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 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

Observation and its History

Recenze: Lorraine DASTON - Elizabeth LUNBECK, E., Histories of Scientific Observation. Chicago - London: University of Chicago Press 2011, 460 pp

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

The Invalidity of the Laplace Law for Biological Vessels and of Estimating Elastic Modulus from Total Stress vs. Strain: a New Practical Method

The quantification of the stiffness of tubular biological structures is often obtained, both in vivo and in vitro, as the slope of total transmural hoop stress plotted against hoop strain. Total hoop stress is typically estimated using the "Laplace law." We show that this procedure is fundamentally flawed for two reasons: Firstly, the Laplace law predicts total stress incorrectly for biological vessels. Furthermore, because muscle and other biological tissue are closely volume-preserving, quantifications of elastic modulus require the removal of the contribution to total stress from incompressibility. We show that this hydrostatic contribution to total stress has a strong material-dependent nonlinear response to deformation that is difficult to predict or measure. To address this difficulty, we propose a new practical method to estimate a mechanically viable modulus of elasticity that can be applied both in vivo and in vitro using the same measurements as current methods, with care taken to record the reference state. To be insensitive to incompressibility, our method is based on shear stress rather than hoop stress, and provides a true measure of the elastic response without application of the Laplace law. We demonstrate the accuracy of our method using a mathematical model of tube inflation with multiple constitutive models. We also re-analyze an in vivo study from the gastro-intestinal literature that applied the standard approach and concluded that a drug-induced change in elastic modulus depended on the protocol used to distend the esophageal lumen. Our new method removes this protocol-dependent inconsistency in the previous result.Comment: 34 pages, 13 figure