A note on closedness of the sum of two closed subspaces in a Banach space

Let $X$ be a Banach space, and $M,N$ be two closed subspaces of $X$. We present several necessary and sufficient conditions for the closedness of $M+N$ ($M+N$ is not necessarily direct sum)

On completeness of the space of weighted pseudo almost automorphic functions

In this paper, we prove that for every $\rho\in \mathbb{U}_{\infty}$, the space of weighted pseudo almost automorphic functions is complete under the supremum norm. This gives an affirmative answer to a key and fundamental problem for weighted pseudo almost automorphic functions, and fills a gap in the proof of [J. Funct. Anal. 258, No. 1, 196-207 (2010)].Comment: 8 page

Optimal Power Allocation for Artificial Noise under Imperfect CSI against Spatially Random Eavesdroppers

In this correspondence, we study the secure multiantenna transmission with artificial noise (AN) under imperfect channel state information in the presence of spatially randomly distributed eavesdroppers. We derive the optimal solutions of the power allocation between the information signal and the AN for minimizing the secrecy outage probability (SOP) under a target secrecy rate and for maximizing the secrecy rate under a SOP constraint, respectively. Moreover, we provide an interesting insight that channel estimation error affects the optimal power allocation strategy in opposite ways for the above two objectives. When the estimation error increases, more power should be allocated to the information signal if we aim to decrease the rate-constrained SOP, whereas more power should be allocated to the AN if we aim to increase the SOP-constrained secrecy rate.Comment: 7 pages, 6 figure

Improving Anti-Eavesdropping Ability without Eavesdropper's CSI: A Practical Secure Transmission Design Perspective

This letter studies the practical design of secure transmissions without knowing eavesdropper's channel state information (ECSI). An ECSI-irrelevant metric is introduced to quantize the intrinsic anti-eavesdropping ability (AEA) that the transmitter has on confronting the eavesdropper via secrecy encoding together with artificial-noise-aided signaling. Non-adaptive and adaptive transmission schemes are proposed to maximize the AEA with the optimal encoding rates and power allocation presented in closed-form expressions. Analyses and numerical results show that maximizing the AEA is equivalent to minimizing the secrecy outage probability (SOP) for the worst case by ignoring eavesdropper's receiver noise. Therefore, the AEA is a useful alternative to the SOP for assessing and designing secure transmissions when the ECSI cannot be prior known.Comment: 4 pages, 2 figures, to be published on IEEE Wireless Communications Letters (WCL

Delivery-Secrecy Tradeoff for Cache-Enabled Stochastic Networks: Content Placement Optimization

Wireless caching has been widely recognized as a promising technique for efficient content delivery. In this paper, by taking different file secrecy levels into consideration, physical-layer security oriented content placement is optimized in a stochastic cache-enabled cellular network. We propose an analytical framework to investigate the nontrivial file delivery-secrecy tradeoff. Specifically, we first derive the closed-form expressions for the file hit and secrecy probabilities. The global optimal probabilistic content placement policy is then analytically derived in terms of hit probability maximization under file secrecy constraints. Numerical results are demonstrated to verify our analytical findings and show that the targeted file secrecy levels are crucial in balancing the file delivery-secrecy tradeoff.Comment: 5 pages, 4 figures, accepted to be published in IEEE Transactions on Vehicular Technolog

Secure and Energy-Efficient Transmissions in Cache-Enabled Heterogeneous Cellular Networks: Performance Analysis and Optimization

This paper studies physical-layer security for a cache-enabled heterogeneous cellular network comprised of a macro base station and multiple small base stations (SBSs). We investigate a joint design on caching placement and file delivery for realizing secure and energy-efficient transmissions against randomly distributed eavesdroppers. We propose a novel hybrid "most popular content" and "largest content diversity" caching placement policy to distribute files of different popularities. Depending on the availability and placement of the requested file, we employ three cooperative transmission schemes, namely, distributed beamforming, frequency-domain orthogonal transmission, and best SBS relaying, respectively. We derive analytical expressions for the connection outage probability and secrecy outage probability for each transmission scheme. Afterwards, we design the optimal transmission rates and caching allocation successively to achieve a maximal overall secrecy throughput and secrecy energy efficiency, respectively. Numerical results verify the theoretical analyses and demonstrate the superiority of the proposed hybrid caching policy.Comment: 13 pages in double-column, 9 figures, accepted for publication on IEEE Transactions on Communication

