TITLE: Does this representation have a name?
QUESTION [2 upvotes]: Let $G$ be a group acting on a set $X$. Let $F(X) = \{f: X \to \mathbb C\}$ be the set of complex valued functions on $X$. This is a complex vector space.
Then $G$ acts on $F(X)$ linearly via the following operation:
$$ \pi (g) f(x) := f(g^{-1}x)$$
where $\pi : G \to (F(X)\to F(X))$ is the representation. Or at least that's what I gathered from the lecture.
Now my question is: does this representation have a name? The lecturer called it a ''quantization of a group action'' but when I tried to google that I found nothing.

REPLY [4 votes]: If a group $G$ acts on a set $X$, the induced action on $\mathbb{C}[X]$, the free complex vector space on $X$, is called the permutation representation associated to the $G$-set $X$. The induced action on $\mathbb{C}^X$ is the dual of the permutation representation. When $X$ is finite, these two representations are isomorphic, but when $X$ is infinite they don't even have the same dimension in general. (When $X$ is infinite the dual representation is probably not what you're looking for, and you want to cut it down in some way.)
Understanding permutation representations reduces quickly to the case that $X$ is transitive, hence that $X = G/H$ for some subgroup $H$. Then $\mathbb{C}[G/H]$ and $\mathbb{C}^{G/H}$ are two versions of the induced representation $\text{Ind}_H^G(1)$, where $1$ is the trivial representation of $H$. When $H$ has finite index in $G$, these two representations are isomorphic, but again, in general the dual representation is probably not what you're looking for, and you want to cut it down in some way.


text:
\begin{document}

\title{A Comprehensive Performance Analysis for mm-Wave Massive MIMO Hybrid Beamforming under PA Nonlinearities}

\author{Murat~Babek~Salman,~\IEEEmembership{Student~Member,~IEEE,}
         Gokhan~Muzaffer~Guvensen,~\IEEEmembership{Member,~IEEE,}

\thanks{Murat~Babek~Salman and G. M. Guvensen are with the Department
of Electrical and Electronics Engineering, Middle East Technical University, Ankara, 06800 Turkey e-mail: mbsalman@metu.edu.tr, guvensen@metu.edu.tr.}
}

\maketitle

\begin{abstract}

In this paper, we develop a framework to investigate the performances of different hybrid beamforming architectures for massive multiple input multiple output (MIMO) systems impaired by power amplifier (PA) nonlinearities. Indirect learning architecture based on feedback after anti-beamforming is adopted in design of digital pre-distortion (DPD) in order to compensate for the nonlinear distortion caused by PA. In addition, we propose a novel analog beamformer design for partially connected array architecture based on generalized eigen-beamformer (GEB) approach. In literature, the effects of nonlinear PA's on the out-of-band (OOB) radiation and achieved signal-to-interference-plus-noise ratio (SINR) are investigated. However, these studies are limited to fully digital or partially connected hybrid beamforming architectures while deploying Bussgang decomposition on a PA basis without considering the array architecture type in performance analysis. In this study, we derived an analytical bit-error-rate (BER) expression based on spatio-frequency Bussgang decomposition in matrix form, and lower bound for mismatched decoding capacity via Generalized Mutual Information (GMI) is obtained under PA nonlinearity for different hybrid massive MIMO architectures. Analytical results show that the nonlinear distortion significantly affects the system performance, and DPD can reduce these effects to some extend. Finally, obtained analytical BER expression is verified via numerical results.
\end{abstract}

\begin{IEEEkeywords}
Massive MIMO, hybrid beamforming, nonlinear PA distortion, performance analysis.
\end{IEEEkeywords}

\IEEEpeerreviewmaketitle

\section{Introduction}

\IEEEPARstart{M}{ultiple}-input multiple-output (MIMO) systems operating at milimeter-wave (mm-Wave) frequencies are expected to meet constantly growing data rate requirements that cannot be met by current systems \cite{mm_wave:1,mm_wave:2,mm_wave:3,mm_wave:4}. Radio signals, at mm-Wave frequencies, are exposed to severe absorption losses \cite{mm_wave:1,mm_wave:2}, hence, large scale antenna arrays, also called as massive MIMO systems, are used to increase the multiplexing gain so that effects of these losses are reduced \cite{mm_wave:3,mMIMO:1}.  However, using a large number of antennas has certain limitations. In conventional MIMO systems, which are employed for sub-6 GHz frequency bands, fully digital precoding can be implemented; however, doing so is quite costly for massive MIMO systems due to need of excessive number of RF chains, which consume immense amount of power \cite{FullDig,HybridBF:1,HybridBF:2}.

To reduce power consumption and implementation complexity, hybrid beamforming based array architectures have been developed. In these architectures, beamforming is divided into two stages, where precoding is performed in digital domain and analog beamformer connects $D$ RF chains to $N_t$ transmit antennas for $D<N_t$. One common approach is to implement analog beamformer as a partially connected structure, where each RF chain is connected to a subarray composed of a series of phase shifters \cite{HybridBF:3}, and digital precoder is designed based on the reduced dimensional effective channel. Joint spatial division and multiplexing (JSDM) is also proposed as a two stage beamforming framework \cite{JSDM} and \cite{JSDM2}. In JSDM, analog beamformer is designed based on spatial characteristics of user channels, which are characterized by their channel covariance matrices (CCM)'s, and users having similar CCM's are grouped so that they can be processed jointly. Analog beamformer is designed to eliminate inter-group interferences and decouples the signals of different groups. Digital precoder, on the other hand, is designed based on instantaneous channel in reduced dimension to suppress intra-group interference. In \cite{GEB:1} and \cite{GEB:2}, fully connected generalized eigen beamformer (GEB) is proposed as a statistical analog beamformer, which is shown to be optimal for several criteria. In proposed GEB, sub-beamformers are formed for each group such that beampattern contains deep nulls for angular sectors (AS)'s of other groups. This type of beamformer is suitable for the wideband massive MIMO channels, which are expected to be sparse both in the angle and delay domains \cite{JSDM2}.

In massive MIMO systems, low-cost PA's operating close to saturation are considered to be employed in future 5G systems in order to increase the power efficiency \cite{NonIdealArr}. It is observed that PA's exhibit nonlinear behavior around saturation region, which yields nonlinear distortion of transmitted signal. In this paper, we investigate the effects of PA nonlinearities on mm-wave Massive MIMO systems and examine compensation methods for such effects.
\subsection{Related Literature}
Nonlinear distortion affects the system performance in several aspects, which are studied in literature \cite{OOBRad:1,OOBRad:2,SpatCharDist,DPD:1,DPD:2,DPD:3,AIR1:1,dinis_ber,abc,SER1,SER2,valkSINR}. In \cite{OOBRad:1,OOBRad:2,SpatCharDist}, out-of-band (OOB) radiation of massive MIMO systems are investigated with a particular focus on spatial characteristics of the radiation and a framework is developed for such analysis. These works consider fully digital beamforming and states that distortion effects are reduced as the number of antennas increases. Also, it was shown that as the number of users increases distortion power becomes isotropic and worst case happens, when a single user is present, where distortion is also concentrated towards that user. In \cite{DPD:1,DPD:2,DPD:3}, hybrid beamforming architecture is considered and both OOB radiation pattern and digital predistortion (DPD) techniques are studied. Different DPD design methods are proposed based on single PA feedback and anti-combining feedback architectures. It was shown that OOB radiation can be significantly reduced by DPD per RF chain. However, in \cite{MBS} we showed that DPD per RF chain cannot mitigate OOB radiation problem for fully connected hybrid architecture. Since, input of each PA is the combination of all RF chains, DPD per RF chain is not sufficient.

Furthermore, effects of nonlinear distortion on the capacity and error probability of the system are studied in  \cite{AIR1:1,dinis_ber,abc,SER1,SER2} and \cite{Capacity:1}. In  \cite{AIR1:1} and \cite{Capacity:1}, analysis on achievable information rate (AIR) is presented showing that capacity of massive MIMO systems is reduced due to nonlinear distortion. However, these studies consider the spectral efficiency in terms of unconstrained Shannon capacity and system is limited to be fully digital. In \cite{MBS}, we extended analysis to a more general framework where mismatched generalized mutual information (GMI) metric, which is  presented in \cite{Capacity:2,GMI:1,GMI:2,Zglgn}, is adopted to evaluate the performance of the system where fully connected hybrid architecture is considered. In literature, bit/symbol error probability (BER/SER) performances of fully digital massive MIMO systems are also studied by using narrowband Bussgang decomposition per antenna (scalar Bussgang with no frequency selectivity). In \cite{dinis_ber,abc,SER1,SER2}, both numerical and analytical BER/SER analysis are presented and it was shown that nonlinear distortion significantly increases BER of the system.

\subsection{Contributions}
In this paper, an extensive performance comparison between different massive MIMO array beamforming structures in JSDM framework under PA nonlinearity is carried out through complete analysis when proper DPD at base station (BS) or post-equalization at user side is utilized. A general analysis framework, covering different massive MIMO array types such as fully digital, fully connected hybrid/partially connected hybrid architectures, is proposed in order to quantify the performance losses in terms of radiation patterns, AIR, and BER under PA nonlinearities for multicarrier downlink transmission with higher order QAM constellations. To the author’s knowledge, in the recent literature, there is no such a comprehensive analytical investigation and comparison made before.

Under the proposed framework, the contribution is two-fold. First, the nonlinearity due to PAs is modeled via vectorial wideband Bussgang decomposition in spatio-frequency domain, while taking the effect of hybrid beamforming structure into account. That is to say, the nonlinear system, from RF chain inputs including DPD units to PA outputs following the analog beamformer, is modeled in upsampled multidimensional signal domain. Here, the proposed modeling considers the spatial correlation among different transmit antennas at different subcarriers due to hybrid massive MIMO structure and PA memory.

Second, based on the general spatio-frequency Bussgang decomposition, peculiar to hybrid array connection type, BER and AIR analysis are fulfilled for different conventional DPD techniques (based on memory polynomial model \cite{GMP}) and post frequency compensation methods at user terminal (UT). The effect of different beamforming types on the BER is demonstrated clearly both via the simulation and provided analysis. Moreover, mismatched decoding capacity via GMI is exploited to find a lower bound for AIR in case of mismatches (depending on array architecture) due to nonlinearity. It is observed that the hybrid array connection type is highly effective on the success of DPD and radiation patterns.

As a side contribution, a novel GEB based partially connected subarray structure is proposed in order to compromise between the OOB pattern, nulling performance and BER.

The remainder of this paper is organized as follows. In Section \ref{SysMod}, system and massive MIMO wideband channel models are presented. In Section \ref{TransSche}, transmission scheme and beamforming structures, which are employed in this study, are introduced. Section \ref{CompMeth} presents the compensation methods that are used to reduce the effects of PA nonlinearities. In Section \ref{PerfMeas}, performance measures, which are used in assessment of the performances, are presented and simulation results are given in Section \ref{NumRes}. Lastly, concluding remarks are stated in Section \ref{conclds}. Throughout the paper, scalars are notated with italic letters as $x$, vectors are notated with bold lower case letters as $\bf x$, matrices are notated with bold capital letters as $\bf X$. Conjugate transpose is denoted as $(\cdot)^H$, $\operatorname{Tr}(\cdot)$ denotes trace, and ${\bf X}[a,b]$ denotes the $(a,b)^{th}$ entry of matrix $\bf X$.

\section{System Model} \label{SysMod}

In the considered system, a base station, which is equipped with $\mathit{D} $ RF chains and $\mathit{N_{t}}$ antennas, serves $\mathit{U}$ single antenna users in time domain duplex (TDD) mode so that channel reciprocity can be exploited. Assuming a block fading frequency selective channel, which is slowly varying compared to signalling interval, received signal at the $\mathit{u^{th}}$ user can be given in time domain as
\begin{equation}
r^{({u})}_{n}=\sum_{l=0}^{L-1} [{\bf{h}}{_{l}^{({u})}}]{^H} {\bf{y}}_{n-l} + \nu^{({u})}_{n}, \label{rec_eq1}
\end{equation}

where, $\mathit{r_n^{({u})}}$ is the received signal, $\mathit{{\bf{h}}_{l}^{({u})}} \in \mathbb{C}^{N_{t} \times 1} $ is the $\mathit{l^{th}}$ delay of the channel impulse response of the $u^{th}$ user, $\mathit{{\bf{y}}_{n}}$  $\in \mathbb{C}^{N_{t} \times 1}$ is the transmitted signal, which is corrupted by the nonlinear PA, and $\mathit{\nu_n^{({u})}}$ is the additive white Gaussian noise (AWGN) with variance $N_o$.

