TITLE: $\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is an ideal of $\mathfrak{J}_{ii}$
QUESTION [1 upvotes]: Let $\mathfrak{J}$ be a Jordan algebra, and $\mathfrak{J} = \sum \mathfrak{J}_{ij}$ the Peirce decomposition of $\mathfrak{J}$ relative to orthogonal idempotents $e_i$ with sum $1$. Prove that $\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is an ideal of $\mathfrak{J}_{ii}$.
I proved $\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is contained in $\mathfrak{J}_{ii}$, and just I need to prove that $a \cdot x \in \mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$, with $a = a_{ii} \in \mathfrak{J}_{ii}$ and $x=(b_{ij}c_{ij})e_i \in \mathfrak{J}_{ij}^{\;\;.2}$. In other words, $a_{ii}[(b_{ij}c_{ij})e_i] \in \mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$.
Thanks.

REPLY [1 votes]: In Jacobson book, Structures and Representations of Jordan Algebras, has 13 properties on page 121.


TITLE: Reducing multi-variable functions to a composition of 1- or 2-variable functions
QUESTION [13 upvotes]: There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta function $B(z; a, b)$, the Lerch transcendent $\Phi(z, s, a)$, the Weierstrass elliptic function $\wp(z;g_2,g_3)$, hypergeometric-family functions, etc.
Is it possible to express each (or at least some) of these functions as a composition of several analytic functions of 1 or 2 complex variables?
Or, if we restrict their domain to reals, it is possible to express them as a composition of several inifinitely differentiable (with respect to each variable) functions of 1 or 2 real variables?
The same question applies to functions of 2 variables (e.g. polylogarithms, incomplete elliptic integrals, Hurwitz zeta function, Bessel-family functions, etc.): Is it possible to represent them as a composition of several infinitely differentiable functions of 1 variable and the single fixed function of 2 variables $(x,y)\mapsto x+y$?

To give an example when the answer to the last question is positive, consider the complete beta function $B(a,b)$. It can be represented as
$$B(a,b)=\exp\big((\ln\Gamma(a)+\ln\Gamma(b))+(-\ln\Gamma(a+b))\big)$$
that is a composition of the 2-variable sum function and several infinitely differentiable 1-variable functions $x\mapsto\exp(x)$, $x\mapsto\ln\Gamma(x)$ and $x\mapsto -x$.

REPLY [2 votes]: I found an answer for essentially the same question asked earlier at MathOverflow.
TL;DR
There exist analytic functions $f$ of 3 variables such that they cannot be represented as a composition of continuously differentiable functions of two variables. The general problem in the class $C^\infty$ seems to be still open.


TITLE: Let $f(x) = \operatorname{max}(\sin x; \cos x)$ for all $x \in \Bbb R$.
QUESTION [2 upvotes]: I was thinking about the following problem:

Let $f(x) = \operatorname{max}(\sin x; \cos x)$ for all $x \in \Bbb R$. Then
a. $f$ is differentiable on $\Bbb R$.
b. $f$ is nowhere differentiable.
c. $f$ is differentiable except at $0$.
d. $f$ is differentiable except at a countable set of points.
e. none of the above.

My Try:  Let $\sin x \ge \cos x$ for some $x \in \Bbb R$ .Then $f(x)=\sin x $ which is differentiable at $0$ and so option
(c) can not be right.That means (b) is also false. I think option (a) is the correct choice. But I am also confused about
option (d).
Similar case will follow if we take  $\sin x \le \cos x$ for some $x \in \Bbb R$ .Then $f(x)=\cos x $.
Can someone give  explanation? Thanks in advance for your time.

REPLY [6 votes]: Try drawing the graphs of $y=\sin x$ and $y=\cos x$. Then you must choose the graph that lies above the other. It will be pretty clear that the function $f$ has countably many points (the solutions of $\sin x = \cos x$) where you glue $\sin x$ on one ice and $\cos x$ on the opposite side. To answer (d), you should check if this gluing is differentiable.
Here is the graph of $f$.


TITLE: 3 x 3 linear system organization
QUESTION [0 upvotes]: How to organize this 3x3 linear system in order to solve it with determinants afterwards.

REPLY [0 votes]: First you bring all the constant terms on the right hand side:
$$ \begin{cases}3x-8y+z-10=0 \\ 9x+8y-2z=-17 \\ 2z+9-4y=30
\end{cases} \iff \begin{cases}3x-8y+z=10 \\ 9x+8y-2z=-17 \\ -4y+2z=21
\end{cases}$$
Now you can write the matrix associated with the linear system, where the last column represents the constant terms:
$$\begin{pmatrix}
3&-8&1&10\\
9&8&-2&-17\\
0&-4&2&21
\end{pmatrix}$$
At this point you can solve the system.


TITLE: Multivariable limit exists?
QUESTION [1 upvotes]: Does the limit
$$\lim_{(x,y)\rightarrow (0,0)} \frac{y^4}{x^\beta(x^2+y^4)}$$ exists for $\beta>0$? I don't think it exists but how do you prove it rigorously. Thanks

REPLY [0 votes]: $$\lim\limits_{(x,y)\to (0,0)} \frac{y^4}{x^\beta\left(x^2+y^4\right)}$$
Using polar coordinates, we have
$$\lim\limits_{r\to 0^+} \frac{(r\sin\phi)^4}{(r\cos\phi)^\beta\left((r\cos\phi)^2+(r\sin\phi)^4\right)}$$
$$=\lim\limits_{r\to 0^+} \frac{r^4\sin^4\phi}{r^\beta\left(\cos^\beta\phi\right)\left(r^2\cos^2\phi+r^4\sin^4\phi\right)}$$
$$=\lim\limits_{r\to 0^+} \frac{r^4\sin^4\phi}{r^\beta r^2\left(\cos^\beta\phi\right)\left(\cos^2\phi+r^2\sin^4\phi\right)}$$
$$=\lim\limits_{r\to 0^+} \frac{r^2\sin^4\phi}{r^\beta\left(\cos^\beta\phi\right)\left(\cos^2\phi+r^2\sin^4\phi\right)}$$
$$=\lim\limits_{r\to 0^+} \frac{\sin^4\phi}{r^{\beta-2}\left(\cos^\beta\phi\right)\left(\cos^2\phi+r^2\sin^4\phi\right)}$$
Now let's attempt to find a bound that is independent of $\phi$
$$\left|\frac{\sin^4\phi}{\left(\cos^\beta\phi\right)\left(\cos^2\phi+r^2\sin^4\phi\right)}\right|\leq \left|\frac{1}{\left(\cos^\beta\phi\right)\left(\cos^2\phi+r^2\sin^4\phi\right)}\right| $$
The right hand side is unbounded, which implies that the limit is dependent on $\phi$. Therefore
$$\lim\limits_{(x,y)\to (0,0)} \frac{y^4}{x^\beta\left(x^2+y^4\right)}\Rightarrow \mbox{does not exist}$$


TITLE: What exactly does "smooth" mean in the sense of the Cayley-Salmon "27 Lines on Cubic" Theorem?
QUESTION [1 upvotes]: I see that the Cayley-Salmon Theorem states that a smooth cubic surface over an algebraically closed field contains exactly 27 lines. But what exactly does "smooth" mean here? Alternatively, what would be an example of a homogeneous cubic polynomial in four variables which does not generate a "smooth" surface? My background is very elementary, and if answers could try to restrict the discussion to varieties and keep the scheme-theoretic language to a minimum, it would be much appreciated.

REPLY [1 votes]: I will assume varieties to be irreducible. One way to define smoothness is the following (for affine varieties, but you can define it similarly for projective varieties):
Let $X \subset \mathbb{A}^n$ be a variety and let $I(X) = (f_1,\dots,f_r) \in k[x_1,\dots,x_n]$. A point $x \in X$ is smooth if $\text{rank}(J(a)) \geq n - \text{dim}(X) = \text{codim}(X)$, where $J(a) = \left( \frac{\partial f_i}{\partial x_j}(a)\right)_{i,j}$ denotes the Jacobian matrix. If every point $x \in X$ is smooth, we say that $X$ is smooth.
If you allow varieties to be reducible you need to replace $\text{dim}(X)$ in this definition. For more information check for example this link.
You can take $X = V(xyz + xyw + xzw + yzw) \subset \mathbb{P}^3$ to get an example of a non-smooth cubic hypersurface with singular points $[1:0:0:0],\dots,[0:0:0:1] \in X$.


TITLE: The jelly bean box problem
QUESTION [0 upvotes]: I believe this is a standard graph theory problem, but I am not sure. I am having a lot of trouble with it though. Give it a go
You have n jelly beans. You want to ship them all to a friend. For 1 ≤ i ≤ m, you can buy any number of boxes, where each box can hold Bi jelly beans. The smallest box fits one bean (b1 = 1). Every box you use must be fully packed. Each box costs a dollar. The goal is to ship your beans with the smallest cost. What is the time and space complexity of finding the optimal solution?
Thanks a lot!

REPLY [1 votes]: I believe what you're describing is equivalent to the Change-making problem.
If the answer to your question is given by the function $p(n)$, then $p$ satisfies the recursion
$$p(n) = 1 + \min_{i=1}^m p(n-b_i),$$ with $p(0) = 0$.
We let $p(x < 0) = \infty$, say, so that we do not consider any values $n-b_i$ which are negative in the above minimum.
Using a dynamic programming solution, you can solve this problem in $O(nm)$ time with $O(\max_i b_i)$ space.


TITLE: Is this a well-known bound for the derived length of a finite group?
QUESTION [13 upvotes]: Let $cd(G)$ be the set of degrees of the irreducible complex characters of the finite group $G$.
It is conjectured that if $G$ is solvable then $dl(G)\leq |cd(G)|$ and it is a result by Gluck that $dl(G) \leq 2|cd(G)|$.
I have managed to show that $dl(G)\leq 2|cd(G)|-3$ and I was wondering if this is a well-known bound and whether even better bounds are known.
Edit Forgot to mention that this is for $|cd(G)|\geq 3$.

REPLY [8 votes]: Taketa proved that $dl(G) \le |cd(G)|$ for M-groups, so this is called the "Taketa inequality", and it is conjectured that it holds for all solvable groups. I think that perhaps G. Seitz first posed that conjecture, but it is sometimes attributed to me. I was the first person to prove any bound at all, and then Gluck squeezed a better bound out of my method. (My bound was something like $3|cd(G)|$.)
Although no one has succeeded in proving the "Taketa inequality" for all solvable groups, examples suggest that much better bounds might exist. Perhaps some sort of logarithmic bound is the correct one. The situation seems to be wide open even for $p$-groups, where the Taketa inequality definitely does hold because $p$-groups are M-groups. I think that it is unknown, for example, if a $p$-group $G$ with $|cd G| = 4$, can have derived length as large as $4$. (If this is known for $4$, I am sure it is not known for $5$.)


TITLE: Can a photographic film on a flat slab create a well visible elongated picture of a far and faint astronomical object?
QUESTION [0 upvotes]: Let say we see a light source with apparent cross section area 1mm squared. If we position a flat slab with inductive high light sensibility at a very small angle to that source one dimension of the source is elongated on that slab so the 1mm of the source length is spread as let say 10cm on the slab. After that operation there is an another operation slightly complicated but I will try to explain:that slab should, after the first operation is done, be used to shine another slab that is positioned in the same way to the first slab as the first slab was to the astronomical source of light. The dimension of the source elongated by the first slab should be paralel with the second slab so we can elongate the other dimension of the picture of the source of light, to finaly get the source picture with real aspect ratio but 10cm squared. My question is this possible not only theoretically? If If so then 100×100 10000 times enlargement!

REPLY [2 votes]: This is very different from how modern imaging systems work.  In general, an imaging system uses a lens or a curved mirror to cause all of the light coming from a particular direction to pass through a particular point; you put your photoreactive detector at that point and learn that that particular direction corresponds to "blue" or "bright green" or "dark."  Usually the focal points for light from different directions are near each other in a "focal plane" that's approximately parallel to the lens; introducing large angles means in general that you have to care about details, like the lens thickness and "chromatic aberration," that are less important in a straight-on configuration.
Your idea of enlarging an image by having an image plane not perpendicular to the light direction might work in a "pinhole camera," for which there isn't really a focal plane.  But a pinhole camera isn't a good low-light detection system, since it works by throwing nearly all of the light away.


text:
\begin{document}

	\maketitle
	\pagestyle{plain}
	\thispagestyle{plain}

	\begin{abstract}
		Next generation wireless base stations and access points will transmit and receive using extremely massive numbers of antennas. A promising  technology for realizing such massive arrays in a dynamically controllable and scalable manner with reduced cost and power consumption utilizes surfaces of radiating metamaterial elements, known as {\em metasurfaces}. To date, metasurfaces are mainly considered in the context of wireless communications as passive reflecting devices, aiding conventional transceivers in shaping the propagation environment. This article presents an alternative application of metasurfaces for wireless communications as active reconfigurable antennas with advanced analog signal processing capabilities for next generation transceivers. We review the main characteristics of metasurfaces used for radiation and reception, and analyze their main advantages as well as their effect on the ability to reliably communicate in wireless networks. As current studies unveil only a portion of the potential of metasurfaces, we  detail a list of exciting research and implementation challenges which arise from the application of metasurface antennas for wireless transceivers.
	\end{abstract}

	\vspace{-0.4cm}
	\section{Introduction}
	\vspace{-0.1cm}

The increasingly demanding objectives for 6th Generation (6G) communications have spurred recent research activities on novel transceiver hardware architectures and relevant communication algorithms. Such hardware architectures comprise large numbers of ElectroMagnetic (EM) radiating elements, thus paving the way for \ac{mmimo} communications.
A \ac{mmimo} system with arbitrarily large number of antenna elements can provide substantial gains in spectral efficiency with relatively simple signal processing algorithms. This potential has motivated the incorporation of \ac{mmimo} technology in the 5G New Radio (NR) interface
, and \ac{mmimo} transceivers with extremely large number of antennas are considered to continue being one of the key technologies for 6G communications \cite{saad2019vision}.

