Critical circle maps near bifurcation

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 1/3. Moreover, the rotation number is a Hölder continuous function of the parameter.Partially supported by KBN grant Iteracje i Fraktale #210909101.Partially supported by NSF Grant #DMS-9206793 and the Sloan Research Fellowship.     

An Asymptotic Ratio in the Complete Binary Tree

Let T _n be the complete binary tree of height n considered as the Hasse-diagram of a poset with its root 1 _n as the maximum element. For a rooted tree T, define two functions counting the embeddings of T into T _n as follows A(n;T)=|{S T _n  : 1 _n ∈S, S≅T}|, and B(n;T)=|{S T _n :1 _n ∉S, S≅T}|. In this paper we investigate the asymptotic behavior of the ratio A(n;T)/B(n;T), and we show that lim  _n→∞[A(n;T)/B(n;T)]=2^ℓ;−1−1, for any tree T with ℓ leaves. This revised version was published online in June 2006 with corrections to the Cover Date.     

Complexity of Weighted Approximation over

We study approximation of univariate functions defined over the reals. We assume that the rth derivative of a function is bounded in a weighted L_p norm with a weight ψ. Approximation algorithms use the values of a function and its derivatives up to order r−1. The worst case error of an algorithm is defined in a weighted L_q norm with a weight ρ. We study the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of function and derivative evaluations needed to obtain error . We provide necessary and sufficient conditions in terms of the weights ψ and ρ, as well as the parameters r, p, and q for the weighted approximation problem to have finite complexity. We also provide conditions which guarantee that the complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted integration problem since its complexity is equivalent to the complexity of the weighted approximation problem for q=1.     

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chieX Chandler Davis.     

Uniqueness of minimal Fourier-type extensions in L_1-spaces

We give a characterization of uniqueness of finite rank Fourier-type minimal extensions in $L_1$ -norm. This generalizes the main result obtained by Lewicki (Proceedings of the Fifth International Conference on Function Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 213, pp. 337–345, 1998) to the case of $n$ -circular sets in $\mathbb{C }^n$ .     

Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains

In this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. Finally, we prove the existence of a pullback attractor for a non-autonomous 2D Navier–Stokes model in an unbounded domain, a case in which the theory of uniform attractors does not work since the non-autonomous term is quite general. To cite this article: T. Caraballo et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Improving the dielectric constant of Al2O3 by cerium substitution for high-k MIM applications

Process compatible high-k dielectric thin films are one of the key solutions to develop high performance metal–insulator–metal (MIM) structures for future microelectronic devices. Engineered cerium–aluminate (Ce_xAl_2–xO₃) thin films were deposited on titanium nitride metal electrodes by electron-beam co-evaporation of ceria and alumina in a molecular beam deposition chamber. X-ray photoelectron spectroscopy clearly reveals that Ce cations can be stabilized in the 3+ valence state in Ce_xAl_2–xO₃ up to x = 0.7 by accommodation in the alumina host matrix. Higher Ce content was observed to result in cerium dioxide segregation in cerium aluminate matrix, probably due to the chemical tendency of Ce cations to exist rather in the 4+ than in the 3+ state. Electrical characterization of the X-ray amorphous Ce_0.7Al_1.3O₃ films reveals a dielectric constant value of about 11 and leakage current lower than 10^?4 A/cm². No parasitic low-k interface formation between the high-k Ce_0.7Al_1.3O₃ film and the TiN metal electrode is detected.     

The power of adaptive algorithms for functions with singularities

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n^−r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n⁻¹ for integration and n^−1/p for the L^p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n^−r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L^∞ approximation. However, they hold for the Skorohodmetric, which we argue to be more appropriate than L^∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed.     

A Study of the Activity of Recombinant Mn-Superoxide Dismutase in the Presence of Gold and Silver Nanoparticles

Applied Biochemistry and Biotechnology - Superoxide dismutase (SOD) is one of the best characterized enzyme maintaining the redox state in the cell. A bacterial expression system was used to...