Addressing the minimum fleet problem in on-demand urban mobility.

Information and communication technologies have opened the way to new solutions for urban mobility that provide better ways to match individuals with on-demand vehicles. However, a fundamental unsolved problem is how best to size and operate a fleet of vehicles, given a certain demand for personal mobility. Previous studies1-5 either do not provide a scalable solution or require changes in human attitudes towards mobility. Here we provide a network-based solution to the following 'minimum fleet problem', given a collection of trips (specified by origin, destination and start time), of how to determine the minimum number of vehicles needed to serve all the trips without incurring any delay to the passengers. By introducing the notion of a 'vehicle-sharing network', we present an optimal computationally efficient solution to the problem, as well as a nearly optimal solution amenable to real-time implementation. We test both solutions on a dataset of 150 million taxi trips taken in the city of New York over one year 6 . The real-time implementation of the method with near-optimal service levels allows a 30 per cent reduction in fleet size compared to current taxi operation. Although constraints on driver availability and the existence of abnormal trip demands may lead to a relatively larger optimal value for the fleet size than that predicted here, the fleet size remains robust for a wide range of variations in historical trip demand. These predicted reductions in fleet size follow directly from a reorganization of taxi dispatching that could be implemented with a simple urban app; they do not assume ride sharing7-9, nor require changes to regulations, business models, or human attitudes towards mobility to become effective. Our results could become even more relevant in the years ahead as fleets of networked, self-driving cars become commonplace10-14.

Self-organized topological state with Majorana fermions.

Most physical systems known to date tend to resist entering the topological phase and must be fine-tuned to reach that phase. Here, we introduce a system in which a key dynamical parameter adjusts itself in response to the changing external conditions so that the ground state naturally favors the topological phase. The system consists of a quantum wire formed of individual magnetic atoms placed on the surface of an ordinary s-wave superconductor. It realizes the Kitaev paradigm of topological superconductivity when the wave vector characterizing the emergent spin helix dynamically self-tunes to support the topological phase. We call this phenomenon a self-organized topological state.

Electromagnetic response of Weyl semimetals.

It has been suggested recently, based on subtle field-theoretical considerations, that the electromagnetic response of Weyl semimetals and the closely related Weyl insulators can be characterized by an axion term θE·B with space and time dependent axion angle θ(r,t). Here we construct a minimal lattice model of the Weyl medium and study its electromagnetic response by a combination of analytical and numerical techniques. We confirm the existence of the anomalous Hall effect expected on the basis of the field theory treatment. We find, contrary to the latter, that chiral magnetic effect (that is, ground state charge current induced by the applied magnetic field) is absent in both the semimetal and the insulator phase. We elucidate the reasons for this discrepancy.