\subsection{mm-Wave Wideband Massive MIMO Channel Model}
In this study, channel under consideration is taken to be sparse in both angular and temporal domains since massive MIMO systems are considered for millimeter (mm) wave frequencies, where channel sparsity is pronounced. By exploiting sparsity, multipath components stemming from the same angular sector can be grouped together so that JSDM can be used to reduce the effective dimension of the problem. In JSDM framework, users that reside in the same angular sector can be processed jointly since they share similar spatial statistics. Spatial characteristics of the multipath components (MPC)'s of each user of group $g$ are described by their CCM's, $\mathit{{\bf{R}}_{l}^{(g_u)}\in \mathbb{C}}^{N_t\times N_t} $, which can be expressed according to one-ring scattering model as \cite{JSDM2}

\begin{equation}
{\bf{R}}_l^{(g_u)} =\frac{{\gamma_l^{(g_u)}}}{2\Delta} \int_{\theta^{g_u,l}_c-\Delta}^{\theta^{g_u,l}_c+\Delta} {\bf{a}}(\theta)[{\bf{a}}(\theta)]^H d\theta,
\label{ccm_eq}
\end{equation}

where $ {\gamma_l^{(g_u)}} $ is the gain of the corresponding channel, which can also be inferred as power delay profile (pdp), and $ \operatorname{Tr}\{{\bf{R}}_l^{(g_u)}\}={\gamma_l^{(g_u)}}$,  $\theta^{g_u,l}_c$ is the center angle of arrival (AoA) for $l^{th}$ MPC of $u^{th}$ user of group $g$, and $ {\bf{a}}(\theta) \triangleq \frac{1}{\sqrt{N_t}} [1 \; e^{j\pi sin(\theta)} \; \ldots\; e^{j (N_t-1) \pi sin(\theta)}]^T$, is the unit norm steering vector for the azimuth angle $\theta$, which corresponds to uniform linear array (ULA) with half wavelength spacing. The spatial channel vector for $l^{th}$ MPC of the $u^{th}$ user in group $g$ is distributed as $ {\bf{h}}_l^{(g_u)} \sim \mathcal{C}\left({\kappa_l^{(g_u)}}{{\bf{a}}(\theta_c^{g_u,l}),{\bf{R}}_l^{(g_u)}} \right) $, where $\theta_c^{g_u,l}$ is the center AoA of the $l^{th}$ MPC of user $g_u$, ${\kappa_l^{(g_u)}}$ is a non-random complex constant with uniform phase distribution, and the channels of different users are independent. In the given channel model, it is assumed that each MPC is Rician distributed where a strong MPC exists at the center of the angular sector with Rician factor, $\left|{{\kappa^{(g_u)}_l}}\right|^2/{{\gamma_l^{(g_u)}}}$  \cite{ricianRef}. Note that, we can construct CCM for each group as, ${\bf R}_l^{(g)} \triangleq \sum_{u=1}^{U_g} {\bf R}_l^{(g_u)}$ \footnote{The design of user grouping algorithm is out of scope of this paper. An efficient procedure can be found in \cite{JSDM} and \cite{UserGr}.}. Then, received signal of the $u^{th}$ user of $g^{th}$ group can be expressed  as $r^{(g_{u})}_{n}=\sum_{l=0}^{L_g-1} \left[{\bf{h}}{_{l}^{(g_{u})}}\right]{^H} {\bf{y}}_{n-l} + \nu^{(g_{u})}_{n}$. Besides, it is assumed that instantaneous channel state information (CSI) is acquired perfectly during the uplink training.

\subsection{Nonlinear PA Distortion}

In order to study the effects of the PA nonlinearities on the performance of the hybrid beamforming system, modelling the nonlinear PA is mandatory. Output of the nonlinear PA can be expressed in terms of arbitrary basis functions as

\begin{equation}
y^{(m)}_{n}=\sum_{\varpi=-\Pi+1}^{\Pi-1}  \sum_{\upsilon =0}^{\Upsilon-1} \alpha_{\varpi,\upsilon}^m\phi_{\varpi,\upsilon}(\bar{x}_{n-\varpi}^{(m)}),\label{pa_output}
\end{equation}

where $ y^{(m)}_n$ is the transmitted signal at the $ m^{th}$ antenna branch, $\phi_{\varpi,\upsilon}(\cdot)$ are the basis functions, $\bar{x}_{n-\varpi}^{(m)}$ is the input of nonlinearity and $\alpha$'s are the basis coefficients. The basis functions can be any function such as Memory Polynomial (MP) \cite{GMP} or Hermite Polynomials \cite{SpatCharDist} basis. In this work we consider MP expansion where the basis functions are $\phi_{\varpi,\upsilon}(\bar{x}_{n-\varpi})= {\bar{x}}(n-\varpi)|{\bar{x}}(n-\varpi)|^{2\upsilon}$.

\input{block_diagram.tex}

\section{A Generic Transmission Scheme} \label{TransSche}
A generic transmitter structure based on hybrid bemforming architecture is shown in Fig. \ref{sys_arch}. In the considered structure, $ U $ users are grouped into $ G $ groups, each having $U_g$ users. In addition, $ D_g$ RF chains are assigned to each group, $ \sum_{g=1}^{G} D_g = D$, where $D<N_t$ is the total number of available RF chains. Main motivation behind the use of hybrid beamforming is that reduced number of RF chains is sufficient in hybrid structure as opposed to fully digital structure, which requires as many RF chains as the number of antennas. Even though, the considered structure defines a hybrid beamforming architecture, it is easily extended to fully digital architecture as $ G=1$ and $D = N_t$. Transmitted upsampled signal can be formed by utilizing
Orthogonal frequency-division multiplexing (OFDM) modulation as

\begin{equation}
{\bf \bar{x}}_n=\frac{1}{\sqrt{K}}\sum_{k=0}^{ \mu K-1} {\bf{B}}{\bf{W}}_k {\bf{d}}_k e^{j\frac{2\pi}{\mu K}kn}, \label{tx_sig_eq}
\end{equation}

where $K$ is the number of QAM modulated active subcarriers, $\mu$ is the oversampling factor, $ {\bf B}\in {\mathbb{C}}^{N_t\times D} $ is the analog beamformer matrix, $ {\bf W}_k\in {\mathbb{C}}^{D\times U} $ is the digital precoder matrix for $k^{th}$ subcarrier, $ \mathbb{\bf{d}}_k = [d_k^1, d_k^2, ..., d_k^{U}]^T \in \mathbb{C}^{U \times 1}$ is composed of $\mathit{i.i.d.}$ QAM modulated data symbols with autocorrelation, $ \mathit{\mathbb{E}\{{\bf{d}}_k{\bf{d}}_{k-l}^H\} = {\bf{I}} \delta_l}$, and constellation order $M$.

Due to impracticality of joint design of digital and analog beamformers, JSDM framework is adopted. In this framework, designs of beamformers are decoupled such that analog beamformer is formed based on second order statistics of the channels, while the digital precoder is designed by using reduced rank instantaneous effective channel. Digital precoder constitutes the first stage of the beamforming, which is used to isolate the intra-group users.

\subsection{Design of Digital Precoder}
In this work, well-known zero-forcing (ZF) precoding is employed, which is designed based on the effective channel matrix of each group, $\mathit{{\bf{H}}_{eff,l}^{(g)}} \triangleq [{\bf{B}}^{(g)}]^H{\bf{H}}_{l}^{(g)}$ ,where $ {\bf{B}}^{(g)} \in \mathbb{C}^{N_t \times D_g }$ for $ g = 1,...,G $ is the sub-beamforming matrix for group $ g $ and $ {\bf{H}}_{l}^{(g)} = [{\bf{h}}_{l}^{(g_1)}, ..., {\bf{h}}_{l}^{(g_{U_g})} ]$ is the full dimension channel matrix for group $g$. Analog beamforming matrix is formed as the combination of the sub-beamformers for each group as, $ {\bf{B}} = [{\bf{B}}^{(1)}, {\bf{B}}^{(2)}, ..., {\bf{B}}^{(G)}]$. In general, channel estimation is performed in reduced dimension; however, we assume that full rank channel is available at the BS since channel estimation is not in our scope. Frequency domain regularized ZF precoder with regularization parameter $\delta$ for a fixed analog beamformer is obtained as

\begin{equation}
{\bf {W}}_k^{(g)} \triangleq {{\bf{\Omega}}^{(g)}_{eff,k}}\left([{{\bf{\Omega}}^{(g)}_{eff,k}}]^H {{{\bf{\Omega}}^{(g)}_{eff,k}}} + \delta {\bf I}_{U_g}\right)^{-1},\label{ZF_ofdm}
\end{equation}

where ${\bf{\Omega}}^{(g)}_{eff,k}= [{\bf{B}}^{(g)}]^H {\bf{\Omega}}^{(g)}_k$ is effective channel matrix in frequency domain and $ {\bf{\Omega}}_{k}^{(g)} = \sum_{l=0}^{\mu K-1}{\bf{H}}{_{l}^{(g)}}e^{-j2\pi l \frac{k}{\mu K}} $ is the frequency domain response for subcarrier $k $ of group $ g $.

Overall digital precoding matrix has the block diagonal form ${\bf{W}}_k \triangleq  \operatorname{bdiag}\{ \sqrt{c^{(g)}} {\bf{W}}_{k}^{(g)} \}_{g=1}^{G}$, where $c^{(g)} $ is the power scaling factor, then digitally precoded upsampled transmitted signal can be expressed as, ${\bf \bar{x}}_n=\frac{1}{\sqrt{K}}\sum_{k=0}^{ \mu K-1} \sum_{g=1}^{ G} \sqrt{c^{(g)}} {\bf{B}}^{(g)}{\bf{W}}^{(g)}_k {\bf{d}}_k^{(g)} e^{j\frac{2\pi}{\mu K}kn}$, where ${\bf d}_k^{(g)} \triangleq [d_k^{(g_1)}\; \ldots \; d_k^{(g_{U_g})}]^T$. In the considered scenario, per group power constraint is adopted to obtain  $c^{(g)}  = E_s/(G P^{(g)}) $, where $E_s = \mathbb{E}\{ |{\bf \bar{x}}_n|^2 \}$ is the total transmit power, and ${ P^{(g)}} $ is
\begin{equation}
{ P^{(g)}} \triangleq  \frac{1}{K} \sum_{k=0}^{K-1} \operatorname{Tr}\left\{ {\bf B}^{(g)} {\bf{W}}_{k}^{(g)} \left[{\bf{W}}_{k}^{(g)}\right]^H \left[{\bf B}^{(g)}\right]^H\right\}.
\label{pow_sc}
\end{equation}

Based on the power constraint expression in \eqref{pow_sc}, it can be inferred that transmit power is kept constant for each channel realization.

\subsection{Design of Analog Beamformer}
In hybrid structures, analog beamforming is applied to radiate the energy in the direction of the group of the intended user. Analog beamformers can be classified into two groups based on the antenna array structure, namely "fully connected" and "partially connected" arrays. In fully connected array architecture, shown in Fig. \ref{array_arch}(a), each antenna input is combination of all RF chains, while in partially connected array architecture, shown in Fig. \ref{array_arch}(b), each antenna input is dedicated to a single RF chain.

\input{BF_archs.tex}

\subsubsection{Fully Connected Array}

For this architecture, angle only generalized eigen beamformer (AO-GEB), which is shown to be nearly optimal for several criteria in \cite{GEB:1}, is employed. Due to frequency domain equalization performed in OFDM modulation, CCM of the frequency domain response should be used in design of analog beamformer. Since each MPC of a group is independent, we can show that $  \mathbb{E}\{ {\bf{\Omega}}_{k}^{(g)} [{\bf{\Omega}}_{k}^{(g)}]^H\} = \sum_{l=0}^{L_g} {\bf{R}}_{l}^{(g)}$. Hence, summation of CCM's of MPC's is used in GEB design. For each group, a sub-beamformer is formed as the first dominant $D_g$ eigenvectors of the generalized eigenvalue problem \cite{GEB:1}
\begin{equation}
{\bf B}^{(g)} = \operatorname{eigs}({\bf R}_{sum}^{(g)},{\bf R}_{sum},D_g), \label{ao_geb}
\end{equation}

where ${\bf{R}}_{sum}^{(g)} \triangleq  \frac{E_s }{G} \sum_{l=0}^{L_g-1}{\bf{R}}^{(g)}_l$ and ${\bf{R}}_{sum} \triangleq \frac{E_s}{G}\sum_{g=1}^{G} \sum_{l=0}^{L_g-1}{\bf{R}}^{(g)}_l + N_o \bf I$. In GEB, objective is to maximize the radiated power in the direction of the intended group while minimizing that of the other groups, and GEB here is a kind of statistical Capon beamformer for general-rank signal models \cite{GEB:3}. Despite the fact that fully connected array architecture performs close to fully digital beamforming, it suffers from high complexity of implementation.

\subsubsection{Partially Connected Array}

Partially connected array architecture is a feasible structure to reduce the complexity of the array. For this architecture, analog beamforming matrix has the block diagonal form ${\bf B}^{(g)} = \operatorname{bdiag}\{{\bf b}_1^{(g)} , {\bf b}_2^{(g)}, ...,{\bf b}_{D_g}^{(g)}\}$, where $ {\bf b}_d^{(g)} = [b_d^1,b_d^2,...,b_d^{N_s}]^T \in \mathbb{C}^{N_s \times 1 }$ is the sub-beamformer for the antenna array with $N_s$ antennas of $ d^{th}$ RF chain and ${\bf B} = \operatorname{bdiag} \{{\bf B}^{(1)}, \; \ldots, \; {\bf B}^{(G)}\}$. Phase only beamforming is a common approach, in which beamforming is applied via phase shifters where  $|b_d^{m}| =1$ $ \forall d,m $. In this work, DFT beamformer is employed for phase only subarray, which steers toward the AoA of the strong MPC of each user without considering inter-group interference \cite{GEB:2}.