With the widespread deployment of Internet of Things (IoT) devices,
the number of  nodes connected over wireless media is expected to reach the order of tens of billions in the next few years. To address these massive connectivity, high peak device rates and increased  throughput requirements, future wireless networks are expected to transit into dense deployments of coordinating extreme \ac{mmimo} transceivers (namely, \acp{bs} and access points), particularly in urban and indoor environments. While the theoretical gains of densely deployed \ac{mmimo} systems are still being unveiled \cite{bjornson2019massive},
implementing such systems in practice is a challenging task. In particular, realizing \acp{bs} with hundreds, or even thousands, of antenna elements being able to simultaneously serve multiple users, gives rise to a multitude of practical difficulties for conventional  sub-6 GHz bands as well  millimeter wave and THz bands. Among those challenges are  high fabrication cost, increased power consumption, constrained physical size and shape, and deployment limitations.

Over the last few years, metamaterials have emerged as a powerful technology with a broad range of applications, including wireless communications. Metamaterials comprise a class of artificial materials whose physical properties, and particularly their permittivity and permeability, can be engineered to exhibit various desired characteristics  \cite{smith2004metamaterials}. When deployed in planar structures (a.k.a. metasurfaces), their effective parameters can be tailored to realize a desired transformation on the transmitted, received, or impinging EM waves \cite{DSmith-2017PRA}. Such structures have been lately  envisioned  as a revolutionary means to transform any naturally passive wireless communication environment (the set of objects between a transmitter and a receiver constitute the wireless environment) to an active one \cite{huang2019reconfigurable,di2019smart}.
Their extremely small hardware footprint enables their cost-effective embedding in various 3D components of the environment (e.g., building facades and room walls/ceilings).

\acp{dma} have been recently proposed as an efficient realization of massive antenna arrays for wireless communications \cite{Yoo2018TCOM, shlezinger2019dynamic}. They provide beam tailoring capabilities \textcolor{NewColor}{and facilitate processing of the transmitted and received signals in the analog domain in a flexible and dynamically configurable manner using simplified transceiver hardware. In addition, \ac{dma}-based architectures require much less power and cost compared with conventional antenna arrays (i.e., those based on patch arrays and phase shifters) eliminating the need for complicated corporate feed and / or active phase shifters.} \acp{dma} may comprise a large number of tunable metamaterial antenna elements that can be packed in small physical areas \cite{Akyildiz-2016NCN} for a wide range of operating frequencies. This feature makes them an appealing technology for the extreme \ac{mmimo} transceivers of 6G wireless networks. In contrast to passive metasurfaces that have received extensive attention recently \cite{huang2019reconfigurable,di2019smart}, the potential and capabilities of metasurfaces as active \ac{mmimo} antenna arrays, as well as their associated challenges and future research directions, have not yet been properly treated in the wireless communication literature.

In this article, we discuss the promising application of \acp{dma} as \ac{mmimo} transceivers for future
wireless communications. We commence with a brief introduction on metasurfaces considered  for wireless communications, highlighting the differences between their nearly passive and dynamic counterparts. We then present the unique properties of \acp{dma} as transceiving structures with Analog and Digital (A/D) processing capabilities. We  elaborate on the relationship between \acp{dma} and conventional hybrid A/D \ac{mmimo} architectures, which are based on phase shifters for analog processing \cite{ioushua2019family} and discuss

the main advantages and drawbacks of using \ac{dma}-based \acp{bs} for \ac{mmimo} communications. A simulation study evaluating the ability of \acp{dma} to reliably communicate \textcolor{NewColor}{with a reduced number of Radio Frequency (RF) chains by exploiting their inherent analog signal processing flexibility} is detailed. Next, we present experimental results demonstrating the strong potential benefits of \acp{dma} for wireless communication. We conclude the article with a description of open problems in this area of research and a discussion of future directions towards unveiling the potential of \acp{dma} for 6G wireless communications.

\section{Metasurfaces for Wireless Communications}
\label{sec:Metasurfaces}
\textcolor{NewColor}{
Metamaterials comprise a class of artificial materials in which macroscopic, structured elements mimic the atoms or molecules. Each metamaterial element behaves as an electric or magnetic polarizable dipole, the collection of which can often be characterized by an effective permittivity and permeability. These dipole moments can be engineered so as to achieve desired EM properties} \cite{smith2004metamaterials}. The underlying idea behind metamaterials is to introduce tailored inclusions in a host medium to emulate diverse targeted responses.
\textcolor{NewColor}{
When metamaterial elements are distributed over a planar surface, they are often referred to as metasurfaces} \cite{HMIMOS}. Such 2-Dimensional (2D) structures support the individual tuning of each metamaterial element, allowing the metasurface to carry out different functionalities, such as radiation, reflection, beamforming, and reception of propagating waves \cite{DSmith-2017PRA}. \textcolor{NewColor}{The ability to stack a large number of elements in a limited surface area allows metasurfaces to achieve highly directed signalling enabling holographic beamforming \cite{bjornson2019massive}.}

The properties of each element can often be externally controlled, yielding a dynamic metasurface, also referred to as a reconfigurable intelligent surface \cite{huang2019reconfigurable}.

Two main applications of dynamically tuned metasurfaces have been considered recently in the context of wireless communications, as illustrated in Fig.~\ref{fig:PassiveVsActive1}. These are passive reflective surfaces \cite{ huang2019reconfigurable} and active transceiver antenna arrays \cite{shlezinger2019dynamic}. The former application builds upon the capability of metasurfaces to generate reconfigurable reflection patterns. In this case, metasurfaces deployed in urban settings can facilitate and improve  communication between the \ac{bs} and multiple users by effectively modifying EM signal propagation. \textcolor{NewColor}{The addition of a metasurface enables the communications system as a whole to overcome} harsh non-line-of-sight conditions and improve coverage, when the metasurface is placed within small distances from the BS or the users, without increasing transmission power.
	\begin{figure*}
	\centering
	\includegraphics[width = \linewidth]{PassiveVsActive3.png}
	\vspace{-0.6cm}
	\caption{Two applications of reconfigurable metasurfaces in the downlink (gray arrows) and uplink (blue arrows) of multi-user mMIMO wireless communications: a) as nearly passive reflective surfaces; and b) as active transceiver antenna arrays.
	}

	\label{fig:PassiveVsActive1}
\end{figure*}
Nearly passive reconfigurable intelligent surfaces require some level of control to alter the impinging EM wave in light of the dynamic wireless environment. This is achieved by embedding a digital control unit, which is capable of tuning the metamaterial elements to obtain desired reflection patterns. Such reflective surfaces do not implement conventional relaying techniques (i.e., neither power amplification nor baseband processing) \cite{huang2019reconfigurable}, but only reflect the impinging signal in a controllable manner.

\acp{dma} comprise an additional recent usage of metasurfaces for communication as mMIMO antenna structures. This application exploits their ability to realize planar, compact, and low cost dynamically tunable massive antenna arrays \cite{shlezinger2019dynamic}, which can be deployed in current and future BSs and access points. As such, a DMA-based BS will consist of a multitude of radiating metamaterial elements that can transmit and receive communication signals over the wireless channel. By dynamically tuning  the EM properties of the DMA elements, one can control the analog beampattern for transmission and reception. Unlike passive reflective metasurfaces, which are  the focus of considerable research attention lately, the usage of metasurfaces as active mMIMO antennas is a relatively new area of research. To understand the potential of \acp{dma} for mMIMO systems, we discuss in the following section the architecture, properties, advantages, and drawbacks of using \acp{dma} for transmitting and receiving communication signals.

\section{\acp{dma} for Massive MIMO Communications}
\label{sec:DMAs}

In this section, we discuss the application of metamaterial-based planar antenna arrays in wireless communication systems. We first detail the architecture of DMAs and their main characteristics in the context of communications, and then, elaborate on its operation when deployed for mMIMO \acp{bs}. Finally, some representative numerical evaluations are presented together with experimental results.

 \subsection{\ac{dma} Hardware Architecture}
 \label{subsec:DMAsModel}

 \acp{dma} consist of a multitude of reconfigurable metamaterial radiating elements that can be used both as transmit and receive antennas. Those elements are placed on a waveguide through which the signals to be transmitted, and the received waveforms intended for information decoding, are transferred. The transceiver digital processor, which generates the outgoing signals and processes the received signals, is connected to the waveguide through dedicated input and output ports, respectively. In general, \acp{dma} can use 2D waveguides connected to several input/output ports \cite{Yoo2018TCOM}. However, the common \ac{dma} architecture, on which we focus in this article, consists of multiple separate waveguide-fed element arrays, referred to as {\em 1-Dimensional (1D) waveguides}, each connected to a single input/output port, as illustrated in Fig. \ref{fig:DMA_Diagram1}.

Such waveguides can accommodate a large number of radiating elements, which are commonly sub-wavelength spaced, allowing each input/output port to feed a multitude of possibly coupled radiators. Since this waveguide is typically designed to be single mode and the wave can only propagate along one line, its analysis is much easier than 2D waveguides (such as parallel plate waveguides), where a scattered wave from each element propagates in all directions. Furthermore, ensuring isolation between different ports is easier in 1D waveguides than in multiple ports of a 2D waveguide. A common implementation of 1D waveguides is based on \textit{microstrips}, as illustrated in Fig.~\ref{fig:DMA_Diagram1}.

	\begin{figure*}
	\centering
	\includegraphics[width = \linewidth]{DMA_Diagram3.png}
	\vspace{-0.6cm}
	\caption{An $N$-element \ac{dma} consisting of $M$ micropstrips, where each microstrip is implemented as an 1D waveguide. The upper right part and the lower right part of the figure illustrate their operation during transmission  and reception, respectively, along with their equivalent signal path models.  }

	\label{fig:DMA_Diagram1}
\end{figure*}

When \acp{dma} are deployed as receive antennas, the signals captured at each metamaterial element propagate through the corresponding waveguide to the output port, where they are acquired and forwarded to the digital unit for baseband processing.

In a \ac{dma}-based transmit antenna array, the signals to be radiated from its metamaterial elements are fed to each waveguide via its input port.	The relationships among the radiating signals and those captured/fed at the input/output port of each waveguide are determined by the following two properties arising from the \ac{dma} architecture:
	\begin{enumerate}
		\item \label{itm:P1} Each metamaterial element acts as a resonant electrical circuit whose frequency response is described by a Lorentzian equation \cite{DSmith-2017PRA}. The parameters of the resonant circuit, i.e., its oscillator strength, damping factor, and resonance frequency, are externally controllable and can be configured in run-time for each element individually. An illustration of the normalized magnitude of the element response achieved for several different resonant frequencies is depicted in Fig.~\ref{fig:Lorentz1Mag}. This figure demonstrates that the elements can be tuned to exhibit diverse responses, varying from bandpass to frequency flat filters.

		\begin{figure}
	    \centering
		\scalebox{0.58}{\includegraphics{LorentzMag3.pdf}}

		\caption{The normalized element response magnitudes for different settings of the Lorentzian resonant frequency (Res. freq) as functions of the operating frequency.}
		\label{fig:Lorentz1Mag}
	\end{figure}

		\item \label{itm:P2} In each waveguide, the signal has to travel between the feed port and each metamaterial element. Consequently, the signals propagating along the waveguide accumulate different phases for each metamaterial element.
	\end{enumerate}
	An illustration of the resulting equivalent model for each DMA waveguide, which arises from the aforementioned properties, is depicted in the  right part of Fig.~\ref{fig:DMA_Diagram1}. As shown in this figure, the input/output port is connected to a set of radiating elements via a reconfigurable filter encapsulating the joint effect of the propagation inside the waveguide and the externally controllable frequency response of the metamaterial element.

 \subsection{\ac{dma}-Based mMIMO Transceivers}
 \label{subsec:DMAsBSs}
Similar to conventional massive antenna arrays, \acp{dma} interface the electronic circuitry of a \ac{mmimo} transceiver with the EM waveforms propagating over the wireless channel. However, their application in \ac{mmimo} BSs induces several unique characteristics, which follow from the \ac{dma} structure discussed in the previous section. First, the number of independent data streams that can be processed by a \ac{dma}-based BS in the digital domain is typically much smaller than the number of its metamaterial antenna elements. In particular, a \ac{dma} consisting of $M$ waveguides (e.g., the 1D waveguides in Fig.~\ref{fig:DMA_Diagram1} termed as microstrips), each embedded with $L$  metamaterial antenna elements, allows the transceiver to process $M$ data streams, while utilizing in total $N=ML$ metamaterial elements. Consequently, \ac{dma}-based transceivers implement a form of hybrid A/D beamforming, since part of the processing of the transmitted and received signals is carried out in the analog domain, as an inherent byproduct of the waveguide-fed metamaterial array architecture. Such hybrid processing, which requires additional dedicated hardware in conventional \ac{mmimo} architectures, allows the \ac{bs} to utilize an amount of metamaterial elements that is much larger than the number of the digitally processed data streams. This inherent expansion upon transmission and compression in reception is typically desirable in \ac{mmimo} \acp{bs} as a method to reduce the number of costly RF chains \cite{ioushua2019family}, while achieving increased beamforming gain, as well as to facilitate efficient operation under low quantization requirements \cite{shlezinger2018hardware}.

In DMA-based transceivers, each metamaterial antenna element can exhibit a broad range of frequency responses with various combinations of amplitude and phase values, ranging from frequency selective to frequency flat profiles. These profiles are externally controllable for each element separately. This implies that \ac{dma}-based mMIMO \acp{bs} implement dynamically reconfigurable and frequency selective hybrid A/D beamforming by tuning their  elements to impart different levels of attenuation and phase shift on the transmitted and received signals. Consequently, \ac{dma}-based BSs can be treated as hybrid A/D beamforming systems which do not require additional dedicated analog combining circuitry, while offering increased flexibility, compared to conventional hybrid architectures consisting of interconnections of frequency static phase shifters and switches \cite{ioushua2019family}.

Furthermore, DMAs provide similar beamforming capabilities to those achievable with typical phased array antennas, but with much lower power consumption and cost. To better illustrate this point, we note that the radiation pattern formed by a DMA is the superposition of the radiated field from many metamaterial radiators, whose complex amplitude are determined by two factors: the tunable resonance response of the elements and the phase accumulated by the guided wave. Using simple holographic techniques, the tuning states of the metamaterial elements can be designed to form beams in any direction of interest. Since the tuning of metamaterial elements is usually accomplished with simple components, such as varactors, it requires minimal additional power for beam steering. This is in contrast to many common antenna arrays which use active phase shifters that consume significant power. The beamforming capability of DMAs is heavily dependent on the subwavelength spacing of metamaterial elements. This is due to the fact that the metamaterial elements amplitude and phase cannot be tuned separately. To augment this limited design space, one can utilize the phase accumulation of the feed wave as it travels between different elements. A densely sampled waveguide thus provides enough degrees of freedom to form any desired beamforming pattern.

