# Volume 14 (2010) No. 2

## Volume 14 (2010) No. 2

### A note on distributional equations in discounted risk processes

Gerardo Hernández del Valle and Carlos G. Pacheco González

**Abstract:**

In this paper we give an account of the classical discounted risk processes and their limiting distributions. For the models considered, we set the Markov chains embedded in the continuous-time processes; we also set distributional equations for the limit distributions. Additionally, we mention some applications regarding ruin probabilities and optimal premium.

### Cohomology groups of configuration spaces of pairs of points in real projective spaces

Jesús González and Peter Landweber

**Abstract:**

The Stiefel manifold of $V_{m+1,2}$ of 2-frames in $\mathbb{R}^{m+1}$ is acted upon by the orthogonal group $O(2)$. By restriction, there are corresponding actions of the dihedral group of order 8, $D_8$, and of the rank-2 elementary 2-group $\mathbb{Z}_2 \times \mathbb{Z}_2$. We use the Cartan-Leray spectral sequences of these actions to compute the integral homology and cohomology groups of the configuration spaces $B(P^m, 2)$ and $F(P^m, 2)$ of (unordered and ordered) pairs of points on the real projective spaces $P^m$.

### An upper bound on the size of irreducible quadrangulations

Gloria Aguilar Cruz and Francisco Javier Zaragoza Martinez

**Abstract:**

Let $S$ be a closed surface with Euler genus $\gamma (S)$. A quadrangulation $G$ of a closed surface $S$ is irreducible if it does not have any contractible face. Nakamoto and Ota gave a linear upper bound for the number $n$ of vertices of $G$ in terms of $\gamma (S)$. By extending Nakamoto and Ota's method we improve their bound to $n \leq 159.5 \gamma (S) - 46$ for any closed surface $S$.