Global exponential convergence to variational traveling waves in cylinders

We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for "pushed" fronts.Comment: 23 page

Interaction of vortices in viscous planar flows

We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter \nu, and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex system is well-posed on the interval [0,T]. Under these assumptions, we prove that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This allows to estimate the self-interactions, which play an important role in the convergence proof.Comment: 39 pages, 1 figur

Phase Slips and the Eckhaus Instability

We consider the Ginzburg-Landau equation, $\partial_t u= \partial_x^2 u + u - u|u|^2$, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it global} theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.Comment: 22 pages, Postscript, A

Orbital stability of periodic waves for the nonlinear Schroedinger equation

The nonlinear Schroedinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure

Renormalizing Partial Differential Equations

In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]

Probability density of quantum expectation values

We consider the quantum expectation value \mathcal{A}=\ of an observable A over the state |\psi\> . We derive the exact probability distribution of \mathcal{A} seen as a random variable when |\psi\> varies over the set of all pure states equipped with the Haar-induced measure. The probability density is obtained with elementary means by computing its characteristic function, both for non-degenerate and degenerate observables. To illustrate our results we compare the exact predictions for few concrete examples with the concentration bounds obtained using Levy's lemma. Finally we comment on the relevance of the central limit theorem and draw some results on an alternative statistical mechanics based on the uniform measure on the energy shell.Comment: Substantial revision. References adde

Human papillomavirus (HPV) contamination of gynaecological equipment.

OBJECTIVE: The gynaecological environment can become contaminated by human papillomavirus (HPV) from healthcare workers' hands and gloves. This study aimed to assess the presence of HPV on frequently used equipment in gynaecological practice. METHODS: In this cross-sectional study, 179 samples were taken from fomites (glove box, lamp of a gynaecological chair, gel tubes for ultrasound, colposcope and speculum) in two university hospitals and in four gynaecological private practices. Samples were collected with phosphate-buffered saline-humidified polyester swabs according to a standardised pattern, and conducted twice per day for 2 days. The samples were analysed by a semiquantitative real-time PCR. Statistical analysis was performed using Pearson's χ(2) test and multivariate regression analysis. RESULTS: Thirty-two (18%) HPV-positive samples were found. When centres were compared, there was a higher risk of HPV contamination in gynaecological private practices compared with hospitals (OR 2.69, 95% CI 1.06 to 6.86). Overall, there was no difference in the risk of contamination with respect to the time of day (OR 1.79, 95% CI 0.68 to 4.69). When objects were compared, the colposcope had the highest risk of contamination (OR 3.02, 95% CI 0.86 to 10.57). CONCLUSIONS: Gynaecological equipment and surfaces are contaminated by HPV despite routine cleaning. While there is no evidence that contaminated surfaces carry infectious viruses, our results demonstrate the need for strategies to prevent HPV contamination. These strategies, based on health providers' education, should lead to well-established cleaning protocols, adapted to gynaecological rooms, aimed at eliminating HPV material

Modulational Instability in Equations of KdV Type

It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly vary in large space and time scales. In the 1970's, Whitham developed an asymptotic (WKB) method to study the effects of small "modulations" on nonlinear periodic wave trains. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham's formal theory. We discuss recent advances in the mathematical understanding of the dynamics, in particular, the instability of slowly modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic

Coherent vortex structures and 3D enstrophy cascade

Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities - 1/2-H\"older coherence of the vorticity direction - coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.Comment: 15 pp; final version -- to appear in CM

Why dig looted tombs? Two examples and some answers from Keushu (Ancash highlands, Peru)

Looted tombs at Andean archaeological sites are largely the result of a long tradition of trade in archaeological artefacts coupled with the 17th century policy of eradicating ancestor veneration and destroying mortuary evidence in a bid to “extirpate idolatry”. On the surface, looted funerary contexts often present abundant disarticulated and displaced human remains as well as an apparent absence of mortuary accoutrements. What kind of information can archaeologists and biological anthropologists hope to gather from such contexts? In order to gauge the methodological possibilities and interpretative limitations of targeting looted tombs, we fully excavated two collective funerary contexts at the archaeological site of Keushu (district and province of Yungay, Ancash, Peru; c. 2000 B.C.-A.D. 1600), which includes several dozen tombs, many built under large boulders or rock shelters, all of which appear disturbed by looting. The first is located in the ceremonial sector and excavation yielded information on four individuals; the second, in the funerary and residential sector, held the remains of seventy individuals - adults and juveniles. Here, we present and discuss the recovered data and suggest that careful, joint excavations by archaeologists and biological anthropologists can retrieve evidence of past mortuary practices, aid the biological characterisation of mortuary populations and help distinguish between a broad range of looting practices and post-depositional processes