 \subsection{Numerical and Experimental Results}
 \label{subsec:DMA_Prototype}
 The \ac{dma} architecture facilitates the implementation of massive amounts of metamaterial elements in simple, energy-efficient, compact, and low-profile design configurations, irrespective of the operating frequency. The planar physical shape of \acp{dma} makes them suitable for installations in urban and indoor environments, and the inherent expansion/compression induced by their waveguide-fed architecture reduces the number of required expensive RF chain circuits.
However, this expansion/compression, which stems from the fact that the transceiver can access the signals only at each waveguide's input/output ports, reduces the  achievable performance compared to a fully-digital transceiver which is capable of simultaneously accessing the signals transmitted and received from each antenna element separately. In particular, this performance reduction follows from the DMA requirement for fewer digital streams than antenna elements, and is a common characteristic of hybrid A/D beamforming architectures.

To evaluate the effect of the aforementioned properties on the performance of \ac{dma}-based \ac{mmimo} \acp{bs}, we depict in Fig.~\ref{fig:SEvsSNR_IFreq} the achievable uplink sum-rate performance of narrowband communications at $3.5$GHz carrier frequency as a function of the operating \ac{snr}. The achievable sum-rate is computed by treating the uplink wireless channel as a multiple access channel, and thus the number of RF chains is not restricted to be larger than the number of users.
In this figure, the same performance with a fully-connected hybrid A/D architecture is included, as well as the sum-capacity of the uplink channel when the number of \ac{bs} antennas equals the number $N$ of receiving elements and when it equals the number $M$ of receive RF chains. The corresponding simulation scenario, based on the setup detailed in \cite[Sec. IV]{shlezinger2019dynamic}, consists of $64$ user terminals uniformly distributed in a single cell of $400$m radius that simultaneously communicate in the uplink direction with a mMIMO \ac{bs} having $N=160$ antenna elements. For the DMA architecture, $L=10$ unit metamaterial elements are coated upon each of the $M=16$ waveguides (microstrips in this case). With the fully connected hybrid A/D beamforming architecture, all antenna elements are attached via phase-shifter networks to each of the $M=16$ RF chains.
As observed in Fig.~\ref{fig:SEvsSNR_IFreq}, the DMA architecture yields sum-rate performance closer to the sum-capacity with $M=N$ than conventional hybrid A/D architectures based on fully-connected networks of phase shifters. This advantage follows from the previously discussed capability of \acp{dma} to implement different forms of analog signal processing. The  rate gains of \acp{dma} with $M$ RF chains and $N=ML$ elements over the sum-capacity when the \ac{bs} has $M$ antennas indicates that, connecting each RF chain to a microstrip with multiple elements can notably improve the communication rate compared to using each RF chain to feed a single antenna. Similar results have been reported for the downlink case
as well as
for communications with low-resolution analog-to-digital converters \cite{wang2020dynamic}.

 	\begin{figure}
		\centering
		\scalebox{0.48}{\includegraphics{SEvsSNR_IFreq2_16_64_2.pdf}}

		\caption{Achievable uplink sum-rate performance in bps/Hz versus the SNR in dB for a $160$-antenna mMIMO BS with $16$ RF chains serving simultaneously $64$ users within a cell with $400$m radius \textcolor{NewColor}{based on \cite{shlezinger2019dynamic}}. For the DMA architecture, each of the $M=16$ microstrips includes $L=10$ unit metamaterial elements. In the fully connected hybrid A/D beamforming architecture, the phase-shifter network interconnects each antenna element to all RF chains. }
		\label{fig:SEvsSNR_IFreq}
	\end{figure}

Despite the growing interest in using reconfigurable metasurfaces (either as an intelligent reflector or a transceiver) to augment wireless communication systems, very few experimental studies have been conducted. Most experimental studies focus on verifying the possibility to model each tunable metamaterial radiator as a polarizable dipole and, optimize their tuning states to form desired beams. For instance, the \ac{dma} configuration illustrated in Fig.~\ref{fig:DMA_Diagram1} consists of an array of microstrip, each comprised of a 1D waveguide-fed metasurface. An experimental implementation of such a dynamic 1D waveguide-fed metasurface, based on \cite{sleasman2017experimental,sleasman2016design}, is shown in Fig.~\ref{fig:DMA_sample}. This waveguide is fashioned with metamaterial elements with two different (interleaved) resonant frequencies. Each element is loaded with two PIN diodes, which render the metamaterial element radiating and non-radiating, as shown in Fig.~\ref{fig:DMA_sample}. A close-up view of a fabricated metamaterial is also shown. In addition, the figure includes experimental results for the ability of the metasurfaces to form beams in different directions, showcasing the potential of DMA-based mMIMO transceivers in forming multiple directed beams.

 	\begin{figure}
		\centering
		\scalebox{0.5}{\includegraphics{Comm_magazine.png}}

		\caption{\textcolor{NewColor}{An experimental implementation of a dynamic 1D waveguide-fed metasurface \cite{sleasman2017experimental,sleasman2016design}:} a) Detailed circuitry of a reconfigurable metamaterial element. b) The simulated response of the metamaterial element, where the impact of the PIN diodes to render the element in radiating ($e_{1,\rm on}$ and $e_{2,\rm on}$) and non-radiating ($e_{1,\rm off}$ and $e_{2,\rm off}$) states are shown. c) A close-up view of a sample 1D DMA with metamaterial elements having two different resonance frequencies. d) Beamforming capability of the fabricated 1D DMA.}
		\label{fig:DMA_sample}
	\end{figure}

\section{Open Research Challenges}
\label{sec:Challenges}
As previously discussed, the hardware architecture of \acp{dma} enables  efficient implementation of extreme mMIMO transceivers, which renders this concept a strong candidate technology for the physical layer of 6G communications. However, \acp{dma} come with certain design challenges that need to be carefully addressed, while giving rise to new opportunities. In the following, we list some of the most important related research challenges to date.

\textbf{Frequency-Selective Analog Beamforming:} Current algorithmic designs for \acp{dma} focus either on narrowband communications or ignore their capability to dynamically configure the frequency-selective profile of each unit metamaterial element. This unique property, which does not exist in any conventional hybrid A/D architecture, provides increased flexibility for the design of wideband operation by matching the spectral behavior of each element to optimize the equivalent wideband channel. Consequently, the true potential of \ac{dma}-based mMIMO systems in achieving reliable and ultra high rate communications is not yet fully explored.

\textbf{Wireless Channel Estimation and Tracking:} To date, studies on \ac{dma}-based communications assume that the transceiver has full channel knowledge. In practice, however, the channel coefficients need to be efficiently estimated, which is a challenging task with hybrid A/D receivers. When channel estimation is carried out in a time-division duplexing manner, \acp{dma} offer the possibility of tuning their elements to facilitate channel estimation via pilot signals, and to adapt in a manner which optimizes data reception in light of the estimated channel. The design and analysis of efficient algorithms for \ac{dma}-based \acp{bs}, which have to estimate features of the wireless channel and reliably communicate have not yet been properly treated in the literature.

\textbf{Hardware Design and Experimentation:} A large body of fabricated designs and experimental work is still required in order to transit the DMA concept into an established technology for future extreme mMIMO \acp{bs} with hundreds or thousands of metamaterial elements. The experimental studies detailed in the previous section  are an important first step which demonstrates the feasibility of this concept. Yet, the performance and other implications of using \ac{dma}-based \acp{bs} need be tested in a broad range of wireless setups. To date, most experimental studies of \acp{dma} are geared toward imaging and radar systems.

In particular, several key properties of \acp{dma} should be thoroughly tested to fully understand their potential for mMIMO communications. First, one  needs to quantify the level of correlation among the signals received by different metamaterial elements on the same waveguide as well as different waveguides. The impact of this correlation on the overall response needs to be investigated. In the example discussed in the previous section, the tuning of the metamaterial elements is optimized based on certain communication criteria. However, the resulting pattern may introduce more correlation or result in low \ac{snr} (for example, in a scenario where the resulting pattern is directed toward directions other than the intended users). The co-design of the metamaterial elements tuning states with the resulting pattern is another important factor that needs to be examined. When used as a transmitter, the possibility of the high amplitude RF signal (the carrier) to push the tunable component of each metamaterial element into a nonlinear regime needs to be investigated. This nonlinearity can cause intersymbol or intercarrier interference.
Finally, although conceptually feasible, \acp{dma} for communications in the high millimeter wave (around 60 GHz) and THz (above 100 GHz) bands are still in relatively early stages of research.

\textbf{Hybrid Passive and Active Metasurfaces:}
As discussed in the introduction, the emerging concept of reconfigurable metasurfaces, with both its passive and active counterparts, is lately gaining increased interest for both controllable reflection and transmission/reception. It is reasonable to envision hybrid passive and active reconfigurable metasurfaces, notably strengthening the design flexibility for such surfaces and their performance gains in programmable wireless environments. For instance, having such a hybrid metasurface acting as a receiving device can significantly facilitate channel estimation \cite{GA2020_RIS}, which is currently a major challenge and a source of substantial overhead in purely passive metasurfaces. In addition, hybrid metasurfaces will enable more advanced relaying strategies, overcoming the dominating impact of pathloss in their passive versions.

\textbf{Use Cases and Applications:} The use cases and applications where \acp{dma} can provide substantial improvement compared to current architectures have not yet been thoroughly identified. For example, their planar shape and reduced size for a small number of RF chains facilitate their deployment in indoor environments, like buildings, malls, train stations, and airports. In such setups,  \acp{dma} are expected to communicate with multiple users in close line-of-sight conditions, possibly operating in the near-field regime. Such near-field scenarios bring forth the possibility of beam focusing,

namely, the ability to focus the signal towards a specific location, instead of a specific direction as in far-field conditions via conventional beamforming, allowing multiple orthogonal links to exist in the same indoor or outdoor area.

\section{Conclusion}
\label{sec:Conclusion}
\acp{dma} are an attractive radiating technology for next generation wireless systems, allowing to realize dynamically controllable \ac{mmimo} antennas of reduced cost and power consumption compared to conventional arrays. In this article we surveyed some of the key properties of \acp{dma} in the context of \ac{mmimo} systems, including their operation model during their transmission and reception, tunable frequency selective profile, beam steering capabilities, as well as advantages and drawbacks compared to conventional arrays. We concluded with a set of key open research directions, which are expected to pave the way in unveiling the full potential of using active metasurfaces in 6G wireless communications.

	\bibliographystyle{IEEEtran}
	\bibliography{IEEEabrv,refs}

\end{document}

config: arxiv


TITLE: What parameter is necessary for the AIC criterion applied to linear regression models?
QUESTION [1 upvotes]: I did a linear regression model (OLS) and a spatial autoregressive model (Spatial lag). I read that for comparing these models I need to use the Akaike information criterion (AIC). The formula is the following
$$
AIC=-2log L(\hat\theta)+2k
$$
where $\theta$ is the vector of model parameters, $L(\hat\theta)$ is the likelihood of the candidate model given the data when evaluated at the maximum likelihood estimate of $\theta$ and $k$ is the number of estimated parameters in the candidate model.
But I don't understand what parameters $\theta$ are necessary to use AIC in a linear regression context. For example, if I have a model like this $Y=\beta_0+\beta_1X_1+\beta_2X_2+...+\beta_nX_n+\epsilon$ where $\epsilon$ is a normal variable with zero mean and variance $\sigma$ How do I calculate its AIC?

REPLY [1 votes]: The parameters $\theta$ are anything you need to estimate in your model. In the case of OLS, it would be the regression coefficients $\beta_0,\ldots,\beta_n$, so you have $k=n+1$.
The log likelihood for OLS is simply the quadratic function that comes from the Gaussian-nkise assumption:
$$\log L(\theta) = -\frac{N}{2}\log \sigma^2-\sum_{k=1}^N -\frac{1}{2\sigma^2}( y_k -\beta_0-\beta_1 x_{k,1} -\cdots -\beta_n x_{k,n})^2 .$$


TITLE: Conditional distribution of jointly Gaussian random variables where one is degenerate
QUESTION [0 upvotes]: If random variables $x \in \mathbb R^n$ and $y \in \mathbb R^m$ have the joint Gaussian distribution
$$
\begin{bmatrix} x \\ y \end{bmatrix} \sim
\mathcal N \left(
\begin{bmatrix} a \\ b \end{bmatrix},
\begin{bmatrix} A & C \\ C^{\top} & B \end{bmatrix}
\right),
$$
then the conditional distributions of $x$ and $y$ are given as
$$
x \, \vert \, y \sim \mathcal N \left( a + C B^{-1} \left( y - b \right), A - C B^{-1} C^{\top} \right) \quad (*)
$$
provided that $B^{-1}$ exists.
Question: Suppose that we know two random vectors $x$ and $y$ are jointly Gaussian with mean and covariance as stated above. What can we say about the conditional distributions when $B^{-1}$ doesn't exist? I am mainly interested in the case where $B$ is restricted to be block diagonal with the submatrix in the upper-leftmost block being a $k \times k$ (with $k < m$) invertible matrix and there are zeros everywhere else (e.g. $B = \begin{bmatrix} \Sigma & \bf 0 \\ \bf 0 & \bf 0 \end{bmatrix}$ where $\Sigma^{-1}$ exists), but still would like to know if there is an answer when $B$ lacks this type of restriction. In this case, what can we say about the:

shape? Can we still say that $x \,\vert\, y$ is Gaussian?
center? Can we find an expression that represents the conditional mean $\mathbb{E} \left[ x \, \vert\, y \right]$?
spread? Can we find an expression that represents the conditional variance $\text{Var} \left[ x \, \vert \, y \right]$?

