SharpL ∞-error estimates for semilinear elliptic problems with free boundaries

Summary A numerical scheme to approximate a semilinear PDE involving a (singular) maximal monotone graph is analyzed inL . A preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute; the discrete maximum principle is thus avoided. Sharp pointwise error estimates are derived for both the smoothing and the discretization procedures. An optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence. These error estimates for solutions, in conjunction with nondegeneracy properties of continuous problems, provide sharp interface error estimates. Two model examples are discussed: the obstacle problem and a combustion equation.This work was partially supported by Consiglio Nazionale delle Ricerche of Italy while the author was in residence at the Istituto di Analisi Numerica del C.N.R. di Pavia     

The Markov structure of population growth

Some problems concerning the development of general models of population growth are examined, with particular reference to Markovian modelling. The tenet of the paper is that such models can be more realistically attempted by focusing on temporal rather than spatial modifications. Specifically, the time concept considered is linked to the dependent structure inherent in the partial order of descent from mother to child. The author attempts to develop "a general theory of populations of individuals under what might be called free reproduction. Hence, the only dependence assumed between individuals is that from mothers to children." He suggests that the results "can be translated into assertions about evolution in real, physical time and also about the final, stable or balanced, composition of populations, over ages, types, family structure, and many other aspects of populations."     

Symmetric Intersections of Rauzy Fractals

In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions.     

The hydrogen-ion selective glass electrode

The properties and mechanism of operation of hydrogen-ion selective glass electrodes are reviewed and a model using information from recent research is presented. In the gel layer of a pH-glass protons or hydronium ions are bound to negative charges in a silicon network cross-linked with bi- or tervalent metal ions. The composition of the gel layer is discussed; it is known that the outer part contains water. The formation of a gel layer and its destruction in alkaline solutions are discussed. Between the gel layer and the dry glass there is a transition layer characterized by a very high resistivity. This forms a barrier to ion transport and prevents rapid corrosion of the glass. The ionic mobilities of various ions change drastically with distance inwards towards the dry glass. The alkaline and acid errors are discussed as well as the asymmetry potential. By etching away the gel layers and the transition layers on both sides of the glass and then starting hydration simultaneously on both sides, two symmetric layers can be produced. This will result in an almost complete cancellation of the asymmetry potential. Some applications to precision measurements and measurement in non-aqueous solvents are discussed.     

The electrical performance of thinn +-window layers ina-Si: H solar cells

The electrical properties ofn ⁺-window layers inp-i-n a-Si:H solar cells were characterised as a function ofn ⁺-layer thickness, , by measuring firstly the activation energyE _a of the dark conductivity and secondly the built-in potentialV _bi of the cells.E _a was found to increase with decreasing attaining values as high as 0.8 eV for 5nm; bulk values, e.g.E _a . 2eV in the amorphous andE _a<0.01 eV in the microcrystalline case, were only observed for >20nm and for >200nm, respectively. In contrast,V _bi did not depend on at all and was further found to be consistent with expectations based on the Fermi level positions in bulkn ⁺ andp ⁺-material. As a consequenceE _a in very thin films can no longer be considered as a measure of (E _C –E _F), the distance of the Fermi level from the conduction band edge. The apparent inconsistency inherent to theE _a and theV _bi results can be resolved by assuming that the deposition of then ⁺-material proceeds via the growth and coalescence of small islands.     

Formation and properties of the Fe<Subscript>8</Subscript>V<Subscript>10</Subscript>W<Subscript>16–x</Subscript>Mo<Subscript>x</Subscript>O<Subscript>85</Subscript> type solid solution

Solid solution phases of a formula Fe₈V₁₀W_16–xMo_xO₈₅ where 0≤x≤4, have been obtained, possessing a structure of the compound Fe₈V₁₀W₁₆O₈₅. It was found on the base of XRD and DTA investigations that these solution phases melted incongruently, with increasing the value of x, in the temperature range from 1108 (x=0) to 1083 K (x=4) depositing Fe₂WO₆ and WO₃. The increase of the Mo⁶⁺ ions content in the crystal lattice of Fe₈V₁₀W₁₆O₈₅ causes the lattice parameters a=b contraction with cbeing almost constant. IR spectra of the Fe₈V₁₀W_16–xMo_xO₈₅ solid solution phases have been recorded.     

Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

The standard problem of radiation transfer in a bounded regionG 8543g13263t32/xxlarge8834.gif" alt="sub" align="MIDDLE" BORDER="0"> 8543g13263t32/xxlarge8477.gif" alt="Ropf" align="BASELINE" BORDER="0"> ⁿ can be reformulated as a weakly singular integral equation with an unknown functionu: G8543g13263t32/xxlarge8594.gif" alt="rarr" align="BASELINE" BORDER="0">C(S ^n–1) and a kernelK: ((G × G 8543g13263t32/xxlarge8726.gif" alt="setmn" align="MIDDLE" BORDER="0"> }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with 8543g13263t32/xxlarge8466.gif" alt="Lscr" align="MIDDLE" BORDER="0">(E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL _p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.     

Stratified structure of the Universe and Burgers' equation—a probabilistic approach

Summary The model of the potential turbulence described by the 3-dimensional Burgers' equation with random initial data was developped by Zeldovich and Shandarin, in order to explain the existing Large Scale Structure of the Universe. Most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results (the Gaussian scenario), under suitable moment assumptions on the initial velocity field. These results and some open questions are discussed in Sect. 2, where we concentrate on the Gaussian model and the shot-noise model. In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise field with high amplitudes) which reveals an intermittent large time behaviour, with the velocity determined by the position of the largest initial fluctuation (discounted by the heat kernelg(t,x·)) in a neighborhood ofx. The asymptoties of such local maximum ast8594.gif" alt="rarr" align="BASELINE" BORDER="0"> can be analyzed with the help of the theory of records (Sect. 4). Finally, in Sect. 5 we introduce a global definition of a point process oft-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit ast8594.gif" alt="rarr" align="BASELINE" BORDER="0">.