Kazhdan-Lusztig polynomials and drift configurations

The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for $H_{v,w}(q)$. We introduce \emph{drift configurations} to formulate a new and compatible combinatorial rule for $P_{v,w}(q)$. From our rules we deduce, for these cases, the coefficient-wise inequality $P_{v,w}(q)\preceq H_{v,w}(q)$.Comment: 26 pages. To appear in Algebra & Number Theor

Exploiting Device-to-Device Communications to Enhance Spatial Reuse for Popular Content Downloading in Directional mmWave Small Cells

With the explosive growth of mobile demand, small cells in millimeter wave (mmWave) bands underlying the macrocell networks have attracted intense interest from both academia and industry. MmWave communications in the 60 GHz band are able to utilize the huge unlicensed bandwidth to provide multiple Gbps transmission rates. In this case, device-to-device (D2D) communications in mmWave bands should be fully exploited due to no interference with the macrocell networks and higher achievable transmission rates. In addition, due to less interference by directional transmission, multiple links including D2D links can be scheduled for concurrent transmissions (spatial reuse). With the popularity of content-based mobile applications, popular content downloading in the small cells needs to be optimized to improve network performance and enhance user experience. In this paper, we develop an efficient scheduling scheme for popular content downloading in mmWave small cells, termed PCDS (popular content downloading scheduling), where both D2D communications in close proximity and concurrent transmissions are exploited to improve transmission efficiency. In PCDS, a transmission path selection algorithm is designed to establish multi-hop transmission paths for users, aiming at better utilization of D2D communications and concurrent transmissions. After transmission path selection, a concurrent transmission scheduling algorithm is designed to maximize the spatial reuse gain. Through extensive simulations under various traffic patterns, we demonstrate PCDS achieves near-optimal performance in terms of delay and throughput, and also superior performance compared with other existing protocols, especially under heavy load.Comment: 12 pages, to appear in IEEE Transactions on Vehicular Technolog