My attempt:
Shape:
I heard the phrase "conditional distributions of jointly Gaussian random variables is Gaussian", so it seems reasonable to say that the conditional distribution $x \, \vert \, y$ is Gaussian, but degenerate in the sense that it doesn't have probability density function.
The comments to the question here seems suggests this is true.
Center:
When it comes to the conditional mean, I constructed a toy example.
Suppose $x \in \mathbb{R}$ and $y = (y_1, y_2, y_3)^{\top} \in \mathbb{R}^3$ such that $x \sim \mathcal{N}(1,1)$ and $y \,\vert\, x \sim \mathcal{N} (Ax, \Sigma)$ where
$$A = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \quad \Sigma = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$
Then
$$\begin{bmatrix} x \\ y_1 \\ y_2 \\ y_3 \end{bmatrix}
\sim \mathcal{N} \left( \begin{bmatrix} 1 \\ 1 \\ 2 \\ 0 \end{bmatrix},
\begin{bmatrix} 1 & 1 & 2 & 0 \\ 1 & 2 & 2 & 0 \\ 2 & 2 & 5 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \right)$$
$\mathbb{E} [x \, \vert \, y ]$ should be an affine transformation of $y_1, y_2, y_3$, so
$$\mathbb{E} [x \, \vert \, y ] = a_0 + a_1 (y_1 - 1) + a_2(y_2 - 2) + a_3 y_3.$$
Now, $x - \mathbb{E} [x \, \vert \, y ]$ and $y_i$ are independent (right?) for $i = 1, 2, 3$, so then $\text{Cov} (x - \mathbb{E} [x \, \vert \, y ], y_i) = 0$ for each $i$. Also, using the tower property, it follows that $\mathbb{E} \left[ \mathbb{E} [x \, \vert \, y ]  \right] = a_0 = 1$.
Next,
\begin{align}
0 = \text{Cov} (x - \mathbb{E} [x \, \vert \, y ], y_i) &= \text{Cov} \left( x - \left( a_0 + a_1 (y_1 - 1) + a_2(y_2 - 2) + a_3 y_3 \right), y_i \right) \\
&= \text{Cov} \left( x, y_i \right) - a_0 \text{Cov} \left(1 ,y_i \right) - a_1 \text{Cov} \left( y_1, y_i \right) - a_2 \text{Cov} \left( y_2, y_i \right) - a_3 \text{Cov} \left( y_3, y_i \right) \\
&= \text{Cov} \left( x, y_i \right) - a_1 \text{Cov} \left( y_1, y_i \right) - a_2 \text{Cov} \left( y_2, y_i \right).
\end{align}
I think I can say that $y_3 = 0$, so this leads to the system
\begin{align}
1 - 2a_1 - 2a_2 &= 0 \\
2 - 2a_1 - 5a_2 &= 0
\end{align}
which has solution $(a_0, a_1, a_2)= \left( 1, \frac16, \frac13 \right)$ with $a_3$ being free. Therefore, $\mathbb{E} [x \, \vert \, y ] = \frac16 + \frac16 y_1 + \frac13 y_2$
The solution in this case is the same as $\mathbb{E} [x \,\vert\, y ]$ if $y = (y_1, y_2)^{\top}$, $A = (1, 2)^{\top}$, and $B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.
So it seems that you'd just use $(*)$, but use the only the first $k$ columns of $C$ and the first $k \times k$ square of $B$.
Spread: To find the conditional variance in the toy example, I would need to find $\mathbb{E} \left[ x^2 \,\vert\, y \right]$, but I have not figured out a reasonable approach to compute this expectation. But, perhaps it is similar to the idea of using a restrict version of $C$ and $B$.

REPLY [1 votes]: If a random vector $z$ has joint Gaussian distribution, possibly degenerate, and we partition $z$ into subvectors, say $z=\begin{pmatrix}x \\ y\end{pmatrix}$, then the conditional distribution of $x$ given $y$ also has Gaussian distribution. The conditional mean and variance are a function of the original mean vector and original covariance matrix. In the case where the original covariance matrix is singular, you will need to find a Moore-Penrose pseudoinverse. For the full derivation, please see this answer.


text:
\begin{document}

\title[Superconvergence of simple conforming MFEMs for linear elasticity]
{Superconvergence of simple conforming mixed finite
   elements for linear elasticity on rectangular grids
  in any space dimension
    }
\date{}
\author {Jun Hu}
\address{LMAM and School of Mathematical Sciences, Peking University,
  Beijing 100871, P. R. China.  hujun@math.pku.edu.cn}

\author {Shangyou Zhang}
\address{Department of Mathematical Sciences, University of Delaware,
    Newark, DE 19716, USA.  szhang@udel.edu }

\thanks{The first author was supported by  the NSFC Project 11271035, and  in part by the NSFC Key Project 11031006.}

\begin{abstract}
This paper is to prove superconvergence of   a  family of
  simple conforming mixed finite elements of first order
 for the  linear elasticity problem with the Hellinger--Reissner  variational formulation.
 The analysis is based on three main ingredients: a new interpolation operator, a new expansion method,
 and a new iterative argument for superconvergence analysis.

  \vskip 15pt

\noindent{\bf Keywords.}{
     mixed finite element,  linear elasticity,
     conforming finite element,  superconvergence.}

 \vskip 15pt

\noindent{\bf AMS subject classifications.}
    { 65N30, 73C02.}

\end{abstract}
\maketitle

\section{Introduction}

This paper investigates superconvergence of  simple conforming
    mixed finite elements \cite{HuManZhang2013}
    for linear  elasticity within the Hellinger-Reissner variational principle.
 It is well--known that it is a challenge problem for stable
    discretizations for this problem, which  results from
    a strong coupling of  the symmetry  requirement  on the  discrete stress tensor
    and the usual stable  conditions for mixed finite element methods.
A lot of efforts, see, for instance,  \cite{Amara-Thomas,
   Arnold-Brezzi-Douglas, Arnold-Douglas-Gupta,  Johnson-Mercier, Morley,
   Stenberg-1, Stenberg-2, Stenberg-3},
    have been devoted to developing stable methods of this problem.
But no stable mixed finite element was found
      in the first four decades \cite{Arnold-Winther-conforming}.
Not until the year 2002, were there some advances
   in this direction.
In \cite{Arnold-Winther-conforming} and
   \cite{Arnold-Awanou-Winther}, a sufficient condition is proposed,
   which states  that a discrete exact sequence
   guarantees the stability of the mixed method.
From then,   conforming mixed finite elements on the
  simplical and rectangular triangulations have been  constructed
    \cite{Adams-Cockburn,Arnold-Awanou,Arnold-Awanou-Winther,
    Arnold-Winther-conforming, Awanou, Chen-Wang}; see
   \cite{Arnold-Winther-n, Gopalakrishnan-Guzman-n,HuShi,
    Man-Hu-Shi, Xie, Yi-3D, Yi} for nonconforming mixed finite elements,  and
   \cite{Arnold-Falk-Winther, Boffi-Brezzi-Fortin, Cockburn-Gopalakrishnan-Guzman,
    Gopalakrishnan-Guzman, Guzman} for new weakly symmetric finite elements.
However, most of these elements are difficult to be implemented;
    numerical examples can
    only be found in \cite{Carstensen-Eigel-Gedicke,
    Carstensen-Gunther-Reininghaus-Thiele2008,
   HuManZhang2012, HuManZhang2013, Yi} so far.

In a recent paper \cite{HuManZhang2013}, a new family of simple,
  any space-dimensional, symmetric, conforming mixed finite
  elements for the problem  is proposed.
In these elements,  quadratic polynomials $\{1, x_i, x_i^2\}$
   are used for the normal stresses $\sigma_{ii}$,
      bilinear polynomials $\{1, x_i, x_j, x_ix_j\}$
        for the shear stresses $\sigma_{ij}$,
        and  linear polynomials $\{1, x_i \}$ for
      the displacements $u_i$.
 The stress and displacement spaces of \cite{HuManZhang2013} are
  actually enrichment of those in \cite{HuManZhang2012},
   a family of symmetric nonconforming elements
These are possibly the simplest conforming mixed
  finite element methods.
A first order convergence was established
   for these elements in \cite{HuManZhang2013}.
However, superconvergence
  was observed from numerical examples presented  therein.

Superconvergence  is  one of the most active research fields
   for finite element methods.
A lot of  fundamental  results can be found  for  conforming,
   nonconforming and mixed finite elements
   of model problems in literature, see for instance,  \cite{Duran90,LinLin,LinYan}.
However,  no results can be found  for the mixed finite element methods
    under consideration in literature so far.
 A very recent paper \cite{ShiDongyang14}  analyzed  superconvergence
   of a family of conforming rectangular mixed  finite element methods
     for the two dimensional linear elasticity problem.
However, in the  conclusion,  it was pointed out that the technique therein
      can not be applied to mixed elements under consideration.

The aim of this paper is to prove superconvergence observed
      in \cite{HuManZhang2013}.
 One challenge is that
      the canonical interpolation operators for the stress spaces
     have no commuting properties,
 which are  indispensable  ingredients  for superconvergence analysis
     for mixed finite elements for the Poisson equations,
  see for instance, \cite{Brandts,DouglasWang1989,Duran90,EwingLazarovWang1991,Wang},
      also for the linear elasticity problem \cite{ShiDongyang14},
   and more details in Section 3.1.
Another challenge is that the normal stresses are coupled and
   consequently the superclose analysis  used in \cite{Duran90}
     for the mixed finite element of the Poisson equation can not be extended to
       the present case (see more details in Section 3.1.)
     To overcome these difficulties, we propose a new interpolation operator.
 Compared with the original interpolation
   operator from \cite{HuManZhang2013},  the new  one has a superclose
    property that is accomplished by adopting a new expansion
    which is motivated by a recent paper \cite{HuShi13}.
 Finally we propose an iterative  argument to establish an
     $O(h^{1+1/2})$ superconvergence.

This paper denotes by $H^k(T,X)$ the Sobolev space consisting of
   functions on domain $T\subset\mathbb{R}^n$, taking values in the
   finite-dimensional vector space $X$ and having all derivatives of
     order at most $k$ square-integrable.
For our purposes, the range
   space $X$ will be  $\mathbb{S}$, or $\mathbb{R}^n$, or
   $\mathbb{R}$.
In the latter case we may write simply $H^k(T)$.
$\|\cdot\|_{k,T}$ is the Sololev norm on $H^k(T)$.
Here $\mathbb{S}$ denotes
   the space of symmetric tensors, $H({\rm div},T,\mathbb{S})$,
   consisting of square-integrable symmetric matrix fields with
   square-integrable divergence.
The norm $\|\cdot\|_{H({\rm div},T)}$ reads
$$\|\tau\|_{H({\rm div},T)}^2:=\|\tau\|_{0,T}^2+\|{\rm div}\tau\|_{0,T}^2.$$
$L^2(T,\mathbb{R}^n)$ is the space of vector-valued functions
   which are square-integrable.

\section{The linear elasticity   problem and mixed finite elements}

\subsection{The linear elasticity   problem}
Based on the Hellinger-Reissner principle, the
  $n$-dimensional linear elasticity
  problem within a stress-displacement ($\sigma$-$u$) form reads:
Find $(\sigma,u)\in\Sigma\times V :=H({\rm div},\Omega,\mathbb
{S})
        \times L^2(\Omega,\mathbb{R}^n)$, such that
\an{\left\{ \ad{
  (A\sigma,\tau)+({\rm div}\tau,u)&= 0 && \hbox{for all \ } \tau\in\Sigma,\\
   ({\rm div}\sigma,v)&= (f,v) &\qquad& \hbox{for all \ } v\in V. }
   \right.\lab{eqn1}
}
Here the symmetric tensor space for stress $\Sigma$ and the
   space for vector displacement $V$ are, respectively,
  \a{

  H({\rm div},\Omega,\mathbb {S})
    &:= \Big\{ \p{\sigma_{ij} }_{n \times n} \in H(\d, \Omega)
    \ \Big| \ \sigma_{ij} = \sigma_{ji} \Big\}, \\

     L^2(\Omega,\mathbb{R}^n) &:=
     \Big\{ \p{u_1 & \cdots & u_n}^T
          \ \Big| \ u_i \in L^2(\Omega) \Big\}  .}

The matrix $A$ is defined as
   \a{
      A \sigma &= \frac 1{2\mu} \left(
       \sigma - \frac{\lambda}{2\mu + n \lambda} \operatorname{tr}(\sigma)
        \delta \right) }
  where $\delta$ is the identity matrix of $n\times n$, and $\mu$ and $\lambda$ are the
    Lam\'e constants.

This paper deals with a pure displacement problem with the
  homogeneous boundary condition that $u\equiv 0$ on
  $\partial\Omega$.
The domain is assumed to be a rectangular polyhedron
  in $\mathbb{R}^n$.

\subsection{The n-dimensional conforming mixed finite element space}

We recall  a conforming mixed finite element method proposed in \cite{HuManZhang2013}
 for the problem \meqref{eqn1}. We shall follow the notations used therein.

The rectangular domain $\Omega$ is subdivided by a family of
  rectangular grids  $\mathcal{T}_h$ (with the grid size $h$).
For convenience, the set of all $n-1$ dimensional faces in
  $\mathcal{T}_h$ is denoted by $\mathcal{F}_h$. For all element
  $K\in\mathcal{T}_h$, the set of all $n-1$ dimensional faces of $K$ perpendicular to $x_i$-axis
is denoted by $\mathcal{F}_{x_i,K}$, the set of all $n-2$
dimensional faces of $K$ perpendicular to $x_i$ and $x_j$ axes by
$\mathcal{F}_{x_i,x_j,K}$. Given any face
  $\mathcal{F}\in\mathcal{F}_h$, one fixed unit normal vector $\nu$
  with components $(\nu_1,\nu_2,\cdots,\nu_n)$ is assigned.

