30,447 research outputs found

    Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem

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    We study the existence of a retraction from the dual space Xβˆ—X^* of a (real or complex) Banach space XX onto its unit ball BXβˆ—B_{X^*} which is uniformly continuous in norm topology and continuous in weak-βˆ—* topology. Such a retraction is called a uniformly simultaneously continuous retraction. It is shown that if XX has a normalized unconditional Schauder basis with unconditional basis constant 1 and Xβˆ—X^* is uniformly monotone, then a uniformly simultaneously continuous retraction from Xβˆ—X^* onto BXβˆ—B_{X^*} exists. It is also shown that if {Xi}\{X_i\} is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity Ξ΄i(Ξ΅)\delta_i(\varepsilon) such that inf⁑iΞ΄i(Ξ΅)>0\inf_i \delta_i(\varepsilon)>0 and X=[⨁Xi]c0X= \left[\bigoplus X_i\right]_{c_0} or X=[⨁Xi]β„“pX=\left[\bigoplus X_i\right]_{\ell_p} for 1≀p<∞1\le p<\infty, then a uniformly simultaneously continuous retraction exists from Xβˆ—X^* onto BXβˆ—B_{X^*}. 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 Xβˆ—X^* onto its unit ball implies that a pair (X,C0(K))(X, C_0(K)) has the Bishop-Phelps-Bollob\'as property for every locally compact Hausdorff spaces KK. As a corollary, we prove that (C0(S),C0(K))(C_0(S), C_0(K)) has the Bishop-Phelps-Bollob\'as property if C0(S)C_0(S) and C0(K)C_0(K) are the spaces of all real-valued continuous functions vanishing at infinity on locally compact metric space SS and locally compact Hausdorff space KK respectively.Comment: 15 page

    Scheduling of Multicast and Unicast Services under Limited Feedback by using Rateless Codes

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    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
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