9,073 research outputs found
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 on manifolds. The main goal is
to show a unifying duality between such properties and the existence of
suitable -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 .
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
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