We first  introduce the finite element space locally on a single $n$-rectangle $
   K\in\mathcal{T}_h$:
  \a{
   V(K):=\Big\{ v =(v_1,\dots, v_n) \ \Big| \
       v_i \in P_1(x_i) \Big\},}
   \a{
        \Sigma(K):  =\Big\{ \sigma\in\p{ \sigma_{ij} }_\mathbb{S}
   \  \Big| \ \sigma_{ii}\in P_2(x_i),
    \sigma_{ij}\in Q_1(x_i,x_j),i\neq j \Big\}, }
where $$\begin{array}{rcl} P_1(x_i)&:=&{\rm span}\left\{1,x_i\right\},\\
P_2(x_i)&:=&{\rm span}\left\{1,x_i,x_i^2\right\},\\
Q_1\left(x_i,x_j\right)&:=&{\rm
span}\left\{1,x_i,x_j,x_ix_j\right\}.
\end{array}$$ For example, in 2D ($n=2$), the spaces may be
displayed as
 \a{ \Sigma(K)&=\operatorname{span}
    \p{ \left\{1, x_1, x_1^2 \right\}
            & \left\{1, x_1, x_2, x_1x_2 \right\}
             \\
        \left\{1, x_1, x_2, x_1x_2 \right\}
            & \left\{1, x_2, x_2^2 \right\}
             }, \\
      V(K)& =\operatorname{span}
    \p{ \left\{1, x_1 \right\}  \\
        \left\{1, x_2 \right\}  }.
  }

Due to the $H(\d)$ requirement, $\sigma_{ii}$ has to be continuous
in $x_i$ direction,
   while $\sigma_{ij}$ has to be continuous in both $x_i$ and $x_j$ directions.
Thus, we can specify the local degrees of freedom for the two
finite element spaces on element $K$ as follows,
\a{
 \bullet & \quad \disp\frac{1}{|K|}\int_K u_i~v{\rm d}V,&&\hbox{for all \ }
    v\in P_1(x_i)  & \hbox{ and \ }& i=1,2,\dots, n; && \\
  \bullet & \quad \disp\frac{1}{|F_{x_i, K}|}\int_{F_{x_i, K}}\sigma_{ii}{\rm
          d}F,&&\hbox{for all \ }
          F_{x_i,K}\in\mathcal{F}_{x_i,K}  & \hbox{ and \ }&
            i=1,2,\cdots, n;&& \\
  \bullet & \quad\frac{1}{|K|} \int_K\sigma_{ii}{\rm d}V,&&  &&   i=1,2,\cdots, n;&& \\
  \bullet & \quad\frac{1}{|F_{x_i,x_j,K}|} \int_{F_{x_i,x_j,K}}\sigma _{ij} {\rm
        d}F,&&\hbox{for all \ }
         F_{x_i,x_j,K}\in\mathcal{F}_{x_i,x_j,K}  & \hbox{ and \ }&
           1\leq i<j\leq n.
    }

The global spaces $\Sigma_h$ and $V_h$ can be defined by their
   property
   \an{\lab{Sh}
 \Sigma_h&:=\{~\sigma
    \in H(\d,\Omega,\mathbb{S}) \ | \ \sigma|_K\in\Sigma(K)
            \ \hbox{ for all } K\in\mathcal{T}_h  \}, \\
  \lab{Vh}
   V_h &:= \{v\in L^2(\Omega,\mathbb{R}^n) \ | \
         v|_K\in V(K) \ \hbox{ for all } K\in\mathcal{T}_h \}.
    }

The mixed finite element approximation of Problem (\ref{eqn1})
reads: Find
   $(\sigma_h,~u_h)\in\Sigma_h\times V_h$ such that
 \e{ \left\{ \ad{
    (A\sigma_h, \tau)+({\rm div}\tau, u_h)&= 0 &&
              \hbox{for all \ } \tau \in\Sigma_h,\\
     (\d\sigma_h, v)& = (f, v) &&  \hbox{for all \ } v\in V_h.
      } \right. \lab{DP}
    }

It follows from the definition of $V_h$ and $\Sigma_h$ that
   \a{ \d  \Sigma_h \subset V_h.}
This, in turn, leads to a strong divergence-free space:
 \an{ \lab {kernel}
    Z_h&:= \{\tau_h\in\Sigma_h \ | \ (\d\tau_h, v)=0 \quad
    \hbox{for all } v\in V_h\}\\
    \nonumber
          &= \{\tau_h \in\Sigma_h \ | \  \d \tau_h=0
    \hbox{\ pointwise } \}.
    }
\subsection{Well-posedness of the discrete problem}
The well-posedness of the discrete problem
    \meqref{DP} is proved in \cite{HuManZhang2013}. More precisely, it was shown therein that:

\begin{enumerate}
\item K-ellipticity. There exists a constant $C>0$, independent of the
   meshsize $h$, such that
    \an{ \lab{below} (A\tau, \tau)\geq C\|\tau\|_{H(\d)}^2\quad
       \hbox{for all } \tau \in Z_h, }
    where $Z_h$ is the divergence-free space defined in \meqref{kernel}.

\item  Discrete B-B condition.
    There exists a positive constant $C>0$
            independent of the meshsize $h$, such that
    \an{\lab{inf-sup}
   \inf_{0\neq v\in V_h}   \sup_{0\neq\tau\in\Sigma_h}\frac{({\rm
        div}\tau, v)}{\|\tau\|_{H(\d)}  \|v\|_{0} }\geq
    C .}
\end{enumerate}

In addition, there is a refined  discrete inf--sup condition from \cite{HuManZhang2013} as follows:
\begin{lemma}\lab{lemma1}
For any $v\in V_h$, there exists  an $H(\d)$ field
\a{ \tau =\p{ \tau_{11}  &  0&\cdots&\cdots&0\\
                 0   & \ddots& & &\vdots\\
                \vdots&    &\tau_{ii}& &\vdots \\
                \vdots& & &\ddots& 0 \\
                0&\cdots&\cdots&0&\tau_{nn}}\in \Sigma_h,
                  }
such that
  \an{\lab{equal} \d \tau=v \text{ and }\frac{({\rm
        div}\tau, v)}{\|\tau\|_{H(\d)}}\geq \sqrt{\frac 23}\|v\|_0.
   }
\end{lemma}

\subsection{Error estimate} Since it is very  difficult to show the superclose property of the interpolation operator
 defined in \cite{HuManZhang2013},  for any $\sigma\in H^2(\Omega,\mathbb{S})$, we define a new  interpolation by
\a{

 \Pi_h\sigma
   =\p{ \Pi_{11}\sigma_{11}  & \Pi_{12}\sigma_{12}
   &\cdots&\cdots&\Pi_{1n}\sigma_{1n}\\
                \vdots & \ddots&\vdots&\vdots&\vdots\\
                \Pi_{i1}\sigma_{i1}&\cdots&\Pi_{ii}\sigma_{ii}
    &\cdots&\Pi_{in}\sigma_{in} \\
                \vdots & \vdots&\vdots&\ddots&\vdots\\
                \Pi_{n1}\sigma_{n1}&\Pi_{n2}\sigma_{n2}
    &\cdots&\cdots&\Pi_{nn}\sigma_{nn}}\in\Sigma_h, }
   where $\Pi_{ij}=\Pi_{ji}$, are defined next.
The interpolation operator $\Pi_{ii}$ is defined by, for any
     $K\in\mathcal{T}_h$,
 \a{

    \int_{F_{x_i,K}}\Pi_{ii}\sigma_{ii}dF & =\int_{F_{x_i,K}}\sigma_{ii}dF
        \quad  &&\hbox{for all \ }
   F_{x_i,K}\in\mathcal{F}_{x_i,K},\\

   \int_K\Pi_{ii}\sigma_{ii}dV & =\int_K\sigma_{ii}dV && \hbox{for all \ }
    ~K\in\mathcal{T}_h. }
    Here and  throughout this paper, $\mathcal{F}_{x_i,K}$ denotes the set of $n-1$ dimensional faces of $K$ which are perpendicular
     to the $x_i$ axis and $\mathcal{F}_{x_i}=\cup_{K\in\mathcal{T}_h}\mathcal{F}_{x_i,K}$.

To define $\Pi_{ij}$, we introduce the nodal basis functions on unit element.
  \a{
   \phi_{k}(x,y) :=
    \begin{cases}\ \ \, (x-1)(y-1), & k=0, \\
      -(x-0)(y-1), & k=1, \\
       \ \ \,    (x-0)(y-0), & k=2, \\
      -(x-1)(y-0), &k=3. \end{cases} \\
   }
We also use multi-index notations as follows
 \an{\lab{out2}   \sum_{L_{\text{\sout{$i$}} ,\text{\sout{$j$}} }=1}^N :=
    \underset {\{1\le l_k \le N \mid k\ne i, k\ne j\}  }
     {\sum\cdots\sum }. }

Since $\sigma_{ij}\in H^2(\Omega)$, these  basis functions and the short
   notation allow for  defining the interpolation as follows
    \begin{equation}
   \Pi_{ij} \sigma_{ij}  =
\sum_{L_{\text{\sout{$i$}} , \text{\sout{$j$}} }=1}^N
         \sum_{ l_i,l_j=1}^N  \sum_{k=0}^3
       c^{(ij),k}_{l_1,\cdots,l_n} \phi_k(\frac{x_i}h-(l_i-1) ,
        \frac{x_j}h -(l_j-1) )
    \end{equation}
 where the interpolation parameters satisfy
 \a{ &\quad \ c^{(ij),2}_{l_1,\cdots,(l_i-1),\cdots,(l_j-1),\cdots,l_n}\\
     &= c^{(ij),3}_{l_1,\cdots,l_i,\cdots,(l_j-1),\cdots, l_n}
     = c^{(ij),0}_{l_1,\cdots,l_i,\cdots,l_j,\cdots, l_n}
     = c^{(ij),1}_{l_1,\cdots,(l_i-1),\cdots,l_j,\cdots, l_n}\\
    &=\frac{1}{|F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}|}
     \int_{F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}}\sigma_{ij}dF,
       \qquad 0<l_i,l_j<N, }
where $F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}$ is the unique
      $n-2$ dimensional face at vertex
$$((l_1-1)h,\cdots, (l_i-1)h, \cdots, (l_j-1), \cdots, (l_n-1)h)$$
 which is shared by elements:
 \begin{equation*}
 \begin{split}
 K_1&=[l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n],\\
  K_2&=[l_1,\cdots,(l_i-1),\cdots,(l_j-1),\cdots,l_n],\\
  K_3&=[l_1,\cdots,l_i,\cdots,(l_j-1),\cdots, l_n],\\
  K_4&=[l_1,\cdots,(l_i-1),\cdots,l_j,\cdots, l_n],\\
  \end{split}
  \end{equation*}
   $|F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}|$ is the measure
 of face $F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}$.  If
 $$|F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}|=0,$$
  $$\frac{1}{|F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}|}
    \int_{F_{x_i, x_j, l_1,\cdots, l_i, \cdots, l_j, \cdots, l_n}}\sigma_{ij}dF$$
   is understood as the value of $\sigma_{ij}$ at  vertex
   $$
   ((l_1-1)h,\cdots, (l_i-1)h, \cdots, (l_j-1), \cdots, (l_n-1)h).
   $$ The operator $\Pi_{ij}$
    is different from that defined in \cite{HuManZhang2013}.
Note that the corresponding operator of \cite{HuManZhang2013} can not be used
     for the superconvergence analysis.
For this operator, we have the following error estimates:
    \an{\lab{inter-error1}
  \|\sigma_{ij}-\Pi_{ij}\sigma_{ij}\|_{0,K} & \leq Ch\|\sigma_{ij}\|_{2,K} ,\\
   \lab{inter-error2}
\|\frac{\partial}{\partial x_i}(\sigma_{ij}-\Pi_{ij}\sigma_{ij})\|_{0,K}
   & \leq Ch\|\sigma_{ij}\|_{2,K} , \\
  \lab{inter-error3}
  \|\frac{\partial}{\partial x_j}(\sigma_{ij}-\Pi_{ij}\sigma_{ij})\|_{0,K}
   &\leq Ch\|\sigma_{ij}\|_{2, K}.}

Since the space for the operator $\Pi_{ii}$ contains
 the 1D quadratic polynomials ${\rm span}\{1,x_i,x_i^2\}$, the
 scaling argument and standard approximation,  state
  \an{\lab{IE1}
   |\sigma_{ii}-\Pi_{ii}\sigma_{ii}|_{0,K}  & \leq
    Ch|\sigma_{ii}|_{1,K} , \\
    \lab{IE2} |\frac{\partial}{\partial
    x_i}(\sigma_{ii}-\Pi_{ii}\sigma_{ii})|_{0,K} & \leq
    Ch|\frac{\partial{\sigma_{ii}}}{\partial
     x_i}|_{1,K}, }
 for all $ K\in\mathcal{T}_h$.

