30,447 research outputs found
Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem
We study the existence of a retraction from the dual space of a (real
or complex) Banach space onto its unit ball which is uniformly
continuous in norm topology and continuous in weak- topology. Such a
retraction is called a uniformly simultaneously continuous retraction.
It is shown that if has a normalized unconditional Schauder basis with
unconditional basis constant 1 and is uniformly monotone, then a
uniformly simultaneously continuous retraction from onto
exists. It is also shown that if is a family of separable Banach
spaces whose duals are uniformly convex with moduli of convexity
such that and or
for , then a uniformly simultaneously continuous retraction
exists from onto .
The relation between the existence of a uniformly simultaneously continuous
retraction and the Bishsop-Phelps-Bollob\'as property for operators is
investigated and it is proved that the existence of a uniformly simultaneously
continuous retraction from onto its unit ball implies that a pair has the Bishop-Phelps-Bollob\'as property for every locally compact
Hausdorff spaces . As a corollary, we prove that has the
Bishop-Phelps-Bollob\'as property if and are the spaces of
all real-valued continuous functions vanishing at infinity on locally compact
metric space and locally compact Hausdorff space respectively.Comment: 15 page
Scheduling of Multicast and Unicast Services under Limited Feedback by using Rateless Codes
Many opportunistic scheduling techniques are impractical because they require
accurate channel state information (CSI) at the transmitter. In this paper, we
investigate the scheduling of unicast and multicast services in a downlink
network with a very limited amount of feedback information. Specifically,
unicast users send imperfect (or no) CSI and infrequent acknowledgements (ACKs)
to a base station, and multicast users only report infrequent ACKs to avoid
feedback implosion. We consider the use of physical-layer rateless codes, which
not only combats channel uncertainty, but also reduces the overhead of ACK
feedback. A joint scheduling and power allocation scheme is developed to
realize multiuser diversity gain for unicast service and multicast gain for
multicast service. We prove that our scheme achieves a near-optimal throughput
region. Our simulation results show that our scheme significantly improves the
network throughput over schemes employing fixed-rate codes or using only
unicast communications
- β¦