Ramification of the Gauss Map of Complete Minimal Surfaces in R^3 and R^4 on Annular Ends

Final version. Small improvement of the result of Theorem 3International audienceIn this article, we study the ramification of the Gauss map of complete minimal surfaces in R^3 and R^4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto and Ru for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement of the results on annular ends of complete minimal surfaces of Kao

Multi-Objective Optimization for IRS-Aidded Multi-user MIMO SWIPT Systems

In this paper, we investigate an intelligent reflecting surface (IRS) assisted simultaneous wireless information and power transfer (SWIPT) system in which users equipped with multiple antennas exploit power-splitting (PS) strategies for simultaneously information decoding (ID) and energy harvesting (EH). Different from the majority of previous studies which focused on single objective optimization problems (SOOPs) and assumed the linearity of EH models, in this paper, we aim at studying the multi-objective optimization problem (MOOP) of the sum rate (SR) and the totalharvested energy (HE) subject to the maximum transmit power (TP) constraint, the user quality of service (QoS), and HE requirements at each user with taking a practical non-linear EH (NLEH) model into consideration. To investigate insightful tradeoffs between the achievable SR and total HE, we adopt the modified weighted Tchebycheff method to transform the MOOP into a SOOP. However, solving the SOOPs and modified SOOP is mathematically difficult due to the non-convexity of the object functions and the constraints of the coupled variables of the  base station (BS) transmit precoding matrices (TPMs), the user PS ratios (PSRs), and the IRS phase shift matrix (PSM). To address these challenges, an alternating optimization (AO) framework is used to decompose the formulated design problem into sub-problems. In addition, we apply the majorization-minimization (MM) approach to transform the sub-problems into convex optimization ones. Finally,  numerical simulation results are conducted to verify the tradeoffs between the SR and the total amount of HE. The numerical results also indicate that the considered system using the IRS with optimal phase shifts provides considerable performance improvement in terms of the achievable SR and HE as compared to the counterparts without using the IRS or with the IRS of fixed phase shifts