A summary of these aforementioned estimates
(\ref{inter-error1})--(\ref{IE2}) leads to
\begin{theorem}\lab{theorem-inter}
   For any $\sigma\in H^2(\Omega, \mathbb{S})$, we have that
 \an{ \lab{inter-err4}
   \|\sigma-\Pi_h\sigma\|_0 & \leq Ch\|\sigma\|_2, \\
  \lab{inter-err5}
   \|{\rm
   div}(\sigma-\Pi_h\sigma)\|_0& \leq Ch\|\sigma\|_2. }
   \end{theorem}

The stability of the elements and the standard theory of mixed
  finite element methods, see for instance \cite{Brezzi, Brezzi-Fortin}, give the
  following abstract error estimate: \an{
  \lab{theorem-err1} \|\sigma-\sigma_h\|_{H({\rm
  div})}+\|u-u_h\|_0\leq C \inf\limits_{\tau_h\in\Sigma_h,v_h\in
  V_h}\left(\|\sigma-\tau_h\|_{H({\rm div})}+\|u-v_h\|_0\right).}
Let $P_h$ denotes the projection operator from $V$ to $V_h$,
  which has the error estimate
\an{\lab{proj-error}
   \|v-P_hv\|_0\leq Ch\|v\|_1. }
Choosing $\tau_h=\Pi_h\sigma$ and
  $v_h=P_hu$ in (\ref{theorem-err1}), the estimates
  (\ref{inter-err4}), (\ref{inter-err5}), (\ref{proj-error}) prove
\begin{theorem}\lab{MainError} Let
  $(\sigma, u)\in\Sigma\times V$ be the exact solution of
   problem \meqref{eqn1} and $(\tau_h, u_h)\in\Sigma_h\times
   V_h$ the finite element solution of \meqref{DP}.  Then,
\a{
\|\sigma-\sigma_h\|_{H({\rm div})}
    &\le Ch(\|\sigma\|_2+\|u\|_1),\\

 \|u-u_h\|_0&\le     Ch(\|\sigma\|_2+\|u\|_1).
      }
\end{theorem}

\section{The superclose property of the canonical interpolations}
\subsection{Main difficulties}
By the K--ellipticity  in \eqref{below} and the discrete
   inf--sup condition \eqref{inf-sup}, it is routine to prove that
\begin{equation}\lab{eq3.1}
\begin{split}
&\quad \ \|\Pi_h\sigma-\sigma_h\|_{H(\d)}+\|P_hu-u_h\|_0\\
&\leq C\sup\limits_{0\ne(\tau, v)\in\Sigma_h\times V_h}
   \frac{ (A(\sigma-\Pi_h\sigma), \tau)+(u-P_h u, \d \tau)
     -(\d (\sigma-\Pi_h\sigma), v) }{\|\tau\|_{H(\d)}+\|v\|_0}.
\end{split}
\end{equation}
Note that the inequality \eqref{eq3.1} is the starting point for
    superconvergence analysis of mixed finite element methods, see
 for instance, \cite{Brandts,Duran90} and \cite{ShiDongyang14}.
However, this formulation can not be directly used for mixed finite
    elements under consideration,  the reasons lie in that
  \begin{itemize}
  \item The interpolation operator lacks the usual commuting property, namely,
   $$
   \d \Pi_h\sigma \not =P_h\d \sigma.
   $$
   \item The components of the stress normal are  coupled  through  $A(\sigma-\Pi_h\sigma)$, it is impossible  to prove directly
    the following super-close property
   $$
   (A(\sigma-\Pi_h\sigma), \tau)\leq Ch^2\|\sigma\|_2\|\tau\|_{H(\d)}
   $$
   for a general $\tau\in\Sigma_h$.
  \end{itemize}
  In the sequel, we will need some results on Sobolev spaces. They are formulated
in the following lemma. First of all, define $\partial\Omega_h$ as the subset of points
 having (Euclidian) distance less than $h$ from the boundary:
 $$
 \partial\Omega_h:=\{x\in\Omega|  \exists y\in\partial\Omega: {\rm dist}(x,y)\leq h \}.
 $$
  \begin{lemma}\lab{trace}
  For $v\in H^s(\Omega)$ with $0\leq s\leq 1/2$, it holds
  \begin{equation}
  \|v\|_{0,\partial\Omega_h}\leq Ch^s\|v\|_s.
  \end{equation}
  \end{lemma}

  \subsection{The superclose property of  $(\d (\sigma-\Pi_h\sigma), v)$}
To overcome the first difficulty, we  follow the idea of \cite{HuShi13}
   to adopt a new expansion of the operator $\Pi_h$.
In fact,
 let $\Pi_K=\Pi_h|_K$, we have the following crucial result.
 \begin{lemma} For any $\sigma\in P_2(K, \mathbb{S})$ and $v\in V_K$, it holds that
 \begin{equation}\lab{eq3.2}
 (\d (\sigma-\Pi_K\sigma), v)_K=0.
 \end{equation}
 \end{lemma}
 \begin{proof} We only need to prove the result on the reference element $K=[-1,1]^n$.
For any $\sigma\in P_2(K, \mathbb{S})$, its components $\sigma_{i j}$, $i, j=1, \cdots, n$
    with $i\not= j$, can be expressed as
 \begin{equation*}
 \sigma_{ij}=p_0(x_i, x_j)+x_ip_1^{(ij)}+x_jp_2^{(ij)}+p_3^{(ij)},
 \end{equation*}
 where $p_0(x_i, x_j)$ is a polynomial of degree $2$ in $x_i$ and $x_j$,
   both $p_1^{(ij)}$ and $p_2^{(ij)}$ are homogeneous
    polynomials of degree $1$ of $(n-2)$ of variables $x_k$, $k=1, \cdots, n$ with $k\not=i, j$,
   and $p_3^{(ij)}\in P_2(K)$
     is a polynomial of degree $2$ with respect to $(n-2)$
        variables $x_k$, $k=1, \cdots, n$ with $k\not=i, j$.
The definition of $\Pi_{ij, K}$ leads to
    \begin{equation}\lab{eq3.3}
    \begin{split}
    \sigma_{ij}-\Pi_{ij, K}\sigma_{ij}&=c_{ii}(x_i^2-1)+c_{ij}
              (x_j^2-1)+x_ip_1^{(ij)}+x_jp_2^{(ij)}\\
    &\quad +p_3^{(ij)}-\frac{1}{|F_{x_i, x_j}|}\int_{F_{x_i, x_j}}p_3^{(ij)}dF,
   \end{split}
    \end{equation}
     for two interpolation parameters $c_{ii}$ and $c_{ij}$, where $F_{x_i, x_j}$ is any
    $n-2$ dimensional face  of $K$ which is perpendicular  to the plane $\sspan \{ x_i, x_j\}$.
Here we use the facts that $\int_{F_{x_i, x_j}}p_1^{(ij)}dF=\int_{F_{x_i, x_j}}p_2^{(ij)}dF=0$,
  and that $p_3^{(ij)}$ is a constant function with respect to variables
     $x_i$ and $x_j$, and that $\sum\limits_{k=0}^3\phi_{k}=1$.

    If $i=j$, we have
 \begin{equation*}
 \sigma_{ii}=p_0(x_i)+x_ip_1^{(ii)}+p_2^{(ii)},
 \end{equation*}
 where $p_0(x_i)$ is a polynomial of degree $2$ or less in one variable $x_i$,
   and $p_1^{(ii)}$ is
  a homogeneous polynomial of degree $1$ of $(n-1)$  variables $x_k$,
    $k=1, \cdots, n$ with $k\not=i$, and $p_2^{(ii)}\in P_2(K)$
   is a polynomial of degree $2$ of $(n-1)$ variables $x_k$,
     $k=1, \cdots, n$ with $k\not=i$.
The definition
    of $\Pi_{ii, K}$ leads to
 \begin{equation}\lab{eq3.4}
 \sigma_{ii}-\Pi_{ii, K}\sigma_{ii}=x_i p_1^{(ii)}+p_2^{(ii)}-\frac{1}{|K|}\int_Kp_2^{(ii)}dV.
 \end{equation}

  By \eqref{eq3.3} and \eqref{eq3.4}, we have
   \begin{equation*}
        \frac{\partial( \sigma_{ij}-\Pi_{ij, K}\sigma_{ij})}{\partial x_j}=p_2^{(ij)}+2c_{ij}x_j
      \hbox{ \     and \ }
        \frac{\partial(\sigma_{ii}-\Pi_{ii, K}\sigma_{ii})}{\partial x_i}=p_1^{(ii)}.
 \end{equation*}
 Hence,  the $i$-th component of $\d (\sigma-\Pi_K\sigma)$ can be expressed as
  \begin{equation*}
 (\d (\sigma-\Pi_K\sigma))_i=p_1^{(ii)}+\sum\limits_{i\not=j=1}^n
     ( p_2^{(ij)}+2 c_{ij}x_j).
  \end{equation*}
We can compute
\a{ \int_K (\d (\sigma-\Pi_K\sigma))_i dV &= \int_K p_1^{(ii)} dV+\sum\limits_{i\not=j=1}^n
     \int_K( p_2^{(ij)}   +2 c_{ij}x_j ) dV\\
               &= 0 + \sum\limits_{i\not=j=1}^n
     ( \int_Kp_2^{(ij)} dV +2 c_{ij}\int_Kx_j dV) = 0, \\
    \int_K (\d (\sigma-\Pi_K\sigma))_i x_i dV &= \int_{-1}^1 x_i d x_i
           \int_{K_{n-1}} \bigg( p_1^{(ii)}+\sum\limits_{i\not=j=1}^n
     ( p_2^{(ij)}  +2 c_{ij}x_j  ) \bigg) d V_{n-1} \\
               &= 0  \cdot 0  = 0, }
   Here $K_{n-1}=[0,1]^{n-1}$ is the $n-1$ dimensional cube without variable $x_i$.
  Note that the $i$-th component of $v$ can be written as
  \begin{equation*}
  v_i=a_0+a_1x_i
  \end{equation*}
  for two interpolation parameters $a_0$ and $a_1$.
Thus
$$
  ((\d (\sigma-\Pi_K\sigma))_i, v_i)_K=0,
$$
which completes the proof.
 \end{proof}

 As a consequence of \eqref{eq3.2},  we have the following
     superclose property for the term $(\d (\sigma-\Pi_h\sigma), v)$.
\begin{lemma} Suppose that $\sigma\in H^3(\Omega, \mathbb{S})$. Then it holds that
\begin{equation}\lab{eq3.5}
|(\d (\sigma-\Pi_h\sigma), v)|\leq Ch^2|\sigma|_3\|v\|_0 \text{ for any }v\in V_h.
\end{equation}
\end{lemma}
\begin{proof} Given element $K$, let $I_{2,K}: H(\d, K, \mathbb{S})\rightarrow
    P_2(K, \mathbb{S})$ be the $L^2$ projection operator defined
    as: Given $\tau\in H(\d, K, \mathbb{S})$,
  find $I_{2,K}\tau\in P_2(K, \mathbb{S})$ such that
 \begin{equation*}
 \int_K I_{2,K}\tau q dV=\int_K\tau qdV\text{ for any }q\in P_2(K, \mathbb{S}).
 \end{equation*}
This allows for the following decomposition:
\a{ (\d (\sigma-\Pi_h\sigma), v) & =\sum\limits_{K\in\mathcal{T}_h}
      (\d ((I-\Pi_K)I_{2,K}\sigma+(I-\Pi_K)(I-I_{2,K})\sigma), v)_K\\
      & =\sum\limits_{K\in\mathcal{T}_h}
      (\d ((I-\Pi_K)(I-I_{2,K})\sigma), v)_K, }
   where we applied \eqref{eq3.2}.
    The desired result follows from the stability of $\Pi_h$
        and the approximation property of $I_{2,K}$.
\end{proof}

\subsection{The superclose property of $(A(\sigma-\Pi_h\sigma), \sigma_h-\Pi_h\sigma)$}

  To deal with the second difficulty, we propose to
explore  the strong discrete inf--sup condition presented in Lemma \ref{lemma1}.

\begin{lemma}\lab{lemma3.3} For any $\sigma\in P_1(K, \mathbb{S})$
    and $\tau\in \Sigma_{n,h}$, it holds that
 \begin{equation}\lab{eq3.6}
     |(A (\sigma-\Pi_K\sigma), \tau)_K |
           \leq Ch^2 \|\d\tau\|_{0, K}|\sigma|_{1, K},
 \end{equation}
 where
 $$
     \Sigma_{n,h}=\{\tau=\diag(\tau_{11}, \cdots, \tau_{nn}), \
             \tau\in \Sigma_{h}\}.
 $$
\end{lemma}
\begin{proof} We only need to prove the result on the reference element
    $K=[-1,1]^n$. For any $\sigma\in P_1(K, \mathbb{S})$, its normal
      components can be written as
 \begin{equation*}
    \sigma_{ii}=c_0^{(ii)}+\sum\limits_{j=1}^nc_j^{(ii)}x_j, \quad i=1, \cdots, n,
 \end{equation*}
 where $c_j^{(ii)}$, $j=0, \cdots, n$, are interpolation parameters.
By the definition of the operator $A$,  the $ii$-th component
  of $A(\sigma-\Pi_K\sigma)$ is
  $$
  A(\sigma-\Pi_K\sigma)_{ii}=\frac{1}{2\mu(2\mu+n\lambda)}\bigg((2\mu+(n-1)\lambda)
  \sum\limits_{i\not=j=1}^nc_j^{(ii)}x_j
 -\lambda\sum\limits_{i\not=k=1}^n\sum\limits_{k\not=j=1}^nc_j^{(kk)}x_j\bigg).
  $$
Note that the $i$-th component of $\tau$ can be written as
  \begin{equation*}
  \tau_{ii}=a_0^{(ii)}+a_1^{(ii)}x_i+a_2^{(ii)}x_i^2
  \end{equation*}
  for  parameters $a_0^{(ii)}$,  $a_1^{(ii)}$ and $a_2^{(ii)}$. Therefore,
  \an{\lab{eq3.7-2}
    \ad{(A(\sigma-\Pi_K\sigma)_{ii}, \tau_{ii})_K
   &=-a_1^{(ii)}\frac{\lambda}{2\mu(2\mu+n\lambda)}\sum\limits_{i\not=k=1}^nc_i^{(kk)}(x_i,x_i)\\
  &=-\frac{2^n}{3}\frac{\partial\tau_{ii}}{\partial x_i}(0)
     \frac{\lambda}{2\mu(2\mu+n\lambda)}\sum\limits_{i\not=k=1}^n
     \frac{\partial \sigma_{kk}}{\partial x_i},} }
  \a{ |(A(\sigma-\Pi_K\sigma)_{ii}, \tau_{ii})_K|
    &\leq C \frac{\lambda}{6\mu(2\mu+n\lambda)}\|\frac{\partial\tau_{ii}}{\partial x_i}\|_{0, K}
     \|\sum\limits_{i\not=k=1}^n\frac{\partial \sigma_{kk}}{\partial x_i}\|_{0, K}. }
A summation over all $n$ components leads to
  \begin{equation*}
  \begin{split}
   |(A(\sigma-\Pi_K\sigma), \tau)_K|\leq C\|\d\tau\|_{0,K}|\sigma|_{1,K}.
  \end{split}
  \end{equation*}
  The final result follows from a scaling argument.
\end{proof}

A combination of  the above lemma and \eqref{eq3.5} yields  the following important result.

