This paper proposes signal detection methods for frequency domain equalization (FDE) based overloaded multiuser multiple input multiple output (MU-MIMO) systems for uplink Internet of things (IoT) environments, where a lot of IoT terminals are served by a base station having less number of antennas than that of IoT terminals. By using the fact that the transmitted signal vector has the discreteness and the group sparsity, we propose a convex discreteness and group sparsity aware (DGS) optimization problem for the signal detection. We provide an optimization algorithm for the DGS optimization on the basis of the alternating direction method of multipliers (ADMM). Moreover, we extend the DGS optimization into weighted DGS (W-DGS) optimization and propose an iterative approach named iterative weighted DGS (IW-DGS), where we iteratively solve the W-DGS optimization problem with the update of the parameters in the objective function. We also discuss the computational complexity of the proposed IW-DGS and show that we can reduce the order of the complexity by using the structure of the channel matrix. Simulation results show that the symbol error rate (SER) performance of the proposed method is close to that of the oracle zero forcing (ZF) method, which perfectly knows the activity of each IoT terminal.

Let $\mathbf{H}_{\mathbb{H}}^{n}$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group $\text{Sp}(n,1)$ acts on $\mathbf{H}_{\mathbb{H}}^{n}$ by isometries. A subgroup $G$ of $\text{Sp}(n,1)$ is called Zariski dense if it neither fixes a point on $\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$ nor preserves a totally geodesic subspace of $\mathbf{H}_{\mathbb{H}}^{n}$. We prove that a Zariski dense subgroup $G$ of $\text{Sp}(n,1)$ is discrete if for every loxodromic element $g\in G$ the two-generator subgroup $\langle f,gfg^{-1}\rangle$ is discrete, where the generator $f\in \text{Sp}(n,1)$ is a certain fixed element not necessarily from $G$.

In this paper we consider the discreteness of spectrum for higher-order differential operators in weighted function spaces. Using the method of embedding theorems of weighted Sobolev spaces Hnp in weighted spaces Ls,r, we obtain a new sufficient and necessary condition to ensure that the spectrum is discrete, which can be easily used to judge the discreteness of some differential operators.