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When a plane shock hits a two-dimensional wedge head on, it experiences a reflection-diffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The experimental, computational, and asymptotic analysis has indicated that various patterns occur, including regular reflection and Mach reflection. The von Neumann's conjectures on the transition from regular to Mach reflection involve the existence, uniqueness, and stability of regular shock reflection-diffraction configurations, generated by concave cornered wedges for compressible flow. In this paper, we discuss some recent developments in the study of the von Neumann's conjectures. More specifically, we discuss the uniqueness and stability of regular shock reflection-diffraction configurations governed by the potential flow equation in an appropriate class of solutions. We first show that the transonic shocks in the global solutions obtained in Chen-Feldman [19] are convex. Then we establish the uniqueness of global shock reflection-diffraction configurations with convex transonic shocks for any wedge angle larger than the detachment angle or the critical angle. Moreover, the solution under consideration is stable with respect to the wedge angle. Our approach also provides an alternative way of proving the existence of the admissible solutions established first in [19]. 

Abstract: Impairments such as time varying phase noise (PHN) and carrier frequency offset (CFO) result in loss of synchronization and poor performance of multi-relay communication systems. Joint estimation of these impairments is necessary in order to correctly decode the received signal at the destination. In this paper, we address spectrally efficient multi-relay transmission scenarios where all the relays simultaneously communicate with the destination. We propose an iterative pilot-aided algorithm based on the expectation conditional maximization for joint estimation of multipath channels, Wiener PHNs, and CFOs in decode-and-forward-based multi-relay orthogonal frequency division multiplexing systems. Next, a new expression of the hybrid Cramér-Rao lower bound (HCRB) for the multi-parameter estimation problem is derived. Finally, an iterative receiver based on an extended Kalman filter for joint data detection and PHN tracking is employed. Numerical results show that the proposed estimator outperforms existing algorithms and its mean square error performance is close to the derived HCRB at different signal-to-noise ratios for different PHN variances. In addition, the combined estimation algorithm and the iterative receiver can significantly improve average bit-error rate (BER) performance compared with existing algorithms. In addition, the BER performance of the proposed system is close to the ideal case of perfect channel impulse responses, PHNs, and CFOs estimation.