A proof of the bounded graph conjecture

Summary An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. We prove an old conjecture of Halin, which characterizes the bounded graphs in terms of four forbidden topological subgraphs.Oblatum 17-IV-1991 & 25-X-1991     

On the normal form of a system of differential equations with nilpotent linear part

We consider prenormal forms associated to generic perturbations of the system $\text{x}\text{?}\text{=2y,}\text{y}\text{?}\text{=3x}{}^2$. It is known that they have a formal normal form $\text{x}\text{?}\text{=2y+2x}\text{Δ}{}^1\text{,}\text{y}\text{?}\text{=3x}{}^2\text{+3y}\text{Δ}{}^1$, where $\text{Δ}{}^1\text{=x+A}{}_0\text{(y}{}^2\text{?x}{}^3\text{)}$ [Differential Equations 158 (1) (1999) 152–173]. We show that the series A₀ and the normalizing transformations are divergent, but 1-summable. To cite this article: M. Canalis-Durand, R. Schäfke, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A note on the reflection principle for the biharmonic equation and the Stokes system

In this note, we discuss the reflection principle of the Stokes system in a half space for the threedimensional case, and of the biharmonic equation. Admitting different boundary conditions, we use the reflection principle to prove uniqueness of solutions of the Stokes system or the biharmonic equation in weightedLq-spaces     

The stationary and non-stationary stokes system in exterior domains with non-zero divergence and non-zero boundary values

In an exterior domain Ω??ⁿ, n ? 2, we consider the generalized Stokes resolvent problem in L^q-space where the divergence g = div u and inhomogeneous boundary values u = ψ with zero flux ∫_?Ωψ·N do = 0 may be prescribed. A crucial step in our approach is to find and to analyse the right space for the divergence g. We prove existence, uniqueness and a priori estimates of the solution and get new results for the divergence problem. Further, we consider the non-stationary Stokes system with non-homogeneous divergence and boundary values and prove estimates of the solution in L⁵(0, T;L^q(Ω)) for 1 < s, q < ∞.     

Cogenerators for convex spaces

We give cogenerators for the categories of convex (= finitely superconvex), finitely positively convex, and absolute convex (= finitely totally convex) spaces introduced by Pumplün and Röhrl.Dedicated to our academic teacher Dieter Pumplün on the occasion of his sixtieth birthday.     

High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip

We show that a concentration of light at a metal tip allows near-field optical imaging of single fluorescent dye molecules at very high resolution, despite strong quenching effects. Details as small as 10 nm were observed in the fluorescence patterns of single Cy-3 dyes bound to the termini of DNA. Data evaluation by model fitting determines the positions of the dyes to an accuracy even better than 1 nm and also yields their 3D orientation. The metal tip simultaneously provides high-resolution topographic imaging complementing the optical signal for a detailed surface examination.     

Waveform-controlled near-single-cycle milli-joule laser pulses generate sub-10 nm extreme ultraviolet continua

We demonstrate the generation of waveform-controlled laser pulses with 1?mJ pulse energy and a full-width-half-maximum duration of ～4 fs, therefore lasting less than two cycles of the electric field oscillating at their carrier frequency. The laser source is carrier-envelope-phase stabilized and used as the backbone of a kHz repetition rate source of high-harmonic continua with unprecedented flux at photon energies between 100 and 200?eV (corresponding to a wavelength range between 12-6?nm respectively). In combination we use these tools for the complete temporal characterization of the laser pulses via attosecond streaking spectroscopy.     

Rigidity percolation in particle-laden foams

We study the viscoelastic behavior of aqueous foam mixed with solid noncolloidal particles. We show that adding a tiny amount of grains can enhance the elastic and loss shear moduli by more than 1 order of magnitude. The scaling of these moduli with solid volume fraction is in qualitative agreement with that predicted by an effective-medium rigidity percolation model. We present a simple model, based on capillary attraction, to explain the particle-size dependence of the threshold.