Orbits in symmetric spaces, II

Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or $(0,\infty)$ or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on $f\in E$ so that its orbit $\Omega(f)$ is the closed convex hull of its extreme points. We also give an application to symmetrically normed ideals of compact operators on a Hilbert space

Topological Measure Locating the Effective Crossover between Segregation and Integration in a Modular Network

We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hyper-graph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of coupled phase oscillators. The dynamical crossover is signaled by a peak in the product of the measures of intra-cluster and global synchronization, which we propose as a dynamical measure of complexity. This quantity is much easier to compute than the entropy (of the average frequencies of the oscillators), and displays a behavior which closely mimics that of the dynamical complexity index based on the latter. The proposed toplogical measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity vs total integration and shows how this affects the capability of the network to perform both local and distributed dynamical tasks

Critical exponents and scaling invariance in the absence of a critical point

The paramagnetic-to-ferromagnetic phase transition is believed to proceed through a critical point, at which power laws and scaling invariance, associated with the existence of one diverging characteristic length scale -- the so called correlation length -- appear. We indeed observe power laws and scaling behavior over extraordinarily many decades of the suitable scaling variables at the paramagnetic-to-ferromagnetic phase transition in ultrathin Fe films. However, we find that, when the putative critical point is approached, the singular behavior of thermodynamic quantities transforms into an analytic one: the critical point does not exist, it is replaced by a more complex phase involving domains of opposite magnetization, below as well as $above$ the putative critical temperature. All essential experimental results are reproduced by Monte-Carlo simulations in which, alongside the familiar exchange coupling, the competing dipole-dipole interaction is taken into account. Our results imply that a scaling behavior of macroscopic thermodynamic quantities is not necessarily a signature for an underlying second-order phase transition and that the paramagnetic-to-ferromagnetic phase transition proceeds, very likely, in the presence of at least two long spatial scales: the correlation length and the size of magnetic domains.Comment: 10 pages, added figure 1, rearranged section

Role of surface microgeometries on electron escape probability and secondary electron yield of metal surfaces

The influence of microgeometries on the Secondary Electron Yield (SEY) of surfaces is investigated. Laser written structures of different aspect ratio (height to width) on a copper surface tuned the SEY of the surface and reduced its value to less than unity. The aspect ratio of microstructures was methodically controlled by varying the laser parameters. The results obtained corroborate a recent theoretical model of SEY reduction as a function of the aspect ratio of microstructures. Nanostructures - which are formed inside the microstructures during the interaction with the laser beam - provided further reduction in SEY comparable to that obtained in the simulation of structures which were coated with an absorptive layer suppressing secondary electron emission