Uniqueness and Pseudolocality Theorems of the Mean Curvature Flow

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.Comment: 40 page

Joint Cooperative Spectrum Sensing and MAC Protocol Design for Multi-channel Cognitive Radio Networks

In this paper, we propose a semi-distributed cooperative spectrum sen sing (SDCSS) and channel access framework for multi-channel cognitive radio networks (CRNs). In particular, we c onsider a SDCSS scheme where secondary users (SUs) perform sensing and exchange sensing outcomes with ea ch other to locate spectrum holes. In addition, we devise the p -persistent CSMA-based cognitive MAC protocol integrating the SDCSS to enable efficient spectrum sharing among SUs. We then perform throughput analysis and develop an algorithm to determine the spectrum sensing and access parameters to maximize the throughput for a given allocation of channel sensing sets. Moreover, we consider the spectrum sensing set optimization problem for SUs to maxim ize the overall system throughput. We present both exhaustive search and low-complexity greedy algorithms to determine the sensing sets for SUs and analyze their complexity. We also show how our design and analysis can be extended to consider reporting errors. Finally, extensive numerical results are presented to demonstrate the sig nificant performance gain of our optimized design framework with respect to non-optimized designs as well as the imp acts of different protocol parameters on the throughput performance.Comment: accepted for publication EURASIP Journal on Wireless Communications and Networking, 201

Design and Optimal Configuration of Full-Duplex MAC Protocol for Cognitive Radio Networks Considering Self-Interference

In this paper, we propose an adaptive Medium Access Control (MAC) protocol for full-duplex (FD) cognitive radio networks in which FD secondary users (SUs) perform channel contention followed by concurrent spectrum sensing and transmission, and transmission only with maximum power in two different stages (called the FD sensing and transmission stages, respectively) in each contention and access cycle. The proposed FD cognitive MAC (FDC-MAC) protocol does not require synchronization among SUs and it efficiently utilizes the spectrum and mitigates the self-interference in the FD transceiver. We then develop a mathematical model to analyze the throughput performance of the FDC-MAC protocol where both half-duplex (HD) transmission (HDTx) and FD transmission (FDTx) modes are considered in the transmission stage. Then, we study the FDC-MAC configuration optimization through adaptively controlling the spectrum sensing duration and transmit power level in the FD sensing stage where we prove that there exists optimal sensing time and transmit power to achieve the maximum throughput and we develop an algorithm to configure the proposed FDC-MAC protocol. Extensive numerical results are presented to illustrate the characteristic of the optimal FDC-MAC configuration and the impacts of protocol parameters and the self-interference cancellation quality on the throughput performance. Moreover, we demonstrate the significant throughput gains of the FDC-MAC protocol with respect to existing half-duplex MAC (HD MAC) and single-stage FD MAC protocols.Comment: To Appear, IEEE Access, 201

Pilot Optimization and Channel Estimation for Multiuser Massive MIMO Systems

This paper proposes novel pilot optimization and channel estimation algorithm for the downlink multiuser massive multiple input multiple output (MIMO) system with $K$ decentralized single antenna mobile stations (MSs), and time division duplex (TDD) channel estimation which is performed by utilizing $N$ pilot symbols. The proposed algorithm is explained as follows. First, we formulate the channel estimation problem as a weighted sum mean square error (WSMSE) minimization problem containing pilot symbols and introduced variables. Second, for fixed pilot symbols, the introduced variables are optimized using minimum mean square error (MMSE) and generalized Rayleigh quotient methods. Finally, for $N=1$ and $N=K$ settings, the pilot symbols of all MSs are optimized using semi definite programming (SDP) convex optimization approach, and for the other settings of $N$ and $K$, the pilot symbols of all MSs are optimized by applying simple iterative algorithm. When $N=K$, it is shown that the latter iterative algorithm gives the optimal pilot symbols achieved by the SDP method. Simulation results confirm that the proposed algorithm achieves less WSMSE compared to that of the conventional semi-orthogonal pilot symbol and MMSE channel estimation algorithm which creates pilot contamination.Comment: Accepted in CISS 2014 Conferenc