Remez-Type Inequalities in the Complex Plane

We obtain a sharp upper bound on the modulus of a complex polynomial p(z) at any boundary point of a domain bounded by the Jordan curve without cusps. The area of the subset of G, where is known.     

Optimal approximation of elliptic problems by linear and nonlinear mappings III: Frames

We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space and is a bounded Lipschitz domain; the error is always measured in the H^s-norm. We study the order of convergence of the corresponding nonlinear frame widths and compare it with several other approximation schemes. Our main result is that the approximation order is the same as for the nonlinear widths associated with Riesz bases, the Gelfand widths, and the manifold widths. This order is better than the order of the linear widths iff p<2. The main advantage of frames compared to Riesz bases, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains—also for the upper bounds.     

Numbers with fixed sum of digits in linear recurrent number systems

We study the set of integers with a given sum of digits with respect to a linear recurrent digit system. An asymptotic formula for the number of integers ≤N with given sum of digits is determined, and the distribution in residue classes is investigated, thus generalizing results due to Mauduit and Sárközy. It turns out that numbers with fixed sum of digits are uniformly distributed in residue classes under some very general conditions. Namely, the underlying linear recurring sequence must have the property that there is no prime factor P of the modulus such that all but finitely many members of the sequence leave the same residue modulo P. The key step in the proof is an estimate for exponential sums using known theorems from Diophantine approximation.     

Impurity clustering and ferromagnetic interactions that are not carrier induced in dilute magnetic semiconductors: the case of Cu2O:Co

Current models for ferromagnetism in diluted magnetic semiconductors, such as "p-d exchange" or "double-exchange", rely on the presence of partially filled gap states. We point out a new mechanism, not requiring partially filled states, in which ferromagnetic coupling arises from the occupation of previously unoccupied levels when two transition metal impurities form a close pair. We find from first-principles calculations that this mechanism explains strong ferromagnetic coupling between Co impurities in Cu2O, and at the same time gives rise to Co clustering.     

Direct Fluorescence Monitoring of Coal Organic Matter Released in Seawater

Whenever immersed in seawater after a collier accident, a fossil fuel such as coal could become a source of pollution to the marine environment. To study the effect of such a contamination, four coal samples from different origins were used. A first analysis on those coals enabled us to determine the content of polycylic aromatic hydrocarbons. Seawater was then mixed with coal to study the organic matter released from coal into seawater. Fluorescence was used for its sensitivity to aromatic compounds, with the additional purpose of evaluating the relevance of using an immerse fluorescence probe to monitor water pollution. Excitation–emission matrices were recorded and the excitation–emission wavelength range corresponding to the highest fluorescence intensity was 230 nm/[370 nm; 420 nm]. The samples with coal happened to fluoresce more than the coal-free samples, the difference depending on the coal origin. The fluorescence intensity increased with coal mass, up to some limit. The particle size also influenced the fluorescence intensity, the finest particles releasing more fluorescing substances, due to their higher exchange surface. When seawater percolated through coal, the samples fluoresced highly at the beginning, and then the fluorescence intensity decreased and reached the seawater level. However, even with a 10 ns acquisition time shift, the fluorescence spectra were not specific enough to show the presence of PAHs in the samples, which were too diluted to be detected, whenever released from coal into seawater. The lifetimes of the seawater and of the coal samples were respectively 4.7 and 3.8 ns, indicating that the substances released from coal mainly consisted of short-lived fluorescing substances, such as natural humic or fulvic substances. Consequently, the presence of coal does not seem to be too detrimental to the marine environment, and a direct fluorescence probe could be used to monitor the seawater organic charge increase due to the immersion of coal in seawater.     

The superconducting specific heat jump of UBe13 and a narrow peak in the electronic density-of-states

The superconducting specific heat jump for the s-f; hybridization model of UBe₁₃ of Overhauser and Appel [2] is calculated. The narrow peak in the density-of-states is found to enhance the size of the jump relative to the BCS value of 1.43, but in weak coupling the increase is insufficient to agree with the measured value for UBe₁₃. The Kondo lattice model of Razafimandimby, Fulde and Keller [9] is considered as a possible modification, but in weak coupling a simple estimate shows that this cannot increase the specific heat jump significantly. It is speculated, however, that a consistent picture may still be possible with conventional strong coupling corrections.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.