In addition to phase only subarray, we propose a novel sub-beamformer design method for partially connected array architecture based on generalized eigen beamforming. Since phase only beamformers cannot suppress the inter-group interference sufficiently, we adopt the GEB design, which is shown to be optimal for fully connected array architecture. In the proposed method, each sub-beamformer is obtained as the eigenvectors of the eigenvalue problem
\begin{equation}
[{\bf b}_1^{(g)} , {\bf b}_2^{(g)}, ...,{\bf b}_{D_g}^{(g)}] = \operatorname{eigs}({\bf R}_{sub,sum}^{(g)},{\bf R}_{sub,sum},D_g), \label{sub_geb}
\end{equation}

where ${\bf{R}}_{sub,sum}^{(g)} \triangleq  \frac{E_s}{G} \sum_{l=0}^{L_g-1}{\bf{R}}^{(g)}_{sub,l}$ and ${\bf{R}}_{sub,sum} \triangleq \frac{E_s}{G}\sum_{g=1}^{G} \sum_{l=0}^{L_g-1}{\bf{R}}^{(g)}_{sub,l} + N_o \bf I$, where ${\bf R}_{sub,l}^{(g)} \triangleq \sum_{u=1}^{U_g} {\bf R}_{sub,l}^{(g_u)}$, and $ {\bf{R}}^{(g_u)}_{sub,l} \in \mathbb{C}^{N_s \times N_s } $ is the sub-CCM of $l^{th}$ MPC of user $g_u$ in the reduced dimension, which is obtained by using an array of size $N_s$ placed at the origin. This yields a phase offset in the effective channel matrix; however, digital precoding makes necessary phase correction in digital domain.

\section{Compensation Methods for PA Nonlinearities} \label{CompMeth}

Two methods will be presented in order to compensate the effects of PA nonlinearities. First method is the well known DPD, which aims to predict the distortion caused by the nonlinear PA and predistorts the signal such that overall transmitted signal has the desired form. The other compensation method is based on Bussgang decomposition for each subcarrier at the user terminal. This method corresponds to frequecy domain equalization for the linear channel formed by the memory of PA, which is not considered in design of the digital precoder.

\subsection{DPD for Hybrid Beamforming Architecture}

In this section, we present the DPD design framework for hybrid beamforming architecture, which employs anti-beamforming in order to reduce the dimension of the observation space to the number of RF chains. In literature, there are several methods that address anti-beamforming, which is implemented by reverse phase shifting; however, these works consider phase only subarray networks as analog beamformer. However, this approach is not valid for the proposed GEB since beamforming gains are not unity in general and required anti-beamforming is more complicated. Consider the transmitted signal, $\mathit{{\bf{y}}_n} = [y_{n}^{(1)} , y_{n}^{(2)}, ... , y^{(N_t)}_n]^T $, which can be expressed in terms of MP basis functions as ${\bf{y}}_n = \sum_{\varpi=-\Pi+1}^{\Pi-1}  \sum_{\upsilon =0}^{\Upsilon-1} \alpha_{\varpi,\upsilon} {\bf \bar{x}}_{n-\varpi}   \odot |{\bf \bar{x}}_{n-\varpi}|^{2\upsilon}$, where $ {\bf{\bar{x}}}_n $ is the analog beamformed signal. By expressing $ {\bf{\bar{x}}}_n $ in terms of digital predistorted signal, one can obtain

\begin{equation}
{\bf{y}}_n = \sum_{\varpi=-\Pi+1}^{\Pi-1}  \sum_{\upsilon =0}^{\Upsilon-1} \alpha_{\varpi,\upsilon} [({\bf B\hat{x}}_{n-\varpi}) \odot |{\bf B \hat{x}}_{n-\varpi}|^{2\upsilon}], \label{pa_y_sig}
\end{equation}

where $ \odot $ denotes the Hadamard product and for simplicity all PA's are assumed to be identical. In the proposed architecture, $ N_t \times 1 $ observation vector, $ {\bf y}_n$, is used to design $ D $ DPD units. Hence, it is necessary to project the $ N_t \times 1 $ observations to the reduced dimension of $ D \times 1 $. For this purpose, consider the beamformed signal, $ \mathit{\bf{\bar{x}}}_u = {\bf{B}}\mathit{\bf{\hat{x}}}_n $, where it can be written in terms of sub-beamformers ${\bf{\bar{x}}}_u = {\bf{B}}^{(1)}{\bf{\hat{x}}}^{(1)}+ {\bf{B}}^{(2)}{\bf{\hat{x}}}^{(2)} + \ldots + {\bf{B}}^{(G)}{\bf{\hat{x}}}^{(G)} \label{bf_sign}$.

In this study, anti-beamforming is performed by projecting the observation vector onto the subspaces, which are spanned by the sub-beamformers. DPD's for each group are designed based on the observation vector projected onto corresponding sub-beamformer space. Proposed approach employs pseudo-inverses of the sub-beamformers for the projection via anti-beamforming matrix, $ {\bf B}_{ab} =  [ ([{\bf{B}}^{(1)}]^\#)^T,([{\bf{B}}^{(2)}]^\#)^T , ..., ([{\bf{B}}^{(G)}]^\#)^T ]^T $, where $[{\bf{B}}^{(g)}]^\# = \left([{\bf{B}}^{(g)}]^H {\bf{B}}^{(g)}\right)^{-1}[{\bf{B}}^{(g)}]^H$ is the pseudo inverse matrix of $ {\bf{B}}^{(g)} $. Then the projected observation vector, ${\bf \tilde{x}}_n = {\bf B}_{ab} {\bf{y}}_n$, becomes

\begin{equation}
\begin{split}
{\bf \tilde{x}}_n = \alpha_{0,0}  {\bf B}_{ab} {\bf B\hat{x}}_n +
 \sum_{\substack{\upsilon =1}}^{\Upsilon-1} \alpha_{0,\upsilon} {\bf B}_{ab} [({\bf B\hat{x}}_n) \odot |{\bf B \hat{x}}_{n}|^{2\upsilon}] +
\\
 \sum_{\substack{{\varpi=-\Pi+1,} \\ { \varpi \neq 0}}}^{\Pi-1}  \sum_{\substack{\upsilon =0}}^{\Upsilon-1} \alpha_{\varpi,\upsilon} {\bf B}_{ab} [({\bf B\hat{x}}_{n-\varpi}) \odot |{\bf B \hat{x}}_{n-\varpi}|^{2\upsilon}],
\label{a_bf1}
\end{split}
\end{equation}

where $ {\bf B}_{ab} {\bf B} \approx {\bf I}_D$ since sub-beamformers are designed to be nearly orthogonal to each other. Second and third terms in \eqref{a_bf1} correspond to remaining interference, which depends on the distorted signals on all RF chains due to nonlinearity. Power of the interference term limits the performances of DPD's since each DPD is designed by using a single RF chain. After obtaining the projected values, one can form the error signal as ${\bf e}_n = {\bf \hat{x}}_{n} - {\bf \tilde{x}}_n$. Error signal, ${\bf e}_n$, is used to design the polynomial DPD by using indirect learning architecture (ILA) for $ d^{th}$ RF branch

\begin{equation}
{{\hat{x}}}_n^{(d)} = \sum_{\varpi=-\Pi'+1}^{\Pi'-1}  \sum_{\upsilon =0}^{\Upsilon'-1} w^d_{\varpi,\upsilon} {x}^{(d)}_{n-\varpi} |x^{(d)}_{n-\varpi}|^{2\upsilon},   \label{dpd_exp}
\end{equation}

where, $ w^d_{\varpi,\upsilon}$ are the DPD coefficients and they can be found by using least squares (LS) method proposed in \cite{GMP} as

\begin{equation}
{\bf w}_b^d ={\bf w}_{b-1}^d + \beta ({ \bf X}_d^H { \bf X}_d)^{-1}{ \bf X}_d^H {\bf e}^{(d)}, \label{dpd_coeff}
\end{equation}

where $ { \bf w}_b^d $ is the DPD coefficient vector calculated at $b^{th}$ block, ${ \bf X}_d $ is the observation matrix whose elements are $ \tilde{x}^{(d)}_{n-\varpi} |\tilde{x}^{(d)}_{n-\varpi}|^{2\upsilon}$, $ {\bf e}^{(d)} \triangleq \left[{\bf e}_n[d] \; {\bf e}_{n-1}[d] \;...\; {\bf e}_{n-P+1}[d]\right]^T $ is the error vector for $d^{th}$ RF branch, where ${\bf e}_n[d]$ is the $d^{th}$ element of ${\bf e}_n$, $P$ is the number of samples, and $\beta$ is the step-size for the adaptation. In this study, memory and nonlinearity order for the DPD is chosen as $\Pi' = 4$ and $\Upsilon' = 4$, respectively.

\subsection{Post Equalization at User Terminal}

A linear post equalizer is proposed in order to compensate the memory effects of PA's on the transmit chain as a simpler alternative to DPD. At BS, a pre-equalization is applied by means of digital precoding based on CSI acquired by uplink training. However, there is an undesired linear channel due to memory of PA's, which is ignored during the design of digital precoder. Therefore, an equalization procedure is necessary at the receiver side. In this work, Bussgang decomposition is applied for each sub-carrier at the receiver. Received signal at $g_u^{th}$ user after FFT operation for each subcarrier, $r^{(g_u)}_k $, can be expressed as \cite{SER1}
\begin{equation}
r^{(g_u)}_k = \alpha_k^{(g_u)} d^{(g_u)}_k + \eta^{(g_u)}_k,
\end{equation}

where $\alpha_k^{(g_u)}$ is the frequecy domain Bussgang coefficient and $\eta^{(g_u)}_k$ is the distortion term. Since Bussgang coefficient is defined for each subcarrier, it can be interpreted as the frequency domain channel for the corresponding subcarrier as

\begin{equation}
\alpha_k^{(g_u)}= \frac{\mathbb{E}\{ r_k^{(g_u)}[d_k^{(g_u)}]^{*}\}}{\mathbb{E}\{ |d_k^{(g_u)}|^2 \}}. \label{buss_coeff_post}
\end{equation}

By using scalar channel coefficient given in \eqref{buss_coeff_post}, symbol estimate is obtained as

\begin{equation}
\hat{d}_k^{(g_u)} = \frac{r_k^{(g_u)}}{\alpha_k^{(g_u)}}.
\end{equation}

\section{Performance Measures}\label{PerfMeas}
In this study, performances of the beamforming structures are evaluated via several performance criteria. Firstly, power spectral density (PSD) of the received signal at different locations  are evaluated. Beampatterns for the in-band and out-of-band radiation are presented in order to understand the effect of hybrid beamforming on the radiation pattern for different architectures. Then, by using GMI metric, AIR's for the considered architectures are obtained via the mismatched capacity framework. Lastly, a comprehensive analysis on BER performance is pursued by deriving signal to interference plus noise ratio (SINR) expressions for different array architectures.

\subsection{Radiation Patterns}

Radiation pattern is evaluated by investigating the PSD of the recieved signal at different angles. For this purpose, channel for any direction, $(\theta)$, is approximated as the steering vector, ${\bf a}^H(\theta)$, pointing that direction.

Then PSD,  $S_r^\theta(f) = \sum_{t = - \infty}^{\infty} R^\theta_t e^{-j2 \pi f t}$, is obtained by taking the Fourier Transform of the autocorrelation function, $R^\theta_t = {\mathbb{E} } [r_n^\theta(r_{n-t}^\theta)^H]$, of the artificially generated received signal, $r^\theta_n = {\bf a}^H(\theta) {\bf y}_n$. Using $S_r^\theta(f) $, in-band and out-of-band radiation powers are calculated as

\begin{equation}
P_{ib}(\theta) = \int_{-B/2}^{B/2} S_r^\theta(f) df, \quad P_{ob}(\theta) = max\left \{\int_{-3B/2}^{-B/2} S_r^\theta(f) df, \int_{B/2}^{3B/2} S_r^\theta(f) df \right\},\label{ib}
\end{equation}

where $ P_{ib}(\theta)$ and $ P_{ob}(\theta)$ are the in-band and OOB radiation patterns in $\theta $ direction respectively, and $B = \frac{2\pi}{\mu}$ is the normalized signal bandwidth.

