Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of this work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations $F$ on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable $F$-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the "standard" operators arising from geometry, and improves on partial results in the literature. Applications include new characterizations of the classical maximum principles at infinity (Ekeland, Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation properties for stochastic processes (martingale completeness). Applications to the theory of submanifolds and Riemannian submersions are also discussed.Comment: 67 pages. Final versio

Hierarchical Features of Large-Scale Cortical Connectivity

The analysis of complex networks has revealed patterns of organization in a variety of natural and artificial systems, including neuronal networks of the brain at multiple scales. In this paper, we describe a novel analysis of the large-scale connectivity between regions of the mammalian cerebral cortex, utilizing a set of hierarchical measurements proposed recently. We examine previously identified functional clusters of brain regions in macaque visual cortex and cat cortex and find significant differences between such clusters in terms of several hierarchical measures, revealing differences in how these clusters are embedded in the overall cortical architecture. For example, the ventral cluster of visual cortex maintains structurally more segregated, less divergent connections than the dorsal cluster, which may point to functionally different roles of their constituent brain regions.Comment: 17 pages, 6 figure

Long-Range Connections in Transportation Networks

Since its recent introduction, the small-world effect has been identified in several important real-world systems. Frequently, it is a consequence of the existence of a few long-range connections, which dominate the original regular structure of the systems and implies each node to become accessible from other nodes after a small number of steps, typically of order $\ell \propto \log N$. However, this effect has been observed in pure-topological networks, where the nodes have no spatial coordinates. In this paper, we present an alalogue of small-world effect observed in real-world transportation networks, where the nodes are embeded in a hree-dimensional space. Using the multidimensional scaling method, we demonstrate how the addition of a few long-range connections can suubstantially reduce the travel time in transportation systems. Also, we investigated the importance of long-range connections when the systems are under an attack process. Our findings are illustrated for two real-world systems, namely the London urban network (streets and underground) and the US highways network enhanced by some of the main US airlines routes