Exact gravitational plane waves and two-dimensional gravity

We discuss dynamical aspects of gravitational plane waves in Einstein theory with massless scalar fields. The general analytic solution describes colliding gravitational waves with constant polarization, which interact with scalar waves and, for generic initial data, produce a spacetime singularity at the focusing hypersurface. There is, in addition, an infinite family of regular solutions and an intriguing static geometry supported by scalar fields. Upon dimensional reduction, the theory can be viewed as an exactly solvable two-dimensional gravity model. This provides a new viewpoint on the gravitational dynamics. Finally, we comment on a simple mechanism by which short-distance corrections in the two-dimensional model can remove the singularity.Comment: 8 page

N=2 gauge theories and quantum phases

The partition function of general N = 2 supersymmetric SU(2) Yang-Mills theories on a four-sphere localizes to a matrix integral. We show that in the decompactification limit, and in a certain regime, the integral is dominated by a saddle point. When this takes effect, the free energy is exactly given in terms of the prepotential, $F=-R^2 Re (4\pi i {\cal F})$, evaluated at the singularity of the Seiberg-Witten curve where the dual magnetic variable $a_D$ vanishes. We also show that the superconformal fixed point of massive supersymmetric QCD with gauge group SU(2) is associated with the existence of a quantum phase transition. Finally, we discuss the case of N=2* SU(2) Yang-Mills theory and show that the theory does not exhibit phase transitions.Comment: 23 pages, 4 figure

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