Interpolation of nonlinear maps

Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c > 0$. For any $0<\theta <1$, denote by $X_\theta = [X_0, X_1]_\theta$ and $Y_\theta = [Y_0, Y_1]_\theta$ the complex interpolation spaces and by $B(r, X_\theta)$, $0 \le \theta \le 1,$ the open ball of radius $r>0$ in $X_\theta$, centered at zero. Then for any analytic map $\Phi: B(r, X_0) \to Y_0+ Y_1$ such that $\Phi: B(r, X_0)\to Y_0$ and $\Phi: B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $\Phi$ to $B(c^{-\theta}r, X_\theta)$, $0 < \theta < 1,$ is shown to be a map with values in $Y_\theta$ which is analytic and bounded by $M_0^{1-\theta} M_1^\theta$

Qualitative features of periodic solutions of KdV

In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree

On geodesic exponential maps of the Virasoro group

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metricsμ(k) (k≥ 0) on the Virasoro group Vir and show that for k≥ 2, but not for k=0,1, each of them defines a smooth Fréchet chart of the unital element e ∈Vir. In particular, the geodesic exponential map corresponding to the Korteweg-de Vries (KdV) equation (k=0) is not a local diffeomorphism near the origi

A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.Comment: This is an extended version of the paper that will appear in Math. Annalen. Some typos were corrected, references were updated, title was changed (as in the journal version). 31 page

Interpolation of nonlinear maps

Let (X 0, X 1) and (Y 0, Y 1) be complex Banach couples and assume that X 1 ⊆ X 0 with norms satisfying ‖x‖X 0 ≤ c‖x‖X 1 for some c > 0. For any 0 0 in X θ centered at zero. Then, for any analytic map Φ: B(r, X 0) → Y 0 + Y 1 such that Φ: B(r, X 0) → Y 0 and Φ: B(c −1 r, X 1) → Y 1 are continuous and bounded by constants M 0 and M 1, respectively, the restriction of Φ to B(c −θ r, X χ ), 0 < θ < 1, is shown to be a map with values in Y θ which is analytic and bounded by M 0 1 − θ M 1 θ

A few things I learnt from Jurgen Moser

A few remarks on integrable dynamical systems inspired by discussions with Jurgen Moser and by his work.Comment: An article for the special issue of "Regular and Chaotic Dynamics" dedicated to 80-th anniversary of Jurgen Mose

Investigation on Parameters Affecting the Effectiveness of Photocatalytic Functional Coatings to Degrade NO: TiO 2

This paper deals with the degradation of NO by photocatalytic oxidation using TiO2-based coatings. Tests are conducted at a laboratory scale through an experimental setup inspired from ISO 22197-1 standard. Various parameters are explored to evaluate their influence on photocatalysis efficiency: TiO2 dry matter content applied to the surface, nature of the substrate, and illumination conditions (UV and visible light). This article points out the different behaviors between three kinds of substrates which are common building materials: normalized mortar, denser mortar, and commercial wood. The illumination conditions are of great importance in the photocatalytic process with experiments under UV light showing the best results. However, a significant decrease in NO concentration under visible light is also observed provided that the TiO2 dry matter content on the surface is high enough. The nature of the substrate plays an important role in the photocatalytic activity with rougher substrates being more efficient to degrade NO. However, limiting the roughness of the substrate seems to be of utmost interest to obtain the highest exposed surface area and thus the optimal photocatalytic efficiency. A higher roughness promotes the surface contact between TiO2 and NO but does not necessarily increase the photochemical oxidation

Synthesis and Characterization of Rutile TiO2Nanopowders Doped with Iron Ions

Titanium dioxide nanopowders doped with different amounts of Fe ions were prepared by coprecipitation method. Obtained materials were characterized by structural (XRD), morphological (TEM and SEM), optical (UV/vis reflection and photoluminescence, and Raman), and analytical techniques (XPS and ICP-OES). XRD analysis revealed rutile crystalline phase for doped and undoped titanium dioxide obtained in the same manner. Diameter of the particles was 5–7 nm. The presence of iron ions was confirmed by XPS and ICP-OES. Doping process moved absorption threshold of TiO2into visible spectrum range. Photocatalytic activity was also checked. Doped nanopowders showed normal and up-converted photoluminescence

Unemployment benefits : discursive convergence, distant realities

Production of INCASI Project H2020-MSCA-RISE-2015 GA 691004Unemployment protection systems have certain characteristics in common in Argentina, Uruguay, Spain and Italy: they are compulsory and contributory-proportional, although in Uruguay, it also has a capitalisation supplement. Despite the similarities, they work differently because the context of informal employment chiefly, and unemployment, low salaries and precariousness differ greatly. Consequently, the unemployment protection coverage rate varies. Theories of the Active Welfare State, the Investor State and the reforms of unemployment protection systems have led to a certain modernising language being adopted in these countries: activation, employability, conditionality, lifelong learning, flexibility, which are, among others, words shared with Europe. However, the meanings of these words differ according to the institutional context of each country. In Latin America the welfare state is low institutionalised even almost non-existent, while in Europe it is a diverse institution. Despite this, the four countries share an upward trend in benefit policies, in accordance with the increase in poverty risk