\subsection{Generalized Mutual Information (GMI)}

In this section, mismatched GMI concept, which is used to obtain a lower bound for the achievable rate for the channels with unknown  probability density function (PDF), is presented. In the considered problem, PA can be considered as a nonlinear channel, whose PDF is not known. In \cite{Capacity:2}, constrained capacity for the channels with known PDF is expressed by using GMI as

\begin{equation}
C = \log_2M - E_{d,r}\left[\log_2\left(\frac{\sum_{d'\in A_d} p(r|d')}{p(r|d)}\right)\right],
\end{equation}

where $A_d$ is QAM symbol alphabet, $M$ is the modulation order. $p(r|d)$ is the conditional PDF of the decoded signal $ r$ given that symbol $d$ is sent. In AWGN channel case, $p(r|d)$ has the Gaussian form; however, due to the nonlinearity caused by PA, Gaussian PDF is not valid. Thus, in this study, we employ an approximation of this PDF so that mismatched capacity introduced in\cite{GMI:1,GMI:2,Zglgn}, can be employed. A lower bound on the mismatched decoding capacity, where assumed PDF is utilized, is expressed as

\begin{equation}
C_M = \log_2M - E_{d,r}\left[\log_2\left(\frac{\sum_{d'\in A_d} \tilde {p}(r|d')}{\tilde {p}(r|d)}\right)\right], \label{capacity}
\end{equation}

where $ \tilde {p}(r|d') $ is the mismatched PDF.

In order to write the mismatched PDF, one should relate the decoded symbol to actual symbol. For this purpose, we use Bussgang theorem in order to decompose the signal into desired and distortion terms in frequency domain as, $r_k = \hat{\alpha}_k d_k + \eta_k$, where $\hat{\alpha}_k$ is the Bussgang coefficient per subcarrier and $ \eta_k$ is the distortion term with power $ \sigma^2_{\eta_k} = \mathbb{E}[\eta_k^H\eta_k]$ and it is orthogonal to linear signal term, $\mathbb{E}\left[d_k^H\eta_k\right]=0$. Bussgang coefficient, $\hat{\alpha}_k$ and the distortion variance, $\sigma^2_{\eta_k}$ for each subcarrier $k=0,1,\ldots,K-1$ can be found by Wiener filtering

\begin{equation}
\hat{\alpha}_k = \frac{\mathbb{E}[d_k^Hr_k]}{\mathbb{E}[d_k^Hd_k]},  \qquad
\sigma^2_{\eta_k} = \mathbb{E}[|r_k - \hat{\alpha}_k d_k|^2],
\label{buss_coef}
\end{equation}

and mismatched PDF is expressed in terms of Bussgang coefficient and the distortion power as

\begin{equation}
\tilde {p}(r_k|d_k)  = \frac{1}{\pi \sigma^2_{\eta_k}} \operatorname{exp}\left( - \frac{|r_k-{\hat{\alpha}}_k d_k|^2}{\sigma^2_{\eta_k}} \right), \label{PDFpost}
\end{equation}

which is valid for systems that utilizes equalization at UT's. Also, mismatch PDF can also be written for the systems without any receiver processing by replacing ${\hat{\alpha}}_k$ with $\hat \alpha$, which is a single phase and amplitude correction term defined as
\begin{equation}
\hat{\alpha} = \frac{\sum_{k=0}^{K-1}r_k d_k^*}{\sum_{k=0}^{K-1}|d_k|^2}. \label{buss_coef1}
\end{equation}
Bussgang coefficient defined in \eqref{buss_coef1} can also be used for the systems employing DPD at the transmitter since DPD also compensates the memory effects of the PA's. Hence, equalization at UT is not necessary. Finally, achievable capacity bound can be obtained by inserting mismatch PDF expression \eqref{PDFpost} into \eqref{capacity} after averaging over subcarriers. In \eqref{buss_coef} and \eqref{buss_coef1}, Bussgang coefficients are considered as independent of constellation point due to multiplexing of different symbols in precoders.

\subsection{Bit-Error-Rate Analysis}

A comprehensive analysis on BER performances of the different array structures is carried out to obtain an analytical approximation for probability of bit error. A general Bussgang decomposition is developed for multidimensional signal domain in which Bussgang coefficient has a matrix form rather than a single scalar due to correlation between signals on different antennas. Based on proposed Bussgang decomposition, SINR expression is obtained and BER is expressed as a function of SINR. For this purpose, nonlinear channel is modelled in frequency domain for different array architectures. For the rest of the analysis, digitally precoded, ${\bf x}_k^f$, and analog beamformed, ${\bf y}^f_k$, signals are defined in frequency domain as

\begin{equation}
{\bf x}_k^f = \frac{1}{\sqrt{\mu K}}\sum_{n=0}^{\mu K-1} {\bf x}_n e^{-j\frac{2\pi}{\mu K}nk }, \qquad

{\bf y}_k^f = \frac{1}{\sqrt{\mu K}}\sum_{n=0}^{\mu K-1} {\bf y}_n e^{-j\frac{2\pi}{\mu K}nk }.
\end{equation}

\subsubsection{Fully Connected Array}

Conventional approaches consider Bussgang decomposition on PA basis; however, in this work, we propose a new decomposition scheme where analog beamformer is also taken into account together with PA's. Consequently, a spatio-frequency Bussgang decomposition is performed which is a mapping from reduced dimensional digital precoded signal vector, ${\bf x}_k^f$, to full dimensional transmitted signal, ${\bf y}_k^f$. Motivated by this approach, Bussgang decomposition is applied in matrix-vector form for each frequency bin, $k$, as
\begin{equation}
{\bf y}^f_k = {\bf A}_k{\bf x}_k^f + {\boldsymbol{\eta}}_k, \label{fully_buss}
\end{equation}

where ${\bf A}_k \in \mathbb{C}^{N\times D}$ can be considered as the linearized analog beamforming matrix after nonlinear amplification and $ {\boldsymbol{\eta}}_k \in \mathbb{C}^{N_t \times 1}$ is the nonlinear distortion vector for $k=0,1,...,\mu K-1$. In addition, ${\bf A}_k$ can be written in terms of sub-beamformers as ${\bf A}_k = [{\bf A}_k^{(1)},{\bf A}_k^{(2)},...,{\bf A}_k^{(G)}]$, which will be used later while defining different interference terms. ${\bf A}_k$ can be obtained as the generalization of the scalar Bussgang decomposition to matrix form

\begin{equation}
{\bf A}_k = \mathbb{E}\left[{\bf y}^f_k({\bf x}_k^f )^H\right]\left(\mathbb{E}\left[{\bf x}^f_k({\bf x}_k^f )^H\right]\right)^{-1}.
\end{equation}

One should note that if all the PA's were linear then Bussgang matrix would be analog beamformer matrix, ${\bf A}_k = \bf B$.
Due to correlation between different antennas, distortion terms are also correlated. Hence autocorrelation matrix for the distortion vector, ${\bf R}_{\boldsymbol{\eta}}^k$, is computed as
\begin{equation}
{\bf R}^k_{\boldsymbol{\eta}} \triangleq \mathbb{E} [{\boldsymbol{\eta}}_k {\boldsymbol{\eta}}_k^H]=  \mathbb{E}\left[  \left({\bf y}^f_k  -{\bf A}_k{\bf x}_k^f \right) \left({\bf y}^f_k  -{\bf A}_k{\bf x}_k^f \right)^H \right]. \label{acm}
\end{equation}

\subsubsection{Partially Connected Array}
In partially connected array structure, Bussgang coefficients are defined as vectors for each sub-array since each sub-array is driven by a single RF chain. Hence, transmitted signal for the sub-array of $g_d^{th}$ RF chain can be expressed as
\begin{equation}
{\bf y}^{(g_d),f}_k = {\bf a}^{(g_d)}_k{ x}_k^{(g_d),f} + {\boldsymbol{ \eta}}_k^{(g_d)}, \quad d = 1,\ldots, D_g\quad g = 1,\ldots,G \label{sub_sig}
\end{equation}

where $g_d$ is the $d^{th}$ RF chain for group $g$. Then linearized sub-beamformer can be expressed as
\begin{equation}
{\bf A}_k^{(g)} = {\bf e}_{G,g} \otimes \sum_{d=1}^{D_g} {\bf E}_{D_g,d} \otimes  {\bf a}^{(g_d)}_k, \label{subarraybuss}
\end{equation}

where $\otimes$ denotes Kronecker product, ${\bf e}_{G,g} $ is the $G$ dimensional elementary vector whose elements are $0$ except $g^{th}$ entry which is $1$, and ${\bf E}_{D_g,d} $ is the $D_g$ dimensional elementary matrix whose elements are $0$ except $d^{th}$ diagonal entry which is $1$. By using signal model given in \eqref{sub_sig}, one can define Bussgang vector for partially connected array as

\begin{equation}
 {\bf a}^{(g_d)}_k = \frac{ \mathbb{E}\left[{\bf y}^{(g_d),f}_k\left({x}_k^{(g_d),f} \right)^*\right]}{\mathbb{E}[|x_k^{(g_d),f}|^2]},
\end{equation}

and distortion covariance matrix, ${\bf R}^k_{\boldsymbol{\eta}} = \operatorname{bdiag}\left \{ \operatorname{bdiag} \left \{ {\bf R}^{(g_d),k}_{\boldsymbol{\eta}} \right \}_{d=1}^{D_g} \right\}_{g=1}^G$, has the block diagonal form since only the distortion terms in the same subarray are correlated, where ${\bf R}^{(g_d),k}_{\boldsymbol{\eta}} \triangleq \mathbb{E}\left[{{\boldsymbol{\eta}}}^{(g_d)}_k \left({{\boldsymbol{\eta}}}^{(g_d)}_k \right)^H\right] = \mathbb{E} \left[ ({\bf y}^{(g_d),f}_k - {\bf a}^{(g_d)}_k{ x}_k^{(g_d),f}) ({\bf y}^{(g_d),f}_k - {\bf a}^{(g_d)}_k{ x}_k^{(g_d),f})^H\right] $ is the autocorrelation matrix of the distortion terms of ${g_d}^{th}$ subarray.

\subsubsection{Fully Digital Beamforming}
For fully digital beamforming architectures, the standard Bussgang decomposition can be applied for each antenna in the system

\begin{equation}
{y}^{m,f}_k = {a}^{(m)}_k{ x}_k^{(m),f} + {\eta}^{(m)}_k, \quad k = 0,\ldots, \mu K-1, \label{dig_sig}
\end{equation}

where ${a}^{(m)}_k $ is the Bussgang coefficient for $m^{th}$ antenna defined as in \eqref{buss_coef}.

Having introduced the spatio-frequency Bussgang decomposition in multidimensional domain, equivalent received signal model at UT's can be obtained. Firstly, consider the input to nonlinear channel before the analog beamformer
\begin{equation}
{\bf x}_n = \frac{1}{\sqrt{K}} \sum_{k =0}^{\mu K-1}  \operatorname{bdiag}\{ \sqrt{c^{(g)}} {\bf W}_k^{(g)} \}_{g=1}^G {\bf d}_k e^{j \frac{2 \pi}{\mu K}kn},
\end{equation}

and also define the channel of the user $g_u$ in frequency domain as the $u^{th} $ column of the channel matrix in frequency domain, $ {\boldsymbol{ \omega}}_k^{(g_u)} \triangleq {\boldsymbol{ \Omega}}_k^{(g)}[:,u]$ and digital precoder for the same user is $ {\bf w}_k^{(g_u)} \triangleq{\bf W}_k^{(g)}[:,u]$ for $u = 1,...,U_g$. After taking DFT at UT for user $g_u$, one can express the received signal in frequency domain by using \eqref{fully_buss} as
\begin{equation} \label{rec_UT}
\begin{split}
r_k^{(g_u),f} &= [{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\bf y}_k^f+\nu_k^{(g_u),f}, \qquad  k=0,1,...,K-1 \\
&= [{\boldsymbol{ \omega}}_k^{(g_u)}]^H({\bf A}_k{\bf x}_k^f + {\boldsymbol {\eta}_k})+\nu_k^{(g_u),f},\\
&= [{\boldsymbol{ \omega}}_k^{(g_u)}]^H\left[\sum_{g=1}^G \sqrt{\frac{c^{(g)}}{K}}{\bf A}_k^{(g)}{\bf W}_k^{(g)}{\bf d}_k^{(g)}+{\boldsymbol {\eta}}_k\right]+\nu_k^{(g_u),f}.
\end{split}
\end{equation}

Received signal in \eqref{rec_UT} is decomposed such that each interference term can be measured as
\begin{equation}
\begin{split}
r_k^{(g_u)} &= \underbrace{\sqrt{{\frac{c^{(g)}}{K}}}[{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\bf A}_k^{(g)}{\bf w}_k^{(g_u)}d_k^{(g_u)}}_\text{desired signal term}+ \underbrace{\sqrt{{\frac{c^{(g)}}{K}}}\sum_{\substack{u'=1 \\ u' \neq u}}^{U_g}[{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\bf A}_k^{(g)}{\bf w}_k^{(g_{u'})}d_k^{(g_{u'})}}_\text{intra-group interference}\\
&+\underbrace{\sum_{\substack{g'=1 \\ g'\neq g}}^{G}\sqrt{{\frac{c^{(g')}}{K}}}[{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\bf A}_k^{(g')}{\bf W}_k^{(g')}{\bf d}_k^{(g')}}_\text{inter-group interference}+ \underbrace{[{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\boldsymbol{ \eta}}_k}_\text{nonlinear distortion} + \underbrace{\nu_k^{(g_u),f}}_\text{AWGN}. \label{rec_dec}
\end{split}
\end{equation}

