Wandering intervals and absolutely continuous invariant probability measures of interval maps

For piecewise $C^1$ interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recurrence along critical orbits, we prove the nonexistence of wandering intervals, the existence of absolutely continuous invariant measures, and the bounded backward contraction property. The proofs are based on the method of proving the existence of absolutely continuous invariant measures of unimodal map, developed by Nowicki and van Strien.Comment: 16 pages, 2 figure

Wireless Communications in the Era of Big Data

The rapidly growing wave of wireless data service is pushing against the boundary of our communication network's processing power. The pervasive and exponentially increasing data traffic present imminent challenges to all the aspects of the wireless system design, such as spectrum efficiency, computing capabilities and fronthaul/backhaul link capacity. In this article, we discuss the challenges and opportunities in the design of scalable wireless systems to embrace such a "bigdata" era. On one hand, we review the state-of-the-art networking architectures and signal processing techniques adaptable for managing the bigdata traffic in wireless networks. On the other hand, instead of viewing mobile bigdata as a unwanted burden, we introduce methods to capitalize from the vast data traffic, for building a bigdata-aware wireless network with better wireless service quality and new mobile applications. We highlight several promising future research directions for wireless communications in the mobile bigdata era.Comment: This article is accepted and to appear in IEEE Communications Magazin