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In this paper we improve the known bound for the $X$-rank $R_{X}(P)$ of an element $P\in {\mathbb{P}}^N$ in the case in which $X\subset {\mathbb P}^n$ is a projective variety obtained as a linear projection from a general $v$-dimensional subspace $V\subset {\mathbb P}^{n+v}$. Then, if $X\subset {\mathbb P}^n$ is a curve obtained from a projection of a rational normal curve $C\subset {\mathbb P}^{n+1}$ from a point $O\subset {\mathbb P}^{n+1}$, we are able to describe the precise value of the $X$-rank for those points $P\in {\mathbb P}^n$ such that $R_{X}(P)\leq R_{C}(O)-1$ and to improve the general result. Moreover we give a stratification, via the $X$-rank, of the osculating spaces to projective cuspidal projective curves $X$. Finally we give a description and a new bound of the $X$-rank of subspaces both in the general case and with respect to integral non-degenerate projective curves., 10 pages

Given the toric (or toral) arrangement defined by a root system $\Phi$, we describe the poset of its layers (connected components of intersections) and we count its elements. Indeed we show how to reduce to zero-dimensional layers, and in this case we provide an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of $\Phi$. Then we compute the Euler characteristic and the Poincare' polynomial of the complement of the arrangement, which is the set of regular points of the torus., Comment: 20 pages. Updated version of a paper published in December 2008

This paper focuses on the writings on probability by Giovanni Rizzetti (1675–1751), who was a member of the group of Venetian scholars around Count Jacopo Riccati (1676–1754). We deal with his approach to probability and, more particularly, with his argumentation on Bernoulli's theorem. The research develops Rizzetti's reasonings as contained in his printed works (1724–1729) as well as in certain key manuscripts. The texts of the latter may be found in the appendix. The statements of Rizzetti seem very interesting; they are situated between the empirical law of chance and the law of large numbers.

In this paper we extend the property of a ringA to be of finite real character to loc-etaleA-algebras and to loc-etale neighbourhoods by a suitable representation of these algebras. By this means we get some result on the property to be completely integrally closed for loc-ind-etaleA-algebras and henselizations. We conclude by a counter-example which shows that the properties to be Krull and completely integrally closed are not preserved in the henselization.