By using \eqref{rec_dec}, average received SINR at the UT for a particular channel is obtained as
\begin{equation}
SINR^{(g_u)}_k = \frac{\mathbb{E}_{\bf H}\left[{\frac{c^{(g)}}{K}} \left| [{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\bf A}_k^{(g)}{\bf w}_k^{(g_u)}\right|^2 \; \middle| \; {\bf h}^{(g_u)}\right]}{\mathbb{E}_{\bf H}\left[\sigma^2_{I,N,k} \; \middle| \;{\bf h}^{(g_u)}\right]}\label{SINR}
\end{equation}

where $\sigma^2_{I,N,k}$ is the total interference-plus-noise power expressed as
\begin{equation}
\begin{split}
\sigma^2_{I,N,k}  =&{ {{\frac{c^{(g)}}{K}}}\sum_{\substack{u'=1 \\ u' \neq u}}^{U_g}|[{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\bf A}_k^{(g)}{\bf w}_k^{(g_{u'})}|^2}  +  \sum_{\substack{g'=1 \\ g'\neq g}}^{G}{{\frac{c^{(g')}}{K}}}[{\boldsymbol{ \omega}}_k^{(g_u)}]^H{\bf A}_k^{(g')}{\bf W}_k^{(g')}[{\bf W}_k^{(g')}]^H  [{\bf A}_k^{(g')}]^H{\boldsymbol{ \omega}}_k^{(g_u)}\\
& + [{\boldsymbol{ \omega}}_k^{(g_u)}]^H {\bf R}_{{\boldsymbol{\eta}}}^k{\boldsymbol{ \omega}}_k^{(g_u)}  + N_o.
\end{split}
\end{equation}

By using SINR expression given in \eqref{SINR}, the average BER can be obtained as

\begin{equation}
P_b \approx \frac{1}{UK} \sum_{g=1}^{G} \sum_{u=1}^{U_g} \sum_{k=0}^{K-1} \frac{4}{\log_2(M)}\left( 1- \frac{1}{\sqrt{M}} \right) \mathbb{E}_{{\bf h}^{(g_u)}} \left[ Q\left(\sqrt{\frac{3}{M-1}SINR^{(g_u)}_k} \right) \right],
\end{equation}

where $Q(x)=\frac{1}{\sqrt{2 \pi}} \int_x^{\infty} e^{(-t^2/2)}dt$.

\section{Numerical Results} \label{NumRes}

Numerical results are presented to compare different beamforming architectures under PA nonlinearities in terms of their radiation patterns, AIR and BER performances. Furthermore, analytical expression of BER is verified by the Monte Carlo trials. In this study, all PA's are assumed to be identical but their operating points depend on the gain of analog beamforming. In the simulations, PA model for $\sim 2$ GHz commercially available GaAs PA, which is presented in \cite{WEBSITE:1}, is used. $N_t = 96$ antennas having ULA geometry is placed at BS. There are $D = 6$ RF chains so each sub-array consists $N_s = 16$ antennas in partially connected array structure. In the considered scenario, there are $G=3$ groups, each group having $2$ users and also there is a victim user who is served by another BS, whose CCM is computed and involved in ${\bf R}_{sum}$ to place deep null in that direction. The user and victim distribution is summarized in Table  \ref{table:scenario}. It is assumed that a strong MPC always exists for each angular sector with Rician factor $\left|{{\kappa^{(g_u)}_l}}\right|^2/{{\gamma_l^{(g_u)}}}=10$. Also, the power of the first MPC is $3 \; dB$ larger than the power of second MPC and they are seperated by $2$ symbols duration. In generated OFDM signal, number of subcarriers and IFFT size are $K =550$  and $\mu K = 4096$, respectively, and cyclic prefix length is selected as $N_{cp}=20$.

\begin{table}
\caption{Scenario}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}

\hline
Group  &  \shortstack{ AS of $1^{st}$ MPC of \\ \vspace{0.5mm} \\ User $1$} & \shortstack{ AS of $1^{st}$ MPC of \\ \vspace{0.5mm} \\ User $2$} & \shortstack{ AS of $2^{nd}$ MPC of \\ \vspace{0.5mm} \\ User $1$} &\shortstack{ AS of $2^{nd}$ MPC of \\ \vspace{0.5mm} \\  User $2$} \\
\hline
1 &  ${[-28^\circ , -25^\circ]}$ & ${[-25^\circ , -22^\circ]}$  & ${[-17^\circ , -14^\circ]}$  & ${[-14^\circ , -11^\circ]}$ \\
\hline
2 &  ${[-4^\circ , -1^\circ]}$ & ${[-1^\circ , 2^\circ]}$ &  ${[8.5^\circ , 11.5^\circ]}$ &${[11.5 ^\circ, 14^\circ]}$ \\
\hline
3 &   ${[24^\circ , 27^\circ]}$& ${[21^\circ , 24^\circ]}$ & - & -\\
\hline
Victim &  ${[-39^\circ , -36^\circ]}$ & - & - & -\\
\hline
\end{tabular}
\label{table:scenario}
\end{center}
\end{table}
\emph{Power Spectral Density and Radiation Pattern Analysis:}
Power spectra and radiation patterns of the array structures are obtained in order to assess the severity of the unwanted power emissions in both frequency and spatial domains. In Fig. \ref{AvgPSD}, average PSD's of signals received by the users are shown. It is observed that OOB is a significant problem for all beamforming architectures. However, classical DPD is shown to be a feasible solution for investigated array structures, except fully connected array, since power leakage to adjacent channel is suppressed notably. In fully connected structure, however, classical DPD does not provide any improvement when OOB radiation is considered. It can be observed that DPD provides the best performance for fully digital system since there is a single DPD unit dedicated to each antenna element.

\begin{figure}
\centering
    \includegraphics[scale=0.75]{PSD.eps}
    \caption{Normalized average power spectral density of intended users for different array architrectures (a) fully connected (b) partially connected with GEB (c) partially connected with phase only subarray (d) fully digital beamforming.}
    \label{AvgPSD}
\end{figure}

Furthermore, we investigate spatial distribution of in-band and out-of-band power emissions for different array architectures. Power angular spectrum for in-band radiation in Fig. \ref{ib_pattern}, shows that nonlinear effects do not distort the beampattern of the antenna array. In addition it can be observed that due to nonlinear distortion, null level of the array gets higher, which degrades the inter-group interference suppression capability. However, despite the higher null level, a significant amount of suppression is still achieved compared to phase only subarray.

\begin{figure}
\centering
    \includegraphics[scale=0.75]{ib_pattern.eps}
    \caption{ In-band power radiation pattern from BS for  the interval of -50 and 50 degrees for different array architrectures (a) fully connected (b) partially connected with GEB (c) partially connected with phase only subarray (d) fully digital beamforming. Maximum in-band power is scaled to be 0 dB.}
    \label{ib_pattern}
\end{figure}

In Fig. \ref{oob_pattern}, power angular spectrum for out-of-band radiation is shown. Out-of-band band power is also concentrated in the served user directions. Similar to PSD analysis, DPD does not provide improvement in the OOB radiation in any direction as can be seen in Fig. \ref{oob_pattern}(a), for fully connected architecture. On the other hand, employing DPD at the transmitter significantly decreases OOB emission for partially connected array and fully digital precoding architectures. However, it is also observed that radiated power in fully connected architecture is not much higher than that of partially connected arrays with DPD since its multiplexing gain is larger than the multiplexing gain of partially connected array. The reason is that in fully connected structure all, $ N_t= 96$, are used to transmit the signal to a single user. On the other hand, only a sub-array of size, $N_s = 16$, is dedicated for a single user in partially connected systems. Also, at the UT, desired signal term is combined coherently, while distortion term is combined non-coherently, consequently, effects of the nonlinear distortion is averaged out in fully connected and fully digital architectures compared to partially connected architecture.

\begin{figure}
\centering
    \includegraphics[scale=0.75]{oob_pattern.eps}
    \caption{ Out-of-band power radiation pattern from BS for the interval of -50 and 50 degrees for different array architrectures (a) fully connected (b) partially connected with GEB (c) partially connected with phase only subarray (d) fully digital beamforming. Maximum in-band power is scaled to be 0 dB.}
    \label{oob_pattern}
\end{figure}

\emph{AIR Analysis:}
Having investigated the spatial radiation of the transmitted power, we can now compare the AIR performances of the different beamforming architectures with different processing techniques. For this purpose, GMI analysis is carried out for these structures. Capacity curves are obtained for a constant transmitted power, $E_s$ and $E_s/N_o$, which is the received signal to noise ratio (SNR) per-subcarrier, is adjusted by modifying the receiver noise power. Firstly, from Fig. \ref{GMI}, it can be inferred that  array structures has vital impact on the capacity of the system. Phase only subarray cannot achieve the maximum AIR regardless of the noise level and nonlinear distortion. Even for linear PA, capacity of this architecture is limited due to inter-group interference. However, the other structures can successfully suppress both inter-group and intra-group interferences, which make them superior compared to phase only beamforming. Apart from array structure, nonlinear distortion introduced by PA has notable effect on the performance since systems with linear PA's can reach higher spectral efficiency for the same noise level compared to systems with nonlinaer PA's, until it is limited by the constellation order.

Both systems with fully connected and fully digital array architectures can achieve maximum capacity by employing either equalization at UT or DPD, and these methods exhibit close performances for both architectures. The reason is that nonlinear distortion is already suppressed by the diversity; therefore, linear processing is sufficient. On the other hand, multiplexing gain cannot be fully exploited by partially connected systems due to decrease in the effective antenna per user, which decreases the multiplexing gain. Therefore, maximum capacity cannot be reached by employing linear receiver processing in GEB subarray whereas it is achieved by conventional DPD since it suppresses nonlinear distortion as well. Consequently, classical DPD technique is proven to be a feasible solution. Lastly, one can see that the systems, which do not employ any processing, suffer from significant performance degradation for all array architectures.

\begin{figure}
\centering
    \includegraphics[scale=0.6]{GMI.eps}
    \caption{Average mismatched capacity of users for $256$ QAM for different array architectures (a) fully connected (b) partially connected with GEB (c) partially connected with phase only subarray (d) fully digital beamforming.}
    \label{GMI}
\end{figure}
\emph{BER Analysis:}
In this section, analysis on uncoded BER performances of the array structures are evaluated. Derived analytical BER expression is verifed with  numerical Monte Carlo simulations for $64$ and $256$ QAM constellations.  Fig. \ref{BER} presents theoretical and  numerical BER curves. It can be seen that analytical BER curves are in compliance with the numerical results. However, there is a divergence in case of fully digital beamforming with $256$ QAM, which is acceptable since BER approximation for a higher order modulation is employed and which may not be exact but it sets a bound for BER. Also note that BER results for the system, which does not employ any processing, is not considered in BER analysis due to its poor AIR performance.

Firstly, it can be observed that phase only subarray exhibits significantly higher error floor compared to other architectures. For fully digital beamforming; on the other hand, both DPD and equalization methods perform close to linear PA in case of $64$ QAM; however, post-equalization method suffers from a performance degradation for $256$ QAM.

Similar to fully digital structure, both DPD and equalization methods also perform close to linear PA for $64$ QAM for fully connected structure. However, there exists an error floor for the systems with nonlinear PA for $256$ QAM. From these results, one can also conclude that classical DPD does not provide any improvement for fully connected structure since in this structure, each PA is connected to all RF chains; however, each DPD is designed based on a single RF chain. Therefore, DPD's cannot compensate the nonlinear distortion, they only compensate the memory effects of PA's. This can be also verified by the simulations since equalization at UT and DPD at BS performs very closely.