\begin{lemma} It holds that
\begin{equation}\lab{eq3.7-1}
\|\Pi_h\sigma-\sigma_h\|_0^2+\|P_hu-u_h\|_0^2
    \leq C(A(\Pi_h\sigma-\sigma), \Pi_h\sigma-\sigma_h) +Ch^4\|\sigma\|_3^2.
\end{equation}
\end{lemma}
\begin{proof}
It follows from Lemma \ref{lemma1} that there exists
     $\tau=\diag(\tau_{11}, \cdots, \tau_{nn})\in\Sigma_h$ such that
\begin{equation}\lab{eq3.7}
\d \tau=u_h-P_hu \text{\ and \ }\|\tau\|_{H(\d)}\leq C\|u_h-P_hu\|_0.
\end{equation}
This allows for the following decomposition:
\begin{equation*}
\begin{split}
(u_h-P_hu, u_h-P_hu)&=(u_h-P_hu, \d \tau)=(u_h-u, \d\tau)=(A(\sigma-\sigma_h), \tau)\\
&=(A(\sigma-\Pi_h\sigma), \tau)+(A(\Pi_h\sigma-\sigma_h), \tau).
\end{split}
\end{equation*}
Since $\tau$ is a diagonal matrix, it follows from \eqref{eq3.6}  and \eqref{eq3.7} that
\begin{equation*}
(A(\sigma-\Pi_h\sigma), \tau)\leq Ch^2|\sigma|_1\|\d\tau\|_0\leq Ch^2|\sigma|_1\|u_h-P_hu\|_0.
\end{equation*}
A substitution of this inequality into the previous equation,
   by the Cauchy--Schwarz inequality and \eqref{eq3.7}, leads to
\begin{equation}\lab{eq3.8}
\|u_h-P_hu\|_0\leq C(h^2|\sigma|_1+\|\Pi_h\sigma-\sigma_h\|_0).
\end{equation}
On the other hand, let $\tau=\Pi_h\sigma-\sigma_h$, we have
\begin{equation}
\begin{split}
(A(\Pi_h\sigma-\sigma_h), \tau)&=(A(\Pi_h\sigma-\sigma), \tau)+(A(\sigma-\sigma_h), \tau)\\
&=(A(\Pi_h\sigma-\sigma), \tau)-(u-u_h, \d  \tau)\\
&=(A(\Pi_h\sigma-\sigma), \tau)-(P_hu-u_h, \d  (\Pi_h\sigma-\sigma_h))\\
&=(A(\Pi_h\sigma-\sigma), \tau)-(P_hu-u_h, \d  (\Pi_h\sigma-\sigma)).
\end{split}
\end{equation}
From \eqref{eq3.5} and \eqref{eq3.8} it follows
\begin{equation}
\begin{split}
(A(\Pi_h\sigma-\sigma_h), \tau)&\leq (A(\Pi_h\sigma-\sigma), \tau)
    +Ch^2|\sigma|_3\|P_hu-u_h\|_0\\
&\leq (A(\Pi_h\sigma-\sigma), \tau)+Ch^2|\sigma|_3(h^2|\sigma|_1+\|\Pi_h\sigma-\sigma_h\|_0).
\end{split}
\end{equation}
Since there exists a positive constant $\beta$ such that
$$
\beta\|\Pi_h\sigma-\sigma_h\|_0^2\leq (A(\Pi_h\sigma-\sigma_h), \tau),
$$
an application of the Young inequality leads to
$$
\|\Pi_h\sigma-\sigma_h\|_0^2+\|P_hu-u_h\|_0^2
   \leq C(A(\Pi_h\sigma-\sigma), \tau) +Ch^4\|\sigma\|_3^2,
$$
which completes the proof.
\end{proof}

\begin{lemma} For any $\sigma_{ij}\in P_1(K)$ and $\tau_{ij}\in Q_1(x_i, x_j)$, it holds that
\begin{equation}
(\sigma_{ij}-\Pi_{ij, K}\sigma_{ij}, \tau_{ij})_K=0.
\end{equation}
\end{lemma}
\begin{proof}
We only need to prove the result on the reference element $K=[-1,1]^n$.
Since $\sigma_{ij}\in P_1(K)$ , we have
 \begin{equation*}
    \begin{split}
    \sigma_{ij}-\Pi_{ij, K}\sigma_{ij}&=p^{(ij)},
   \end{split}
    \end{equation*}
 where $p^{(ij)}$ be a homogeneous polynomial of degree $1$ with respect to
      variables $x_k$, $k=1, \cdots, n$ but $k\not=i, j$.
   Any $\tau_{ij}\in Q_1(x_i, x_j)$ can be expressed as
 \begin{equation*}
 \tau_{ij}=a_0+a_1x_i+a_2x_j+a_3x_ix_j,
 \end{equation*}
 for four interpolation parameters $a_k$, $k=0, \cdots, 3$.  On the reference element $K$, it is straightforward to see that
\begin{equation*}
 (\sigma_{ij}-\Pi_{ij, K}\sigma_{ij}, \tau_{ij})=0.
 \end{equation*}
 This completes the proof.
\end{proof}

This lemma and a similar argument of \eqref{eq3.5} can prove the following supercloseness.
\begin{lemma} For any $\tau_{ij}\in \Sigma_{ij,h}
    :=\{e_i^\prime\tau e_j, \tau\in\Sigma_h \}$ it holds that
\begin{equation}\lab{eq3.15}
(\sigma_{ij}-\Pi_{ij}\sigma_{ij}, \tau_{ij})\leq Ch^2|\sigma_{ij}|_2\|\tau_{ij}\|_0,
\end{equation}
provided that $\sigma_{ij}\in H^2(\Omega)$. Here $e_i$ and $e_j$ are the $i$-th and $j$-th  canonical basis of the space $\mathbb{R}^n$, respectively.
\end{lemma}

\begin{lemma}\lab{lemma3.8} Let $(\sigma, u)$ and $(\sigma_h, u_h)$
    be solutions of problems \eqref{eqn1} and \eqref{DP},
  respectively.
Suppose that $\sigma \in H^2(\Omega, \mathbb{S})$ and $u\in H^1(\Omega, \mathbb{R}^n)$.
  Then  there holds that
\begin{equation}\lab{eq3.16}
   (A(\sigma-\Pi_h\sigma), \sigma_h-\Pi_h\sigma)
    \leq Ch^{5/2}(\|\sigma\|_2+\|u\|_1)\|\sigma\|_2.
\end{equation}
\end{lemma}
\begin{proof} Let $\tau=\sigma_h-\Pi_h\sigma$. Given element $K$, let $I_{1,K}: L^2(K)\rightarrow
    P_1(K)$ be the $L^2$ projection operator defined
    as: Given $v \in L^2(K)$,
  find $I_{1,K}v\in P_1(K)$ such that
 \begin{equation*}
 \int_K I_{1,K}v q dV=\int_Kv qdV\text{ for any }q\in P_1(K, \mathbb{S}).
 \end{equation*}
This leads to the following decomposition:
\begin{equation*}
  \begin{split}
  &(A(\sigma-\Pi_h\sigma)_{ii}, \tau_{ii})
  =\sum\limits_{K\in\mathcal{T}_h}(A(\sigma-\Pi_K\sigma)_{ii}, \tau_{ii})_K\\
  &=\sum\limits_{K\in\mathcal{T}_h}(A((I-\Pi_K)I_{1,K}\sigma)_{ii}, \tau_{ii})_K+\sum\limits_{K\in\mathcal{T}_h}(A((I-\Pi_K)(I-I_{1,K})\sigma)_{ii}, \tau_{ii})_K.
  \end{split}
  \end{equation*}
Then it follows from \eqref{eq3.7-2} that
\begin{equation*}
  \begin{split}
  &(A(\sigma-\Pi_h\sigma)_{ii}, \tau_{ii})
  =-\frac{\lambda \, h^2}{24 \mu(2\mu+n\lambda)}\sum\limits_{i\not=k=1}^n
  \sum\limits_{K\in\mathcal{T}_h} \big(\frac{\partial (I_{1,K}\sigma)_{kk}}{\partial x_i}, \frac{\partial\tau_{ii}}{\partial x_i}\big)_K
    +Ch^2\|\sigma\|_2\|\tau_{ii}\|_0.
  \end{split}
  \end{equation*}
 Since $\frac{\partial\tau_{ii}}{\partial x_i}$ is of the form  $a_1^{(ii)}+a_2^{(ii)}x_i$  for  parameters  $a_1^{(ii)}$ and $a_2^{(ii)}$, $\big(\frac{\partial (I_{1,K}x_l^2-x_l^2)}{\partial x_i}, \frac{\partial\tau_{ii}}{\partial x_i}\big)_K=0$ for $l\not=i$.  Therefore,
     $$
  \big(\frac{\partial (I_{1,K}\sigma-\sigma)_{kk}}{\partial x_i}, \frac{\partial\tau_{ii}}{\partial x_i}\big)_K=-\frac{h}{3}\big(\frac{\partial^2 \sigma_{kk}}{\partial x_i^2}, \frac{\partial\tau_{ii}}{\partial x_i}\big)_K+Ch^2|\sigma|_{3,K}|\tau_{ii}|_{1,K}.
 $$
After an elementwise inverse estimate, a combination of these two equations yield
\begin{equation}\lab{eq3.18}
  \begin{split}
  (A(\sigma-\Pi_h\sigma)_{ii}, \tau_{ii})
  &=-\frac{\lambda \, h^2}{24 \mu(2\mu+n\lambda)}\sum\limits_{i\not=k=1}^n
   \big(\frac{\partial  \sigma_{kk}}{\partial x_i}, \frac{\partial\tau_{ii}}{\partial x_i}\big)
    +Ch^2\|\sigma\|_2\|\tau_{ii}\|_0.
  \end{split}
  \end{equation}
Since convergence of terms $\frac{\partial\tau_{ii}}{\partial x_i}$ is  unclear,
    we can not obtain directly supercloseness from the previous equation.
The remedy is to use  convergence of divergence
   $\sum\limits_{\ell=1}^n \frac{\partial\tau_{i\ell}}{\partial x_\ell}$ and continuity
 of $\tau_{i\ell}$ across $n-1$ dimensional interior faces
  which are perpendicular to the axis $x_\ell$.
This idea leads to the following decomposition:
\begin{equation}\lab{eq3.19}
  \begin{split}
\sum\limits_{i\not=k=1}^n\bigg(\frac{\partial \sigma_{kk}}{\partial x_i},
    \frac{\partial\tau_{ii}}{\partial x_i}\bigg)
=\sum\limits_{\ell=1}^n\sum\limits_{i\not=k=1}^n
   \bigg(\frac{\partial \sigma_{kk}}{\partial x_i}, \frac{\partial\tau_{i\ell}}{\partial x_\ell}\bigg)
-\sum\limits_{i\not=\ell=1}^n\sum\limits_{i\not=k=1}^n
 \bigg(\frac{\partial \sigma_{kk}}{\partial x_i}, \frac{\partial\tau_{i\ell}}{\partial x_\ell}\bigg).
  \end{split}
  \end{equation}
  Since $\tau_{i\ell}=\sigma_{i\ell,h}-\Pi_{i\ell}\sigma_{i\ell}$,
   the first term on the right--hand side of \eqref{eq3.19} can be estimated
   by the error estimates  presented in \eqref{inter-err5} for the interpolation operator $\Pi_h$
   and convergence  from Theorem \ref{MainError} for the finite element solution $\sigma_h$.
This yields
  \begin{equation}\lab{eq3.20}
  \sum\limits_{\ell=1}^n\sum\limits_{i\not=k=1}^n
  \bigg(\frac{\partial \sigma_{kk}}{\partial x_i}, \frac{\partial\tau_{i\ell}}{\partial x_\ell}\bigg)
  \leq Ch (\|\sigma\|_2^2+\|u\|_1^2).
  \end{equation}
  To analyze the  second term on the right--hand side of \eqref{eq3.19}, we shall explore the continuity to transfer integrations
  on the volume to integrations on the boundary  and  use Lemma \ref{trace}. In fact,
  since the jump $[\tau_{i\ell}]_F$ across face $F$ vanishes for interior face $F\in \mathcal{F}_{x_\ell}$, we have
\begin{equation}\lab{eq3.21}
\begin{split}
\bigg(\frac{\partial \sigma_{kk}}{\partial x_i}, \frac{\partial\tau_{i\ell}}{\partial x_\ell}\bigg)
&=-\bigg(\frac{\partial^2 \sigma_{kk}}{\partial x_i\partial x_\ell}, \tau_{i\ell}\bigg)+\sum\limits_{F\in\mathcal{F}_{x_\ell}}\int_F \big[ \tau_{i\ell} \big]_F \frac{\partial \sigma_{kk}}{\partial x_i} dF\\
&=-\bigg(\frac{\partial^2 \sigma_{kk}}{\partial x_i\partial x_\ell}, \tau_{i\ell}\bigg)+\sum\limits_{F\in\mathcal{F}_{x_\ell}\cap\partial \Omega}\int_F \tau_{i\ell} \frac{\partial \sigma_{kk}}{\partial x_i} dF.
\end{split}
\end{equation}
In order to use  Lemma \ref{trace}, for any $F\in \mathcal{F}_{x_\ell}\cap\partial \Omega$, let $K_F$ be the unique element
 such that $F$ is one of its $n-1$ dimensional faces. Given $v\in L^2(K_F)$, define the constant projection $\Pi_{K_F}^0v$ by
 $$
 \Pi_{K_F}^0v:=\frac{1}{|K_F|}\int_{K_F}vdV.
 $$
This, the trace theorem, inverse estimate and  triangle inequality lead to
\begin{equation}\lab{eq3.21-1}
\begin{split}
\int_F \tau_{i\ell} \frac{\partial \sigma_{kk}}{\partial x_i} dF&=\int_F \tau_{i\ell}(I-\Pi_{K_F}^0)\frac{\partial \sigma_{kk}}{\partial x_i} dF
+ \Pi_{K_F}^0\frac{\partial \sigma_{kk}}{\partial x_i}\int_F \tau_{i\ell}  dF\\
&\leq C\|\tau_{i\ell}\|_{0, K_F}|\sigma_{kk}|_{2, K_F}+Ch^{-1}\|\Pi_{K_F}^0\frac{\partial \sigma_{kk}}{\partial x_i}\|_{0, K_F}\|\tau_{i\ell}\|_{0, K_F}\\
&\leq C \|\tau_{i\ell}\|_{0, K_F}|\sigma_{kk}|_{2, K_F}+Ch^{-1}\|\frac{\partial \sigma_{kk}}{\partial x_i}\|_{0, K_F}\|\tau_{i\ell}\|_{0, K_F}.
\end{split}
\end{equation}
Summing over $F$ in $\mathcal{F}_{x_\ell}\cap\partial \Omega$ and taking in account
the error estimates  presented in \eqref{inter-err5} for the interpolation operator $\Pi_h$ and convergence  from Theorem \ref{MainError} for the finite element solution $\sigma_h$, we arrive at
\begin{equation*}
\begin{split}
\sum\limits_{F\in\mathcal{F}_{x_\ell}\cap\partial \Omega}\int_F \tau_{i\ell} \frac{\partial \sigma_{kk}}{\partial x_i} dF
&\leq C(\|\sigma\|_2+\|u\|_1)(h|\sigma|_{2, \partial\Omega_h}+|\sigma|_{1, \partial\Omega_h}).
\end{split}
\end{equation*}
Hence it follows from Lemma \ref{trace} that
\begin{equation}\lab{eq3.22}
\begin{split}
\sum\limits_{F\in\mathcal{F}_{x_\ell}\cap\partial \Omega}\int_F \tau_{i\ell} \frac{\partial \sigma_{kk}}{\partial x_i} dF
&\leq C(\|\sigma\|_2+\|u\|_1)(h|\sigma|_{2, \Omega}+h^{1/2}\|\sigma\|_{3/2, \Omega}).
\end{split}
\end{equation}
A summary of \eqref{eq3.18} through \eqref{eq3.22}  shows that
\begin{equation}
\sum\limits_{i=1}^n(A(\sigma-\Pi_h\sigma)_{ii}, \tau_{ii})
  \leq Ch^{5/2}(\|\sigma\|_2+\|u\|_1)\|\sigma\|_2.
\end{equation}
Finally, let $\sigma_n=\diag(\sigma_{11}, \cdots, \sigma_{nn})$,
  $\tau_n=\diag(\tau_{11}, \cdots, \tau_{nn})$, $\sigma_s=\sigma-\sigma_n$
  and $\tau_s=\tau-\tau_n$.
 The previous equation, the estimate \eqref{eq3.15} for the shear stress,
estimates \eqref{inter-err4}--\eqref{inter-err5}, and estimates in Theorem \ref{MainError}, yield
\begin{equation*}
(A(\sigma-\Pi_h\sigma), \tau)=(A(\sigma_n-\Pi_h\sigma_n), \tau_n)+(\sigma_s-\Pi_h\sigma_s, \tau_s)
\leq Ch^{5/2}(\|\sigma\|_2+\|u\|_1)\|\sigma\|_2.
\end{equation*}
This completes the proof.
\end{proof}

