The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator.     

A complete global solution to the pressure gradient equation

We study the domain of existence of a solution to a Riemann problem for the pressure gradient equation in two space dimensions. The Riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum quadrants. The global existence of a smooth solution was established in Dai and Zhang [Z. Dai, T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal. 155 (2000) 277-298] up to the free boundary of vacuum. We prove that the vacuum boundary is the coordinate axes.     

Measures on minimally generated Boolean algebras

We investigate properties of minimally generated Boolean algebras. It is shown that all measures defined on such algebras are separable but not necessarily weakly uniformly regular. On the other hand, there exist Boolean algebras small in terms of measures which are not minimally generated. We prove that under CH a measure on a retractive Boolean algebra can be nonseparable. Some relevant examples are indicated. Also, we give two examples of spaces satisfying some kind of Efimov property.     

The time recovery of the spectroscopic properties of heavily irradiated CdZnTe and CdTe detectors

The time recovery of the spectroscopic capabilities of CdZnTe and CdTe detectors, irradiated with increasing doses of high- and low-energy neutrons, as well as electrons, has been investigated by studying their spectroscopic behavior at different photon energies using leakage current measurements and PICTS (photo-induced current transient spectroscopy) analysis. The detectors were stored at room temperature for up to one year to study the time evolution of their spectroscopic performance and to correlate it with the presence of defective states in the material. We have observed a clear improvement in the material’s detection properties with time, though only in those detectors which have not been severely degraded by the irradiation. The recovery can be associated with a decrease in the concentration of some defective states, thus allowing the assessment of the crucial role these play in determining the charge collection processes in the material and its spectroscopic capabilities. Received: 1 August 2001 / Accepted: 3 August 2001 / Published online: 20 December 2001     

A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals Jλ, λ∈I⊂R. Under suitable assumptions, we locate an open subinterval of values λ in I for which Jλ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.     

Almost sure estimates for the concentration neighborhood of Sinai’s walk

We consider Sinai’s random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is bounded almost surely. We also get that i.o. the maximal distance between two favorite sites is bounded almost surely.     

Bias of the structural quasi-score estimator of a measurement error model under misspecification of the regressor distribution

In a structural measurement error model the structural quasi-score (SQS) estimator is based on the distribution of the latent regressor variable. If this distribution is misspecified, the SQS estimator is (asymptotically) biased. Two types of misspecification are considered. Both assume that the statistician erroneously adopts a normal distribution as his model for the regressor distribution. In the first type of misspecification, the true model consists of a mixture of normal distributions which cluster around a single normal distribution, in the second type, the true distribution is a normal distribution admixed with a second normal distribution of low weight. In both cases of misspecification, the bias, of course, tends to zero when the size of misspecification tends to zero. However, in the first case the bias goes to zero in a flat way so that small deviations from the true model lead to a negligible bias, whereas in the second case the bias is noticeable even for small deviations from the true model.     

On the Hausdorff dimension of generalized Besicovitch-Eggleston sets of d-tuples of numbers

We give a systematic and detailed account of the Hausdorff dimensions of sets of d-tuples of numbers defined in terms of the asymptotic behaviour of the frequencies of strings of digits in their N-adic expansion.     

Symmetric incidence groupoids

In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.This paper was sponsored by Vigoni Program 1999.