In subarray with GEB, improvement in the performance becomes more apparent. For both $256$ and $64$ QAM, equalization at UT cannot prevent error floor. However, by employing DPD, error floors for $64$ and $256$ QAM are significantly reduced. Furthermore, error floor of subarray with GEB for $256$ QAM is slightly lower than that of fully connected architecture.
\begin{figure}
\centering
    \includegraphics[scale=0.35]{BERCURVES.eps}
    \caption{Analytical and simulation results for BER obtained for different array architectures (a) fully connected (b) partially connected with GEB (c) partially connected with phase only subarray (d) fully digital beamforming for $64$ and $256$ QAM}
    \label{BER}
\end{figure}

\section{Conclusions and Future Works} \label{conclds}

In this work, we presented a comprehensive performance analysis framework for evaluation of massive MIMO systems, which are subject to PA nonlinearities for different beamforming architectures.  In addition, anti-beamforming based indirect learning approach is adopted in design of DPD and a post-equalization method is proposed.  GEB based sub-array design is proposed in order to provide sufficient inter-group interference in analog domain, which cannot be achieved by conventional DFT beamformer. Both in-band and out-of-band radiation patterns are considered for different array architectures. Significant amount of OOB radiation results due to nonlinear distortion caused by PA's; however,  in band pattern is not significantly affected. Conventional DPD can achieve OOB reduction for all array architectures except fully connected architecture. DPD design for fully connected architecture that considers the connectivity of this architecture remains as a future work. GMI metric is employed in order to compare array architectures in terms of AIR. Observations on AIR performances are in compliance with that of OOB radiation. It can be observed that fully digital beamforming arcihtecture is robust agianst PA nonlinearities since maximum capacity can be achieved by using only linear equalization at the receiver. In addition, fully connected architecture provides acceptable performance under PA nonlinearities even if its OOB radiation cannot be recduced. On the other hand, DFT beamformer suffers from significant performance degradation even with linear PA due to poor performance of analog beamformer design. Lastly, analytical BER expressions are derived for different array architecutres based on spatio-frequency multidimensional Bussgang decomposition and BER performances of array architectures are compared. Sub-array with DFT beamformer exhibits poor performance while fully digital beamformer provides the best possible performance. Proposed sub-array with GEB outperforms fully connected architecture for higher order modulations; however, for lower order constellations fully connected architecture is more beneficial. As a result, sub-array with GEB provides tradeoff between OOB supression and BER performance. However, fully connected architecture would have promising performance with proper DPD design.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

config: arxiv


TITLE: binary matrix and product between a matrix and its transpose
QUESTION [3 upvotes]: Let be $A$ a square matrix of order $n$ which has only the numbers
$0$ and $1$ as entries such that: $$A+A^{T}+I_n=EE^{T}$$ where $E=(1,1,...,1)^{T}$
Find $n$ for which there is a matrix $A$ such that $AA^{T}$ has no
null entries.

Let be $a_{ij}$ the entry of A on the line $i$ and column $j$, for every $i,j\in \{1,2,...,n\}$.
I have got that all entries on the first diagonal must be $0$. And  if $a_{ij}=1$ then $a_{ij}=0$ and vice versa, where $i\ne j$ and $i,j\in \{1,2,...,n\}$ since we have $$A+A^{T}+I_n=EE^{T}$$
The relation
$$A+A^{T}+I_n=EE^{T}$$ may be re-written equivalently as:
$$A+A^{T}=\begin{pmatrix}0&1&1&...&1\\ 1&0&1&...&1\\ 1&1&0&...&1\\ ...&...&...&...&...\\ 1&1&1&...&0\end{pmatrix}$$
Can somebody give some examples for some particular cases?
Maybe you can write a program to do it for you using backtracking for example.
Maybe we can generalize it for every $n\ge n_0$.
I noticed that $n\ge 6$(I made a program :)) but it is inefficient for numbers greater than 6).
So we know that $$A+A^{T}=\begin{pmatrix}0&1&1&...&1\\ 1&0&1&...&1\\ 1&1&0&...&1\\ ...&...&...&...&...\\ 1&1&1&...&0\end{pmatrix}$$ and $$AA^T=\begin{pmatrix}b_{11}&b_{12}&b_{13}&...&b_{1n}\\ b_{21}&b_{22}&b_{23}&...&b_{2n}\\ b_{31}&b_{31}&b_{33}&...&b_{3n}\\ ...&...&...&...&...\\ b_{n1}&b_{n2}&b_{n3}&...&b_{nn}\end{pmatrix}$$ where $$b_{ij}\ne 0\ , \text{for every } i,j\in \{1,2,...,n\}$$
What do you think of case $n=6$? How can you prove that there are no matices with these properties without checking every possibility?

REPLY [1 votes]: CLAIM: For every $n\geq 7$, we can find $A_{n\times n}$ satisfying the required conditions.
Proof.
It is an induction on $n$. The case $n=7$ was provided by Rammus.
Suppose there is $A_{n\times n}$ satisfying the required conditions. Let us build  $B_{n+1\times n+1}$ satisfying the same conditions.
Let $L_1$ be the first row of $A$. Define $x=(x_1,\ldots,x_n)=(1,1,\ldots,1)-L_1$.
Let $B=\begin{pmatrix} A & x^t\\
L_1 & 0
\end{pmatrix}$.
Note that

$(BB^t)_{ij}\geq (AA^t)_{ij}>0$, for $1\leq i,j\leq n$,
$(BB^t)_{(n+1)j}\geq (AA^t)_{1j}>0$, for $1\leq j\leq n$, and
$(BB^t)_{(n+1)(n+1)}= (AA^t)_{11}>0$.

In addition, $B+B^t+Id_{n+1\times n+1}=\begin{pmatrix} A+A^t+Id_{n\times n} & x^t+L_1^t\\
L_1+x & 1
\end{pmatrix}=1_{n+1\times n+1}$.
The induction is complete. $\square$

The case $n=6$ does not work because each row must contain at least three 1s. See the claim below.
So $\#\{(i,j),\ A_{ij}=1\}\geq 3n$.
By the required conditions $\#\{(i,j),\ A_{ij}=1\}=\frac{n(n-1)}{2}$.  So $\frac{n(n-1)}{2}\geq 3n$, which implies $n\geq 7$.

CLAIM 2: If $n\geq 3$ then the number of 1s on each row of $A$ is at least 3.
Proof. If the row $i$ of $A$ contains only one number $1$, let us say $A_{ij}=1$, then row $i$ and row $j$ of $A$ are orthogonal, because $A_{jj}=0$. This is impossible, since $AA^t_{ij}>0$.
Next, let us assume that the row $i$ of $A$ contains only two numbers 1:
$A_{ij}=1$ and $A_{ik}=1$. Of course, $i$, $j$ and $k$ are distinct.
Since $A_{jj}=0$ , it implies $A_{jk}=1$ $($otherwise rows i and j would be orthogonal, which is impossible by $AA^t_{ij}>0)$.
Again, since $A_{kk}=0$ , it implies $A_{kj}=1$ $($otherwise rows i and k would be orthogonal, which is impossible by $AA^t_{ik}>0)$.
Now $A^t_{jk}+A_{jk}=A_{kj}+A_{jk}=2>1$, which contradicts the equation
$A+A^t+Id=EE^t$.


TITLE: What length would the sides of a triangle over Earth's surface be for the sum of its angles to be 180.1°
QUESTION [9 upvotes]: For simplicity's sake, let the Earth be a perfect sphere. Imagine you are drawing an equilateral triangle over its surface. How long should its sides be, for the sum of its angles to be 180.1 degrees?

REPLY [2 votes]: Consider an equilateral spherical triangle with side lengths $a$ and angles $A$, by the supplemental cosine rule https://en.wikipedia.org/wiki/Spherical_trigonometry#Identities
\begin{eqnarray*}
\cos(A) =- \cos^2(A)+\sin^2(A) \cos(a) \\
\cos(a) =\frac{\cos(A)+\cos^2(A)}{\sin^2(A)} =\frac{\cos(A)}{1-\cos(A)}
\end{eqnarray*}
Now we have $A=60.0333\cdots$, which gives $a=3.6\cdots$.
To obtain the distance on a sphere of radius $r$ calculate $ra\pi/180$.


TITLE: Geometric significance of completion of a ring
QUESTION [8 upvotes]: I am reading about completions in Eisenbud's commutative algebra book.
He mentions that the completion $\hat{R}_m$ can represent the properties of a variety, I imagine with coordinate ring $R$,  in a much smaller neighborhood of $\frak{m}$ than what we can see via localizing at $\frak{m}$. This is an interesting observation I would like to know more about.
Could somebody give me a simple example of how one would use completion in algebraic geometry? How is it employed and what makes it useful?

REPLY [11 votes]: This is not a complete answer to the question "where is completion used?": I just want to give an example of its usefulness. In the following, we work over $\mathbb{C}$.
Let us say that we want to classify singularities of curves: our first task would be to understand when two singularities are the same and when they are not. How to do that? Clearly, we should use a tool that permits us to look at the behavior of the singularity, i.e. how the curve look like near the singular point. So, a first attempt might be to define two singularities on two curves to be the same when the coordinate rings of the curves, localized at the singular points, are isomorphic. Actually, this is not a good definition.
Indeed, take the two curves $C_1$ and $C_2$ in $\mathbb{C}^2$ respectively defined by the ideals $(y^3+y^2-x^2)$ and $(xy)$. Let me remark that these two curves are not isomorphic, as $C_1$ is irreducible whilst $C_2$ is not. Now, both of them have a singularity in the origin, and by drawing a picture we immediately see that the two singularities must be the same in our classification, as near the origin the two curves look pretty similar. But if we take the localization of the coordinate ring of $C_1$ and $C_2$ near the origin we obtain two non-isomorphic local rings (one of them is a domain, the other is not).
Another obvious example is given by regular points: clearly, in our classifications of singularities we would like to consider all the regular points in the same class of singularity, namely "non-singular". But again, if we take localization of coordinate ring at regular points, in general we will obtain non-isomorphic local rings.
The point is that the localized rings give us information about the local behavior of the curve near the singular point in the Zariski topology, and the open sets of the Zariski topology are too big to tell you what is happening really close to the singular point. If we take instead $\mathbb{C}^2$ with the euclidean topology in the example above, we can find an open neighborhood of the origin such that, inside it, the two curves $C_1$ and $C_2$ are topologically (and analytically) isomorphic.
Completion is the algebraic tool that permits us to analyze the behavior of the curve around the point "more locally" than we would be able to do with localization, that only tells us about what is happening around the point by approaching it via smaller and smaller Zariski open neighborhoods. And it works! Indeed, the completion of the local rings of $C_1$ and $C_2$ around the origin are both isomorphic to $\mathbb{C}[[x,y]]/(xy)$, as we were expecting. Also, the Cohen structure theorem tells us that the completion of the local ring of a curve at a regular point is always isomorphic to the ring of formal power series in one variable $\mathbb{C}[[x]]$. Thus, we can now define two singularities of two curves to be the same (more formally, analytically isomorphic) iff the completion of the localization of the coordinate rings of the curves at the singular points are isomorphic. In this way, by what we have observed before, the two singularities of $C_1$ and $C_2$ are the same, namely they are both a node, and regular points are all of the same kind, namely non-singular.
Now you may wonder why we have introduced completion when we could just use the euclidean topology to define classes of singularities. The point here is that we can extend this definition to curves defined over an algebraically closed field $k$. In this context we do not have anymore a euclidean topology and we can only use "algebraically defined" notions, as completions.
A final remark: as stated before, the key observation is that Zariski topology is too coarse for certain scopes. A beautiful idea of the second half of the last century was to consider, instead of the Zariski topology, the so called Grothendieck topologies. These are not topologies in the usual sense (but usual topologies are particular cases of Grothendieck topologies) and they are the right tools to study algebraic varieties in a refined way. In particular, by taking a particular Grothendieck topology called étale topology, which captures a lot more topological information of the algebraic varieties than the Zariski topology, we can talk of the étale neighborhood of a point. Algebraically, this corresponds to taking a refined version of the completion of a local ring, i.e. its strict henselization.


text:
\begin{document}

\title{Smooth structures and Einstein metrics on ${\bcp ^2}\# 5,6,7{\overline{\bcp^2}}$ }

\author{Rare{\c s} R{\u a}sdeaconu}

\address{Rare{\c s} R{\u a}sdeaconu,
         IRMA, 7 Rue Ren{\'e} Descartes, 67184 Strasbourg Cedex,
         FRANCE}

\email{rasdeaco@math.u-strasbg.fr}

\author{Ioana {\c S}uvaina}

\address{Ioana {\c S}uvaina,
         IHES,
         35 Rue de Chartres,
Bures-sur-Yvette, 91440, FRANCE}

\email{ioana@ihes.fr}

\keywords{rational blowing down,
symplectic 4-manifolds, Einstein metrics}

\subjclass[2000]{Primary 70G55; Secondary 32J15}

\date{\today}

\begin{abstract}
We show that each of the topological 4-manifolds $\bcp^2\# k\overline{\bcp^2},$ for $k=6,7$ admits a smooth structure which has an Einstein metric of scalar curvature $s>0$, a
smooth structure which has an Einstein metric with $s<0$ and
infinitely many non-diffeomorphic smooth structures which do not
admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic  non-diffeomorphic to $\bcp^2\# 5\cpb$ which don't admit an Einstein metric. We also exhibit new examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature.
The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.
\end{abstract}

\maketitle

\section{Introduction}
\label{intro}

Recently, new methods were developed \cite{park, abp} to construct symplectic 4-mani-folds with small topology and exotic smooth structures. Moreover, the method proposed by J. Park \cite{park} was later refined \cite{lee-park, ppsh} to produce interesting examples of minimal complex surfaces of general type. In this paper we show how we can use these constructions in regard to the existence or non-existence of Einstein metrics.

 LeBrun and Catanese \cite{leca} showed that there exist homeomorphic, but non-diffeo-morphic manifolds such that one of them admits an Einstein metric of positive sign while the other admits an Einstein metric of negative sign.
The manifold which has positive scalar curvature Einstein metric is $\bcp^2\#8\cpb,$ while the second manifold is a deformation of the Barlow surface.  The key ingredient in their proof was to show that the Barlow surface, a simply connected minimal surface of general type with $K^{2}=1$ can be deformed to a surface with ample canonical bundle. In particular, they showed that there exists a simply connected complex surface of general type with $p_{g}=0, K^{2}=1$ and having ample canonical bundle. The ampleness of the canonical bundle ensures the existence of a K\"ahler-Einstein metric of negative scalar curvature.