Uniqueness of the inverse reaction coefficient problems for nonlocal diffusion models

In this paper, we consider the inverse reaction coefficient problems (IRCPs) for nonlocal diffusion equation and multi-term time-fractional nonlocal diffusion equation from the average nonlocal flux data in external reaction region. Based on the nonlocal maximum principle we established, the uniqueness theorem for IRCPs are proved

Identification of the degradation coefficient for an anomalous diffusion process in hydrology

In hydrology, the degradation coefficient is one of the key parameters to describe the water quality change and to determine the water carrying capacity. This paper is devoted to identify the degradation coefficient in an anomalous diffusion process by using the average flux data at the accessible part of boundary. The main challenges in inverse degradation coefficient problems (IDCP) is the average flux measurement data only provide very limited information and cause the severe ill-posedness of IDCP. Firstly, we prove the average flux measurement data can uniquely determine the degradation coefficient. The existence and uniqueness of weak solution for the direct problem are established, and the Lipschitz continuity of the corresponding forward operator is also obtained. Secondly, to overcome the ill-posedness, we combine the variational regularization method with Laplace approximations (LA) to solve the IDCP. This hybrid method is essentially the combination of deterministic regularization method and stochastic method. Thus, it is able to calculate the minimizer (MAP point) more rapidly and accurately, but also enables captures the statistics information and quantifying the uncertainty of the solution. Furthermore, the existence, stability and convergence of the minimizer of the variational problem are proved. The convergence rate estimate between the LA posterior distribution and the actual posterior distribution in the sense of Hellinger distance is given, and the skewness are introduced for characterizing the symmetry or slope of LA solution, especially the relationship with the symmetry of the measurement data. Finally, the one-dimensional and two-dimensional numerical examples are presented to confirm the efficiency and robustness of the proposed method

Adaptive Full-Duplex Jamming Receiver for Secure D2D Links in Random Networks

Device-to-device (D2D) communication raises new transmission secrecy protection challenges, since conventional physical layer security approaches, such as multiple antennas and cooperation techniques, are invalid due to its resource/size constraints. The full-duplex (FD) jamming receiver, which radiates jamming signals to confuse eavesdroppers when receiving the desired signal simultaneously, is a promising candidate. Unlike existing endeavors that assume the FD jamming receiver always improves the secrecy performance compared with the half-duplex (HD) receiver, we show that this assumption highly depends on the instantaneous residual self-interference cancellation level and may be invalid. We propose an adaptive jamming receiver operating in a switched FD/HD mode for a D2D link in random networks. Subject to the secrecy outage probability constraint, we optimize the transceiver parameters, such as signal/jamming powers, secrecy rates and mode switch criteria, to maximize the secrecy throughput. Most of the optimization operations are taken off-line and only very limited on-line calculations are required to make the scheme with low complexity. Furthermore, some interesting insights are provided, such as the secrecy throughput is a quasi-concave function. Numerical results are demonstrated to verify our theoretical findings, and to show its superiority compared with the receiver operating in the FD or HD mode only

Energy-Efficient Transmission Design in Non-Orthogonal Multiple Access

Non-orthogonal multiple access (NOMA) is considered as a promising technology for improving the spectral efficiency (SE) in 5G. In this correspondence, we study the benefit of NOMA in enhancing energy efficiency (EE) for a multi-user downlink transmission, where the EE is defined as the ratio of the achievable sum rate of the users to the total power consumption. Our goal is to maximize the EE subject to a minimum required data rate for each user, which leads to a non-convex fractional programming problem. To solve it, we first establish the feasible range of the transmitting power that is able to support each user's data rate requirement. Then, we propose an EE-optimal power allocation strategy that maximizes the EE. Our numerical results show that NOMA has superior EE performance in comparison with conventional orthogonal multiple access (OMA)