\begin{theorem} Let $(\sigma, u)$ and $(\sigma_h, u_h)$ be solutions of
    problems \eqref{eqn1} and \eqref{DP},
  respectively.
Suppose that $\sigma \in H^3(\Omega, \mathbb{S})$ and $u\in H^1(\Omega, \mathbb{R}^n)$.
  Then  there holds that
\begin{equation}\lab{eq3.24}
\|\sigma_h-\Pi_h\sigma\|_{H(\d)}^2+\|u_h-P_hu\|_0^2\leq Ch^3(\|\sigma\|_3^2+\|u\|_1^2).
\end{equation}
\end{theorem}
\begin{proof} It is apparent that we can not  derive the desired superconvergence directly
    from  \eqref{eq3.7-1} and \eqref{eq3.16}.
We propose an iterative argument  to show  \eqref{eq3.24}, which consists of the following steps:
\begin{description}

\item[Step 1]
 By  \eqref{eq3.7-1} and \eqref{eq3.16}, we can deduce the following initial superconvergence result:
\begin{equation}\lab{eq3.25-1}
\|\sigma_h-\Pi_h\sigma\|_0^2+\|u_h-P_hu\|_0^2\leq Ch^{2+\frac{1}{2}}(\|\sigma\|_3^2+\|u\|_1^2).
\end{equation}

\item[Step 2]
 We  show an intermediate  superconvergence for  $\|\d (\sigma_h-\Pi_h\sigma)\|_0$
   based on  \eqref{eq3.1}, which for convenience is recalled as follows
 \begin{equation}\lab{eq3.25}
\begin{split}
&\quad \ \|\Pi_h\sigma-\sigma_h\|_{H(\d)}+\|P_hu-u_h\|_0\\
&\leq C\sup\limits_{0\not=(\tau, v)\in\Sigma_h\times V_h}
  \frac{(A(\sigma-\Pi_h\sigma), \tau)+(u- P_hu, \d \tau)-(\d (\sigma-\Pi_h\sigma), v)
      }{\|\tau\|_{H(\d)}+\|v\|_0}.
\end{split}
\end{equation}
Since the second term on the right--hand side of  \eqref{eq3.25} vanishes and the third term is already analyzed in  \eqref{eq3.5}, we only need
 to show  a better bound for the first term. In fact, it follows
   from  \eqref{eqn1}, \eqref{DP} and \eqref{eq3.25-1} that
 \begin{equation}
 \begin{split}
 (A(\sigma-\Pi_h\sigma), \tau)&=(A(\sigma -\sigma_h), \tau)+(A(\sigma_h-\Pi_h\sigma), \tau)\\
  &=(u_h-u, \d \tau)+(A(\sigma_h-\Pi_h\sigma), \tau)\\
  &=(u_h-P_hu, \d \tau)+(A(\sigma_h-\Pi_h\sigma), \tau)\\
  &\leq Ch^{1+\frac{1}{4}}(\|\sigma\|_3+\|u\|_1)\|\tau\|_{\d}.
\end{split}
 \end{equation}
 Consequently,
 \begin{equation}
 \|\Pi_h\sigma-\sigma_h\|_{H(\d)}\leq Ch^{1+\frac{1}{4}}(\|\sigma\|_3+\|u\|_1).
 \end{equation}
We substitute this estimate into \eqref{eq3.20} to get an improved estimate as
\begin{equation}
  \sum\limits_{\ell=1}^n\sum\limits_{i\not=k=1}^n\bigg(\frac{\partial \sigma_{kk}}{\partial x_i}, \frac{\partial(\sigma_{i\ell,h}-\Pi_{i\ell}\sigma_{i\ell})}{\partial x_\ell}\bigg)
  \leq Ch^{1+\frac{1}{4}} (\|\sigma\|_3^2+\|u\|_1^2).
  \end{equation}
 \item[Step 3]
  We establish an improved estimate for  the boundary term by
   putting the estimate $\|\sigma_h-\Pi_h\sigma\|_0$ of \eqref{eq3.25-1} into \eqref{eq3.21-1}:
\begin{equation}
\begin{split}
\sum\limits_{F\in\mathcal{F}_{x_\ell}\cap\partial \Omega}\int_F \tau_{i\ell} \frac{\partial \sigma_{kk}}{\partial x_i} dF
&\leq Ch^{\frac{3}{4}}(\|\sigma\|_3^2+\|u\|_1^2).
\end{split}
\end{equation}
\item[Step 4]
 We replace those corresponding estimates used in the proof of Lemma \ref{lemma3.8}
    by these improved estimates to obtain
\begin{equation}
(A(\sigma-\Pi_h\sigma), \sigma_h-\Pi_h\sigma_h)\leq Ch^{2+\frac{3}{4}}(\|\sigma\|_3^2+\|u\|_1^2).
\end{equation}
\end{description}
Thus we increase the order of convergence in \meqref{eq3.25-1} from $1+1/4$ to
   $1+1/4+1/8$.
Now we go back to Step 1  and repeat this procedure to get another $1/16$ higher
  order of convergence.
The iteration converges with \meqref{eq3.24}.
\end{proof}


TITLE: When a algebra is a sigma algebra
QUESTION [0 upvotes]: I'm trying to prove:

Giving $X$ a non-empty set. Let $A \subset P(X)$, we say that $A$ it's a algebra if and only if

$A$ it's a ring of sets
$X \in A$

Prove that; if $A \subset P(X)$ it's a algebra such that for every succesion of subsets disjoint $(E_n)_{n \in N}$ with every $E_n \in A$ we have $\cup_{n=1}^{\infty} E_n \in A$. Then A it's a $\sigma$-algebra.

I already have that $X, \varnothing \in A$ and for every $E \in A, E^c \in A$.
Just don't see why if $(E_n)_{n \in N} \in A$ then $\cup_{n=1}^\infty E_n \in A$
Thank you.

REPLY [1 votes]: Hint: Let $E_n \in \mathcal A$ for all $n$. Let $F_1=E_1,F_n=E_n\setminus  ( E_1 \cup ...\cup E_{n-1})$ for $n \geq 2$. Then $F_n \in \mathcal A$,$F_n$'s are disjoint. Now verify that $\cup_n E_n =\cup_n F_n$.


text:
\section{Bootstrapping the Quantiles}\label{sec:bootsrapping the quantiles}
The quantiles of the limit distribution of $\dodstat$ under $H_0$ depend on the unknown distribution $U$ and are therefore in general not accessible. One possible approach, which is quite cumbersome, is to estimate the covariance matrix of the Gaussian limit process $G$ from the data and use this to approximate the quantiles required. Alternatively, we suggest to directly bootstrap the quantiles of the limit distribution of $\dodstat$ under $H_0$. To this end, we define and investigate the bootstrap versions of $U_n$, $U_n^{-1}$ and $\mathbb{U}_n^{-1}\coloneqq \sqrt{n}\left( U_n^{-1}-U^{-1}\right)$.

\noindent Let $\mu_n$ denote the empirical measure based on the sample $X_1,\dots,X_n$. Given the sample values, let ${X}_1^*,\dots,{X}_{n_B}^*$ be an independent identically distributed sample of size $n_B$ from $\mu_n$. Then, the bootstrap estimator of $U_n$ is defined as
\begin{align*}
&{U}_{n_B}^*(t):= \frac{2}{n_B(n_B-1)}\sum_{1\leq i<j\leq n_B}\mathds{1}_{\left\lbrace d_\mathcal{X}({X}_i^*,{X}_j^*)\leq t\right\rbrace},
\end{align*}
the corresponding bootstrap empirical $U$-process is for $t\in\R$ given as ${\mathbb{U}}_{n_B}^*(t)=\sqrt{n_B}\big({U}_{n_B}^*(t)-U_n(t) \big) $ and the corresponding bootstrap quantile process for $t\in(0,1)$ as $\left( {\mathbb{U}}_{n_B}^*\right) ^{-1}(t)=\sqrt{n_B}\Big(\big( {U}_{n_B}^*\big) ^{-1}(t)-U_n^{-1}(t) \Big)$.\\

\noindent One can easily verify along the lines of the proof of \Cref{thm:main2} that for $n\to\infty$ it also holds for $\beta\in(0,1/2)$
\begin{equation}\label{eq:bootstrapideabasis}
\int_{\beta}^{1-\beta}\!\left( \mathbb{U}_n^{-1}(t)\right)^2 \,dt\rightsquigarrow\Xi=\int_{\beta}^{1-\beta}\!\mathbb{G}^2(t) \,dt.
\end{equation}
Hence, this suggests to to approximate the quantiles of $\Xi$ by the quantiles of its bootstrapped version
\begin{equation}
{\Xi}_{n_B}^*\coloneqq\int_{\beta}^{1-\beta}\!\left(\left( {\mathbb{U}}_{n_B}^*\right) ^{-1}(t)\right)^2 \,dt.\label{eq:defofhatXinun}\end{equation}

	Let $\beta\in(0,1/2)$, suppose that \Cref{cond:firstcondition} holds, let $\sqrt{n_B}=o(n)$ and let $\xi_{n_B,\alpha}^{(R)}$ denote the empirical bootstrap quantile of $R$ independent bootstrap realizations ${\Xi}_{n_B}^{*,(1)},\dots,{\Xi}_{n_B}^{*,(R)}$. Under these assumptions, we derive (cf. Section C of the supplement \cite{supplement}) that for any $\alpha\in(0,1)$ it follows
\begin{equation}\label{eq:bootstrapconsitency}\lim_{n,n_B,R\to\infty}\prob\left(\int_{\beta}^{1-\beta}\!\left(\mathbb{U}_n^{-1}(t)\right)^2\,dt \geq \xi_{n_B,\alpha}^{(R)} \right)=\alpha. \end{equation}

Because of \eqref{eq:bootstrapideabasis} the statement \eqref{eq:bootstrapconsitency} guarantees the consistency of $\xi_{n_B,\alpha}^{(R)}$ for $n,n_B,R\to\infty$. Hence, a consistent bootstrap analogue of the test defined by the decision rule \eqref{eq:decision rule} is for $\beta\in(0,1/2)$ given by the \textit{bootstrapped Distribution of Distances (DoD)}-test
\begin{equation}\label{eq:bootstraptest}\Phi_{DoD}^*(\mathcal{X}_n,\mathcal{Y}_m)=\begin{cases}
1, &\text{if}~\frac{nm}{n+m}\dodstat>\xi_{n_B,1-\alpha}^{(R)}\vspace*{5mm}\\
0, &\text{if}~\frac{nm}{n+m}\dodstat\leq \xi_{n_B,1-\alpha}^{(R)}.
\end{cases}\end{equation}


TITLE: Highest exponent of $2$ dividing $\binom{n}{k}$
QUESTION [2 upvotes]: What is the highest exponent of $2$ dividing $\binom{n}{k}$?

I know that
$$\binom{n}{k}=\frac{n!}{k!(n-k)!}$$
And therefore the highest exponent of $2$ dividing $\binom{n}{k}$ is
$$\sum_{i=1}^{\infty} \left( \lfloor \frac{n}{2^i}\rfloor -\lfloor \frac{k}{2^i}\rfloor-\lfloor \frac{n-k}{2^i}\rfloor \right)$$
I am stuck here!

REPLY [2 votes]: From Legendre's Theorem

The greatest power of $2$ dividing $n!$ is $2^{n−r}$ where $r$ is the
  number of $1$s in the binary expansion of $n$.

As a result $n!=2^{n-r_n}\cdot q_n$, $k!=2^{k-r_k}\cdot q_k$ and $(n-k)!=2^{n-k-r_{n-k}}\cdot q_{n-k}$. So
$$\frac{n!}{k!(n-k)!}=\frac{2^{n-r_n}q_n}{2^{k-r_k}q_k\cdot 2^{n-k-r_{n-k}}q_{n-k}}=\frac{q_n}{q_kq_{n-k}}\cdot 2^{n-r_n-(k-r_k)-(n-k-r_{n-k})}=\\\frac{q_n}{q_kq_{n-k}}\cdot 2^{r_k+r_{n-k}-r_n}$$