Based on ideas from \cite{park}, Lee and Park \cite{lee-park} and more recently Park, Park, and Shin \cite{ppsh} constructed new examples of simply connected, minimal surfaces of general type with
$p_{g}=0, K^{2}=1,2$ or $3.$ We show that their examples satisfy the ampleness condition:

\begin{thm}\label{amplu}
There exist simply connected complex surfaces of general type, with $p_g=0,~~K^2=2$ or $3,$ and ample canonical bundle.
\end{thm}

In his paper, Park \cite{park}, used his examples and LeBrun's Obstruction Theorem to exhibit a manifold homeomorphic, non-diffeomorphic  to $\bcp^2\#8\cpb$  which does not admit an Einstein metric. We extend his results to $\bcp^2\#k\cpb, k=6,7,$ and we also exhibit an infinite family of homeomorphic manifolds for which no Einstein metric exists.

\begin{thm}
\label{einstein}
Each of the topological 4-manifolds $\bcp^2\# k\overline{\bcp^2},$ for $k=6,7,8$ admits a smooth
structure which has an Einstein metric of scalar curvature $s>0$, a
smooth structure which has an Einstein metric with $s<0$ and
infinitely many non-diffeomorphic smooth structures which do not
admit Einstein metrics.
\end{thm}

We can also prove non-existence results for an even smaller topological manifold:
\begin{prop}
\label{5-blow-ups}
The canonical smooth structure on $M=\bcp^2\# 5\cpb$ admits an Einstein metric of positive scalar curvature, but $M$ also supports infinitely many exotic smooth structures which don't have an Einstein metric.
\end{prop}

It is a well known fact that in dimensions $2$ and $3$ the sign of the Einstein metric is a topological invariant. This led Besse \cite[p.19]{bes} to consider the conjecture that no smooth compact $n-$manifold can admit Einstein metrics with different scalar curvature signs. As we see in the previous theorems, in dimension $4,$ there are homeomorphic, non-diffeomorphic $4-$manifolds which admit Einstein metrics of opposite sign. For higher dimensions, as far as the authors know, LeBrun and Catanese constructed the only known counterexamples to Besse's conjecture. These examples are in dimensions $4k,$ where $k\geq 2.$ In these dimensions, we are able to provide many new examples with the same property:

\begin{prop}\label{high}
Let $N_1=\bcp^2\# 8\cpb, N_2=\bcp^2\# 7\cpb, N_3=\bcp^2\# 6\cpb.$ Then the manifold $N$ obtained by taking the  $k-$fold products, $k\geq 2,$ of arbitrary  $N_1, N_2,$ or $N_3$, admits two Einstein metrics $g_1,g_2$ such that the signs of the scalar curvature are $s_{g_1}=-1, s_{g_2}=+1.$ Moreover, these metrics are K\"ahler-Einstein with respect to two distinct complex structues $J_1,J_2.$
\end{prop}

As a consequence of the K\"ahlerian property of the metrics and of our method of construction, we have the following in peculiar fact: their corresponding volumes are equal $Vol_{g_1}(N)=Vol_{g_2}(N).$

\begin{rmk}
\label{rmk}
Soon after this paper was finished, Jongil Park and his collaborators found \cite{ppsh2} an analogous example of a  complex surface of general type homeomorphic to $\bcp^2\# 5\cpb.$ Using this construction the results of Theorems \ref{amplu}, \ref{einstein} and Proposition \ref{high} can be extended for $\bcp^2\# 5\cpb.$
 \end{rmk}

The paper is organized as follows: in Section \ref{ampl} we discuss the ampleness of the canonical line bundles of some interesting examples of complex surfaces and the relation to the existence of Einstein metrics, while in Section \ref{exot} we introduce the Seiberg-Witten invariants and use them to get results on non-existence of Einstein metrics. In Section \ref{High} we treat the higher dimensional case.

\section{Ampleness of the (anti)canonical bundle: Existence of Einstein metrics}
\label{ampl}

In general the existence or non-existence of Einstein metrics on a given manifold is hard to prove.
In the case $c_1(M)>0,$ in the unobstructed situations, i.e. for certain complex surfaces whose underlying differential structure is $\bcp^2\#k\cpb, 3\leq k\leq8,$ the existence of K\"ahler-Einstein metrics was proved by Siu \cite{siu}, or Tian and Yau \cite{tian-yau}. A complete solution to the existence was given by Tian \cite{tian}:

\begin{thm}[Tian]\label{tian}
A compact complex surface $(M^4,J)$ admits
a compatible K\"ahler-Einstein metric with
$s >  0$ if and only if
its anti-canonical line bundle $K_M^{-1}$ is ample and its
Lie algebra of holomorphic vector fields is
reductive. \end{thm}

In the case when $c_1(M)<0,$  a criterion for the existence of a K\"ahler-Einstein metric was independently found by Aubin \cite{aubin} and Yau \cite{yau}:

\begin{thm}[Aubin-Yau]\label{ay}
A compact complex manifold $(M^4,J)$ admits
a compatible K\"ahler-Einstein metric with
$s < 0$ if and only if its
canonical line bundle $K_{M}$ is ample. When such a metric exists, it is
unique, up to an overall multiplicative constant.
\end{thm}

In this section, we discuss some examples of smooth structures on  $\bcp^2\#k\cpb, 6\leq k\leq8.$ For each such $k,$ the natural smooth structure as the complex projective plane blown-up at $k$ generic points, can be endowed \cite{tian} by a K\"ahler-Einstein metric of positive scalar curvature. On the other hand, Lee and Park \cite{lee-park} and more recently Park, Park and Shin \cite{ppsh} constructed new exotic smooth structures on $\bcp^2\#k\cpb,$ for any  $6\leq k\leq8.$ Moreover, these admit complex structures yielding interesting examples of minimal surfaces of general type. We are going to prove that the canonical bundle of these surfaces is ample, and so they admit K\"ahler-Einstein metrics of negative scalar curvature.

The examples constructed in these papers are
very similar. In this article we are going to treat just the more complicated example \cite{ppsh} of a simply connected  minimal surface of general
type with $p_g=0$ and $K^2=3.$ The other examples
follow similarly. We begin by a brief description of the example of a minimal surface of general
type with $p_g=0$ and $K^2=3,$ referring frequently to \cite{ppsh}.

We start with an appropriate pencil of plane elliptic curves, and we resolve the base locus to obtain a rational elliptic surface $Y$ with the following set of singular fibers: one $I_8-$fiber, one reducible $I_2-$fiber and two nodal fibers. Moreover, this rational elliptic surface admits 4 sections. Now, $Y$ is blown-up 21 times at well chosen points to obtain a new rational surface $Z.$ These points can be chosen in such a way \cite{ppsh}  that $Z$ contains four particular disjoint chains of rational curves consisting of proper transforms of the singular fibers,  proper transform of some sections and some of the exceptional divisors introduced in the blowing-up process. According to \cite{ppsh}, these chains are $G=\sum_{i=1}^{10} G_i,~H=\sum_{i=1}^7 H_i,~I=\sum_{i=1}^6 I_i$  and a chain of length one denoted by ${A}.$ These chains are represented by continuous lines in Figure \ref{figure}.

\begin{figure}[hbtb]
\begin{center}
\hspace*{-1 cm}
\setlength{\unitlength}{1mm}
\includegraphics[width=10cm]{figura.eps}\\
\end{center}
\vspace*{-1 em}
\caption{Manifold $Z=\bcp^{2}\#30\cpb$}
\label{figure}
\end{figure}

They all satisfy Artin's contractibility criterion \cite{km}. By contracting them, we obtain a projective surface $X$ with 4 singularities, each admitting 1-parameter $\QQ-$Gorenstein smoothings. Let $f:Z\to X$ be the contraction map. Now, it is proved \cite{ppsh} that the projective surface $X$  admits a $\QQ-$Gorenstein smoothing, which is a smooth, simply-connected, minimal surface of general type with $K_{X}^{2}=3.$ We denote by $F_{i}, i=1,\cdots 7$ the seven smooth rational curves of self-intersection $-1,$ which are marked by discontinuous lines in Figure \ref{figure} and by $F_8$ the remaining exceptional divisor from the last singular fiber, drawn as a dotted line .  It is easy to see that the Poincar{\'e} duals of the curves $F_{i},i=1,\dots,7,$ together with all of the irreducible components of our 4 chains $G, H, I$ and $A$

form a basis of $H^{2}(Z,\QQ),$ which is torsion-free and 31-dimensional.

First, we prove the ampleness of the canonical divisor of the singular surface $X.$ Note that since $X$ has only 1-parameter $\QQ-$Gorenstein smoothing singularities, its canonical divisor is in fact $\QQ-$Cartier.

A common feature of all of the examples appearing in \cite{lee-park, ppsh} is that the pullback of the canonical divisor of the singular variety $X$ to its minimal resolution is effective. For instance, in the example described above, we can write:
\begin{equation}
\label{simplecan}
f^*K_X\equiv_{\QQ} \sum_{i=1}^8 a_iF_i +\sum_{i=1}^{10} b_iG_i + \sum_{i=1}^7 c_iH_i +
\sum_{i=1}^6 d_i I_i + e{A},
\end{equation}
In \cite{ppsh}, the coefficients are explicitely given, but for our purpose it is important to know only that
$a_i, b_i, c_i, d_i$ and $e$ are strictly positive rational numbers. For convenience, let
$Exc(f)=\sum G_i +\sum H_i+\sum I_i+{A}$ denote the exceptional divisor of $f.$

By a direct computation, in \cite{ppsh} it is proved that $K_X^2=3$ and the nefness of $K_X.$
Suppose now that $K_X$ fails to be ample. Since $K_X$ is already nef, according to the Nakai-Moishezon criterion \cite{km}, there exists an irreducible curve $C\subset X$ such that
$(K_X\cdot C)=0.$

The total transform of $C$ in $Z$ is
\begin{eqnarray}
f^*C\equiv_{\QQ} C'+\sum_{i=1}^{10} x_iG_i + \sum_{i=1}^7 y_iH_i +
\sum_{i=1}^6 z_i I_i + t{A},
\end{eqnarray}
 where $C'$ is the strict transform and $x_i, y_i, z_i, t$ are non-negative
rational numbers. We should note here that $C'$ is not numerically equivalent to $0.$ To give an immediate proof for this assertion, note that $(C\cdot H)>0$ for any ample line bundle $H$ on
the projective surface $X.$ But, if $C'\equiv_{\QQ} 0$ we would have
$$
0=(C'\cdot f^*H)=(f^*C\cdot f^*H)=(C\cdot H),
$$
and this is a contradiction.

Now, a straightforward computation gives:
\begin{align}
(K_X\cdot C)= &(f^*K_X\cdot f^*C)=(f^*K_X\cdot C') \\ \notag
=&\sum_{i=1}^8 a_i(F_i\cdot C')  +\sum_{i=1}^{10} b_i(G_i\cdot C') + \sum_{i=1}^7 c_i(H_i\cdot C') \\ \notag
&+\sum_{i=1}^6 d_i(I_i\cdot C') + e({A}\cdot C').
\end{align}
Now, obviously the intersection number of $C'$ with any component of $Exc(f)$ is greater or equal to $0,$ with equality if and only if $C'$ does not intersect any of the irreducible components of $Exc(f),$ i.e. $C$ does not pass through the singular points of $X.$ Hence, it follows that
$$\sum_{i=1}^{10} b_i(G_i\cdot C') + \sum_{i=1}^7 c_i(H_i\cdot C') +\sum_{i=1}^6 d_i(I_i\cdot C') + e({A}\cdot C')\geq 0.
$$

Thus $\sum_{i=1}^8 a_i(F_i\cdot C')\leq 0.$ In this case, either there is an $i_0\in \{1,...,8\}$  such that $(C'\cdot F_{i_0})<0,$   or $(C'\cdot F_i)=0$ for all $i=1,...,8,$ and $C'$ does not meet any component of $Exc(f).$ In the first case $C'$ must coincide with $F_{i_0}.$ But the computations in \cite{ppsh,lee-park} show that $(f^*K_X\cdot F_i)>0$ for all $i=1,...,8,$ which is impossible by our assumption. Therefore $C'$ must have vanishing intersection number with all of the $F_i$'s and also with the all of the components of $Exc(f).$ But, as the Poincar\'e duals of the irreducible components of $Exc(f)$ and of the $F_i'$s generate $H^2(Z, \QQ),~C'$ must be numerically trivial on $Z,$ and this is a contradiction. Hence, $X$ has ample canonical bundle.

As a consequence of  Theorem \ref{ay} we get:
\begin{cor}\label{ample}
There exist  smooth complex structures on $\bcp^2\# k\overline{\bcp^2},$ for $k=6,7,8$ which have ample canonical line bundles and hence admitting
K\"ahler-Einstein metrics of negative scalar curvature.
\end{cor}
\begin{proof}
As we have already mentioned earlier, we only discuss the case of $\bcp^2\# 6\overline{\bcp^2}.$ The other cases follow similarly.

