In this note we prove the measurability of the family of non-degenerate conic sections in the projective space ${P}^{4}$, obtained by cutting a non-degenerate quadratic cone by a linear variety of dimension two not containing the cone vertex.

In this paper we consider a random variable $h$ arising from an intersection problem between a fixed convex body ${K}_{0}$ and a system of random independent and uniformly distributed ovaloids in ${E}_{3}$.

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

We study the geometry of a rational surface of Kodaira type IV by giving the nature of its integral curves of self-intersection less than zero, in particular we show that they are smooth and rational. Hence, under a reasonable assumption, we prove the finite generation of its monoid of effective divisor classes and in almost all cases its anticanonical complete linear system is of projective dimension zero and of self- intersection strictly negative. Furthermore, we show that if this condition is...

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