On an argument of J.--F. Cardoso dealing with perturbations of joint diagonalizers

B. Afsari has recently proposed a new approach to the matrix joint diagonalization, introduced by J.--F. Cardoso in 1994, in order to investigate the independent component analysis and the blind signal processing in a wider prospective. Delicate notions of linear algebra and differential geometry are involved in the works of B. Afsari and the present paper continues such a line of research, focusing on a theoretical condition which has significant consequences in the numerical applications.Comment: 9 pages; the published version contains significant revisions (suggested by the referees

On the Connectivity of the Sylow Graph of a Finite Group

The Sylow graph $\Gamma(G)$ of a finite group $G$ originated from recent investigations on the so--called $\mathbf{N}$--closed classes of groups. The connectivity of $\Gamma(G)$ was proved only few years ago, involving the classification of finite simple groups, and the structure of $G$ may be strongly restricted, once information on $\Gamma(G)$ are given. The first result of the present paper deals with a condition on $\mathbf{N}$--closed classes of groups. The second result deals with a computational criterion, related to the connectivity of $\Gamma(G)$.Comment: 8 pp. with Appendix; Fundamental revisions have been don

On the tensor degree of finite groups

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x \otimes y= 1_{_{G \otimes G}}$ in the nonabelian tensor square $G \otimes G$ of $G$. This number, divided by $|G|^2$, is called the tensor degree of $G$ and has connection with the exterior degree, introduced few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335--343]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.Comment: 10 pages, accepted in Ars Combinatoria with revision

On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

We contribute to an original problem studied by Hamilton and others, in order to understand the behaviour of maximal solutions of the Ricci flow both in compact and non-compact complete orientable Riemannian manifolds of finite volume. The case of dimension two has peculiarities, which force us to use different ideas from the corresponding higher dimensional case. We show the existence of connected regions with a connected complementary set (the so-called "separating regions"). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile. This is possible via an argument of compactness in geometric measure theory. Indeed we develop a definitive theory, which allows us to circumvent the shortening curve flow approach of previous authors at the cost of some applications of geometric measure theory and Ascoli-Arzela's Theorem.Comment: Example 5.4 is new; Theorem 4.5 is reformulated; 29 pages; 7 figure