We know that a  $\QQ-$Gorenstein smoothing of $X$ exists \cite{ppsh} and gives a smooth exotic structure on  $\bcp^2\# 6\overline{\bcp^2}.$

As we showed, the singular projective surface $X$ has ample $\QQ-$ Cartier canonical divisor.
But, ampleness is an open property \cite{km}, so the canonical bundle of these smoothings is automatically ample. This will allow us now to apply Aubin-Yau's Theorem \ref{ay} to prove the existence of K\"ahler-Einstein metrics of negative scalar curvature on the minimal surfaces of general type described above.
\end{proof}

\begin{rmk}
\label{barlow}
The examples of complex surfaces with ample canonical bundle we found are new to our knowledge, when $K^{2}=2$ or $3.$ When $K^{2}=1,$ LeBrun and Catanese \cite{leca} were able to find a complex surface with ample canonical bundle homeomorphic to  $\bcp^2\# 8\overline{\bcp^2},$ as a deformation of the Barlow surface. As was pointed out by \cite{lee-park}, it is not known whether the surfaces we found in Corollary \ref{ample} are deformation equivalent to the Barlow surface, or even diffeomorphic to it.
\end{rmk}

\section{Exotic structures: Non-Existence of Einstein metrics}
\label{exot}
The classical obstruction to the existence of an Einstein metric on an oriented, compact, smooth, Riemannian, four-manifold $(M,g)$ is the Hitchin-Thorpe Inequality:
\begin{equation}\label{ht}
(2\chi\pm 3\tau)(M) \geq 0
\end{equation}
with equality when the Einstein manifold $(M,g)$ is covered by the hyperk\"ahler $K3$ surface or by the flat four-torus, or by the above with the reversed orientations. Here $\chi(M), \tau(M)$ denote the Euler characteristic and the signature of the manifold $M,$ respectively.
Using the Seiberg-Witten theory, LeBrun \cite{lno,lric} was able to find new obstruction to the existence of Einstein metrics. The novelty of his results is that the existence of Einstein metrics depends on the differential structure considered. The techniques developed by LeBrun rely on the existence of a non-trivial solution of the Seiberg-Witten equations.  We briefly introduce the needed notions of the Seiberg-Witten theory and state the main obstruction theorem.

For simplicity, we consider only the case when $H_1(M,\ZZ)$ has no $2-$torsion. Then there is a one-to-one correspondence between the set of $Spin^c$ structures and the set $\{c\in H^2(M,\ZZ)| c_1\equiv w_2(M) \mod 2\}$ of characteristic classes in $H^2(M,\ZZ).$ The Seiberg-Witten invariant is defined \cite{mo} as an integer valued function
\begin{equation}
SW_M:\{c\in H^2(M,\ZZ)~|~~ c\equiv w_2(M)\mod 2\} \longrightarrow\ZZ  \notag
\end{equation}
We call $c$ a basic class if $SW_M(c)\neq 0.$

The Seiberg-Witten invariant is a diffeomorphism invariant in the case $b_+>1.$ In the case of $b_+=1,$ the invariant depends on the choice of an orientation of $H^2(M,\ZZ)$ and $H^1(M,\RR).$ But, as we are interested in manifolds for which the existence of an Einstein metric is topologically unobstructed,  the inequality (\ref{ht}) must be strictly satisfied.
Hence, in the case when the $SW_M(c)\neq0,$ we have $c^2\geq (2\chi+3\tau)(M) >0.$ But for any arbitrary metric $g,~c^2=c_+^2-c_-^2>0$ where we denote by $c_{\pm}$ the (anti)-self-dual part of the harmonic representative 2-form. Thus $c_+\neq0,$ and in this special situation  (see \cite[Thm 6.9.2]{mo} ) the Seiberg-Witten invariant does not depend on the choice of metric.

We are now ready to state LeBrun's obstruction, see \cite[Thm. 3.3]{lric}:

\begin{thm}[LeBrun]\label{obstr}
Let $X$ be a compact oriented $4-$manifold with a non-trivial Seiberg-Witten invariant and with $(2\chi+3\tau)(X)>0.$ Then
\begin{equation*}
M=X\#k\cpb
\end{equation*}
does not admit an Einstein metric if $k\geq \frac13(2\chi+3\tau)(X).$
\end{thm}

In the last years there was a lot of work done on constructing exotic structures with small topology. In this paper we will use a construction due to Akhmedov, Baykur and Park \cite{abp}. Similar results could also be found in papers of above authors or of Baldridge and Kirk.

Here we are interested in some exotic structures on $M=\bcp^2\#3\cpb.$ In \cite[Section 3.4]{abp}, the authors construct a minimal symplectic manifold $X$ homeomorphic to $M,$ but not diffeomorphic. To give the reader a better understanding of this new manifold, we sketch its construction. We start with two product  manifolds $Y=T^2\times \Sigma_2$ and $T^4$, where  $T^n$ is a torus of dimension $n$ and $\Sigma _2$ is a Riemann surface of genus $2.$ If we consider the diagonal class $[T^2\times point] + [point\times T^2]\in H_2(T^4,\ZZ),$ then a smooth representative in this class is a Riemann surface of genus $2$ with self-intersection $2.$ Blowing up two points on the smooth representative gives $\Sigma'\subset (T^4\#2\cpb),$ a smooth Riemann surface of genus $2$ and $0-$self-intersection. If we endow the manifolds $Y$ and $T^4\#2\cpb$ with the canonical symplectic structures, we can choose the representative $\Sigma'$ to be a symplectic submanifold. We can now take the symplectic sum $X'=Y\#_{\Sigma_2}(T^4\#2\cpb)$ along the two genus $2$ symplectic submanifolds $point\times \Sigma_2 \subset Y$ and $\Sigma' \subset (T^4\#2\cpb).$ $X'$ is a symplectic manifold with topological invariants: Euler characteristic $\chi(X')=6$ and signature $\tau(X')=-2.$ As $X'$ is not simply connected we need to make six Luttinger surgeries along well chosen tori (see \cite{abp}), which yield a new simply connected, minimal, symplectic manifold $X,$ with the same topological invariants as $X'.$
Notice that the definition of the Seiberg-Witten invariant in \cite{abp} is for the Poincar\'e dual of the characteristic class. The results are nevertheless the same.
The manifold $X$ has a unique basic class $\beta=c_1(K_X)\in H_2(X,\ZZ),$ such that its Seiberg-Witten invariant $SW_X(\beta)=1.$ Moreover, $X$ contains a null-homologous torus $\Lambda,$ with a preferred simple loop $\lambda.$
In Section 4, \cite{abp}, it is shown that $0-$surgery on $\Lambda$ with respect to $\lambda$ yields a symplectic manifold $X_0,$ which also has an unique basic class $\beta_0=c_1(K_{X_0}).$ Then $1/n-$surgery$,n\geq1$ on $\Lambda$ with respect to $\lambda$  generates a family of manifolds $X_n$ which are homeomorphic to $X_1=X.$ Corresponding to the basic class $\beta$ of $X$ there is a unique basic class $\beta _n$ of $X_n$ for which the Seiberg-Witten invariant is non-zero,  and it can be computed as follows:

$$SW_{X_n}(\beta _n)=SW_X(\beta)+(n-1)SW_{X_0}(\beta _0)=1+(n-1)=n.$$

Hence, the family $X_n, n\geq1$ consists of homeomorphic, pairwise non-diffeomorphic manifolds.
As $\beta_n$ is the unique basic class on $X_n,$ and the Seiberg-Witten invariant of $X_n, n>1$ is not $\pm1,$ the manifold $X_n$ does not admit a symplectic structure. Of course, its basic class satisfies  $\beta_n^2=\beta^2=6=(2\chi+3\tau)(X)=(2\chi+3\tau)(X_n).$

Let $M_{n,k}=X_n\#k\cpb, k=2,3,4,5.$ Then, the conditions of Theorem \ref{obstr} are satisfied and we have the following:
\begin{prop}\label{non-Einstein}
The manifolds $\bcp^2\#l\cpb$ for $l=5,6,7,8$ support infinitely many non-diffeomorphic exotic smooth structures none of which admits an Einstein metric.
\end{prop}
\begin{proof}
The only thing that needs to be argued is that the manifolds $\{M_{n,k}\}_n$ are homeomorphic but pairwise not diffeomorphic. For this we use again the Seiberg-Witten invariant and the commutativity of our construction. First we blow-up $k=l-3$ points on $X=X_1$ and denote by $E_1,\dots E_{k}$ the exceptional divisors.  This gives a new symplectic manifold, which has basic classes $\beta\pm c_1(E_1)\pm \dots\pm c_1(E_{k}).$ $0-$surgery on $\Lambda$ with respect to $\lambda$ gives the manifold $X_0\#k\cpb$ with corresponding basic classes $\beta_0\pm c_1(E_1)\pm \dots\pm c_1(E_{k}).$ We have a similar relation for the Seiberg-Witten invariant of $M_{n,k}$:
\begin{align}
SW_{M_{n,k}}(\beta _n\pm c_1(E_1)\pm \dots\pm c_1(E_{k}))&=SW_X(\beta\pm c_1(E_1)\pm \dots\pm c_1(E_{k})) \notag \\
&+(n-1)SW_{X_0}(\beta _0\pm c_1(E_1)\pm \dots\pm c_1(E_{k}))\notag \\
&=1+(n-1)=n. \notag
\end {align}
Hence, for any fixed $k\in\{2,3,4,5\},$ the manifolds $M_{n,k}$ are homeomorphic to $\bcp^2\#(k+3)\cpb,$ but pairwise not diffeomorphic. Moreover, if $n>1$ they don't support a symplectic structure.
\end {proof}

\begin{rmk}
We would like to note that since for all of the above smooth structures the Seiberg-Witten invariant is non-trivial the manifolds don't admit Riemannian metrics of positive constant scalar curvature. Moreover, as the $c_1^2(M_{n,k})>0$ there is a bound \cite{lno} on the scalar curvature:

$$\int_{M_{n,k}} s_g^2 d\mu \geq 32 \pi^2c_1^2({M_{n,k}})>0$$

where $d\mu$ is the volume form with respect to $g.$ Hence the manifolds $M_{n,k}$ don't admit non-negative constant scalar curvature metrics, and their Yamabe invariant is negative.
\end{rmk}

The results of Theorem \ref{einstein} are immediate by putting together the results from \ref{ay}, \ref{ample}, \ref{non-Einstein}.

\section{Higher Dimensional Manifolds}\label{High}

Let $S_{1}$ be a complex surface with ample canonical bundle and $c_{1}^{2}=1.$ As an example of such surface, we can either take the one provided by Corollary \ref{ample}, or the example \cite{leca} constructed by LeBrun and Catanese. Let $S_2, S_3$ also be complex surfaces of general type, with ample canonical line bundle and $c_1^2=2,3,$ respectively, as in Corollary \ref{ampl}. We are now ready to give the proof of Proposition \ref{high}.

\begin{proof}
Let $N=N_{i_1}\times\dots\times N_{i_k}$ and let $S=S_{i_1}\times\dots\times S_{i_k},$ where $i_k$ is either $1,2$ or $3.$ The manifolds $S_i$ and $N_i$ are homeomorphic, hence by a theorem of Wall \cite{wall}, they are h-cobordant. These h-cobordisms induce an h-cobordism between $N$ and $S.$ But as $N,S$ are simply connected manifolds of dimension greater than $5$, Smale's h-cobordism theorem tells us that they are diffeomorphic. We know that $N_i$ and $S_i$ admit K\"ahler-Einstein metrics of positive, negative scalar curvature, respectively. We can rescale these metrics  such that the scalar curvatures are $\pm1.$ On $N,S$ we consider the product metrics associated to the corresponding $i_k.$ As we take products of K\"ahler-Einstein with the same scalar curvature, the new metrics are going to be K\"ahler-Einstein. The product complex structures on $N, S$ are of Kodaira dimension $-\infty$ and $4k,$ respectively.
\end{proof}

\begin{rmk}
As the dimension considered increases, the number of manifolds constructed in Proposition \ref{high}  is
$\frac12 (k+1)(k+2).$\end{rmk}

\section*{Acknowledgements} The authors would like to thank  Claude LeBrun for suggesting this problem and for his remarks and suggetions on an early version of this paper. This paper was written while the first author was a CNRS post-doc at IRMA, Strasbourg, and the second was visiting IHES as a IPDE fellow. We would like to thank these institutions for their hospitality.

\bibliographystyle{alpha}


TITLE: Existence of square root of an invertible matrix
QUESTION [0 upvotes]: Does there exist a non-zero non-invertible matrix having square root?
I know if a matrix is invertible,then it has a square root. I want to know whether its converse is true. I wasn't able to find one counterexample, so I want to know if its converse is true or false.

REPLY [0 votes]: I don't think this has a square-root
$$\left[\begin{array}{cc}0&1\\ 0&0\end{array}\right]$$