Problems associated with interferences in the analysis of serum for polychlorinated biphenyls

During a recent survey to determine serum concentrations of polychlorinated biphenyls (PCBs) among people living around New Bedford, MA, U.S.A., an unidentified contaminant precluded the quantification of some early eluting Webb and McCall peaks. Loss of data is estimated to have reduced reported serum levels by 12%. Efforts to identify the contaminant by gas chromatography with an electron-capture detector, a Hall electrolytic condutivity detector, and mass spectrometer were not successful. Researchers ascertained, however, that the contaminant is not a PCB, it does not contain halogens, but it may contain phthalates. Vacutainer tubes and closures for serum storage bottles are suspected sources of contamination.     

A theory of anti-selfdual Lagrangians: dynamical case

We consider the class of time-dependent anti-selfdual Lagrangians, which – just like the stationary case announced in Ghoussoub [C. R. Acad. Sci. Paris, Ser. I 340 (2005)] – enjoys remarkable permanence properties and provides variational formulations and resolutions for several initial-value parabolic equations including gradient flows and other dissipative systems. Even though these evolutions do not fit in the standard Euler–Lagrange theory, we show that their solutions can be obtained as minima – but also as zeroes – of action functionals of the form ${\int }_0^{T}L(t\text{,}u(t)\text{,}\dot{u}(t)+{\Lambda }_{t}u(t)\mathrm{d}t)$ where L is a time-dependent anti-selfdual Lagrangian and where ${\Lambda }_t$ is a flow of skew-adjoint operators. To cite this article: N. Ghoussoub, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Optimal mass transport and symmetric representations of their cost functions

We consider Monge–Kantorovich problems corresponding to general cost functions \(c(x,y)\) but with symmetry constraints on a Polish space \(X\times X\) . Such couplings naturally generate anti-symmetric Hamiltonians on \(X\times X\) that are \(c\) -convex with respect to one of the variables. In particular, if \(c\) is differentiable with respect to the first variable on an open subset \(X\) in \( \mathbb {R}^d\) , we show that for every probability measure \(\mu \) on \(X\) , there exists a symmetric probability measure \(\pi _0\) on \(X\times X\) with marginals \(\mu \) , and an anti-symmetric Hamiltonian \(H\) such that \(\nabla _2H(y, x)=\nabla _1c(x,y)\) for \( \pi _0\) -almost all \((x,y) \in X \times X.\) If \(\pi _0\) is supported on a graph \((x, Sx)\) , then \(S\) is necessarily a \(\mu \) -measure preserving involution (i.e., \(S^2=I\) ) and \(\nabla _2H(x, Sx)=\nabla _1c(Sx,x)\) for \(\mu \) -almost all \(x \in X.\) For monotone cost functions such as those given by \(c(x,y)=\langle x, u(y)\rangle \) or \(c(x,y)=-|x-u(y)|^2\) where \(u\) is a monotone operator, \(S\) is necessarily the identity yielding a classical result by Krause, namely that \(u(x)=\nabla _2H(x, x)\) where \(H\) is anti-symmetric and concave-convex.     

Ionic diffusion on a lattice: Effects of the order-disorder transition on the dynamics of non-equilibrium systems

We examine the behaviour of the concentration profiles of particles with repulsive interactions diffusing on a host lattice. At low temperature, the diffusion process is strongly influenced by the presence of ordered domains. We use mean field equations and Monte-Carlo simulations to describe the various effects which influence the kinetic behaviour. An effective diffusion coefficient is determined analytically and is compared with the simulations. Finite gradient effects on the ordered domains and on the diffusion are discussed. The kinetics studied is relevant for superionic conductors, for intercalation and also for the diffusion of particles adsorbed on a substrate. Received: 26 June 1997 / Revised: 18 September 1997 / Accepted: 10 November 1997     

Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows

Anti-selfdual Lagrangians on a state space lift to path space provided one adds a suitable selfdual boundary Lagrangian. This process can be iterated by considering the path space as a new state space for the newly obtained anti-selfdual Lagrangian. We give here two applications for these remarkable permanence properties. In the first, we establish for certain convex–concave Hamiltonians on a–possibly infinite dimensional–symplectic space H ², the existence of a solution for the Hamiltonian system that connects in a given time T > 0, two Lagrangian submanifolds. Another application deals with the construction of multiparameter flows, including those generated by vector fields that represent superpositions of skew-adjoint operators with gradients of convex potentials. Our methods are based on the new variational calculus for anti-selfdual Lagrangians developed in [5–7].     

Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term. Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal where this work was initiated. Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering Research Council of Canada.     

Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem ? Δu = λf(x)/(1 ? u)² on a bounded domain Ω ? ?^N, which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of 11 relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f. We also show the remarkable fact that powerlike profiles f(x) ? |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N ≥ 8 and as long as As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp. © 2007 Wiley Periodicals, Inc.     

In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography

An ultrahigh-speed spectral domain optical coherence tomography (SD-OCT) system is presented that achieves acquisition rates of 29,300 depth profiles/s. The sensitivity of SD-OCT and time domain OCT (TD-OCT) are experimentally compared, demonstrating a 21.7-dB improvement of SD-OCT over TD-OCT. In vivo images of the human retina are presented, demonstrating the ability to acquire high-quality structural images with an axial resolution of 6 microm at ultrahigh speed and with an ocular exposure level of less than 600 microW.