based on Poisson processes. We first develop general results for what we call CFTP times with ambiguities. These are analogous to classical coupling (from the past) times, except that the coupling property holds only provided that some ambiguities concerning the stochastic evolution of the system are resolved. If these ambiguities are rare enough on average, CFTP times with ambiguities can be used to build actual CFTP times, whose properties can be controlled in terms of those of the original CFTP time with ambiguities. We then prove a general perturbation result, which can be stated informally as follows. Start with an interacting particle system possessing a CFTP time whose definition involves the exploration of an exponentially integrable number of points in the graphical construction, and which satisfies the positive rates property. Then consider a perturbation obtained by adding new transitions to the original dynamics. Our result states that, provided that the perturbation is small enough (in the sense of small enough rates), the perturbed interacting particle system too possesses a CFTP time (with nice properties such as an exponentially decaying tail). The proof consists in defining a CFTP time with ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed dynamics. Finally, we discuss examples of particle systems to which this result can be applied. Concrete examples include a class of neighbor-dependent nucleotide substitution model, and variations of the classical voter model, illustrating the ability of our approach to go beyond the case of weakly interacting particle systems.Comment: This paper is an extended and revised version of an earlier manuscript available as arXiv:0712.0072, where the results were limited to perturbations of RN+YpR nucleotide substitution model

Gain scheduled flight control law for a flexible aircraft : A practical approach

Abstract: This paper presents a gain-scheduling method applied to flight control law design. The method is a stabilitypreservinginterpolation technique ofexisting controllers under observer-state feedback form. Application is made on a flexible civil aircraft example considering multiple scheduling parameters. Although the interpolation technique gives powerful a priori stability guarantees, the sufficient condition to satisfy leads to conservative results in practice. We thus use a fixed observer model and check stability andperformance thanks to μ-analysis. Provided results are really satisfactory for a final controller of little complexity

Coupling times with ambiguities for particle systems and applications to context-dependent DNA substitution models

We define a notion of coupling time with ambiguities for interacting particle systems, and show how this can be used to prove ergodicity and to bound the convergence time to equilibrium and the decay of correlations at equilibrium. A motivation is to provide simple conditions which ensure that perturbed particle systems share some properties of the underlying unperturbed system. We apply these results to context-dependent substitution models recently introduced by molecular biologists as descriptions of DNA evolution processes. These models take into account the influence of the neighboring bases on the substitution probabilities at a site of the DNA sequence, as opposed to most usual substitution models which assume that sites evolve independently of each other.Comment: 33 page

Fluctuations of the front in a one-dimensional model for the spread of an infection

We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent nearest-neighbor continuous-time symmetric random walks on the integer lattice $\mathbb{Z}$ with jump rates $D_R$ for red particles and $D_B$ for blue particles, the interaction rule being that blue particles turn red upon contact with a red particle. The initial condition consists of i.i.d. Poisson particle numbers at each site, with particles at the left of the origin being red, while particles at the right of the origin are blue. We are interested in the dynamics of the front, defined as the rightmost position of a red particle. For the case $D_R=D_B$, Kesten and Sidoravicius established that the front moves ballistically, and more precisely that it satisfies a law of large numbers. Their proof is based on a multi-scale renormalization technique, combined with approximate sub-additivity arguments. In this paper, we build a renewal structure for the front propagation process, and as a corollary we obtain a central limit theorem for the front when $D_R=D_B$. Moreover, this result can be extended to the case where $D_R>D_B$, up to modifying the dynamics so that blue particles turn red upon contact with a site that has previously been occupied by a red particle. Our approach extends the renewal structure approach developed by Comets, Quastel and Ram\'{{\i}}rez for the so-called frog model, which corresponds to the $D_B=0$ case.Comment: Published at http://dx.doi.org/10.1214/15-AOP1034 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org